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https://arxiv.org/abs/0912.1436 | On the number of zeros of multiplicity r | Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of variables and |S|. In the present work we take into account what is the leading monomial. This allows us to consider more general point ensembles and most importantly it allows us to produce much more detailed information about the number of zeros of multiplicity r than can be deduced from the generalized Schwartz-Zippel bound. We present both upper and lower bounds. | \section{Introduction}\label{secintro}
In this paper we consider multivariate polynomials over an arbitrary
field ${\mathbf{F}}$. Our studies focus on the zeros of given
prescribed multiplicity, a concept to be defined more formally below.
The definition of multiplicity that we will use relies on the Hasse
derivative. This derivative
coincides with the usual analytic
derivative in the case of polynomials over the reals. Before recalling the definition of the
Hasse derivative let us fix some notation. Assume we are given a vector of
variables $\vec{X}=(X_1, \ldots ,X_m)$ and a vector $\vec{k}=(k_1,
\ldots , k_m)\in {\mathbf{N}}_0^m$ then we will write
$\vec{X}^{\vec{k}}=X_1^{k_1} \cdots X_m^{k_m}$. We will always assume
that
$\vec{X}$ and $\vec{Z}$ are vectors of $m$ variables.
\begin{Definition}
Given $F(\vec{X})\in {\mathbf{F}}[\vec{X}]$ and
$\vec{k} \in {\mathbf{N}}_0^m$ the $\vec{k}$'th
Hasse derivative of $F$, denoted by $F^{(\vec{k})}(\vec{X})$ is the
coefficient of $\vec{Z}^{\vec{k}}$ in $F(\vec{X}+\vec{Z})$. In other words
$$F(\vec{X}+\vec{Z})=\sum_{\vec{k}} F^{(\vec{k})}(\vec{X})\vec{Z}^{\vec{k}}.$$
\end{Definition}
The concept of multiplicity for univariate polynomials is generalized
to multivariate polynomials in the following way.
\begin{Definition}\label{defmult}
For $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]\backslash \{ \vec{0} \}$ and
$\vec{a}\in {\mathbf{F}}^m$ we define the multiplicity of $F$ at $\vec{a}$
denoted by ${\mbox{mult}}(F,\vec{a})$ as follows. Let $M$ be an
integer such that for every $\vec{k}=(k_1, \ldots ,
k_m) \in {\mathbf{N}}_0^m$ with $k_1+\cdots +k_m < M$, $F^{(\vec{k})}(\vec{a})=0$
holds, but for some $\vec{k}=(k_1, \ldots ,
k_m) \in {\mathbf{N}}_0^m$ with $k_1+\cdots +k_m = M$,
$F^{(\vec{k})}(\vec{a})\neq 0$ holds, then
${\mbox{mult}}(F,\vec{a})=M$. If $F=0$ then we define ${\mbox{mult}}(F,\vec{a})=\infty$.
\end{Definition}
It is of evident interest to investigate for multivariate polynomials $F$
and a finite ensemble
of points the following questions:
\begin{itemize}
\item[Q1] How many zeros can $F$ have in total when counted with multiplicity?
\item[Q2] How many zeros of a given prescribed multiplicity can $F$ have?
\end{itemize}
Clearly, assuming finite ensembles of points is not a restriction
when $\mathbf{F}$ is a finite field
${\mathbf{F}}_q$. We note that the above questions have important implications in a number of applications, see~\cite{dvir}
and~\cite{pw_expanded}. What we would like to have for certain natural
ensembles of points is bounds on the
number of points in terms of the total degree of $F$ or even better in terms
of ${\mbox{lm}}(F)$. Here, ${\mbox{lm}}(F)$ denotes the leading
monomial of $F$ with respect to some fixed monomial
ordering.\\
The related problem of bounding the number of zeros (counted without multiplicity) has been
dealt with using two completely different approaches. On the one hand a tight bound in terms of the leading
monomial has be derived using the footprint bound from Gr\"{o}bner
basis theory (see~\cite{clo} and \cite{gh}). On the other hand a tight bound in term of the total degree, known
as the Schwartz-Zippel bound, was derived using only very simple
combinatorial arguments \cite{schwartz}, \cite{zippel}. To answer
partly
question Q2 in
terms of the total degree Pellikaan and Wu in~\cite{pw_expanded} followed the footprint
bound approach. Later a generalized Schwartz-Zippel bound
that deals
with question Q1 in terms of the total degree was suggested by Augot,
El-Khamy, McEliece, Parvaresh, Stepanov, Vardy
in~\cite{manyauthors} for the case of two variables, and by Augot,
Stepanov in \cite{augot} for arbitrarily many variables. The bound was
proven to be correct in a recent paper by Dvir, Kopparty, Saraf and Sudan~\cite{dvir}. The generalized Schwartz-Zippel bound goes as follows.
\begin{Theorem}\label{SZ-lemma}
Let $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]$ be a non-zero polynomial of
total degree $d$. Then for any finite set $S \subseteq {\mathbf{F}}$
\begin{eqnarray}
\sum_{\vec{a}\in S^n}{\mbox{mult}}(F,\vec{a}) \leq d | S |^{{m-1}}. \nonumber
\end{eqnarray}
\end{Theorem}
As a corollary we get an immediate partial answer to question Q2 in
terms of the total degree of $F$.
\begin{Corollary}\label{corsz}
Let $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]$ be a
non-zero polynomial of total degree $d$ and let $S \subseteq
{\mathbf{F}}$ be finite.
The number
of zeros of $F$ of multiplicity at least $r$ from $S^m$ is at most
$$\frac{d}{r}|S |^{m-1}.$$
\end{Corollary}
In the present paper we take the Schwartz-Zippel
approach. We use the methods from~\cite{dvir}, but rather than
taking into account only information about the total degree and
allowing only point ensembles $S^n$ we
\begin{itemize}
\item use information about the leading monomial with respect to a
lexicographic ordering.
\item consider the more general point ensembles $S_1\times \cdots
\times S_m$, the sets $S_i$ all being
finite.
\end{itemize}
In Section~\ref{secleading} we easily translate Theorem~\ref{SZ-lemma} into this setting and
derive an immediate translation of Corollary~\ref{corsz}. As will be
shown in Section~\ref{seclin},
Theorem~\ref{SZ-lemma} and its translation are
tight for all products of univariate linear
terms. A similar result by no means holds for
Corollary~\ref{corsz} and its translation. Actually, a refinement of the methods
from~\cite{dvir} yields for dramatic improvements to
Corollary~\ref{corsz} and its translation. In its most general
form in Section~\ref{sec-cor-sz-gen-bet} we state
an algorithm to upper bound the number of zeros of multiplicity at
least $r$. Using this algorithm we then derive in Section~\ref{sectwovar}
closed formulas in the case where the number of variables is two and
the multiplicity is arbitrary. Section~\ref{secismall} further presents a
simple closed formula for the case of arbitrary many variables where, however, the
powers $i_1, \ldots , i_m$ in the leading monomial
${\mbox{lm}}(F)=X_1^{i_1}\cdots X_m^{i_m}$ are all small. In
Section~\ref{seclin} we consider the case when the polynomial is a
product of univariate linear terms. Such polynomials are easy to
analyze and by doing this we get in appendix~\ref{applin} an algorithm to
produce lower bounds on the maximal attainable number of zeros of
multiplicity at least $r$. Section~\ref{secequal} describes various
conditions under which our upper bound equals our lower bound. Having improved on
the results in~\cite[Section 2]{dvir} we conclude the paper by showing
in Appendix~\ref{seccomp} that
Corollary~\ref{corsz} is stronger than the corresponding result
given by Pellikaan and Wu in~\cite{pw_expanded}. From this we can conclude that the results found in the
present paper are the strongest known. The present paper comes with a
webpage~\cite{hmpage} where a large number of experimental results
are presented.
\section{Using information about the leading monomial}\label{secleading}
In the following we modify the method from~\cite[Section 2]{dvir}. One could choose to prove the results of the present
section using the original method, however, the modification will
be needed in the section to follow. For simplicity we stick to the
modified method in both sections. Throughout the paper
$S_1,\ldots, S_m \subseteq {\mathbf{F}}$ are finite sets and we write $s_1=|S_1 | , \ldots , s_m=|S_m|$. In the following the monomial
ordering $\prec$ on the set of monomials in variables $X_1,
\ldots , X_m$ will always be the lexicographic ordering with $X_m\prec \cdots \prec X_1$.\\
We start our investigations by recalling two results from~\cite[Section
2]{dvir}. The first corresponds to~\cite[Lemma 5]{dvir}.
\begin{Lemma}\label{lem5}
Consider $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]$ and
$\vec{a} \in {\mathbf{F}}^m$. For any $\vec{k}=(k_1, \ldots ,k_m)
\in {\mathbf{N}}_0^m$ we have
$${\mbox{mult}}(F^{(\vec{k})},\vec{a}) \geq mult(F,\vec{a})-(k_1+ \cdots +k_m).$$
\end{Lemma}
The next result that we recall corresponds to the last part of
\cite[Proposition 6]{dvir}.
\begin{Proposition}\label{propsammensat}
Given $F(X_1, \ldots , X_m) \in {\mathbf{F}}[X_1, \ldots
,X_m]$ and
$$Q(Y_1, \ldots , Y_l)=(Q_1(\vec{Y}), \ldots ,Q_m(\vec{Y}))
\in {\mathbf{F}}[Y_1, \ldots , Y_l]^m$$ let $F \circ Q$ be the
polynomial $F(Q_1(\vec{Y}), \ldots , Q_m(\vec{Y}))$. For any
$\vec{a}\in {\mathbf{F}}^l$ we have $${\mbox{mult}}(F \circ Q,\vec{a})
\geq {\mbox{mult}}(F,Q(\vec{a})).$$
\end{Proposition}
We get the following Corollary, which is closely related
to~\cite[Corollary 7]{dvir}.
\begin{Corollary}\label{cor8}
Let $F(X_1, \ldots , X_m) \in {\mathbf{F}}[X_1, \ldots , X_m]$ and
$
\vec{b}_1, \ldots ,\vec{b}_{m-1},\vec{c} \in {\mathbf{F}}^m$ be
given. Write $F^\ast(T_1, \ldots ,T_{m-1}) =F(T_1\vec{b}_1 + \cdots +
T_{m-1}\vec{b}_{m-1}+\vec{c})$.
For any $(t_1, \ldots , t_{m-1}) \in {\mathbf{F}}^{m-1}$ we
have
\begin{multline*}
{\mbox{mult}}(F^\ast(T_1, \ldots , T_{m-1}),(t_1, \ldots , t_{m-1})) \\
\geq{\mbox{mult}}(F(X_1, \ldots ,X_m),
t_1\vec{b}_1+\cdots +t_{m-1}\vec{b}_{m-1}+\vec{c}).
\end{multline*}
\end{Corollary}
We now write
$$F(X_1, \ldots , X_m)=\sum_{j_1, \ldots ,j_{m-1}}
X_1^{j_1}\cdots X_{m-1}^{j_{m-1}}F_{j_1, \ldots
j_{m-1}} (X_m).$$
Let $X_1^{i_1} \cdots X_m^{i_m}$ be the leading monomial of $F$ with
respect to $\prec$. Then
due to the definition of $\prec$, $F_{i_1, \ldots , i_{m-1}}(X_m)$ is a (univariate) polynomial
of degree $i_m$.
For $a_m \in {\mathbf{F}}$ define
$$
r(a_m)={\mbox{mult}}(F_{i_1, \ldots , i_{m-1}}(X_m),a_m).$$
Clearly,
\begin{equation}
\sum_{a_m \in S_m}r(a_m) \leq i_m. \label{eqfoerste}
\end{equation}
We have
$$
F^{(0,\ldots , 0,r(a_m))}(X_1, \ldots , X_m)=\sum_{j_1, \ldots ,
j_{m-1}}
X_1^{j_1} \cdots X_{m-1}^{j_{m-1}}F_{j_1, \ldots
,j_{m-1}}^{(r(a_m))}(X_m)$$
and due to the definition of $\prec$ and to the definition of
$r(a_m)$ we have
\begin{equation}
{\mbox{lm}}_{\prec}(F^{(0,\ldots , 0,r(a_m))}(X_1, \ldots ,
X_{m-1},a_m))=X_1^{i_1}\cdots X_{m-1}^{i_{m-1}}. \label{eqnaeste}
\end{equation}
Applying first Lemma~\ref{lem5} with $\vec{k}=(0,\ldots ,0,r(a_m))$ and
afterwards
Corollary~\ref{cor8} with
$\vec{b}_1=(1,0,\ldots ,0), \ldots ,
\vec{b}_{m-1}=(0, \ldots , 0,1,0)$, $\vec{c}=(0, \ldots , 0,a_m)$ and $t_1=a_1, \ldots ,
t_{m-1}=a_{m-1}$ we get the following result which is
closely related to a result in~\cite[Proof of Lemma 8]{dvir}:
\begin{align}
&{\mbox{mult}}\big(F(X_1, \ldots ,X_m),(a_1, \ldots , a_m)\big)\nonumber \\
&\leq (0+ \cdots +0+r(a_m))+{\mbox{mult}}\big(F^{(0,\ldots
,0,r(a_m))}(X_1, \ldots ,X_m),(a_1, \ldots , a_m)\big)\nonumber\\
&\leq r(a_m) +{\mbox{mult}}\big(F^{(0,\ldots ,0,r(a_m))}(X_1, \ldots
,X_{m-1},a_m),(a_1, \ldots ,a_{m-1})\big). \label{eqtredie}
\end{align}
We are now in the position that we can prove the main result of this section.
\begin{Theorem}\label{prop-sz-gen}
Let $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]$ be a non-zero polynomial and
let ${\mbox{lm}}(F)=X_1^{i_1} \cdots X_m^{i_m}$ be its leading
monomial with respect to a lexicographic ordering. Then for
any finite sets $S_1, \ldots ,S_m \subseteq {\mathbf{F}}$
\begin{eqnarray}
\sum_{\vec{a}\in S_1 \times \cdots \times S_m}{\mbox{mult}}(F,\vec{a})
\leq i_1s_2\cdots s_m+s_1i_2s_3 \cdots s_m+\cdots +s_1\cdots s_{m-1}i_m.\nonumber
\end{eqnarray}
\end{Theorem}
\begin{proof}
We prove the theorem for the monomial
ordering $\prec$. Dealing with general
lexicographic orderings is simply a question of relabeling the
variables. Clearly the theorem holds for
$m=1$. For $m>1$ we consider~(\ref{eqtredie}). Assuming the
theorem holds when the number of variables is smaller than $m$ we
get by applying~(\ref{eqfoerste}) and~(\ref{eqnaeste}) the following estimate
\begin{align}
&\sum_{\vec{a} \in S_1 \times \cdots \times
S_m}{\mbox{mult}}(F,\vec{a}) \nonumber \\
&\leq i_ms_1 \cdots s_{m-1}+s_m(i_1s_2
\cdots s_{m-1}+\cdots +i_{m-1}s_1\cdots s_{m-2})\nonumber \\
&=i_1s_2\cdots s_m+i_2s_1 s_3\cdots s_m+\cdots i_ms_1\cdots
s_{m-1}\nonumber
\end{align}
as required.
\end{proof}
We have the following immediate generalization of Corollary~\ref{corsz}.
\begin{Corollary}\label{cor-sz-gen}
Let $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]$ be a non-zero polynomial and
let ${\mbox{lm}}(F)=X_1^{i_1} \cdots X_m^{i_m}$ be its leading
monomial with respect to a lexicographic ordering. Assume $S_1, \ldots
,S_m \subseteq {\mathbf{F}}$ are finite sets.
Then over
$S_1 \times \cdots \times S_m $ the number
of zeros of multiplicity at least $r$ is less than or equal to
\begin{eqnarray}
\big( i_1s_2\cdots s_m+s_1i_2s_3\cdots s_m+\cdots +s_1\cdots
s_{m-1}i_m \big) /r.\nonumber
\end{eqnarray}
\end{Corollary}
{\section{Improvements to Corollary~\ref{cor-sz-gen} \label{sec-cor-sz-gen-bet}
}}
In this section we shall see that a further analysis allows for
dramatic improvements to Corollary~\ref{cor-sz-gen}.
Let $X_1^{i_1}\cdots X_m^{i_m}$ be the leading monomial of $F(\vec{X})\in {\mathbf{F}}[\vec{X}]$ with respect
to $\prec$. Recall from~(\ref{eqtredie}) the bound
\begin{align}
&{\mbox{mult}}(F(X_1, \ldots ,X_m),(a_1, \ldots , a_m))\nonumber \\
& \leq r(a_m) +{\mbox{mult}}(F^{(0,\ldots ,0,r(a_m))}(X_1, \ldots
,X_{m-1},a_m),(a_1, \ldots ,a_{m-1})).\label{eqnoget}
\end{align}
Here, $r(a_m)$ are numbers that when summed over all possible $a_m \in S_m$
give at most $i_m$ and the leading monomial of $F^{(0,\ldots ,0,r(a_m))}(X_1, \ldots
,X_{m-1},a_m)$ with respect to $\prec$ is $X^{i_1} \cdots
X_{m-1}^{i_{m-1}}$. Our analysis suggests the
following recursive definition of a function to bound the number of
zeros of multiplicity $r$.\\
\begin{Definition}\label{defD}
Let $r \in {\mathbf{N}}, i_1, \ldots , i_m \in {\mathbf{N}}_0$. Define
$$D(i_1,r,s_1)=\min \big\{\big\lfloor \frac{i_1}{r} \big\rfloor,s_1\big\}$$
and for $m \geq 2$
\begin{multline*}
D(i_1, \ldots , i_m,r,s_1, \ldots ,s_m)=
\\
\begin{split}
\max_{(u_1, \ldots ,u_r)\in A(i_m,r,s_m) }&\bigg\{ (s_m-u_1-\cdots -u_r)D(i_1,\ldots ,i_{m-1},r,s_1,
\ldots ,s_{m-1})\\
&\quad+u_1D(i_1, \ldots , i_{m-1},r-1,s_1, \ldots ,s_{m-1})+\cdots
\\
&\quad +u_{r-1}D(i_1, \ldots ,i_{m-1},1,s_1, \ldots , s_{m-1})+u_rs_1\cdots
s_{m-1} \bigg\}
\end{split}
\end{multline*}
where
\begin{multline}
A(i_m,r,s_m)= \nonumber \\
\{ (u_1, \ldots , u_r) \in {\mathbf{N}}_0^r \mid u_1+ \cdots
+u_r \leq s_m {\mbox{ \ and \ }} u_1+2u_2+\cdots +ru_r \leq i_m\}.\nonumber
\end{multline}
\end{Definition}
\begin{Theorem}\label{prorec}
For a polynomial $F(\vec{X})\in {\mathbf{F}}[\vec{X}]$ let $X_1^{i_1}\cdots X_m^{i_m}$ be its leading monomial with
respect to $\prec$ (this is the lexicographic ordering with $X_m\prec \cdots \prec X_1$). Then $F$ has at most $D(i_1, \ldots , i_m,r,s_1,
\ldots ,s_m)$ zeros of multiplicity at least $r$ in $S_1\times \cdots
\times S_m$. The corresponding recursive algorithm produces a number
that is at most equal to the number found in
Corollary~\ref{cor-sz-gen} and is at most equal to $s_1 \cdots s_m$.
\end{Theorem}
\begin{proof}
The proof of the first part of the proposition is an induction proof. The result clearly holds for
$m=1$. Given $m>1$ assume it holds for $m-1$.
For $d=1, \ldots ,
r-1$ let $u_d$ be the number of $a_m$'s with $r(a_m)=d$ and let $u_r$
be the number of $a_m$'s with $r(a_m) \geq r$. The number of $a_m$'s
with $r(a_m)=0$ is $s_m-u_1-\cdots -u_r$. The boundary conditions that
$u_1+\cdots +u_r \leq s_m$ and $u_1+2u_2+\cdots +ru_r \leq i_m$ are
obvious. For every $a_m$ with $r(a_m)=d$, $d=0, \ldots , r-1$ for
$(a_1, \ldots , a_m)$ to be a zero of multiplicity at least $r$ the
last expression in~(\ref{eqnoget}) must be at least $r-d$. For
$a_m$ with $r(a_m)\geq r$ all choices of $a_1, \ldots , a_{m-1}$ are
legal. This proves the first part of the proposition. As both
Corollary~\ref{cor-sz-gen} and the above proof rely on~(\ref{eqnoget}), Theorem~\ref{prorec} cannot produce a
number greater than what is found in Corollary~\ref{cor-sz-gen}. The
condition $u_1+\cdots +u_r \leq s_m$ and the definition of
$D(i_1,r,s_1)$ imply the last result.
\end{proof}
The next remark shows that we need only apply the algorithm to a
restricted set of exponents $(i_1,\ldots ,i_m)$.
\begin{Remark}\label{rembig}
Given $(i_1, \ldots , i_m, r, s_1, \ldots ,s_m)$ with $\lfloor
i_1/s_1\rfloor+\cdots +\lfloor i_m/s_m\rfloor \geq r$ then there exist
polynomials with the leading monomial being $X_1^{i_1} \cdots X_m^{i_m}$ such that all points in $S_1 \times \cdots \times S_m$ are
zeros of multiplicity at least $r$.
Hence, we need only apply
the algorithm to cases that do not satisfy the above inequality. In Section~\ref{seclin}, Example~\ref{exbig}, we will
explain this fact in more detail.
\end{Remark}
In a series of experiments we found that the above algorithm produces
numbers that are often much lower than the minimum of the corresponding result from
Corollary~\ref{cor-sz-gen} and $s_1\cdots s_m$. In the
webpage~\cite{hmpage} we list all results of our experiments. Here,
we only mention a
few.
\begin{Example}\label{exny1}
In this example we bound the number of zeros of multiplicity
$3$ or more for polynomials in two
variables. Both $S_1$ and $S_2$ are assumed to be of size $5$. Table~\ref{tabny1} shows
information obtained from our algorithm for the exponents $i_1,i_2$
not treated by Remark~\ref{rembig}. Table~\ref{tabny2} illustrates the
improvement on the bound $\lfloor \min\{
(i_1+i_2)5/3,5^2\}\rfloor$. Here, the first expression comes from
Corollary~\ref{cor-sz-gen} and the last expression is the number of
points in $S_1 \times S_2$. Observe, that the tables are not symmetric
meaning that $D(i_1,i_2,3,5,5)$ does not always equal $D(i_2,i_1,3,5,5)$.
\begin{table}
\centering
\caption{$D(i_1,i_2,3,5,5)$}
\newcommand{~~}{~~}
\begin{tabular}{@{}c@{}cc@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
&&\multicolumn{15}{c}{$i_1$}\\
&&0 &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &13 &14\\
\addlinespace
\multirow{15}{*}{$i_2$}
&\multicolumn{1}{c}{0 }&0 &0 &0 &5 &5 &5 &10&10&10&15&15&15&20&20&20\\
&\multicolumn{1}{c}{1 }&0 &0 &1 &5 &6 &6 &11&11&12&16&17&17&21&21&21\\
&\multicolumn{1}{c}{2 }&0 &1 &2 &7 &8 &9 &13&13&14&17&19&19&22&22&22\\
&\multicolumn{1}{c}{3 }&5 &5 &5 &9 &9 &10&14&14&16&18&21&21&23&23&23\\
&\multicolumn{1}{c}{4 }&5 &5 &6 &9 &11&13&16&16&18&19&23&23&24&24&24\\
&\multicolumn{1}{c}{5 }&5 &6 &7 &11&12&14&17&17&20&20\\
&\multicolumn{1}{c}{6 }&10&10&10&13&14&17&19&19&21&21\\
&\multicolumn{1}{c}{7 }&10&10&11&13&15&18&20&20&22&22\\
&\multicolumn{1}{c}{8 }&10&11&12&15&17&21&22&22&23&23\\
&\multicolumn{1}{c}{9 }&15&15&15&17&18&22&23&23&24&24\\
&\multicolumn{1}{c}{10}&15&15&16&17&20\\
&\multicolumn{1}{c}{11}&15&16&17&19&21\\
&\multicolumn{1}{c}{12}&20&20&20&21&22\\
&\multicolumn{1}{c}{13}&20&20&21&21&23\\
&\multicolumn{1}{c}{14}&20&21&22&23&24\\
\bottomrule
\end{tabular}
\label{tabny1}
\end{table}
\begin{table}
\centering
\caption{Improvements found in Example~\ref{exny1}}
\newcommand{~~}{~\hspace*{0.851em}}
\begin{tabular}{@{}c@{}cc@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
&&\multicolumn{15}{c}{$i_1$}\\
&&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\
\addlinespace
\multirow{15}{*}{$i_2$}
&\multicolumn{1}{c}{0}&0&1&3&0&1&3&0&1&3&0&1&3&0&1&3\\
&\multicolumn{1}{c}{1}&1&3&4&1&2&4&0&2&3&0&1&3&0&2&4\\
&\multicolumn{1}{c}{2}&3&4&4&1&2&2&0&2&2&1&1&2&1&3&3\\
&\multicolumn{1}{c}{3}&0&1&3&1&2&3&1&2&2&2&0&2&2&2&2\\
&\multicolumn{1}{c}{4}&1&3&4&2&2&2&0&2&2&2&0&2&1&1&1\\
&\multicolumn{1}{c}{5}&3&4&4&2&3&2&1&3&1&3\\
&\multicolumn{1}{c}{6}&0&1&3&2&2&1&1&2&2&4\\
&\multicolumn{1}{c}{7}&1&3&4&3&3&2&1&3&3&3\\
&\multicolumn{1}{c}{8}&3&4&4&3&3&0&1&3&2&2\\
&\multicolumn{1}{c}{9}&0&1&3&3&3&1&2&2&1&1\\
&\multicolumn{1}{c}{10}&1&3&4&4&3\\
&\multicolumn{1}{c}{11}&3&4&4&4&4\\
&\multicolumn{1}{c}{12}&0&1&3&4&3\\
&\multicolumn{1}{c}{13}&1&3&4&4&2\\
&\multicolumn{1}{c}{14}&3&4&3&2&1\\
\bottomrule
\end{tabular}
\label{tabny2}
\end{table}
\end{Example}
\newpage
\begin{Example}\label{exny2}
In this example we bound the number of zeros of multiplicity
$3$ or more for polynomials in four
variables. The sets $S_1$, $S_2$, $S_3$ and $S_4$ are all assumed to
be of size $6$. Table~\ref{tabny3} shows
information obtained from our algorithm for a small sample of values
$(i_1,i_2,i_3=3,i_4=5)$. Table~\ref{tabny4} illustrates the
improvement on the bound $\min\{
(i_1+i_2+i_3+i_4)6^3/3,6^4\}$. Here, the first expression comes from
Corollary~\ref{cor-sz-gen} and the last expression is the number of
points in $S_1 \times S_2\times S_3\times S_4$.
\begin{table}
\centering
\caption{$D(i_1,i_2,i_3=3,i_4=5,3,6,6,6,6)$}
\newcommand{~~}{~\hspace*{0.851em}}
\begin{tabular}{@{}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
& &\multicolumn{8}{c}{$i_1$} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\addlinespace
\multirow{8}{*}{$i_2$}
&0&468&486&504& 642& 666& 720& 912& 912\\
&1&486&501&536& 651& 705& 764& 964& 964\\
&2&504&536&574& 700& 759& 840&1024&1024\\
&3&642&651&666& 771& 816& 908&1077&1077\\
&4&666&684&732& 807& 880& 984&1140&1140\\
&5&720&750&816& 876& 952&1056&1197&1197\\
&6&912&912&960& 976&1024&1134&1260&1260\\
&7&912&928&980&1008&1060&1155&1263&1263\\
\bottomrule
\end{tabular}
\label{tabny3}
\end{table}
\begin{table}
\centering
\caption{Improvements found in Example~\ref{exny2}}
\newcommand{~~}{~\hspace*{0.851em}}
\begin{tabular}{@{}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
& &\multicolumn{8}{c}{$i_1$} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\addlinespace
\multirow{8}{*}{$i_2$}
&0&108&162&216&150&198&216&96 &168\\
&1&162&219&256&213&231&244&116&188\\
&2&216&256&290&236&249&240&128&200\\
&3&150&213&270&237&264&244&147&219\\
&4&198&252&276&273&272&240&156&156\\
&5&216&258&264&276&272&240&99 &99 \\
&6&96 &168&192&248&272&162&36 &36 \\
&7&168&224&244&288&236&141&33 &33 \\
\bottomrule
\end{tabular}
\label{tabny4}
\end{table}
\end{Example}
\newpage
\begin{Example}\label{exangle}
Let $s_1=\cdots =s_m=q$. Our experiments listed in~\cite{hmpage} show that the value $D(i_1, \ldots ,
i_m,r,q,\ldots ,q)$
often improves dramatically on the previous known bounds. We here list the
maximal attained improvement for a selection of fixed values of $m, q,
r$. We do this relatively to the number of points in $S_1 \times
\cdots \times S_m$. In other words we list in Table~\ref{tabny5} the value
$$\bigg(\max_{i_1, \ldots , i_m} \{\min\{ (i_1+\cdots
i_m)q^{m-1}/r,q^m\}-D(i_1, \ldots , i_m,r,q,\ldots ,q)\}\bigg)/q^m$$
for various choices of $m, q, r$.
\begin{table}
\centering
\caption{Maximum improvements relative to $q^m$; truncated}
\newcommand{~~}{~~}
\begin{tabular}{@{}c@{}c@{~~~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{}}
\toprule
$m$&& \multicolumn{8}{c}{2} & \multicolumn{8}{c}{3} & \multicolumn{4}{c}{4} \\
\cmidrule(r){3-10} \cmidrule(r){11-18} \cmidrule{19-22}
$r$&&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}\\
\addlinespace
\multirow{7}{*}{$q$}
&\multicolumn{1}{c}{2}&0&25 &0&25 &0&25 &0&25 &0&25 &0&375&0&375&0&375&0&312&0&375\\
&\multicolumn{1}{c}{3}&0&222&0&222&0&222&0&222&0&296&0&296&0&296&0&296&0&296&0&333\\
&\multicolumn{1}{c}{4}&0&187&0&187&0&187&0&187&0&281&0&25 &0&25 &0&265&0&316&0&289\\
&\multicolumn{1}{c}{5}&0&24 &0&16 &0&16 &0&2 &0&256&0&256&0&232&0&24 &0&307&0&288\\
&\multicolumn{1}{c}{6}&0&222&0&194&0&166&0&166&0&277&0&25 &0&231&0&212&0&293&0&287\\
&\multicolumn{1}{c}{7}&0&204&0&204&0&163&0&142&0&279&0&244&0&227&0&209&0&299&0&276\\
&\multicolumn{1}{c}{8}&0&234&0&203&0&171&0&140&0&275&0&25 &0&214&\multicolumn{2}{l}{?}&0&299&0&275\\
\bottomrule
\end{tabular}
\label{tabny5}
\end{table}
The experiments also show a distinct average improvement. This is
illustrated in Table~\ref{tabendnuenny1} where for fixed $q, r, m$ we list
the mean value of
\begin{equation}
\frac{ \min \{(i_1+\cdots + i_m)q^{m-1},q^m\}-D(i_1, \ldots ,
i_m,r,q, \ldots ,q)}{\min \{(i_1+\cdots +
i_m)q^{m-1},q^m\}}.\label{eqangle}
\end{equation}
The average is taken over the set of exponents $(i_1, \ldots ,
i_m)\neq \vec{0}$ where
$\lfloor i_1/q\rfloor+\cdots +\lfloor i_m/q\rfloor
< r$
holds.
\begin{table}[!h]
\centering
\caption{The mean value of (\ref{eqangle}); truncated}
\newcommand{~~}{~~}
\begin{tabular}{@{}c@{}c@{~~~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{}}
\toprule
$m$&& \multicolumn{8}{c}{2} & \multicolumn{8}{c}{3} & \multicolumn{4}{c}{4} \\
\cmidrule(r){3-10} \cmidrule(r){11-18} \cmidrule{19-22}
$r$&&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}\\
\addlinespace
\multirow{7}{*}{$q$}
&\multicolumn{1}{c}{2}&0&363&0&273&0&337&0&291&0&301&0&300&0&342&0&307&0&248&0&260\\
&\multicolumn{1}{c}{3}&0&217&0&286&0&228&0&236&0&194&0&224&0&213&0&214&0&158&0&177\\
&\multicolumn{1}{c}{4}&0&191&0&197&0&232&0&195&0&158&0&169&0&180&0&172&0&125&0&135\\
&\multicolumn{1}{c}{5}&0&155&0&167&0&174&0&197&0&139&0&145&0&148&0&153&0&110&0&116\\
&\multicolumn{1}{c}{6}&0&148&0&160&0&156&0&154&0&128&0&132&0&132&0&131&0&100&0&105\\
&\multicolumn{1}{c}{7}&0&128&0&137&0&138&0&138&0&119&0&122&0&121&0&119&0&093&0&098\\
&\multicolumn{1}{c}{8}&0&126&0&127&0&134&0&126&0&114&0&115&0&113&\multicolumn{2}{l}{?}&0&089&0&093\\
\bottomrule
\end{tabular}
\label{tabendnuenny1}
\end{table}
\end{Example}
\begin{Example}
In Example~\ref{exny1} for any total degree $d$ there exists a choice of
$i_1,\ldots , i_m$ with $i_1+\cdots +i_m=d$ such that
$$D(i_1,\cdots , i_m,r,q,\ldots , q)=\min\{ dq^{m-1}/r,q^m\}.$$
However, there are cases where such a result does not hold. Going through
all possible choices of $i_1,i_2,i_3$ with $i_1+i_2+i_3=12$ we see
that the largest obtained value of $D(i_1,i_2,i_3,3,8,8,8)$ equals
$224$ whereas $\min\{12\cdot 8^2/3,8^3\}=256$.
\end{Example}
The next two examples are of a theoretical nature.
\begin{Example}\label{ex1}
Given an arbitrary monomial ordering let ${\mbox{lm}}(F)=X_1^{i_1} \cdots X_m^{i_m}$ with $i_1 \leq s_1, \ldots , i_m
\leq s_m$. Using results from Gr\"{o}bner
basis theory we can deduce that $F$ can have no more than
\begin{equation}
s_1\cdots s_m-(s_1-i_1)\cdots (s_m-i_m) \label{eqrlig1}
\end{equation}
zeros (of multiplicity $1$ or more) over $S_1 \times \cdots \times S_m$. (See~\cite{hyperbolic} and
\cite{secondweight} for the case of $S_1=\cdots=S_m={\mathbf{F}}_q$.)
This result is known to be sharp meaning that polynomials exist with
this many zeros.
It is interesting to observe that~(\ref{eqrlig1}) follows as
an immediate corollary to Theorem~\ref{prorec} in the case where
the monomial ordering $\prec$ is the pure lexicographic ordering with
$X_m\prec \cdots \prec X_1$. In contrast~(\ref{eqrlig1})
only equals the result in Corollary~\ref{cor-sz-gen} when
${\mbox{lm}}(F)$ is univariate; in general the
two bounds can differ very much. In
Section~\ref{secismall} we will see that for the case of the monomial
ordering being $\prec$
(\ref{eqrlig1}) can be viewed as a special case
of a more general result.
\end{Example}
\begin{Example}
Consider that the leading monomial is
univariate, i.e.\ ${\mbox{lm}}(F)=X_t^{i_t}$ for some $t \in \{1, \ldots
, m\}$. Theorem~\ref{prorec} tells us that $F$ can have at most
$$s_1 \cdots s_{t-1} \lfloor \frac{i_t}{r}\rfloor s_{t+1} \cdots s_m$$
zeros of multiplicity $r$ or more. In contrast Corollary~\ref{cor-sz-gen} only
gives us the bound
$$\lfloor s_1\cdots s_{t-1} \frac{i_t}{r}s_{t+1}\cdots s_m \rfloor.$$
For $m >1$ the bounds are the same only when $r$ divides $i_t$. Assume in
larger generality that $i_{t_1}, \ldots ,i_{t_v}$, $t_u < t_w$ for $u
< w$ are the non-zero elements in $\{i_1, \ldots , i_m\}$. Then
$$D(i_1, \ldots , i_m,r,s_1, \ldots ,s_m)=
\bigg( \prod_{i_d=0}s_d \bigg) D(i_{t_1},\ldots,i_{t_v},r,s_{t_1}, \ldots , s_{t_v}).$$
\end{Example}
{\section{The case of two variables}\label{sectwovar}}
\noindent In this section we derive closed formulas for the case of two variables and
the multiplicity being arbitrary. By
Remark~\ref{rembig} the following Proposition covers all non-trivial cases.\\
\begin{Proposition}\label{protwovar}
For $k=1, \ldots , r-1$, $D(i_1,i_2,r,s_1,s_2)$ is upper bounded by\\
$\begin{array}{cl}
{\mbox{(C.1)}}& {\displaystyle{s_2\frac{i_1}{r}+\frac{i_2}{r}\frac{i_1}{r-k}}}\\
&{\mbox{if \ }}(r-k)\frac{r}{r+1}s_1 \leq i_1 < (r-k)s_1
{\mbox{ \ and \ }} 0\leq i_2 <ks_2\\
{\mbox{(C.2)}}&
{\displaystyle{s_2\frac{i_1}{r}+((k+1)s_2-i_2)(\frac{i_1}{r-k}-\frac{i_1}{r})+(i_2-ks_2)(s_1-\frac{i_1}{r})}}\\
& {\mbox{if \ }}(r-k)\frac{r}{r+1}s_1 \leq i_1 < (r-k)s_1 {\mbox{ \
and \ }} ks_2\leq i_2 <(k+1)s_2\\
{\mbox{(C.3)}}&
{\displaystyle{s_2\frac{i_1}{r}+\frac{i_2}{k+1}(s_1-\frac{i_1}{r})}}\\
&{\mbox{if \ }} (r-k-1)s_1 \leq i_1 < (r-k)\frac{r}{r+1}s_1 {\mbox{ \
and \ }} 0 \leq i_2 < (k+1)s_2.
\end{array}
$\\
Finally,\\
$\begin{array}{cl}
{\mbox{(C.4)}}& {\displaystyle{D(i_1,i_2,r,s_1,s_2)=s_2\lfloor \frac{i_1}{r} \rfloor
+i_2(s_1-\lfloor \frac{i_1}{r} \rfloor )}}\\
& {\mbox{if \ }} s_1(r-1) \leq i_1 < s_1r {\mbox{ \ and \ }} 0 \leq i_2 < s_2.
\end{array}
$\\
The above numbers are at most equal to $\min\{(i_1s_2+s_1i_2)/r, s_1s_2 \}$.
\end{Proposition}
\begin{proof}First we consider the values of $i_1, i_2,
r,s_1,s_2$ corresponding to one of the cases (C.1), (C.2), (C.3). Let
$k$ be the largest number (as in Proposition~\ref{protwovar})
such that $i_1 < (r-k)s_1$. Indeed $k \in \{1, \ldots ,
r-1\}$. We have
\begin{multline}
\label{eqsnabelen}
D(i_1,i_2,r,s_1,s_2) \leq\\
\max_{(u_1, \ldots ,u_r)\in B(i_2,r,s_2)} \bigg\{
s_2\frac{i_1}{r}+u_1(\frac{i_1}{r-1}-\frac{i_1}{r})+\cdots
+u_k(\frac{i_1}{r-k}-\frac{i_1}{r})\\
+u_{k+1}(s_1-\frac{i_1}{r})+ \cdots +u_r(s_1-\frac{i_1}{r})\bigg\}
\end{multline}
where
\begin{multline*}
B(i_2,r,s_2)=\{(u_1, \ldots , u_r) \in {\mathbf{Q}}^r \mid 0 \leq u_1,
\ldots ,u_r,\\
u_1+\cdots +u_r \leq s_2, u_1+2u_2+
\cdots +ru_r \leq i_2\}.
\end{multline*}
We observe, that
$$k(\frac{i_1}{r-l}-\frac{i_1}{r})\leq l
(\frac{i_1}{r-k}-\frac{i_1}{r})$$
holds for $l \leq k$. Furthermore, we have the biimplication
\begin{eqnarray}
(r-k)\frac{r}{r+1} s_1
\leq i_1
\Leftrightarrow (k+1)(\frac{i_1}{r-k}-\frac{i_1}{r}) \geq
k(s_1-\frac{i_1}{r}). \nonumber
\end{eqnarray}
Therefore, if the conditions in (C.1) are satisfied then~(\ref{eqsnabelen}) takes on its maximum when
$u_k=\frac{i_2}{k}$ and the remaining $u_i$'s equal $0$. If the
conditions in (C.2) are satisfied then (\ref{eqsnabelen}) takes on its
maximum at $u_k=(k+1)s_2-i_2$, $u_{k+1}=(i_2-ks_2)$ and the remaining
$u_i$'s equal $0$. If the conditions in (C.3) are satisfied
then~(\ref{eqsnabelen}) takes on its maximal value at
$u_{k+1}=\frac{i_2}{k+1}$ and the remaining $u_i$'s equal $0$.\\
Finally, if $s_1(r-1) \leq i_1 < s_1r$ and $0 \leq i_2 \leq s_2$ then
$D(i_1,i_2,r,s_1,s_2)$ is the maximal value of
$$s_2\lfloor \frac{i_1}{r} \rfloor +u_1(s_1-\lfloor \frac{i_1}{r}
\rfloor)+ \cdots +u_r(s_1-\lfloor \frac{i_1}{r}
\rfloor)$$
over $B(i_2,r,s_2)$. The maximum is attained for $u_1=i_2$ and all other
$u_i$'s equal $0$. The proof of the last result follows the proof of
the last part of Theorem~\ref{prorec}.
\end{proof}
\begin{Remark}
Experiments show (see~\cite{hmpage}) that
the numbers produced by Proposition~\ref{protwovar} are often much
smaller than $\min \{ (i_1s_2 +s_1i_2)/r,s_1s_2\}$. However, there are
cases where they are the identical. This happens for example when
$i_1=s_1(r-1)$ and $r$ divides $s_1$ and $s_2$. In the
proof of (C.1), (C.2), (C.3) we allowed $u_1, \ldots , u_r$ to be
rational numbers rather than integers. Therefore we cannot expect the upper bounds in
Proposition~\ref{protwovar} to equal the true value of
$D(i_1,i_2,r,s_1,s_2)$ in general. Our experiments show that the
bounds in (C.1), (C.2), (C.3) are sometimes close to
$D(i_1,i_2,r,s_1,s_2)$ but not always. Hence the best information is
found by actually applying the algorithm from the previous section.
\end{Remark}
{\section{When $i_1, \ldots ,i_m$ are small}\label{secismall}}
\noindent
Having already four different cases when $m=2$ the situation gets
rather complicated when we have more variables. Assuming however that
all exponents $i_1, \ldots , i_m$ in the leading monomial are small we
can give a very simple formula. Whereas the formula is simple we must
admit that the precise definition of $i_1, \ldots , i_m$ being small
is a little involved. It goes as follows.
\begin{Definition}\label{defconda}
Let $m \geq 2$. We say that $(i_1, \ldots , i_m,r,s_1, \ldots , s_m)$ satisfies Condition A if the following hold
$$
\begin{array}{rl}
(A.1)& i_1, \ldots ,i_m \leq s_m\\
(A.2)&s(s_1-\frac{i_1}{l}) \cdots (s_{m-2}-\frac{i_{m-2}}{l}) \leq l
(s_1-\frac{i_1}{s})\cdots (s_{m-2}-\frac{i_{m-2}}{s})\\
& {\mbox{ \ for all \ }}
l=2, \ldots , r, s=1, \ldots l-1.\\
(A.3)&s(s_1-\frac{i_1}{r}) \cdots (s_{m-1}-\frac{i_{m-1}}{r}) \leq r
(s_1-\frac{i_1}{s})\cdots (s_{m-1}-\frac{i_{m-1}}{s})\\
&{\mbox{ \ for all \ }} s=1, \ldots
, r-1.
\end{array}
$$
\end{Definition}
\begin{Example}\label{exny}
If $r=1$ then (A.2) and (A.3) do not apply and with a reference to
Remark~\ref{rembig} (A.1) is a natural requirement.
\end{Example}
\begin{Example}\label{exsmart}
For $m=2$ and $r$ arbitrary condition (A.2) does not apply and
condition (A.3)
simplifies to
$$i_1 \leq \frac{r^2s-rs^2}{r^2-s^2}s_1$$
for all $s$ with $1 \leq s < r$. The minimal upper bound on
$i_1$ is attained for $s=1$. Hence, in case of two variables Condition A reads $i_1
\leq \frac{r}{r+1}s_1$, $i_2 \leq s_2$.
\end{Example}
\begin{Example}\label{exsmalll}
For $r=2$ conditions (A.2), (A.3) simplifies all together to
$$\big(s_1-\frac{i_1}{2}\big) \cdots\big(s_{m-1}-\frac{i_{m-1}}{2}\big) \leq
2\big(s_1-i_1\big) \cdots\big(s_{m-1}-i_{m-1}\big).$$
For $r=2$, $m=3$ and $s_1=s_2=s_3=q$ Condition A therefore reads
$$\frac{3}{2}(I_1+I_2)-\frac{7}{4}I_1I_2 \leq 1, {\mbox{ \ \ }} I_3
\leq 1$$
where $I_1=i_1/q$, $I_2=i_2/q$ and $I_3=i_3/q$. For $r=2$, $m=4$ and $s_1=s_2=s_3=s_4=q$ Condition A reads
$$\frac{3}{2}(I_1+I_2+I_3)-\frac{7}{4}(I_1I_2+I_1I_3+I_2I_3)+\frac{15}{8}I_1I_2I_3 \leq 1, {\mbox{ \ \ }} I_4 \leq 1$$
where $I_4=i_4/q$. This is illustrated in Figure~\ref{figimp}.
\begin{figure}
\begin{center}
\input{impl-bw}
\end{center}
\caption{The surface $\frac{3}{2}(I_1+I_2+I_3)-\frac{7}{4}(I_1I_2+I_1I_3+I_2I_3)+\frac{15}{8}I_1I_2I_3 = 1$}
\label{figimp}
\end{figure}
\end{Example}
In Example~\ref{ex1} we discussed a well known bound on the number of
zeros of
multiplicity at least $r=1$. With Example~\ref{exny} in mind the last part of the following Proposition can be viewed as
a generalization of this bound.
\begin{Proposition}\label{prosmall}
Assume that $(i_1, \ldots , i_m,r,s_1, \ldots , s_m)$, $m\geq 2$ satisfies Condition
A. If $r \geq 2$ then
\begin{equation}
i_1 \leq \frac{r}{r+1}s_1, \ldots , i_{m-1} \leq \frac{r}{r+1}s_{m-1}.\label{eqeqeq}\end{equation}
For general $r$ we have
\begin{eqnarray}
D(i_1, \ldots ,i_m,r,s_1,\ldots , s_m) \leq s_1\cdots s_m-(s_1-\frac{i_1}{r})\cdots
(s_m-\frac{i_m}{r})\label{eqendnuenstjerne}
\end{eqnarray}
which is at most equal to $\min \{ (i_1s_2\cdots s_m+\cdots +s_1\cdots
s_{m-1}i_m)/r, s_1\cdots s_m\}$.
\end{Proposition}
\begin{proof}
We start by noting that (A.2) implies
$$
(s_1-\frac{i_1}{l}) \cdots (s_{t-1}-\frac{i_{t-1}}{l}) \leq l
(s_1-\frac{i_1}{s})\cdots (s_{t-1}-\frac{i_{t-1}}{s})
$$
for all
$t=2,\ldots , m-1$, $l=2, \ldots , r$, $s=1, \ldots l-1$. A similar
thing holds regarding (A.3) and if we combine this fact with the
result in Example~\ref{exsmart} we get
(\ref{eqeqeq}) for $r \geq 2$.
Now let $(i_1, \ldots ,i_m,r,s_1, \ldots ,s_m)$ with $m>1$ be
such that Condition A holds. We give an induction proof
that
\begin{equation}
\begin{array}{r}
D(i_1, \ldots , i_t,l,s_1,\ldots ,s_t) \leq s_1\cdots s_t-(s_1-\frac{i_1}{l})
\cdots (s_t-\frac{i_{t}}{l})\\
{\mbox{ \ for all \ }}1 \leq t < m , 1 \leq
l\leq r
\end{array}
\label{eqminus1}
\end{equation}
For $t=1$ the
result is clear. Let $1 < t<m$ and assume the result holds when $t$ is
substituted with $t-1$. According to Definition~\ref{defD} we have
\begin{multline*}
D(i_1, \ldots , i_t,l,s_1, \ldots ,s_t)=
\\
\begin{split}
\max_{(u_1, \ldots ,u_l)\in A(i_t,l,s_t) }\bigg\{& (s_t-u_1-\cdots -u_l)D(i_1,\ldots ,i_{t-1},l,s_1,
\ldots ,s_{t-1})\\
&+u_1D(i_1, \ldots , i_{t-1},l-1,s_1, \ldots ,s_{t-1})+\cdots
\\
&+u_{l-1}D(i_1, \ldots ,i_{t-1},1,s_1, \ldots ,
s_{t-1})+u_ls_1\cdots s_{t-1} \bigg\}
\end{split}
\end{multline*}
where
\begin{eqnarray}
A(i_t,l,s_t)&=&\{(u_1, \ldots ,u_l) \in {\mathbf{N}}_0^l \mid
u_1+2u_2+\cdots + l u_l \leq i_t\}\nonumber
\end{eqnarray}
follows from~(\ref{eqeqeq}). By the above assumptions this implies that
\begin{multline*}
D(i_1, \ldots ,i_t,l,s_1, \ldots ,s_t) \leq
\\
\shoveleft
\max_{(u_1, \ldots ,u_l)\in B(i_t,l,s_t) } \bigg\{ s_t\big(
s_1 \cdots s_{t-1}-(s_1-\frac{i_1}{l})\cdots(s_{t-1}-\frac{i_{t-1}}{l})\big)\\
\begin{split}
&+u_1\big((s_1-\frac{i_1}{l})\cdots(s_{t-1}-\frac{i_{t-1}}{l})-(s_1-\frac{i_1}{l-1})\cdots(s_{t-1}-\frac{i_{t-1}}{l-1}) \big)\\
&+ \cdots
\\
&+u_{l-1}\big(
(s_1-\frac{i_1}{l})\cdots(s_{t-1}-\frac{i_{t-1}}{l})-(s_1-\frac{i_1}{1})\cdots(s_{t-1}-\frac{i_{t-1}}{1})
\big)\\
&+u_{l}\big((s_1-\frac{i_1}{l})\cdots(s_{t-1}-\frac{i_{t-1}}{l})\big) \bigg\}
\end{split}
\end{multline*}
where $$B(i_{t},l,s_t)=\{(u_1, \ldots , u_l) \in {\mathbf{Q}}^l \mid 0
\leq u_1, \ldots , u_l {\mbox{ \ and \ }} u_1+2u_2+\cdots +lu_l \leq i_t\}.$$
As $t<m $ condition (A.2) applies and tells us that the maximal value
is attained for $u_1=\cdots =u_{l-1}=0$ and
$u_l=\frac{i_t}{l}$. This concludes the induction proof of (\ref{eqminus1}).\\
To show (\ref{eqendnuenstjerne}) we apply similar arguments
to the case
$t=m$
but use condition (A.3) rather than condition (A.2).
The proof of the last result in the proposition follows the proof of
the last part of Theorem~\ref{prorec}.
\end{proof}
\begin{Remark}
Experiments show (see~\cite{hmpage}) that the
bound in Theorem~\ref{prorec} is very often much better than $\min \{
(i_1s_2\cdots s_m+\cdots + s_1 \cdots s_{m-1}i_m)/r,s_1\cdots s_m\}$,
however, they also reveal that in many cases one can get more information about the number of zeros by actually
applying the algorithm from Section~\ref{sec-cor-sz-gen-bet}.
\end{Remark}
\begin{Example}
This is a continuation of Example~\ref{exsmart} where we translated
Condition A into bounds on $i_1$ and $i_2$ in the case of
two variables. Applying in turn Proposition~\ref{prosmall} and
(C.3) in Proposition~\ref{protwovar} with $k=r-1$ we see that the two
bounds produce the very same values when $m=2$.
\end{Example}
{\section{Products of univariate linear terms}\label{seclin}}
\noindent
In this section we study the situation where $F(\vec{X})$ is
a product of univariate linear terms. First we note that
equivalently to Defintion~\ref{defmult} one can define the
multiplicity of a polynomial as
follows.
\begin{Definition}
Let $F(\vec{X}) \in {\mathbf{F}}[\vec{X}]\backslash \{0\}$ and
$\vec{a}=(a_1, \ldots , a_m) \in {\mathbf{F}}^m$.
Consider the ideal
\begin{eqnarray}
J_t=\langle (X_1-a_1)^{p_1}\cdots(X_m-a_m)^{p_m} \mid p_1+\cdots
+p_m=t \rangle \subseteq {\mathbf{F}}[X_1, \ldots , X_m]. \nonumber
\end{eqnarray}
We have
${\mbox{mult}}(F,\vec{a})=r$ if $F\in J_r \backslash
J_{r+1}$. If $F=0$ we have ${\mbox{mult}}(F,\vec{a})=\infty$.
\end{Definition}
The above definition makes it particularly simple to calculate the number of
zeros of multiplicity at least $r$ when $F$ is a product of univariate
linear terms. In the following write
$$S_j=\{\alpha_1^{(j)}, \ldots , \alpha_{s_j}^{(j)} \}$$
for $j=1, \ldots , m$.
\begin{Proposition}\label{pronuogsaa}
Consider
$$F(\vec{X})=\prod_{u=1}^{m} \prod_{v=1}^{s_u}(X_u-\alpha_v^{(u)})^{r_{v}^{(u)}}.$$
The multiplicity of $(\alpha_{j_1}^{(1)}, \ldots ,
\alpha_{j_m}^{(m)})$ in $F(\vec{X})$ equals
\begin{equation}
r_{j_1}^{(1)}+\cdots +r_{j_m}^{(m)}. \label{eqHjubi}
\end{equation}
\end{Proposition}
\begin{proof}
Without loss of generality assume
$j_1=\cdots =j_m=1$. Clearly, the multiplicity is greater than or equal to
$r=r_1^{(1)}+\cdots +r_1^{(m)}$. Using Gr\"{o}bner basis theory we now
show that it is not larger. We substitute
${\mathcal{X}}_i=X_i-\alpha_1^{(i)}$ for $i=1, \ldots ,m$ and observe
that by Buchberger's S-pair criteria
$${\mathcal{B}}=\{{\mathcal{X}}_1^{r_1} \cdots {\mathcal{X}}_m^{r_m}
\mid r_1+\cdots +r_m=r+1\}$$
is a Gr\"{o}bner basis (with respect to any fixed monomial
ordering).
The support of $F({\mathcal{X}}_1, \ldots , {\mathcal{X}}_m)$ contains
a monomial of the form ${\mathcal{X}}_1^{i_1} \cdots
{\mathcal{X}}_m^{i_m}$ with $i_1+\cdots +i_m=r$.
Therefore the remainder of $F({\mathcal{X}}_1, \ldots ,
{\mathcal{X}}_m)$ modulo ${\mathcal{B}}$ is non-zero.
It is well known that if a
polynomial is reduced modulo a Gr\"{o}bner basis then the remainder is
zero if and only if it belongs to the ideal generated by the elements
in the basis.
\end{proof}
We now show that Theorem~\ref{prop-sz-gen} is tight. It follows of
course that so is Theorem~\ref{SZ-lemma} (a fact that has not been
stated in the literature).
\begin{Proposition}
Let $S_1, \ldots , S_m \subseteq {\mathbf{F}}$ be finite sets. If
$F(\vec{X})\in {\mathbf{F}}[\vec{X}]$ is a product
of univariate linear factors then the number of
zeros of $F$ counted with multiplicity reach the generalized Schwartz-Zippel bound
(Theorem~\ref{prop-sz-gen}).
\end{Proposition}
\begin{proof}
Consider the polynomial
$$F(\vec{X})=\prod_{u=1}^m \prod_{v=1}^{s_u}\big(X_u-\alpha_v^{(u)}\big)^{r_v^{(u)}}.$$
Write $i_u=\sum_{v=1}^{s_u}r_v^{(u)}$, $u=1, \ldots , m$. We have
\begin{align*}
\sum_{\vec{a} \in S_1 \times \cdots \times S_m}
{\mbox{mult}}(F,\vec{a})
&=\sum_{t=1}^{s_1}(s_2 \cdots s_m )r_t^{(1)}+\cdots +
\sum_{t=1}^{s_m}(s_1 \cdots s_{m-1} )r_t^{(m)}
\\
&=i_1s_2 \cdots s_{m} +\cdots +s_1
\cdots s_{m-1} i_m.
\qedhere
\end{align*}
\end{proof}
\begin{Example}\label{exbig}
Let $(i_1, \ldots , i_m, r, s_1,
\ldots ,s_m)$ be such that $\lfloor
i_1/s_1\rfloor+\cdots +\lfloor i_m/s_m\rfloor \geq r$.
As mentioned in Remark~\ref{rembig} there exist
polynomials with the leading monomial being $X_1^{i_1} \cdots X_m^{i_m}$ such that all points in $S_1 \times \cdots \times S_m$ are
zeros of multiplicity at least $r$. To see this define $r_1=\lfloor i_1/s_1 \rfloor, \ldots ,
r_m=\lfloor i_m/s_m \rfloor$. Multiplying
$$\prod_{u=1}^m
\prod_{v=1}^{s_u}\big(X_u-\alpha_v^{(u)}\big)^{r_u}$$
by an appropriate monomial we get a polynomial having the prescribed
leading monomial (with respect to any monomial ordering). Clearly, all points in the ensemble are zeros of multiplicity at least $r$.
\end{Example}
\begin{Definition}
Given $(i_1, \ldots , i_m,r, s_1, \ldots , s_m)$ consider the set of
polynomials that are products of univariate linear terms and have
$X_1^{i_1} \cdots X_m^{i_m}$ as leading monomial. By $H(i_1, \ldots
,i_m,r,s_1, \ldots , s_m)$ we denote the maximal number of zeros of
multiplicity at least $r$ that a polynomial from the above set can
have over $S_1 \times \cdots \times S_m$.
\end{Definition}
Based on Proposition~\ref{pronuogsaa} it is straightforward to
describe an iterative algorithm that finds $H(i_1, \ldots
, i_m,r,s_1, \ldots ,s_m)$. For the convenience of the reader we
include such an algorithm in Appendix~\ref{applin}.\\
In the previous sections we considered the general set of polynomials
$F$ with ${\mbox{lm}}_{\prec}(F)=X_1^{i_1} \cdots X_m^{i_m}$. We
gave upper bounds on the maximal attainable number of zeros of
multiplicity $r$ or more. It is clear that $H(i_1, \ldots , i_m,r,s_1,
\ldots ,s_m)$ serves as a lower bound on the maximal attainable number
of zeros of multiplicity $r$ or more.
In particular
$H(i_1, \ldots , i_m,r,s_1, \ldots , s_m) \leq D(i_1, \ldots ,
i_m,r,s_1, \ldots , s_m)$ holds. Experiments show (see~\cite{hmpage})
that the two functions are sometimes quite close. In the next
section we present various conditions under which the two functions
attain the same value. Clearly, when this happens we know what is the maximal number of
zeros of multiplicity at least $r$ that any polynomial with leading monomial
$X_1^{i_1} \cdots X_m^{i_m}$ can have over $S_1 \times \cdots \times S_m$.
\begin{Example}\label{exny4}
This is a continuation of Example~\ref{exny1} where we studied the
upper bound
$D(i_1,i_2,3,5,5)$ for relevant choices of $i_1, i_2$. Using the
algorithm in Appendix~\ref{applin} we calculated the corresponding
values of the lower bound $H(i_1,i_2,3,5,5)$. We list in Table~\ref{tabny6} the
difference $D(i_1,i_2,3,5,5)-H(i_1,i_2,3,5,5)$. We see that for many
choices of $i_1,i_2$ the upper bound equals the lower bound.
\begin{table}
\centering
\caption{Difference between upper and lower bound in Example~\ref{exny4}}\label{tabny6}
\newcommand{~~}{~\hspace*{0.851em}}
\begin{tabular}{@{}c@{}cc@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
&&\multicolumn{15}{c}{$i_1$}\\
&&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14\\
\addlinespace
\multirow{15}{*}{$i_2$}
&\multicolumn{1}{c}{0} &0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\
&\multicolumn{1}{c}{1} &0&0&0&0&1&0&1&1&1&1&2&1&1&1&0\\
&\multicolumn{1}{c}{2} &0&0&0&2&2&2&3&2&2&2&3&2&2&1&0\\
&\multicolumn{1}{c}{3} &0&0&0&0&0&1&1&1&3&1&4&3&2&2&0\\
&\multicolumn{1}{c}{4} &0&0&0&0&2&3&3&3&2&2&3&2&2&1&0\\
&\multicolumn{1}{c}{5} &0&0&0&2&2&3&2&2&0&0\\
&\multicolumn{1}{c}{6} &0&0&0&0&1&2&3&2&1&0\\
&\multicolumn{1}{c}{7} &0&0&0&0&2&3&3&3&1&0\\
&\multicolumn{1}{c}{8} &0&0&0&2&1&1&2&1&2&0\\
&\multicolumn{1}{c}{9} &0&0&0&0&1&2&2&1&1&0\\
&\multicolumn{1}{c}{10}&0&0&0&0&0\\
&\multicolumn{1}{c}{11}&0&0&0&1&0\\
&\multicolumn{1}{c}{12}&0&0&0&0&0\\
&\multicolumn{1}{c}{13}&0&0&0&0&0\\
&\multicolumn{1}{c}{14}&0&0&0&0&0\\
\bottomrule
\end{tabular}
\end{table}
\end{Example}
\begin{Example}\label{exny5}
This is a continuation of Example~\ref{exny2} where we studied
$D(i_1,i_2,i_3=3,i_4=5,3,6,6,6,6,)$ for a collection of values
$i_1,i_2$. In Table~\ref{tabny7} we list the difference between these
upper bounds and the lower bounds $H(i_1,i_2,i_3=3,i_4=5,3,6,6,6,6)$.
\begin{table}
\centering
\caption{Difference between upper and lower bound in Example~\ref{exny5}}
\newcommand{~~}{~\hspace*{0.851em}}
\begin{tabular}{@{}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{}}
\toprule
& &\multicolumn{8}{c}{$i_1$} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\addlinespace
\multirow{8}{*}{$i_2$}
&0& 72& 60& 48& 96& 90&114&216&186\\
&1& 60& 50& 60& 80&109&143&248&217\\
&2& 48& 60& 53&104&133&179&265&190\\
&3& 96& 80& 70&100&120&143&213&150\\
&4& 90& 88&106&111&112&114&168&120\\
&5&114&129&155&111& 82& 81&117& 84\\
&6&216&196&201&112& 52& 54& 72& 54\\
&7&186&181&146& 81& 40& 42& 57& 42\\
\bottomrule
\end{tabular}
\label{tabny7}
\end{table}
\end{Example}
\begin{Example}
Let $s_1=\cdots =s_m=q$. Our experiments listed in~\cite{hmpage} show that $D(i_1, \ldots ,
i_m,r,q,\ldots ,q)$ is often close to $H(i_1, \ldots , i_m,r,q,\ldots
,q)$. In Table~\ref{tabheart} we list the mean value of
\begin{equation}
\frac{D(i_1, \ldots , i_m,r,q,\ldots , q)-H(i_1, \ldots , i_m,r,q,\ldots , q)}{\frac{1}{2}\big(D(i_1, \ldots , i_m,r,q,\ldots , q)+H(i_1, \ldots , i_m,r,q,\ldots , q)\big)}.\label{eqheart}
\end{equation}
The average is taken over the set of exponents with
$\lfloor i_1/q\rfloor+\cdots +\lfloor i_m/q\rfloor
< r$ and $D(i_1,\ldots,i_m,r,q,\ldots, q)\neq 0$.
\begin{table}[!h]
\centering
\caption{Mean value of (\ref{eqheart}); rounded up}
\newcommand{~~}{~~}
\begin{tabular}{@{}c@{}c@{~~~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{~~}r@{.}l@{}}
\toprule
$m$&& \multicolumn{8}{c}{2} & \multicolumn{8}{c}{3} & \multicolumn{4}{c}{4} \\
\cmidrule(r){3-10} \cmidrule(r){11-18} \cmidrule{19-22}
$r$&&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}&\multicolumn{2}{l}{4}&\multicolumn{2}{l}{5}&\multicolumn{2}{l}{2}&\multicolumn{2}{l}{3}\\
\addlinespace
\multirow{7}{*}{$q$}
&\multicolumn{1}{c}{2}&0&044&0&066&0&08 &0&088&0&048&0&085&0&106&0&120&0&041&0&081\\
&\multicolumn{1}{c}{3}&0&039&0&048&0&068&0&075&0&046&0&067&0&092&0&104&0&038&0&064\\
&\multicolumn{1}{c}{4}&0&044&0&057&0&054&0&067&0&049&0&075&0&083&0&098&0&037&0&068\\
&\multicolumn{1}{c}{5}&0&042&0&057&0&061&0&060&0&045&0&073&0&086&0&092&0&034&0&064\\
&\multicolumn{1}{c}{6}&0&041&0&057&0&065&0&066&0&043&0&072&0&087&0&095&0&031&0&061\\
&\multicolumn{1}{c}{7}&0&040&0&054&0&063&0&066&0&042&0&069&0&085&0&094&0&030&0&058\\
&\multicolumn{1}{c}{8}&0&038&0&053&0&062&0&067&0&040&0&067&0&083&\multicolumn{2}{l}{?}&0&029&0&056\\
\bottomrule
\end{tabular}
\label{tabheart}
\end{table}
\end{Example}
{\section{Some conditions for $H = D$ to hold}\label{secequal}}
As the polynomial ring in one variable is a unique factorization
domain we get $H(i_1,r,s_1)=D(i_1,r,s_m)$ for all choices of $i_1, r,
s_1$. Experiments suggest (see~\cite{hmpage}) that for two
variables we have a similar equality for certain systematic choices of
$i_1, i_2$. For other choices of $i_1, i_2$ the picture is more
blurred. The results of our experiments further suggest that it might not be
an easy task
to say much about which values of $(i_1, \ldots . i_m, r, s_1, \ldots
, s_m)$ causes equality when $m \geq 3$. We summarize our
findings below.
\begin{Proposition}
For $\frac{r}{r+1}s_1 \leq i_1 < s_1$, $(r-1)s_2 \leq i_2 <rs_2$ we have
\begin{eqnarray}
H(i_1, i_2,r,s_1,s_2)=D(i_1,i_2,r,s_1,s_2) =rs_2i_1+i_2s_1-i_1i_2-(r-1)s_1s_2.\nonumber
\end{eqnarray}
\end{Proposition}
\begin{proof}
The value of $D$ is upper bounded by (C.2)
in Proposition~\ref{protwovar}. The value of $H$ is lower bounded by
studying the zeros of
\begin{multline*}
(X_2-\alpha_1^{(2)})^r\cdots
(X_2-\alpha_w^{(2)})^r(X_2-\alpha_{w+1}^{(2)})^{r-1} \cdots
\\
(X_2-\alpha_{s_2}^{(2)})^{r-1}(X_1-\alpha_1^{(1)})\cdots
(X_1-\alpha_{s_1}^{(1)})
\end{multline*}
where $w=i_2-(r-1)s_2$.
\end{proof}
We leave the proofs of the following two results for the reader.
\begin{Proposition}
Assume $r \leq s_1$. If $0 \leq i_1 < r$ and $0 \leq i_2 < rs_2$ holds
then
$$H(i_1,i_2,r,s_1,s_2)=D(i_1,i_2,r,s_1,s_2)=\lfloor i_2/r\rfloor
s_2+\delta$$
where $\delta=i_1-(r-w)+1$ if $r-w \leq i_1$ and $\delta=0$ otherwise.
\end{Proposition}
\begin{Proposition}
If $H(i_1, \ldots , i_m,r,s_1, \ldots ,s_m)=D(i_1, \ldots , i_m,r,s_1,
\ldots ,s_m)$ then
\begin{align*}
&H(i_1, \ldots , i_t,0,i_{t+1}, \ldots , i_m,r,s_1, \ldots ,
s_t,s^{\prime},s_{t+1}, \ldots ,s_m)\\
&=D(i_1, \ldots , i_t,0,i_{t+1}, \ldots , i_m,r,s_1, \ldots ,
s_t,s^{\prime},s_{t+1}, \ldots ,s_m)\\
&=s^{\prime}D(i_1, \ldots , i_m,r,s_1, \ldots ,s_m).
\end{align*}
\end{Proposition}
\begin{Proposition}
Assume $(i_1, \ldots ,i_m,r,s_1, \ldots , s_m)$ satisfies Condition A
(Definition~\ref{defconda}) and that $r$ divides $i_1, \ldots ,
i_m$. Then
\begin{align}
H(i_1, \ldots , i_m,r,s_1, \ldots ,s_m)
&=D(i_1, \ldots , i_m,r,s_1, \ldots ,s_m) \nonumber \\
&=s_1\cdots s_m-(s_1-\frac{i_1}{r}) \cdots (s_m-\frac{i_m}{r}). \nonumber
\end{align}
\end{Proposition}
\begin{proof}
Consider
$$\prod_{j=1}^m\prod_{v=1}^{i_j/r} (X_j-\alpha_v^{(j)})^r$$
and apply Proposition~\ref{prosmall}.
\end{proof}
{\section{Acknowledgments}\label{hallihallo}}
The authors wish to thank T.\ Mora, P.\ Beelen, D.\ Ruano
and T.\ H{\o}holdt for pleasant discussions. Thanks to L.\ Grubbe Nielsen
for linguistical assistance.
| {
"timestamp": "2009-12-21T16:13:22",
"yymm": "0912",
"arxiv_id": "0912.1436",
"language": "en",
"url": "https://arxiv.org/abs/0912.1436",
"abstract": "Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of variables and |S|. In the present work we take into account what is the leading monomial. This allows us to consider more general point ensembles and most importantly it allows us to produce much more detailed information about the number of zeros of multiplicity r than can be deduced from the generalized Schwartz-Zippel bound. We present both upper and lower bounds.",
"subjects": "Number Theory (math.NT); Commutative Algebra (math.AC); Rings and Algebras (math.RA)",
"title": "On the number of zeros of multiplicity r",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9702399094961359,
"lm_q2_score": 0.7310585669110203,
"lm_q1q2_score": 0.7093021977961232
} |
https://arxiv.org/abs/1603.07261 | A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets | The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating function form of these polynomial sets is given in Theorem 2.2. As applications of the Theorem 2.2, we revisit the d-orthogonal polynomial sets exist in the literature and discover new d-orthogonal polynomial sets. Moreover, we obtain the d-dimensional functional vector ensuring the d-orthogonality of these new polynomial sets. | \section{Introduction}
Recently, the generalization of orthogonal polynomials called "$d
-orthogonal polynomials" have attracted so much attention from many authors.
The well-known properties of orthogonal polynomials such as recurrence
relations, Favard theorem, generating function relations and differential
equations have found correspondence in this new notion. New polynomial sets
which contain classical orthogonal polynomials have been created so far. Let
us give a brief summary of d-orthogonal polynomials.
Let $\mathcal{P}$ be the vector space of polynomials with complex
coefficients and $\mathcal{P}^{^{\prime }}$be the vector space of all linear
functionals on $\mathcal{P}$ called the algebraic dual. $\left\langle
u,f\right\rangle $ is the representation of the effect of any linear
functional $u\in \mathcal{P}^{^{\prime }}$ to the polynomial $f\in \mathcal{
}$. Let $\left\{ P_{n}\right\} _{n\geq 0}$ be a polynomial set ($\deg \left(
P_{n}\right) =n$ for all non-negative integer $n$), and corresponding dual
sequence $\left( u_{n}\right) _{n\geq 0}$ for polynomials taken from this
set can be given by
\begin{equation*}
\left\langle u_{n},P_{k}\right\rangle =\delta _{nk}\text{ \ \ \ , \ \ \
n,k=0,1,...,\text{\ }
\end{equation*
where $\delta _{nk}$ is the Kronecker delta.
A polynomial set $\left\{ P_{n}\right\} _{n\geq 0}$ in $\mathcal{P}$ is said
to be $d$-orthogonal polynomial set with respect to the $d$-dimensional
functional vector $\Gamma =^{t}\left( u_{0},u_{1},...,u_{d-1}\right) $ if
the following orthogonality conditions are hold
\begin{equation}
\left\{
\begin{array}{cc}
\left\langle u_{k},P_{n}P_{m}\right\rangle =0\text{ ,} & m>nd+k\text{ ,} \\
\left\langle u_{k},P_{n}P_{nd+k}\right\rangle \neq 0\text{ ,} & n\geq 0\text{
,
\end{array
\right. \label{1}
\end{equation
where $n\in
\mathbb{N}
_{0}=\left\{ 0,1,2,...\right\} $, $d$ is a positive integer and $k\in
\left\{ 0,1,...,d-1\right\} $ $($see $\left[ 1-2\right] )$. Characterization
of these polynomials by recurrence relations and Favard type theorem was
also given in $\left[ 1-2\right] $. A polynomial set $\left\{ P_{n}\right\}
_{n\geq 0}$ is a $d$-orthogonal polynomial set if and only if it fulfills a
\left( d+1\right) $-order recurrence relation of the typ
\begin{equation}
xP_{n}\left( x\right) =\sum\limits_{k=0}^{d+1}\alpha _{k,d}\left( n\right)
P_{n-d+k}\left( x\right) \text{ \ \ \ , \ \ }n\in
\mathbb{N}
_{0}\text{ \ ,} \label{2}
\end{equation
with the regularity conditions $\alpha _{d+1,d}\left( n\right) \alpha
_{0,d}\left( n\right) \neq 0$ , $n\geq d$ and by convention $P_{-n}\left(
x\right) =0$ , $n\geq 1$. Taking $d=1$ in $\left( \ref{1}\right) $ and
\left( \ref{2}\right) $ leads us to the celebrated notion of orthogonal
polynomials $\left( \text{see \cite{3}}\right) $.
The recurrence relation of order $d+1$ has been the main reason of deriving
many $d$-orthogonal polynomial sets as an extension of known ones in
orthogonal polynomials. Classical orthogonal polynomials such as Laguerre,
Hermite and Jacobi polynomials, discrete orthogonal polynomials like
Charlier, Meixner polynomials and so on were extended to the $d-
orthogonality notion and many basic properties linking with these
polynomials were stated by various authors $\left( \left[ 4-23\right]
\right) $. Especially, in \cite{14}, the authors described a useful method
for checking whether if a given polynomial set $\left\{ P_{n}\right\}
_{n\geq 0}$ is $d$-orthogonal or not. This method will be described and used
in the sequel.
A polynomial set $\left\{ P_{n}\right\} _{n\geq 0}$ is called Sheffer
polynomial set if and only if it has the generating function of the form
\begin{equation}
A\left( t\right) e^{xH\left( t\right) }=\sum\limits_{n=0}^{\infty
}P_{n}\left( x\right) \frac{t^{n}}{n!}\text{ \ } \label{3}
\end{equation
where $A\left( t\right) $ and $H\left( t\right) $ have the power series
expansions as following
\begin{equation*}
A\left( t\right) =\sum\limits_{k=0}^{\infty }a_{k}t^{k}\text{ \ , \
a_{0}\neq 0\text{ , }H\left( t\right) =\sum\limits_{k=0}^{\infty
}h_{k}t^{k+1}\text{ \ , \ }h_{0}\neq 0\text{ .}
\end{equation*
This means that $A\left( t\right) $ is invertible and $H\left( t\right) $
has a compositional inverse. There are numerous polynomial sets belong to
the class of Sheffer polynomials $\left( \text{see }\left[ 24-25\right]
\right) $. Note that, for $H\left( t\right) =t$, we meet the definition of
Appell polynomial sets \cite{25} from the aspect of generating functions.
That is to say, Appell polynomials can be defined by generating function of
the typ
\begin{equation*}
A\left( t\right) e^{xt}=\sum\limits_{n=0}^{\infty }P_{n}\left( x\right)
\frac{t^{n}}{n!}\text{ \ \ }
\end{equation*
with $A\left( t\right) =\sum\limits_{k=0}^{\infty }a_{k}t^{k}$ $\left(
a_{0}\neq 0\right) $. In this contribution, our aim is to find the exact
form of $d$-orthogonal polynomial sets which are at the same time Sheffer
polynomial set. Then, we try to derive new $d$-orthogonal polynomial sets
and find some of them's $d$-dimensional functional vector for which promises
the $d$-orthogonality. Also, we revisit some known $d$-orthogonal polynomial
sets exist in the literature.
\section{Main Results}
Characterization problems related to Sheffer polynomial set have a deep
history. Both Meixner \cite{26} and Sheffer \cite{27} interested in the same
problem: what is the all possible forms of polynomial sets which are at the
same time orthogonal and Sheffer polynomials. They stated that $A\left(
t\right) $ and $H\left( t\right) $ of $\left( \ref{3}\right) $ should
satisfy the following conditions
\begin{eqnarray*}
\frac{1}{H^{^{\prime }}\left( t\right) } &=&\left( 1-\alpha t\right) \left(
1-\beta t\right) \text{ \ \ \ ,} \\
\frac{A^{^{\prime }}\left( t\right) }{A\left( t\right) } &=&\frac{\lambda
_{2}t}{\left( 1-\alpha t\right) \left( 1-\beta t\right) }\text{ \ \ .}
\end{eqnarray*
If we discuss all possible cases in view of these two conditions, then we
face with the known orthogonal polynomial sets listed below:
\textbf{Case 1: }$\alpha =\beta =0\Rightarrow $ Hermite polynomials.
\textbf{Case 2: }$\alpha =\beta \neq 0\Rightarrow $ Laguerre polynomials.
\textbf{Case 3: }$\alpha \neq 0$, $\beta =0\Rightarrow $ Charlier
polynomials.
\textbf{Case 4: }$\alpha \neq \beta $, $\left( \alpha \text{, }\beta \in
\mathbb{R}
\right) \Rightarrow $ Meixner polynomials.
\textbf{Case 5: }$\alpha \neq \beta $, $\left( \text{complex conjugate of
each other}\right) \Rightarrow $ Meixner-Pollaczek polynomials.
For more information see \cite{28}. Similar investigation was made in \cit
{29} for $2$-orthogonal polynomials. In order to solve a characterization
problem for $d$-orthogonality, we need the following lemma.
\begin{lemma}
$\left( \text{\cite{14}}\right) $ Let $\left\{ P_{n}\right\} _{n\geq 0}$ be
a polynomial set defined by the following relation
\begin{equation*}
G\left( x,t\right) =A\left( t\right) G_{0}\left( x,t\right)
=\sum\limits_{k=0}^{\infty }P_{n}\left( x\right) \frac{t^{n}}{n!}
\end{equation*
with $G_{0}\left( 0,t\right) =1$ and let $\hat{X}_{t}$ and $\sigma :=\hat{T
_{x}$ be the transform operator of the multiplication operator by $x$ and $t
, respectively. Thus,
\begin{equation}
\left\{
\begin{array}{c}
\hat{X}_{t}G\left( x,t\right) =xG\left( x,t\right) \text{ \ ,} \\
\hat{T}_{x}G\left( x,t\right) =tG\left( x,t\right) \text{ \ .
\end{array
\right. \label{4}
\end{equation
Then, $\left\{ P_{n}\right\} _{n\geq 0}$ is a $d$-orthogonal polynomial set
if and only if $\hat{X}_{t}\in \vee _{d+2}^{\left( -1\right) }$ where the
action of the set of operators $\tau \in \vee _{r}^{\left( -1\right) }$,
\left( r\geq 2\right) $, to $t^{n}$ i
\begin{equation*}
\left\{
\begin{array}{cc}
\tau \left( 1\right) =\sum\limits_{k=1}^{r-1}\alpha _{k-1}^{\left( k\right)
}t^{k-1}\text{ \ ,} & \\
\tau \left( t^{n}\right) =\sum\limits_{k=0}^{r-1}\alpha _{n}^{\left(
k\right) }t^{n+k-1}\text{ \ ,} & n\geq 1\text{ \ .
\end{array
\right.
\end{equation*
Here, $r$ complex sequences $\left\{ \alpha _{n}^{\left( k\right) }\right\}
_{n\geq 0}$ appear for $k=0,1,...,r-1$ with the condition $\alpha
_{n}^{\left( 0\right) }\alpha _{n}^{\left( r-1\right) }\neq 0$. Moreover,
the $d$-dimensional functional vector which ensures the $d$-orthogonality is
given by
\begin{equation}
\left\langle u_{i},f\right\rangle =\frac{1}{i!}\left[ \frac{\sigma ^{i}}
A\left( \sigma \right) }f\left( x\right) \right] _{x=0}\text{ \ , \ \
i=0,1,...,d-1\text{ \ , \ }f\in \mathcal{P}\text{ \ .} \label{5}
\end{equation}
\end{lemma}
From $\left( \ref{4}\right) $, It is obvious that $\sigma :=\hat{T}_{x}$ is
the lowering operator of $\left\{ P_{n}\right\} _{n\geq 0}$. Now, we can
state our main theorem.
\begin{theorem}
Let $\gamma _{d}\left( t\right) =\sum\limits_{k=0}^{d}\beta _{k}t^{k}$ be a
polynomial of degree $d$ $\left( \beta _{d}\neq 0\right) $ and $\sigma
_{d+1}\left( t\right) =\sum\limits_{k=0}^{d+1}\alpha _{k}t^{k}$ be a
polynomial of degree less than or equal to $d+1$. The only polynomial sets
\left\{ P_{n}\right\} _{n\geq 0}$ ,which are $d$-orthogonal and also Sheffer
polynomial set, are generated by
\begin{equation}
\exp \left[ \int\limits_{0}^{t}\frac{\gamma _{d}\left( s\right) }{\sigma
_{d+1}\left( s\right) }ds\right] \exp \left[ x\int\limits_{0}^{t}\frac{1}
\sigma _{d+1}\left( s\right) }ds\right] =\sum\limits_{n=0}^{\infty
}P_{n}\left( x\right) \frac{t^{n}}{n!} \label{6}
\end{equation
with the condition
\begin{equation}
\alpha _{0}\left( n\alpha _{d+1}-\beta _{d}\right) \neq 0\text{ \ },\text{ \
\ }n\geq 1\text{ \ \ .} \label{8}
\end{equation}
\end{theorem}
\begin{proof}
Let $\left\{ P_{n}\right\} _{n\geq 0}$ be a Sheffer polynomial set defined
by the generating function $\left( \ref{3}\right) $. Taking the derivative
of the both sides of the following equality
\begin{equation*}
G\left( x,t\right) =A\left( t\right) e^{xH\left( t\right) }
\end{equation*
with respect to $t$ leads t
\begin{equation*}
\left[ \frac{1}{H^{^{\prime }}\left( t\right) }D_{t}-\frac{A^{^{\prime
}}\left( t\right) }{A\left( t\right) H^{^{\prime }}\left( t\right) }\right]
G\left( x,t\right) =xG\left( x,t\right)
\end{equation*
where $D_{t}=\frac{d}{dt}$. Thus, from $\left( \ref{4}\right)
\begin{equation*}
\hat{X}_{t}=\frac{1}{H^{^{\prime }}\left( t\right) }D_{t}-\frac{A^{^{\prime
}}\left( t\right) }{A\left( t\right) H^{^{\prime }}\left( t\right) }\text{ \
.}
\end{equation*
If $\left\{ P_{n}\right\} _{n\geq 0}$ is a $d$-orthogonal polynomial set,
according to Lemma 2.1, $\hat{X}_{t}$ should belong to the set of operators
\vee _{d+2}^{\left( -1\right) }$. This means that following equalities must
be satisfie
\begin{equation}
\left\{
\begin{array}{c}
\frac{1}{H^{^{\prime }}\left( t\right) }=\sum\limits_{k=0}^{d+1}\alpha
_{k}t^{k}=\sigma _{d+1}\left( t\right) \\
\frac{A^{^{\prime }}\left( t\right) }{A\left( t\right) H^{^{\prime }}\left(
t\right) }=\sum\limits_{k=0}^{d}\beta _{k}t^{k}=\gamma _{d}\left( t\right)
\text{ \ , \ \ }\beta _{d}\neq 0\text{ \ ,
\end{array
\right. \label{7}
\end{equation
with the conditions $\left( \ref{8}\right) $. Solving equations $\left( \re
{7}\right) $ allows us to get the desired result given by $\left( \ref{6
\right) $.
Conversely, Let $\left\{ P_{n}\right\} _{n\geq 0}$ be a Sheffer polynomial
set generated by $\left( \ref{6}\right) $ with the conditions $\left( \ref{8
\right) $. Thus
\begin{equation*}
G\left( x,t\right) =\exp \left[ \int\limits_{0}^{t}\frac{\gamma _{d}\left(
s\right) }{\sigma _{d+1}\left( s\right) }ds\right] \exp \left[
x\int\limits_{0}^{t}\frac{1}{\sigma _{d+1}\left( s\right) }ds\right] \text{
\ \ .}
\end{equation*
If we apply derivative operator $D_{t}$ to the both sides of the above
equality, then we obtai
\begin{equation*}
\left[ \sigma _{d+1}\left( t\right) D_{t}-\gamma _{d}\left( t\right) \right]
G\left( x,t\right) =xG\left( x,t\right) \text{ \ \ .}
\end{equation*
In view of Lemma 2.1 and the conditions $\left( \ref{8}\right) $
\begin{equation*}
\hat{X}_{t}=\sigma _{d+1}\left( t\right) D_{t}-\gamma _{d}\left( t\right)
\in \vee _{d+2}^{\left( -1\right) }\text{ \ ,}
\end{equation*
so $\left\{ P_{n}\right\} _{n\geq 0}$ is a $d$-orthogonal polynomial set.
\end{proof}
\begin{remark}
This characterization of $d$-orthogonal Sheffer polynomial sets seems to be
new in this notion. For $d=1$, these results reduce to the ones obtained for
orthogonal polynomials which are summarized in the beginning of this section.
\end{remark}
Next, as applications of Theorem 2.2, we revisit some known $d$-orthogonal
polynomial sets which are at the same time Sheffer polynomial sets. Also, we
derive new $d$-orthogonal polynomial sets and find their $d$-dimensional
functional vector.
$\left( i\right) $ \textbf{Laguerre type }$d$\textbf{-orthogonal polynomial
sets}
During the last decade, authors have paid so much attention to extend
Laguerre polynomials to $d$-orthogonality. Thus, there are several
extensions of Laguerre polynomials in the context of $d$-orthogonality
\left( \text{see \cite{7}, \cite{14} and \cite{22}}\right) $. Now, taking
Theorem 2.2 into account, we revisit some of them which are explicitly
obtained from the couple of polynomials $\left[ \gamma _{d}\left( t\right)
\text{, }\sigma _{d+1}\left( t\right) \right] $.
\textbf{Application 1: }Let $\left\{ P_{n}\right\} _{n\geq 0}$ be a Sheffer
polynomial set generated by $\left( \ref{6}\right) $ with the following
couple of polynomials
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{d+1}\left( t\right)
\right] =\left[ -\left( \alpha +1\right) \left( 1-t\right) ^{d}\text{,
\frac{-1}{d}\left( 1-t\right) ^{d+1}\right] \text{ \ \ .}
\end{equation*
where $\alpha \neq -1$. After some calculations, thanks to Theorem 2.2, we
obtain a $d$-orthogonal polynomial set of the form
\begin{equation}
\left( 1-t\right) ^{-\left( \alpha +1\right) d}\exp \left\{ -x\left[ \left(
1-t\right) ^{-d}-1\right] \right\} =\sum\limits_{n=0}^{\infty }P_{n}\left(
x\right) \frac{t^{n}}{n!} \label{9}
\end{equation
with the conditions $\frac{n}{d}+\alpha +1\neq 0$. The $d$-orthogonality of
this polynomial set deeply investigated in \cite{22}. Also, the authors
stated basic properties of these polynomials.
\textbf{Application 2: }Assume that $\left\{ P_{n}\right\} _{n\geq 0}$ is a
Sheffer polynomial set which has the generating function of the form $\left(
\ref{6}\right) $ due to the couple of polynomials given belo
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{2}\left( t\right) \right]
=\left[ -\left( 1-t\right) ^{2}\pi _{d-1}^{^{\prime }}\left( t\right)
-\left( \alpha +1\right) \left( 1-t\right) \text{, }-\left( 1-t\right) ^{2
\right] \text{ \ \ .}
\end{equation*
Here $\pi _{d-1}\left( t\right) =\sum\limits_{k=0}^{d-1}a_{k}t^{k}$ with
a_{d-1}\neq 0$. According to Theorem 2.2, necessary computations lead us to
get $d$-orthogonal polynomial sets of the typ
\begin{equation}
e^{\pi _{d-1}\left( t\right) }\left( 1-t\right) ^{-\alpha -1}\exp \left(
\frac{-xt}{1-t}\right) =\sum\limits_{n=0}^{\infty }P_{n}\left( x\right)
\frac{t^{n}}{n!}\text{ \ \ .} \label{10}
\end{equation
$a_{d-1}\neq 0$ is enough to satisfy the conditions $\left( \ref{8}\right)
. These $d$-orthogonal polynomial sets found and studied in \cite{14}.
\begin{remark}
These two polynomial sets $\left( \ref{9}\right) $ and $\left( \ref{10
\right) $ are the generalizations of Laguerre polynomials to the $d
-orthogonal polynomials since we meet Laguerre polynomials for $d=1$.
\end{remark}
Now, we present a new $d$-orthogonal polynomial set for $d\geq 2$. It seems
that this polynomial set is a Laguerre type $d$-orthogonal polynomial set
but the difference is Laguerre polynomials are not generated hence $d\geq 2$.
\textbf{Application 3: }Suppose that $\left\{ P_{n}\right\} _{n\geq 0}$ is a
Sheffer polynomial set possessing the generating function $\left( \ref{6
\right) $ relate to the couple of polynomials
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{3}\left( t\right) \right]
=\left[ -\left( 1-t\right) ^{3}\pi _{d-2}^{^{\prime }}\left( t\right)
-\left( \alpha +1\right) \left( 1-t\right) ^{2}\text{, }-\left( 1-t\right)
^{3}\right] \text{ \ \ }
\end{equation*
where $\pi _{d-2}\left( t\right) =\sum\limits_{k=0}^{d-2}a_{k}t^{k}$ with
a_{d-2}\neq 0$. Then, taking Theorem 2.2 into account, we have a new $d
-orthogonal polynomial set generated by
\begin{equation*}
e^{\pi _{d-2}\left( t\right) }\left( 1-t\right) ^{-\alpha -1}\exp \left(
\frac{t^{2}-2t}{2\left( 1-t\right) ^{2}}\right) =\sum\limits_{n=0}^{\infty
}P_{n}\left( x\right) \frac{t^{n}}{n!}\text{ \ \ }
\end{equation*
for $d\geq 2$. Similarly, $a_{d-2}\neq 0$ guarantees the conditions $\left(
\ref{8}\right) $. Now, we deal with the case $d=2$ i.e.: a new $2
-orthogonal polynomial set. Thu
\begin{equation}
\left( 1-t\right) ^{-\alpha -1}\exp \left( x\frac{t^{2}-2t}{2\left(
1-t\right) ^{2}}\right) =\sum\limits_{n=0}^{\infty }P_{n}\left( x\right)
\frac{t^{n}}{n!}\text{ \ } \label{11}
\end{equation
with conditions $\alpha +n+1\neq 0$ , $n\geq 0$. Before finding the
corresponding linear functionals $u_{0}$ and $u_{1}$ of this new $2
-orthogonal polynomial set, we need to state following useful lemma.
\begin{lemma}
$\left( \text{\cite{30}}\right) $ Let $A\left( t\right) $ and $H\left(
t\right) $ be two power series given as in $\left( \ref{3}\right) $ and
\begin{equation*}
H^{\ast }\left( t\right) =\sum\limits_{k=0}^{\infty }h_{k}^{\ast }t^{k+1}
\end{equation*
is the compositional inverse of $H\left( t\right) $ such tha
\begin{equation*}
H\left( H^{\ast }\left( t\right) \right) =H^{\ast }\left( H\left( t\right)
\right) =t\text{ \ .}
\end{equation*
$\left( i\right) $ The lowering operator $\sigma :=\hat{T}_{x}$ of the
polynomial set $\left\{ P_{n}\right\} _{n\geq 0}$ generated by $\left( \re
{3}\right) $ is given with
\begin{equation*}
\sigma =H^{\ast }\left( D\right) \text{ \ \ , \ \ }D=\frac{d}{dx}\text{ \ .}
\end{equation*
$\left( ii\right) $ The lowering operator $\sigma :=\hat{T}_{x}$ of the
polynomial set $\left\{ P_{n}\right\} _{n\geq 0}$ generated b
\begin{equation*}
A\left( t\right) \left( 1+\omega H\left( t\right) \right) ^{\frac{x}{\omega
}=\sum\limits_{n=0}^{\infty }P_{n}\left( x\right) \frac{t^{n}}{n!}
\end{equation*
is given by
\begin{equation*}
\sigma =H^{\ast }\left( \Delta _{\omega }\right) \text{ \ \ , \ \ \ }\Delta
_{\omega }\left[ f\left( x\right) \right] =\frac{f\left( x+\omega \right)
-f\left( x\right) }{\omega }\text{.}
\end{equation*}
\end{lemma}
\begin{theorem}
The polynomial set $\left\{ P_{n}\right\} _{n\geq 0}$ generated by $\left(
\ref{11}\right) $ are $2$-orthogonal for $\alpha >-1$ with respect to the
following linear functional
\begin{eqnarray*}
\left\langle u_{0},f\right\rangle &=&\int\limits_{0}^{\infty }\Psi _{\alpha
,f}\left( x\right) e^{-x}dx\text{ \ \ \ , \ \ }f\in \mathcal{P}\text{ \ ,} \\
\left\langle u_{1},f\right\rangle &=&\int\limits_{0}^{\infty }\left[ \Psi
_{\alpha ,f}\left( x\right) -\Psi _{\alpha +1,f}\left( x\right) \right]
e^{-x}dx\text{ \ \ \ , \ \ }f\in \mathcal{P}\text{ \ ,}
\end{eqnarray*
where
\begin{equation*}
\Psi _{\alpha ,f}\left( x\right) =\frac{x^{\alpha }}{\Gamma \left( \alpha
+1\right) }\sum\limits_{k=0}^{\infty }\frac{f^{\left( k\right) }\left(
0\right) }{k!\left( \frac{\alpha +2}{2}\right) _{k}}\left( \frac{x^{2}}{2
\right) ^{k}\text{ \ \ \ ,}
\end{equation*
$\Gamma $ is the widely known Gamma function and $\left( a\right) _{n}$ is
the Pochhammer's symbol defined by the rising factoria
\begin{eqnarray*}
\left( a\right) _{n} &=&a\left( a+1\right) ...\left( a+n-1\right) \text{ \ \
, \ \ }n\geq 1\text{ \ ,} \\
\left( a\right) _{0} &=&1\text{ \ \ .}
\end{eqnarray*}
\end{theorem}
\begin{proof}
$\left( \ref{5}\right) $ yields that
\begin{equation*}
\left\langle u_{i},f\right\rangle =\frac{1}{i!}\left[ \frac{\sigma ^{i}}
A\left( \sigma \right) }f\left( x\right) \right] _{x=0}\text{ \ , \ \ }i=0,
\text{ \ , \ }f\in \mathcal{P}\text{ \ ,}
\end{equation*
where $\sigma $ is the lowering operator of $2$-orthogonal polynomial set
generated by $\left( \ref{11}\right) $. The lowering operator $\sigma $ of
this polynomial set is
\begin{equation*}
H\left( t\right) =\frac{t^{2}-2t}{2\left( 1-t\right) ^{2}}\Rightarrow \sigma
=H^{\ast }\left( D\right) =1-\left( 1-2D_{x}\right) ^{-1/2}
\end{equation*
where we use Lemma 2.5. Then, for $i=0$ and $A\left( t\right) =\left(
1-t\right) ^{-\alpha -1}$, we obtai
\begin{eqnarray*}
\left\langle u_{0},f\right\rangle &=&\left[ \left( 1-2D_{x}\right) ^{-\left(
\frac{\alpha +1}{2}\right) }f\left( x\right) \right] _{x=0} \\
&=&\sum\limits_{k=0}^{\infty }\frac{\left( \frac{\alpha +1}{2}\right)
_{k}2^{k}}{k!}f^{\left( k\right) }\left( 0\right) \\
&=&\sum\limits_{k=0}^{\infty }\frac{\Gamma \left( \alpha +2k+1\right) }
\Gamma \left( \alpha +1\right) \left( \frac{\alpha +2}{2}\right) _{k}2^{k}
\frac{f^{\left( k\right) }\left( 0\right) }{k!} \\
&=&\int\limits_{0}^{\infty }\left[ \frac{x^{\alpha }}{\Gamma \left( \alpha
+1\right) }\sum\limits_{k=0}^{\infty }\frac{f^{\left( k\right) }\left(
0\right) }{k!\left( \frac{\alpha +2}{2}\right) _{k}}\left( \frac{x^{2}}{2
\right) ^{k}\right] e^{-x}dx\text{ \ \ \ \ .}
\end{eqnarray*
Furthermore, we calculate in a similar manner for $i=1
\begin{eqnarray*}
\left\langle u_{1},f\right\rangle &=&\left[ \sum\limits_{r=0}^{1}\binom{1}{
}\left( -1\right) ^{r}\left( 1-2D_{x}\right) ^{-\left( \frac{\alpha +r+1}{2
\right) }f\left( x\right) \right] _{x=0} \\
&=&\sum\limits_{r=0}^{1}\binom{1}{r}\left( -1\right)
^{r}\sum\limits_{k=0}^{\infty }\frac{\left( \frac{\alpha +r+1}{2}\right)
_{k}2^{k}}{k!}f^{\left( k\right) }\left( 0\right) \\
&=&\sum\limits_{r=0}^{1}\binom{1}{r}\left( -1\right)
^{r}\sum\limits_{k=0}^{\infty }\frac{\Gamma \left( \alpha +r+2k+1\right) }
\Gamma \left( \alpha +r+1\right) \left( \frac{\alpha +r+2}{2}\right)
_{k}2^{k}}\frac{f^{\left( k\right) }\left( 0\right) }{k!} \\
&=&\int\limits_{0}^{\infty }\left[ \sum\limits_{r=0}^{1}\binom{1}{r}\left(
-1\right) ^{r}\frac{x^{\alpha +r}}{\Gamma \left( \alpha +r+1\right)
\sum\limits_{k=0}^{\infty }\frac{f^{\left( k\right) }\left( 0\right) }
k!\left( \frac{\alpha +r+2}{2}\right) _{k}}\left( \frac{x^{2}}{2}\right) ^{k
\right] e^{-x}dx\text{ \ .}
\end{eqnarray*
This finishes the proof.
\end{proof}
$\left( ii\right) $ \textbf{Hermite type }$d$\textbf{-orthogonal polynomial
sets}
Hermite type $d$-orthogonal polynomials were the first example of $d
-orthogonal polynomials which obtained constructively by Douak \cite{16}. He
discovered these polynomials as solution of the problem: Find all $d
-orthogonal polynomial sets which are at the same time Appell polynomials.
\textbf{Application 4: }Let $\left\{ P_{n}\right\} _{n\geq 0}$ be a Sheffer
polynomial set due to the generating function $\left( \ref{6}\right) $ and
the following couple of polynomials
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{0}\left( t\right) \right]
=\left[ \pi _{d+1}^{^{\prime }}\left( t\right) \text{, }1\right]
\end{equation*
where $\pi _{d+1}\left( t\right) =\sum\limits_{k=0}^{d+1}a_{k}t^{k}$ with
a_{d+1}\neq 0$. Theorem 2.2 allows us to present a $d$-orthogonal set
generated by
\begin{equation}
e^{\pi _{d+1}\left( t\right) }\exp \left( xt\right)
=\sum\limits_{n=0}^{\infty }P_{n}\left( x\right) \frac{t^{n}}{n!}\text{ \ \
} \label{12}
\end{equation
under the conditions $a_{d+1}\neq 0$ from $\left( \ref{8}\right) $. The only
polynomials which are $d$-orthogonal and Appell polynomials at the same time
are generated by $\left( \ref{12}\right) $. Also, these polynomial sets are
the generalization of Gould-Hopper polynomials \cite{32}. For $d=1$, we meet
again the Hermite polynomials. The properties of these polynomial sets were
intensively studied by Douak in \cite{16}.
$\left( iii\right) $ \textbf{Charlier type }$d$\textbf{-orthogonal
polynomial sets}
Also, Charlier polynomials extended to the notion of $d$-orthogonality. Now,
we revisit these polynomial sets owing to Theorem 2.2.
\textbf{Application 5: }Suppose that $\left\{ P_{n}\right\} _{n\geq 0}$ is a
Sheffer polynomial set generated by $\left( \ref{6}\right) $ associated with
the couple of polynomials
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{1}\left( t\right) \right]
=\left[ \left( 1+\omega t\right) \pi _{d}^{^{\prime }}\left( t\right) \text
, }1+\omega t\right]
\end{equation*
where $\pi _{d}\left( t\right) =\sum\limits_{k=0}^{d}a_{k}t^{k}$ with
a_{d}\neq 0$. From Theorem 2.2, we have a $d$-orthogonal polynomial set of the
form
\begin{equation}
e^{\pi _{d}\left( t\right) }\left( 1+\omega t\right) ^{\frac{x}{\omega
}=\sum\limits_{n=0}^{\infty }P_{n}\left( x\right) \frac{t^{n}}{n!}\text{ \
\ } \label{13}
\end{equation
with the conditions $a_{d}\neq 0$ from $\left( \ref{8}\right) $. The $d
-orthogonal polynomial set generated by $\left( \ref{13}\right) $ was found
in \cite{13}. A similar characterization problem stated in $\left( ii\right)
$ for discrete case was solved by the authors. It is obvious that the
generating function relation $\left( \ref{13}\right) $ yields Charlier
polynomials for $\left( d,\omega \right) =\left( 1,1\right) $.
$\left( iii\right) $ \textbf{Meixner type }$d$\textbf{-orthogonal polynomial
sets}
Another important member of discrete orthogonal polynomial sets called
Meixner polynomials were also generalized in the context of $d
-orthogonality \cite{14}. The authors discovered these polynomials by means
of a form of generating function and they found the linear functions $u_{0}$
and $u_{1}$ for the case $d=2$. Following application of Theorem 2.2 shows
that these polynomial sets can be obtained by the special case of the couple
of polynomials $\left[ \gamma _{d}\left( t\right) \text{, }\sigma
_{d+1}\left( t\right) \right] $.
\textbf{Application 6: }Assume that $\left\{ P_{n}\right\} _{n\geq 0}$ is a
Sheffer polynomial set having the generating function of the type $\left(
\ref{6}\right) $ for the couple of polynomials
\begin{equation*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{2}\left( t\right) \right]
=\left[ \frac{1}{c-1}\left( c-t\right) \left( 1-t\right) \pi
_{d-1}^{^{\prime }}\left( t\right) +\frac{\beta }{c-1}\left( c-t\right)
\text{, }\frac{1}{c-1}\left( c-t\right) \left( 1-t\right) \right] \text{ .}
\end{equation*
Here $\pi _{d-1}\left( t\right) =\sum\limits_{k=0}^{d-1}a_{k}t^{k}$ with
a_{d-1}\neq 0$ and $c\neq \left\{ 0,1\right\} $. Thanks to Theorem 2.2 and
this couple of polynomials, $\left( \ref{6}\right) $ generates the $d
-orthogonal polynomial sets given belo
\begin{equation}
e^{\pi _{d-1}\left( t\right) }\left( 1-t\right) ^{-\beta }\left( 1+\frac{c-
}{c}\frac{t}{1-t}\right) ^{x}=\sum\limits_{n=0}^{\infty }P_{n}\left(
x\right) \frac{t^{n}}{n!}\ \ \text{.} \label{14}
\end{equation
$a_{d-1}\neq 0$ and $c\neq \left\{ 0,1\right\} $ are sufficient enough the
conditions $\left( \ref{8}\right) $ hold true. One can find detailed
information of these polynomial sets in \cite{14}. It is easily seen that we
face with the Meixner polynomial set by taking $d=1$ in $\left( \ref{14
\right) $. Recently, a generalization of $d$-orthogonal Meixner polynomial
sets via quantum calculus has been given in \cite{31}. Next, we express a
new Meixner type $d$-orthogonal polynomial set and we \ find its $d
-dimensional functional vector.
\textbf{Application 7: }Let\textbf{\ }$\left\{ P_{n}\right\} _{n\geq 0}$ be
a Sheffer polynomial set generated by $\left( \ref{6}\right) $ according to
the couple of polynomials
\begin{eqnarray*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{d+1}\left( t\right)
\right] &=&\left[ \frac{dc\beta }{c-1}\left[ \left( 1-t\right) ^{d}+\frac
c-1}{dc}\left[ 1-\left( 1-t\right) ^{d}\right] \right] ,\right. \\
&&\left. \frac{c}{c-1}\left( 1-t\right) \left[ \left( 1-t\right) ^{d}+\frac
c-1}{dc}\left[ 1-\left( 1-t\right) ^{d}\right] \right] \right]
\end{eqnarray*
with the restrictions
\begin{equation}
\left\{
\begin{array}{c}
c\neq \left\{ 0,\frac{1}{1-d},1\right\} \\
\beta \neq -\frac{n}{d}\text{ \ , \ }n\geq 0\text{
\end{array
\right. \text{ \ .} \label{15}
\end{equation
After some computations, Theorem 2.2 allows us to introduce the following new
d$-orthogonal polynomial set with
\begin{equation}
\left( 1-t\right) ^{-\beta d}\left( 1+\frac{c-1}{dc}\left[ \frac{1}{\left(
1-t\right) ^{d}}-1\right] \right) ^{x}=\sum\limits_{n=0}^{\infty
}P_{n}\left( x\right) \frac{t^{n}}{n!}\ \ \text{.} \label{16}
\end{equation
The conditions $\left( \ref{8}\right) $ are satisfied from restrictions
\left( \ref{15}\right) $. It seems that this Meixner type $d$-orthogonal
polynomial set is the first explicit one among others in the literature.
\begin{theorem}
The $d$-dimensional functional vectors, which the $d$-orthogonality of the
polynomial set generated by $\left( \ref{16}\right) $ holds, are
\begin{equation}
\left\langle u_{r},f\right\rangle =\frac{1}{r!}\sum\limits_{i=0}^{r}\binom{
}{i}\left( -1\right) ^{i}\sum\limits_{j=0}^{\infty }\frac{\left( \beta
\frac{i}{d}\right) _{j}\left( \frac{dc}{1-c}\right) ^{j}}{\left( 1-\frac{dc}
c-1}\right) ^{\beta +\frac{i}{d}+j}j!}f\left( j\right) \label{19}
\end{equation
where $r=0,1,...,d-1$ and $f\in \mathcal{P}$.
\end{theorem}
\begin{proof}
Lemma 2.5 helps us to find the lowering operators of the $d$-orthogonal
polynomial set generated by $\left( \ref{16}\right) $ with
\begin{equation*}
H\left( t\right) =\frac{c-1}{dc}\left[ \frac{1}{\left( 1-t\right) ^{d}}-
\right] \Rightarrow \sigma =H^{\ast }\left( \Delta \right) =1-\left( 1-\frac
dc\Delta }{1-c}\right) ^{-\frac{1}{d}}
\end{equation*
where $\Delta f\left( x\right) =f\left( x+1\right) -f\left( x\right) $.
Thus, by using Lemma 2.1 and $\left( \ref{5}\right) $, we conclude that for
r=0,1,...,d-1$ and $f\in \mathcal{P}
\begin{eqnarray}
\left\langle u_{r},f\right\rangle &=&\frac{1}{r!}\left[ \frac{\sigma ^{r}}
A\left( \sigma \right) }f\left( x\right) \right] _{x=0} \notag \\
&=&\frac{1}{r!}\left[ \sum\limits_{i=0}^{r}\binom{r}{i}\left( -1\right)
^{i}\left( 1-\frac{dc\Delta }{1-c}\right) ^{-\left( \beta +\frac{i}{d
\right) }f\left( x\right) \right] _{x=0} \notag \\
&=&\frac{1}{r!}\sum\limits_{i=0}^{r}\binom{r}{i}\left( -1\right)
^{i}\sum\limits_{k=0}^{\infty }\frac{\left( \beta +\frac{i}{d}\right) _{k}}
k!}\left( \frac{dc}{1-c}\right) ^{k}\Delta ^{k}f\left( 0\right) \text{ \ \ .}
\label{17}
\end{eqnarray
Substituting the fact
\begin{equation*}
\Delta ^{k}f\left( 0\right) =\sum\limits_{j=0}^{k}\left( -1\right) ^{k-j
\binom{k}{j}f\left( j\right)
\end{equation*
into $\left( \ref{17}\right) $ and after shifting indices, we obtai
\begin{equation}
\left\langle u_{r},f\right\rangle =\frac{1}{r!}\sum\limits_{i=0}^{r}\binom{
}{i}\left( -1\right) ^{i}\sum\limits_{j=0}^{\infty }\left\{
\sum\limits_{k=0}^{\infty }\frac{\left( \beta +\frac{i}{d}\right)
_{k+j}\left( -1\right) ^{k}\left( \frac{dc}{1-c}\right) ^{k}}{k!}\right\}
\frac{\left( \frac{dc}{1-c}\right) ^{j}f\left( j\right) }{j!}\text{ .}
\label{18}
\end{equation
The equality $\left( \ref{18}\right) $ leads us to get the desired result by
applying the following property of the Pochhammer's symbo
\begin{equation*}
\left( \beta +\frac{i}{d}\right) _{k+j}=\left( \beta +\frac{i}{d}\right)
_{j}\left( \beta +\frac{i}{d}+j\right) _{k}\text{ \ \ .}
\end{equation*}
\end{proof}
\begin{remark}
For $d=1$, $\left( \ref{16}\right) $ reduces to the well known generating
function of Meixner polynomial set and $\left( \ref{19}\right) $ becomes the
following linear functional for Meixner polynomials
\begin{equation}
\left\langle u_{0},f\right\rangle =\left( 1-c\right) ^{\beta
}\sum\limits_{j=0}^{\infty }\frac{\left( \beta \right) _{j}\text{ }c^{j}}{j
}f\left( j\right) \label{20}
\end{equation
with $0<c<1$ and $\beta >0$. Meixner polynomial set is orthogonal with
respect to the linear functional given by $\left( \ref{20}\right) $.
\end{remark}
\textbf{Application 8: }Suppose that $\left\{ P_{n}\right\} _{n\geq 0}$ is a
Sheffer polynomial set represented by $\left( \ref{6}\right) $ associated to
the couple of polynomials
\begin{eqnarray*}
\left[ \gamma _{d}\left( t\right) \text{, }\sigma _{3}\left( t\right) \right]
&=&\left[ -\frac{c}{c-1}\left( 1-t\right) \left[ \left( 1-t\right) ^{2}
\frac{c-1}{2c}\left[ \left( 1-t\right) ^{2}-1\right] \right] \pi
_{d-2}^{^{\prime }}\left( t\right) \right. \\
&&\left. -\frac{\beta c}{c-1}\left[ \left( 1-t\right) ^{2}+\frac{c-1}{2c
\left[ \left( 1-t\right) ^{2}-1\right] \right] ,\right. \\
&&\left. -\frac{c}{c-1}\left( 1-t\right) \left[ \left( 1-t\right) ^{2}+\frac
c-1}{2c}\left[ \left( 1-t\right) ^{2}-1\right] \right] \right]
\end{eqnarray*
where $\pi _{d-2}\left( t\right) =\sum\limits_{k=0}^{d-2}a_{k}t^{k}$ with
a_{d-2}\neq 0$ and $c\neq \left\{ 0,\frac{1}{3},1\right\} $. Taking Theorem
2.2 into account, we derive a new $d$-orthogonal polynomial set for $d\geq 2$
wit
\begin{equation}
e^{\pi _{d-2}\left( t\right) }\left( 1-t\right) ^{-\beta }\left( 1+\frac{c-
}{2c}\frac{t^{2}-2t}{\left( 1-t\right) ^{2}}\right)
^{x}=\sum\limits_{n=0}^{\infty }P_{n}\left( x\right) \frac{t^{n}}{n!}
\label{21}
\end{equation
The conditions $\left( \ref{8}\right) $ are satisfied since $a_{d-2}\neq 0$
and $c\neq \left\{ 0,\frac{1}{3},1\right\} $. These $d$-orthogonal
polynomial sets $\left( \ref{21}\right) $ can not generate an orthogonal
polynomial set since $d\geq 2$. But for $d=2$, one can study the properties
of this Meixner type $2$-orthogonal polynomial set.
\subsection{Concluding Remarks}
Theorem 2.2 is the generalization of the characterization problem related to
the orthogonality of Sheffer polynomial set. The version of this problem
corresponding to $d=1$ and $d=2$ already exist in the literature. Then, it
is expected to find similar results for $d$-orthogonality. Although the
results obtained in Theorem 2.2 are expected and natural, this theorem
motivates us to derive new $d$-orthogonal polynomial sets as mentioned in
this paper. One can generate more $d$-orthogonal polynomial sets which are
Sheffer polynomial set at the same time with the help of Theorem 2.2.
| {
"timestamp": "2016-03-24T01:11:17",
"yymm": "1603",
"arxiv_id": "1603.07261",
"language": "en",
"url": "https://arxiv.org/abs/1603.07261",
"abstract": "The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating function form of these polynomial sets is given in Theorem 2.2. As applications of the Theorem 2.2, we revisit the d-orthogonal polynomial sets exist in the literature and discover new d-orthogonal polynomial sets. Moreover, we obtain the d-dimensional functional vector ensuring the d-orthogonality of these new polynomial sets.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9702399026119352,
"lm_q2_score": 0.7310585669110202,
"lm_q1q2_score": 0.7093021927633691
} |
https://arxiv.org/abs/2003.08083 | Additive Representations of Natural Numbers | Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number. | \section{Introduction}
In 1742, Goldbach conjectured that every even integer greater than two is the sum of two primes.
In 1966, Chen \cite{Chen66,Chen78} proved that every sufficiently large even integer is the sum of one prime and a product of at most two primes\footnote{Yamada \cite{Yamada2015} has shown that Chen's theorem holds for even integers larger than $\exp(\exp(38))$.}, and in 2013, Helfgott \cite{helfgott2013} proved the ternary Goldbach conjecture, which states every odd integer larger than five is the sum of three primes.
Since a complete proof of the Goldbach conjecture remains out of reach, we consider results where we relax one of the primes to be a square-free number instead. To this end, Dudek \cite{dudek2017} proved the following version of Estermann's result \cite{estermann1931}.
\begin{thm}[Dudek, 2017]\label{thm:dudek}
Every integer greater than two is the sum of a prime and a square-free number.
\end{thm}
A simple extension of Theorem \ref{thm:dudek} is given in Corollary \ref{thm:prime_prime_sq_free_theorem}.\footnote{To establish Corollary \ref{thm:prime_prime_sq_free_theorem}, observe that if $n > 4$, then $n-2 >2$, so Theorem \ref{thm:dudek} implies that $n-2 = p + \eta$ for at least one prime $p$ and square-free number $\eta$.}
\begin{cor}\label{thm:prime_prime_sq_free_theorem}
Every integer greater than four may be written as the sum of two primes and a square-free number.
\end{cor}
As an extension of Theorem \ref{thm:dudek}, Yau \cite{yau2019} established a uniform bound for the number of representations of an integer as a prime in a fixed residue class plus a square-free number.
Instead of placing constraints on the prime, we will impose an additional condition on the divisors of the square-free numbers in Theorem \ref{thm:dudek}.
That is, suppose $q$ is prime and $n > n_0$ where $n_0$ is small. Does there exist at least one prime $p$ and a square-free integer $\eta$ which is co-prime to $q$ and
\begin{equation}\label{eqn:referenceme_coprime_q}\tag{Q1}
n = p + \eta \, ?
\end{equation}
Moreover, does there exist at least one pair of primes $p_1$, $p_2$ and a square-free integer $\eta$ such that $(\eta, q)=1$ and
\begin{equation}\label{eqn:referenceme_two_coprime_q}\tag{Q2}
n = p_1 + p_2 + \eta \, ?
\end{equation}
We have answered \eqref{eqn:referenceme_coprime_q} and \eqref{eqn:referenceme_two_coprime_q} for primes $2\leq q < 10^5$ in the following results. Our results are stated for the best possible range of $n$.
\begin{thm}\label{thm:one_prime_two}
Every even integer greater than three can be written as the sum of a prime and an odd square-free number.\footnote{For sufficiently large even integers, Theorem \ref{thm:one_prime_two} is also a consequence of Chen's theorem.}
\end{thm}
\begin{thm}\label{thm:one_prime_three}
Every integer greater than two except for eleven can be written as the sum of a prime and a square-free number which is co-prime to three.
\end{thm}
\begin{thm}\label{thm:one_prime_general}
Suppose $3 < q < 10^5$ is prime. Every integer greater than two can be written as the sum of a prime and a square-free number co-prime to $q$.
\end{thm}
\begin{cor}\label{thm:two_prime_general}
Suppose $2 \leq q < 10^5$ is prime. Every integer greater than four can be written as the sum of two primes and a square-free number co-prime to $q$.
\end{cor}
Computations suggest that one could establish similar results for composite $q$. The authors also believe that a similar method of proof is plausible, if the auxiliary results in Section \ref{ssec:aux_results} can be extended in the appropriate manner.
In future work, it may also be interesting to investigate the quantity $\max{S_q}$, where an \textit{exception set} $S_q$ contains all the integers which do not have a representation as a prime plus a square-free number co-prime to $q$ for any integer $q > 1$. For example, Theorem \ref{thm:one_prime_three} implies $\max{S_{3}} = 11$, and a search over the first $10^8$ integers suggests that $\max{S_{15}} = 23$, $\max{S_{\prod_{i=2}^{35} p_i}} = 355$. Here, $p_i$ denotes the $i^\text{th}$ prime.
\begin{comment
For $q \geq 2$, let $S_q$ denote the set of $n\in\mathbb{N}$ such that $n$ cannot be represented as a sum of a prime and a square-free integer co-prime to $q$. We call $S_q$ an \textit{exception set} for $q$ and formally write
$$S_q = \left\{n \in \mathbb{N} : p \leq n \mbox{ such that } (n-p, q) > 1\text{ or }\mu(n-p) =0\right\}.$$
Computations suggest that $S_q$ will be finite when $q$ is odd, but exceptions still exist. For example, a search over $n \leq 10^8$ implies that that $\{1,2,11\}\subset S_3$. Indeed, Theorem \ref{thm:one_prime_three} confirms that $S_3 = \{1,2,11\}$. Similarly, the largest exceptions we could find in the sets $S_{15}$ and $S_{\prod_{i=2}^{35} p_i}$ were 23 and 355 (respectively).
\end{comment}
\subsection*{Outline of the paper}
In Section \ref{sec:aux_results}, we provide all of the necessary notation and auxiliary results which will be needed throughout the paper.
In Section \ref{sec:analytic_bounds}, we will prove an important lemma.
Finally, in Section \ref{sec:main_results}, we will prove the main results of this paper.
\subsection*{Acknowledgements}
We would like to thank Tim Trudgian for his comments and bringing this project to our attention. We would also like to thank Nathan Ng and Stephan Garcia for their feedback.
\section{Notation and Auxiliary results}\label{sec:aux_results}
\subsection{Notation}
Throughout, $p$ will denote a prime number and $n$ will denote an integer. Further, $\varphi$ denotes the Euler-phi function, $\mu$ denotes the M\"{o}bius function,
\begin{align*}
\theta(x)&=\sum_{p\leq x} \log p,\quad
\theta(x;q,a)=\sum_{\substack{{p\leq x}\\{p \equiv a \imod{q}}}} \log p,\quad
\mu_2(n) = \sum_{a^2|n}\mu(a),\\
R(n) &= \sum_{p \leq n} \mu_2(n-p) \log{p} = \sum_{a \leq n^{\frac{1}{2}}} \mu(a) \theta(n;a^2,n),
\end{align*}
in which $\mu_2(n)=1$ if $n$ is square-free, and $\mu_2(n)=0$ otherwise. We also use $\mu^2$ as a square-free identifier function and $(a,b)$ as the greatest common divisor function.
\subsection{Auxiliary results}\label{ssec:aux_results}
In Section \ref{sec:analytic_bounds}, we will determine estimates for $R(n)$. To do this, we will appeal to the following estimate, which follows from the work of Bennett et al. \cite{bennett2018}.
\begin{prop}\label{prop:bennett_prop}
For each square $a^2\in [2^2, 316^2]$ and integer $n$ which is co-prime to $a$, there exist explicit constants $c_{\theta}(a^2)$ and $x_{\theta}(a^2) \leq 4.81\cdot 10^9$ such that
$$\left| \theta(x;a^2,x) - \frac{x}{\varphi(a^2)} \right| < c_{\theta}(a^2)\frac{x}{\log x}$$
for all $x\geq x_{\theta}(a^2)$.
\end{prop}
\begin{proof}
For each $3\leq q\leq 10^5$ and integers $a$ such that $(a,q) = 1$, Bennett et al. \cite[Theorem 1.2]{bennett2018} provide explicit constants $c_{\theta}(q)$ and $x_{\theta}(q) \leq 8\cdot 10^9$ such that
$$\left| \theta(x;q,a) - \frac{x}{\varphi(q)} \right| < c_{\theta}(q)\frac{x}{\log x}$$
for all $x\geq x_{\theta}(q)$. Analysis on the values of $c_{\theta}(q)$ and $x_{\theta}(q)$ from the tables\footnote{The tables are available at \url{https://www.nt.math.ubc.ca/BeMaObRe/}.} provided for \cite{bennett2018} at each square $q = a^2$ in this range will yield the constants $c_{\theta}(a^2)$ and demonstrate that the maximum value of $x_{\theta}(a^2)$ is $\num{4800162889}\leq 4.81\cdot 10^9$.
\end{proof}
\begin{comment}
\begin{thm}[Dusart]\label{thm:dusart_thm}
We have
\begin{equation}\label{eqn:dusart_k}
|\theta(n) - n | \leq \eta_k \frac{n}{\log^k n},
\end{equation}
where $n\geq n_k$ such that:
\begin{center}
\begin{tabular}{lll}
$k$ & $\eta_k$ & $n_k$ \\
\hline
$1$ & $0.001$ & \num{908994923} \\
$2$ & $0.05$ & \num{122568683} \\
$3$ & $0.5$ & \num{767135587} \\
$4$ & $151.3$ & $2$
\end{tabular}
\end{center}
\end{thm}
\end{comment}
We will also make use of the following estimate for $\theta(x)$ from Broadbent et al. \cite[Theorem 1]{broadbent2020}.
\begin{thm}[Broadbent et al.]\label{thm:lethbridge_thm}
For $x > e^{20} \approx 3.59\cdot10^9$, we have
\begin{equation}\label{eqn:lethbridge_k}
\lvert \theta(x) - x \rvert \leq 0. 375 \frac{x}{\log^3 x}.
\end{equation}
\end{thm}
\begin{comment}
\textit{Remark.}
The estimates in Proposition \ref{prop:bennett_prop} and Theorem \ref{thm:lethbridge_thm} will hold for all $n \geq 4.81\cdot 10^9$. Therefore, we can construct a bound for $n \geq 4.81\cdot 10^9$ for our lower bound in Lemma \ref{thm:analytic_bound}.
\end{comment}
\section{An Important Lemma}\label{sec:analytic_bounds}
Theorem \ref{thm:dudek} is true as long as $R(n) > 0$ for all $n \geq 3$.
To prove Theorem \ref{thm:dudek}, Dudek found an explicit lower bound for $R(n)$ in \cite[Section 2.3]{dudek2017}, which inferred that $R(n) > 0$ for all $n \geq 10^{10}$, then computationally checked $R(n) > 0$ for $n \in [3, 10^{10})$.
We will prove Lemma \ref{thm:analytic_bound} and use it to prove the main results of this paper. The main benefit of this (over the lower bound in \cite[Section 2.3]{dudek2017}) is that we will need to manually verify our main results for a smaller range of $n$.
\begin{lem}\label{thm:analytic_bound}
Suppose $A\in (0, 1/2)$ and $n \geq 4.81\cdot 10^9$, then
\begin{align}
\frac{R(n)}{n}
> 0.37395 - \frac{0.95}{\log n} - \frac{0.375}{\log^3 n} &- 0.0096 \left(\frac{1+2A}{1-2A}\right)\nonumber\\
&- \log n\left(n^{-2A} + n^{-A} - n^{A - 1} + n^{-\frac{1}{2}}\right), \label{eqn:analytic_bound_result}
\end{align}
where $0.37395$ is Artin's constant, rounded to 5 decimal places.
\end{lem}
The improvements we obtain come from Proposition \ref{prop:bennett_prop} (which is wider-reaching than the results from Ramar\'{e}--Rumely \cite{ramare1996} which Dudek used), and Theorem \ref{thm:lethbridge_thm}.
\subsection{Set-up}
Trivially, if $(a,n) > 1$, then $\theta(n;a^2,n) \leq \log n$. Therefore,
$$R(n) > \sum_{\substack{a\leq n^{\frac{1}{2}}\\(a,n)=1}}\mu(a)\theta(n;a^2,n) - n^{\frac{1}{2}}\log n = \Sigma_1 + \Sigma_2 + \Sigma_3 - n^\frac{1}{2}\log{n},$$
in which $A \in \left(0,1/2\right)$ will be chosen later,
\begin{align*}
\Sigma_1 = \sum_{\substack{a \leq 316 \\ (a,n) = 1}} \mu(a)\theta(n;a^2,n),\quad
\Sigma_2 &= \sum_{\substack{316 < a \leq n^A \\ (a, n) = 1}} \mu(a)\theta(n;a^2,n),\mbox{ and}\\
\Sigma_3 &= \sum_{\substack{n^A < a \leq n^\frac{1}{2} \\ (a, n) = 1}} \mu(a)\theta(n;a^2,n).
\end{align*}
We will bound $\Sigma_1 + \Sigma_2$ and $\Sigma_3$ separately.
\subsection{Bounding $\Sigma_1 + \Sigma_2$}\label{subsec:first}
We start by listing important bounds, which we will use to deduce a lower bound for $\Sigma_1 + \Sigma_2$.
First, we observe by computation that
\begin{equation}\label{eq:uno}
\sum_{2\leq a\leq 316} c_{\theta}(a^2) = 0.9474935 < 0.95.
\end{equation}
Second, suppose that $c$ denotes Artin's constant. It follows from computations by Wrench \cite{wrench1961} that
\begin{equation}\label{eq:dos}
\sum_{(a,n) = 1} \frac{\mu(a)}{\varphi(a^2)} > \prod_p \left(1 - \frac{1}{p(p - 1)}\right) = c > 0.37395.
\end{equation}
Third, in the range $316 < a \leq n^A$, the Brun--Titchmarsh theorem \cite{montgomery1973} yields
\begin{equation}\label{eq:tres_Brun_Titchmarsh}
\theta(n;a^2,n) = \frac{n}{\varphi(a^2)} + \varepsilon \left(\frac{1+2A}{1-2A}\right)\frac{n}{\varphi(a^2)},
\end{equation}
such that $|\varepsilon | < 1$.
Finally, we may observe that
\begin{equation}\label{eq:quattro}
\sum_{\substack{a>316\\(a,n)=1}}\frac{\mu(a)}{\varphi(a^2)}
\leq \sum_{a=1}^\infty \frac{\mu^2(a)}{\varphi(a^2)} - \sum_{a\leq 316}\frac{\mu^2(a)}{\varphi(a^2)}
< 0.0096.
\end{equation}
To deduce the upper bound, we observed that an upper bound for the infinite sum is $1.95$ \cite{ramare1995} and computed the finite sum manually. Computations suggest that \eqref{eq:quattro} may be numerically improved, but this is unnecessary for our purposes.
To bound $\Sigma_1$, combine Proposition \ref{prop:bennett_prop}, Theorem \ref{thm:lethbridge_thm}, \eqref{eq:uno} and \eqref{eq:dos} to yield
\begin{align*}
\Sigma_1
&> n \left( \sum_{\substack{2\leq a \leq 316 \\ (a,n) = 1}} \frac{\mu(a)}{\varphi(a^2)} - \sum_{\substack{2\leq a \leq 316 \\ (a,n) = 1}} \frac{c_{\theta}(a^2)\mu(a)}{\log n} + 1 - \frac{0.375}{\log^3 n}\right)\\
&> n \left( \sum_{(a,n)=1} \frac{\mu(a)}{\varphi(a^2)} - \sum_{\substack{a > 316 \\ (a,n) = 1}} \frac{\mu(a)}{\varphi(a^2)} - \sum_{\substack{2\leq a\leq 316\\(a,n)=1}}\frac{c_{\theta}(a^2)}{\log n} - \frac{0.375}{\log^3 n}\right)\\
&> n \left( 0.37395 - \sum_{\substack{a > 316 \\ (a,n) = 1}} \frac{\mu(a)}{\varphi(a^2)} - \frac{0.95}{\log n} - \frac{0.375}{\log^3 n}\right),
\end{align*}
since the $+1$ gets absorbed into the left-most sum. Next, use \eqref{eq:tres_Brun_Titchmarsh} to see that
\begin{align*}
\Sigma_2
&> n\left(\sum_{\substack{316 < a \leq n^A \\ (a, n) = 1}} \frac{\mu(a)}{\varphi(a^2)} - \left(\frac{1 + 2A}{1 - 2A}\right) \sum_{\substack{316 < a \leq n^A \\ (a, n) = 1}} \frac{\mu^2(a)}{\varphi(a^2)} \right).
\end{align*}
Finally, using the preceding observations and \eqref{eq:quattro}, $\Sigma_1 + \Sigma_2$ is larger than
\begin{align*}
n &\left(0.37395 - \sum_{\substack{a > n^A \\ (a, n) = 1}} \frac{\mu(a)}{\varphi(a^2)} - \frac{0.95}{\log n} - \frac{0.375}{\log^3 n}-\left(\frac{1+2A}{1-2A}\right)\sum_{\substack{316<a\leq n^A\\(a,n)=1}}\frac{\mu^2(a)}{\varphi(a^2)} \right)\\
&\quad > n \left(0.37395 - \frac{0.95}{\log n} - \frac{0.375}{\log^3 n} -\left(\frac{1+2A}{1-2A}\right)\sum_{\substack{a>316\\(a,n)=1}}\frac{\mu^2(a)}{\varphi(a^2)} \right)\\
&\quad > n \left(0.37395 - \frac{0.95}{\log n} - \frac{0.375}{\log^3 n} - 0.0096 \left(\frac{1+2A}{1-2A}\right) \right).
\end{align*}
\subsection{Final steps}
Following a similar logic to Dudek \cite{dudek2017}, with less waste, we used a trivial bound for $\theta(n;a^2,n)$ to bound $|\Sigma_3|$ and obtain
\begin{equation}\label{eqn:Sigma_3}
\Sigma_3 > - n \log n\left(n^{-2A} + n^{-A} - n^{A - 1}\right).
\end{equation}
Combining our preceding observations, we have established \eqref{eqn:analytic_bound_result} for all $n\geq 4.81\cdot 10^9$.
\section{Main Results}\label{sec:main_results}
In this section, we will establish all of the main results of this paper.
We treat Theorem \ref{thm:one_prime_two} separately, but to prove the remaining results, we will use Lemma \ref{thm:analytic_bound} to establish them for large $n$ and the algorithm described in Section \ref{sec:computation} to verify the results for small $n$.
Suppose that $2\leq q\leq 10^5$ is prime and $R_q(n)$ denotes the weighted number of representations of $n$ as the sum of a prime and a square-free number coprime to $q$. Then, we have
$$R_q(n) = \sum_{\substack{p \leq n \\ p \not\equiv n \imod{q}}} \mu^2(n-p)\log{p} = R(n) - \sum_{\substack{p \leq n \\ p \equiv n \imod{q}}} \mu^2(n-p)\log{p}.$$
Therefore, to show $R_q(n) > 0$, it suffices to demonstrate
\begin{equation}\label{eqn:rvsrq}
R(n) > \sum_{\substack{p \leq n \\ p \equiv n \imod{q}}} \mu^2(n-p)\log{p}.
\end{equation}
\subsection{Proof of Theorem \ref{thm:one_prime_two}}\label{ssec:two}
It is an equivalent problem to establish $R_2(n)>0$ for all even $n > 2$. First, we observe that for each \textit{odd} $n$, $n-p$ is even for all odd primes $p$, hence
\begin{equation*}
R_2(n) = \begin{cases} \log(n-2) & \mbox{if }\mu^2(n - 2) = 1,\\0 & \mbox{if }\mu(n - 2) = 0. \end{cases}
\end{equation*}
There are infinitely many odd choices for $n$ such that $\mu(n - 2) = 0$, thence our restriction to \textit{even} $n$ in this case.
Suppose that $n$ is \textit{even}, then $R_2(n) > 0$ if and only if $R(n) > \theta(n;2,n)$.
If $(n, q) > 1$, then $\theta(n;q,n) \leq \log{q}$. Therefore it suffices to show that $R(n) > \log 2$.
If $n \geq 4$, then Theorem \ref{thm:dudek} guarantees that there exists at least one prime $p \in (2,n)$ such that $\mu^2(n-p)=1$. Note that $p\neq 2$, because $n-2 \geq 2$ is even for even $n$, so $(n-2,2)\neq 1$.
It follows that there exists a prime $p \in (2,n)$ such that $R(n) > \log{p} > \log{2}$.
\qed
\subsection{Proof of Theorem \ref{thm:one_prime_general} for large $n$}\label{ssec:one_gen}
Suppose $3 < q \leq 10^5$ is prime, then Theorem \ref{thm:one_prime_general} holds for $n \geq 8\cdot 10^9$ if and only if $R_q(n) > 0$. Using \eqref{eqn:rvsrq}, it suffices to show that $R(n) > \theta(n;q,n)$.
By Lemma \ref{thm:analytic_bound}, we need to show that there exists $A\in (0,1/2)$ such that
\begin{align}
0.37395 - \frac{0.95}{\log{n}} &- \frac{0.375}{\log^3{n}} - 0.0096 \left(\frac{1+2A}{1-2A}\right)\nonumber\\
&- \left(n^{-2A} + n^{- A} - n^{A-1} + n^{-\frac{1}{2}}\right)\log{n} > \frac{\theta(n;q,n)}{n}\label{eqn:sufficientcheck}.
\end{align}
We may use Proposition \ref{prop:bennett_prop} to estimate $\theta(n;q,n)$ for each $3 \leq q \leq 10^5$. It follows that $A = 0.33$ implies \eqref{eqn:sufficientcheck} for all primes $q$ in our assumed range.
\qed
\subsection{Proof of Theorem \ref{thm:one_prime_three} for large $n$}\label{ssec:one_three}
We use a similar method to the preceding proof, although we must consider the case $q=3$ separately because it is clear that $1/\varphi(3) = 1/2 > 0.37395$. So, we will need to consider a stronger version of \eqref{eqn:sufficientcheck}.
Observe that
\begin{equation}\label{eqn:mod3breakdown}
\sum_{\substack{p \leq n \\ p \equiv n \imod{3}}}\mu^2(n-p)\log{p} = \sum_{\substack{p \leq n \\ p \equiv n \imod{3}}} \log{p} - \sum_{\substack{p \leq n \\ p \equiv n \imod{3} \\ \mu(n-p) = 0}}\log{p}.
\end{equation}
An inclusion-exclusion argument yields
\begin{align*}
\sum_{\substack{p \leq n \\ p \equiv n \imod{3} \\ \mu(n-p) = 0}}\log{p}
&> \sum_{\substack{p \leq n \\ p \equiv n \imod{9} \textrm{ or } \\ p \equiv n \imod{12} \textrm{ or } \\ p \equiv n \imod{75} } } \log{p} \\
&= \theta(n;9,n) + \theta(n;12,n) + \theta(n;75,n) \\
&\qquad - \theta(n;36,n) - \theta(n;225,n) - \theta(n;300,n) + \theta(n;900,n).
\end{align*}
Therefore, \eqref{eqn:mod3breakdown} yields
\begin{align}
\sum_{\substack{p \leq n \\ p \equiv n \imod{3}}}&\mu^2(n-p)\log{p} < \theta(n;3,n) - \theta(n;9,n) - \theta(n;12,n) - \theta(n;75,n)\\
&\qquad + \theta(n;36,n)+ \theta(n;225,n) + \theta(n;300,n) - \theta(n;900,n).\label{eqn:allthetas}
\end{align}
Using the explicit bounds from Bennett et al. \cite[Theorem 1.2]{bennett2018} to estimate each $\theta(n;q,n)$ term in \eqref{eqn:allthetas} according to these values establishes
\begin{equation}\label{eqn:modupperbound}
\sum_{\substack{p \leq n \\ p \equiv n \imod{3}}}\mu^2(n-p)\log{p} < \frac{19}{120}n + 0.00592\frac{n}{\log{n}}.
\end{equation}
\begin{comment}
, we note that
\begin{align*}
\max\{x_\theta(3), x_\theta(9), x_\theta(12), x_\theta(36), x_\theta(75), x_\theta(225), x_\theta(300), x_\theta(900)\} < 8\cdot 10^9,\\
\frac{1}{\varphi(3)} - \frac{1}{\varphi(9)} - \frac{1}{\varphi(12)} - \frac{1}{\varphi(75)} + \frac{1}{\varphi(36)} + \frac{1}{\varphi(225)} + \frac{1}{\varphi(300)} - \frac{1}{\varphi(900)} = \frac{19}{120},\\
c_\theta(3) + c_\theta(9) + c_\theta(12) + c_\theta(75) + c_\theta(36) + c_\theta(225) +c_\theta(300) + c_\theta(900) < 0.00592.
\end{align*}
\end{comment}
We may compare \eqref{eqn:modupperbound} with Lemma \ref{thm:analytic_bound}, and thereby establish \eqref{eqn:rvsrq} whenever
\begin{align}
0.37395 - \frac{0.95}{\log n} &- \frac{0.375}{\log^3{n}} - 0.0096 \left(\frac{1+2A}{1-2A}\right) \label{eqn:finalcheck}\\
&- \left( n^{-2A} + n^{- A} + n^{A-1} - n^{-\frac{1}{2}}\right)\log n > \frac{19}{120} + \frac{0.00592}{\log{n}}.\nonumber
\end{align}
Choosing $A = 0.33$ will verify that \eqref{eqn:finalcheck} holds for $n \geq 8\cdot 10^9$.
\qed
\subsection{Proof of Corollary \ref{thm:two_prime_general} for large $n$}\label{subsec:extension}
Suppose that $2\leq q\leq 10^5$ is prime. We will consider the cases $q=2$ and $q \geq 3$ separately. In the former case, we did not require the computations (which will be outlined in section \ref{sec:computation}) to verify that the result is true for small $n$, so we consider a larger range for $n$ in this case.
If $q = 2$ and $n > 4$ is \textit{even}, then $n - 2 > 2$ is also even. Therefore, there exists a prime $p_1$ and odd square-free number $\eta_1$ such that $n - 2 = p_1 + \eta_1$ by Theorem \ref{thm:one_prime_two}. Moreover, if $q = 2$ and $n > 5$ is \textit{odd}, then $n - 3 > 2$ is even. Therefore, there exists a prime $p_2$ and odd square-free number $\eta_2$ such that $n - 3 = p_2 + \eta_2$ by Theorem \ref{thm:one_prime_two}. To complete Corollary \ref{thm:two_prime_general} at $q = 2$, observe that $5 = 2 + 2 + 1$.
If $q \geq 3$ and $n \geq 8\cdot 10^9$, then suppose that
\begin{equation*}
T(n)
:= \sum_{\substack{p\leq n\\n - p \not\in \{1,2,11\}}}\mu^2(n-p)\log p
> R(n) - 3 \log n .
\end{equation*}
If $T(n) > 0$ then there exists at least one prime $p_3$ such that $n-p_3 > 2$ and $n-p_3\neq 11$. Hence, Corollary \ref{thm:two_prime_general} is also true by corollary of Theorem \ref{thm:one_prime_general} for $q > 3$, and by corollary of Theorem \ref{thm:one_prime_three} for $q=3$.
Therefore, it suffices to show $T(n) > 0$ in the desired range of $n$.
It follows from Lemma \ref{thm:analytic_bound} that
\begin{align}
\frac{T(n)}{n}
> 0.37395 - \frac{0.95}{\log n} &- \frac{0.375}{\log^3 n} - 0.0096 \left(\frac{1+2A}{1-2A}\right)\nonumber\\
&- \log n\left(n^{-2A} + n^{-A} - n^{A - 1} + n^{-\frac{1}{2}} + \frac{3}{n}\right).\label{eqn:final_appl_3_large_n}
\end{align}
Now, \eqref{eqn:final_appl_3_large_n} with $A=0.385$ implies the result for large $n$.
\qed
\subsection{Computations}\label{sec:computation}
To complete each of our main results, we verified each result for small $n$, wherever necessary. We did this computationally, by slightly adapting the algorithm used by Dudek in \cite[p.~239]{dudek2017}. Our computations took just short of 7 hours on a machine equipped with 3.20 GHz CPU, using Maple${}^\mathrm{TM}$ \footnote{Maple is a trademark of Waterloo Maple, inc.}.
If $3 < n \leq 4\cdot10^{18}$ is even, we know by Oliveira e Silva et al. \cite{oliveira2014} that $n$ is the sum of two primes. Unless $n = q+q$ for some prime $q \in \left[3, 10^5\right]$, we are done. When $n = q + q$, it is a simple task to verify that it has at least one other representation as a prime plus a square-free co-prime to $q$. Hence, we only need to consider odd integers between $3$ and $8\cdot10^{8}$.
As in Dudek's algorithm, we pre-compute a set $S$ of square-free numbers up to $2\cdot10^7$. We break the problem up, considering $n$ in intervals of the form
\[I_a = \left[a\cdot10^7, (a+1)\cdot10^7\right),\]
where $a$ is an integer between 1 and 800. For each such $a$, we compute decreasing lists $P_a = \left(p_1, p_2,\ldots, p_{100}\right)$ of the 100 largest primes in $I_{a-1}$. Starting with the smallest odd $n$ in $I_a$, we check if $n-p_i$ is in $S$ as $i$ ranges from 1 to 100. Each time this check is successful, we compute the $\gcd$ of $n-p_i$ with all previous successful $n-p_j$, moving on to $n+2$ when this $\gcd$ equals 2 (that is, when there is a representation with a square-free number co-prime to every prime $q \in \left[3, 10^5\right]$). If there were any $n$ for which the largest 100 primes did not produce all the appropriate representations, we could have checked these cases separately with more primes. However, our program did not return any such $n$.
For the initial interval $n \in \left(2, 10^7\right)$, a similar check can be used. Relevant representations can easily be found for $n$ up to $10^6$, with the exception of $n = 2$ and $n = 11$ (which is an exception only when $q = 3$). Then, letting $P_0$ be the set of the 100 largest primes less than $10^6$, we perform the same check as we did for the other intervals to $n \in (10^6, 10^7)$, finding no new exceptions.
To verify Corollary \ref{thm:two_prime_general}, we adapted the algorithm above. Note that we only need to check the even $n$ in this scenario, since the result follows directly from the ternary Goldbach conjecture for odd $n$ \cite{helfgott2013}. Importantly, we used $S' = S \setminus \{1,2,11\}$ in place of $S$ and no exceptions were found for $n$ between 5 and $8\cdot10^9$.
\bibliographystyle{amsplain}
| {
"timestamp": "2020-11-12T02:08:47",
"yymm": "2003",
"arxiv_id": "2003.08083",
"language": "en",
"url": "https://arxiv.org/abs/2003.08083",
"abstract": "Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.",
"subjects": "Number Theory (math.NT)",
"title": "Additive Representations of Natural Numbers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9914225158534695,
"lm_q2_score": 0.7154240079185319,
"lm_q1q2_score": 0.7092874698325634
} |
https://arxiv.org/abs/1701.08504 | A generalization of the practical numbers | A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and asking whether all integers in a given interval can be expressed as sums of $f(d)$'s, where the $d$'s are distinct divisors of $n$. We will refer to such $n$ as `$f$-practical.' In this paper, we introduce the $f$-practical numbers for the first time. We give criteria for when all $f$-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct $f$-practical sets with any asymptotic density, and prove a series of results related to the distribution of $f$-practical numbers for many well-known arithmetic functions $f$. | \section{Introduction}
Srinivasan first introduced the practical numbers as integers $n$ for which every number between $1$ and $n$ is representable as a sum of distinct divisors of $n$. In her Ph.D. thesis, the second author adapted this concept to study the degrees of divisors of $x^n-1$. Recall that $x^n - 1 = \prod_{d \mid n} \Phi_d(x)$ where $\Phi_d(x)$ is the $d^{th}$ cyclotomic polynomial with $\deg \Phi_d(x) = \varphi(d)$. By applying Euler's totient function on the divisors of $n$, the second author categorized the numbers $n$ for which $x^n-1$ has a divisor in $\Z[x]$ of every degree smaller than $n$, calling these integers ``$\varphi$-practical.'' The aim of the present paper is to generalize much of the existing literature on practical and $\varphi$-practical numbers.
\begin{definition}
Let $f:\N\to \N$ be a multiplicative function. We define
\begin{align*}
S_f(n) = \sum_{d\mid n} f(d).
\end{align*}
Therefore we have $S_f = f\ast \mathbbold{1}$, where $\mathbbold{1}(n)$ denotes the arithmetic function that is identically $1$.
\end{definition}
We note that the function $S_f(n)$ is multiplicative, since it is the Dirichlet convolution of two multiplicative functions.
\begin{definition}\label{def:fpractical}
Let $f : \N \to \N$ be a multiplicative function for which for every prime $p$ and every positive integer $k$ satisfies $f(p^{k-1}) \leq f(p^k)$. A positive integer $n$ is called $f$-practical if for every positive integer $m\leq S_f(n)$ there is a set $\mathcal{D}$ of divisors of $n$ for which
\begin{align*}
m = \sum_{d\in \mathcal{D}} f(d)
\end{align*}
holds. \end{definition}
\begin{example} If $f = I$ (the identity function), this is equivalent to the definition of practical. \end{example}
\begin{example} If $f = \varphi$, this is precisely the definition of $\varphi$-practical. \end{example}
One can check whether all integers in an interval can be expressed as subsums of numbers from a particular set by applying the following naive algorithm (cf. \cite[Theorem A.1]{thesis}):
\begin{prop}\label{thm:thesisA1}
Let $w_1\leq w_2\leq\dots\leq w_k$ be positive integers with $\sum_{i=1}^k w_i = s$. Then, every integer in $[0,s]$ can be represented as a sum of some subset of the $w_i$'s if and only if $w_{i+1} \leq 1 + w_1 +\dots +w_i$ holds for every $i<k$.
\end{prop}
\noindent It is not so ``practical'' to use this algorithm to determine whether an integer $n$ is practical. Instead, it is useful to have a criterion in terms of the prime factors of $n$. In \cite{Stewart} Stewart gave such a criterion for constructing practical numbers and proved that every practical number can be obtained in this fashion. Stewart's criterion for a number to be practical serves as a key lemma in many subsequent papers on the practical numbers. The second author showed in \cite{thesis} that there are $\phi$-practical numbers which cannot be constructed in such a manner. In the present paper, we examine the functions for which this means of constructing $f$-practical numbers is possible and obtain the following theorem.
\begin{thm}\label{thm:stewartlike}
Let $f$ be a multiplicative function. All $f$-practical numbers are constructable by a Stewart-like criterion if and only if for every prime $p$ for which there is a coprime integer $n$ with $f(p) \leq 1+\sum_{d\mid n}f(d)$ the inequality
\begin{align*}
f(p^{k+1})\leq f(p)f(p^k)
\end{align*}
holds for every integer $k\geq 0$.
\end{thm}
One of the aims of this paper is to study the distribution of $f$-practical numbers for various arithmetic functions $f$. We show that it is possible to construct $f$-practical sets with any asymptotic density. In fact:
\begin{thm}\label{thm:densitiesaredense}
The densities of the $f$-practical sets are dense in $[0,1].$
\end{thm}
The $f$-practical sets that are the most interesting to study are those that are neither finite nor all of $\N$. Intuitively, if the values of $f(n)$ are too large relative to $n$, then some integers in the interval $[1, S_f(n)]$ will always be skipped, resulting in a finite set of $f$-practical numbers. On the other hand, if the values of $f(n)$ are too small relative to $n$, then every integer winds up being $f$-practical. Thus, the arithmetic functions which produce non-trivial $f$-practical sets are those that behave like the identity function, i.e., those for which $f(p^k) \approx (f(p))^k$ at all prime powers. Examples of functions satisfying this condition include $I$, $\varphi$, $\lambda$ (the Carmichael $\lambda$-function), and $\varphi^*$ (the unitary totient function).
\begin{thm}\label{thm:fpracticalchebyshev}
Let $f = \phi^*$. Let $F_{f}(X)$ be the number of $f$-practical numbers less than or equal to $X$. Then there exist positive constants $l_{f}$ and $u_{f}$ such that
\begin{align*}
l_{f}\frac{X}{\log X}\leq F_{f}(X)\leq u_{f}\frac{X}{\log X}
\end{align*}
for all $X\geq2$.
\end{thm}
The result also holds for $f = I$ and $f = \varphi$. For these functions, the distributions have been well-studied. In a 1950 paper, Erd\H{o}s \cite{Erdos} claimed that the practical numbers have asymptotic density $0$. Subsequent papers by Hausman and Shapiro \cite{H&S}, Margenstern \cite{Margenstern}, Tenenbaum \cite{Tenenbaum}, and Saias \cite{Saias} led to sharp upper and lower bounds for the count of practical numbers in the interval $[1, X]$. A recent paper of Weingartner \cite{Weingartner} showed that the count of practical numbers is asymptotically $c X/\log X$, for some positive constant $c$. In her PhD thesis \cite{thesis}, the second author proved sharp upper and lower bounds for the count of $\varphi$-practical numbers. This work was improved to an asymptotic in a subsequent paper with Pomerance and Weingartner \cite{ptw}.
For $f = \lambda$, computational evidence seems to suggest that $X/\log X$ is the correct order of magnitude for the $f$-practicals. Indeed, we prove that the upper bound from Theorem \ref{thm:fpracticalchebyshev} holds when $f = \lambda$. However, we have not been able to obtain a sharp lower bound.
The proofs of the aforementioned theorems rely heavily on the fact that the functions $f$ are multiplicative (or nearly multiplicative, in the case of the Carmichael $\lambda$-function). We are also able to prove density results for certain non-multiplicative $f$. For example, we classify the additive functions $f$ for which all positive integers are $f$-practical. We also examine some $f$-practical sets where $f$ is neither additive nor multiplicative.
The paper is organized according to the following scheme. In Section \ref{f-practicalcriterion}, we provide a method for constructing infinite families of $f$-practical numbers. In Section \ref{stewart-like}, we classify the set of all $f$-practical numbers that can be completely determined via a Stewart-like condition on the sizes of the prime factors. In Section \ref{densities}, we give examples of $f$-practical sets with various densities and show that the densities themselves are dense in $[0,1]$. In Section \ref{boundsforcertainf}, we prove upper and lower bounds for the sizes of the sets of $f$-practical numbers for certain arithmetic functions $f$. Much of the work in the aforementioned sections applies only to functions which are multiplicative or nearly multiplicative. In Section \ref{non-multiplicativef}, we give density results for $f$-practicals for some well-known non-multiplicative functions $f$.
Throughout this paper, we will use $n$ to denote an integer and $p$ to denote a prime number. Moreover, we will use $P(n)$ to represent the largest prime factor of $n$.
\section{$f$-practical construction for multiplicative $f$}\label{f-practicalcriterion}
In this section, we develop the basic machinery for constructing infinite families of $f$-practical numbers. Following the definition of weakly $\phi$-practical numbers (see \cite[Definition 4.4]{Thompson}), we introduce the concept of weakly $f$-practical numbers.
\begin{definition}\label{def:weaklyfpractical}
Let $n = p_1^{e_1} \dots p_k^{e_k}$, where $f(p_1) \leq f(p_2) \leq \cdots \leq f(p_k)$. We define $m_i = \prod_{j=1}^i p_j^{e_j}$ for every non-negative integer $i<k$. We call $n$ weakly $f$-practical if for every $i$
\begin{align*}
f(p_{i+1}) \leq S_f(m_i) + 1
\end{align*}
holds.
\end{definition}
\begin{thm}\label{thm:f-practicalisweakly}
Every $f$-practical number is also weakly $f$-practical.
\end{thm}
\begin{proof}
Let $n = \prod_{i=1}^k p_i^{e_i}$ be $f$-practical with $f(p_1) \leq f(p_2) \leq \dots \leq f(p_k)$. Now let $m_i = \prod_{j=1}^ip_j^{e_j}$ for every $i<k$. Suppose $n$ is not weakly $f$-practical. Then there must be an $i<k$ so $f(p_{i+1}) > S_f(m_i) +1$ holds. For every $j>i+1$ we have $f(p_j) \geq f(p_{i+1})$. Therefore for every divisor $d$ of $n$ which does not divide $m_i$ we have $f(d)\geq f(p_{i+1})$ since $f$ is multiplicative and $d$ must be divisible by some $p_j$ with $j>i$. Because the sum of $f(t)$ over all $t\mid m_i$ is exactly $S_f(m_i)$ there is no possibility to express $S_f(m_i) +1$ as a sum of $f(d)$ for $d\mid n$. This contradicts the fact that $n$ is $f$-practical.
\end{proof}
\begin{remark}
In this proof we did not use the condition that $f(ab) = f(a)f(b)$ but the implied and weaker fact that $f(ab) \leq f(a)f(b)$. Hence this proof also holds, for example, for the Carmichael $\lambda$-function. In fact, since $\lambda(a) \leq \phi(a)$ for any integer $a$ and $\lambda(p) = \phi(p)$, every weakly $\lambda$-practical number is also weakly $\phi$-practical, because $\phi(p_{i+1}) = \lambda(p_{i+1}) \leq S_\lambda(m_i) +1 \leq S_\phi(m_i)+1$.
\end{remark}
\begin{cor}\label{cor:smallprime}
If $n$ is weakly $f$-practical and $p\leq P(n)$ then $pn$ is also weakly $f$-practical.
\end{cor}
The following theorem gives a necessary and sufficient condition for a product of a prime power and an $f$-practical number to be $f$-practical itself. For some functions $f$ (like the identity function) this gives a way to construct all $f$-practical numbers (cf. \cite[Corollary 1]{Stewart}). This is not the case for all functions. For example, for the $\phi$ function, there are numbers that are $\phi$-practical that are not the product of a $\phi$-practical number and a prime power, as is the case with $45=3^2\cdot5$.
\begin{thm}\label{thm:f-practicalconstruct}
Let $n$ be $f$-practical. Let $p$ be prime with $(p,n) = 1$. Then $np^k$ is $f$-practical if and only if $f(p^i) \leq S_f(np^{i-1}) +1$ for all $1\leq i\leq k$.
\end{thm}
\begin{proof}
If $f(p^i)> S_f(np^{i-1}) +1$ for some $i\geq 1$, then $S_f(np^{i-1})+1$ is not representable as a sum of $f(d)$'s with $d\mid n$, since for every divisor $d$ of $np^k$ which does not divide $np^{i-1}$ we have $f(d)> S_f(np^{i-1})+1$.
We show by induction on $k$ that $np^{k}$ is $f$-practical for all $k$. Assume $np^{k-1}$ is $f$-practical for some $k\geq 1$. For $k=1$ we have $np^{k-1}=n$ which is given to be $f$-practical. We examine the intervals between $af(p^k)$ and $af(p^k) +S_f(np^{k-1})$ for $a = 0,1,\dots, S_f(n)$. Since
\begin{align*}
(a+1) f(p^k) \leq af(p^k) + S_f(np^{k-1}) +1
\end{align*}
holds the intervals overlap or are contiguous. Because we have
\begin{align*}
S_f(n)f(p^k) + S_f(np^{k-1}) = \sum_{d\mid n}f(dp^k) + \sum_{d\mid np^{k-1}}f(d) = S_f(np^k),
\end{align*}
1 and $S_f(np^k)$ are included in these intervals. Thus, every integer $m$ with $1\leq m\leq S_f(np^k)$ is representable as $af(p^k) + b$ with $0\leq a \leq S_f(n)$ and $0\leq b\leq S_f(np^{k-1})$. As a result, we have two sets $\mathcal{D}$ and $\mathcal{T}$ of divisors of $n$ and $np^{k-1}$ respectively, so
\begin{align*}
a = \sum_{d\in\mathcal{D}}f(d) && b = \sum_{t\in \mathcal{T}}f(t).
\end{align*}
Therefore we have
\begin{align*}
m = f(p^k) \sum_{d\in\mathcal{D}} f(d)+ \sum_{t\in \mathcal{T}} f(t)= \sum_{d\in\mathcal{D}} f(dp^k)+ \sum_{t\in \mathcal{T}} f(t).
\end{align*}
Since $t\mid np^{k-1}$ and $p^k\mid dp^k$ the $t$'s and $dp^k$'s are distinct. Hence, if we take $\mathcal{E} = \{dp^k: d\in \mathcal{D}\} \cup\mathcal{T}$, we can write
\begin{align*}
m= \sum_{e\in \mathcal{E}}f(e)
\end{align*}
where every $e$ divides $np^k$. Therefore $np^k$ is $f$-practical.
\end{proof}
\begin{cor}
Every squarefree integer is $f$-practical if and only if it is weakly $f$-practical.
\end{cor}
\begin{proof}
By Theorem \ref{thm:f-practicalisweakly}, every squarefree $f$-practical number is also weakly $f$-practical.
Let $n= p_1\dots p_k$ be weakly $f$-practical and squarefree. For every $i=0,1,\dots, k$ we define $m_i=p_1\dots p_i$. Since $n$ is weakly $f$-practical, we have $f(p_{i+1}) \leq S_f(m_i) +1$ for every $i\geq k$. Since 1 is $f$-practical we get from Theorem \ref{thm:f-practicalconstruct} that every $m_i$ is $f$-practical and $n$ is also $f$-practical.
\end{proof}
\begin{thm}\label{thm:everyint}
Every integer $n\in \N$ is $f$-practical if and only if \begin{align}\label{eq:everyint}
f(p^{k}) \leq S_f(p^{k-1})+1
\end{align}
holds for every prime $p$ and every integer $k\geq1$.
\end{thm}
\begin{proof}
If the inequality holds we can use the fact that $S_f$ is multiplicative to show that
\begin{align*}
S_f(np^{k-1}) +1 = S_f(n)S_f(p^{k-1}) +1 \geq S_f(n)(f(p^k)-1) +1
\end{align*}
for every integer $n$ coprime to $p$. Furthermore for $f(p^k)\not=1$ the inequality $S_f(n)(f(p^k)-1)+1\geq f(p^k)$ is equivalent by multiplication by $f(p^k)-1$ to $S_f(n) \geq 1$ which holds for every $n$ since $f(1) = 1$ and $f(m)\geq 0$ for every $m\in\N$. In addition for $f(p^k) = 1$ we have $S_f(n)(f(p^k)-1) +1 = 1 =f(p^k)$. Hence the inequality
\begin{align*}
S_f(np^{k-1}) +1 \geq S_f(n)(f(p^k)-1) +1 \geq f(p^k)
\end{align*}
holds for every prime $p$, any integer $k\geq 1$ and any integer $n\geq0$ coprime to $p$. Thereby the condition for Theorem \ref{thm:f-practicalconstruct} holds for any $n$. Because $1$ is always $f$-practical we can construct every integer greater than 1 as a product of prime powers which are $f$-practical which implies that this integer is also $f$-practical, if $f$ satisfies \eqref{eq:everyint} for every $p$ and $k$.
If there exists a $k\geq 1$ so $f(p^{k}) > S_f(p^{k-1})+1$ holds, $p^k$ is not $f$-practical since $S_f(p^{k-1})+1\leq S_f(p^k)$ is not representable as a sum of $f(d)$'s for some $d\mid p^k$. Hence not every integer greater than $0$ is $f$-practical.
\end{proof}
\section{Classifying functions with Stewart-like criteria}\label{stewart-like}
Stewart gave a way to construct every practical number as a product of practical numbers and prime powers (\cite[Corollary 1]{Stewart}). As shown in the previous section, this is not possible for the $\phi$-practical numbers. We will now categorize the functions for which this means of construction is possible.
\begin{definition}\label{def:iffconvenient}
A function $f$ is \textbf{convenient} if and only if every weakly $f$-practical number is also $f$-practical.
\end{definition}
The following theorem gives an explicit way to check whether a function $f$ is convenient.
\begin{thm}
Let $P_f$ be the set of the prime numbers which are $f$-practical. It is easy to see, that these are exactly the primes $p$ with $f(p)\leq 2$. Then $f$ is convenient if an integer $n$ is $f$-practical if and only if $n$ is of the form
\begin{align*}
n = p_1^{a_1}\dots p_k^{a_k}q_1^{b_1}\dots q_l^{e_l}
\end{align*}
with primes $p_1,\dots,p_k\in P_f$ and $q_1,\dots,p_l\not\in P_f$ with exponents $a_1,\dots, a_k \geq 1$ and $b_1,\dots, b_l \geq 1$ (respectively) satisfying the following conditions
\begin{align*}
f(p_1)\leq \dots \leq f(p_k)<f(q_1)\leq \dots \leq f(q_l) && k>0 \\
f(q_{i+1}) \leq S_f(p_1^{a_1}\dots p_k^{a_k}q_1^{b_1}\dots q_i^{b_i}) +1 &&\text{ for } i=0,1,\dots, l-1.
\end{align*}
\end{thm}\label{def:convenient}
\begin{proof} We have shown in Theorem \ref{thm:f-practicalisweakly} that, for every $f$, each $f$-practical number is also weakly $f$-practical. Hence a function $f$ is convenient if and only if the set of weakly $f$-practical numbers is identical to the set of $f$-practical numbers. \end{proof}
Stewart's condition shows that the identity function is convenient, whereas $\phi$ is inconvenient as the number $75=3 \cdot 5^2$ fulfills every condition and thereby is weakly $\varphi$-practical but not $\phi$-practical.
\begin{thm}
A function is convenient if and only if for every prime $p$ and $f$-practical integer $m$ which is coprime to $p$ the inequality $f(p) \leq S_f(m)+1$ implies $f(p^{k+1}) \leq S_f(mp^k)$.
\end{thm}
\begin{proof}
Let $p$ be a prime for which $f(p)\leq S_f(m) +1$. Therefore $mp^{k+1}$ should be $f$-practical if $f$ is convenient for every $k\geq 0$. If there is an integer $k$ for which $f(p^{k+1})> S_f(mp^k) +1$ then $S_f(mp^k)+1$ is not representable, which is a contradiction. Therefore, for a convenient function $f$, the inequality $f(p)\leq S_f(m) +1$ always implies that $f(p^{k+1}) \leq S_f(mp^k) +1$ for every integer $k$.
If we have $f(p^{k+1})\leq S_f(mp^k) +1$ for every $k$ and if $f(p)\leq S_f(m) +1$ holds for coprime $p$ and $m$, we can use Theorem \ref{thm:f-practicalconstruct} to show that $mp^k$ is $f$-practical for every $k$. Therefore every integer $n$ which fulfills the conditions of Definition \ref{def:convenient} is $f$-practical. Since it has already been shown that every $f$-practical fulfills this condition, we have that $f$ is convenient.
\end{proof}
\begin{thm}
A function $f$ is convenient if and only if for every prime $p$ for which there is a coprime integer $m$ with $f(p)\leq S_f(m) +1$ the inequality
\begin{align*}
f(p^{k+1}) \leq f(p)f(p^k)
\end{align*}
holds.
\end{thm}
\begin{proof}
Let $p$ be a prime and $m\in\N$ with $\gcd(m,p)=1$ and $f(p)\leq S_f(m) +1$. Assume that the above inequality holds for $p$. We then show $f(p^{k+1})\leq S_f(mp^k) +1$ by induction over $k$. The base case is fulfilled for $k=0$ since we have $f(p) \leq S_f(m)+1$. Assume that $f(p^{i+1})<S_f(mp^i)+1$ for all $i<k$. We obtain
\begin{align*}
f(p^{k+1})&\leq f(p)f(p^k) \leq (S_f(m)+1)f(p^k) = S_f(m)f(p^k) + f(p^k) \\
&\leq S_f(m) f(p^k) + S_f(mp^{k-1}) +1 = S_f(m)f(p^k) + S_f(m)S_f(p^{k-1}) +1 \\
&= S_f(m)(S_f(p^{k-1} +f(p^k))) +1 = S_f(mp^k) +1.
\end{align*}
Therefore, we have $f(p^{k+1}) \leq f(p)f(p^k)$ for some $p$ for which there is an integer $m$ coprime to $p$ with $f(p)\leq S_f(m) +1$, so $f$ is convenient.
Assume $f$ is convenient. Hence, for all primes $p$ and integers $m$ coprime to $p$ with $f(p)\leq S_f(m)+1$, we have $f(p^{k+1}) \leq S_f(mp^k)+1$ for every $k$. For every such $p$ we also have
\begin{align*}
f(p^{k+1}) &\leq S_f(mp^k) +1 = S_f(m)S_f(p^k) +1 = S_f(m)(f(p^k) + S_f(p^{k-1})) +1 \\
&= f(p^k)S_f(m) + S_f(p^{k-1}m)+1 \leq f(p^k)(f(p)-1) +S_f(mp^{k-1}) +1 \\
&= f(p^k)f(p) +S_f(mp^{k-1}) +1 -f(p^k)\\
&\leq f(p^k)f(p)
\end{align*}
Therefore every convenient $f$ fulfills the above condition.
\end{proof}
\begin{cor}
Let $f$ be convenient. Suppose there is at least one prime $p$ with $1\leq f(p)\leq 2$. Then, for every $k$ primes $f(p_1) \leq f(p_2) \leq \dots \leq f(p_k)$ where there is at least one prime $p_j$ with $1\leq f(p_i)\leq 2$ for all $i \leq j$, there are $k$ integers $E_1,E_2,\dots,E_k$ so that, for every $k$ integers $e_1,\dots,e_k$ with $e_i\geq E_i$, the integer $p_1^{e_1}\dots p_k^{e_k}$ is $f$-practical.
\end{cor}
\section{$f$-practical sets with various densities}\label{densities}
As we remarked in the introduction, the set of practical numbers and the set of $\varphi$-practical numbers both have asymptotic density $0$. In this section, we examine the densities of other $f$-practical sets. First, we give some natural examples with asymptotic density $1$.
\begin{example}
Let $\tau$ be the count-of-divisors function. Every positive integer is $\tau$-practical. This follows from Theorem \ref{thm:everyint} due to the fact that $\tau(p^k) = k+1 \leq \frac{k(k+1)}{2}+1 = S_f(p^{k-1}) +1 \leq S_f(n)S_f(p^{k-1}) +1 = S_f(np^{k-1}) +1$ holds for every prime $p$ and positive integer $k$.
\end{example}
If we take the inequality \eqref{eq:everyint} as an equality for every $p$ and $k$ we obtain following function.
\begin{example}
Let $v_p(n)$ denote the $p$-adic valuation of $n$. The function $h:\N\to\N$ is defined by
\begin{align*}
h(n) = 2^{\sum_p v_p(n)}.
\end{align*}
Since we have $v_p(ab) = v_p(a) +v_p(b)$ this function is multiplicative. It satisfies the condition of Theorem \ref{thm:everyint} and therefore every positive integer is $h$-practical.
\end{example}
The following lemma shows that one can construct $f$-practical sets with any density.
\begin{lemma}\label{thm:denseconstruction}
For each $n\in \N$, there is a function $f_n$ such that the asymptotic density of $f_n$-practical numbers in $\N$ is $1-\frac{\phi(n)}{n}$.
\end{lemma}
\begin{proof}
We define the multiplicative function $f_n$ by $f_n(1) = 1$ and
\begin{align*}
f_n(p^k) = \begin{cases}
2 &\text{ if } p|n\\
3 &\text{ else }
\end{cases}.
\end{align*}
By Definition \ref{def:iffconvenient}, this function is convenient. So, by definition, the $f_n$-practical numbers are exactly $1$ and the natural numbers divisible by a prime which also divides $n$, since $f(q) = 3 \leq S_f(m) +1$ for every prime $q$ which does not divide $n$ and every $m>1$. Therefore the density of the $f_n$-practical numbers is the density of the numbers not coprime to $n$, which is $1-\frac{\phi(n)}{n}$.
\end{proof}
From this lemma, we can deduce the following theorem:
\begin{thm} The densities of the $f$-practical sets are dense in $[0, 1].$
\end{thm}
\begin{proof} From Lemma \ref{thm:denseconstruction}, for any integer $n \in \N$, we can construct a set of $f$-practical numbers with density $1 - \frac{\varphi(n)}{n}$. By \cite[\S 5.17]{Schoenberg}, the values of $\frac{\varphi(n)}{n}$ are dense in $[0, 1].$ Thus, the complementary values $1 - \frac{\varphi(n)}{n}$ must be dense in $[0, 1]$ as well. \end{proof}
\section{Chebyshev bounds for certain $f$-practical sets}\label{boundsforcertainf}
In this section, we demonstrate how the machinery developed in \cite{Thompson} can be used to prove Chebyshev-type bounds for other $f$-practical sets with $f(p^k) \approx (f(p))^k$. We investigate two particular examples with this property: $f = \varphi^*, \lambda.$
\subsection{The $\phi^*$ function}
A divisor $d$ of an integer $n$ is unitary if $\gcd(d, n/d) = 1.$ The unitary totient function $\phi^*$ counts the number of positive integers $k\leq n$ for which the greatest unitary divisor of $n$ which is also a divisor of $k$ is $1$. Therefore we have for all prime $p$ and all integers $k\geq 1$
\begin{align*}
\phi^*(p^k) = p^k-1.
\end{align*}
Following the second author's proofs in \cite{thesis} we can now establish an upper bound for the number of $\phi^*$-practical integers.
\begin{lemma}\label{lem:evenphistar}
Every even weakly $\phi^*$-practical number is practical.
\end{lemma}
\begin{proof}
Let $n$ be an even $\phi^*$-practical number. Then, with the notation of Definition \ref{def:weaklyfpractical}, for every $0\leq i < \omega(n)$, the inequality $\phi^*(p_{i+1}) = p_{i+1}^{e_i+1}-1\leq 1+ S_{\phi^*}(m_i)$ holds. For every integer $k>1$, we have $\phi^*(k)< k$ and $S_{\phi^*}(k) < \sigma(k)$, since the inequalities hold for the prime powers and the functions are multiplicative. Therefore, for every $m_i>1$, we have $p_{i+1}^{e_{i+1}} = \phi^*(p_{i+1}^{e_{i+1}}) +1 \leq S_{\phi^*}(m_i) + 2 \leq \sigma(m_i) +1$. Since $n$ is even, we have $m_0=1$ and $m_1=2$. Hence, we obtain by induction over $i$ and by \cite[Theorem 1]{Stewart} that $n$ is practical.
\end{proof}
Now we can use Saias' upper bound for the number $PR(X)$ of practical numbers less than or equal to $X$ (\cite[Th\'eor\`eme 2]{Saias}) to prove the next result following the second author's proof in \cite[Theorem 4.8]{thesis}.
\begin{thm}\label{thm:upperphistar}
There exists a positive constant $u_{\phi^*}$ such that
\begin{align*}
F_{\phi^*}(X) \leq u_{\phi^*}\frac{X}{\log X}
\end{align*}
holds for any $X\geq 2$.
\end{thm}
\begin{proof}
Let $
X$ be a positive number and let $n$ be a $\phi^*$-practical number in the interval $(0,X]$. Therefore $n$ is also weakly $f$-practical. If $n$ is even it is also practical. If $n$ is odd there is a unique integer $l$ so that $2^ln$ is in $(X,2X]$. Corollary \ref{cor:smallprime} implies that $2^ln$ is also weakly $\phi^*$-practical for $n>1$ and it is easy to see that every power of 2 is weakly $\phi^*$-practical. Thus, we obtain
\begin{align*}
F_{\phi^*} (X) &= \#\{n\leq X: n \text{ is } \phi^*\text{-practical}\}\\
&= \#\{n\leq X: n \text{ is even and } \phi^*\text{-practical}\} +\#\{n\leq X: n \text{ is odd and } \phi^*\text{-practical}\}\\
&\leq \#\{n\leq X: n \text{ is practical}\} + \#\{X<m\leq 2X: m \text{ is practical}\}\\
&= \mathrm{PR}(2X).
\end{align*}
As proven by Saias \cite[Theorem 2]{Saias} there exists a positive constant $u_{I}$ so that
\begin{align*}
\mathrm{PR}(X) \leq u_I \frac{X}{\log X}
\end{align*}
holds. By choosing $u_{\phi^*}=2u_I$ we obtain the desired result.
\end{proof}
Since we have $\phi(p) = \phi^*(p)$ for every prime $p$ the squarefree $\phi^*$-practical numbers are exactly the squarefree $\phi$-practical numbers. As shown by the second author in \cite[Lemma 4.17 and Theorem 4.21]{thesis} there exists a lower bound $c\frac{X}{\log X}$ for the number of squarefree $\phi$-practical numbers less than or equal to $X$. Since the squarefree $\phi$-practical and $\phi^*$-practical numbers are the same, we thereby obtain a lower bound for the number of squarefree $\phi^*$-practical numbers less than or equal to $X$.
\begin{thm}
There exists a positive constant $l_{\phi^*}$ so that
\begin{align*}
l_{\phi^*} \frac{X}{\log X} \leq \#\{n\leq X: n \text{ is squarefree and $\phi^*$-practical}\} \leq F_{\phi^*}(X)
\end{align*}
holds for every $X\geq 2$.
\end{thm}
\subsection{The Carmichael $\lambda$ function}
The Carmichael function $\lambda(n)$ denotes the least integer $m$ for which we have
\begin{align*}
a^m \equiv 1 \mod n
\end{align*}
for every integer $a$ coprime to $n$. We will use $\lambda^\star$ to denote the set of positive integers that are $f$-practical when $f = \lambda$. We use the $\star$ notation to emphasize that this notion of ``$\lambda$-practical'' differs from the definition of $\lambda$-practical given by the second author in \cite{lambda}, which can be stated as follows:
\begin{definition}\label{def:lambdapractical} An integer $n$ is $\lambda$-practical if and only if we can write every $m$ with $1 \leq m \leq n$ in the form $m = \sum_{d \mid n} \lambda(d) m_d$, where $m_d$ is an integer with $0 \leq m_d \leq \frac{\varphi(d)}{\lambda(d)}.$ \end{definition}
The values of $n$ satisfying this definition of $\lambda$-practical are precisely those for which the polynomial $x^n-1$ has a divisor of every degree between $1$ and $n$ in $\mathbb{F}_p[x]$ for all primes $p$. The sets of $\lambda$-practical numbers and $\lambda^\star$-practical numbers do not coincide. For example, $156$ satisfies the definition of $\lambda$-practical in \cite{lambda} but it does not satisfy our definition of $\lambda^\star$-practical. However, it turns out that every $\lambda^\star$-practical number is $\lambda$-practical. We will prove a slightly more general theorem.
\begin{thm}\label{sortedlist} Suppose that $w_1,...,w_t$ and $u_1,...,u_t$ are positive integers, with $w_1 > w_2 > \cdots > w_t$. Let $\mathcal{S} = \sum_{i=1}^t w_i$ and $\mathcal{T} = \sum_{i=1}^t u_i w_i$. Suppose that each positive integer up to $S$ is a subset sum of $w_i$'s. Then each $m \leq \mathcal{T}$ can be written in the form $$m = \sum_{i=1}^t a_i w_i,$$ where $0 \leq a_i \leq u_i$.\end{thm}
\begin{proof} Let $\mathcal{W}$ be a list of $w_i$'s, with $u_i$ instances of each $w_i$, written in decreasing order. Let $k \leq \mathcal{T}$. Starting with the first entry, iteratively subtract elements of $\mathcal{W}$ from $k$, removing each element from the list after it is subtracted to create a new list with one fewer entry. Terminate the process upon arriving at some $k'$ that is either $0$ or smaller than the largest remaining $w_i$, which we will denote $w_j$. By hypothesis, since $k' < w_j \leq S$ then $k'$ is a subset sum of $w_{j+1},...,w_t$. Now, if we add $k'$ to all of the $w_i$'s which were previously subtracted, then $k$ is representable as
\begin{align*}
k=\sum_{i=1}^t a_i w_i,
\end{align*}
with $0 \leq a_i \leq u_i$, as claimed.
\end{proof}
\begin{cor}
Every $\lambda^\star$-practical number is also $\lambda$-practical.
\end{cor}
\begin{proof}
The result follows from applying Theorem \ref{sortedlist} with $t = \tau(n)$; $w_1,...,w_t$ the sorted list of values of $\lambda(d)$ with $d \mid n$; $u_1,...,u_t$ the corresponding values of $\varphi(d)/\lambda(d)$; $\mathcal{S} = S_\lambda(n)$; and $\mathcal{T} = n$. \end{proof}
We can use the upper bound for the count of $\lambda$-practical numbers given by \cite[Proposition 5.1]{lambda} to deduce the following theorem.
\begin{thm}
There exists a positive constant $u_{\lambda^\star}$ such that
\begin{align*}
F_{\lambda^\star}(X) \leq u_{\lambda^\star} \frac{X}{\log X}
\end{align*}
holds.
\end{thm}
Unfortunately, we have been unable to prove a reasonable lower bound for $F_{\lambda^\star}(X)$. It is not clear from our computations (see Tables \ref{table:lambdastar1} and \ref{table:lambdastar2}) whether $X/\log X$ is the correct order of magnitude for the $\lambda^\star$-practicals.
\begin{table}
\centering
\begin{minipage}{0.4\textwidth}
\begin{tabular}{ | l | l | c |}
\hline
$X$ & $F_\lambda^\star(X)$ & $F_\lambda^\star(X)/(X/\log X)$ \\ \hline
$10^1$ & 6 & 1.381551 \\
$10^2$ & 28 & 1.289448 \\
$10^3$ & 164 & 1.132872 \\
$10^4$ & 1015 & 0.934850 \\
$10^5$ & 7128 & 0.820641 \\
$10^6$ & 52326 & 0.722910 \\
$10^7$ & 409714 & 0.660381 \\
\hline
\end{tabular}
\caption{Ratios for $\lambda^\star$-practicals}\label{table:lambdastar1}
\end{minipage}
\hspace{0.4 in}
\begin{minipage}{.5\textwidth}
\begin{tabular}{ | l | l | c | }
\hline
$X$ & $F_\lambda^\star(X)$ & $F_\lambda^\star(X)/(X/\log X)$\\ \hline
$1 \cdot 10^6$ & 52326 & 0.722910\\
$2 \cdot 10^6$ & 96667 & 0.701254\\
$3 \cdot 10^6$ & 139139 & 0.691712\\
$4 \cdot 10^6$ & 179854 & 0.683526\\
$5 \cdot 10^6$ & 219598 & 0.677458\\
$6 \cdot 10^6$ & 258656 & 0.672819\\
$7 \cdot 10^6$ & 297202 & 0.669189\\
$8 \cdot 10^6$ & 335181 & 0.665961\\
$9 \cdot 10^6$ & 372779 & 0.663246\\
$1 \cdot 10^7$ & 409714 & 0.660381\\
\hline
\end{tabular}\caption{A closer look at the range from $10^6$ to $10^7$}\label{table:lambdastar2}
\end{minipage}\hfill
\end{table}
\section{$f$-practicals for non-multiplicative $f$}\label{non-multiplicativef}
In this section, we remove the condition that $f$ is multiplicative and study the corresponding $f$-practical sets for several well-known non-multiplicative functions.
\subsection{additive functions}
The naive criterion of Proposition \ref{thm:thesisA1} is of much better use for additive functions than for multiplicative functions. For the additive functions we want to consider, we require $f(p)\geq 1$ for every prime $p$.
\begin{lemma}\label{lem:naiveadditive}
Let $n=\prod_{i=1}^k p_i^{e_i}$ be a positive integer with prime $p_i$. Then $n$ is $f$-practical for an additive function $f$ if and only if
\begin{align*}
f(p_i^e) \leq 1 + \sum_{\substack {d|n\\f(d) < f(p_i^e)}} f(d)
\end{align*}
holds for every $1\leq i\leq k$ and $e\leq e_i$.
\end{lemma}
\begin{proof}
It follows immediately from Proposition \ref{thm:thesisA1} that this inequality is necessary for $n$ to be $f$-practical. Now let $t = \prod_{i=1}^k p_i^{a_i}$ be a divisor of $n$. Since $f$ is additive we have $f(t) > f(p_i^{a_i})$ for every $i$ if $t$ is not a prime power, i.e., we must have $a_i>0$ for at least two different $i$. We thereby obtain
\begin{align*}
f(t) = f(\prod_{i=1}^k p_i^{a_i}) = \sum_{i=1}^k f(p_i^{a_i}) \leq 1 + \sum_{\substack{d|n\\f(d) < f(t)}} f(d)
\end{align*}
which implies that $n$ is $f$-practical.
\end{proof}
\begin{cor}\label{cor:additiveconvenient}
For additive functions $f$, every integer $n$ is $f$-practical if and only if $f(p^k) \leq 1+\sum_{i=0}^{k-1} f(p^i)$ holds for every prime $p$ and any positive integer $i$.
\end{cor}
\begin{remark} Let $\omega(n)$ denote the number of distinct prime factors of an integer $n$, and let $\Omega(n)$ denote the number of prime factors of $n$ with multiplicity. Both functions are additive but not multiplicative. Corollary \ref{cor:additiveconvenient} shows that every positive integer $n$ is $\omega$-practical and $\Omega$-practical. \end{remark}
\begin{remark} Let $f = v_p(n)$, the exact power of $p$ dividing $n$. The fact that the set of $f$-practicals encompasses all natural numbers follows from Corollary \ref{cor:additiveconvenient}. One can also prove that all natural numbers are $v_p$-practical via a simple combinatorial argument: we can write $$S_{v_p}(n) = \frac{{v_p(n)}(v_p(n) + 1)}{2} \cdot \tau\left(\frac{n}{p^{v_p(n)}}\right),$$ where $\frac{{v_p(n)}(v_p(n) + 1)}{2} = 1 + 2 + \cdots + v_p(n)$ is the sum of all of the valuations at powers of $p$ and $\tau(\frac{n}{p^{v_p(n)}})$ represents the number of identical copies of the valuations $1,2,...,v_p(n)$, which come from multiplying the powers of $p$ by each of the divisors of $n$ that are coprime to $p$. Every integer $m$ in the interval $[1, S_{v_p}(n)]$ can be represented as $m = v_p(n) q + r$ for some $r, q$ satisfying $0 \leq r < v_p(n)$ and $0 \leq q \leq \tau(n/p^{v_p(n)})$. \end{remark}
As the next example demonstrates, there are also some natural examples of additive functions for which the set of $f$-practicals does not coincide with the full set of natural numbers.
\begin{example}
Let $a_1(n) = \sum_{p|n} p$, the sum of the distinct primes dividing $n$. For every $n>1$ and every $1<d|n$ there is a prime $p>1$ which divides $d$. Therefore we have $a_1(d) \geq p >1$. Hence the number $1< a_1(n) \leq S_{a_1}(n)$ is not representable. In particular, this shows that $1$ is the only $a_1$-practical number.
\end{example}
\subsection{Functions which are neither additive nor multiplicative}
We can also define $f$-practical numbers for functions $f$ which are neither multiplicative nor additive. One such example is the sum-of-proper-divisors function, which is defined as follows:
\begin{definition} Let $s: \N \to \N$ be given by $s(n)= \sigma(n) - n.$\end{definition}
The function $s$ is used in the study of perfect numbers. Namely, if $s(n) = n$ then $n$ is perfect. If $s(n) > n$, we say that $n$ is abundant. Since $s$ is an arithmetic function, we can use the $f$-practical definition to define $s$-practical numbers. We begin by demonstrating that there are infinitely many $s$-practical numbers. To show this, we will need the following lemma.
\begin{lemma}\label{lem:calcs}
For two coprime integers $a,b$ we have $$s(ab) = s(a)s(b) + as(b) + bs(a).$$
\end{lemma}
\begin{proof} Observe that
\begin{align*}
s(ab) &= \sigma(ab) -ab = \sigma(a)\sigma(b) -ab = (s(a)+a)(s(b) +b) -ab \\
&= s(a)s(b) + as(b) + bs(a).
\end{align*}
\end{proof}
\begin{thm} There are infinitely many $s$-practical numbers.\end{thm}
\begin{proof}
Every prime is $s$-practical, since we have
\begin{align*}
S_s(p) = \sum_{d|p}s(d) = s(1) + s(p) = 0 + 1= 1
\end{align*}
for prime $p$.
\end{proof}
\begin{comment}We proceed by constructing an infinite family of $s$-practical numbers. Namely, we will show that for every integer $k\geq 4$, the number $2^k\cdot 3$ is $s$-practical. We can check that $2^4\cdot 3 = 48$ is $s$-practical. The rest follows by induction over $k$. We have $s(2^{k+1}) = 2^{k+1} - 1$ which is smaller than $s(2^k\cdot 3) = s(2^{k+1}) + 3s(2^k)= 2^{k+1} +3\cdot 2^k -4$. We also get
\begin{align*}
s(2^{k+1}\cdot 3) &= s(2^{k+2}) + 3s(2^{k+1}) = 2^{k+2} +3\cdot 2^{k+1}-4\\
&\leq s(2^{k+1}) +s(2^k\cdot 3) + s(2^k) + s(2^{k-1}\cdot 3) + s(2) +s(3) + s(4) +1
\end{align*}
Therefore Proposition \ref{thm:thesisA1} holds for all new divisors and for the old ones.
\end{comment}
The function $s$ is not multiplicative, which prevents us from using the machinery developed in previous sections to prove upper and lower bounds for the count of $s$-practical numbers. However, it is still possible to show that the $s$-practical numbers arising from integers $n$ with $n \leq 2S_s(n)$ have asymptotic density $0$. To prove this, we will follow an argument that was used by the second author in \cite{Thompson} to show that the $\varphi$-practical numbers have asymptotic density $0$. We note that Erd\H{o}s \cite{Erdos} was the first to claim that the practical numbers have asymptotic density $0$. Although he did not write down a proof, it is likely that he had a similar argument in mind.
\begin{thm}\label{thm:abundantdensity}
If $n^{1/2} \leq S_s(n)$ then the $s$-practicals have asymptotic density $0$.
\end{thm}
\begin{proof}
We have $ \tau(n)\leq 2^{\Omega(n)}$. Because $\Omega(n)$ has normal order $\log\log n$, for all $n$ except a set with asymptotic density $0$ we have
\begin{align*}
\tau(n)\leq 2^{\Omega(n)} \leq 2^{(1+\epsilon)\log\log(n)} = (\log n)^{(1+\epsilon)\log 2} \leq (\log n)^{0.7}
\end{align*}
if we fix $\epsilon=1/1000$. But for every $s$-practical $n$ it has to be the case that $ S_s(n)\leq 2^{\tau(n)-1}$ since there are at most $2^{\tau(n)}$ different numbers which can be represented as the sum of $s(d)$'s where the $d$'s are some of the $\tau(n)$ divisors of $n$ and each number between $1$ and $S_s(n)$ has to be representable as such a sum in order for $n$ to be $s$-practical. Because we have $s(1)=0$, half of these possible sums coincide, as there is no difference between the sums with and without $s(1)$. From the hypothesis $n^{1/2} \leq S_s(n)$, we hence obtain
\begin{align*}
\frac{1}{2} \log n \leq \log S_s(n)\leq \tau(n)\log 2 < \tau(n) \leq (\log n)^{0.7}.
\end{align*}
But for all $n \geq e^{8\sqrt[3]{2}}$, we have
\begin{align*}
\frac 12(\log n)^{0.3} \geq \frac12 \left(\sqrt[3]{1024}\right)^{0.3} = \frac 12 \cdot 1024^{0.1} = 1.
\end{align*}
So for almost all $n$ the inequality $\frac12\log n \geq (\log n)^{0.7}$ holds. Therefore the set of $s$-practical values of $n$ with $n^{1/2} \leq S_s(n)$ has asymptotic density 0.
\end{proof}
\begin{lemma}\label{lemma:density1ineq} The inequality $n^{1/2} \leq S_s(n)$ holds for almost all $n$. \end{lemma}
\begin{proof} If $n$ is composite, its least prime factor $p$ satisfies $p \leq n^{1/2}$, so $n/p$ is a proper divisor of $n$ that is $\geq n^{1/2}$. Thus, we have $$S_s(n) \geq s(n) \geq \frac{n}{p} \geq n^{1/2}.$$ Since the set of composite numbers has asymptotic density $1$, it follows that $S_s(n) \geq n^{1/2}$ holds for almost all $n$. \end{proof}
We can deduce the following corollary from Theorem \ref{thm:abundantdensity} and Lemma \ref{lemma:density1ineq}.
\begin{cor} The set of $s$-practical numbers has asymptotic density $0$. \end{cor}
\begin{comment}
The standard definition of weakly $f$-practicals is not helpful for the $s$ function, since we have $s(p) = 1$ for every prime $p$ and so every positive integer ist weakly $s$-practical. Therefore we want to somehow adapt the definition to our needs.
\begin{definition}\label{def:weaklyspractical}
We call a number $n = \prod_{i=1}^k p_i^{e_i}$ weakly $s$-practical if for every $1\leq i \leq k$ the inequality
\begin{align*}
p_i \leq 1+ S_s\left(\prod_{j=1}^{i-1}p_i^{e_i}\right) + k - i+1
\end{align*}
holds.
\end{definition}
\begin{thm}\label{thm:weaklyisspractical}
Every $s$-practical number is either a prime or weakly $s$-practical.
\end{thm}
\begin{proof}
Let $n=\prod_{i=1}^k p_i^{e_i}$ be a $s$-practical number which is not prime. If $n$ were a prime power $p^e$, with $e\geq2$, then the divisors of $p^e$ are all of the form $p^i$ with $0\leq i \leq e$. We have $s(p^i) >s(p^j)$ for $i>j$. Therefore we need $s(p^2) = p+1 \leq 1+ s(0) +s(p) = 2$ which is not possible. Therefore we have $k\geq2$. Now assume that there ist an $i$ with
\begin{align*}
p_i& > 1+ S_s\left(\prod_{j=1}^{i-1}p_i^{e_i}\right)+k -i +1= 1+ S_s\left(\prod_{j=1}^{i-1}p_i^{e_i}\right)+ \sum_{j=i}^k s(p_j).
\end{align*}
Now on the right side is the sum of all divisors $d$ of $n$ with $s(d)\leq p_i$. To proof this take a divisor $d$ of $n$ which is not in this sum. Hence $d$ is not prime and has at least one prime divisor $p$ with $p>p_i$. If $d$ is a prime power then we have $s(d) = s(p^e) \geq s(p^2) = p+1 > p$. Otherwise we can write $d = p^{v_p(d)} \cdot a$ for some $a>1$ coprime to $p$. Then we obtain by Lemma \ref{lem:calcs}
\begin{align*}
s(d) = s(p^{v_p(d)}a) = s(p^{v_p(d)})s(a) + p^{v_p(d)} s(a) + s(p^{v_p(d)}) a \geq 1\cdot 1 + p \cdot 1 = p+1 > p.
\end{align*}
Since $p$ is bigger than the sum of all $s(d)$ with $s(d)\leq p$ and smaller than $s(d)$ for any other $s$, $p$ is not representable as sum of $s(d)$'s for divisors $d$ of $n$. This a contradiction, therefore the inequality holds for all $i$ and $n$ is weakly $s$-practical.
\end{proof}
\end{comment}
\begin{comment}
In the spirit of the classical work of Schoenberg \cite{Schoenberg} et al. on $\frac{\varphi(n)}{n}$, it is natural to consider whether the function $\frac{S_s(n)}{n}$ possesses a distribution function. A key ingredient will be the following theorem of Erd\H{o}s and Wintner (see, for example, \cite[Theorem 4.1]{tenenbaumbook}:
\begin{thm}\label{thm:erdoswintner} A real additive function $f(n)$ has a limiting distribution if and only if the following three series converge simultaneously for at least one value of the positive real numbers $R$:
\begin{enumerate*}
\item[(a)] $\displaystyle\sum_{|f(p)| > R} \frac{1}{p};$
\item[(b)] $\displaystyle\sum_{|f(p)| \leq R} \frac{f(p)^2}{p};$
\item[(c)] $\displaystyle\sum_{|f(p)| \leq R} \frac{f(p)}{p}.$
\end{enumerate*}
\end{thm}
We can, in fact, deduce that the distribution function for $\frac{S_s(n)}{n}$ is continuous by appealing to a recent theorem of Lebowitz-Lockard and Pollack \cite[Theorem 1]{llp}:
\begin{thm}\label{thm:llp} Let $f_1,...,f_k$ be multiplicative arithmetic functions taking values in the nonzero real numbers and satisfying the following conditions:
\begin{enumerate}
\item $f_k$ does not cluster around $0$
\item for all $i < j$ with $i, j \in \{1,2,...,k\}$, the function $f_i/f_j$ is nonclustering.
\item for each $i$, whenever $p$ and $p'$ are distinct primes, we have $f_i(p) \neq f_i(p')$. Then for all nonzero $c_1,...,c_k \in \mathbb{R}$, the arithmetic function $F:= c_1 f_1 + \cdots c_k f_k$ is nonclustering.
\end{enumerate}
\end{thm}
\begin{thm} The function $\frac{S_s(n)}{n}$ has a continuous distribution function. \end{thm}
\begin{proof}
We can write $$S_s(n) = \sum_{d \mid n} (\sigma(d) - d) = S_{\sigma}(n) - \sigma(n).$$ Thus, $$\frac{S_s(n)}{n} = \frac{S_\sigma(n)}{n} - \frac{\sigma(n)}{n}$$ is a difference of two multiplicative functions.
Let $f_1 = \frac{S_\sigma(n)}{n}$, $f_2 = \frac{\sigma(n)}{n}$, and $F = f_1 + (-1)f_2$. It is easy to check that $\log f_1$ and $\log f_2$ are additive functions satisfying conditions (a) - (c) of Theorem \ref{thm:erdoswintner} (one can take, for example, $R=1$), so both $\log f_1$ and $\log f_2$ have distribution functions. It follows that $f_1$ and $f_2$ have distribution functions as well. From \cite[Proposition 5]{llp} the function $F$, as a linear combination of $f_1$ and $f_2$, has a distribution function. To show that the distribution function for $F$ is continuous, we need to show that $F$ satisfies the conditions of Theorem \ref{thm:llp}. Applying Theorem \ref{thm:erdoswintner} to $\log f_2$ and $\log (f_1/f_2)$ shows that $f_2$ and $f_1/f_2$ are non-clustering. Thus, conditions (1) and (2) of Theorem \ref{thm:llp} are satisfied. Moreover, the functions $\frac{\sigma(p)}{p}$ and $\frac{S_\sigma(p)}{p}$ are strictly increasing as they run over prime inputs, so condition (3) is satisfied. Therefore, by Theorem \ref{thm:llp}, $F$ is non-clustering. Since a distribution function for an arithmetic function $F$ is continuous precisely when $F$ is non-clustering, it follows that $F$ is continuous.
\end{proof}
\end{comment}
\section*{Acknowledgements}
This research was initiated when the first author was a student in the intern program at the Max-Planck-Institut f\"{u}r Mathematik and while the second author was a visiting researcher there. Both authors would like to thank the Max-Planck-Institut f\"{u}r Mathematik for making this collaboration possible. Portions of this work were completed while the second author was in residence at the Mathematical Sciences Research Institute, during which time she was supported by the National Science Foundation under Grant No. DMS-1440140. The second author is also supported by an AMS Simons Travel Grant. Both authors are grateful to Carl Pomerance for posing the question answered in Theorem \ref{thm:densitiesaredense} and for suggesting the generalized version of Theorem \ref{sortedlist} that is presented in this paper. The authors are also grateful to Paul Pollack for helpful comments which led to an improvement in their computation of the asymptotic density of $s$-practicals.
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| {
"timestamp": "2017-03-24T01:02:13",
"yymm": "1701",
"arxiv_id": "1701.08504",
"language": "en",
"url": "https://arxiv.org/abs/1701.08504",
"abstract": "A positive integer $n$ is practical if every $m \\leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and asking whether all integers in a given interval can be expressed as sums of $f(d)$'s, where the $d$'s are distinct divisors of $n$. We will refer to such $n$ as `$f$-practical.' In this paper, we introduce the $f$-practical numbers for the first time. We give criteria for when all $f$-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct $f$-practical sets with any asymptotic density, and prove a series of results related to the distribution of $f$-practical numbers for many well-known arithmetic functions $f$.",
"subjects": "Number Theory (math.NT)",
"title": "A generalization of the practical numbers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9914225148397244,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7092874630918207
} |
https://arxiv.org/abs/1506.08869 | Carries and the arithmetic progression structure of sets | If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $m=p$, $p$ prime, the only set $A$ which induces two distinct carries, i. e. with $A+A \subseteq \{ x, y \}+A$ for some $x, y \in \mathbb{Z}/p^2\mathbb{Z}$, is the arithmetic progression $[0, p-1]$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $m^2$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference. | \section{Introduction}
If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues
modulo $m$. The most popular choices are the integers in $[0, m-1]$ and the integers in $(-m/2, m/2]$.
When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a\in A$ such that
\[ a_1+a_2 \equiv a \pmod m , \] which will be the last digit of the
sum, and $(a_1 + a_2 -a)/m$ will be the \emph{carry}. Diaconis, Shao and Soundararajan in the nice paper \cite{diaconis2014carries} and Alon in \cite{alon} show that the above two popular sets
both have an extremal property: $(-m/2, m/2]$ minimizes the number of pairs $a_1, a_2$ for which there is a nonzero carry, while
$[0, m-1]$ minimizes the number of distinct carries, and both examples are unique up to certain linear transformations.
The second extremal property is essentially equivalent to the following statement:
\emph{Let $A\subset \setZ_{m^2} $ be a set which forms a complete set of residues modulo $m$. If $A+A \subset A + \{x,y\}$ with some
$x,y\in \setZ_{m^2} $, then $A$ is an arithmetic progression.}
In \cite{diaconis2014carries} this is proved for the case $m$ prime.
From now on we call a set $A\subset\setZ_q$ a \emph{digital set,} if $m=|A|$ satisfies $m|q$, and $A$ is a complete set of residues modulo $m$.
A more general claim could sound as follows:
\emph{Let $A\subset \setZ_{m^2} $ be a digital set with $|A| =m$. If $|A+A| \leq 2m$,
then $A$ is an arithmetic progression.}
In \cite{Grynkiewicz} we find a complete description of finite sets in commutative groups satisfying $|A+A| \leq 2|A|$. This could be used to deduce the
above claim. This deduction is not immediate, however, as this description contains a lot of subcases.
We remark also that in \cite{Hamidoune} Hamidoune, Serra and Z{\'e}mor prove a result somehow similar to our Theorem \ref{main}, albeit with a restriction on $k$ and with different hypotheses, which could be used to prove the claim.
The aim of this paper is to provide a further generalization of the following form:
\emph{Let $A\subset \setZ_{m^2} $ be a digital set with $|A| =m$. For every set $B$
such that $ 1 < |B| < m^2-m$ we have $ |A+B| > m + |B|$, with certain exactly described exceptions.}
Digital sets of cardinality $m$ exist in $\setZ_q$ whenever $m|q$.
For our arguments we need a stronger assumption, which is, however, more general than the case $q=m^2$, namely, that
$m$ and $q$ are composed of the same primes, and the exponent of each prime in $q$ is strictly greater than in $m$.
This is a natural restriction, as otherwise there are digital sets that are either contained in a nontrivial subgroup,
or are unions of cosets of a nontrivial subgroup.
As we are looking for estimates that depend only on the cardinality of the other set $B$, it is comfortable to express this
in terms of the \emph{impact function} of the set $A$:
\[ \xi(n) = \xi_A(n) = \min_{|B| =n} |A+B| , \]
defined for integers $n$ that can serve as cardinality of a set; if we are in $ \setZ_q$, this means $|B| \leq q$.
Some values of $\xi$ are determined by the size of $A$: we have $\xi(0)=0$, $\xi(1)=m$, $\xi(n)=q$ for $q-m < n \leq q$ and
$\xi(q-m)=q-1$ by a familiar pigeonhole argument. A nontrivial estimate may exist for $1<n<q-m$. The case $n=2$ can be interpreted
via the arithmetic progression structure of $A$. Given any $t\in \setZ_q \setminus \{0 \} $, $A$ can be decomposed as the union of
some cosets of the subgroup generated by $t$ and some arithmetic progressions of difference $t$. Let $\alpha_t(A)$ be the number of
arithmetic progressions in this decomposition. We have clearly
\[ | A + \{x, x+t\}| = m + \alpha_t(A) \]
for every $x$, hence
\[ \xi(2) = \min_t \alpha_t(A) . \]
Thus, $\xi(2)>m+2$ holds unless $A$ is the union of at most two arithmetic progressions (as we shall soon see, digital sets
do not contain nontrivial cosets). Hence the strongest result of this kind that may hold (save the bound 15) sounds as follows.
\bteo\label{digsetteo}
Let $q$ and $m$ be positive integers composed of the same primes such that the exponent of each prime in $q$
is strictly greater than in $m$. Let $A\subset \setZ_{q} $ be a digital set with $|A| =m > 15$.
We have
\[ \xi_A(n) > m+n \]
for $1<n<q-m$, unless $A$ is the union of at most two arithmetic progressions with a common difference.
\eteo
A description of sets satisfying $|A+A| \leq 2m$ could be achieved by analyzing unions of two arithmetic progressions,
a task not difficult which allows us to generalize the aforementioned result found in \cite{diaconis2014carries}.
\begin{Cor}\label{digsetcorollary}
Let $q$ and $m$ be positive integers composed of the same primes such that the exponent of each prime in $q$
is strictly greater than in $m$. Let $A\subset \setZ_{q} $ be a digital set with $|A| =m > 15$ such that $2A \subseteq \{x,y\}+A$ for some $x, y \in \mathbb{Z}_q$. Then there exist $c \in (\mathbb{Z}_q)^\times$ and $d \in q\mathbb{Z}_q$ such that either $cA+d = \{ 0, 1, \dots, q-1\}$ or $cA+d = \{ 1, 2, \dots, q\}$.
\end{Cor}
In the first part of the paper we prove a somewhat weaker result. It turns out that the key to the conjecture above
would be to understand (i) the cases when $\xi(2)=\xi(3)$, (ii) the cases when the decomposition of our set into the
minimal $\xi(2)$ arithmetic progressions is not unique. The second part of the paper is devoted to these questions,
including the proof of Theorem \ref{digsetteo}.
Our main result is as follows.
\begin{Th} \label{main}
Let $q$ and $m$ be positive integers composed of the same primes such that the exponent of each prime in $q$
is strictly greater than in $m$. Let $A\subset \setZ_{q} $ be a digital set with $|A| =m$ and $\xi=\xi_A$
its impact function. Let $k$ be a nonnegative integer. If the inequality
\[ \xi(n) \geq n+m+k \]
holds in the range
\[ 2 \leq n \leq \frac{3+\sqrt{16k+1}}{2} \]
and $m > m_0(k)$,
then it holds in the range
\[ 2 \leq n \leq q-m-k-1. \]
\end{Th}
\begin{Cor} [Case $k=0$.]
Let $q$ and $m$ be positive integers composed of the same primes such that the exponent of each prime in $q$
is strictly greater than in $m$, and $m\geq5$. Let $A\subset \setZ_{q} $ be a digital set with $|A| =m$.
If $A$ is not an arithmetic progression, then $ \xi(n) \geq n+m$ in the range
\[ 2 \leq n \leq q-m-1. \]
\end{Cor}
\begin{Cor} [Case $k=1$.]
Let $q$ and $m$ be positive integers composed of the same primes such that the exponent of each prime in $q$
is strictly greater than in $m$, and $m\geq10$. Let $A\subset \setZ_{q} $ be a digital set with $|A| =m$.
If $\xi(2) \geq m+3$ (that is, $A$ is not a union of at most two arithmetic progressions of a common difference) and
$\xi(3) \geq m+4$,
then $ \xi(n) \geq n+m+1$ in the range
\[ 2 \leq n \leq q-m-2. \]
\end{Cor}
\section{Proof of Theorem \ref{main}.}
We fix the following assumptions: $q$ and $m$ are positive integers composed of the same primes
such that the exponent of each prime in $q$ is strictly greater than in $m$, $p$ is the smallest prime divisor of $q$,
and $A$ is our digital set with $|A| =m$.
First we consider adding a subgroup to $A$.
\begin{Lemma} \label{sg}
Let $H$ be a subgroup of $\setZ_q$, $H \neq \{ \emptyset \}$, $H \neq \setZ_q$.
\begin{enumerate}[(i)]
\item For every $t$ we have
\begin{equation}\label{sg1} |A \cap (H+t)| \leq \frac{\min (m, |H| ) }{p} \leq \frac{\min (m, |H| ) }{2} \end{equation}
\item For every nonempty subset $A'$ of $A$ we have
\begin{equation}\label{sg2} |A'+H| \geq p |A'| \geq 2 |A'| . \end{equation}
\item We have
\begin{equation}\label{sg3}|A+H| \geq ( m|H|, q) \geq
\begin{cases}
p \, \max(m, |H| ) \geq (p-1)m + |H|, \\ \min \left(q, \frac{4}{3}m + |H|. \right)
\end{cases}
\end{equation}
\end{enumerate}
\end{Lemma}
\begin{proof}
Write $|H| =n$. We have $n|q$, $1<n<q$ and
\[ H = \left\{ 0, \frac{q}{n}, \frac{2q}{n}, \ldots, \frac{(n-1)q}{n} \right\} . \]
Some of these numbers are congruent modulo $m$, namely, if $m | (jq/n)$, then after $j$ steps the
residues modulo $m$ are repeating. Clearly
\[ m \Bigm| \frac{jq}{n} \Longleftrightarrow mn \Bigm| jq \Longleftrightarrow \frac{mn}{(mn,q)} \Bigm| j .\]
Hence
\[ |A \cap (H+t)| \leq \frac{mn}{(mn,q)} =
\frac{m}{(m,q/n )} = \frac{n}{(n,q/m )} . \]
Since both $m$ and $q/m$ contain all prime divisors of $q$, both denominators are divisible by at least one prime factor of $q$,
hence both are $ \geq p$. This shows \kp{sg1}.
To show \kp{sg2}, let let $z$ be the number of cosets of $H$ that intersect $A'$. In each intersection we have
\[ |A' \cap (H+t)| \leq |A \cap (H+t)| \leq n/p ,\]
so $|A'| \leq zn/p$ while $|A'+H| = zn$.
To prove \kp{sg3}, observe that as any coset of $H$ contains at most $m/(m,q/n)$ elements of $A$,
hence $A$ must intersect at least $(m,q/n)$ cosets, which together have $n(m,q/n) = (mn,q)$ elements.
Since
\begin{equation}\label{lcm1} (mn, q) = n(m, q/n) \geq pn \end{equation}
and
\begin{equation}\label{lcm2} (mn, q) = m(n, q/m) \geq pm, \end{equation}
we immediately get the bound in the upper line. It is stonger than the lower line unless $p=2$.
If $p=2$, then \kp{lcm1} becomes
\[ (mn, q) = n(m, q/n) \geq 2n, \]
and \kp{lcm2} can be strengthened to
\[ (mn, q) = m(n, q/m) \geq 3m ,\]
unless $(n, q/m)=2$. If both inequalities hold, then their arithmetic mean yields the stronger bound $(3/2)m+n$.
If the second inequality fails, then $n$ is a power of 2, say $n=2^j$. If $j=1$, then we have
\[ (mn, q) = (2m, q) =2m \geq(4/3)m + n = \frac{4}{3}m + 2, \]
as $m\geq 3$.
If $j\geq2$, then
$q/m$ must contain 2 exactly in the first power, say
$q=2^sq'$, $m=2^{s-1} m'$ with odd $q', m'$.
If $q'=m'=1$, then $q|mn$ and $|A+H| = q$. Otherwise $m' \geq 3$, consequently
$m \geq 3 \cdot 2^{s-1} \geq (3/2) n $ and
\[ (mn, q) = 2m \geq \frac{4}{3}m + n .\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}.]
We want to estimate $\xi(n)$ in the range $2 \leq n \leq q-m-k-1$. Let $n$ be the number in this interval where
$\xi(n)-n$ assumes its minimum, and if there are several such values, we take $n$ to be the smallest of them.
Write $\xi(n)-n=m+r$. If $r=k$, we are done, so we suppose that $r \leq k-1$.
Let $B$ be a set such that $|B| =n$, $|A+B| = m+n+r$. We shall bound $n$ from above in
several stages.
The set $D = G \setminus (A+B)$ satisfies $|D| = q-(m+n+r)$ and
\[ (A-D) \subset G \setminus (-B ), \]
hence $|A-D| \leq q-n = |D| + m+r$. The minimality of $|B|$ implies $|B|\leq |D|$, that is,
\[ n \leq \frac{q-(m+r)}2 . \]
The same argument can be used to show that
\[ \xi(n)=q \ \text{for } q-m+1 \leq n \leq q \]
(as already mentioned) and
\begin{equation}\label{x1}
\xi(n)=q-1 \ \text{for } q-m-k-1 \leq n \leq q-m. \end{equation}
This shows that the range $2 \leq n \leq q-m-k-1$ in the Theorem is best possible.
Next we show that $A+B$ is aperiodic. To this end we
use \emph{Kneser's theorem:} for any finite sets $A,B$ in a commutative group $G$ we have
\[ |A+B| \geq |A+H| + |B+H| - |H| ,\]
where
\[ H = \{ t\in G: A+B+t=A+B \} ,\]
the group of periodes of $A+B$. If $H= \setZ_q$, then we get $|A+B| \geq |A+H|=q$ and we are done.
If $H \neq \{ \emptyset \}$, $H \neq \setZ_q$, then we apply Lemma \ref{sg} to conclude
\[ |A+H| \geq \frac{4}{3}m + |H|\]
and so
\[ |A+B| \geq\frac{4}{3}m + |B+H| \geq\frac{4}{3}m + |B| \geq m +k+n \]
as wanted (here we use the bound $m \geq 3k$).
Next we show that $B$ is a Sidon set, that is, for every $t \neq 0$ we have
$ | B\cap (B+t)| \leq 1$.
Suppose the contrary. Fix a $t$ such that $ | B\cap (B+t)| \geq 2$ and write
\[ B_1 = B \cap(B+t), \ B_2 = B \cup (B+t) . \]
These sets satisfy
\[ |B_1| + |B_2| = 2 |B| = 2n, \]
\[ A+B_1 \subset (A+B) \cap (A+B+t) , \]
\[ A+B_2 = (A+B) \cup (A+B+t) , \]
consequently
\begin{equation}\label{s1} |A+B_1| + |A+B_2| \leq 2 |A+B| = 2(m+n+r). \end{equation}
$B_1$ must be a proper subset of $B$, since otherwise $B$ and a fortiori $A+B$ would be periodic.
Consequently we have
\begin{equation}\label{s2}
|A+B_1| > m + |B_1| + r \end{equation}
by the minimality of $|B|$. The set $B_2$ satisfies
\[ |B_2| = 2n- |B_1| \leq 2n-2 \leq q-(m+r+2). \]
If $ |B_2| \leq q-m-k-1$, then we have
\begin{equation}\label{s3} |A+B_2| \geq m + |B_2| + r \end{equation}
by the definition of $r$. If
\[ q-m-k-1 < |B_2| \leq q-(m+r+2), \]
then
\[ |A+B_2| \geq q-1 > m + |B_2| + r \]
by \kp{x1}, so \kp{s3} holds anyway. By adding \kp{s2} and \kp{s3} we obtain
\[ |A+B_1| + |A+B_2| > 2m +|B_1| + |B_2| + 2r = 2(m+n+r), \]
which contradicts \kp{s1}.
Since $B$ is a Sidon set, we have (see \cite{Ruzsalineq})
\[ |A+B| \geq \frac{mn^2}{m+n-1} .\]
This inequality holds for every set of $m$ elements and it is nearly best in this generality; to use the special properties of $A$
we will need another approach.
Comparing this lower bound with the value $m+n+r$ yields the inequality
\[ mn^2 \leq (m+n+r) (m+n-1) \leq (m+n+k-1) (m+n-1) .\]
This is a quadratic inequality in $n$ and it gives the bound
\[ n\leq \frac{b+\sqrt{b^2 +4ac}}{2a}, \ a=m-1, \ b=2m+k-2, \ c = (m-1)(m+k-1). \]
For large $m$ this is asymptotic to $\sqrt{m}$; in particular, there is an $m_0$ depending on $k$ such that
\[ \beta = \frac{|A+B|}{|A|} = \frac{m+n+r}{m} < \sqrt{2} \]
for $m>m_0$. Such a bound is easily found in the particular cases $k=0,1$;
if $k=0$, it holds for $m \geq 5$, if $k=1$, it holds for $m\geq 10$.
Pl\"unnecke's theorem (see \cite{Ruzsapluennecke}) implies the existence of a nonempty subset $A'$ of $A$ such that
\begin{equation}\label{b2} |A'+ 2B | \leq \beta^2 |A'| < 2 |A'| . \end{equation}
We shall compare this to the Kneser bound
\[ |A'+ 2B | \geq |A'+H| + |2B+H| - |H|, \]
where $H$ is the group of periodes of $A'+ 2B$. If $H$ is a nontrivial subgroup, then
\[ |A'+H| \geq 2 |A'| \]
by \kp{sg2}; this also holds trivially if $H=\setZ_q$, and this contradicts \kp{b2}.
If $H=\{ \emptyset \}$, then Kneser's bound reduces to
\[ |A'+ 2B | \geq |A'| + |2B| - 1 = |A'| + \frac{n(n+1)}2 -1 , \]
as $|2B| = n(n+1)/2$ by the Sidon property. A comparison with the upper estimate \kp{b2} gives
\[ |A'|+ \frac{n(n+1)}2 -1 \leq \left(\frac{m+n+r}{m} \right)^2 |A'|, \]
\[ \frac{n(n+1)}2 -1 \leq |A'|\left( \left(\frac{m+n+r}{m} \right)^2 -1 \right) \]
\[ \leq m \left( \left(\frac{m+n+r}{m} \right)^2 -1 \right) = \frac{(2m+n+r)(n+r)}{m} \leq \frac{(2m+n+k-1)(n+k-1)}{m}. \]
This is again a quadratic inequality in $n$ and it gives the bound
\[ n\leq \frac{b+\sqrt{b^2 +4ac}}{2a}, \ a=m-2, \ b=3m+4k-4, \ c =2m+ 2(k-1)(2m+k-1). \]
As $m \to \infty$, this bound tends to $\left(3+\sqrt{16k+1}\right)/{2}$. The bound $m_0$ after which we can claim this
bound for $n$ depends on the fractional part of the square root inside, but it is easily found in the particular cases $k=0,1$;
if $k=0$, it holds for $m \geq 4$, if $k=1$, it holds for $m\geq 9$.
\end{proof}
\begin{section}{Arithmetic progression structure of sets}
We are interested in studying sets $A \subseteq \mathbb{Z}_q$ containing no nontrivial cosets such that $\xi_A(2)=\xi_A(3).$
If such an equality were to hold, then there exist non zero elements $d_1\neq d_2$ such that
$$|A| + k = \xi_A(3) = |A + \{ 0, d_1, d_2 \}| \geq |A+ \{ 0, d_1 \}| \geq \xi_A(2) = \xi_A(3),$$
so that $|A+\{ 0, d_1, d_2\}| = |A+ \{0, d_1\}| =|A+ \{0, d_2\}|=|A+ \{d_1, d_2\}|$.
In particular, this tells us that the set $A$ can be written as the union of $k$ arithmetic progressions of difference $d_1$ or $d_2$, and there exists 3 distinct elements $x_i \in \mathbb{Z}_q$ such that $$ \bigcup_{i=1}^3 (A+x_i) = (A+x_a )\cup (A+x_b) $$ for any choice of distinct $a, b \in \{1,2,3\}$.
In the sequel we will study these two problems.
Given integers $a, b$ we let the interval $[a,b] \subseteq \mathbb{Z}_q$ be the image of $[a, b']$ under the natural projection $\varphi: \mathbb{Z} \to \mathbb{Z}_q$, where $b'$ is the minimal integer such that $b' \equiv b$ mod $q$ and $a \leq b'$.
Let $|x| = \min\{|x+kq| : k \in \mathbb{Z}\}$ for $x\in \mathbb{Z}_q$ be the seminorm measuring the distance of an element in $\mathbb{Z}_q$ from zero.
If $\xi_A(2) =|A|+k$, is the decomposition of $A$ as the union of $k$ arithmetic progression unique up to a sign?
In other words, can there be two proper decomposition of $A$ as a
$$A=\cup_{i=1}^k P_i = \cup_{i=1}^k Q_i,$$
$$P_i= \{a_i, a_i+d_1, \dots, a_i + k_i d_1 \}, \quad Q_i = \{a'_i, a'_i + d_2, \dots, a'_i + k_i d_2 \}$$
with $d_1 \neq \pm d_2$, $d_1, d_2 \in (-q/2, q/2]$?
If $A$ is an arithmetic progression of difference $d$ itself, so that $k=1$, the only possibility is clearly $d_1= \pm d_2$.
Suppose now $k=2$. Very small (or, by taking their complement, very large) sets $A$ with $|A| \leq 4$ may have multiple representation as union of two arithmetic progression, as happens for sets of the form $A=\{ a, a+d, b, b+d\}$.
On the other hand, we can easily provide examples of different minimal arithmetic progression decompositions if the ratio $|d_1/d_2|$ or $|d_2/d_1|$ is less or equal to $2$, as happens for sets of the form $A=[a, b] \cup \{ b+2\}$ or $A=\{a-2\} \cup [a, b]$
The following Theorem states that these are the only kinds of sets having multiple minimal arithmetic progression decompositions.
\bteo\label{k2mv1}
Let $A \subseteq \mathbb{Z}_q$, $4 < |A| < q-4$. Assume that $q$ is odd, $q>100$ and $A$ is not contained in a coset of any nontrivial subgroup of $\mathbb{Z}_q$.
If $\xi_A(2)=|A+\{ 0, d \}| =|A|+2$, then the only elements $x \in \mathbb{Z}_q$ with $|A+\{0, x\}| = |A|+2$ are $\pm d$, unless $A$ is a dilation of sets of the form $[a, b] \cup \{ b+2\}$ or $\{a-2\} \cup [a, b]$ for suitable $a, b \in \mathbb{Z}_q$.
\eteo
\bdimo
{\bfseries Case 1}: $(d_1, q)=(d_2, q)=1$.
Let $A$ be a set with $\xi_A(2)=|A|+2$ having a double decomposition as the union of two proper arithmetic progression of difference $d_1$ or $d_2$. Dilating $A$ by $d_2^{-1}$ we can assume that $A$ is the union of two disjoint intervals in $\mathbb{Z}_q$. Also, by taking the complementary of $A$, we can assume $|A| < q/2$. (This may fail if the differences are not coprime to $q$; then possibly the complement is the union of the same number of arithmetic progressions and some cosets of the subgroup generated by the difference.)
Let $A=I_1 \cup I_2 = P_1 \cup P_2$ where $P_i$ are arithmetic progressions with common difference $1<d<q/2$ and $I_i = [a_i, b_i]$.
Let $d=\frac{q+1}{2}-x$ for a positive integer $x<\frac{q-1}{2}$. Either $d^{-1}$ or $-d^{-1}$ must be conguent to $\frac{q+1}{2}-y$ for a positive integer $y<\frac{q-1}{2}$. Then
$$\pm4 \equiv 4d(\pm d^{-1}) \equiv (2x-1)(2y-1)\mbox{\quad mod $q$}, $$ which implies that either $x$ or $y$ must be greater than $\frac{\sqrt{q-4}+1}{2} \geq \frac{\sqrt{q}}{2}$.
Hence we can also assume $1< d<(q-\sqrt{q})/2$.
We say that a progression $P_i = \{ a+kd : k =0, \dots, l \}$ jumps from $I_1$ to $I_2$ at $l$ if $a+(l-1)d \in I_1 \cap P_i$ and $a+ld \in I_2 \cap P_i$.
We now split the proof into two cases.
{\bfseries Subcase 1}: $d=2$.
Since $|A|<q/2$, neither $P_1$ nor $P_2$ can jump from $I_1$ to $I_2$ or viceversa more than once. Then it's easy to see that the only possibility is that $A$ behaves as in the statement of the theorem.
{\bfseries Subcase 2:} $d>2$.
Since $A < q/2$ there must be a gap between the intervals $I_1$ and $I_2$ of length $g>q/4$.
Let $|I_1| \leq |I_2|$ and $a_1 - b_2 - 1 \equiv g$ mod $q$, so that $|I_2| > 2$ and hence $I_2$ contains $3$ consecutive elements. Then at least one of the $P_i$'s must jump from $I_2$ to $I_1$ and then to $I_2$ again, implying that $d > g >q/4 > |I_1|$, and that at least one element $x'$ in $I_1$ satisfies $ x' \pm d \in A$.
There are at most 4 elements $x \in A$, the starting and ending points of the $P_i$'s, such that $\{ x+d, x-d \} \not\subseteq A$. So we can find an element $y \in [a_1, a_1+4] \cap A \subseteq I_1$ such that $y \pm d \in A$, either by taking $y=x'$ if $|I_1| < 5$ or $y$ as a point in the middle of an arithmetic progression if $|I_1| \geq 5$.
Since $|I_1| < d$ we have $y \pm d \in I_2$, and so the interval $[y+d, y-d]$ must be contained in $I_2$.
Take now an element $z \in [y-d-7, y-d-5] \subseteq [y+d, y-d]$ which is not the ending element of $P_1$ or $P_2$, so that $z + d \in A$, to obtain a contradiction since $z+d \in [y-7, y-5] \subseteq [a_1 -7, a_1 - 1]\subseteq A^c$. (Here we need that $2d+7 \leq q$, which follows from the assumption on the size of $q$ and the above inequality for $d$.)
\edimo
To proceed to the case of not coprime differences we need a simple lemma which allows us to normalize the differences of the arithmetic progressions.
\blem\label{redd2}
For arbitrary integers $a, q$ there exists an integer $a'$, $a' \equiv a$ mod $q$ and $a' = a_1 a_2$, with $a_1 | q$ and $(a_2, q)=1$
\elem
\bdimo
Let $I = \{ p : p \mbox{ prime}, v_p(a) = v_p(q)\}$.
Define $a':= q\prod_{p \in I} p + a$. Then $a' \equiv a$ mod $q$, $v_p(a') = v_p(q)$ for all primes $p \in I$ and $v_p(a') = \min(v_p(a), v_p(q)) \leq v_p(q)$ for all primes $p \not\in I$.
\end{proof}
\bdimo
Let $q=\prod p_i^{r_i}$ be the decomposition of $q$ as a product of powers of distinct primes.
Let $A=P_1 \cup P_2 = Q_1 \cup Q_2$, with $P_i$'s arithmetic progressions of difference $d_1$ and $Q_i$'s of difference $d_2$, with $P_i = \{\alpha_i, \alpha_i + d, \dots \}$.
{\bfseries Case 2}: $(d_1,q) = 1 < (d_2, q)$.
After a dilation we can assume $d_1=1$, and so there are three consecutive elements $\{\gamma, \gamma +1, \gamma+2 \}$ contained in $A$.
However, since $2\nmid q$, we have that $d=(d_2, q) > 2$ and so the union of $Q_1$ and $Q_2$ can cover at most two of these three elements, which is a contradiction.
{\bfseries Case 3}: $(d_1, q), (d_2, q) > 1$.
After a dilation, thanks to Lemma \ref{redd2}, we can assume $d_1 | q$.
If $\alpha_1 \equiv \alpha_2$ mod $d_1$ then $A$ is contained in a single coset of the subgroup generated by $d_1$, contrary to the assumption.
If $\alpha_1 \not\equiv \alpha_2$ mod $d_1$ then $P_i = \{x \in A : x\equiv \alpha_i \mbox{ mod $d_1$}\}$.
If $d_1 | d_2$ then we also get $Q_i= \{x \in A : x\equiv \alpha_{\varphi(i)} \mbox{ mod $d_1$}\}$ for a permutation $\varphi:\{1,2\} \to \{1,2\}$, and the result follows immediately.
If $d_1 \nmid d_2$ then, letting $Q_1=\{q_1, q_2, q_3, \dots \}$ be an arithmetic progression with at least three elements, we have $q_1 + d_2 \not\equiv q_1$ mod $d_1$ and so $q_1 + 2d_2 \equiv q_1$ mod $d_1$, which implies that $2|q$, again a contradiction.
\edimo
Trying to prove results similar to Theorem \ref{k2mv1} for higher $k$ becomes a harder task, since new families of exceptions have to be considered, as already shown for $k=3$ by the set
$$A=[1,a] \cup \left(\left[(p-1)/2,(p-1)/2+a-1\right] \setminus \left\{ 1+(p+1)/2 \right\}\right),$$ which is the union of 3 intervals as well as of 3 arithmetic progressions of difference $d=1+\frac{p+1}{2}$.
For $k>2$ we also still find the same families of sets having more than one decomposition which we found for $k=2$: sets $A$ with $|A| \leq k^2$ or with $|d_1/d_2| \leq k$.
In the former case, $|A| \leq k^2$, there exists an arithmetic progression in its decomposition having cardinality less or equal than $k$, so after removing its points from $A$ we obtain a set $\widetilde{A}$ with $|\widetilde{A}| \geq |A|-k$ and $\xi_{\widetilde{A}}(2) - |\widetilde{A}| \leq k-1$.
In the latter case, $|d_1/d_2| \leq k$, after multiplying the set $A$ by $\pm d_2^{-1}$, we have that $A=I_1 \cup \dots \cup I_k = P_1\cup \dots \cup P_k$ for intervals $I_i$'s and arithmetic progressions $P_i$'s of difference $d\leq k$.
Since at least one of these arithmetic progressions must jump from one interval to another there exists a gap between two intervals of length less or equal than $k$, and so, by adding those points to $A$ we obtain a set $\widetilde{A}$ with $|\widetilde{A}| \leq |A|+k$ and $\xi_{\widetilde{A}}(2) - |\widetilde{A}| \leq k-1$.
The common point between these two kinds of sets and the multitude of other types of examples one can produce as $k$ grows, is that even though they both are the union of $k$ $d$-arithmetic progression, they are actually obtained by sets $\widetilde{A}$ which are the union of $k-1$ $d$-arithmetic progressions by removing or adding up to $k$ elements.
To exclude these sets, we give the following definition.
\begin{Def}\label{def1}
$A$ has $k$ stable components if $\xi_A(2) = |A| + k$, and for any $d$ such that $|(A+d) \setminus A|=k$, any set $\widetilde{A}$ obtained by $A$ by removing or adding up to $k$ elements satisfies $|(\widetilde{A}+d) \setminus \widetilde{A}| \geq k$.
\end{Def}
Moreover, if we work in the composite number modulus case, new sets having multiple representation as union of a minimal number of arithmetic progressions can be found, because of the presence of nontrivial cosets in this setting.
Of course, the union of $k$ disjoint cosets has a lot of representations as the union of $k$ arithmetic progressions, but it is not hard to find other less trivial sets which satisfy this property.
For example, for suitable $k,q$, $k\mid q$,
\begin{equation}\label{mcontro}A = [0,2k-1] \bigcup_{i=1} ^{k-1} [q/k+i, q/k+(k+1)+i]\subseteq \mathbb{Z}_q
\end{equation}
is a set of $k$ stable components which is not the union of cosets but still is the union of either $k$ intervals or $k$ arithmetic progressions of difference $d=q/k +1$.
Nevertheless, this set $A$ has high density in some coset of $\mathbb{Z}_q$, namely $\langle q/k \rangle$.
In the following Theorem we show that the essential uniqueness of the decomposition of a set into $k$ arithmetic progressions still holds for sets of $k$ stable components and with low density into any coset of $\mathbb{Z}_q$.
Moreover, it will be clear from the proof that the only sets of $k$ stable components with such multiple decompositions will be of the same kind as the set in \eqref{mcontro}.
\bteo
Let $A \subseteq \mathbb{Z}_q$ be the union of $k$ arithmetic progressions of difference $d_1$ and $d_2$, $|A\cap(H+t)|<|H|/2$ for any nonzero coset $H+t$ of $\mathbb{Z}_q$ , and $A$ has $k$ stabel components.
Then $d_1 = \pm d_2$.
\eteo
\bdimo
Since we are going to prove $d_1 = \pm d_2$, and every arithmetic progression of difference $d$ is also an arithmetic progression of difference $-d$, during the course of the proof we are going to choose suitable signs for $d_i$ in order to simplify the notations.
Let $A=P_1 \cup \dots \cup P_k = Q_1 \cup \dots \cup Q_k$ with $P_i$'s being arithmetic progressions of difference $d_1$ and $Q_i$'s of difference $d_2$.
We denote by $S_i$ and $E_i$, $i=1, 2$ the starting and ending points of the arithmetic progressions of difference $d_i$ forming $A$, i.e.
$$S_i = \{ x \in A: x-d_i \not\in A\}, \qquad E_i= \{ x \in A : x+d_i \not\in A\},$$
with $|S_i|=|E_i|= k$.
Given $x, y$, we will write $x \sim_i y$ for $i=1, 2$ if $x, y \in A$ and they both belong to the same arithmetic progression of difference $d_i$.
Since $A$ has $k$ stable components, the following properties hold:
\begin{enumerate}[(i)]
\item\label{pr1} $|P_i|, |Q_i| \geq k+1$ $\forall i=1,\dots,k$, for if otherwise, by removing a short arithmetic progression, we would obtain a contradiction with Definition \ref{def1}.
\item\label{pr2} If $P_i = \{ a + l d_1: l=0, \dots, M_i-1\}, P_j = \{ a + (M_i+ l) d_1 : l = N, \dots, N+ M_j\}, N >0,$ are two different components contained in the same coset $a + \left< d_1 \right>$, then $N \geq k+1$, for otherwise, by adding the elements $\{ a + ld_1 : l= M_i, \dots, M_i +N-1\}$ to $A$ we would obtain a contradiction with Definition \ref{def1}.
A similar statement holds for $Q_i, Q_j$ and $d_2$ instead of $P_i, P_j$ and $d_1$.
\item\label{pr3} $\forall i \exists j : (P_i + d_2) \cap A \subseteq P_j$. In fact, if $P_i \subseteq a + \left< d_1 \right>$ and $(P_i +d_2) \cap A \cap P_{k_l} \neq \emptyset$ for two different components $P_{k_1}$ and $P_{k_2}$, then we have $P_i + d_2 \subseteq a + d_2 + \left< d_1 \right>$, which implies that both $P_{k_1}$ and $P_{k_2}$ are contained in the same coset of $\left< d_1 \right>$. Then, because of \eqref{pr2}, the set $P_i + d_2$ contains at least $k+1$ elements not belonging to $A$, and hence $|E_2| \geq k+1$, a contradiction.
A similar statement hold for $Q_i$ and $d_1$ instead of $P_i$ and $d_2$.
\item\label{pr4} $\forall i \exists j : (P_i - d_2) \cap A \subseteq P_j$ and $\forall i \exists j : (Q_i - d_1) \cap A \subseteq Q_j$, by an argument similar to \eqref{pr3}.
\end{enumerate}
Thanks to Lemma \ref{redd2} we can assume, after a dilation, $d_1, d_2 \in [0, q-1]$, $d_2 \mid q$.
Let $d=(d_1, d_2)$, $d_i = d_i' d$ for $d=1, 2$, $q = dq'$ and $\mathcal{A}_i = \{x \in A : x \equiv i \mbox{ mod $d$}\}$.
Clearly, if $P_j \cap \mathcal{A}_i \neq \emptyset$, then $P_j \subseteq \mathcal{A}_i$, and the same holds for the $Q_j$'s, so that every $\mathcal{A}_i$ is the union of $r_{1, i}$ $d_1$-arithmetic progressions and $r_{2,i}$ $d_2$-arithmetic progressions.
We are going to show that the ratio $r_{1,i}/r_{2,i}$ is constant for every $i$.
Let $A_i = \frac{\mathcal{A}_i - i}{d} \subseteq \mathbb{Z}_{q'}$.
Clearly every set $A_i$ inherits from $A$ the same stability properties (relative to $k$) and the condition of density into cosets.
We use the same notation above for subsets of $\mathbb{Z}_{q'}$.
\bclaim $ r_{2, i} \geq d_2'$.
\eclaim
\bdimo[Proof of claim]
Since $d_2'|q'$, $x\sim_2 y$ implies $x \equiv y$ mod $d_2'$.
Given $s \in S_1$, if by contradiction $m' >d_2' > r_{2,i}$ then the set $B=\{ s, s+ d_1', \dots, s + r_{2,i} d_1' \} \subseteq A_i$ has cardinality $r_{2,i}+1$ since $r_{2,i} \leq k$.
For $j \in [0, r_{2,i}] \subseteq [0, d_2'-1]$, $jd_1' \equiv 0$ mod $d_2'$ can only happen for $j=0$ by the coprimality of $d_1'$ and $d_2'$.
Hence $B$ intersects $r_{2,i}+1$ distinct $d_2'$-arithmetic progression, which leads to the contradiction.
\edimo
Let $X=\{ x \in [k]: xd'_1 \equiv 0 \mbox{ mod $d'_2$}\}$.
From $d'_2 \leq r_{2,i}\leq k$ we get $X \neq \emptyset$.
For every $x \in X$ let $\beta_+(x)$ be the minimal positive integer such that $xd'_1 \equiv \beta_+(x)d'_2$ mod $q'$, and $\beta_-(x)$ be the minimal positive integer such that $-xd'_1 \equiv \beta_-(x)d'_2$ mod $q'$. Let $\beta(x) = \min (\beta_+(x), \beta_-(x))$ and $\beta(\alpha) = \min_{x\in X} \beta(x), \alpha \in [k]$.
After changing $d'_1$ with $-d'_1$ we can assume $\beta(\alpha) = \beta_+(\alpha)$.
Let $S_1 = \{ s_1, \dots, s_{r_{2,i}}\}$. For every $i$ define $l_i$ to be the minimal integer such that $s_i + l_id'_1 \sim_2 s_i$. Clearly $l_i \in X$, and one of the following must happen, according to which one between $\beta_+(l_i)$ and $\beta_-(l_i)$ is minimal:
\begin{enumerate}[(i)]
\item $s_i + ld'_2 \in A_i$ for $l \in [0, \beta_+(l_i)]$
\item $s_i - ld'_2 \in A_i$ for $l \in [0, \beta_-(l_i)]$
\end{enumerate}
\boss\label{oss123}
Because of \eqref{pr3} all the elements $x$ such that $x \sim_{1,2} s_i$ are of the form $s_i + l l_i d'_1$ for some $l \geq 0$. In particular, $\beta(\frac{x-s_i}{d'_1}) > \beta(l_i)$, otherwise $|A_i \cap (s_i + \langle d'_2\rangle)| > \frac{|\langle d'_2\rangle|}{2}$.
Moreover, all those elements $x$ belong to the same semicircle $[s_i, s_i + m'/2)$ or $(s_i - m'/2, s_i]$.
\eoss
Suppose $\beta(l_i) > k$ for all $i$. Suppose $\beta(l_1) = \beta_+(l_1)$ and $\beta(l_1) = \min_{i=1, \dots, r_{2,i}} (\beta(l_i))$. Then the set $\{ s_1, s_1 + d'_2, \dots, s_1 + l_1 d'_1 = s_1 + \beta(l_1)d'_2 \} \subseteq A_i$ intersects at least $k+1$ different $d'_1$-arithmetic progression, leading to a contradiction.
A similar argument works if $\beta(l_1) = \beta_-(l_1)$).
But then, since $\beta(l_1), l_1 \leq k$, we get that for every $j$, $s_j + l_1 d'_1 = s_j + \beta(l_1) d'_2 \sim_{1,2} s_j$. Moreover, Remark \ref{oss123} tells us that $l_1=l_j = \alpha$ for all $j$.
Split the set $A_i$ into $M$ equivalence classes under the relation $P_{j_1} \sim P_{j_2}$ if there are $p_1 \in P_{j_1}$, $p_2 \in P_{j_2}$, with $p_1 \sim_2 p_2$. This is well defined by \eqref{pr3}.
Each equivalence class is composed by $\alpha$ $d'_2$-arithmetic progressions, so that $r_{2,i} = M \alpha$.
If $x, x+ \alpha d'_1 \in A_i$, there does not exists a $y \in \{ x+ l d'_2, l \in [0, \beta(\alpha)) \}$ with $y \sim_1 x$, and hence $k \geq r_{1,i} \geq \beta(\alpha)$. On the other hand, we already know that $x \sim_1 x+ \alpha d'_1$, and so $r_{1,i} = M \beta(\alpha)$.
In particular, the ratio $r_{1,i}/r_{2,i} = \beta(\alpha) / \alpha$, a constant not depending on $i$.
Since $A$ is the union of $k$ $d_1$-arithmetic progressions and $k$ $d_2$-arithmetic progressions, we must have $\beta(\alpha)=\alpha$.
We now show that this leads to $d'_1 = d'_2$, which concludes the proof since, after dilating the set $A$ so that $d_2 \mid q$, we did already choose between $d_1$ and $-d_1$ in order to simplify the notation.
Going back to $A_i$ we have $\alpha d'_1 \equiv \alpha d'_2$ mod $q'$, and so, for $D=(\alpha, q')$ we get $\frac{q'}{D} \mid \frac{\alpha}{D}(d'_1 - d'_2)$ and so $d'_1 = d'_2 + j \frac{q'}{D}$ for some $j\geq 0$.
Assume by contradiction that $D> j > 0$.
We already know that $B=\{s_1, s_1+ d'_2, \dots, s_1 + \alpha d'_2 = s_1 + \alpha d'_1 \} \subseteq A_i$.
Let $D'$ be the additive order of $j\frac{q'}{D}$ in $\mathbb{Z}_{q'}$, $D' \leq D \leq \alpha \leq k$.
Then $s_1 + D' d'_1 = s_1 + D' d'_2 \in B$ and $s_1 + D' d'_1 \sim_2 s_1$, so that $D' = \alpha$
Moreover, $s_1 + l d'_1 \in A_i$ for $0\leq l \leq \alpha$.
By \eqref{pr1} and $\alpha \leq k$ we have that at least one between
$$ld'_1 - ld'_2 = l j \frac{q'}{D} \quad \mbox{ or }\quad ld'_1 + (\alpha - l) d'_2 = \alpha d'_2 + l j\frac{q}{D}$$ belong in $A_i$, and so at least one of the two cosets $\langle j\frac{q'}{D}\rangle$ and $\alpha + \langle j\frac{q'}{D}\rangle$, both having cardinality $\alpha$, intersects $A_i$ in more than half of its elements, which leads to a contradiction with our hypothesis of low density in cosets.
Hence $j=0$ and $d'_1 = d'_2$.
\edimo
\end{section}
\begin{section}{Sets $A$ with $\xi_A(2)=\xi_A(3)$}
Let $A \subseteq \mathbb{Z}_q$ be a set which does not contain any non trivial cosets, with $|A|=m$, $\xi_A(2)=\xi_A(3)$. Then there are $d_1 \neq d_2$ such that
\begin{equation}\label{eqmodq}A+ \{0,d_1,d_2\}=A+ \{0,d_1\}=A+ \{0,d_2\}=A+\{d_1,d_2\}, \end{equation}
After a dilation, applying Lemma \ref{redd2}, we can assume $d_1, d_2 \in [0, q-1]$ and $d_1 | q$. Let $H = \langle d_1 \rangle$ be the subgroup generated by $d_1$, so that $|H|=q/d_1$.
As usual, write $A=P_1 \cup \dots \cup P_k = Q_1 \cup \dots \cup Q_k$ as the union of $k$ $d_1$-arithmetic progressions $P_i$'s as well as $k$ $d_2$-arithmetic progressions $Q_i$'s, with
$$P_i = \{ a_i + j d_1; j = 0, \dots, j_i \}, \quad a_i + j_i d_1= b_i,$$
$$Q_i = \{ \alpha_i + l d_2; l = 0, \dots, l_i \}, \quad \alpha_i + l_i d_2= \beta_i$$
Since
$$A+ \{ 0,d_1\} = A \amalg \{ b_i + d_1\}_{i=1, \dots, k} $$
$$A+ \{ 0,d_2\} = A \amalg \{ \beta_i + d_2\}_{i=1, \dots, k} $$
we have
\begin{equation}\label{bbeta}\{ b_i + d_1\}_{i=1, \dots, k} = \{ \beta_i + d_2\}_{i=1, \dots, k}.\end{equation}
Suppose that set $A$ has non empty intersection with $z$ cosets of $H$.
Let $\{G_i\}_{i=1, \dots, k}$ be the set of maximal $d_1$-arithmetic progressions contained in those $z$ cosets of $H$ such that $G_i \subseteq A^c$.
In particular, after a reordering, we can assume $G_i = \{x_i + h d_1, h = 0, \dots, h_i\}$, with $x_i = b_i + d_1$ and $x_i + h_i d_1 = a_{\varphi(i)}-d_1$ for a permutation $\varphi: [k] \to [k]$.
Note that $a_i \in A+\{d_1, d_2\} \setminus A+d_1$, for otherwise $A$ would contain a full coset of $H$.
Hence
\begin{equation}\label{aia}a_i - d_2 \in A\end{equation}
and from \eqref{bbeta} and \eqref{aia} we deduce that
$$(G_i - d_2) \cap A = \{ \beta_{\tau(i)}\}$$
for a permutation $\tau: [k] \to [k]$. Moreover, either $|G_i|=1$ or $(G_i - d_2) \cap A^c = G_j$ for another $G_j$ with $|G_j| = |G_i|-1$.
We can then define a partial order $\leq$ on the $G_i$'s by $G_a \leq G_b$ if and only if $\exists i \geq 0$ such that
$$G_a = (G_b - i d_2) \cap G_b - i(d_2-d_1).$$
A $G_i$ which is maximal for this partial order satisfies $G_i + d_2 \subseteq \{ \alpha_i\}_{i=1, \dots, k} \subseteq A$, and so $|G_i| \leq k$, leading to
\begin{equation}\label{eqmodq}|A| \geq z|H| - \frac{k(k+1)}{2}.\end{equation}
We have then proved the following:
\bteo\label{teo01dq}
Let $A \subseteq \mathbb{Z}_q$ be a set not containing any nontrivial cosets and which satisfies
$$\xi_A(2)=\xi_A(3).$$
Then there exists a $d_1|q$ such that $A$ intersects $z$ cosets of $H=\langle d_1 \rangle$ and, after a dilation, $A$ is of the form
$$\mathbb{Z}_q \setminus \left(\coprod_i \mathcal{G}_i \coprod \coprod_{j=1}^{d_1-z} (t_j + H )\right), $$
where $\mathcal{G}_i$ are chains $\mathcal{G}_i = \{ \{ g_i\} = G_{i, 1} \leq \dots \leq G_{i, j_i}\}$ with
\begin{enumerate}[(i)]
\item\label{cco1} $|G_{i, j_i}| \leq \xi_A(3) - |A|$,
\item $|G_{i, j-1}| = |G_{i, j}| -1$,
\item $g_i -d_2 \in A$,
\item\label{cco4} $(G_{x,y }+\{0,d_1\}) \cap (G_{w,z} + \{0, d_1\}) = \emptyset$ for $(x,y)\neq(w,z)$.
\end{enumerate}
\eteo
Restricting ourselves to the case $q=p$ prime, it is an interesting question to study the minimal cardinality of $A$ in order to have $\xi_A(2)=\xi_A(3)$.
A rectification argument (see \cite{BiluLevRuzsa} and \cite{Levrect}) shows that $|A| > \log_4(p)$.
Since every element $a \in A+\{ 0, d_1, d_2\}$ must belong to at least two sets $A+x$, $x\in \{0, d_1, d_2\}$, as long as $|A| < 2/3 p$ we have
$$k = |A+\{0, d_1, d_2\}|-|A| \leq \frac{|A|}{2}.$$
This, combined with the bound in \eqref{eqmodq}, gives
$$|A| \geq \sqrt{8p +25}-5. $$
Let $\mu(p) = \min (|A|: A \subseteq \mathbb{Z}_p \mbox{ and $A$ satisfies $\xi_A(2)=\xi_A(3)$ } )$. We conjecture the following:
\bconj\label{conj}
$$\lim_{p\to \infty}\frac{\mu(p)}{p} > 0.$$
\econj
In the following we will show that $\liminf_{p\to \infty}\frac{\mu(p)}{p} \leq \frac{5}{18}$.
To do this we construct sets $B \subseteq [0, 2^{2m}]$ of cardinality $|B| = \frac{13}{18}2^{2m}+ o(2^{2m})$ which is the union of disjoint chains satisfying conditions \eqref{cco1}-\eqref{cco4} in Theorem \ref{teo01dq}.
Since by \cite{Pintz} there exists a prime $p$ in $[2^{2m}, 2^{2m}+2^{21m/20}]$, the complement of the image of the canonical projection of $B$ into $\mathbb{Z}_p$ will have density asymptotic to $5/18$ as required.
Let $d=2^m$, $\mathcal{G}_l = \{ \{0\} \leq [d-1, d] \leq \dots \leq [d(l-1) - (l-1) , d(l-1)] \}$ for $l\leq d$ and $H_i = (id-d, id]$. Let $\varphi(\mathcal{G}_l) = d+ \mathcal{G}_{l-1}$ be the chain of intervals obtained from $\mathcal{G}_l$ by removing the first element in each of its intervals.
If $C= \cup_{i \in I} \mathcal{G}_{l_i} + x_i$ and $\mathcal{G}_a \cap \mathcal{G}_b = \emptyset$ for all $a,b \in I$, then the set $B=\cup_{i \in I} \varphi(\mathcal{G}_{l_i})+x_i$ satisfies the conditions of Theorem \ref{teo01dq}.
Let
$$C=C_0 \coprod_{l=1}^{m-1} \coprod_{i=1}^{m-l} B_i^{(l)},$$
where
\begin{eqnarray*}
C_0&=& \mathcal{G}_{2^m},\\
B_i^{(l)} &=& 2^m(2^{m+1-l} -2^{m+2-l-i} - 1) + 2^{m+1-l-i} + \mathcal{G}_{2^{m+1-l-i}}.
\end{eqnarray*}
If we denote by $B_{i,k}^{(l)}$ the $k$-th interval of the chain, $0 \leq k \leq 2^{m+1-l-i}-1$, we have that
\begin{multline*}B_{i,k}^{l} = [2^m(2^{m+1-l} -2^{m+2-l-i} -1+k) + 2^{m+1-l-i} -k,\\ 2^{m}(2^{m+1-l} -2^{m+2-l-i} -1+k) + 2^{m+1-l-i} ]
\end{multline*}
Suppose now that $B_{i,k}^{(l)} \cap B_{i',k'}^{(l')} \neq \emptyset$. Then, since $B_{i,k}^{(l)} \subseteq H_{2^{m+1-l} -2^{m+2-l-i} +k}$, for $\alpha(l, i, k) = 2^{m+1-l} -2^{m+2-l-i} +k$, we must have $\alpha(l, i, k) = \alpha(l', i', k')$.
{\bfseries Case 1:} $i, i' \geq 2$.
In this case we have $\alpha(l, i, k) \in [2^{m-l}, 2^{m+1-l})$ and since any two of these intervals are disjoint, we must have $l = l'$, which implies that $$k-2^{m+2-l-i} = k'-2^{m+2-l'-i'} \in [-2^{m+2-l-i'}, -2^{m+1-l-i'}].$$ Again, since any two of these intervals are disjoint, we must have $i=i'$, which immediately gives $k=k'$.
{\bfseries Case 2:} $i=1$.
In this case from the equality $\alpha(l, i, k)=\alpha(l', i', k')$ we have $$k = 2^{m+1-l'}-2^{m+2-l'-i'}+k'.$$
If $i' \geq 2$, then the left hand side is in $[0, 2^{m-l})$, while the right hand side belongs to $[2^{m-l'}, 2^{m+1-l'})$. From this we get that $m-l' < m-l$ and so $\max(B_{i,k}^{l}) > \max{B_{i', k'}^{l'}}$.
Moreover, we have
$$2^{m-l}-k = 2^{m-l} - 2^{m+1-l'} + 2^{m+2-l'-i'} - k' > 2^{m+1-l'-i'}$$
since $k' < 2^{m+1-l'-i'}$, so that $\max{B_{i', k'}^{l'}} < \min(B_{i,k}^{l})$ and $B_{i,k}^{(l)} \cap B_{i,k}^{(l')} = \emptyset$.
If also $i'=1$, then $k=k'$ and, if $l < l'$, we have $k \leq 2^{m-l'}-1 \leq 2^{m-l-1} -1$, so that $2^{m-l}-k \geq 2^{m-l-1} +1 \geq 2^{m-l'} +1$, and so $B_{i,k}^{(l)} \cap B_{i,k}^{(l')} = \emptyset$.
Since $|\varphi(\mathcal{G}_l)| = \frac{l(l-1)}{2}$, for $B=\varphi(C_0) \coprod_{l=1}^{m-1} \coprod_{i=1}^{m-l} \varphi(B_i^{(l)})$, we have
\begin{eqnarray*}
|B| &=& \frac{2^m(2^m-1)}{2} + \sum_{l=1}^{m-1} \sum_{i=1}^{m-l} \frac{2^{m+1-l-i}(2^{m+1-l-i}-1)}{2} = \frac{13}{18}2^{2m} + o(2^{2m})
\end{eqnarray*}
as required.
Go back to the general case of composite modulus $q$.
An analogue of Conjecture \ref{conj} cannot hold in this case, as we can just take a set $A' \subseteq \mathbb{Z}_{q'}$ with $\xi_{A'}(2)=\xi_{A'}(3)$ and consider the set $A = A' \times \{0\} \subseteq \mathbb{Z}_{q'} \times \mathbb{Z}_{q''}=\mathbb{Z}_q$ for any coprime $q', q''$ with $q=q'q''$.
We can now finish the proof of Theorem \ref{digsetteo}and Corollary \ref{digsetcorollary}.
If $A$ is a digital set with $\xi_A(2) = m+3 = \xi_A(3) = |A+\{0, d_1, d_2\}|$ as in Theorem \ref{teo01dq}, we have by Lemma \ref{sg} and \eqref{eqmodq} that
$$|A| \geq z|H| -6,$$
with $z \in \{2,3 \}$.
Therefore there exists a coset $t+H$ of $H=\langle d_1 \rangle$ such that
$$\frac{|H|}{2} \geq |A \cap (t+H)| \geq |H|-3.$$
This means that $q/d_1 = |H| \leq 6$, and so $l d_1 \equiv 0$ mod $q$ for some $1 \leq l \leq 6$, and any arithmetic progression of difference $d_1$ forming $A$ could not have more that $5$ elements, implying that $m \leq 15$.
To prove Corollary \ref{digsetcorollary}, thanks to Theorem \ref{digsetteo} we are left to cosider the case of $A=P_1 \cup P_2$ a proper union of two arithmetic progressions of common difference $d$, and $2A \subseteq \{x, y\} +A$.
Once we establish that such a set cannot be a digital set, we are done since the only possibilities for a single arithmetic progression to be a digital set with minimal number of distinct carries are clearly the ones stated in the corollary.
Consider at first the case $(d, q) > 1$. After a dilation, thanks to Lemma \ref{redd2}, we can assume $d|q$. Since $A$ is a digital set, we must have $d=2$ and hence $2|q$.
Moreover, by \ref{sg1} we have $|P_1|=|P_2|= m/2$, and $P_i = \alpha_i + 2\cdot[0, m/2-1]$, $i=1,2$ , where $\alpha_1 \not\equiv \alpha_2$ mod $2$.
Then
$$2A= (2\alpha_1 + 2\cdot[0, m-1]) \cup (\alpha_1 + \alpha_2 + 2\cdot[0,m-1]) \cup (2\alpha_2 + 2\cdot[0, m-1]).$$
By the parity of $\alpha_1$ and $\alpha_2$, we must have $$|(2\alpha_1 + 2\cdot[0,m-1]) \cup (2\alpha_2 + 2\cdot[0,m-1])| \leq m+1,$$
which implies without loss of generality, since $2m \leq q$, that $2\alpha_1 \in \{2\alpha_2, 2\alpha_2 + 2, 2\alpha_2 + 4 \}$.
Once again, since $\alpha_1 \not\equiv \alpha_2$ mod $2$, and $A$ is not an arithmetic progression, this leaves us with the only choice $\alpha_1 = \alpha_2 + 1 + q/2$, and so, up to translation,
$$A=2\cdot\left[0, \frac{m}{2}-1\right] \cup \left(\frac{q}{2}+1+2\cdot \left[0, \frac{m}{2}-1\right]\right),$$
which is a single arithmetic progression of difference $q/2+1$.
Assume now $(d, q)=1$, so that, after a dilation and a translation, we can assume that $A$ is of the form
$$A = [0, a-1] \cup [bm+a, (b+1)m -1],$$
with $a \geq m-a$, $1 \leq b \leq q/m -2$.
Then $2A = B_1 \cup B_2 \cup B_3$, where
$$
B_1 = [0, 2a-2], \quad B_2 = [bm+a, (b+1)m+a-2], \quad B_3 = [2bm + 2a, 2(b+1)m -2]
$$
are all non empty sets.
A routine check shows that $B_1 \cap B_2 = \emptyset$, and $|B_1|+|B_2|= 2a+m-2 \leq 2m$ implies $m/2 \leq a \leq (m+2)/2$ and $|B_3 \cap (B_1 \cup B_2)^c| \leq 2$.
For these possible values of $a$, we must have $B_3 \subseteq B_1$, so that $m(2b+1) \equiv 0$ mod $q$, and since all primes dividing $m$ must divide $q/m$, we have $2 \nmid q$ and so
$$a=\frac{m+1}{2},bm= \frac{q-m}{2} \Longrightarrow A=\left[ 0, \frac{m-1}{2}\right] \cup \left[ \frac{q+1}{2}, \frac{q+m-2}{2}\right].$$
Once again, this is a single arithmetic progression of difference $(q+1)/2$.
\end{section}
\section*{}
| {
"timestamp": "2015-07-01T02:01:33",
"yymm": "1506",
"arxiv_id": "1506.08869",
"language": "en",
"url": "https://arxiv.org/abs/1506.08869",
"abstract": "If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \\in A$, we find the unique $a \\in A$ such that $a_1 + a_2 \\equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\\mathbb{Z}/m \\mathbb{Z} \\subseteq \\mathbb{Z}/m^2\\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis, Shao and Soundararajan proved that, when $m=p$, $p$ prime, the only set $A$ which induces two distinct carries, i. e. with $A+A \\subseteq \\{ x, y \\}+A$ for some $x, y \\in \\mathbb{Z}/p^2\\mathbb{Z}$, is the arithmetic progression $[0, p-1]$, up to certain linear transformations. We present a generalization of the result above to the case of generic modulus $m^2$, and show how this is connected to the uniqueness of the representation of sets as a minimal number of arithmetic progression of same difference.",
"subjects": "Number Theory (math.NT)",
"title": "Carries and the arithmetic progression structure of sets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.991422514586288,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7092874629105063
} |
https://arxiv.org/abs/1602.08698 | Equal Sums of Like Powers with Minimum Number of Terms | This paper is concerned with the diophantine system, $\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\, r=1,\,2,\,\ldots,\,k, $ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as small as possible. We define $\beta(k)$ to be the minimum value of $s_1+s_2$ for which there exists a nontrivial solution of this diophantine system. We find nontrivial integer solutions of this diophantine system when $k < 6$, and thereby show that $\beta(2) =4,\;\, \beta(3) = 6,\;\, 7 \leq \beta(4) \leq 8$ and $8 \leq \beta(5) \leq 10$. | \section{Introduction}
\hspace*{0.25in}The Tarry-Escott problem of degree $k$ consists of finding two distinct sets of integers $\{x_1,\,x_2,\,\ldots,\,x_s\}$ and $\{y_1,\,y_2,\,\ldots,\,y_s\}$ such that
\begin{equation}
\sum_{i=1}^s x_i^r=\sum_{i=1}^s y_i^r,\quad
r=1,\,2,\,\ldots,\,k.
\label{tepsk}
\end{equation}
It is well-known that for a non-trivial
solution of (\ref{tepsk}) to exist, we must have $s\geq (k+1)$ \cite[p.\ 616]{Dor2}. Solutions of (\ref{tepsk}) with the minimum possible value of $s$, that is, with $s=k+1,$ are known as ideal solutions of the problem.
This paper is concerned with finding solutions in integers of the related diophantine system,
\begin{equation}
\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\quad
r=1,\,2,\,\ldots,\,k, \label{sysgenk}
\end{equation}
where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is minimum.
Without loss of generality, we may take $s_1 \leq s_2$. A solution of the system of equations \eqref{sysgenk} will be said to be trivial if $y_i=0$ for $s_2-s_1$ values of $i$ and the remaining integers $y_i$ are a permutation of the integers $x_i$.
We define $\beta(k)$ to be the minimum value of $s_1+s_2$ for which there exists a nontrivial solution of the diophantine system \eqref{sysgenk}.
According to a well-known theorem of Frolov \cite[p.\ 614]{Dor2}, if $x_i=a_i,\,y_i=b_i, \;i=1,\,2,\,\ldots,\,s$ is any solution of the diophantine system \eqref{tepsk}, and $d$ is an arbitrary integer, then $x_i=a_i+d,\;y_i=b_i+d,\;i=1,\,2,\,\dots,\,s$,
is also a solution of \eqref{tepsk}. Taking $d=-a_1$, we immediately get a solution of \eqref{sysgenk} with $s_1=s-1,\;s_2=s$. Thus, if an ideal solution of \eqref{tepsk} is known for any specific value of $k$, then $\beta(k) \leq 2k+1.$
Ideal solutions of \eqref{tepsk} are known for $k=2,\,3,\,\ldots,\,9$ (\cite{Bor}, \cite{Che}, \cite{Cho1}, \cite{Cho2}, \cite[pp.\ 52,\,55-58]{Di1}, \cite{Dor1}, \cite[pp.\ 41-54]{Glo}, \cite{Let}, \cite{Smy}) as well as for $k=11$ \cite{Cho3}. Thus, for these values of $k$, we have $\beta(k) \leq 2k+1$, and in particular, we get
\begin{equation}
\beta(2) \leq 5,\;\;\beta(3) \leq 7,\;\;\beta(4) \leq 9,\;\;\beta(5) \leq 11.
\end{equation}
In this paper, we find parametric solutions of (\ref{sysgenk}) when $2 \leq k \leq 5$, and show that,
\begin{equation}
\beta(2) =4,\quad \beta(3) = 6,\quad 7 \leq \beta(4) \leq 8,\quad 8 \leq \beta(5) \leq 10. \label{estimatebeta}
\end{equation}
\section{A preliminary lemma}
\noindent {\bf Lemma 1:} If there exists a nontrivial solution of the diophantine system \eqref{sysgenk}, then
\begin{equation}
{\rm max}(s_1,\,s_2) \geq k+1. \label{betakest1}
\end{equation}
Further, if $k \geq 4$, then,
\begin{equation}
{\rm min}(s_1,\,s_2) \geq 2. \label{betakest2}
\end{equation}
\noindent {\bf Proof:} Any solution of the diophantine system \eqref{sysgenk} with ${\rm max}(s_1,\,s_2) < k+1$, implies the existence of a solution of \eqref{sysgenk} with $s_1=s_2 < k+1$, and by a theorem of Bastien (as quoted by Dickson \cite[p. 712]{Di2}), such a solution is necessarily trivial. Thus we must have the relation \eqref{betakest1}.
When $k \geq 4$, if we assume a solution of the diophantine system \eqref{sysgenk} with $s_1=1$, it is easily seen on eliminating $x_1$ from the two relations obtained by taking $r=2$ and $r=4$ in \eqref{sysgenk} that the resulting condition can be satisfied only if all but one of the numbers $y_i$ are 0, and the solution is necessarily trivial. Thus we must have the relation \eqref{betakest2}.
\noindent {\bf Corollary 1:} For any arbitrary positive integer $k$,
\begin{equation}
\beta(k) \geq k+2.
\label{betaestgen}
\end{equation}
Further, when $k \geq 4$,
\begin{equation}
\beta(k) \geq k+3.
\label{betaestspl}
\end{equation}
\noindent{\bf Proof:} We have trivially ${\rm min}(s_1,\,s_2) \geq 1$. The corollary now follows immediately from the lemma.
We have already seen that if there exists a solution of the diophantine system \eqref{sysgenk} with $s_1=s_2$, Frolov's theorem immediately gives another solution of \eqref{sysgenk} with $s_1=s-1,\;s_2=s$. Thus for $s_1+s_2$ to be minimum, we can always take $s_1 < s_2$. Accordingly, we will henceforth always consider the diophantine system \eqref{sysgenk} with $s_1 < s_2$.
We note that all the equations of the diophantine system \eqref{sysgenk} are homogeneous, and therefore, any solution of \eqref{sysgenk} in rational numbers may be multiplied through by a suitable constant to obtain a solution of \eqref{sysgenk} in integers.
\section{Determination of $\beta(k)$}
\hspace*{0.25in}It is trivially true that $\beta(1)=3$. In the next four subsections, we will find solutions of the diophantine system \eqref{sysgenk} when $k=2,\,3,\,4$ and $5$ respectively, and prove the results stated in \eqref{estimatebeta}.
\noindent {\bf 3.1} It follows from Cor. 1 that $\beta(2) \geq 4$. We will show that $\beta(2) = 4$ by solving the diophantine system \eqref{sysgenk} with $s_1=1,\;s_2=3$ and $k=2$. On eliminating
$x_1$ from the two equations of this diophantine system, we get,
\begin{equation}
y_1y_2+y_2y_3+y_3y_1=0. \label{sysgen1cond1}
\end{equation}
The complete solution of Eq.~\eqref{sysgen1cond1} is readily obtained and this immediately yields the following simultaneous identities:
\begin{equation}
(p^2+pq+q^2)^r=(p^2+pq)^r+(pq+q^2)^r+(-pq)^r,\;\;r=1,\,2,
\end{equation}
where $p$ and $q$ are arbitrary parameters. This shows that $\beta(2) = 4$.
\noindent {\bf 3.2} We now consider the diophantine system \eqref{sysgenk} when $k=3$. It follows from Cor. 1 that $\beta(3) \geq 5$. We will prove that $\beta(3)=6$.
If there exists a nontrivial solution of the system of equations,
\begin{equation}
\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\quad
r=1,\,2,\,3, \label{sysgen3}
\end{equation}
with $s_1 < s_2$, it follows from Lemma 1 that $s_2 \geq 4$. Thus the only case to consider when $s_1+s_2=5$ is with $s_1=1$ and $s_2=4$ when the diophantine system \eqref{sysgen3} may be written as,
\begin{align}
x_1&=y_1+y_2+y_3+y_4, \label{tep3r1}\\
x_1^2&=y_1^2+y_2^2+y_3^2+y_4^2, \label{tep3r2} \\
x_1^3&=y_1^3+y_2^3+y_3^3+y_4^3. \label{tep3r3}
\end{align}
Eliminating $x_1$ from Eqs.~\eqref{tep3r1} and \eqref{tep3r2}, we get the condition,
\begin{equation}
(y_1+y_2+y_3)y_4+y_1y_2+y_1y_3+y_2y_3=0,\label{restep3r1r2}
\end{equation}
while eliminating $x_1$ from Eqs.~\eqref{tep3r1} and \eqref{tep3r3}, we get the condition,
\begin{equation}
(y_1+y_2+y_3)y_4^2+(y_1+y_2+y_3)^2y_4+(y_2+y_3)(y_1+y_3)(y_1+y_2)=0. \label{restep3r1r3}
\end{equation}
If $y_1+y_2+y_3=0$, it follows from \eqref{tep3r1} that $x_1=y_4$ and now \eqref{tep3r2} implies that $y_1=0,\,y_2=0,\,y_3=0$, and we thus get a trivial solution of the system of equations \eqref{tep3r1}, \eqref{tep3r2} and \eqref{tep3r3}. If $y_1+y_2+y_3 \neq 0$, we eliminate $y_4$ from Eqs.~\eqref{restep3r1r2} and \eqref{restep3r1r3}, and get,
\begin{equation}
(y_1^2+y_1y_2+y_2^2)y_3^2+y_1y_2(y_1+y_2)y_3+y_1^2y_2^2=0. \label{restepr123}
\end{equation}
Eq.~\eqref{restepr123} can have a rational solution for $y_3$ only if its discriminant, that is, $-(3y_1^2+2y_1y_2+3y_2^2)y_1^2y_2^2$, is a perfect square. It follows that either $y_1$ or $y_2$, or both of them, must be 0, and in each case, it readily follows that the only solution of Eqs.~\eqref{tep3r1}, \eqref{tep3r2} and \eqref{tep3r3} is the trivial solution.
Since the diophantine system \eqref{sysgen3} has only the trivial solution when $s_1+s_2 = 5$, it follows that,
\begin{equation}
\beta(3) > 5. \label{beta3est1}
\end{equation}
We will now obtain nontrivial solutions of the diophantine system \eqref{sysgen3} with $s_1=2$ and $s_2=4$, that is, of the system of equations,
\begin{align}
x_1+x_2&=y_1+y_2+y_3+y_4,\label{sysgen3eq1} \\
x_1^2+x_2^2&=y_1^2+y_2^2+y_3^2+y_4^2,\label{sysgen3eq2}\\
x_1^3+x_2^3&=y_1^3+y_2^3+y_3^3+y_4^3.\label{sysgen3eq3}
\end{align}
If $(a,\,b,\,c)$ is any Pythagorean triple satisfying the relation $a^2+b^2=c^2$, it is easily seen that a solution in integers of the simultaneous equations \eqref{sysgen3eq1}, \eqref{sysgen3eq2} and \eqref{sysgen3eq3} is given by $(x_1,\,x_2,\,y_1,\,y_2,\,y_3,\,y_4)=(c,\,-c,\,a,\,-a,\,b,\,-b)$.
Next we obtain a more general parametric solution of the simultaneous equations \eqref{sysgen3eq1}, \eqref{sysgen3eq2} and \eqref{sysgen3eq3}. We will use a parametric solution of the simultaneous diophantine equations,
\begin{equation}
\begin{aligned}
x+y+z&=u+v+w,\\
x^3+y^3+z^3&=u^3+v^3+w^3,
\end{aligned}
\end{equation}
given in \cite[Theorem 1, p.\ 61]{Cho3}. From this solution, on writing $x_1=x,\,x_2=y,\,y_1=u,\,y_2=v,\,y_3=w,\,y_4=-z$, we immediately derive the following solution of the simultaneous equations \eqref{sysgen3eq1} and \eqref{sysgen3eq3} in terms of arbitrary parameters $p,\,q,\,r$ and $s$:
\begin{equation}
\begin{aligned}
x_1&= pq-pr+qr-(p-q-r)s,\\
x_2& = -pq+pr+qr+(p-q+r)s, \\
y_1& = pq+pr-qr+(p-q+r)s, \\
y_2& = pq-pr+qr+(p+q-r)s, \\
y_3& = -pq+pr+qr-(p-q-r)s,\\
y_4& = -pq-pr+qr-(p+q-r)s.
\end{aligned}
\label{valpqrssysgen3}
\end{equation}
Substituting the above values of $x_i,\;y_i$ in \eqref{sysgen3eq2}, we get, after necessary transpositions, the following quadratic equation in $s$:
\begin{multline}
2(p+q-r)^2s^2+(12p^2q-4p^2r-4pq^2-4pqr\\
+4pr^2+4q^2r-4qr^2)s+2(pq+pr-qr)^2=0. \label{sysgen3eq2a}
\end{multline}
On taking $r=p+q$, the coefficient of $s^2$ in Eq,~\eqref{sysgen3eq2a} vanishes, and we can then readily solve Eq.~\eqref{sysgen3eq2a} for $s$, and we thus obtain the following solution of the simultaneous equations \eqref{sysgen3eq1}, \eqref{sysgen3eq2} and \eqref{sysgen3eq3} in terms of arbitrary parameters $p$ and $q$:
\begin{equation}
\begin{aligned}
x_1 &= (3p^4-2p^3q-p^2q^2+q^4)q,\;\; &x_2 &= (p^4-p^2q^2-2pq^3+3q^4)p, \\
y_1 &= (p^4-p^2q^2+2pq^3-q^4)p, &y_2 &= 2pq(p-q)(p^2-pq-q^2), \\
y_3 &= -(p^4-2p^3q+p^2q^2-q^4)q, &y_4 &= 2pq(p-q)(p^2+pq-q^2).
\end{aligned}
\label{solsysgen3}
\end{equation}
As a numerical example, taking $p=2,\;q=1$, we get the solution,
\[29^r+22^r=30^r+4^r+(-3)^r+20^r,\quad r=1,\,2,\,3.
\]
We note that more parametric solutions of the system of equations \eqref{sysgen3eq1}, \eqref{sysgen3eq2} and \eqref{sysgen3eq3} can be obtained by solving Eq.~\eqref{sysgen3eq2a} in different ways, for example, by choosing $p, \,q,\,r$ such that the constant term in Eq.~\eqref{sysgen3eq2a} vanishes and then solving this equation for $s$, or by choosing $p, \,q,\,r$ such that the discriminant of Eq.~\eqref{sysgen3eq2a}, considered as a quadratic equation in $s$, becomes a perfect square, and then solving this equation for $s$.
As we have obtained nontrivial solutions of the system of equations \eqref{sysgen3eq1}, \eqref{sysgen3eq2} and \eqref{sysgen3eq3}, we get $\beta(3) \leq 6$, and on combining this result with \eqref{beta3est1}, we get,
\begin{equation}
\beta(3)=6. \label{valbeta3}
\end{equation}
\noindent {\bf 3.3} We will now obtain parametric solutions of the diophantine system,
\begin{equation}
\sum_{i=1}^3 x_i^r=\sum_{i=1}^5 y_i^r,\quad
r=1,\,2,\,3,\,4. \label{sysgen4}
\end{equation}
We write,
\begin{equation}
\begin{aligned}
x_1 &= 4uv+w+1,& x_2 &= -4uv+w-1,\\
x_3 &= -8u^2+8uv+4u-2,& y_1 &= 4u-2,\\
y_2 &= -4u,& y_3 &= 4uv+w-1,\\
y_4 &= -4uv+w+1,& y_5 &= -8u^2+8uv+4u,
\end{aligned}
\label{sysgen4valxy}
\end{equation}
where $u,\,v,\,w$ are arbitrary parameters.
It is readily verified that the values of $x_i,\,y_i$ given by \eqref{sysgen4valxy} satisfy Eq.~\eqref{sysgen4} when $r=1$ and $r=2$. Further, on substituting these values of $x_i,\,y_i$ in Eq.~\eqref{sysgen4} and taking $r=3$, we get the condition,
\begin{equation}
4u^3-8u^2v+4uv^2-4u^2+4uv-vw+u-v=0. \label{sysgen4condr3}
\end{equation}
On solving Eq.~\eqref{sysgen4condr3}, we get,
\begin{equation}
w=(4u^3-8u^2v+4uv^2-4u^2+4uv+u-v)/v.\label{sysgen4valw}
\end{equation}
Finally, we substitute the values of $x_i,\,y_i$ given by \eqref{sysgen4valxy} in Eq.~\eqref{sysgen4}, and take $r=4$, and use the value of $w$ given by \eqref{sysgen4valw} to get the condition,
\begin{equation}
u^2(2u-1)^2\{24uv^2-2(4u-1)^2v+3u(2u-1)^2\}=0. \label{sysgen4condr4}
\end{equation}
While equating the first two factors of Eq.~\eqref{sysgen4condr4} to 0 leads to trivial results, on equating the last factor to 0, we get a quadratic equation in $v$ which will have a rational solution if its discriminant $4(-32u^4+32u^3+24u^2-16u+1)$ becomes a perfect square. We thus have to solve the diophantine equation,
\begin{equation}
t^2=-32u^4+32u^3+24u^2-16u+1. \label{ecquartic}
\end{equation}
Now Eq.~\eqref{ecquartic} is a quartic model of an elliptic curve, and we use the birational transformation given by,
\begin{equation}
\begin{aligned}
t &= (X^3-36X^2+36X-72Y+432)/(4X+Y-12)^2,\\
u& = (X-12)/(4X+Y-12),
\end{aligned}
\label{birat1}
\end{equation}
and,
\begin{equation}
\begin{aligned}
X &=(4u^2-8u+t+1)/(2u^2),\\
Y& = (8u^3+12u^2-4ut-12u+t+1)/(2u^3),
\end{aligned}
\end{equation}
to reduce Eq.~\eqref{ecquartic} to the Weierstrass model which is given by the cubic equation,
\begin{equation}
Y^2=X^3-36X. \label{ecweier}
\end{equation}
It is readily seen from Cremona's well-known tables \cite{Cre}
that \eqref{ecweier} is an elliptic curve of rank 1 and its Mordell-Weil basis is given by the rational point $P$ with co-ordinates $(X,\,Y)=(-3,\,9).$ There are thus infinitely many rational points on the elliptic curve \eqref{ecweier} and these can be obtained by the group law. Using the relations \eqref{birat1}, we can find infinitely many rational solutions of Eq.~\eqref{ecquartic} and thus obtain infinitely many integer solutions of the diophantine system \eqref{sysgen4}.
While the point $P$ leads to a trivial solution of the diophantine system \eqref{sysgen4}, the point $2P$ yields the solution,
\[
(-74)^r+124^r+78^r=126^r+(-72)^r+(-20)^r+70^r+24^r,\;\;r=1,\,2,\,3,\,4,
\]
and the point $3P$ leads to the solution
\begin{multline*}
(-40573)^r+66494^r+118981^r=(-15181)^r+119510^r+63756^r\\
+(-37835)^r+14652^r,\;\;r=1,\,2,\,3,\,4.
\end{multline*}
In view of the above solutions of the diophantine system \eqref{sysgen4}, it follows that $\beta(4) \leq 8$, and on combining with the result $\beta(4) \geq 7$ which follows from Cor. 1, we get,
\[
7 \leq \beta(4) \leq 8.
\]
\noindent {\bf 3.4} We will now obtain parametric solutions of the diophantine system,
\begin{equation}
\sum_{i=1}^4 x_i^r=\sum_{i=1}^6 y_i^r,\quad
r=1,\,2,\,3,\,4,\,5. \label{sysgen5}
\end{equation}
We will use a parametric solution of the diophantine system,
\begin{equation}
\sum_{i=1}^{6} X_i^r=\sum_{i=1}^{6} Y_i^r,\quad
r=1,\,2,\,3,\,4,\,5, \label{tep5}
\end{equation}
to obtain two parametric solutions of the diophantine system \eqref{sysgen5}.
A solution of the simultaneous equations $\sum_{i=1}^{3} X_i^r=\sum_{i=1}^{3} Y_i^r,\;r=2,\,4$, in terms of arbitrary parameters $m,\,n,\,x$ and $y$, given in
\cite[p. 102]{Cho2}, is as follows:
\begin{equation}
\begin{aligned}
X_1&=(m+2n)x-(m-n)y,\quad &X_2&=-(2m+n)x-(m+2n)y,\\
X_3&=(m-n)x+(2m+n)y,&Y_1&=(m-n)x-(m+2n)y,\\
Y_2&=-(2m+n)x-(m-n)y,&Y_3&=(m+2n)x+(2m+n)y,
\end{aligned}
\label{soltep5}
\end{equation}
It immediately follows that a parametric solution of the diophantine system \eqref{tep5} is given by \eqref{soltep5} and
\begin{equation}
\begin{aligned}
X_4&=-X_3,\;\;&X_5&=-X_2,\;\;&X_6&=-X_1,\\
Y_4&=-Y_3,&Y_5&=-Y_2,&Y_6&=-Y_1.
\end{aligned}
\label{valX456}
\end{equation}
We choose the parameters $x$ and $y$ such that $X_3=0$ and immediately obtain the following parametric solution of the diophantine system \eqref{sysgen5}:
\begin{equation}
\begin{aligned}
x_1 &= m^2+mn+n^2,\quad & x_2 &= m^2+mn+n^2,\\ x_3 &= -m^2-mn-n^2, &x_4 &= -m^2-mn-n^2,\\ y_1 &= m^2-n^2, &y_2 &= -m^2-2mn,\\ y_3 &= 2mn+n^2,& y_4 &= -2mn-n^2,\\ y_5 &= m^2+2mn, &y_6 &= -m^2+n^2,
\end{aligned}
\label{sol1sysgen5}
\end{equation}
where $m$ and $n$ are arbitrary parameters.
To obtain a second solution of the diophantine system \eqref{sysgen5}, we again use the parametric solution of the diophantine system \eqref{tep5} given by \eqref{soltep5} and \eqref{valX456}. We now choose the parameters $x,\,y$ such that we get $X_2=X_3$, and then apply the theorem of Frolov mentioned in the Introduction, taking $d=-X_3$. We thus get a solution of the diophantine system \eqref{sysgen5} which may be written as follows:
\begin{equation}
\begin{aligned}
x_1 &= 3m^2+3mn+3n^2,& x_2 &= 2m^2+2mn+2n^2,\\ x_3 &= -m^2-mn-n^2,& x_4 &= 2m^2+2mn+2n^2,\\ y_1 &= 3m^2+3mn,& y_2 &= -3mn,\\ y_3 &= 3mn+3n^2,& y_4 &= 2m^2-mn-n^2,\\ y_5 &= 2m^2+5mn+2n^2,& y_6 &= -m^2-mn+2n^2,
\end{aligned}
\label{sol2sysgen5}
\end{equation}
where $m$ and $n$ are arbitrary parameters.
As a numerical example, taking $m=2,n=1$ in \eqref{sol2sysgen5}, we get the solution
\[
21^r+14^r+(-7)^r+14^r=18^r+(-6)^r+9^r+5^r+20^r+(-4)^r,\;\;r=1,\,2,\,3,\,4,\,5.
\]
The two parametric solutions \eqref{sol1sysgen5} and \eqref{sol2sysgen5} of the diophantine system \eqref{sysgen5} are rather special since in both of them, the ratios $x_i/x_j$ of the integers on the left-hand side are all fixed. We now show that there exist infinitely many other solutions of the diophantine system \eqref{sysgen5} that are not generated by these parametric solutions.
We write
\begin{equation}
\begin{aligned}
x_1&=uv^2+(6u^3-12u^2+32u-32)v\\
& \quad \quad +9u^5-36u^4-336u^2+96u^3+240u,\\
x_2& = (2u-2)v^2+(12u^3-48u^2+40u-16)v\\
& \quad \quad +18u^5-126u^4-288u^2+264u^3+96,\\
x_3&=-x_1,\quad x_4=-x_2,\\
y_1 &= (2u-2)v^2+(12u^3-48u^2+48u)v\\
& \quad \quad +18u^5-126u^4-144u^2+288u^3+96u-96,\\
y_2 &= uv^2+(6u^3-12u^2-32u+32)v\\
& \quad \quad +9u^5-36u^4+240u^2-96u^3-144u,\\
y_3 &= 2v^2+(24u^2-40u+16)v\\
& \quad \quad +54u^4-264u^3+96u+192u^2-96,\\
y_4&=-y_3,\quad y_5=-y_2,\,\quad y_6=-y_3.
\end{aligned}
\label{valxysysgen5}
\end{equation}
With these values of $x_i,\,y_i$, it is readily seen that \eqref{sysgen5} is identically satisfied for $r=1,\,3,\,5$. Further,
\begin{align}
\sum_{i=1}^4 x_i^2-\sum_{i=1}^6 y_i^2&=-8(9u^4-72u^3+24u^2+96u-48-v^2)^2,\\
\sum_{i=1}^4 x_i^4-\sum_{i=1}^6 y_i^4&=-8(9u^4-72u^3+24u^2+96u-48-v^2)^4.
\end{align}
It follows that a solution of the diophantine system \eqref{sysgen5} will be given by \eqref{valxysysgen5} if we choose $u,\,v$ such that,
\begin{equation}
v^2=9u^4-72u^3+24u^2+96u-48. \label{sysgen5ecquartic}
\end{equation}
Now Eq.~\eqref{sysgen5ecquartic} represents the quartic model of an elliptic curve, and the birational transformation given by,
\begin{equation}
\begin{aligned}
u& = (6X+2Y-12)/(3X-24),\\
v& = (4X^3-96X^2+84X-144Y+832)/\{3(X-8)^2\},
\end{aligned} \label{sysgen5birat1}
\end{equation}
and,
\begin{equation}
\begin{aligned}
X&=(9u^2-36u+3v+4)/8,\\
Y&=(27u^3-162u^2+9uv+36u-18v+72)/16,
\end{aligned}\label{sysgen5birat2}
\end{equation}
reduces Eq.~\eqref{sysgen5ecquartic} to the Weierstrass form of the elliptic curve which is as follows:
\begin{equation}
Y^2=X^3-21X-20. \label{sysgen5ecweier}
\end{equation}
We again refer to Cremona's database of elliptic curves,
and find that \eqref{sysgen5ecweier} represents an elliptic curve of rank 1 and its Mordell-Weil basis is given by the rational point $P$ with co-ordinates $(X,\,Y)=(-3,\,4).$ There are thus infinitely many rational points on the elliptic curve \eqref{ecweier} and these can be obtained by the group law. Using the relations \eqref{sysgen5birat1}, we can find infinitely many rational solutions of Eq.~\eqref{sysgen5ecquartic} and thus obtain infinitely many solutions of the diophantine system \eqref{sysgen5}.
While the point P leads to a trivial solution of the diophantine system \eqref{sysgen5}, the point 2P yields the solution,
\begin{multline*}
241^r+218^r+(-241)^r+(-218)^r=266^r+143^r+120^r\\
+(-266)^r+(-143)^r+(-120)^r,\;\;r=1,\,2,\,3,\,4,\,5.
\end{multline*}
The solutions of the diophantine system \eqref{sysgen5} obtained in this Section show that $\beta(5) \leq 10$, and on combining with the result $\beta(5) \geq 8$ which follows from Cor. 1, we get,
\[ 8 \leq \beta(5) \leq 10.
\]
\section{Concluding Remarks}
\hspace*{0.25in} It would be of interest to determine the precise values of $\beta(4)$ and $\beta(5)$. Further, it would be interesting to find integer solutions of the diophantine system \eqref{sysgenk} with $k>5$ and $s_1+s_2 < 2k+1$.
\begin{center}
\Large
Acknowledgment
\end{center}
I wish to thank the Harish-Chandra Research Institute, Allahabad for providing me with all necessary facilities that have helped me to pursue my research work in mathematics.
| {
"timestamp": "2016-03-01T02:11:04",
"yymm": "1602",
"arxiv_id": "1602.08698",
"language": "en",
"url": "https://arxiv.org/abs/1602.08698",
"abstract": "This paper is concerned with the diophantine system, $\\sum_{i=1}^{s_1} x_i^r=\\sum_{i=1}^{s_2} y_i^r,\\, r=1,\\,2,\\,\\ldots,\\,k, $ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as small as possible. We define $\\beta(k)$ to be the minimum value of $s_1+s_2$ for which there exists a nontrivial solution of this diophantine system. We find nontrivial integer solutions of this diophantine system when $k < 6$, and thereby show that $\\beta(2) =4,\\;\\, \\beta(3) = 6,\\;\\, 7 \\leq \\beta(4) \\leq 8$ and $8 \\leq \\beta(5) \\leq 10$.",
"subjects": "Number Theory (math.NT)",
"title": "Equal Sums of Like Powers with Minimum Number of Terms",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9914225133191064,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.709287462003934
} |
https://arxiv.org/abs/2204.05250 | Revisiting and improving upper bounds for identifying codes | An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\ell)/2$, where $n$ is the order and $\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $\frac{5n+2\ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars. | \section{Introduction}
An identifying code of a graph is a subset of its vertices that allows to distinguish all pairs of vertices by means of their neighbourhoods within the identifying code. This concept is related to other similar notions that deal with domination-based identification of the vertices/edges of a graph or hypergraph, such as locating-dominating sets~\cite{S87}, separating systems~\cite{BS07,B72}, or test covers~\cite{MS85}, to name a few. These types of problems have natural applications in fault-detection in networks~\cite{KCL98,UTS04}, biological diagnosis~\cite{MS85} or machine learning~\cite{BGL19}.
A lot of the research in the area has been dedicated to understanding the behaviour of these types of problems for graphs of specific graph classes, by proving lower and upper bounds on the smallest cardinality of a solution. One of the simplest classes of graphs to consider is the one of trees, and indeed a large number of papers in the area consider identifying codes of trees, see for example~\cite{A10,BCHL05,BCMMS07,CS11,GGNUV01,HHH06,JR18,NLG16,RRM19,S87}. We improve and generalize some of these results. As some of the best known bounds (see Theorem \ref{2lTreeBound}), which are tight for some trees, have been around for more than twenty years, it is quite surprising that we manage to improve the upper bound for the smallest size of identifying codes of trees by a notable amount, with a quite simple proof. Moreover, our bounds do not only hold for trees, but also for larger classes of graphs, in particular, bipartite graphs without twins of degree at least~2. We also apply the new bound to get a new tight bound for graphs of girth at least~5.
\paragraph{Notations and definitions.} We consider connected finite undirected graphs on at least three vertices. Let us first define some basic notations. A vertex $u\in V(G)$ is said to be a \textit{leaf}, that is, a pendant vertex, if it has degree~1. A vertex $v\in V(G)$ is said to be a \textit{support vertex} if it is adjacent to a leaf.
We denote by $L(G)$ the set of leaves and by $S(G)$ the set of support vertices in graph $G$. Moreover, we denote the number of leaves and support vertices by $\ell(G)=|L(G)|$ and $s(G)=|S(G)|$, respectively. A graph is \textit{bipartite} if it does not contain any odd cycles and it has \textit{girth} $g$ if the length of its smallest cycle is~$g$. The \emph{$2$-corona} $H\circ_2$ of a graph $H$ (defined in~\cite[Section 1.3]{bookTD}) is the graph of order~$3|V(H)|$ obtained from $H$ by adding a vertex-disjoint copy of a path $P_2$ for each vertex $v$ of $H$ and adding an edge joining $v$ to one end of the added path. We define the \emph{$k$-corona} $H\circ_k$ of $H$ in an analogous way with $P_k$, for any $k\geq1$.
We denote by $N(v)\subseteq V(G)$ the \textit{open neighbourhood} of vertex $v$ and by $N[v]=N(v)\cup\{v\}$, its \textit{closed neighbourhood}. If $C$ is a set of vertices, or a \textit{code}, and $v$, a vertex, we denote the intersection between $N[v]$ and code $C$ by the $I$-set of $v$, $I(v)=N[v]\cap C$. Identifying codes were defined over twenty years ago in~\cite{KCL98} by Karpovsky et al. and since then they have been studied in a large number of articles, see~\cite{biblio} for an online bibliography. A set $C\subseteq V(G)$ is called an \textit{identifying code} of $G$ if for each pair of distinct vertices $u,v\in V(G)$, we have that (i) they are \textit{covered}/\textit{dominated}, that is, $I(u)\neq \emptyset$ and $I(v)\neq \emptyset$ and (ii) they are \textit{distinguished}/\textit{separated}, that is, their $I$-sets are distinct, that is, $$I(u)\neq I(v).$$
The vertices of the code are called \emph{codewords}. Two distinct vertices are \emph{open twins} if their open neighbourhoods are the same, and \emph{closed twins} if their closed neighbourhoods are the same. A graph admits an identifying code if and only if it has no pair of closed twins~\cite{KCL98}; in that case we say the graph is \emph{identifiable}. Note that any connected bipartite (in fact, triangle-free) graph is identifiable, with the exception of the complete graph of order~2. We say that a graph is \textit{twin-free} if it contains neither open nor closed twins. Twins are important for identifying codes, indeed closed twins cannot be identified, and for any set of mutually open twins, at most one can be absent from the identifying code.
For an identifiable graph $G$, we denote by $\gamma^{\text{\tiny{ID}}}(G)$ the smallest size of an identifying code of $G$.
Identifying codes and related concepts have been extensively studied for trees;
in particular, lower and upper bounds involving the number of leaves and support vertices have been proposed. Among these, the following two (due to Gimbel et al. and Rahbani et al.) are the currently best known upper bounds.
\begin{theorem}[\protect{\cite[Theorem 15]{GGNUV01}}]\label{2lTreeBound}
Let $T$ be a tree on $n\geq3$ vertices. Then $\gamma^{\text{\tiny{ID}}}(T)\leq \frac{n+2\ell(T)-2}{2}$.
\end{theorem}
\begin{theorem}[\protect{\cite[Theorem 11]{RRM19}}]\label{3/5TreeBound}
Let $T$ be a tree on $n\geq3$ vertices. Then $\gamma^{\text{\tiny{ID}}}(T)\leq \frac{3n+2\ell(T)-1}{5}$. Equality holds if and only if $T=P_4$.
\end{theorem}
The bound in Theorem~\ref{2lTreeBound} is better when the tree has few leaves, while the bound in Theorem~\ref{3/5TreeBound} is better when there are many leaves. Both bounds are tight for the 4-vertex path $P_4$, for which $\gamma^{\text{\tiny{ID}}}(P_4)=3$. Moreover, Theorem~\ref{2lTreeBound} is tight for any path on at least four vertices since $\gamma^{\text{\tiny{ID}}}(P_n)=\left\lceil\frac{n+1}{2}\right\rceil$ as proved in~\cite{BCHL04,GGNUV01}. However, we will see that tightness only holds for this case and on some trees of odd order with three leaves.
The following bound for graphs of girth at least~5 with no leaves was also proved by Balbuena et al.
\begin{theorem}[\protect{\cite[Theorem 13]{BFH15}}]\label{The:ID5/7delta}
Let $G$ be a graph of order $n$ and girth at least $5$ with minimum degree $\delta(G)\geq2$. Then $\gamma^{\text{\tiny{ID}}}(G)\leq 5n/7$.
\end{theorem}
\paragraph{Our results.}
Inspired by the aforementioned results from the literature, we present improved (and tight) upper bounds. Our bounds not only improve on the known results for trees, but also hold for a larger class of graphs: bipartite graphs which do not have any twins of degree~2 or greater. Observe that this class of graphs contains, for example, the class of $C_4$-free bipartite graphs.
In particular, we show in Theorem~\ref{TheShift} that the bound $\gamma^{\text{\tiny{ID}}}(G)\leq\frac{n+\ell(G)}{2}$ holds for every bipartite graph $G$ of order $n$ which does not have any twins of degree~2 or greater. This bound is never larger than either of the bounds of Theorems~\ref{2lTreeBound} or~\ref{3/5TreeBound}. In fact, Theorem~\ref{2lTreeBound} is only tight when $\ell(T)\leq 3$. (Otherwise, one can check that our bound is smaller, indeed then we have $\lfloor(n+\ell(T))/2\rfloor<\lfloor(n+2\ell(T)-2)/2\rfloor$.) The bound of Theorem~\ref{3/5TreeBound} can be modified into the equivalent form $\gamma^{\text{\tiny{ID}}}(T)\leq (3n-3\ell(T)-1)/5+\ell(T)$. If we similarly modify the bound of Theorem~\ref{TheShift} to $\gamma^{\text{\tiny{ID}}}(T)\leq (n-\ell(T))/2+\ell(T)$, then we can clearly observe the improvement provided by our bound.
Moreover, as opposed to the existing bounds, our bound is tight for a very rich class of graphs, in particular, for many trees: paths, stars, and more complicated examples that will be described in Section~\ref{Sec:Remarks}. Our proof is also rather simple.
We also extend the bound $\gamma^{\text{\tiny{ID}}}(G)\leq n - s(G)$ (even holding for identifying codes that are also total dominating sets), which was known to hold for trees~\cite{HHH06}, to a class that includes all triangle-free graphs. We also show that the slightly larger bound $\gamma^{\text{\tiny{ID}}}(G)\leq n-s(G)+1$ holds for all identifiable graphs. Both bounds are tight.
We then apply two of the above upper bounds to \emph{twin-free} bipartite graphs, showing that for such a graph $G$ of order $n$, we always have $\gamma^{\text{\tiny{ID}}}(G)\leq 2n/3$ (unless $G$ is the path $P_4$). Moreover, we characterize those graphs reaching this bound, as the 2-coronas of bipartite graphs.
Finally, we extend the bound of Theorem~\ref{The:ID5/7delta} to all graphs of girth at least~5, showing that for such a graph $G$ of order $n$, we have $\gamma^{\text{\tiny{ID}}}(G)\leq\frac{5n+2\ell(G)}{7}$. This bound is tight for all stars and for the cycle $C_7$. We also present a new infinite family of graphs of girth at least $5$ which has largest known ratio $\gamma^{\text{\tiny{ID}}}(G)/n$ for graphs with $\delta(G)\geq2$.
We present our upper bounds for identifying codes in bipartite graphs in Section~\ref{SecBounds}. The application to twin-free graphs is presented in Section~\ref{sec:twinfree}. The bound for graphs of girth at least~5 is proved in Section~\ref{sec:girth5}.
We conclude in Section~\ref{sec:conclu}.
\paragraph{Further related work.}
For a tree $T$ of order $n\geq 4$, the lower bounds $\gamma^{\text{\tiny{ID}}}(T)\geq\frac{3(n-1)}{7}$~\cite{BCHL05,GGNUV01}, $\gamma^{\text{\tiny{ID}}}(T)\geq\frac{2n-s(T)+3}{4}$~\cite{RRM19} and $\gamma^{\text{\tiny{ID}}}(T)\geq\frac{3n+\ell(T)-s(T)+1}{7}$~\cite{BCMMS07} have been proved.
We note that a polynomial-time algorithm to compute $\gamma^{\text{\tiny{ID}}}(T)$ for a tree $T$ has been provided in~\cite{A10}, however the problem is hard to approximate within any sub-logarithmic factor even for bipartite graphs with no 4-cycles~\cite{BLLPT15} (bipartite graphs wihout $4$-cycles do not have any twins of degree~2 or greater). Identifying codes in graphs of girth at least~5 were also considered in~\cite{BFH15,FP11}.
Many upper bounds for trees similar to those of Theorems~\ref{2lTreeBound} and \ref{3/5TreeBound} have been obtained for related graph parameters.
In particular, tight upper bounds on trees for identifying codes that are also total dominating sets have been considered in~\cite{JR18,NLG16}. Similar results have also have been proved for the related locating-dominating sets (where one only needs to distinguish pairs of vertices that are not part of the solution set)~\cite{BCMMS07,BDLP21,JR18} and their total dominating variant~\cite{CS11}.
Bounds for twin-free graphs have been studied for locating-dominating sets. It was proved in~\cite{GGM14} that every twin-free bipartite graph and every twin-free graph with no $4$-cycles has a locating-dominating set of size at most $n/2$; the bound is tight for infinitely many trees, which are characterized in~\cite{Heia}.
It is conjectured that this bound holds for all twin-free graphs~\cite{GGM14}.
\section{Two upper bounds}\label{SecBounds}
In this section, we present our main result in Theorem~\ref{TheShift}. However, first we give some upper bounds which are useful when the number of support vertices in $G$ is large.
\subsection{A first pair of upper bounds using the number of support vertices}
Our first lemma holds for identifying codes that are also \emph{total dominating sets}, that is, every vertex of the graph has a neighbour in the dominating set~\cite{bookTD}. A graph admits no total dominating set only if it has isolated vertices. For an identifiable graph $G$ with no isolated vertices, we denote by $\gamma_t^{\text{\tiny{ID}}}(G)$ the smallest size of a total dominating identifying code of $G$. Total dominating identifying codes have usually been called \emph{differentiating-total dominating sets} in the literature, see for example~\cite{GGNUV01,HHH06,JR18,NLG16}.
The following lemma has previously been proven for trees in~\cite{HHH06}. We extend it to a larger class which contains, for example, all triangle-free graphs (note that for every triangle-free graph $G$, $G-L(G)$ is identifiable, unless $G-L(G)$ contains a component isomorphic to $P_2$).
\begin{lemma}\label{LemSupportBip}
Let $G$ be a connected graph on $n\geq4$ vertices that is not the path $P_4$, such that $G-L(G)$ is identifiable or $G$ is triangle-free.
Then $$\gamma_t^{\text{\tiny{ID}}}(G)\leq n-s(G).$$
\end{lemma}
\begin{proof}
Let us choose for each support vertex $u\in S(G)$ exactly one adjacent leaf $v\in L(G)$ and say that these vertices form the set $C'$. Hence, $|C'|=s(G)$. Next, we form the code $C=V(G)\setminus C'$. We show that $C$ is a total dominating identifying code in $G$. We have $|C|=n-s(G)$.
Observe that each non-codeword $v$ is a leaf. Moreover, if $v,u\in C'$ and $I(v)=I(u)=\{w\}$, then $w\in S(G)$ but we have chosen two vertices adjacent to $w$ into $C'$, a contradiction. Since $C'\subseteq L(G)$, $G[C]$ is a connected induced subgraph of $G$. Moreover, as $n\geq4$ and $G$ is not $P_4$, we have $|I(c)|\geq2$ for each codeword $c\in C$. Thus, codewords and non-codewords have different $I$-sets. Furthermore, we have $I(c)\neq I(c')$ for two codewords $c\neq c'$ since $|V(G[C])|\geq3$ and there are no triangles in $G[C]$ or $G[C]$ is identifiable and hence, each closed neighbourhood is unique in $G[C]$. Finally, $C$ is total dominating since $G[C]$ is connected.
\end{proof}
Lemma~\ref{LemSupportBip} is tight for example for 3-coronas of graphs (but for these graphs the regular identifying code number is smaller). It is also tight for both identifying codes and total dominating identifying codes, by considering for example the 1-corona of any triangle-free graph of order at least~3, or any star of order at least~3.
Moreover, we require the stated restrictions in the claim. For example, for the 1-corona $K_m\circ_1$ of a complete graph of order $m\geq3$ (for which $K_m\circ_1 - L(K_m\circ_1)$ is far from identifiable), we have $\gamma_t^{\text{\tiny{ID}}}(K_m\circ_1)=2m-1$ and not $2m-s(K_m\circ_1)=m$. Indeed, all vertices of the $m$-clique need to be in the code to totally dominate the leaves. Moreover, for any two vertices of the clique, one of their two leaf neighbours needs to be in the code to identify them, and hence $\gamma_t^{\text{\tiny{ID}}}(K_m\circ_1)\geq 2m-1$. Moreover, $\gamma_t^{\text{\tiny{ID}}}(K_m\circ_1)\leq 2m-1$ by considering the whole vertex set except one leaf as a code.
The same example shows that the statement of Lemma~\ref{LemSupportBip} is also not true for identifying codes on general graphs, as $\gamma^{\text{\tiny{ID}}}(K_m\circ_1)=m+1$ and not $2m-s(K_m\circ_1)=m$. Indeed, as above, we need at least $m-1$ leaf codewords, to distinguish the vertices of the $m$-clique. Moreover, to distinguish leaves from their neighbour support vertex, we need at least two codewords inside the $m$-clique. This implies $\gamma^{\text{\tiny{ID}}}(K_m\circ_1)\geq m+1$. Conversely, consider $L(G)$ together with any two vertices in the $m$-clique, and remove a leaf neighbour of one of the clique code vertices: this set forms an identifying code of size $m+1$.
In the following theorem, we show that the previous construction is actually the worst case for identifying codes: in this case, a very similar upper bound as the one of Lemma~\ref{LemSupportBip} is true.
\begin{theorem}
Let $G$ be a connected identifiable graph on $n\geq3$ vertices. Then $$\gamma^{\text{\tiny{ID}}}(G)\leq n-s(G)+1.$$
\end{theorem}
\begin{proof}
When $s(G)\leq 2$, the claim is clear since $\gamma^{\text{\tiny{ID}}}(G)\leq n-1$ for any connected identifiable graph on at least three vertices~\cite{GM07}. Hence, we may assume that $s(G)\geq3$ which implies $n\geq 2s(G)\geq6$. Let us then prove the claim by induction on $n$. Let $s(G)\geq3$. If $G'=G-L(G)$ is identifiable, then $\gamma^{\text{\tiny{ID}}}(G)\leq n-s(G)$ by Lemma~\ref{LemSupportBip}. Thus, we may assume that $N_{G'}[u]=N_{G'}[v]$ for some distinct vertices $u,v$ in $V(G')$. Since $G$ is identifiable, we have $u$ or $v\in S(G)$. Assume that $u\in S(G)$ and let $L_u=N(u)\cap L(G)$. Let us form graph $G_u=G-u-L_u$.
Observe that $G_u$ is connected since $N_{G'}[v]=N_{G'}[u]$. Moreover, since $s(G)\geq3$ and $G$ is identifiable, also $G_u$ is identifiable. Indeed, if $N_{G_u}[x]=N_{G_u}[y]$ for some vertices $y$ and $x$, then $u$ separates them in $G$, let us say, $u\in N_G[x]\setminus N_G[y]$. However, we have $N_{G'}[u]=N_{G'}[v]$. Hence, $v\in N_{G_u}[x]\setminus N_{G_u}[y]$, a contradiction, or $y$ is a leaf in $G$ which is adjacent to $v$ but now $N_{G_u}[v]=N_{G_u}[y]$ and $G_u$ is a path on two vertices which is impossible. Notice that we have $s(G_u)\geq s(G)-1\geq2$, since we removed at most one support vertex. Since $s(G_u)\geq2$, $G_u$ has order at least~4.
By the induction hypothesis, $G_u$ has an identifying code of cardinality at most $(n-1-|L_u|)-s(G_u)+1\leq n-|L_u|-s(G)+1$. Let $C_u$ be an identifying code with such cardinality. Let us first consider the case with $|L_u|\geq2$. Now, we may take $C=C_u\cup L_u$ and it is an identifying code of $G$ with at most $n-s(G)+1$ vertices, and we are done. However, if $|L_u|=1$ and let us say that $u'$ is the leaf in $L_u$, then just by adding $u'$ to $C_u$, we might have $I(u)=I(u')$. Let us divide this into two cases. First assume that, for $C_u$, $I(v)\not\subseteq L(G)$, now $I(v)\cap N(u)\neq \emptyset$ and $C_u\cup\{u'\}$ is an identifying code in $G$ of cardinality at most $n-s(G)+1$, as required. Hence, we may assume that $I(v)\subseteq L(G)$ for $C_u$. In this case we have $|I(v)|=|N(v)\cap L(G)|\geq2$. Now, to form code $C$, we may shift one codeword from $N(v)\cap L(G)$ to $v$ and add $u$ to the code. The resulting code is identifying in $G$ and has cardinality at most $n-s(G)+1$. \end{proof}
\subsection{A second upper bound using the number of leaves}
Now, we prove an upper bound for identifying codes in some bipartite graphs based on the number of leaves. In particular, this bound is an improvement for trees. Notice that the graphs in the following theorem contain $C_4$-free bipartite graphs.
\begin{theorem}\label{TheShift}
Let $G$ be a connected bipartite graph on $n\geq3$ vertices which does not have any twins of degree~2 or greater. We have $$\gamma^{\text{\tiny{ID}}}(G)\leq\frac{n+\ell(G)}{2}.$$
\end{theorem}
\begin{proof}
Let $G$ be a connected bipartite graph on $n\geq3$ vertices which does not have any twins of degree~2 or greater. Let us fix a non-leaf vertex $x\in V(G)\setminus L(G)$ as the root of the graph. We present the bipartite graph as a layered graph, so that vertex $u$ is in layer $i$ if $d(x,u)=i$. Observe that adjacent vertices have different distances to the root $x$ since $G$ is bipartite. Our goal is to construct two identifying codes, and to show that at least one of them has the claimed cardinality.
Let us first construct code $C'_e$ by choosing $u\in C'_e$ if $d(x,u)$ is even, or if $u\in L(G)$. Next we will shift some codewords to construct an identifying code $C_e$. Observe that if $u\in L(G)$ has an odd distance to root $x$, then there is an adjacent support vertex $v\in S(G)\cap I(u)$. Let us first have $C_e=C'_e$. We modify code $C_e$ by shifting some codewords in leaves according to following rule. If $N(v)\cap L(G)=\{u\}$ and $d(x,u)$ is odd, then we remove $u$ from $C_e$ and we add some vertex $v'\in N(v)$ with $d(v',x)=d(v,x)-1$ to $C_e$ (if $v=x$, we instead add some non-leaf vertex adjacent to $x$ to $C_e$). We illustrate codes $C_e'$ and $C_e$ together with the shift in the left graph of Figure~\ref{TreeExamp}. We next prove that $C_e$ is an identifying code in $G$.
If $w\not\in L(G)$ is in layer $i$ where $i$ is odd, then $w$ has at least two adjacent codewords and $N(w)\subseteq I(w)$. Thus, $w$ has a unique $I$-set since if $I(w')=I(w)$ for some vertex $w'\neq w$, then $w'$ is a non-leaf in an odd layer and $N(w')\subseteq I(w)$. Thus, $w$ and $w'$ are twins of degree at least~2, a contradiction. Let us then consider the case where $w\in L(G)$ and $d(w,x)$ is odd; assume $u\in S(G)$ is the adjacent support vertex. Now, if $|N(u)\cap L(G)|\geq2$, then $I(w)=\{w,u\}$ and $|I(u)|\geq3$. Thus, $I(w)$ is unique. If $|N(u)\cap L(G)|=1$, then $w\not\in C_e$ due to shifting and $I(w)=\{u\}$. However, $|I(u)|\geq2$ and hence, $I(w)$ is again unique. Let us then consider the case where $d(w,x)$ is even. Now, $w\in C_e$ and hence, if $I(w)=I(w')$, then $w'\in N(w)$. But then $d(w',x)$ is odd and thus, $I(w')$ is also unique by our earlier arguments. Thus, $C_e$ is an identifying code in $G$.
As the second code, we construct $C'_o$ similarly as we constructed $C_e'$, except that we add vertices in odd layers, that is, we have $u\in C'_o$ if $d(x,u)$ is odd, or if $u\in L(G)$. Then, we again use the shifting to obtain the code $C_o$. This time, we shift some codewords away from some leaves in even layers. Let $u\in L(G)$ be a leaf with $d(u,x)$ even and $v\in S(G)\cap N(u)$. Thus, the distance between vertices $v$ and $x$ is odd and $v\in C'_o$. Moreover, let, again, $v'\in N(v)$ be the vertex adjacent $v$ with $d(v',x)=d(v,x)-1$. Now, if $N(v)\cap L(G)=\{u\}$, then we remove $u$ from $C_o$ and add $v'$ to $C_o$. Codes $C_o'$ and $C_o$ are illustrated in the right graph of Figure~\ref{TreeExamp} and they can be compared with the codes $C_e'$ and $C_e$. The proof that the code $C_o$ is identifying is similar to the proof for $C_e$.
Thus, we have $\gamma^{\text{\tiny{ID}}}(G)\leq \min \{|C_e|,|C_o|\}\leq \min \{|C'_e|,|C'_o|\}$. Moreover, $C'_e$ and $C'_o$ both contain every leaf and at least one of them contains at most half of the non-leaf vertices. Thus, $\gamma^{\text{\tiny{ID}}}(G)\leq\min \{|C'_e|,|C'_o|\}\leq \ell(G)+(n-\ell(G))/2=(n+\ell(G))/2.$
\end{proof}
\begin{figure}[!htpb]
\centering
\begin{tikzpicture}
\draw[dashed] (-4.2,-0.5) -- (11.5,-0.5);
\draw[dashed] (-4.2,-1.5) -- (11.5,-1.5);
\draw[dashed] (-4.2,-2.5) -- (11.5,-2.5);
\node[code node](x) at (0,0) {$ x $};
\node[main node](1) at (-1.2,-1) {};
\node[code node](11) at (-2.4,-2) {};
\node[code node](12) at (-0.5,-2) {};
\node[code node](111) at (-3.6,-3) {};
\node[code node](112) at (-2.0,-3) {};
\node[square code node](2) at (1,-1) {};
\node[code node](21) at (2,-2) {};
\node[square main node](211) at (3,-3) {};
\node[] at (-4.1,-0) {$0$};
\node[] at (-4.1,-1) {$1$};
\node[] at (-4.1,-2) {$2$};
\node[] at (-4.1,-3) {$3$};
\path[draw,thick]
(x) edge node {} (1)
(1) edge node {} (11)
(1) edge node {} (12)
(11) edge node {} (111)
(11) edge node {} (112)
(x) edge node {} (2)
(2) edge node {} (21)
(21) edge node {} (211)
(211)[->,bend right=55, looseness=0.9] edge node {} (2)
;
\node[square code node2](Rx) at (8,0) {$ x $};
\node[code node](R1) at (6.8,-1) {};
\node[main node](R11) at (5.6,-2) {};
\node[square main node](R12) at (7.5,-2) {};
\node[code node](R111) at (4.4,-3) {};
\node[code node](R112) at (6.0,-3) {};
\node[code node](R2) at (9,-1) {};
\node[main node](R21) at (10,-2) {};
\node[code node](R211) at (11,-3) {};
\path[draw,thick]
(Rx) edge node {} (R1)
(R1) edge node {} (R11)
(R1) edge node {} (R12)
(R11) edge node {} (R111)
(R11) edge node {} (R112)
(Rx) edge node {} (R2)
(R2) edge node {} (R21)
(R21) edge node {} (R211)
(R12)[->,bend right=25, looseness=0.85] edge node {} (Rx)
;
\end{tikzpicture}\centering
\caption{Gray circle and white square vertices form the codes $C'_e$ (left) and $C'_o$ (right). Arrows and squared vertices depict the shifts in the forming of identifying codes $C_e$ and $C_o$ which contain gray circled and squared vertices.}\label{TreeExamp}
\end{figure}
\subsection{Remarks and consequences}\label{Sec:Remarks}
Observe that the conditions (being bipartite and not having twins of degree~2 or greater) in the statement of Theorem~\ref{TheShift} are necessary. For example, we have $\gamma^{\text{\tiny{ID}}}(C_4)=\gamma^{\text{\tiny{ID}}}(C_5)=3$, and when $n\geq 7$ is odd, we have $\gamma^{\text{\tiny{ID}}}(C_n)=\lceil\frac{n}{2}\rceil+1$~\cite{GGNUV01}; for a complete bipartite graph $K_{k_1,k_2}$ of order $n$, with $k_1,k_2\geq2$, we have $\gamma^{\text{\tiny{ID}}}(K_{k_1,k_2})=n-2$.
The upper bound given by Theorem~\ref{TheShift} is tight for quite a rich class of graphs that includes many structurally different graphs. Those graphs include, for example, any path or even-length cycle with at least six vertices~\cite{BCHL04}, or any star on at least three vertices (for a star $S_n$ of order $n$, we have $\gamma^{\text{\tiny{ID}}}(S_n)=n-1$). The bound is also tight for any \emph{spider graph} where the length of each leg is odd (that is, a star whose edges are subdivided an even number of times) and $2$-coronas of bipartite graphs discussed in Theorem \ref{The:2/3}, as well as some other trees like the ones presented in Figure~\ref{TightTrees}. It seems difficult to obtain a full characterization of this family (even for trees), given the diversity of these examples.
Note that the same bound as the one in Theorem~\ref{TheShift} has been proved for trees, in~\cite{CS11}, for locating-total dominating sets, and the trees reaching the bound are characterized therein. (A \emph{locating-total dominating set} is a set $D$ of vertices such that each vertex of $G$ has a neighbour in $D$, and any two vertices not in $D$ are separated by $D$.) However, the extremal families are not the same: for example, the trees of Figure~\ref{TightTrees} have locating-total dominating sets smaller than the bound (examples of such sets are formed by the square vertices). Moreover, the upper bound for locating-total dominating sets cannot be generalized in the same way to bipartite graphs since the cycle $C_6$ requires at least four codewords in the case of locating-total domination.
\begin{figure}[!htpb]
\centering
\begin{tikzpicture}
\draw[dashed] (-4.2,-0.5) -- (11.5,-0.5);
\draw[dashed] (-4.2,-1.5) -- (11.5,-1.5);
\draw[dashed] (-4.2,-2.5) -- (11.5,-2.5);
\node[code node](x) at (0,0) {$ x $};
\node[square main node](1) at (-1,-1) {};
\node[square code node](11) at (-1.5,-2) {};
\node[code node](12) at (-0.5,-2) {};
\node[square main node](2) at (1,-1) {};
\node[code node](21) at (0.5,-2) {};
\node[square code node](22) at (1.5,-2) {};
\node[] at (-4.1,-0) {$0$};
\node[] at (-4.1,-1) {$1$};
\node[] at (-4.1,-2) {$2$};
\node[] at (-4.1,-3) {$3$};
\path[draw,thick]
(x) edge node {} (1)
(1) edge node {} (11)
(1) edge node {} (12)
(x) edge node {} (2)
(2) edge node {} (21)
(2) edge node {} (22)
;
\node[square main node2](Rx) at (8,0) {$ x $};
\node[code node](R1) at (6.8,-1) {};
\node[square main node](R11) at (5.6,-2) {};
\node[square code node](R111) at (5.1,-3) {};
\node[code node](R112) at (6.1,-3) {};
\node[square code node](R0) at (8,-1) {};
\node[code node](R2) at (9,-1) {};
\node[square main node](R21) at (10,-2) {};
\node[code node](R211) at (9.5,-3) {};
\node[square code node](R212) at (10.5,-3) {};
\path[draw,thick]
(Rx) edge node {} (R1)
(R1) edge node {} (R11)
(Rx) edge node {} (R0)
(R11) edge node {} (R111)
(R11) edge node {} (R112)
(Rx) edge node {} (R2)
(R2) edge node {} (R21)
(R21) edge node {} (R211)
(R21) edge node {} (R212);
\end{tikzpicture}\centering
\caption{The gray vertices form optimal identifying codes in these two trees, whose sizes are equal to the bound presented in Theorem~\ref{TheShift}. The squared vertices form optimal locating-total dominating sets.}\label{TightTrees}
\end{figure}
We get the following corollary of Lemma~\ref{LemSupportBip} and Theorem~\ref{TheShift}.
\begin{corollary}\label{CorTreeBound}
Let $G$ be a connected bipartite graph on $n\geq5$ vertices which does not have any twins of degree~2 or greater. We have $$\gamma^{\text{\tiny{ID}}}(G)\leq\min\left\{\frac{n+\ell(G)}{2},n-s(G)\right\}.$$
\end{corollary}
\section{An application to twin-free bipartite graphs}\label{sec:twinfree}
We next apply our bounds to obtain upper bounds for twin-free graphs. Similar upper bounds have been studied in the context of location-domination, see~\cite{FH16,Heia,GGM14}
\begin{corollary}\label{cor:2n/3}
Let $G$ be a twin-free bipartite graph on $n\geq3$ vertices that is not $P_4$. Then $$\gamma^{\text{\tiny{ID}}}(G)\leq\frac{2n}{3}.$$
\end{corollary}
\begin{proof}
Since $G$ is twin-free, we have $s(G)=\ell(G)$. Hence, Corollary~\ref{CorTreeBound} gives $\gamma^{\text{\tiny{ID}}}(G)\leq \min\{(n+\ell(G))/2,n-\ell(G)\}$. Since $(n+\ell(G))/2=n-\ell(G)$, when $\ell(G)=n/3$, we attain the upper bound of $\gamma^{\text{\tiny{ID}}}(G)\leq 2n/3$.
\end{proof}
Note that the conditions in the statement of Corollary~\ref{cor:2n/3} are best possible, in the sense that there exist twin-free non-bipartite graphs that need $n-1$ vertices in any of their identifying codes, such as the complements of half-graphs, see~\cite{FGKNPV1l}. Those graphs have very large cliques, but there are twin-free graphs with small clique number that also have large identifying codes: for any $\Delta\geq 3$, arbitrarily large twin-free graphs of order $n$, maximum degree~$\Delta$ and optimal identifying codes of size $\frac{(\Delta-1)n}{\Delta}$ have been presented in~\cite{FP11}. The condition on twin-freeness cannot be relaxed, as stars of order $n$ also have no identifying code smaller than $n-1$, and all other complete bipartite graphs need $n-2$ vertices in their identifying codes. Moreover, this bound does not hold for non-bipartite graphs of girth at least~5, since $\gamma^{\text{\tiny{ID}}}(C_7)=5$.
We next show that the bound of Corollary~\ref{cor:2n/3} is tight.
\begin{proposition}\label{prop:coronas}
Let $H$ be any connected graph of order at least~$2$. Then, the 2-corona $H\circ_2$ is twin-free and $\gamma^{\text{\tiny{ID}}}(H\circ_2)=2n/3$, where $n$ is the order of $H\circ_2$.
\end{proposition}
\begin{proof}
For any vertex $v$ of $H$, let $v_1$ be the vertex adjacent to $v$ that was not in $H$, and let $v_2$ be the leaf adjacent to $v_1$. To separate $v_1$ from $v_2$, $v$ needs to belong to any identifying code of $H\circ_2$. Moreover, to dominate $v_2$, one of $v_1,v_2$ needs to belong to any identifying code. This shows that $\gamma^{\text{\tiny{ID}}}(H\circ_2)\geq 2n/3$.
For the upper bound, one can consider the set containing $v$ and $v_2$ for each vertex $v$ of $H$: this is an identifying code of $H\circ_2$. If $H$ has at least three vertices, the set containing $v$ and $v_1$ for each vertex $v$ of $H$ also works.
\end{proof}
In the following theorem, we characterize all the twin-free
bipartite graphs achieving the upper bound of $2n/3$ of Corollary~\ref{cor:2n/3} by showing that they are exactly the $2$-coronas of
bipartite graphs.
\begin{theorem}\label{The:2/3}
Let $G$ be a connected twin-free bipartite graph on $n$ vertices
with $\gamma^{\text{\tiny{ID}}}(G)=2n/3$. Then $G$ is the 2-corona $H\circ_2$ of some bipartite graph $H$.
\end{theorem}
\begin{proof}
Since $G$ is twin-free, we have $s(G)=\ell(G)$. Together with Corollary \ref{CorTreeBound} and the fact that $\gamma^{\text{\tiny{ID}}}(G)=2n/3$, this means that $V(G)$ can be partitioned into three equal-sized parts: $$s(G)=\ell(G)=|V(G)\setminus(L(G)\cup S(G))|=n/3.$$ We will call the vertices in $V(G)\setminus(L(G)\cup S(G))$ \textit{central vertices}.
Let us first assume that $v$ is a central vertex without an adjacent support vertex. Notice that $v$ has at least two adjacent central vertices. Let us now construct set $C'=V(G)\setminus (L(G)\cup\{v\})$. Clearly, $v$ and each leaf are distinguished. Moreover, since $G$ is bipartite, all the codewords are distinguished (in other words, $G-\{v\}-L(G)$ is identifiable), unless there exists a 2-vertex component $w, w'$ in $G-\{v\}-L(G)$. One of them must be a central vertex adjacent to $v$, and the other, a support vertex. Assume that $w\in N(v)$ and let $w'$ be the support vertex in $G$. Now, we may just shift the codeword from $w'$ to the leaf adjacent to $w'$ in $G$. Code $C$ we get in this way is identifying in $G$ and $|C|=2n/3-1$, a contradiction. Hence, we may next assume that each central vertex is adjacent to a support vertex.
Let us now assume, that there exists a support vertex $u$ such that it has at least two adjacent central vertices $u_1$ and $u_2$. Then, we have $n/3$ central vertices but at most $n/3-1$ support vertices with exactly one adjacent central vertex. Hence, there exists a central vertex $v$ such that each support vertex adjacent to $v$ has also another adjacent central vertex as a neighbour.
Let us again choose $C'=V(G)\setminus (L(G)\cup\{v\})$. Notice that $|I(v)|\geq2$, thus $v$ and each leaf are distinguished. Moreover, we may apply the same argument on $G-\{v\}-L(G)$ as above and we may form $C$ in the same way. %
Again, $C$ is identifying in $G$ and it has cardinality $2n/3-1$, a contradiction.
Therefore, each support vertex is adjacent to exactly one central vertex and each central vertex is adjacent to exactly one support vertex. Finally we only need to show that no two support vertices are adjacent. If $u,w\in S(G)$, $d(u,w)=1$ and the central vertex adjacent to $u$ is $v$, then the code $G-\{v\}-L(G)$ is again identifying in $G$, unless there exists a component of size two in $G-\{v\}-L(G)$ but in that case we can apply the earlier argument of shifting a codeword from a support vertex to a leaf to get an identifying code.
This means that $(G-L(G)-S(G))\circ_2=G$ and the claim follows.\end{proof}
\section{Graphs of girth at least~5}\label{sec:girth5}
In this section we prove our upper bound for graphs of girth at least~5, which generalizes Theorem~\ref{The:ID5/7delta}. It is natural to study these graphs here since bipartite graphs of girth at least $5$ are contained in Theorem~\ref{TheShift} and trees can be considered as graphs with unbounded girth. Notice that the new upper bound is tight for $C_7$ and stars. Moreover, it is the best possible bound, using only the order of graph $n$ and the number of leaves $\ell(G)$, in the sense that every non-leaf vertex has to increase the upper bound by $5/7$ as witnessed by the graph $C_7$ and each leaf has to increase the upper bound by $1$ as we have seen in the case of star graphs. Besides a new upper bound, we also present a new infinite family of graphs of girth at least $5$ which has large identifying codes.
\begin{theorem}\label{The:ID5/7}
Let $G$ be an identifiable graph of girth at least $5$ without isolated vertices. Then $\gamma^{\text{\tiny{ID}}}(G)\leq \frac{5n+2\ell(G)}{7}$
\end{theorem}
\begin{proof}
It is sufficent to prove the claim for connected graphs as each connected component can be considered independently. Note that a graph of girth at least~5 is identifiable if and only if no connected component is a $P_2$. Thus, let $G$ be a connected identifiable graph on $n$ vertices of girth at least $5$ without isolated vertices.
We prove the claim by induction. Note that the upper bound can be written as $(n-\ell(G))5/7+\ell(G)$, which will be used in this way in the proof. Assume first that $n=3$ ($n\geq 3$ since $G$ is identifiable and is not an isolated vertex). In this case, $G$ is $P_3$ and we have $\gamma^{\text{\tiny{ID}}}(G)=2$. Let us then assume that the claim holds for all $n$ with $n\leq k$ and let us consider $n=k+1$. Observe that if $G$ has minimum degree at least~$2$ or if $G$ is bipartite, then we are done by Theorems~\ref{TheShift} and~\ref{The:ID5/7delta}. Hence, we may assume that there exists a vertex $v\in V(G)$ such that $v$ belongs to a cycle or there are two disjoint paths from $v$ to two cycles, and there is a cut-edge $vu$ such that $G-vu$ is disconnected and one of the components is a tree $T_u$ on $n_u\geq 1$ vertices which contains vertex $u$. We will perform a case analysis based on the structure of this tree and the surroundings of $v$. The basic idea is to apply Theorem~\ref{TheShift} on $T_u$ and use the induction hypothesis on $G_u=G-T_u$. Let us denote by $T_v$ the tree $G[V(T_u)\cup\{v\}]$, $G_v=G-T_v$ and let $C_u$ and $C_v$ be an optimal identifying code in $G_u$ and $G_v$, respectively. We denote the graphs' orders by $n_u$ and $n_v$, respectively.
We will use the following observations throughout this proof. (i) If a non-codeword is dominated by two codewords, then it has a unique $I$-set, since $G$ has girth at least~5. Indeed, if any other vertex has those two codewords in its neighbourhood, then we have a 4-cycle. (ii) Similarly, we notice that if a codeword has three or more vertices in its $I$-set, then this $I$-set is unique.
We will next consider several cases.
\medskip
\noindent\textbf{Case 1.} Let us assume that $T_u=K_1$, that is, it is a single leaf-vertex $u$ attached to $v$. Moreover, let us assume that $v\not\in C_u$. We have one vertex less and one leaf less, thus, by the induction hypothesis, $|C_u|\leq (n-\ell(G))5/7+\ell(G)-1$ and hence, $C= C_u\cup\{u\}$ has cardinality at most $ (n-\ell(G))5/7+\ell(G)$. Moreover, $C$ is an identifying code since $I(u)=\{u\}$ and if $I(x)=\{u\}$, then $x$ is not dominated by $C_u$, a contradiction. If $v\in C_u$ and $I(v)\neq\{v\}$, then we may again consider code $C=C_u\cup\{u\}$. If $I(v)=\{v\}$, then $v$ has a non-codeword neighbour $v'$ in $G_u$. Now, instead of $u$, we add $v'$ to code $C_u$, that is, $C=C_u\cup\{v'\}$. Observe that since $C_u$ is an identifying code in $G_u$, we have $|I(v')|\geq3$ in $G$ and thus, $u$ is the only vertex with $I(u)=\{v\}$ and thus it is separated from all other vertices. All other vertex pairs are separated by the vertices of $C_u$.
\medskip
\noindent\textbf{Case 2.} Let us now assume that $T_u$ is a star $K_{1,n_u-1}$ where $n_u\geq3$. Let us assume first that $u$ is the central vertex of the star. In this case, $\ell(G)=\ell(G_u)+n_u-1$. Thus, we may just add all the $n_u-1$ leaves of $T_u$ to $C_u$ and the resulting code is an identifying code of the claimed cardinality. If $u$ is one of the leaves and $u'$ is the central vertex, then $\ell(G)=\ell(G_u)+n_u-2$. However, we also increase the number of non-leaf vertices by 2 when we transform $G_u$ into $G$ and hence, we may add $\lfloor n_u-2+10/7\rfloor$, that is $n_u-1=\ell(T_u)$, codewords to $C_u$. If $n_u\geq4$ ($T_u$ is not a path), then we may add to $C_u$ each vertex in $V(T_u)$ except $u$ to form an identifying code. If $n_u=3$, then $T_u$ is the path $P_3$. If $v\in C_u$, then we add vertices $u$ and $u'$ to $C_u$ and if $v\not\in C_u$, then we add the two leaves of $T_u$ to $C_u$. In both cases we obtain an identifying code of the claimed size.
\medskip
\noindent\textbf{Case 3.} Let us now assume that $n_u-\ell(T_u)\geq2$ and $\gamma^{\text{\tiny{ID}}}(T_u)\leq\lfloor\frac{n_u-\ell(T_u)}{2}\rfloor+\ell(T_u)-1$. Let us denote by $C'_u$ the optimal identifying code in $T_u$. If $u\not\in C'_u$ or $v\not\in C_u$, then we may just consider $C=C_u\cup C'_u$ as the identifying code and we are done. If $v\in C_u$ and $u\in C'_u$, then we may be required to do some modifications to the code as we may have $I(v)=I(u)$. Since we have $\ell (G)\geq\ell(G_u)+\ell(T_u)-1$ depending on whether $u\in L(T_u)$, we may just add some codeword $u'$ adjacent to $u$ and we get an identifying code $C_u\cup C'_u\cup\{u'\}$ of claimed cardinality. Indeed, we have
\begin{align*}
&5\frac{n-\ell(G)}{7}+\ell(G)\\
\geq&5\frac{(n-n_u)-\ell(G_u)+n_u+1-\ell(T_u)}{7}+\ell(G_u)+\ell(T_u)-1\\
\geq&|C_u|+\frac{n_u-\ell(T_u)}{2}+\ell(T_u)-1+\frac{3(n_u-\ell(T_u))+10}{14}\\
>&|C_u|+|C_u'|+1.
\end{align*}
In the first inequality we use $\ell(G)\geq \ell(G_u)+\ell(T_u)-1$. We can do this since $5\frac{n-\ell(G)}{7}+\ell(G)=\frac{5n+2\ell(G)}{7}$. In the second inequality, we use the induction hypothesis with $|C_u|\leq 5\frac{n-n_u-\ell(G_u)}{7}+\ell(G_u)$. In the third inequality, we use our assumption $|C_u'|\leq \gamma^{\text{\tiny{ID}}}(T_u)\leq\lfloor\frac{n_u-\ell(T_u)}{2}\rfloor+\ell(T_u)-1$ and the assumption $n_u-\ell(T_u)\geq2$ to show that $\frac{3(n_u-\ell(T_u))+10}{14}>1.$ Hence, we may add the new codeword adjacent to $u$ and we are done.
\medskip
\noindent\textbf{Case 4.} Let us now assume that $n_u-\ell(T_u)\geq2$ and $\gamma^{\text{\tiny{ID}}}(T_u)=\lfloor\frac{n_u-\ell(T_u)}{2}\rfloor+\ell(T_u)$. Notice that in this case Theorem~\ref{TheShift} provides a tight bound and we may assume that $C'_u$ has the structure provided by the proof. In particular we may assume that for any non-codeword $w\in V(T_u)\setminus L(T_u)$, we have $|I(w)|\geq2$. Moreover, we notice that Theorem~\ref{TheShift} actually offers two identifying codes with cardinalities $\lfloor\frac{n_u-\ell(T_u)}{2}\rfloor+\ell(T_u)$ and $\lceil\frac{n_u-\ell(T_u)}{2}\rceil+\ell(T_u)$. Moreover, if a vertex $w$ is a non-codeword vertex in one these codes, then it is a codeword in the other one, and if $w$ is a codeword in both of them, then it has an adjacent codeword in at least one of these codes.
Let us assume first that $v\not\in C_u$ or $u\not\in C'_u$. Then we may consider the identifying code $C=C_u\cup C'_u$. Since $n_u-\ell(T_u)\geq2$, this code has the desired cardinality. Thus, we may assume from now on that $u\in C'_u$ and $v\in C_u$.
Assume then that $u\not\in L(T_u)$ and $n_u-\ell(T_u)$ is even. In this case Theorem~\ref{TheShift} offers two identifying codes with equal cardinalities and we may choose the code in which $u$ is either a non-codeword or has adjacent codeword(s) and we are done.
When $n_u-\ell(T_u)$ is odd, we have to do some calculations since the two codes have cardinalities $\frac{n_u-1-\ell(T_u)}{2}+\ell(T_u)$ and $\frac{n_u+1-\ell(T_u)}{2}+\ell(T_u)$, respectively. However, even the larger of these two codes is small enough. Indeed, $\frac{n_u+1-\ell(T_u)}{2}+\ell(T_u)+5\frac{n-n_u-\ell(G_u)}{7}+\ell(G_u)=5\frac{n-\ell(G)}{7}+\ell(G)+\frac{3(\ell(T_u)-n_u)+7}{14}.$ Since $n_u-\ell(T_u)$ is odd and at least 3, the last sum term is negative and hence, we may use either of the two codes also in this case. Notice that since $u\not\in L(T_u)$, we used $\ell(G)=\ell(G_u)+\ell(T_u)$.
Now, we are left with the case where $u\in L(T_u)$. Thus, $\ell(G)=\ell(G_u)+\ell(T_u)-1$. Moreover, we have $\lfloor\frac{n_u-\ell(T_u)}{2}\rfloor+\ell(T_u)\leq \lfloor5\frac{n_u+1-\ell(T_u)}{7}\rfloor+\ell(T_u)-1$ when $n_u-\ell(T_u)\geq2$. Thus, code $C_u\cup C'_u$ has the claimed cardinality. Let $u'$ be the support vertex adjacent to $u$ in $T_u$. If $u'\in C'_u$, then we may use $C_u\cup C'_u$ as our identifying code. Hence, let us assume that $u'\not\in C'_u$. Notice that if $u'$ has an adjacent leaf other than $u$, then by the construction of $C'_u$, $|I(u')|\geq3$ and we can shift the codeword in $u$ to $u'$. In fact, each neighbour of $u'$ is a codeword in $C'_u$ since the structure of $C'_u$ is as in the proof of Theorem $6$ and hence, we may assume from now on that $\deg(u')=2$. If we have $n_u-\ell(T_u)$ even, then there is also another identifying code of the same cardinality as $C'_u$ in $T_u$ by the proof of Theorem~\ref{TheShift}. Moreover, this other code will have $u'$ as a codeword.
Let us assume from now on that $n_u-\ell(T_u)$ is odd. Consider the tree $T'=T_u-u$. We have $\ell(T_u)=\ell(T')$, since $\deg(u')=2$, and hence, $T'$ has an even number of non-leaves. If the optimal identifying code in $T'$ has cardinality $\frac{n_u-1-\ell(T_u)}{2}+\ell(T_u)$, then there exist two codes of this size and in one of the codes, let us say in $C_{T'}$, $u'$ is a codeword. Now $C_u\cup C_{T'}$ is an identifying code of $G$. On the other hand, if $T'$ has a smaller identifying code $C'_{T'}$, then $C_u\cup C'_{T'}\cup\{u'\}$ is an identifying code of $G$ of the correct size.
\medskip
\noindent\textbf{Case 5.} We are left with the case where $T_u$ is a path on two vertices. By applying the previous cases if possible, we may assume that each leaf of $G$ is part of a $P_2$ which is joined with a single edge to some vertex similar to $v$, belonging to a cycle or connected to at least two cycles. In particular, every support vertex has degree~2. Since $P_2$ is not identifiable, $C'_u$ does not exist. Let $u'$ be the leaf neighbour of $u$ in $T_u$.
Observe that if $v\in C_u$, then we can consider the code $C_u\cup\{u'\}$, which is identifying and of the correct size. Thus, we assume that $v\not\in C_u$. In particular, if multiple $P_2$'s are connected to $v$, $v$ is forced to be a codeword in $C_u$ to separate a leaf and its support vertex, and we are done. Hence, we may assume that there is at most one support vertex adjacent to $v$ in $G$. This implies that $G_v$ is identifiable.
If we have $\ell(G_v)<\ell(G)$, that is, $\ell(G_v)=\ell(G)-1$, then we may consider the graphs $G_v$ and the path on three vertices formed by $v,u$ and $u'$. If $C_v\cap N(v)=\emptyset$, then we may consider code $C=C_v\cup \{v,u'\}$ and if $C_v\cap N(v)\neq\emptyset$, then we may consider code $C=C_v\cup \{v,u\}$. Both of these codes have the claimed cardinality and they are identifying codes.
Let us then assume that $\ell(G_v)=\ell(G)$. In other words, exactly one vertex in $N_{G}(v)$ is a leaf in $G_v$, that is, it has degree~2 in $G$. Let us denote this vertex by $v_1$ and by $v'_1$ the second neighbour of $v_1$. If $\deg(v'_1)\geq3$, then $\ell(G_v-v_1)=\ell(G_v)-1=\ell(G)-1$. Let $C_{v_1}$ be an optimal identifying code in $G_v-v_1$ (note that this graph is identifiable). By induction we have $|C_{v_1}|\leq 5(n_v-1-\ell(G_v)+1)/7+\ell(G_v)-1=5(n-3-\ell(G))/7+\ell(G)-1$. We may now consider the code $C=C_{v_1}\cup \{v,u,u'\}$. This code has cardinality at most $5(n-3-\ell(G))/7+\ell(G)-1+3< 5(n-\ell(G))/7+\ell(G)$ and is an identifying code. Thus, we may assume that $\deg(v'_1)=2$. Moreover, let us denote by $C_{v'_1}$ the optimal identifying code in $G_{v'_1}=G_v-v_1-v'_1$ (note that this graph is also identifiable). Observe that $\ell(G_{v'_1})\leq\ell(G_v)$. We now consider the code $C=C_{v'_1}\cup\{u,v,v_1\}$, which is identifying. It has cardinality $|C|\leq 5(n-5-\ell(G_{v'_1}))/7+\ell(G_{v'_1})+3\leq 5(n-\ell(G_{v}))/7+\ell(G_{v})=5(n-\ell(G))/7+\ell(G)$.
From now on we may assume that in $G_v$ we have leaves $v_1$ and $v_2$ which are of degree~2 and adjacent to $v$ in $G$.
We know that $\deg(v)\geq3$. Let us construct graph $T^*_v$ by starting from $T_v$ and adding to $T_v$ vertex $v_1$, and iteratively, any vertex of degree~2 adjacent to the previous vertex. We do this until we reach a vertex (denoted by $w$) that does not have degree~2.
It is possible that $w=v$, if we have a suitable cycle. We denote by $P_t$ the path from $v_1$ to the leaf in $T^*_v$ that is adjacent to $w$ in $G$. Let $P_t$ be the path on vertices $x_1,x_2,\dots, x_t$ where consecutive vertices are adjacent and $x_1=v_1$, and let us denote graph $G_P=G-P_t$ with an optimal identifying code $C_P$ (note that this graph is identifiable). Notice that $|V(G_P)|\geq3$ since it at least contains vertices $v, u$ and $u'$. Moreover, if $|V(G_P)|=3$, then $G$ consists of a cycle, leaf and a support vertex. Now we may choose some optimal identifying code for the cycle so that $v$ is a codeword which gives us a case which we have already considered. Hence, $|V(G_P)|>3$ and $\ell(G_P)=\ell(G)$.
Let us further split this case into subcases, based on the value of $t$. Assume first that $t\geq5$. Recall that $\gamma^{\text{\tiny{ID}}}(P_t)=\lceil (t+1)/2\rceil$ since $t\geq5$~\cite{BCHL04}. Notice that $v\in C_P$ as it is the only vertex which can separate $u$ and $u'$. Assume first that $t$ is even. In that case $|C_P|+\gamma^{\text{\tiny{ID}}}(P_t)\leq 5(n-t-\ell(G_P))/7+\ell(G_P)+t/2+1=5(n-\ell(G))/7+\ell(G)+(14-3t)/14$. We will use code $\{x_2,x_4,\dots,x_{t-2},x_{t-1},x_t\}$ for the path. Together with $C_P$ it has the claimed cardinality and is identifying in $G$ (notice that $I(x_1)=\{v,x_2\}$).
Let us then consider the case where $t\geq5$ is odd. As above, we get that $|C_P|+\gamma^{\text{\tiny{ID}}}(P_t)\leq 5(n-t-\ell(G_P))/7+\ell(G_P)+(t+1)/2=5(n-\ell(G))/7+\ell(G)+(7-3t)/14$. We may use set $\{x_2,x_4,\dots,x_{t-3},x_{t-2},x_{t-1}\}$ together with $C_P$. Therefore, we may now assume that $t\leq4$.
When $t=4$, we can add two codewords to $C_P$ to form a code in $G$ since $4\cdot5/7\geq2$ and $\ell(G)=\ell(G_P)$. Recall that $v\in C_P$ and that $x_1=v_1$. If $w\not\in C_P$, we add $x_2,x_4$ on $P_t$ to $C_P$. If $w\in C_P$, then, if $u'\in C_P$, we shift it to $u$ and after that add codewords $x_1,x_3$. Notice that to dominate $u'$ at least one of $u$ and $u'$ is a codeword of $C_P$.
When $t=3$, we can again add two codewords since $3\cdot 5/7\geq2$. The codewords we add are $x_1$ and $x_2$.
When $t=2$, we can add at most one codeword. Again we shift the codeword possibly in $u'$ to $u$ and after that add vertex $x_1$ as a codeword.
Therefore, we are left with the case where $t=1$. Consider the graph $G^*=G-v_1-u'$ (which is identifiable) with optimal identifying code $C^*$. We have $\ell(G)=\ell(G^*)$ since $v_1$ has degree~2 and has no adjacent vertices of degree~2. Hence, if we can construct an identifying code $C$ for graph $G$ by adding at most one codeword to $C^*$, then $C$ has the claimed cardinality. Observe that at least one of $u$ and $v$ belongs to code $C^*$ to dominate $u$. Moreover, we have $|I(v)|\geq2$ as one vertex is needed to separate $u,v$.
Let us assume first that $v\not\in C^*$ and $u\in C^*$. In that case, we can consider code $C^*\cup\{v\}$ for $G$. Now $I(u')=\{u\}$ and if $I(v_1)=I(z)$, then $v\in I(z)$ and hence, $|I(z)|\geq2$ since $z$ was dominated by $C^*$. Since $z$ is $2$-dominated, $v_1$ and $z$ are separated. Thus, $C$ is an identifying code.
Consider then the case with $v,u\in C^*$. In this case, we consider code $C=C^*\cup\{v_1\}$. Clearly $I(u')$ is unique and $v_1$ and $v$ are the only vertices with $v_1$ and $v$ in their $I$-sets. Since $u\in I(v)$, also $I(v_1)$ is unique. Hence, $C$ is an identifying code.
Finally, we are left with the case $v\in C^*$ and $u\not \in C^*$. We consider the code $C=C^*\cup\{u\}$. Again $I(u')$ is unique. Moreover, in $G^*$ vertex $u$ is the only vertex whose $I$-set is $\{v\}$. Thus, if $I(v_1)=\{v\}$, then $v_1$ is separated from every other vertex and if $|I(v_1)|\geq2$, then $I(v_1)$ is clearly unique. Thus, $C$ is an identifying code of claimed cardinality in $G$. As this was final case, we have now proven the claim.\end{proof}
In~\cite{BFH15}, the authors have constructed an infinite family of connected graphs without leaves which have girth at least~5 and $\gamma^{\text{\tiny{ID}}}(G)=3(n-1)/5$, where $n$ is the order. To date, this infinite family of graphs features the largest known ratio between $\gamma^{\text{\tiny{ID}}}(G)$ and the number of vertices, among graphs without leaves and with girth at least $5$ (apart from some small examples such as the $7$-cycle). The interest to these kinds of constructions is due to the fact that Theorem~\ref{The:ID5/7delta} is tight only for the $7$-cycle and so, perhaps there exists a way to improve the bound for connected graphs by excluding the $7$-cycle as a single exception. New constructions which increase the ratio $\gamma^{\text{\tiny{ID}}}(G)/n$ give new limits to how much the bound of Theorem~\ref{The:ID5/7delta} could possibly be improved. In the following proposition, we give a new infinite family of such graphs which offers the largest known ratio for $\gamma^{\text{\tiny{ID}}}(G)/n$.
\begin{figure}[h]
\centering
\begin{tikzpicture}
\draw[thick, dotted] (-0.5,6) -- (3.5,6);
\draw[thick, dotted] (0.5,4) -- (2.5,4);
\node[main node](x) at (0,8) {};
\node[code node](1) at (-4,7) {$x_1$};
\node[code node](2) at (-1,7) {$x_2$};
\node[code node](3) at (4,7) {$x_k$};
\node[code node](10) at (-4,6) {$v_1$};
\node[code node](11) at (-3,5) {};
\node[main node](12) at (-3,4) {};
\node[code node](13) at (-3,3) {};
\node[main node](16) at (-5,5) {};
\node[code node](15) at (-5,4) {};
\node[main node](14) at (-5,3) {};
\node[code node](20) at (-1,6) {$v_2$};
\node[code node](21) at (0,5) {};
\node[main node](22) at (0,4) {};
\node[code node](23) at (0,3) {};
\node[main node](26) at (-2,5) {};
\node[code node](25) at (-2,4) {};
\node[main node](24) at (-2,3) {};
\node[code node](30) at (4,6) {$v_k$};
\node[code node](31) at (3,5) {};
\node[main node](32) at (3,4) {};
\node[code node](33) at (3,3) {};
\node[main node](36) at (5,5) {};
\node[code node](35) at (5,4) {};
\node[main node](34) at (5,3) {};
\path[draw,thick]
(x) edge node {} (1)
(x) edge node {} (2)
(x) edge node {} (3)
(1) edge node {} (10)
(10) edge node {} (11)
(11) edge node {} (12)
(12) edge node {} (13)
(13) edge node {} (14)
(14) edge node {} (15)
(15) edge node {} (16)
(16) edge node {} (10)
(2) edge node {} (20)
(20) edge node {} (21)
(21) edge node {} (22)
(22) edge node {} (23)
(23) edge node {} (24)
(24) edge node {} (25)
(25) edge node {} (26)
(26) edge node {} (20)
(3) edge node {} (30)
(30) edge node {} (31)
(31) edge node {} (32)
(32) edge node {} (33)
(33) edge node {} (34)
(34) edge node {} (35)
(35) edge node {} (36)
(36) edge node {} (30)
;
\end{tikzpicture}\centering
\caption{Graph $G$ of girth $7$ with no leaves, on $8k+1$ vertices with $\gamma^{\text{\tiny{ID}}}(G)=5k$. Gray vertices form an optimal identifying code.}\label{5/8Example}
\end{figure}
\begin{proposition}\label{Prop5/8}
For each integer $k\geq1$, there exists a connected graph $G$ on $n=8k+1$ vertices with $\gamma^{\text{\tiny{ID}}}(G)=5k=\frac{5(n-1)}{8}$.
\end{proposition}
\begin{proof}
Let $k\geq1$ be an integer and let us construct graph $G$ by taking a star $K_{1,k}$ and by attaching a unique $7$-cycle to each leaf of the star with a single edge. The resulting graph has $k+1+7k=8k+1$ vertices. Let us denote the leaves of $K_{1,k}$ by $x_1,\dots, x_k$ and vertices in the $7$-cycles adjacent to them by $v_1,\dots, v_k$. Graph $G$ is illustrated in Figure \ref{5/8Example}.
Let us now consider the identifying code number of $G$. Let $C$ be an optimal identifying code. We claim that there are at least five code vertices among the eight vertices in $x_i$ and the $7$-cycle attached to it.
Recall that we have $\gamma^{\text{\tiny{ID}}}(C_7)=5$. Thus, if $x_i\not\in C$, then we have at least five code vertices in the $7$-cycle. Assume then that $x_i\in C$ and that there exists a set $C'$ of three vertices in the $7$-cycle such that $C'\cup\{x_i\}$ dominates and distinguishes all of these vertices. Let $v_i$ be the vertex in the $7$-cycle $C_7$ adjacent to $x_i$. If $I(v_i)=\{x_i\}$, then $C'$ is an identifying code for $C_7-v_i$. However, $C_7-v_i$ is a $6$-path $P_6$ and we have $\gamma^{\text{\tiny{ID}}}(P_6)=4$, a contradiction. Hence, $C'$ is a dominating set of $C_7$. Since $C'\cup\{x_i\}$ distinguishes every vertex in $C_7$ and $\gamma^{\text{\tiny{ID}}}(C_7)=5$, we have $N[v_i]\cap C'=N[w]\cap C'$ for some $w\in V(C_7)$ and there are no other vertices with the same $I$-sets. Let $u$ be a vertex in $N[w]\setminus N[v_i]$. Now $C'\cup \{u\}$ is an identifying code of size $4$ in $C_7$, a contradiction. Thus, $\gamma^{\text{\tiny{ID}}}(G)\geq 5k$.
Finally, we show that $\gamma^{\text{\tiny{ID}}}(G)\leq 5k$. To construct an identifying code $C$, we choose vertices $x_i$ and $v_i$ for each $i$, and for each vertex $v_i$ we choose one adjacent vertex $w_i$ in the cycle and then we choose two additional code vertices $u_1$ and $u_2$ in each cycle so that $I(u_i)=\{u_i\}$. This code is depicted with gray vertices in Figure~\ref{5/8Example}. One can easily check that the code is indeed identifying.
\end{proof}
\section{Concluding remarks}\label{sec:conclu}
We have improved several bounds from the literature, both in terms of the values of the bounds, and/or in terms of the generality of the considered graph classes. Our bounds confirm the known facts that certain structural graph features such as leaves, twins or short cycles are crucial for a graph to have a large identifying code. By considering the number of leaves on graphs other than trees, our bounds enable us to quantify the effect of these structures on the identifying code number.
In Section \ref{SecBounds}, we have given a new tight upper bound $\gamma^{\text{\tiny{ID}}}(G)\leq (n+\ell(G))/2$ for bipartite graphs without twins of degree two or greater. We have characterized all twin-free graphs attaining this bound. However, it would be interesting to see a characterization for all graphs attaining the bound.
Our bound $\gamma^{\text{\tiny{ID}}}(G)\leq\frac{5n+2\ell(G)}{7}$ from Theorem~\ref{The:ID5/7} for graphs of girth at least~5 is tight for stars, the path $P_4$ and the 7-cycle. However, we do not know of any other tight examples. Perhaps this bound can be extended by considering other structural properties of the graph, to give a bound that is tight for a more diverse class of graphs. Perhaps it can also be improved by excluding the 7-cycle as an exception? We have shown in Proposition~\ref{Prop5/8} that there are arbitrarily large connected twin-free graphs of girth~7 without leaves with $\gamma^{\text{\tiny{ID}}}(G)=5(n-1)/8$, hence such an improved bound could not be less than that.
More generally, it would be interesting to see whether other bounds can be proved for graphs of larger girth. For example, perhaps a stronger version of the bound $\gamma^{\text{\tiny{ID}}}(G)\leq\frac{5n+2\ell(G)}{7}$ (for graphs of girth at least $5$) of Theorem~\ref{The:ID5/7} can be proved for graphs of girth at least~$g$ with $g\geq 9$. As we have $\gamma^{\text{\tiny{ID}}}(C_g)=(g+1)/2+1$ for odd $g$, such an upper bound cannot be less than that.
| {
"timestamp": "2022-04-12T02:47:25",
"yymm": "2204",
"arxiv_id": "2204.05250",
"language": "en",
"url": "https://arxiv.org/abs/2204.05250",
"abstract": "An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\\ell)/2$, where $n$ is the order and $\\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $\\frac{5n+2\\ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars.",
"subjects": "Combinatorics (math.CO)",
"title": "Revisiting and improving upper bounds for identifying codes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795095031688,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.7092379530074165
} |
https://arxiv.org/abs/1606.01443 | Filters in the partition lattice | Given a filter $\Delta$ in the poset of compositions of $n$, we form the filter $\Pi^{*}_{\Delta}$ in the partition lattice. We determine all the reduced homology groups of the order complex of $\Pi^{*}_{\Delta}$ as ${\mathfrak S}_{n-1}$-modules in terms of the reduced homology groups of the simplicial complex $\Delta$ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank--Hanlon--Robinson and Wachs on the $d$-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes $a$ or~$b$. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression $a, a + d, \ldots, a + (a-1) \cdot d$, extending work of Browdy. | \section{Introduction}
\label{section_introduction}
In his physics dissertation
Sylvester~\cite{Sylvester} considered
the even partition lattice,
that is, the poset of all set partitions where the blocks have
even size.
He computed the M\"obius function of this lattice and
showed that it equals, up to a sign, the tangent number.
Stanley then introduced the $d$-divisible partition lattice. This is
the collection of all set partitions with blocks having size divisible by $d$, denoted by~$\Pi_{n}^d$.
He showed that the M\"obius function is, up to a sign,
the number of permutations in
the symmetric group~${\mathfrak S}_{n-1}$
with descent set $\{d,2d, \ldots, n-d\}$;
see~\cite{exponential_structures}.
Calderbank, Hanlon and Robinson~\cite{Calderbank_Hanlon_Robinson}
continued this work by studying the top
homology group of
the order complex~$\triangle(\Pi_{n}^{d} - \{\hat{1}\})$
and gave an explicit description of
the ${\mathfrak S}_{n-1}$-action on this homology group
in terms of a Specht module.
However, they were unable to obtain the other homology
groups and asked Wachs if it was possible that the complex
$\triangle(\Pi_{n}^{d} - \{\hat{1}\})$ was shellable, which
would imply that the other homology groups are trivial.
Wachs~\cite{Wachs}
proved that this was indeed the case by showing
that the poset $\Pi_{n}^{d} \cup \{\hat{0}\}$ is
$EL$-shellable,
and thus the homotopy type
of the complex
$\triangle(\Pi_{n}^{d} - \{\hat{1}\})$ is a wedge
of spheres of the same dimension.
Additionally, Wachs gave a different proof for the ${\mathfrak S}_{n-1}$-action
on the top homology of~$\Pi_{n}^d$,
as well as matrices for the action of~${\mathfrak S}_{n}$ on this homology.
Ehrenborg and Jung~\cite{Ehrenborg_Jung}
further generalized the $d$-divisible partition lattice by defining
a subposet~$\Pi^{*}_{\vec{c}}$ of the partition lattice for
a composition $\vec{c}$ of~$n$. The subposet
reduces to the $d$-divisible partition lattice when the composition $\vec{c}$
is given by $\vec{c}=(d,d,\dots,d)$.
Their work consists of three main results.
First, they showed that the M\"obius function of $\Pi^{*}_{\vec{c}} \cup \{\hat{0}\}$
equals, up to a given sign,
the number of
permutations in
${\mathfrak S}_{n}$ ending with the element $n$
having descent composition~$\vec{c}$.
Second, they showed that the order complex
$\triangle(\Pi^{*}_{\vec{c}} - \{\hat{1}\})$ is
homotopy equivalent to
a wedge of
spheres of the same dimension. Lastly,
they proved that the action of ${\mathfrak S}_{n-1}$
on the top homology group of $\triangle(\Pi^{*}_{\vec{c}} - \{\hat{1}\})$ is given by
the Specht module corresponding to the composition~$\vec{c}-1$.
In the current paper we continue this research program
by considering a
more general class of filters in the partition lattice.
Let~$\Delta$ be a filter in the poset of compositions.
Since the poset of compositions is isomorphic to
a Boolean algebra, the filter~$\Delta$ under the reverse
order is a lower order ideal and hence can be viewed
as the face poset of a simplicial complex.
We define the associated filter $\Pi^{*}_{\Delta}$ in the partition
lattice. This extends the definition of~$\Pi^{*}_{\vec{c}}$.
In fact, when $\Delta$ is a simplex generated
by the composition~$\vec{c}$ the two definitions agree.
Our main result is that we can determine all the reduced
homology groups of the order complex
$\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
in terms of the reduced homology groups of links
in $\Delta$ and of Specht modules
of border shapes;
see Theorem~\ref{theorem_main_theorem_order_complex}.
The proof proceeds by
showing that if the result holds
for the two complexes~$\Delta$, $\Gamma$
and also for their intersection $\Delta \cap \Gamma$,
then it holds for their union $\Delta \cup \Gamma$.
Furthermore, the proof relies on
Mayer--Vietoris sequences
to construct the isomorphism of
Theorem~\ref{theorem_main_result}.
As our main tool, we use Quillen's fiber lemma to translate
topological data
from the filter~$Q^{*}_{\Delta}$
to the filter~$\Pi^{*}_{\Delta}$.
We also present a second proof of our main result,
Theorem~\ref{theorem_main_result},
using an equivariant poset fiber theorem of
Bj\"orner, Wachs and Welker~\cite{Bjorner_Wachs_Welker}.
Even though this approach is concise, it does not yield
an explicit construction of the isomorphism of
Theorem~\ref{theorem_main_result}.
In particular, our hands on approach using Mayer--Vietoris
sequences reveals how the homology groups of
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$ are changing as
the complex $\Delta$ is built up.
Once again, the Ehrenborg--Jung
result on $Q^{*}_{\vec{c}}$
is needed
to apply the poset fiber theorem.
Our main result yields explicit expressions
for the reduced homology groups of the complex
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$,
most notably when $\Delta$ is homeomorphic to a ball
or to a sphere.
The same holds when $\Delta$ is a shellable complex.
We are able to describe
the homotopy type of the order complex
$\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
using the homotopy fiber theorem
of~\cite{Bjorner_Wachs_Welker}.
Again, when~$\Delta$ is homeomorphic to a ball
or to a sphere, we obtain that $\Pi^{*}_{\Delta}$ is
a wedge of spheres.
We are also able to lift discrete Morse matchings from~$\Delta$
and its links to form a discrete Morse matching
on the filter of ordered set partitions~$Q^{*}_{\Delta}$.
In Sections~\ref{section_examples}
through~\ref{section_partition_filter_a_b}
we give a plethora of examples of our results.
We consider the case when the complex~$\Delta$
is generated by a knapsack partition
to obtain a previous result of
Ehrenborg and Jung.
In Section~\ref{section_Frobenius}
we study the case when
$\Lambda$ is a semigroup of positive integers
and we consider the filter of partitions whose
block sizes belong to the semigroup~$\Lambda$.
When $\Lambda$ is generated by
the arithmetic progression
$a, a+d, a+2d, \ldots$ we are able to describe
the reduced homology groups of the associated
filter in the partition lattice.
The particular case when $d$ divides $a$
was studied by Browdy~\cite{Browdy},
where the filter $\Lambda$ consists of partitions whose block sizes
are divisible by $d$ and are greater than or equal to~$a$.
Finally, in Section~\ref{section_partition_filter_a_b}
we study the filter
corresponding to the semigroup generated by
two relative prime integers.
Here we are able to give explicit results for
the top and bottom reduced homology groups.
Other previous work in this area is due to
Bj\"orner and Wachs~\cite{Bjorner_Wachs_non_pure_I}.
Additionally, Sundaram
studied the subposet of the partition
lattice defined by a set of forbidden block sizes
using plethysm and the Hopf trace formula;
see~\cite{Sundaram_Hopf,Sundaram}.
We end the paper by posing questions for further study.
\section{Integer and set partitions}
\label{section_integer_set_partitions}
We define an integer partition $\lambda$ to be a finite multiset of
positive integers. Thus
the multiset
$\lambda = \{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}\}$
is a partition of $n$ if
$\lambda_{1} + \lambda_{2} + \cdots + \lambda_{k} = n$.
Sometimes it will be necessary to consider the
multiplicity of the elements of the partition~$\lambda$.
We then write
$$ \lambda
=
\{\lambda_{1}^{m_{1}}, \lambda_{2}^{m_{2}},
\ldots, \lambda_{p}^{m_{p}}\}
, $$
where we tacitly assume that $\lambda_{i} \neq \lambda_{j}$
for two different indices $i \neq j$.
Let $I_{n}$ be the set of all integer partitions of $n$.
We form a poset on these integer partitions where
the cover relation is given by adding two parts.
In terms of multisets the cover relation is
$$
\{\lambda_{1}, \lambda_{2},\lambda_{3}, \ldots, \lambda_{k}\}
\prec
\{\lambda_{1} + \lambda_{2},\lambda_{3}, \ldots, \lambda_{k}\} . $$
Note that the partition $\{1,1, \ldots, 1\}$ is the minimal element
and $\{n\}$ is the maximal element in the partial order.
Let $\Pi_{n}$ denote the poset of all set partitions of
$[n] = \{1,2, \ldots, n\}$
where the partial order is given by merging blocks,
that is,
$$ \{B_{1}, B_{2}, B_{3}, \ldots, B_{k}\}
\prec
\{B_{1} \cup B_{2}, B_{3}, \ldots, B_{k}\} . $$
The poset $\Pi_{n}$ is in fact a lattice,
called the partition lattice.
Let $|\pi|$ denote the number of blocks of the partition~$\pi$.
Furthermore, for a set partition
$\pi = \{B_{1}, B_{2}, \ldots, B_{k}\}$
define its type to be
the integer partition of $n$ given by the multiset
$\type(\pi) = \{|B_{1}|, |B_{2}|, \ldots, |B_{k}|\}$.
The symmetric group ${\mathfrak S}_{n}$ acts on
subsets of $[n]$ by relabeling the elements.
Similarly, the symmetric group ${\mathfrak S}_{n}$ acts on
the partition lattice by relabeling the elements of the blocks.
For $\pi = \{B_{1}, B_{2}, \ldots, B_{k}\}$ a set partition
the action is given by
$\alpha \cdot \pi = \{\alpha(B_{1}), \alpha(B_{2}), \ldots, \alpha(B_{k})\}$.
Finally, when we speak about the action of the symmetric
group ${\mathfrak S}_{n-1}$, we view
the group ${\mathfrak S}_{n-1}$ as the subgroup
$\{\alpha \in {\mathfrak S}_{n} \: : \: \alpha_{n} = n\}$
of the symmetric group~${\mathfrak S}_{n}$.
\section{Compositions and ordered set partitions}
\label{section_compositions_ordered_sets}
A composition
$\vec{c} = (c_{1}, c_{2}, \ldots,c_{k})$ of $n$
is an ordered list of positive integers such that
$c_{1} + c_{2} + \cdots + c_{k} = n$.
Let $\Comp(n)$ be the set of all compositions of $n$.
We make $\Comp(n)$ into a poset by
introducing the cover relation
given by adding adjacent entries, that is,
$$ (c_{1},\ldots,c_{i},c_{i+1},\ldots,c_{k})
\prec
(c_{1},\ldots,c_{i}+c_{i+1},\ldots,c_{k}) . $$
The poset $\Comp(n)$ is isomorphic to
the Boolean algebra on $n-1$ elements.
Note that
$(1,1, \ldots, 1)$ and $(n)$
are the minimal and maximal elements of $\Comp(n)$, respectively.
Define the type of a composition
$\vec{c} = (c_{1}, c_{2}, \ldots, c_{k})$
to be the integer partition
$\type(\vec{c}\,) = \{c_{1}, c_{2}, \ldots,c_{k}\}$ of~$n$.
Furthermore, let $|\vec{c}\,|$ denote the number of parts
of the composition~$\vec{c}$.
For a composition
$\vec{c} = (c_{1}, c_{2}, \ldots, c_{k})$
of $n$, the {\em multinomial coefficient}
is given by
$$ \binom{n}{\vec{c}}
=
\binom{n}{c_{1}, c_{2}, \ldots, c_{k}}
=
\frac{n!}{c_{1}! \cdot c_{2}! \cdots c_{k}!} . $$
For $\alpha\in{\mathfrak S}_{n}$, let the descent set of $\alpha$,
denoted by $\Des(\alpha)$, be the subset of $[n-1]$
given by $\Des(\alpha)=\{i\in[n-1]\: : \: \alpha(i)>\alpha(i+1)\}$.
Throughout this paper it will be more
convenient to consider $\Des(\alpha)$ as a composition of $n$,
namely,
if $\Des(\alpha)=\{i_{1} < i_{2} < \cdots < i_{k}\}$,
then we consider $\Des(\alpha)$ as a composition of $n$
given by $\Des(\alpha)=(i_{1},i_{2}-i_{1}, \ldots, i_{k}-i_{k-1},n-i_{k})$.
Note that the identity permutation
$(1,2,\ldots,n)$ has descent composition~$(n)$.
Let $\beta_{n}(\vec{c}\,)$ be the number of permutations
$\alpha$ in ${\mathfrak S}_{n}$ such that $\Des(\alpha)=\vec{c}$.
Likewise, define~$\beta_{n}^{*}(\vec{c}\,)$
to be the number of permutations $\alpha$ in ${\mathfrak S}_{n}$ with
descent composition $\vec{c}$ and $\alpha(n)=n$.
Observe that
\begin{equation}
\binom{n-1}{c_{1}, \ldots, c_{k-1}, c_{k}-1}
=
\sum_{\onethingatopanother
{\vec{d} \in \Comp(n)}
{\vec{c}\,\leq\vec{d}}}
\beta_{n}^{*}(\vec{d}\,)
.
\label{equation_beta_star_binomial}
\end{equation}
An {\em ordered set partition} $\sigma = (C_{1}, C_{2}, \ldots, C_{p})$
of~$[n]$ is a
list of non-empty blocks such that
the set
$\{C_{1}, C_{2}, \ldots, C_{p}\}$ is a partition of the set $[n]$,
where the order of the blocks now matters.
Let $|\sigma|$ denote the number of blocks
in the ordered set partition~$\sigma$.
Let $Q_{n}$ be the set of all ordered set partitions on the set $[n]$.
Introduce a partial order on~$Q_{n}$ where the cover relation
is joining adjacent blocks, that is,
$$
(C_{1}, \ldots, C_{i}, C_{i+1}, \ldots, C_{p})
\prec
(C_{1}, \ldots, C_{i} \cup C_{i+1}, \ldots, C_{p}) .
$$
Observe that the poset~$Q_{n}$ has the maximal element~$([n])$,
along with $n!$ minimal elements,
namely the ordered set partitions
$(\{\alpha_{1}\}, \{\alpha_{2}\}, \ldots, \{\alpha_{n}\})$, one for each
permutation $\alpha_{1} \alpha_{2} \cdots \alpha_{n} \in {\mathfrak S}_{n}$.
Moreover, every interval in~$Q_{n}$
is isomorphic to a Boolean algebra.
Define the {\em type} of an ordered set partition
$\sigma = (C_{1}, C_{2}, \ldots, C_{k})$
to be the composition of~$n$ given
by $\type(\sigma) = (|C_{1}|, |C_{2}|, \ldots, |C_{k}|)$.
\begin{definition}
\label{definition_sigma}
For a permutation $\alpha \in {\mathfrak S}_{n}$
and a composition
$\vec{d} = (d_{1}, d_{2}, \ldots, d_{k})$ of~$n$,
let~$\sigma(\alpha,\vec{d}\,)$ denote
the unique ordered set partition in $Q_{n}$
of type~$\vec{d}$ whose elements are given,
in order, by the permutation~$\alpha$,
that is,
$$
\sigma(\alpha,\vec{d}\,)
=
(\{\alpha(1), \ldots, \alpha(d_{1})\},
\{\alpha(d_{1}+1), \ldots, \alpha(d_{2})\},
\ldots,
\{\alpha(d_{k-1}+1), \ldots, \alpha(n)\})
.
$$
\end{definition}
Finally, the symmetric group ${\mathfrak S}_{n}$ acts on
ordered set partitions by relabeling, that is
$$ \alpha \cdot (C_{1}, C_{2}, \ldots, C_{k})
=
(\alpha(C_{1}), \alpha(C_{2}), \ldots, \alpha(C_{1})) . $$
\section{Topological considerations}
\label{section_topological_considerations}
Let $P$ be a poset.
Recall the order complex of $P$, denoted~$\triangle(P)$,
is the simplicial
complex whose $i$-dimensional faces
are the chains in $P$ with $i+1$ elements.
If $P$ has a minimal element~$\hat{0}$
or a maximal element~$\hat{1}$,
then $\triangle(P)$ is a contractible complex.
Thus we will be removing these elements to
ensure interesting topology.
Recall a simplicial complex~$\Delta$
is a finite collection of sets such that
the empty set belongs to~$\Delta$
and
$\Delta$~is closed under inclusion.
We will find it easier to view a simplicial complex
as a partially ordered set $\Delta$ such that
(i) $\Delta$ has a unique minimal element~$\hat{0}$
and
(ii) every interval $[\hat{0},x]$ for $x \in \Delta$ is
isomorphic to a Boolean algebra.
A poset $P$ satisfying these conditions is called
a \emph{simplicial poset}.
Notice that a poset $P$ is simplicial if $P$ is the face poset
of a simplicial complex.
Furthermore, note that the second condition in the
definition of a simplicial poset makes the poset~$\Delta$ ranked
since every saturated chain between the minimal element~$\hat{0}$
and an element~$x$ has the same length.
Thus the {\em dimension} of an element $x$ is defined
by its rank minus one, that is,
$\dim(x) = \rho(x) - 1$.
A filter in a poset $P$ is an upper order ideal.
Hence if $F$ is a filter in $P$,
then the dual filter~$F^{*}$ in the dual poset~$P^{*}$
is now a lower order ideal.
In particular,
if $\Delta \subseteq \Comp(n)$ is a filter,
since upper order ideals in~$\Comp(n)$
are isomorphic to Boolean algebras,
the dual of $\Delta$ is a simplicial poset
in the dual space~$\Comp(n)^{*}$, which
has cover relation given by splitting rather than merging.
To emphasize that we have dualized, we use~$\leq^{*}$
to denote the order relation in the dualized~$\Comp(n)$.
Lastly, the {\em link} of a face $F$ in
a simplicial complex $\Delta$ is given by
$\link_{F}(\Delta) = \{G \in \Delta \: : \: F \cup G \in \Delta, \:
F \cap G = \emptyset\}$.
However, working with the poset definition of a simplicial
complex, we have the following equivalent definition of the link.
The link is the principle filter generated by the face~$x$,
that is,
$\link_{x}(\Delta) = \{y \in \Delta \: : \: x \leq y\}$.
One advantage of this definition is that we do not have
to relabel the faces when considering the link.
From now on our simplicial complex~$\Delta$
will be a filter in the composition lattice~$\Comp(n)$,
with the dual order~$\leq^{*}$.
Let $C_{k}(\Comp(n))$ be the linear span over~$\mathbb{C}$ of
all compositions of~$n$ into $k+2$ parts.
We obtain a chain complex by defining the
boundary map as follows.
Define the map
$\partial_{k,j}:C_{k}(\Comp(n)) \longrightarrow C_{k-1}(\Comp(n))$ by
$$ \partial_{k,j}(c_{1},\ldots,c_{j},c_{j+1},\ldots,c_{k+2})
=
(c_{1},\ldots,c_{j}+c_{j+1},\ldots,c_{k+2}) . $$
Then the boundary map on $\Comp(n)$ is given by
$\partial_{k} = \sum_{j=1}^{k+1} (-1)^{j-1} \cdot \partial_{k,j}$
Consider the dual order on the set of ordered set partitions~$Q_{n}$.
For $\Delta \subseteq \Comp(n)$ a complex, let
$Q_{\Delta}=\{\tau \in Q_{n} \,:\, \type(\tau)\in\Delta\}$.
The filter $Q_{\Delta}$
is also a simplicial poset, so we refer to~$Q_{\Delta}$ as a complex.
Define $C_{k}(Q_{n})$ to be the linear span over~$\mathbb{C}$ of all ordered
set partitions of $[n]$ with $k+2$ blocks.
The boundary map
$\partial_{k} : C_{k}(Q_{n}) \longrightarrow C_{k-1}(Q_{n})$
on $Q_{n}$ is given by
$\partial_{k}(\sigma(\alpha,\vec{d}\,))
=
\sigma(\alpha,\partial_{k}(\vec{d}\,))$,
where $\partial_{k}(\vec{d}\,)$
is the boundary map applied to the composition $\vec{d}$
in $C_{k}(\Comp(n))$, and
where $\sigma(\alpha,\vec{c}\,)$
is given in Definition~\ref{definition_sigma}.
This boundary map is inherited by the subcomplex~$Q_{\Delta}$.
Finally, for simplicial complexes $\Delta$ and~$\Gamma$
in $\Comp(n)$ and $\Comp(m)$ respectively, their {\em join}
is defined to be poset
$$ \Delta * \Gamma
=
\{\vec{c} \circ \vec{d}
\: : \:
\vec{c} \in \Delta, \vec{d} \in \Gamma \} , $$
where $\circ$ denote the concatenation of compositions.
Note that the join $\Delta * \Gamma$ has
the composition~$(n,m)$ as its minimal element.
Furthermore, we have the following basic lemma
on Morse matchings of joins of complexes.
\begin{lemma}
\label{lemma_critical_cell_join}
Let $\Delta$ and $\Gamma$ be two complexes
in $\Comp(m)$ and $\Comp(m)$ respectively,
each having a discrete Morse matching.
Let~$\Delta^{c}$ and
$\Gamma^{c}$ be the sets of critical cells of $\Delta$ and~$\Gamma$, respectively.
Then the join~$\Delta * \Gamma$ has a Morse
matching where the critical cells are
$$
\{\vec{c} \circ \vec{d}
\: : \:
\vec{c} \in \Delta^{c}, \vec{d} \in \Gamma^{c} \} . $$
\label{lemma_join_critical_cells}
\end{lemma}
\begin{proof}
Define a matching of the join $\Delta * \Gamma$ as follows.
If $\vec{c} \prec \vec{c}\,^{\prime}$ is an edge in the discrete Morse
matching of~$\Delta$ and
$\vec{d} \in \Gamma$
then match
$\vec{c} \circ \vec{d} \prec \vec{c}\,^{\prime} \circ \vec{d}$.
If $\vec{c}$ is a critical cell of $\Delta$
and
$\vec{d} \prec \vec{d}\,^{\prime}$ is an edge in the discrete Morse
matching of $\Gamma$
then match
$\vec{c} \circ \vec{d} \prec \vec{c} \circ \vec{d}\,^{\prime}$.
It is straightforward to verify that this matching is acyclic
and that the set of critical cells is as described.
\end{proof}
\section{Border strips and Specht modules}
\label{section_border_strips}
A border strip $B$ is a connected skew-shape which does not contain
a two by two square.
For each composition $\vec{c} = (c_{1}, c_{2}, \ldots, c_{k})$
there is a unique border strip
such that the number of boxes in the $i$th row is given
by $c_{i}$ and every two adjacent rows overlap in
one position.
Denote this border strip by~$B(\vec{c}\,)$.
See Figure~\ref{figure_stabilizer} for an example.
In an analogous fashion, for a composition $\vec{c} \in \Comp(n)$
we define the border shape~$A(\vec{c}\,)$ to be the skew-shape
whose $i$th row has length~$c_i$ such that
the rows of $A(\vec{c}\,)$ are non-overlapping.
Let $R_{i}$ be the interval
$[c_{1} + \cdots + c_{i-1} + 1, c_{1} + \cdots + c_{i-1} + c_{i}]$.
The row stabilizer of the border strip $B(\vec{c}\,)$
is the subgroup
${\mathfrak S}_{R_{1}} \times {\mathfrak S}_{R_{2}} \times \cdots \times {\mathfrak S}_{R_{k}}$
of the symmetric group~${\mathfrak S}_{n}$.
Since the poset $\Comp(n)$ of all compositions of $n$ is
a isomorphic to Boolean algebra,
every composition has a complementary
composition $\vec{c}^{\,c}$.
To obtain the complement of composition
write every part of the composition as a sum of $1$s
where we separate the parts with commas.
Then the complement is obtained by
exchanging the plus signs and the commas.
Similarly, the column stabilizer is defined as
the row stabilizer of the border strip of the complementary
composition. More precisely, let
$(d_{1}, d_{2}, \ldots, d_{p})$
be the complementary composition~$\vec{c}\,^{c}$
and let
$K_{i}$ be the interval
$[d_{1} + \cdots + d_{i-1} + 1, d_{1} + \cdots + d_{i-1} + d_{i}]$.
Then the column stabilizer is the subgroup
${\mathfrak S}_{\vec{c}}^{C}
=
{\mathfrak S}_{K_{1}} \times {\mathfrak S}_{K_{2}} \times \cdots \times {\mathfrak S}_{K_{p}}$.
See Figure~\ref{figure_stabilizer}.
We now review some basic representation theory of the symmetric group. For a less terse introduction,
see~\cite[Chapter~3]{Sagan}.
A border strip tableau~$t$ of shape~$\vec{c}$
is a filling of the border strip~$B(\vec{c}\,)$.
We say a tableau $t$ is {\em standard} if the entries of~$t$
are increasing along the rows from left to right
and increasing down the columns.
A border strip tabloid, denoted~$[t]$,
is a border strip tableau under row equivalence.
Define the permutation module, $M^{B(\vec{c}\,)}$,
to be the vector space with basis elements given
by all tabloids of shape~$B(\vec{c}\,)$.
A polytabloid is defined
by the alternating sum
$e_{t}
=
\sum_{\gamma \in {\mathfrak S}^{C}_{\vec{c}}}
(-1)^{\gamma} \cdot [\gamma \cdot t]$,
where ${\mathfrak S}^{C}_{\vec{c}}$
is the column stabilizer of the tableau~$t$ of shape~$\vec{c}$.
Lastly, the Specht module,
denoted $S^{B(\vec{c}\,)}$, is the subspace of $M^{B(\vec{c}\,)}$ generated by polytabloids.
The dimension of
the Specht module~$S^{B(\vec{c}\,)}$
is given by the descent set statistics~$\beta_{n}(\vec{c}\,)$,
while the dimension of the permutation module~$M^{B(\vec{c}\,)}$
is given by the multinomial coefficient~$\binom{n}{\vec{c}}$.
We now define two operations on compositions.
The motivation comes from the associated
Specht and permutation modules.
For a composition
$\vec{c} = (c_{1},\ldots,c_{k-1},c_{k})$
let $\vec{c}-1$ denote the composition
$(c_{1},\ldots,c_{k-1},c_{k}-1)$
if $c_{k} \geq 2$,
and otherwise let $\vec{c}-1$ denote the empty composition.
Similarly,
let $\vec{c}/1$ denote the composition
$(c_{1},\ldots,c_{k-1},c_{k}-1)$
if $c_{k} \geq 2$,
and otherwise let $\vec{c}/1$ denote
the composition $(c_{1},\ldots,c_{k-1})$.
Note that
if $\vec{c}$ is a composition of~$n$
then
$\vec{c}/1$ is always a composition of~$n-1$.
For a composition $\vec{c}$ of $n$
let $B^{*}(\vec{c}\,)$ denote the border strip~$B(\vec{c} - 1)$.
All our results of this paper
are stated in terms of
the Specht modules $S^{B^{*}(\vec{c}\,)}$
where the group action is by~${\mathfrak S}_{n-1}$.
We think of this Specht module as a submodule
of $S^{B(\vec{c}\,)}$ spanned by all standard Young tableaux
where the northeastern-most box is filled with~$n$.
Note that when the composition ends with the entry~$1$,
there are no such standard Young tableaux, and hence
$S^{B^{*}(\vec{c}\,)}$ is the zero module.
For a composition~$\vec{c}$
define the two shapes
$B^{\#}(\vec{c}\,)=B(\vec{c}/1)$
and
$A^{\#}(\vec{c}\,)=A(\vec{c}/1)$.
Observe that the permutation module~$M^{B^{\#}(\vec{c}\,)}$
is a submodule of~$M^{B(\vec{c}\,)}$
That is, the span of the tabloids in~$M^{B(\vec{c}\,)}$
where the tabloids has the filling~$n$ in the northeastern-most box
is the module~$M^{B^{\#}(\vec{c}\,)}$.
Furthermore,
the dimensions of
the Specht module~$S^{B^{*}(\vec{c}\,)}$
and
the permutation module~$M^{B^{\#}(\vec{c}\,)}$
are~$\beta^{*}_{n}(\vec{c}\,)$ and~$\binom{n}{\vec{c}/1}$, respectively.
Additionally, we have the decomposition
$$
M^{B^{\#}(\vec{c}\,)}
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \leq \vec{d}} S^{B^{*}(\vec{d}\,)} ,
$$
which is the representation theoretic analogue of
equation~\eqref{equation_beta_star_binomial}.
See Lemma~\ref{lemma_permutation_module_isomorphism}
for a proof.
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\put(20,20){\line(0,-1){20}}
\put(30,10){\line(0,1){20}}
\put(40,10){\line(0,1){20}}
\end{picture}
\end{center}
\caption{The border strip $B(\vec{c}\,)$ associated with
the composition $\vec{c} = (2,3,1)$.
The row stabilizer is the group
${\mathfrak S}_{\vec{c}} =
{\mathfrak S}_{[1,2]} \times {\mathfrak S}_{[3,5]} \times {\mathfrak S}_{[6,6]}$.
Note that $\vec{c} = (1+1,1+1+1,1)$
so that the complementary composition is
$\vec{c}^{\,c} = (1,1+1,1,1+1) = (1,2,1,2)$.
Hence the column stabilizer is
${\mathfrak S}_{\vec{c}}^{C} =
{\mathfrak S}_{[1,1]} \times {\mathfrak S}_{[2,3]} \times {\mathfrak S}_{[4,4]}
\times {\mathfrak S}_{[5,6]}$.}
\label{figure_stabilizer}
\end{figure}
\section{The ordered partition filter $Q_{\Delta}^{*}$}
\label{section_the_main_result}
We now introduce the ordered partition filter~$Q_{\Delta}^{*}$.
This filter will serve us as an important stepping stone
to understanding the topology of
general filters in the partition lattice.
The transition from~$Q_{\Delta}^*$ to the partition lattice
uses Quillen's Fiber Lemma; see
Section~\ref{section_filters_in_the_set_partition_lattice}.
Note that by considering the reverse orders
in $\Comp(n)$ and in $Q_{n}$ we obtain two simplicial
posets.
Hence for~$\Delta$ a non-empty filter in~$\Comp(n)$,
we view~$\Delta$ as a simplicial complex under
the reverse order~$\leq^{*}$. See the discussion in
Section~\ref{section_topological_considerations}.
\begin{definition}
Let $\Delta$ be a filter in $\Comp(n)$, that is,
$\Delta$ is a simplicial complex consisting of compositions of~$n$.
Define the ordered partition filter~$Q^{*}_{\Delta}$ to be all
ordered set partitions whose type is in
the complex~$\Delta$ and whose last block contains the
element $n$, that is,
$$
Q^{*}_{\Delta}
= \{\sigma = (C_{1}, C_{2}, \ldots, C_{k}) \in Q_{n}
\: : \: \type(\sigma) \in \Delta, \:
n \in C_{k}\} . $$
\end{definition}
Note that we view $Q^{*}_{\Delta}$
as a simplicial complex. Our purpose is to study the reduced homology groups
of this complex.
Recall that the link of a composition $\vec{c}$
in~$\Delta$ is the filter
$$ \link_{\vec{c}}(\Delta)
=
\{\vec{d} \in \Delta \: : \: \vec{d}\leq^{*}\vec{c}\,\} , $$
where $\leq^{*}$ is the reverse of
the partial order of $\Comp(n)$.
Since $\link_{\vec{c}}(\Delta)$ is now a simplicial poset with
minimal element~$\vec{c}$,
we have a dimension shift from $\Delta$ to $\link_{\vec{c}}(\Delta)$
given by
\begin{equation}
\dim_{\link_{\vec{c}}(\Delta)}(\vec{d}\,)
=
\dim_{\Delta}(\vec{d}\,)-|\vec{c}\,|+1
\label{equation_dimension}
\end{equation}
for $\vec{d}\in\link_{\vec{c}\,}(\Delta)$.
\begin{remark}
{\rm
The symmetric group ${\mathfrak S}_{n-1}$ acts on $Q_{\Delta}^{*}$
by permutation, whereas
the action of~${\mathfrak S}_{n-1}$ on the complex $\Delta$
is the trivial action.
Furthermore, the type map from
$Q_{\Delta}^{*}$ to $\Delta$ respects this action,
since the two ordered set partitions
$\sigma$ and $\tau \cdot \sigma$
have the same type.
}
\label{remark_action}
\end{remark}
A special case of $Q^{*}_{\Delta}$ is when
the simplicial complex $\Delta$ is a simplex,
that is, $\Delta$ is generated by one composition~$\vec{c}$.
This case was studied by Ehrenborg and Jung
in~\cite{Ehrenborg_Jung}.
Their results are given below.
\begin{theorem}[Ehrenborg--Jung]
Let $\vec{c}$ be a composition of $n$ into $k$ parts.
Then the complex~$Q^{*}_{\vec{c}}$ is a wedge
of $\beta^{*}_{n}(\vec{c}\,)$ spheres of dimension~$k-2$.
Furthermore,
the top homology group~$\widetilde{H}_{k-2}(Q^{*}_{\vec{c}})$
is isomorphic to
the Specht module~$S^{B^{*}(\vec{c}\,)}$
as an ${\mathfrak S}_{n-1}$-module.
This isomorphism
$\phi : S^{B^{*}(\vec{c}\,)} \longrightarrow \widetilde{H}_{k-2}(Q^{*}_{\vec{c}})$
is given by
$$ \phi\left(e_{t}\right)
=
\sum_{\gamma \in {\mathfrak S}^{C}_{\vec{c}}}
(-1)^{\gamma} \cdot \sigma(\alpha\cdot\gamma,\vec{c}\,) ,
$$
where the permutation $\alpha \in {\mathfrak S}_{n}$
is obtained by reading the entries of
the tabloid~$t$
from southwest to northeast
and attaching the element~$n$ at the end.
\label{theorem_Ehrenborg_Jung}
\end{theorem}
Note that Ehrenborg and Jung formulated their result
in terms of pointed set partitions. That is, our
notation~$Q^{*}_{\vec{c}}$
is~$\Delta_{\vec{d}}$ in their notation, where
$\vec{d} = (c_{1}, \ldots, c_{k-1},c_{k}-1)$.
They allow the last entry of a composition to be zero
and similarly the last entry of an ordered set partition
to be empty.
Moreover, our notation~$\Pi^{*}_{\vec{c}}$
is in their notation~$\Pi^{\bullet}_{\vec{d}}$.
We can now state the main result of this section.
\begin{theorem}
Let $\Delta$ be a simplicial complex of compositions of $n$.
Then the $i$th reduced homology group
of the simplicial complex~$Q^{*}_{\Delta}$
is given by
$$
\widetilde{H}_{i}(Q^{*}_{\Delta})
\cong
\bigoplus_{\vec{c}\, \in \Delta}
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
\otimes
S^{B^{*}(\vec{c}\,)}. $$
Furthermore, this isomorphism holds as
${\mathfrak S}_{n-1}$-modules.
\label{theorem_main_result}
\end{theorem}
We will prove Theorem~\ref{theorem_main_result}
in Sections~\ref{section_homomorphism_phi}
through~\ref{section_the_building_step}.
\section{The homomorphism $\phi^{\Delta}_{i}$}
\label{section_homomorphism_phi}
In this section and the next two sections we present
a proof of Theorem~\ref{theorem_main_result}.
The major step
is to show that if Theorem~\ref{theorem_main_result}
holds for $\Delta$, $\Gamma$,
and the intersection $\Delta \cap \Gamma$,
then it also holds for
the union~$\Delta \cup \Gamma$.
This step requires Mayer--Vietoris sequences.
When
$\Delta$ is generated by
a single composition~$\vec{c}$ in~$\Comp(n)$,
the result follows from
Theorem~\ref{theorem_Ehrenborg_Jung}.
Finally, since any simplicial complex is a union of
simplices, Theorem~\ref{theorem_main_result}
will hold for arbitrary simplicial complexes~$\Delta$ in~$\Comp(n)$.
We begin by defining the isomorphism of
Theorem~\ref{theorem_main_result} explicitly.
Throughout the paper we will let $i_{\vec{c}}$
denote the shift $i-|\vec{c}\,|+1$.
\begin{definition}
\label{definition_D_chain_complex}
Let $D_{i}^{\vec{c}\,}(\Delta)$ be the tensor product
$C_{i_{\vec{c}\,}}(\link_{\vec{c}\,}(\Delta))\otimes M^{B^{\#}(\vec{c}\,)}$
where
$C_{j}(\link_{\vec{c}\,}(\Delta))$ is the $j$th chain group of the link
$\link_{\vec{c}\,}(\Delta)$.
Let $D^{\vec{c}\,}(\Delta)$
be the chain complex whose $i$th chain group
is $D^{\vec{c}\,}_{i}(\Delta)$ and whose boundary map is
$\partial \otimes \id$.
Lastly, let $D(\Delta)$ be the chain complex with $i$th chain group
$\bigoplus_{\vec{c}\,\in\Delta}D^{\vec{c}}_{i}(\Delta)$
with the differential
$\bigoplus_{\vec{c}\in\Delta} \partial \otimes \id$.
\end{definition}
\begin{definition}
\label{definition_E_chain_complex}
Define the chain complex $E^{\vec{c}}(\Delta)$
analogous to $D^{\vec{c}}(\Delta)$ of
Definition~\ref{definition_D_chain_complex} above by
replacing the permutation module $M^{B^{\#}(\vec{c}\,)}$
with the Specht module~$S^{B^{*}(\vec{c}\,)}$.
We also have the corresponding chain complex~$E(\Delta)$
with the same differential.
\end{definition}
\begin{lemma}
\label{lemma_chain_homology}
The homology of the chain complexes
$D(\Delta)$ and $E(\Delta)$ are given by
\begin{align*}
\widetilde{H}_{i}(D(\Delta))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\in\Delta}
\widetilde{H}_{i_{\vec{c}}\,}(\link_{\vec{c}\,}(\Delta))\otimes M^{B^{\#}(\vec{c}\,)} , \\
\widetilde{H}_{i}(E(\Delta))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\in\Delta}
\widetilde{H}_{i_{\vec{c}}\,}(\link_{\vec{c}\,}(\Delta))\otimes S^{B^{*}(\vec{c}\,)} .
\end{align*}
\end{lemma}
\begin{proof}
The homology of the chain complex $D^{\vec{c}}(\Delta)$
is given by
$\ker(\partial_{i_{\vec{c}} } \otimes \id)/\im(\partial_{i_{\vec{c}} +1}
\otimes \id)
\cong
(\ker(\partial_{i_{\vec{c}} }) \otimes M^{B^{\#}(\vec{c}\,)})/
(\im(\partial_{i_{\vec{c}} +1}) \otimes M^{B^{\#}(\vec{c}\,)})
\cong
\ker(\partial_{i_{\vec{c}} })/\im(\partial_{i_{\vec{c}} +1})
\otimes M^{B^{\#}(\vec{c}\,)}
\cong
\widetilde{H}_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}$.
The analogous result holds for $E^{\vec{c}}(\Delta)$
and the lemma follows by taking direct sums.
\end{proof}
For the rest of this section we let $t$ denote a tabloid
in the permutation module $M^{B^{\#}(\vec{c}\,)}$
and $\alpha \in {\mathfrak S}_{n}$
is the permutation obtained by reading the entries of
the tabloid $t$ in increasing order from
southwest to northeast and adjoining the element $n$
at the end.
\begin{definition}
\label{definition_group_action}
The ${\mathfrak S}_{n-1}$-action on $D^{\vec{c}}_{i}(\Delta)$
is given by
$\tau \cdot (\vec{d} \otimes t)=\vec{d} \otimes (\tau \circ t)$,
for $\tau \in {\mathfrak S}_{n-1}$ and $\vec{d} \otimes t$
a basis element of
$D^{\vec{c}}_{i}(\Delta)=C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}$.
\end{definition}
Notice that Definition~\ref{definition_group_action}
states that ${\mathfrak S}_{n-1}$ acts on $D^{\vec{c}}_{i}(\Delta)$
by acting trivially on the chain group
$C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta))$
and by relabeling on $M^{B^{\#}(\vec{c}\,)}$.
\begin{definition}
For a simplicial complex $\Delta$ and
a composition $\vec{c}$ in $\Delta$
define the map
$$ \phi^{\Delta,\vec{c}}_{i}
: C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}
\longrightarrow
C_{i}(Q^{*}_{\Delta}) , $$
on basis elements by
$\phi_{i}^{\Delta,\vec{c}}(\vec{d} \otimes t)=\sigma(\alpha,\vec{d}\,)$.
\end{definition}
Since $\vec{d}\in C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta))$ is a basis element,
we know that $\vec{d}$ is a simplex of $\link_{\vec{c}}(\Delta)$ of dimension $i_{\vec{c}}=i-|\vec{c}\,|+1$,
and thus by the dimension shift in
equation~\eqref{equation_dimension},
we have that $|\vec{d}\,|=i+2$, so that
$\phi_{i}^{\Delta,\vec{c}}(\vec{d}\,)=\sigma(\pi,\vec{d}\,)$
is an ordered partition of dimension~$i$. Lastly,
since tabloids in~$M^{B^{\#}(\vec{c}\,)}$ have $n$ in the last block,
we are guaranteed that
$\phi_{i}^{\Delta,\vec{c}}(\vec{d}\,)\in C_{i}(Q_{\Delta}^{*})$.
\begin{lemma}
\label{lemma_equivariance}
The map
$\phi_{i}^{\Delta,\vec{c}}:
C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}
\longrightarrow
C_{i}(Q^{*}_{\Delta})$
respects the ${\mathfrak S}_{n-1}$-action.
\end{lemma}
\begin{proof}
Let $\tau \in {\mathfrak S}_{n-1}$ and $\vec{d} \otimes t$
be a basis element of
$C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}$.
Then we have
\begin{align*}
\phi_{i}^{\Delta,\vec{c}}(\tau\cdot(\vec{d}\otimes t))
&
= \phi_{i}^{\Delta,\vec{c}}(\vec{d}\otimes (\tau \cdot t))
= \sigma(\tau \cdot \alpha,\vec{d}\,)
= \tau \cdot \sigma(\alpha,\vec{d}\,)
= \tau \cdot \phi_{i}^{\Delta,\vec{c}}(\vec{d} \otimes t).
\qedhere
\end{align*}
\end{proof}
\begin{lemma}
\label{lemma_commute_diagram_1}
The map $\phi^{\Delta,\vec{c}}_{i}$ is an equivariant chain map
between the complexes
$D^{\vec{c}\,}(\Delta)$ and~$C_{i}(Q^{*}_{\Delta})$. That is, the following diagram commutes:
$$
\xymatrixcolsep{5pc}
\xymatrix{
D_{i}^{\vec{c}\,}(\Delta)
\ar[d]^{\phi^{\Delta,\vec{c}}_{i}} \ar[r]^{\partial \otimes \id} &
D_{i-1}^{\vec{c}\,}(\Delta)
\ar[d]^{\phi^{\Delta,\vec{c}}_{i-1}} \\
C_{i}(Q^{*}_{\Delta})
\ar[r]^{\partial} &
C_{i-1}(Q^{*}_{\Delta})}
$$
\end{lemma}
\begin{proof}
The boundary map~$\partial$ of~$\Comp(n)$
as well as the boundary map~$\partial$ of~$Q_{\Delta}^{*}$
are given in
Section~\ref{section_topological_considerations}.
Let
$\vec{d} \otimes t \in
C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}$.
Tracing first right then down we obtain:
$$\phi_{i-1}^{\Delta,\vec{c}} \circ (\partial \otimes \id)(\vec{d} \otimes t)
=
\phi_{i-1}^{\Delta,\vec{c}}(\partial(\vec{d}\,) \otimes t)
=
\sigma(\alpha,\partial(\vec{d}\,)).$$
Next, we trace down then right to obtain the same result:
$$
\partial\circ\phi_{i}^{\Delta,\vec{c}}(\vec{d} \otimes t)
=
\partial(\sigma(\alpha,\vec{d}\,))
=
\sigma(\alpha,\partial(\vec{d}\,)).$$
The equivariance of $\phi_{i}^{\Delta,\vec{c}}$ is a consequence of Lemma~\ref{lemma_equivariance}.
\end{proof}
\begin{lemma}
The map $\phi_{i}^{\Delta, \vec{c}}$
induces a map
$$\phi_{i}^{\Delta, \vec{c}}:
\widetilde{H}_{i_{\vec{c}}}(\link_{\vec{c}\,}(\Delta))\otimes M^{B^{\#}(\vec{c}\,)}
\longrightarrow
\widetilde{H}_{i}(Q^{*}_{\Delta}) $$
given by
$\phi_{i}^{\Delta}(\overline{\vec{d}\,} \otimes t)
=
\overline{\sigma(\alpha,\vec{d}\,)}$,
for $\vec{d}\in C_{i_{\vec{c}}}(\link_{\vec{c}\,}(\Delta))$ a cycle.
\label{lemma_passing_to_homology}
\end{lemma}
\begin{proof}
Since $\phi_{i}^{\Delta, \vec{c}}$ is
an equivariant chain map
between the chain complexes
$D^{\vec{c}\,}(\Delta)$
and~$C_{i}(Q^{*}_{\Delta})$
by
Lemma~\ref{lemma_commute_diagram_1},
the result follows.
\end{proof}
For the rest of the paper, the use of the bar to indicate the quotient
in passing from the chain space to the homology group
will be suppressed for ease of notation.
\begin{definition}
Define the map
$\phi^{\Delta}_{i}$
from
$D_{i}(\Delta)
=
\bigoplus_{\vec{c}\,\in\Delta} D^{\vec{c}}_{i_{\vec{c}}}(\Delta)$
to $C_{i}(Q^{*}_{\Delta})$
by adding
all the
$\phi^{\Delta,\vec{c}}_{i}$ maps together,
that is,
\begin{equation}
\label{equation_phi}
\phi^{\Delta}_{i}
=
\sum_{\vec{c} \in \Delta}
\phi^{\Delta,\vec{c}}_{i} .
\end{equation}
\end{definition}
Observe that $\phi^{\Delta}_{i}$ restricts
to a map from
$E_{i}(\Delta)$
to $C_{i}(Q^{*}_{\Delta})$.
Therefore $\phi_{i}^{\Delta}$ also induces a map
from
$\widetilde{H}_{i}(E(\Delta))
=
\bigoplus_{\vec{c}\in\Delta}
\widetilde{H}_{i_{\vec{c}}\,}(\link_{\vec{c}\,}(\Delta))\otimes S^{B^{*}(\vec{c}\,)}$
to $\widetilde{H}_{i}(Q^{*}_{\Delta})$
using Lemma~\ref{lemma_passing_to_homology}.
\section{The main theorem}
\label{section_base_case}
We can now explicitly state the isomorphism of
Theorem~\ref{theorem_main_result}.
First we introduce notation for the
right-hand side of this theorem.
\begin{definition}
Let
$K_{i}(\Delta)$
denote the direct sum
$\bigoplus_{\vec{c}\, \in \Delta}
\widetilde{H}_{i_{\vec{c}\,}}(\link_{\vec{c}\,}(\Delta))
\otimes S^{B^{*}(\vec{c}\,)}$.
\label{definition_K}
\end{definition}
A sharpening of
Theorem~\ref{theorem_main_result}
is the following result.
\begin{theorem}
Let $\Delta$ be a subcomplex of $\Comp(n)$.
Then the map
$$\phi^{\Delta}_{i}:K_{i}(\Delta)\longrightarrow \widetilde{H}_{i}(Q^{*}_{\Delta})$$
is an ${\mathfrak S}_{n-1}$-equivariant isomorphism.
\label{theorem_main}
\end{theorem}
Note that Lemma~\ref{lemma_chain_homology} tells us that the homology of the complex~$E(\Delta)$ is~$K(\Delta)$,
that is,
for all~$i$ we have
$\widetilde{H}_{i}(E(\Delta))\cong_{{\mathfrak S}_{n-1}}K_{i}(\Delta)$.
Lemma~\ref{lemma_commute_diagram_1} implies that
equation~\eqref{equation_phi} is a well-defined map
from
the homology of~$E(\Delta)$
to the homology groups~$\widetilde{H}_{i}(Q_{\Delta}^{*})$.
We first prove Theorem~\ref{theorem_main}
in the case when $\Delta$ is a simplex.
This is the case when $\Delta$ is generated by one
composition.
\begin{proposition}
Assume that $\Delta$ is a filter in $\Comp(n)$
generated by one composition,
that is, $\Delta$ is a simplex.
Then Theorem~\ref{theorem_main} holds for~$\Delta$.
\label{proposition_simplex}
\end{proposition}
\begin{proof}
Suppose that $\Delta\subseteq \Comp(n)$ is generated by the
composition $\vec{d}=(d_{1},d_{2}, \ldots, d_{k})$.
Theorem~\ref{theorem_Ehrenborg_Jung}
states that $Q_{\Delta}^{*}$ only has
reduced homology in dimension $k-2$.
Additionally, it states that the action
of ${\mathfrak S}_{n-1}$ on the top homology of~$Q_{\Delta}^{*}$
is given by the border
shape Specht module~$S^{B^{*}(\vec{c}\,)}$, that is,
$\widetilde{H}_{k-2}(Q_\Delta)\cong_{{\mathfrak S}_{n-1}} S^{B^{*}(\vec{c}\,)}$.
Next we show that
$\phi_{i}^{\Delta}:K_{i}(\Delta)\longrightarrow\widetilde{H}_{i}(Q_{\Delta}^{*})$
is an isomorphism for all $i$.
When $i \neq k-2$ both sides are the trivial module,
that is,
$K_{i}(\Delta) = 0 = \widetilde{H}_{i}(Q_{\Delta}^{*})$
and the map $\phi_{i}^{\Delta}$ is directly
an isomorphism.
Now assume that $i=k-2$.
Since all the links
$\link_{\vec{c}}(\Delta)$
for $\vec{c} <^{*} \vec{d}$ are contractible,
we have
$$
K_{k-2}(\Delta)
= \bigoplus_{\vec{c}\in\Delta}
\widetilde{H}_{k-2-|\vec{c}\,|+1}(\link_{\vec{c}}(\Delta))\otimes S^{B^{*}(\vec{c}\,)}
= \widetilde{H}_{-1}(\link_{\vec{d}\,}(\Delta)) \otimes S^{B^{*}(\vec{d}\,)} .
$$
Notice that $\link_{\vec{d}\,}(\Delta)$ consists only of
the composition $\vec{d}$ itself,
so that the $(-1)$-dimensional
reduced homology group
$\widetilde{H}_{-1}(\link_{\vec{d}\,}(\Delta))$
is the homology of the chain space
$C_{-1}(\link_{\vec{d}\,}(\Delta))$,
which is the one dimensional vector space
with the generator~$\vec{d}$.
Therefore, the map
$\phi_{k-2}^{\Delta}:
\widetilde{H}_{-1}(\link_{\vec{d}\,}(\Delta))\otimes S^{B^{*}(\vec{d})}
\longrightarrow
\widetilde{H}_{k-2}(Q^{*}_{\Delta})$
is given by
\begin{align*}
\vec{d} \otimes e_{t}
& =
\vec{d} \otimes \left(\sum_{\gamma \in {\mathfrak S}^{C}_{\vec{d}}}
(-1)^{\gamma} \cdot [\gamma\cdot t]\right)
\longmapsto
\sum_{\gamma\in {\mathfrak S}^{C}_{\vec{d}}}
(-1)^{\gamma}
\cdot
\sigma(\alpha\cdot\gamma,\vec{d}\,) .
\end{align*}
But this is an isomorphism by
Theorem~\ref{theorem_Ehrenborg_Jung}.
\end{proof}
As a direct corollary we have that
Theorem~\ref{theorem_main}
holds for the empty simplex~$\{(n)\}\subseteq\Comp(n)$.
\begin{corollary}
Theorem~\ref{theorem_main} holds for
the empty simplicial complex,
that is, the simplicial complex consisting only of the composition~$(n)$.
\label{corollary_empty}
\end{corollary}
\begin{proof}
Apply Proposition~\ref{proposition_simplex}
to the simplicial complex $\Delta$
generated by the composition~$(n)$ in $\Comp(n)$.
\end{proof}
\section{The building step}
\label{section_the_building_step}
A simplicial complex which is not a simplex
is the union of smaller simplicial complexes.
We now prove that Theorem~\ref{theorem_main}
holds for the complex $\Delta \cup \Gamma$,
assuming that Theorem~\ref{theorem_main} holds for
the simplicial complexes~$\Delta$,
$\Gamma$, as well as
the intersection~$\Delta \cap \Gamma$.
We build up in the isomorphism~$\phi_{i}^{\Delta\cup\Gamma}$
between
$K_{i}(\Delta\cup\Gamma)$
and
$\widetilde{H}_{i}(Q^{*}_{\Delta\cup\Gamma})$
from
the associated isomorphisms holding for the smaller
complexes.
\begin{lemma}
\label{lemma_link}
The following two identities hold for the link:
$$
\link_{\vec{c}\,}(\Delta\cap\Gamma)
=
\link_{\vec{c}\,}(\Delta)\cap\link_{\vec{c}\,}(\Gamma)
\:\:\:\: \text{ and } \:\:\:\:
\link_{\vec{c}\,}(\Delta\cup\Gamma)
=
\link_{\vec{c}\,}(\Delta)\cup\link_{\vec{c}\,}(\Gamma) .
$$
\end{lemma}
\begin{lemma}
\label{lemma_Q_split}
The following two identities hold for the ordered set
partition poset:
$$
Q^{*}_{\Delta \cap \Gamma}
=
Q^{*}_{\Delta} \cap Q^{*}_{\Gamma}
\:\:\:\: \text{ and } \:\:\:\:
Q^{*}_{\Delta \cup \Gamma}
=
Q^{*}_{\Delta} \cup Q^{*}_{\Gamma} .
$$
\end{lemma}
The proofs of these two lemmas are straightforward
and are omitted.
Before we begin the proof of Theorem~\ref{theorem_main},
let us remind ourselves of
Definition~\ref{definition_D_chain_complex}.
For each composition~$\vec{c}$ in~$\Delta$
we have the chain complex~$D^{\vec{c}}(\Delta)$
whose $i$th chain group is
$D^{\vec{c}}_{i}(\Delta)
=
C_{i_{\vec{c}}}(\link_{\vec{c}}(\Delta)) \otimes M^{B^{\#}(\vec{c}\,)}$.
Furthermore,
$D(\Delta)$ is the chain complex obtained by
taking the direct sum
of~$D^{\vec{c}}(\Delta)$,
where $\vec{c}$ ranges over all
compositions in~$\Delta$.
We now begin the proof of Theorem~\ref{theorem_main}.
\begin{lemma}
\label{lemma_exact_intersection}
For $\vec{c}\in\Delta\cap\Gamma$ the following diagram is commutative, and its rows are exact.
$$
\xymatrix{
0\ar[r]
&D^{\vec{c}}_{i}(\Delta\cap\Gamma)\ar[d]^{\phi_{i}^{\Delta\cap\Gamma,\vec{c}}}\ar[r]
&D^{\vec{c}}_{i}(\Delta)\oplus D^{\vec{c}}_{i}(\Gamma)\ar[d]^{\phi_{i}^{\Delta,\vec{c}}\oplus\phi_{i}^{\Gamma,\vec{c}}}\ar[r]
&D^{\vec{c}}_{i}(\Delta\cup\Gamma)\ar[d]^{\phi_{i}^{\Delta\cup\Gamma,\vec{c}}}\ar[r]
&0\\
0\ar[r]&C_{i}(Q^{*}_{\Delta\cap\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta}) \oplus C_{i}(Q^{*}_{\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta\cup\Gamma})\ar[r]
&0}
$$
\end{lemma}
\begin{proof}
The horizontal maps in the above diagram are given by
the construction of the Mayer--Vietoris sequence applied to
$\link_{\vec{c}\,}(\Delta\cup\Gamma)=\link_{\vec{c}\,}(\Delta)\cup\link_{\vec{c}\,}(\Gamma)$
in the top row,
and
$Q^{*}_{\Delta\cup\Gamma}=Q^{*}_{\Delta}\cup Q^{*}_{\Gamma}$
in the bottom row.
The top horizontal maps have also been tensored with
the identity map on the Specht modules.
As the Specht module is free, both the top and bottom rows of the diagram remain exact.
We show commutativity of the left square, as the right square
is analogous.
Let
$\vec{d}\otimes\alpha\in C_{i_{\vec{c}}}(\Delta\cap\Gamma)\otimes M^{B^{\#}(\vec{c}\,)}$
be a basis element.
First we trace right then down to obtain:
$$
\vec{d}\otimes\alpha
\longmapsto
(\vec{d}\otimes\alpha)\oplus-(\vec{d}\otimes\alpha)
\longmapsto
\sigma(\alpha,\vec{d}\,)\oplus-\sigma(\alpha,\vec{d}\,) .
$$
We obtain the same result by first tracing
down then right:
\begin{align*}
\vec{d}\otimes\alpha
& \longmapsto
\sigma(\alpha,\vec{d}\,)
\longmapsto
\sigma(\alpha,\vec{d}\,)\oplus-\sigma(\alpha,\vec{d}\,) .
\qedhere
\end{align*}
\end{proof}
\begin{lemma}
\label{lemma_exact_intersection_4}
For each $\vec{c}\in\Delta-\Gamma$,
we have the commutative diagram with exact rows:
$$
\xymatrix{
0\ar[r]
&0\ar[r]\ar[d]^{0}
&D_{i}^{\vec{c}}(\Delta)\ar[r]^{\id}\ar[d]^{\phi_{i}^{\Delta}\oplus 0}
&D_{i}^{\vec{c}}(\Delta)\ar[r]\ar[d]^{\phi_{i}^{\Delta\cup\Gamma}}
&0\\
0\ar[r]
&C_{i}(Q^{*}_{\Delta\cap\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta})\oplus C_{i}(Q^{*}_{\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta\cup\Gamma})\ar[r]
&0}
$$
\end{lemma}
\begin{proof}
The left-hand square is trivially commutative.
We show the right-hand square commutes
by first tracing right then down:
$$ \vec{c} \otimes \pi
\longmapsto
\vec{c} \otimes \pi
\longmapsto
\sigma(\pi,\vec{c}\,) . $$
Now we trace down then right:
\begin{align*}\vec{c}\otimes\pi\longmapsto\sigma(\pi,\vec{c}\,)\oplus 0\longmapsto\sigma(\pi,\vec{c}\,)+0=\sigma(\pi,\vec{c}\,)
\end{align*}
Exactness of the rows in the diagram follows from
Lemma~\ref{lemma_exact_intersection},
as the bottom row has remained unchanged.
\end{proof}
\begin{lemma}
\label{lemma_exact_intersection_3}
The following diagram is commutative, and its rows are exact.
$$
\xymatrix{
0\ar[r]
&D_{i}(\Delta\cap\Gamma)\ar[d]^{\phi_{i}^{\Delta\cap\Gamma}}\ar[r]
&D_{i}(\Delta)\oplus D_{i}(\Gamma)\ar[d]^{\phi_{i}^{\Delta}\oplus\phi_{i}^{\Gamma}}\ar[r]
&D_{i}(\Delta\cup\Gamma)\ar[d]^{\phi_{i}^{\Delta\cup\Gamma}}\ar[r]
&0\\
0\ar[r]
&C_{i}(Q^{*}_{\Delta\cap\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta})\oplus C_{i}(Q^{*}_{\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta\cup\Gamma})\ar[r]
&0}
$$
\end{lemma}
\begin{proof}
The proof is to take
direct sums of the previous two short exact sequences.
First, take the direct sum of the diagram in
Lemma~\ref{lemma_exact_intersection}
for each $\vec{c}\in\Delta\cap\Gamma$.
Next, take the resulting short exact sequence of chain complexes
and take its direct sum with the diagram in
Lemma~\ref{lemma_exact_intersection_4}
for each $\vec{c}\in\Delta-\Gamma$.
Finally, switch $\Delta$ and $\Gamma$ in
Lemma~\ref{lemma_exact_intersection_4}
and take the direct sum of the resulting diagram with the diagram from
Lemma~\ref{lemma_exact_intersection_4}
for each $\vec{c}\in\Gamma-\Delta$. Observe that the second row of
the diagram remains the same throughout this process.
Also, note that the top row is exact
as it is the direct sum of exact sequences.
All together, this yields the desired commutative diagram.
\end{proof}
\begin{proposition}
\label{proposition_D_to_E}
The following diagram is commutative, and its rows are exact.
$$
\xymatrix{
0\ar[r]
&E_{i}(\Delta\cap\Gamma)\ar[d]^{\phi_{i}^{\Delta\cap\Gamma}}\ar[r]
&E_{i}(\Delta)\oplus E_{i}(\Gamma)\ar[d]^{\phi_{i}^{\Delta}\oplus\phi_{i}^{\Gamma}}\ar[r]
&E_{i}(\Delta\cup\Gamma)\ar[d]^{\phi_{i}^{\Delta\cup\Gamma}}\ar[r]
&0\\
0\ar[r]
&C_{i}(Q^{*}_{\Delta\cap\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta})\oplus C_{i}(Q^{*}_{\Gamma})\ar[r]
&C_{i}(Q^{*}_{\Delta\cup\Gamma})\ar[r]
&0}
$$
\end{proposition}
\begin{proof}
Since $E_{i}(\Delta)$ is a subspace of $D_{i}(\Delta)$,
it follows from Lemma~\ref{lemma_exact_intersection_3}
that the diagram is commutative.
Furthermore, that the
second row is exact also follows from this lemma.
It remains to show that the first row is exact.
However, this follows by the same reasoning that the first row
of Lemma~\ref{lemma_exact_intersection_3} is exact,
but with the permutation module~$M^{B^{\#}(\vec{c}\,)}$
replaced with the Specht module~$S^{B^{*}(\vec{c}\,)}$.
\end{proof}
\begin{proposition}
Assume that Theorem~\ref{theorem_main}
holds for the simplicial complexes
$\Delta$, $\Gamma$, and
the intersection $\Delta \cap \Gamma$.
Then Theorem~\ref{theorem_main}
also holds for the union~$\Delta \cup \Gamma$.
\label{proposition_the_building_step}
\end{proposition}
\begin{proof}
Consider the diagram of short exact sequences of chain complexes
given in Proposition~\ref{proposition_D_to_E}.
Use the zig-zag lemma to obtain
the Mayer--Vietoris sequence:
$$
\xymatrix{
\cdots\ar[r]
&K_{i}(\Delta\cap\Gamma)\ar[d]^{\phi_{i}^{\Delta\cap\Gamma}}\ar[r]
&K_{i}(\Delta)\oplus K_{i}(\Gamma)\ar[d]^{\phi_{i}^{\Delta}\oplus\phi_{i}^{\Gamma}}\ar[r]
&K_{i}(\Delta\cup\Gamma)\ar[d]^{\phi_{i}^{\Delta\cup\Gamma}}\ar[r]
&\cdots\\
\cdots\ar[r]
&\widetilde{H}_{i}(Q^{*}_{\Delta\cap\Gamma})\ar[r]
&\widetilde{H}_{i}(Q^{*}_{\Delta}) \oplus \widetilde{H}_{i}(Q^{*}_{\Gamma})\ar[r]
&\widetilde{H}_{i}(Q^{*}_{\Delta\cup\Gamma})\ar[r]
&\cdots\\}
$$
The assumption that
Theorem~\ref{theorem_main}
holds for
the complexes
$\Delta \cap \Gamma$, $\Delta$, and $\Gamma$
implies that
$\phi_{i}^{\Delta\cap\Gamma}$
and
$\phi_{i}^{\Delta} \oplus \phi_{i}^{\Gamma}$
are isomorphisms.
The five-lemma now implies that
$\phi_{i}^{\Delta\cup\Gamma}$ is also an isomorphism.
Furthermore, $\phi_{i}^{\Delta\cup\Gamma}$ is an ${\mathfrak S}_{n-1}$-equivariant map
by Lemma~\ref{lemma_equivariance}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem_main}.]
Since every simplicial complex $\Delta$ is
the union of simplexes,
Proposition~\ref{proposition_the_building_step}
implies that it is enough to prove
Theorem~\ref{theorem_main}
for simplexes and the empty simplex.
This was done in
Proposition~\ref{proposition_simplex}
and
Corollary~\ref{corollary_empty}.
\end{proof}
\section{Alternate Proof of Theorem~\ref{theorem_main_result}}
As mentioned in the introduction,
we now give an alternate proof of Theorem~\ref{theorem_main_result}
using a poset fiber theorem of
Bj\"orner, Wachs and Welker~\cite{Bjorner_Wachs_Welker}.
\begin{theorem}
Let $\Delta$ be a simplicial complex of compositions of $n$.
Then the $i$th reduced homology group
of the simplicial complex~$Q^{*}_{\Delta}$
is given by
$$
\widetilde{H}_{i}(Q^{*}_{\Delta})
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\, \in \Delta}
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
\otimes
S^{B^{*}(\vec{c}\,)}. $$
\end{theorem}
\begin{proof}
Consider
the two posets
$\Delta$ and~$Q^{*}_{\Delta}$
with the reverse order $\leq^{*}$
and the poset map
$$ \type : Q^{*}_{\Delta} - \{([n])\} \longrightarrow \Delta - \{(n)\} . $$
Observe that the type map respects the action of
the symmetric group~${\mathfrak S}_{n-1}$.
Now the inverse image
$\type^{-1}(\Delta_{\leq^{*} \vec{c}})$
is the filter~$Q^{*}_{\vec{c}}$.
Since $Q^{*}_{\vec{c}}$ only has reduced homology
in dimension $|\vec{c}\,| - 2$ by
Theorem~\ref{theorem_Ehrenborg_Jung},
we have that
the fiber
$\triangle(\type^{-1}(\Delta_{\leq^{*} \vec{c}}))$
is
$(|\vec{c}\,| - 3)$-acyclic,
where $|\vec{c}\,| - 3$ is the length of the longest chain
in $\type^{-1}(\Delta_{<^{*} \vec{c}})$.
Hence
Theorem~9.1 of~\cite{Bjorner_Wachs_Welker}
applies.
Since~${\mathfrak S}_{n-1}$ acts trivially on $\Delta$
(see Remark~\ref{remark_action}),
we have that the stabilizer
$\Stab_{{\mathfrak S}_{n-1}}(\vec{c}\,)$
is in fact the whole group ${\mathfrak S}_{n-1}$.
Thus there is no representation to induce and we have
\begin{align*}
\widetilde{H}_{i}(Q^{*}_{\Delta})
& \cong_{{\mathfrak S}_{n-1}}
\widetilde{H}_{i}(\Delta)
\oplus
\bigoplus_{\vec{c}\, \in \Delta - \{(n)\}}
\widetilde{H}_{|\vec{c}\,|-2}(\type^{-1}(\Delta_{\leq^{*} \vec{c}}))
\otimes
\widetilde{H}_{i - |\vec{c}\,| + 1}((\Delta - \{(n)\})_{>^{*} \vec{c}}) \\
& \cong_{{\mathfrak S}_{n-1}}
\widetilde{H}_{i}(\Delta)
\oplus
\bigoplus_{\vec{c}\, \in \Delta - \{(n)\}}
S^{B^{*}(\vec{c}\,)}
\otimes
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta)) ,
\end{align*}
where the first summand corresponds to $\vec{c} = (n)$
and the trivial representation $S^{B^{*}(n)}$, proving the result.
\end{proof}
\section{Filters in the set partition lattice}
\label{section_filters_in_the_set_partition_lattice}
In Theorem~\ref{theorem_main_result} we characterized
each homology group of~$Q^{*}_{\Delta}$,
a subspace of ordered set partitions.
We will now translate the topological data we have gathered on $Q_{\Delta}^{*}$ into data on the usual partition lattice~$\Pi_{n}$.
Recall that $Q_{\Delta}^{*}$ is the collection of ordered set partitions
containing the element $n$ in the last block, whose type is contained
in the simplicial complex $\Delta\subseteq \Comp(n)$.
Define the \emph{forgetful map}
$f:Q^{*}_{\Delta} \longrightarrow \Pi_{n}$
given by removing the order between blocks, that is,
$f((C_{1},C_{2},\ldots,C_{k})) = \{C_{1},C_{2},\ldots,C_{k}\}$.
\begin{definition}
Let $\Pi^{*}_{\Delta}\subseteq \Pi_{n}$
be the image of $Q_{\Delta}^{*}$ under the forgetful map~$f$.
\end{definition}
\begin{lemma}
\label{lemma_integer_filter}
Suppose that $F$ is a filter in the integer partition lattice.
Let $\Delta_{F}$ be the
filter of compositions given by
$\{\vec{c} \in \Comp(n) : \type(\vec{c}\,)\in F\}$.
Then the associated filter~$\Pi^{*}_{\Delta_{F}}$
in the partition lattice is given by
$\{\pi \in \Pi_{n} : \type(\pi)\in F\}$.
\end{lemma}
\begin{proof}
Choose $\pi \in \Pi_{n}$ such that $\type(\pi)\in F$,
with $\pi=\{B_{1},B_{2}, \ldots, B_{k}\}$
where we assume $n \in B_{k}$.
The ordered set partition $\tau=(B_{1},B_{2}, \ldots, B_{k})$
is an element of $Q_{\Delta_{n}}^*$,
since $\type(\tau) = \type(\pi) \in F$.
Hence $\pi$ is in the image of the forgetful map~$f$.
The other direction is clear.
\end{proof}
\begin{remark}
{\rm
In general, taking the image of a filter $\Delta\subseteq\Comp(n)$
under the map $\type$ does not define a filter
in the integer partition lattice~$I_{n}$.
For example,
consider the simplex~$\Delta$ in~$\Comp(6)$ generated by~$(3,2,1)$.
Note that
$\type(\Delta)$ consists of the four partitions
$\{\{3,2,1\}, \{3+2,1\}, \{3,2+1\}, \{3+2+1\}\}
=
\{\{3,2,1\}, \{5,1\}, \{3,3\}, \{6\}\}$.
This is not a filter in $I_{6}$ since it does not contain
the partition~$\{4,2\}$.
}
\end{remark}
\begin{lemma}
\label{lemma_forgetful_equivariant}
The forgetful map $f:Q_{\Delta}^{*}\longrightarrow\Pi^{*}_{\Delta}$
respects the ${\mathfrak S}_{n-1}$-action.
\end{lemma}
\begin{proof}
Let $\alpha \in {\mathfrak S}_{n-1}$
and $\sigma=(C_{1}, \ldots, C_{k})\in Q_{\Delta}^{*}$.
Then we have that
\begin{align*}
f(\alpha \cdot \sigma)
& =
f((\alpha(C_{1}),\ldots,\alpha(C_{k})))
=
\{\alpha(C_{1}),\ldots,\alpha(C_{k})\}
=
\alpha \cdot f(\sigma).
\qedhere
\end{align*}
\end{proof}
The ${\mathfrak S}_{n-1}$ action on $\Pi^{*}_{\Delta}$
extends to the chains in the order complex
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$.
For a statement of the equivariant version of the Quillen Fiber Lemma,
see~\cite[Theorem~5.2.2]{Wachs_III}.
\begin{proposition}
\label{proposition_forgetful}
The forgetful map
$f : Q^{*}_{\Delta} - \{\hat{1}\}
\longrightarrow \Pi^{*}_{\Delta} - \{\hat{1}\} = P$
satisfies the condition of
Quillen's Equivariant Fiber Lemma, that is,
for a partition $\pi = \{B_{1}, B_{2}, \ldots, B_{k}\}$
in~$P$,
the order complex
$\triangle(f^{-1}(P_{\geq \pi}))$ is the barycentric subdivision of a cone, and is
therefore contractible and acyclic.
\end{proposition}
\begin{proof}
Let $B_{k}$ be the block of the partition~$\pi$
that contains the element~$n$.
Note that because every ordered partition
in~$Q_{\Delta}^{*}$ must have the element $n$ in its last block,
we must have that each ordered set partition in the fiber
$f^{-1}(\pi)$ has the set~$B_{k}$ as its last block.
Furthermore,
the last block of each ordered set partition in $f^{-1}(P_{\geq \pi})$
contains the block~$B_{k}$.
We claim that $f^{-1}(P_{\geq \pi})$ is a cone
with apex $([n] - B_{k}, B_{k})$.
Let $\sigma \in f^{-1}(P_{\geq \pi})$ be the ordered set partition
$\sigma=(C_{1}, \ldots, C_{p-1}, C_{p})$.
Note that the number of blocks of $\sigma$,
is greater than or equal to~$2$ as we have removed
the maximal element~$\hat{1}$ from~$Q_{\Delta}^{*}$.
If $C_{p} = B_{k}$ then the face $\sigma$
contains the vertex $([n] - B_{k}, B_{k})$.
If $C_{p} \supsetneq B_{k}$ then both
$\sigma$ and the vertex $([n] - B_{k}, B_{k})$
are contained in
the face
$(C_{1}, \ldots, C_{p-1}, C_{p} - B_{k}, B_{k})$
in~$f^{-1}(P_{\geq \pi})$.
Thus
$f^{-1}(P_{\geq \pi})$
is the face poset of a cone with vertex $([n]-B_{k},B_{k})$, and therefore $\triangle(f^{-1}(P_{\geq\pi})$
is the barycentric subdivision of a cone and hence contractible and acyclic.
\end{proof}
Combining Proposition~\ref{proposition_forgetful}
with Theorem~\ref{theorem_main_result},
we have the following result for the homology of the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$.
\begin{theorem}
\label{theorem_main_theorem_order_complex}
The $i$th reduced homology group of the order complex of $\Pi^{*}_{\Delta}-\{\hat{1}\}$ as an ${\mathfrak S}_{n-1}$-module
is given by
$$ \widetilde{H}_{i}(\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \in \Delta}
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
\otimes
S^{B^{*}(\vec{c}\,)}. $$
\end{theorem}
\begin{remark}
{\rm
Suppose that $\link_{\vec{c}\,}(\Delta)$
has reduced homology in dimension $j$.
By Theorem~\ref{theorem_main_theorem_order_complex}
this reduced homology contributes to
dimension $j+|\vec{c}\,|-1$
of the reduced homology of the order complex
of~$\Pi^{*}_{\Delta}-\{\hat{1}\}$.
}
\label{remark_dimension_shift}
\end{remark}
We end the section with a discussion of Morse matchings
in the link~$\link_{\vec{c}\,}(\Delta)$.
Assume that the link $\link_{\vec{c}}(\Delta)$
has a discrete Morse matching
with critical cell~$\vec{d}$, which also contributes
to the reduced homology of $\link_{\vec{c}}(\Delta)$.
For instance, this case occurs if $\vec{d}$ is
a facet.
Similarly, $\vec{d}$ will contribute to the reduced homology
of $\link_{\vec{c}}(\Delta)$ if $\vec{d}$ is a homology
facet of a shelling.
In either case, the critical cell~$\vec{d}$ contributes to
the reduced homology of
$\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
in dimension
$\dim_{\link_{\vec{c}}(\Delta)}(\vec{d}\,) + |\vec{c}\,| - 1
=
\dim_{\Delta}(\vec{d}\,)
= |\vec{d}\,| - 2$,
by equation~\eqref{equation_dimension}.
Note that this dimension is independent of the
composition~$\vec{c}$.
\section{Consequences of the main result}
\label{section_consequences}
As the title of this section suggests, we will now derive results from Theorem~\ref{theorem_main_theorem_order_complex} using topological data from~$\Delta$.
\begin{theorem}
\label{theorem_manifold_homology}
Assume that $\Delta$ is homeomorphic to a $k$-dimensional manifold with or without boundary.
Then the reduced homology of
the order complex $\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
is given by
$$
\widetilde{H}_{i}(\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\widetilde{H}_{i}(\Delta)
\otimes
1_{{\mathfrak S}_{n-1}} \:\:\:\:\: \text{ for } i < k, $$
and the top dimensional homology is given by
$$
\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\widetilde{H}_{k}(\Delta)
\otimes
1_{{\mathfrak S}_{n-1}}
\oplus
\bigoplus_{\vec{c}\, \in\, \Int({\Delta})}
S^{B^{*}(\vec{c}\,)},
$$
where $1_{{\mathfrak S}_{n-1}}$ is the trivial representation of ${\mathfrak S}_{n-1}$, and the direct sum is over the interior faces of the manifold~$\Delta$.
Moreover, these isomorphisms hold as
${\mathfrak S}_{n-1}$-modules.
\end{theorem}
\begin{proof}
Since $\Delta$ is homeomorphic to a $k$-dimensional manifold,
we may apply the comment preceding Proposition~3.8.9 of~\cite{EC1},
which states that for any $\vec{c}\in\Delta$,
where $\vec{c}$ is not the empty composition~$(n)$,
we have that $\link_{\vec{c}\,}(\Delta)$ has the homology groups
of a sphere of dimension $k-|\vec{c}\,|+1$
if $\vec{c}$ is on the interior of~$\Delta$,
or the homology groups of a ball of dimension $k-|\vec{c}\,|+1$
if $\vec{c}$ is on the boundary.
Hence if $\vec{c}$ is on the boundary of $\Delta$ it does not contribute
to the reduced homology of $\Delta(\Pi^{*}_{\Delta}-\{\hat{1}\})$.
If instead $\vec{c}$ is in the interior of~$\Delta$
then
by Remark~\ref{remark_dimension_shift}
it will contribute to the
reduced homology group of
dimension $(k-|\vec{c}\,|+1)+|\vec{c}\,|-1=k$.
This is the top homology of the complex.
Finally, observe that the composition~$(n)$
contributes to all homology groups
of $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
when $\Delta$ has nontrivial homology,
and that the Specht module
$S^{B^{*}(n)}$ is the trivial representation~$1_{{\mathfrak S}_{n-1}}$.
\end{proof}
We now give two immediate corollaries of
Theorem~\ref{theorem_manifold_homology},
when $\Delta$ is
homeomorphic to a sphere or a ball.
\begin{corollary}
\label{corollary_sphere}
Suppose that $\Delta$ is homeomorphic to a sphere of dimension $k$. Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$ only has homology in dimension $k$ given by
$$\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\in\Delta}S^{B^{*}(\vec{c}\,)}.$$
\end{corollary}
\begin{corollary}
\label{corollary_ball}
Suppose that $\Delta$ is homeomorphic to a ball of dimension $k$. Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$ only has homology in dimension $k$ given by
$$\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\,\in\Int(\Delta)}S^{B^{*}(\vec{c}\,)}.$$
\end{corollary}
Next we obtain a result about
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
when $\Delta$ is shellable.
\begin{proposition}
\label{proposition_pure_shellable}
Suppose that $\Delta$ is a shellable complex of dimension $k$.
Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
only has reduced homology in dimension $k$
given by
$$\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\,\in \Delta}
\widetilde{\beta}_{k - |\vec{c}\,| + 1}(\link_{\vec{c}}(\Delta)) \cdot
S^{B^{*}(\vec{c}\,)} . $$
\end{proposition}
\begin{proof}
Note that the face $\vec{c}$ has dimension
$|\vec{c}\,|-2$. Hence the link
$\link_{\vec{c}}(\Delta)$ has dimension
$k-\dim(\vec{c}\,)-1 = k - |\vec{c}\,| + 1$,
by equation~\eqref{equation_dimension}.
Since the link is shellable, all of its reduced homology
occurs
in dimension $k - |\vec{c}\,| + 1$ and this contributes
only to the reduced homology of dimension~$k$
of $\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$ by
Remark~\ref{remark_dimension_shift}.
Lastly, the betti number $\widetilde{\beta}_{k-|\vec{c}\,|+1}$
is explained by the fact that a shellable complex has the homotopy type
of a wedge of spheres of the same dimension.
\end{proof}
\section{The representation ring}
\label{section_representation_ring}
The {\em representation ring} $R(G)$ of a group $G$
is the free abelian group with
generators given by representations $V$ of~$G$
modulo the subgroup generated by $V + W - V \oplus W$.
Elements of the representation ring are called virtual representations
because summands can have negative coefficients. For finite groups,
complete reducibility implies $R(G)$ is just the free abelian group
generated by
the irreducible representations~$V$ of~$G$.
\begin{remark}
\label{remark_trivial_action}
{\rm
Suppose $G$ acts trivially on the space $V$.
Then $V \otimes W \cong_{G} \dim(V) \cdot W$
in the representation ring~$R(G)$.
}
\end{remark}
\begin{proof}
Since $G$ acts trivially on $V$ we know that
$V \cong_{G} \mathbb{C}^{\dim(V)}$.
Thus,
$V\otimes W \cong_{G} \mathbb{C}^{\dim(V)} \otimes W
\cong_{G} \dim(V) \cdot W.$
\end{proof}
In the representation ring we can compute the
alternating sum of the homology groups
of $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$,
which we do in the following proposition.
This can be seen as ${\mathfrak S}_{n-1}$-analogue of
the reduced Euler characteristic.
\begin{proposition}
\label{proposition_virtual}
As virtual ${\mathfrak S}_{n-1}$-representations we have that
$$ \bigoplus_{i \geq -1} (-1)^{i} \cdot \widetilde{H}_{i}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
\cong
\bigoplus_{\vec{c}\, \in \Delta}
(-1)^{|\vec{c}\,| - 1} \cdot
\widetilde{\chi}(\link_{\vec{c}\,}(\Delta))
\cdot
S^{B^{*}(\vec{c}\,)} .
$$
\end{proposition}
\begin{proof}
We begin the proof by applying alternating sums to both sides of
Theorem~\ref{theorem_main_theorem_order_complex}.
\begin{align*}
\bigoplus_{i \geq -1} (-1)^{i} \cdot \widetilde{H}_{i}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
& \cong
\bigoplus_{i \geq -1} (-1)^{i} \cdot
\bigoplus_{\vec{c}\, \in \Delta}
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
\otimes
S^{B^{*}(\vec{c}\,)} \\
& \cong
\bigoplus_{\vec{c}\, \in \Delta}
\bigoplus_{i \geq -1} (-1)^{i} \cdot
\widetilde{\beta}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
\cdot
S^{B^{*}(\vec{c}\,)} \\
& \cong
\bigoplus_{\vec{c}\, \in \Delta}
(-1)^{|\vec{c}\,| - 1} \cdot
\bigoplus_{j \geq -1} (-1)^{j} \cdot
\widetilde{\beta}_{j}(\link_{\vec{c}\,}(\Delta))
\cdot
S^{B^{*}(\vec{c}\,)} \\
& \cong
\bigoplus_{\vec{c}\, \in \Delta}
(-1)^{|\vec{c}\,| - 1} \cdot
\widetilde{\chi}(\link_{\vec{c}\,}(\Delta))
\cdot
S^{B^{*}(\vec{c}\,)} ,
\end{align*}
where the second step is by
Remark~\ref{remark_trivial_action},
since ${\mathfrak S}_{n-1}$ acts trivially on
$\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))$.
In the last step we used that the alternating sum of
the Betti numbers is the reduced Euler characteristic.
\end{proof}
The next lemma is straightforward to prove using jeu-de-taquin;
see~\cite{Sagan} or~\cite[A.1.2]{EC2}.
\begin{lemma}
\label{lemma_permutation_module_isomorphism}
The permutation module~$M^{B^{\#}(\vec{c}\,)}$
is isomorphic to the direct sum over
all border strip Specht modules~$S^{B^{*}(\vec{d}\,)}$
for~$\vec{d}\leq^{*}\vec{c}$,
that is,
$$ M^{B^{\#}(\vec{c}\,)}
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{d}\,\leq^{*}\,\vec{c}}S^{B^{*}(\vec{d}\,)} . $$
\end{lemma}
\begin{proof}
Recall that the border strip of shape $A^{\#}(\vec{c}\,)$ was defined
in Section~\ref{section_border_strips}.
We have the isomorphism
$S^{A^{\#}(\vec{c}\,)} \cong_{{\mathfrak S}_{n-1}} M^{A^{\#}(\vec{c}\,)}$
because the rows of the shape~$A(\vec{c}/1)$ are non-overlapping,
thus polytabloids of shape~$A(\vec{c}/1)$
are tabloids of shape~$A(\vec{c}/1)$.
Additionally, we have
$M^{A^{\#}(\vec{c}\,)}\cong_{{\mathfrak S}_{n-1}}M^{B^{\#}(\vec{c}\,)}$,
since tabloids are defined as row equivalence classes of tableaux
and $A(\vec{c}/1)$ and $B(\vec{c}/1)$ have the same rows.
Combining these two $\mathfrak{S}_{n-1}$-isomorphisms
yields $M^{B^{\#}(\vec{c}\,)}\cong S^{A^{\#}(\vec{c}\,)}$.
Now consider the $k-1$ empty boxes
situated to the left of every row in
the Specht module defined by the shape~$A^{\#}(\vec{c}\,)$,
but above the last box of the previous row.
For each of these boxes
perform
a jeu-de-taquin slide into this box.
For each slide, there are two alternatives.
If the slide is horizontal, it moves
the upper row one step to the left such that
the two rows overlap in one position.
If the slide is vertical then every
entry in the lower row moves one step up.
After performing all the $k-1$ slides the result
is a border shape of shape~$B^{\#}(\vec{c}\,)$,
where the composition~$\vec{c}$ is less
than or equal to the composition $\vec{d}$
in the dual order.
\end{proof}
Proposition~\ref{proposition_virtual}
can also be proved using the Hopf trace formula;
see~\cite[Theorem 2.3.9]{Wachs_III}.
\begin{proof}[Second proof of Proposition~\ref{proposition_virtual}.]
Recall that
$\widetilde{H}_{i}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})) \cong \widetilde{H}_{i}(Q_{\Delta}^{*})$.
By applying the Hopf trace formula we have that
\begin{align*}
\bigoplus_{i \geq -1}
(-1)^{i} \cdot \widetilde{H}_{i}(Q^{*}_{\Delta})
& \cong
\bigoplus_{i \geq -1}
(-1)^{i} \cdot C_{i}(Q^{*}_{\Delta}) \\
& \cong
\bigoplus_{\vec{d} \in \Delta}
(-1)^{|\vec{d}\,|} \cdot M^{B^{\#}(\vec{d}\,)} \\
& \cong
\bigoplus_{\vec{d} \in \Delta}
(-1)^{|\vec{d}\,|} \cdot
\bigoplus_{\vec{c}\, \leq^{*} \vec{d}}
S^{B^{*}(\vec{c}\,)} \\
& \cong
\bigoplus_{\vec{c} \in \Delta}
\sum_{\onethingatopanother{\vec{d} \geq^{*} \vec{c}}{\vec{d} \in \Delta}}
(-1)^{|\vec{d}\,|} \cdot
S^{B^{*}(\vec{c}\,)} \\
& \cong
\bigoplus_{\vec{c} \in \Delta}
(-1)^{|\vec{c}\,| - 1} \cdot
\sum_{\onethingatopanother{\vec{d} \geq^{*} \vec{c}}{\vec{d} \in \Delta}}
(-1)^{|\vec{d}\,| - |\vec{c}\,| - 1} \cdot
S^{B^{*}(\vec{c}\,)} .
\end{align*}
Notice that in the second isomorphism
we have used that the chain space~$C_{i}(Q_{\Delta}^{*})$ has basis given by
all ordered set partitions into $i+2$ parts
with type in~$\Delta$.
This is equivalent
to the direct sum over all permutation modules~$M^{B^{\#}(\vec{d}\,)}$ where $\vec{d}\in\Delta$
is a composition of~$n$ into $i+2$~parts. The remaining step is to observe that the inner sum of the last line
is given by
the reduced Euler characteristic~$\widetilde{\chi}(\link_{\vec{c}}(\Delta))$.
\end{proof}
We observe that in the case when
the order complex $\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
has all its reduced homology concentrated in
one dimension,
the second proof of Proposition~\ref{proposition_virtual}
which uses the Hopf trace formula
gives a shorter proof of our main result
Theorem~\ref{theorem_main_theorem_order_complex}.
Lastly, by taking dimension on both sides of
Proposition~\ref{proposition_virtual}
we obtain the reduced Euler characteristic
of~$\triangle(\Pi^{*}_{\Delta} -\{\hat{1}\})$.
\begin{corollary}
\label{corollary_Euler}
The reduced Euler characteristic of $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
is given by
$$
\widetilde{\chi}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
=
\sum_{\vec{c}\, \in \Delta}
(-1)^{|\vec{c}\,| - 1} \cdot
\widetilde{\chi}(\link_{\vec{c}\,}(\Delta))
\cdot
\beta^{*}_{n}(\vec{c}\,) .
$$
\end{corollary}
This corollary extends Theorem~3.1
from~\cite{Ehrenborg_Readdy_I}.
\section{The homotopy type of $\Pi^{*}_{\Delta}$}
We turn our attention to the homotopy type
of the order complex $\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$.
By combining the poset fiber theorems
of Quillen~\cite{Quillen}
and Bj\"orner, Wachs and
Welker~\cite{Bjorner_Wachs_Welker}
we obtain the next result.
Recall that $*$ denotes the (free) join of complexes.
\begin{theorem}
The order complex of $\Pi^{*}_{\Delta}-\{\hat{1}\}$
is homotopy equivalent to
the complex of ordered set partitions~$Q_{\Delta}^{*}$,
that is,
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}) \simeq Q_{\Delta}^{*}$.
Furthermore, the following
homotopy equivalence holds:
$$
Q^{*}_{\Delta}
\: \simeq \:
\triangle(\Delta - \{(n)\})
\: \vee \:
\{ Q^{*}_{\vec{c}} * \link_{\vec{c}}(\Delta)
\: : \:
\vec{c} \in \Delta - \{(n)\} \} ,
$$
where $\vee$ denotes identifying each vertex
$\vec{c}$ in $\triangle(\Delta - \{(n)\}$
with any vertex in~$Q^{*}_{\vec{c}}$.
In the case when the complex~$\Delta$ is connected
then
the homotopy equivalence simplifies to
$$
Q^{*}_{\Delta}
\simeq
\bigvee_{\vec{c} \in \Delta}
Q^{*}_{\vec{c}} * \link_{\vec{c}}(\Delta) .
$$
\end{theorem}
\begin{proof}
The first homotopy equivalence
follows by applying Quillen's fiber lemma
to the forgetful map $f$.
This yields
$\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}) \simeq \triangle(Q_{\Delta}^{*}-\{\hat{1}\})
\cong Q_{\Delta}^{*}$, since
$\triangle(Q_{\Delta}^{*} - \{\hat{1}\})$
is the barycentric subdivision of~$Q_{\Delta}^{*}$.
The second homotopy equivalence in both cases follows
by Theorem~1.1
in~\cite{Bjorner_Wachs_Welker}, with the same
reasoning as in the proof of Theorem~\ref{theorem_main_result}.
Furthermore,
when $\vec{c} = (n)$ then
the complex~$Q^{*}_{\vec{c}}$ is the empty complex,
which is the identity for the join.
\end{proof}
\begin{corollary}
Let $\Delta$ be a connected
simplicial complex.
Assume furthermore that each link
(including $\Delta$)
$\link_{\vec{c}}(\Delta)$ is a wedge of spheres.
Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
is also a wedge of spheres.
Furthermore, the number
of $i$-dimensional spheres is given by the
sum
\begin{equation}
\sum_{\vec{c} \in \Delta}
\beta^{*}_{n}(\vec{c}\,)
\cdot
\widetilde{\beta}_{i - |\vec{c}\,| + 1}(\link_{\vec{c}\,}(\Delta)) ,
\label{equation_i_spheres}
\end{equation}
where $\widetilde{\beta}_{j}$ denotes the $j$th reduced Betti number.
\label{corollary_every_link_wedge_spheres}
\end{corollary}
Next we have the homotopy versions
of Corollaries~\ref{corollary_sphere}
and~\ref{corollary_ball}.
To prove the next two corollaries,
we are again using
the comment preceding Proposition~3.8.9 of~\cite{EC1}
to determine the reduced Betti numbers of the links.
\begin{corollary}
\label{corollary_homotopy_sphere}
Suppose that $\Delta$ is homeomorphic to a sphere of dimension $k$.
Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
is a wedge of $k$-dimensional spheres
and the number of spheres is given by the sum:
$$ \sum_{\vec{c}\in\Delta} \beta^{*}_{n}(\vec{c}\,). $$
\end{corollary}
\begin{corollary}
\label{corollary_homotopy_ball}
Suppose that $\Delta$ is homeomorphic to a ball of dimension $k$.
Then the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
is a wedge of $k$-dimensional spheres
and the number of spheres is given by the sum:
$$ \sum_{\vec{c}\,\in\Int(\Delta)} \beta^{*}_{n}(\vec{c}\,). $$
\end{corollary}
We end this section with a discussion of how
we can lift discrete Morse matchings
from the links of~$\Delta$ to the complex
of order set partitions~$Q^{*}_{\Delta}$.
\begin{definition}
For an ordered set partition
$\sigma = (C_{1}, C_{2}, \ldots, C_{k})$ of $n$,
where
$C_{i} = \{c_{i,1} < c_{i,2} < \cdots< c_{i,j_{i}}\}$
and
$|C_{i}| = j_{i}$,
define the permutation
$\perm(\sigma)\in{\mathfrak S}_{n}$ to be the elements
of the blocks listed in the order of the blocks,
that is,
$$
\perm(\sigma)
=
c_{1,1}, c_{1,2}, \ldots, c_{1,j_{1}},
c_{2,1}, c_{2,2}, \ldots,
c_{k,j_{k}} . $$
\end{definition}
Define the descent set of an ordered
set partition $\sigma$ to be
$\Des(\sigma) = \Des(\perm(\sigma))$.
Observe that the descent composition of an
ordered set partition is an order preserving
map from the poset of ordered set partitions~$Q_{n}$
to the poset of compositions~$\Comp_{n}$, that is,
$\Des: Q_{n}^{*} \longrightarrow \Comp(n)$ is a poset map.
\begin{lemma}
\label{lemma_patchwork}
Let $\Delta$ be a filter in the composition poset $\Comp(n)$.
For the order preserving map
$\Des : Q^{*}_{\Delta} \longrightarrow \Delta$
the poset fiber $\Des^{-1}(\vec{c}\,)$
is the (poset) direct sum of $\beta_{n}^{*}(\vec{c}\,)$
copies of the poset
$\link_{\vec{c}\,}(\Delta)
=
\{\vec{d} \in \Delta \: : \: \vec{d} \leq^{*} \vec{c}\,\}$.
\end{lemma}
\begin{proof}
Let $\sigma$
be an ordered set partition and assume that
the $i$th block $C_{i}$ is the disjoint union
of the two non-empty sets $X$ and $Y$ such that
$\max(X) < \min(Y)$.
Observe now that the two ordered set partitions~$\sigma$
and
$(\ldots, C_{i-1}, X, Y, C_{i+1}, \ldots)$
have the same descent composition,
since there is no descent
between blocks~$X$ and~$Y$.
Let $\vec{c}$ be a composition in the filter $\Delta$.
For any ordered set partition $\tau$
in the fiber $\Des^{-1}(\vec{c}\,)$
we know that $\tau$ has descent composition $\vec{c}$,
that is, $\Des(\tau)=\vec{c}$.
As $\tau$ can only have descents between blocks,
we know the minimal elements of
$\Des^{-1}(\vec{c}\,)$ have the form $\sigma(\alpha,\vec{c}\,)$
for $\alpha\in{\mathfrak S}_{n}$ satisfying $\Des(\alpha)=\vec{c}$
and $\alpha_{n}=n$.
To remain in the same
fiber as these minimal elements,
we are free to break blocks
as in previous paragraph,
hence
$$ \Des^{-1}(\vec{c}\,)
=
\{\sigma(\alpha,\vec{d}\,)
\: : \:
\vec{c} \leq^{*} \vec{d},
\: \vec{d} \in \Delta,
\: \Des(\alpha)=\vec{c},\,\alpha_{n}=n\}
. $$
Notice that the poset $\link_{\vec{c}\,}(\Delta)$
is isomorphic to
the poset
$\{\sigma(\alpha,\vec{d}\,)\: : \: \vec{d}\leq\vec{c}, \, \vec{d} \in \Delta\}$
for a fixed permutation $\alpha\in{\mathfrak S}_{n}$
satisfying $\Des(\alpha)=\vec{c}$ and $\alpha_{n}=n$.
Finally, for a composition
$\vec{e} \in \link_{\vec{c}\,}(\Delta)$
and a permutation $\beta \in {\mathfrak S}_{n}$
different from $\alpha$
such that
$\Des(\beta)=\vec{c}$ and $\beta_{n}=n$,
consider the two ordered set partitions
$\sigma(\beta,\vec{e}\,)$ and $\sigma(\alpha,\vec{d}\,)$,
where $\vec{d} \in \link_{\vec{c}\,}(\Delta)$.
By examining the first increasing run in the permutations
$\alpha$ and $\beta$ where their elements differ, we conclude
that the two ordered set partitions
$\sigma(\beta,\vec{e}\,)$ and $\sigma(\alpha,\vec{d}\,)$
are incomparable.
Thus the fiber~$\Des^{-1}(\vec{c}\,)$ is a direct sum of
copies of the poset $\link_{\vec{c}\,}(\Delta)$,
one for each permutation $\alpha$ in ${\mathfrak S}_{n}$ satisfying
$\Des(\alpha)=\vec{c}$ and $\alpha(n)=n$.
\end{proof}
\begin{theorem}
Let $\Delta$ be a simplicial complex of compositions
such that every link $\link_{\vec{c}\,}(\Delta)$ has a
Morse matching where the critical cells are facets
of the link $\link_{\vec{c}\,}(\Delta)$. Then the simplicial
complex $Q_{\Delta}^{*}$ has a Morse matching, where
the number of $i$-dimensional critical cells is given by
equation~\eqref{equation_i_spheres}.
\label{theorem_wedge_of_spheres}
\end{theorem}
\begin{proof}
Apply the
Patchwork Theorem~\cite[Theorem~11.10]{Kozlov_book}
to the poset map
$\Des:Q_{\Delta}^{*} \longrightarrow \Delta$.
By Lemma~\ref{lemma_patchwork},
each fiber is a direct sum of
links of $\Delta$,
each of which has a Morse matching.
Each critical cell is a facet.
Hence $Q_{\Delta}^{*}$ is homotopy equivalent to a wedge of spheres,
and thus the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$ is
also a wedge of spheres.
The number of $i$-dimensional critical cells
of $Q_{\Delta}^{*}$ in the fiber $\Des^{-1}(\vec{c}\,)$
is the number of critical cells of dimension $i-|\vec{c}\,|+1$
in the link $\link_{\vec{c}\,}(\Delta)$ times the number of
copies of the link, that is $\beta^{*}_{n}(\vec{c}\,)$.
By summing over all compositions $\vec{c}$ in $\Delta$
the result follows.
\end{proof}
Now suppose that $\Delta$ is a non-pure shellable complex
in the sense of~\cite{Bjorner_Wachs_non_pure_I}.
Then each link
in $\Delta$ is also shellable, and thus for each link there exists
a discrete Morse matching whose critical
cells are facets of the link;
see Chapter~12 of~\cite{Kozlov_book}.
\begin{corollary}
\label{shellable_order_complex}
If $\Delta$ is a non-pure shellable complex
then Theorem~\ref{theorem_wedge_of_spheres}
applies and the simplicial complex~$Q_{\Delta}^{*}$ has
a Morse matching where the number of $i$-dimensional
critical cells is given by equation~\eqref{equation_i_spheres}.
\end{corollary}
\begin{proof}
This follows directly from two observations:
(i)
a non-pure shellable complex has a Morse matching with all critical cells being facets,
(ii)
each link of a non-pure shellable complex
is non-pure shellable.
See Section~12.1 in~\cite{Kozlov_book}.
\end{proof}
\section{Examples}
\label{section_examples}
In this section we use
Theorem~\ref{theorem_main_theorem_order_complex} and its consequences from Section~\ref{section_consequences}
to derive results about various filters~$\Pi^{*}_{\Delta}$.
\begin{example}
\label{example_ball}
{\rm
Let $\vec{d}$ be a composition of $n$
into $k+2$ parts and let $\Delta$ be
the simplex generated by~$\vec{d}$.
Since the simplex is homeomorphic to a $k$-dimensional ball, by Corollary~\ref{corollary_ball}
we have that the $k$th reduced homology group
is given by
\begin{align*}
\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
&
\cong_{{\mathfrak S}_{n-1}}
S^{B^{*}(\vec{d}\,)} ,
\end{align*}
since the only face of $\Delta$ in
the interior of $\Delta$ is the facet $\vec{d}$.
This example illustrates Theorems~5.3
and~7.4
in~\cite{Ehrenborg_Jung}.
Moreover,
this is the base case of the authors' proof of
Theorem~\ref{theorem_main_result} using
the Mayer--Vietoris sequence.
}
\end{example}
\begin{example}
\label{example_sphere}
{\rm
Let $\vec{d}$ be a composition
into $k+3$ parts and let $\Delta$ be
the boundary of the simplex generated by~$\vec{d}$,
that is, $\Delta$ is homeomorphic to a $k$-dimensional
sphere. Then $\Delta$ is shellable
and the order complex $\triangle(\Pi^{*}_{\Delta} -\{\hat{1}\})$
is a wedge of $k$-dimensional spheres.
Now by Corollary~\ref{corollary_sphere}
we have that the $k$th reduced homology group
is given by
$$
\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} <^{*} \vec{d}} S^{B^{*}(\vec{c}\,)}
\cong_{{\mathfrak S}_{n-1}}
M^{B^{\#}(\vec{d}\,)}
/
S^{B^{*}(\vec{d}\,)} .
$$
Note that we have used
Lemma~\ref{lemma_permutation_module_isomorphism}
to express the permutation module $M^{B^{\#}(\vec{d}\,)}$
as a direct sum of Specht modules.
}
\end{example}
\begin{example}
\label{example_k_r}
{\rm
Let $\vec{d}$ be a composition of $n$ into $k+r$ parts, where $r\geq1$.
Let $\Delta$ be the $k$-skeleton of the simplex generated
by the composition~$\vec{d}$.
Note that $\Delta$ is shellable, so by
Corollary~\ref{shellable_order_complex}
the order complex $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$ is a wedge of
$k$-dimensional spheres. By
Proposition~\ref{proposition_pure_shellable} we have
the following calculation in the representation ring:
\begin{align}
\widetilde{H}_{k}(\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\}))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\substack{\vec{c}\,<\vec{d}\\ |\vec{c}\,|\leq k+2}}
\binom{k+r-|\vec{c}\,|-1}{k-|\vec{c}\,|+2}\cdot S^{B^{*}(\vec{c}\,)} .
\label{equation_k_r}
\end{align}
Here we have used that
$\widetilde{\beta}_{k-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta))
= (-1)^{k-|\vec{c}\,|+1} \cdot \widetilde{\chi}(\link_{\vec{c}\,}(\Delta))$
since $\link_{\vec{c}\,}(\Delta)$ is shellable.
Lastly, we also used a basic identity on
the alternating sum of binomial coefficients,
which arises in computing the Euler characteristic of the link.
}
\end{example}
\begin{example}
[The $d$-divisible partition lattice with minimal elements removed]
\label{example_boundary_simplex}
{\rm
Let $n$ be a multiple of $d$. Consider the boundary
of the simplex
generated by the composition~$(d,d,\ldots,d)$ of~$n$.
Then~$\Delta$ is a $(n/d-3)$-dimensional simplicial complex,
and $\Pi^{*}_{\Delta}$ is the $d$-divisible partition lattice
without its minimal elements.
By applying Example~\ref{example_sphere}
we obtain that
$\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
is a wedge of $(n/d-3)$-dimensional spheres
and
the reduced homology group is given by
$\widetilde{H}_{n/d-3}(\Pi^{*}_{\Delta}-\{\hat{1}\})
\cong_{{\mathfrak S}_{n-1}}
M^{B^{\#}(d,\ldots,d,d)}
/
S^{B^{*}(d,d,\ldots,d)}$.
}
\end{example}
Setting $d=1$ in the last example shows that the action of
${\mathfrak S}_{n-1}$ on the reduced homology group of
$\triangle(\Pi_{n} - \{\hat{0},\hat{1}\})$ is
$M^{B^{\#}(1,\ldots,1,1)} = M^{B(1,\ldots,1)}$,
which is the regular representation
of~${\mathfrak S}_{n-1}$.
\begin{example}[The truncated $d$-divisible partition lattice]
{\rm
To generalize Example~\ref{example_boundary_simplex}
and specialize Example~\ref{example_k_r},
let $n = (k+r) \cdot d$
and consider the $k$-skeleton of the simplex generated by
the composition $(d,d, \ldots, d)$ of~$n$.
Here $\Pi^{*}_{\Delta}$
consists of all set partitions in the $d$-divisible partition lattice
with at most $k+2$ parts.
Directly we have that
the order complex
$\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$
is a wedge of $k$-dimensional spheres
and
its $k$-dimensional reduced homology is
given by equation~\eqref{equation_k_r}.
}
\end{example}
\begin{example}
{\rm
An integer partition
$\lambda =
\{\lambda_{1}^{m_{1}}, \lambda_{2}^{m_{2}}, \ldots, \lambda_{p}^{m_{p}}\}$
of the non-negative integer~$n$ is called a {\em knapsack partition}
if all the sums
$\sum_{i=1}^{p} e_{i} \cdot \lambda_{i}$,
where $0 \leq e_{i} \leq m_{i}$, are distinct.
In other words, $\lambda$ is a knapsack partition if
$$
\left|
\left\{\sum_{i=1}^{p} j_{i}\cdot\lambda_{i}
: 0 \leq j_{i} \leq m_{i}\right\}
\right|
=
\prod_{i=1}^{p} (m_{i}+1) .
$$
For a knapsack partition~$\lambda$ into $k-1$ parts
of $n-m$, where $m < n$,
define~$\Delta_{\lambda,m}$
to be the simplicial complex~$\Delta_{\lambda,m}$
which has the facets
$(c_{1}, c_{2}, \ldots, c_{k-1}, c_{k})$
with $\type(c_{1}, c_{2}, \ldots, c_{k-1}) = \lambda$
and the last part~$c_{k}$ is~$m$.
The complex~$\Delta_{\lambda,m}$
is homeomorphic to a $(k-2)$-dimensional ball;
see the proof of Theorem~4.4 in~\cite{Ehrenborg_Readdy_I}.
Applying Corollary~\ref{corollary_ball},
we obtain the following result:
$$ \widetilde{H}_{k-2}(\triangle(\Pi^{*}_{\Delta_{\lambda,m}}-\{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \in \Int(\Delta_{\lambda,m})} S^{B^{*}(\vec{c}\,)} . $$
Furthermore,
the set of interior faces of $\Delta_{\lambda,m}$
is given by compositions
$\vec{c}$ in~$\Delta_{\lambda,m}$
such that when each part of $\vec{c}$ is written as a sum of parts of
$\lambda$, those parts are distinct.
This example is Theorem~10.3
in~\cite{Ehrenborg_Jung}. Moreover, $\Delta_{\lambda,m}$
is shellable, so Theorem~\ref{theorem_wedge_of_spheres}
yields a Morse matching
of $Q_{\Delta_{\lambda,m}}^*$;
see Theorem~8.2 of~\cite{Ehrenborg_Jung}.
}
\end{example}
\section{The Frobenius complex}
\label{section_Frobenius}
\newcommand{\TableOne}{
\begin{table}
\caption{The reduced homology groups of the
order complex $\triangle(\Pi_{n}^{\langle 3,5,7 \rangle} - \{\hat{1}\})$
for the even cases $n = 8$, $10$, $12$ and $14$.
Instead of writing out the notation
$S^{B^{*}(\vec{d},\vec{b}\,)}$ for the Specht modules,
we have drawn the associated border shapes.
Observe that when a row has three boxes,
there is overlap with the row above.
}
\label{table_Pi_3_5_7_even}
\begin{tabular}{c|c|c}
$n$ & $\widetilde{H}_{0}$ & $\widetilde{H}_{2}$ \\
\hline
8
&
$\Yboxdim{5pt}
\young(::\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
&
0 \\
\hline
10
&
$\Yboxdim{5pt}
\young(::\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
$
&
0
\\
\hline
\vphantom{\rule{0 mm}{8 mm}}
12
\vphantom{\rule{0 mm}{8 mm}}
&
$\Yboxdim{5pt}
\young(:::::\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
$
&
$\Yboxdim{5pt}
\young(::::::\hfil\hfil,::::\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
$
\\
\hline
\vphantom{\rule{0 mm}{8 mm}}
14
\vphantom{\rule{0 mm}{8 mm}}
&
$\Yboxdim{5pt}
\young(:::::::\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
$
&
$\Yboxdim{5pt}
\young(::::::\hfil\hfil\hfil\hfil,::::\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil,::::\hfil\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
$
\\
&
&
$\Yboxdim{5pt}
\oplus\,
\young(:::::::::\hfil\hfil,:::::::\hfil\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil,:::::::\hfil\hfil\hfil,:::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
\end{tabular}
\end{table}
}
\newcommand{\TableTwo}{
\begin{table}
\caption{The reduced homology groups of the
order complex $\triangle(\Pi_{n}^{\langle 3,5,7 \rangle} - \{\hat{1}\})$
for the odd cases $n = 9$, $11$, $13$ and $15$.}
\label{table_Pi_3_5_7_odd}
\begin{tabular}{c|c|c}
$n$ & $\widetilde{H}_{1}$ & $\widetilde{H}_{3}$ \\
\hline
9
&
$\Yboxdim{5pt}
\young(::::\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
$
&
0 \\
\hline
11
&
$\Yboxdim{5pt}
\young(::::\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil,:::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
&
0 \\
\hline
13
&
$\Yboxdim{5pt}
\young(:::::::\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil\hfil\hfil,:::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
\oplus
\young(::::::::::\hfil\hfil,:::::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
&
0 \\
&
$\Yboxdim{5pt}
\oplus\,
\young(::::\hfil\hfil\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil,:::::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
$
&
0 \\
\hline
&
$\Yboxdim{5pt}
\young(:::::::\hfil\hfil\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil\hfil\hfil,::\hfil\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil)
\oplus
\young(:::::::\hfil\hfil\hfil\hfil\hfil\hfil,:::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
&
\\
15
&
$\Yboxdim{5pt}
\oplus\,
\young(::::::::::::\hfil\hfil,:::::\hfil\hfil\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
\oplus
\young(:::::::::\hfil\hfil\hfil\hfil,:::::::\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
\oplus
\young(::::::::::::\hfil\hfil,:::::::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil\hfil\hfil)
$
&
$\Yboxdim{5pt}
\young(::::::::\hfil\hfil,::::::\hfil\hfil\hfil,::::\hfil\hfil\hfil,::\hfil\hfil\hfil,\hfil\hfil\hfil)
$
\\
&
$\Yboxdim{5pt}
\oplus\,
\young(::::::::::\hfil\hfil\hfil\hfil,:::::\hfil\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil\hfil)
$
&
\end{tabular}
\end{table}
}
We now consider a different class of examples stemming
from~\cite{Clark_Ehrenborg}.
Let~$\Lambda$ be a semigroup of positive integers,
that is, a subset of the positive integers which is closed
under addition.
Let~$\Delta_{n}$ be the collection of all compositions of $n$
whose parts belong to $\Lambda$,
that is,
$$
\Delta_{n}
=
\{ (c_{1}, \ldots, c_{k}) \in \Comp(n) \:\: : \:\:
c_{1}, \ldots, c_{k} \in \Lambda \} .
$$
Since $\Lambda$ is closed under addition,
we obtain that $\Delta_{n}$ is a filter in the poset of
compositions~$\Comp(n)$ and hence we view it as
a simplicial complex.
This complex is known as the Frobenius complex;
see~\cite{Clark_Ehrenborg}.
Moreover, since $\Lambda$ is a semigroup,
the collection of integer partitions of $n$
with parts in~$\Lambda$ is a filter, therefore,
using
Lemma~\ref{lemma_integer_filter}
the associated filter in
the partition lattice is given by
$$ \Pi_{n}^{\Lambda}
=
\{ \{B_{1}, \ldots, B_{k}\} \in \Pi_{n}
\:\: : \:\: |B_{1}|, \ldots, |B_{k}| \in \Lambda\} . $$
Let $\Psi_{n}$ be the generating function
$$ \Psi_{n}
=
\sum_{i \geq -1} \widetilde{\beta}_{i}(\Delta_{n}) \cdot t^{i+1} . $$
Observe that for a composition $\vec{c}$ in $\Delta_{n}$
we have that
the link $\link_{\vec{c}}(\Delta_{n})$ is given by
the join
\begin{equation}
\link_{\vec{c}}(\Delta_{n})
=
\Delta_{c_{1}} * \Delta_{c_{2}} * \cdots * \Delta_{c_{k}} .
\label{equation_Frobenius_link}
\end{equation}
We can apply the K\"unneth theorem to obtain
that the $i$th reduced Betti number
of the link is given by
$$
\widetilde{\beta}_{i}(\link_{\vec{c}\,}(\Delta_{n}))
=
[t^{i+1}] \Psi_{c_{1}} \cdot \Psi_{c_{2}} \cdots \Psi_{c_{k}} .
$$
Using Theorem~\ref{theorem_main_theorem_order_complex},
the $i$th reduced Betti number of
the order complex $\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\})$
is given in the representation ring of ${\mathfrak S}_{n-1}$ by
\begin{align*}
\widetilde{H}_{i}(\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\}))
\cong_{{\mathfrak S}_{n-1}}
\sum_{\vec{c}} [t^{i_{\vec{c}\,}+1}]
\Psi_{c_{1}} \cdot
\Psi_{c_{2}} \cdots
\Psi_{c_{k}}
\cdot
S^{B^{*}(\vec{c}\,)} ,
\end{align*}
where the sum is over all compositions
$\vec{c} = (c_{1}, c_{2}, \ldots, c_{k})$ of $n$.
A more explicit approach is possible when the
complex $\Delta_{n}$ has a discrete Morse matching.
By combining equation~\eqref{equation_Frobenius_link},
Lemma~\ref{lemma_join_critical_cells},
and a Morse matching from~\cite{Clark_Ehrenborg},
we create a Morse matching on every link.
We will see this method in the remainder of this section.
We continue by studying one concrete example.
Let $a$ and $d$ be two positive integers.
Let~$\Lambda$ be the semigroup
generated by the arithmetic progression
\begin{align*}
\Lambda
& =
\langle a, a+d, a+2d, \ldots \rangle . \\
\intertext{Since for $j \geq a$ we have that
$a + j \cdot d = d \cdot a + a + (j-a) \cdot d$,
the semigroup is generated by
the finite arithmetic progression
}
\Lambda
& =
\langle a, a+d, a+2d, \ldots, a+(a-1)d\, \rangle .
\end{align*}
Clark and Ehrenborg proved that the Frobenius complex $\Delta_{n}$
is a wedge of spheres of different dimensions;
see~\cite[Theorem~5.1]{Clark_Ehrenborg}.
Observe that their result
is formulated in terms of sets,
instead of compositions.
However,
the two notions are equivalent via the
natural bijection given by
sending
a composition $(c_{1}, c_{2}, \ldots, c_{k})$ of~$n$
to the subset
$\{c_{1},c_{1}+c_{2}, \ldots, c_{1} + \cdots + c_{k-1}\}$
of the set~$[n-1]$.
To state their result, let $A$ be the set
$\{a+d, a+2d, \ldots, a+(a-1) \cdot d\}$.
\begin{proposition}
For $n$ in the semigroup $\Lambda$,
there is a discrete Morse matching on the
Frobenius complex $\Delta_{n}$ such that
the critical cells are compositions $\vec{c} = (c_{1}, \ldots, c_{k})$
characterized by
\begin{itemize}
\item[(i)]
All but the last entry of the composition
belongs to the set $A$,
that is,
$c_{1}, \ldots, c_{k-1} \in A$.
\item[(ii)]
The last entry $c_{k}$ belongs to
$\{a\} \cup A$.
\end{itemize}
Furthermore, all the critical cells are facets.
\label{proposition_Clark_Ehrenborg}
\end{proposition}
\begin{proof}
When $a$ and $d$ are relative prime,
that is, $\gcd(a,d) = 1$, this result is
Lemma~5.10
in~\cite{Clark_Ehrenborg}.
When $a$ and $d$ are not relative prime,
the result follows by scaling down
the three parameters~$a$, $d$ and~$n$ by
$a^{\prime} = a/\gcd(a,d)$,
$d^{\prime} = d/\gcd(a,d)$
and
$n^{\prime} = n/\gcd(a,d)$.
Now the result applies the semigroup
$\Lambda^{\prime}
=
\langle a^{\prime}, a^{\prime}+d^{\prime}, a^{\prime}+2d^{\prime},
\ldots\rangle$
and its associated Frobenius complex~$\Delta^{\prime}_{n^{\prime}}$.
However, this complex is isomorphic to $\Delta_{n}$ by
sending the composition
$\vec{c} = (c_{1}, \ldots, c_{k})$
in~$\Delta^{\prime}_{n^{\prime}}$
to the composition
$\gcd(a,d) \cdot \vec{c} =
(\gcd(a,d) \cdot c_{1}, \ldots, \gcd(a,d) \cdot c_{k})$
in~$\Delta_{n}$.
\end{proof}
\begin{corollary}
The order complex
$\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\})$ is
a wedge of spheres.
\end{corollary}
\begin{proof}
Since $\Delta_{n}$ has a discrete Morse matching
where each critical cell is a facet, $\Delta_{n}$ is
homotopy equivalent to a wedge of spheres.
Furthermore, by equation~\eqref{equation_Frobenius_link}
we know that every link of $\Delta_{n}$ is a wedge of
spheres. Finally, by
Corollary~\ref{corollary_every_link_wedge_spheres}
we obtain the result.
\end{proof}
Next we need to extend
Lemma~\ref{lemma_permutation_module_isomorphism}
to collect Specht modules together.
We call the sum $c_{1} + c_{2} + \cdots + c_{j}$
an {\em initial sum} of a composition $\vec{c} = (c_{1}, c_{2}, \ldots, c_{k})$
for $1 \leq j \leq k$.
\begin{definition}
\label{definition_d_b}
For an interval
$[\vec{d}, \vec{b}\,]$ in the lattice of compositions $\Comp(n)$
let
$B^{*}(\vec{d},\vec{b}\,)$
be the skew-shape
where the row lengths are given by
$d_{1}, d_{2}, \ldots, d_{r-1}, d_{r}-1$
and
if the initial sum $d_{1} + \cdots + d_{j}$
is equal to an initial sum of the composition $\vec{b}-1$,
then $j$th row and the $(j+1)$st row overlap in one column. All other rows
of $B^{*}(\vec{d},\vec{b}\,)$ are non--overlapping.
\end{definition}
As an example, if $\vec{d}=(2,5,4,1,3,2)$ and $\vec{b}=(2+5+4,1+3,2)$,
then $B^*(\vec{d},\vec{b}\,)$ is the border strip with row lengths
$2,5,4,1,3$ and $2-1=1$ which overlaps between the rows of length $4$ and $1$
and the rows of length $3$ and~$1$.
Note that
$B^{*}(\vec{d},(n)) = A^{*}(\vec{d}\,)$.
The proof of the next lemma
is the same as the proof of
Lemma~\ref{lemma_permutation_module_isomorphism},
that is, it uses
jeu-de-taquin
moves where two adjacent rows do not overlap.
\begin{lemma}
\label{lemma_interval_in_composition_poset}
Let $\vec{b}$ and $\vec{d}$ be two compositions
in $\Comp(n)$ such that $\vec{d} \leq \vec{b}$.
Then Specht module~$S^{B^{*}(\vec{d},\vec{b})}$
is given by the direct sum
$$
S^{B^{*}(\vec{d},\vec{b})}
\cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{d} \leq \vec{c} \leq \vec{b}} S^{B^{*}(\vec{c}\,)} .
$$
\end{lemma}
In order to state the main result for the semigroup
$\Lambda = \langle a, a + d, a + 2 d, \ldots \rangle$
and the associated filter in the partition lattice,
we need one last definition.
\begin{definition}
For a composition~$\vec{d}$ of $n$ with entries in the set
$\{a\} \cup A$
let $\vec{b}(\vec{d}\,)$ be the composition
greater than or equal to~$\vec{d}$
obtained by adding runs of entries of $\vec{d}$
together where each run ends with the entry $a$.
\end{definition}
As an example, for
$a = 3$, $d = 2$ we have $A = \{5,7\}$.
Hence for the composition
$\vec{d} = (5,3, 7,5,3, 3, 7,5)$
we obtain
$\vec{b}(\vec{d}\,)
= (5+3, 7+5+3, 3, 7+5)
= (8, 15, 3, 12)$.
\begin{remark}
{\rm
Observe that the skew-shape
$S^{B^{*}(\vec{d}, \vec{b}(\vec{d}\,))}$
has the row lengths $d_{1}, \ldots, d_{r-1}, d_{r}-1$
and satisfies the condition that
$d_{i} = a$ if and only if there is overlap between
$i$th and $(i+1)$st rows. See Definition~\ref{definition_d_b}.
}
\label{remark_a_is_special}
\end{remark}
\TableOne
\begin{theorem}
Let $a$ and $d$ be two positive integers
and let $\Pi_{n}^{\Lambda}$ be the
filter in the partition lattice~$\Pi_{n}$
where each partition $\pi$ consists of blocks
whose cardinalities belong to the semigroup~$\Lambda$
generated by the arithmetic progression
$a, a + d, \ldots, a + (a-1) \cdot d$.
Then the $i$th reduced homology group of
the order complex $\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\})$
is given by the direct sum
\begin{align*}
\widetilde{H}_{i}(\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\}))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{d}}
S^{B^{*}(\vec{d}, \vec{b}(\vec{d}\,))} ,
\end{align*}
where the sum is over all compositions~$\vec{d}$ into $i+2$
parts such that every entry belongs to the set
$\{a\} \cup A = \{a, a+d, a + 2 \cdot d, a + (a-1) \cdot d\}$.
\label{theorem_a_d}
\end{theorem}
\begin{proof}
Let $\vec{c}$ be a composition in the complex $\Delta_{n}$.
Using the Morse matching
given by
Proposition~\ref{proposition_Clark_Ehrenborg}
and Lemma~\ref{lemma_critical_cell_join},
we obtain that a critical cell $\vec{d}$
in the link
$\link_{\vec{c}}(\Delta_{n})
=
\Delta_{c_{1}} * \Delta_{c_{2}} * \cdots * \Delta_{c_{k}}$
is a composition
$\vec{d} \leq \vec{c}$
where the entries of $\vec{d}$ belong to the set $\{a\} \cup A$.
Furthermore,
in the run of entries of $\vec{d}$ that sums to the entry~$c_{i}$
of the composition~$\vec{c}$, only the last entry of the run
is allowed to be equal to $a$.
Using Theorem~\ref{theorem_main_theorem_order_complex}
we have
\begin{align*}
\widetilde{H}_{i}(\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\}))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \in \Delta_{n}}
\widetilde{H}_{i-|\vec{c}\,|+1}(\link_{\vec{c}\,}(\Delta_{n}))
\otimes
S^{B^{*}(\vec{c}\,)}. \\
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \in \Delta_{n}}
\bigoplus_{\vec{d}}
S^{B^{*}(\vec{c}\,)},
\end{align*}
where the inner sum consists of critical compositions~$\vec{d}$
satisfying the conditions discussed in the previous paragraph
and with $|\vec{d}\,| = i+2$.
By changing the order of summation
we obtain
\begin{align*}
\widetilde{H}_{i}(\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\}))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{d}}
\bigoplus_{\vec{c}} S^{B^{*}(\vec{c}\,)} ,
\end{align*}
where
the outer direct sum is over all compositions~$\vec{d}$
of $n$ into $i+2$ parts where each part is in the set $\{a\} \cup A$
and
the inner direct sum is over all compositions~$\vec{c}$
greater than $\vec{d}$ obtained by
adding runs of entries of $\vec{d}$ where an entry equal to $a$
can only be at the end of a run.
The inner direct sum is hence given by
the Specht module~$S^{B^{*}(\vec{d},\vec{b}(\vec{d}))}$
by
Remark~\ref{remark_a_is_special}
and
Lemma~\ref{lemma_interval_in_composition_poset},
and therefore the result follows.
\end{proof}
\TableTwo
\begin{corollary}
The order complex $\triangle(\Pi_{n}^{\Lambda} - \{\hat{1}\})$
only has non-vanishing reduced homology
in dimension $i$ when
$n \equiv (i+2) \cdot a \bmod d$.
\end{corollary}
\begin{proof}
Since all entries in the set $\{a\} \cup A$ are
congruent to $a$ modulo $d$, we have
$n = \sum_{j=1}^{i+2} d_{j} \equiv (i+2) \cdot a \bmod d$.
\end{proof}
In Tables~\ref{table_Pi_3_5_7_even}
and~\ref{table_Pi_3_5_7_odd}
we have explicitly calculated the
reduced homology groups
for the order complex
$\triangle(\Pi^{\langle 3,5,7 \rangle}_{n} - \{\hat{1}\})$
for $8 \leq n \leq 15$,
that is, the case $a=3$ and $d=2$.
In this case the previous corollary
implies that the order complex only
has non-vanishing homology in dimensions
of the same parity as~$n$.
\begin{example}
{\rm
When the integer $d$ divides the integer $a$,
the homology groups of $\Pi_{n}^{\Lambda}$ have been studied.
In this case, the filter $\Pi_{n}^{\Lambda}$ consists of all partitions
where the block sizes are divisible by~$d$
and the block sizes are greater than or equal to $a$.
This filter was studied by Browdy~\cite{Browdy},
and our Theorem~\ref{theorem_a_d} reduces to
Browdy's result; see Corollary~5.3.3 in~\cite{Browdy}.
}
\end{example}
\begin{example}
{\rm
The previous example is particularly nice when $d=1$.
The semigroup~$\Lambda$ is given by
$\Lambda = \{n \in \mathbb{P} \: : \: n \geq a\}$
and the filter $\Pi_{n}^{\Lambda}$
consists of all partitions where
$1, 2, \ldots, a-1$ are forbidden block sizes.
In this case it follows by Billera and
Meyers~\cite{Billera_Myers} that $\Delta_{n}$ is non-pure shellable.
Additionally, Bj\"orner and Wachs~\cite{Bjorner_Wachs_non_pure_I}
gave an $EL$-labelling of $\Pi_{n}^{\Lambda} \cup \{\hat{0}\}$.
This order complex was also
considered by Sundaram in Example~4.4 in~\cite{Sundaram_Hopf}.
}
\end{example}
\section{The partition filter $\Pi_{n}^{\langle a,b \rangle}$}
\label{section_partition_filter_a_b}
Let $a$ and $b$ be two relatively prime integers greater than $1$.
Let $\Pi_{n}^{\langle a,b \rangle}$ be the filter in $\Pi_{n}$ generated by all partitions
whose block sizes are all $a$ or $b$.
As an example, $\Pi_{n}^{\langle 2,3 \rangle}$
consists of all partitions in $\Pi_{n}$ with no singleton blocks.
The corresponding complex $\Delta_{n}$
in $\Comp(n)$ consists of all compositions of~$n$
whose parts are contained in the set
$\langle a,b\rangle=\{i\cdot a+j\cdot b\::\:a,b\in\mathbb{N}\}$.
When $a=2$ and $b=3$ the complex $\Delta_{n}$ is known as
the complex of sparse sets;
see~\cite{Clark_Ehrenborg,Kozlov_sparse_sets}.
Following Theorem~4.1 in~\cite{Clark_Ehrenborg},
we define the set
$A = \{ n\in{\mathbb P} : n\equiv0,a,b\text{ or }a+b \bmod ab\}$
and the function $h : A \longrightarrow \mathbb{Z}_{\geq -1}$
as follows:
\begin{equation}
h(n) =
\begin{cases} \frac{2n}{ab}-2 &\mbox{if } n \equiv 0 \bmod{ab}, \\
\frac{2(n-a)}{ab}-1 &\mbox{if } n \equiv a \bmod{ab}, \\
\frac{2(n-b)}{ab}-1 &\mbox{if } n \equiv b \bmod{ab}, \\
\frac{2(n-a-b)}{ab} &\mbox{if } n \equiv a+b \bmod{ab}.
\end{cases}
\label{equation_h}
\end{equation}
Then Theorem 4.1 in~\cite{Clark_Ehrenborg}
states that $\Delta_{n}$ is either homotopy equivalent to
a sphere or is contractible, according to
$$\Delta_{n}\simeq\begin{cases} S^{h(n)} &\mbox{if } n\in A,\\
\mbox{point}&\mbox{otherwise.}
\end{cases}
$$
Using equation~\eqref{equation_Frobenius_link}
we see that if $\vec{c} \in \Delta_{n}$
has any part not in $A$, then $\link_{\vec{c}\,}(\Delta_{n})$ is contractible.
If each part of $\vec{c}$ is in~$A$, then
$\link_{\vec{c}\,}(\Delta_{n})
\simeq S^{h(c_{1})}*\cdots*S^{h(c_{k})}=S^{h(\vec{c}\,)}$,
where we define
$h(\vec{c}\,)=k-1+\sum_{j=1}^{k} h(c_{j})$
for compositions~$\vec{c}$ with all parts in $A$, since the join of
an $n$-dimensional sphere and an $m$-dimensional sphere
is an $(n+m+1)$-dimensional sphere.
Note that
$h(\vec{c}\,)$ is undefined for all other compositions.
For a composition $\vec{c}=(c_{1}, \ldots, c_{k})$ of $n$
with all of its parts in $A$,
let $\dim(\vec{c}\,)$ denote the dimension of the reduced homology
of $\triangle(\Pi_{n}^{\langle a,b \rangle} - \{\hat{1}\})$
to which the composition~$\vec{c}$ contributes.
That is,
$\dim(\vec{c}\,)$ is given by
\begin{equation}
\dim(\vec{c}\,) = h(\vec{c}\,) + k-1 = \sum_{i=1}^{k} h(c_{i}) + 2k-2.
\label{equation_composition_dimension}
\end{equation}
We can apply
Theorem~\ref{theorem_main_theorem_order_complex}
to obtain
\begin{theorem}
Let $2\leq a<b$ with $\gcd(a,b)=1$.
Then the $i$th reduced homology group of
$\triangle(\Pi_{n}^{\langle a,b \rangle}-\{\hat{1}\})$
is given by the direct sum of Specht modules
$\bigoplus_{\vec{c}\,\in F_{i}} S^{B^{*}(\vec{c}\,)}$,
where $F_{i}$ is the collection of compositions~$\vec{c}$ of $n$
where all the parts are in the set $A$ with
$\dim(\vec{c}\,) = i$.
\end{theorem}
\begin{proof}
We directly have
\begin{align*}
\widetilde{H}_{i}(\Delta(\Pi_{n}^{\langle a,b \rangle} - \{\hat{1}\}))
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c} \in \Delta}
\widetilde{H}_{i_{\vec{c}}}(\link_{\vec{c}\,}(\Delta_{n})) \otimes S^{B^{*}(\vec{c}\,)} \\
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\in F_{i}}
\widetilde{H}_{i_{\vec{c}\,}}(S^{h(\vec{c}\,)}) \otimes S^{B^{*}(\vec{c}\,)}\\
& \cong_{{\mathfrak S}_{n-1}}
\bigoplus_{\vec{c}\in F_{i}} S^{B^{*}(\vec{c}\,)} .
\qedhere
\end{align*}
\end{proof}
We now describe the top and
bottom reduced homology of
the order complex $\Delta(\Pi_{n}^{\langle a,b \rangle} - \{\hat{1}\})$.
We begin with the top homology.
\begin{proposition}
\label{proposition_a_b_top}
Let $2\leq a<b$ with $\gcd(a,b)=1$.
Let $r$ be the unique integer such that
$0 \leq r < a$ and $n\equiv rb\bmod a$.
Then the top homology of $\triangle(\Pi_{n}^{\langle a,b \rangle}-\{\hat{1}\})$, which occurs
in dimension $(n-r(b-a))/a-2$,
is given by the direct sum of Specht modules
$\bigoplus_{\vec{c}\,\in R} S^{B^{*}(\vec{c}\,)}$,
where $R$ is the collection of compositions~$\vec{c}$ of~$n$ where
exactly $r$ of the parts are equal to $b$ or $a+b$,
and the remaining parts are all equal to~$a$.
\end{proposition}
\begin{proof}
We present two procedures that will
change a composition~$\vec{c}$ into
another composition~$\vec{c}\,^{\prime}$ such that
the dimension of contribution from~$\vec{c}\,^{\prime}$ is greater than
the contribution of~$\vec{c}$,
that is, $\dim(\vec{c}\,) < \dim(\vec{c}\,^{\prime})$.
The compositions which we cannot
improve with this procedure are those described in the statement of the
proposition.
We now describe the first replacement
procedure.
If the composition~$\vec{c}$ has a part of the form
\begin{itemize}
\item[(i)] $jab$, replace it with $jb$ $a$'s,
\item[(ii)] $jab+a$, replace it with $(jb+1)$ $a$'s,
\item[(iii)] $jab+b$, replace it with $jb$ $a$'s and one $b$,
\item[(iv)] $jab+a+b$, replace it with $(jb+1)$ $a$'s and one $b$,
\end{itemize}
to obtain a new composition~$\vec{c}\,^{\prime}$.
We claim that $\dim(\vec{c}\,^{\prime}) - \dim(\vec{c}\,) = (b-a) \cdot j$.
We check the computation in the case (iv),
the other three cases are similar.
The difference
$\dim(\vec{c}\,^{\prime}) - \dim(\vec{c}\,)$ only depends on the parts affected
and the number of them.
Hence
\begin{align*}
\dim(\vec{c}\,^{\prime}) - \dim(\vec{c}\,)
& =
[(jb+1) \cdot h(a) + h(b) + 2(jb+2)] - [h(jab+a+b) + 2] \\
& =
[jb+2] - [2j + 2] = (b-a) \cdot j > 0 ,
\end{align*}
using that $h(a) = h(b) = -1$ and $h(jab+a+b) = 2j$.
Hence this procedure increases the dimension.
Iterating this procedure
we obtain
a new composition
with all the parts
of the form $a$, $b$ and~$a+b$.
The second replacement procedure is as follows.
Assume that there are $a$ parts of the composition~$\vec{c}$
that are different from~$a$.
Assume that $p$ of these parts are equal to $a+b$,
and hence $a-p$ of them are equal to $b$.
Replace these $a$ parts with $b+p$ parts equal to $a$
to obtain a new composition~$\vec{c}\,^{\prime}$.
\begin{align*}
\dim(\vec{c}\,^{\prime}) - \dim(\vec{c}\,)
& =
[(b+p) \cdot h(a) + 2 (b+p)]
-
[p \cdot h(a+b) + (a-p) \cdot h(b) + 2a] \\
& =
[b+p] - [a+p] = b-a > 0.
\end{align*}
Hence the new composition~$\vec{c}\,^{\prime}$ contributes
to a homology of dimension
$b-a > 0$ greater than the composition~$\vec{c}$ does.
Iterating the last procedure,
we are left with a composition~$\vec{c}$ where the number of parts
different from $a$ is at most $a-1$.
By considering the equation
$c_{1} + \cdots +c_{k} = n$ modulo~$a$,
we obtain the number of parts
different from $a$ is given by the integer $r$ from the statement of the proposition.
Additionally, switching between one part of $a+b$ and the two parts
$a$ and $b$ does not change the dimension of the contribution of the composition.
Finally, we compute the contribution of the composition
$(\underbrace{a, \ldots, a}_{(n-br)/a}, \underbrace{b, \ldots, b}_{r})$
to obtain the desired dimension.
\end{proof}
\begin{corollary}
Let $2\leq a<b$ with $\gcd(a,b)=1$.
Assume that $n$ is divisible by $a$.
Then the top homology of $\triangle(\Pi_{n}^{\langle a,b \rangle}-\{\hat{1}\})$,
which occurs
in dimension $n/a -2$,
is the Specht module~$S^{B^{*}(a,a, \ldots, a)}$.
\end{corollary}
\begin{proof}
When $a$ divides $n$, then the integer $r$ of Proposition~\ref{proposition_a_b_top}
is $0$. Thus the only contribution to reduced homology in dimension $n/a-2$ is
given by $(a,a, \ldots, a)$.
\end{proof}
We now turn our attention to the bottom reduced homology.
\begin{proposition}
\label{proposition_a_b_bottom}
Let $3\leq a<b$ with $\gcd(a,b)=1$.
Let $r$ and $s$ be the two unique integers
such that
$$ n \equiv rb \bmod a,
\:\:\:\: 0 \leq r < a,
\:\:\:\: n \equiv sa \bmod b
\:\:\:\: \text{ and } \:\:\:\: 0 \leq s < b . $$
Then the bottom reduced homology of $\triangle(\Pi_{n}^{\langle a,b \rangle}-\{\hat{1}\})$
occurs in dimension $2\cdot\frac{n-sa-rb}{ab}+r+s-2$, and
is given by the direct sum of Specht modules $S^{B^{*}(\vec{c}\,)}$
over all compositions~$\vec{c}$
such that
the number of parts of $\vec{c}$
of the form $j\cdot ab+a$ and $j\cdot ab+a+b$
is~$s$
and
the number of parts of the form $j\cdot ab+b$ and $j\cdot ab+a+b$
is~$r$.
\end{proposition}
\begin{proof}
Just as in Proposition~\ref{proposition_a_b_top}, we will define
replacement procedures, where our goal now
is to decrease the dimension of the homology that our composition contributes to, rather than increase it, as was the case in Proposition~\ref{proposition_a_b_top}.
The first procedure takes $b$ parts of the composition~$\vec{c}$
of the form $jab+a$ and $jab+a+b$ and
subtracts~$a$ from each of these $b$~parts,
and adjoins a new part $ab$.
Notice that
the resulting new composition $\vec{c}\,^{\prime}$
remains a composition of $n$.
Observe that
$h(jab) = h(jab+a) -1$,
$h(jab+b) = h(jab+a+b) -1$,
and $h(ab) = 0$.
Hence the dimension $\vec{c}\,^{\prime}$ contributes to
is
$\dim(\vec{c}\,^{\prime}) = \sum_{i=1}^{k+1} h(c_{i}^{\prime}) + 2(k+1) - 2
= \sum_{i=1}^{k} h(c_{i}) - b + 2(k+1) - 2
= \dim(\vec{c}\,) - b+2 < \dim(\vec{c}\,)$.
There is one small caveat.
In the procedure, replacing a part $a$ with $0$
we obtain a weak composition, that is, we can introduce zero entries.
Note the natural extension of the function~$h$
satisfies $h(0) = -2$.
Assume that $\vec{c}\,^{\prime}$ has a zero entry, say in its last entry,
and let $\vec{c}\,^{\prime\prime}$ be the (weak) composition with this last entry removed.
Then we have that
$\dim(\vec{c}\,^{\prime}) = \sum_{i=1}^{k+1} h(c_{i}^{\prime}) + 2(k+1) - 2
= \sum_{i=1}^{k} h(c_{i}^{\prime\prime}) + 2k - 2
= \dim(\vec{c}\,^{\prime\prime})$.
Thus zero entries can be removed without
changing the dimension.
The second procedure
is symmetric to the first in the two parameters $a$ and~$b$.
That is, it takes $a$~parts of the composition $\vec{c}$
of the form $jab+b$ and $jab+a+b$ and
subtracts~$b$ from each of these $a$ parts and
adjoins a new part~$ab$. Now we have
$\dim(\vec{c}\,^{\prime}) = \dim(\vec{c}\,) - a+2 < \dim(\vec{c}\,)$,
using the fact that $a \geq 3$.
Iterating these two procedures we obtain a composition
which
has at most $b-1$ parts of the form $jab+a$ and $jab+a+b$,
and
at most $a-1$ parts of the form $jab+b$ and $jab+a+b$.
Hence this composition satisfies the condition of the statement
of the proposition.
Finally, one has to observe that all such composition contribute
to the same dimension.
\end{proof}
\begin{corollary}
Let $3 \leq a < b$, $\gcd(a,b) = 1$
and let $n$ be divisible by~$ab$.
Then the bottom reduced homology
of the order complex
$\triangle(\Pi_{n}^{\langle a,b \rangle} - \{\hat{1}\})$
is given by
the permutation module
$M^{B^{\#}(ab, \ldots, ab, ab)} = M^{B(ab, \ldots, ab, ab-1)}$.
\end{corollary}
\begin{proof}
We have $r=s=0$. Hence the compositions only
have parts of the form $j \cdot ab$. The result follows
from Lemma~\ref{lemma_permutation_module_isomorphism}.
\end{proof}
We end with a complete description in the case when $a=2$.
\begin{proposition}
Let $b$ be odd and greater than or equal to~$3$.
Then the $i$th reduced homology of
$\triangle(\Pi_{n}^{\langle 2,b \rangle}-\{\hat{1}\})$ is given by
the direct sum
of Specht modules $S^{B^{*}(\vec{c}\,)}$
over all compositions $\vec{c}$
with all parts congruent to $0$ or $2$ modulo~$b$,
where exactly $(b(i+2) - n)/(b-2)$
entries of $\vec{c}$ are congruent to $2$ modulo~$b$.
The bottom reduced homology occurs in dimension
$\lceil n/b \rceil - 2$.
Furthermore, when $b$ divides~$n$
the bottom reduced homology is given by
the permutation module $M^{B^{\#}(b,\ldots,b,b)}=M^{B(b,\ldots,b,b-1)}$.
\end{proposition}
\begin{proof}
Since $a=2$
the expression for $h(n)$ in equation~\eqref{equation_h}
reduces to
$h(n)=\lceil n/b \rceil - 2$
and the set~$A$ reduces to
$\{n \in {\mathbb P} \: : \: n \equiv 0,2 \bmod b\}$.
Let $\vec{c}$ be a composition of $n$ into $k$ parts,
where each part belongs to the set $A$.
Furthermore, assume that $\vec{c}$
has $s$ entries congruent to $2$ modulo~$b$.
The contribution of $\vec{c}$ to the reduced homology of
$\triangle(\Pi_{n}^{\langle 2,b \rangle}-\{\hat{1}\})$,
given by equation~\eqref{equation_composition_dimension}, is in dimension
$$
\dim(\vec{c}\,)
=
\sum_{i=1}^{k} h(c_{i}) + 2k - 2
=
\sum_{i=1}^{k} \left\lceil\frac{c_{i}}{b}\right\rceil - 2
=
\frac{\sum_{i=1}^{k} c_{i} + s \cdot (b-2)}{b} - 2
=
\frac{n + s \cdot (b-2)}{b} - 2 .
$$
Solving for $s$ in this equation yields the desired expression.
For real numbers $x$ and $y$ we have
the inequality
$\lceil x \rceil + \lceil y \rceil \geq \lceil x+y \rceil$.
Hence we obtain the lower bound on the dimension of the homology:
$\dim(\vec{c}\,)
=
\sum_{i=1}^{k} \left\lceil\frac{c_{i}}{b}\right\rceil - 2
\geq
\left\lceil\frac{n}{b}\right\rceil - 2$.
When $b$ divides~$n$ the only way to obtain equality in the previous inequality is when
all the parts of the composition are divisible by~$b$.
The bottom reduced homology group is then the direct sum over
all compositions~$\vec{c}$ of $n$ where each part is divisible by $b$,
that is, $(b,b,\ldots,b) \geq^{*} \vec{c}$.
Hence we obtain the permutation module
$M^{B^{\#}(b,\ldots,b,b)} = M^{B(b,\ldots,b,b-1)}$
by Lemma~\ref{lemma_permutation_module_isomorphism}.
\end{proof}
\section{Concluding remarks}
Using Theorem~\ref{theorem_main_theorem_order_complex}
we have been able to classify the action of ${\mathfrak S}_{n-1}$ on
the top homology of $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
for any complex $\Delta\subseteq\Comp(n)$.
In the case when $\triangle(\Pi^{*}_{\Delta} - \{\hat{1}\})$ is shellable,
is there an $EL$-labelling of $\Pi^{*}_{\Delta}\cup\{\hat{0}\}$
that realizes this shelling order?
Is there a way we can classify the ${\mathfrak S}_{n}$-action
on the homology groups of $\triangle(\Pi^{*}_{\Delta}-\{\hat{1}\})$
rather than the ${\mathfrak S}_{n-1}$-action?
Browdy described the matrices
representing the action of ${\mathfrak S}_{n}$
on the cohomology groups of the filter
with block sizes belonging
to the arithmetic progression
$k \cdot d, (k+1) \cdot d, \ldots$;
see~\cite[Section~5.4]{Browdy}.
The partition lattice is naturally associated
with the symmetric group, that is, the Coxeter group
of type~$A$.
Miller~\cite{Miller} has extended the results about the
filter $\Pi^{*}_{\vec{c}}$ to other root systems.
Hence it is natural to ask if our results for the filter~$\Pi^{*}_{\Delta}$
can be extended to other root systems.
Is there a non-pure shelling of the
Frobenius complex generated by $a$ and~$b$?
Alternatively, is there a Morse matching for this
Frobenius complex such that all the critical cells
are facets? While we do have this property
for $\Lambda$ defined by an arithmetic progression as in Section~\ref{section_Frobenius},
unfortunately the general matching given
in~\cite{Clark_Ehrenborg} does not have this property.
Lastly, all of our results are based upon $\Delta$ being a filter
in the composition lattice $\Comp(n)$. What if we remove
the filter constraint?
That is, let $\Omega$ be an arbitrary collection of compositions of $n$ not containing
the extreme composition~$(n)$.
Define~$Q^{*}_{\Omega}$ to be all ordered set partitions
$\sigma = (C_{1}, C_{2}, \ldots, C_{k})$
such that $\type(\sigma) \in \Omega$ and
containing $n$ in the last block $C_{k}$.
Let $\Pi_{\Omega}$ be the image of $Q^{*}_{\Omega}$
under the forgetful map $f$. What can be said
about the homology groups and the homotopy type of
the order complex $\triangle(\Pi_{\Omega})$?
We need to understand the topology
of the links~$\link_{\vec{c}\,}(\Omega)$, even though these
links are not themselves simplicial complexes.
\section*{Acknowledgements}
The authors thank Bert Guillou and Kate Ponto
for their homological guidance and expertise.
They also thank Sheila Sundaram
and Michelle Wachs for essential references.
The authors thank Margaret Readdy for
her comments on an earlier draft.
Both authors were partially supported by
National Security Agency grant~H98230-13-1-0280.
The first author wishes to thank the
Princeton University Mathematics Department
where this work began.
{\small
| {
"timestamp": "2016-06-07T02:08:30",
"yymm": "1606",
"arxiv_id": "1606.01443",
"language": "en",
"url": "https://arxiv.org/abs/1606.01443",
"abstract": "Given a filter $\\Delta$ in the poset of compositions of $n$, we form the filter $\\Pi^{*}_{\\Delta}$ in the partition lattice. We determine all the reduced homology groups of the order complex of $\\Pi^{*}_{\\Delta}$ as ${\\mathfrak S}_{n-1}$-modules in terms of the reduced homology groups of the simplicial complex $\\Delta$ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank--Hanlon--Robinson and Wachs on the $d$-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes $a$ or~$b$. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression $a, a + d, \\ldots, a + (a-1) \\cdot d$, extending work of Browdy.",
"subjects": "Combinatorics (math.CO)",
"title": "Filters in the partition lattice",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795091201805,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.7092379527322034
} |
https://arxiv.org/abs/1712.09488 | Positive Solutions of p-th Yamabe Type Equations on Infinite Graphs | Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on $G$, where $h$ and $g$ are known, $2<\alpha\leq p$. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on $G$. | \section{Introduction}
Recently, the investigations of discrete weighted Laplacians and various equations on graphs have attracted much attention (cf. \cite{GLY,GLY2,GLY3,Ge1,Ge2,Ge3,Ge4,ZL}). Grigor'yan, Lin and Yang \cite{GLY3} first studied a Yamabe type equation on a finite graph $G$ as follows
\begin{equation}\label{YE1}
-\Delta u+hu=|u|^{\alpha-2}u,\quad \alpha>2
\end{equation}
where $\Delta$ is a usual discrete graph Laplacian, and $h$ is a positive function defined on the vertices of $G$. They show that the above equation (\ref{YE1}) always has a positive solution. Inspired by their work, Ge and Jiang \cite{Ge4} studied the following Yamabe type equation on an infinite graph, that is
\begin{equation}\label{alpha>p}
-\Delta_pu+hu^{p-1}=gu^{\alpha-1},\ u>0
\end{equation}
where $\Delta_{p}$ is $p$-th discrete graph Laplacian.\par
Now, we recall the main result in \cite{Ge4}.
\begin{theorem}\label{GJ}(\cite{Ge4}, Theorem 1)
Consider the $p$-th Yamabe equation (\ref{alpha>p}) on a connected, infinite and locally finite graph $G$ with $\alpha>p\geq2$. Assume $g\geq0$ and $g$ is bounded from above, $h$ satisfies $\inf_{x\in V}h(x)>0$ and $\inf_{x\in V}h(x)\mu(x)>0$. Further assume $h^{-1}\in L^{\delta}(V)$ for some $\delta>0$ (or $h(x)\rightarrow\infty$ when $x\rightarrow\infty$), then (\ref{alpha>p}) has a positive solution.
\end{theorem}
From this result, one still needs to know:\\
\centerline{Can one solve the $p$-th Yamabe equation (\ref{alpha>p}) under the assumption $2<\alpha\leq p$ ?}\par
The main purpose of this paper is to answer the above question. This paper is organized as follows: In section 2, we give some notations on graph and state main results. Existence of positive solutions in an infinite graph is proved in section 3.
\section{Settings and main results}
First let's recall basic definitions for weighted graphs. Let $G=(V,E)$ be a locally finite graph, where $V$ denotes the vertex set and $E$ denotes the edge set, $\omega:V\times V\ni(x,y)\mapsto\omega_{xy}\in[0,\infty)$ be an edge weight function satisfying
\begin{itemize}
\item $\omega_{xy}=\omega_{yx},\quad\quad \forall x,y \in V,$
\item $\sum_{y\in V}\omega_{xy}<\infty, \quad\quad \forall x\in V,$
\end{itemize}
and $\mu: V\ni x \mapsto \mu(x)\in(0,\infty)$ be a measure on $V$ of full support, and for $x,y\in V, \{x,y\}\in E$ if and only if $\omega_{xy}>0$, in symbols $x\thicksim y$. Alternatively, $\omega_{xy}$ can be considered as a positive function on the set $E$ of edges, that is extended to be $0$ on non-edge pairs $(x,y)$. Note that $G=(V,E)$ possibly possesses self-loops.
We say that a graph is locally finite if for any $x\in V$, there holds $\sum_{y\sim x}1<\infty$. Throughout this paper, we denote $C_{G,h,\cdots}$ as some positive constant depending only on the information of $G,H,\cdots$. Note that the information of $G$ contains $V,E,\mu$ and $\omega$.
For any function $u:V\rightarrow\mathbb{R}$, the $\mu$-Laplacian (or Laplacian for short)
of $u$ is defined as
\begin{equation*}
\Delta u(x)=\frac{1}{\mu(x)}\sum_{y\sim x}w_{xy}(u(y)-u(x)).
\end{equation*}
Denote $C(V)$ as the set of all real functions defined on $V$, then when $V$ is an infinite (a finite) set, $C(V)$ is an infinite (a finite) dimensional linear space with the usual functions additions and scalar multiplications.
When $p\geq2$, the genelized discrete graph $p$-Laplacian $\Delta_p:C(V)\rightarrow C(V)$ is defined as
\begin{equation}\label{p-LDef}
\Delta_p f(x)=\frac{1}{\mu (x)}\sum_{y\thicksim x}\omega_{xy}\left|f(y)-f(x)\right|^{p-2}(f(y)-f(x)),
\end{equation}
for $f\in C(V)$ and $x\in V$.
With respect to the vertex weight $\mu$, the integral of $f$ over $V$ is defined by
$$\int_Vfd\mu=\sum\limits_{x\in V}\mu(x)f(x),$$
for any $f\in C(V)$. Set $\mathrm{Vol}(G)=\int_Vd\mu$. Note that $f$ may not be integrable generally. Denote $L^q(V)$ as the space of all $q$-th integrable functions on $V$.
We define a space of functions
\begin{equation}
\mathcal{H}=\left\{u\in L^p(V): \int_V \left(|\nabla_p u|^p+h|u|^p\right)d\mu <+\infty\right\}
\end{equation}
with a norm
\begin{equation}\|u\|_{\mathcal{H}}=\left(\int_V\left(|\nabla_p u|^p+h|u|^p\right)d\mu\right)^{1/p},
\end{equation}
where $|\nabla_p u|$ is defined as
\begin{equation}\label{ptd}
|\nabla_p u(x)|=\Big(\frac{1}{2\mu(x)}\sum_{y\thicksim x}\omega_{xy}\big|u(y)-u(x)\big|^p\Big)^{1/p},
\end{equation}
which implies the following identity
$$\int_V |\nabla_p u|^p d\mu=\sum_{y\thicksim x}\omega_{xy}\big|u(y)-u(x)\big|^p.$$
Now we state our main result as follows
\begin{theorem}\label{theorem2}
Let $G=(V,E)$ be a connected, locally finite and infinite graph. Assume that its weight satisfies $\omega_{xy}=\omega_{yx}$ for all $y\thicksim x\in V,$ and that its measure $\mu(x)>0$ for all $x\in V$. Suppose $2<\alpha\leq p$, $g\geq0$ and $g$ is bounded from above, $h$ satisfies $\inf_{x\in V}h(x)>0$ and $\inf_{x\in V}h(x)\mu(x)>0$. Further assume $h^{-1}\in L^{\delta}(V)$ for some $0<\delta<\frac{1}{p-2}$, then the
equation
\begin{equation}\label{equation2}
-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}
\end{equation}
has a positive solution.
\end{theorem}
\begin{remark}
Grigor'yan, Lin and Yang \cite{GLY3} studied the following equation
$$-\Delta_pu+h|u|^{p-2}u=f(x,u),\ p>1$$
and established some existence results under certain assumptions of $f(x,u)$. However, it is remarkable that the definition of discrect graph $p$-Laplacian $\Delta_p$ in \cite{GLY3} is different from our definition in (\ref{p-LDef}) when $p\neq2$. Moreover, under their assumptions, $f(x,u)$ can not choose $gu^{\alpha-1}$ for $2<\alpha\leq p$.
\end{remark}
Combining Theorem \ref{GJ} with Theorem \ref{theorem2}, we have
\begin{corollary}
Let $G=(V,E)$ be a connected, locally finite and infinite graph. Assume that its weight satisfies $\omega_{xy}=\omega_{yx}$ for all $y\thicksim x\in V,$ and that its measure $\mu(x)>0$ for all $x\in V$. Suppose $\alpha>2$, $p\geq2$, $g\geq0$ and $g$ is bounded from above, $h$ satisfies $\inf_{x\in V}h(x)>0$ and $\inf_{x\in V}h(x)\mu(x)>0$. Further assume $h^{-1}\in L^{\delta}(V)$ for some $0<\delta<\frac{1}{p-2}$, then the
equation
\begin{equation*}
-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}
\end{equation*}
has a positive solution.
\end{corollary}
\section{Proof of Theorem \ref{theorem2}}
For each $u\in \mathcal{H}$, we define a functional
$$J(u)=\int_V (|\nabla_pu|^p+h|u|^p)d\mu.$$
It is easy to see that $J(u)=\|u\|_{\mathcal{H}}^p$ and $J$ is continuously differentiable. For any positive $\theta$, set
\begin{equation*}
G(x,s)=\begin{cases}g(x)\theta s^{\alpha}\ &s\geq0,\\0 & s<0.\end{cases}
\end{equation*}
It is continuously differentiable with respect to $s$ with $\partial_s G(x,s)=\alpha g(x)\theta s^{\alpha-1} $ when $s\geq0$ and $\partial_s G(x,s)=0$ when $s<0$. We write $G'(x, s)=\partial_sG(x,s)$ for short.
Now consider the following functional
\begin{equation}\label{K}
K(u)=\int_V G(x,u)d\mu,\ u\in \mathcal{H}.
\end{equation}
\begin{claim}
The function $K$, defined in (\ref{K}), is continuously differentiable on $\mathcal{H}$ for all $|u|\leq C$, where $C$ is some constant independent of $u$.
\end{claim}
\begin{proof}
By direct calculation, the Fr\'{e}chet derivative of $K(u)$ is a $K'(u)\in \mathcal{H^*}$ with
$$\mathcal{H}\ni v\mapsto K'(u)(v)=\int_V G'(x,u(x))v(x)d\mu.$$
For any $x,y\geq0$, $1\leq a<\infty$, we have an elementary inequality
$$\big|x^a-y^a\big|\leq a|x-y|\big(x^{a-1}+y^{a-1}\big).$$
Note that $\alpha>2$. As $g$ is bounded, by the above elementary inequality, we have
$$|G'(x,u_1(x))-G'(x,u_2(x))|\leq \tilde{C}_{g,\alpha,\theta}|u_1^{\alpha-1}(x)-u_2^{\alpha-1}(x)|\leq C_{g,\alpha,\theta}|u_1(x)-u_2(x)|,$$
where we using the condition $|u|\leq C$ in last inequality.\par
For $\xi\in\mathcal{H}$, it follows
$$|(K'(u_1)-K'(u_2))\xi|\leq C_{g,\alpha,\theta}\int_V|u_1-u_2||\xi|d\mu.$$
Since $h^{-1}\in L^{\delta}(V)$ for some $0<\delta<\frac{1}{p-2}$, there exists some constant $C_{G,h}$, such that
$$\Big(\int_V \frac {1}{h^{\delta}}d\mu\Big)^{\delta}\leq C_{G,h}\ .$$
Denote $\inf_{x\in V}h(x)=C_h>0$. Note that $\frac{1}{p-2}-\delta>0$. For any $x\in V$, we have
\begin{equation}\label{gj}
\frac{1}{h^{1/(p-2)}(x)}\leq \frac{1}{h^{\delta}(x)}\Big(\frac{1}{\inf_{x\in V}h(x)}\Big)^{\frac{1}{p-2}-\delta}\leq C_{G,\delta,p,h}\frac{1}{h^{\delta}(x)},
\end{equation}
it follows
\begin{equation*}
\int_V |u_1-u_2|^{\frac{p}{p-1}}d\mu\leq \Big(\int_V \frac{1}{h^{1/(p-2)}}d\mu\Big)^{\frac{p-2}{p-1}}\Big(\int_V h|u_1-u_2|^pd\mu\Big)^{\frac{1}{p-1}}\leq C_{G,\delta,p,h}\|u_1-u_2\|_{\mathcal{H}}^{\frac{p}{p-1}}
\end{equation*}
Moreover,
$$\int_V |u_1-u_2||\xi|d\mu\leq \Big(\int_V |u_1-u_2|^{\frac{p}{p-1}}d\mu\Big)^{\frac{p-1}{p}}\|\xi\|_{\mathcal{H}}
\ .$$
Thus, we have
$$|(K'(u_1)-K'(u_2))\xi|\leq C_{G,h,p,g,\delta,\theta}\|\xi\|_{\mathcal{H}}\|u_1-u_2\|_{\mathcal{H}} \ .$$
Therefore
$K':\mathcal{H}\rightarrow\mathcal{H^*}$, the Frechet derivative of $K$ satisfies
$$\|K'(u_1)-K'(u_2)\|_{\mathcal{H^*}}\leq C_{G,h,p,g,\delta,\theta}\|u_1-u_2\|_{\mathcal{H}}\ , $$
This implies that $K'$ is continuous on $\mathcal{H}$, i.e. $K$ is continuously differentiable on $\mathcal{H}$.\quad\quad$\Box$
\end{proof}
Now, we consider the functional $J(u)$ under the constraint $K(u)=1$. Since $J(u)\geq0$,
$$\gamma=inf\{J(u): u\in\mathcal{H},\ K(u)=1\}$$
is well defined. Obviously, $\gamma\geq0$. We can also choose a sequence $\{u_i\}_{i\geq1}$ in $\mathcal{H}$ with $J(u_i)\rightarrow\gamma$, $J(u_i)<\gamma+1$ and $K(u_i)=1$.
Denote $C(G, h)=\inf_{x\in V}h(x)\mu(x)>0$. At each vertex $x\in V$, we have
\begin{equation}\label{BD}
C(G, h)\big|u_i(x)\big|^p\leq h(x)\mu(x)\big|u_i(x)\big|^p\leq \int_V h|u_i|^pd\mu<J(u_i)\leq \gamma+1.
\end{equation}
This means $\big|u_i(x)\big|\leq C_{G,h,p,\gamma}$ for all $x\in V$ and all $i\geq1$. In other words, $\{u_i\}_{i \geq 1}$ are uniformly bounded.
Hence, there exists some $\bar{u}$ such that up to a
subsequence, $u_i \rightarrow \bar{u}$ on $V$. We may well denote this subsequence as $u_i$. Because $G$ is locally finite, $|\nabla_p u_i|\rightarrow|\nabla_p \bar{u}|$ at each vertex $x$.
According to Fatou's lemma, we obtain
\begin{equation}\label{bound}
\int_V(h|\bar{u}|^p+|\nabla_p \bar{u}|^p)d\mu \leq\gamma,
\end{equation}
\begin{equation}\label{2}
K(\bar{u})=\int_V G(x,\bar{u})d\mu\leq 1,
\end{equation}
which implies $\bar{u}\in\mathcal{H}$.
\begin{claim}
$\bar{u}$, as above, is not identically zero on $V$.
\end{claim}
\begin{proof}
Let $x_0 \in V$ be fixed. For any $\epsilon>0$, $h^{-1}\in L^{\delta}(V)$ for some $0<\delta<\frac{1}{p-2}$, there exists some $R>0$ such that
\begin{equation}\label{control}
\Big(\int_{dist(x,x_0)>R} \frac {1}{h^{\delta}}d\mu\Big)^{\delta}\leq {\epsilon}.
\end{equation}
Denote $\inf_{x\in V}h(x)=C_h>0$, and we discuss in two cases. First if $p>\alpha>2$, we gain $\frac{\alpha}{p-\alpha}-\delta>0$. Hence, for any $x\in V$
$$\frac{1}{h^{{\alpha}/(p-\alpha)}(x)}\leq \frac{1}{h^{\delta}(x)}\Big(\frac{1}{\inf_{x\in V}h(x)}\Big)^{\frac{\alpha}{p-\alpha}-\delta'}\leq C_{\alpha,\delta,p,h}\frac{1}{h^{\delta}(x)}.$$
As $g$ is bounded from above and $\|u_i\|_{\mathcal{H}}=J(u_i)^{\frac{1}{p}}$
, using the definition of $G(x,s)$ and the inequality ((\ref{BD})), we obtain
\begin{align*}
\int_{dist(x,x_0)>R}G(x,u_i)d\mu\leq& C_{\alpha,g,\theta}\int_{dist(x,x_0)>R,u_i(x)>0}u_i^{\alpha}d\mu
\\ \leq& C_{\alpha,g,\theta} \Big(\int_{dist(x,x_0)>R,u_i(x)>0}\frac{1}{h^{{\alpha}/(p-\alpha)}}d\mu\Big)^{\frac{p-\alpha}{\alpha}}
\|u_i\|^{\alpha}_{\mathcal{H}}
\\ \leq&C_{\alpha,g,\delta,p,h,\theta}\ {\epsilon}^{\frac{(p-\alpha)\delta}{\alpha}} \|u_i\|^{\alpha}_{\mathcal{H}}
\\ \leq&C_{\alpha,g,\delta,p,h,\theta}\ {\epsilon}^{\frac{(p-\alpha)\delta}{\alpha}}(\gamma+1)^{\frac{\alpha}{p}}.
\end{align*}
Using $K(u_i)=1$, for $p>\alpha>2$, we have
\begin{equation}\label{p>}
\int_{dist(x,x_0)\leq R}G(x,u_i)d\mu\geq1-C_{\alpha,g,\delta,p,h,\theta}\ {\epsilon}^{\frac{p-\alpha}{\alpha\delta}} (\gamma+1)^{\frac{\alpha}{p}}.
\end{equation}
Then let $i\rightarrow\infty$, and we have
$$K(\bar{u})=\int_VG(x,\bar{u})d\mu\geq \int_{dist(x,x_0)\leq R}G(x,\bar{u})d\mu\geq1-C_{\alpha,g,\delta,p,h,\theta}\ {\epsilon}^{\frac{p-\alpha}{\alpha\delta}} (\gamma+1)^{\frac{\alpha}{p}}.$$
Let $\epsilon\rightarrow0$, we obtain $K(\bar{u})\geq1$.\par
Then in the case of $p=\alpha$, because $\{u_i\}_{i \geq 1}$ are uniformly bounded, using (\ref{gj}), we get
\begin{align*}
\int_{dist(x,x_0)>R}G(x,u_i)d\mu\leq& C_{g,\theta}\int_{dist(x,x_0)>R,u_i(x)>0}u_i^pd\mu\\ \leq&C_{p,g,\theta}\int_{dist(x,x_0)>R,u_i(x)>0}u_i^{\frac{p}{p-1}}d\mu
\end{align*}
Moreover, using (\ref{gj}), (\ref{BD}) and (\ref{control}), we obtain
\begin{align*}
\int_{dist(x,x_0)>R} u_i^{\frac{p}{p-1}}d\mu\leq& \Big(\int_{dist(x,x_0)>R}\frac{1}{h^{1/(p-2)}}d\mu\Big)^{\frac{p-2}{p-1}}
\Big(\int_{dist(x,x_0)>R}hu_i^p d\mu\Big)^{\frac{1}{p-1}}\\ \leq&
C_{G,\delta,p,h}{\epsilon}^{\frac{(p-2)\delta}{p-1}}\|u_i\|_{\mathcal{H}}^{\frac{p}{p-1}}\\ \leq&
C_{G,\delta,p,h}{\epsilon}^{\frac{(p-2)\delta}{p-1}}(\gamma+1)^{\frac{1}{p-1}}.
\end{align*}
Then, for $p=\alpha>2$, using $K(u_i)=1$, we have
\begin{equation}\label{p=}
\int_{dist(x,x_0)\leq R}G(x,u_i)d\mu\geq1- C_{G,\delta,p,h}{\epsilon}^{\frac{(p-2)\delta}{p-1}}(\gamma+1)^{\frac{1}{p-1}}.
\end{equation}
Let $i\rightarrow\infty$ again, we have
$$K(\bar{u})=\int_VG(x,\bar{u})d\mu\geq \int_{dist(x,x_0)\leq R}G(x,\bar{u})d\mu\geq1- C_{G,\delta,p,h}{\epsilon}^{\frac{(p-2)\delta}{p-1}}(\gamma+1)^{\frac{1}{p-1}}.$$
Let $\epsilon\rightarrow0$, we obtain $K(\bar{u})\geq1$. In a word, we obtain $K(\bar{u})\geq1$ for $p\geq\alpha>2$. By (\ref{2}), we get $K(\bar{u})=1$, which implies that $\bar{u}$ is not identically zero.\quad\quad$\Box$
\end{proof}
\begin{claim}\label{positive}
$\bar{u}$, as above, is positive everywhere on $V$.
\end{claim}
\begin{proof}
From (\ref{bound}), we can prove $|\bar{u}|\leq C_{G,h,p,\gamma}$ which is totally similar to prove $|u_{i}(x)|\leq C_{G,h,p,\gamma}$. Then $\big|\bar{u}+t\varphi\big|\leq C_{G,h,p,\gamma}$ as $t\rightarrow0$, for any $\varphi\in \mathcal{H}$.
We calculate the Euler-Lagrange equation at $\bar{u}$ under the constraint condition $K(\bar{u})=1$. For any $\varphi\in \mathcal{H}$, there holds
\begin{align*}
0=&\frac{d}{dt}\Big|_{t=0}\Big\{J(\bar{u}+t\varphi)-\lambda\Big(\int_V G(x,\bar{u}+t\varphi)d\mu-1\Big)\Big\}\\=&\int_V \Big(-p\Delta_p\bar{u}+ph|\bar{u}|^{p-2}\bar{u}-\lambda G'(x,\bar{u})\Big)\varphi d\mu.
\end{align*}
Hence, we get
\begin{equation}\label{EL}
-p\Delta_p\bar{u}+ph|\bar{u}|^{p-2}\bar{u}=\lambda G'(x,\bar{u}).
\end{equation}
Noting that
$$\int_V (-\bar{u}\Delta_p\bar{u})d\mu=\int_V |\nabla_p u|^p d\mu,$$
multiplying $\bar{u}$ on both sides of the equation (\ref{EL}), and taking integration, we can see $\lambda>0$. If $\bar{u}(x)<0$, at some vertex $x\in V$, then by the equation $(\ref{EL})$, we see
$$\Delta_p \bar{u}(x)<0.$$
However, by the definition of $\Delta_p$, there is a $y\thicksim x$ with $\bar{u}(y)<\bar{u}(x)<0$. In view of the connectedness of the graph $G=(V,E)$, by inductively, we obtain a sequence $x=x_1\thicksim x_2\thicksim x_3\thicksim\cdots$, such that $$\bar{u}(x_i)<\bar{u}(x_{i-1})<\cdots<\bar{u}(x_1)<0.$$
This is impossible because $\mu(x)\geq\mu_{min}>0$, $\inf_{x\in V}h(x)>0$, and $h|\hat{u}|^p$ is integrable. Hence $\bar{u}$ is nonnegative on $V$.
If $\bar{u}$ is not positive everywhere on $V$, we can always find two vertices $x,y$ with $y\thicksim x$, $\bar{u}(x)=0$, $\bar{u}(y)>0.$ Then it follows $\Delta_p \bar{u}(x)>0$ by the definition of $\Delta_p$, which contradicts to the equation (\ref{EL}). Hence $\bar{u}$ is positive everywhere on $V$.\quad\quad$\Box$
\end{proof}
\begin{claim}
The equation (\ref{equation2}) has a strictly positive solution.
\end{claim}
By Claim $1,2,3$, we know that $\bar{u}$, as above, is positive everywhere on $V$, and it satisfies
\begin{equation}\label{change2}
-p\Delta_p\bar{u}+ph{\bar{u}}^{p-1}=\lambda\alpha\theta g \bar{u}^{\alpha-1}.
\end{equation}
If $p>\alpha>2$, choose $\theta=1$ and set $$u=\Big(\frac{p}{\alpha\lambda}\Big)^{\frac{1}{p-\alpha}}\bar{u}$$ in (\ref{change2}), then we have
\begin{equation}\label{Y-E}
-\Delta_pu+hu^{p-1}=gu^{\alpha-1}.
\end{equation}
If $p=\alpha$, choose $\theta=\frac{1}{\lambda}$ and set $u=\bar{u}$ in (\ref{change2}), we also obtain (\ref{Y-E}).
This implies that $u$ is a positive solution to the $p$-th Yamabe equation (\ref{equation2}).$\hfill\Box$
\noindent \textbf{Acknowledgements:} The first author would like to thank Professor Yanxun Chang for constant guidance and encouragement. The second author would like to thank Professor Gang Tian and Huijun Fan for constant encouragement and support. Both authors would also like to thank Professor Huabin Ge for many helpful conversations. The first author is supported by National Natural Science Foundation of China under Grant No. 11431003. The second author is supported by National Natural Science Foundation of China under Grant No. 11401578.
| {
"timestamp": "2018-01-17T02:05:06",
"yymm": "1712",
"arxiv_id": "1712.09488",
"language": "en",
"url": "https://arxiv.org/abs/1712.09488",
"abstract": "Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\\Delta_pu+h|u|^{p-2}u=gu^{\\alpha-1}$$ on $G$, where $h$ and $g$ are known, $2<\\alpha\\leq p$. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on $G$.",
"subjects": "Analysis of PDEs (math.AP); Combinatorics (math.CO)",
"title": "Positive Solutions of p-th Yamabe Type Equations on Infinite Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795087371921,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.70923795245699
} |
https://arxiv.org/abs/2210.11270 | Factorisation in the semiring of finite dynamical systems | Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS $A$ is cancellative if one can divide by it unambiguously, i.e. $AX = AY$ implies $X = Y$. We prove that an FDS $A$ is cancellative if and only if it has a fixpoint. Secondly, we prove that if an FDS $A$ has a $k$-th root (i.e. $B$ such that $B^k = A$), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs. |
\section{Introduction}
Finite dynamical systems are commonly used to model systems with a finite number of states that
evolve deterministically and at discrete time steps. Multiple models have been proposed for various settings, such as Boolean networks \cite{BoolNets, Goles90}, reaction systems \cite{ReactionSystems}, or sandpile models \cite{BTW87}, with applications to biology \cite{Tho73, TD90, BCRCGD13}, chemistry \cite{ReactionSystems}, or information theory \cite{GR11,GRR15}.
The dynamics of an FDS are easily described via its graph, which consists of a collection of cycles containing the periodic states, to which are attached tree-like structures containing the transient states. As such, two families of FDSs are of particular interest: permutations only have disjoint cycles in their graphs, while the so-called dendrons, where all states eventually converge towards the same fixed point, only have a tree in their graphs. Therefore, any FDS can be viewed as a collection of dendrons attached to a given permutation.
Given two FDSs $A$ and $B$, we can either \textit{add} them (that is, create a system
that behave like $A$ when it starts in a state of $A$, and like $B$
when it starts in a state of $B$) or \textit{multiply} them (that is,
create a system that corresponds to $A$ and $B$ evolving in parallel). Thus, the set $\mathbb{D}$ of FDSs, endowed with the sum and product above, forms a semiring.
Since the introduction of the semiring of finite
dynamical systems (FDSs) in \cite{PolEqFDS} as an abstract way of
studying FDSs, some research has been devoted to understand more
thoroughly the multiplicative structure of this semiring \cite{Dor17, ComposingBehaviors, Couturier, GE20}.
We can highlight three important problems related to the multiplicative structure of $\mathbb{D}$.
\begin{enumerate}
\item Perhaps the most obvious problem is factorisation: given an FDS $C$,
can we find two non-trivial FDSs $A$ and $B$ (with fewer states than $C$) such that $C=A\times B$? This corresponds to whether the system modelled by $C$ is actually composed of two independent parts working in parallel.
In \cite{Dor17, ComposingBehaviors}, it is shown that the answer is usually negative: the proportion of reducible FDSs of size $n$ vanishes when $n\rightarrow\infty$. Moreover, unlike for integers, the semiring $\mathbb{D}$ does not have unique factorisation into irreducible elements. Worse yet, this is true when we restrict ourselves to permutations or to dendrons. This adds another layer of difficulty for problems related to factorisation in the semiring of FDSs.
\item Another important problem is division: given $C$ and $A$ such that $C = AB$ for some $B$, can we find $B$? Or in other words, if $C$ is indeed composed of two parts, and we know one part, what is the other?
This problem is particularly interesting, as the FDS $B$ may not be unique: there exist many examples of FDSs $A,B,D$ such that $AB = AD$.
\item The third problem is $k$-th root: given an FDS $A$ and an integer $k$, is there $B$ such that $B^k = A$, and how many such roots exist? Until now, very little is known about this problem; for instance there was no result asserting that the solution $B$ should be unique.
\end{enumerate}
In this paper, we establish important connections between FDSs and infinite, periodic trees. In particular, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. The unrolling preserves all the information about the transient dynamics of an FDS, and preserves the product operation. However, the product on trees (and in particular, on unrollings of FDSs) is much better behaved than its counterpart for FDSs and hence allows us to prove our main results.
This paper makes four main contributions towards the understanding of the three problems listed above.
\begin{enumerate}
\item \label{itemConnected}
An FDS is connected if its graph is connected; in other words, it has only one periodic cycle. We first prove a fundamental property of connected FDSs. For any FDS $A$, if $X$ and $Y$ are connected and $AX = AY$, then $X = Y$. Intuitively, this means that division is unambiguous when we know the quotient is connected.
\item \label{itemCancellative}
Intuitively, a cancellative FDS is one those that can be unambiguously divided by. Formally, $A$ is cancellative if $AB = AC$ implies $B = C$. Our first and major result is the characterisation of cancellative FDSs: they are exactly those with a fixpoint.
\item \label{itemAlgorithm}
Our proof methods involve working with (possibly infinite) trees, and going back and forth between FDSs and trees. As a bi-product, we obtain an algorithm for division of dendrons. That is, given two dendrons $A$ and $B$, the algorithm determines the dendron $C$ such that $A = BC$ or returns a failure if no such dendron exist. It is easily shown that this algorithm runs in time polynomial in the size of the input.
\item \label{itemRoots}
Our result on cancellative FDSs has an important consequence: many polynomials in $\mathbb{D}[X]$ are injective. We then prove that the polynomial $P(X) = X^k$ is injective, i.e. if an FDS has a $k$-th root then it is unique.
\item \label{itemLDK}
Throughout the paper, we investigate the structure of division and factorisation in dendrons. We further this investigation by exhibiting large monoids of dendrons with unique factorisation.
\end{enumerate}
While writing up this paper, we have discovered the related paper \cite{DFPR22}. This work and \cite{DFPR22} have two main similarities: both have independently introduced the unrolling construction, and both have proved the cancellative nature of the product on infinite trees (\Cref{InfTreeDiv} in this work, Theorem 3.3 in \cite{DFPR22}). However, we would like to stress the significant differences between these two papers. First, the respective proofs of the result mentioned above are completely different: ours is based on a lexicographic order on trees, while theirs is based on counting tree homomorphisms. Second, and more importantly, both papers consider completely different problems about FDSs. As such, the last four main contributions of this work (items \ref{itemCancellative} to \ref{itemLDK} in the list above) are novel and do not appear in the literature so far. Third, \cite{DFPR22} proposes the following conjecture (Conjecture 3.1): let $A$ and $B$ be two connected FDSs, then for all FDSs $X$ and $Y$, if $AX = B$ and $AY = B$, then $X = Y$. Our first main contribution (item \ref{itemConnected} in the list above) was added once we were aware of \cite{DFPR22}; it is actually a more general result than their conjecture.
The rest of the paper is organised as follows. \Cref{sec:Defns} introduces all the necessary
definitions to work on the semiring of FDSs. Then,
\Cref{sec:Cancellative} shows that the cancellative elements of the
semiring of finite dynamical systems are exactly those with a
fixpoint. From this, we give in \Cref{sec:Algos} a
polynomial-time algorithm for division in dendrons. \Cref{kroot} then proves the unicity of $k$-th roots. Then
\Cref{sec:LD_K} constructs a class of monoids with unique
factorisation on each of them. Finally, some avenues for further work are proposed in
\Cref{sec:Directions}.
\section{General definitions}\label{sec:Defns}
A finite dynamical system (FDS) is a function from a finite set into itself. We denote $\mathbb{D}$ the set of all FDSs.
Given an FDS $A$, we denote $S_A$ the finite set on which it acts.
Given two FDSs $A$ and $B$, we can assume that $S_A\cap S_B=\varnothing$ (if
that is not the case, we can simply rename the elements of one of
those sets). Then, we define their \textit{sum} as follows:
$$
\begin{array}{llll}
A+B: & S_A\sqcup S_B & \rightarrow & S_A\sqcup S_B \\
& x & \mapsto &
\begin{cases}
A(x) & \text{ if $x\in S_A$} \\
B(x) & \text{ otherwise.}
\end{cases}
\end{array}
$$
Given two FDSs $A$ and $B$, we define their \textit{product} as
follows:
$$
\begin{array}{cccc}
AB: & S_A\times S_B & \rightarrow & S_A\times S_B \\
& (a,b) & \mapsto & (A(a), B(b)).
\end{array}
$$
Defining the size of an FDS $A$ as $|A|=|S_A|$, we see that:
$|A+B|=|A|+|B|$ and $|AB|=|A||B|$.
When multiplying two FDSs $A$ and $B$ (for example, in \Cref{FDSmul}),
we get $AB$ along with a labelling of the states of $AB$ by pairs of
states of $A$ and of $B$. That is, we get an isomorphism $S_{AB}\simeq
S_A\times S_B$ that respects the product structure. However, we shall
consider FDSs up to isomorphism, hence we do not get this labelling
along with the FDS in general. The problem of factorising an FDS $C$,
for example, just means labelling the states $S_C$ with $S_A\times
S_B$, where $A,B\in\mathbb{D}$, such that this labelling respects the product
$AB=C$. A formalization of this idea of labelling the states of a
product with the Cartesian product of the state sets of its factors is
provided below:
\begin{defn}
Given a sequence $(A_i)_{i\in I}\in \mathbb{D}^I$ for some finite set $I$ and a product $B=\prod_{i\in I} A_i$ we say that the function
$\phi:S_B\mapsto \prod_{i\in I}S_{A_i}$ is a \textit{product
isomorphism for the product $B=\prod_{i\in I}A_i$} if:
\begin{enumerate}
\item it is a bijection, and
\item for any sequence of states $(s_i)_{i\in I}\in \prod_{i\in
I}S_{A_i}$, we have: $B(\phi^{-1}((s_i)_{i\in I})) =
\phi^{-1}((A_i(s_i))_{i\in I})$.
\end{enumerate}
\end{defn}
We remark that computing the product gives such a product isomorphism.
\begin{prop}[\cite{PolEqFDS}]
The set of FDSs, with the above sum and product, forms a semiring \cite{HW98},
with additive identity the empty function and multiplicative
identity the function $1:\{1\}\rightarrow\{1\}, 1\mapsto 1$.
\end{prop}
For FDSs, we can adopt a graph-theoretical point of view, by
associating to an FDS $A\in \mathbb{D}$ an oriented graph
$\mathcal{G}_A=(V,E)$ where $V=S_A$ and $E=\{(x,y)\in S_A^2:y=A(x)\}$.
Then, for $A,B\in\mathbb{D}$, $\mathcal{G}_{A+B}$ is the disjoint union of
$\mathcal{G}_A$ and $\mathcal{G}_B$, and $\mathcal{G}_{AB}$ is the direct product
$\mathcal{G}_A\times G_B$ (see the corresponding section in
\cite{HandbookProds}). In the following, we will often identify an FDS
and its graph, and thus, implicitly quotient $\mathbb{D}$ by graph
isomorphism, that is, we consider that $A=B$ if and only if $\mathcal{G}_A$
and $\mathcal{G}_B$ are isomorphic, as $A$ and $B$ have the same dynamics in
this case.
\begin{defn}
Given $A,B\in\mathbb{D}$, we say that $B$ is a sub-FDS of $A$ if $\mathcal{G}_B$
is a subgraph of $\mathcal{G}_A$.
\end{defn}
Since FDSs take their values in a finite set, the structure of their
graphs is simple: they consist of some cycles, on the states of which,
trees (with arrows going upwards, towards the root) are connected, as
in the example of \Cref{FDSmul}. This leads us to several definitions
that are useful to study FDSs.
\begin{figure}
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\draw[->] (D3) -- (E1);
\draw[->] (E1) -- (F2);
\draw[->] (F2) -- (A3);
\draw[->] (A3) -- (B1);
\draw[->] (B1) -- (C2);
\tikzyshift{-4cm}
\tikzxshift{-8cm}
\node[draw,circle] (A7) at (-1.5,0) {$A7$};
\node[draw,circle] (B8) at (-0.5*1.5,1.5*0.86) {$B8$};
\node[draw,circle] (C7) at (0.5*1.5,1.5*0.86) {$C7$};
\node[draw,circle] (D8) at (1.5,0) {$D8$};
\node[draw,circle] (E7) at (0.5*1.5,-1.5*0.86) {$E7$};
\node[draw,circle] (F8) at (-0.5*1.5,-1.5*0.86) {$F8$};
\draw[->] (C7) -- (D8);
\draw[->] (D8) -- (E7);
\draw[->] (E7) -- (F8);
\draw[->] (F8) -- (A7);
\draw[->] (A7) -- (B8);
\draw[->] (B8) -- (C7);
\tikzxshift{6cm}
\node[draw,circle] (B7) at (-1.5,0) {$B7$};
\node[draw,circle] (C8) at (-0.5*1.5,1.5*0.86) {$C8$};
\node[draw,circle] (D7) at (0.5*1.5,1.5*0.86) {$D7$};
\node[draw,circle] (E8) at (1.5,0) {$E8$};
\node[draw,circle] (F7) at (0.5*1.5,-1.5*0.86) {$F7$};
\node[draw,circle] (A8) at (-0.5*1.5,-1.5*0.86) {$A8$};
\draw[->] (B7) -- (C8);
\draw[->] (C8) -- (D7);
\draw[->] (D7) -- (E8);
\draw[->] (E8) -- (F7);
\draw[->] (F7) -- (A8);
\draw[->] (A8) -- (B7);
\node[draw,circle] (G1) [above=of B2] {$G1$};
\draw[->] (G1) -- (B2);
\node[draw,circle] (G2) [above=of B3] {$G2$};
\draw[->] (G2) -- (B3);
\node[draw,circle] (G3) [above=of B1] {$G3$};
\draw[->] (G3) -- (B1);
\node[draw,circle] (G4) [above right=of B1] {$G4$};
\draw[->] (G4) -- (B1);
\node[draw,circle] (G5) [above left=of B1] {$G5$};
\draw[->] (G5) -- (B1);
\node[draw,circle] (G6) [above left=of B3] {$G6$};
\draw[->] (G6) -- (B3);
\node[draw,circle] (G7) [left=of B8] {$G7$};
\draw[->] (G7) -- (B8);
\node[draw,circle] (G8) [below left=1cm and 0.2cm of B7] {$G8$};
\draw[->] (G8) -- (B7);
\node[draw,circle] (G9) [below left=0.5cm and 2cm of B8] {$G9$};
\draw[->] (G9) -- (B8);
\node[draw,circle] (A4) [above right=3cm and 0.5cm of B1] {$A4$};
\draw[->] (A4) -- (B1);
\node[draw,circle] (B4) [above right=of C1] {$B4$};
\draw[->] (B4) -- (C1);
\node[draw,circle] (C4) [below right=0.6cm and 0.2cm of D1] {$C4$};
\draw[->] (C4) -- (D1);
\node[draw,circle] (D4) [right=of E1] {$D4$};
\draw[->] (D4) -- (E1);
\node[draw,circle] (E4) [left=of F1] {$E4$};
\draw[->] (E4) -- (F1);
\node[draw,circle] (F4) [left=of A1] {$F4$};
\draw[->] (F4) -- (A1);
\node[draw,circle] (G4) [above left=3cm and 0.5cm of B1] {$G4$};
\draw[->] (G4) -- (B1);
\node[draw,circle] (A5) [above right=3cm and 1.7cm of B1] {$A5$};
\draw[->] (A5) -- (B1);
\node[draw,circle] (B5) [above=of C1] {$B5$};
\draw[->] (B5) -- (C1);
\node[draw,circle] (C5) [above right=0.6cm and 0.2cm of D1] {$C5$};
\draw[->] (C5) -- (D1);
\node[draw,circle] (D5) [below right=of E1] {$D5$};
\draw[->] (D5) -- (E1);
\node[draw,circle] (E5) [below right=of F1] {$E5$};
\draw[->] (E5) -- (F1);
\node[draw,circle] (F5) [above left=of A1] {$F5$};
\draw[->] (F5) -- (A1);
\node[draw,circle] (G5) [above left=3cm and 1.7cm of B1] {$G5$};
\draw[->] (G5) -- (B1);
\node[draw,circle] (A6) [above right=3cm and 0.5cm of B3] {$A6$};
\draw[->] (A6) -- (B3);
\node[draw,circle] (B6) [above=of C3] {$B6$};
\draw[->] (B6) -- (C3);
\node[draw,circle] (C6) [above right=0.6cm and 0.2cm of D3] {$C6$};
\draw[->] (C6) -- (D3);
\node[draw,circle] (D6) [right=of E3] {$D6$};
\draw[->] (D6) -- (E3);
\node[draw,circle] (E6) [below left=of F3] {$E6$};
\draw[->] (E6) -- (F3);
\node[draw,circle] (F6) [above left=of A3] {$F6$};
\draw[->] (F6) -- (A3);
\node[draw,circle] (G6) [above left=3cm and 1.5cm of B3] {$G6$};
\draw[->] (G6) -- (B3);
\end{tikzpicture}
}
\caption{Product of two FDSs.}
\label{FDSmul}
\end{figure}
\begin{defn}
Let $A\in\mathbb{D}$. A state $s\in S_A$ is said to be a \textit{cycle
state} if it is on a cycle of $\mathcal{G}_A$, or, equivalently, if there
exists $n>0$ such that $A^n(s)=s$. We denote $S^C_A$ the set of
cycle states of $A$. Otherwise, $s$ is said to be a \textit{tree
state}.
We define a function $\operatorname{depth}_A: S_A\rightarrow\mathbb{N}$ that gives the
\textit{depth} of any state of $A$, and is defined recursively as
follows:
\begin{eqnarray*}
\forall s\in S^C_A, &&\operatorname{depth}_A(s)=0 \\
\forall s\in S_A\setminus S^C_A, &&\operatorname{depth}_A(s) = \operatorname{depth}_A(A(s))+1.
\end{eqnarray*}
Furthermore, for any $k\in\mathbb{N}$, we define the \textit{truncature of
$A$ at depth $k$}, denoted $[A]_k$, as the sub-FDS of $A$ which
contains all the states of $A$ at depth at most $k$.
\end{defn}
A very useful and simple result is the following:
\begin{lem}
For any $A,B\in\mathbb{D}$ and $k\in\mathbb{N}$, $[AB]_k=[A]_k[B]_k$.
\end{lem}
Of particular interest are the FDSs we call \textit{dendrons}, that
is, connected FDSs (\textit{i.e.} with a connected graph) with a
fixpoint. Those FDS can be seen as rooted trees with arrows pointing
towards the root, with a loop on the root. We denote $\mathbb{D}_D$ the set
of dendrons (remark that it is not a semiring, since the sum of two
dendrons is not a dendron).
Let's now focus on those two types of parts of FDSs: trees (which,
when summed, form forests) and cycles (which, when summed, form
permutations).
\subsection{Forests}
We introduce forests as a way to have a product between FDSs that has
an inductive definition that works level by level. In FDSs, the state
set of a product is the Cartesian product of the state sets of the
factors. This makes the identification of states of an unlabelled FDS
difficult. For forests, the pairs of states which end up in the
product are those of even depth. Finally, \Cref{ProdBot} is the reason
forests are useful: their product is compatible with that of FDSs.
\begin{defn}
By \textit{tree}, we shall mean an in-tree \cite[p.21]{BG09a}, i.e. an oriented connected acyclic graph
with a special vertex called its \textit{root}, such that every edge is oriented towards the root. The trees we consider may be infinite, but the degree of each vertex shall always be finite.
We denote the root of a tree $\bott T$ as
$\operatorname{root}(\bott T)$.
A \textit{forest} is a disjoint union of trees. The set of forests
is denoted $\mathbb{F}$, and that of trees is denoted $\mathbb{F}_T$. In the
following, we denote forests in bold face to distinguish them from
FDSs.
If $\bott T\in\mathbb{F}_T$, and if there is a single infinite path
starting from the root of $\bott T$, we can extract the sequence
$tseq(\bott T)$ of trees anchored on that path. If this sequence is
periodic, we say that $\bott T$ is \textit{periodic}, and that
$\bott T$ is of \textit{tree period} the period of the sequence. We
denote the set of periodic trees as $\mathbb{F}_P$.
\end{defn}
We consider trees as dendrons which have had their fixpoint
transformed into a sink, and extend the notations from dendrons
whenever they make sense. In particular, we denote $S_{\bott A}$ the
set of vertices of the forest $\bott{A}$. Moreover, the parent of a
vertex $x\in S_{\bott A}$ is denoted $\bott{A}(x)$. For an FDS $A \in
\mathbb{D}$ and a state $s\in S_A$, we say that $\bott T$ is the tree
\textit{anchored on} $s$ if the tree of the tree state predecessors of
$s$ in the graph is $\bott T$; we naturally extend this notation to any forest $\bott A$. By convention, the depth of an infinite dendron is $\infty$, while the depth of an empty dendron is $-1$.
Given a tree $\bott T$, we define $\mathcal{D}(\bott T)$ to be the multiset
containing the subtrees anchored on the children of the root of
$\bott T$.
Now, we define a sum and a product operation on forests in order to
endow the set of forests with a semiring structure.
The \textit{sum} of two forests $\bott A,\bott B$ (for which we can
assume $S_{\bott A}\cap S_{\bott B}=\varnothing$) is the forest $\bott C$
defined as the disjoint union of the graphs $\bott A$ and $\bott B$.
Let $\bott A, \bott B\in\mathbb{F}_T$. Then the \textit{product} of $\bott A$
and $\bott B$ is $\bott{A}\bott B=(V,A)$ with
\begin{align*}
V = S_{\bott A\bott B} &= \{(a,b)\in S_{\bott A}\times S_{\bott B}: \operatorname{depth}_{\bott A}(a)=\operatorname{depth}_{\bott B}(b)\},\\
A &= \{((a,b), (\bott{A}(a),\bott{B}(b))): (a,b)\in S_{\bott A\bott B}\}.
\end{align*}
This
product is almost the same as that on FDSs but here only states of
same depth get multiplied together.
It will often prove useful to use multisets with the following
product. Given two multisets $\mathcal{A}$ and $\mathcal{B}$, their
\textit{product} $\mathcal{A}\mathcal{B}$ is
$\{\{ab:a\in\mathcal{A},b\in\mathcal{B}\}\}$.
The following lemma explains why trees are interesting: the product is
done level by level. Moreover, the root does not behave differently
than the other states (as it does on dendrons), which means that this
product is much easier to work with.
\begin{lem}\label{LevelByLevelProduct}
If $\bott A,\bott B,\in\mathbb{F}_T$ are finite, then: $$\mathcal{D}(\bott A\bott
B)=\mathcal{D}(\bott A)\mathcal{D}(\bott B)=\{\{\bott T\times\bott{T'}:\bott T\in
\mathcal{D}(\bott A), \bott{T'}\in\mathcal{D}(\bott B)\}\}.$$
\end{lem}
\begin{proof}
The proof is by induction on the depths of $\bott A$ and $\bott B$.
The case for trees with depth $\leq 1$ is trivial. The depth $1$
vertices of $\bott A\bott B$ form the set $\{(a,b)\in S_{\bott
A}\times S_{\bott B}: \operatorname{depth}_{\bott A}(a)=\operatorname{depth}_{\bott
B}(b)=1\}$. We simply show that the tree $\bott T_{(a,b)}$
anchored on $(a,b)$ in $\bott A\bott B$ is the product of the tree $\bott T_{a}$
anchored on $a$ in $\bott A$ with the tree $\bott T_{b}$ anchored
on $b$ in $\bott B$. By induction, we know that $\mathcal{D}(\bott
T_{(a,b)})=\mathcal{D}(\bott T_{a})\mathcal{D}(\bott T_{b})$, so $\bott
T_{(a,b)}=\bott T_{a}\bott T_{b}$. This concludes.
\end{proof}
It is easy to verify that the set of forests becomes a semiring with these operations:
\begin{lem}
The set $\mathbb{F}$ of forests becomes a semiring when endowed with the
sum and product defined above. Its additive identity is $\bott 0$, the empty
tree with $(V=\varnothing, A=\varnothing)$, while its multiplicative identity is the
rooted infinite directed path $\bott P_\infty$ with $(V=\mathbb{N},
A=\{(n+1, n)|n\in\mathbb{N}\})$.
\end{lem}
A straightforward inductive proof gives the following lemma:
\begin{lem}\label{DepthBotTreeState}
If $\bott A, \bott B\in\mathbb{F}$, then for $a\in S_{\bott A}, b\in
S_{\bott B}$ such that $(a,b)\in S_{\bott A\bott B}$, we have
$\operatorname{depth}_{\bott A\bott B}((a,b))=\operatorname{depth}_{\bott A}(a)=\operatorname{depth}_{\bott
B}(b)$.
\end{lem}
\begin{lem}\label{DepthTrees}
If $\bott A, \bott B\in\mathbb{F}$, then $\operatorname{depth}(\bott A\bott
B)=\min(\operatorname{depth}(\bott A), \operatorname{depth}(\bott B))$.
\end{lem}
\begin{proof}
We have:
$$S_{\bott A\bott B}=\{(a,b)\in S_{\bott A}\times S_{\bott B}:
\operatorname{depth}_{\bott A}(a)=\operatorname{depth}_{\bott B}(b)\}.$$
From \Cref{DepthBotTreeState}, a state $(a,b)\in S_{\bott A}\times
S_{\bott B}$ has depth at most $\min(\operatorname{depth}_{\bott
A}(a),\operatorname{depth}_{\bott B})$. Moreover, if $k=\min(\operatorname{depth}(\bott A),
\operatorname{depth}(\bott B))$ and we let $a\in S_{\bott A},b\in S_{\bott B}$ two
states that have both depth $k$ in their respective trees, then
$(a,b)$ has depth $k$ too.
\end{proof}
\subsection{Permutations}
For every $k\geq 1$, we denote $C_k$ the \textit{cycle of length $k$}
defined as the FDS whose graph is the directed cycle of length $k$. We
say that $A\in\mathbb{D}$ is a \textit{permutation} if the function $A$ is
bijective. In that case, all the states of $A$ are cycle states. We
denote the semiring of permutations $\mathbb{D}_P$ (it has multiplicative
identity $C_1$ and additive identity the empty function). In
particular, for any $A\in\mathbb{D}$, $[A]_0\in \mathbb{D}_P$.
We introduce two shortened notations: $a\vee
b=\text{lcm}(a,b)$ and $a\wedge b=\text{gcd}(a,b)$. In \cite{PolEqFDS}, the following very useful and simple result is
proven:
\begin{lem}\label{LemCycleProduct}
$C_a\times C_b=(a\wedge b)C_{a\vee b}$.
\end{lem}
We now extend it to arbitrary products of cycles. For any multiset $J$ of positive integers, we denote $\bigwedge J = \bigwedge_{j \in J} j$ and $\bigvee J = \bigvee_{j \in J} j$; if $J$ is empty, then those terms are equal to $1$.
\begin{lem} \label{LemDelta}
Let $J$ be a multiset of positive integers. Then $\prod_{j \in J} C_j = \delta_J C_{\bigvee J}$, where $\delta_J$ is recursively defined as $\delta_\emptyset = 1$ and for any $a \in \mathbb{N}$
\[
\delta_{ J \cup \{a\} } = (a \wedge \bigvee J) \delta_J.
\]
\end{lem}
\begin{proof}
The proof is by induction on the cardinality of $J$. The result is clear when $J$ is empty. Assume it is true for $J$, and let $a \in \mathbb{N}$. Then
\[
\prod_{ j \in J \cup \{a\} } C_j = C_a \prod_{j \in J} C_j = \delta_J C_a C_{\bigvee J} = \delta_J ( a \wedge \bigvee J ) C_{\bigvee J \cup \{a\}}.
\]
\end{proof}
Given a permutation
$A\in \mathbb{D}_P$, such that the length of each of its cycles is a multiple
of some $k\in\mathbb{N}$, and $k$ trees $\bott T_0, \dots, \bott
T_{k-1}\in\mathbb{F}_T$, we denote $A(\bott T_0, \dots, \bott T_{k-1})$ the FDS
obtained by taking each cycle of $A$, traversing it by following the
arrows, and anchoring on the $i$-th state encountered the dendron
$T_{i\mod k}$. This is pictured in \Cref{CycleTreezation}.
We use the following notation: for $A\in \mathbb{D}$, and for all $i\in\mathbb{N}$,
we denote $\lambda_i^A$ the number of cycles of length $i$ in $A$.
Finally, given an FDS $A\in\mathbb{D}$, and a set $L\subseteq\mathbb{N}$, we
define the \textit{$L$-support} of $A$, denoted $\operatorname{supp}_L(A)$, as the FDS
made of the connected components of $A$ with cycle size in $L$.
The following two results will prove useful to understand the product
of a permutation with a dendron.
\begin{lem}
For any $\ell, k \geq 1$ and trees $\bott T_1, \dots, \bott T_k\in\mathbb{F}_T$,
$C_k(\bott T_1, \dots, \bott T_k)\times C_\ell=(C_kC_\ell)(\bott T_1, \dots, \bott T_k)$.
\end{lem}
\begin{proof}
Take a product isomorphism for the product $C_k\times
C_\ell=(k\wedge\ell)C_{k\vee \ell}$, and write $S_{(k\wedge
\ell)}C_{k\vee\ell}\simeq S_{C_k}\times S_{C_\ell}$ accordingly.
Then, take $(i,j)\in S_{(k\wedge \ell)C_{k\vee\ell}}$. Let's show
that the tree that is anchored on this state in $S_{C_k(\bott T_1, \dots,
\bott T_k)\times C_\ell}$ is the tree that is anchored on $i$ in
$C_k(\bott T_1, \dots, \bott T_k)$, say $\bott T_i$. Indeed, since $C_\ell$ has no tree
states, the tree states over $(i,j)$ have a first component with is
a tree state, and a second one which is a cycle state. But since
each state of $C_\ell$ has exactly one predecessor, and the tree
anchored on $(i,j)$ is indeed $\bott T_i$. This proves the result.
\end{proof}
\begin{cor}\label{CorMultPermTrees}
For any $A\in\mathbb{D}_P$ and trees $\bott T_1, \dots, \bott T_k\in\mathbb{F}_T$, $C_k(\bott T_1,
\dots, \bott T_k)\times A = (C_kA)(\bott T_1, \dots, \bott T_k)$.
\end{cor}
\begin{proof}
Let's write $A=\sum_{i\in\mathbb{N}}\lambda^A_iC_i$. Then,
\[
C_k(\bott T_1, \dots,
\bott T_k)\times A=\sum_{i\in\mathbb{N}}\lambda^A_i (C_kC_i)(\bott T_1, \dots, \bott T_k) =
(C_kA)(\bott T_1, \dots, \bott T_k)
\]
from the previous lemma.
\end{proof}
\begin{figure}
\begin{subfigure}{0.3\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (1) {\phantom{$1$}};
\node[draw,circle] (2) [above right=of 1] {\phantom{$2$}};
\node[draw,circle] (3) [above left=of 1] {\phantom{$3$}};
\draw[->] (2) -- (1);
\draw[->] (3) -- (1);
\end{tikzpicture}
\end{center}
\caption{$\bott T_1$.}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (1) {\phantom{$1$}};
\node[draw,circle] (2) [above=of 1] {\phantom{$2$}};
\node[draw,circle] (3) [above=of 2] {\phantom{$3$}};
\draw[->] (2) -- (1);
\draw[->] (3) -- (2);
\end{tikzpicture}
\end{center}
\caption{$\bott T_2$.}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\resizebox{1\textwidth}{!}{
\begin{tikzpicture}
\node[draw,circle] (1) {\phantom{$1$}};
\node[draw,circle] (2) [below=of 1] {\phantom{$2$}};
\draw[->] (1) edge[bend left] (2);
\draw[->] (2) edge[bend left] (1);
\node[draw,circle] (1A1) [above left=of 1] {\phantom{$1$}};
\node[draw,circle] (1A2) [above right=of 1] {\phantom{$1$}};
\draw[->] (1A1) -- (1);
\draw[->] (1A2) -- (1);
\node[draw,circle] (1B1) [below=of 2] {\phantom{$1$}};
\node[draw,circle] (1B2) [below=of 1B1] {\phantom{$1$}};
\draw[->] (1B1) -- (2);
\draw[->] (1B2) -- (1B1);
\tikzxshift{4cm}
\node[draw,circle] (3) {\phantom{$3$}};
\node[draw,circle] (4) [right=of 3] {\phantom{$4$}};
\node[draw,circle] (5) [below=of 4] {\phantom{$5$}};
\node[draw,circle] (6) [left=of 5] {\phantom{$6$}};
\draw[->] (3) -- (4);
\draw[->] (4) -- (5);
\draw[->] (5) -- (6);
\draw[->] (6) -- (3);
\node[draw,circle] (5A1) [below left=of 5] {\phantom{$1$}};
\node[draw,circle] (5A2) [below right=of 5] {\phantom{$1$}};
\draw[->] (5A1) -- (5);
\draw[->] (5A2) -- (5);
\node[draw,circle] (3A1) [above left=of 3] {\phantom{$1$}};
\node[draw,circle] (3A2) [above right=of 3] {\phantom{$1$}};
\draw[->] (3A1) -- (3);
\draw[->] (3A2) -- (3);
\node[draw,circle] (2B1) [below left=of 6] {\phantom{$1$}};
\node[draw,circle] (2B2) [below=of 2B1] {\phantom{$1$}};
\draw[->] (2B1) -- (6);
\draw[->] (2B2) -- (2B1);
\node[draw,circle] (4B1) [above right=of 4] {\phantom{$1$}};
\node[draw,circle] (4B2) [above=of 4B1] {\phantom{$1$}};
\draw[->] (4B1) -- (4);
\draw[->] (4B2) -- (4B1);
\end{tikzpicture}
}
\caption{$A(\bott T_1, \bott T_2)$.}
\end{subfigure}
\caption{The FDS $A(\bott T_1, \bott T_2)$ for $A=C_2+C_4$, and two
trees $\bott T_1, \bott T_2$.}
\label{CycleTreezation}
\end{figure}
\section{Cancellative finite dynamical systems} \label{sec:Cancellative}
It is known that the division operation can sometimes not yield a
unique result; a well-known example is: $C_2^2=2C_2$. We can also show
that if we have $AB=AC$ for $A,B,C\in\mathbb{D}$, even setting $[B]_0=[C]_0$
does not guarantee that $B=C$, since, given two different trees $\bott
T_1$ and $\bott T_2$, we have the identity
$C_2(2\bott T_1+C_2(\bott T_2))=C_2(C_2(\bott T_1)+2\bott T_2)$. We therefore consider the elements for which division is unambiguous, defined as follows.
\begin{defn}[Cancellative element]
An FDS $A\in\mathbb{D}$ is said \textit{cancellative} if for all
$B,C\in\mathbb{D}$, $AB=AC\implies B=C$.
\end{defn}
In this section, we prove that an FDS is cancellative if and only if
it has a fixpoint. We approach this theorem in steps. First, we
introduce an order on trees, based on a code for trees. Then, we move
on to show that we can transform FDSs into forests in a way that works
well with both the product on forests and on FDSs. Finally, we show
that result.
\subsection{Order on trees}
It will prove very useful to
have a total order on finite trees that is compatible with the
product. That is, if $\bott T_1,\bott T_2,\bott T_3,\bott
T_4\in\mathbb{F}_T$, and $\bott T_1 < \bott T_2$ and $\bott T_3\leq\bott T_4$,
we want to have $\bott T_1\bott T_3<\bott T_2\bott T_4$. This will be
guaranteed by \Cref{CompleteCodeOrder}.
To do so, we define a code $\mathcal{C}_f$ from finite trees to $\mathbb{N}^*$ (the
set of finite sequences of nonnegative integers), and we say that
$\bott T_1<\bott T_2\iff \mathcal{C}_f(\bott T_1)<_{\text{lex}}\mathcal{C}_f(\bott
T_2)$ (where $<_{\text{lex}}$ is the lexicographical order).
The code is computed as follows, using two mutually recursive
functions. We consider for a moment that trees are ordered: the
children of a node are stored in an ordered list, say, from left to
right. That is, for a tree $\bott T$, $\mathcal{D}(\bott T)$ is now a tuple
rather than a multiset. Then, we define a procedure $collect$ which
takes a finite tree, sorts it (using the function $sort$ defined
below), and then traverses level by level, following the
order of the predecessors, starting from depth $0$, and outputs a
tuple of the number of predecessors of each node encountered.
We also define a procedure $sort$ that takes a finite tree $\bott T$,
begins by calling $collect$ on each of the subtrees anchored on direct
predecessors of the root, and then order those predecessors from left
to right by increasing return value of $collect$. Finally, $\mathcal{C}_f(\bott
T)=collect(\bott T)$.
A pseudocode implementation of $sort$ and $collect$ is found in \Cref{SortAndCollect}.
\begin{figure}
\begin{subfigure}[B]{0.5\textwidth}
\begin{center}
\begin{procedure}[H]
\If{$|\bott T|>1$}{
$\bott T\leftarrow sort(\bott T)$\;
}
$t=[]$\;
\For{$i\in\iitv{0,\operatorname{depth}(\bott T)}$}{
\For{$v$ in $\bott T$'s depth $i$, from left to right}{
$d\leftarrow$ number of children of $v$\;
$t\leftarrow t :: d$\;
}
}
\Return $t$\;
\end{procedure}
\end{center}
\caption{$collect(\bott T)$ ($::$ is the concatenation operator)}
\end{subfigure}
\begin{subfigure}[B]{0.5\textwidth}
\begin{center}
\begin{procedure}[H]
$(T_1, \dots, T_n) := \mathcal{D}(\bott T)$\;
\For{$i\in\iitv{1,n}$}{
$c_i\leftarrow collect(T_i)$\;
}
$(U_1, \dots, U_n)\leftarrow$ sort $(T_1, \dots, T_n)$ by increasing
$(c_1, \dots, c_n)$\;
Let $\bott{T'}$ such that $\mathcal{D}(\bott{T'})=(U_1, \dots, U_n)$\;
\Return $\bott{T'}$.
\end{procedure}
\end{center}
\caption{$sort(\bott T)$}
\end{subfigure}
\caption{The two mutually recursive functions for computing $\mathcal{C}_f$.}
\label{SortAndCollect}
\end{figure}
\begin{expl}
The tree in \Cref{fig:tildeExTree} has the code $(2, 0, 2, 0, 0)$;
its states are traversed in the following order: $A, B, C, D, E$ in
the topmost call to $collect$.
\end{expl}
\begin{lem}\label{CodePrefixFree}
The code $\mathcal{C}_f$ is prefix-free. That is, if $\bott T, \bott{T'}\in
\mathbb{F}_T$ are such that $\mathcal{C}_f(\bott T)$ is a prefix of $\mathcal{C}_f(\bott{T'})$,
we have $\mathcal{C}_f(\bott T)=\mathcal{C}_f(\bott{T'})$.
\end{lem}
\begin{proof}
Given a code $c$, and an index $i$, write $\delta(c,i)=\sum_{j=1}^i
c_j - i$. This is the number of vertices that have been announced as
children of vertices in $c_1, \dots, c_i$ but which are not themselves
in $c_1, \dots, c_i$. Thus, if we are reading a code $c$, and we have
read the $i$ first elements, we know that we must read at least
$\delta(c,i)$ other elements. Moreover, remark that if
$\delta(c,i)=0$, then we are at the end of the code, since we have
already read the children of every vertex.
Now, suppose that $c:=\mathcal{C}_f(\bott{T})$ is a prefix of
$c':=\mathcal{C}_f(\bott{T'})$. Let $i=|c|$: we have $\delta(c,i)=0$ since
$c$ is completely read once we have read the $i$ first elements.
Moreover, we must have $\delta(c',i)=\delta(c,i)$ since
$c'_1 \dots c'_i = c_1 \dots c_i$. So, $\delta(c',i)=0$ too, and thus, $c=c'$.
\end{proof}
We now say that, for two trees $\bott T,\bott{T'}$, we have $\bott
T\leq_f \bott{T'}$ if $\mathcal{C}_f(\bott{T})\leq_\text{lex} \mathcal{C}_f(\bott{T'})$.
We claim that this defines a total order on trees. Reflexivity and
transitivity are trivial, and its antisymmetry is guaranteed by the
following lemma:
\begin{lem}\label{OrderAntisymmetry}
For any two finite trees $\bott T,\bott{T'}$, $\mathcal{C}_f(\bott
T)=\mathcal{C}_f(\bott{T'})\implies \bott T=\bott{T'}$.
\end{lem}
\begin{proof}
We just show that we can reconstruct $\bott T$ from $\mathcal{C}_f(\bott
T)=collect(\bott T)$. We can ignore the call to $sort(\bott T)$ in
$collect(\bott T)$: we can consider that the tree $\bott T$ we will
recover is already sorted. To shorten notations, let's write
$c:=\mathcal{C}_f(\bott T)$.
First, we can partition $c$ into levels. Indeed, remark that if we
know that the indices corresponding to states at depth $d$ form the
set $\iitv{k,\ell}$, then we know that the number of states at depth
$d+1$ is $\sum_{j=k}^\ell c_j$, and so the states at depth $d+1$
correspond to indices $\iitv{\ell+1, \ell+\sum_{j=k}^\ell c_j}$. So,
we can now iterate on the levels of $c$: let's write for convenience
$k_d,\ell_d\in\mathbb{N}$ the first and last indices of states at depth $d$.
The first level, corresponding to depth $0$, is easy to reconstruct:
simply create the root. For our induction, we also create the
predecessors of the root, of which we know the number, so the
induction begins at depth $1$.
Now, suppose we have uniquely reconstructed $\bott T$ up to depth
$d$, and that we want to reconstruct level $d+1$. We traverse our
reconstructed depth $1$ from left to right, and simultaneously
traverse $c_{k_d}, \dots, c_{\ell_d}$. The $j$-th state we encounter
has degree $c_{k_d+j}$, so we create $c_{k_d+j}$ children for that
state. Thus, the level at depth $d+1$ is reconstructed uniquely too.
Thus, we reconstruct $\bott T$, and this concludes the proof.
\end{proof}
The results which make this code useful are the following lemma and its
corollary.
\begin{lem}\label{TreeOrderProduct}
For all finite trees $\bott T_1, \bott T_2, \bott T_3\in\mathbb{F}_T$, we
have $[\bott T_1]_{\operatorname{depth}(\bott T_3)}<_f[\bott T_2]_{\operatorname{depth}(\bott T_3)}\implies \bott T_1\bott T_3<_f\bott
T_2\bott T_3$.
\end{lem}
\begin{proof}
Let's prove this by induction on $\bott T_1$ and $\bott T_2$'s depth. It's
trivial at depth $0$. Take $\bott T_1,\bott T_2$ of depth $\leq
k+1$, with $k$ such that the result stands for trees of depth $\leq
k$.
Because of \Cref{CodePrefixFree}, since $\bott T_1<_f\bott T_2$,
$\mathcal{C}_f(\bott T_1)$ cannot be a prefix of $\mathcal{C}_f(\bott T_2)$.
Thus, there exists an index $i$ such that $\mathcal{C}_f(\bott
T_1)_i<\mathcal{C}_f(\bott T_2)_i$ and for all $j<i$, we have $\mathcal{C}_f(\bott
T_1)_j=\mathcal{C}_f(\bott T_2)_j$. Let $x\in S_{\bott T_1}$
be the vertex at index $i$ in $\mathcal{C}_f(\bott T_1)$, and let $y\in
S_{\bott T_2}$ be the vertex at index $i$ in $\mathcal{C}_f(\bott T_2)$. In
the following, for a tree $\bott T$ and a vertex $u\in S_{\bott T}$,
we denote by ${\mathcal{C}_f}_{\bott T}(u)$ the code of the subtree with
root $u$ in $\bott T$. Since the codes share the same prefix of
length $i-1$, $\operatorname{depth}_{\bott T_1}(x)=\operatorname{depth}_{\bott T_2}(y)$ (as seen
in the proof of \Cref{OrderAntisymmetry}, this shared prefix of
length $i-1$ holds all the information necessary to reconstruct
everything above $x$ and $y$). Let's denote $d$ this depth. Because $[\bott T_1]_{\operatorname{depth}(\bott T_3)}<_f[\bott T_2]_{\operatorname{depth}(\bott T_3)}$, we have $d<\operatorname{depth}(\bott T_3)$.
It is clear that we have $[\bott T_1]_{d-1}=[\bott T_2]_{d-1}$, so
in particular, $[\bott T_1\bott T_3]_{d-1}=[\bott T_2\bott
T_3]_{d-1}$. Let $z$ be the root of the tree with minimal code in $\bott T_3$ at depth $d$. Now, we show that the first difference between the
codes of $\bott T_1\bott T_3$ and $\bott T_2\bott T_3$ is at the
index $j$ corresponding to the vertex $(x,z)$ in $\mathcal{C}_f(\bott T_1\bott
T_3)$, and to the vertex $(y,z)$ in $\mathcal{C}_f(\bott T_2\bott T_3)$. There
might be multiple possibilities for $z$; we can assume that we take
the one which gives the minimum $j$.
Indeed, assume that a vertex of the form $(x,t)$ for some vertex $t$
of $\bott T_3$ appears in $\mathcal{C}_f(\bott T_1\bott T_3)$ at depth $d$
before index $j$. Since it appears before vertex $(x,z)$, by
induction hypothesis, it means that the code of the subtree anchored
on $t$ must be smaller than that of the subtree anchored on $z$. By
minimality of $z$, this means that ${\mathcal{C}_f}_{\bott
T_3}(z)={\mathcal{C}_f}_{\bott T_3}(t)$. Since we have chosen $z$ to be
the first occurence of this code at this depth, we must have $t=z$.
So, $(x,z)$ is the first vertex in $\mathcal{C}_f(\bott T_1\bott T_3)$ in
which $x$ appears. A similar reasoning shows that no vertex
involving $y$ appears before index $j$ in $\mathcal{C}_f(\bott T_2\bott
T_3)$. Since every element before $x$ is shared between $\mathcal{C}_f(\bott
T_1)$ and $\mathcal{C}_f(\bott T_2)$, this means that the first difference
between $\mathcal{C}_f(\bott T_1\bott T_3)$ and $\mathcal{C}_f(\bott T_2\bott T_3)$
is at or after index $j$.
At index $j$, the number of predecessors $\mathcal{C}_f(\bott T_1\bott
T_3)_j$ is $\operatorname{npreds}_{\bott T_1}(x)\operatorname{npreds}_{\bott T_3}(z)$ while
$\mathcal{C}_f(\bott T_1\bott T_3)_j$ is $\operatorname{npreds}_{\bott
T_2}(y)\operatorname{npreds}_{\bott T_3}(z)$. Since $\operatorname{npreds}_{\bott
T_1}(x)<\operatorname{npreds}_{\bott T_2}(y)$, this shows that $\bott T_1\bott
T_3<_f\bott T_2\bott T_3$.
\end{proof}
\begin{cor}\label{CompleteCodeOrder}
For all finite trees $\bott T_1, \bott T_2, \bott T_3,\bott
T_4\in\mathbb{F}_T$, if $[\bott T_1]_{\operatorname{depth}(\bott T_3)}<_f[\bott T_2]_{\operatorname{depth}(\bott T_3)}$ and $[\bott T_3]_{\operatorname{depth}(\bott T_2)}\leq_f [\bott
T_4]_{\operatorname{depth}(\bott T_2)}$, we have $\bott T_1\bott T_3<_f\bott T_2\bott T_4$.
\end{cor}
\begin{proof}
By \Cref{TreeOrderProduct}, we have $\bott T_1\bott T_3<_f\bott
T_2\bott T_3$. If $\bott T_3=\bott T_4$, we can conclude now.
Otherwise, $\bott T_3<_f\bott T_4$, and we have, by
\Cref{TreeOrderProduct}, $\bott T_2\bott T_3<_f\bott T_2\bott T_4$. Combining the two inequalities, we
get: $\bott T_1\bott T_3<_f \bott T_2\bott T_4$.
\end{proof}
We are now ready for the recovery algorithm on finite trees.
\begin{lem}\label{FiniteTreeDiv}
If $\bott A,\bott B,\bott C\in\mathbb{F}_T$, $\bott A$ is finite, and
$\bott A\bott B=\bott A\bott C$, then $[\bott B]_{\operatorname{depth}(\bott
A)}=[\bott C]_{\operatorname{depth}(\bott A)}$.
\end{lem}
\begin{proof}
Since $<_f$ is a complete order, if $[\bott B]_{\operatorname{depth}(\bott
A)}\neq[\bott C]_{\operatorname{depth}(\bott A)}$, we can assume without loss of generality that we are in the case $[\bott B]_{\operatorname{depth}(\bott A)}<[\bott C]_{\operatorname{depth}(\bott A)}$. In that case, by \Cref{TreeOrderProduct}, we have $\bott A\bott B<_f\bott A\bott C$. This concludes.
\end{proof}
\begin{lem}\label{InfTreeDiv}
If $\bott A,\bott B, \bott C\in \mathbb{F}_T$ and $\bott A$ is infinite,
and $\bott A\bott B=\bott A\bott C$, then $\bott B=\bott C$.
\end{lem}
\begin{proof}
For every $d\in\mathbb{N}$, we have $[\bott A]_d[\bott B]_d=[\bott
A]_d[\bott C]_d$, and thus, from \Cref{FiniteTreeDiv}, $[\bott
B]_d=[\bott C]_d$. This implies that $\bott B=\bott C$.
\end{proof}
\begin{cor}\label{UniqTreeDiv}
If $A,B,C\in\mathbb{D}_D$, and $AB=AC$, then $B=C$.
\end{cor}
\begin{proof}
If $AB=AC$, then $\tildeop A\tildeop B=\tildeop A\tildeop C$. Using
\Cref{InfTreeDiv}, this means that $\tildeop B=\tildeop C$. Thus,
$B=C$.
\end{proof}
We can now extend the order on possibly infinite trees; this will be of use for our results on the unicity of $k$-th roots. For a tree
$\bott T$, define its code as $\mathcal{C}(\bott T):=(\mathcal{C}_f([\bott
T]_i))_{i\in \mathbb{N}}$, and say that $\bott T\leq\bott U$ if and only if
$\mathcal{C}(\bott T)\leq_{\text{lex}} \mathcal{C}(\bott U)$.
\begin{lem}\label{GeneralRestrictedProdOrder}
For all trees $\bott T_1, \bott T_2, \bott T_3\in\mathbb{F}_T$, if $\bott
T_1<\bott T_2$, we have $\bott T_1\bott T_3<\bott T_2\bott T_3$.
\end{lem}
\begin{proof}
If $\bott T_1<\bott T_2$, then $\bott T_1\neq\bott T_2$. In
particular, there is a minimal depth $d$ such that $[\bott
T_1]_d\neq[\bott T_2]_d$. Since for every $i<d$, we have $[\bott T_1]_i=[\bott T_2]_i$, we have $\mathcal{C}(\bott T_1\bott T_3)_1, \dots, \mathcal{C}(\bott T_1\bott
T_3)_{d-1}=\mathcal{C}(\bott T_2\bott T_3)_1, \dots, \mathcal{C}(\bott T_2\bott
T_3)_{d-1}$.
What is left to prove is that $\mathcal{C}(\bott T_1\bott T_2)_d<\mathcal{C}(\bott T_1\bott T_3)_d$, that is $\mathcal{C}_f([\bott T_1\bott T_3]_d)<\mathcal{C}_f([\bott T_1\bott T_3]_d)$. This follows from the fact that $\mathcal{C}_f([\bott T_1]_d)<\mathcal{C}_f([\bott T_2]_d)$ and \Cref{TreeOrderProduct}.
\end{proof}
\begin{cor}\label{GeneralProdOrder}
For all trees $\bott T_1, \bott T_2, \bott T_3,\bott T_4\in\mathbb{F}_T$,
if $\bott T_1<\bott T_2$ and $\bott T_3\leq \bott T_4$, we have
$\bott T_1\bott T_3<\bott T_2\bott T_4$.
\end{cor}
\begin{proof}
By \Cref{GeneralRestrictedProdOrder}, we have $\bott T_1\bott
T_3<\bott T_2\bott T_3$. If $\bott T_3=\bott T_4$, we can conclude
now. Otherwise, $\bott T_3<\bott T_4$, and we have, by
\Cref{GeneralRestrictedProdOrder}, $\bott T_1\bott T_3<\bott
T_2\bott T_3$ and $\bott T_2\bott T_3<\bott T_2\bott T_4$.
Combining the two, we get: $\bott T_1\bott T_3< \bott T_2\bott
T_4$.
\end{proof}
\subsection{Transforming an FDS into a forest}
In this subsection, we introduce a way of converting a general FDS into a
forest, since the product on forests works level by level. We do as follows:
\begin{defn}
Let $A = C_n(\bott T_1, \dots, \bott T_n)$ be a connected FDS. For any $a\in S_A$, we write $A^{-k}(a):=\{s\in S_A:A^k(s)=a\}$. Then, for each $a\in [A]_0$, we set $$S_a:=\{(s,k):s\in A^{-k}(a),k\in\mathbb{N}\}$$
and
$$E_a:=\{((s,k),(A(s),k-1)):(s,k)\in S_a\}.$$
\end{defn}
\begin{lem} \label{LemTa}
The directed graph $\bott T_a(A)$ with vertex set $S_a$ and edge set $E_a$ defined above is a tree. Moreover $S_a$ and $S_b$ are disjoint for all $a \ne b$.
\end{lem}
\begin{proof}
Take $a\in[A]_0$. We will show that $\bott T_a(A)$ is a tree of root $(a,0)$. First, $\bott T_a(A)$ is acyclic because $k$ necessarily decreases following any arc, which also shows that $\bott T_a(A)$ is correctly oriented. Furthermore, if $b\in S_a$, then there exists $k$ such that $A^k(b)=a$, and thus we have the following path from $a$ to $b$:
$$
(b,k) \rightarrow (A(b), k-1) \rightarrow \dots \rightarrow (A^{k-1}(b), 1) \rightarrow (a,0).
$$
which has all of its edges in $E_a$. So, $T_a(A)$ is a well-defined tree, and $\tildeop{A}$ is a forest.
Now, we show that if $a,b\in S_{[A]_0}$ and $a\neq b$, then $S_a\cap S_b=\varnothing$. Suppose that $(s,k)\in S_a\cap S_b$. Then, $A^k(s)=a=b$, which is the desired contradiction. Thus, we may write $S_{\tildeop A}=\bigcup_{a\in S_{[A]_0}}S_a$ without renaming.
\end{proof}
We thus define the \textit{unrolling} of $A$ as $\tildeop{A}:=\sum_{a\in S_{[A]_0}} \bott T_a(A)$, with $S_{\tildeop A}=\bigcup_{a\in S_{[A]_0}}S_a$.
We can then extend this to general FDSs, by writing:
$\tildeop{A+B}=\tildeop{A}+\tildeop{B}$. Note that the unrolling is not injective. Indeed, for
instance, $\tildeop{C_3}=\tildeop{3C_1}$. This is not true even for FDSs with the same periodic part: if $\bott T$ and $\bott U$ are two distinct trees and $X = 2 C_1( \bott T ) + C_2( \bott U, \bott U )$ and $Y = 2 C_1( \bott U ) + C_2( \bott T, \bott T )$, then $\tildeop{ X } = \tildeop{ Y }$. However, in the connected case, we have injectivity.
\begin{lem} \label{UnrollingInjective}
Let $X,Y \in \mathbb{D}$. If $X$ and $Y$ are connected and $[X]_0 = [Y]_0$, then $\tildeop{ X } = \tildeop{ Y } \implies X = Y$.
\end{lem}
\begin{proof}
Let $X = C_x(\bott T_1, \dots, \bott T_x)$. Then $\tildeop{ X }$ has $x$ infinite trees $\bott X_1, \dots, \bott X_x$, each a periodic shift of the previous one:
\[
tseq(\bott X_1) = (\bott T_1, \bott T_2, \dots, \bott T_x), \dots, tseq(\bott X_x) = (\bott T_x, \bott T_1, \dots, \bott T_{x-1}).
\]
We have $Y = C_x( \bott U_1, \dots, \bott U_x )$, and similarly $\tildeop{ Y }$ consists of the trees $\bott Y_1, \dots, \bott Y_x$ where
\[
tseq(\bott Y_1) = (\bott U_1, \bott U_2, \dots, \bott U_x), \dots, tseq(\bott U_x) = (\bott U_x, \bott U_1, \dots, \bott U_{x-1}).
\]
Then $\bott U_1 \in \{ \bott T_1, \dots, \bott T_x \}$, without loss say $\bott U_1 =\bott T_1$, then $\bott U_y = \bott T_y$ for all $1 \le y \le x$ and $X = Y$.
\end{proof}
\begin{expl}
See \Cref{fig:tildeExTrees} and \Cref{fig:tildeExConn}.
\begin{figure}
\hspace{-2.4cm}\noindent\makebox[\textwidth]{%
\begin{subfigure}[B]{0.4\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (A) {$A$};
\node[draw,circle] (B) [above left=of A] {$B$};
\node[draw,circle] (C) [above right=of A] {$C$};
\node[draw,circle] (D) [above left=of C] {$D$};
\node[draw,circle] (E) [above right=of C] {$E$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A);
\draw[->] (B) -- (A);
\draw[->] (E) -- (C);
\draw[->] (A) edge [loop below] (A);
\end{tikzpicture}
\end{center}
\caption{A dendron $T$.}
\label{fig:tildeExTree}
\end{subfigure}
\begin{subfigure}[B]{0.5\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (A0) {$A^0$};
\node[draw,circle] (B) [above left=1.3cm and 1cm of A0] {$B^1$};
\node[draw,circle] (C) [above=of A0] {$C^1$};
\node[draw,circle] (D) [above left=1.3cm and 1cm of C] {$D^2$};
\node[draw,circle] (E) [above=of C] {$E^2$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A0);
\draw[->] (B) -- (A0);
\draw[->] (E) -- (C);
\node[draw,circle] (A1) [above right=1.2cm and 3cm of A0] {$A^1$};
\node[draw,circle] (B) [above left=1.3cm and 1cm of A1] {$B^2$};
\node[draw,circle] (C) [above=of A1] {$C^2$};
\node[draw,circle] (D) [above left=1.2cm and 1cm of C] {$D^3$};
\node[draw,circle] (E) [above=of C] {$E^3$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A1);
\draw[->] (B) -- (A1);
\draw[->] (E) -- (C);
\draw[->] (A1) -- (A0);
\node[draw,circle] (A2) [above right=1.2cm and 3cm of A1] {$A^2$};
\node[draw,circle] (B) [above left=1.3cm and 1cm of A2] {$B^3$};
\node[draw,circle] (C) [above=of A2] {$C^3$};
\node[draw,circle] (D) [above left=1.3cm and 1cm of C] {$D^4$};
\node[draw,circle] (E) [above=of C] {$E^4$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A2);
\draw[->] (B) -- (A2);
\draw[->] (E) -- (C);
\draw[->] (A2) -- (A1);
\node (etc) [above right=1.3cm and 3cm of A2] {$\iddots$};
\draw[->] (etc) -- (A2);
\end{tikzpicture}
\end{center}
\caption{The tree $\tildeop{T}$, rooted in $A^0$.}
\end{subfigure}
}
\caption{The $\tildeop{\cdot}$ operation on a dendron.}
\label{fig:tildeExTrees}
\end{figure}
\begin{figure}
\begin{subfigure}[B]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (A) {$A$};
\node[draw,circle] (F) [below=of A] {$F$};
\node[draw,circle] (G) [below=of F] {$G$};
\node[draw,circle] (B) [above left=of A] {$B$};
\node[draw,circle] (C) [above right=of A] {$C$};
\node[draw,circle] (D) [above left=of C] {$D$};
\node[draw,circle] (E) [above right=of C] {$E$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A);
\draw[->] (B) -- (A);
\draw[->] (E) -- (C);
\draw[->] (A) to[bend left] (F);
\draw[->] (F) to[bend left] (A);
\draw[->] (G) -- (F);
\end{tikzpicture}
\end{center}
\caption{A connected FDS $S$.}
\end{subfigure}
\begin{subfigure}[B]{0.7\textwidth}
\begin{center}
\begin{tikzpicture}
\node[draw,circle] (A0) {$A^0$};
\node[draw,circle] (B) [above left=1.2cm and 1.5cm of A0] {$B^1$};
\node[draw,circle] (C) [above=of A0] {$C^1$};
\node[draw,circle] (D) [above left=1.2cm and 1.5cm of C] {$D^2$};
\node[draw,circle] (E) [above=of C] {$E^2$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A0);
\draw[->] (B) -- (A0);
\draw[->] (E) -- (C);
\node[draw,circle] (F1) [above right=1.2cm and 1.5cm of A0] {$F^1$};
\node[draw,circle] (G2) [above=of F1] {$G^2$};
\draw[->] (G2) -- (F1);
\draw[->] (F1) -- (A0);
\node[draw,circle] (A2) [above right=1.2cm and 1.5cm of F1] {$A^2$};
\node[draw,circle] (B) [above left=1.2cm and 1cm of A2] {$B^3$};
\node[draw,circle] (C) [above=of A2] {$C^3$};
\node[draw,circle] (D) [above left=of C] {$D^4$};
\node[draw,circle] (E) [above=of C] {$E^4$};
\draw[->] (D) -- (C);
\draw[->] (C) -- (A2);
\draw[->] (B) -- (A2);
\draw[->] (E) -- (C);
\draw[->] (A2) -- (F1);
\node (etc) [above right=of A2] {$\iddots$};
\draw[->] (etc) -- (A2);
\tikzxshift{7cm}
\node[draw,circle] (F0) {$F^0$};
\node[draw,circle] (G1) [above=of F0] {$G^1$};
\draw[->] (G1) -- (F0);
\node[draw,circle] (A1p) [above right=1.2cm and 1.5cm of F0] {$A^1$};
\node[draw,circle] (Bp) [above left=1.2cm and 1cm of A1p] {$B^2$};
\node[draw,circle] (Cp) [above=of A1p] {$C^2$};
\node[draw,circle] (Dp) [above left=1.2cm and 1cm of Cp] {$D^3$};
\node[draw,circle] (Ep) [above=of Cp] {$E^3$};
\node (etcp) [above right=of A1p] {$\iddots$};
\draw[->] (Dp) -- (Cp);
\draw[->] (Cp) -- (A1p);
\draw[->] (Bp) -- (A1p);
\draw[->] (Ep) -- (Cp);
\draw[->] (A1p) -- (F0);
\draw[->] (etcp) -- (A1p);
\end{tikzpicture}
\end{center}
\caption{The forest $\tildeop{S}$, with roots $A^0$ and $F^0$.}
\end{subfigure}
\caption{The $\tildeop{\cdot}$ operation on a connected FDS.}
\label{fig:tildeExConn}
\end{figure}
\end{expl}
The following lemma explains why the unrolling operation makes sense: it
is compatible with the product. The proof is rather technical, but the
intuition for this result is simple. A cycle behaves very much like an
infinite path in terms of predecessors,and the unrolling converts the cycle into an infinite path
that behaves similarly. Moreover, the reason we create multiple
infinite trees for each cycle is to avoid problems with cases where
the product of two connected FDSs gives a non-connected FDS.
\begin{lem}\label{ProdBot}
For any $A,B\in\mathbb{D}$, we have: $\tildeop{A}\tildeop{B}=\tildeop{AB}$.
\end{lem}
\begin{proof}
We show this result for connected $A$ and $B$ as the the other cases follow by distributivity. Thus, we write $A=C_m(\bott T_0,\dots,\bott T_{m-1})$ and $B=C_n(\bott U_0, \dots, \bott U_{n-1})$. Now, we can write:
\begin{eqnarray*}
S_{\tildeop A}&=&\bigcup_{a\in[A]_0}\{(s,k):s\in A^{-k}(a),k\in\mathbb{N}\} \\
S_{\tildeop B}&=&\bigcup_{b\in[B]_0}\{(s,k):s\in B^{-k}(b),k\in\mathbb{N}\} \\
\end{eqnarray*}
Now, the product $\tildeop A\tildeop B$ has the following state set:
\begin{eqnarray*}
S_{\tildeop A\tildeop B}&=&\{(a,b)\in S_{\tildeop A}\times S_{\tildeop B}:\operatorname{depth}_{\tildeop A}(a)=\operatorname{depth}_{\tildeop B}(b)\} \\
&\simeq& \{((s_a,k_a),(s_b,k_b))\in S_{\tildeop A}\times S_{\tildeop B}:k_a=k_b\} \\
&\simeq& \bigcup_{(a,b)\in[A]_0\times [B]_0}\{(s_a, s_b, k)\in S_{A}\times S_{B}\times \mathbb{N}: s_a\in A^{-k}(a), s_b\in B^{-k}(b);k\in\mathbb{N}\}.
\end{eqnarray*}
Now, let's show that this is isomorphic to $S_{\tildeop{AB}}$ (remember that $S_{AB}=S_A\times S_B$):
\begin{eqnarray*}
S_{\tildeop{AB}} &=& \bigcup_{c\in [AB]_0}\{(s,k)\in S_{AB}\times\mathbb{N}:s\in AB^{-k}(c),k\in\mathbb{N}\} \\
&=& \bigcup_{(a,b)\in[A]_0\times [B]_0}\{((s_a,s_b),k)\in S_{AB}\times\mathbb{N}:(s_a,s_b)\in AB^{-k}((a,b)),k\in\mathbb{N}\} \\
&=& \bigcup_{(a,b)\in[A]_0\times [B]_0}\{((s_a,s_b),k)\in S_{AB}\times\mathbb{N}:s_a\in A^{-k}(a), s_b\in B^{-k}(b),k\in\mathbb{N}\}.
\end{eqnarray*}
The last step comes from the following identity: for $c=(a,b)\in S_{AB}=S_A\times S_B$, we have $AB^{-k}(c) = A^{-k}(a)\times B^{-k}(b)$. Thus, we have shown that $S_{\tildeop A\tildeop B} \simeq S_{\tildeop{AB}}$.
Now, what is left to do is show that the edges are also isomorphic. Thus, we must show that for any $(s_a,s_b,k), (s'_a,s'_b,k')\in S_{\tildeop A\tildeop B}$, we have $(s_a,s_b,k)\rightarrow (s'_a,s'_b,k')$ in $\tildeop A\tildeop B$ if and only if we have $(s_a,s_b,k)\rightarrow (s'_a,s'_b,k')$ in $\tildeop{AB}$.
In the end of the proof, we denote $x\rightarrow{C} y$ the existence of an edge from $x$ to $y$ in the forest or FDS $C$ (if $C$ is an FDS, $x\rightarrow{C} y$ means $y=C(x)$). Now, we can reason by equivalence:
\begin{eqnarray*}
&& (s_a,s_b,k)\xrightarrow{\tildeop A\tildeop B} (s'_a,s'_b,k') \\
&\iff& k'=k-1 \wedge (s_a,k)\xrightarrow{\tildeop A} (s'_a,k') \wedge (s_b,k)\xrightarrow{\tildeop B} (s'_b,k') \\
&\iff& k'=k-1 \wedge s_a\xrightarrow{A} s'_a \wedge s_b\xrightarrow{B} s'_b \\
&\iff& k'=k-1 \wedge (s_a,s_b)\xrightarrow{AB} (s'_a,s'_b) \\
&\iff& (s_a,s_b,k)\xrightarrow{\tildeop{AB}} (s'_a,s'_b,k').
\end{eqnarray*}
This concludes.
\end{proof}
We can now show that division is unambiguous when restricted to connected FDSs.
\begin{thm} \label{thmConnected}
For any FDS $A \in \mathbb{D}$, if $X, Y \in \mathbb{D}$ are connected, then
\[
AX = AY \implies X = Y.
\]
\end{thm}
\begin{proof}
Suppose $AX = AY$. Let $[X]_0 = C_x$ and $[Y]_0 = C_y$, then $|[AX]_0| = x |[A]_0|$ and $|[AY]_0| = y |[A]_0|$ show that $x = y$, that is $[X]_0 = [Y]_0$. Thus,
\[
AX = AY
\implies \tildeop{AX} = \tildeop{AY}
\xRightarrow{ \text{ \Cref{ProdBot} } } \tildeop{A} \tildeop{X} = \tildeop{A} \tildeop{Y}
\xRightarrow{ \text{ \Cref{InfTreeDiv} } } \tildeop{X} = \tildeop{Y}
\xRightarrow{ \text{ \Cref{UnrollingInjective} } } X = Y.
\]
\end{proof}
We remark that \Cref{thmConnected} implies \cite[Conjecture 3.1]{DFPR22}. Indeed, if $A$ and $B$ are connected and $AX = AY = B$, then $X$ and $Y$ are connected, thus $X = Y$.
\subsection{Cancellative FDSs are those with a fixpoint}
Using the results of the previous part, we have the following lemma:
\begin{lem}\label{FixpointIsCancellative}
If $A\in\mathbb{D}$, and $A$ has a fixpoint, then $A$ is cancellative.
\end{lem}
\begin{proof}
Take $B,D\in\mathbb{D}$ such that $AB=D$. Let's show that we can recover
$B$ by induction on the size of $D$. The base case is trivial: if
$|D|=0$, then $D=0$ and since $A$ has a fixpoint, $A\neq 0$, so
$B=0$.
Denote $\ell$ the size of the smallest cycle of $D$. Since $A$ has a
cycle of length $1$, it means that the smallest cycle of $B$ is of
length $\ell$ too. Let $L \subseteq \mathbb{N}$ be the set of divisors of $\ell$. We denote $A' = \operatorname{supp}_L(A)$, and similarly $B' = \operatorname{supp}_L(B)$ and $D' = \operatorname{supp}_L(D)$.
Then we have $A'B'=D'$. Indeed, cycles of length $\ell$ in $D$
come from a product of a cycle of length $a$ in $A$ and length $b$
in $B$, such that $a\vee b=\ell$. In particular, this implies that
$a|\ell$, and since $b\geq\ell$ because $\ell$ is the smallest cycle
length in $B$, this implies $b=\ell$.
So, we have $A'B'=D'$, which implies
$\tildeop{A'}\tildeop{B'}=\tildeop{D'}$. Take the smallest tree in
$\tildeop{A'}$, denote it $\bott T_A$, and take the smallest tree in
$\tildeop{D'}$, denote it $\bott T_D$. Then, there is a tree $\bott
T_B$ in $\tildeop{B'}$ such that $\bott T_A\bott T_B=\bott T_D$, by
\Cref{CompleteCodeOrder} and minimality of $\bott T_A$ and $\bott
T_D$.
This means that by \Cref{InfTreeDiv}, we find $\bott T_B$ by
dividing $\bott T_D$ by $\bott T_A$. Moreover, since $\bott T_B$ is
in $\tildeop{B'}$, we know that it comes from a cycle of length
$\ell$ in $B$. So, we set $E=C_\ell(tseq(\bott T_B)_1,\dots,tseq(\bott T_B)_\ell)$ the ``reconstruction'' of this cycle. The useful
property of $E$ is that it is part of $B$. Thus, the equation
becomes $A(B-E)=D-AE$ (those two subtractions are well-defined since $E$ is a connected component of $B$, and $AE$ is a connected component of $D$), which involves a product strictly smaller
than $D$.
\end{proof}
Now, we show that if an FDS has no fixpoint, then it is not
cancellative.
\begin{lem}\label{LemChinese}
Let $\mathcal{A}$ be a finite set of integers greater than $1$. Then there exist $X \ne X' \in \mathbb{D}_P$ such that $C_a X = C_a X'$ for all $a \in \mathcal{A}$.
\end{lem}
\begin{proof}
Recall the sequence $\delta_J$ from Lemma \ref{LemDelta}. For all $I \subseteq \mathcal{A}$, let $\alpha_I = \delta_\mathcal{A}\prod_{a\in \mathcal{A}} a$ and $\alpha'_I = \alpha_I+(-1)^{|I|}\delta_I\prod_{a\in A\setminus
I}a$.
Since $\alpha_I, \alpha'_I \ge 0$, we can then define the FDSs $X=\sum_{I\subseteq \mathcal{A}}\alpha_IC_{\bigvee I}$ and $X' = \sum_{I \subseteq \mathcal{A}} \alpha'_I C_{\bigvee I}$. We remark that the number of fixpoints in $X$ and $X'$ are $\alpha_\varnothing$ and $\alpha'_\varnothing$, respectively. Since $\alpha'_\varnothing = \alpha_\varnothing + \prod_{a \in \mathcal{A}} a \ne \alpha_\varnothing$, $X$ and $X'$ are distinct FDSs.
Let $b \in \mathcal{A}$. For all $I \subseteq \mathcal{A} \setminus \{b\}$, let $J = I \cup \{b\}$.
Then we have
\begin{eqnarray*}
C_{b}(\alpha'_IC_{\bigvee I} + \alpha'_{J}C_{\bigvee J}) &=&
(\alpha'_I(b\wedge \bigvee I) + \alpha'_Jb)C_{\bigvee J} \\
&=& ((\alpha_I+(-1)^{|I|}\delta_I\prod_{a\in \mathcal{A} \setminus I}a)(b\wedge \bigvee I) +
(\alpha_J-(-1)^{|I|}\delta_J\prod_{a\in A\setminus J}a)b)C_{\bigvee J} \\
&=& ( \alpha_I (b \wedge \bigvee I) + \alpha_J b ) C_{\bigvee J} \\
&& + \left[ \left( (-1)^{|I|} \delta_I \prod_{a \in A \setminus I} a \right) (b \wedge \bigvee I) - \left( (-1)^{|I|} \delta_I (b \wedge \bigvee I) \prod_{a \in A \setminus I} a \right) \right] C_{\bigvee J} \\
&=& C_{b}(\alpha_IC_{\bigvee I} + \alpha_{J}C_{\bigvee J}).
\end{eqnarray*}
Therefore,
\[
C_b X' = \sum_{ I \subseteq \mathcal{A} \setminus \{b\} } C_{b}(\alpha'_IC_{\bigvee I} +
\alpha'_{J}C_{\bigvee J}) \\
= \sum_{ I \subseteq \mathcal{A} \setminus \{b\} } C_{b}(\alpha_IC_{\bigvee I} + \alpha_{J}C_{\bigvee J}) \\
= C_b X.
\]
\end{proof}
This lemma above combined with \Cref{FixpointIsCancellative} gives:
\begin{thm} \label{Cancellative}
An FDS is cancellative if and only if it has a fixpoint.
\end{thm}
\begin{proof}
The case where the FDS has a fixpoint is handled by \Cref{FixpointIsCancellative}. Suppose $A$ has no fixpoint and let $\mathcal{A}$ be the set of all cycle lengths of $A$. Following \Cref{LemChinese}, there exist $X, X' \in \mathbb{D}_P$ such that $C_a X = C_a X'$ for all $a \in \mathcal{A}$. Let $B = C_a( \bott T_1, \dots, \bott T_2 )$ be a connected component of $A$, where $a \in \mathcal{A}$. According to \Cref{CorMultPermTrees}, we have $BX = BX'$. Summing over all connected components of $A$, we finally obtain $AX = AX'$.
\end{proof}
From now on, we define $\mathbb{D}^*$ to be the set of cancellable
FDSs. Its algebraic structure is that of a cancellative
subsemiring of $\mathbb{D}$, but $\mathbb{D}^*$ does not have an additive identity.
\section{Polynomial-time algorithm for tree and dendron division}\label{sec:Algos}
The proof of \Cref{FiniteTreeDiv} is actually in the form of a
polynomial division algorithm on trees, as formalised in
\Cref{divideAlgo}.
\begin{figure}
\begin{center}
\begin{procedure}[H]
$\mathcal{M}_{\bott A}\leftarrow \mathcal{D}(\bott A)$\;
$\mathcal{M}'\leftarrow\varnothing$\;
\While{$\mathcal{M}_{\bott A}\neq\varnothing$}{
$d\leftarrow \operatorname{depth}(\bott A)-1$\;
$\mathcal{T}_{\bott C}\leftarrow \{\{[\bott X]_d:\bott X\in\mathcal{D}(\bott C), \operatorname{depth}(\bott X)\geq d\}\}$\;
$\mathcal{T}_{\bott A}\leftarrow \{\{[\bott Y]_d:\bott Y\in\mathcal{M}_{\bott A},\operatorname{depth}(\bott Y)\geq d\}\}$\;
$\bott t_{\bott C} \leftarrow \argmin_{\bott X\in\mathcal{T}(\bott C)}
\mathcal{C}_f(X)$\;
$\bott t_{\bott A} \leftarrow \argmin_{\bott Y\in\mathcal{T}_{\bott A}}
\mathcal{C}_f(Y)$\;
$\bott t_{\bott B}\leftarrow divide(\bott t_{\bott C},\bott t_{\bott A})$\;
\If{$\bott t_{\bott B}=\bot$ or $\bott t_{\bott B}\mathcal{D}(\bott A)\centernot\subseteq \mathcal{M}_{\bott A}$}{
\Return $\bot$\;
}
$\mathcal{M}_{\bott A}\leftarrow \mathcal{M}_{\bott A} \setminus \bott t_{\bott B}\mathcal{D}(\bott
C)$\;
$\mathcal{M}'\leftarrow \mathcal{M}'\cup \{\bott t_{\bott B}\}$\;
}
Let $\bott B$ such that $\mathcal{D}(\bott B)=\mathcal{M}'$\;
\Return $\bott B$\;
\end{procedure}
\end{center}
\caption{$divide(\bott C,\bott A)$ to divide $\bott C$ by $\bott A$,
for finite $\bott C$ and $\bott A$.}
\label{divideAlgo}
\end{figure}
\begin{lem}
The $divide$ algorithm is correct: for all $\bott A,\bott B, \bott C\in \mathbb{F}_T$, $\bott{A}\bott{B}=\bott{C}\implies [\bott{B}]_{\operatorname{depth}(\bott A)} =
divide(\bott C, \bott A)$, and $[\bott C]_{\operatorname{depth}(\bott A)}\centernot|\bott A\implies divide(\bott C,\bott A)=\bot$.
\end{lem}
\begin{proof}
In the case in which $\bott A\bott B=\bott C$, we show that we can recover uniquely $[\bott B]_{\operatorname{depth}(\bott A)}$ from $\bott A$ and $\bott A\bott B$ by induction on $\operatorname{depth}(\bott A)$.
The base case is for $\operatorname{depth}(\bott A)=-1$, in which $\bott A=\bott
0$ is the empty tree. Then, the result is trivial since $[\bott
B]_{-1}=\bott 0$ for any $\bott B\in\mathbb{F}_T$.
Now, for the general case, we do an induction on the size of the
product $\bott C=\bott A\bott B$. The base case for $\bott C= \bott 0$ is
trivial. Let's write $\{\{\bott T_1, \dots, \bott T_n\}\}=\mathcal{D}(\bott
A)$ with $\bott T_1\leq_f \dots\leq_f \bott T_n$, $\{\{\bott U_1, \dots,
\bott U_k\}\}=\mathcal{D}(\bott B)$ with $\bott U_1\leq_f \dots\leq_f \bott
U_k$, and finally, write $\{\{\bott V_1, \dots, \bott
V_{nk}\}\}=\mathcal{D}(\bott C)$ with $\bott V_1\leq_f \dots\leq_f \bott
V_{nk}$. We remark that to recover $[\bott B]_{\operatorname{depth}(\bott
A)}$, all we need is to recover $[\bott U_j]_{\operatorname{depth}(\bott A) - 1}$ for all $1 \le j \le k$.
Let $d$ be $\operatorname{depth}(\bott C)-1$ as in the algorithm. Then let $\bott t_{\bott A}$ (respectively $\bott t_{\bott B}$, $\bott t_{\bott C}$) be the maximum tree in $\mathcal{D}(\bott A)$ (respectively $\mathcal{D}(\bott B)$, $\mathcal{D}(\bott C)$) of depth $d$.
We can then write $\bott t_{\bott A}\bott t_{\bott B}=\bott t_{\bott C}$ without loss of generality. Since $\bott t_{\bott A}$ has
depth $d<\operatorname{depth}(\bott A)$, the outer induction hypothesis shows that $divide(\bott t_{\bott C},\bott t_{\bott A})=[\bott t_{\bott B}]_d$.
There are two cases. If $\operatorname{depth}(\bott B)\leq\operatorname{depth}(\bott A)$, then
$d=\operatorname{depth}(\bott B)$ by \Cref{DepthTrees} and so $[\bott t_{\bott B}]_d=\bott t_{\bott B}$. Otherwise, if $\operatorname{depth}(\bott B)>\operatorname{depth}(\bott A)$, then $\operatorname{depth}(\bott C)=\operatorname{depth}(\bott A)$ by \Cref{DepthTrees}
and so $\bott t_{\bott B}=[\bott t_{\bott B}]_{\operatorname{depth}(\bott A)-1}$, which is a depth $1$ subtree of $[\bott B]_{\operatorname{depth}(\bott A)}$. So, in both cases, $\bott t_{\bott B}$ is a depth $1$ subtree of $[\bott B]_{\operatorname{depth}(\bott A)}$.
Now that we have $\bott t_{\bott B}$, the algorithm computes $\bott t_{\bott B}\mathcal{D}(\bott A) = \{\{\bott t_{\bott B}\bott T_1, \dots, \bott t_{\bott B}\bott T_n\}\}$, which are $n$ subtrees of $\bott C$, and removes them from $\bott C$. Finally, the next loop iteration corresponds to applying the internal induction hypothesis to the identity $\bott A\bott{B'}=\bott{D'}$ where
$$
\mathcal{D}(\bott{B'}) = \mathcal{D}([\bott B]_{\operatorname{depth}(\bott A)}) \setminus \{\bott t_{\bott B}\}
$$
and
$$
\mathcal{D}(\bott{D'}) = \mathcal{D}(\bott{D})\setminus\bott t_{\bott B}\mathcal{D}(\bott A).
$$
To conclude, if we are in the case where $[\bott C]_{\operatorname{depth}(\bott A)}\centernot|\bott A$, we need to show that if $divide([\bott C]_{\operatorname{depth}(\bott A)},\bott A)$ does not return $\bot$ but some tree $\bott B$, then $\bott A\bott B=[\bott C]_{\operatorname{depth}(\bott A)}$ which is a contradiction. To do so, remark that by construction during the while loop, $\mathcal{D}(\bott A)\mathcal{D}(\bott B)=\mathcal{D}([\bott C]_{\operatorname{depth}(\bott A)})$, which means that $\bott A\bott B=[\bott C]_{\operatorname{depth}(\bott A)}$.
\end{proof}
This algorithm only works on trees. But the following lemma allows one
to use it on dendrons, using the truncature of their unrollings. First, we need the following definition, adapting the definition of product isomorphism for forests:
\begin{defn}
Given a product $\bott B = \prod_{i\in I} \bott A_i$ for some finite set $I$, a family $(\bott A_i)_{i\in I}\in \mathbb{F}^I$, and denoting $S_{\prod_{i\in I} \bott A_i}=\bigcup_{k\in\mathbb{N}}\{(a_i)_{i\in I}\in\prod_{i\in I} S_{\bott A_i}:\operatorname{depth}_{\bott A_i}(a_i)=k\}$, we say that the function $\psi: S_{\bott B}\mapsto S_{\prod_{i\in I} \bott A_i}$ is a \textit{forest product isomorphism for the product} $\bott B = \prod_{i\in I} \bott A_i$ if:
\begin{enumerate}
\item it is a bijection,
\item for any $b\in S_{\bott B}$, $\psi(b)$ is a root if and only if $b$ is a root, and
\item for any families of non-root states $(s_i)_{i\in I},(s'_i)_{i\in I}\in S_{\prod_{i\in I}\bott A_i}$, we have: $\psi^{-1}((s_i)_{i\in I})\rightarrow \psi^{-1}((s'_i)_{i\in I})$ is an edge of $\bott B$ if and only if for each $i\in I$, $s_i\rightarrow s'_i$ is an edge of $\bott A_i$.
\end{enumerate}
\end{defn}
As for the first definition of a product isomorphism, if there is a tree product isomorphism between $\bott B$ and $\prod_{i\in I} \bott A_i$, this means that $\bott B=\prod_{i\in I} \bott A_i$. A simple inductive proof shows that:
\begin{lem}\label{TreeProdIsoDepth}
Given a tree product isomorphism $\psi$ for a product $\bott B=\prod_{i\in I} \bott A_i$ is such that for any $(a_i)_{i\in I}\in S_{\prod_{i\in I} \bott A_i}$ and $b\in S_{\bott B}$, such that $\psi(b)=(a_i)_{i\in I}$, we have $\operatorname{depth}_{\bott B}(b)=\operatorname{depth}_{\prod_{i_\in I} \bott A_i}((a_i)_{i\in I})$.
\end{lem}
\begin{lem}\label{LemTruncUnrolling}
Let $A,B,C\in\mathbb{D}_D$, and let $k\geq\operatorname{depth}(A)$. Then $A=BC$ if and
only if $[\tildeop A]_k=[\tildeop B]_k[\tildeop C]_k$.
\end{lem}
\begin{proof}
Remember that we already know that $A=BC\iff \tildeop A=\tildeop
B\tildeop C$. Now, one direction is trivial: if $A=BC$, then $\tildeop
A=\tildeop B\tildeop C$ so $[\tildeop A]_k=[\tildeop B]_k[\tildeop
C]_k$ for every $k$. Now, we assume that $[\tildeop A]_k=[\tildeop
B]_k[\tildeop C]_k$ for some $k \ge \operatorname{depth}(A)$ and we show that $\tildeop A=\tildeop
B\tildeop C$.
Now, we want to create a tree product isomorphism
$\phi: S_{\tildeop A}\rightarrow S_{\tildeop B\tildeop C}$ for the product $\tildeop A=\tildeop B\tildeop C$. To do so,
we start from the tree product isomorphism $\psi: S_{[\tildeop A]_k}\rightarrow
S_{[\tildeop B]_k[\tildeop C]_k}$ for the product $[\tildeop A]_k=[\tildeop
B]_k[\tildeop C]_k$.
We can extend $\psi$ to $\phi$ easily. For all
$(a,d)\in S_A\times\mathbb{N}$ where $d\geq \operatorname{depth}_A(a)$, set
$\phi(a,d)=((b,d),(c,d))$ where
$\psi(a,\operatorname{depth}_A(a))=((b,\operatorname{depth}_A(a)),(c,\operatorname{depth}_A(a)))$. This is a
well-defined function since $\psi(a,\operatorname{depth}_A(a))$ will always exist
as $k\geq\operatorname{depth}(A)$.
Let's prove that this is a valid tree product isomorphism. First, $\phi$ is bijective. Indeed, suppose that $\psi(a,d)=\psi(a',d')$. Denote $\psi(a,d)=((b,d),(c,d))$ and $\psi(a',d')=((b',d'),(c',d'))$. We directly have $(b,c,d)=(b',c',d')$. This means that $\operatorname{depth}_A(a)=\operatorname{depth}_A(a')$, by definition of $\phi$, because $b$ and $c$ are at the same depth as $a$ (this follows from \Cref{TreeProdIsoDepth}, since $\psi$ is a tree product isomorphism).This means that $\psi(a,\operatorname{depth}_A(a))=\psi(a',\operatorname{depth}_A(a))$, which implies $a=a'$ by bijectivity of $\psi$.
Now, for any $(a,d)\in S_A\times\mathbb{N}$ such that $d\geq\operatorname{depth}_A(a)$, $\phi(a,d)=((b,d),(c,d))$ is a root if and only if $d=0$ and $b$ and $c$ are roots. Because of the definition of $\psi$, $b$ and $c$ are roots if and only if $a$ is a root in $A$, since $\psi$ is a tree product isomorphism.
For the last property we need to check, we write $x\xrightarrow{\bott C} y$ to mean that there is an edge from $x\in S_{\bott C}$ to $y\in S_{\bott C}$ in $\bott C$.
Finally, we show that for all $((b,d),(c,d)),((b',d'),(c',d'))\in S_{\tildeop{B}\tildeop C}$, we have: $\phi^{-1}(((b,d),(c,d)))\xrightarrow{\tildeop A} \phi^{-1}(((b',d'),(c',d')))$ if and only if $(b,d)\xrightarrow{\tildeop B} (b',d')$ and $(c,d)\xrightarrow{\tildeop C} (c',d')$. Indeed, following the definition of $\phi$ from $\psi$, we can write $\phi^{-1}(((b,d),(c,d)))=(a,d)\in S_{\tildeop A}$ and $\phi^{-1}(((b',d'),(c',d')))=(a',d')\in S_{\tildeop A}$.
Since $\psi$ is a tree product isomorphism, there is an edge $(a,d)\xrightarrow{\tildeop A} (a',d')$ if and only if there is an edge $((b,\operatorname{depth}_A(a)), (c,\operatorname{depth}_A(a)))\xrightarrow{[\tildeop B]_k[\tildeop C]_k} ((b',\operatorname{depth}_A(a)), (c',\operatorname{depth}_A(a)))$, that is, if and only if there is an edge $((b,d),(c,d))\xrightarrow{\tildeop B\tildeop C} ((b',d),(c',d))$, which is equivalent to the existence of $(b,d)\xrightarrow{\tildeop B} (b',d')$ and $(c,d)\xrightarrow{\tildeop C} (c',d')$.
This proves that $\tildeop A=\tildeop B\tildeop C$, which in turn
proves that $A=BC$, and concludes.
\end{proof}
\begin{thm}
Given $A,B\in\mathbb{D}_D$, we can find $C\in\mathbb{D}_D$ such that $A=BC$ or
prove that it does not exist in polynomial time in the sizes of $A$
and $B$.
\end{thm}
\begin{proof}
Given $A,B\in\mathbb{D}_D$, let $k=\operatorname{depth}(A)$. Then, call $divide([\tildeop
A]_k, [\tildeop B]_k)$. If this function returns $\bot$, then
there is no $X\in \mathbb{F}_T$ such that $[\tildeop A]_k=[\tildeop B]_kX$,
which shows that there is no $C\in\mathbb{D}_D$ such that $A=BC$ by
\Cref{LemTruncUnrolling}.
Otherwise, if this function returns some $X\in\mathbb{F}_T$, then we have
$[\tildeop A]_k=[\tildeop B]_kX$ with $\operatorname{depth}(X)=k$. Now, remark
that if there is some $C\in\mathbb{D}_D$ such that $A=BC$, we have
$\operatorname{depth}(C)\leq k$ and thus $[\tildeop A]_k=[\tildeop B]_k[\tildeop
C]_k$, so by \Cref{FiniteTreeDiv}, we have $X=[\tildeop C]_k$.
Therefore, if the function returns an $X\in\mathbb{F}_T$, either $X$ is of
the form $[\tildeop C]_k$ for some $C\in \mathbb{D}_D$, and then we recover
$C$ such that $A=BC$ from the reverse direction of
\Cref{LemTruncUnrolling}, or $X$ is not of that form, and by
\Cref{FiniteTreeDiv}, there is no $C\in\mathbb{D}_D$ such that $A=BC$.
The $divide$ algorithm is indeed in polynomial time since a call to $divide(\bott T, \bott U)$ ends up making at most one call to $divide(\bott V,\bott W)$ for $\bott V$ some subtree of $\bott T$ and $\bott W$ some subtree of $\bott U$. Since every operation in a call to $divide$ is in polynomial time, this concludes.
\end{proof}
\section{Unicity of $k$-th roots} \label{kroot}
Using \Cref{Cancellative}, we can prove a simple result above polynomials,
which in particular states that a polynomial with a coefficient of
degree $1$ which is cancellative is injective.
\begin{prop} \label{Polynomials}
Let $P=\sum_{i=0}a_iX^i\in\mathbb{D}[X]$ and $A,B\in\mathbb{D}$ such that
$P(A)=P(B)$. Then, we have $A=B$ if $a_1\in\mathbb{D}^*$ or if for some
$i>1$, $a_i\in\mathbb{D}^*$ and $A\in\mathbb{D}^*$.
\end{prop}
\begin{proof}
Write $P(X)=\sum_{i=0}^da_iX^i$. We can assume that $a_0=0$, and we
still have $P(A)=P(B)$. Let $D=\sum_{i=1}^da_i\sum_{j=0}^{i-1}
A^{i-1-j}B^{j}$. Then:
\begin{eqnarray*}
AD &=& \sum_{i=1}^da_i\sum_{j=0}^{i-1} A^{i-j}B^j \\
&=& \sum_{i=1}^da_i\left(A^i + \sum_{j=1}^{i-1} A^{i-j}B^j\right) \\
&=& P(A) + \sum_{i=1}^da_i\sum_{j=1}^{i-1} A^{i-j}B^j \\
&=& P(B) + \sum_{i=1}^da_i\sum_{j=1}^{i-1} A^{i-j}B^j \\
&=& \sum_{i=1}^da_i\left(B^i + \sum_{j=1}^{i-1} A^{i-j}B^j\right) \\
&=& \sum_{i=1}^da_i\sum_{j=1}^{i} A^{i-j}B^j \\
&=& \sum_{i=1}^da_i\sum_{j=0}^{i-1} A^{i-1-j}B^{j+1} \\
&=& BD.
\end{eqnarray*}
In the case where $a_1$ has a fixpoint, remark that the term for
$i=1$ in $D=\sum_{i=1}^da_i\sum_{j=0}^{i-1} A^{i-1-j}B^{j}$ is simply
$a_1$, and so, $D$ has a fixpoint. Otherwise, in the case where
there is $i>1$ such that $a_i$ with a fixpoint, and $A$ has a
fixpoint, the term in the sum for that $i$ is: $a_i\sum_{j=0}^{i-1}
A^{i-1-j}B^{j}$, in which we find the term $a_iA^{i-1}$, which has a
fixpoint, so $D\in\mathbb{D}^*$.
Since $D\in\mathbb{D}^*$, $AD=BD$ implies $A=B$.
\end{proof}
A general characterisation of injective polynomials would be very
interesting. It seems unlikely that the condition $a_1\in\mathbb{D}^*$ is necessary
since that would mean that if $a_1\notin\mathbb{D}^*$ then, even if every
other coefficient is in $\mathbb{D}^*$, one could find $A\neq B$ such that
$P(A)=P(B)$.
In the rest of this section, we show that for any $k \ge 1$, the polynomial $P(X) = X^k$ is injective.
\begin{thm} \label{roots}
For all $k \ge 1$ and $A,B\in\mathbb{D}$, if $A^k=B^k$, then $A=B$.
\end{thm}
Our first step is to prove the injectivity of the mapping $\bott X\mapsto \bott X^k$ on $\mathbb{F}$. Given a forest $\bott F \in \mathbb{F}$, let $\mathcal{R}( \bott F ) \in \mathbb{F}_T$ be the tree obtained by joining all the trees of $\bott F$ to a new common root. More formally, if $\mathcal{F}( \bott F )$ is the multiset of trees of $\bott F$, then $\mathcal{D}( \mathcal{R}( \bott F ) ) = \mathcal{F}$.
\begin{lem}\label{LemTreeIsForest}
For any forest $\bott F \in \mathbb{F}$ and any $k \ge 1$, we have $\mathcal{R}^k( \bott F ) = \mathcal{R}( \bott F )$.
\end{lem}
\begin{proof}
By \Cref{LevelByLevelProduct}, we have $\mathcal{D}( \mathcal{R}^k( \bott F ) ) = \mathcal{F}^k( \bott F)$. Now, it is clear that $\mathcal{F}^k( \bott F ) = \mathcal{F}( \bott F^k )$. This concludes.
\end{proof}
\begin{lem}
The mapping $\bott X\mapsto \bott X^k$ is injective on $\mathbb{F}$.
\end{lem}
\begin{proof}
We first prove that the mapping $\bott X\mapsto \bott X^k$ is injective on $\mathbb{F}_T$. Let $\bott T_1, \bott T_2 \in \mathbb{F}_T$ with $\bott T_1 < \bott T_2$. Then by induction on $k$, \Cref{GeneralProdOrder} shows that $\bott T_1^k < \bott T_2^k$.
We now prove injectivity on $\mathbb{F}$. Let $\bott A,\bott B\in\mathbb{F}$, such that $\bott A^k=\bott B^k$.
By \Cref{LemTreeIsForest}, we have $\mathcal{R}^k(\bott A) = \mathcal{R}^k(\bott B)$. By injectivity on $\mathbb{F}_T$, we obtain $\mathcal{R}(\bott A) = \mathcal{R}(\bott B)$, which implies $\bott A = \bott B$.
\end{proof}
Our second step is to prove the result for bijective FDSs.
\begin{lem}\label{lensRecovery}
Let $A,B\in\mathbb{D}$. If $A^k=B^k$, then $[A]_0=[B]_0$.
\end{lem}
\begin{proof}
Remark that $A^k=B^k$ implies $[A]_0^k=[B]_0^k$. All that's left to
show is that if $A,B\in\mathbb{D}_P$ and $A^k=B^k$, then $A=B$.
Take $D \in\mathbb{D}_P$, and write $D =\sum_{i} \lambda^A_iC_i$. Assume
there exists $B=\sum_i \lambda^B_i C_i$ such that $B^k = D$.
For all $i \in \mathbb{N}$, let $F_i = \{ L = (l_j)_{j \in \iitv{1,k}}:\bigvee_j l_j=i\}$ denote the possible ways a product of $k$ cycles $C_{l_1} \times \dots \times C_{l_k}$ is equal to some scalar multiple of $C_i$.
For any sequence $L = (l_j)$, we abuse notation and identify $L$ with the multiset of its entries; we can then use the notation $\delta_L$. By \Cref{LemDelta}, we obtain for all $i \in \mathbb{N}$
\[
\sum_{L \in F_i} \delta_L \prod_{j=1}^k \lambda^B_{l_j} = \lambda^A_i.
\]
This is a set of triangular positive polynomial equations (as the equation for $i$ only involves $\lambda^B_1, \dots, \lambda^B_i$), thus it has at most one solution. Therefore, if $B$ exists, it is unique.
\end{proof}
Our third and final step proves the theorem.
\begin{lem}\label{SmallestLengthExtractPoly}
Let $P\in\mathbb{N}[X]$ be a polynomial with coefficients in $\mathbb{N}$, and let
$A\in\mathbb{D}$. Then, for any $\ell\in\mathbb{N}$, we have
$\operatorname{supp}_{\leq\ell}(P(A))=P(\operatorname{supp}_{\leq\ell}(A))$.
\end{lem}
\begin{proof}
Write $P=\sum_{i=1}^d a_iX^i$. If $A=\sum_{j=1}^n A_j$ where each
$A_j$ is connected, then the products that appear in $P(A)$ are the
$a_i\prod_{k=1}^i A_{\beta_k}$ for each $i\in\iitv{1,n}$ and
$\beta = (\beta_k)_{k \in \iitv{1,i}} \in\iitv{1,n}^i$. Remark that for such a product
$a_i\prod_{k=1}^i A_{\beta_k}$ to have a cycle length $\leq\ell$,
every $A_{\beta_k}$ must have cycle length $\leq\ell$. This
concludes.
\end{proof}
\begin{lem}\label{LemFF_TToDD}
The mapping $A\mapsto
A^k$ is injective on $\mathbb{D}$.
\end{lem}
\begin{proof}
Given $A^k$, we find $[A]_0$ and thus we know the lengths of the
cycles of $A$ by \Cref{lensRecovery}; denote them
$\ell_1<\dots<\ell_n$. We show by induction on $i\in \iitv{0,n}$ that we can
recover $\operatorname{supp}_{\leq \ell_i}(A)$ from $A^k$ (with an implicit
$\ell_0=0$, such that $\operatorname{supp}_{\leq \ell_0}(A)=0$, to make for a
trivial base case and avoid repetition).
Take some $i \in \iitv{1,n-1}$ such that the induction hypothesis
stands for $i$. We show that it also stands for $i+1$. By
\Cref{SmallestLengthExtractPoly},
$\operatorname{supp}_{\leq\ell_{i+1}}(A^k)=(\operatorname{supp}_{\leq\ell_{i+1}}(A))^k$. By the
lemma's hypothesis, we recover $\tildeop{\operatorname{supp}_{\leq\ell_{i+1}}(A)}$
from $\tildeop{(\operatorname{supp}_{\leq\ell_{i+1}}(A))^k}$. Now, since we have
$\tildeop{\operatorname{supp}_{\leq\ell_i}(A)}$ from the induction hypothesis, we
recover
$$\tildeop{\operatorname{supp}_{\ell_{i+1}}(A)}=\tildeop{\operatorname{supp}_{\leq\ell_{i+1}}(A)}\setminus\tildeop{\operatorname{supp}_{\leq\ell_{i}}(A)}.$$
It is straightfoward to reconstruct $\operatorname{supp}_{\ell_{i+1}}(A)$ from
$\tildeop{\operatorname{supp}_{\ell_{i+1}}(A)}$ since we know there every tree in
$\tildeop{\operatorname{supp}_{\ell_{i+1}}(A)}$ comes from a connected component
of cycle length $\ell_{i+1}$. And thus, we recover
$\operatorname{supp}_{\leq\ell_{i+1}}(A)=\operatorname{supp}_{\ell_{i+1}}(A)+\operatorname{supp}_{{\leq\ell_i}(A)}$,
which concludes the induction.
\end{proof}
\section{A family of monoids with unique factorisation}\label{sec:LD_K}
The $C_2^2=2C_2$ identity shows that factorisation into irreducible
FDSs is not unique on $\mathbb{D}$. Moreover, it is shown in \cite{Couturier}
that factorisation is also not necessarily unique on $\mathbb{D}_D$, for
example with the identity presented in \Cref{CexCouturier}. We
can however exhibit an example of an interesting class of trees in
which every element has a unique factorisation in irreducible FDSs.
Although our example might not be useful in practice, it is interesting as a
generalisation of the simpler result that shows that factorisation is
unique on the multiplicative monoid generated by products of paths
(which is called $LD_1$ with the notations below).
\begin{figure}
\begin{center}
\begin{tikzpicture}[c/.style={shape=circle, draw=black}]
\node[c] (C1) {}; \node[c] (C2) [above of=C1] {}; \draw[->] (C2)
-- (C1); \draw[->] (C1) edge[loop below] (C1);
\tikzset{xshift=1.5cm}
\node (times1) at (0, 0.2) {$\times$};
\tikzset{xshift=1.5cm}
\node[c] (B1) {};
\node[c] (B2) [above left of=B1] {};
\node[c] (B3) [above right of=B1] {};
\node[c] (B4) [above left of=B3] {};
\node[c] (B5) [above right of=B3] {};
\node[c] (B6) [above of=B3] {};
\draw[->] (B2) -- (B1);
\draw[->] (B3) -- (B1);
\draw[->] (B4) -- (B3);
\draw[->] (B5) -- (B3);
\draw[->] (B6) -- (B3);
\draw[->] (B1) edge[loop below] (B1);
\tikzset{xshift=1.5cm}
\node (eq1) at (0, 0.2) {$=$};
\tikzset{xshift=1.5cm}
\node[c] (A1) {};
\node[c] (A2) [above left of=A1] {};
\node[c] (A3) [above right of=A1] {};
\draw[->] (A2) -- (A1);
\draw[->] (A3) -- (A1);
\draw[->] (A1) edge[loop below] (A1);
\tikzset{xshift=1.5cm}
\node (times2) at (0, 0.2) {$\times$};
\tikzset{xshift=1.5cm}
\node[c] (D1) {};
\node[c] (D2) [above of=D1] {};
\node[c] (D3) [above left of=D2] {};
\node[c] (D4) [above right of=D2] {};
\draw[->] (D2) -- (D1);
\draw[->] (D3) -- (D2);
\draw[->] (D4) -- (D2);
\draw[->] (D1) edge[loop below] (D1);
\end{tikzpicture}
\end{center}
\caption{A dendron that admits two different factorisations in irreducible factors.}
\label{CexCouturier}
\end{figure}
\begin{defn}
A \textit{rhizome} is a path from a leaf to the fixpoint in a dendron.
The length of a rhizome is its number of transitions, that is its
number of non-fixpoint states.
\end{defn}
According to our terminology, the depth of a dendron is the length of its longest rhizome.
\begin{defn}
An FDS $A\in\mathbb{D}$ is a \textit{linear dendron} if it is a dendron
such that only its fixpoint may have more than one predecessor. A
linear dendron has $K$ rhizomes if its fixpoint has $K$ non-fixpoint
predecessors.
A \textit{star} $S_n$ is a linear dendron of depth $1$ and $n$ states, while a \textit{path} $P_n$ is a linear dendron with only one rhizome and $n+1$ states.
\end{defn}
We are now in position to show that most linear dendrons are irreducible. We remark that the semigroup of stars is isomorphic to that of the positive integers: $S_{ab} = S_a \times S_b$. Therefore, composite stars have a unique factorisation in $\mathbb{D}$.
\begin{prop}\label{irreducible_dendrons}
The only reducible linear dendrons are the stars with a composite number of states.
\end{prop}
\begin{proof}
The case of stars is straightforward. Let $T$ be a linear dendron of depth $k > 1$. Then any rhizome of maximum length of $T$ contains a state with exactly one predecessor: the state at depth $1$ of the rhizome.
Suppose $T$ is reducible towards a contradiction, say $T = A \times B$. The depth of either $A$ or $B$ is at least $k$, say $P_k$ is a subdendron of $A$. Moreover, $P_1$ is a subdendron of $B$. Thus, $P_k\times P_1$ is a subdendron of $A\times B=T$. It's easy to see that
$P_k\times P_1$ contains a path of depth $k$ states with more than
one predecessor each (except the leaf at the end). This is a rhizome
of maximal length in $T$ in which no state has exactly one
predecessor. This concludes.
\end{proof}
\begin{defn}
For all $K\in\mathbb{N}$, we define $LD_K$ the multiplicative monoid
generated by linear dendrons with $K$ rhizomes.
\end{defn}
Based on \Cref{irreducible_dendrons}, if $P \in LD_K$ has a unique factorisation in $LD_K$, then it has a unique factorisation in $\mathbb{D}$. Thus, we focus on factorisation in $LD_K$.
Let $P \in LD_K$ be factorised as $P = F_1 \times \dots \times F_N$ where $F_j$ is a linear dendron for each $1 \le j \le N$. Each state $s \in S_P$ can be expressed as $s = (s_1, \dots, s_N)$ where $s_j \in S_{F_j}$ for all $j$. Some of those $s_j$'s could be fixed points; let $I(s) = \{ j : F_j(s_j) = s_j \}$. Then the number of predecessors of $s$ is either $0$ if any $s_j$ is a leaf, or equal to $(K+1)^{|I(s)|}$ otherwise. This suggests the following notation.
\begin{defn}
Let $P \in LD_K$ and $i\in\mathbb{N}$. A state $s$ of $P$ is \textit{$i$-fixed} if it
has $(K+1)^i$ predecessors.
\end{defn}
\begin{lem}
Any $i$-fixed state has a unique $i$-fixed predecessor; all other predecessors are either leaves or $j$-fixed for some $j < i$.
\end{lem}
\begin{proof}
Let $s = (s_1, \dots, s_N)$ be $i$-fixed and without loss let $I(s) = \iitv{1,i}$. Remark that $s_1, \dots, s_i$ are fixed points, while $s_{i+1}, \dots, s_N$ have a unique predecessor each, say $t_{i+1}, \dots, t_N$ respectively. Then any predecessor of $s$ is of the form $u = (u_1, \dots, u_i, t_{i+1}, \dots, t_N)$ where $u_l$ is a predecessor of $s_l$ for all $1 \le l \le i$. Therefore $u$ is at most $i$-fixed, with equality if and only if $u = (s_1, \dots, s_i, t_{i+1}, \dots, t_N)$.
\end{proof}
Now that we have all the necessary definitions, we can introduce the
following lemma, which enables a partial recovery of some factors from
a product of linear dendrons. This is the core lemma, and it is from
it that we can finally recover every factor.
\begin{lem}[Linear extraction lemma]\label{LemExtract}
Let $P = F_1 \times \dots \times F_N \in LD_K$ and let $s$ be a depth $1$, codepth $\ell$, $i$-fixed state of $P$. Consider the tree anchored on $s$ in $P$ and remove the unique $i$-fixed predecessor of $s$ and all its antecedents. Then the obtained dendron is
$E_s = [ \prod_{j \in I(s)} F_j ]_\ell$.
\end{lem}
\begin{proof}
Without loss, let $I(s) = \iitv{1,i}$. Denote $s = (s_1, \dots, s_n)$ and for all $i+1 \le j \le N$ and $d \in \mathbb{N}$ let $t^d_j$ be the unique state of $F_j$ satisfying $F_j^d( t^d_j ) = s_j$. All the states in the dendron $E_s$ are either $s$ or of the form $u = (u_1, \dots, u_i, t^d_{i+1}, \dots, t^d_N)$, where $d$ is the depth of $u$ in $E_s$ and $(u_1, \dots, u_i) \ne (s_1, \dots, s_i)$. By removing the coordinates $i+1, \dots, N$ from each state, we see that $E_s$ is a sub-FDS of $F_1 \times \dots \times F_i$.
All that is left is to show that we do indeed get the truncature at
depth $\ell$. Remark that the codepth $\ell$ of $s$ is the length of
the smallest path among the rhizomes anchored at $s_{i+1}, \dots, s_N$
in their respective factors, minus $1$. Thus, the sub-FDS of
$F_1\times \dots\times F_i$ we obtain is indeed truncated at depth $\ell$.
\end{proof}
Let $P = F_1 \times \dots \times F_N$ where all the factors have depth $k+1$. Let $\mathfrak{A}=\{[F_i]_k:i\in\iitv{1,N}\}$ be the collection of truncated factors and for each $B \in \mathfrak{A}$, denote its multiplicity $n_B = |\{ i \in\iitv{1,N} : [F_i]_k = B \}|$. We denote $D_i$ the set of $i$-fixed depth 1 states of codepth $k$ of $P$.
\begin{lem} \label{extraction_powers}
For all $B \in \mathfrak{A}$, there exists $s \in D_i$ with $E_s = B^i$ if and only if $i \le n_B$.
\end{lem}
\begin{proof}
Without loss, let $B\in\mathfrak{A}$ such that $B = F_1 = \dots = F_{n_B}$.
Let $i \le n_B$ and consider a state $s = (s_1, \dots, s_N)$ of $P$ where $s_1, \dots, s_i$ are fixed points of $B$, while for every $i+1 \le j \le N$, $s_i$ is a depth $1$ state on
a path of depth $k+1$. Then $s \in D_i$,
and the extraction lemma extracts $B^n$ from $s$. Conversely, if $E_s = B^j$, then $B^j$ divides $[P]_k$ and hence $j \le n_B$.
\end{proof}
We now show
that factorisation is unique on products of linear dendrons which share the same
depth.
\begin{lem}
A product of elements of $LD_K$ which have the same depth $k$ is
uniquely factorisable.
\end{lem}
\begin{proof}
We do this by induction on the depth $k$. For $k=0$, this lemma is
obvious (the factorisation is $C_1$). Take some $k$ such that the
lemma stands for depth $k$. We show that the lemma is also true
for depth $k+1$. The proof is in four steps. First, we identify
the number of factors, then we recover the set of their depth $k$
truncatures, then we recover the multiset of these truncatures and
finally, we recover the full, untruncated factors. Take $P$ a
product of elements of $LT_K$.
\head{Number of factors} We recover $N$ the number of factors of $P$
by remarking that its fixpoint is $N$-fixed: thus by counting its
number of predecessors, we can recover $N$ from $P$ and write
$P=F_1\times \dots \times F_N$.
\head{Set of truncatures}
According to \Cref{extraction_powers}, by applying the extraction lemma to all the elements of $D_1$, we recover all the factors $B \in \mathfrak{A}$.
\head{Multiset of truncatures} By \Cref{extraction_powers}, for all $B \in \mathfrak{A}$, $n_B = \max\{ i : \exists s \in D_i E_s = B^i \}$. As such, applying the extraction lemma on $D_i$ for $1 \le i \le N$ then yields $n_B$ for all $B \in \mathfrak{A}$.
\head{Untruncated factors} As of now, we have all the
factors and their multiplicity, but they are truncated at depth $k$. To fully reconstruct the linear dendron $F_i$ of depth $k+1$ from $[F_i]_k$, all we need is the number $f_i$ of paths of depth $k+1$ in $F_i$. We now show how to determine this number.
Fix $B\in \mathfrak{A}$. Let's denote $f_1, \dots, f_{n_B}$ the
number of paths of depth $k+1$ of and let $G_1, \dots, G_{n_B}$ be
the elements of $\phi(B)$. For any $n\in\iitv{0,n_B}$, let's count
in $P$ the number of states of $D_{N-n}$ from which the extraction
lemma extracts $[P]_k/B^n$.
Each of these states
corresponds to an $n$-uple of depth 1 states of $G_1, \dots,
G_{n_B}$ (each in a distinct factor) on which a path of depth $k+1$
is anchored. As such, there are
$p_n:=\sum_{\substack{I\subseteq\iitv{1,n_B}\\|I|=n}}\prod_{i\in I} f_i$
of them (given the set of factors of $G_1, \dots, G_{n_B}$ of index in $I$, the number of depth $1$ states of codepth $k+1$ is $\prod_{i\in I} f_i$). Finding that number for all $n\in\iitv{0,n_B}$ makes it
possible to express the $f_1, \dots, f_{n_B}$ as the roots of a
polynomial of degree $n_B$ and thus, allows one to find them. Here
is how we proceed. Write $R(X) = \sum_{m=0}^{n_B} (-1)^mp_mX^{n_B-m}$. By
Vieta's relations, we know that the $n_B$ roots of $R$ are $f_1,
\dots, f_{n_B}$.
\end{proof}
Now, we show that we can always get to this case:
\begin{thm}
Factorisation is unique on $LD_K$.
\end{thm}
\begin{proof}
Let $P = F_1 \times \dots \times F_N \in LD_K$ have depth $k+1$. Let $I = \{ 1 \le i \le N : \operatorname{depth}(F_i) \le k \}$ be the set of indices of factors with no paths of depth $k+1$. Now, let $S$ be the set of
depth 1 states belonging to a rhizome in $P$ of depth $k+1$. For all $s \in S$, since $s$ has depth $1$ and codepth $k$, $s_i$ is a fixpoint for all $i \in I$ and hence $s$ is at least $|I|$-fixed. Conversely, if $s \in S$ such that $s_j$ has codepth $k$ for all $j \notin I$, then $s$ is $|I|$-fixed.
Using the extraction lemma on such a state $s$, we recover
$[\prod_{i\in I} F_i]_k = \prod_{i\in I} F_i$.
Let's divide $S$ by $\prod_{i\in I} F_i$. The result is unique by
\Cref{UniqTreeDiv}. So, we get $\prod_{j\notin I} F_j$ the product
of the factors of depth $k+1$, and $\prod_{i\in I} F_i$ the product
of the factors of depth at most $k$. An induction on the second
subproduct means that we can extract all the subproducts of shared
depth, and apply the previous lemma on each of them.
\end{proof}
\section{Conclusion}\label{sec:Directions}
In this article, we have obtained results which may lead to a deeper understanding of the structure of the semiring of FDSs $\mathbb{D}$. In particular, we have characterised the cancellative elements of $\mathbb{D}$, shown how to perform division of dendrons in polynomial time, proved that $k$-th roots are unique, and we have exhibited a family of monoids with unique factorisation. While this sheds some light on the structure of $\mathbb{D}$, there are still many questions.
An interesting direction is the complexity of division on general FDSs, or on cycles. Contrary to the situation on trees, this algorithmic problem may not be in $\mathsf{P}$. On the other hand, it is clearly in $\mathsf{NP}$. The question of knowing whether it is $\mathsf{NP}$-complete is still open, as a reduction (if it exists) does not seem obvious at all.
Another important direction to better understand the structure of $\mathbb{D}$ is the study of primality, defined as follows: $A\in \mathbb{D}$ is prime if and only if for every $B,C\in\mathbb{D}$, $A|BC$ implies $A|B$ or $A|C$. Most of the work on this has been done in \cite{Couturier}, in which Couturier proves that for an FDS to be prime, it must be a dendron. Still, as of now, no example of a prime FDS is known, and no finite-time algorithm to check primality is known.
One could also be interested in more practical applications of FDS factorisation. Imagine for example a "grey box" (some deterministic mechanism that does not display its internal workings, but displays its state such that two different states can always be recognized) that is observed by a probe that records the evolution of its state, until this state falls into a cycle, at which point the probe launches the process again, and so on. Thus, the probe reconstructs the FDS $S$ governing the evolution of the grey box's state. We are interested in a way to know, with the current partial recovery of $S$, how many more states we need to add at the minimum in order to get a factorisable system. This is useful because suppose that the probabilistic model of exploration shows that there is a $90\%$ chance that the probe has recovered at least $90\%$ of the states of $S$. Then, if we know that, say, in order to get a factorisable recovered system, we need to add at least $30\%$ more states than the ones we already have recovered, we know that with probably at least $90\%$, the grey box is not factorisable, that is, it does not contain two independent mechanisms running in parallel.
| {
"timestamp": "2022-10-21T02:14:29",
"yymm": "2210",
"arxiv_id": "2210.11270",
"language": "en",
"url": "https://arxiv.org/abs/2210.11270",
"abstract": "Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS $A$ is cancellative if one can divide by it unambiguously, i.e. $AX = AY$ implies $X = Y$. We prove that an FDS $A$ is cancellative if and only if it has a fixpoint. Secondly, we prove that if an FDS $A$ has a $k$-th root (i.e. $B$ such that $B^k = A$), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.",
"subjects": "Discrete Mathematics (cs.DM); Dynamical Systems (math.DS); Rings and Algebras (math.RA)",
"title": "Factorisation in the semiring of finite dynamical systems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795083542036,
"lm_q2_score": 0.7185944046238982,
"lm_q1q2_score": 0.7092379521817767
} |
https://arxiv.org/abs/2009.11069 | Towards accelerated rates for distributed optimization over time-varying networks | We study the problem of decentralized optimization over time-varying networks with strongly convex smooth cost functions. In our approach, nodes run a multi-step gossip procedure after making each gradient update, thus ensuring approximate consensus at each iteration, while the outer loop is based on accelerated Nesterov scheme. The algorithm achieves precision $\varepsilon > 0$ in $O(\sqrt{\kappa_g}\chi\log^2(1/\varepsilon))$ communication steps and $O(\sqrt{\kappa_g}\log(1/\varepsilon))$ gradient computations at each node, where $\kappa_g$ is the global function number and $\chi$ characterizes connectivity of the communication network. In the case of a static network, $\chi = 1/\gamma$ where $\gamma$ denotes the normalized spectral gap of communication matrix $\mathbf{W}$. The complexity bound includes $\kappa_g$, which can be significantly better than the worst-case condition number among the nodes. |
\section{Introduction}
In this work, we study a sum-type minimization problem
\begin{align}\label{eq:initial_problem}
f(x) = \frac{1}{n}\sum_{i=1}^n f_i(x) \to \min_{x\in\mathbb{R}^d}.
\end{align}
Convex functions $f_i$ are stored separately by nodes in communication network, which is represented by an undirected graph $\mathcal{G} = (V, E)$. This type of problems arise in distributed machine learning, drone or satellite networks, statistical inference \cite{nedic2017fast} and power system control \cite{ram2009distributed}. The computational agents over the network have access to their local $f_i$ and can communicate only with their neighbors, but still aim to minimize the global objective in \eqref{eq:initial_problem}.
The basic idea behind approach of this paper is to reformulate problem \eqref{eq:initial_problem} as a problem with linear constraints. Let us assign each agent in the network a personal copy of parameter vector $x_i$ and introduce
\begin{align*}
{\bf X} = \begin{pmatrix} x_1^\top \\ \vdots \\ x_n^\top \end{pmatrix} \in \mathbb{R}^{n\times d},~
F({\bf X}) = \sum_{i=1}^n f_i(x_i).
\end{align*}
Now we equivalently rewrite problem \eqref{eq:initial_problem} as
\begin{align}\label{eq:problem_linear_constraints}
F({\bf X}) = \sum_{i=1}^n f_i(x_i) \to \min_{x_1 = \ldots = x_n}.
\end{align}
This reformulation increases the number of variables, but induces additional constraints at the same time. Problem \eqref{eq:problem_linear_constraints} has the same optimal value as problem \eqref{eq:initial_problem}.
Let us denote the set of consensus constraints $\mathcal{C} = \{x_1 = \ldots = x_n\}$. Also, for each ${\bf X}\in\mathbb{R}^{d\times n}$ denote average of its columns $\overline x = \frac{1}{n}\sum_{i=1}^n x_i$ and introduce its projection onto constraint set.
\begin{align*}
\overline{\bf X} = \frac{1}{n} \mathbf{1}_n\onevec^\top {\bf X} = \Pi_{\mathcal{C}}({\bf X}) = \begin{pmatrix} \overline x^\top \\ \vdots \\ \overline x^\top \end{pmatrix}.
\end{align*}
Note that $\mathcal{C}$ is a linear subspace in $\mathbb{R}^{n\times d}$, and therefore projection operator $\Pi_\mathcal{C}(\cdot)$ is linear.
Decentralized optimization methods aim at minimizing the objective function and maintaining consensus accuracy between nodes. The optimization part is performed by using gradient steps. At the same time, keeping every agent's parameter vector close to average over the nodes is done via communication steps. Alternating gradient and communication updates allows both to minimize the objective and control consensus constraint violation.
In centralized scenario, there exists a server which is able to communicate with every agent in the network. In particular, a common parameter vector is maintained at all of the nodes. However, in decentralized setting it is only possible to ensure that agent's vectors are approximately equal with desired accuracy. The algorithm studied in this paper runs a sequence of communication rounds after every optimization step. We refer to this series of communications as \textit{consensus subroutine}. Such information exchange allows to reach approximate consensus between nodes after each gradient update, while the accuracy is controlled by the number of communication rounds.
On the one hand, a method which employs a consensus subroutine after each gradient update mimics a centralized algorithm. The difference is that in presence of a master node all computational entities have access to a common variable, while in decentralized case consensus constraints are satisfied only with nonzero accuracy. On the other hand, consensus subroutine may be interpreted as an inexact projection onto the constraint set $\mathcal{C}$. Every communication round is a step towards the projection. Therefore, our approach fits the inexact oracle framework, which has been studied in \cite{devolder2014first,devolder2013first}. We note that a similar approach to decentralized optimization is studied in \cite{jakovetic2014fast}.
We aim at building a first-order method with trajectory lying in neighborhood of $\mathcal{C}$. A simple example would be GD with inexact projections.
\begin{align}\label{eq:example_gd}
{\bf X}^{k+1} \approx \Pi_\mathcal{C}({\bf X}^k - \gamma\nabla F({\bf X}^k)) = \overline{\bf X}^k - \gamma \overline{\nabla F}({\bf X}^k),
\end{align}
where $\nabla F({\bf X}^k) = (\nabla f_1(x_1^k)\ldots \nabla f_n(x_n^k))^\top$ denotes the gradient of $F$.
Algorithm with update rule \ref{eq:example_gd} can be viewed as a gradient descent with inexact oracle. If the oracle was exact, the update rule would write as
\begin{align*}
\overline{\bf X}^{k+1} = \overline{\bf X}^k - \gamma\overline{\nabla F}(\overline{\bf X}^k),
\end{align*}
thus making the method trajectory stay precisely in $\mathcal{C}$. In this particular example, inexact gradient $\overline{\nabla F}({\bf X}^k)$ approximates exact gradient $\overline{\nabla F}(\overline{\bf X}^k)$.
Throughout the paper, $\angles{\cdot, \cdot}$ denotes the inner product of vectors or matrices. Correspondingly, by $\norm{\cdot}$ we denote a $2$-norm for vectors or Frobenius norm for matrices.
\subsection{Related work}
A decentralized algorithm makes two types of steps: local updates and information exchange. The complexity of such methods depends on objective condition number $\kappa$ and a term $\chi$ representing graph connectivity.
Local steps may use gradient \cite{Nedic2017achieving,scaman2018optimal,Pu2018,Qu2017,shi2015extra,ye2020multi,li2018sharp} or sub-gradient \cite{Nedic2009} computations. In primal-only methods, the agents compute gradients of their local functions and alternate taking gradient steps and communication procedures. Under cheap communication costs, it may be beneficial to replace a single consensus iteration with a series of information exchange rounds. Such methods as MSDA \cite{scaman2017optimal}, D-NC \cite{Jakovetic} and Mudag \cite{ye2020multi} employ multi-step gossip procedures.
Typically, non-accelerated methods need $O(\kappa\chi\log(1/\varepsilon))$ iterations to yield a solution with $\varepsilon$-accuracy \cite{rogozin2019projected}. Nesterov acceleration may be employed to improve dependence on $\kappa$ or $\chi$ and obtain algorithms with $O(\sqrt{\kappa\chi}\log(1/\varepsilon))$ complexity. In order to achieve this, one may distribute accelerated methods directly \cite{Qu2017,ye2020multi,li2018sharp,Jakovetic,dvinskikh2019decentralized} or use a Catalyst framework \cite{li2020revisiting}. Accelerated methods meet the lower complexity bounds for decentralized optimization \cite{hendrikx2020optimal,li2020optimal,scaman2017optimal}.
Consensus restrictions $x_1 = \ldots = x_n$ may be treated as linear constraints, thus allowing for a dual reformulation of problem \eqref{eq:initial_problem}. Dual-based methods include dual ascent and its accelerated variants \cite{scaman2017optimal,Wu2017,Zhang2017,uribe2020dual}. Primal-dual approaches like ADMM \cite{arjevani2020ideal,wei2012distributed} are also implementable in decentralized scenarios.
In \cite{scaman2018optimal}, the authors developed algorithms for non-convex objectives and provided lower complexity bounds for non-convex case, as well.
Time-varying networks open a new venue in research. Changing topology requires new approaches to decentralized methods and a more complicated theoretical analysis. The first method with provable geometric convergence was proposed in \cite{Nedic2017achieving}. Such primal algorithms as Push-Pull Gradient Method \cite{Pu2018} and DIGing \cite{Nedic2017achieving} are robust to network changes and have theoretical guarantees of convergence over time-varying graphs. Recently, a dual method for time-varying architectures was introduced in \cite{Maros2018}.
\subsection{Summary of contributions}
Our approach uses multi-step gossip averaging, and the analysis is based on the inexact oracle framework.
The proposed algorithm (Algorithm \ref{alg:decentralized_agd}) requires $O(\sqrt{\kappa_g} \chi \log^2(1/\varepsilon))$, where $\kappa_g$ denotes the (global) condition number of $f$ and $\chi$ is a term characterizing graph connectivity, which is defined later in the paper. For a static graph, $\chi = 1 / \gamma$, where $\gamma$ denotes the normalized eigengap of communication matrix associated with the network. Our result has an accelerated rate on function condition number and is derived for time-varying networks.
Our complexity estimate includes the condition number $\kappa_g$, i.e. \textit{global} strong convexity and smoothness constants instead of \textit{local} ones. If the data among the nodes is strongly heterogeneous, it is possible that $\kappa_g \gg \kappa_l$, where $\kappa_l$ is the local condition number \cite{scaman2017optimal,hendrikx2020optimal,tang2019practicality}. Therefore, the method with complexity depending on $\kappa_g$ may perform significantly better. A recently proposed method Mudag \cite{ye2020multi} has a complexity depending on $\sqrt{\kappa_g}$, as well, but the method is designed for time-static graphs.
The lower bound for number of communications is $\Omega(\sqrt{\kappa_l \chi}\log(1/\varepsilon))$ \cite{scaman2017optimal}. Our result is obtained for time-varying graphs and has a worse dependence on $\chi$. On the other hand, we derive a better dependence on condition number, i.e. we use $\kappa_g$ instead of $\kappa_l$. However, this by no means breaks the lower bounds. First, our result includes $\log^2(1/\varepsilon)$ instead of $\log(1/\varepsilon)$. Second, a function in \cite{scaman2017optimal} on which the lower bounds are attained has $\kappa_g\sim \kappa_l$. Namely, for the bad function it holds $\kappa_g\ge \kappa_l/16$ (see Appendix A.1 in \cite{scaman2017optimal} for details).
\section{Inexact oracle framework}
In this section, we describe the inexact oracle construction for objective function $f$.
\subsection{Preliminaries}
Initially we recall the definition of $(\delta, L, \mu)$-oracle from \cite{devolder2013first}. Let $h(x)$ be a convex function defined on a convex set $Q\subseteq\mathbb{R}^m$. We say that $(h_{\delta,L,\mu}(x), s_{\delta,L,\mu}(x))$ is a $(\delta, L, \mu)$-model of $h(x)$ at point $x\in Q$ if for all $y\in Q$ it holds
\begin{align}\label{eq:inexact_oracle_def_devolder}
\frac{\mu}{2}\norm{y - x}^2 \le h(y) - \cbraces{h_{\delta,L,\mu}(x) + \angles{s_{\delta,L,\mu}(x), y - x}} \le \frac{L}{2} \norm{y - x}^2 + \delta.
\end{align}
\subsection{Inexact oracle for f}
Consider $\overline x, \overline y\in\mathbb{R}^{d}$ and define
\begin{align*}
\overline{\bf X} = \begin{pmatrix} \overline x^\top \\ \vdots \\ \overline x^\top \end{pmatrix},~ \overline{\bf Y} = \begin{pmatrix} \overline y^\top \\ \vdots \\ \overline y^\top \end{pmatrix} \in \mathcal{C}.
\end{align*}
Let ${\bf X}\in\mathbb{R}^{n\times d}$ be such that $\Pi_\mathcal{C} ({\bf X}) = \overline{\bf X}$ and $\norm{\overline{\bf X} - {\bf X}}^2\le \delta'$.
\begin{lemma}\label{lem:inexact_oracle}
Define
\begin{align*}
\delta &= \frac{1}{2n}\cbraces{\frac{L_{l}^2}{L_{g}} + \frac{2L_{l}^2}{\mu_{g}} + L_{l} - \mu_{l}} \delta', \addtocounter{equation}{1}\tag{\theequation}\label{eq:delta_inexact_oracle} \\
f_{\delta, L, \mu}(\overline x, {\bf X}) &= \frac{1}{n} \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{1}{2}\cbraces{\mu_{l} - \frac{2L_{l}^2}{\mu_{g}}}\norm{\overline{\bf X} - {\bf X}}^2}, \\
g_{\delta, L, \mu}(\overline x, {\bf X}) &= \frac{1}{n}\sum_{i=1}^n \nabla f_i(x_i).
\end{align*}
Then $(f_{\delta,L,\mu}(\overline x, {\bf X}), g_{\delta,L,\mu}(\overline x, {\bf X}))$ is a $(\delta,2L_{g},\mu_{g}/2)$-model of $f$ at point $\overline x$, i.e.
\begin{align*}
\frac{\mu_{g}}{4}\norm{\overline y - \overline x}^2 \le f(\overline y) - f_{\delta, L, \mu}(\overline x, {\bf X}) - \angles{g_{\delta, L, \mu}(\overline x, {\bf X}), \overline y - \overline x} \le L_{g}\norm{\overline y - \overline x}^2 + \delta.
\end{align*}
\end{lemma}
\begin{proof}
We aim at obtaining estimates for $F(\overline{\bf Y})$ similar to \eqref{eq:inexact_oracle_def_devolder}. First, we get a lower bound on $F(\overline{\bf Y})$.
\begin{align*}
F(\overline{\bf Y})
&\ge F({\bf X}) + \sbraces{\angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{\mu_{l}}{2}\norm{{\bf X} - \overline{\bf X}}^2} + \sbraces{\angles{\overline{\nabla F}(\overline{\bf X}), \overline{\bf Y} - \overline{\bf X}} + \frac{\mu_{g}}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2} \\
&= \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{\mu_{l}}{2}\norm{{\bf X} - \overline{\bf X}}^2} + \angles{\overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} \\
&\qquad + \angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} + \frac{\mu_{g}}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2. \addtocounter{equation}{1}\tag{\theequation}\label{eq:inexact_oracle_lower_bound_1}
\end{align*}
Let us lower bound the term $\angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}}$ using Young inequality $\angles{a, b}\le \frac{\norm{a}^2}{2p} + \frac{p}{2}\norm{b}^2,~ p > 0$.
\begin{align*}
\angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}}
&\ge -\frac{1}{2p}\norm{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X})}^2 - \frac{p}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2 \\
&\ge -\frac{L_{l}^2}{2p}\norm{\overline{\bf X} - {\bf X}}^2 - \frac{p}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2.
\end{align*}
Returning to \eqref{eq:inexact_oracle_lower_bound_1} and noting that $\angles{\overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} = \angles{\nabla F({\bf X}), \overline{\bf Y} - \overline{\bf X}}$, we get
\begin{align*}
F(\overline{\bf Y}) &\ge \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{1}{2}\cbraces{\mu_{l} - \frac{L_{l}^2}{p}}\norm{{\bf X} - \overline{\bf X}}^2} \\
&\qquad + \angles{\nabla F({\bf X}), \overline{\bf Y} - \overline{\bf X}} + \frac{\mu_{g} - p}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2 \addtocounter{equation}{1}\tag{\theequation}\label{eq:inexact_oracle_lower_bound_2}
\end{align*}
Second, we get an upper estimate on $F(\overline{\bf Y})$.
\begin{align*}
F(\overline{\bf Y})
&\le \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{L_{l}}{2}\norm{\overline{\bf X} -{\bf X}}^2} + \sbraces{\angles{\overline{\nabla F}(\overline{\bf X}), \overline{\bf Y} - \overline{\bf X}} + \frac{L_{g}}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2} \\
&= \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{L_{l}}{2}\norm{\overline{\bf X} -{\bf X}}^2} + \angles{\overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} \\
&\qquad + {\angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}}} + \frac{L_{g}}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2. \addtocounter{equation}{1}\tag{\theequation} \label{eq:inexact_oracle_upper_bound_1}
\end{align*}
Analogously, we estimate the term $\angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}}$ with Young inequality.
\begin{align*}
\angles{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}}
&\le \frac{1}{2q}\norm{\overline{\nabla F}(\overline{\bf X}) - \overline{\nabla F}({\bf X})}^2 + \frac{q}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2 \\
&\le \frac{L_{l}^2}{2q}\norm{\overline{\bf X} - {\bf X}}^2 + \frac{q}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2,~ q > 0.
\end{align*}
Plugging it into \eqref{eq:inexact_oracle_upper_bound_1} and once again using $\angles{\overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} = \angles{\nabla F({\bf X}), \overline{\bf Y} - \overline{\bf X}}$ yields
\begin{align*}
F(\overline{\bf Y})&\le \sbraces{F({\bf X}) + \angles{\nabla F({\bf X}), \overline{\bf X} - {\bf X}} + \frac{1}{2}\cbraces{\mu_{l} - \frac{L_{l}^2}{p}}\norm{{\bf X} - \overline{\bf X}}^2} \\
&\qquad + \angles{\nabla F({\bf X}), \overline{\bf Y} - \overline{\bf X}} + \frac{L_{g} + q}{2}\norm{\overline{\bf Y} - \overline{\bf X}}^2 \\
&\qquad + \frac{1}{2}\cbraces{\frac{L_{l}^2}{q} + \frac{L_{l}^2}{p} + L_{l} - \mu_{l}}\norm{{\bf X} - \overline{\bf X}}^2 \addtocounter{equation}{1}\tag{\theequation}\label{eq:inexact_oracle_upper_bound_2}
\end{align*}
Consequently, smoothness and strong convexity constants for inexact oracle are $L = L_{g} + q,~ \mu = \mu_{g} - p$, respectively. Choosing $q$ and $p$ allows to control condition number $L/\mu$. Letting $q = L_{g},~ p = \mu_{g} / 2$ leads to
\begin{subequations}\label{eq:L_mu_def}
\begin{align}
L &= 2L_{g}, \\
\mu &= \mu_{g} / 2.
\end{align}
\end{subequations}
Noting that
\begin{align*}
&F(\overline{\bf X}) = nf(\overline x),~ F(\overline{\bf Y}) = nf(\overline y), \\
&\norm{\overline{\bf Y} - \overline{\bf X}}^2 = n\norm{\overline y - \overline x}^2, \\
&\angles{\overline{\nabla F}({\bf X}), \overline{\bf Y} - \overline{\bf X}} = n\angles{g_{\delta, L, \mu}(\overline x), \overline y - \overline x}
\end{align*}
and combining \eqref{eq:inexact_oracle_lower_bound_2} and \eqref{eq:inexact_oracle_upper_bound_2} leads to
\begin{align*}
\frac{\mu}{2}\norm{\overline y - \overline x}^2 \le f(\overline y) - f_{\delta, L, \mu}(\overline x, {\bf X}) - \angles{g_{\delta, L, \mu}(\overline x, {\bf X}), \overline y - \overline x} \le \frac{L}{2}\norm{\overline y - \overline x}^2 + \delta,
\end{align*}
which concludes the proof.
\end{proof}
\section{Algorithm and results}
We take Algorithm 2 from \cite{stonyakin2020inexact} as a basis for our method. The algorithm is designed for the inexact oracle model and achieves an accelerated rate.
\begin{algorithm}[H]
\caption{Decentralized AGD with consensus subroutine}
\label{alg:decentralized_agd}
\begin{algorithmic}[1]
\REQUIRE{Initial guess ${\bf X}^0\in \mathcal{C}$, constants $L, \mu > 0$, ${\bf U}^0 = {\bf X}^0$, $\alpha^0 = {\bf A}^0 = 0$}
\FOR{$k = 0, 1, 2,\ldots$}
\STATE{Find $\alpha^{k+1}$ as the greater root of \\$(A^k + \alpha^{k+1})(1 + A^k \mu) = L(\alpha^{k+1})^2$}
\STATE{$A^{k+1} = A^k + \alpha^{k+1}$}
\STATE{$\displaystyle {\bf Y}^{k+1} = \frac{\alpha^{k+1} {\bf U}^k + A^k {\bf X}^k}{A^{k+1}}$}
\STATE{\label{alg_step:agd_step}${\bf V}^{k+1} = \frac{\mu{\bf Y}^{k+1} + (1 + A^k\mu){\bf U}^k}{1 + A^k\mu + \mu} - \frac{\alpha^{k+1}\nabla F({\bf Y}^{k+1})}{1 + A^k\mu + \mu}$}
\STATE{\label{alg_step:consensus_update}
$
{\bf U}^{k+1} = \text{Consensus}({\bf V}^{k+1}, T^k)
$
}
\STATE{$\displaystyle {\bf X}^{k+1} = \frac{\alpha^{k+1} {\bf U}^{k+1} + A^k {\bf X}^k}{A^{k+1}}$}
\ENDFOR
\end{algorithmic}
\end{algorithm}
\vspace{-0.5cm}
\begin{algorithm}[H]
\caption{Consensus}
\label{alg:consensus}
\begin{algorithmic}
\REQUIRE{Initial ${\bf X}^0\in\mathcal{C}$, number of iterations $T$.}
\FOR{$t = 1, \ldots, T$}
\STATE{${\bf X}^{t+1} = {\bf W}^t {\bf X}^t$}
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Consensus}
We consider a sequence of non-directed communication graphs $\braces{\mathcal{G}^k = (V, E^k)}_{k=0}^\infty$ and a sequence of corresponding mixing matrices $\braces{{\bf W}^k}_{k=0}^\infty$ associated with it. We impose the following
\begin{assumption}\label{assum:mixing_matrix}
Mixing matrix sequence $\braces{{\bf W}^k}_{k=0}^\infty$ satisfies the following properties.
\begin{itemize}
\item (Decentralized property) If $(i, j)\notin E_k$, then $[{\bf W}^k]_{ij} = 0$.
\item (Double stochasticity) ${\bf W}^k \mathbf{1}_n = \mathbf{1}_n,~ \mathbf{1}_n^\top{\bf W}^k = \mathbf{1}_n^\top$.
\item (Contraction property) There exist $\tau\in\mathbb{Z}_{++}$ and $\lambda\in(0, 1)$ such that for every $k\ge \tau - 1$ it holds
\begin{align*}
\norm{{\bf W}_{\tau}^k {\bf X} - \overline{\bf X}} \le (1 - \lambda)\norm{{\bf X} - \overline{\bf X}},
\end{align*}
where ${\bf W}_\tau^k = {\bf W}^k \ldots {\bf W}^{k-\tau+1}$.
\end{itemize}
\end{assumption}
The contraction property in Assumption \ref{assum:mixing_matrix} generalizes several assumptions in the literature.
\begin{itemize}
\item Time-static connected graph: ${\bf W}^k = {\bf W}$. In this classical case we have $\lambda = 1 - \sigma_2({\bf W})$, where $\sigma_2({\bf W})$ denotes the second largest singular value of ${\bf W}$.
\item Sequence of connected graphs: every $\mathcal{G}_k$ is connected. In this scenario $\lambda = 1 - \underset{k\ge 0}{\sup}~\sigma_2({\bf W}^k)$.
\item $\tau$-connected graph sequence (i.e. for every $k\ge 0$ graph $\mathcal{G}^k_\tau = (V, E^k\cup E^{k+1}\cup\ldots\cup E^{k+\tau-1})$ is connected \cite{Nedic2017achieving}). For $\tau$-connected graph sequences it holds $1 - \lambda = \underset{k\ge 0}{\sup}~\sigma_{\max}({\bf W}_\tau^k - \frac{1}{n}\mathbf{1}_n\onevec^\top)$.
\end{itemize}
A stochastic variant of this contraction property is also studied in \cite{koloskova2020unified}.
During every communication round, the agents exchange information according to the rule
\begin{align*}
x_i^{k+1} = w_{ii}^k + \sum_{(i, j)\in E^k} w_{ij}^k x_j^k.
\end{align*}
In matrix form, this update rule writes as ${\bf X}^{k+1} = {\bf W}^k {\bf X}^k$. The contraction property in Assumption \ref{assum:mixing_matrix} is needed to ensure geometric convergence of Algorithm \ref{alg:consensus} to the average of nodes' initial vectors, i.e. to $\overline x^0$. In particular, the contraction property holds for $\tau$-connected graphs with Metropolis weights choice for ${\bf W}^k$, i.e.
\begin{align*}
[{\bf W}^k]_{ij} =
\begin{cases}
1 / (1 + \max\{d^k_i, d^k_j\}) &\text{if }(i, j)\in E^k, \\
0 &\text{if } (i, j)\notin E^k, \\
1 - \displaystyle\sum_{(i, m)\in E^k} [{\bf W}^k]_{im} &\text{if } i = j,
\end{cases}
\end{align*}
where $d^k_i$ denotes the degree of node $i$ in graph $\mathcal{G}^k$.
\subsection{Complexity result for Algorithm \ref{alg:decentralized_agd}}
This paper focuses on smooth strongly convex objectives. Our analysis is bounded to the following
\begin{assumption}\label{assum:str_convex_smooth}
For every $i = 1,\ldots, n$, function $f_i$ is differentiable, $\mu_i$-strongly convex and $L_i$-smooth ($\mu_i,~ L_i > 0$).
\end{assumption}
Under this assumption it holds
\begin{itemize}
\item (local constants) $F(X)$ is $\mu_{l}$-strongly convex and $L_{l}$-smooth on $\mathbb{R}^{n\times d}$, where $\displaystyle \mu_{l} = \min_i\mu_i,~ L_{l} = \max_i L_i$.
\item (global constants) $F(X)$ is $\mu_{g}$-strongly convex and $L_{g}$-smooth on $\mathcal{C}$, where $\mu_{g} = \frac{1}{n}\sum_{i=1}^n\mu_i,~ L_{g} = \frac{1}{n}\sum_{i=1}^n L_i$.
\end{itemize}
The global conditioning may be significantly better than local (see i.e. \cite{scaman2017optimal} for details). Our analysis shows that performance of Algorithm \ref{alg:decentralized_agd} depends on global constants.
In the next theorem, we provide computation and communication complexities of Algorithm \ref{alg:decentralized_agd}.
\begin{theorem}\label{th:total_iterations_strongly_convex}
Choose some $\varepsilon > 0$ and set
\begin{align*}
T_k = T = \frac{\tau}{2\lambda}\log\frac{D}{\delta'},~ \delta' = \frac{n\varepsilon}{32} \frac{\mu_{g}^{3/2}}{L_{g}^{1/2} L_{l}^2}.
\end{align*}
Also define
\begin{subequations}\label{eq:log_coefs_strongly_conv}
\begin{align*}
D_1 &= \frac{L_{l}}{L_{g}^{1/2}\mu_{g}} \sbraces{ 8\sqrt{2}L_{l}\norm{\overline u^0 - x^*}\cbraces{\frac{L_{g}}{\mu_{g}}}^{3/4} + \frac{4\sqrt{2}\norm{\nabla F({\bf X}^*)}}{\sqrt{n}}\cbraces{\frac{L_{g}}{\mu_{g}}}^{1/4} } \\
D_2 &= \frac{L_{l}}{L_{g}^{1/2}\mu_{g}} \sbraces{ 3\sqrt{\mu_{g}} + 4\sqrt{2n}\cbraces{\frac{L_{g}}{\mu_{g}}}^{1/4} }
\end{align*}
\end{subequations}
Then Algorithm \ref{alg:decentralized_agd} requires
\begin{align}\label{eq:agd_computational_complexity}
N = 2\sqrt{\frac{L_{g}}{\mu_{g}}} \log\cbraces{\frac{\norm{\overline u^0 - x^*}^2}{2\varepsilon L_{g}}}
\end{align}
gradient computations at each node and
\begin{align}\label{eq:agd_communication_complexity}
N_{tot} = N\cdot T = 2\sqrt{\frac{L_{g}}{\mu_{g}}} \frac{\tau}{\lambda} \cdot \log\cbraces{\frac{2L_{g}\norm{\overline u^0 - x^*}^2}{\varepsilon}} \log\cbraces{\frac{D_1}{\sqrt\varepsilon} + D_2}
\end{align}
communication steps to yield ${\bf X}^N$ such that
\begin{align*}
&f(\overline x^N) - f(x^*)\le \varepsilon,~ \norm{{\bf X}^N - \overline{\bf X}^N}^2\le \delta'.
\end{align*}
\end{theorem}
We provide the proof of Theorem \ref{th:total_iterations_strongly_convex} in Appendix \ref{app:total_iterations_strongly_convex}.
The number of gradient computations in \eqref{eq:agd_computational_complexity} reaches the lower bounds for non-distributed optimization up to a constant factor. Number of communication steps includes an additional term of $\tau/\lambda$, which characterizes graph connectivity.
\section{Numerical experiments}
We consider the logistic regression problem with L2 regularizer:
\begin{align*}
f(x) = \frac{1}{n}\sum\limits_{i=1}^n \log\left(1 + \exp(-b_i\angles{a_i, x})\right) + \frac{\theta}{2}\norm{x}_2^2.
\end{align*}
Here $a_1,\ldots,a_n\in\mathbb{R}^d$ denote the data points, $b_1,\ldots,b_n\in\{-1, 1\}$ denote class labels and $\theta > 0$ is a penalty coefficient.
Also we run experiments on a least-squares task:
\begin{align*}
f(x) = \frac{1}{2}\norm{{\bf A} x - b}_2^2.
\end{align*}
The blocks of data matrix ${\bf A}$ and vector $b$ are distributed among the agents in the network.
The simulations are run on LIBSVM datasets \cite{Chang2011}. We compare the performance of Algorithm \ref{alg:decentralized_agd} (DAccGD in legend of plots) to EXTRA \cite{shi2015extra}, DIGing \cite{Nedic2017achieving}, Mudag \cite{ye2020multi} and APM-C \cite{li2020decentralized}.
Logistic regression is carried out on a9a data-set, inner iterations are set to $T = 5$ for Mudag and DAccGD. Generation of random geometric graph goes on $20$ nodes.
\begin{figure}[H]
\centering
\includegraphics[width = \textwidth]{./figures/a9a.png}
\caption{a9a (logistic regression), $100$ nodes}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width = \textwidth]{./figures/cadata_scaled.png}
\caption{cadata (least squares), $20$ nodes}
\end{figure}
\section{Conclusion}
This paper studies an inexact oracle-based approach to decentralized optimization. The paper focuses on a specific case of strongly convex smooth functions, but the inexact oracle framework introduced in \cite{devolder2014first} is also applicable to non-strongly convex functions. The development of this framework in \cite{stonyakin2020inexact} also enables to generalize the results of this article to composite optimization problems and distributed algorithms for saddle-point problems and variational inequalities.
Another interesting application of inexact oracle approach lies in stochastic decentralized algorithms. Consider a class of $L$-smooth $\mu$-strongly convex objectives with gradient noise of each $f_i$ being upper-bounded by $\sigma^2$. For this class of problems, lower complexity bounds write as \cite{arjevani2015communication}
\begin{align*}
O(\sqrt{L/\mu\chi}\log(1/\varepsilon)) &\qquad\text{stochastic oracle calls per node} \\
O\cbraces{\max\cbraces{\frac{\sigma^2}{m\mu\varepsilon}, \sqrt{L/\mu}}\log(1/\varepsilon)} &\qquad\text{communication steps}.
\end{align*}
At the moment, there exist methods which are optimal either in the number of oracle calls or in the number of communication steps. We believe that approach of this paper combined with a specific batch-size choice described in \cite{dvinskikh2020accelerated} allows to develop a decentralized algorithm reaching both optimal complexities.
\section{Acknowledgements}
The authors are grateful to Dmitriy Metelev and Fedor Stonyakin for their thoughtful proofreading of the paper.
\bibliographystyle{unsrt}
| {
"timestamp": "2022-04-21T02:04:37",
"yymm": "2009",
"arxiv_id": "2009.11069",
"language": "en",
"url": "https://arxiv.org/abs/2009.11069",
"abstract": "We study the problem of decentralized optimization over time-varying networks with strongly convex smooth cost functions. In our approach, nodes run a multi-step gossip procedure after making each gradient update, thus ensuring approximate consensus at each iteration, while the outer loop is based on accelerated Nesterov scheme. The algorithm achieves precision $\\varepsilon > 0$ in $O(\\sqrt{\\kappa_g}\\chi\\log^2(1/\\varepsilon))$ communication steps and $O(\\sqrt{\\kappa_g}\\log(1/\\varepsilon))$ gradient computations at each node, where $\\kappa_g$ is the global function number and $\\chi$ characterizes connectivity of the communication network. In the case of a static network, $\\chi = 1/\\gamma$ where $\\gamma$ denotes the normalized spectral gap of communication matrix $\\mathbf{W}$. The complexity bound includes $\\kappa_g$, which can be significantly better than the worst-case condition number among the nodes.",
"subjects": "Optimization and Control (math.OC)",
"title": "Towards accelerated rates for distributed optimization over time-varying networks",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795075882267,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.70923795163135
} |
https://arxiv.org/abs/1603.04637 | On the Randomization of Frolov's Algorithm for Multivariate Integration | We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube.In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^{-r-1/2} \, (\log n)^{(d-1)/2}$ in expectation and by $n^{-r} \, (\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on $H^{r,\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesis is a review of the mentioned upper bounds and their proofs. | \section{Introduction}
Many applications deal with multivariate functions $f$ which are smooth
in the sense that certain weak derivatives $\diff^\alpha f$ exist and are
square-integrable, functions from a \textit{Sobolev space}.
Which derivatives $\diff^\alpha f$ of $f$ are known to be existent and
square-integrable highly depends on the actual problem.
Classically, $\alpha$ covers the range of all vectors
in $\ensuremath{\mathbb{N}}_0^d$ with $\Vert\alpha\Vert_1\leq s$ for some $s\in\ensuremath{\mathbb{N}}$.
The corresponding Sobolev space is called \textit{isotropic Sobolev space of smoothness $s\in\ensuremath{\mathbb{N}}$}.
For instance, the solutions of elliptic partial differential
equations in general and Poisson's equation in particular, have this form.
They typically appear in electrostatics or continuum mechanics.
But often $f$ is known to satisfy a stronger smoothness condition:
Derivatives $\diff^\alpha f$ for each $\alpha\in\ensuremath{\mathbb{N}}_0^d$
with $\Vert\alpha\Vert_\infty\leq s$ exist and are square-integrable.
This is typically the case, if $f$ is a tensor product of $s$-times
differentiable functions of one variable: $f(x_1,\dots,x_d)=f_1(x_1)\cdot\hdots\cdot f_d(x_d)$.
We say that $f$ is from a \textit{Sobolev space of dominating mixed smoothness $s$}.
For example, solutions of the electronic Schrödinger equation are of this
form.\\
We are concerned with the numerical integration of such functions and refer to
\cite{hartri} and \cite{giltru} for a treatise on elliptic partial
differential equations and their connection with Sobolev spaces
and to \cite{yserentant} for further information about
electronic wave functions.
More precisely,
we want to use linear quadrature rules to approximate the integral $I_d(f)$ of integrable,
real valued functions $f$ in $d$ real variables, with a particular interest
in functions with dominating mixed smoothness $s$.
A linear \textit{quadrature rule}, \textit{algorithm} or \textit{method} $A_n$
is given by a finite number $n$ of weights $a_1,\dots,a_n\in\ensuremath{\mathbb{R}}$ and nodes
$x^{(1)},\dots,x^{(n)}\in\ensuremath{\mathbb{R}}^d$, and the rule
\[
A_n(f)=\sum\limits_{j=1}^{n} a_j\, f\braces{x^{(j)}}
.\]
All these numbers and vectors can be deterministic or random variables.
Since $n$ counts the number of function values computed by $A_n$,
it is a measure for the cost of $A_n$, commonly
referred to as \textit{information cost} of the algorithm.
The error of $A_n$ associated with the integration of $f$ is
$\abs{A_n(f)-I_d(f)}$.
We are interested in sequences $\braces{A_n}_{n\in\ensuremath{\mathbb{N}}}$ of quadrature rules
whose error decreases fast with respect to growing information cost $n$.
In this sense, numerical integration of functions
with dominating mixed smoothness $s$ is significantly easier
than the integration of functions with isotropic smoothness $s$,
especially if the number $d$ of variables is large:
It turns out that the convergence order $n^{-s-1/2}$ can be achieved for the expected error,
while $n^{-s/d-1/2}$ is the best possible rate in the isotropic case.\\
From now on, for the sake of distinction, we will use $s$ as a parameter for isotropic
smoothness and $r$ as a parameter for dominating mixed smoothness.
The smoothness parameters $r$ and $s$ and the dimension $d$
are arbitrary natural numbers, with the single condition that $s>d/2$.
But they are considered to be fixed in the sense that
any constant in this thesis is merely a constant with respect to the
information cost $n$ and may depend on $r,s$ and $d$.\\
Let us end this introductory section with an outline of the thesis.
We start with a brief compilation of the definitions and fundamental properties
of the above mentioned Sobolev spaces.
In Section~\ref{basicquadrulesection}, we will present a familiy of deterministic quadrature rules
for the integration of compactly supported, continuous functions.
Among those rules is Frolov's algorithm, which will be examined in
Section \ref{frolovsrulesection}.
With respect to the information cost $n$, its integration error for functions $f$ with
dominating mixed smoothness $r$ and compact support in the open unit cube $(0,1)^d$
is bounded above by a constant multiple of
$n^{-r} \, (\log n)^{(d-1)/2}$ times the corresponding norm of $f$.
The order $n^{-r} \, (\log n)^{(d-1)/2}$ is optimal.
For functions with support in $(0,1)^d$ and isotropic smoothness $s$
the order $n^{-s/d}$ is achieved, which is also optimal.
In Section~\ref{randomdilationsection}, we will add random dilations to Frolov's algorithm
and examine the integration error of the resulting algorithm for the same types of functions.
We will see that in both cases the random dilations improve the order of
the algorithm's error by $1/2$ in expectation, while
not changing it in the worst case.
The additional random shift introduced in Section \ref{randomshiftsection}
makes the algorithm unbiased and,
in case of functions with dominating mixed smoothness $r$ and compact support in $(0,1)^d$,
further improves the order of its expected error by a logarithmic term.
Section \ref{transformationsection} shows
that the condition of having support in $(0,1)^d$ can be dropped
by applying a suitable change of variables to the above algorithms.
The resulting algorithms satisfy the error bounds from above
for any function on $[0,1]^d$ with dominating mixed smoothness $r$
or isotropic smoothness $s$.
Beyond that, the change of variables preserves unbiasedness.
\newpage
\section{The Function Spaces}
\label{functionclassessec}
For natural numbers $r$ and $d$ the Sobolev space $\ensuremath{H^{r,{\rm mix}}(\IR^d)}$ of dominating
mixed smoothness $r$ is the real vector space
\[
\ensuremath{H^{r,{\rm mix}}(\IR^d)} = \set{f\in L^2(\ensuremath{\mathbb{R}}^d)\mid \diff^\alpha f
\in L^2(\ensuremath{\mathbb{R}}^d) \text{ for every } \alpha \in \set{0,\dots,r}^d}
\]
of $d$-variate, real valued functions, equipped with the scalar product
\[
\mixscalar{f}{g}= \sum\limits_{\alpha \in \set{0,\dots,r}^d}
\lscalar{\diff^\alpha f}{\diff^\alpha g}
.\]
The scalar product induces the norm
\[
\mixnorm{f}= \left(\sum\limits_{\alpha \in \set{0,\dots,r}^d}
\lnorm{\diff^\alpha f}^2\right)^{1/2}.
\]
It is known that $\ensuremath{H^{r,{\rm mix}}(\IR^d)}$ is a Hilbert space and its elements can be considered
to be continuous functions.
In this thesis, the Fourier transform is the unique continuous linear operator
$\mathcal{F}: L^2(\ensuremath{\mathbb{R}}^d)\to L^2(\ensuremath{\mathbb{R}}^d)$
satisfying
\[
\mathcal{F}f(y) = \int_{\ensuremath{\mathbb{R}}^d} f(x)\, e^{-2\pi i \scalar{x}{y}} \, \d x
\]
for integrable $f:\ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}$ and $y\in\ensuremath{\mathbb{R}}^d$.
The space $\ensuremath{H^{r,{\rm mix}}(\IR^d)}$ contains exactly those
functions $f\in L^2(\ensuremath{\mathbb{R}}^d)$ with ${\mathcal{F}f \cdot h_r^{1/2} \in L^2(\ensuremath{\mathbb{R}}^d)}$
for the Fourier transform $\mathcal{F}f$ of $f$ and the weight function
\[
h_r: \ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}^+,\quad h_r(x)= \sum\limits_{\alpha\in\{0,\dots,r\}^d}
\prod\limits_{j=1}^{d} |2\pi x_j|^{2\alpha_j}
= \prod\limits_{j=1}^{d} \sum\limits_{k=0}^{r} |2\pi x_j|^{2k}.
\]
In terms of its Fourier transform, the norm of $f\in\ensuremath{H^{r,{\rm mix}}(\IR^d)}$ is given by
\[
\mixnorm{f}^2= \int\limits_{\ensuremath{\mathbb{R}}^d} \left|\mathcal{F}f(x)\right|^2\cdot h_r(x) \, \d x.
\]
Analogously, the isotropic Sobolev space $\ensuremath{H^s(\IR^d)}$ of smoothness $s\in\ensuremath{\mathbb{N}}$ is
\[
\ensuremath{H^s(\IR^d)} = \set{f\in L^2(\ensuremath{\mathbb{R}}^d)\mid \diff^\alpha f \in L^2(\ensuremath{\mathbb{R}}^d)
\text{ for every } \alpha \in \ensuremath{\mathbb{N}}_0^d\text{ with } \Vert\alpha\Vert_1\leq s}
,\]
equipped with the scalar product
\[
\isoscalar{f}{g}= \sum\limits_{\Vert\alpha\Vert_1\leq s}
\lscalar{\diff^\alpha f}{\diff^\alpha g}
\]
and its induced norm $\isonorm{f}$.
For $\alpha\in\ensuremath{\mathbb{N}}_0^d$, we will frequently use the abbreviation
$\abs{\alpha}=\Vert\alpha\Vert_1=\sum_{j=1}^{d}\abs{\alpha_j}$.
The space $\ensuremath{H^s(\IR^d)}$ is a Hilbert space, too. In the following, we will assume that $s$
is greater than $d/2$.
Then $\ensuremath{H^s(\IR^d)}$ also consists of continuous functions, exactly those functions
$f\in L^2(\ensuremath{\mathbb{R}}^d)$ with ${\mathcal{F}f \cdot v_s^{1/2}\in L^2(\ensuremath{\mathbb{R}}^d)}$
for the Fourier transform $\mathcal{F}f$ of $f$ and the weight function
\[
v_s: \ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}^+,\quad v_s(x)= \sum\limits_{\abs{\alpha}\leq s}
\prod\limits_{j=1}^{d} |2\pi x_j|^{2\alpha_j}
\asymp \left( 1+\Vert x\Vert_2^2\right)^s
.\]
In terms of its Fourier transform, the norm of $f\in\ensuremath{H^s(\IR^d)}$ is given by
\[
\isonorm{f}^2= \int\limits_{\ensuremath{\mathbb{R}}^d} \left|\mathcal{F}f(x)\right|^2\cdot v_s(x) \, \d x
.\]
Furthermore, let $\ensuremath{C_c(\IR^d)}$ be the real vector space of all continuous real valued functions
with compact support in $\ensuremath{\mathbb{R}}^d$. The spaces $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ and $\ensuremath{\mathring{H}^s([0,1]^d)}$ of
functions in $\ensuremath{H^{r,{\rm mix}}(\IR^d)}$ or $\ensuremath{H^s(\IR^d)}$ with compact support in the
unit cube are subspaces of $\ensuremath{C_c(\IR^d)}$.
They can also be considered as subspaces of the Hilbert space
\[
\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}= \set{f\in L^2([0,1]^d) \mid \diff^\alpha
f \in L^2([0,1]^d) \text{ for every } \alpha \in \set{0,\dots,r}^d}
,\]
equipped with the scalar product
\[
\mixscalarunit{f}{g}= \sum\limits_{\alpha \in \set{0,\dots,r}^d}
\lscalarunit{\diff^\alpha f}{\diff^\alpha g} ,
\]
or the Hilbert space
\[
\ensuremath{{H^s([0,1]^d)}}= \set{f\in L^2([0,1]^d)
\mid \diff^\alpha f \in L^2([0,1]^d) \text{ for } \alpha
\in \ensuremath{\mathbb{N}}_0^d\text{ with } \abs{\alpha}\leq s}
,\]
with the scalar product
\[
\isoscalarunit{f}{g}= \sum\limits_{\abs{\alpha}\leq s}
\lscalarunit{\diff^\alpha f}{\diff^\alpha g}
.\]
\newpage
\section{The Basic Quadrature Rule}
\label{basicquadrulesection}
We introduce a family of deterministic and linear quadrature rules.
This family is fundamental to our studies.
All the algorithms to be presented are based on the following definition.
\medskip
\noindent
{\bf Algorithm.} \
Let $S\in\ensuremath{\mathbb{R}}^{d\times d}$ be invertible and $v$ be a vector in $\ensuremath{\mathbb{R}}^d$.
We define
\[
Q_S^v(f)=\frac{1}{|\det S|}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} f\left(S^{-\top}(m+v)\right)
\]
for any admissible input function $f:\ensuremath{\mathbb{R}}^d\to\ensuremath{\mathbb{R}}$. We call $v$
\emph{shift parameter} and denote by $Q_S$ the algorithm $Q_S^v$ for shift
parameter $v=0$.
\medskip
The matrix $S^{-\top}$ is the transpose of the inverse of $S$.
For now, $S$ can be any invertible matrix. But
later on, it will be a fixed matrix $B$ multiplied with a number
$n^{1/d}$ and a \textit{dilation matrix} $\hat{u}=\mathop{\mathrm{diag}}(u_1,\dots,u_d)$ for
a \textit{dilation parameter} $u\in\ensuremath{\mathbb{R}}^d$.
The dilation parameter $u\in\ensuremath{\mathbb{R}}^d$ and shift parameter $v\in\ensuremath{\mathbb{R}}^d$
are also arbitrary. As we go along, they will be chosen as
independent random variables $U$ and $V$
that are uniformly distributed in $[1,2^{1/d}]^d$ and $[0,1]^d$, respectively.\\
The rule $Q_S^v$ adds up the values of $f$ at the lattice points
$\braces{S^{-\top}\braces{m+v}}$, $m\in\ensuremath{\mathbb{Z}}^d$, in the
corner of each parallelepiped $\braces{S^{-\top}\braces{m+v+[0,1]^d}}$
weighted with the volume $\abs{\det S}^{-1}$ of this parallelepiped.
The value $Q_S^v(f)$ hence can be thought of as a Riemann sum of $f$
over $\ensuremath{\mathbb{R}}^d$ with respect to the partition
$\set{S^{-\top}\braces{m+v+[0,1]^d}\mid m\in\ensuremath{\mathbb{Z}}^d}$.\\
Admissible input functions are, for instance, functions $f$
with compact support.
For such functions the above sum is a finite sum.
To integrate $f$, the algorithm $Q_S^v$ uses the nodes $S^{-\top}(m+v)$,
where $m\in\ensuremath{\mathbb{Z}}^d$ is a lattice point in the
compact set $\braces{S^\top\left(\mathop{\mathrm{supp}} f\right)-v}$ of volume
$\braces{\det(S)\cdot \lambda^d\left(\mathop{\mathrm{supp}} f\right)}$.
Here, $\lambda^d$ is the Lebesgue measure in $\ensuremath{\mathbb{R}}^d$.
The indicated volume is the approximate number of function values computed by $Q_S^v$.
In particular, the number of nodes of $Q_{aS}^v$ for growing $a\geq 1$ is of order $a^d$.
The following simple lemma gives an exact upper bound, see~\cite{skriganov} for other bounds.
\begin{lemma}
\label{anlemma}
Suppose $f: \ensuremath{\mathbb{R}}^d \to \ensuremath{\mathbb{R}}$
is supported in an axis-parallel
cube of edge length $l>0$. For any invertible matrix $S\in\ensuremath{\mathbb{R}}^{d\times d}$,
$v\in\ensuremath{\mathbb{R}}^d$ and $a\geq 1$ the quadrature rule $Q_{aS}^v$ uses at most
$
\left(l\cdot\Vert S\Vert_1+1\right)^d\cdot a^d
$
function values of $f$.
\end{lemma}
\begin{proof}
By assumption, $f$ has compact support in $\frac{l}{2}\cdot [-1,1]^d+x_0$
for some $x_0\in\ensuremath{\mathbb{R}}^d$. The number of computed function values is the
number of points $m\in\ensuremath{\mathbb{Z}}^d$ for which $(aS)^{-\top}(m+v)$ is
in $\mathop{\mathrm{supp}} f$ and hence bounded by the size of
\[\begin{split}
M&=\set{m\in\ensuremath{\mathbb{Z}}^d \mid (aS)^{-\top}(m+v)\in \frac{l}{2}\cdot [-1,1]^d+x_0}\\
&= \set{m\in\ensuremath{\mathbb{Z}}^d \mid m+\left(v-aS^\top x_0\right)\in \frac{al}{2}\cdot S^\top [-1,1]^d}
.\end{split}\]
Since $\Vert S^\top x\Vert_\infty\leq \Vert S^\top\Vert_\infty
=\Vert S\Vert_1$ for $x\in[-1,1]^d$,
\[
M \subseteq \set{m\in\ensuremath{\mathbb{Z}}^d \mid m+\left(v-aS^\top x_0\right)\in
\left[-\frac{al}{2}\Vert S\Vert_1,\frac{al}{2}\Vert S\Vert_1\right]^d}
\]
and $\vert M\vert \leq \left(al\Vert S\Vert_1+1\right)^d$.
With $1\leq a$ we get the estimate of Lemma~\ref{anlemma}.
\end{proof}
The error of this algorithm for integration on $\ensuremath{C_c(\IR^d)}$ can be expressed in terms of the Fourier transform.
\begin{lemma}
\label{errorlemma}
For any invertible matrix $S\in\ensuremath{\mathbb{R}}^{d\times d}$, $v\in\ensuremath{\mathbb{R}}^d$ and $f\in C_c(\ensuremath{\mathbb{R}}^d)$
\[
\left| Q_S^v(f)-I_d(f)\right|
\leq \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \left| \mathcal{F}f(Sm)\right|
.\]
\end{lemma}
\begin{proof}
The function $g=f\circ S^{-\top}(\cdot +v)$ is continuous with compact support.
Hence, the Poisson summation formula and an affine linear substitution $x=S^\top y-v$ yield
\[\begin{split}
Q_S^v(f)&=\frac{1}{\abs{\det S}}\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} g(m)
= \frac{1}{\abs{\det S}}\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \mathcal{F}g(m)\\
&= \frac{1}{\abs{\det S}}\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}
\int\limits_{\ensuremath{\mathbb{R}}^d} f\left(S^{-\top}(x+v)\right)\cdot e^{-2\pi i\langle x,m\rangle}
\, \d x \\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \int\limits_{\ensuremath{\mathbb{R}}^d} f\left(y\right)
\cdot e^{-2\pi i\langle S^\top y-v,m\rangle} \, \d y \\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \mathcal{F}f(Sm)\cdot e^{2\pi i\langle v,m\rangle}
,\end{split}\]
if the latter series converges absolutely, see \cite[pp.\,356]{koch}. If not,
the stated inequality is obvious.
This proves the statement,
since $I_d(f)=\mathcal{F}f(S\cdot 0)\cdot e^{2\pi i\langle v,0\rangle}$.
\end{proof}
\newpage
\section{Frolov's Deterministic Algorithm on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$}
\label{frolovsrulesection}
It is known how to choose the matrix $S$ in the rule $Q_S^v$ to get a good
deterministic quadrature rule on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$.
Let the matrix $B\in \ensuremath{\mathbb{R}}^{d\times d}$ satisfy the following three conditions:
\begin{itemize}
\item[(a)] $B$ is invertible,
\item[(b)] $\left|\prod\limits_{j=1}^{d}(Bm)_j\right|\geq 1$,
for any $m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}$,
\item[(c)] For any $x,y\in\ensuremath{\mathbb{R}}^d$ the box $[x,y]$ with
volume $V=\prod\limits_{j=1}^{d}|x_j-y_j|$ contains at most $V+1$
lattice points $Bm$, $m\in\ensuremath{\mathbb{Z}}^d$,
\end{itemize}
where $[x,y]=\set{z\in\ensuremath{\mathbb{R}}^d\mid z_j
\text{ is inbetween of }x_j\text{ and }y_j\text{ for }j=1,\dots,d}$.
Such a matrix shall be called a \textit{Frolov matrix}.
Property (b) says that for $n>0$ every point of the lattice $n^{1/d}B\ensuremath{\mathbb{Z}}^d$
but zero lies in the set $D_n$ of all vectors $x\in\ensuremath{\mathbb{R}}^d$
with $\prod_{j=1}^{d}\abs{x_j}\geq n$,
the complement of a hyperbolic cross.
\begin{minipage}[h!]{.6\linewidth}
\includegraphics[width=.95\linewidth]{hyperboliccross-eps-converted-to}
\end{minipage}
\begin{minipage}[h!]{.35\linewidth}
This graphic shows the lattice $n^{1/d}B\ensuremath{\mathbb{Z}}^d$ for $d=2$, $n=9$ and the Frolov matrix
\[B=\begin{pmatrix}
1 & 2-\sqrt{2}\\
1 & 2+\sqrt{2}
\end{pmatrix}
.\]
Except zero, every lattice point lies inside $D_9$.
\end{minipage}
It is known that one can construct such a matrix $B$ in the following way.
Let $p\in\ensuremath{\mathbb{Z}}[x]$ be a polynomial of degree $d$ with leading
coefficient 1 which is irreducible over $\ensuremath{\mathbb{Q}}$ and has $d$ different
real roots $\zeta_1,\hdots,\zeta_d$. Then the matrix
\[B=\left(\zeta_i^{j-1}\right)_{i,j=1}^d\]
has the desired properties, as shown in \cite[p.\,364]{temlyakovbuch} and \cite{ullrich}.
In arbitrary dimension $d$ we can choose $p(x)=(x-1)(x-3)\cdot\hdots\cdot(x-2d+1)-1$,
see \cite{frolov} or \cite{ullrich}, but there are many other possible choices.
For example, if $d$ is a power of two, we can set $p(x)=2\cos\left(d\cdot\arccos
(x/2)\right)=2\,T_d(x/2)$, where $T_d$ is the Chebyshev
polynomial of degree $d$, see \cite[p.\,365]{temlyakovbuch}.
Then the roots of $p$ are explicitly given by $\zeta_j
=2\cos\left(\frac{2j-1}{2d}\pi\right)$ for $j=1,\hdots,d$.\\
From now on, let $B$ be an arbitrary but fixed, $d$-dimensional Frolov matrix.
Constants may depend on the choice of $B$.
\medskip
\noindent
{\bf Algorithm.} \
For any natural number $n$, we consider the quadrature rule
$Q_{n^{1/d} B}$ from Section \ref{basicquadrulesection} with shift parameter zero.
This deterministic algorithm is usually referred to as \textit{Frolov's algorithm}.
\medskip
For input functions $f$ with support in $[0,1]^d$ the number of function values computed by $Q_{n^{1/d}B}$
is of order $n$. To be precise, Lemma \ref{anlemma} says that $Q_{n^{1/d}B}$ uses at most
$\left(\Vert B\Vert_1+1\right)^d\cdot n$ function values of $f$.
K.\,K.\,Frolov has already seen in 1976 that the algorithm $Q_{n^{1/d}B}$
is optimal on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ in the sense of order of convergence.
It satisfies the following error bound.
\begin{thm}
\label{frolovboundtheorem}
There is some $c>0$ such that for
every $n\geq 2$ and $f\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[
\abs{Q_{n^{1/d}B}(f)-I_d(f)} \leq\, c \, n^{-r}
\, (\log n)^\frac{d-1}{2} \, \mixnormunit{f}
.\]
\end{thm}
See also \cite{frolov} and \cite{ullrich} or my Bachelor thesis for a proof of this error
bound and its optimality.
In fact, this error bound holds uniformly for $Q_{n^{1/d}\hat{u}B}^v$ for any $u\in[1,2^{1/d}]^d$ and $v\in[0,1]^d$,
which is the statement of Theorem~\ref{mixthmworstcase} in Section~\ref{randomdilationsection1}.
Theorem \ref{frolovboundtheorem} is only a special case.\\
But Frolov's algorithm is also optimal among deterministic quadrature rules
on $\ensuremath{\mathring{H}^s([0,1]^d)}$ in the sense of order of convergence.
It satisfies:
\begin{thm}
\label{frolovboundisotheorem}
There is some $c>0$ such that for
every $n\geq 2$ and $f\in \ensuremath{\mathring{H}^s([0,1]^d)}$
\[
\abs{Q_{n^{1/d}B}(f)-I_d(f)} \leq\, c \, n^{-s/d}
\, \isonormunit{f}
.\]
\end{thm}
This is a special case of Theorem~\ref{isothmworstcase} in Section~\ref{randomdilationsection1}.
See \cite{ln} for a proof of the optimality of this order.
\newpage
\section{The Effect of Random Dilations}
\label{randomdilationsection}
We study the impact of random dilations on Frolov's algorithm $Q_{n^{1/d}B}$.
\medskip
\noindent
{\bf Algorithm.} \
For any natural number $n$ and shift parameter $v\in\ensuremath{\mathbb{R}}^d$
we consider the method $Q_{n^{1/d}\hat{U}B}^v$
from Section \ref{basicquadrulesection}
with a dilation parameter $U$ that is uniformly distributed in the box $[1,2^{1/d}]^d$.
\medskip
For input functions $f$ from $\ensuremath{\mathring{H}^s([0,1]^d)}$ or $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ the
information cost of $Q_{n^{1/d}\hat{U}B}^v$ is roughly between $\det(B) \cdot n$
and $2\cdot\det(B) \cdot n$. More precisely, it uses at most
$2\cdot\left(\Vert B\Vert_1+1\right)^d\cdot n$ function values of $f$.\\
\subsection{Worst Case Errors}
\label{randomdilationsection1}
In the worst case, the error of this method has the same order
of convergence like Frolov's algorithm, both for $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
and $\ensuremath{\mathring{H}^s([0,1]^d)}$.
\begin{thm}
\label{mixthmworstcase}
There is a constant $c>0$ such that for any shift parameter $v\in\ensuremath{\mathbb{R}}^d$,
$n\geq 2$ and $f\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[\begin{split}
\sup\limits_{u\in [1,2^{1/d}]^d} \abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)} \leq\, c \, n^{-r}
\, (\log n)^\frac{d-1}{2} \, \mixnormunit{f}
.\end{split}\]
\end{thm}
\begin{proof}
Let $Q_{n^{1/d}\hat{u}B}^v$ be an arbitrary realization
of the algorithm $Q_{n^{1/d}\hat{U}B}^v$ under consideration.
By Lemma~\ref{errorlemma} and Hölder's inequality,
\[\begin{split}
&\abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)}^2
\leq \left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \left|
\mathcal{F}f(n^{1/d}\hat{u}Bm)\right|\right)^2\\
&\leq \left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} h_r(n^{1/d}\hat{u}Bm)^{-1}\right)
\cdot\left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} h_r(n^{1/d}\hat{u}Bm)\cdot
\abs{\mathcal{F}f(n^{1/d}\hat{u}Bm)}^2\right)
.\end{split}\]
We first prove that the first factor in this product is bounded above by a constant multiple of $n^{-2r}
\, (\log n)^{d-1}$.\\
Consider the auxiliary set $N(\beta)=\{x\in\ensuremath{\mathbb{R}}^d \mid \lfloor 2^{\beta_j-1}\rfloor \leq|x_j|<2^{\beta_j},1\leq j\leq d\}$
for $\beta\in\ensuremath{\mathbb{N}}_0^d$ and $G_n^\beta=\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid n^{1/d}\hat{u}Bm\in N(\beta)}$.
The domain $\ensuremath{\mathbb{Z}}^d\setminus\set{0}$ of summation is the disjoint union of all $G_n^\beta$ over $\beta\in\ensuremath{\mathbb{N}}_0^d$.\\
For $|\beta|\leq \log_2 n$, the points $x$ in $N(\beta)$ satisfy
$\prod_{j=1}^{d}\abs{x_j}<2^{\abs{\beta}}\leq n$. But the second property of the Frolov matrix $B$ yields
$\prod_{j=1}^d \abs{n^{1/d}u_j(Bm)_j}\geq n$ for any $m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}$.
Hence, $G_n^\beta$ is empty for $|\beta|\leq \log_2 n$.
For $\abs{\beta} >\log_2 n$, any $m\in G_n^\beta$ satisfies
\[
h_r(n^{1/d}\hat{u}Bm)\geq \prod_{j=1}^d \left(1+\lfloor 2^{\beta_j-1}\rfloor^{2r}\right)
\geq \prod_{j=1}^d 2^{2r(\beta_j-1)} = 2^{2r(|\beta|-d)}
\]
and hence $h_r(n^{1/d}\hat{u}Bm)^{-1}\leq 2^{2r(d-|\beta|)}$.
Because of the third property of the Frolov matrix, we obtain
\[\begin{split}
\abs{G_n^\beta} \leq \abs{\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid \abs{(Bm)_j}
< \frac{2^{\beta_j}}{n^{1/d}}}} \leq 2^{d+\abs{\beta}} n^{-1} +1 \leq 2^{d+1+\abs{\beta}}
n^{-1}
.\end{split}\]
That yields
\[\begin{split}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} &h_r\left(n^{1/d}\hat{u}Bm\right)^{-1}\\
&= \sum\limits_{\beta\in\ensuremath{\mathbb{N}}_0^d} \sum\limits_{m\in G_n^\beta} h_r(n^{1/d}\hat{u}Bm)^{-1}\\
&= \sum\limits_{|\beta|>\log_2 n}\, \sum\limits_{m\in G_n^\beta} h_r(n^{1/d}\hat{u}Bm)^{-1}\\
&\overset{\star}{\leq} \sum\limits_{|\beta|>\log_2 n}\, 2^{2r(d-|\beta|)}
\cdot n^{-1}\cdot 2^{d+1+|\beta|}\\
&= \sum\limits_{k=\lceil \log_2 n\rceil}^\infty\, 2^{2r(d-k)}
\cdot n^{-1}\cdot 2^{d+1+k}\cdot \abs{\set{\beta\in\ensuremath{\mathbb{N}}_0^d\mid |\beta|=k}}\\
&\leq 2^{2rd+d+1} \cdot n^{-1} \sum\limits_{k=\lceil \log_2 n\rceil}^{\infty}
2^{(1-2r)k}\cdot (k+1)^{d-1}\\
&= 2^{2rd+d+1} \cdot n^{-1}
\sum\limits_{k=0}^{\infty} 2^{(1-2r)(k+\lceil \log_2 n\rceil)}\cdot
\left(k+1+\lceil \log_2 n\rceil\right)^{d-1}\\
&\leq 2^{2rd+d+1} \cdot n^{-1} \cdot n^{1-2r}
\cdot \sum\limits_{k=0}^{\infty} 2^{(1-2r)k}\cdot 2^{d-1}
\cdot (k+1)^{d-1}\cdot\lceil \log_2 n\rceil^{d-1}\\
&\leq 2^{2rd+2d} \cdot n^{-2r}
\cdot \sum\limits_{k=0}^{\infty} 2^{(1-2r)k} (k+1)^{d-1}
\left( 2\cdot\frac{\log n}{\log 2}\right)^{d-1}\\
&= \left( 2^{2rd+3d-1}\, (\log 2)^{1-d}
\sum\limits_{k=0}^{\infty} \left(2^{1-2r}\right)^k (k+1)^{d-1} \right)
\cdot n^{-2r}\, (\log n)^{d-1}
.\end{split}\]
This is the desired estimate, since $2^{1-2r}<1$.\\
We now show that the second factor in the above inequality is bounded above by a constant
multiple of $\mixnormunit{f}^2$. This proves the theorem.
For $x\in\ensuremath{\mathbb{R}}^d$ we have
\[
h_r(x)\cdot \abs{\mathcal{F}f(x)}^{2}
=\sum\limits_{\alpha\in\set{0,\dots,r}^d}\abs{\mathcal{F}D^\alpha f(x)}^2
.\]
The function $g_\alpha=D^\alpha f\circ(n^{1/d}\hat{u}B)^{-\top}$ has compact support
in the parallelepiped $(n^{1/d}\hat{u}B)^\top[0,1]^d$.
Consider the set
$J_n$ of all $k\in\ensuremath{\mathbb{Z}}^d$ for which $\braces{k+[0,1]^d}$ has a nonempty intersection
with $(n^{1/d}\hat{u}B)^\top[0,1]^d$. Then
\[\begin{split}
\abs{\mathcal{F}D^\alpha f \braces{n^{1/d}\hat{u}Bm}}^2
&= \left| \int_{\ensuremath{\mathbb{R}}^d} D^\alpha f(y)\cdot
e^{-2\pi i \langle n^{1/d}\hat{u}Bm,y\rangle} \d y\right|^2\\
&= \left|\frac{1}{\det(n^{1/d}\hat{u}B)}\int_{\ensuremath{\mathbb{R}}^d} g_\alpha(x)\cdot
e^{-2\pi i\langle m,x\rangle} \d x\right|^2\\
&= \left|\frac{1}{\det(n^{1/d}\hat{u}B)}\sum\limits_{k\in J_n} \langle g_\alpha(x),
e^{2\pi i\langle m,\cdot\rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2\\
&\leq \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{k\in J_n}
\left|\langle g_\alpha,e^{2\pi i\langle m,\cdot \rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2
.\end{split}\]
Thus we obtain
\[\begin{split}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} &h_r(n^{1/d}\hat{u}Bm)\cdot \abs{\mathcal{F}f(n^{1/d}\hat{u}Bm)}^2
\leq \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \sum\limits_{\alpha\in\set{0,\dots,r}^d}
\abs{\mathcal{F}D^\alpha f(n^{1/d}\hat{u}Bm)}^2\\
&\leq \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}
\sum\limits_{\alpha\in\set{0,\dots,r}^d}
\sum\limits_{k\in J_n} \left|\langle g_\alpha,e^{2\pi i\langle m,
\cdot \rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{\alpha\in\set{0,\dots,r}^d}
\sum\limits_{k\in J_n} \Vert g_\alpha\Vert_{L^2\left(k+[0,1]^d\right)}^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2}
\sum\limits_{\alpha\in\set{0,\dots,r}^d} \lnorm{g_\alpha}^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}} \sum\limits_{\alpha\in\set{0,\dots,r}^d}
\lnorm{D^\alpha f}^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}} \mixnormunit{f}^2
.\end{split}\]
Since both $\abs{J_n}$ and $\abs{\det(n^{1/d}\hat{u}B)}$ are of order $n$, their ratio is bounded
by a constant and the above inequality yields the statement.
\end{proof}
\begin{thm}
\label{isothmworstcase}
There is a constant $c>0$ such that for any shift parameter $v\in\ensuremath{\mathbb{R}}^d$,
$n\in\ensuremath{\mathbb{N}}$ and $f\in \ensuremath{\mathring{H}^s([0,1]^d)}$
\[\begin{split}
\sup\limits_{u\in [1,2^{1/d}]^d} \abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)} \leq\, c \, n^{-s/d}
\, \isonormunit{f}
.\end{split}\]
\end{thm}
\begin{proof}
Let $Q_{n^{1/d}\hat{u}B}^v$ be an arbitrary realization
of the algorithm $Q_{n^{1/d}\hat{U}B}^v$ under consideration.
By Lemma~\ref{errorlemma} and Hölder's inequality,
\[\begin{split}
&\abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)}^2
\leq \left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}
\left| \mathcal{F}f\braces{n^{1/d}\hat{u}Bm}\right|\right)^2\\
&\leq \left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} v_s\braces{n^{1/d}\hat{u}Bm}^{-1}\right)
\cdot\left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} v_s\braces{n^{1/d}\hat{u}Bm}
\cdot \abs{\mathcal{F}f\braces{n^{1/d}\hat{u}Bm}}^2\right)
.\end{split}\]
The first factor in this product is bounded above by a constant multiple of
$n^{-2s/d}$: Since
\[
v_s\braces{n^{1/d}\hat{u}Bm} \geq \Vert n^{1/d}\hat{u}Bm\Vert_2^{2s}
\geq n^{2s/d} \cdot \Vert Bm\Vert_2^{2s}
\geq n^{2s/d}\cdot \Vert B^{-1}\Vert_2^{-2s}\cdot
\Vert m\Vert_2^{2s}
,\]
we have
\[
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} v_s\braces{n^{1/d}\hat{u}Bm}^{-1}
\leq n^{-2s/d}\cdot \Vert B^{-1}\Vert_2^{2s}\cdot
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}\Vert m\Vert_2^{-2s}
,\]
where this last series converges for $2s>d$.
We show that the second factor in the above inequality is bounded above
by a constant multiple of $\isonormunit{f}^2$. This proves the theorem.
For any $x\in\ensuremath{\mathbb{R}}^d$ we have
\[
v_s(x)\cdot \abs{\mathcal{F}f(x)}^{2}
=\sum\limits_{\abs{\alpha}\leq s}\abs{\mathcal{F}D^\alpha f(x)}^2
.\]
The function $g_\alpha=D^\alpha f\circ(n^{1/d}\hat{u}B)^{-\top}$ has compact support
in the parallelepiped $(n^{1/d}\hat{u}B)^\top[0,1]^d$.
Again consider the set
$J_n$ of all $k\in\ensuremath{\mathbb{Z}}^d$ for which $\braces{k+[0,1]^d}$ has a nonempty intersection
with $(n^{1/d}\hat{u}B)^\top[0,1]^d$.\\
We have the estimate
\[\begin{split}
\abs{\mathcal{F}D^\alpha f\braces{n^{1/d}\hat{u}Bm}}^2
&= \left| \int_{\ensuremath{\mathbb{R}}^d} D^\alpha f(y)\cdot
e^{-2\pi i \langle n^{1/d}\hat{u}Bm,y\rangle} \d y\right|^2\\
&= \left|\frac{1}{\det(n^{1/d}\hat{u}B)}\int_{\ensuremath{\mathbb{R}}^d} g_\alpha(x)\cdot
e^{-2\pi i\langle m,x\rangle} \d x\right|^2\\
&= \left|\frac{1}{\det(n^{1/d}\hat{u}B)}\sum\limits_{k\in J_n}
\langle g_\alpha(x),e^{2\pi i\langle m,\cdot \rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2\\
&\leq \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{k\in J_n}
\left|\langle g_\alpha,e^{2\pi i\langle m,\cdot \rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2
.\end{split}\]
Thus we obtain
\[\begin{split}
&\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} v_s\braces{n^{1/d}\hat{u}Bm}\cdot \abs{\mathcal{F}f\braces{n^{1/d}\hat{u}Bm}}^2
\leq \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \sum\limits_{\abs{\alpha}\leq s}
\abs{\mathcal{F}D^\alpha f\braces{n^{1/d}\hat{u}Bm}}^2\\
&\leq \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}
\sum\limits_{\abs{\alpha}\leq s}\,
\sum\limits_{k\in J_n} \left|\langle g_\alpha,e^{2\pi i\langle m,
\cdot \rangle}\rangle_{L^2\left(k+[0,1]^d\right)}\right|^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{\abs{\alpha}
\leq s}\,
\sum\limits_{k\in J_n} \Vert g_\alpha\Vert_{L^2\left(k+[0,1]^d\right)}^2
= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}^2} \sum\limits_{\abs{\alpha}
\leq s}
\lnorm{g_\alpha}^2\\
&= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}} \sum\limits_{\abs{\alpha}
\leq s}
\lnorm{D^\alpha f}^2
= \frac{\abs{J_n}}{\abs{\det(n^{1/d}\hat{u}B)}} \isonormunit{f}^2
.\end{split}\]
Since both $\abs{J_n}$ and $\abs{\det(n^{1/d}\hat{u}B)}$ are of order $n$, their ratio is bounded
by a constant and the above inequality yields the statement.
\end{proof}
\subsection{Expected Errors}
\label{randomdilationsection2}
In expectation, the random dilations improve the order of the error
of Frolov's algorithm by $1/2$ for
both $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ and $\ensuremath{\mathring{H}^s([0,1]^d)}$.
These results are based on the following general error bound for
continuous functions with compact support. Recall that
$D_n$ is the set of all $x\in\ensuremath{\mathbb{R}}^d$
with $\prod_{j=1}^{d}\abs{x_j}\geq n$.
\begin{thm}
\label{keyprop}
There is a constant $c>0$ such that for every $n\in\ensuremath{\mathbb{N}}$, shift parameter $v\in\ensuremath{\mathbb{R}}^d$ and $f\in\ensuremath{C_c(\IR^d)}$
\[\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^v(f)-I_d(f)} \leq\, c \, n^{-1} \cdot \int_{D_n} \abs{\mathcal{F}f(x)}\, \d x
.\]
\end{thm}
\begin{proof}
Thanks to Lemma~\ref{errorlemma}
and the monotone convergence theorem we have
\[\begin{split}
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^v(f)-I_d(f)}
&\leq \ensuremath{\mathbb{E}}
\left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \abs{\mathcal{F}f\braces{n^{1/d}\hat{U}Bm}} \right)\\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \ensuremath{\mathbb{E}} \abs{\mathcal{F}f\braces{n^{1/d}\hat{U}Bm}}
.\end{split}\]
Since each $n^{1/d}\hat{U}Bm$ is uniformly distributed in the
box $[n^{1/d}Bm,(2n)^{1/d}Bm]$ with volume
$\left(2^{1/d}-1\right)^d\cdot\abs{\prod_{j=1}^dn^{1/d} (Bm)_j}$, this series equals
\[\begin{split}
&\frac{1}{\left(2^{1/d}-1\right)^d} \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}\,
\int\limits_{[n^{1/d}Bm,(2n)^{1/d}Bm]} \frac{\abs{\mathcal{F}f(x)}}{\prod_{j=1}^d
\abs{n^{1/d}(Bm)_j}} \, \d x \\
&\leq \frac{1}{\left(2^{1/d}-1\right)^d}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}\, \int\limits_{[n^{1/d}Bm,(2n)^{1/d}Bm]}
\frac{\abs{\mathcal{F}f(x)}}{\prod_{j=1}^d 2^{-1/d}\abs{x_j}} \, \d x\\
&= \frac{2}{\left(2^{1/d}-1\right)^d}\cdot \int_{\ensuremath{\mathbb{R}}^d}
\frac{\abs{\mathcal{F}f(x)}}{\prod_{j=1}^d \abs{x_j}}\cdot
\abs{\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid x\in [n^{1/d}Bm,(2n)^{1/d}Bm]}} \, \d x
\end{split}\]
\[\begin{split}
&= \frac{2}{\left(2^{1/d}-1\right)^d}\cdot \int_{\ensuremath{\mathbb{R}}^d}
\frac{\abs{\mathcal{F}f(x)}}{\prod_{j=1}^d \abs{x_j}}\cdot
\abs{\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid Bm\in
\left[\frac{x}{(2n)^{1/d}},\frac{x}{n^{1/d}}\right]}}\, \d x
.\end{split}\]
Thanks to the properties of the Frolov matrix $B$, if $\prod_{j=1}^d \abs{x_j}<n$,
the latter set is empty and otherwise contains no more
than $\prod_{j=1}^d \abs{\frac{x_j}{n^{1/d}}}+1\leq 2 n^{-1} \prod_{j=1}^d \abs{x_j}$ points.
Thus, we arrive at
\[
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^v(f)-I_d(f)} \leq
\frac{4}{\left(2^{1/d}-1\right)^d}\cdot n^{-1} \int_{D_n} \abs{\mathcal{F}f(x)} \, \d x
\]
and the theorem is proven.
\end{proof}
Additional differentiability properties of the function $f\in\ensuremath{C_c(\IR^d)}$ result in decay
properties of its Fourier transform $\mathcal{F}f$. This leads to estimates of the
integral $\int_{D_n} \abs{\mathcal{F}f(x)} \, \d x$.
Hence, the general upper bound for the error of $Q_{n^{1/d}\hat{U}B}^v(f)$ in Theorem
\ref{keyprop} adjusts to the differentiability of $f$.
Two such examples are functions from $\ensuremath{\mathring{H}^{r,{\rm mix}}(\IR^d)}$ and $\ensuremath{\mathring{H}^s(\IR^d)}$.
\begin{lemma}
\label{intlemmamix}
There is some $c>0$ such that
for each $n\geq 2$ and $f\in\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[
\int_{D_n} \abs{\mathcal{F}f(x)} \, \d x \leq c \, n^{-r+1/2}
\, \left(\log n\right)^{\frac{d-1}{2}} \, \mixnormunit{f}
.\]
\end{lemma}
\begin{proof}
Applying Hölder's inequality and a linear substitution $x=n^{1/d}By$
to the above integral, we get
\[\begin{split}
&\left( \int_{D_n} \abs{\mathcal{F}f(x)} \, \d x \right)^2
= \left( \int_{D_n} h_r(x)^{-1/2} \cdot
\abs{\mathcal{F}f(x)} h_r(x)^{1/2}\, \d x \right)^2\\
&\leq \braces{\int_{D_n} h_r(x)^{-1} \, \d x} \mixnorm{f}^2
= n \abs{\det B} \left(\int_{G} h_r(n^{1/d}By)^{-1} \, \d y \right) \mixnorm{f}^2
\end{split}\]
with $G=B^{-1}D_1$ being the set of all $y\in\ensuremath{\mathbb{R}}^d$ with
$\prod_{j=1}^{d}\abs{(By)_j}\geq 1$. It it thus sufficient to prove that the
integral $\int_{G} h_r(n^{1/d}By)^{-1} \, \d y$ is bounded by a constant
multiple of $n^{-2r}\, \left(\log n\right)^{d-1}$.\\
Again consider the auxiliary set $N(\beta)=\{x\in\ensuremath{\mathbb{R}}^d \mid \left[2^{\beta_j-1}\right]
\leq|x_j|<2^{\beta_j},1\leq j\leq d\}$ for $\beta\in\ensuremath{\mathbb{N}}_0^d$
and the set $G_n^\beta=\set{y\in G\mid n^{1/d}By\in N(\beta)}$.
Similar to the proof of Theorem \ref{mixthmworstcase},
the domain $G$ of integration is the disjoint union of all $G_n^\beta$ over $\beta\in\ensuremath{\mathbb{N}}_0^d$,
where $G_n^\beta=\emptyset$, if $\abs{\beta}\leq \log_2 n$, and otherwise
the integrand is bounded above by $2^{2r(d-\abs{\beta})}$ for $y\in G_n^\beta$.
On the other hand,
\[\begin{split}
&\lambda^d(G_n^\beta)\leq \lambda^d\left((n^{1/d}B)^{-1}N(\beta)\right)
= n^{-1}\cdot |\det B|^{-1}\cdot \lambda^d(N(\beta)) \\
&= n^{-1}\cdot |\det B|^{-1}\cdot 2^d \cdot \prod_{j=1}^d\left(2^{\beta_j}-
\left[2^{\beta_j-1}\right]\right)
\leq n^{-1}\cdot |\det B|^{-1}\cdot 2^d\cdot 2^{|\beta|}
.\end{split}\]
Like in the proof of Theorem \ref{mixthmworstcase}, we obtain
\[\begin{split}
&\int\limits_{G} h_r(n^{1/d}By)^{-1} \, \d y
= \sum\limits_{\beta\in\ensuremath{\mathbb{N}}_0^d}\, \int_{G_n^\beta} h_r(n^{1/d}By)^{-1} \, \d y \\
&= \sum\limits_{|\beta|>\log_2 n}\ \int_{G_n^\beta} h_r(n^{1/d}By)^{-1} \, \d y \\
&\leq \sum\limits_{|\beta|>\log_2 n} 2^{2r(d-|\beta|)}
\cdot n^{-1}\cdot |\det B|^{-1}\cdot 2^{d+\beta}\\
&= |\det B|^{-1}\cdot 2^{-1}\cdot \sum\limits_{|\beta|>\log_2 n} 2^{2r(d-|\beta|)}
\cdot n^{-1}\cdot 2^{d+1+\abs{\beta}}\\
&\overset{\star}{\leq}
\left( 2^{2rd+3d-2} |\det B|^{-1} (\log 2)^{1-d}
\sum\limits_{k=0}^{\infty} \left(2^{1-2r}\right)^k (k+1)^{d-1} \right)
\, n^{-2r}\, (\log n)^{d-1}
,\end{split}\]
where the constant is finite, since $2^{1-2r}<1$.
\end{proof}
Combining Theorem~\ref{keyprop} and Lemma~\ref{intlemmamix} yields:
\begin{thm}
\label{mixthm}
There is a constant $c>0$ such that
for every $n\geq 2$, shift parameter $v\in\ensuremath{\mathbb{R}}^d$ and $f\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^v(f)-I_d(f)} \leq\, c \, n^{-r-1/2}
\, (\log n)^\frac{d-1}{2} \, \mixnormunit{f}
.\]
\end{thm}
If, however, the integrand is from the space $\ensuremath{\mathring{H}^s([0,1]^d)}\subseteq\ensuremath{C_c(\IR^d)}$,
the following lemma holds.
\begin{lemma}
\label{intlemmaiso}
There is some $c>0$
such that for each $n\in\ensuremath{\mathbb{N}}$ and $f\in\ensuremath{\mathring{H}^s([0,1]^d)}$
\[
\int_{D_n} \abs{\mathcal{F}f(x)} \, \d x \leq c \, n^{-s/d+1/2} \, \isonormunit{f}
.\]
\end{lemma}
\begin{proof}
Like in Lemma~\ref{intlemmamix}, we apply Hölder's inequality and get
\[\begin{split}
&\left( \int_{D_n} \abs{\mathcal{F}f(x)} \, \d x \right)^2
=\left( \int_{D_n} \abs{\mathcal{F}f(x)} v_s(x)^{1/2}
\cdot v_s(x)^{-1/2} \, \d x \right)^2\\
&\leq \left(\int_{D_n} v_s(x)^{-1} \, \d x \right)\cdot \isonorm{f}^2
\leq \tilde{c}\cdot \left(\int_{D_n} \left(1+\Vert x\Vert_2^2
\right)^{-s} \, \d x \right)\cdot \isonormunit{f}^2
,\end{split}\]
for some $\tilde{c}>0$.
Since $\Vert x\Vert_2\geq \max\set{\abs{x_j}\mid j=1,\dots,n}\geq n^{1/d}$ for $x\in D_n$,
the set $D_n$ is a subset of $\set{x\in\ensuremath{\mathbb{R}}^d:\,\Vert x\Vert_2\geq n^{1/d}}$
and the latter integral in the above integral is less than
\[\begin{split}
&\int\limits_{\set{x\in\ensuremath{\mathbb{R}}^d:\,\Vert x\Vert_2\geq n^{1/d}}}
\left(1+\Vert x\Vert_2^2\right)^{-s} \, \d x
= \int\limits_{n^{1/d}}^{\infty}\int\limits_{S_{d-1}}
\braces{1+R^2}^{-s}\cdot R^{d-1} \, \d \sigma \, \d R \\
&= \sigma\left(S_{d-1}\right) \int\limits_{n^{1/d}}^{\infty}
\braces{1+R^2}^{-s}\cdot R^{d-1} \, \d R
\leq \sigma\left(S_{d-1}\right) \int\limits_{n^{1/d}}^{\infty} R^{-2s+d-1} \, \d R
\leq \hat{c} \cdot n^{-2s/d+1}
,\end{split}\]
for some $\hat{c}>0$, since $-2s+d-1<-1$.
\end{proof}
In this case, combining Theorem~\ref{keyprop} and Lemma~\ref{intlemmaiso} yields:
\begin{thm}
\label{isothm}
There is a constant $c>0$ such that for
every $n\in\ensuremath{\mathbb{N}}$, shift parameter $v\in\ensuremath{\mathbb{R}}^d$ and $f\in \ensuremath{\mathring{H}^s([0,1]^d)}$
\[\begin{split}
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^v(f)-I_d(f)} \leq\, c \, n^{-s/d-1/2} \, \isonormunit{f}
.\end{split}\]
\end{thm}
We remark that the Frolov properties of the matrix $B$ are not needed to get this
estimate on $\ensuremath{\mathring{H}^s([0,1]^d)}$, although they are essential for the upper bound
on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ from Theorem~\ref{mixthm}. As seen in the proof of Lemma~\ref{intlemmaiso}
and contrarily to Lemma~\ref{intlemmamix},
we do not need that the lattice points of $n^{1/d}B\ensuremath{\mathbb{Z}}^d\setminus\set{0}$
lie in $D_n$, but only that they lie in the
bigger set $\set{x\in\ensuremath{\mathbb{R}}^d\mid \Vert x\Vert_2\geq n^{1/d}}$. For example, the
identity matrix would do. But if $B$ is a Frolov matrix, $Q_{n^{1/d}\hat{u}B}^v$ works
universally for $\ensuremath{\mathring{H}^{r,{\rm mix}}(\IR^d)}$ and $\ensuremath{\mathring{H}^s(\IR^d)}$. Furthermore, the Frolov properties
of $B$ prevent extremely large jumps of the number of nodes of
$Q_{n^{1/d}\hat{u}B}^v=Q_{n^{1/d}\hat{u}B}^v$ for small changes of
the dilation parameter $u\in[1,2^{1/d}]^d$.
\newpage
\section{Further Improvements through Random Shifts}
\label{randomshiftsection}
Now we also choose the
shift parameter $v$ in $Q_{n^{1/d}\hat{U}B}^v$ at random.
\medskip
\noindent
{\bf Algorithm.} \
For any natural number $n$
we consider the method $Q_{n^{1/d}\hat{U}B}^V$
from Section \ref{basicquadrulesection}
with independent dilation parameter $U$, uniformly distributed in $[1,2^{1/d}]^d$,
and shift parameter $V$, uniformly distributed in $[0,1]^d$.
\medskip
For input functions $f$ from $\ensuremath{\mathring{H}^s([0,1]^d)}$ or $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ the
information cost of the method $Q_{n^{1/d}\hat{U}B}^V$ is again of order $n$.\\
The first advantage of this method is its unbiasedness.
\begin{prop}
\label{Munbiased}
Let $S\in\ensuremath{\mathbb{R}}^{d\times d}$ be an
invertible matrix. For any $f\in L^1\braces{\ensuremath{\mathbb{R}}^d}$, the method $Q_S^V$ satisfies
\[
\ensuremath{\mathbb{E}} \braces{Q_S^V(f)}=I_d(f)
.\]
In particular, the method $Q_{n^{1/d}\hat{U}B}^V$ is well-defined and unbiased on $L^1\braces{\ensuremath{\mathbb{R}}^d}$.
\end{prop}
\begin{proof}
By the monotone convergence theorem,
\[\begin{split}
&\ensuremath{\mathbb{E}} \braces{\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \frac{1}{\abs{\det S}}
\abs{f\left( S^{-\top}(m+V) \right)}}
=\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \ensuremath{\mathbb{E}} \braces{\frac{1}{\abs{\det S}}
\abs{f\left( S^{-\top}(m+V) \right)}} \\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \frac{1}{\abs{\det S}}
\int_{[0,1]^d} \abs{f\left( S^{-\top}(m+x) \right)} \, \d x \\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}\ \int_{S^{-\top}\braces{m+[0,1]^d}}
\abs{f(y)} \, \d y
= \int_{\ensuremath{\mathbb{R}}^d} \abs{f(y)} \, \d y\,
< \infty
.\end{split}\]
The series $Q_S^V(f)= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \frac{1}{\abs{\det S}}
f\left( S^{-\top}(m+V) \right)$ hence converges absolutely almost surely
and is dominated by the integrable function
$\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \frac{1}{\abs{\det S}}
\abs{f\left( S^{-\top}(m+V) \right)}$.
We can thus apply Lebesgue's dominated convergence theorem to get
\[\begin{split}
\ensuremath{\mathbb{E}} \braces{Q_{S}^V(f)}
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \frac{1}{\abs{\det S}}
\int_{[0,1]^d} f\left( S^{-\top}(m+x) \right) \, \d x\\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}\ \int_{S^{-\top}\braces{m+[0,1]^d}}
f(y) \, \d y
= \int_{\ensuremath{\mathbb{R}}^d} f(y) \, \d y
= I_d(f)
,\end{split}\]
for the expected value of the general algorithm at $f$.\\
For the method $Q_{n^{1/d}\hat{U}B}^V$, Fubini's theorem yields
\[\begin{split}
\ensuremath{\mathbb{E}} \left(Q_{n^{1/d}\hat{U}B}^V(f)\right)
= \ensuremath{\mathbb{E}}_U \ensuremath{\mathbb{E}}_V \left(Q_{n^{1/d}\hat{U}B}^V(f)\right)
= \ensuremath{\mathbb{E}}_U \braces{I_d(f)}
= I_d(f)
,\end{split} \]
as claimed. In particular, $Q_{n^{1/d}\hat{U}B}^V(f)$ is almost surely absolutely convergent.
\end{proof}
The worst case error of this method, too, has
the order $n^{-r}\braces{\log n}^{(d-1)/2}$ on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
and $n^{-s/d}$ on $\ensuremath{\mathring{H}^s([0,1]^d)}$. This is a direct consequence
of Theorem~\ref{mixthmworstcase} and Theorem~\ref{isothmworstcase}.
\begin{cor}
\label{mixthmworstcase2}
There is some $c>0$ such that for any
$n\geq 2$ and $f\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[\begin{split}
\sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)} \leq\, c \, n^{-r}
\, (\log n)^\frac{d-1}{2} \, \mixnormunit{f}
.\end{split}\]
\end{cor}
\begin{cor}
\label{isothmworstcase2}
There is some $c>0$ such that for any
$n\in\ensuremath{\mathbb{N}}$ and $f\in \ensuremath{\mathring{H}^s([0,1]^d)}$
\[\begin{split}
\sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{Q_{n^{1/d}\hat{u}B}^v(f)-I_d(f)} \leq\, c \, n^{-s/d}
\, \isonormunit{f}
.\end{split}\]
\end{cor}
The second advantage of this method is the slightly better
convergence order of its expected error on $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$.
As proven by Mario Ullrich in \cite{ullrichneu},
the expected error $\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}$
of $Q_{n^{1/d}\hat{U}B}^V$ in $f\in\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ is bounded
above by a constant multiple of $n^{-r-1/2} \, \mixnormunit{f}$
instead of a constant multiple of
$n^{-r-1/2}\,(\log n)^\frac{d-1}{2} \, \mixnormunit{f}$.
The proof even shows that the quantity $\braces{\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2}$
satisfies this bound.
This is a stronger statement, as implied by Hölder's inequality.\\
In Lemma~\ref{errorlemma}, the absolute error of $Q_S^v$ for integration
on $\ensuremath{C_c(\IR^d)}$ was expressed in terms of the Fourier transform.
The same can be done for
the expected quadratic error of $Q_S^V$.
\begin{lemma}
\label{errorlemma2}
For any invertible matrix
$S\in\ensuremath{\mathbb{R}}^{d\times d}$ and $f\in \ensuremath{C_c(\IR^d)}$ we have
\[
\ensuremath{\mathbb{E}} \abs{Q_S^V(f)-I_d(f)}^2
= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \abs{\mathcal{F}f(Sm)}^2
.\]
\end{lemma}
\begin{proof}
Since the expected value of $Q_S^V(f)$ is $I_d(f)$, we have
\[
\ensuremath{\mathbb{E}} \abs{Q_S^V(f)-I_d(f)}^2
= \mathrm{Var} \braces{Q_S^V(f)}
= \ensuremath{\mathbb{E}} \braces{Q_S^V(f)^2} - I_d(f)^2
.\]
The algorithm $Q_S^v(f)$ considered as a function of $v\in[0,1]^d$
is a finite sum of functions
$|\det S|^{-1}\,f\left(S^{-\top}(k+\cdot)\right)$ in $L^2\braces{[0,1]^d}$
and hence itself in $L^2\braces{[0,1]^d}$.
Parseval's identity states
\[\begin{split}
\ensuremath{\mathbb{E}} \braces{Q_S^V(f)^2}
= \lnormunit{Q_S^{\braces{\cdot}}(f)}^2
= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d}\abs{\lscalarunit{Q_S^{\braces{\cdot}}(f)}{e^{2\pi i \scalar{m}{\cdot}}}}^2
.\end{split}\]
For each index $m\in\ensuremath{\mathbb{Z}}^d$ we have the equality
\[\begin{split}
\lscalarunit{Q_S^{\braces{\cdot}}(f)}{e^{2\pi i \scalar{m}{\cdot}}}
&= \abs{\det S}^{-1} \sum\limits_{k\in\ensuremath{\mathbb{Z}}^d}
\int_{[0,1]^d} f\left(S^{-\top}(k+v)\right)\, e^{-2\pi i \scalar{m}{v}}\d v\\
&= \abs{\det S}^{-1}
\int_{\ensuremath{\mathbb{R}}^d} f\left(S^{-\top}v\right)\, e^{-2\pi i \scalar{m}{v}}\d v\\
&= \int_{\ensuremath{\mathbb{R}}^d} f\left(v\right)\, e^{-2\pi i \scalar{Sm}{v}}\d v\\
&= \mathcal{F}f(Sm)
.\end{split}\]
We arrive at
\[
\ensuremath{\mathbb{E}} \abs{Q_S^V(f)-I_d(f)}^2
= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} \abs{\mathcal{F}f(Sm)}^2 - I_d(f)^2
= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \abs{\mathcal{F}f(Sm)}^2
,\]
which is what had to be proven.
\end{proof}
Now follows an analogue of Theorem~\ref{keyprop}
for expected quadratic errors.
Like before, the general error bound for
continuous functions with compact support
adjusts to additional smoothness
properties.
\begin{thm}
\label{keyprop2}
There is some $c>0$ such that for every $n\in\ensuremath{\mathbb{N}}$ and $f\in\ensuremath{C_c(\IR^d)}$
\[
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2
\leq\, c \, n^{-1} \, \Vert\mathcal{F}f\Vert_{L^2\braces{D_n}}^2
.\]
\end{thm}
\begin{proof}
By Lemma~\ref{errorlemma2}
and the monotone convergence theorem,
\[\begin{split}
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2
&= \ensuremath{\mathbb{E}}_U \ensuremath{\mathbb{E}}_V \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2\\
&= \ensuremath{\mathbb{E}}_U \left(\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \abs{\mathcal{F}f(n^{1/d}\hat{U}Bm)}^2 \right)\\
&= \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}} \ensuremath{\mathbb{E}}_U \abs{\mathcal{F}f(n^{1/d}\hat{U}Bm)}^2
.\end{split}\]
Since each $n^{1/d}\hat{U}Bm$ is uniformly distributed in the
box $[n^{1/d}Bm,(2n)^{1/d}Bm]$ with volume
$\left(2^{1/d}-1\right)^d\cdot\abs{\prod_{j=1}^d n^{1/d} (Bm)_j}$, this series equals
\[\begin{split}
&\frac{1}{\left(2^{1/d}-1\right)^d} \sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}\,
\int\limits_{[n^{1/d}Bm,(2n)^{1/d}Bm]} \frac{\abs{\mathcal{F}f(x)}^2}{\prod_{j=1}^d
\abs{n^{1/d}(Bm)_j}} \, \d x \\
&\leq \frac{1}{\left(2^{1/d}-1\right)^d}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}}\, \int\limits_{[n^{1/d}Bm,(2n)^{1/d}Bm]}
\frac{\abs{\mathcal{F}f(x)}^2}{\prod_{j=1}^d 2^{-1/d}\abs{x_j}} \, \d x\\
&= \frac{2}{\left(2^{1/d}-1\right)^d}\cdot \int_{\ensuremath{\mathbb{R}}^d}
\frac{\abs{\mathcal{F}f(x)}^2}{\prod_{j=1}^d \abs{x_j}}\cdot
\abs{\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid x\in [n^{1/d}Bm,(2n)^{1/d}Bm]}} \, \d x\\
&= \frac{2}{\left(2^{1/d}-1\right)^d}\cdot \int_{\ensuremath{\mathbb{R}}^d}
\frac{\abs{\mathcal{F}f(x)}^2}{\prod_{j=1}^d \abs{x_j}}\cdot
\abs{\set{m\in\ensuremath{\mathbb{Z}}^d\setminus\set{0}\mid Bm\in
\left[\frac{x}{(2n)^{1/d}},\frac{x}{n^{1/d}}\right]}}\, \d x
.\end{split}\]
Thanks to the properties of the Frolov matrix $B$, if $\prod_{j=1}^d \abs{x_j}<n$,
the latter set is empty and otherwise contains no more
than $\prod_{j=1}^d \abs{\frac{x_j}{n^{1/d}}}+1\leq 2 n^{-1} \prod_{j=1}^d \abs{x_j}$ points.
Thus, we arrive at
\[
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)} \leq
\frac{4}{\left(2^{1/d}-1\right)^d}\cdot n^{-1} \int_{D_n} \abs{\mathcal{F}f(x)}^2 \, \d x
\]
and the theorem is proven.
\end{proof}
Finally, we can prove the stated upper bound
for the expected quadratic error of the
method $Q_{n^{1/d}\hat{U}B}^V$.
\begin{thm}
\label{mixthm2}
There is some $c>0$ such that
for every $n\geq 2$ and $f\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$
\[
\braces{\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2} \leq\, c \, n^{-r-1/2}
\, \mixnormunit{f}
.\]
\end{thm}
\begin{proof}
If $c_0$ is the constant of Theorem~\ref{keyprop2}, we have the upper bound
\[\begin{split}
\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2
&\leq c_0 \, n^{-1} \, \Vert\mathcal{F}f\Vert_{L^2\braces{D_n}}^2\\
&= c_0\, n^{-1} \, \int_{D_n} h_r(x)^{-1}\cdot \left|\mathcal{F}f(x)\right|^2\, h_r(x)\, \d x\\
&\leq\, c_0 \, n^{-1} \, \Vert h_r^{-1}\Vert_{L^\infty\braces{D_n}} \cdot \int_{\ensuremath{\mathbb{R}}^d}\left|\mathcal{F}f(x)\right|^2\, h_r(x)\, \d x
\end{split}\]
for the expected quadratic error.\\
Since $h_r(x)\geq n^{2r}$ for $x\in D_n$, we obtain the estimate
\[
\braces{\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2} \leq c_0^{1/2}\, n^{-r-1/2}
\, \mixnormunit{f}
.\]
which proves the theorem.
\end{proof}
The method $Q_{n^{1/d}\hat{U}B}^V$ is also optimal for $\ensuremath{\mathring{H}^s([0,1]^d)}$.
This can be derived from Theorem~\ref{keyprop2} using the same short argument
from the proof of Theorem~\ref{mixthm2}.
The upper bound for $\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}$
is also a direct consequence of Theorem~\ref{isothm}.
\begin{thm}
\label{isothm2}
There is some $c>0$ such that
for every $n\in\ensuremath{\mathbb{N}}$ and $f\in \ensuremath{\mathring{H}^s([0,1]^d)}$
\[
\braces{\ensuremath{\mathbb{E}} \abs{Q_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2} \leq\, c \, n^{-s/d-1/2}
\, \isonormunit{f}
.\]
\end{thm}
See \cite{ln} for a proof of the optimality of this order.
\newpage
\section{Transformations to $\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$}
\label{transformationsection}
We can transform the above methods $Q_{n^{1/d}B}$ and $Q_{n^{1/d}\hat{U}B}^V$
such that their errors satisfy the same upper bounds for the full spaces
$\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$ and $\ensuremath{{H^s([0,1]^d)}}$, that the original algorithms satisfy
for the subspaces $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ and $\ensuremath{\mathring{H}^s([0,1]^d)}$.
This is done by a standard method, which was already used in
\cite[pp.\,359]{temlyakov} to transform Frolov's deterministic algorithm.
It preserves the unbiasedness of the algorithm $Q_{n^{1/d}\hat{U}B}^V$.
To that end let $\psi:\ensuremath{\mathbb{R}}\to\ensuremath{\mathbb{R}}$ be an infinitely differentiable function
such that $\psi|_{(-\infty,0)}=0$, $\psi|_{(1,\infty)}=1$
and $\psi|_{(0,1)}:(0,1)\to(0,1)$ is a diffeomorphism. For example, we can choose
\[
h(x)=\begin{cases}
e^\frac{1}{(2x-1)^2-1} & \text{if } x\in(0,1),\\
0 & \text{else,}
\end{cases}
\quad\quad
\psi(x)=\frac{\int_{-\infty}^x h(t) \, \d t}{\int_{-\infty}^{\infty} h(t) \, \d t}
\]
for $x\in\ensuremath{\mathbb{R}}$. Like $h$ also $\psi$ is infinitely differentiable and apparently
satisfies $\psi|_{(-\infty,0)}=0$ and $\psi|_{(1,\infty)}=1$.
Since the derivative of $\psi$ is strictly positive on $(0,1)$,
it is strictly increasing and a bijection of $(0,1)$ and its inverse function is smooth.
\begin{minipage}[h!]{.49\linewidth}
\includegraphics[width=\linewidth]{plotofh-eps-converted-to}
\end{minipage}
\begin{minipage}[h!]{.46\linewidth}
\includegraphics[width=\linewidth]{plotofpsi-eps-converted-to}
\end{minipage}
Given such $\psi$, the map $\Psi:\ensuremath{\mathbb{R}}^d\to\ensuremath{\mathbb{R}}^d$ with
$\Psi(x)=(\psi(x_1),\hdots,\psi(x_d))^{\top}$ is a diffeomorphism on $(0,1)^d$
with inverse $\Psi^{-1}(x)=(\psi^{-1}(x_1),\hdots,\psi^{-1}(x_d))^{\top}$
and $|D\Psi(x)|\overset{\psi'\geq 0}{=}\det D\Psi(x)=\prod\limits_{i=1}^{d}\psi'(x_i)$.
If $A_n$ is any linear quadrature formula for integration on the unit cube
with nodes $x^{(j)}\in[0,1]^d$ and weights $a_j\in\ensuremath{\mathbb{R}}$,
where $j=1,\hdots,n$, we define the transformed quadrature formula
$\widetilde{A}_n$ by choosing the nodes and weights to be
\[
\tilde{x}^{(j)}=\Psi(x^{(j)})$ \text{\ \ \ \ and \ \ \ \ }
$\tilde{a_j}=a_j\cdot|D\Psi(x^{(j)})|.
\]
Thus, $\widetilde{Q}_S^v$ for $v\in\ensuremath{\mathbb{R}}^d$ and invertible $S\in\ensuremath{\mathbb{R}}^{d\times d}$ takes the form
\[
\widetilde{Q}_S^v(f)=\frac{1}{\abs{\det S}}
\sum\limits_{m\in\ensuremath{\mathbb{Z}}^d} f\left(\Psi\left(S^{-\top}(m+v)\right)\right)
\cdot \left|D\Psi\left(S^{-\top}(m+v)\right)\right|
\]
for any input function $f:[0,1]^d\to \ensuremath{\mathbb{R}}$. Note that
$\left|D\Psi\left(S^{-\top}(m+v)\right)\right|$ is zero for any index
$m\in\ensuremath{\mathbb{Z}}^d$ with $S^{-\top}(m+v)\not\in [0,1]^d$.
\medskip
\noindent
{\bf Algorithms.}
For any $n\in\ensuremath{\mathbb{N}}$ we consider the transformed versions
$\widetilde{Q}_{n^{1/d}B}$ and $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ of the algorithms
$Q_{n^{1/d}B}$ and $Q_{n^{1/d}\hat{U}B}^V$ from Section \ref{frolovsrulesection}
and Section \ref{randomshiftsection}.
\medskip
These algorithms are well defined for any input function $f:[0,1]^d\to\ensuremath{\mathbb{R}}$.
The information costs of $\widetilde{Q}_{n^{1/d}B}$ and $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ are of order $n$.
By Lemma \ref{anlemma}, they are at most
$2\cdot\left(\Vert B\Vert_1+1\right)^d\cdot n$.
\begin{prop}
\label{MSchlangeunbiased}
The method $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ is well-defined and unbiased on $L^1([0,1]^d)$.
\end{prop}
\begin{proof}
Let $f\in L^1([0,1]^d)$. By the Change of Variables Theorem,
$f_0=f\circ \Psi \cdot \left|D\Psi\right|$ is also integrable on $[0,1]^d$
and satisfies
\[
I_d(f)=I_d(f_0)\quad \text{and}\quad \widetilde{Q}_{n^{1/d}\hat{u}B}^v(f) = Q_{n^{1/d}\hat{u}B}^v(f_0)
\]
for any realization $\widetilde{Q}_{n^{1/d}\hat{u}B}^v$ of $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$.
This yields
\[
\ensuremath{\mathbb{E}} \left(\widetilde{Q}_{n^{1/d}\hat{U}B}^V(f)\right) = \ensuremath{\mathbb{E}} \left( Q_{n^{1/d}\hat{U}B}^V(f_0)\right) = I_d(f_0) = I_d(f)
\]
by Proposition~\ref{Munbiased}.
\end{proof}
Most notably, $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ satisfies the following error bounds on $\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$.
\begin{thm}
\label{MSchlangebounds}
There is some $c>0$ such that for every $n\geq 2$ and $f\in\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$
\[\begin{split}
\braces{\ensuremath{\mathbb{E}}\abs{\widetilde{Q}_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2}
&\leq c\, n^{-r-1/2}\,\mixnormunit{f}
\quad \text{and}\\
\sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{\widetilde{Q}_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}
&\leq\, c \, n^{-r}\, (\log n)^\frac{d-1}{2}
\, \mixnormunit{f}
.\end{split}\]
\end{thm}
\begin{proof}
Recall that
$\widetilde{Q}_{n^{1/d}\hat{u}B}^v(f) = Q_{n^{1/d}\hat{u}B}^v(f_0)$ and $I_d(f)=I_d(f_0)$
for any function $f\in L^1(\ensuremath{\mathbb{R}}^d)$ and $f_0=f\circ \Psi \cdot \left|D\Psi\right|$.
Since $\psi'(x)=0$ for $x\not\in(0,1)$, we have
$\diff^\alpha f_0|_{\partial[0,1]^d}=0$ for each $\alpha\in\{0,\dots,r\}^d$
and hence $f_0\in \ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ for any $f\in\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$.\\
That implies the estimates
\[\begin{split}
\braces{\ensuremath{\mathbb{E}} \abs{\widetilde{Q}_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2}
&= \braces{\ensuremath{\mathbb{E}} \abs{ Q_{n^{1/d}\hat{u}B}^v(f_0)-I_d(f_0)}^2}^{1/2}\\
&\leq c \cdot n^{-r-1/2}\cdot \mixnormunit{f_0}, \\
\sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{\widetilde{Q}_{n^{1/d}\hat{u}B}^v(f)-I_d(f)}
&= \sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{ Q_{n^{1/d}\hat{u}B}^v(f_0)-I_d(f_0)}\\
&\leq c \cdot n^{-r} (\log n)^\frac{d-1}{2} \cdot \mixnormunit{f_0}
,\end{split}\]
if $c>0$ is the maximum of the constants of
Theorem~\ref{mixthm2} and Corollary~\ref{mixthmworstcase2}.
That proves the statement,
since there is some $c_0>0$ such that every $f\in\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$
satisfies $\mixnormunit{f_0}\leq c_0\, \mixnormunit{f}$.
This can be proven as follows.
The partial derivatives of $f_0$ take the form
\[
D^\alpha f_0(x)=\frac{\partial^{\abs{\alpha}}}{\partial
x_1^{\alpha_1}\cdots\partial x_d^{\alpha_d}}\ f(\Psi(x))
\cdot\prod\limits_{i=1}^{d}\psi'(x_i)
= \sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d}
D^\beta f(\Psi(x))\cdot S_{\alpha,\beta}(x)
\]
for $\alpha\in\{0,1,\hdots,r\}^d$, where $S_{\alpha,\beta}(x)$ is a finite
sum of finite products of terms $\psi^{(j)}(x_i)$ with $i\in\{1,\hdots,d\},
j\in\{1,\hdots,rd+1\}$ and does not depend on $f$. It is therefore continuous
and bounded by some $c_{\alpha,\beta}>0$.
A special case of Cauchy's inequality says that
the square of a sum is at most the sum of the squares
times the number of addends.
Using these facts, we get
\[\begin{split}
\lnormunit{D^\alpha f_0}^2
&\leq \left(\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d}
\lnormunit{(D^\beta f \circ \Psi) \cdot S_{\alpha,\beta}}\right)^2\\
&\leq \left(\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d}
c_{\alpha,\beta} \cdot\lnormunit{D^\beta f \circ \Psi}\right)^2\\
&\leq (r+1)^d\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,
\hdots,\alpha_d} c_{\alpha,\beta}^2 \cdot\lnormunit{D^\beta f \circ \Psi}^2\\
&= (r+1)^d\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d}
c_{\alpha,\beta}^2 \int_{(0,1)^d} |D^\beta f (\Psi (x))|^2 \,\mathrm{d}x\\
&= (r+1)^d\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d}
c_{\alpha,\beta}^2 \int_{\Psi\left((0,1)^d\right)} |D^\beta f (\Psi (\Psi^{-1}(x))|^2
\cdot |D\Psi^{-1}(x)| \, \mathrm{d}x\\
&\leq (r+1)^d\sup\limits_{x\in(0,1)^d} |D\Psi^{-1}(x)|
\sum\limits_{\beta_1,\hdots,\beta_d=0}^{\alpha_1,\hdots,\alpha_d} c_{\alpha,\beta}^2
\cdot \lnormunit{D^\beta f}^2
\end{split}\]
\[\begin{split}
&\leq c_\alpha \cdot \mixnormunit{f}^2\
,\end{split}\]
for some $c_\alpha>0$ and
\[
\mixnormunit{f_0}^2
=\sum\limits_{\alpha\in\{0,1,\hdots,r\}^d}\lnormunit{D^\alpha f_0}^2
\leq \tilde{c} \, \mixnormunit{f}^2
,\]
if $\tilde{c}$ is the sum of the constants $c_\alpha$ for $\alpha\in\set{0,1,\dots,r}^d$.
\end{proof}
The corresponding error bounds for $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ on $\ensuremath{{H^s([0,1]^d)}}$ are proven in the exact same manner.
\begin{thm}
\label{MSchlangebounds2}
There is some $c>0$ such that for every $n\in\ensuremath{\mathbb{N}}$ and $f\in\ensuremath{{H^s([0,1]^d)}}$
\[\begin{split}
\braces{\ensuremath{\mathbb{E}}\abs{\widetilde{Q}_{n^{1/d}\hat{U}B}^V(f)-I_d(f)}^2}^{1/2}
&\leq c\, n^{-s/d-1/2}\,\isonormunit{f}
\quad \text{and}\\
\sup\limits_{(u,v)\in[1,2^{1/d}]^d\times[0,1]^d} \abs{\widetilde{Q}_{n^{1/d}\hat{u}B}^v(f)-I_d(f)}
&\leq\, c \, n^{-s/d}
\, \isonormunit{f}
.\end{split}\]
\end{thm}
The optimality of this order of convergence of the expected error on $\ensuremath{{H^s([0,1]^d)}}$
was already stated by N.\,S.\,Bakhvalov in 1962, see \cite{bakhvalov}.
A proof can be found in \cite{ln}.
The optimality of the upper bound of Theorem~\ref{mixthm2}
for arbitrary dimensions can be derived from Bakhvalov's result
for the one-dimensional case.\\
Since the transformed version $\widetilde{Q}_{n^{1/d}B}$ of Frolov's deterministic algorithm is a particular realization
of the method $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$, Theorem~\ref{MSchlangebounds} and \ref{MSchlangebounds2} imply the
following error bounds.
\begin{cor}
There is some $c>0$ such that for any $n\geq 2$ and $f\in\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$
\[
\abs{\widetilde{Q}_{n^{1/d}B}(f)-I_d(f)} \leq\, c \, n^{-r}
\, (\log n)^\frac{d-1}{2} \, \mixnormunit{f}
\]
and for any $n\in\ensuremath{\mathbb{N}}$ and $f\in\ensuremath{{H^s([0,1]^d)}}$
\[
\abs{\widetilde{Q}_{n^{1/d}B}(f)-I_d(f)} \leq\, c \, n^{-s/d}
\, \isonormunit{f}
.\]
\end{cor}
It is also not hard to see, that the error bounds for $Q_{n^{1/d}\hat{U}B}^v$ from Theorem~\ref{mixthmworstcase},
\ref{isothmworstcase}, \ref{mixthm} and \ref{isothm}
on the classes $\ensuremath{\mathring{H}^{r,{\rm mix}}([0,1]^d)}$ and $\ensuremath{\mathring{H}^s([0,1]^d)}$ are inherited by
the method $\widetilde{Q}_{n^{1/d}\hat{U}B}^v$ on the classes $\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$ and $\ensuremath{{H^s([0,1]^d)}}$
in the same way.\\
To sum up, both the expected error and the worst case error
of the method $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$ have an optimal
rate of convergence
on both $\ensuremath{{H^{r,{\rm mix}}([0,1]^d)}}$ and $\ensuremath{{H^s([0,1]^d)}}$.
In addition, the method is unbiased.
It is also worth stressing that the algorithm is universal:
It does not depend on the smoothness $r$ or $s$ of the input function in any way
and hence no prior knowledge of it is needed to run $\widetilde{Q}_{n^{1/d}\hat{U}B}^V$.
Nonetheless, the convergence rate of its error perfectly adjusts to that smoothness.
The same is valid for the algorithms $\widetilde{Q}_{n^{1/d}B}$ and $\widetilde{Q}_{n^{1/d}\hat{U}B}^v$.
\newpage
| {
"timestamp": "2016-03-17T01:09:50",
"yymm": "1603",
"arxiv_id": "1603.04637",
"language": "en",
"url": "https://arxiv.org/abs/1603.04637",
"abstract": "We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\\in\\mathbb{N}$ over the $d$-dimensional unit cube.In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \\, (\\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^{-r-1/2} \\, (\\log n)^{(d-1)/2}$ in expectation and by $n^{-r} \\, (\\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on $H^{r,\\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesis is a review of the mentioned upper bounds and their proofs.",
"subjects": "Numerical Analysis (math.NA)",
"title": "On the Randomization of Frolov's Algorithm for Multivariate Integration",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795125670755,
"lm_q2_score": 0.7185943985973773,
"lm_q1q2_score": 0.7092379492610702
} |
https://arxiv.org/abs/2110.08492 | Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed | An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements displaced by any non-identity automorphism. In this paper we confirm Tom Tucker's "Infinite Motion Conjecture" that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of a sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. | \section{Introduction}
A graph is \emph{locally finite} if every vertex has finite degree.
A graph is \emph{asymmetric} if it has no nontrivial
automorphisms. A \emph{coloring} of the vertices of a graph
is asymmetric if no non-identity automorphisms of the graph
preserves the coloring. This author established in 1977
that every regular tree of (finite or infinite) degree $\kappa \ge 2$
admits an \emph{asymmetric 2-coloring}~\cite{trees}. While this
result is trivial for locally finite (and therefore countable)
trees (the point of~\cite{trees} was that it holds for arbitrary
infinite cardinals $\kappa$, and the difficulty begins at
strongly inaccessible cardinals), several interesting questions
about asymmetric colorings of locally finite graphs have
been asked and partly answered (see, e.\,g.,
\cite{wong,trofimov,cuno14,watkins15,lehner16,lehner17,imrich-tucker17,
cuno14,pilsniak18,pilsniak19}).
A particularly intriguing
conjecture was formulated by Thomas W. Tucker in 2011~\cite{tucker}.
The \emph{degree} of a permutation is the number of elements it
moves. The \emph{minimal degree} of a permutation group is the
minimum of the degrees if its non-identity elements.
The \emph{motion} of a graph is the minimal degree of its
automorphism group. Tucker's \emph{Infinite Motion Conjecture}
states that \emph{every connected, locally finite graph with infinite
motion admits an asymmetric 2-coloring.} A number of recent papers
obtained partial result on this conjecture,
including~\cite{trofimov,cuno14,imrich-tucker17}.
Notably, Florian Lehner confirmed
the conjecture for graphs with intermediate ($\exp(O(\sqrt{n})$)
growth~\cite{lehner16}. Lehner, Monika Pil{\'s}niak, and
Marcin Stawiski confirmed the conjecture for graphs with
maximum degree $\le 5$~\cite{pilsniak18}.
The same authors proved that if the maximum degree of the
graph is $k$ then an asymmetric coloring with $O(\sqrt{k}\log k)$
colors exists~\cite{pilsniak19}.
The main result of our paper confirms Tucker's conjecture
in full generality. Along the way, we raise and partially
solve a number of questions regarding group-theoretic
properties of colorings for finite permutation groups.
\begin{theorem}[Tucker's Infinite Motion Conjecture confirmed]
\label{thm:main}
Let $X$ be a locally finite connected graph with infinite motion.
Then $X$ admits an asymmetric 2-coloring.
\end{theorem}
In other words, the conclusion says that
the set of vertices has a subset,
not fixed setwise by any non-identity automorphism of $X$.
Given the fact that the automorphism group of a
connected locally finite \emph{rooted} graph is the
inverse limit of a sequence of finite permutation groups,
the proof boils down to the study of inverse systems of
epimorphisms among a sequence of finite permutation
groups. The motion condition translates to
the \emph{disjointness} of the domains of the groups in the system.
Our result in this context is the following;
this is the main technical result of the paper.
\begin{theorem}[asymmetric coloring of inverse limit] \label{thm:main1}
Let $\mathcal G$ be the inverse limit of an
infinite sequence of finite
permutation groups with disjoint domains, viewed as
a permutation group acting on the union of those domains.
Then $\mathcal G$ admits an asymmetric 2-coloring.
\end{theorem}
In other words, the conclusion says that
the domain of $\mathcal G$ has a subset,
not fixed setwise by any non-identity element of $\mathcal G$.
Our proof builds on a line of work on asymmetric
colorings of finite primitive permutation groups,
started by David Gluck (1983)~\cite{gluck}
and Peter Cameron, Peter M. Neumann, and Jan Saxl (1984)~\cite{saxl},
a theory to which we contribute in this paper.
By a counting argument (see Prop.~\ref{prop:motionlemma}),
also used by Gluck, Cameron et al.
proved that in addition to the symmetric and alternating
groups, there are only a finite number of
primitive groups that do not admit an asymmetric
2-coloring (see Theorem~\ref{thm:saxl}).
Seress classified the exceptions (1997)~\cite{seress97}.
Our proof depends on Seress's classification.
The result of~\cite{saxl} depends on the
Classification of Finite Simple Groups (CFSG)
through a result by Cameron~\cite{cameron81}.
Seress uses detailed explicit knowledge of CFSG.
While Cameron's result, used in~\cite{saxl},
has recently been given a remarkable elementary
combinatorial proof by Sun and Wilmes~\cite{sunwilmes}
(see Sec.~\ref{sec:cfsg-free}),
through Seress's work, our paper continues to depend
on CFSG. We suspect, though, that our proof can be
modified to avoid dependence on CFSG.
The automorphism group of a 2-coloring is the same as the
setwise stablizer of a subset of the permutation domain.
Our method is that we approximate asymmetry by
gradually simplifying the structure of the groups
involved in the inverse limit, by 2-coloring the
underlying sets of an infinite, coinfinite subset
of the finite groups that participate in the inverse limit.
The hard part is to reduce all groups
to solvable groups. To this end, we introduce the concept
of \emph{solvable coloring:} a coloring that is preserved
only by a solvable subgroup of the automorphism group.
This concept, intermediate on the way to asymmetry, may
deserve a systematic study. Once reduced to solvable
groups, we reduce the groups to bounded derived length,
and finally we show how to reduce the derived length,
until, at derived length zero, the group vanishes and asymmetry occurs.
Now we state our key lemma, which may be of independent interest.
\begin{theorem}[Reducing simple image] \label{thm:simple}
Let $G\le\sym(\Omega)$ be a finite permutation group and
$T$ a finite nonabelian simple group. Let
$\varphi: G\twoheadrightarrow T$ be an epimorphism. Then
there exists a subset $\Delta\subseteq\Omega$
such that $\varphi(G_{\Delta}) < T$, where $G_{\Delta}$
denotes the setwise stabilizer of $\Delta$.
\end{theorem}
We shall need the following two additional results
about colorings for finite permutation groups.
\begin{theorem}[Bounded orbits for solvable groups] \label{thm:bded-orb}
There is a constant $C$ such that the following holds.
Let $G\le\sym(\Omega)$ be a solvable finite permutation group.
Then there exists a subset $\Delta\subseteq\Omega$
such that every orbit of $G_{\Delta}$ has length $\le C$.
\end{theorem}
We use this result only to infer that
$G_\Delta$ has bounded derived length.
Along the way to proving Theorem~\ref{thm:bded-orb},
we show that every solvable permutation group
has an asymmetric 5-coloring (Lemma~\ref{lem:solv5color}).
(The bound 5 is tight, see Remark~\ref{rem:5colorstight}.)
The following observation will allow us to
move from bounded derived length to asymmetry.
\begin{prop} \label{prop:derivedlength-intro}
Every nontrivial solvable permutation group
admits a 2-coloring that reduces its derived length.
\end{prop}
\subsection{Structure of the paper}
We briefly review the author's view of the
history of combinatorial symmetry breaking
in Sec.~\ref{sec:history}. Rudimentary
definitions follow in Sec.~\ref{sec:def1}.
We reduce Tucker's conjecture to the study of
epimorphisms of finite permutation groups in
Sections~\ref{sec:inverse}--\ref{sec:approximation}.
This includes a review of inverse systems of group
homomorphisms in Sec.~\ref{sec:inverse}.
The program of gradual structural reduction of the groups
is formalized in Theorem~\ref{thm:program}.
After Sec.~\ref{sec:approximation}, all groups considered
are finite. Further details of group theoretic notation
are reviewed in Sec.~\ref{sec:defs}.
The first phase of our program of structural reductions,
going from general to solvable groups,
occupies Sections~\ref{sec:solvable} to~\ref{sec:red-to-solvable}.
This includes the proof of Theorem~\ref{thm:simple}
for primitive groups in Sec.~\ref{sec:primitive}
and for general permutation groups in Sec.~\ref{sec:general2simple}.
We achieve bounded derived length in Section~\ref{sec:bded-derived-length}
by proving Theorem~\ref{thm:bded-orb}.
The reduction of the derived length (Prop.~\ref{prop:derivedlength-intro})
is the subject of Sec.~\ref{sec:reducing-derived-length}.
This completes the proof of the Infinite Motion Conjecture.
In Section~\ref{sec:effective} we state a finite version of
Theorem~\ref{thm:main1} (Theorem~\ref{thm:finite}) and
state two conjectures in search for more effective versions
of this result.
In Section~\ref{sec:cfsg-free} we address the question of CFSG-free
proofs and combinatorial generalizations, notably to the motion
of primitive coherent configurations.
The techniques introduced in this paper give rise
to a number of new questions and potential areas
of research; some of these are listed in the
concluding section of this paper, Sec.~\ref{sec:open}.
\section{A brief history of combinatrial symmetry breaking}
\label{sec:history}
The study of asymmetry dates back to the 1939 paper by
Roberto Frucht~\cite{frucht39} which established
that every finite group is isomorphic to the automorphism
group of a graph. Frucht introduced asymmetric ``gadgets''
to code colors and thereby eliminate unwanted automorphisms.
A decade later Frucht proved the same result for 3-regular
graphs~\cite{frucht49}; as a tool, he constructed
an asymmetric 3-regular graph that today bears his name.
Analogues of Frucht's Theorem were found to hold for
many other classes of structures, notably including
Steiner Triple Systems~\cite{mendelsohn} and algebraic number
fields~\cite{fried}. A theory was developed to construct
graphs whose automorphism group was a prescribed regular
permutation group (GRR theory \cite{hetzel, godsil, DRR, morris}).
In all these cases,
it is easy to construct structures on which the given group acts;
the problem is the elimination of unwanted symmetry by coding
asymmetry into the structure to get exactly the desired group of
automorphisms.
In the 1960s, this line of work was extended by the Prague school
of category theory to endomorphisms~\cite{pultr65}. A key aspect of
this generalization was the construction of \emph{rigid graphs},
i.\,e., graphs with no nontrivial
endomorphisms~\cite{pultr65,vopenka65,lambek69}.
It was in this context that this author considered the
simplest possible asymmetric graphs
in the infinite case, establishing that for any two
(finite or infinite) cardinal numbers $\kappa > \lambda \ge 2$
there exists an asymmetric tree with only these two
degrees~\cite{trees}.
The key auxiliary result was that a regular tree
of (finite or infinite) degree $\kappa \ge 2$ admits
an \emph{asymmetric 2-coloring}---the first result
on asymmetric colorings.
In 1983/84, the idea of asymmetric 2-colorings was introduced
in group theory in a pair of independent papers,
by Gluck~\cite{gluck} and by Cameron--Neumann--Saxl~\cite{saxl}.
Gluck found that all but a finite number of solvable primitive
groups admit an asymmetric 2-coloring, and gave the exact
list of exceptions (all of them of degree $\le 9$).
Cameron--Neumann--Saxl proved
that all but a finite number
of primitive permutation groups other than $A_n$ and $S_n$
admit an asymmetric 2-coloring. Subsequently
Seress~\cite{seress97} classified all the exceptions;
Seress's paper is one of our key references.
The motivation of Gluck, Cameron et al., and Seress
was in classical questions of the theory of
permutation groups and partly, in questions of computational
group theory through the closely related concept of \emph{bases}
of permutation groups (see Seress's monograph~\cite{seressbook}).
These papers discovered the relationship of asymmetric
colorings with the \emph{minimal degree} of the permutation
group, i.\,e., the minimum number of elements not fixed by a
non-identity element of the group. This is a classical concept,
studied since the time of Jordan~\cite{jordan} and
Bochert~\cite{bochert} in the 19th century
(see~\cite[Sec. 3.3]{dixon-mortimer}).
A \emph{base} of a permutation group $G\le\sym(\Omega)$
is a subset $\Delta\subseteq\Omega$ such that the
pointwise stabilizer $G_{(\Delta)}$ is the identity.
This classical ``symmetry breaking'' concept, evidently
closely related to the notion of asymmetric colorings,
gained interest in computational group theory through
the work of Charles Sims in the 1960s~\cite{sims}
(see~\cite{seressbook}). The significance of this concept
to asymptotic group theory comes from the observation
that if $G\le S_n$ has a base of size $b$ then $|G|\le n^b$.
Bounding the orders of \emph{primitive} permutation groups
was a central question of 19th century group theory
(see \cite{praeger} for some of this history).
In 1979, this author found a graph theoretic method to bound the
base size of primitive permutation groups in the more general
context of \emph{primitive coherent configurations} (PCCs)--certain
highly regular colorings of the edges of the complete directed graph
(see Sec.~\ref{sec:PCC}). Entirely omitting group theory,
this method nevertheless produced a nearly tight upper
bound on the base size ($O(\sqrt{n}\log n)$), and therefore
on the order, of any primitive group other than the
alternating and the symmetric groups, solving a then century-old
problem~\cite{annals}. The key technical result was a
symmetry-breaking tool:
how many vertices can distinguish between a given pair of vertices,
see Theorem~\ref{thm:annals}. As a byproduct, a lower bound on
the minimal degree of primitive groups and on the motion of
PCCs follows (Theorem~\ref{thm:annals} and
Obs.~\ref{obs:uni-motion}); the bound is tight
within a constant factor.
Significant progress over this result occurred in 2015 when
Xiaorui Sun and John Wilmes extended the result to a classification
of all primitive groups of minimum base size greater than essentially
$n^{1/3}$, while building a structure theory of
primitive coherent configurations along the way.
This remarkable result, previously
only known through CFSG~\cite{cameron81}, can be used to give
an elementary proof of the Cameron--Neumann--Saxl result mentioned
(see Sec.~\ref{sec:cfsg-free}).
It would presumably play a prominent role in a CFSG-free
proof of Tucker's Conjecture.
The \emph{Graph Isomorphism problem} has been a chief producer
and consumer of cost-conscious symmetry breaking techniques.
Both~\cite{annals} and~\cite{sunwilmes} were partly motivated by
the complexity of this problem.
Many new details about this connection, in the context of
coherent configurations, emerged in~\cite{quasipoly},
where a quasipolynomial-time algorithm for testing
graph isomorphism is described. Comments on
measuring the cost appear in Sec.~\ref{sec:open}.
\medskip\noindent
{\bf Terminology.} \quad
A much-cited 1996 paper by
Michael Albertson and Karen Collins \cite{collins}
introduced asymmetric colorings under the name
``distinguishing coloring.'' Although this terminology
has found a large following, I find it unfortunate
for multiple reasons and will continue to use the term
``distinguish'' in more natural meanings, including in this paper
(see right before Theorem~\ref{thm:annals}, and
Problem~\ref{pr:cameron-schemes}
in Sec.~\ref{sec:open}). The term ``asymmetric coloring'' was
introduced in~\cite{trees} (1977).
The term ``minimal degree'' of a permutation group has been in
use at least since Wielandt's 1964 book~\cite{wielandt}, and
has since been used in a large body of literature in the theory
of permutation groups. However, this term is ill-suited for
applications to the automorphism group of a graph, where it
could be confused with the minimum degree of the graph---an
unrelated concept. Therefore I have adopted the term
``motion'' of a graph or other structure, meaning the minimal
degree of its automorphism group, as suggested in~\cite{russell}.
\section{Definitions, notation: group actions and coloring}
\label{sec:def1}
Structural group theoretic definitions and notation
will be reviewed in Section~\ref{sec:defs}. The beautiful
monograph~\cite{dixon-mortimer} covers most of the
group theory we need. In this section we only deal
with the most rudimentary concepts.
\begin{definition}[Coloring]
Let $\Sigma$ be an ordered set; we refer to the elements
of $\Sigma$ as ``colors.'' A $\Sigma$-coloring of a
set $\Omega$ is a function $\gamma: \Omega\to \Sigma$.
If $|\Sigma|=k$, we speak of a $k$-coloring and
often use $\Sigma :=[k]=\{1,2,\dots, k\}$ as the set of colors.
We identify the subsets of $\Omega$
with 2-colorings, using a fixed ordered pair of colors.
\end{definition}
\begin{notation}[Symmetric and alternating groups]
For a set $\Omega$, we write $\sym(\Omega)$ and $\alt(\Omega)$
for the symmetric and the alteranting group, resp., on $\Omega$.
We also write $S_n$ for the generic symmetric group of
degree $n$ and $A_n$ for the alternating group of degree $n$.
\end{notation}
\begin{definition}[Group action] \label{def:action}
Let $G$ be a group, $\Omega$ a set. A \emph{$G$-action}
on $\Omega$, denoted $G\curvearrowright\Omega$,
is a homomorphism $\varphi: G\to\sym(\Omega)$.
We refer to $\Omega$ as the \emph{permutation domain.}
The \emph{image} of the action $G\stackrel{\varphi}{\curvearrowright}\Omega$
is the group $\varphi(G)\le\sym(\Omega)$.
For $\pi\in G$ and $x\in\Omega$ we write $\pi(x)$ to
denote $(\varphi(\pi))(x)$ if the action $\varphi$ is clear
from the context.
\end{definition}
\begin{definition}[Coloring for a group action]
Let $\varphi: G\curvearrowright\Omega$ be a group action.
By a coloring for $\varphi$ we mean a coloring of $\Omega$.
For a coloring $\gamma:\Omega\to\Sigma$ and $\pi\in G$
we write $\pi(\gamma)$ to denote the coloring
$(\pi(\gamma))(x)=\gamma(\pi^{-1}(x))$ $(x\in\Omega)$.
\end{definition}
\begin{definition}[Stabilizers]
Let $\varphi: G\curvearrowright\Omega$ be a group action.
Let $x\in\Omega$. The \emph{stabilizer} of $x\in\Omega$
is the subgroup $G_x=\{\pi\in G\mid \pi(x)=x\}$.
For $\Delta\subseteq\Omega$ we write
$G_\Delta=\{\pi\in G\mid \pi(\Delta)=\Delta\}$
for the \emph{setwise stabilizer} of $\Delta$
and $G_{(\Delta)}=\bigcap_{x\in\Delta} G_x$
for the \emph{pointwise stabilizer} of $\Delta$.
The \emph{stabilizer of the coloring} $\gamma$ is the
subgroup $G_\gamma = \{\pi\in G\mid \pi(\gamma)=\gamma\}$.
We refer to $G_\gamma$ as the group of \emph{$G$-automorphisms
of the coloring $\gamma$.} In other words, the $G$-automorphisms
of the coloring $\gamma$ are those elements of $G$ that
preserve $\gamma$.
\end{definition}
\begin{obs}[Subsets vs. 2-colorings] \label{obs:set-stab}
If $\Sigma=\{a,b\}$ where $a < b$
and $\gamma:\Omega\to\Sigma$ is a 2-coloring
then $G_\gamma=G_\Delta$ where $\Delta=\gamma^{-1}(b)$.
\hfill $\Box$
\end{obs}
\begin{definition}[Permutation group]
A \emph{permutation group} acting on $\Omega$ is a subgroup of
$\sym(\Omega)$. We view permutation groups
as faithful actions. A \emph{coloring} for $G$
is a coloring of $\Omega$. (This means a coloring for this
faithful action).
\end{definition}
\begin{definition}[Group theoretic properties of colorings] \label{def:asy}
Let $G\curvearrowright\Omega$ be an action. We say that a coloring $\gamma$
for $G$ is \emph{asymmetric} if $G_\gamma=1$.
We also say that such a coloring of $\Omega$ is $G$-asymmetric.
Analogously we speak of \emph{$G$-asymmetric subsets}
of the domain. We call the minimum number of colors
required by an asymmetric coloring for $G$ the
\emph{asymmetric coloring number} of $G$.
We say that $\gamma$ is a \emph{solvable coloring} if $G_\gamma$
is a solvable group. We also express this circumstance by saying
that the coloring $\gamma$ \emph{results in a solvable group}.
We call the minimum number of colors required by a solvable coloring
the \emph{solvable coloring number} of $G$.
We may analogously ascribe other group theoretic properties
to $\gamma$. For example, $\gamma$ can result in a 2-group
(i.\,e., $G_{\gamma}$ is a 2-group) or in a solvable group
with derived length $\le 3$, etc.
We define the corresponding concepts
for subsets of the domain (asymmetric subsets,
solvable subsets, subsets resulting in a 2-group, etc.)
via the correspondence to 2-colorings (Obs.~\ref{obs:set-stab}).
We say that $G$ \emph{admits} a coloring with certain property
(e.\,g., a solvable $k$-coloring) if there is a coloring
of the domain with the given property.
\end{definition}
Note that an asymmetric coloring exists if and only
if the action of $G$ is faithful.
\begin{definition}[Support, minimal degree]
Let $G\le\sym(\Omega)$ be a permutation group.
The \emph{support} of $\sigma\in G$ is the set
$\supp(\sigma):=\{x\in\Omega\mid \sigma(x)\neq x\}$.
The \emph{minimal degree} of $G$ is
$\mu(G):=\min_{\sigma\in G\setminus\{1\}}|\supp(\sigma)|$\,,
the minimum number of elements moved (not fixed) by
any non-identity element of $G$. If $|G|=1$
then its minimal degree is ``super-infinity,'' denoted $\infty$
and thought of as being greater than any cardinal number.
\end{definition}
For instance, for $n\ge 3$, $\mu(S_n)=2$ and $\mu(A_n)=3$.
\begin{definition}[Coloring of structures]
Let $\mathfrak X=(\Omega,\mathcal R)$ be a structure (such as a graph).
We say that a coloring $\gamma:\Omega\to\Sigma$ of the
underlying set (set of vertices) has a certain property
\wrt $\mathfrak X$ (such as being asymmetric or solvable)
if it has the corresponding property \wrt $\aut(\mathfrak X)$.
\end{definition}
\begin{definition}[Motion]
The \emph{motion} of a structure $\mathfrak X$ is the minimal degree
of $\aut(\mathfrak X)$.
\end{definition}
The term ``motion'' in this meaning was introduced
in \cite{russell}.
\medskip\noindent
Let $G\le \sym(\Omega)$ be a permutation group.
We say that $G$ is a \emph{finite permutation group}
if its domain $\Omega$ is finite.
\section{Inverse systems of epimorphisms}
\label{sec:inverse}
Inverse systems can be indexed by an arbitrary poset. In this paper,
we only consider inverse systems of infinite \emph{sequences} of groups,
so the index set is $\mathbb N=\{0,1,\dots\}$, the set of natural numbers,
and the infinite subsets of $\mathbb N$, under the natural ordering.
\begin{convention}
Throughout this paper, the letters $I$ and $J$ will denote infinite
subsets of $\mathbb N$, with the natural ordering.
\end{convention}
\begin{definition}[Inverse system of homomorphisms of finite groups]
Let $(G_i\mid i\in I)$ be an infinite sequence of finite groups.
For $i \le j$ $(i,j\in I)$ let $\varphi_{i,j} : G_j \to G_i$ be a
homomorphism such that for $i \le j\le k$ $(i,j,k\in I)$, the
\emph{compatibility condition} $\varphi_{i,j}\varphi_{j,k} = \varphi_{i,k}$ holds.
The homomorphism $\varphi_{ii}$ is the identity on $G_i$.
The $\varphi_{i,j}$ are called the \emph{transition homomorphisms}.
The groups $G_i$, together with the transition homomorphisms,
form an \emph{inverse system}, denoted $(G_i, \varphi_{i,j})_I$,
or simply $(G_i, \varphi_{i,j})$ if the index set $I$ is clear from
the context. We say that the system is \emph{epimorphic} if all
transition homomorphisms are surjective.
\end{definition}
\begin{definition}[Inverse limit, strands] \label{def:strand}
Let us consider a sequence $(g_i \mid i\in I)$ of group
elements, $g_i\in G_i$.
Given the inverse system above, we call such a
sequence a \emph{strand} if for all $i \le j$ we have
$g_i = \varphi_{i,j}(g_j)$. The inverse limit
$\mathcal G =\varprojlim G_i$ is the subgroup of the
direct product of the $G_i$ consisting of the strands.
Viewing the $G_i$ as discrete groups and endowing their
direct product with the product topology, $\mathcal G$ is
a closed subgroup of the direct product (a profinite group).
\end{definition}
\begin{fact} \label{fact:aut-is-limit}
Let $X$ be a connected, locally finite, infinite
graph, with possibly colored vertices. Let $x_0$ be a vertex.
Then $\aut(X)_{x_0}$ (the stabilizer of $x_0$ in $\aut(X)$)
is the inverse limit of a sequence of finite permutation groups.
\end{fact}
\begin{proof}[Idea of proof]
The finite permutation groups in question are the
automorphism groups of the balls of each radius about $x_0$.
We can alternatively also take the restrictions of $\aut(X)$
to those balls; this would correspond to the epimorphic
reduction discussed below.
\end{proof}
\begin{fact}
Let $(G_i, \varphi_{i,j})_I$ be an \emph{epimorphic}
inverse system with limit $\mathcal G =\varprojlim G_i$.
Let $\pi_i: \mathcal G \to G_i$ denote the projection to the $i$-th
coordinate. Then each $\pi_i$ is an epimorphism.
\hfill $\Box$
\end{fact}
\begin{fact}[Epimorphic reduction] \label{fact:epiredux}
Let $(G_i, \varphi_{i,j})_I$ be an inverse system of finite groups
with inverse limit $\mathcal G = \varprojlim G_i$.
Then there exist subgroups $H_i\le G_i$ such that
$(H_i, (\varphi_{i,j})_{|H_j})_I$ is an \emph{epimorphic}
inverse system such that $\mathcal G = \varprojlim H_i$. We call
this system the \emph{epimorphic reduction} of the system
$(G_i, \varphi_{i,j})_I$. The epimorphic reduction is unique.
\end{fact}
\begin{proof}
Let $\pi_i: \mathcal G \to G_i$ denote the projection to the $i$-th
coordinate. Let $H_i = \pi_i(\mathcal G)$.
Then the system consisting of the groups $H_i$
with the transition maps $(\varphi_{i,j})_{|H_j}$
is clearly the unique inverse system satisfying
the conditions.
\end{proof}
\begin{prop}[Sublimit]
Let $J\subseteq I\subseteq\mathbb N$
and let $(G_i,\varphi_{i,j})_I$ be an inverse system
of finite groups. Let $\mathcal G=\varprojlim_{i\in I} G_i$
and let $\mathcal H=\varprojlim_{j\in J} G_j$. Then the projection
$\mathcal G\twoheadrightarrow\mathcal H$ is an isomorphism. \hfill $\Box$
\end{prop}
\begin{definition}[$\Lambda$-reduction]
Let $(G_i,\varphi_{i,j})_I$
be an inverse system as above,
with inverse limit $\mathcal G$, and
let $\Lambda=(L_i \mid i\in I)$ where $L_i\le G_i$.
The \emph{$\Lambda$-reduction} $\mathcal L$ of $\mathcal G$ consists
of those strands $(g_i : i\in I)$
where $g_i\in L_i$ for all $i$.
\end{definition}
\begin{fact}
$\mathcal L$ is a closed subgroup of $\mathcal G$ and
$\mathcal L = \varprojlim_{i\in I} L_i$.
\end{fact}
Let us now consider an inverse system $(G_i,\varphi_{i,j})_I$
where each $G_i$ is a finite permutation group,
$G_i\le \sym(\Omega_i)$.
Critically, we assume the $\Omega_i$ are \emph{disjoint}
(an assumption that is patently \emph{false} in the proof of
Fact~\ref{fact:aut-is-limit}).
\begin{definition}[Action of inverse limit of permutation groups
with disjoint domains] \label{def:limperm}
Let $(G_i,\varphi_{i,j})_I$ be an inverse system of finite
permutation groups $G_i\le \sym(\Omega_i)$.
Assume the $\Omega_i$ are \emph{disjoint}.
Let $\Omega = \bigsqcup_{i\in I}\Omega_i$.
We view the direct product $\prod_{i\in I} G_i$
as a permutation group acting on $\Omega$ (coordinatewise).
As a consequence, the inverse limit $\mathcal G = \varprojlim_{i\in I} G_i$
is also a permutation group acting on $\Omega$.
\end{definition}
\begin{obs}[Coloring of inverse limit of
permutation groups with disjoint domains]
Under the assumptions of Def.~\ref{def:limperm},
let $\gamma$ be a coloring of the set
$\Omega=\bigsqcup_{i\in I}\Omega_i$\,.
In accordance with our conventions,
we say that $\gamma$ is a \emph{coloring for the inverse limit}
$\mathcal G = \varprojlim G_i$.
Let $\gamma_i$ be the coloring $\gamma$ restricted to $\Omega_i$.
Let $L_i=(G_i)_{\gamma_i}$ and set $\Lambda = (L_i : i\in I)$.
Let $\mathcal L\le\mathcal G$ denote the $\Lambda$-reduction of $\mathcal G$.
Then $\mathcal L = \mathcal G_\gamma$. In particular,
$\gamma$ is an \emph{asymmetric coloring} for $\mathcal G$ if and only if
$\mathcal L = 1$. \hfill $\Box$
\end{obs}
The Infinite Motion Conjecture will easily follow from the
following result, previously stated as Theorem~\ref{thm:main1}.
\begin{theorem}[main technical result] \label{thm:main2}
Let $\mathcal G$ be the inverse limit of an infinite sequence of finite
permutation groups with disjoint domains.
Then $\mathcal G$ admits an asymmetric 2-coloring.
\end{theorem}
The disjointness condition illuminates the role of the
\emph{infinite motion assumption} in Tucker's Conjecture.
The latter will permit us to replace the \emph{balls}
in the proof of Fact~\ref{fact:aut-is-limit}
by the \emph{spheres}, which are disjoint
(see Lemma~\ref{lem:invertible} below).
\section{Reduction of Tucker's Conjecture to Theorem~\ref{thm:main1}}
\label{sec:tucker-reduction}
First we reduce the problem to the case of rooted graphs,
where a designated vertex (the ``root'') is fixed by
all automorphisms.
\begin{notation}[Spheres, balls]
Let $\rho$ denote the distance metric in the connected graph $X$
with vertex set $V$.
For a vertex $v\in V$, let $S_d(v)=\{w\in V\mid \rho(v,w)=d\}$
denote the \emph{sphere} of radius $d$ about $v$ and
$B_d(v)=\{w\in V\mid \rho(v,w)\le d\}$
the \emph{ball} of radius $d$ about $v$.
\end{notation}
\begin{definition}[Twins]
Let us call the vertices $u\neq v$ of the graph $X$
\emph{twins} if the transposition $(u,v)$ is an automorphism of $X$.
We say that the graph $X$ \emph{twin-free} if there are
no twins in $X$.
\end{definition}
\begin{obs}
If $X$ has infinite motion then it is twin-free.
\hfill $\Box$
\end{obs}
\begin{definition}[Special subset]
Let $X$ be a connected locally finite graph with vertex set $V$.
Let $x_0\in V$. We call a subset $\Delta\subseteq V$ \emph{special}
(with respect to $x_0$)
if $\Delta\,\cap\, B_1(x_0)=\emptyset$ and
$S_{2d}(x_0)\subseteq\Delta$ for all $d\ge 1$.
\end{definition}
\begin{lemma}[Designated root] \label{lem:root}
Let $X$ be a connected, twin-free graph with vertex set $V$.
Let $x_0\in V$ and let $\Delta$ be a special subset of $V$
(with respect to $x_0$).
Then $\aut(X)_\Delta$ fixes $x_0$.
\end{lemma}
\begin{proof}
Let $Z=\{v\in V\mid B_1(v)\,\cap\,\Delta =\emptyset\}$.
Then $x_0\in Z\subseteq B_1(x_0)$. Moreover, for each
$z\in Z$ we have $B_1(z)\subseteq B_1(x_0)$.
Since $X$ is twin-free, we in fact have $B_1(z)\subset B_1(x_0)$.
Consequently $x_0$ is the unique vertex in $Z$ adjacent
to all other vertices in $Z$.
\end{proof}
Using Theorem~\ref{thm:main1} we shall show the following.
\begin{theorem} \label{thm:rooted}
Let $X$ be a locally finite connected
graph with infinite motion and let $x_0$ be a vertex.
Then $X$ admits an asymmetric special subset \wrt $x_0$.
\end{theorem}
Note that here we do not make it an assumption that
all automorphisms fix $x_0$. The automorphisms that
fix a special subset will automatically fix $x_0$
by Lemma~\ref{lem:root}.
For a locally finite connected rooted graph $X=(V,E,x_0)$
we use the following notation.
Let $\mathcal G=\aut(X)$. Here we made $x_0$ a constant in
the language of the structure $X$, so this group fixes $x_0$ by
definition.
Consequently, $\mathcal G$ also fixes each sphere about $x_0$ (setwise).
Let $B_i=B_i(x_0)$ and $S_i=S_i(x_0)$.
Let $H_i$ denote the restriction of $\mathcal G$ to
$B_i$ and $G_i$ the restriction of $\mathcal G$ to $S_i$
(so $H_i\le\sym(B_i)$ and $G_i\le\sym(S_i)$).
For $i\in I$ let $\pi_i: H_i \twoheadrightarrow G_i$ denote
the restriction from $B_i$ to $S_i$ (an epimorphism)
and for $i\le j$ let $\psi_{i,j}: H_j\twoheadrightarrow H_i$ denote the
restriction from $B_j$ to $B_i$ (again, an epimorphism).
\begin{lemma} \label{lem:invertible}
Let $X$ be a locally finite connected
rooted graph with infinite motion.
Then each restriction epimorphism $\pi_i$
(from the ball $B_i$ to the sphere $S_i$)
is an isomorphism.
\end{lemma}
\begin{proof}
Let $\sigma\in\ker\pi_i$. So $\sigma$ acts on $B_i$ and
fixes $S_i$ pointwise. Let $\widehat \sigma$ denote
the extension of $\sigma$ to $V$ obtained by fixing
all vertices in $V\setminus B_i$. So $\widehat \sigma$
is an automorphism of $X$ that fixes all vertices
outside the finite set $B_i$ and therefore,
by the infinite motion assumption, it fixes
all vertices in $B_i$ as well, so $\sigma=1$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:rooted}
from Theorem~\ref{thm:main1}]
By Lemma~\ref{lem:invertible}, we can define,
for $i \le j$, the $G_j\twoheadrightarrow G_i$ epimorphism
$\varphi_{i,j}=\pi_j\psi_{i,j}\pi_i^{-1}$.
Let $\mathcal G$ denote the inverse limit of the
system $(G_i,\varphi_{i,j})_{2\mathbb N +3}$.
Recall that the domain of $G_i$ is the sphere $S_i$,
so the domains of these permutation groups
are disjoint.
Let $\widehat\mathcal G$ denote the restriction of
$\mathcal G$ to the set $A=\bigsqcup\{S_i \mid i\in 2\mathbb N +3\}$.
By Theorem~\ref{thm:main1}, the group $\widehat\mathcal G$ admits
an asymmetric subset $\Gamma$. Let now
$\Delta = \Gamma \cup \bigcup \{S_j\mid j\in 2\mathbb N+2\}$.
Then $\Delta$ is a special subset and $(\mathcal G)_{\Delta}$
fixes $A$ pointwise. But then the epimorphisms $\varphi_{i,j}$
ensure that all of $V$ is fixed pointwise.
\end{proof}
\section{Approximation process}
\label{sec:approximation}
We say that a class $\mathcal Q$ of groups is \emph{HS-closed}
if $\mathcal Q$ is closed under subgroups and homomorphic
images. (This includes being closed under isomorphisms.)
\begin{obs}
Let $(G_i, \varphi_{i,j})_I$ be an \emph{epimorphic} system of
finite groups. If $\mathcal Q$ is an HS-closed class and infinitely
many of the $G_i$ belong to $\mathcal Q$ then all of them belong
to $\mathcal Q$. \hfill $\Box$
\end{obs}
\begin{definition}[Color-reduction between classes of permutation groups]
Let $\mathcal Q$ and $\mathcal R$ be HS-closed classes of finite permutation groups.
We say that $\mathcal Q$ is \emph{$k$-color-reducible} to $\mathcal R$
if the following holds for all pairs of permutation groups
$G,H\in\mathcal Q$. If $\varphi:H\twoheadrightarrow G$ is an epimorphism and
$G\notin \mathcal R$ then there exists a $k$-coloring $\gamma$
for $H$ such that $\varphi(H_\gamma) < G$.
\end{definition}
\begin{lemma}[Color-reduction of inverse limits] \label{lem:reduction}
Let $\mathcal Q$ and $\mathcal R$ be HS-closed classes of finite groups.
Assume $\mathcal Q$ is $2$-color-reducible to $\mathcal R$.
Let $(G_i,\varphi_{i,j})_I$ be an epimorphic inverse system
of finite permutation groups $G_i\in \mathcal Q$; let $\Omega_i$ denote
the permutation domain of $G_i$. Assume the $\Omega_i$ are disjoint.
Let $\mathcal G = \varprojlim G_i$. Let $J$ be an infinite subset of $I$.
Then there exists a subset $\Delta\subseteq \bigcup_{i\in J} \Omega_i$
such that for all $i\in I$ we have $\pi_i(\mathcal G_{\Delta})\in\mathcal R$.
\end{lemma}
\begin{proof}
For a subset $K\subseteq I$ let us use the notation
$\Omega(K):=\bigcup_{i\in K} \Omega_i$.
For $i\in I$, in increasing order, we shall inductively designate
a finite subset $J_i\subset J$ and a subset
$\Delta_i\subseteq \Omega(J_i)$ such that the $J_i$ are
disjoint, $i < j$ for all $j\in J_i$, and $\pi_i(\mathcal G_{\Delta_i})\in\mathcal R$.
It is clear then, that $\Delta :=\bigcup_{i\in I} \Delta_i$ accomplishes
our goal.
Suppose the $J_{\ell}$ have already been constructed for $\ell < i$.
Let $K_i$ be the complement in $J$ of the set
$\{t\in J\mid t\le i\}\cup\bigcup \{J_{\ell}\mid \ell < i\}$.
We perform the following algorithm to construct $J_i$ and $\Delta_i$.
The algorithm will gradually reduce $G_i$ until it becomes an
element of $\mathcal R$, using the 2-color-reducibility of $\mathcal Q$ to
$\mathcal R$. The variable $F$ stores the current
group $G_i$. The $\nextt(K_i,m)$ operation produces the
smallest element of $K_i$ that is greater than $m$.
\medskip\noindent
\indent 01 \quad $F :=G_i$\\
\indent 02 \quad $J_i := \emptyset$ \\
\indent 03 \quad $m := 0$ \\
\indent 04 \quad {\bf while}\ $F\notin \mathcal R$ \\
\indent 05 \quad \qquad $m:=\nextt(K_i,m)$\\
\indent 06 \quad \qquad $J_i := J_i \cup \{m\}$ \\
\indent 07 \quad \qquad let $\Psi_m \subseteq\Omega_m$ such that \\
\indent 08 \quad \qquad\qquad $\varphi_{im}((\varphi_{im}^{-1}(F))_{\Psi_m}) < F$ \\
\indent 09 \quad {\bf end}({\bf while}) \\
\indent 10 \quad $\Delta_i := \bigcup_{j\in J_i} \Psi_j$ \\
\indent 11 \quad {\bf return}\ $J_i$ and $\Delta_i$
\medskip\noindent
Explanation of lines 07--08. Both $G_i$ and $G_m$ belong to $\mathcal Q$,
and therefore their subgroups $F$ and $\varphi_{im}^{-1}(F)$, resp., also
belong to $\mathcal Q$. The restriction of the epimorphism
$\varphi_{im} : G_m\to G_i$ to $\varphi_{im}^{-1}(F)$ is an epimorphism
from $\varphi_{im}^{-1}(F)$ onto $F$. Therefore, if $F\notin \mathcal R$,
by the 2-color-reducibility of $\mathcal Q$ to $\mathcal R$ there exists
$\Psi_m\subseteq \Omega_m$ as required in line 08.
\medskip\noindent
Since $F$ is reduced in every round of the {\bf while}-loop, the
process terminates in a finite number of steps, and on termination,
$F\in\mathcal R$, as desired.
\end{proof}
\medskip\noindent
We shall use Lemma~\ref{lem:reduction} in each successive
step in a chain of HS-closed classes which we now list.
\begin{itemize}
\item \quad ${\mathscr{Gr}} = \{\text{ all finite groups }\}$
\item \quad ${\mathscr{Sol}} = \{\text{ all finite solvable groups }\}$
\item \quad ${\mathscr{Der}_k} = \{\text{ all finite solvable groups
of derived length }\le k\,\}$
\end{itemize}
For some constant $k_0$ we shall descend along the chain
\begin{equation} \label{eq:chain-of-classes}
{\mathscr{Gr}} \supset {\mathscr{Sol}} \supset
{\mathscr{Der}_{k_0}} \supset {\mathscr{Der}_{k_0-1}}
\supset \dots \supset {\mathscr{Der}_1} \supset {\mathscr{Der}_0}\,.
\end{equation}
\medskip\noindent
Note that $\mathscr{Der}_0$ consists only of the
trivial group, so once that class has been reached,
we have found an asymmetric coloring.
So we have reduced the proof of Tucker's Conjecture to the
following result.
\begin{theorem} \label{thm:program}
There exists a positive integer $k_0$ such that the
following holds.\\
Let $\mathcal Q\supset\mathcal R$ be a pair of consecutive terms in
the chain~\eqref{eq:chain-of-classes}. Then
$\mathcal Q$ is 2-color-reducible to $\mathcal R$.
\end{theorem}
The rest of the paper describes the proof of this result.
\section{Definitions, notation: group theory} \label{sec:defs}
For the rest of this paper, {\bf all groups will be finite},
except where expressly stated otherwise.
We use the notation $[n]=\{1,\dots, n\}$ for integers $n\ge 0$.\\
For groups $G,H$, the notation $H\le G$ indicates that $H$ is
a subgroup, and $H < G$ indicates a proper subgroup.
The notation $N\,\triangleleft\, G$ indicates a (not necessarily proper)
normal subgroup.
An \emph{epimorphism} (surjective homomorphism)
from $G$ onto $H$ is indicated as $G\twoheadrightarrow H$.
For $K\subseteq G$ a subset of the group $G$ we
denote the centralizer of $K$ in $G$ by $\mathbb C_G(K)$.
The center of $G$ is $Z(G)= \mathbb C_G(G)$.
Let $G$ be a group.
The \emph{commutator} of $h,k\in G$ is the element
$[h,k]=h^{-1}k^{-1}hk$. If $H,K\le G$ then
$[H,K]$ denotes the subgroup generated by all commutators
$[h,k]$ for $h\in H$, $k\in K$.
The \emph{commutator subgroup} or \emph{derived subgroup} of
the group $G$ is $[G,G]$, also denoted $G'$. The members of
the \emph{derived series} are denoted $G^{(k)}$ where
$G^{(0)}=G$ and $G^{(k+1)}=(G^{(k)})'$. A group $G$ is
\emph{perfect} if $G'=G$. Every finite group $G$ contains
a unique largest perfect subgroup, called the
\emph{perfect core} of $G$, reached when the
derived series stabilizes: $G^{(k+1)}=G^{(k)}$.
We denote the perfect core of $G$ by $G^{(\infty)}$.\
The group $G$ is solvable if and only if its perfect core is
the identity.
The \emph{socle} of a group $G$, denoted $\soc(G)$,
is the product of its minimal normal subgroups.
An \emph{almost simple} group is a group $G$ of the form
$N\,\triangleleft\, G\le \aut(N)$ where $N$ is a nonabelian simple group.
In this case, $N$ is the unique minimal normal subgroup of $G$
and therefore it is the socle of $G$.
The group $\inn(G)$ of inner automorphisms of $G$
consists of the conjugations by elements of $G$.
The \emph{outer automorphism group} of a group $G$
is the quotient $\out(G)=\aut(G)/\inn(G)$.
\emph{Schreier's Hypothesis} states that the
outer automorphism groups of all finite simple groups
are solvable. This is equivalent to saying that the
perfect core of an almost simple group is simple.
Schreier's Hypothesis is a known consequence of the
Classification of Finite Simple Groups.
For a group action $G\curvearrowright\Omega$ (see Def.~\ref{def:action})
we usually reserve the letter $n$ for $|\Omega|$, the \emph{degree}
of the action. For additional definitions and notation about group
actions, see Section~\ref{sec:def1}.
\section{Solvable colorings of primitive groups}
\label{sec:solvable}
In this section we build one of our main tools for the
proof of Tucker's conjecture.
\begin{definition}
For a permutation group $G\le \sym(\Omega)$, let $\solv(G)$
denote the minimum number $k$ of colors such that $G$
admits a solvable $k$-coloring. We call $\solv(G)$ the
\emph{solvable coloring number} of $G$.
\end{definition}
\begin{obs} \label{obs:ps-closed}
Let $G,H \le\sym(\Omega)$.
\begin{itemize}
\item[(a)] If $H\le G$ then $\solv(H)\le \solv(G)$.
\item[(b)] Let the orbits of $G$ be $\Omega_1,\dots,\Omega_k$
and let $G_i$ be the restriction of $G$ to $\Omega_i$.
Then $\solv(G)=\max_i \solv(G_i)$.
\item[(c)] For $x\in\Omega$, let $G(x)$ denote the action of
the stabilizer $G_x$ on $\Omega\setminus {x}$. Then
$\solv(G(x))\le \solv(G)$.
\end{itemize}
\end{obs}
\begin{proof} (a) A solvable coloring for $G$ is also a solvable
coloring for any subgroup of $G$.\quad (b) We can color
each orbit independently, observing that $G$ is a subdirect
product of the $G_i$. \quad (c) First, $\solv(G_x)\le\solv(G)$
by (a). Second, $\solv(G(x))=\solv(G_x)$ by (b).
\end{proof}
\begin{remark} \label{rem:ps-closed}
This observation would remain true if we replaced ``solvable groups''
by any PS-closed class of groups (closed under direct products
and subgroups) such as nilpotent groups, $2$-groups, groups
with composition factors of order $\le c$ or
solvable groups of derived length $\le c$ for some constant $c$.
\end{remark}
\bigskip
\begin{theorem}[Solvable coloring number of primitive groups]
\label{thm:prim3col}
Let $G\le \sym(\Omega)$ be a primitive permutation group
of degree $n=|\Omega|$.
\begin{itemize}
\item[(I)] \ If $G$ is solvable then $\solv(G)=1$.
\item[(II)] \ If $\alt(\Omega)\le G\le \sym(\Omega)$ then
$\solv(G) = \lceil n/4 \rceil$.
\item[(III)]\ In all other\footnote{In
a previous version of this paper, I listed $M_{24}$ as a
possible exception. I am grateful to
Saveliy Skresanov~\cite{skresanov}
for pointing out that the case of $M_{24}$ was settled by
Chang Choi in 1972~\cite{choi}. In the previous version I
proved $\solv(M_{24})\le 3$, which suffices for our main results.}
cases, $\solv(G)=2$.
\end{itemize}
\end{theorem}
Items (I) and (II) are straightforward. (Regarding (II),
each color must occur at most 4 times.)
We discuss the Mathieu groups in Sec.~\ref{sec:mathieu}.
The bulk of Section~\ref{sec:solvable} concerns the proof
of item (III).
It was shown by Cameron, Neumann, and Saxl~\cite{saxl} in 1984
by a simple counting argument that all but a finite number of
primitive groups admit an asymmetric 2-coloring. Gluck~\cite{gluck}
(1983) and Seress~\cite{seress97} (1997) classified the exceptions:
Gluck for the case when $G$ is solvable and Seress for
the non-solvable case. Seress gives their combined
list of 43 groups~\cite[Theorem 2]{seress97}.
We shall rely on Seress's list of non-solvable
exceptions.
\begin{remark}
For the proof of our main result, solvable colorings
of almost simple primitive groups are not required.
We include this part of the result for completeness.
In particular, the hard part of Seress's result,
the classification of those almost simple primitive
groups that admit an asymmetric 2-coloring, is not
required.
\end{remark}
We begin with a straightforward but useful observation.
\begin{obs} \label{obs:ptwise}
Let $H\le \sym(\Omega)$ be a permutation group
and $\Phi\subseteq\Omega$ an $H$-invariant subset
such that
(a) the image of the action $H\curvearrowright\Phi$
is solvable and (b) the pointwise stabilizer
$H_{(\Phi)}$ is solvable.
Then $H$ is solvable. In particular, condition (a)
is met if $|\Phi|\le 4$. \hfill $\Box$
\end{obs}
First we consider three particular classes of primitive groups:
affine groups, the projective linear groups, and the
Mathieu groups. Our proof for these classes is self-contained
and does not rely on Seress's work. (See the footnote for $M_{24}$.)
\subsection{Affine groups}
\label{sec:affine2col}
\begin{prop}
Let $G$ be a primitive permutation group with an
elementary abelian normal subgroup. Then $G$ admits
a solvable 2-coloring.
\end{prop}
\begin{proof}
We need to find a subset $\Delta\subseteq\Omega$
such that $G_{\Delta}$ is solvable.
\medskip\noindent
Let $N$ be the elementary abelian normal subgroup; so
$n=|N|=|\Omega|=p^d$ for some prime $p$ and $d\ge 1$.
$\Omega$ can be viewed as the $d$-dimensional vector
space over $\mathbb F_p$, and $G\le\agl(d,p)$ (the affine
group acting on $\Omega$).
\medskip\noindent
If $d=1$ then $G$ is solvable, so $\Delta=\emptyset$ will do.
\medskip\noindent
Let now $d\ge 2$. Let $e_0=0$ and let $e_1,\dots,e_d$
be a basis of $\Omega$.
\medskip\noindent
If $d=2$ then let $\Delta=\{e_0,e_1\}$.
The pointwise stabilizer
$G_{(\Delta)}$ consists of linear transformations
of $\Omega$, described by triangular matrices (in the basis $\{e_1,e_2\}$),
hence $G_{(\Delta)}$ is solvable. An application of
Obs.~\ref{obs:ptwise} shows that $G_{\Delta}$ is solvable.
\medskip\noindent
For $d\ge 3$ we observe that
$G$ preserves affine relations, i.\,e., relations of the form
$\sum \alpha_i x_i=0$ where $x_i\in\Omega$, $\alpha_i\in\mathbb F_p$,
and $\sum \alpha_i=0$.
\medskip\noindent
By a quadruple we shall mean a set of four elements.
We shall say that quadruple $Q\subseteq\Omega$
satisfies the equation $f(x_1,x_2,x_3,x_4)=0$
if there is a bijection $\beta: \{1,2,3,4\}\to Q$
such that $f(\beta(1),\beta(2),\beta(3),\beta(4))=0$.
\medskip\noindent
If $3\le d \le 6$ then let
$\Delta = \{e_0,e_1,e_2,e_1+e_2, e_3, \dots, e_d\}$.
There is exactly one quadruple of
elements of $\Delta$ satisfying the equation
$x_1+x_2-x_3-x_4=0$, namely, $\{e_1,e_2,e_0,e_1+e_2\}$.
This means $G_{\Delta}$ fixes this quadruple (setwise)
and also the set $\{e_3,\dots,e_d\}$ (setwise).
On the other hand, $G_{(\Delta)}=1$. So an application
of Obs.~\ref{obs:ptwise} to $G_{\Delta}$
shows that $G_{\Delta}$ is solvable.
\medskip\noindent
Assume $d\ge 7$ and let $k=\lfloor (d-1)/2\rfloor$.
Let $\Delta = \{(-1)^{i+1} e_i\mid 0\le i\le d\}\cup
\{e_{2i-1}+e_{2i}+e_{2i+1}\mid 1\le i\le k\}$.
Consider the set $\mathcal H$ of those quadruples
in $\Delta$ that satisfy the equation
$x_1-x_2+x_3-x_4=0$. These are exactly the quadruples
$Q_i:=\{e_{2i-1}, -e_{2i}, e_{2i+1}, e_{2i-1}+e_{2i}+e_{2i+1}
\mid 1\le i\le k\}$.
Let us now consider the graph with vertex set $\Delta$ where
two vertices are adjacent if there is a quadruple in $\mathcal H$
in which both of them participate. This graph is invariant
under $G_{\Delta}$. The graph has one or two isolated
vertices ($e_0$ and, if $d$ is even, $e_d$)
and otherwise consists of a chain of
4-cliques, each one sharing one vertex with the next one.
The automorphism group of this graph is easy to determine;
it has a normal $2$-subgroup of index $9$,
therefore it is solvable. On the other hand, $G_{(\Delta)}=1$,
so $G_{\Delta}$ acts faithfully on our graph and therefore
it is solvable.
\end{proof}
\subsection{Projective linear groups}
\label{sec:proj2col}
\begin{prop}
For $d\ge 2$ and a prime power $q$, the projective linear group
$L_d(q)$ in its natural action on the $(d-1)$-dimensional projective
space over $\mathbb F_q$ admits a solvable 2-coloring.
\end{prop}
\begin{proof}
For $d=2$, the stabilizer of any point is the $1$-dimensional
affine group, which is solvable.
For $d\ge 3$, let $e_i$ denote the standard unit vectors in
$\mathbb F_q^d$ (the $i$-th coordinate $1$, the other coordinates $0$).
For $v\in \mathbb F_q^d\setminus\{0\}$ we write
$[v]=\{\lambda v\mid \lambda\in\mathbb F_q^{\times}\}$ for the equivalence
class representing a point in $PG(d-1,q)$ by its homogeneous coordinates.
For $d=3$, let $\Delta=\{[e_1], [e_2], [e_3], [e_1+e_2+e_3]\}$.
The pointwise stabilizer of $\Delta$ in $N := L_3(q)$ is the identity;
therefore, the setwise stabilizer is $N_{\Delta}\le \sym(\Delta)$
which is solvable.
For $d\ge 4$, let
$\Delta=\{[e_i]\mid 1\le i\le d\}\cup
\{[e_i+e_{i+1}]\mid 1\le i\le d-1\}\cup
\{[f]\}$
where $f=\sum_{i=1}^d e_i\}$.
The pointwise stabilizer of $\Delta$ is the idenity, so we
only need to consider what permutations of $\Delta$ are
feasible under $L_d(q)$. The action of $L_d(q)$ preserves
the underlying matroid (i.\,e., it maps linearly independent
sets to linearly independent sets).
The only linearly dependent triples in $\Delta$ are the triples
of the form $\{[e_i], [e_{i+1}], [e_i+e_{i+1}]\}$, and in
the case of $d=4$, the triple $\{[e_1+e_2], [e_3+e_4], [f]\}$.
So the 3-uniform hypergraph $\mathcal H$ formed by these triples is preserved
by $L_d(q)$. Let us find the degree of each element of $\Delta$
in this hypergraph (i.\,e., how many times each element of $\Delta$
appears in these triples). Let
$\Delta_j = \{u\in\Delta\mid \deg_{\mathcal H}(u)=j\}$.
Each $\Delta_i$ is fixed (setwise) by $N_{\Delta}$.
For $d=4$ we have $\Delta_1=\{[e_1], [e_4], [e_2+e_3], [f]\}$
and $\Delta_2 =\{[e_2], [e_3], [e_1+e_2], [e_3+e_4]\}$.
Therefore $N_{\Delta}\le \sym(\Delta_1)\times \sym(\Delta_2)$, solvable.
For $d\ge 5$ we have $\Delta_0=\{[f]\}$,\
$\Delta_1 = \{[e_1], [e_d]\}\cup \{[e_i+e_{i+1}]\mid 1\le i\le d-1\}$,
and $\Delta_2 = \{[e_2],\dots,[e_{d-1}]\}$.
Let us define the graph $R$ on vertex set $\Delta$ by making
$u, v\in\Delta$ adjacent if $u\neq v$ and $\{u,v\}$ is a
subset of a triple in $\mathcal H$. The induced subgraph $R[\Delta_2]$
is the path $[e_2]--\dots--[e_{d-1}]$, which has only 2 automorphisms,
so the pointwise stabilizer of $\Delta_2$ has index $\le 2$ in
$N_{\Delta}$. Moreover, this pointwise stabilizer also fixes
each point that has two neighbors in this path, so it
can only swap the pair $([e_1],[e_1+e_2])$ and the pair
$([e_d],[e_{d-1}+e_d])$. In sumary, the order of
$N_{\Delta}$ divides $8$, so $N_{\Delta}$ is solvable.
\end{proof}
\subsection{Mathieu groups}
\label{sec:mathieu}
\begin{prop}[Choi] \label{prop:mathieu}
Each of the five Mathieu groups admits a solvable 2-coloring.
\end{prop}
\begin{proof}
We give a simple direct proof in the cases other than
$M_{24}$ and refer Choi~\cite{choi} for $M_{24}$.
Let $G$ be one of the Mathieu groups
$M_{23}$, $M_{22}$, $M_{12}$ and $M_{11}$.
So we can write
$G=M_{m+k}$ where $m\in\{10, 21\}$ and $k=1,2$.
Let $G$ act on $\Omega$ where $|\Omega|=m+k$.
Let $\Delta\subset\Omega$ be a set of $2+k$ elements,
so $|\Delta|\le 4$.
Then the order of the pointwise stabilizer of $\Delta$ is
$|G_{(\Delta)}|=48$ if $m=21$ and $8$ if $m=10$.
Therefore $G_{(\Delta)}$ is solvable, so by
Obs.~\ref{obs:ptwise}, $G_{\Delta}$ is solvable.
The remaining case, $G=M_{24}$, was settled by
Chang Choi in 1972~\cite{choi}.
Choi classified all setwise stabilizers of $M_{24}$.
He found a set of size 8
he denotes by $8'''$ such that $G_{8'''}$ has an
elementary abelian normal subgroup of order 16
with quotient $S_4$ \cite[Prop. 4.1]{choi}.
He also found a set of size 10 he denotes by $10'''$
such that $G_{10'''}\cong S_3\times S_4$ \cite[Prop. 6.3]{choi}.
\end{proof}
\begin{remark} \label{rem:mathieu}
A set equivalent to $10'''$ was found by
Saveliy Skresanov\footnote{Saveliy V. Skresanov,
Sobolev Institute of Mathematics, Novosibirsk.}~\cite{skresanov}
using the GAP computer algebra system, thus providing
independent verification of the fact that $\solv(M_{24}=2$.
Saveliy kindly agreed that I share his code.
\small{
\begin{verbatim}
gap> G := MathieuGroup(24);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23),
(3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16), (1,24)(2,23)
(3,12)(4,16)(5,18)(6,10)(7,20)(8,14)(9,21)(11,17)(13,22)(15,19) ])
gap> S := [1..10];
[ 1 .. 10 ]
gap> StructureDescription(Stabilizer(G, S, OnSets));
"S4 x S3"
\end{verbatim}
}
\end{remark}
\begin{remark}
Of course $\solv(M_{24})=2$ implies $\solv(M_{23})=\solv(M_{22})=2$
(see Obs.~\ref{obs:ps-closed}),
so those observations should also be attributed to Choi.
Let us note that $\solv(M_{23})=2$ immediately implies
$\solv(M_{24})\le 3$, which suffices for our main results.
Indeed, even the weaker statement $\solv(M_{24})\le 5$
would suffice: the only place in the proof of
Theorem~\ref{thm:simple1} (and consequently in the
proof of Theorem~\ref{thm:main})
where a bound on $\solv(M_{24})$ is used
is Case~2a of the proof of Theorem~\ref{thm:simple1}.
So the main results do not depend on Choi's
classification theorem.
\end{remark}
\subsection{Proof of Theorem~\ref{thm:prim3col}}
We refer to Seress's list of primitive groups that do not
admit an asymmetric 2-coloring~\cite[Theorem 2]{seress97}.
Seress lists 43 groups (this includes Gluck's list
of solvable exceptions); we organize the list into
four categories.
We write $n=|\Omega|$ for the degree of $G$.
\begin{theorem}[Seress] \label{thm:seress}
Let $G\le\sym(\Omega)$ be primitive, $G\ngeq \alt(\Omega)$.
Assume $G$ does not admit an asymmetric 2-coloring.
Then $n\le 32$ and $G$ falls into one of the following categories.
\begin{itemize}
\item[(a)] $G$ is solvable,
\item[(b)] $G$ is an affine group, i.\,e., $G$ has an elementary
abelian normal subgroup (so $n$ is a prime power),
\item[(c)] $G$ has degree $n\le 8$\,,
\item[(d)] $G$ is almost simple (so $N\le G\le \aut(N)$ for some
nonabelian simple group $N$).
\end{itemize}
\end{theorem}
In Seress's list, Case (d) falls into the following subcategories.
The list indicates each group $G$ by the pair
$(n,N)$ where $n$ is the degree of $G$ (size of $|\Omega|$)
and $N$ is the (unique, simple) minimal normal subgroup of $G$.
\begin{itemize}
\item[(i)] Mathieu groups in their natural action,
$(n,M_n)$ for $n=11, 12, 22, 23, 24$
\item[(ii)] projective linear groups $L_d(q)$ for some pairs
$(d,q)$ where $d\ge 2$ and $q$ is a prime power,
in their natural action on the $(d-1)$-dimensional
projective space over $\mathbb F_q$ $(n=(q^d-1)/(q-1))$
\item[(iii)] $(10,A_5)$, $(10, A_6)$, $(12, M_{11})$, $(15, A_8)$.
\end{itemize}
\begin{proof}[Proof of Theorem~\ref{thm:prim3col}]\quad \\
Given a primitive group $G\le \sym(\Omega)$ such that
$G\ngeq\alt(\Omega)$, we need to show that $G$ admits
a solvable 2-coloring, except if $G=M_{24}$ in its natural
action then we provide a solvable 3-coloring.
\medskip\noindent
Case (o).\ If $G$ admits an asymmetric 2-coloring, that
is more than sufficient for us. Now we need to eliminate
the finite number of exceptions.
\medskip\noindent
In Case (a), the constant coloring $(\Delta=\emptyset)$
works.
\medskip\noindent
Case (b).\ This case was settled in Sec.~\ref{sec:affine2col}.
\medskip\noindent
Case (c).\ Now $5\le |\Omega|\le 8$. Let $\Delta$ be any
4-subset of $\Omega$. Then, applying Obs.~\ref{obs:ptwise}
to $H:= G_{\Delta}$, it follows that $G_{\Delta}$ is solvable.
\medskip\noindent
Case (d).\ Now $G$ is almost simple, so it has a
unique minimal normal subgroup $N$ which is
nonabelian simple. In particular, the quotient
$G/N \le\out(N)$ is solvable by Schreier's
hypothesis. It follows that it suffices
to find $\Delta\subseteq\Omega$ such that $M:=N_{\Delta}$
is solvable. Indeed, let $L:=G_{\Delta}$. Now $L$ is
solvable because $L/(L\,\cap\, N)$ is a subgroup of $G/N$.
\medskip\noindent
Subcase (i) was settled in Section~\ref{sec:mathieu}.
\medskip\noindent
Subcase (ii).\
In Section~\ref{sec:proj2col} we have shown that all projective
linear groups, in their natural action, admit solvable
2-colorings.
\medskip\noindent
Subcase (iii).
\medskip\noindent
$A_5$ is a minimal simple group, so all proper subgroups
are solvable; we can take any nontrivial subset of $\Omega$
for $\Delta$.
\medskip\noindent
For $(10,A_6)$, the stabilizer of a point has order $36$
and is therefore solvable.
\medskip\noindent
$A_8\cong L_4(2)$, and Seress's example $(15,A_8)$ is the
standard action of $L_4(2)$ on $PG(3,2)$, covered under Subcase (ii).
\medskip\noindent
In the case of $G\cong M_{11}$ acting as a primitive group on
a set of size $|\Omega|=12$, take any four elements, $x,u,v,w\in \Omega$
such that the successive pointwise stabilizers strictly decrease:
$G > G_x > G_{(x,u)} > G_{(x,u,v)} > G_{(x,u,v,w)}$. Let
$\Delta =\{x,u,v,w\}$. Now $|G_x|=|G|/12=660=2^2\cdot 3\cdot 5\cdot 11$.
At each subsequent step, the order drops by at least a prime
factor, in total by at least a factor of $2\cdot 2\cdot 3=12$,
so the order of the pointwise stabilizer of $\Delta$ is
$|G_{(\Delta)}|\le 660/12=55$.
Therefore $G_{(\Delta)}$ is solvable. By Obs.~\ref{obs:ptwise},
it follows that $G_{\Delta}$ is solvable.
\end{proof}
\section{Proof of Theorem~\ref{thm:simple} for primitive groups}
\label{sec:primitive}
We restate this case.
\begin{lemma}
Let $G\le\sym(\Omega)$ be a primitive group,
$T$ a nonabelian simple group, and $\varphi : G\twoheadrightarrow T$
an epimorphism. Then there exists a subset
$\Delta\subseteq \Omega$ such that
$\varphi(G_{\Delta}) < T$.
\end{lemma}
\begin{proof}
\noindent
Case 1. \ $G$ is almost simple.
In this case we claim that for any nontrivial subset
$\Delta\subseteq\Omega$ ($\Delta\neq\emptyset$ and
$\Delta\neq\Omega$) we have $\varphi(G_{\Delta}) < T$.
\medskip\noindent
Indeed, in this case $G$ has a unique minimal normal
subgroup $N$ which is nonabelian simple. In particular,
the quotient $G/N=\out(N)$ is solvable by Schreier's
hypothesis. Let $\ker(\varphi)=K$.
Since $G/K\cong T$, it follows that $K\ngeq N$ and
therefore $K=1$, hence $G=N\cong T$. So for any
nontrivial $\Delta$ we have $|G_\Delta| < |G|=|T|$ and
therefore $\varphi(G_{\Delta}) < T$.
\medskip\noindent
Case 2.\ $G$ is not almost simple. In particular,
$G\ngeq \alt(\Omega)$.
In this case the result is an immediate consequence of
Theorem~\ref{thm:prim3col}. Indeed, now we are in
Case (III)
of Theorem~\ref{thm:prim3col}, so
$G$ admits a solvable 2-coloring, i.\,e., there is a subset
$\Delta\subseteq\Omega$ such that $G_{\Delta}$ is solvable,
and therefore $G_{\Delta}$ has no epimorphism on $T$.
\end{proof}
\begin{remark}
Note that while this proof rests on Seress's classification
of the primitive groups that do not admit an asymmetric 2-coloring,
it avoids any reference to the most difficult part of
Seress's work, the classification of the almost simple groups.
\end{remark}
\section{Reducing the image: Proof of Theorem~\ref{thm:simple}}
\label{sec:general2simple}
\subsection{Three lemmas}
The following lemma may be folklore.
It appears as \cite[Lemma 8.1.1]{quasipoly}
along with an elegant proof, supplied by P\'eter P. P\'alfy
and reproduced below for completeness.
As remarked there, the result can also be derived
from \cite[Lemma 2.8]{meierfrankenfeld}.
\begin{lemma}[Subdirect product lemma] \label{lem:subdirect}
Let $G$ be a subdirect product of the finite groups $H_i$
($i=1,\dots,r$).
Let $\pi_i : G\twoheadrightarrow H_i$ be the corresponding projections.
Let $\varphi: G\twoheadrightarrow T$ be an epimorphism, where
$T$ is a nonabelian simple group. Let $K=\ker(\varphi)$
and $M_i=\ker(\pi_i)$.
Then $(\exists i)(M_i\le K)$.
\end{lemma}
\begin{proof}[Proof by P\'eter P. P\'alfy]
For subgroups $G_1,\dots,G_k\le G$ we use the
notation
\[ [G_1,\dots,G_k] =[\dots [[G_1,G_2],G_3],\dots, G_k]\,.\]
Assume for a contradiction that
$K \ngeq M_i$ for all $i$. Then $M_iK = G$ (because $K$ is a maximal
normal subgroup). It follows that
$[G,\dots,G]=[M_1K,\dots,M_mK]\le K[M_1,\dots,M_m]\le
K\left(\bigcap_{i=1}^m M_i\right) = K$,
so $[G/K,\dots,G/K]=1$, a contradiction because $G/K\cong T$ is
nonabelian simple.
\end{proof}
\begin{obs}[Perfect core lemma] \label{obs:core}
Let $\varphi: G\twoheadrightarrow H$ be an epimorphism of finite groups.
Then $\varphi(G')=H'$.
Consequently, $\varphi(G^{(\infty)})=H^{(\infty)}$.
\hfill $\Box$
\end{obs}
\begin{lemma}[Three normal subgroups lemma] \label{lem:3normal}
Let $H$ be a finite group and $A,B,C$ three normal subgroups
satisfying the following conditions.
\begin{itemize}
\item[(i)] $AB=AC=BC=H$.
\item[(ii)] $H/A$ and $H/B$ are nonabelian simple.
\end{itemize}
Then $B\ngeq A\,\cap\, C$.
\end{lemma}
\begin{proof}
Let $S=H/A$ and $T=H/B$. So $S$ and $T$
are nonabelian simple groups.
Without loss of generality we may assume $A\,\cap\, B\,\cap\, C=1$.
Assume for a contradiction that $B\ge A\,\cap\, C$. So
$1 = A\,\cap\, B\,\cap\, C=A\,\cap\, C$.
Since $A\,\cap\, C=1$ and $AC=H$, we have $H=A\times C$.
Therefore $C\cong H/A\cong S$ and $A\le \mathbb C_H(C)$.
We claim that $B\,\cap\, C=1$. Indeed, given that
$C\cong S$ is simple, the alternative would be
$C\le B$. But this would mean $H=BC=B$,
impossible by Assumption (ii).
Since $B\,\cap\, C=1$, we have $B\le \mathbb C_H(C)$.
Combining this with $A\le \mathbb C_H(C)$ we obtain that
$H=AB\le \mathbb C_H(C)$, i.\,e., $C\le Z(H)$.
But then $C$ must be abelian,
contradicting the isomorphism $C\cong S$.
\end{proof}
\subsection{The proof of Theorem~\ref{thm:simple}}\quad
\medskip\noindent
We restate and augment Theorem~\ref{thm:simple}.
\begin{theorem} \label{thm:simple1}
Let $G\le \sym(\Omega)$, where $\Omega$ is a finite set.
Let $\varphi: G\twoheadrightarrow T$ be an
epimorphism where $T$ is a nonabelian simple group. Then
$(\exists \Delta \subseteq\Omega)(\varphi(G_\Delta) < T)$.
Moreover, $\Delta$ can be chosen
to be a subset of one of the orbits of $G$.
\end{theorem}
\begin{proof}
Let $n=|\Omega|$.
We fix $T$ and proceed by induction on $n$.
The statement is vacuously true if $G$ is solvable;
in particular, if $n\le 4$. Assume now $n\ge 5$.
Let $K=\ker(\varphi)$.
1. First assume $G$ is intransitive, with orbits
$\Omega_1,\dots,\Omega_{r}$. Let $H_i$ denote
the restriction of $G$ to $\Omega_i$, so $H_i\le\sym(\Omega_i)$
and $G$ is a subdirect product of the $H_i$.
Let $M_i$ denote the kernel of the epimorphism $\pi_i:G\twoheadrightarrow H_i$,
i.\,e., the pointwise stabilizer of $\Omega_i$ in $G$.
By the Subdirect product lemma (Lemma~\ref{lem:subdirect}), there
is an $i\le r$ such that $M_i\le K$. This means that $\varphi$ induces
an epimorphism $\overline{\varphi}: H_i\twoheadrightarrow T$.
Note that $\varphi$ is
the composition of $\pi_i$ and $\overline{\varphi}$\,:
\begin{equation} \label{eq:orbits}
\xymatrix{
G \ar[rr]^{\varphi} \ar[dr]^{\pi_i} & & T \\
& H_i \ar[ur]^{\overline{\varphi}}
}
\end{equation}
By induction on $n$,
we find a subset $\Delta\subseteq \Omega_i$ such that
$\overline{\varphi}((H_i)_\Delta) < T$.
By restricting $G$ to $\Omega_i$ we see that
$(H_i)_\Delta=(\pi_i(G))_\Delta = \pi_i(G_\Delta)$.
\medskip\noindent
Now by diagram~\eqref{eq:orbits},
$\varphi(G_\Delta)=\overline{\varphi}(\pi_i(G_\Delta))=\overline{\varphi}((H_i)_\Delta) < T$,
as desired.
\medskip\noindent
2. Assume now that $G$ is transitive, imprimitive.
Let $\mathcal B=\{B_1,\dots,B_k\}$
be a minimal system of imprimitivity (the $B_i$ are maximal blocks
of imprimitivity). So the induced action
$\psi: G \curvearrowright\mathcal B$ is primitive.
Let $N$ be the kernel of this action; so $N$ fixes each block $B_i$
(setwise) and we can naturally identify $\psi(G)$ with $G/N$.
\medskip\noindent
Case 1.\quad $K\ge N$. In this case we have the commutative diagram
\begin{equation} \label{eq:case1}
\xymatrix{
G \ar[rr]^{\varphi} \ar[dr]^{\psi} & & T \\
& G/N \ar[ur]^{\overline{\varphi}}
}
\end{equation}
So we have the epimorphism $\overline{\varphi} : G/N \twoheadrightarrow T$
and $G/N$ is a primitive subgroup of $\sym(\mathcal B)$.
Therefore, by the primitive case of Theorem~\ref{thm:simple},
proved in Section~\ref{sec:primitive}, there exists
$\overline{\Delta}\subseteq \mathcal B$
such that $\overline{\varphi}((G/N)_{\overline{\Delta}}) < T$.
Let $\Delta\subseteq\Omega$ be any subset that,
for all $i$, intersects $B_i$ if and
only if $B_i\in\overline{\Delta}$. Then
$\psi(G_{\Delta})\le (G/N)_{\overline{\Delta}}$ and therefore
$\varphi(G_{\Delta})\le \overline{\varphi}((G/N)_{\overline{\Delta}}) < T$
and we are done.
\medskip\noindent
Case 2.\quad $K\ngeq N$. Since $K$ is a maximal normal subgroup of $G$,
this is equivalent to saying that $KN=G$ and therefore
$N/(N\,\cap\, K)\cong G/K \cong T$, so $\varphi(N)=T$.
In particular it follows that $N$ is not solvable and
therefore
\begin{equation} \label{eq:block5}
|B_i|\ge 5\,.
\end{equation}
\medskip\noindent
Moreover, $G/N = \psi(G)=\psi(KN)=\psi(K)\psi(N)=\psi(K)$.
\medskip\noindent
Let $N_i$ denote the restriction of $N$ to $B_i$, so $N_i\le\sym(B_i)$.
So $N$ is a subdirect product of the $N_i$. Let $M_i$ denote the
kernel of the epimorphism $\rho_i : N\twoheadrightarrow N_i$, i.\,e., the pointwise
stabilizer of $B_i$ in $N$. By the Subdirect product lemma
(Lemma~\ref{lem:subdirect}), there is an $i\le k$ such that
\begin{equation} \label{eq:Mi}
M_i\le K\,.
\end{equation}
This means that $\varphi$ induces an epimorphism
$\sigma_i: N_i\twoheadrightarrow T$.
Let $\widetilde\varphi: N\twoheadrightarrow T$ denote the restriction of $\varphi$ to $N$.
Note that $\widetilde\varphi$ is the composition of $\rho_i$ and $\sigma_i$:
\begin{equation} \label{eq:case2}
\xymatrix{
N \ar[rr]^{\widetilde\varphi} \ar[dr]^{\rho_i} & & T \\
& N_i \ar[ur]^{\sigma_i}
}
\end{equation}
We may assume $i=1$.
(Actually, given that $G$ acts transitively on the blocks and
therefore on the $M_i$, we in fact have that $M_i \le K$ for all $i$,
hence the diagram~\eqref{eq:case2} holds for all $i$. But we shall
not need this fact.)
Since $N \le G_{B_1}$, we have $\varphi(G_{B_1})=T$.
We now split the proof into two cases.
\medskip\noindent
Case 2a.
\quad The action $G/N\curvearrowright \mathcal B$ admits
a solvable 5-coloring.
\medskip\noindent
By Theorem~\ref{thm:prim3col}, this case covers all primitive groups
except the alternating groups $A_k$ in their natural action for $k \ge 21$.
(Actually, we can claim a solvable 3-coloring in all these cases
except $A_k$ for $k\ge 13$. We shall not use the full force of
Theorem~\ref{thm:prim3col}.)
\medskip\noindent
By definition, in this case there exists a 5-coloring
$\gamma_0 : \mathcal B\to [5]$ such that the group $(G/N)_{\gamma_0}$
is solvable. Let $\gamma_1$ be the lifting of $\gamma_0$
to $\Omega$, i.\,e., for $x\in B_i$ we set $\gamma_1(x)=\gamma_0(B_i)$.
So under $\gamma_1$, each block is monochromatic.
Moreover, $\psi(G_{\gamma_1})=(G/N)_{\gamma_0}$.
Therefore, $N\cdot G_{\gamma_1}/N$ is solvable.
\medskip\noindent
Let now $\Delta_1\subseteq B_1$ be such that
$\sigma_1((N_1)_{\Delta_1}) < T$.
Such $\Delta_1$ exists by induction on $n$. Let $t:=|\Delta_1|$.
Let $\beta : [5] \to \{0,1,\dots,5\}\setminus \{t\}$
be an injection.
For $i=2,\dots,k$ let $\Delta_i\subseteq B_i$
be an arbitrary subset of size $\beta(\gamma_0(B_i))$.
For this we need $|B_i|\ge 5$, which holds by Eq.~\eqref{eq:block5}.
\medskip\noindent
Let $\Delta = \bigcup_{i=1}^k \Delta_i$. What we have done
was coding the colors of $\gamma_0$ by the sizes of the subsets
$\Delta_i$, taking care to have $\Delta_1$ the only one among the
$\Delta_i$ to have size $t$.
\medskip\noindent
We observe that $G_{\Delta}\le G_{\gamma_1}$
and therefore $NG_{\Delta}/N$ is solvable.
\medskip\noindent
Claim A. \quad $\varphi(G_{\Delta}) < T$.
\begin{proof}
Suppose for a contradiction that $\varphi(G_{\Delta}) = T$.
Then by Obs.~\ref{obs:core}, the perfect core of
$G_{\Delta}$ still maps onto $T$: \quad
$\varphi((G_{\Delta})^{(\infty)}) = T$. But
$(NG_{\Delta})^{(\infty)}/N=1$ (because $NG_{\Delta}/N$ is solvable),
hence $(G_{\Delta})^{(\infty)}\le N$, so
$\varphi((G_{\Delta})^{(\infty)})=\sigma_1\rho_1(G_{\Delta})^{(\infty)})
\le \sigma_1((N_1)_{\Delta_1}) < T$ by the choice of $\Delta_1$,
a contradiction, proving Claim~A and thereby completing the proof
of the Theorem in Case 2a.
\end{proof}
\medskip\noindent
Case 2b.\quad $G/N=\alt(\mathcal B)$ or $\sym(\mathcal B)$ and $k\ge 21$.
\medskip\noindent
Let $G_1 :=G_{B_1}$. If $\varphi(G_1) < T$ then select $\Delta := B_1$
and we are done. Henceforth we assume $\varphi(G_1)=T$.
The kernel of the epimorphism $\varphi_1 : G_1\twoheadrightarrow T$
(the restriction of $\varphi$ to $G_1$) is $K_1 :=K\,\cap\, G_1$.
\medskip\noindent
Let $G^*$ denote the restriction of $G_1$ to $B_1$, so $G^*\le\sym(B_1)$.
Let $L$ denote the kernel of the epimorphism
$\pi : G_1\twoheadrightarrow G^*$, so $L$ is the pointwise stabilizer
of $B_1$ in $G_1$, \quad $L=(G_1)_{(B_1)}$.
\medskip\noindent
Case 2b1.\quad $L \le K_1$.\quad
\medskip\noindent
Now $\varphi_1 : G_1\twoheadrightarrow T$ factors across
$G^*$:\quad $\varphi_1 = \tau\pi$, hence $\tau(G^*)=T$.
\medskip\noindent
Let $\Delta \subseteq B_1$ be such that $\tau((G^*)_{\Delta}) < T$.
Such a subset exists by induction on $n$.
Note that $\Delta$ is a nontrivial subset of $B_1$
since $\tau(G^*)=T$.
\medskip\noindent
Claim B. \quad $\varphi(G_{\Delta}) < T$.
\begin{proof}
Since $\Delta\subseteq B_1$ and $\Delta\neq\emptyset$,
we have $G_{\Delta}\le G_1$, so $G_{\Delta}=(G_1)_{\Delta}$.
Observe also that $\pi(G_1)_{\Delta}=(G^*)_{\Delta}$.
Therefore $\varphi(G_{\Delta})=\varphi_1((G_1)_{\Delta})=
\tau((G^*)_{\Delta}) < T$.
This proves Claim~B and thereby completes the proof of
Case 2b1.
\end{proof}
\medskip\noindent
Case 2b2.\quad $L\nleq K_1$.
\medskip\noindent
Since $G_1/K_1\cong T$ is simple, $K_1$ is a maximal
normal subgroup in $G_1$, therefore $K_1L=G_1$.
\medskip\noindent
Case 2b2i. \quad $N\le K_1$.
\medskip\noindent
In this case we are in a situation analogous to
Case 1: $\varphi_1$ factors across $N$, i.\,e., there exists
$\overline{\varphi}_1 : G_1/N \to T$ such that
$\varphi_1 = \overline{\varphi}_1\psi_1$ where $\psi_1 : G_1\twoheadrightarrow G_1/N$.
It follows that $\overline{\varphi}: G_1/N\twoheadrightarrow T$ is an epimorphism,
but $G_1/N$ is symmetric or alternating of degree $\ge 20$,
so $\overline{\varphi}$ is an isomorphism. Let $\Delta=B_1\cup \{x\}$
for an arbitrary $x\in B_2$. Now $G_{\Delta}$ fixes
both $B_1$ and $B_2$ setwise, therefore $NG_{\Delta}/N$
has order less than $|NG_1/N|=|T|$, consequently
$|\varphi(G_{\Delta})|=|\overline{\varphi}_1(NG_{\Delta}/N)| < |T|$
and we are done with Case 2b2i.
\medskip\noindent
Case 2b2ii. \quad $N\nleq K_1$ and $G/N = \alt(\mathcal B)$.\\
We claim that this case cannot occur.
\medskip\noindent
Again because $K_1$ is a maximal normal subgroup of $G_1$,
we have $K_1N = G_1$.
\medskip\noindent
Recall that $M_1 = N_{(B_1)}$, the pointwise stabilizer of
$B_1$ in $N$. Moreover, we are in Case~2, so
$M_1\le K$ by Eq.~\eqref{eq:Mi}, and therefore $M_1\le K_1$.
\medskip\noindent
Recall that $L=(G_1)_{(B_1)}$. Therefore $M_1 = N\,\cap\, L$.
So if $L\le N$ then $L=M_1\le K_1$,
contradicting the assumption that put us in Case 2b2.
Therefore $L\nleq N$.
But now $N$ is a maximal normal subgroup in $G_1$
(since $G_1/N\cong A_{k-1}$), so $NL=G_1$.
\medskip\noindent
Summarizing, we have $NK_1=NL=K_1L=G_1$ and $G_1/N\cong A_{k-1}$
and $G_1/K_1\cong T$; these quotients are nonabelian simple.
Setting $H:=G_1$, $A:=N$, $B:=K_1$, and $C:=L$, all assumptions
of Lemma~\ref{lem:3normal} are satisfied. The conclusion is
that $K_1\ngeq N\,\cap\, L$. But $N\,\cap\, L=M_1$ and we know that
$M_1\le K_1$, a contradiction, proving that Case 2b2ii cannot
occur.
\medskip\noindent
Case 2b2iii. \quad $N\nleq K_1$ and $G/N = \sym(\mathcal B)$.
\medskip\noindent
In this case we have $\psi(G)= \sym(\mathcal B)$. Let
$\widetilde{G} = \psi^{-1}(\alt(\mathcal B))$. Replace $G$ by
$\widetilde{G}$; then we land in Case~2b2ii, which is impossible.
This completes the proof of this last case and with it
the proof of Theorems~\ref{thm:simple1} and~\ref{thm:simple}.
\end{proof}
\section{Reduction to the solvable case}
\label{sec:red-to-solvable}
First we extend Theorem~\ref{thm:simple} to all non-solvable
target groups.
\begin{theorem}[Reducing non-solvable image]\
\label{thm:nonsolvable}
Let $G\le \sym(\Omega)$. Let $\varphi: G\twoheadrightarrow H$ be an epimorphism
where $H$ is a non-solvable group. Then
$(\exists \Delta \subseteq\Omega)(\varphi(G_\Delta) < H)$.
\end{theorem}
\begin{obs}\ \label{obs:inf-epi}
If $\varphi : G\twoheadrightarrow H$ is an epimorphism then
$\varphi(G')=H'$. It follows that $\varphi(G^{(\infty)})=H^{(\infty)}$.
\hfill $\Box$
\end{obs}
\begin{proof}[Proof of Theorem~\ref{thm:nonsolvable}]
Since $H$ is not solvable,
there is an epimorphism $\psi : H^{(\infty)}\twoheadrightarrow T$ where
$T$ is nonabelian simple. Composing $\psi$ with $\varphi$ we obtain
an epimorphism $\xi : G^{(\infty)} \twoheadrightarrow T$. By Theorem~\ref{thm:simple}
there exists
$\Delta\subseteq\Omega$ such that $\xi((G^{(\infty)})_\Delta) < T$
and therefore $\varphi((G^{(\infty)})_\Delta) < H^{(\infty)}$.
We claim that this set $\Delta$ satisfies the conclusion
of the Lemma, i.\,e., $\varphi(G_\Delta) < H$.
Let $L=G_\Delta$. Observe that
$L^{(\infty)}\le G^{(\infty)}\,\cap\, L=(G^{(\infty)})_\Delta$.
Therefore $\varphi(L^{(\infty)})\le \varphi((G^{(\infty)})_\Delta)
< H^{(\infty)}$. It follows (by Observation~\ref{obs:inf-epi})
that $\varphi(L) < H$.
\end{proof}
\section{Reduction to bounded derived length}
\label{sec:bded-derived-length}
\begin{theorem} \label{thm:bded-orbits}
There exists a constant $c$ such that the following holds.
If $G\le\sym(\Omega)$ is solvable then there exists
$\Delta\subset\Omega$ such that all orbits
of $G_{\Delta}$ have length $\le c$. In particular,
the derived length of $G_{\Delta}$ is less than $2c$.
\end{theorem}
The bound $2c$ comes from \cite{chains} where it is shown
that the length of any subgroup chain in $S_n$ is less than $2n$.
A far stronger bound on the length of the derived chains
of permutation groups is known;
Dixon~\cite{dixon} has shown that the derived length of
a solvable group of degree $n$ is at most $(5\log_3 n)/2$.
However, we do not need any of this, all we need is that
groups of bounded order have bounded derived length, which
is obvious.
We devote the rest of this section to proving Theorem~\ref{thm:bded-orbits}.
We say that a $k$-coloring $\gamma: \Omega\to [k]$
is \emph{uniform} if $|\gamma^{-1}(i)|=n/k$ for each $i\in [k]$.
Note that in a non-uniform $k$-coloring we permit some of the colors
not to be used.
\begin{lemma} \label{lem:solvprim5color}
Let $G\le \sym(\Omega)$ be a primitive solvable group.
Then $G$ admits a non-uniform asymmetric 5-coloring.
\end{lemma}
\begin{proof}
Gluck proves \cite[Theorem 1]{gluck} that for $n\ge 10$,
$G$ admits an asymmetric 2-coloring.
We can view such a coloring as a non-uniform 3-coloring
by adding an empty color.
(In fact, Gluck proves the existence of a non-uniform
2-coloring for $n\ge 10$.)
To address the remaining cases directly, we observe that,
since $G$ is primitive and solvable, we are in the affine case:
$\Omega$ can be identified with the vector space $\mathbb F_p^d$
for some prime $p$ and $d\ge 1$, and $G\le \agl(d,p)$.
Let us assign distinct colors to $0$ and the $d$ vectors in
a basis, and one more color to the rest of the space.
This coloring is clearly asymmetric, and it uses
at most $d+2$ colors. If $n\le 9$ then $d\le 3$, so
we are using at most $5$ colors.
This coloring is not uniform for any value $n\le 9$.
(Note that we throw in one or more empty color classes
for $n\le 4$.)
\end{proof}
\begin{remark}
The bound $5$ in the Lemma is tight, as shown by $G=S_4$.
\end{remark}
Now we extend this result to all solvable permutation groups,
dropping the condition of non-uniformity.
\begin{lemma} \label{lem:solv5color}
Let $G\le \sym(\Omega)$ be a solvable group.
Then $G$ admits an asymmetric 5-coloring.
\end{lemma}
\begin{remark} \label{rem:5colorstight}
The bound $5$ in the Lemma is tight, as shown by the wreath
product $G=S_4\wr H$ where $H$ is any nontrivial permutation
group. ($H$ is the top group in the wreath product.)
\end{remark}
\begin{definition}
Let $G\le\sym(\Omega)$ and let $\gamma_1,\gamma_2 :\Omega\to\Sigma$
be two colorings. We say that a permutation $\pi\in\sym(\Omega)$
is a \emph{$G$-isomorphism} of $\gamma_1$ to $\gamma_2$ if
for all $x\in\Omega$, $\gamma_2(\pi(x))=\gamma_1(x)$.
\end{definition}
\begin{obs} \label{obs:permute-colors}
If a group $G$ admits a non-uniform asymmetric $k$-coloring
then by permuting the colors, we obtain at least $k$
non-$G$-isomorphic non-uniform $k$-colorings.
\hfill $\Box$
\end{obs}
\begin{proof}[Proof of Lemma~\ref{lem:solv5color}]
Without loss of generality we may assume $G$ is transitive.
(Otherwise, select a coloring of each orbit and combine them.)
We proceed by induction on $n=|\Omega|$.
If $G$ is primitive, we are done by
Lemma~\ref{lem:solvprim5color}.
Let now $G$ be imprimitive and
let $\mathcal B=\{B_1,\dots,B_k\}$ be a maximal system
of imprimitivity (the blocks are minimal). Let $|B_i|=t$.
Let $G_i$ denote the restriction of the setwise stabilizer
$G_{B_i}$ to $B_i$, so $G_i\le\sym(B_i)$. Since $B_i$
is a minimal block, $G_i$ is primitive. The $G_i$ are
equivalent permutation groups (equivalent under the
action of $G$). Let $\delta_{1,1},\dots,\delta_{1,5}$ be five
non-$G_1$-isomorphic non-uniform $G_1$-asymmetric colorings
of $B_1$, and let $\delta_{i,1},\dots,\delta_{i,5}$ be
colorings of $B_i$ obtained by applying an element
$\sigma_i$ to our colorings of $B_1$ where $\sigma_i(B_1)=B_i$.
(Let $\sigma_1$ be the identity.) Let $\widehat{\delta}_{i,j}$
be the 6-coloring of $\Omega$ obtained by adding a 6th color
everywhere outside $B_i$. It should be clear that
if $\widehat{\delta}_{i,j}$ is $G$-isomorphic to
$\widehat{\delta}_{i',j'}$ then $j=j'$, simply by counting the
multiplicities of colors.
Let $\widetilde{G}$ denote the image of the $G\curvearrowright\mathcal B$ action,
so $\widetilde{G}\le\sym(\mathcal B)$. By induction,
let $\widetilde{\gamma} : \mathcal B\to [5]$
be a non-uniform asymmetric 5-coloring for $\widetilde{G}$.
We encode these colors by non-isomorphic colorings
of the blocks: We define the 5-coloring
$\gamma : \Omega\to [5]$ by setting, for each
$x\in B_i$,\ $\gamma(x)=\delta_{i,j}(x)$ where $j=\widetilde{\gamma}(B_i)$.
Assume $\sigma\in G$ preserves the coloring $\gamma$. Since
the coloring $\widetilde{\gamma}$ can be reconstructed from $\gamma$,
we infer that $\sigma(B_i)=B_i$ for every $i$. Now
$\sigma_{|B_i}\in G_i$, and the coloring $\gamma|_{B_i}=\delta_{i,j}$
is $G_i$-asymmetric, so $\sigma=1$.
\end{proof}
\begin{lemma} \label{lem:many2colorings}
There exist $\epsilon > 0$
and a threshold $n_1$
such that the following holds. If $G\le \sym(\Omega)$ is a
primitive solvable permutation group of degree $n \ge n_1$
then for every $j$ in the interval $n^{1-\epsilon}\le j \le n/2$
there exists a $G$-asymmetric subset $\Delta_i\subseteq\Omega$
of size $|\Delta_j|=j$.
\end{lemma}
\begin{proof}
The proof consists of drawing the required conclusion from Gluck's
counting argument~\cite{gluck}. We review the argument.
Since $G$ is primitive and solvable, we are in the affine case:
$\Omega$ can be identified with the $d$-dimensional vector
space over $\mathbb F_p$ for some prime $p$ and $d\ge 1$,
and with this identification, $G\le \agl(d,p)$.
It follows that the fixed points of any element of $G$
form an affine subspace of $\Omega$ and therefore the minimal
degree of $G$ is $\ge n/2$ and consequently every
non-identity element $\sigma\in G$ decomposes into at
most $3n/4$ cycles. It follows that the number of subsets
fixed by $\sigma$ is at most $2^{3n/4}$ and therefore
the number of subsets that are not $G$-asymmetric is at most
$|G|\cdot 2^{3n/4}$.
Invoking the P\'alfy--Wolf Theorem~\cite{palfy, wolf}
that states that the order of a
primitive solvable group of degree $n$ is less than $n^C$
for some constant $C$, the stated conclusion follows.
\end{proof}
We shall only use a weak corollary of this result.
\begin{corollary} \label{cor:5asymm}
There exists a constant $c_0$ such that every primitive
solvable permutation group $G$ of degree $\ge c_0$ admits
at least five $G$-asymmetric subsets of different sizes.
\hfill $\Box$
\end{corollary}
We are now ready to prove the main result of this section.
\begin{proof}[Proof of Theorem~\ref{thm:bded-orbits}]
This proof is not by induction, but we follow the scheme
of the inductive step in the proof of Lemma~\ref{lem:solv5color}.
Like there, we may assume $G$ is transitive.
\medskip\noindent
We shall refer to the constant $c_0$ from Cor.~\ref{cor:5asymm}.
We claim that there exists a set $\Delta\subseteq\Omega$
such that the orbits of $G_{\Delta}$ have length $\le c:=3c_0$.
\medskip\noindent
A $G$-asymmetric subset does more than required (orbits of length 1),
so if $G$ is primitive and $n\ge c_0$ then we are done by
Cor.~\ref{cor:5asymm}. In fact, by Gluck~\cite{gluck}, we are
done for all $n\ge 10$.
Let now $G$ be imprimitive and
let $\mathcal B=\{B_1,\dots,B_k\}$ be a system
of imprimitivity. Let $|B_i|=t$.
Let $G_i$ denote the restriction of the setwise stabilizer
$G_{B_i}$ to $B_i$, so $G_i\le\sym(B_i)$.
Let $\widetilde{G}$ denote the image of the $G\curvearrowright\mathcal B$ action,
so $\widetilde{G}\le\sym(\mathcal B)$. Using Lemma~\ref{lem:solv5color},
let $\gamma : \mathcal B\to [5]$ be a $\widetilde{G}$-asymmetric
5-coloring of $\mathcal B$.
\medskip\noindent
Case 1.\quad $4\le t \le 3c_0$. In this case we claim
there exists $\Delta\subseteq\Omega$ such that the orbits
of $G_{\Delta}$ have length $\le t\le 3c_0$.
\medskip\noindent
In this case, let $\Delta_i$ be an arbitrary subset of
$B_i$ of size $\gamma(B_i)-1$. Let $\Delta = \bigcup_{i=1}^k \Delta_i$.
Assume $\sigma\in G$ preserves the set $\Delta$. Since
the coloring $\gamma$ can be reconstructed from $\Delta$
simply by the sizes of the $\Delta_i$,
we infer that $\sigma(B_i)=B_i$ for every $i$.
Therefore, the orbits of $G_{\Delta}$ are subsets of the $B_i$
and therefore have length $\le t < 3c_0$. This completes
the proof in Case 1.
\medskip\noindent
Case 2.\quad $t \ge c_0$ and
the $B_i$ are minimal blocks of imprimitivity.
In this case we claim
there exists an asymmetric $\Delta\subseteq\Omega$,
so the orbits of $G_{\Delta}$ have length 1.
\medskip\noindent
Let $j_1 <\dots < j_5$ be five different sizes of $G_1$-asymmetric
subsets of $B_1$.
Let $\Delta_i\subseteq B_i$ be a $G_i$-asymmetric subset of
size $j_{\gamma(B_i)}$. Let $\Delta = \bigcup_{i=1}^k \Delta_i$.
Assume $\sigma\in G$ preserves the set $\Delta$. Since
the coloring $\gamma$ can be reconstructed from $\Delta$
simply by the sizes of the $\Delta_i$,
we infer that $\sigma(B_i)=B_i$ for every $i$. Now
$\sigma_{|B_i}\in G_i$, and the set $\Delta_i = \Delta\,\cap\, B_i$
is $G_i$-asymmetric, so $\sigma=1$.
This completes the proof in Case 2.
\medskip\noindent
Case 3.\quad In all the remaining cases, we claim
there exists $\Delta\subseteq \Omega$ such that
the orbits of $G_{\Delta}$ have length $\le t\le 3$.
\medskip\noindent
Now there is no block of size $4\le t < 3c_0$ (Case~1)
and also no minimal block of size $t \ge c_0$ (Case~2).
This means all minimal blocks have size $2\le t\le 3$
and the next smallest blocks have size $\ge 3c_0$.
\medskip\noindent
Let $\mathcal B=\{B_1,\dots,B_k\}$ be a system of minimal blocks,
so $2\le |B_i|\le 3$. As before, let $\widetilde{G}$ be the
image of the action $G\curvearrowright\mathcal B$.
\medskip\noindent
Let further $\mathcal D=\{D_1,\dots,D_{\ell}\}$ be a system of
imprimitivity that is a maximal coarsening of $\mathcal B$,
i.\,e., $\mathcal B$ is a refinement of $\mathcal D$ and the $D_j$
are minimal among the blocks of imprimitivity that strictly
include some $B_i$. Let $|D_j|=st$. Let
$\widetilde{D}_j = \{B_i\in\mathcal B\mid B_i\subseteq D_j\}$.
Now the $\widetilde{D}_j$ are minimal blocks for $\widetilde{G}$
and their size is $s \ge c_0$. Therefore, by Case~2,
we have a $\widetilde{G}$-asymmetric set $\widetilde{\Delta}\subseteq \mathcal B$.
\medskip\noindent
Let us lift $\widetilde{\Delta}$ to $\Omega$ by setting
$\Delta = \bigcup \{B_i \mid B_i\in\widetilde{\Delta}\}$.
Now $G_{\Delta}$ fixes each $B_i$ setwise, so
its orbits have length $\le 3$. This
completes the proof of Theorem~\ref{thm:bded-orbits}.
\end{proof}
\section{From bounded derived length to asymmetry:
reducing the derived length}
\label{sec:reducing-derived-length}
We restate Prop.~\ref{prop:derivedlength-intro}
\begin{prop} \label{prop:derivedlength}
Let $G\le \sym(\Omega)$ be a solvable group with derived length $k\ge 1$.
Then there exists a subset $\Delta\subseteq \Omega$ such that
the derived length of $G_{\Delta}$ is at most $k-1$.
\end{prop}
We begin with the abelian case.
\begin{obs}
All abelian permutation groups admit an asymmetric 2-coloring.
\end{obs}
\begin{proof}
We need to construct a $G$-asymmetric subset of
the domain $\Omega$.
Let $R_1,\dots,R_k$ be the orbits of $G$ and let $\Delta$ be a
transversal of the orbits, i.\,e., $\Delta\subseteq\Omega$ and
$|\Delta\,\cap\, R_i|=1$ for all $i$.\\
We claim that $|G_{\Delta}|=1$. Indeed, the restriction of
$G$ to $R_i$ is a transitive abelian group which therefore is
regular, so fixing one of its points fixes the entire orbit
pointwise.
\end{proof}
\begin{proof}[Proof of Prop.~\ref{prop:derivedlength}]
\medskip\noindent
Let $H:=G^{(k-1)}$ be the last nontrivial term in the derived series
of $G$. So $H$ is abelian. Let $\Delta\subseteq\Omega$ be such
that $|H_{\Delta}|=1$. We claim that the derived length of
$G_{\Delta}$ is at most $k-1$.
Indeed, let $L=G_{\Delta}$. So $L^{(k-1)}\le L\,\cap\, G^{(k-1)} =
L\,\cap\, H = H_{\Delta} = 1$.
\end{proof}
This completes the proof of Theorem~\ref{thm:main1} and with it
the proof of our main result, Theorem~\ref{thm:main}.
\section{An effective version?} \label{sec:effective}
In this section we consider finite inverse sequences, of
length $k$, of finite permutation groups. Such a system is defined by
a sequence of $k+1$ groups, $(G_i : i=0,\dots,k)$ and a
sequence of $k$ homomorphisms, $\varphi_i : G_i\to G_{i-1}$, $i=1\dots, k$.
We denote such a system as $(G_i,\varphi_i)_{i\le k}$. Here
$G_i\le \sym(\Omega_i)$. For $i\ge j$, the transition homomorphism
$\varphi_{i,j} : G_i \to G_j$ is defined as the composition of
$\varphi_{\ell,\ell-1}$ for $\ell=i,\dots,j+1$. Now the definition
of inverse limits applies; in particular, strands are defined
as in Def.~\ref{def:strand} and they form the inverse limit
$\mathcal G=\varprojlim G_i$. We say that the inverse system, and
the inverse limit, \emph{ends} at $G_0$.
If the $\Omega_i$ are disjoint then
we view $\mathcal G$ as a permutation group acting on
$\Omega:=\bigcup_{i=0}^k \Omega_i$.
We say that a coloring $\gamma:\Omega\to\Sigma$ is
\emph{zero-asymmetric} if $\mathcal G_{\gamma}$ fixes $\Omega_0$
pointwise. We say that the coloring is \emph{zero-neutral}
if $\Omega_0$ is monochromatic, i.\,e., $|\gamma(\Omega_0)|=1$.
If all the $\varphi_i$ (and therefore all the $\varphi_{i,j}$)
are epimorphisms, we speak of an \emph{epimorphic
inverse sequence.}
Let $\mathscr{Gr}$ denote the class of
finite permutation groups.
\begin{theorem}[asymmetric coloring of inverse limit---finite version]
\label{thm:finite}
There exists a positive integer $c$ and a function
$f: \mathscr{Gr}\to\mathbb N$ such that the following holds.
Let $(G_i,\varphi_i)_{i\le k}$ be an epimorphic inverse sequence
of length $k\ge c$ of finite permutation groups with disjoint domains.
Assume $k\ge c+f(G_c)$. Then the inverse limit of this
system admits a zero-neutral zero-asymmetric 2-coloring.
\end{theorem}
It is easy to see that our main technical result,
Theorem~\ref{thm:main1}, is a consequence of
Theorem~\ref{thm:finite}. (Zero-neutrality is not
needed for this inference.) The proof follows the lines
of the proof of Lemma~\ref{lem:reduction}.
It is also easy to see that our proof of Theorem~\ref{thm:main1}
in effect proved Theorem~\ref{thm:finite}.
\begin{proof}[Sketch of proof of Theorem~\ref{thm:finite}]
Take $c:=2c'+1$ where $c'$ is the constant
denoted by $c$ in Theorem~\ref{thm:bded-orbits}.
Set $f(n):= \lfloor\log_2(n)\rfloor$.
We color the domains $\Omega_i$ one at a time,
reducing the group $G_i$ and thereby $\mathcal G$.
We call such a step a \emph{round},
and we conclude each round by
\emph{epimorphic reduction}
(Fact~\ref{fact:epiredux}).
First we color the domains $\Omega_i$ for $i > c$
to reduce $G_c$ to a solvable group.
While $G_c$ is not solvable, we reduce it to a
proper subgroup by coloring the next $\Omega_i$
(Theorem~\ref{thm:nonsolvable}). This process clearly
terminates in $\le \log_2 |G_c|$ rounds and when it
terminates, $G_c$ is solvable. But then,
all the $G_j$,\ $j\le c$, are solvable as well, by
epimorphic reduction.
Next we color $\Omega_c$, applying Theorem~\ref{thm:bded-orbits}
to $G_c$ and achieving
derived length $\le 2c'=c-1$ for $G_c$ and therefore for all $G_i$
for $i\le c$.
Then, for $i=c-1$ down to $i=1$, we successively reduce
the derived length of $G_i$ to $\le i-1$,
using the procedure of Section~\ref{sec:reducing-derived-length}.
In the end, the derived length of $G_1$ is reduced to zero,
hence the same is true for $G_0$ without having colored
$\Omega_0$, yielding zero-asymmetry and zero-neutrality.
\end{proof}
To make this result more effective, we need to replace
the bound $k\ge c+f(G_c)$ by a bound of that only
depends on $G=G_0$.
\begin{conjecture}[asymmetric coloring of inverse limit---effective version]
\label{conj:finite1}
There exists a function
$g: {\mathscr{Gr}}\to\mathbb N$ such that the following holds.
Let $(G_i,\varphi_i)_{i\le k}$ be an epimorphic inverse sequence of
length $k\ge g(G_0)$ of finite permutation groups with disjoint
domains. Then the inverse limit of this
system admits a zero-neutral zero-asymmetric 2-coloring.
\end{conjecture}
If this conjecture is true, here is a lower bound on the
function $g$. Let $\asy(G)$ denote the asymmetric coloring number
of the permutation group $G$ (Def.~\ref{def:asy}).
\begin{prop}
If Conjecture~\ref{conj:finite1} holds for a function $g$ then
for all finite permutation groups $G$ we have
\begin{equation}
g(G) \ge \log_2 \asy(G).
\end{equation}
\end{prop}
\begin{proof}
Let $G\le \sym(\Omega)$.
Consider the length-$k$ inverse system $(G,G,\dots,G)$ with the identity
serving as transition homomorphisms. So the inverse limit $\mathcal G$
is the diagonal of the direct product $G^{k+1}$, acting on
$\Omega\times \{0,\dots,k\}$. Assume there is a zero-asymmetric
zero-neutral 2-coloring $\gamma$ of $\mathcal G$. Now define the
coloring $\delta$ of $\Omega$ by setting
$\delta(x)=(\gamma(x,1),\dots,\gamma(x,k))$ for $x\in\Omega$.
It should be clear that $\delta$ is an asymmetric coloring of $G$
with $\le 2^k$ colors, so $\asy(G)\le 2^{g(G)}$.
\end{proof}
One might ask, how close is this lower bound to the true
upper bound (if one exists). I would risk the following
bold conjecture.
\begin{conjecture}[asymmetric coloring of inverse limit---polylog bound]
\label{conj:finite2}
There exists a polynomial $p$ such that
Conjecture~\ref{conj:finite1} holds with
$g(G)=p(\log(\asy(G)))$.
\end{conjecture}
Conjecture~\ref{conj:finite2} is true for inverse systems of
solvable groups. Since solvable groups have bounded
asymmetric coloring number (Lemma~\ref{lem:solv5color}),
this means for solvable groups $G$ the quantity $g(G)$ should
be bounded, and indeed it is.
\begin{prop}
There exists a constant $c$ such that the following holds
for all inverse sequences of solvable permutation groups
with disjoint domains. If the length of the sequence
is at least $c$ then the inverse limit admits a
zero-asymmetric zero-neutral 2-coloring.
We can take $c:=2c'+1$ where $c'$ is the constant
denoted by $c$ in Theorem~\ref{thm:bded-orbits}.
\end{prop}
\begin{proof}
We just proved this in the second (solvable) phase of the
proof of Theorem~\ref{thm:finite}.
\end{proof}
\section{Combinatorial relaxation of symmetry: CFSG-free proofs}
\label{sec:cfsg-free}
One of the key facts underlying our result was the following.
\begin{theorem}[\cite{saxl}] \label{thm:saxl}
All but a finite number of primitive permutation groups,
other than the symmetric and alternating groups
in their natural action, admit an asymmetric 2-coloring.
\end{theorem}
The original proof of Theorem~\ref{thm:saxl}
rests on the Classification of Finite Simple Groups (CFSG).
In this section we address the following two questions.
\medskip\noindent
\begin{itemize}
\item[($\alpha$)]
Can one avoid the use of the CFSG in the proof of Theorem~\ref{thm:saxl}?
\item[($\beta$)]
Is there a combinatorial generalization of Theorem~\ref{thm:saxl},
i.\,e., an asymmetric 2-colorability result for
a class of combinatorial structures with no symmetry
assumptions, that includes Theorem~\ref{thm:saxl}?
\end{itemize}
Question ($\alpha$) was already raised by
Cameron \emph{et al.}~\cite{saxl}
and was reiterated by Imrich \emph{et al.}
as~\cite[Question 1]{watkins15}.
We point out that a positive answer to both questions
follows from a recent breakthrough
by Xiaorui Sun and John Wilmes~\cite{sunwilmes}
on the number of automorphisms of primitive coherent
configurations (see Theorems~\ref{thm:sunwilmes}
and~\ref{thm:sunwilmes2} below).
\subsection{CFSG-free proof of the Cameron--Neumann--Saxl Theorem}
\label{subsec:cfsg-free}
Recall that the \emph{line-graph} of a graph $X=(V,E)$
is the graph $L(X)$ with vertex set $E$ where two edges
$e,f\in E$ (as vertices of $L(X)$) are adjacent in $L(X)$
if they share a vertex in $X$. The line-graphs of the cliques
$K_r$ are called \emph{triangular graphs}, denoted $T(r)$.
The graph $T(r)$ has $\binom{r}{2}$ vertices, $r!$ automorphisms,
and motion $2r-4$. The socle has index 2 in $\aut(T(r))$.
The line-graphs of balanced bipartite cliques
$K_{r,r}$ are called \emph{lattice graphs}, denoted $L_2(r)$.
The graph $L_2(r)$ has $r^2$ vertices, $2(r!)^2$ automorphisms,
and motion $2r$. The socle has index 8 in $\aut(T(r))$.
The original proof of Theorem~\ref{thm:saxl} depends on CFSG through
the following result, a special case of much more detailed
result in~\cite{cameron81}. Let $\soc(H)$ denote the socle
of the group $H$. Let us say that $G$ is a \emph{top group}
if either $A_n\le G\le S_n$ or $\soc(H)\le G\le H$ where $H$ is
the automorphism group of a triangular graph or a lattice graph.
\begin{prop}[Cameron + CFSG] \label{prop:cnsbound}
Let $G\le S_n$ be a primitive permutation group.
Assume $|G|\ge 2^{\sqrt{n/2}}$. If $n$ is sufficiently
large then $G$ is a top group.
\end{prop}
The following elementary result by Sun and Wilmes~\cite{sunwilmes},
a consequence of their result on coherent configurations
(Theorem~\ref{thm:sunwilmes2}), implies Prop.~\ref{prop:cnsbound}.
\begin{theorem}[Sun--Wilmes, elementary] \label{thm:sunwilmes}
There exists $c > 0$ such that the following holds.
Let $G\le S_n$ be a primitive but not doubly transitive
permutation group.
Assume $|G|\ge \exp(c(n^{1/3}(\log n)^{7/3}))$.
Then $G$ is a top group.
\end{theorem}
This result implies Prop.~\ref{prop:cnsbound}
except in the case that the group is
doubly transitive. In that case, however,
known elementary combinatorial bounds
show that the order of the group is
quasipolynomially bounded.
\begin{theorem}[\cite{inventiones, pyber-doubly}] \label{thm:doubly}
Let $G\le S_n$ be a doubly transitive group
and assume $G\ngeq A_n$. Then
$|G| \le \exp(O(\log n)^4)$.
\end{theorem}
\noindent
(The result in \cite{inventiones} gives the weaker upper
bound $\exp\exp(O(\sqrt{\log n}))$, which would also suffice
in our context since this quantity is less than $\exp(n^\epsilon)$
for all $\epsilon > 0$ and all sufficiently large $n$, so
it is much smaller than the threshold in Prop.~\ref{prop:cnsbound}.
The improved bound stated above was obtained
in~\cite{pyber-doubly} using the framework of \cite{inventiones}.)
This completes the list of ingredients of an elementary
proof of Theorem~\ref{thm:saxl}.
\hfill $\Box$
\medskip
While a lemma in \cite{saxl} shows that the upper bound
in Theorem~\ref{thm:doubly} on the order of doubly transitive
groups other than $A_n$ and $S_n$
implies a nearly linear lower bound on the minimal
degree of these groups, we should mention that a
stronger, linear lower bound has been known for more
than 120 years. The following result was proved by
Alfred Bochert in the 19th century by a lovely
combinatorial argument~\cite{bochert}.
\begin{theorem}[Bochert, 1897]
Let $G\le S_n$ be a doubly transitive group
and assume $G\ngeq A_n$. Then the
minimal degree of $G$ is
$\mu(G) \ge n/8$. For $n > 216$, the
lower bound improves to $n/4$.
\end{theorem}
\begin{remark}
Cameron's results~\cite{cameron81} classify all primitive permutation
groups of order greater than $n^{\log\log n}$ and naturally include
Theorem~\ref{thm:sunwilmes}. The point here is that the proof by
Sun and Wilmes is elementary: it does not use the CFSG; in fact, it
uses no group theory at all.
\end{remark}
\subsection{Combinatorial relaxation of symmetry: Coherent configurations}
\label{sec:PCC}
The significance of the work of Sun and Wilmes goes far beyond
giving elementary proofs of group theoretic results previously
only known through the CFSG. Their result is purely combinatorial;
it concerns primitive coherent configurations (PCCs): highly regular
colorings of the directed complete graph, with no symmetry assumptions.
We define this very general class of objects now.
\begin{definition}
A \emph{coherent configuration} (CC) is a pair $\mathfrak X=(\Omega,c)$
where $\Omega$ is a set (the set of \emph{vertices}) and
$c:\Omega\times\Omega\to\Sigma$ is a coloring of the
ordered pairs of vertices ($\Sigma$ is the set of colors),
subject to the following regularity constraints.
We assume $c$ is surjective. Below, $x,y,u,v\in\Omega$.
\begin{itemize}
\item[(i)] $(\forall x,y,z)(\text{if }c(x,x)=c(y,z)\text{ then }y=z)$.
\item[(ii)] $(\forall x,y,u,v)(\text{ if }c(x,y)=c(u,v)\text{ then }
c(y,x)=c(v,u))$.
\item[(iii)] There exists a family of $|\Sigma|^3$ non-negative
integers $p_{i,j}^k$, called the \emph{intersection numbers},
such that
$$(\forall x,y)(\text{if }c(x,y)=k\text{ then }
|\{z\ :\ c(x,z)=i,\ c(y,z)=j\}|=p_{i,j}^k)\,.$$
\end{itemize}
The number of colors used is called the \emph{rank} of $\mathfrak X$.
\end{definition}
Let $G\le \sym(\Omega)$ be a permutation group. Let $E_1,\dots,E_r$
denote the \emph{orbitals} of $G$, i.\,e., the orbits of the $G$-action on
$\Omega\times \Omega$. Assigning color $i$ to the elements of $E_i$
we obtain a coloring $c:\Omega\times\Omega\to [r]$. It is easy to see
that the resulting pair $\mathfrak X(G):=(\Omega,c)$ is a CC,
and $G\le\aut(\mathfrak X(G))$. A CC arising in this manner is called
a \emph{Schurian CC}, after Issai Schur who first introduced CCs
in 1933, as a tool
in the study of permutation groups~\cite{schur}. CCs were
subsequently rediscovered several times in different contexts.
They include such much-studied structures as strongly regular
graphs, distance-regular graphs, association schemes.
If $X=(V,E)$ is a graph then the Weisfeiler--Leman color refinement
process~\cite{weisfeiler-leman, weisfeiler-book} (see, e.\,g.,~\cite{quasipoly})
efficiently constructs a CC $\mathfrak X(X)=(V,c)$ such that
$\aut(X)=\aut(\mathfrak X(X))$. In particular, $X$ has an asymmetric
$d$-coloring (of the vertices, as always in this paper) if and only if
$\mathfrak X(X)$ has an asymmetric $d$-coloring. CCs are also
critical ingredients in the recent isomorphism test~\cite{quasipoly}.
That paper includes a detailed introduction to the combinatorial
theory of CCs.
\begin{definition}[Constituents, PCC]
Given a CC $\mathfrak X=(\Omega,c)$,
the digraphs $R_i=(\Omega,c^{-1}(i))$ $(i\in\Sigma)$ are the
\emph{constituent digraphs} of $\mathfrak X$. If one of these
is the diagonal $\diag(\Omega):=\{(x,x)\mid x\in\Omega\}$,
we call $\mathfrak X$ \emph{homogeneous}. Observe that a group
$G$ is transitive if and only if the corresponding Schurian
CC $\mathfrak X(G)$ is homogeneous. We call $\mathfrak X=(\Omega,c)$
a \emph{primitive CC (PCC)} if it is homogeneous and
every non-diagonal constituent is a (strongly) connected
digraph.
\end{definition}
It is not difficult to show that a permutation
group $G$ is primitive if and only if $\mathfrak X(G)$ is a PCC.
We should emphasize that conjecturally and empirically,
most CCs are not Schurian.
Note that for every $n$ there is essentially only one
rank-2 CC, namely, $\mathfrak X(S_n)$, to which we refer as the
$n$-clique, and also as the trivial CC.
\begin{definition}[UPCC]
A \emph{uniprimitive coherent configuration (UPCC)}
is a PCC of rank $\ge 3$ (a nontrivial PCC).
\end{definition}
Now we can state the actual result of Sun and Wilmes.
\begin{theorem}[Sun--Wilmes] \label{thm:sunwilmes2}
Let $\mathfrak X$ be a UPCC with $n$ vertices. If $n$ is sufficiently
large then $|\aut(\mathfrak X)|\le \exp(O(n^{1/3}(\log n)^{7/3}))$,
unless $\mathfrak X$ is the CC corresponding to a triangular graph
or a lattice graph.
\end{theorem}
Finally, we are in the position to address Question~($\beta$)
above, by generalizing Theorem~\ref{thm:saxl} to UPCCs.
\begin{theorem} \label{thm:uni-asymm}
All sufficiently large UPCCs admit an asymmetric 2-coloring.
\end{theorem}
Like much of the literature about asymmetric colorings,
we shall rely on the following lemma,
implicit in the counting argument used by
Gluck~\cite{gluck} and Cameron et al.~\cite{saxl} and
made explicit by Russell and Sundaram a decade and a
half later~\cite{russell}.
\begin{prop}[Motion Lemma, \cite{gluck}, \cite{saxl}, \cite{russell}]
\label{prop:motionlemma}
Let $G\le S_n$ be a permutation group of minimal degree $\mu$.
If $d^{\mu/2} \ge |G|$ then $G$ admits an asymmetric $d$-coloring.
\hfill $\Box$
\end{prop}
In order to take advantage of this lemma, we need
and upper bound on the order of the automorphism
group of $\mathfrak X$, and a lower bound on the motion of $\mathfrak X$.
The former is provided by the Sun--Wilmes result,
Theorem~\ref{thm:sunwilmes2}.
We take the latter from
a 1981 paper of this author~\cite{annals}.
Since the result is not explicitly stated in~\cite{annals},
let me show how it follows immediately from the
main technical result of that paper. (This fact
has been known since immediately after the
publication of~\cite{annals}.)
Let $\mathfrak X=(\Omega,c)$ be a UPCC.
Following~\cite{annals}, we say that vertex $z$
\emph{distinguishes} vertices $x$ and $y$ if $c(z,x)\neq c(z,y)$.
Let $D(x,y)$ denote the set of vertices $z$ that distinguish
$x$ and $y$. The core technical result of~\cite{annals} is the
following.
\begin{theorem}[\cite{annals}] \label{thm:annals}
Let $\mathfrak X$ be a UPCC with $n$ vertices. Then, for every pair
$x,y$ of distinct vertices, $|D(x,y)|\ge (\sqrt{n}-1)/2$.
\end{theorem}
All we need to add to this result is the following
observation.
\begin{obs} \label{obs:uni-motion}
Let $\mathfrak X$ be a UPCC with $n$ vertices. Then the motion
of $\mathfrak X$ is $\ge \min_{x\neq y} |D(x,y)|$.
\end{obs}
\begin{proof}
Let $T=\supp(\sigma)$ for some $\sigma\in\aut(X)$
such that the size of $T$ is the motion of $\mathfrak X$.
Let $x\in T$ and $y=\sigma(x)$. Then $y\neq x$ by
definition. We claim that $T\supseteq D(x,y)$.
Indeed, if $z\in\Omega\setminus T$ then
$c(z,x)=c(\sigma(z),\sigma(x))=c(z,y)$, so $z\not\in D(x,y)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:uni-asymm}]
Let $\mu=\mu(\mathfrak X)$ be the motion of $\mathfrak X$.
We have $\mu \ge (\sqrt{n}-1)/4$ by Theorem~\ref{thm:annals}
and Obs.~\ref{obs:uni-motion}. So
$2^{\mu/2} \ge 2^{(\sqrt{n}-1)}/4$.
For sufficiently large $n$, this quantity is greater than the
Sun--Wilmes bound (Theorem~\ref{thm:sunwilmes2}),
which is $\exp(C(n^{1/3}(\log n)^{7/3}))$ for some constant $C$.
\end{proof}
\section{Open problems}
\label{sec:open}
Theorem~\ref{thm:finite} describes a finite version of our main
technical result, Theorem~\ref{thm:main1}. I would be most
interested in more effective versions of this result, and
specifically in Conjectures~\ref{conj:finite1} and~\ref{conj:finite2}
(Polylog bound conjecture).
Below I list a number of additional problems and directions
of study.
All groups in this section, except in Problems (1), (2), and (11),
are finite.
\medskip\noindent
{\bf Terminology.}\quad Given a permutation group $G\le\sym(\Omega)$,
recall that we say that a coloring $\gamma$ ``results in a 2-group''
if $\gamma$ is a coloring of the permutation domain $\Omega$
and $G_\gamma$ (the color-preserving subgroup of $G$)
is a 2-group. And we can substitute any class of groups
in such a statement for ``2-groups,'' so for instance
the statement that ``a coloring results in a group with
derived length $\le 3$'' should have a clear meaning.
PS-closed classes of groups (classes closed under direct
products and subgroups), such as those mentioned above,
are of particular interest because of their monotonicity
properties described in Obs.~\ref{obs:ps-closed} and
Remark~\ref{rem:ps-closed}.
\begin{enumerate}
\item
\begin{itemize}
\item[(a)]
Give a CFSG-free proof of Theorem~\ref{thm:main1}
and thereby to the Infinite Motion Conjecture.\\
I expect that the results mentioned in the preceding section,
and in particular the Sun--Wilmes Theorem
(Theorem~\ref{thm:sunwilmes}), will play a role.
\item[(b)] How much of Theorem~\ref{thm:simple}
(``Reducing simple image'') can be salvaged without CFSG?
\end{itemize}
\item
\begin{itemize}
\item[(a)]
Does there exist a constant $C$ such that the following
strengthening of the Infinite Motion Conjecture holds?\\
\emph{Let $X$ be a connected locally finite rooted graph with
infinite motion. Then $X$ has an asymmetric 2-coloring (red/blue)
that is overwhelmingly blue in the sense that
every sphere about the root gets at most $C$ red vertices.}\\
This question is motivated by the consideration of the ``cost''
of coloring, as defined below. It is easy to see
that the statement is true for locally finite rooted trees
without vertices of degree~1.
\item[(b)]
Does there exist a constant $C$ such that the following
strengthening of Theorem~\ref{thm:simple}
(``Reducing simple image'') holds?\\
\emph{Let $G\le \sym(\Omega)$, where $\Omega$ is a finite set.
Let $\varphi: G\twoheadrightarrow T$
be an epimorphism where $T$ is a nonabelian simple group. Then
there exists a subset $\Delta \subseteq\Omega$ of size
$|\Delta|\le C$ such that $\varphi(G_\Delta) < T$.}\\
We note that $C=1$ will not suffice, as the example
$G=\mathbb Z_p^d \rtimes T\le \agl(d,p)$ shows, where the
semidirect product is defined by a nontrivial $d$-dimensional
irreducible representation of $T$ over $\mathbb F_p$.
\end{itemize}
\item
Recall Theorem~\ref{thm:nonsolvable} (reducing non-solvable image):
\vspace{0.1cm}
\noindent
\emph{Let $G\le \sym(\Omega)$ where $\Omega$ is a finite set.
Let $H$ be a group and $\varphi: G\twoheadrightarrow H$ an epimorphism.
Then $(\exists \Delta \subseteq\Omega)(\varphi(G_\Delta) < H)$,
assuming $H$ is not solvable.}
\vspace{0.1cm}
\begin{itemize}
\item[(a)] We note that the condition that ``$H$ is not solvable''
cannot be replaced by the condition ``$|H| > 1$,'' as shown
by the example $G=D_k$ (the dihedral group of order $2k$ acting
naturally on $k$ elements) and $H=\mathbb Z_2$ where the epimorphism
$\varphi$ is defined by the natural epimorphism $D_k \twoheadrightarrow D_k/\mathbb Z_k$,
where $3\le k\le 5$.
\item [(b)] Question. Does there exist a number $C$ such that
the following holds?\\
\emph{Let $G\le \sym(\Omega)$ where $\Omega$ is a finite set.
Let $H$ be a group. Let $\varphi: G\twoheadrightarrow H$ be an epimorphism.
Assume $|H|\ge C$.
Then $(\exists \Delta \subseteq\Omega)(\varphi(G_\Delta) < H)$.}
\item[(c)] Does the conclusion of (3)(b) follow if we only require
$|G| \ge C$ and $|H|\ge 2$ ?
\end{itemize}
\item
\begin{itemize}
\item[(a)]
Given a sequence $n_0,\dots,n_k$ of positive integers,
consider the \emph{balanced rooted tree} of height $k$
where the vertices at distance $j$ from the root
have $n_j$ children. So the automorphism group is
the wreath product of the symmetric groups of degree $n_j$.
What is the asymmetric coloring number of these trees?
\item[(b)] More generally,
how does the \emph{position of symmetric and
alternating groups in a structure tree} (hierarchy of
blocks of imprimitivity) of a transitive group
affect the asymmetric coloring number?
\end{itemize}
\item
A systematic study of \emph{solvable colorings}
for permutation groups would be of interest. Recall that
these are colorings of the permutation domain that
result in a solvable group.
More specific questions on this subject follow.
\begin{itemize}
\item[(a)]
Within various classes of permutation groups, characterize those
that do not admit a solvable 2-coloring.\\
Among primitive groups, the only groups that do not admit
a solvable 2-coloring are the symmetric and alternating
groups of degree $\ge 9$ in their natural action.
\item[(b)]
The wreath product $S_8\wr S_5$ does not admit a solvable
2-coloring. Let us now consider the \emph{transitive permutation
groups without alternating composition factors}.
Can we characterize, which of these do not admit a solvable
2-coloring?
\item[(c)]
The automorphism group of every tournament is solvable.
This statement is equivalent to the Feit--Thompson Theorem.
Can we prove without using heavy group theory that
\emph{all tournaments have a solvable $k$-coloring
for some fixed value $k$?} Or is such a statement still
equivalent to Feit--Thompson?
\end{itemize}
\item
A general theme is, what kind of structural
reductions of the group can be achieved by a bounded number
of colors. Here is a specific question of this type.
\begin{itemize}
\item[(a)]
Does there exist a number $g_0$ such that the following
holds: \emph{Every permutation group admits a 2-coloring that
kills all non-alternating composition factors of order
$\ge g_0$}, i.\,e., after the coloring, all composition
factors will either be alternating or of order $ < g_0$.
\end{itemize}
\item
Some questions of this type arose in this
paper when starting from a solvable group.
\begin{itemize}
\item[(a)] What is the smallest $c$ such that
\emph{every solvable permutation group admits a
2-coloring that results in derived length $\le c$ ?}
Such a $c$ exists by Theorem~\ref{thm:bded-orb}.
\item[(b)] What is the smallest $C$ such that
\emph{every solvable permutation group admits a
2-coloring that reduces the length of all orbits to $\le C$ ?}
Such a $C$ exists by Theorem~\ref{thm:bded-orb}.
The group $S_4\wr S_4$ shows that $C$ cannot be less than 4.
\item[(c)] Does every solvable group have a 3-coloring
that results in a 2-group?\\
Two colors do not suffice for this, as the
example of $S_4\wr K$ shows for any solvable group $K$
that is not a 2-group.
\item[(d)] Does every solvable permutation group have a 2-coloring
that results in an \emph{abelian-by-2-group}, i.\,e., in
a group that has an abelian normal subgroup with the
quotient being a 2-group?
\item[(e)]
Sometimes instead of solvability of the automorphism group,
we can assume something about the underlying structure.
We discussed tournaments above. Another example:
If $X$ is a connected cubic graph
then it has a \emph{low-cost} (see below) 2-coloring
that results in a 2-group: just color a pair of
adjacent vertices red, the rest blue.
For the same reason, if $X$ is a connected graph
such that every vertex has degree $\le k$ then
the same low-cost 2-coloring results in a group
with bounded composition factors. (Every composition
factor is a subgroup of $S_{k-1}$.) This fact
was used by Gene Luks to revolutionize the
theory of Graph Isomorphism testing in
1980~\cite{luks-bded}.
\end{itemize}
\item
An important direction of study is the extension of
known results about the minimal degree of primitive groups
(often obtained via CFSG)
to the motion of \emph{strongly regular graphs,}
\emph{distance-regular graphs,} and \emph{primitive coherent
configurations (PCCs)}. Some work in this direction has already been
done, see, e.\,g., \cite{annals, itcs14}, and the profound results
in \cite{sunwilmes, kivva21a, kivva21b, kivva-rank4}. A conjecture of this
author that motivates Kivva's work~\cite{kivva21a,kivva21b,kivva-rank4}
is the following.
\begin{conjecture}
Let $\mathfrak X$ be a PCCs
with $n$ vertices. If $\mathfrak X$ is not a Cameron scheme
then the motion of $\mathfrak X$ is $\ge cn$
for some positive constant $c$.
\end{conjecture}
The conjecture is motivated by a
1991 result by Liebeck and Saxl~\cite{liebecksaxl-mindeg}
that says that the statement is true with $c=1/3$
in the Schurian case.
The Cameron schemes are Schurian. They correspond to
primitive permutation groups $G$ acting on $n=\binom{m}{k}^r$
elements for some $m\ge 5$, $1\le k \le m-1$, and $r\ge 1$,
as a subgroup of $S_m\wr S_r$ containing $A_m^r$
where $S_m$ acts on the $k$-subsets of an $m$-set
and $S_r$ acts on the ordered $r$-tuples of such subsets.
In his monumental work,
Kivva \emph{confirmed the conjecture for PCCs of rank $\le 4$}
\cite{kivva-rank4}
and for \emph{distance-regular graphs of bounded diameter}
\cite{kivva21a,kivva21b}.
Given the combinatorial nature of the question,
no group theory is involved in his proofs.
\item
\emph{Combinatorial symmetry breaking is a key aspect
of the Graph Isomorphism problem} (see~\cite{quasipoly}).
From this perspective,
the cost of breaking the symmetry is not the number of
colors used but the \emph{entropy of the distribution
of colors:} if color $i$ occurs $k_i$ times on a set of size
$n=\sum_{i=1}^s k_i$ then we are looking at the quantity
$H(k_1/n,\dots,k_s/n)=-\sum (k_i/n)\log_2 (k_i/n)$.
If the $s$ colors are uniformly distributed ($k_i=n/s$ for all $i$) then
the entropy is $\log_2 s$. On the other hand, if one color dominates
and all the orther colors occupy just a small portion of the domain
then the entropy is close to zero. For instance, in the case
of a 2-coloring, which is equivalent to fixing a subset, we wish
that subset to be as small as possible. The size of that
set as a cost measure was introduced by Debra Boutin in
2008~\cite{boutin} (see also \cite{boutinimrich}) from
the philosophical consideration that,
given that in most cases of interest, an asymmetric 2-coloring
exists, a more refined measure of the cost of symmetry breaking is
needed. This measure of cost represents a \emph{convergence with the
classical concept of minimum bases of permutation groups.}
A \emph{base} $\Delta\subseteq\Omega$ is a subset such that the
pointwise stabilizer is trivial: $G_{(\Delta)}=1$.
Bases have been introduced in computational group
theory (Sims~\cite{sims}) in the 1960s with the express purpose
of breaking all symmetry, but the concept also has great
theoretical significance (see below).
A base of size $b$ gives a coloring with $b+1$ colors:
individual colors for each element of the base, and a single color
for the rest. Bases provide the \emph{lowest entropy} among all colorings
with $b+1$ colors. But bases may not use the optimal number
of colors for the type of questions we are considering here; for
instance, the base size for $S_8$ is 8, but
$S_8$ has a solvable 2-coloring.
In any case, the sizable literature about minimal bases of
permutation groups will be particularly relevant in the
context of the refined cost measures. Here is a selection
of relatively recent papers on minimum bases:
\cite{liebeck84, pyber,seress97,
gluckseress, liebeckshalev, burness11, burness17}.
One of the recent motivators of the area has been \emph{Pyber's
base size conjecture} (1993)~\cite[p. 207]{pyber}, resolved
in~\cite{duyhan} (2018) and made effective in \cite{halasi} (2019).
The latter paper also includes a nice overview of the subject.
\item
\label{pr:cameron-schemes}
Symmetry-breaking by coloring of primitive coherent configurations
is at the heart of the study of the \emph{Graph Isomorphism}
problem~\cite{quasipoly}. One of the types of problems that
arise there is to distinguish the Johnson, Hamming, and
Cameron schemes from all other primitive coherent configurations
by showing that for all other configurations,
symmetries are destroyed at much lesser cost (see~\cite{quasipoly}).
Here the ``cost'' refers to the refined cost measures
explained in the previous item.
\item
\label{finite-subdegree}
Let me highlight an interesting generalization of Tucker's
Conjecture, proposed by Imrich, Smith, Tucker, and Watkins,
that states that
\emph{a closed permutation group $G$, acting on a countably
infinite set, with infinite minimal degree and finite subdegrees,
admits an asymmetric 2-coloring} (``Infinite Motion Conjecture
for Permutation Groups'') \cite[Sec 4]{watkins15}. Here ``closed''
means a closed subgroup of the symmetric group in the permutation
topology, where a neighborhood basis of the identity consists of
the pointwise stabilizers of finite subsets of the permutation domain.
The subdegrees are the lengths of the orbits of the stabilizer of a point.
\end{enumerate}
Let me conclude with a conjecture I have been entertaining for decades.
We have seen that epimorphisms to the alternating groups give
us a lot of trouble. (Such epimorphisms have also defined
the bottleneck for Luks's graph isomorphism test that
caused three decades without progress on that problem,
see \cite{quasipoly}.)
I believe the conjecture below relates
to our subject, although I cannot draw a formal connection.
Answering this author's question, in 1983,
Martin Liebeck~\cite{liebeck83} proved
that if $X$ is a graph and $\aut(X)\cong A_k$ then $n$ (the
number of vertices) must grow exponentially as a function
of $k$. Specifically, he showed
that $n\ge 2^k-k-2$ for all $k\ge 13$ and that
this lower bound is tight for $k\equiv 0$ or $1\pmod 4$.
\begin{conjecture}
There exists a constant $C > 1$ such that the following holds.
Let $X$ be a graph with $n$ vertices. Assume $\aut(X)$ has
an epimorphism onto the alternating group $A_k$ $(k\ge 3)$.
Then $n \ge C^k$.
\end{conjecture}
\section*{Acknowledgments}
First and foremost, I would like to thank
Wilfried Imrich, my friend of over half a century,
for bringing the Tucker conjecture to my attention
and also for his insistence that I attend
the BIRS/CMO workshop on
``Symmetry Breaking in Discrete Structures.''
I wish to thank the BIRS-affiliated
\emph{Casa Matem\'atica Oaxaca (CMO)} for their hospitality
during the workshop, September 16--21, 2018.
During the workshop I had the opportunity
to learn about current research in this area,
including work on Tom Tucker's conjecture addressed in this paper.
It is a pleasure to acknowledge productive conversations
with Wilfried Imrich, Florian Lehner, Monika Pil\'sniak,
Tom Tucker, and other participants of the meeting.
My special thanks are due to my recent student John Wilmes
and my current student Bohdan Kivva, for all the insights
they shared with me over the years, some of which
turned out to be relevant to this paper.
I thank Saveliy Skresanov~\cite{skresanov}
for pointing out Choi's work~\cite{choi}
and his own GAP search
(see Prop.~\ref{prop:mathieu} and Remark~\ref{rem:mathieu}).
Last but not least, I'd like to pay tribute to Jan Saxl,
whose lifelong influence on my work started in 1979
when his article with Cheryl Praeger~\cite{praeger}
inspired my entry into the theory of primitive
permutation groups~\cite{annals}, and continued
until recently with work on our joint paper~\cite{bps}.
His imprint is discernible throughout this article.
\medskip
The research presented in this paper was supported in part
by NSF Grant CCF-1718902. The views expressed in the paper
are solely the author's and have not been evaluated or
endorsed by the NSF.
| {
"timestamp": "2021-11-16T02:11:31",
"yymm": "2110",
"arxiv_id": "2110.08492",
"language": "en",
"url": "https://arxiv.org/abs/2110.08492",
"abstract": "An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements displaced by any non-identity automorphism. In this paper we confirm Tom Tucker's \"Infinite Motion Conjecture\" that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of a sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains.",
"subjects": "Group Theory (math.GR)",
"title": "Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795118010989,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7092379487106436
} |
https://arxiv.org/abs/1912.02225 | Intrinsic Topological Transforms via the Distance Kernel Embedding | Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform and Euler Characteristic Transform, both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms that depend only on the intrinsic geometry of a shape, and hence are invariant to the choice of embedding. To that end, given an abstract metric measure space, we define an integral operator whose eigenfunctions are used to compute sublevel set persistent homology. We demonstrate that this operator, which we call the distance kernel operator, enjoys desirable stability properties, and that its spectrum and eigenfunctions concisely encode the large-scale geometry of our metric measure space. We then define a number of topological transforms using the eigenfunctions of this operator, and observe that these transforms inherit many of the stability and injectivity properties of the distance kernel operator. |
\section{Introduction}
\input{intro}
\section{The Distance Kernel}
\label{sec:distkernel}
\input{distkernel}
\section{The Distance Kernel Embedding}
\label{sec:distkernelembed}
\input{distkernelembed}
\section{Finite Approximation of the Distance Kernel Embedding}
\label{sec:discretization}
\input{discretization}
\section{Stability of the Distance Kernel Embedding}
\label{sec:stability}
\input{stability}
\section{Inverse Results}
\label{sec:stabinv}
\input{stabinv}
\section{Error Bounds}
\label{sec:embeddingconstants}
\input{embeddingconstants}
\section{The Intrinsic PHT/ECT}
\label{sec:intrinsicpht}
\input{intrinsicpht}
\section{Experiments}
\label{sec:experiments}
\input{experiments}
\section{Conclusion}
\label{sec:conclusion}
\input{conclusion}
\bibliographystyle{plain}
\subsection*{Distribution of Eigenvalues and Hausdorff Distances between DKEs}
The goal of this section is to illustrate the results of Sections~\ref{sec:discretization}, \ref{sec:stability}, and \ref{sec:stabinv}. In the following experiments, we compute the DKE for a variety of discrete samples on the torus and 2-sphere with metric induced by their embedding in Euclidean space, the 3-sphere, $L(7,1)$ Lens space, and $L(7,4)$ Lens space, with spherical geometry. The measures on these samples are uniform.
These spaces have distinct integer homology, except for the two Lens spaces that have the same homotopy type but are not homeomorphic, and therefore not isometric. This makes $L(7,1)$ and $L(7,4)$ difficult to distinguish by purely topological methods. We see that the DKE (and, therefore, the resulting topological transforms) is capable of distinguishing these Lens spaces.
\smallskip
\noindent
\emph{Spectra of various manifolds.}
In Figure~\ref{fig:eig}, we have plotted the first $8$ eigenvalues of five discrete metric spaces, sampled from each of these five manifolds, normalized by the number of points in each sample. We can observe the following: (1) the two Lens spaces have relatively similar eigenvalues, (2) the $2$- and $3$-sphere have many similar eigenvalues, but their first and fourth eigenvalues are significantly different, and (3) the torus has the most distinct spectrum.
\smallskip
\noindent
\emph{Spectra of Lens spaces for various samples.} In Figure \ref{fig:Lenspec}, we compare the spectra of a number of different random i.i.d. samples of the two Lens spaces $L(7,1)$ and $L(7,4)$. To be precise, for each Lens space we compute the spectra of two distinct random samples with $2000$ points, and a third sample with $5000$ points. The spectra for the different samples of the same Lens space are virtually impossible to distinguish, and only two curves---one for the spectrum of $L(7,1)$, and one of the spectrum of $L(7,4)$---are visible in Figure \ref{fig:Lenspec}. This attests to the stability of the eigenvalues of the distance kernel operator under random i.i.d. sampling, in line with Theorem \ref{thm:approxDKE}.
Notably, the two Lens spaces are distinguished by the first, third, and fourth eigenvalues of their distance kernel operators. $L(7,1)$ and $L(7,4)$ having same homotopy type, this illustrates the ability of the operator to capture geometric information and distinguish between non-isometric spaces.
\smallskip
\noindent
\emph{Hausdorff distance between DKEs.} Finally, in Figure \ref{fig:Haus}, we compare the Hausdorff distances between various pairs of distance kernel embeddings. We observe the following:
(1) The two samples of the same size coming from the $L(7,1)$ Lens space are the closest in Hausdorff distance, and that distance is close to zero up to dimension $k=4$. Indeed, if we had taken samples of sufficiently high resolution, we would see the Hausdorff distances going to zero for larger values of $k$, as proven in Theorem \ref{thm:approxDKE}.
(2) The second closest pair of spaces are the Lens spaces $L(7,1)$ and $L(7,4)$, that have same homotopy type and have both spherical geometry.
(3) The third closest pair of spaces are the Lens space $L(7,1)$ and the 3-Sphere, both with spherical geometry (Lens spaces are constructed as quotients of 3-Spheres).
(4) The manifold that appears to be most distinct from the rest is the torus.
(5) For all pairs of manifolds, the Hausdorff distance stabilizes at around $k=10$, after which eigenvalues are close to $0$.
Figure \ref{fig:Haus} illustrates that manifolds sharing topological or geometrical properties lead to more similar DKE. However, the DKEs of distinct samples of $L(7,1)$ remain substantially closer in Hausdorff distance compare to the DKE of samples of $L(7,4)$, with same homotopy type and same geometry locally, and the 3-sphere, with same local geometry.
\medskip
In conclusion, these experiments illustrate that the spectra and embedding of the distance kernel operator can be approximated by finite samples, as predicted by Theorem \ref{thm:approxDKE}. Moreover, by combining the DKE with the Hausdorff metric on Euclidean space, we obtain a pseudo-metric on the space of compact metric measure spaces that succeeds in distinguishing a variety of diverse manifolds.
\begin{figure}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{EigenvaluesDKO.png}
\caption{Eigenvalues of the DKO for a variety of\\ spaces, normalized by the number of points\\ in the sample.}
\label{fig:eig}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{EigenvaluesLensSamples.png}
\caption{A comparison of the eigenvalues of various samples, at different resolutions, of these two Lens spaces.}
\label{fig:Lenspec}
\end{subfigure}
\vspace{10mm}
\begin{subfigure}[t]{1\textwidth}
\centering
\includegraphics[scale = 0.7]{Hausdorff.png}
\caption{A comparison of Hausdorff distances between various samples of 2- and 3-manifolds.}
\label{fig:Haus}
\end{subfigure}%
\caption{Spectra and DKE for samples of various manifolds.}
\end{figure}
\subsection*{Distribution of mass and $L^{\infty}$ norms of $\phi_i$ and $\Phi_{k}$.}
Complementing the prior simulations, we now explore the distribution of mass of (the absolute value) our eigenfunctions $\phi_i$. This is important for estimating the $L^{\infty}$ norm of these coordinate functions in the continuous case (in the finite case, the fact that $\|\phi_i\|_{L^2} = 1$ forces all the entries of the vector to have norm at most $1$). Our stability theory (Section \ref{sec:stability}) tells us that the histograms of the continuous eigenfunctions is approximated by the histograms of the discrete eigenfunctions. We can estimate the $L^{\infty}$ norm of $\phi_{i}$ by considering the histogram of values of $|\phi_{i}(x)|$ (i.e. its distribution of mass) as follows. If $|\phi_{i}(x)|$ assumes values close to its maximum on a large (in measure) subset of $X$, the fact that $\|\phi_i\|_{L^2} = 1$ ensures that this maximum cannot be too large, relative to the volume of $X$. Whether or not this is the case can be determined by inspecting this histogram. Figures \ref{fig:T1eighist}, \ref{fig:SS1eighist}, \ref{fig:SSS1eighist}, and \ref{fig:A1eighist} show the histogram of absolute values for the tenth and twentieth eigenfunctions of a range of shapes. Observe that for all of these histograms, the eigenfunctions assumes within 20\% of their maximum value on a subset of measure at least $\vol(X)/2$. This tells us that the $L^{\infty}$ norms of these eigenfunctions cannot be too big.
We can carry out the same analysis for the distance kernel embedding $\Phi_{k}$. In Figures \ref{fig:T1hist500}, \ref{fig:SS1hist500}, \ref{fig:SSS2hist500}, and \ref{fig:A1hist500}, we see the histograms of the magnitudes of these embedding vectors for a variety of shapes. We see here in every example that the magnitude of the embedding vector is never below 50\% of the maximum, and is often much greater. As before, this rules out the possibility of large embedding vectors, relative to the volume of our manifold.
\begin{figure}[htb!]
\includegraphics[scale=0.5]{simulations/T1_eig10_hist} \includegraphics[scale=0.5]{simulations/T1_eig20_hist}
\caption{The histogram of values, together with the maximum and minimum values, for the tenth and twentieth eigenvalues of a sample from the torus (1158 points).}
\label{fig:T1eighist}
\end{figure}
\begin{figure}[htb!]
\includegraphics[scale=0.5]{simulations/SS1_eig10_hist} \includegraphics[scale=0.5]{simulations/SS1_eig20_hist}
\caption{The histogram of values, together with the maximum and minimum values, for the tenth and twentieth eigenvalues of a sample from the 2-sphere (564 points).}
\label{fig:SS1eighist}
\end{figure}
\begin{figure}[htb!]
\includegraphics[scale=0.5]{simulations/SSS1_eig10_hist} \includegraphics[scale=0.5]{simulations/SSS1_eig20_hist}
\caption{The histogram of values, together with the maximum and minimum values, for the tenth and twentieth eigenvalues of a sample from the 3-sphere (488 points).}
\label{fig:SSS1eighist}
\end{figure}
\begin{figure}[htb!]
\includegraphics[scale=0.5]{simulations/A1_eig10_hist} \includegraphics[scale=0.5]{simulations/A1_eig20_hist}
\caption{The histogram of values, together with the maximum and minimum values, for the tenth and twentieth eigenvalues of a sample from the $L(11,3)$ Lens space(980 points).}
\label{fig:A1eighist}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{simulations/T1_dim500_hist}
\caption{The histogram of the magnitude of the\\ embedding vectors for a sample from the\\ torus (1158 points) in dimension $k=500$.}
\label{fig:T1hist500}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{simulations/SS1_dim500_hist}
\caption{The histogram of the magnitude of the\\ embedding vectors for a sample from the 2-sphere\\ (564 points) in dimension $k=500$.}
\label{fig:SS1hist500}
\end{subfigure}
\vspace{10mm}
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{simulations/SSS2_dim500_hist}
\caption{The histogram of the magnitude of the\\ embedding vectors for a sample from the\\ 3-sphere (1079 points) in dimension $k=500$.}
\label{fig:SSS2hist500}
\end{subfigure}%
~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\includegraphics[scale = 0.5]{simulations/A1_dim500_hist}
\caption{The histogram of the magnitude of the\\ embedding vectors for a sample from the $L(11,3)$ Lens space (980 points) in dimension $k=500$.}
\label{fig:A1hist500}
\end{subfigure}
\caption{}
\end{figure}
\clearpage
\subsection*{Embedding Bounds}
In this section, we consider some simulations for the quantites $\|E_{k}\|_{\infty}$ and $\max_{x \in X} \|\Phi_{k}(x)\|_{2}$ appearing in Theorem \ref{thm:invstabmeas}, as well as the explicit upper bounds presented in Lemmas \ref{lem:errorbound} and \ref{lem:embedbound}. Let us introduce some notations used in our table of results:
\begin{itemize}
\item $A = \|E_{X,k}\|_{\infty}$ is our additive constant.
\item $B = \max_{x \in X} \|\Phi_{k}(x)\|_{2}$ is our multiplicative constant.
\item A\_Bound is the bound on $A$ given in Lemma \ref{lem:errorbound}.
\item B\_Bound is the bound on $B$ given in Lemma \ref{lem:embedbound}.
\end{itemize}
We compute these quantities across a range of dimensions for the torus, 2-sphere, and 3-sphere. We also note the diameters of these spaces, for comparison. In each table, we observe the same phenomenon. The worst-case bounds on B and A grow in the embedding dimension. However, the multiplicative constant B stabilizes at a small value (relative to $1$), and the additive constant A goes to zero, quickly becoming much smaller than the diameter.
\begin{table}[htb!]
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Dimension & 2 & 5 & 10 & 50 & 200 \\ \hline
B\_Bound & 7.098810562 & 11.95365386 & 18.41320952 & 50.85533019 & 91.71628432 \\ \hline
B & 2.785385644 & 2.866424003 & 2.937811041 & 3.061283988 & 3.127787041 \\ \hline
A\_bound & 50.42347825 & 126.1558938 & 292.5051042 & 2313.723784 & 4249.850655 \\ \hline
A & 2.831494962 & 0.863195908 & 0.904480257 & 0.188507231 & 0.080790547 \\ \hline
\end{tabular}
\caption{Simulations of the constants A and B, and their upper bounds, for a sample from the torus (1158 points, diam 7.52945)}
\end{table}
\begin{table}[htb!]
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Dimension & 2 & 5 & 10 & 50 & 200 \\ \hline
B\_Bound & 4.415703664 & 8.423681386 & 12.7145307 & 41.13417364 & 44.98760706 \\ \hline
B & 1.655840841 & 1.703608832 & 1.711791639 & 1.747744118 & 1.767800296 \\ \hline
A\_Bound & 19.05590903 & 63.08320762 & 141.1412916 & 1115.905333 & 1115.912121 \\ \hline
A & 1.529116741 & 0.369015434 & 0.291928199 & 0.07784093 & 0.018153383 \\ \hline
\end{tabular}
\caption{Simulations of the constants A and B, and their upper bounds, for a sample from the 2-sphere (564 points, diam 3.10087)}
\end{table}
\begin{table}[htb!]
\centering
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Dimension & 2 & 5 & 10 & 50 & 200 \\ \hline
B\_Bound & 4.658734251 & 7.958344597 & 14.99125753 & 33.83567486 & 34.71946412 \\ \hline
B & 1.769845855 & 1.895902581 & 1.942573759 & 1.978396949 & 2.011362641 \\ \hline
A\_Bound & 21.43974332 & 56.15815487 & 215.9544069 & 653.2712051 & 653.2911455 \\ \hline
A & 1.138554951 & 0.364063761 & 0.290519127 & 0.123081673 & 0.032296336 \\ \hline
\end{tabular}
\caption{Simulations of the constants A and B, and their upper bounds, for a sample from the 3-sphere(488 points, diam 3.06147)}
\end{table}
\subsection*{Background}
The results of this section assume familiarity with certain tools and results in applied topology, namely those pertaining to persistence modules and persistence diagrams. For an introduction to these topics, the reader can consult the articles of Ghrist \cite{ghrist2008barcodes} and Carlsson \cite{carlsson2009topology}. A more formal and comprehensive treatment can be found in the texts by Edelsbrunner and Harer \cite{edelsbrunner2010computational}, Ghrist \cite{ghrist2014elementary}, and Oudot \cite{oudot2015persistence}.
\subsection{Existence Results}
In this section, we introduce a large family of topological transforms. These topological transforms can be defined for any homological degree, and so we work with graded persistence diagrams: families of persistence diagrams indexed by degree. Thus, the notation $PH(X,f)$ will correspond to the graded persistence diagram of the sublevel set filtration of the function $f$ on the space $X$, and the notation $\mathbf{GrDiag}$ will refer to the space of graded persistence diagrams.\\
Our first results demonstrate that, under certain general hypotheses, persistence diagrams and their associated Betti and Euler curves exist.
\begin{prop}
\label{prop:pers}
Let $(X,\dist_{X},\mu_X)$ be a compact metric measure space homeomorphic to the geometric realization of a finite simplicial complex. Then, any finite linear combination $f = \sum_{i=1}^{n}c_i \phi_i$ of eigenfunctions of $D^X$ with nonzero eigenvalue has a well-defined sublevel set persistence diagram $PH(X,f)$.
\end{prop}
\begin{proof}
Lemma \ref{lem:eiglip} states that the eigenfunctions of $D^X$ are Lipschitz, and hence continuous. Since $f$ is a finite linear combination of continuous functions, it, too, is continuous. Lastly, we appeal to Theorem 3.33 in \cite{chazal2016structure}, which asserts the existence of persistence diagrams of continuous functions on geometric realizations of finite simplicial complexes.
\end{proof}
In addition to persistent homology, we also define the Betti and Euler curves of the pair $(X,f)$.
\begin{definition}
Let $(X,\dist_{X},\mu_X)$ be a compact metric measure space. For any homological degree $k \geq 0$, and any finite linear combination $f = \sum_{i=1}^{n}c_i \phi_i$ of eigenfunctions of $D^X$ with nonzero eigenvalue, we view our graded persistence diagrams as graded barcodes and (tentatively) define the degree-$k$ Betti curve to be the sum of the indicator functions of the intervals in the degree-$k$ persistent homology of $(X,f)$:
\[\beta_{k}(X,f) = \sum_{I \in PH_{k}(X,f)} 1_{I}. \]
The Euler curve is then (tentatively) defined to be the alternating sum of these Betti curves:
\[\chi(X,f) = \sum_{k=0}^{\infty} (-1)^{k} \beta_{k}(X,f). \]
\end{definition}
Justifying the existence of these curves requires introducing some more ideas from the theory of persistent homology.
\begin{definition}
For a point $x$ in a persistence diagram, define $\operatorname{per}(x)$ to be the distance from $x$ to the diagonal. For a real-valued function $f: X \to \mathbb{R}$ on a triangulable, compact metric space $X$, exponent $q > 0$, and threshold parameter $t \geq 0$, we compute the sublevel set persistence and define:
\[\operatorname{Pers}_{q}(f,t) = \sum_{\operatorname{per}(x) > t} \operatorname{per}(x)^q. \]
This is the sum, over all homological degrees, of the $q$th powers of the persistence of points in $PH(X,f)$ with persistence more than $t$. When $t=0$, this quantity is called the \emph{ degree-$q$ total persistence of $f$.} Note that the parameter $q$ refers to the exponent in the sum, not the homological degree of the persistence diagram.
\end{definition}
Bounding the total degree-$q$ persistence of $PH(X,f)$ necessitates placing some restrictions on $X$ and $f$.
\begin{definition}
A triangulable metric space $X$ has \emph{polynomial combinatorial complexity} if there are constants $C_0$ and $d$ such that, for any radius parameter $r>0$, there exists a triangulation $T$ of $X$ where every triangle has diameter at most $r$, and $T$ has at most $C_0/r^d$ simplices.
\end{definition}
In~\cite{cohen2010lipschitz}, the authors note that the bilipschitz image of a $d$-dimensional Euclidean simplicial complex is always of polynomial combinatorial complexity. They also prove that Lipschitz functions on such spaces have finite degree-$q$ total persistence, for $q$ sufficiently large.
\begin{theorem}[{\cite[Section 2.3]{cohen2010lipschitz}}]
\label{thm:boundedtotper}
Let $f: X \to \mathbb{R}$ be a Lipschitz function on a metric space $X$ of $(C_0,d)$-polynomial combinatorial complexity which is homeomorphic to the geometric realization of a finite simplicial complex. Let $q = d + \delta$ for some constant $\delta > 0$. Then we have the following bound on degree-$q$ total persistence of the sublevel set filtration:
\[\operatorname{Pers}_{q}(f,0) \leq C_0 \operatorname{Lip}(f)^{d}\operatorname{Amp}(f)^\delta \cdot \left(1 + \frac{d+ \delta}{\delta} \right), \]
where $\operatorname{Lip}(f)$ is the Lipschitz constant of $f$ and $\operatorname{Amp}(f) = \max f - \min f$.
\end{theorem}
This motivates the following definition:
\begin{definition}[ {\cite[Section 2.3]{cohen2010lipschitz}} ]
\label{def:boundedtotal}
A metric space $X$ \emph{implies bounded degree-$q$ total persistence} if there is a constant $C_{X}$ such that $\operatorname{Pers}_{q}(f,0) \leq C_{X}$ for any real-valued Lipschitz function $f$ with $\operatorname{Lip}(f) \leq 1$.\\
Note that if $X$ implies bounded degree-$q$ total persistence, then $\operatorname{Pers}_{q}(f,0) \leq \operatorname{Lip}(f)^{q} C_{X}$ for any Lipschitz function $f$, as scaling a function only changes the resulting sublevel set persistence by scaling the endpoints of the intervals in its barcode.
\end{definition}
We now prove the existence of Betti and Euler curves for eigenfunctions with nonzero eigenvalue.
\begin{prop}
\label{prop:betticurves}
Suppose that $X$ is homeomorphic to the geometric realization of a finite simplicial complex that implies bounded degree-$q$ total persistence. Let $p=1/q$. Then for any homological degree $k$, the sum defining $\beta_{k}(X,f)$ converges in $L^p$. Moreover, the sum defining $\chi(X,f)$ is finite, so the Euler curve exists as a function in $L^p$.
\end{prop}
\begin{proof}
This proof makes use of Lemma \ref{lem:eiglip}, that eigenfunctions of $D^{X}$ with nonzero eigenvalue are Lipschitz. We would like to show that the potentially infinite sum of indicator functions arising in the definition of $\beta_{k}(X,f)$ converges in $L^p$ for $0 < p < \infty$. The Weierstrass M-test guarantees convergence if:
\[\sum_{I \in PH_{k}(X,f)} \|1_{I}\|_{L^p} < \infty. \]
Note that the length of an interval $I$ in a barcode is twice the persistence of the corresponding point $x$ in the associated persistence diagram. Thus we have:
\[\| 1_I \|_{L^p} = \operatorname{len}(I)^{1/p} = (2 \operatorname{per}(x))^{1/p} .\]
We can thus rewrite the above sum:
\[\sum_{I \in PH_{k}(X,f)} \|I\|_{L^p} = 2^{1/p} \sum_{x \in PH_{k}(X,f)} \operatorname{per}(x)^{1/p}.\]
Since $p = 1/q$ and $f$ is Lipschitz, we can leverage the fact that $X$ implies bounded degree-$q$ total persistence to show:
\[\sum_{x \in PH_{k}(X,f)} \operatorname{per}(x)^{1/p} = \sum_{x \in PH_{k}(X,f)} \operatorname{per}(x)^{q} \leq \sum_{j=0}^{\infty} \sum_{x \in PH_{j}(X,f)} \operatorname{per}(x)^{1/p} = \operatorname{Pers}_{q}(f,0) < \infty. \]
Thus, we have shown that $\beta_{k}(X,f)$ exists and is well-defined.\\
To demonstrate that the sum defining $\chi(X,f)$ is finite, we merely note that, as $X$ is homeomorphic to the geometric realization of a finite simplical complex, there is a finite degree $K$ such that no subset of $X$ has nontrivial homology in degree $k \geq K$. Thus we can write:
\[\chi(X,f) = \sum_{k=0}^{K} (-1)^k \beta_{k}(X,f). \]
\end{proof}
We now define the topological transforms of interest, beginning with those previously studied and then introducing our new contributions. We continue to assume all the assumptions needed to guarantee the existence of the objects in question: our eigenspaces have multiplicity zero, our scheme for identifying ``positive" eigenfunctions does not fail, we only take eigenfunctions with nonzero eigenvalue, the metric measure space is homeomorphic to the geometric realization of a finite simplicial complex and, to guarantee the existence of the appropriate Euler curves, the metric space implies bounded degree-$q$ total persistence.\\
\begin{definition}
Let $\mathbb{S}^{k}$ be the $k$-dimensional sphere, and $L(\mathbb{R}^{k+1},\mathbb{R})$ the space of linear maps from $\mathbb{R}^{k+1}$ to $\mathbb{R}$. Define the map $\Theta : \mathbb{S}^{k} \to L(\mathbb{R}^{k+1},\mathbb{R})$ which sends $v \in \mathbb{S}^{k}$ to the map $x \mapsto \langle x, v \rangle$.
\end{definition}
\begin{definition}(\cite{curry2018many})
Let $X \subset \mathbb{R}^d$ be a compact, definable set\footnote{See \cite{curry2018many} \S2 for the definition of definable sets.}. For every $v \in \mathbb{S}^{d-1}$, the sublevel set persistence diagram and Euler curve of the pair $(X,\Theta(v))$ exist. The \emph{Persistent Homology Transform} is the map $PHT(X):\mathbb{S}^{d-1} \to \mathbf{GrDiag}$ defined by:
\[PHT(X)(v) = PHT(X,\Theta(v)).\]
If one computes Euler curves instead of persistence diagrams, one obtains the \emph{Euler Characteristic Transform} $ECT(X)$.
\end{definition}
\begin{theorem}(\cite{curry2018many,ghrist2018persistent})
\label{thm:PHTinj}
The PHT and ECT are injective for all $k$. That is, let $X,Y \subset \mathbb{R}^{d}$ be compact, definable subsets. If $PHT(X) = PHT(Y)$ or $ECT(X) = ECT(Y)$, then $X=Y$ as sets.
\end{theorem}
We now define our new topological transforms, of which there are two types: (1) intrinsic transforms, which compute the persistence diagrams and Euler curves of linear combinations of eigenfunctions, and (2) embedded transforms, which are the composition of the PHT or ECT with the distance kernel embedding.
\begin{convention}
Recall that the coordinate functions $\alpha_{i}$ defining the distance kernel embedding are complex-valued. For the purpose of computing topological invariants, we would like our height functions to be real-valued. We will thus replace each function $\alpha_{i}$ with its real and imaginary parts: $\alpha_{i}^{R} = \operatorname{Re}(\alpha_{i})$ and $\alpha_{i}^{I} = \operatorname{Im}(\alpha_{i})$, and likewise replace the embedding $\Phi_{k}$ by two $\mathbb{R}^{k}$ valued maps: $\Phi_{k}^{R} = \operatorname{Re}(\Phi_{k})$ and $\Phi_{k}^{I} = \operatorname{Im}(\Phi_{k})$. Note that, for any index $i$, one of $\alpha_{i}^{R}$ or $\alpha_{i}^{I}$ is identically zero, and the other is equal to $\sqrt{|\lambda_i}|\phi_{i}$.
\end{convention}
\begin{definition}
\label{def:transforms}
For $k$ finite, the \emph{embedded persistence kernel transform} e-$PKT_{k}(X)$ is the PHT applied to the image of the embedding $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X)) \subset \mathbb{R}^{2k}$, which takes as input vectors in $\mathbb{S}^{2k-1}$ and takes values in $\mathbf{GrDiag}$. The \emph{intrinsic persistence kernel transform} i-$PKT_{k}(X)$ is the map from $\mathbb{S}^{2k-1}$ to $\mathbf{GrDiag}$ that maps $(u,v) \in \mathbb{S}^{2k-1} \subset \mathbb{R}^{k} \times \mathbb{R}^{k}$ to the sublevel set persistence of the pair:
\[\left(X,\sum_{i=0}^{k} u_{i}\alpha_{i}^{R} + v_{i}\alpha_{i}^{I} \right).\]
Using Euler curves in place of persistent homology gives rise to the embedded and intrinsic \emph{Euler kernel transforms}, noted e-$EKT_{k}$ and i-$EKT_{k}$, respectively.
\end{definition}
\begin{remark}
The difference between the intrinsic and embedded transforms is that the intrinsic transforms are defined using the metric measure space $X$ as the base topological space, whereas the embedded transforms use its image $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X)) \subset \mathbb{R}^{2k}$. When the mapping $(\Phi_{k}^{R},\Phi_{k}^{I})$ is injective (which is equivalent to the injectivity of $\Phi_{k}$) the two transforms are equivalent, as $X$ is homeomorphic to $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X))$. Otherwise, they contain different information, as $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X))$ is homeomorphic to the quotient of $X$ obtained by identifying points $x,x' \in X$ whenever $\phi_{i}^{R}(x)= \phi_{i}^{R}(x')$ and $\phi_{i}^{I}(x)= \phi_{I}^{R}(x')$ for all $i = 1, \cdots, k$. Note, in particular, that the topological type of $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X))$ may change as $k$ increases.\\
To give a simplified example of the failure of the intrinsic and embedding transforms to give the same result, set $k=1$ and suppose that the first coordinate function $\alpha_{1}$ is constant. Then the set $\Phi_{1}(X) \subset \mathbb{R}^2$ is simply a point, and for any vector direction, the PHT or ECT give the persistence diagram or Euler curve of a point. However, the sublevel set filtration of the pair $(X,\alpha_{1})$ gives the empty set until the constant value of $\alpha_{1}$ is reached, at which point the entire space $X$ appears all at once, and its homology groups/Euler characteristic appear in the persistence diagram or Euler curve.
\end{remark}
\subsection{Stability and Inverse Results}
We will make use of the well-known functional stability result in the theory of persistence.
\begin{theorem}[Functional Stability, \cite{chazal2016structure,cohen2005stability}]
\label{thm:funcstab}
Let $X$ be a topological space, and let $f,g:X \to \mathbb{R}$ be two functions whose sublevel sets have finite-dimensional homology groups. Then $(X,f)$ and $(X,g)$ give rise to pointwise finite dimensional persistence modules with well-defined graded persistence diagrams $PH(X,f)$ and $PH(X,g)$. Moreover, writing $d_{B}$ for the graded bottleneck distance, we have:
\[d_{B}(PH(X,f),PH(X,g)) \leq \|f-g\|_{\infty}.\]
\end{theorem}
We now state our first stability result: stability of the i-$PKT_{k}$ and e-$PKT_{k}$ with respect to the vector $v \in \mathbb{S}^{2k-1}$.
\begin{prop}
\label{prop:iPKTlip}
Suppose that $X$ is homeomorphic to the geometric realization of a finite simplicial complex. If we equip $\mathbf{GrDiag}$ with the graded bottleneck distance and the sphere $\mathbb{S}^{2k-1}$ with the $\ell^1$ distance, then both the i-$PKT_{k}$ and e-$PKT_{k}$ are Lipschitz continuous.
\end{prop}
\begin{proof}
We begin with the i-$PKT_{k}$. Our embedding functions $\alpha_i^{R}$ and $\alpha_{i}^{I}$ are continuous functions on a compact space, hence they are uniformly bounded by a constant $C$. Then, if $(u,v), (w,z) \in \mathbb{S}^{2k-1} \subset \mathbb{R}^{k} \times \mathbb{R}^{k}$ are vectors with $\|(u,v) - (w,z)\|_{1} = \| u-w\|_{1} + \| v-z\|_{1} \leq \epsilon$, we have:
\begin{align*}
\|\sum_{i=1}^{k} (u_{i}\alpha_{i}^{R} + v_{i}\alpha_{i}^{I}) - \sum_{i=1}^{k}(w_{i}\alpha_{i}^{R} + z_{i}\alpha_{i}^{I}) \|_{\infty} &= \|\sum_{i=1}^{k} (u_i - w_i)\alpha_i^{R} + (v_i - z_i)\alpha_i^{I} \|_{\infty}\\
& \leq \sum_{i=1}^{k} |u_i - w_i|\|\alpha_i^{R} \|_{\infty} + \sum_{i=1}^{k} |v_i - z_i|\|\alpha_i^{I} \|_{\infty}\\
& \leq C \sum_{i=1}^{k} \left(|u_i - w_i| + |v_i - z_i|\right)\\
& \leq C\varepsilon.
\end{align*}
Thus, by Theorem \ref{thm:funcstab}, we have:
\[d_{B}\left(PH \left(X,\sum_{i=1}^{k} (u_{i}\alpha_{i}^{R} + v_{i}\alpha_{i}^{I})\right),PH \left(X,\sum_{i=1}^{k} (w_{i}\alpha_{i}^{R} + z_{i}\alpha_{i}^{I})\right)\right) \leq C\varepsilon.\]
For the e-$PKT_{k}$, the boundedness of our eigenfunctions implies that $(\Phi_{k}^{R}(X),\Phi_{k}^{I}(X))$ sits inside the $\ell^\infty$-ball of radius $C$ in $\mathbb{R}^{2k}$. Then, if $(u,v),(w,z) \in \mathbb{S}^{2k-1}$ are vectors with $\| (u,v) - (w,z)\|_{1} \leq \epsilon$, we deduce, for all $(x,y) \in (\Phi_{k}^{R}(X),\Phi_{k}^{I}(X))$:
\begin{align*}
|\Theta(u,v)(x,y) - \Theta(w,z)(x,y)| & =|\langle u,x \rangle + \langle v,y \rangle - \langle w,x\rangle - \langle z,y\rangle|\\
& = |\langle u - w,x\rangle + \langle v-z,y \rangle|\\
& = |\langle (u-w,v-z),(x,y)\rangle| \\
\mbox{(H\"{o}lder's Inequality)} &\leq \|(x,y)\|_{\infty}\|(u-w,v-z)\|_{1}\\
& \leq C\varepsilon.
\end{align*}
And thus, again by Theorem \ref{thm:funcstab},
\[d_{B}(PH(X,\Theta(u,v)),PH(X,\Theta(w,z))) \leq C\varepsilon.\]
\end{proof}
The next stability result, which applies to the i-$ECT_{k}$ and e-$ECT_{k}$, relies on the fact that the map from a persistence diagram to its Euler curve is stable:
\begin{lemma}
\label{clm:bar2euler}
Let $B_1$ and $B_2$ be graded persistence diagrams. Suppose that $B_1$ and $B_2$ have at most $L$ points each. Define the following operation that sends a graded persistence diagram to the Euler curve associated to its barcode:
\[\chi(B) = \sum_{I \in B} (-1)^{\operatorname{deg}(I)}1_{I}, \]
where we write $\operatorname{deg}(I)$ to indicate the homological degree of an interval $I$. Set $\chi_{1} = \chi(B_1)$ and $\chi_{2} = \chi(B_2)$. Then for any $0<p<1$,
\[\|\chi_{1} - \chi_{2}\|_{p} \leq (4L)^{q}d_{B}(B_1,B_2)^{q},\]
and for any $p \geq 1$,
\[\|\chi_{1} - \chi_{2}\|_{p} \leq (2^{1 + q}L) d_{B}(B_1,B_2)^{q}, \]
where $q = 1/p$.
\end{lemma}
\begin{proof}
By the definition of the Bottleneck distance, there are decompositions:
\[B_1 = \{I_{i}^{1}\}_{i=1}^{n_1} \cup \{J_{i}^{1}\}_{i=1}^{n_2} \]
\[B_2 = \{I_{i}^{2}\}_{i=1}^{n_1} \cup \{J_{i}^{2}\}_{i=1}^{n_3}, \]
where the intervals $I_{i}^{1}$ are paired with the intervals $I_{i}^{2}$ (both have the same degree), and the intervals $J_{i}^{1}$ and $J_{i}^{2}$ are paired with the diagonal. This means that, for $i \in \{1, \cdots, n_1\}$, the function $1_{I^{1}_{i}} - 1_{I^{2}_{i}}$ is supported on a set of measure at most $2d_{B}(B_1,B_2)$. Moreover, as the difference of two indicator functions, $1_{I^{1}_{i}} - 1_{I^{2}_{i}}$ is bounded in absolute value by one. Additionally, all the intervals $\{J_{i}^{1}\}_{i=1}^{n_2}$ and $\{J_{i}^{2}\}_{i=1}^{n_3}$ have length at most $2d_{B}(B_1,B_2)$. Note that $n_1 + n_2 + n_3 \leq (n_1 + n_2) + (n_1 + n_3) \leq 2L$. Thus:
\[\chi_{1} - \chi_{2} = \sum_{i=1}^{n_1}(-1)^{\operatorname{deg}(I_{i}^{\ast})}(1_{I_{i}^{1}} - 1_{I_{i}^{2}}) + \sum_{i=1}^{n_2}(-1)^{\operatorname{deg}(J_{i}^{1})}1_{J_{i}^{1}} - \sum_{i=1}^{n_3}(-1)^{\operatorname{deg}(J_{i}^{2})}1_{J_{i}^{2}}, \]
where the $\ast$ indicates that the choice of $I_{i}^{1}$ or $I_{i}^{2}$ does not matter, as both intervals have the same homological degree. We now split our analysis into two cases: (a) $0< p<1$, and (b) $p \geq 1$.\\
(a) In this case, we use the modified Minkowski's inequality $\|f-g\|_{p}^{p} \leq \|f\|_{p}^{p} + \|g\|_{p}^{p}$. We deduce:
\begin{align*}
\|\chi_{1} - \chi_{2}\|_{p}^{p} &\leq \sum_{i=1}^{n_1}\|1_{I_{i}^{1}} - 1_{I_{i}^{2}}\|_{p}^{p} + \sum_{i=1}^{n_2}\|1_{J_{i}^{1}}\|_{p}^{p} + \sum_{i=1}^{n_3}\|1_{J_{2}^{1}}\|_{p}^{p}\\
& \leq (n_1 + n_2 + n_3)\|1_{[0,2d_{B}(B_1,B_2)]}\|_{p}^{p} \\
& \leq 2L \|1_{[0,2d_{B}(B_1,B_2)]}\|_{p}^{p}\\
& = 2L(2d_{B}(B_1,B_2))\\
& = 4Ld_{B}(B_1,B_2).
\end{align*}
Hence, $\|\chi_{1} - \chi_{2}\|_{p} \leq (4L)^{q}d_{B}(B_1,B_2)^{q}$.
(b) In this case, we can use Minkowski's inequality to deduce:
\begin{align*}
\|\chi_{1} - \chi_{2}\|_{p} &\leq \sum_{i=1}^{n_1}\|1_{I_{i}^{1}} - 1_{I_{i}^{2}}\|_{p} + \sum_{i=1}^{n_2}\|1_{J_{i}^{1}}\|_{p} + \sum_{i=1}^{n_3}\|1_{J_{2}^{1}}\|_{p}\\
& \leq (n_1 + n_2 + n_3)\|1_{[0,2d_{B}(B_1,B_2)]}\|_{p} \\
& \leq 2L \|1_{[0,2d_{B}(B_1,B_2)]}\|_{p}\\
& = 2L \sqrt[p]{2d_{B}(B_1,B_2)}\\
& = 2^{1 + 1/p}L \sqrt[p]{d_{B}(B_1,B_2)}\\
& = (2^{1 + q}L) d_{B}(B_1,B_2)^{q}.
\end{align*}
\end{proof}
\begin{cor}
\label{cor:ECTlip}
Suppose that $X$ is homeomorphic to the geometric realization of a finite simplicial complex which implies bounded degree-$q$ total persistence, for some $q > 0$. Suppose further that there is a uniform bound on the number of points in the persistence diagrams obtained when evaluating the i-$PKT_{k}$ and e-$PKT_{k}$ at an arbitrary vector $(u,v) \in \mathbb{S}^{2k-1}$. If we equip the sphere $\mathbb{S}^{2k-1}$ with the $\ell^1$ distance, and the space of Euler curves with the $L^{1/q}$ distance, the i-$ECT_{k}$ and e-$ECT_{k}$ are $q$-H\"{o}lder continuous on $\mathbb{S}^{2k-1}$.
\end{cor}
\begin{proof}
The result follows from Proposition \ref{prop:iPKTlip} and Lemma \ref{clm:bar2euler}.
\end{proof}
We conclude this section with a coarse injectivity result, this time only for our embedded transforms. Note that, when $\Phi_{k}$ is injective, and our transforms agree, this result also holds for the intrinsic transforms.
\begin{theorem}
\label{thm:ePKTcoarseinj}
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be compact metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$ respectively. Let $k \in \mathbb{N}$ be a positive integer, and suppose that $\Phi_{k}(X)$ and $\Phi_{k}(Y)$ are definable. Then if either e-PKT$_{k}(X)$ = e-PKT$_{k}(Y)$ or e-EKT$_{k}(X)$ = e-EKT$_{k}(Y)$, we have:
\[d_{GH}(X,Y) \leq \| E_{X,k} \|_{\infty} + \| E_{Y,k} \|_{\infty}.\]
\end{theorem}
\begin{proof}
This follows from the injectivity of the PHT and ECT, as demonstrated in Theorem \ref{thm:PHTinj}, together with Theorem \ref{thm:invstabmeas}, where we take $\epsilon = 0$.
\end{proof}
The condition that the distance kernel embeddings be definable is always satisfied when the spaces are finite. Demonstrating definability more generally will require techniques and results from the theory of o-minimal geometry, and will therefore constitute a line of inquiry quite distinct from the focus of this paper.
\subsection{Prior Work}
In \cite{turner2014persistent}, Turner et. al. defined the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT). These transforms take as input a sufficiently regular subset of Euclidean space $S \subseteq \mathbb{R}^d$, and associate to every vector $v \in \mathbb{S}^{d-1}$ the persistence diagram, or Euler characteristic curve, of the sublevel-set filtration of $S$ induced by the function $f_{v}:S \to \mathbb{R}$:
\[f_{v}(x) = v \cdot x.\]
It was subsequently proven in \cite{curry2018many} and \cite{ghrist2018persistent} that these topological transforms are injective in all dimensions\footnote{By ``injective," we mean that two subsets of Euclidean space have the same transform if and only if they are identical. Thus, the transform is injective on the space of admissible subsets.}. Moreover, it was shown in \cite{curry2018many} and \cite{belton2018learning} that, for certain families of embedded shapes, these topological transforms can be computed in finitely many steps\footnote{That is, finitely many directions determine the entire transform, and these directions can be identified with finitely many geometric computations}. Complimenting these theoretical results, Crawford et. al. \cite{crawford2016topological} demonstrated how to use these topological transforms to build an improved classifier for glioblastoma patient outcomes.
In \cite{oudot2017barcode}, Oudot and Solomon defined a topological transform for intrinsic metric spaces $(X,d_X)$. This transform associates to every basepoint $x_0 \in X$ the extended persistence diagram of the function $f_{x_0}:X \to \mathbb{R}$:
\[f_{x_0}(x) = d_{X}(x_0,x).\]
The resulting invariant, called the Intrinsic Persistent Homology Transform (IPHT), is the collection of all persistence diagrams arising from basepoints in $X$. By computing Euler characteristic curves instead of persistence diagrams, one obtains the Intrinsic Euler Characteristic Transform (IECT). This invariant was first studied, in the case of metric graphs, by Dey, Shi, and Wang in \cite{dey2015comparing}, where they proved stability and computability results and ran some experiments. The main result of \cite{oudot2017barcode} demonstrated that these invariants are injective\footnote{Similarly to the PHT and ECT, this injectivity means that two graphs having the same transform must be isometric.} on an appropriately generic subset of the space of metric graphs.
We refer the reader to \cite{oudot2018inverse} for a survey of inverse problems in applied topology
As this paper is concerned with both applied topology and spectral geometry, let us now consider some results, both classical and modern, in the latter field. To begin, the data of a weighted graph can be encoded via its adjacency matrix, and the spectral theory of these matrices is deep and of great utility, seeing application in, e.g., graph clustering and Google's PageRank algorithm. Another matrix associated to a graph is its Laplacian, whose eigendecomposition forms the basis for the Laplacian Eigenmaps technique studied by Belkin and Niyogi in \cite{belkin2002laplacian,belkin2003laplacian,belkin2007convergence}. If one performs spectral analysis on the centered Gram matrix of a distance matrix, the resulting eigenvectors give the classical Multi-Dimensional Scaling embedding. Tenenbaum et. al. \cite{tenenbaum2000global} defined an extension of Multi-Dimensional Scaling for point clouds, called IsoMap, by building a neighborhood graph on the points, defining distances using shortest paths in this graph, and applying Multi-Dimensional Scaling to the resulting distance matrix. Diffusion map embeddings arise from studying the spectral theory of the diffusion operator, as defined by Coifman and Lafon in \cite{coifman2006diffusion}. Lastly, the X-ray transform of \cite{john1938ultrahyperbolic} takes as input a continuous, compactly supported function $f$ on $\mathbb{R}^d$, and outputs a function on the space of lines in $\mathbb{R}^d$ that encodes the corresponding line integral of $f$.
Like the embeddings of \cite{belkin2003laplacian} and \cite{coifman2006diffusion}, we study the eigenfunctions and eigenvalues of an operator defined on our shape. However, our goal is not to find a low-dimensional representation of a noisy data set, but rather to use our eigenfunctions to compute Hausdorff distances, persistence diagrams, and Betti curves. As we are interested in injectivity results, we want these eigenfunctions to encode the large-scale geometry of our shape. The results that follow demonstrate that using the distance function as an integral kernel produces an operator well suited to this task.
\subsection{Results}
In Section \ref{sec:distkernel}, we define the distance kernel operator on metric measure spaces, proving that it is a compact, self-adjoint operator to which the spectral theorem can be applied (Propositions \ref{prop:selfadjoint} and \ref{prop:compact}). We then establish conventions for choosing eigenfunctions for our embedding.
In Section \ref{sec:distkernelembed}, we use the eigenfunctions $\{\phi_i\}$ of the distance kernel operator to define complex-valued coordinates $\alpha_i = \sqrt{\lambda_i}\phi_i$ on our metric-measure space. We then prove that, for non-zero eigenvalues, these eigenfunctions, and their associated coordinates, are Lipschitz, with constant inversely proportional to the magnitude of the eigenvalue (Lemma \ref{lem:eiglip}). We then define a $\mathbb{C}^k$-valued map $\Phi_{k}$, called the distance kernel embedding, whose component functions are the $\alpha_{i}$, and study its injectivity. When infinitely many eigenfunctions are used, we show that the embedding is pointwise injective\footnote{This injectivity is defined for pairs of points on a fixed space $X$.}:
\begin{mr}[Theorem \ref{lem:psiembed}]
Let $(X,\dist_{X},\mu_X)$ be a compact, strictly positive\footnote{See Definition \ref{def:strictlypositive}.} metric measure space. Then the map $\Phi : X \to \mathbb{C}^{\infty}$ is injective.
\end{mr}
When finitely many eigenfunctions are used, we show that the fiber of a vector in $\mathbb{C}^k$ has bounded diameter, with an upper bound on the diameter decreasing in the embedding dimension $k$. This result holds for Riemannian manifolds equipped with their volume measure and, more generally, metric measure spaces satisfying a regularity condition called $(a,b)$-standardness\footnote{See Definition \ref{def:abstandard}}:
\begin{mr}[Corollary \ref{cor:Rcoarseinj}]
There exists a function $N(r,n): \mathbb{R}_{>0} \times \mathbb{N} \to \mathbb{N}$ with the following property.
Let $X$ be a complete $d$-dimensional Riemannian manifold with positive injectivity radius $R$. For every $r \leq R/2$, and $k \geq N(r,n)$, if $\Phi_{k}(x) =\Phi_{k}(y)$ then $\dist_{X}(x,y) \leq 3r$.
\end{mr}
\begin{mr}[Corollary \ref{cor:ABcoarseinj}]
There exists a function $N(s,a,b): \mathbb{R}_{>0} \times \mathbb{R}_{>0} \times \mathbb{R}_{>0} \to \mathbb{N}$ with the following property.
Let $(X,\dist_{X},\mu_X)$ be a compact $(a,b)$-standard metric measure space with threshold parameter $r$. For every $s \leq r$ and $k \geq N(s,a,b)$, if $\Phi_{k}(x) =\Phi_{k}(y)$ then $\dist_{X}(x,y) \leq 3s$.
\end{mr}
Section \ref{sec:discretization} demonstrates that the spectral properties of the distance kernel operator can be approximated, within arbitrary precision, by finite, random i.i.d. samples. The main result is the following:
\begin{mr}[Theorem~\ref{thm:approxDKE}]
For any compact metric measure space $(X,\dist,\mu)$ with $(a,b)$-standard Borel measure, and $X_n$ an i.i.d. sample of X of size $n$,
\[
\dist_H^{L^2}(\Phi_k(X),\hat{\Phi}_k(X_n)) \xrightarrow[]{\text{a.s}}0 \ \text{as} \ n \to +\infty
\]
where $\dist_H^{L^2}$ is the Hausdorff distance for the $L^2$ norm in $\C^k$, and $\hat{\Phi}_k(X_n)$ is the \emph{empirical distance kernel embedding}\footnote{See Section~\ref{sec:discretization} for the appropriate construction} defined on the sample $X_n$.
\end{mr}
Section \ref{sec:stability} proves that the distance kernel embedding is stable for compact Riemannian manifolds. More precisely, Main Result \ref{mr:stab} proves that there is a choice of metric on the space of Riemannian manifolds, namely the (modified) Gromov Hausdorff Prokhorov distance\footnote{See Definitions~\ref{def:modifiedPdistance} and~\ref{def:modifiedGHPdistance}}, such that the Hausdorff distance between the $k$-dimensional distance kernel embeddings of two Riemannian manifolds is bounded by a function $F$ that depends on their distance in this metric as well as their $k$ largest (in absolute value) eigenvalues, and where the function $F$ goes to zero as the distance between the manifolds goes to zero.
\begin{mr}[Theorem \ref{thm:stability}]
\label{mr:stab}
Let $(X,\dist_X,\mu)$ and $(Y,\dist_Y,\eta)$ be compact finite dimensional Riemannian manifolds with their volume measures, such that $\mu(X) = \eta(Y)$. Let $|\lambda_1| > \ldots > |\lambda_k| > 0$ and $|\nu_1| > \ldots > |\nu_k| > 0$ be the $k$ largest eigenvalues in absolute value of operators $D^X$ and $D^Y$ respectively, all non-trivial, of multiplicity one, and with distinct absolute values. Let $\Phi_k\colon X \to \mathbb{C}^k$ and $\Psi_k\colon Y\to \mathbb{C}^k$ be the DKE for $D^X$ and $D^Y$ respectively.
\medskip
Define the intertwining $\Delta_k^{X,Y}$ of the spectra of $D^X$ and $D^Y$, measuring the separation between the spectra of the operators, by:
\[
\Delta_k^{X,Y} = \min \left\{ |\lambda_i^2-\nu_j^2| \ : \ 1 \leq i,j \leq k, i \neq j \right\},
\]
and let:
\[
|\tau_1| := \min \{|\lambda_1|,|\nu_1|\}, \ \ \text{and} \ \ |\tau_k| := \max \{|\lambda_k|,|\nu_k|\}.
\]
\medskip
Then,
\[
\dist_H^{L^2}(\Phi_k(X),\Psi_k(Y)) \leq \displaystyle\sqrt{k}\left( 4\sqrt{2} \frac{(\varepsilon+|\tau_1|)}{\Delta^{X,Y}_k}\sqrt{|\tau_1|}\right) \cdot \varepsilon +
\displaystyle\sqrt{k}\left( 2\sqrt{2} \sqrt{\frac{(\varepsilon + |\tau_1|)}{|\tau_k|}}\right) \cdot \sqrt{\varepsilon},
\]
where $\varepsilon := \dist_{G\bar{P}}(X,Y)$ stands for the modified Gromov-Prokhorov distance between $X$ and $Y$ (see Definitions~\ref{def:modifiedPdistance} and~\ref{def:modifiedGHPdistance} below).
\end{mr}
In Section \ref{sec:stabinv}, we study the global injectivity and stability properties of the distance kernel embedding. The first result is that, under mild regularity assumptions, the embedding encodes the entire metric data of the space:
\begin{mr}[Lemma \ref{lem:PhiInj}]
Fix a topological space $X$. Let $\mu_{1}$ and $\mu_{2}$ be strictly positive measures for the Borel $\sigma$-algebra on $X$, with $\mu_1$ absolutely continuous with respect to $\mu_2$, and $\dist_{1}$ and $\dist_{2}$ metrics on $X$ making $X_1 = (X,\dist_1,\mu_1)$ and $X_2 = (X,\dist_2,\mu_2)$ metric measure spaces. Let $D_1$ and $D_2$ be the resulting integral operators. If $\Phi(X_1) = \Phi(X_2)$, then $\dist_1 = \dist_2$.
\end{mr}
Moving forward, we show that if the distance kernel embeddings of two metric measure spaces are at small Hausdorff distance from each other, then these spaces are close in the Gromov-Hausdorff distance:
\begin{definition}
For a compact metric measure space $(X, \dist_{X}, \mu_{X})$ and a positive integer $k$, define the error function (where $[\cdot,\cdot]$ is a bilinear form defined in Section \ref{sec:stabinv}):
\[E_{X,k}(x,x') = |[\Phi_{k}(x),\Phi_{k}(x')] - \dist_{X}(x,x')|.\]
This error measures the extent to which the eigenfunction expansion of $\dist_{X}$ fails to approximate the metric.
\end{definition}
\begin{mr}[Theorem \ref{thm:invstabmeas}]
\label{mr:invstabmeas}
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be compact metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$ respectively. Take $k \in \mathbb{N}$ to be a positive integer, and let $\varepsilon = \dist_{H}^{L^2}(\Phi_{k}(X),\Phi_{k}(Y))$. Then
\[d_{GH}(X,Y) \leq 2\varepsilon \min \left\{ \max_{x \in X} \| \Phi_{k}(x) \|_{2}, \max_{y \in Y} \| \Phi_{k}(y) \|_{2}\right\} + \| E_{X,k} \|_{\infty} + \| E_{Y,k} \|_{\infty} + \varepsilon^2. \]
\end{mr}
Simulations conducted in Section \ref{sec:experiments} demonstrate the constants appearing in this result are well-behaved, with the multiplicative error remaining small, and the additive error going to $0$, as $k$ goes to infinity.
We then study the embedding in detail in the special case of finite metric measure spaces, culminating with the following simplified version of Theorem \ref{thm:invstabmeas}:
\begin{mr}[Theorem \ref{thm:invstab}]
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be finite metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$, and let $\theta = \min \{\min_{x \in X}\mu_{X}(x),\min_{y \in Y}\mu_{X}(y)\}$. Take $k \leq |X|,|Y|$, and suppose that $\dist_{H}^{L^{2}}(\Phi_{k}(X),\Phi_{k}(Y)) \leq \varepsilon$. Then,
\[\dist_{GH}(X,Y) \leq 2\varepsilon \frac{\min (\sqrt{|\lambda_{1}|},\sqrt{|\nu_{1}|})}{\theta} + \varepsilon^2 + \frac{|\lambda_{k+1}| + |\nu_{k+1}|}{\theta}. \]
\end{mr}
The quality of the bound in Main Result~\ref{mr:invstabmeas} depends on the $L^2$ norms of the embedding vectors and the $L^\infty$ norms of the error functions. In Section~\ref{sec:embeddingconstants}, explicit bounds on these quantities are given in terms of the geometry of the metric measure spaces. These worst-case bounds are in fact pessimistic, and the simulations considered at the end of the paper suggest that they can be improved upon greatly.
In Section~\ref{sec:intrinsicpht}, we study the topological features of the spectrum of the distance kernel operator. We demonstrate that our eigenfunctions have persistence diagrams and, under certain regularity assumptions, Betti and Euler curves:
\begin{mr}[Proposition \ref{prop:pers}]
Let $(X,\dist_{X},\mu_X)$ be a compact metric measure space homeomorphic to the geometric realization of a finite simplicial complex. Then, any finite linear combination $f = \sum_{i=1}^{n}c_i \phi_i$ of eigenfunctions of $D^X$ with nonzero eigenvalue has a well-defined sublevel set persistence diagram $PH(X,f)$.
\end{mr}
\begin{mr}[Proposition \ref{prop:betticurves}]
Suppose now that $X$ is homeomorphic to the geometric realization of a finite simplical complex which implies bounded degree-$q$ total persistence\footnote{See Definition \ref{def:boundedtotal}. This condition ensures the existence of our Euler curves}. Let $p=1/q$. Then for any homological degree $k$, the sum defining $\beta_{k}(X,f)$ converges in $L^p$. Under the same hypothesis, the sum defining $\chi(X,f)$ is finite, so that the Euler curve also exists as a function in $L^p$.
\end{mr}
In Definition \ref{def:transforms}, we define two types of topological transforms: (1) intrinsic transforms, namely the i-PKT and i-EKT, that compute the persistence diagrams and Euler curves of linear combinations of eigenfunctions, and (2) embedded transforms, namely the e-PKT and e-EKT, that amount to computing the PHT and ECT on the image of the distance kernel embedding. Like the PHT and ECT, all of these transforms give maps from a sphere of the appropriate dimension to a space of topological summaries (either persistence diagrams or Euler curves). We then prove that the i-PKT and e-PKT are Lipschitz continuous, whereas their Euler Characteristic counterparts are H\"{o}lder continuous:
\begin{mr}[Proposition \ref{prop:iPKTlip}]
Suppose that $X$ is homeomorphic to the geometric realization of a finite simplicial complex, and the barcode space is equipped with the bottleneck distance. Then, both the i-$PKT_{k}$ and e-$PKT_{k}$ are Lipschitz continuous on $\mathbb{S}^{2k-1}$.
\end{mr}
\begin{mr}[Corollary \ref{cor:ECTlip}]
Let $q \leq 1$, and suppose that $X$ is homeomorphic to the geometric realization of a finite simplicial complex which implies bounded degree-$q$ total persistence. Suppose further that there is a uniform bound on the number of points in the persistence diagrams obtained when evaluating the i-$PKT_{k}$ and e-$PKT_{k}$ at an arbitrary vector $(u,v) \in \mathbb{S}^{2k-1}$. If we equip the sphere $\mathbb{S}^{2k-1}$ with the $\ell^1$ distance, and the space of Euler curves with the $L^{1/q}$ distance, the i-$ECT_{k}$ and e-$ECT_{k}$ are $q$-H\"{o}lder continuous on $\mathbb{S}^{2k-1}$.
\end{mr}
Lastly, as a consequence of Main Result \ref{mr:invstabmeas}, we show that the e-PKT and e-EKT have bounded inverses. To obtain this technical result, we need to assume that our distance kernel embeddings are \emph{definable} subsets of Euclidean space. The notion of definability comes from the theory of o-minimal geometry, and is necessary for the application of techniques from Euler calculus. C.f. \cite{curry2018many} \S2 for precise definitions. Finite embeddings are always definable.
\begin{mr}[Theorem \ref{thm:ePKTcoarseinj}]
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be compact metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$ respectively. Let $k \in \mathbb{N}$ be a positive integer, and suppose that $\Phi_{k}(X)$ and $\Phi_{k}(Y)$ are definable. Then if either e-PKT$_{k}(X)$ = e-PKT$_{k}(Y)$ or e-EKT$_{k}(X)$ = e-EKT$_{k}(Y)$, we have:
\[d_{GH}(X,Y) \leq \| E_{X,k} \|_{\infty} + \| E_{Y,k} \|_{\infty}.\]
\end{mr}
The final section of the paper, Section \ref{sec:experiments}, consists of a variety of numerical experiments serving as proofs of concept for the distance kernel embedding. We consider a number of discrete metric spaces sampled from Lens spaces, tori, 2-spheres, and 3-spheres, and we look at the Hausdorff distances between their embeddings, the distribution of their eigenvalues, the distribution of mass of the scaled eigenvectors, the distribution of magnitudes of their embedding vectors, the values of the additive and multiplicative error bounds appearing in Theorem \ref{thm:invstabmeas}, and the values of the estimates for these bounds appearing in Lemmas \ref{lem:errorbound} and \ref{lem:embedbound}. We observe that, for our data sets, the values of the additive and multiplicative errour bounds in Theorem \ref{thm:invstabmeas} are much smaller than the estimates provided in Section \ref{sec:embeddingconstants}, and that the distance kernel embedding suffices to distinguish a variety of manifolds, including two Lens spaces with same homotopy type.
\subsection{Injectivity}
In this section, we explore the injectivity of the distance kernel embedding on the space of metric measure spaces. Our first result is that, under mild regularity assumptions, two metric measure spaces have the same distance kernel embedding if and only if there is an isometry between them.
\begin{obs}
\label{obs:hom}
Corollary \ref{cor:DKThom} states that $\Phi$ is a homeomorphism when $(X,\dist_X,\mu_X)$ is a strictly positive metric measure space. Therefore, if $\Phi(X,\dist_X,\mu_X) = \Phi(Y,\dist_{Y},\mu_Y)$ for a pair of strictly positive metric measure spaces, then there is a homeomorphism from $X$ to $Y$ preserving all the eigenfunctions. Thus, in the following lemma, we fix an underlying topological space and allow the metric and measure to vary.
\end{obs}
\begin{lemma}
\label{lem:PhiInj}
Fix a compact topological space $T$. Let $\mu$ and $\mu'$ be strictly positive measures for the Borel $\sigma$-algebra on $T$, with $\mu$ absolutely continuous with respect to $\mu'$, and $\dist$ and $\dist'$ metrics on $X$, both inducing the topology $T$, making $X = (T,\dist,\mu)$ and $X' = (T,\dist',\mu')$ metric measure spaces. Let $D$ and $D'$ be the resulting integral operators. If $\Phi(X) = \Phi(X')$, then $\dist = \dist'$.
\end{lemma}
\begin{proof}
By assuming that both metric measure spaces induce the same topology, we can work with a single $\sigma$-algebra: their common Borel $\sigma$-algebra. This will prove essential in the following proof, where we take various unions and complements of measurable sets for $\mu$ and $\mu'$, respectively.\\
The equality $\Phi(X) = \Phi(X')$ implies that $D$ and $D'$ have the same scaled eigenfunctions $\alpha_i$. The distance functions $\dist, \dist'$ thus have the same eigenfunction expansion:
\[(x_1, x_2) \mapsto \sum_{i=1}^{\infty}\alpha_{i}(x_1)\alpha_{i}(x_2). \]
This converges to $\dist$ in $L_2 (\mu \otimes \mu)$ and to $\dist'$ in $L_2 (\mu' \otimes \mu')$ to $\dist'$. Let us denote by $S_n$ the partial sums of this expansion:
\[S_n = \sum_{i=1}^{n}\alpha_{i}(x_1)\alpha_{i}(x_2). \]
It is a standard result in measure theory that any $L^{2}$-convergent sequence admits a subsequence that converges pointwise a.e.\footnote{See Theorem 2.15(c) in \cite{folland2009guide} for the implication that convergence in measure implies the existence of pointwise a.e. convergent subsequence. Chebyshev's inequality proves that $L^2$ convergence implies convergence in measure.} Thus, one can extract a subsequence $S_{n_{k}}$ that converges to $\dist$ pointwise on $(T \times T) \setminus N_1$, for some set $N_1 \subset T$ such that $(\mu \otimes \mu)(N_1) = 0$. We can then extract a further subsequence $S_{n_{k_j}}$ that converges pointwise to $\dist'$ on $((T \times T) \setminus N_1) \setminus N_2$, where $(\mu' \otimes \mu')(N_2) = 0$. Since $\mu$ is absolutely continuous to $\mu'$, if we set $N = N_1 \cup N_2$ then $(\mu \otimes \mu)(N) = 0$. Since $\mu$ is strictly positive, the set $N$ cannot contain any open sets, hence $N^{c}$ is dense in $T \times T$. We see then that $\dist = \dist'$ on a dense subset of $T \times T$; since these functions are both continuous in the same topology $T$, they are equal everywhere.
\end{proof}
For finite metric measure spaces, Lemma \ref{lem:PhiInj} only requires a finite-dimensional embedding to hold.
\begin{cor}
Let $X = (X,\dist_{X},\mu_X)$ and $Y = (Y, \dist_{Y}, \mu_Y)$ be a pair of finite metric measure spaces, with $|X|, |Y| \leq k$. If $\Phi_{k}(X) = \Phi_{k}(Y)$, then $X$ and $Y$ are isometric.
\end{cor}
\begin{proof}
From Observation \ref{obs:hom}, we see that $|X| = |Y|$, so $X$ and $Y$ are homeomorphic when equipped with the discrete topology, which is the same topology induced by any choice of metric on $X$ or $Y$. The result follows then from Lemma \ref{lem:PhiInj}, by noting that all the ``higher" eigenfunctions vanish and therefore are equal.
\end{proof}
\subsection{Inverse Stability}
These injectivity results tell us that distinct metric spaces have distinct embeddings. However, we would like to assert something stronger: spaces with similar embeddings are also geometrically similar. To that end, we need to introduce some new definitions and technical lemmas.
\begin{definition}
For vectors $v,w \in \mathbb{C}^k$, define the following bilinear form:
\[[v,w] = \sum_{i=1}^{k} v_{i}w_{i} \in \mathbb{C}. \]
This form is symmetric but not a dot product.
\end{definition}
The utility of the bilinear form $[\cdot, \cdot]$ comes from the following equality:
\begin{equation}
\label{eqn:expansion}
[\Phi_{k}(x),\Phi_{k}(x')] = \sum_{i=1}^{k}\alpha_{i}(x)\alpha_{i}(x') = \sum_{i=1}^{k}\left(\sqrt{\lambda_{i}}\phi_{i}(x)\right)\left(\sqrt{\lambda_{i}}\phi_{i}(x')\right)= \sum_{i=1}^{k} \lambda_{i}\phi_{i}(x)\phi_{i}(x').
\end{equation}
That is, when applied to the distance kernel embedding it gives the first $k$ terms of the eigenfunction expansion of the distance function $\dist_{X}$.
\begin{lemma}
The bilinear form $[,]$ satisfies the following Cauchy-Schwarz inequality:
\[|[v,w]| \leq \| v \|_{2}\| w \|_{2}. \]
\end{lemma}
\begin{proof}
Let $v = (v_1, \cdots, v_k) \in \mathbb{C}^k$ and $w = (w_1, \cdots, w_k) \in \mathbb{C}^k$. By definition,
\[[v,w] = \sum_{i=1}^{k}v_{i}w_{i}. \]
Using the triangle inequality for complex numbers,
\[|[v,w]| = \left|\sum_{i=1}^{k}v_{i}w_{i}\right| \leq \sum_{i=1}^{k}|v_{i}||w_{i}| = \langle \tilde{v}, \tilde{w} \rangle, \]
where $\tilde{v},\tilde{w} \in \mathbb{R}^k$ are obtained from $v$ and $w$ by taking component-wise moduli. Note that
\[\| v \|_{2}^2 = \sum_{i=1}^{k}|v_{i}|^2 = \|\tilde{v}\|_{2}^2. \]
\[\| w \|_{2}^2 = \sum_{i=1}^{k}|w_{i}|^2 = \|\tilde{w}\|_{2}^2. \]
Thus $v$ and $\tilde{v}$ have the same magnitude, as do $w$ and $\tilde{w}$. To complete the proof, we apply the ordinary Cauchy-Schwarz inequality to $\tilde{v}$ and $\tilde{w}$,
\[\langle \tilde{v}, \tilde{w} \rangle \leq \|\tilde{v}
\|_2 \|\tilde{w}\|_2 = \| v \|_{2}\| w \|_{2}. \]
\end{proof}
The following lemma asserts that pairs of nearby vectors have similar bilinear products.
\begin{lemma}
\label{lem:haustodist}
Let $v_{1},v_{2},w_{1},w_{2} \in \mathbb{C}^k$ be vectors such that $\|v_1 - w_1\|_{2} \leq \varepsilon$ and $\|v_2 - w_2\|_{2} \leq \varepsilon$. Then \[|[v_1,v_2] - [w_1,w_2]| \leq \varepsilon \min \left\{\|v_1\|_{2} + \|v_2\|_{2}, \|w_1\|_{2} + \|w_2\|_{2}\right\} + \varepsilon^2.\]
\end{lemma}
\begin{proof}
By bilinearity,
\[[w_1,w_2] = [v_1,v_2] + [v_1,(w_2 - v_2)] + [(w_1 -v_1),v_2] + [(w_1 - v_1),(w_2 - v_2)].\]
Thus,
\[|[v_1,v_2] - [w_1,w_2]| \leq | [v_1,(w_2 - v_2)]| + |[(w_1 -v_1),v_2]| + |[(w_1 - v_1),(w_2 - v_2)]|.\]
By a symmetric argument, switching $v_1$ and $v_2$ with $w_1$ and $w_2$, one obtains:
\[|[v_1,v_2] - [w_1,w_2]| \leq | [w_1,(v_2 - w_2)]| + |[(v_1 -w_1),w_2]| + |[(v_1 - w_1),(v_2 - w_2)]|.\]
The result then follows by applying the Cauchy-Schwarz to each term on the right-hand sides of both inequalities, and taking the minimum of the two sums.
\end{proof}
Before stating our first inverse stability result, we need the following definition:
\begin{definition}
For a compact metric measure space $(X, \dist_{X}, \mu_{X})$ and a positive integer $k$, define the error function:
\[E_{X,k}(x,x') = |[\Phi_{k}(x),\Phi_{k}(x')] - \dist_{X}(x,x')|\]
By equation (\ref{eqn:expansion}), this measures the extent to which the eigenfunction expansion of $\dist_{X}$, truncated at $k$, approximates the metric.
\end{definition}
\begin{theorem}
\label{thm:invstabmeas}
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be compact metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$. Take $k \in \mathbb{N}$ to be a positive integer, and let $\varepsilon = \dist_{H}^{L^2}(\Phi_{k}(X),\Phi_{k}(Y))$. Then
\[d_{GH}(X,Y) \leq 2\varepsilon \min \left\{ \max_{x \in X} \| \Phi_{k}(x) \|_{2}, \max_{y \in Y} \| \Phi_{k}(y) \|_{2}\right\} + \| E_{X,k} \|_{\infty} + \| E_{Y,k} \|_{\infty} + \varepsilon^2. \]
\end{theorem}
\begin{proof}
Let $C$ be an optimal Hausdorff correspondence between $\Phi_{k}(X)$ and $\Phi_{k}(Y)$. Let $(x,x') \in X \times X$ and $(y,y') \in Y \times Y$ with $(\Phi_{k}(x),\Phi_{k}(y)),(\Phi_{k}(x'),\Phi_{k}(y')) \in C$. Lemma \ref{lem:haustodist}, together with the bounds $\|\Phi_{k}(x)-\Phi_{k}(y)\|_{L^2} \leq \varepsilon$ and $\|\Phi_{k}(x')-\Phi_{k}(y')\|_{L^2} \leq \varepsilon$ , gives
\[|[\Phi_{k}(x),\Phi_{k}(x')] - [\Phi_{k}(y),\Phi_{k}(y')] | \leq 2\varepsilon \min \left\{ \max_{x \in X} \| \Phi_{k}(x) \|_{2}, \max_{y \in Y} \| \Phi_{k}(y) \|_{2} \right\} + \varepsilon^2.\]
Using the triangle inequality, we can replace $[\Phi_{k}(x),\Phi_{k}(x')]$ with $\dist_{X}(x,x')$ and $[\Phi_{k}(y),\Phi_{k}(y')]$ with $\dist_{Y}(y,y')$, at the cost of adding an additive error of at most $\| E_{X,k} \|_{\infty}$ and $\| E_{Y,k} \|_{\infty}$ respectively, giving the inequality:
\[ |\dist_{X}(x,x') - \dist_{Y}(y,y')| \leq 2\varepsilon \min \left\{ \max_{x \in X} \| \Phi_{k}(x) \|_{2}, \max_{y \in Y} \| \Phi_{k}(y) \|_{2}\right\} + \| E_{X,k} \|_{\infty} + \| E_{Y,k} \|_{\infty} + \varepsilon^2. \]
The result follows.\end{proof}
\begin{remark}
In Section \ref{sec:embeddingconstants}, we provide analytic estimates on the constants appearing in Theorem \ref{thm:invstabmeas}, in terms of the geometry of the spaces $X$ and $Y$. These estimates are pessimistic and fall short of the positive results obtained in the simulations of Section \ref{sec:experiments}.
\end{remark}
\begin{remark}
The theorem is stated in terms of the Gromov-Hausdorff distance rather than its measure-theoretic variants, such as the Gromov-Hausdorff-Prokhorov distance. This is because scaling the measure does not affect the embedding. Thus, we can say that our embedding uses the measure to prioritize certain distance functions on our space, but it does not record the measure itself.
\end{remark}
Theorem~\ref{thm:invstabmeas} applies to compact metric spaces in general. In the following section, we reconsider these results for the finite case, demonstrating how the embedding is computed and concluding with a simplified, more precise form of Theorem~\ref{thm:invstabmeas}.
\subsection*{Finite Metric Measure Spaces}
In the finite case, we have a metric measure space $(X,\dist_{X},\mu_X)$ with $|X| = n$. We assume that $\mu_{X}$ has full support, so that $\mu_{X}(x) > 0$ for all $x \in X$. Define the matrix\footnote{We intentionally use the same symbol $D$ to refer to both the operator and its associated matrix.} $D_{ij} = \dist_{X}(x_i,x_j)\mu_{X}(x_j)$, so that if $f: X \to \mathbb{R}$ is a function, and $v \in \mathbb{R}^n$ is the vector $v_{i} = f(x_i)$, then $(Df)(x_j) = \sum_{i=1}^{n}\mu_{X}(x_i)\dist_{X}(x_i,x_j)f(x_i) = (Dv)_{j}$. Define the diagonal matrix $Q_{ii} = \mu_{X}(x_i)$, giving rise to the inner product $Q(v,w) = v^T Q w$.
\begin{lemma}
The operator $D$ is self-adjoint with respect to $Q$.
\end{lemma}
\begin{proof}
Let $v,w \in \mathbb{R}^n$. Compute:
\begin{align*}
\langle Dv, w \rangle_{Q} &= \sum_{i}(Dv)_{i}w_{i}\mu_{X}(x_i)\\
& = \sum_{i}(\sum_{j}v_j D_{ij} )w_{i}\mu_{X}(x_i)\\
& = \sum_{i,j}\dist(x_i,x_j)v_j w_i \mu_{X}(x_j)\mu_{X}(x_i)\\
& = \sum_{j}(\sum_{i} \dist(x_j,x_i)\mu_{X}(x_i)w_i) v_j \mu_{X}(x_j)\\
& = \sum_{j}(\sum_{i} D_{ji} w_i) v_j \mu_{X}(x_j)\\
& = \sum_{j} (Dw)_{j}v_{j}\mu_{X}(x_j)\\
& = \langle v, Dw \rangle_{Q}.
\end{align*}
\end{proof}
Thus, by the spectral theorem, $D$ has real eigenvalues $\{\lambda_1, \cdots, \lambda_n \}$, ordered by decreasing absolute value, and a basis of $Q$-orthonormal real eigenvectors $\{e_1, \cdots e_n\}$. As earlier, we maintain the convention that each eigenvalue has multiplicity one.\\
Let $A$ be the $n \times n$ matrix whose $i$th column is $e_i$. $Q$-orthonormality of the eigenbasis means that:
\[A^{T}QA = I.\]
Since $\mu_{X}(x_i)>0$ for all $i$, the diagonal matrix $Q$ is invertible. Moreover, since $A$ is orthonormal for the inner product induced by $Q$, it too is invertible. We can thus deduce:
\[ A^{T}QA = I \Longrightarrow AA^{T} = Q^{-1}.\]
Denoting the $i$th row of the matrix $A$ by $r_i$, this tells us that:
\begin{equation}
\label{eqn:rownorm}
\| r_i \|_2 = 1/\sqrt{\mu_{X}(x_i)}.
\end{equation}
As before, we define the functions $\alpha_{i}(x_j) = \sqrt{\lambda_{i}}(e_i)_{j}$ with the convention that the square root of a negative eigenvalue is the imaginary number with positive imaginary part, giving rise to the embedding $\Phi = (\alpha_1, \cdots, \alpha_n)$. If $V$ is the diagonal matrix whose $(ii)$th entry is $\sqrt{\lambda_i}$, then $\Phi$ maps $x_i$ to the $i$th row of $AV$. We now show how to recover the geometry of $X$ from its embedding. For $x_{i} \in X$, let $\dist_{i} : X \to \mathbb{R}$ be the ``distance to $x_i$" function, thought of as a vector in $\mathbb{R}^n \subset \mathbb{C}^n$. Observe that
\begin{align*}
\dist_{i} &= \sum_{l=1}^{n}\langle e_l, \dist_i \rangle_{Q} e_l\\
& = \sum_{l=1}^{n} \left(\sum_{j=1}^{n}(e_{l})_{j}Q_{jj}(d_{i})_{j} \right) e_l\\
& = \sum_{l=1}^{n}\left(\sum_{j=1}^{n}(e_{l})_{j}\mu_{X}(x_j)(d_{i})_{j} \right) e_l\\
& = \sum_{l=1}^{n} (De_l)_{i} e_{l}\\
& = \sum_{l=1}^{n} \lambda_{l}(e_{l})_{i}e_{l}.
\end{align*}
Hence, using the same bilinear form $[\cdot, \cdot]$ as earlier,
\[\dist(x_i,x_j) = (\dist_{i})_{j} = \sum_{l=1}^{n}\lambda_{l}(e_l)_{i}(e_{l})_{j} = \sum_{l=1}^{n} (\sqrt{\lambda_l} e_{l})_{i} (\sqrt{\lambda_l} e_{l})_{j} = [ \Phi(x_i), \Phi(x_j) ].\]
\begin{example}
Let $X$ consist of two points,$x_1$ and $x_2$, with $\dist(x_1,x_2) = 1$. Let $\mu (x_1) = 1$ and $\mu (x_2) = 4$. Our matrix $D$ is then
\[\begin{pmatrix}
0 & 4\\
1 & 0
\end{pmatrix}.\]
The eigenvalues of this matrix are $\pm 2$. Let
\[e_1 = \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{2\sqrt{2}}
\end{pmatrix} \,\,\,\,\,\,\,\, e_2 = \begin{pmatrix}
\frac{1}{\sqrt{2}}\\
\frac{-1}{2\sqrt{2}}
\end{pmatrix}. \]
These are eigenvectors with eigenvalue $+2$ and $-2$ respectively. Define the inner product matrix
\[Q = \begin{pmatrix}
1 & 0\\
0 & 4
\end{pmatrix}.\]
Observe that
\[e_1^T Q e_1 = e_2^T Q e_2 = 1 \cdot \frac{1}{2} + 4 \cdot \frac{1}{8} = 1 \]
\[e_1^T Q e_2 = e_2^T Q e_1 = 1 \cdot \frac{1}{2} - 4 \cdot \frac{1}{8} = 0, \]
so that $\{e_1,e_2\}$ is $Q$-orthonormal. We then have that
\[\Phi(x_1) = \langle \frac{\sqrt{2}}{\sqrt{2}}, \frac{\sqrt{-2}}{\sqrt{2}} \rangle = \langle 1, \sqrt{-1} \rangle \]
\[\Phi(x_2) = \langle \frac{\sqrt{2}}{2\sqrt{2}}, \frac{-\sqrt{-2}}{2\sqrt{2}} \rangle = \langle \frac{1}{2}, \frac{-\sqrt{-1}}{2} \rangle, \]
and
\[\dist_{X}(x_1,x_2) = [\Phi(x_1),\Phi(x_2)] = \frac{1}{2} + \frac{1}{2} = 1 \]
\[\dist_{X}(x_1,x_1) = [\Phi(x_1),\Phi(x_1)] = 1 - 1 = 0 \]
\[\dist_{X}(x_2,x_2) = [\Phi(x_2),\Phi(x_2)] = \frac{1}{4} - \frac{1}{4} = 0. \]
\end{example}
For finite metric spaces, we have the following explicit bound on the norms of our embedding vectors:
\begin{prop}
\label{prop:embedboundtopeig}
We have $\| \Phi_{k}(x)\|_{2} \leq \sqrt{|\lambda_1|}/\mu_{X}(x)$ for all $x \in X$.
\end{prop}
\begin{proof}
Denote by $s_i$ the complex vector whose $j$th component $(s_i)_j$ is equal to $\sqrt{\lambda_{j}}(r_i)_j$. That is, $s_i$ is obtained by rescaling the entries of the row vector $r_i$ by the square roots of the eigenvalues corresponding to the columns of those entries. For $m \in \{1, \cdots ,n\}$, let $s_{i,m}$ be the vector obtained by keeping only the first $m$ coordinates of $s_{i}$, and define $r_{i,m}$ similarly. By definition, $s_{i,k} = \Phi_{k}(x_i)$.\\
By our ordering of eigenvalues, $|\lambda_{1}| \geq |\lambda_{j}|$ for all $j$. Thus, each component of $s_{i,k}$ has modulus no larger than the corresponding component of $\sqrt{\lambda_{1}}r_{i,k}$. From this we deduce that $\|\Phi_{k}(x_i)\|_{2} = \|s_{i,k}\|_{2} \leq \|\sqrt{\lambda_{1}} r_{i,k}\|_{2} = |\sqrt{\lambda_{1}}|\|r_{i,k}\|_{2} = \sqrt{|\lambda_{1}|}\|r_{i,k}\|_{2}$. Since $r_{i,k}$ is obtained by truncating $r_i$, we have $\|r_{i,k}\|_{2} \leq \|r_{i}\|_{2}$. We thus deduce that $\|\Phi_{k}(x_i)\|_{2}\leq \sqrt{|\lambda_{1}|}\|r_{i}\|_{2}$. Combining this inequality with Equation~(\ref{eqn:rownorm}) completes the proof.
\end{proof}
We also have the following bound on the error function:
\begin{prop}
\label{prop:truncbound}
Applying the bilinear form $[\cdot,\cdot]$ to the truncated embedding $\Phi_{k}$ gives an approximation of the distance function $\dist$ whose error is bounded by the largest (in absolute value) omitted eigenvalue and the distribution of $\mu_{X}$:
\[E_{X,k}(x_i,x_j) = |\dist(x_i,x_j) - [ \Phi_{k}(x_i), \Phi_{k}(x_j) ]| \leq \frac{|\lambda_{k+1}|}{\sqrt{\mu_{X}(x_i)\mu_{X}(x_j)}}.\]
\end{prop}
\begin{proof}
Define the vectors $s_i$ as in the proof of Proposition \ref{prop:embedboundtopeig}. For $m \in \{1,2, \cdots ,n, n+1\}$, let $s_{i}^{m}$ be the vector obtained by throwing away the first $(m-1)$ coordinates of $s_{i}$, so that $s_{i}^{1} = s_{i}$ and $s_{i}^{(n+1)}$ is empty. Define $r_{i}^{m}$ similarly. Observe that:
\[|\dist(x_i,x_j) - [ \Phi_{k}(x_i), \Phi_{k}(x_j) ]| = \left|\sum_{l=k+1}^{n} \lambda_{l}(e_l)_{i}(e_l)_{j} \right| = |\langle s_{i}^{(k+1)}, s_{j}^{(k+1)} \rangle|,\]
where the inner product on the right-hand side is the ordinary, Euclidean one. The Cauchy-Schwarz inequality tells us that:
\[|\langle s_{i}^{(k+1)}, s_{j}^{(k+1)} \rangle| \leq \|s_{i}^{(k+1)}\|_{2}\|s_{j}^{(k+1)}\|_{2}.\]
We now bound the quantities on the right-hand side of the above inequality. By our ordering of eigenvalues, $|\lambda_{k+1}| \geq |\lambda_{l}|$ for all $l \geq k+1$. Thus every component of the vector $\sqrt{\lambda_{k+1}}r_{i}^{(k+1)}$ has a larger modulus than the corresponding component of $s_{i}^{(k+1)}$. We deduce that $\|s_{i}^{(k+1)}\|_{2} \leq \|\sqrt{\lambda_{k+1}}r_{i}^{(k+1)}\|_{2}= |\sqrt{\lambda_{k+1}}|\|r_{i}^{(k+1)}\|_{2} = \sqrt{|\lambda_{k+1}|} \|r_{i}^{(k+1)}\|_{2}$. We can similarly deduce that $\|s_{j}^{(k+1)}\|_{2} \leq \sqrt{|\lambda_{k+1}|} \|r_{j}^{(k+1)}\|_{2}$. Since $r_{i}^{(k+1)}$ and $r_{j}^{(k+1)}$ are truncations of $r_i$ and $r_j$, respectively, we deduce that:
\[\|s_{i}^{(k+1)}\|_{2}\|s_{j}^{(k+1)}\|_{2} \leq (\sqrt{|\lambda_{k+1}|})^{2}\|r_{i}^{(k+1)}\|_{2}\|r_{j}^{(k+1)}\|_{2} \leq |\lambda_{k+1}| \|r_{i}\|_{2}\|r_{j}\|_{2}. \]
Recalling from Equation~(\ref{eqn:rownorm}) that $\|r_{i}\|_{2} = 1/\sqrt{\mu_{X}(x_i)}$ and $\|r_{j}\|_{2} = 1/\sqrt{\mu_{X}(x_j)}$, the proof follows.
\end{proof}
We now provide a finite analogue of Theorem \ref{thm:invstabmeas}:
\begin{theorem}
\label{thm:invstab}
Let $(X,\dist_{X}, \mu_X)$ and $(Y,\dist_{Y}, \mu_Y)$ be finite metric measure spaces, with eigenvalues $\{\lambda_{i}\}$ and $\{\nu_{i}\}$, and let $\theta = \min \{\min_{x \in X}\mu_{X}(x),\min_{y \in Y}\mu_{X}(y)\}$. Take $k \leq |X|,|Y|$, and suppose that $\dist_{H}^{L^{2}}(\Phi_{k}(X),\Phi_{k}(Y)) \leq \varepsilon$. Then,
\[\dist_{GH}(X,Y) \leq 2\varepsilon \frac{\min (\sqrt{|\lambda_{1}|},\sqrt{|\nu_{1}|})}{\theta} + \varepsilon^2 + \frac{|\lambda_{k+1}| + |\nu_{k+1}|}{\theta}. \]
\end{theorem}
\begin{proof}
The result holds trivially for $\theta = 0$, so let us assume $\theta > 0$.\\
Suppose that the Hausdorff distance between $\Phi_{k}(X)$ and $\Phi_{k}(Y)$ is realized by a correspondence $C \subset \Phi_{k}(X) \times \Phi_{k}(Y)$. This induces a correspondence on $X \times Y$. Suppose that $x$ is paired with $y$ and $x'$ is paired with $y'$. If we write $v_{1} = \Phi_{k}(x), v_{2} = \Phi_{k}(x')$, $w_{1} = \Phi_{k}(y)$, and $w_{2} = \Phi_{k}(y')$, this means that $\|v_1 - w_1\|_{2} \leq \varepsilon$ and $\|v_2 - w_2\|_{2} \leq \varepsilon$. Then, Proposition \ref{prop:truncbound} tells us that $|\dist_{X}(x,x') - [v_1,v_2]| \leq |\lambda_{k+1}|/\theta$ and $|\dist_{Y}(y,y') - [w_1,w_2]| \leq |\nu_{k+1}|/\theta$. By Proposition \ref{prop:embedboundtopeig}, the $L^2$ norms of $v_{1}$ and $v_{2}$ are at most $\sqrt{|\lambda_{1}|}/\theta$, and similarly the $L^2$ norms of $w_{1}$ and $w_{2}$ are at most $\sqrt{|\nu_{1}|}/\theta$.\\
Combining these various bounds with Lemma \ref{lem:haustodist}, and applying the triangle inequality, we deduce:
\begin{align*}
|\dist_{X}(x,x') - \dist_{Y}(y,y')|
& \leq |\dist_{X}(x,x') - [v_1,v_2]| + |[v_1,v_2] - [w_1,w_2]| + |[w_1,w_2] - d_{Y}(y,y')|\\
& \leq \frac{|\lambda_{k+1}|}{\theta} + \varepsilon\min \left\{\|v_1\|_{2} + \|v_2\|_{2}, \|w_1\|_{2} + \|w_2\|_{2}\right\} + \varepsilon^2 + \frac{|\nu_{k+1}|}{\theta}\\
& \leq \varepsilon \min \left\{2\frac{\sqrt{|\lambda_{1}|}}{\theta},2 \frac{\sqrt{|\nu_{1}|}}{\theta}\right\} + \varepsilon^2 + \frac{|\lambda_{k+1}| + |\nu_{k+1}|}{\theta}\\
& = 2\varepsilon \frac{\min (\sqrt{|\lambda_{1}|},\sqrt{|\nu_{1}|})}{\theta} + \varepsilon^2 + \frac{|\lambda_{k+1}| + |\nu_{k+1}|}{\theta}.
\end{align*}
\end{proof}
When $k \geq |X|$, $\mu_{k+1} = 0$, and similarly for $Y$ and $\nu_{k+1}$. Thus, the final additive error term in Theorem \ref{thm:invstab} disappears in this setting. Note that in the case of uniform unit atomic measures, i.e. $\mu_X(x_i) = 1 = \mu_{Y}(y_j) \,\, \forall i,j$, we have $\theta = 1$.
| {
"timestamp": "2020-04-01T02:03:23",
"yymm": "1912",
"arxiv_id": "1912.02225",
"language": "en",
"url": "https://arxiv.org/abs/1912.02225",
"abstract": "Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform and Euler Characteristic Transform, both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms that depend only on the intrinsic geometry of a shape, and hence are invariant to the choice of embedding. To that end, given an abstract metric measure space, we define an integral operator whose eigenfunctions are used to compute sublevel set persistent homology. We demonstrate that this operator, which we call the distance kernel operator, enjoys desirable stability properties, and that its spectrum and eigenfunctions concisely encode the large-scale geometry of our metric measure space. We then define a number of topological transforms using the eigenfunctions of this operator, and observe that these transforms inherit many of the stability and injectivity properties of the distance kernel operator.",
"subjects": "Algebraic Topology (math.AT); Metric Geometry (math.MG); Spectral Theory (math.SP)",
"title": "Intrinsic Topological Transforms via the Distance Kernel Embedding",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795114181106,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7092379484354303
} |
https://arxiv.org/abs/1809.08554 | An explicit solution for a multimarginal mass transportation problem | We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to addition of constants). | \section{Introduction}
\subsection{Notation}
Assume we are given $n$ polish spaces $X_1$, $X_2$, \dots, $X_n$, equipped with probability measures $\mu_i$ on $X_i$ and a cost function $c: X_1 \times \dots \times X_n \to \mathbb{R}$.
In multimarginal Monge-Kantorovich problem (called primal problem throughout this paper) we seek to minimize $$\int_{X_1 \times \dots \times X_n}
c(x_1, x_2, ..., x_n)~d\mu(x_1, x_2, \dots, x_n)$$
over the set $\Pi(\mu_1, \mu_2, \dots, \mu_n)$ of positive joint measures $\mu$ on the product space
$X_1 \times \dots \times X_n$ whose marginals are the $\mu_i$. See \cite{Villani, bogachev} for an account in the optimal transportation problem with two marginals and \cite{brendan_pass}.
With the primal problem people also consider the dual problem. Under conditions above we are concerned with the supremum of $$\sum_{i = 1}^n \int_{X_i}f_i(x_i)~d\mu_i$$
where supremum is taken over all sets of functions $\{f_i\}$ such that $\sum_{i = 1}^n f_i(x_i) \le c(x_1, \dots, x_n)$ for any $x_i \in X_i$.
It is easy to show that minimum in primal problem is less or equal to the supremum in dual problem. Under some conditions it is true that this numbers are equal \cite{Villani, brendan_pass, Kellerer}.
We do not need a full power of duality here. This paper relies on the following easy fact.
\begin{lemma}[Complementary Slackness Condition]\label{lem:slackness_conditions}
Let $\mu \in \Pi(\mu_1, \dots, \mu_n)$ be a joint measure and $f_1, f_2, \dots, f_n$ be a tuple of functions such that $\sum_{i = 1}^n f_i(x_i) \le c(x_1, \dots, x_n)$.
If there is a set $M \subset X_1 \times X_2 \times \dots \times X_n$ such that on $M$ one has $\sum_{i = 1}^n f_i(x_i) = c(x_1, \dots, x_n)$ with the additional property $\mu(M)=1$, then $\mu$ is a primal solution and $f_i$ is a dual solution.
\end{lemma}
The aim of this paper is to describe an example of explicit solution to the mass transportation problem on $[0, 1]^3$ ($X_1 = X_2 = X_3 = [0, 1]$) with one-dimensional Lebesgue measure projections and the cost function $c(x, y, z) = xyz$. In this paper we call the measures on $[0, 1]^3$ with Lebesgue projections onto the axes $(3, 1)-$stochastic measures.
In fact, we will construct the primal and dual solutions for any cost function $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ such that the function $tC'(t)$
strictly increases on the segment $[0, 1]$.
\subsection{Motivation}
Our problem appears to be the simplest generalization of the classical Monge--Kantorovich problem with one-dimensional marginals and quadratic cost function. It seems to be never considered in the literature, though other generalizations mentioned in \cref{subsection:relatedproblems} received some attention.
Note that the particular cost function $(x-y)^2$ (equivalently $- xy$) is mostly used in the classical Monge--Kantorovich theory. A natural replacement of $-xy$ for the case of three variables is $-xyz$. For the cost function $-xyz$ the solution to the primal problem with the same marginals admits a simple structure: it is concentrated on the main diagonal of $[0, 1]^3$ (this can be viewed as a ``continuous rearrangement inequality'' or ``Hardy-Littlewood inequality''). Unlike this, solutions for $xyz$ are non-trivial, that is why we are interested in the cost function $xyz$.
\subsection{Main results}
In this paper we construct the set $M$ which is $c-$monotone for the cost function $c(x, y, z) = xyz$. The set $M$ is the union of three segments and one 2-dimensional part as below:
\begin{align*}
&M_x = \{(t, 1 - 2t, 1 - 2t) \mid 0 \le t \le l\},\\
&M_y = \{(1 - 2t, t, 1 - 2t) \mid 0 \le t \le l\},\\
&M_z = \{(1 - 2t, 1 - 2t, t) \mid 0 \le t \le l\},\\
&M_2 = \{(x, y, z) \mid l \le x, y, z \le r = 1 - 2l, xyz = lr^2\}, \\
&M = M_x \cup M_y \cup M_z \cup M_2,
\end{align*}
where $l \approx 0.0945$, $r \approx 0.8119$ are some transcendent constants.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{Set}
\caption{Set $M$}
\end{figure}
Initially, we got an explicit construction of this set from heuristic considerations (see \cref{section:heuristic}). In \cref{section:primal_construction} we see that the integral $\int xyz~d\mu$ is the same for any $(3, 1)-$stochastic measure $\mu$ such that $\mathrm{supp}(\mu) \subset M$ (see \cref{prop:same_integral}). After that we explicitly construct a $(3, 1)-$stochastic measure $\pi$ concentrated on the set $M$ (see the proof of \cref{thm:osexists}). The proof contains nontrivial construction and technical computations. The constructed measure is the primal solution of the related transport problem.
To prove that the measure $\pi$ is the primal solution in \cref{section:dual} we solve a related dual problem for a cost function $c(x, y, z) = C(xyz)$. Our proof works for $C:[0, 1] \to \mathbb{R}$ such that the function $tC'(t)$
strictly increases on the segment $[0, 1]$. Naturally that means $C(xyz)=\widehat{C}(\ln x + \ln y + \ln z)$ where $\widehat{C}$ is a bounded continuously differentiable convex function on $(-\infty, 0]$.
The following theorem gives an explicit construction for the dual solution (see \cref{thm:primal_dual}). Thus, together with \cref{thm:osexists} it gives a characterization of both primal and dual solution.
\begin{theorem}[Main result]\label{thm:intro_dual}
Suppose that $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ and the function $tC'(t)$
strictly increases on the segment $[0, 1]$. Set:
$$\hat{f}(s) = \int_{0}^{s}\lambda(t)C'(t\lambda(t))~dt,$$ where the function $\lambda$ is as in \cref{def:lambda_definition}. Then for any constants $C_x$, $C_y$, $C_z$ such that $$C_x + C_y + C_z = C(0) - 2\int_{0}^{1}\lambda(t)C'(t\lambda(t))~dt$$ the following inequality holds
$$(\hat{f}(x) + C_x) + (\hat{f}(y) + C_y) + (\hat{f}(z) + C_z) \le c(x, y, z)$$
with equality on $M$.
Using the complementary slackness conditions (see \cref{lem:slackness_conditions}) we conclude that for any cost function $C(xyz)$, for which the conditions above are satisfied, any $(3, 1)-$stochastic measure $\pi$ with $\mathrm{supp}(\pi) \subset M$ is a primal solution for a multimarginal mass transportation problem and functions $\hat{f}$ defined in \cref{thm:intro_dual} is a dual solution.
\end{theorem}
In \cref{subsection:explicit_dual_xyz} the explicit form of the dual solution for the cost function $c(x, y, z) = xyz$ is specified. It has the following form (see \cref{prop:explicit_dual_xyz}):
\begin{align*}
&\hat{f}(x) = \begin{cases}
c \ln l - {1 \over 3}(c\ln c - c) + {1 \over 6}((2x - 1)^3 - (2l - 1)^3), &\text{if } 0 \le x \le l,\\
c\ln x - {1 \over 3}(c\ln c - c), &\text{if } l \le x \le r,\\
c\ln r - {1 \over 3}(c\ln c - c) + {1 \over 4}(x^2 - r^2) - {1 \over 6}(x^3 - r^3), &\text{if } r \le x \le 1,
\end{cases}\\
&f(t) = g(t) = h(t) = \hat{f}(t),
\end{align*}
for constants $l, r, c$.
In \cref{section:uniqueness} we prove that for any cost function $c(x, y, z) = C(xyz)$ dual solution is unique up to adding constants and measure zero.
Structural results (see \cite{Pass12, brendan_pass}) allow us to estimate the local dimension $d$ of $M$. We apply this results in \cref{section:inertion} to see that $d$ is bounded by $2$. The dimension of the support is important for computations and was studied in details in \cite{Friesecke}. It is interesting that the local dimension of $M$ is not constant as $M$ admits one-dimensional parts and a two-dimensional part.
This two-dimensional part is a source of non-uniqueness for the primal problem. After the logarithmic change of coordinates cost function $C(xyz)$ becomes convex in sum of coordinates, Lebesgue measure on axis becomes an exponential distribution and two-dimensional part of $M$ becomes a triangle on a plane $x+y+z=\mathrm{const}$. This resembles the situation in \cite[Lemma 4.3]{dim-ger-nen} where the authors consider the multimarginal problem with the same cost function and Lebesgue marginals. They prove that the plan is optimal if and only if it is concentrated on a plane $x+y+z=\mathrm{const}$.
The cost function $xyz$ violates the standard uniqueness assumption, the so-called twist condition (see \cite{KP, Pass11, brendan_pass}). The primal problem admits many solutions. In particular, we show that there exist solutions which are singular with respect to the Hausdorff measure on $M$.
We also propose the following
\begin{conjecture}\label{conj:hausdorff}
There exists a solution which is concentrated on a set which has Hausdorff dimension less than $2$.
\end{conjecture}
This conjecture is motivated by \cite[Theorem 4.6]{dim-ger-nen} where the authors construct a primal solution with a fractal support.
\subsection{Related problems}\label{subsection:relatedproblems}
Our example contributes to the list of several known explicit examples and to the list of cost functions where the structure of solutions is investigated in details.
Here are some other examples.
\begin{enumerate}
\item Cost function $$-\sum_{i \ne j} x_i x_j .$$
This cost function is related to the geodesic barycenter problem (see \cite{Carlier, AguehC}).
\item
Determinantal cost \cite{CarNaz}.
\item
Coulomb cost \cite{CFK} (see \cite{CPM} for generalizations). The motivation for this problem comes from mathematical physics.
\item
$\min(x_1, \ldots, x_n)$ (more generally, minumum of affine functions) \cite{KL}.
\item
Convex function of $x_1+ \ldots + x_n$ (see \cite{dim-ger-nen}).
\end{enumerate}
Some other examples can be found in \cite{brendan_pass}.
Also, our problem is closely related to $(3, 2)$ problem, studied in \cite{main_work}. In particular, our example can be considered as a solution to the primal $(3, 2)$-problem with the same cost function $xyz$ and the corresponding 2-dimensional projections.
In the $(3, 2)$-problem we consider a modification of the transportation problem. Namely, we deal with the space of measures with fixed projections onto $$X_1 \times X_2, \ X_2 \times X_3, \ X_1 \times X_3.$$
The main result of \cite{main_work} describes a solution to the $(3, 2)$-problem on $[0, 1]^3$ with the cost function $xyz$ ($-xyz$) and two-dimensional Lebesgue measure projections. It turns out that in strong contrast with the classical transportation problem the solution is supported by the fractal set (Sierpi\'nski tetrahedron) $$z = x \oplus y,$$ where $\oplus$ is bitwise addition.
Let us also mention another related important modification: Monge--Kantorovich problem with linear constraints, which has been introduced and studied in \cite{zaev}.
\section{An heuristic description of $M$}\label{section:heuristic}
In this subsection we collect some informal observations related to our main construction.
In particular, we briefly analyze the cyclical monotonicity property of the support set of our primal solution and describe how to approach the problem numerically.
Let $M$ be a full measure set for the primal solution. Since all the marginals and the cost function are symmetric with respect to the coordinate axes interchange, we may assume without loss of generality that $M$ is also symmetric in this sense.
The set $M$ can be chosen to be $c-$cyclically monotone. This is well known for two marginals, for many marginals we refer to the work \cite{griessler}. In particular that means that for any $(x_1, y_1, z_1), (x_2, y_2, z_2) \in M$ one has
\begin{align}\label{eq:c_monotonicity}
\begin{split}
c(x_1, y_1, z_1) + c(x_2, y_2, z_2) \le c(x_2, y_1, z_1) + c(x_1, y_2, z_2),\\
c(x_1, y_1, z_1) + c(x_2, y_2, z_2) \le c(x_1, y_2, z_1) + c(x_2, y_1, z_2),\\
c(x_1, y_1, z_1) + c(x_2, y_2, z_2) \le c(x_1, y_1, z_2) + c(x_2, y_2, z_1).
\end{split}
\end{align}
The \cref{alg:primal_solution_approximation_general} constructing an approximation to a primal solution is based on the inequality above.
\begin{algorithm}
\caption{Primal solution approximation (general version)}
\label{alg:primal_solution_approximation_general}
\begin{algorithmic}[1]
\STATE{Generate three samples: $x_1, x_2, \dots, x_n$ from $\mu_1$, $y_1, y_2, \dots, y_n$ from $\mu_2$, $z_1, z_2, \dots, z_n$ from $\mu_3$; $n$ is a parameter, $\mu_i$ are the marginals in the primal problem.}
\STATE{Define $S := \{(x_k, y_k, z_k) \text{ for } 1 \le k \le n\}$.}
\WHILE{$S$ doesn't satisfy \cref{eq:c_monotonicity}}
\STATE{Take two points $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ from $S$.}
\IF{$c(a_1, b_1, c_1) + c(a_2, b_2, c_2) > c(a_2, b_1, c_1) + c(a_1, b_2, c_2)$}
\STATE{replace $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ with $(a_2, b_1, c_1)$ and $(a_1, b_2, c_2)$ is $S$}
\ELSIF{$c(a_1, b_1, c_1) + c(a_2, b_2, c_2) > c(a_1, b_2, c_1) + c(a_2, b_1, c_2)$}
\STATE{replace $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ with $(a_1, b_2, c_1)$ and $(a_2, b_1, c_2)$ is $S$}
\ELSIF{$c(a_1, b_1, c_1) + c(a_2, b_2, c_2) > c(a_1, b_1, c_2) + c(a_2, b_2, c_1)$}
\STATE{replace $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ with $(a_1, b_1, c_2)$ and $(a_2, b_2, c_1)$ is $S$}
\ENDIF
\ENDWHILE
\STATE{$S$ is an approximation of the primal solution.}
\end{algorithmic}
\end{algorithm}
In our case $c(x, y, z) = xyz$, so
\begin{align*}
&x_1y_1z_1 + x_2y_2z_2 \le x_1y_1z_2 + x_2y_2z_1,\\
&(x_1y_1 - x_2y_2)(z_1 - z_2) \le 0.
\end{align*}
It follows that if $(x_1, y_1, z_1), (x_2, y_2, z_2) \in M$ then
\begin{align}\label{cond:weak_monotonicity}
\begin{split}
&z_1 < z_2 \Rightarrow x_1y_1 \ge x_2y_2 \text{ and by the symmetry}\\
&y_1 < y_2 \Rightarrow x_1z_1 \ge x_2z_2,\\
&x_1 < x_2 \Rightarrow y_1z_1 \ge y_2z_2.
\end{split}
\end{align}
\begin{algorithm}
\setcounter{ALC@unique}{0}
\caption{Primal solution approximation (faster version)}
\label{alg:primal_solution_approximation_faster}
\begin{algorithmic}[1]
\STATE{Generate three samples: $x_1, x_2, \dots, x_n$ from $\mu_1$, $y_1, y_2, \dots, y_n$ from $\mu_2$, $z_1, z_2, \dots, z_n$ from $\mu_3$; $n$ is a parameter, $\mu_i$ are the marginals in the primal problem.}
\STATE{Define $S := [(x_k, y_k, z_k) \text{ for } 1 \le k \le n]$.}
\WHILE{$S$ doesn't satisfy \cref{cond:weak_monotonicity}}
\STATE{Sort $S$ by the first coordinate in the ascending order. Denote by $(a_k, b_k, c_k)$ the $k-$th item of $S$ after sorting, $1 \le k \le n$.}\label{alg:step_sort_1}
\STATE{Update $S := [(a_k, b_{\sigma(k)}, c_{\sigma(k)}) \text{ for } 1 \le k \le n]$ where $\sigma \in S_n$ and the sequence $b_{\sigma(k)}c_{\sigma(k)}$ is descending.}\label{alg:step_sort_2}
\STATE{Repeat \cref{alg:step_sort_1} and \cref{alg:step_sort_2} for the second and third coordinates.}
\ENDWHILE
\STATE{$S$ is an approximation of the primal solution.}
\end{algorithmic}
\end{algorithm}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{M_sorted}
\caption{The final set $S$ for $n = 200000$.}
\label{fig:sorted_M}
\end{figure}
This allows us to improve the performance of \cref{alg:primal_solution_approximation_general} using sortings. That leads us to a much faster version, namely \cref{alg:primal_solution_approximation_faster}. We were able to run an \cref{alg:primal_solution_approximation_faster} for $n=2 \times 10^5$.
Despite the fact that this algorithm does not necessarily converges to the solution
for all admissible data, our numerical experiments demonstrate that the algorithm works well in
many cases. A proof of convergence for a suitable set of admissible data must be investigated.
On \cref{fig:sorted_M} it is shown a scatter plot of $S$ after the completion of the algorithm. As one can see on this graph, the set $M$ consists of four parts. There exist real values $0 < l < r < 1$ such that if $l \le x, y, z \le r$ and $(x, y, z) \in M$ then $(x, y, z)$ lies on 2-dimensional part $M_2$. If $0 \le x \le l$ and $(x, y, z) \in M$ then $(x, y, z)$ lies on a 1-dimensional curve $M_x = (p(t), p_y(t), p_z(t))$, $0 \le t \le 1$. By the virtue of the symmetry $p_y(t) = p_z(t) = q(t)$. Also $q(0) = 1$, $q(1) = r$ and $q(t)$ strictly decrease; $p(0) = 0$, $p(1) = l$ and $p(t)$ strictly increase.
By the virtue of the symmetry, if $(x, y, z) \in M$ and $0 \le y \le l$, then this point lies on a curve $M_y = (q(t), p(t), q(t))$, and if $0 \le z \le l$ then $(x, y, z)$ lies on a curve $M_z = (q(t), q(t), p(t))$.
Let $\nu$ be a primal solution and $\nu_x$, $\nu_y$, $\nu_z$ be the restrictions of $\nu$ to $M_x, M_y$ and $M_z$ accordingly. Suppose $F_x(a) = \nu_x(\{(p(t), q(t), q(t)) \mid 0 \le t \le a\})$. Define $F_y$ and $F_z$ in a similar way. By the virtue of the symmetry we can assume that $F_x(a) = F_y(a) = F_z(a) = F(a)$ for any $0 \le a \le 1$.
For any $0 \le a \le 1$ one has
\begin{multline*}
\nu(\{0 \le x \le p(a)\}) = \nu_x(\{(p(t), q(t), q(t)) \mid 0 \le t \le a\}) = F_x(a) \\
= {1 \over 2}(F_y(a) + F_z(a)) = {1 \over 2}\nu(\{q(a) \le x \le 1\}).
\end{multline*}
Since all the marginals are Lebesgue measures on the segments $[0, 1]$, one has $2p(a) = 1 - q(a)$ for any $0 \le a \le 1$.
Thus $$M_x = (t, 1 - 2t, 1-2t)$$
$$ M_y = (1 - 2t, t, 1 - 2t)$$
and $$M_z = (1 - 2t, 1-2t, t),$$ $0 \le t \le l$; $r = 1 - 2l$. That means that 1-dimensional parts of the set $M$ are segments.
The set $M_x$ is $c-$cyclically monotone. In particular, if $1 - 2t_1 < 1 - 2t_2$, then $t_1(1 - 2t_1) \ge t_2(1 - 2t_2)$ or, equivalently, the function $t(1 - 2t)$ increases on the set $[0, l]$. The derivative of this function is $1 - 4t \ge 0$ for any $0 \le t \le l$. That means that $0 < l \le {1 \over 4}$.
Let us describe the set $M_2$. As we know from the general duality theory, there exist functions $$f, g, h: [l, r] \to \mathbb{R}$$ satisfying $f(x) + g(y) + h(z) \le xyz$ and the equality holds provided $(x, y, z) \in M_2$. Again by symmetry we can assume that $f(x) = g(x) = h(x)$ for any $l \le x \le r$.
Suppose that $f$ is continuously differentiable. Let $(x, y, z) \in M_2$ and $l \le x, y, z \le r$. Then that point is an inner maximum point of the function $$F(x, y, z) = f(x) + f(y) + f(z) - xyz.$$ That means that
$$\nabla F = (f'(x) - yz, f'(y) - xz, f'(z) - xy)^T = \vec{0}.$$
So if $(x, y, z) \in M_2$ then $xf'(x) = yf'(y) = zf'(z) = xyz$. From \cref{fig:sorted_M} we see that if we fix $x=l$, then for any $l \le y \le r$ there exists $l \le z \le r$ such that $(x, y, z) \in M_2$. Then the function $tf'(t)$ is equal to a constant $C=lf'(l)$ for any $l \le t \le r$. In this case if $(x, y, z) \in M_2$ then $xyz = xf'(x) = C$. Since $(l, r, r) \in M_2$ the constant $C$ is equal to $lr^2$.
\section{Solving the primal problem}\label{section:primal_construction}
Summarizing the facts about the set $M$, which supports the primal solutions, we realize that one can try to find $M$
in the following form:
\begin{align*}
&M_x = \{(t, 1 - 2t, 1 - 2t) \mid 0 \le t \le l\},\\
&M_y = \{(1 - 2t, t, 1 - 2t) \mid 0 \le t \le l\},\\
&M_z = \{(1 - 2t, 1 - 2t, t) \mid 0 \le t \le l\},\\
&M_2 = \{(x, y, z) \mid l \le x, y, z \le r = 1 - 2l, xyz = lr^2\},
\end{align*}
where $l$ is an unknown parameter; $0 \le l < {1 \over 4}$.
\begin{proposition}\label{prop:same_integral}
An integral $\int xyz~d\nu(x, y, z)$ is the same for any probability measure $\nu$ such that $\Pr_x(\nu) = \Pr_y(\nu) = \Pr_z(\nu) = \lambda$ where $\lambda$ is the Lebesgue measure on the segment $[0, 1]$ and $\mathrm{supp}(\nu) \subset M$.
\end{proposition}
\begin{proof}
Let $\nu_x$, $\nu_y$, $\nu_z$ and $\nu_2$ be restrictions of $\nu$ to $M_x$, $M_y$, $M_z$ and $M_2$ respectively. Since the projection of $\nu_x$ on the first marginal is a restriction of $\lambda$ to the segment $[0, l]$, one has
$$
\int_{M_x} xyz~d\nu_x(x, y, z) = \int x(1 - 2x)(1-2x)~d\nu_x(x, y, z) = \int_0^lx(1-2x)^2~dx
$$
Similarly
$$
\int_{M_y} xyz~d\nu_y(x, y, z) = \int_{M_z} xyz~d\nu_z(x, y, z) = \int_0^lx(1-2x)^2~dx.
$$
Finally, the projection of $\nu_2$ on the first marginal is a restriction of $\lambda$ to the segment $[l, r]$. So
$$
\int_{M_2}xyz~d\nu_2 = lr^2\cdot \nu_2(\{l \le x \le r\}) = lr^2(r - l).
$$
Consequently $\int xyz~d\nu(x, y, z) = 3\int_0^lx(1 - 2x)^2~dx + lr^2(r - l)$ and this integral does not depend on $\nu$.
\end{proof}
So we only have to find any measure with desired projections such that its support is contained in $M$. In \cref{thm:primal_dual} we find an appropriate triple of functions and by \cref{lem:slackness_conditions} we rigorously prove that any $(3, 1)$-stochastic measure on $M$ is indeed a primal solution.
First, we define a measure on the three one-dimensional segments. Let $L = \sqrt{l^2 + 2(1 - r)^2}$ be the lengths of these segments. We set on every segment a uniform measure with density $\frac{l}{L}$.
Clearly, projections of two segments coincide with $[r, 1]$, the densities are equal to $\frac{L}{1 - r} \cdot \frac{l}{L} = \frac{1}{2}$. Their sum is the Lebesgue measure on $[r, 1]$.
The projection of the third interval is a measure on $[0, l]$, its density equals $\frac{L}{l} \cdot \frac{l}{L} = 1$.
After this, it remains to determine the measure on the remaining two-dimensional set such that its projection on each of the axes is uniform.
Let us make the following change of coordinates: $$u := {\ln x - \ln l \over \ln r - \ln l}, v := {\ln y - \ln l \over \ln r - \ln l}, w := {\ln z - \ln l \over \ln r - \ln l}.$$
Two-dimensional set
$$xyz = c, l \le x, y, z \le r$$ admits the following parametrization: $$u + v + w = 2, 0 \le u, v, w \le 1.$$
One has the following relations:
\begin{align*}
&dx = de^{u(\ln r - \ln l) + \ln l} = l\ln \left(r \over l \right) \left({r \over l}\right)^udu = l\ln(\alpha) \alpha^u du,\\
&dy = l\ln(\alpha) \alpha^v dv,\\
&dz = l\ln(\alpha) \alpha^w dw,
\end{align*}
where $\alpha = \frac{r}{l}$.
Clearly, the problem is reduced to the following problem: find a measure on the triangle $u + v + w = 2, 0 \le u, v, w \le 1$ with exponential projections onto the axes.
\subsection{Necessary conditions for existence of a measure on the triangle with given projections}
\
One can put the problem into a more general setting.
When there exists a measure $\mu$ on the triangle $$\Delta = \{x + y + z = 2,~0 \le x, y, z \le 1\}$$ with given projections $\mu_x$, $\mu_y$, $\mu_z$?
In what follows we are only interested in the case $\mu_x = \mu_y = \mu_z = \pi$. A necessary condition is given in the following lemma.
\begin{lemma} \label{inequality}
Let function $f : [0, 1] \to \mathbb{R}$ satisfy $f(x) + f(y) + f(z) \le 0$ for $x + y + z = 2$ and there exist a measure $\mu$ on $\Delta$, whose projections
onto the axes are equal to $\pi$. Then $\int_0^1 f(x) d\pi \le 0$.
\end{lemma}
\begin{proof}
We compute $\int_\Delta (f(x) + f(y) + f(z))~d\mu$. On the one hand, it is nonpositive, since at each point $f(x) + f(y) + f(z) \le 0$. On the other hand,
$$
\int_\Delta (f(x) + f(y) + f(z))d\mu = 3\int_0^1 f(x)d\pi(x) \le 0.
$$
\end{proof}
In particular, for the function $f(x)=x-\frac{2}{3}$ one has $f(x) + f(y) + f(z) = 0$ for $x + y + z = 2$. So we get
\begin{equation} \label{main_traingle_equality}
\int_0^1 \left(x - {2 \over 3}\right) d\pi(x) = 0.
\end{equation}
Check this for $d\pi = \alpha^xdx$:
$$
\int_0^1 \left(x - {2 \over 3}\right)d\pi = \int_0^1 \left(x - {2 \over 3}\right)\alpha^xdx = {{\alpha(\ln\alpha - 3) + 3 + 2\ln\alpha} \over 3\ln^2\alpha}
$$
Thus, $\alpha$ must satisfy
\begin{equation} \label{alpha_equation}
{\alpha(\ln\alpha - 3) + 3 + 2\ln\alpha} = 0.
\end{equation}
Apply the relation $\alpha = {1 - 2l \over l}$:
$$
\alpha(\ln\alpha - 3) + 3 + 2\ln\alpha
= {\ln(1 - 2l) - \ln l - 3 + 9l \over l} = 0.
$$
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Plot_L.pdf}
\caption{A graph of a function $y(x) = 9x + \ln(1 - 2x) - \ln x - 3$}
\label{fig:plot_l}
\end{figure}
It is seen from \cref{fig:plot_l} that the function $\ln(1 - 2l) - \ln l - 3 + 9l$ has exactly one root lying in the interval $\left(0, {1 \over 4}\right)$, namely $l\approx 0.0945$. So $r \approx 0.8109$ and $\alpha \approx 8.577$.
Let us prove that there is a unique root of $h$ lying inside $\left(0, {1 \over 3}\right)$. To this end we find the derivative of $h(l) = 9l + \ln(1 - 2l) - \ln l - 3$ and show it is negative for $l < \frac{1}{6}$ and positive for $\frac{1}{6} < l < \frac{1}{3}$. Indeed,
$$
h'(l) = (9l + \ln(1 - 2l) - \ln l - 3)' = -\frac{(3l - 1)(6l - 1)}{l(1-2l)},
$$
and it is easy to check the signs.
For $l\to+0$ one has $$h(l) \to \infty.$$ For $l = {1 \over 6}$ there holds
$$
h\left(\frac{1}{6}\right) = 2\ln{2} - {3 \over 2} < 0,
$$
since $\ln 2 \approx 0.69 < {3 \over 4}$.
For $l = {1 \over 3}$ there holds
$$
h\left(\frac{1}{3}\right)= 3 + \ln\left(1 - {2 \over 3}\right) - \ln\left(1 \over 3\right) - 3 = 0.
$$
It follows that on the interval {$\left(0, {1 \over 4}\right)$} function $h(l)$ has exactly one root {and this root is less than $1 \over 6$.}
Assumption of \cref{inequality} is satisfied for the following broad class of functions.
\begin{lemma}
Let $f(x): [0, 1] \to \mathbb{R}$ be convex on $\left[0, {2 \over 3}\right]$ and $f(2x) + 2f(1 - x) = 0$ for $0 \le x \le {1 \over 3}$. Then $f(x) + f(y) + f(z) \le 0$ for $x + y + z = 2$.
\end{lemma}
\begin{proof}
Assume that $f(x) + f(y) + f(z) > 0$ for some $x$, $y$ and $z$ satisfying $x + y + z = 2$.
Let among $x$, $y$, $z$ be at least two numbers (say, $x \le y$) less than $\frac{2}{3}$.
Replace these numbers by $x', y'$ in such a way that $x+y=x'+y'$, $[x', y'] \subset\left[0, \frac{2}{3}\right]$ and either $x'=0$ or $y'=\frac{2}{3}$.
By convexity $f(x') + f(y') + f(z) \ge f(x) + f(y) + f(z) > 0$.
If $x'=0$, then $y' \le \frac{2}{3}$, and $z \le 1$, thus $x'+y' + z < 2$. Hence $y' =\frac{2}{3}$.
Thus from the very beginning one can assume that $x \le\frac{2}{3}$ and $y, z \ge \frac{2}{3}$.
If $x = y = z = \frac{2}{3}$, then $f(x) + f(y) + f(z) = f\left(2 \cdot \frac{1}{3}\right) + 2f\left(1 - \frac{1}{3}\right) = 0$.
Repeating the same trick and using concavity of $f$ on $\left[\frac{2}{3}, 1\right]$ one can reduce the problem to the case $y=z$. But for any triple $x, y = z = 1 - \frac{x}{2}$ there holds $f(x) + f(y) + f(z) = 0$, which contradicts the assumption $f(x) + f(y) + f(z) > 0$.
\end{proof}
\subsection{Description of projections of measure classes on the triangle}
We will consider special classes of measures on $\Delta$ and describe their projections onto the axes.
First, consider the Lebesgue measure on $\Delta$. It can be normalized in such a way that the measure of the whole triangle is equal to $\frac{1}{2}$. We denote the normalized measure by $\lambda_\Delta$.
Projecting it to any hyperplane $\{x=0\}$, $\{y=0\}$, $\{z=0\}$, we get a triangle with the usual Lebesgue measure. In what follows we shall consider the densities with respect to this normalized measure.
\begin{definition}
Let $\mu$ be a measure on $\Delta$ absolutely continuous with respect to $\lambda_\Delta$. For any point $(x, y, z) \in \Delta$ define $M(x, y, z) = \min(1 - x, 1 - y, 1 - z)$. We call a measure $\mu$ \textbf{\textit{layered}} if for any $t$ the density of $\mu$ is constant on a set $M(x, y, z) = t$, that is density depends only on $M(x, y, z)$.
\end{definition}
It is easy to see that $M$ is proportional to the distance from the point to the nearest side of $\Delta$. Therefore, points with constant $M$ form a triangle homothetic to the original one, with the same center. It is also easy to see that due to the symmetry of the layered measure, its projection on all three axes will be the same. Also note that $M$ takes values only in $\left[0, \frac{1}{3}\right]$.
\begin{definition}
We say that a function $p: \left[0, \frac{1}{3}\right] \to \mathbb{R}$ \textit{\textbf{generates}} a layered measure $\mu$ if $\frac{d\mu}{d\lambda_\Delta}(x, y, z) = p\left(M(x, y, z)\right)$.
\end{definition}
Let us find the projections of a layered measure $\mu$ generated by $p$ to the coordinate axes.
\begin{proposition}
\label{prop:layered_projection}
Let $\mu$ be a layered measure generated by a function $p$. Let $p_*: [0, 1] \to \mathbb{R}_+$ be the density of the projection of this measure onto an axis. Then
\begin{align*}
p_*(x) =
\begin{cases}
2\int_0^\frac{x}{2}p(t)dt,&\text{ if } x \le \frac{2}{3},\\
(3x - 2)p(1 - x) + 2\int_0^{1 - x}p(t)dt,&\text{ if } x \ge \frac{2}{3}
\end{cases}
\end{align*}
\end{proposition}
\begin{proof}
Denote the projection of $\mu$ onto the hyperplane $xy$ by $\mu_{xy}$. It is concentrated on the triangle $T$ with vertices $(0, 1)$, $(1, 0)$ and $(1, 1)$. Its density with respect to the Lebesgue measure on the plane at the point $(x, y)$ lying inside $T$ is $$p( M(x, y, 2 - x - y)) = p(\min(1 - x, 1 - y, x + y - 1)).$$
Define $\mu_x$ as the projection of $\mu$ onto $x$, or, what is the same, the projection of the measure $\mu_{xy}$ onto $x$. Then the measure of $[0, x_0]$ on the one hand is $\int_{0}^{x_0} p_*(x) dx$, and on the other hand is equal to the measure of the part of the triangle $T$ where the $x$ coordinate belongs to $[0, x_0]$. Thus, we have established the equality $\int_{0}^{x_0} p_*(x) dx = \int_{0}^{x_0} \int_{1-x}^1 p(M(x, y, 2 - x - y)) dx dy$. Differentiating both sides of this equality with respect to $x_0$, we obtain $p_*(x) = \int_{1 - x}^1 p(\min(1 - x, 1 - y, x + y - 1)) dy$.
Assume $x \le \frac{2}{3}$. Then:
\begin{align*}
\min(1 - x, 1 - y, x + y - 1) = \begin{cases}
x + y - 1, &\text{ for } y \in \left[1 - x, 1 - \frac{x}{2}\right],\\
1 - y, &\text{ for } y \in \left[1 - \frac{x}{2}, 1\right].
\end{cases}
\end{align*}
From here we get:
\begin{align*}
p_*(x) &= \int_{1 - x}^1 p(\min(1 - x, 1 - y, x + y - 1)) dy \\
&= \int_{1 - {x \over 2}}^1 p(1 - y) dy + \int_{1 - x}^{1 - {x \over 2}} p(x + y - 1) dy = 2\int_0^{x \over 2}p(t)~dt.
\end{align*}
Analogously for $x \ge \frac{2}{3}$:
\begin{align*}
\min(1 - x, 1 - y, x + y - 1) = \begin{cases}
x + y - 1, &\text{ for } y \in \left[1 - x, 2 - 2x\right],\\
1 - x, &\text{ for } y \in \left[2 - 2x, x\right],\\
1 - y, &\text{ for } y \in \left[x, 1\right].
\end{cases}
\end{align*}
After this we calculate $p_*(x)$:
\begin{align*}
p_*(x) &= \int_{1 - x}^1 p(\min(1 - x, 1 - y, x + y - 1)) dy \\
&= \int_{1 - x}^{2 - 2x}p(x + y - 1)dy + \int_{2 - 2x}^{x}p(1 - x)dy + \int_{x}^{1}p(1 - y)dy \\
&= 2\int_0^{1 - x}p(t)dt + (3x - 2)p(1-x).
\end{align*}
\end{proof}
Next we define \textit{median measure}.
\begin{definition}
\textit{The median subset} of $\Delta$ is the set $$\{(x, y, z) \in \Delta \mid x = y \ge z\} \cup \{(x, y, z) \in \Delta \mid y = z \ge x\} \cup \{(x, y, z) \in \Delta \mid x = z \ge y\}.$$ From a geometric point of view, this is a union of three segments in $\Delta$ from vertices to the center of the triangle $\Delta$.
\end{definition}
Projections of any segment from the median set are $\left[0, \frac{2}{3}\right]$ and $\left[\frac{2}{3}, 1\right]$.
On these segments one can define a measure proportional to the Lebesgue measure such that the measure of each segment is $\frac{2}{3}$. In what follows, we shall consider all the densities on the median set with respect to this measure.
\begin{definition}
\textit{Median measure} $\mu$, generated by a density function $q: [0, {2 \over 3}] \to \mathbb{R}_+$, is a measure with density on the median set that its density on each of the segments is equal to $q(t)$ at the points $(t, t, 2-2t)$, $(t, 2-2t, t)$, $(2-2t, t, t)$ with respect to the
reference measure described above.
\end{definition}
It is easy to verify the following assertion:
\begin{proposition} \label{prop:median_projection}
Let $\mu$ be a median measure generated by $q$. Let $q_*(x)$ be the density of the projection of this measure onto an arbitrary axis. Then
\begin{align*}
q_*(x) = \begin{cases}
q(x), &\text{ for } x < {2 \over 3},\\
4q\left(2 - 2x\right), &\text{ for } x > {2 \over 3}.
\end{cases}
\end{align*}
\end{proposition}
This implies, in particular, the following identity
\begin{equation}\label{eq:median_measure_projection}
4q_*(2x) = q_*(1 - x), \ x < \frac{1}{3}.
\end{equation}
The converse is also true: if nonnegative $q_*$ satisfies \cref{eq:median_measure_projection}, then there is a median measure which projection onto arbitrary axis coincides with $q_*$.
\subsection{Combining measures}
Let $\pi$ be a measure on the segment $[0, 1]$ with density $f$. We are concerned with $f(x)=\alpha^x$, but we will only use the fact that $f(x)$ is continuously differentiable, increasing, convex and satisfies $\int_0^1 \left(t - \frac{2}{3}\right)f(t) dt = 0$. The last means that measure with density $f$ satisfies \cref{main_traingle_equality}.
We want to find a measure $\mu$ that is the sum of the layered measure $\mu_p$ generated by a function $p$ and the median measure $\mu_q$ generated by a function $q$, whose projection on each of the axes coincides with $\pi$.
We subtract $\mu_p$ from $\mu$ and look at the projection of $\mu - \mu_p$ on the axes with the density $q_*(x)$. By \cref{prop:layered_projection}, the projection is equal to
\begin{align*}q_*(x)=
\begin{cases}
f(x) - 2\int_0^\frac{x}{2}p(t)dt, &\text{ for } {x \le \frac{2}{3}},\\
f(x) - (3x - 2)p(1 - x) - 2\int_0^{1 - x}p(t)dt, &\text{ for } x \ge \frac{2}{3}.
\end{cases}
\end{align*}
In order for $q_*(x)$ to be a density of the projection of a median measure, it suffices that $q_*(x) \ge 0$ and $4q_*(2x) = q_*(1 - x)$ for $x \le \frac{1}{3}$. Using the identities on $q_*(x)$ given above, we obtain the equivalent equation:
\begin{equation} \label{eq:diffur}
4\left(f(2x) - 2 \int_0^x p(t)dt\right) = f(1 - x) - (1 - 3x)p(x) - 2\int_0^xp(t)dt
\end{equation}
Assuming $P(x) = \int_0^x p(t)dt$, we get the following equation
$$
4\left(f(2x) - 2P(x)\right) = f(1 - x) - (1 - 3x)P'(x) - 2P(x)
$$
This is a differential equation of the first degree, its solutions have the form
$$
P(x) = \frac{c_1 + \int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt}{(1 - 3x)^2}.
$$
Using $P(0) = 0$ we get $c_1 = 0$,
$$
P(x) = \frac{1}{(1 - 3x)^2} \int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt.
$$
Now suppose that $f$ is continuously differentiable. We find $p(x)$ using integration by parts:
\begin{align*}
p(x) &= P'(x) \\
&= \frac{(f(1 - x)-4f(2x))(1 - 3x)^2 + 6\int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt}{(1 - 3x)^3} \\
&= \frac{1}{(1 - 3x)^3}\left(f(1) - 4f(0) - \int_0^x(1 - 3t)^2(f'(1-t) + 8f'(2t))dt\right).
\end{align*}
To prove that $p(x)$ and corresponding $q(x)$ generate a nonnegative density, we need to check that $p(x) \ge 0$ for $x \in \left[0, \frac{1}{3}\right]$, $p\left(\frac{1}{3}\right)$ is well-defined and $f(2x)-2P(x) = q_*(2x) = q(2x) \ge 0$ for $x \in \left[0, \frac{1}{3}\right]$, where $q(x)$ generates the median measure.
\begin{lemma}\label{lem:1_3_well_defined}
Suppose that $f : [0, 1] \to \mathbb{R}$ is a continuously differentiable monotonically increasing function and $\int_0^1 \left(t - \frac{2}{3}\right)f(t) dt = 0$. Then the function
$$
I(x) = f(1) - 4f(0) - \int_0^x(1 - 3t)^2(f'(1-t) + 8f'(2t))dt
$$
is nonnegative on $\left[0, {1 \over 3}\right]$ and $I\left(1 \over 3\right) = 0$.
\end{lemma}
\begin{proof}
Since $f$ is increasing, $f' \ge 0$ and the integrand $(1 - 3t)^2(f'(1-t) + 8f'(2t))$ is nonnegative. So the integral increases and $I(x)$ monotonically decreases to $I\left(\frac{1}{3}\right)$. Integrating by parts we get
\begin{align*}
I\left(1 \over 3\right) &= f(1) - 4f(0) - \int_0^\frac{1}{3}(1 - 3t)^2(f'(1-t) + 8f'(2t)) dt \\
&= \int_0^\frac{1}{3} (4f(2t) - f(1-t)) d(1-3t)^2\\
&= 6\int_0^\frac{1}{3} (1 - 3t)(f(1 - t) - 4f(2t)) dt\\
&= 18\int_0^1 \left(t - \frac{2}{3}\right)f(t) dt \\
&= 0.
\end{align*}
\end{proof}
Using this lemma one can check that $p(x)$ is nonnegative and well-defined.
\begin{proposition}\label{prop:pcorrect}
Suppose that $f(x)$ satisfies the conditions of \cref{lem:1_3_well_defined}. Then the function
$$
p(x) = \frac{1}{(1 - 3x)^3}\left(f(1) - 4f(0) - \int_0^x(1 - 3t)^2(f'(1-t) + 8f'(2t))dt\right)
$$
is nonnegative and $\lim_{x \to {1 \over 3}}p(x) = f'\left(2 \over 3\right)$.
\end{proposition}
\begin{proof}
Using the function $I(x)$ from \cref{lem:1_3_well_defined} we can rewrite the function $p(x)$ as follows:
$$
p(x) = \frac{I(x)}{(1 - 3x)^3}.
$$
$I(x)$ is nonnegative, so is $p(x)$. Let us check that $p\left(1 \over 3\right)$ is well-defined.
Since $I\left(1 \over 3\right) = 0$ one can apply the L'Hospital rule to $p(x)$:
\begin{align*}
\lim_{x \to \frac{1}{3}}p(x) &= \lim_{x \to \frac{1}{3}}\frac{I(x)}{(1 - 3x)^3} = \lim_{x \to \frac{1}{3}}-\frac{I'(x)}{9(1 - 3x)^2}\\
&=\lim_{x \to \frac{1}{3}}\frac{(1 - 3x)^2(f'(1-x) + 8f'(2x))}{9(1 - 3x)^2}\\
&=\frac{1}{9}\lim_{x \to \frac{1}{3}}(f'(1-x) + 8f'(2x)) = f'\left({2 \over 3}\right).
\end{align*}
\end{proof}
Now we will check that the function $q$ is nonnegative as well, so it generates the measure with nonnegative density.
\begin{proposition}\label{prop:qcorrect}
Suppose that $f(x)$ satisfies the conditions of \cref{lem:1_3_well_defined} and $f(x)$ is convex on $[0, 1]$. Then the function
$$
q(2x) = f(2x) - 2P(x) = f(2x) - \frac{2}{(1 - 3x)^2} \int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt
$$
is nonnegative.
\end{proposition}
\begin{proof}
Write the function $q$ in the following form:
\begin{align*}
q(2x) &= f(2x) - \frac{2}{(1 - 3x)^2} \int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt \\
&= f(2x) + \frac{1}{3(1 - 3x)^2}\int_0^x(f(1-t) - 4f(2t))d(1 - 3t)^2\\
&= f(2x) + \frac{(1 - 3t)^2(f(1 - t) - 4f(2t))|_0^x - \int_0^x(1 - 3t)^2(f'(1 - t) + 8f'(2t))dt}{3(1 - 3x)^2}\\
&= \frac{(1 - 3x)^2(f(1 - x) - f(2x)) - I(x)}{3(1 - 3x)^2}.
\end{align*}
To check that $q \ge 0$ it suffices to check that the numerator $n(x) = (1 - 3x)^2(f(1 - x) - f(2x)) - I(x)$ is nonnegative. From \cref{lem:1_3_well_defined} $n\left(\frac{1}{3}\right) = 0$. So we check that $n$ is decreasing.
\begin{align*}
n'(x) &= 6(1 - 3x)(f'(2x)(1 - 3x) - (f(1 - x) - f(2x))),\\
n'(x) &\le 0 \Leftrightarrow f'(2x) \le \frac{f(1 - x) - f(2x)}{1 - 3x}.
\end{align*}
The last equality holds since $f$ is convex.
\end{proof}
Summarizing the last two propositions we obtain the following theorem:
\begin{theorem}
For any continuously differentiable, increasing and convex function $f: [0, 1]$ satisfying $\int_0^1 \left(t - \frac{2}{3}\right)f(t) dt = 0$, there exists a measure on $\Delta$ with projections onto the axes have densities $f(x)$.
\end{theorem}
All the assumptions can be applied to $f(x) = \alpha^x$, where $\alpha$ is a solution of \cref{alpha_equation}.
Also we find $p(x)$ and $q(x)$ explicitly.
\begin{align*}
p(x) &= -\frac{1}{(1 - 3x)^3}\int_0^x(1 - 3t)^2(f'(1-t) + 8f'(2t))dt \\
&=\frac{\alpha^{1-x}-4 \alpha^{2 x}}{1-3 x}-6\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3x)^2 \ln\alpha}-6\frac{3 \alpha^{2 x}+3\alpha-\alpha
\ln\alpha-3-2\ln\alpha-3 \alpha^{1-x}}{(1-3 x)^3 \ln ^2\alpha}\\
&= \frac{\alpha^{1-x}-4 \alpha^{2 x}}{1-3 x}-6\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3x)^2 \ln\alpha}-18\frac{\alpha^{2 x}-\alpha^{1-x}}{(1-3 x)^3 \ln ^2\alpha},
\end{align*}
\begin{align*}
q(2x) &= f(2x) - 2P(x) = f(2x) - {2 \over (1 - 3x)^2}\int_0^x(1 - 3t)(f(1-t) - 4f(2t))dt \\
&=\alpha^{2 x}+2\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3 x) \ln
\alpha}+2\frac{3 \alpha^{2 x}+3\alpha-\alpha \ln\alpha-3
-2 \ln\alpha-3\alpha^{1-x}}{(1-3 x)^2 \ln ^2\alpha} \\
&= \alpha^{2 x}+2\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3 x) \ln
\alpha}+6\frac{\alpha^{2 x}-\alpha^{1-x}}{(1-3 x)^2 \ln ^2\alpha}.
\end{align*}
The last identities follow from \cref{alpha_equation}.
Now we are ready to present the main theorem of this section:
\begin{theorem}
\label{thm:osexists}
There exists a $(3, 1)$-stochastic measure concentrated on the set $M$.
\end{theorem}
\begin{proof}
Let us collect all the details of the proof together and describe our measure explicitly. Set $M$ contains segments connecting points $(0, 1, 1)$ and $(l, r, r)$, $(1, 0, 1)$ and $(r, l, r)$, $(1, 1, 0)$ and $(r, r, l)$. This segments have length $L = \sqrt{l^2 + 2(1 - r)^2}$.
Define measure $\mu_{lin}$ as a sum of Lebesgue measures on this segments divided by$\frac{l}{L}$.
The projections of two segments coincide with $[r, 1]$, the densities are equal to $\frac{L}{1 - r} \cdot \frac{l}{L} = \frac{1}{2}$. Their sum is the Lebesgue measure on $[r, 1]$.
The projection of the third interval is a measure on $[0, l]$, its density equals $\frac{L}{l} \cdot \frac{l}{L} = 1$.
The mapping $$u = {\ln x - \ln l \over \ln r - \ln l}, \ v = {\ln y - \ln l \over \ln r - \ln l}, \ w = {\ln z - \ln l \over \ln r - \ln l}$$
transforms the two-dimensional part of $M$ into triangle $\Delta$.
We equip $\Delta$ with the layered measure $\mu_p$ generated by $$p(x) = \frac{\alpha^{1-x}-4 \alpha^{2 x}}{1-3 x}-6\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3x)^2 \ln\alpha}-18\frac{\alpha^{2 x}-\alpha^{1-x}}{(1-3 x)^3 \ln ^2\alpha},$$ and the median measure $\mu_q$ generated by $$q(2x) = \alpha^{2 x}+2\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3 x) \ln\alpha}+6\frac{\alpha^{2 x}-\alpha^{1-x}}{(1-3 x)^2 \ln^2\alpha}.$$
Then by \cref{prop:layered_projection} the projection of $\mu_p$ coincides with
\begin{align*}
\begin{cases}
2\int_0^\frac{x}{2}p(t)dt, &\text{ for } {x \le \frac{2}{3}},\\
(3x - 2)p(1 - x) + 2\int_0^{1 - x}p(t)dt, &\text{ for } x \ge \frac{2}{3}.
\end{cases}
\end{align*}
Since $p$ is a solution of \cref{eq:diffur} for $f(x) = \alpha^x$ we can conclude that for
\begin{align*}q_*(x) =
\begin{cases}
f(x) - 2\int_0^\frac{x}{2}p(t)dt, &\text{ for } {x \le \frac{2}{3}},\\
f(x) - (3x - 2)p(1 - x) - 2\int_0^{1 - x}p(t)dt, &\text{ for } x \ge \frac{2}{3}.
\end{cases}
\end{align*}
there holds $4q_*(2x) = q_*(1 - x)$. Thus by \cref{prop:median_projection} $q_*(x)$ is the projection of $\mu_q$ generated by $q(2x) = f(2x) - 2\int_0^\frac{x}p(t)dt = \alpha^{2 x}+2\frac{2 \alpha^{2 x}+\alpha^{1-x}}{(1-3 x) \ln
\alpha}+6\frac{\alpha^{2 x}-\alpha^{1-x}}{(1-3 x)^2 \ln ^2\alpha}$.
By \cref{prop:pcorrect} and \cref{prop:qcorrect} this construction is well-defined. Projections of $\mu_p + \mu_q$ on axes coincide with $\alpha^x$ in coordinates $u, v, w$ and with the uniform measure on $[l, r]$ in initial coordinates.
Thus the projections of $\mu = \mu_p + \mu_q + \mu_{lin}$ coincide with Lebesgue measure on $[0, 1]$.
\end{proof}
\section{The dual solution construction}\label{section:dual}
To prove that the measure $\mu$ from \cref{thm:osexists} is the primal solution it is enough to find a triple of functions $f, g, h: [0, 1] \to \mathbb{R}$ such that $f(x) + g(y) + h(z) \le c(x, y, z)$ and equality holds on the set $M$ by \cref{lem:slackness_conditions}. In this case the triple $(f, g, h)$ will be a dual solution of the related problem. In this section we will construct the dual solution for a wide class of cost functions.
We will construct the dual solution for $c(x, y, z) = \widehat{C}(\ln x + \ln y + \ln z)$ where $\widehat{C}$ is a bounded continuously differentiable strictly convex function on $(-\infty, 0]$. Our function $c(x, y, z) = xyz$ is a partial case for $\widehat{C}(t) = \exp(t)$. At the same time we will use the more convenient equivalent description. Namely, $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ and the function $tC'(t)$ strictly increases on the segment $[0, 1]$.
\subsection{Another description of the support of primal solutions}
\begin{definition}\label{def:lambda_definition}
Set $c = lr^2$. Define a function $\lambda: [0, 1] \to \mathbb{R}$ as follows
\begin{align*}
\lambda(x)=\begin{cases}
(1-2x)^{2} & \text{if } x \in [0, l)\\
\frac{c}{x} & \text{if } x \in [l, r),\\
\frac{1}{2}x(1-x) & \text{if } x \in [r, 1].
\end{cases}
\end{align*}
\end{definition}
\begin{lemma}
The function $\lambda$ defined above is continuous and strictly decreases.
\end{lemma}
\begin{proof}
It suffices to check the continuity at points $l$ and $r$. For this it suffices to check that $(1 - 2l)^{2} = \frac{c}{l}$ and $\frac{c}{r} = \frac{1}{2}r(1-r)$.
All these equalities are trivial.
Let us check that the derivative of $\lambda$ is negative everywhere except of the points $l$ and $r$: in these points $\lambda$ has no derivatives.
If $x \in (0, l)$ then $\lambda'(x) = 2(2x-1) < 0$, since $x < l < \frac{1}{2}$. If $x \in (l, r)$ then
$\lambda'(x)=-\frac{c}{x^{2}} < 0$ since $c > 0$. If $x \in (r, 1)$ then $\lambda'(x) = 1 - \frac{1}{2}x < 0$ since $x > r > \frac{1}{2}$.
It follows from this that $\lambda$ strictly decreases.
\end{proof}
\begin{proposition}\label{prop:description_M}
Suppose that $M$ is the (hypothetical) primal solution support as in the previous sections. Then a point $(x, y, z)$ is contained in $M$ if and only if the following equalities hold
$\lambda(x)=yz$, $\lambda(y)=xz$, $\lambda(z)=xy$.
\end{proposition}
\begin{proof}
$\Leftarrow$ Suppose that $\kappa(x)=x\lambda(x)$. If $\lambda(x)=yz$,
$\lambda(y)=xz$ and $\lambda(z)=xy$ then $\kappa(x)=\kappa(y)=\kappa(z)=xyz$.
The function $\kappa(x)$ is continuous and has a continuous derivative on intervals $(0, l)$, $(l, r)$ and $(r, 1)$. If $x \in (0, l)$ then $\kappa'(x) = (1-2x)^{2} - 2x(1 - 2x) = (1 - 2x)(1 - 4x) > 0$
since $x < l < \frac{1}{4}$. On the segment $[l, r]$ $\kappa$ is constant: $\kappa(x) = lr^2 = c$.
If $x \in (r, 1)$ then $\kappa'(x) = x(1-x) - \frac{1}{2}x^2=x\left(1 - \frac{3}{2}x\right) < 0$ since $x > r > \frac{2}{3}$. So $\kappa(x)$ strictly increases on the segment $[0, l]$, is constant on $[l, r]$, and strictly decreases on $[r, 1]$.
Note in addition that $\kappa(0) = \kappa(1) = 0$. Thus, the equation $\kappa(x) = c_0$ for $0 \le x \le 1$
\begin{enumerate}
\item has no root if $c_0 < 0$ or $c_0 > c$;
\item has exactly two roots if $0 \le c_0 < c$: one of them lies on the interval $[0, l)$ and another one lies on the interval $(r, 1]$;
\item holds on whole segment $[l, r]$ if $c_0 = c$.
\end{enumerate}
If $\lambda(x) = yz$, $\lambda(y) = xz$, and $\lambda(z) = xy$ then $\kappa(x)=\kappa(y)=\kappa(z)=xyz$ and one of the following cases occurs:
\begin{enumerate}
\item $x, y, z \in [l, r]$. In this case $c = \kappa(x) = \kappa(y) = \kappa(z) = xyz$ so $(x, y, z) \in M$.
\item $x = y = z \in[0, l)$. Then $\lambda(x)=x^2$. On the other hand if $x\in[0, l)$ then $\lambda(x)=(1-2x)^2$. The equation $(1-2x)^{2}=x^{2}$
has two solutions $x = 1$ and $x = \frac{1}{3}$. But these values are not feasible because
$x \in [0, l)$ and $l < \frac{1}{6}$. So, this case is not possible.
\item $x=y=z\in(r, 1]$. Similarly in this case $\lambda(x)=x^2$. On the other hand if $x \in (r, 1]$ then $\lambda(x)=\frac{1}{2}x(1-x)$.
Equation $\frac{1}{2}x(1-x)=x^2$ has two solutions $x = 0$ and $x = \frac{1}{3}$, but they do not belong
$(r, 1]$ for any $r > \frac{1}{2}$. So, this case is not possible.
\item $x = y \in [0, l)$, $z\in(r, 1]$ and similar cases obtained by permutations of coordinates. One has $x(1-2x)^2 = \kappa(x) = \kappa(z) = \frac{1}{2}z^2(1-z)$.
The function $\kappa(z)$ strictly decreases on the interval $(r, 1]$, hence for a fixed $x$ there exists at most one $z$ satisfying this equality. But $z=1-2x \in (r, 1]$ and $\kappa(z) = \frac{1}{2}z^2(1-z) = \frac{1}{2}(1-2x)^2 \cdot 2x = \kappa(x)$.
This means that $z=1-2x$. In this case $x(1-2x) = \frac{1}{2} z(1-z) = \lambda(z)=xy = x^2$.
Hence $x=0$ or $x=\frac{1}{3}$. But for $x=0$ one has $1 = \lambda(x) = yz = xz = 0$. The value $x=\frac{1}{3}$
is not suitable because $x \in [0, l)$ and $l < {1 \over 6}$. So, this case is not possible.
\item $x \in [0, l)$, $y = z \in (r, 1]$ and similar cases obtained by permutations of coordinates. Arguing as above, we get $\kappa(x) = \kappa(z)$,
$x \in [0, l)$, $z \in (r, 1]$ so $y = z = 1-2x$. The points $(x, 1-2x, 1-2x)$ are contained in $M$ for any $x \in [0, l)$.
\end{enumerate}
So, the only possible cases are cases 1 and 5. In these cases $(x, y, z) \in M$.
$\Rightarrow$ The set $M$ consists of four parts: $M = M_x \cup M_y \cup M_z \cup M_2$. If $(x, y, z) \in M_x$, then $y = z = 1 - 2x$. Hence $\lambda(x) = (1 - 2x)^2$ and $yz = (1 - 2x)^2$. $\lambda(y) = \lambda(1 - 2x) = {1 \over 2}(1 - 2x)\cdot 2x = x(1 - 2x) = xz$ since $r \le 1 - 2x \le 1$. Similarly $\lambda(z) = xy$.
Hence if $(x, y, z) \in M_x$, then $\lambda(x) = yz$, $\lambda(y) = xz$ and $\lambda(z) = xy$. By symmetry, these conditions hold for any $(x, y, z) \in M_y$ and for any $(x, y, z) \in M_z$.
If $(x, y, z) \in M_2$, then $l \le x, y, z \le r$ and $xyz = c$. This means that $\lambda(x) = {c \over x} = yz$, $\lambda(y) = {c \over y} = xz$, $\lambda(z) = {c \over z} = xy$.
\end{proof}
\subsection{The construction of the dual solution}
If $M$ is indeed a support of the primal solution and $f, g, h$ is a dual solution, then by complementary slackness $f(x)+g(y)+h(z)$ is equal to $c(x, y, z)$ on almost all points of $M$. This will help us to guess the form of $f, g, h$.
\begin{lemma}\label{thm:weak_uniqueness}
Assume that $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$
and the triple of functions
$$f, g, h: [0, 1] \to \mathbb{R}
$$ satisfies inequality $f(x) + g(y) + h(z) \le c(x, y, z)$ and $f(x) + g(y) + h(z) = c(x, y, z)$ for all $(x, y, z) \in M$.
Then the functions $f, g, h$ are continuously differentiable and $f'(x)=\lambda(x)C'(x\lambda(x))$,
$g'(y)=\lambda(y)C'(y\lambda(y))$, $h'(z)=\lambda(z)C'(z\lambda(z))$.
\end{lemma}
\begin{proof}
For any $x_0$ there exist $y_0$ and $z_0$ such that $(x_0, y_0, z_0) \in M$. This means that $f(x_0) + g(y_0) + h(z_0) = c(x_0, y_0, z_0) = C(x_0\lambda(x_0))$.
In addition, for any $x$ one has
$$f(x) + g(y_0) + h(z_0) \le c(x, y_0, z_0) = C(x\lambda(x_0)).
$$
Hence for any $x_0, x \in [0, 1]$ one has
$f(x) - f(x_0) \le C(x\lambda(x_0))-C(x_0\lambda(x_0))$.
Passing to the limit $x \to x_0$ one gets
\begin{align*}
C(x\lambda(x_0))-C(x_0\lambda(x_0)) = (x - x_0) \cdot \lambda(x_0)C'(x_0\lambda(x_0)) + o(|x - x_0|).
\end{align*}
Interchanging $x_0$ and $x$ one gets $f(x_0) - f(x) \le C(x_0\lambda(x)) - C(x\lambda(x))$. By the mean value theorem, $C(x_0\lambda(x)) - C(x\lambda(x)) = (x_0 - x)\lambda(x)C'(\xi(x))$, where $\xi(x) \in [x_0\lambda(x), x\lambda(x)]$. If $x \to x_0$, then $\xi(x) \to x_0\lambda(x_0)$ and
\begin{align*}
C(x_0\lambda(x))-C(x\lambda(x)) & = (x_0 - x) \lambda(x)C'(x_0\lambda(x_0)) + o(|(x_0 - x) \lambda(x)|)
\\& = (x_0 - x) \lambda(x)C'(x_0\lambda(x_0)) + o(|x - x_0|)
\\& = (x_0 - x) \lambda(x_0)C'(x_0\lambda(x_0)) + o(|x - x_0|).
\end{align*}
This means that
\begin{multline*}
\lambda(x_0)C'(x_0\lambda(x_0)) \cdot (x - x_0) + o(|x - x_0|) \\
\le f(x) - f(x_0) \\
\le \lambda(x_0)C'(x_0\lambda(x_0)) \cdot (x - x_0) + o(|x - x_0|).
\end{multline*}
Hence $f(x)$ has a derivative at the point $x = x_0$ and it is equal to $\lambda(x_0)C'(x_0\lambda(x_0))$. This function is continuous since $\lambda$ and $C'$ are continuous.
One can check in the same way the statements of the theorem for the functions $g$ and $h$.
\end{proof}
\begin{theorem}
Suppose that $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ and the function $U(t) = tC'(t)$ strictly increases on the segment $[0, 1]$.
Suppose that $\hat{f}(s) = \int_{0}^{s} \lambda(t)C'(t\lambda(t))~dt$.
Then the arg max of the function $\hat{f}(x) + \hat{f}(y) + \hat{f}(z) - c(x, y, z)$ contains the set $M$.
\end{theorem}
\begin{proof}
Assume that $T(x, y, z) = \hat{f}(x) + \hat{f}(y) + \hat{f}(z) - c(x, y, z) = \hat{f}(x) + \hat{f}(y) + \hat{f}(z) - C(xyz)$.
If $(x, y, z) \in M$ then
\begin{equation*}
\nabla T(x, y, z) =
\begin{pmatrix}
\lambda(x)C'(x\lambda(x)) - yzC'(xyz)\\
\lambda(y)C'(y\lambda(y)) - xzC'(xyz)\\ \lambda(z)C'(z\lambda(z)) - xyC'(xyz)
\end{pmatrix}= \vec{0}.
\end{equation*}
Hence, all values of $T$ on the set $M$ are the same since $M$ is path-connected.
The function $T$ is continuous on the compact set $[0, 1]^{3}$, so the function $T$ reaches its maximum at some point $(x_0, y_0, z_0)$. Then either
$x_0$ lies on the boundary of the segment $[0, 1]$ or $\frac{\partial T}{\partial x}(x_0, y_0, z_0) = 0$.
For any $x > 0$ the following equality holds
\[
\frac{\partial T}{\partial x}(x, y_0, z_0) = \lambda(x)C'(x\lambda(x))-y_0z_0C'(xy_0z_0)=\frac{U(x\lambda(x)) - U(xy_0z_0)}{x}.
\]
Assume that $x_0 = 0$. By the mean value theorem
for any $x > 0$ there exists $0 < \xi(x) < x$ such that
\begin{multline*}
T(x, y_0, z_0) - T(x_0, y_0, z_0) = x\frac{\partial T}{\partial x}(\xi(x), y_0, z_0)\\
=\frac{x}{\xi(x)}\left(U[\xi(x)\lambda(\xi(x))]-U[\xi(x)y_0z_0]\right).
\end{multline*}
One has
$T(x, y_0, z_0) \le T(x_0, y_0, z_0)$ since $(x_0, y_0, z_0)$ is a maximum point of $T$. Hence, $U[\xi(x)\lambda(\xi(x))] \le U[\xi(x)y_0z_0]$ and $\xi(x)\lambda(\xi(x)) \le \xi(x)y_0z_0$ since $U$ strictly increases. This means that $\lambda(\xi(x)) \le y_0z_0$ for all $x > 0$. If $x \to 0$ then $\lambda(\xi(x)) \to \lambda(0) = 1$. Thus $y_0z_0 \ge 1 \Rightarrow \lambda(x_0) = 1 = y_0z_0$.
Suppose that $x_0 = 1$. In this case $\frac{\partial T}{\partial x}(x_{0},y_{0},z_{0})$
must be nonnegative. But $\frac{\partial T}{\partial x}(x_0, y_0, z_0) = \frac{U(x_0\lambda(x_0)) - U(x_0y_0z_0)}{x_0} = U(0) - U(y_0z_0)$.
The function $U(t)$ strictly increases, hence $y_0z_0 = 0$. This implies $0 = \lambda(x_{0}) = y_0z_0$.
Otherwise one has $\frac{\partial T}{\partial x}(x_0, y_0, z_0) = \frac{1}{x_0}(U(x_0\lambda(x_0))-U(x_0y_0z_0)) = 0$. The function $U(t)$ strictly increases. Hence $x_0\lambda(x_0)=x_0y_0z_0$ and $\lambda(x_0)=y_0z_0$.
Consequently, if the function $T$ has maximum at the point $(x_0, y_0, z_0)$, one gets $\lambda(x_0) = y_0z_0$. Similarly, one can prove that $\lambda(y_0) = x_0z_0$
and $\lambda(z_0) = x_0y_0$. Hence by \cref{prop:description_M} $(x_{0},y_{0},z_{0}) \in M$. Since $T$ is constant on $M$, one has $M \subset \arg\max T$.
\end{proof}
Summarizing the results from the last two sections we get
\begin{theorem}\label{thm:primal_dual}
Suppose that $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ and the function $tC'(t)$
strictly increases on the segment $[0, 1]$. Set:
$$\hat{f}(s) = \int_{0}^{s}\lambda(t)C'(t\lambda(t))~dt.$$ Then for any constants $C_x$, $C_y$, $C_z$ such that $$C_x + C_y + C_z = C(0) - 2\int_{0}^{1}\lambda(t)C'(t\lambda(t))~dt$$ the following inequality holds
$$(\hat{f}(x) + C_x) + (\hat{f}(y) + C_y) + (\hat{f}(z) + C_z) \le c(x, y, z)$$
with equality on $M$.
This means by \cref{lem:slackness_conditions} that the triple $(\hat{f} + C_x,~\hat{f} + C_y,~\hat{f} + C_z)$ is the dual solution for the cost function $c(x, y, z)$ and any probability measure $\mu$ such that $\Pr_X(\mu) = \Pr_Y(\mu) = \Pr_Z(\mu) = \lambda$ and $\mathrm{supp}(\mu) \subset M$ is the primal solution to the related problem.
Moreover such a measure $\mu$ exists by \cref{thm:osexists}.
\end{theorem}
We note that any primal solution is universal in a sense it is the same for the cost functions of type $C(xyz)$ where $tC'(t)$ is strictly increasing on $[0, 1]$. It is important for the proof that $M$ is path-connected. Numerical experiments for other marginals show that sometimes the support of a primal solution is not necessarily path-connected. For example for a measure $SF$ on $[0, 5]$ given by a density
\begin{align*}
\rho_{SF}(t) = \begin{cases}
\frac{1}{15} \text{, if } t \in [0, 1] \cup [2, 3] \cup [4, 5],\\
\frac{2}{5} \text{, if } t \in (1, 2) \cup (3, 4),
\end{cases}
\end{align*}
primal solution (more precisely the result of \cref{alg:primal_solution_approximation_faster}) for the cost function $c(x, y, z) = xyz$ is pictured on \cref{fig:SF}.
\begin{figure}[h]
\centering \label{fig:SF}
\includegraphics*[scale=0.3]{Snowflake} \\
\caption{Primal solution for marginals $SF$}
\end{figure}
\subsection{Construction for the cost function $c(x, y, z) = xyz$}\label{subsection:explicit_dual_xyz}
Suppose that $$c(x, y, z) = xyz = C(xyz),$$
where $C(t) = t$, $0 \le t \le 1$. The function $C(t)$ is continuously differentiable and $tC'(t) = t$ strictly increases. \cref{thm:primal_dual} implies that any probability measure $\mu$ with projections $\Pr_X(\mu) = \Pr_Y(\mu) = \Pr_Z(\mu) = \lambda$ and $\mathrm{supp}(\mu) \subset M$ is the primal solution to the related problem; in particular the probability measure from \cref{thm:osexists} is the primal solution. Also, we can construct explicitly the dual solution in this case.
Consider the following functions:
\begin{align*}
&f_1(x) = c \ln l - {1 \over 3}(c\ln c - c) + {1 \over 6}((2x - 1)^3 - (2l - 1)^3),\\
&f_2(x) = c\ln x - {1 \over 3}(c\ln c - c),\\
&f_3(x) = c\ln r - {1 \over 3}(c\ln c - c) + {1 \over 4}(x^2 - r^2) - {1 \over 6}(x^3 - r^3).
\end{align*}
These functions satisfy the following identities:
$$f_1(l) = f_2(l),$$
$$f_2(r) = f_3(r),$$
$$f_1'(l) = f_2'(l),$$
$$f_2'(r) = f_3'(r).$$
The first and the second equality are easy to check directly. For the third and the fourth compute $f_1'(x) = (2x - 1)^2$, $f_2'(x) = \frac{c}{x}$, $f_3'(x) = \frac{1}{2}(x - x^2)$.
$f_1'(l) = (2l - 1)^2 = r^2 = \frac{c}{l} = f_2'(l)$, $f_2'(r) = lr = \frac{1}{2}r(1 - r) = f_3'(r)$.
Define:
\begin{align*}
f(x) = g(x) = h(x) = \begin{cases}
f_1(x) \text{, if } 0 \le x \le l,\\
f_2(x) \text{, if } l \le x \le r,\\
f_3(x) \text{, if } r \le x \le 1,
\end{cases}
\end{align*}
It follows immediately from the properties checked above of the functions $f_1, f_2, f_3$ that $f$ is continuous and continuously differentiable on $[0, 1]$ and $f'(x) = \lambda(x)$.
\begin{proposition}\label{prop:explicit_dual_xyz}
The triple of functions $(f, g, h)$ defined above is a dual solution of related problem for the cost function $c(x, y, z) = xyz$.
\end{proposition}
\begin{proof}
Since $f'(x) = \lambda(x)$ it follows that
$$
f(x) = g(x) = h(x) = \int_0^x\lambda(x)~dx + C_f = \int_0^x\lambda(x)C'(x\lambda(x))~dx + C_f
$$
for some constant $C_f$. By \cref{thm:primal_dual} it is enough to check that $f(0) + f(1) + f(1) = c(0, 1, 1) = 0$.
\begin{align*}
f(0) = f_1(0) &= c\ln l - {1 \over 3}(c\ln c - c) - {1 \over 6}(2l - 1)^3 - {1 \over 6},\\
f(1) = f_3(1) &= c\ln r - {1 \over 3}(c\ln c - c) - {1 \over 4}r^2 + {1 \over 6}r^3 + {1 \over 12},\\
f(0) + 2f(1) &= c\ln(lr^2) - (c\ln c - c) + 2 \cdot{1 \over 12} - {1 \over 6} - {1 \over 2}r^2
+ {1 \over 3}r^3 - {1 \over 6}(2l - 1)^3\\
&= c - {1 \over 2}r^2 + {1 \over 2}r^3 = c - {1 - r \over 2}r^2 = c - lr^2 = 0.
\end{align*}
So the triple $(f, g, h)$ is the dual solution for the cost function $c(x, y, z)=xyz$.
\end{proof}
\section{Uniqueness of the dual solution}\label{section:uniqueness}
\begin{theorem}\label{thm:uniqueness}
Suppose that $c(x, y, z) = C(xyz)$ for some continuously differentiable function $C:[0, 1] \to \mathbb{R}$ and the function $tC'(t)$
strictly increases on the segment $[0, 1]$. Then the triple $(f, g, h)$ is a dual solution if and only if there exist constants $C_f$, $C_g$, $C_h$ such that
$$
C_f + C_g + C_h = C(0) - 2\int_{0}^{1}\lambda(t)C'(t\lambda(t))~dt,
$$
and
$$f(x) \le \int_{0}^{x}\lambda(t)C'(t\lambda(t))~dt + C_f,$$
$$ g(y) \le \int_{0}^{y}\lambda(t)C'(t\lambda(t))~dt + C_g,$$
$$h(z) \le \int_{0}^{z}\lambda(t)C'(t\lambda(t))~dt + C_h.$$
where equality is achieved almost everywhere.
\end{theorem}
\begin{proof}
$\Leftarrow$ Suppose that $\widetilde{f}(x) = \int_{0}^{x}\lambda(t)C'(t\lambda(t))~dt + C_f$, $\widetilde{g}(y) = \int_{0}^{y}\lambda(t)C'(t\lambda(t))~dt + C_g$ and $\widetilde{h}(z) = \int_{0}^{z}t\lambda(t)C'(t\lambda(t))~dt + C_h$. Then the triple $(\widetilde{f}, \widetilde{g}, \widetilde{h})$ is the dual solution by \cref{thm:primal_dual}. Also $f(x) + g(y) + h(z) \le \widetilde{f} + \widetilde{g} + \widetilde{h} \le c(x, y, z)$ and $\int_0^1f(x) + g(x) + h(x)~dx = \int_0^1\widetilde{f} + \widetilde{g} + \widetilde{h}~dx$ so the triple $(f, g, h)$ is the dual solution.
$\Rightarrow$ For any dual solution $(f, g, h)$ there exists a triple $(\widetilde{f}, \widetilde{g}, \widetilde{h})$ such that $f \le \widetilde{f}$, $g \le \widetilde{g}$, $h \le \widetilde{h}$ and $\widetilde{f}(x) = \inf_{y, z}(c(x, y, z) - \widetilde{g}(y) - \widetilde{h}(z))$, $\widetilde{g}(y) = \inf_{x, z}(c(x, y, z) - \widetilde{f}(x) - \widetilde{h}(z))$, $\widetilde{h}(z) = \inf_{x, y}(c(x, y, z) - \widetilde{f}(x) - \widetilde{g}(y))$. One can prove this by applying the Legendre transformation subsequently to $f$, $g$, $h$.
For any $x$, $y$, $z$ inequality $\widetilde{f}(x) + \widetilde{g}(y) + \widetilde{h}(z) \le c(x, y, z)$ holds since $\widetilde{f}(x) = \inf_{y, z}(c(x, y, z) - \widetilde{g}(y) - \widetilde{h}(z))$. Also $$
\int_0^1\widetilde{f}(x)~dx + \int_0^1\widetilde{g}(y)~dy + \int_0^1\widetilde{h}(z)~dz \ge \int_0^1f(x)~dx + \int_0^1 g(y)~dy + \int_0^1 h(z)~dz$$ since $f \le \widetilde{f}$, $g \le \widetilde{g}$ and $h \le \widetilde{h}$. This means that the triple $(\widetilde{f}, \widetilde{g}, \widetilde{h})$ is a dual solution and $\widetilde{f} = f$, $\widetilde{g} = g$, $\widetilde{h} = h$ almost everywhere.
A function $F[y, z]: [0, 1] \to \mathbb{R}$, $F[y, z](x) = c(x, y, z) - \widetilde{g}(y) - \widetilde{h}(z)$ is a Lipschitz continuous function since ${\partial \over \partial x}c(x, y, z)$ is a well-defined continuous function on the cube $[0, 1]^3$. This means that $\widetilde{f}(x)$ is a Lipschitz continuous function since $\widetilde{f}$ is an infimum of the family of Lipschitz continuous functions $F[y, z]$ with common constant $\max_{x, y, z}{\partial \over \partial x}c(x, y, z)$. In particular this means that $\widetilde{f}$ is continuous on the segment $[0, 1]$. Similarly, the functions $\widetilde{g}$ and $\widetilde{h}$ are continuous.
For any primal solution $\mu$ equality $\widetilde{f}(x) + \widetilde{g}(y) + \widetilde{h}(z) = c(x, y, z)$ holds $\mu$-almost everywhere. The set of equality points is closed, because $f$, $g$ and $h$ are continuous. This means that $\widetilde{f}(x) + \widetilde{g}(y) + \widetilde{h}(z) = c(x, y, z)$ on the support of $\mu$. For the primal solution $\mu$ from \cref{section:primal_construction} $\mathrm{supp}(\mu) = M$. So the equality $\widetilde{f}(x) + \widetilde{g}(y) + \widetilde{h}(z) = c(x, y, z)$ holds on the set $M$.
By \cref{thm:weak_uniqueness} the functions $\widetilde{f}$, $\widetilde{g}$ and $\widetilde{h}$ are continuously differentiable and $\widetilde{f}'(x) = \lambda(x)C'(x\lambda(x))$, $\widetilde{g}'(y) = \lambda(y)C'(y\lambda(y))$, $\widetilde{h}'(z) = \lambda(z)C'(z\lambda(z))$. This means that $\widetilde{f}(x) = \hat{f}(x) + C_f$, $\widetilde{g}(y) = \hat{f}(y) + C_g$ and $\widetilde{h}(z) = \hat{f}(z) + C_h$ for some constants $C_f$, $C_g$ and $C_h$. Since $(0, 1, 1) \in M$ the equality holds $C_f + C_g + C_h = c(0, 1, 1) - \hat{f}(0) - \hat{f}(1) - \hat{f}(1) = C(0) - 2\int_{0}^{1}\lambda(t)C'(t\lambda(t))~dt$.
\end{proof}
\section{A priori estimates for the dimension}\label{section:inertion}
Following \cite{brendan_pass} let us introduce the following sets of matrices
$$g_{\{x\}} = g_{\{y, z\}} =
\begin{pmatrix}
0 & z & y \\
z & 0 & 0 \\
y & 0 & 0
\end{pmatrix},
$$
$$g_{\{y\}} = g_{\{x, z\}} =
\begin{pmatrix}
0 & z & 0 \\
z & 0 & x \\
0 & x & 0
\end{pmatrix},
$$
$$g_{\{z\}} = g_{\{x, y\}} =
\begin{pmatrix}
0 & 0 & y \\
0 & 0 & x \\
y & x & 0
\end{pmatrix}.
$$
Further, $G$ is a linear combination of $g_p$ with nonnegative coefficients :
$$ G = \left\{ \left.
\begin{pmatrix}
0 & (\alpha + \beta)z & (\alpha + \gamma)y \\
(\alpha + \beta)z & 0 & (\beta + \gamma)x \\
(\alpha + \gamma)y & (\beta + \gamma)x & 0
\end{pmatrix}
\right| \alpha, \beta, \gamma \ge 0
\right\}.
$$
By Theorem 2.1.2 from \cite{brendan_pass} the supports of solutions to the primal problem are locally contained inside a manifold of dimension
$$ d = 3 - \text{positive index of inertia of $g$}$$
for any $g \in G$. This index is computed below.
\begin{proposition}
The quadratic form given by
$$ g =
\begin{pmatrix}
0 & a & b \\
a & 0 & c \\
b & c & 0
\end{pmatrix}
$$
with non-negative $a$, $b$ and $c$ has positive index of inertia at most $1$.
\end{proposition}
\begin{proof}
Consider two cases.
\textbf{First case.} Let $a, b, c > 0$. Then principal upper left minors are $\Delta_0 = 1$, $\Delta_1 = 0$, $\Delta_2 = -a^2 < 0$ and $\Delta_3 = 2abc > 0$. So number of sign changes in sequence of principal upper left minors is $2$ and negative index of inertia is $2$. This means that the positive index of inertia is at most $1$.
\textbf{Second case.} Without loss of generality $c = 0$. Then $g$ has the form $2a xy + 2b xz = \frac{1}{2}(x + (ay + bz))^2 - \frac{1}{2}(x - (ay + bz))^2$. Thus the positive index of inertia is at most $1$.
\end{proof}
We see that the local dimension of our solution is indeed not bigger than $2$, but unfortunately this bound does not help to determine the local dimension of our solution without solving problem explicitly.
\section{Extreme points}
We show in this section that the extreme points of the primal solutions are singular to the surface (Hausdorff)
measure on $M$. Applying logarithmic transformation from the proof of \cref{thm:osexists}
and noticing that this is a (locally) bi-Lipschitz transformation
one can easily verify that it is sufficient to prove the claim for the triangle $\Delta$.
Further, projecting $\Delta$ onto the $xy$-hyperplane we reduce the proof of the statement to the proof
of the following fact:
\begin{theorem}
Let $\mu_x, \mu_y$, and $\mu_{x+y}$ be one-dimensional probability measures on
the axes $x, y$ and on the line $l_{x+y} = \{(x,y) \in \mathbb{R}^2 \colon x=y\}$
respectively. We assume that $\mu_x, \mu_y$ and $\mu_{x+y}$ are compactly supported. Let $\Pi$ be the set of probability measures with projections
$$
\mu_x = {\rm Pr}_x(\pi), \ \mu_y = {\rm Pr}_y(\pi), \ \mu_{x+y} = {\rm Pr}_{x+y}(\pi),
$$
where ${\rm Pr}_x$, ${\rm Pr}_y$ are projection onto $x, y$, and ${\rm Pr}_{x+y}$
is the projection onto $l_{x+y}$: ${\rm Pr}_{x+y}(x,y) = x + y$.
Assume that $\Pi$ is nonempty and $\pi \in \Pi$ is an extreme point. Then
there exists a set $S$ of Lebesgue measure zero such that $\pi(S)=1$.
\end{theorem}
\begin{proof}
Without loss of generality let us assume that $\pi$ is supported by $X = [0,1]^2$.
Let us consider the set of tuples of $6$ points
\begin{align*}
N = \Bigl\{ & \Bigl((x_1, y_2), (x_1, y_3), (x_2, y_1), (x_2, y_3), (x_3, y_2), (x_3, y_1) \Bigr) \colon \ x_1 < x_2 < x_3,
\\& y_1 < y_2 < y_3, x_1 + y_2 = x_2 + y_1, x_1 + y_3 = x_3 + y_1, x_2 + y_3 = x_3 + y_2
\Bigr\} \subset X^6.
\end{align*}
For arbitrary $\Gamma \in N$ let us set
$$
\Gamma_+ = \{(x_1, y_2), (x_2, y_3), (x_3,y_1)\}, \ \Gamma_{-} = \{(x_1, y_3), (x_2, y_1), (x_3,y_2)\}.
$$
Note that $\Gamma = \Gamma_{-} \sqcup \Gamma_{+}$ and uniform distributions on the sets
$\Gamma_+$ and $\Gamma_-$ have the same projections onto the both axes and $l_{x+y}$.
Let us show that there exists a set $S \subset X$ with the properties: $\pi(S)=1$, $S$ does not contain any subset of $6$ points in $N$.
According to a Kellerer's result (see \cite{Kellerer}) the following alternative holds:
\begin{itemize}
\item There exists a measure $\gamma$ on $X^6$ with the property $\gamma(N)>0$, such that
$\mathrm{Pr}_i \gamma \le \pi$, $1 \le i \le 6$.
\item For $1 \le i \le 6$ there exists a set $N_i \subset [0,1]^2 = X$ with the property $\pi(N_i)=0$ and
$$
N \subset \cup_{i=1}^6 X \times \ldots \times N_i \times \ldots X.
$$
\end{itemize}
In the second case
$$
S=X \setminus\cup_{i=1}^6 N_i
$$
will be a desired set. We will prove it later.
First we prove that the first case is impossible. We can assume that $\pi(X^6 \backslash N) = 0$ and $\pi$ is still nonzero.
Suppose that $\Gamma = \Bigl((x_1, y_2), (x_1, y_3), (x_2, y_1), (x_2, y_3), (x_3, y_2), (x_3,y_1) \Bigr)$ is an arbitrary point of $N$ and $B_\Gamma \subset X^6$ is ball with a center at $\Gamma$ and a radius of $\varepsilon < {1 \over 2}\min(x_2 - x_1, x_3 - x_2, y_2 - y_1, y_3 - y_2)$. Also suppose that $\tilde{\gamma} = \gamma|_{B_\Gamma}$ is a (possibly zero) measure on $X^6$ and $\gamma_i = \mathrm{Pr}_i \tilde{\gamma}$ are measures on $X$. If $\gamma(B_\Gamma) >0$ then full measure sets for $\gamma_i$ are pairwise disjoint. In this case measures $\delta_- = {1 \over 3}(\gamma_1 + \gamma_4 + \gamma_6)$ and $\delta_+ = {1 \over 3}(\gamma_2 + \gamma_3 + \gamma_5)$ are distinct and have the same projections onto the axes and diagonal $l_{x + y}$.
\begin{lemma}
$\delta_- = {1 \over 3}(\gamma_1 + \gamma_4 + \gamma_6)$ and $\delta_+ = {1 \over 3}(\gamma_2 + \gamma_3 + \gamma_5)$ have the same projections onto the axes $x$, $y$ and the line $l_{x+y}$.
\end{lemma}
\begin{proof}
The functions $\Pr_x \circ \Pr_1$ and $\Pr_x \circ \Pr_2$, $\Pr_x \circ \Pr_3$ and $\Pr_x \circ \Pr_4$, $\Pr_x \circ \Pr_5$ and $\Pr_x \circ \Pr_6$, coincide on $N$. So the images of $\pi$ under this projections coincide. That means $\Pr_x(\gamma_1) = \Pr_x(\gamma_2)$, $\Pr_x(\gamma_3) = \Pr_x(\gamma_4)$, $\Pr_x(\gamma_5) = \Pr_x(\gamma_6)$.
Analogously
$\Pr_y(\gamma_1) = \Pr_y(\gamma_5)$,
$\Pr_y(\gamma_2) = \Pr_y(\gamma_4)$,
$\Pr_y(\gamma_3) = \Pr_y(\gamma_6)$ and
$\Pr_{x+y}(\gamma_1) = \Pr_{x+y}(\gamma_3)$,
$\Pr_{x+y}(\gamma_2) = \Pr_{x+y}(\gamma_6)$,
$\Pr_{x+y}(\gamma_4) = \Pr_{x+y}(\gamma_5)$.
\end{proof}
Also $\delta_- \le \pi$ and $\delta_+ \le \pi$ since $\gamma_i =\mathrm{Pr}_i \tilde{\gamma} \le \mathrm{Pr}_i \gamma \le \pi$.
Hence $\pi_1 = \pi + \delta_+ - \delta_-$ and $\pi_2 = \pi - \delta_+ + \delta_-$ are nonnegative measures and have the same projections as $\pi$. So $\pi = {1 \over 2}(\pi_1 + \pi_2)$ is not an extreme point.
That means that for any $\Gamma \in N$ the measure of $B_\Gamma$ with respect to $\gamma$ is 0. Hence $\gamma(N) = 0$ which contradicts the assumption.
Thus we get that there exists a set $S$ with $\pi(S) = 1$ such that $S$ does not contain the sets of the type
\begin{align*}
\Bigl\{ & (x_1, y_2), \ (x_1, y_3), \ (x_2, y_1), \ (x_2, y_3), \ (x_3, y_2), \ (x_3,y_1) , \ \ x_1 < x_2 < x_3, \ y_1 < y_2 < y_3,
\\& x_1 + y_2 = x_2 + y_1, \ x_1 + y_3 = x_3 + y_1,\ x_2 + y_3 = x_3 + y_2
\Bigr\}.
\end{align*}
Let us show that $S$ has Lebesgue measure zero. Assuming the contrary, let us apply the Lebesgue's density theorem.
According to this theorem for almost all $(x, y) \in S$ and every $\varepsilon>0$ there exists a $r$-neighborhood $U$ of $(x, y)$
such that $\lambda(U \cap S) > (1 - \varepsilon) \lambda(U)$.
On the other hand, for all $\alpha$ and $\beta$ the tuple of points
\begin{align*} \Bigl\{(x+\alpha, y+\beta), {}& (x+\alpha, y+\frac{r}{10}+\beta), (x+\frac{r}{10}+\alpha, y+\beta),
(x+\frac{r}{10}+\alpha, y+\frac{2r}{10}+\beta), \\& (x+\frac{2r}{10}+\alpha, y+\frac{r}{10}+\beta), (x+\frac{2r}{10}+\alpha, y+\frac{2r}{10}+\beta)
\Bigr\}
\end{align*}
belongs to $M$. Hence at least one of these points does not belong to $S$.
If $0 \le \alpha, \beta \le \frac{r}{10}$, all these points belong to the $r$-neighborhood of $(x, y)$,
hence the measure of the set $U \setminus S)$ is at least $\frac{r^2}{100}$.
Choosing $\varepsilon < \frac{1}{100\pi}$ one gets a contradiction with the Lebesgue's density theorem.
\end{proof}
\begin{remark}
\cref{conj:hausdorff} says that there exist extreme measures with Hausdorff dimension less than $2$.
Numerical experiments reveal certain empirical evidence of this. Nevetheless, we were not able to verify this conjecture. In general, it is not true that sets which do not contain given configurations of points have dimension strictly less than the ambient space (see \cite{Maga, Mathe}).
An example of a low-dimensional solution is given in \cite[Theorem 4.6]{dim-ger-nen}.
\end{remark}
| {
"timestamp": "2020-02-25T02:10:29",
"yymm": "1809",
"arxiv_id": "1809.08554",
"language": "en",
"url": "https://arxiv.org/abs/1809.08554",
"abstract": "We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to addition of constants).",
"subjects": "Optimization and Control (math.OC)",
"title": "An explicit solution for a multimarginal mass transportation problem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795106521339,
"lm_q2_score": 0.7185943985973773,
"lm_q1q2_score": 0.7092379478850038
} |
https://arxiv.org/abs/1607.00854 | Lecture Notes on the ARV Algorithm for Sparsest Cut | One of the landmarks in approximation algorithms is the $O(\sqrt{\log n})$-approximation algorithm for the Uniform Sparsest Cut problem by Arora, Rao and Vazirani from 2004. The algorithm is based on a semidefinite program that finds an embedding of the nodes respecting the triangle inequality. Their core argument shows that a random hyperplane approach will find two large sets of $\Theta(n)$ many nodes each that have a distance of $\Theta(1/\sqrt{\log n})$ to each other if measured in terms of $\|\cdot \|_2^2$.Here we give a detailed set of lecture notes describing the algorithm. For the proof of the Structure Theorem we use a cleaner argument based on expected maxima over $k$-neighborhoods that significantly simplifies the analysis. | \section{Introduction}
Let $G = (V,E)$ be a complete, undirected graph on $|V| = n$ nodes and let $c : E \to \setR_{\geq 0}$
be a cost function on the edges. For a subset $S \subseteq V$ of nodes, let $\delta(S) := \{ \{ i,j\} \in E \mid |\{ i,j\} \cap S| = 1\}$ be the induced \emph{cut}. We abbreviate $c(\delta(S)) := \sum_{e \in \delta(S)} c(e)$ as the
cost of the cut.
The \emph{(Uniform) Sparsest Cut} problem is then to find the cut that minimizes the \emph{cost-over-separated-pairs} ratio:
\[
\min \left\{ \frac{c(\delta(U))}{|U| \cdot |V \setminus U|} \mid \emptyset \subset U \subset V \right\}.
\]
There is also a non-uniform version of the problem where each pair $i,j \in V$ has an associated
\emph{demand} $d(i,j) \geq 0$ and one aims for the cut minimizing the ratio $c(\delta(S)) / d(\delta(S))$.
We will now see the celebrated algorithm by Arora, Rao and Vazirani~\cite{DBLP:conf/stoc/AroraRV04}
that finds a $O(\sqrt{\log n})$-approximation in polynomial time.
For the algorithm we will not try to optimize any constant.
To fix some notation, we will denote any vector in bold font, like $\bm{v}_i \in \setR^m$.
If we write $i \sim V$, then we mean that $i$ is a uniform random node from $V$.
We denote $N(0,1)$ as the \emph{$1$-dimensional Gaussian distribution} with mean 0 and variance 1.
In particular, a random variable $g \sim N(0,1)$ has \emph{density} $\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$.
If we write $\bm{g} \sim N^m(0,1)$, then we mean that $\bm{g}$ is an \emph{$m$-dimensional Gaussian}.
Recall that the vector $\bm{g} = (g_1,\ldots,g_m)$ can be generated by sampling each coordinate independently
with $g_i \sim N(0,1)$. In reverse, for any pair of orthonormal vectors $\bm{u},\bm{v} \in \setR^m$,
the inner products $\left<\bm{g},\bm{u}\right>,\left<\bm{g},\bm{v}\right>$ are independently distributed from $N(0,1)$.
\section{A semidefinite program}
Sparsest Cut is an unusual problem in the sense that it minimizes the ratio of two functions.
Let us assume for the sake of simplicity that we guessed the cost $C^*$ and the size $S^*$ of an optimum cut,
say with $S^* \leq \frac{n}{2}$.
Then we define a \emph{semidefinite programming relaxation}
\begin{eqnarray*}
\begin{array}{rclll}
\displaystyle \sum_{i \in V} \|\bm{v}_i\|_2^2 &=& S^* & & (I) \\
\displaystyle \sum_{\{i,j\} \in {V \choose 2}} \|\bm{v}_i - \bm{v}_j\|_2^2 &=& S^* \cdot (n-S^*) & & (II) \\
\displaystyle \sum_{\{i,j\} \in E} c_{ij}\|\bm{v}_i - \bm{v}_j\|_2^2 &=& C^* & & (III) \\
\displaystyle \| \bm{v}_i - \bm{v}_j\|_2^2 &\leq& \|\bm{v}_i - \bm{v}_k\|_2^2 + \|\bm{v}_k - \bm{v}_j\|_2^2 & \forall i,j,k \in V \cup \{0\} & (IV) \\
\displaystyle \|\bm{v}_i\|_2^2 &\leq& 1 & \forall i \in V & (V) \\
\displaystyle \|\bm{v}_{0}\|_2^2 &=& 0 & & (VI)
\end{array}
\end{eqnarray*}
where we use an artificial index $0$ with $\bm{v}_0 := \bm{0}$, so that the triangle inequality
also holds for the origin.
\begin{lemma}
If there is cut $U^* \subseteq V$ of cost $C^*$ and size $S^*$, then the above SDP has a solution.
\end{lemma}
\begin{proof}
One could choose 1-dimensional vectors by defining
\[
\bm{v}_i := \begin{cases} 1 & \textrm{if }i \in U^* \\ 0 & \textrm{if }i \notin U^*. \end{cases}
\]
Then the only non-trivial case is verifying the \emph{triangle inequalities}
in $(IV)$. These are satisfied by our choice of $\bm{v}_i \in \{ 0,1\}$
since if $\|\bm{v}_i - \bm{v}_j\|_2^2 = 1$, then $i$ and $j$ have to be on different sides of the cut $U^*$ and
any node $k$ has to be either not on the side of $i$ or not on the side of $j$.
\end{proof}
We can solve the semi-definite program (SDP) in polynomial time~\cite{GroetschelLovaszSchrijver88}; let $\{\bm{v}_i\}_{i \in V} \subseteq \setR^m$
be the solution.
Due to the triangle inequalities (IV) we can define a \emph{metric} $d : V \times V \to \setR_{\geq 0}$
by setting $d(i,j) := \|\bm{v}_i - \bm{v}_j\|_2^2$. Note that while $\|\cdot \|_2$ is always a metric, $\|\cdot \|_2^2$ is
not a metric on all points sets. For sets of nodes $A,B \subseteq V$
we define $d(i,A) := \min\{ d(i,j) : j \in A\}$ and $d(A,B) := \min\{ d(i,j) : i \in A, j \in B\}$.
\section{A ball rounding scheme}
Given that family $\{\bm{v}_i\}_{i \in V}$ of SDP vectors, there are several natural rounding
procedures that would come to mind. For example one could try the \emph{hyperplane rounding}
that Goemans and Williamson~{\cite{DBLP:conf/stoc/GoemansW94} have used for MaxCut.
The natural algorithm for Sparsest Cut
would be to take a random Gaussian $\bm{g} \sim N^m(0,1)$ and set $U := \{ i \in V \mid \left<\bm{g},\bm{v}_i\right> \geq 0\}$. Assume for the
sake of simplicity that we are in the \emph{balanced} case of Sparsest Cut
with $S^* = \Theta(n)$. Then an edge $(i,j) \in E$ has a contribution to the objective function of $\Theta(d(i,j) / n^2)$. On the other hand, the probability that $(i,j)$ is separated is roughly proportional to the
Euclidean distance $\|\bm{v}_i - \bm{v}_j\|_2$ and even if the hyperplane generates perfectly balanced cuts,
the expected contribution of an edge $(i,j)$ to the hyperplane cut would be $\Theta(\sqrt{d(i,j)} / n^2)$.
In other words, short edges would be separated far too likely.
The second best idea that one might have, would be to select a node $i \in V$ and take a random cut $U := \{ j \in V \mid d(i,j) \leq r\}$ where $r \sim [0,1]$. Now every edge is cut with a probability not exceeding $d(i,j)$.
On the other hand, this argument seems to not give any guarantee on the size of $U$ and $V \setminus U$,
hence the objective function can be arbitrarily bad again.
But a slight fix of this rounding argument can work. We only need a large ``core'' of nodes
so that the remaining nodes still have a decently large distance to it.
\begin{lemma} \label{lem:LargeCoreRounding}
Suppose we have a set of nodes $A \subseteq V$ with $|A| \geq \alpha n$ and $\sum_{i \in V} d(i,A) \geq \beta \cdot S^*$.
Then the best cut of the form $\{ i \in V \mid d(i,A) \leq r\}$ is a $\frac{2}{\alpha \beta}$-approximation.
\end{lemma}
\begin{proof}
Recall that the optimum value of the objective function is $\frac{C^*}{S^* \cdot (n-S^*)}$.
Suppose we sample $r \sim [0,1]$ and take $U := U(r) := \{ i \in V \mid d(i,A) \leq r\}$ as a random cut.
\begin{center}
\psset{unit=1.4cm}
\begin{pspicture}(-2,-1.5)(2,1.5)
\psellipse[fillcolor=lightgray,fillstyle=solid,linestyle=dashed](0,0)(1,1.5)
\psellipse[fillcolor=gray,fillstyle=solid](0,0)(0.5,1)
\pnode(0.5,0){A} \pnode(1.0,0){B}
\ncline{<->}{A}{B} \naput{$r$}
\rput[c](0,0){$A$}
\rput[l](0.8,1){$U := U(r)$}
\end{pspicture}
\end{center}
Then for an edge $(i,j) \in E$, say with $d(i,A) \leq d(j,A)$, we have
\[
\Pr_{r \sim [0,1]}[(i,j) \in \delta(U)] = \Pr_{r \sim [0,1]}[ d(i,A) \leq r \leq d(j,A) ]
\stackrel{\textrm{triangle inequality}}{\leq} d(i,j).
\]
Hence the expected cost of the cut $U$ is
\[
\mathop{\mathbb{E}}_{r \sim [0,1]}[c(\delta(U))] \leq \sum_{(i,j) \in E} d(i,j) = C^*.
\]
Note that in any case $|U| \geq |A| \geq \alpha n$. We know that
$d(i,A) \leq 2$ for all $i \in V$ and hence $\Pr_{r \sim [0,1]}[i \notin U] = \Pr_{r \sim [0,1]}[r < d(i,A)] \geq \frac{1}{2}d(i,A)$.
Hence
\[
\mathop{\mathbb{E}}_{r \sim [0,1]}[|V \setminus U|] \geq \frac{1}{2} \sum_{i \in V} d(i,A) \geq \frac{\beta}{2} \cdot S^*.
\]
Then $\mathop{\mathbb{E}}_{r \sim [0,1]}[|U| \cdot |V \setminus U|] \geq \frac{\alpha \beta}{2} \cdot S^* \cdot n$.
In other words, the random cut seems to have the right expected nominator and denominator to satisfy the claim.
But this is \emph{not} enough to argue that their ratio satisfies $\mathop{\mathbb{E}}_{r \sim [0,1]}[\frac{c(\delta(U))}{|U| \cdot |V \setminus U|}] \leq \frac{2}{\alpha \beta} \cdot \frac{C^*}{S^* \cdot (n-S^*)}$. The following insight comes to rescue:
\begin{quote}
{\bf Fact.} Let $\bm{a},\bm{b} \in \setR_{\geq 0}^m$ be non-negative numbers and $\pazocal{D}$ be a distribution over
indices in $[m]$. Then
\[
\min_{i\in \{ 1,\ldots,m\}} \left\{ \frac{\bm{a}_i}{\bm{b}_i} \right\} \leq \frac{\mathop{\mathbb{E}}_{i \sim \pazocal{D}}[\bm{a}_i]}{\mathop{\mathbb{E}}_{i \sim \pazocal{D}}[\bm{b}_i]}.
\]
\end{quote}
Now, this fact implies that \emph{best} choice of $U$ (over all $r \in [0,1]$) will indeed satisfy the claim and the
lemma is proven.
\end{proof}
We should also remark that if $A$ is given, we can find the cut $U$ in polynomial time
as we only need to try out at most $n^2$ many values of $r$.
\section{The case of heavy clusters}
Let $B(i,r) := \{ j \in V \mid d(i,j) \leq r\}$ be the ``ball'' of radius $r$ around node $i$.
A slight annoyance of the ARV algorithm is that it requires a case split. If we can find a
\emph{cluster center} $i^* \in V$,
then we can use Lemma~\ref{lem:LargeCoreRounding} to get a constant factor approximation
by just taking a ball around the center $i^*$.
\begin{lemma}
Suppose there is a node $i^* \in V \cup \{0\}$ with $|B(i^*,\frac{1}{8} \cdot \frac{S^*}{n})| \geq \frac{n}{4}$. Then in polynomial time
one can find a cut that gives a $O(1)$-approximation.
\end{lemma}
\begin{proof}
We set $A := B(i^*, \frac{1}{8} \cdot \frac{S^*}{n})$.
Then by assumption $|A| \geq \frac{n}{4}$.
Moreover, bounding the average distance of pairs of nodes from above and from below gives
\[
\frac{1}{2} \cdot \frac{S^*}{n} \leq \frac{S^*}{n} \cdot \underbrace{\frac{n-S^*}{n}}_{\geq 1/2} \stackrel{(II)}{\leq} \mathop{\mathbb{E}}_{i,j \sim V}[d(i,j)] \stackrel{\textrm{triangle ineq.}}{\leq} 2\cdot \mathop{\mathbb{E}}_{i \sim V} \Big(d(i,A)+\frac{1}{8} \cdot \frac{S^*}{n}\Big).
\]
This can be rearranged to $\mathop{\mathbb{E}}_{i \sim V}[d(i,A)] \geq \frac{1}{8} \cdot \frac{S^*}{n}$.
We obtain a $64$-approximation by applying Lemma~\ref{lem:LargeCoreRounding}.
\end{proof}
\section{An algorithm for the main case}
From now on we make the assumption that no cluster exists:
\[
\left|B\Big(i,\frac{1}{8} \cdot \frac{S^*}{n}\Big)\right| < \frac{n}{4} \quad \forall i \in V \cup \{ 0\}.
\]
We will prove that in this case, there are sets $L,R \subseteq V$ of size $|L|,|R| \geq \Omega(n)$
with $d(L,R) \geq \Delta \cdot \frac{S^*}{n}$ for $\Delta := \Theta(1/\sqrt{\log n})$. Then choosing Lemma~\ref{lem:LargeCoreRounding} with $A := L$
will give a $O(\frac{1}{\Delta})$-approximation.
Before we start proving this, we want to further simplify the situation.
Note that by $(I)$ we have $\mathop{\mathbb{E}}_{i \sim V}[d(i,\bm{0})] = \frac{S^*}{n}$, and hence at most half the nodes can have a distance of
more than $2 \cdot \frac{S^*}{n}$ to $\bm{0}$. Moreover we have $|B(\bm{0},\frac{1}{8} \cdot \frac{S^*}{n})| \leq \frac{n}{4}$.
Then we only loose a constant factor if we delete
those nodes and assume that $\frac{1}{8} \cdot \frac{S^*}{n} \leq d(i,\bm{0}) \leq 2 \cdot \frac{S^*}{n}$ for all remaining nodes.
Next we scale the vectors $\bm{v}_i$ by a factor
of $\sqrt{n / S^*}$, which scales the distances $d(i,j)$ by a factor of $\frac{n}{S^*}$.
After this transformation it suffices to prove the following structure theorem:
\begin{theorem}[ARV Structure Theorem]
Given any set of $|V| = n$ vectors $\{\bm{v}_i\}_{i \in V} \subseteq \setR^m$ with $\frac{1}{8} \leq \|\bm{v}_i\|_2^2 \leq 2$ and
$|B(i,\frac{1}{8})| \leq \frac{3}{4}n$ for all $i \in V$ that satisfy the triangle inequalities
\[
\|\bm{v}_i - \bm{v}_j\|_2^2 \leq \|\bm{v}_i - \bm{v}_k\|_2^2 + \|\bm{v}_k - \bm{v}_j\|_2^2 \quad \forall i,j,k \in V.
\]
Then there is a polynomial time algorithm that with constant
probability finds sets $L,R \subseteq V$ of size $|L|,|R| \geq \Omega(n)$ with $d(L,R) \geq \Delta$
for $\Delta := \Theta(1/\sqrt{\log n})$.
\end{theorem}
From now on, we complete ignore the cost function and only use the properties given in
the Structure Theorem.
Such sets $L$ and $R$ with $d(L,R) \geq \Delta$ are called \emph{$\Delta$-separated}.
Let $c'>0$ be a small enough constants.
The algorithm to produce the $\Delta$-separated sets is as follows:
\begin{center}
\psframebox{%
\begin{minipage}{15cm}
\noindent {\bf Well-separated sets algorithm} \vspace{1mm} \hrule \vspace{2mm}
\noindent {\bf Input: } Vectors $\{\bm{v}_i\}_{i \in V} \subseteq \setR^m$ satisfying the triangle inequality with $\frac{1}{8} \leq \|\bm{v}_i\|_2^2 \leq 2$ and $|B(i,\frac{1}{8})| \leq \frac{3}{4}n$ for all $i \in V \cup \{0\}$. \vspace{0.5mm} \\
\noindent {\bf Output: } Either $\Delta$-separated sets $L',R'$ of size $|L'|,|R'| \geq \frac{c'}{2}n$ or $\texttt{FAIL}$ \vspace{1mm} \hrule \vspace{1mm}
\begin{enumerate*}
\item[(1)] Pick a random Gaussian $\bm{g} \sim N^m(0,1)$
\item[(2)] Set $L := \{ i \in V \mid \left<\bm{v}_i,\bm{g}\right> \leq -1\}$ and $R := \{ i \in V \mid \left<\bm{v}_i,\bm{g}\right> \geq 1\}$
\item[(3)] If either $|L| \leq c'n$ or $|R| \leq c'n$ then $\texttt{FAIL}$
\item[(4)] Compute any inclusion-wise maximal matching $M(\bm{g}) \subseteq \{ (i,j) \in L \times R \mid d(i,j) \leq \Delta\}$
\item[(5)] If $|M(\bm{g})| > \frac{c'}{2}n$ then $\texttt{FAIL}$
\item[(6)] Return $L' := \{ i \in L \mid i\textrm{ not covered by }M(\bm{g})\}$ and $R' := \{ i \in V \mid i \textrm{ not covered by } M(\bm{g})\}$
\end{enumerate*}
\end{minipage}
}
\end{center}
Observe that any pair $i \in L'$ and $j \in R'$ that remains will have $d(i,j) > \Delta$
as otherwise the matching $M(\bm{g})$ would not have been maximal. Also, if the algorithm
reaches (6), then $\min\{ |L'|,|R'| \} \geq \frac{c'}{2}n$.
\begin{center}
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\end{pspicture}
\end{center}
The first step is to argue that with constant probability both $L$ and $R$ have size at least $\Omega(n)$.
\begin{lemma}
There is an absolute constant $c' >0$ so that $\Pr_{\bm{g} \sim N^m(0,1)}[\min\{ |L|,|R|\} \geq c'n] \geq c'$.
\end{lemma}
\begin{proof}
We will prove that $\mathop{\mathbb{E}}[|L| \cdot |R|] \geq \Omega(n^2)$ which then implies the claim.
Fix any $i \in V$ and select one of the at least $\frac{1}{4}n$ nodes $j$ with $d(i,j) \geq \frac{1}{8}$.
Let $\bm{w}$ be the orthogonal projection of $\bm{v}_j$ on $\bm{v}_i$, see figure below.
Let $\alpha \in [0,\frac{\pi}{2}]$ be the angle between $\bm{v}_i-\bm{v}_j$ and $\bm{w}$ and let $\beta \in [0,\frac{\pi}{2}]$ be the angle between $\bm{v}_j$ and $\bm{w}$.
Due to the triangle inequalities, the angle spanned by the points $\bm{0}$, $\bm{v}_i$ and $\bm{v}_j$ is non-obtuse
and $\alpha+\beta \leq \frac{\pi}{2}$.
Then we have either $\alpha \leq \frac{\pi}{4}$ and
\[
\|\bm{w}\|_2 = \underbrace{\cos(\alpha)}_{\geq 1/\sqrt{2}} \cdot \underbrace{\|\bm{v}_i - \bm{v}_j\|_2}_{\geq 1/\sqrt{8}} \geq \frac{1}{4}
\]
or otherwise we have $\beta \leq \frac{\pi}{4}$ and
\[
\|\bm{w}\|_2 = \underbrace{\cos(\beta)}_{\geq 1/\sqrt{2}} \cdot \underbrace{\|\bm{v}_j\|_2}_{\geq 1/\sqrt{8}} \geq \frac{1}{4}
\]
Either way, $\|\bm{w}\|_2 \geq \frac{1}{4}$.
Since $\bm{v}_i \perp \bm{w}$, the inner products $\left<\bm{g},\bm{v}_i\right>$ and
$\left<\bm{g},\bm{w}\right>$ are independent random variables and we can estimate
\begin{eqnarray*}
\Pr[i \in L\textrm{ and }j \in R] &\geq& \Pr[-2 \leq \left<\bm{g},\bm{v}_i\right> \leq -1\textrm{ and }\left<\bm{g},\bm{w}\right> \geq 3] \\
&=& \Pr\Big[ -\underbrace{\frac{1}{\|\bm{v}_i\|_2}}_{\in [\frac{1}{\sqrt{8}},\frac{1}{\sqrt{2}} ]} \leq \left<\bm{g},\frac{\bm{v}_i}{\|\bm{v}_i\|_2}\right> \leq -\frac{2}{\|\bm{v}_i\|_2}\Big] \cdot \Pr\Big[\left<\bm{g},\frac{\bm{w}}{\|\bm{w}\|_2}\right> \geq \underbrace{\frac{3}{\|\bm{w}\|_2}}_{\leq 12}\Big] > 0
\end{eqnarray*}
which is some tiny, yet absolute constant.
Note that in case the latter event happens, then indeed
\[
\left<\bm{g},\bm{v}_j\right> = \underbrace{\left<\bm{g},\bm{v}_i\right>}_{\geq -2} \cdot \underbrace{\frac{\left<\bm{v}_i,\bm{v}_j\right>}{\|\bm{v}_i\|_2^2}}_{\in [0,1]} + \underbrace{\left<\bm{g},\bm{w}\right>}_{\geq 3} \geq 1.
\]
\begin{center}
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\cnode*(1,0.7){2.5pt}{vj} \nput[labelsep=2pt]{120}{vj}{$\bm{v}_j$}
\ncline{->}{origin}{vi}
\ncline{->}{origin}{vj}
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\nput[labelsep=12pt]{165}{vj}{$\alpha$}
\nput[labelsep=12pt]{-160}{vj}{$\beta$}
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\pnode(-1.5,0){R2} \ncline{<->}{origin}{R2}\naput[labelsep=2pt,npos=0.6]{$\sqrt{2}$}
\end{pspicture}
\end{center}
\end{proof}
This implies that with a constant probability, the algorithm does not fail in (3).
The main technical part lies in proving that $\mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[|M(\bm{g})|] \leq cn$
where we can make the constant $c$ as small as we want, at the expense of a smaller value of $\Delta$.
If we choose $c := (\frac{c'}{2})^2$, then $\Pr[|M(\bm{g})| > \frac{c'}{2}] \leq \frac{c'}{2}$
and the success probability of the algorithm is at least $\frac{c'}{2}$.
\section{The proof of the Structure Theorem}
The following geometric theorem by Arora, Rao and Vazirani is the heart of their
$O(\sqrt{\log n})$-approximation for Sparsest Cut. To be precise, the original ARV result~\cite{DBLP:conf/stoc/AroraRV04}
only showed this theorem for $\Delta = \Theta((\log n)^{-2/3})$ and needed a lot of extra work
to get the $O(\sqrt{\log n})$-approximation. The claim as it is stated here was first proven by Lee~\cite{DBLP:conf/soda/Lee05}.
For an edge set $E'$, let $\beta(E')$ be the size of the maximum matching.
\begin{theorem}[\cite{DBLP:conf/stoc/AroraRV04,DBLP:conf/soda/Lee05}] \label{thm:StructureTheoremBoundOnMatchingSize}
For any constant $c>0$ there is a choice of $\Delta := \Theta_c(1 / \sqrt{\log n})$ so that the following holds:
Let $\{ \bm{v}_i\}_{i \in V} \subseteq \setR^m$ be a set of $|V| = n$ vectors satisfying the triangle inequality
\[
\|\bm{v}_i - \bm{v}_j\|_2^2 \leq \|\bm{v}_i - \bm{v}_k\|_2^2 + \|\bm{v}_k - \bm{v}_j\|_2^2 \quad \forall i,j,k \in V.
\]
For a vector $\bm{g} \in \setR^m$ define
\[
E(\bm{g}) := \left\{ (i,j) \in V \times V \mid \left<\bm{v}_j - \bm{v}_i,\bm{g}\right> \geq 2\textrm{ and } \|\bm{v}_i - \bm{v}_j\|_2^2 \leq \Delta \right\}.
\]
Then $\mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[\beta(E(\bm{g}))] \leq cn$.
\end{theorem}
Here we think of $E(\bm{g})$ as directed edges.
Let $M(\bm{g})$ be a maximum matching attaining $\beta(E(\bm{g}))$.
We will assume the existence of such a matching $M(\bm{g})$ and lead this to a contradiction.
By inducing on a subgraph and reducing the constant $c$ one can even assume that for \emph{every} node
the probability of having an outgoing edge is at least $c$ and the same is true for ingoing edges.
First, there is no harm in assuming that $M(\bm{g})$ has the reverse edges of $M(-\bm{g})$,
which implies that each node has an outgoing edge with the same probability as it has an incoming edge.
\begin{lemma}
Assume that Theorem~\ref{thm:StructureTheoremBoundOnMatchingSize} is false for vectors $\{ \bm{v}_i\}_{i\in V} \subseteq \setR^m$.
Then there is a subset $V' \subseteq V$ of size $|V'| \geq cn$ and a matching $M'(\bm{g}) \subseteq M(\bm{g})$ on $V'$ so that
every node $i \in V'$ has an outgoing edge in $M'(\bm{g})$ with probability at least $\frac{c}{8}$ and an ingoing edge with probability at least $\frac{c}{8}$.
\end{lemma}
\begin{proof}
For each node $i \in V$ define $p(i) := \Pr_{\bm{g} \sim N^m(0,1)}[i\textrm{ has outgoing edge in }M(\bm{g})]$.
If there is a node $i$ with $p(i) \leq \frac{c}{8}$, then we imagine to delete the node from the graph and remove
from $M(\bm{g})$ any edge containing node $i$. Note that this decreases the expected size of the matching
by at most $2 \cdot \frac{c}{8}$.
We continue this procedure until no such node exists anymore. Let $V'$ be the remaining set of nodes
with $M'(\bm{g}) := M(\bm{g}) \cap (V' \times V')$. Then $\mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[|M'(\bm{g})|] \geq \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[|M(\bm{g})|] - n \cdot \frac{c}{4} \geq \frac{c}{2}n$.
Then there must be at least $|V'| \geq cn$ many nodes left.
\end{proof}
After changing the constants and adapting the value of $n$, we assume to have $n$ nodes and
for every node $i \in V$, the matching $M(\bm{g})$ has an outgoing and an incoming edge with
probability at least $c$.
We call an edge $(i,j)$ \emph{$\Delta$-short} if $d(i,j) \leq \Delta$.
For a node $i \in V$, let $\Gamma(i) := \{ j \in V \mid d(i,j) \leq \Delta\}$ be the \emph{neighborhood} of $i$
with respect to the graph of $\Delta$-short edges. Moreover, let $\Gamma_k(i) := \Gamma_{k-1}(\Gamma(i))$
be the nodes that can be reached from $i$ via at most $k$ many $\Delta$-short edges.
\begin{lemma} \label{lem:DistanceOfkNeighborhood}
For any $k \in \setZ_{\geq 0}$ and $i' \in \Gamma_k(i)$ one has $\|\bm{v}_i - \bm{v}_{i'}\|_2 \leq \sqrt{k\Delta}$.
\end{lemma}
\begin{proof}
We have $\|\bm{v}_i - \bm{v}_{i'}\|_2^2 = d(i,i') \leq k \cdot \Delta$ by the SDP triangle inequality. Taking square roots
gives the claim.
\end{proof}
At the heart of the arguments lies the fact that the value of Lipschitz functions is well concentrated.
Recall that a function $F : \setR^m \to \setR$ is called \emph{$L$-Lipschitz}, if
$|F(\bm{x}) - F(\bm{y})| \leq L \cdot \|\bm{x} - \bm{y}\|_2$ for all $\bm{x},\bm{y} \in \setR^m$.
\begin{lemma}[Concentration for Lipschitz Functions (Sudakov-Tsirelson, Borrell)] \label{lem:ConcentrationForLipschitzFunctions}
Let $F : \setR^m \to \setR$ be an $L$-Lipschitz function with
Gaussian mean $\mu := \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[F(\bm{g})]$. Then $\Pr_{\bm{g} \sim N^m(0,1)}[|F(\bm{g}) - \mu| \geq \alpha] \leq 2e^{-\frac{1}{2} \cdot (\alpha/L)^2}$ for any $\alpha \geq 0$.
\end{lemma}
We define a function
\[
F_{i,k}(\bm{g}) := \max\{ \left<\bm{g},\bm{v}_i - \bm{v}_j\right> \mid j \in \Gamma_k(i) \}.
\]
In other words, $F_{i,k}(\bm{g})$ gives the maximum inner product $\left<\bm{g},\bm{v}_i-\bm{v}_j\right>$ over all nodes $j \in V$ that are
within $k$ many $\Delta$-short edges of node $i$.
Note that $F_{i,k}(\bm{g}) \geq \left<\bm{g},\bm{v}_i-\bm{v}_i\right> = 0$ for all $\bm{g} \in \setR^m$ as $i \in \Gamma_k(i)$.
\begin{lemma} \label{lem:LipschitzPropertyOfFik}
The function $F_{i,k} : \setR^m \to \setR$ is $\sqrt{k \cdot \Delta}$-Lipschitz.
\end{lemma}
\begin{proof}
Fix $\bm{g},\bm{g}' \in \setR^m$ and assume for the sake of symmetry that $F(\bm{g}) \geq F(\bm{g}')$.
Let $j,j' \in \Gamma_k(i)$ be the nodes attaining $F(\bm{g})$ and $F(\bm{g}')$, resp.
Then
\[
|F(\bm{g}) - F(\bm{g}')| = \left<\bm{g},\bm{v}_{i}-\bm{v}_{j}\right> - \left<\bm{g}',\bm{v}_{i} - \bm{v}_{j'}\right>
\leq \left<\bm{g}-\bm{g}',\bm{v}_{i}-\bm{v}_j\right>
\stackrel{\textrm{Cauchy-Schwarz}}{\leq} \|\bm{g} - \bm{g}'\|_2 \cdot \underbrace{\|\bm{v}_{i} - \bm{v}_j\|_2}_{\leq \sqrt{k \cdot \Delta}\textrm{ by Lem.~\ref{lem:DistanceOfkNeighborhood}}}.
\]
Here we used in the first inequality that $j' \in \Gamma_k(i)$ maximizes $\left<\bm{g}',\bm{v}_{i}-\bm{v}_{j'}\right>$.
\end{proof}
Now, let $\mu_{i,k} := \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[F_{i,k}(\bm{g})]$ be the expected maximum inner product over $k$-neighbors
of node $i \in V$.
One useful argument will be a nice relation between expectations of neighbors:
\begin{lemma} \label{lem:BehauviorOfMuKIForNeighbors}
For any node $i \in V$ and $i' \in \Gamma(i)$ and $k \in \setZ_{\geq 0}$ one has $\mu_{i',k+1} \geq \mu_{i,k}$.
\end{lemma}
\begin{proof}
We have
\begin{eqnarray*}
\mu_{i',k+1} &=& \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}\Big[\max \Big\{ \left<\bm{v}_{i'} - \bm{v}_{i},\bm{g}\right> + \left<\bm{v}_i - \bm{v}_{j},\bm{g}\right>\mid j \in \Gamma_{k+1}(i') \Big\}\Big] \\
&\stackrel{\Gamma_{k+1}(i') \supseteq \Gamma_k(i)}{\geq}& \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}\left[\max\left\{ \left<\bm{v}_i-\bm{v}_{j},\bm{g}\right> \mid j \in \Gamma_k(i) \right\} \right] + \underbrace{\mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[\left<\bm{v}_{i'} - \bm{v}_{i},\bm{g}\right>]}_{=0} = \mu_{i,k}.
\end{eqnarray*}
\end{proof}
On the other hand, we can get the following upper bound:
\begin{lemma} \label{lem:UpperBoundOnMuKI}
For any $k \in \setZ_{\geq 0}$ and any $i \in V$ one has $\mu_{i,k} \leq 10\sqrt{\log n} \cdot \sqrt{k \Delta}$.
\end{lemma}
\begin{proof}
For any $j \in \Gamma_k(i)$ we have $\|\bm{v}_i - \bm{v}_j\|_2 \leq \sqrt{k\Delta}$ and generously
\[
\Pr_{\bm{g} \sim N^m(0,1)}\Big[\left<\bm{v}_i-\bm{v}_j,\bm{g}\right> \geq 8\sqrt{\log n} \cdot \sqrt{k\Delta}\Big] \leq \int_{8\sqrt{\log n}}^{\infty} \frac{1}{\sqrt{2\pi}}e^{-x^2/2} dx \leq \frac{1}{2n}.
\]
That means $\Pr_{\bm{g} \sim N^m(0,1)}[F_{i,k}(\bm{g}) \leq 8\sqrt{\log n} \cdot \sqrt{k\Delta}] \geq 1/2$. Again, $F_{i,k}$ is $\sqrt{k\Delta}$-Lipschitz,
hence $\Pr_{\bm{g} \sim N^m(0,1)}[|F_{i,k}(\bm{g}) - \mu_{i,k}| \geq 2\sqrt{k\Delta}] \leq \frac{1}{2}$ and $2 \leq 2\sqrt{\log(n)}$ for $n \geq 2$.
\end{proof}
\subsection{Extend or expand}
The core argument to get to a contradiction is the following:
\begin{lemma} \label{lem:ExpansionOrExtensionLemma}
Let $\Delta>0$, $\delta \in \setR$ and $k \in \{ 0,\ldots,\frac{1}{100} \log(1/c) \cdot \frac{1}{\Delta} \}$ be parameters
and $U \subseteq \{ i \in V \mid \mu_{i,k} \geq \delta \}$ be a set of nodes. Then
\begin{enumerate}
\item[(A)] either there is a subset $U' \subseteq \Gamma(U)$ so that $|U'| \geq \frac{c}{4} \cdot |U|$ and $\mu_{i,k+1} \geq \delta+1$ for all $i \in U'$.
\item[(B)] or the neighborhood $U' := \Gamma(U)$ satisfies $|U'| \geq \frac{4}{c} \cdot |U|$ and
$\mu_{i,k+1} \geq \delta$ for all $i \in U'$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $|\Gamma(U)| \geq \frac{4}{c} \cdot |U|$, then every node in $i \in \Gamma(U)$ has $\mu_{i,k+1} \geq \delta$ by Lemma~\ref{lem:BehauviorOfMuKIForNeighbors}
and we are in case (B).
So suppose that $|\Gamma(U)| < \frac{4}{c} \cdot |U|$. Consider the random matching
\[
\tilde{M}(\bm{g}) := \Big\{ (i,j) \in M(\bm{g}) \mid i \in U\textrm{ and } F_{i,k}(\bm{g}) \geq \delta-\frac{1}{2} \Big\}
\]
that is the restriction of $M(\bm{g})$ to edges that are going out of $U$ and where $F_{i,k}(\bm{g})$ is
large enough.
Note that $\mu_{i,k} = \mathop{\mathbb{E}}_{\bm{g} \sim N^m(0,1)}[F_{i,k}(\bm{g})] \geq \delta$ for all $i \in U$ and
$F_{i,k}$ is $\sqrt{k\Delta}$-Lipschitz. Hence by Lemma~\ref{lem:LipschitzPropertyOfFik} we have
\[
\Pr_{\bm{g} \sim N^m(0,1)}\Big[F_{i,k}(\bm{g}) < \delta - \frac{1}{2}\Big] \leq 2\exp\Big(-\frac{1}{8k\Delta}\Big) < \frac{c}{2} \quad \forall i \in U.
\]
This implies that each node in $U$ will have an outgoing edge in $\tilde{M}(\bm{g})$
with probability at least $\frac{c}{2}$.
Now, define $U' := \{ j \in \Gamma(U) \mid \Pr[j\textrm{ has incoming edge from }\tilde{M}(\bm{g})] \geq \frac{c^2}{16} \}$.
Since $\tilde{M}(\bm{g})$ is a matching we have
\[
\frac{c}{2} \cdot |U| \leq \mathop{\mathbb{E}}[|\tilde{M}(\bm{g})|] \leq \frac{c^2}{16} \cdot \underbrace{|\Gamma(U) \setminus U'|}_{\leq (4/c) \cdot |U|} + |U'| \leq \frac{c}{4} \cdot |U| + |U'|
\]
which implies that $|U'| \geq \frac{c}{4} \cdot |U|$. Now fix a node $j \in U'$.
It remains to argue that $\mu_{j,k+1} \geq \delta+1$ for all $j \in U'$.
First, condition on the event that $\tilde{M}(\bm{g})$ has an edge incoming to $j$.
We denote that edge by $(i(\bm{g}),j) \in \tilde{M}(\bm{g})$
with $i(\bm{g}) \in U$\footnote{Here we write $i(\bm{g})$ to indicate that this node will depend on the choice of $\bm{g}$.}.
We know by the definition of $\tilde{M}(\bm{g})$ that $\left<\bm{v}_{j} - \bm{v}_{i(\bm{g})},\bm{g}\right> \geq 2$. Moreover,
we know that there is a node $h(\bm{g}) \in \Gamma_{k}(i(\bm{g}))$ so that $\left<\bm{v}_{i(\bm{g})} - \bm{v}_{h(\bm{g})},\bm{g} \right> \geq \delta - \frac{1}{2}$.
\begin{center}
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\psellipse[fillcolor=lightgray,fillstyle=solid](0,0)(0.5,1.1)
\psellipse[fillcolor=lightgray,fillstyle=solid](2,0)(0.5,1.6)
\psellipse[fillcolor=gray,fillstyle=solid](2,0)(0.4,0.75)
\pspolygon[linestyle=dashed,linearc=5pt,fillcolor=gray,fillstyle=none](0.6,-0.25)(-1.8,-1.25)(-1.8,0.75)
\cnode*(0,0.75){2.5pt}{v1}
\cnode*(0,0.25){2.5pt}{v2}
\cnode*(0,-0.25){2.5pt}{v3}
\cnode*(0,-0.75){2.5pt}{v4}
\cnode*(2,1.0){2.5pt}{w1}
\cnode*(2,0.5){2.5pt}{w2}
\cnode*(2,0){2.5pt}{w3}
\cnode*(2,-0.5){2.5pt}{w4}
\cnode*(2,-1.0){2.5pt}{w5}
\cnode*(-1.5,-0.75){2.5pt}{u1}
\rput[c](-1.5,0.9){$\Gamma_k(i(\bm{g}))$}
\nput[labelsep=2pt]{80}{u1}{$h(\bm{g})$}
\nput[labelsep=2pt]{180}{v3}{$i(\bm{g})$}
\nput[labelsep=2pt]{0}{w3}{$j$}
\ncline[arrowsize=5pt]{->}{v3}{w3} \naput[labelsep=0pt]{$\tilde{M}(\bm{g})$}
\ncline[arrowsize=5pt]{->}{v4}{w4}
\rput[c](0,1.3){$U$}
\rput[c](2,1.75){$\Gamma(U)$}
\nput[labelsep=8pt]{30}{w2}{$U'$}
\end{pspicture}
\end{center}
Then
$\left<\bm{v}_{j} - \bm{v}_{h(\bm{g})},\bm{g}\right> = \left<\bm{v}_{j} - \bm{v}_{i(\bm{g})},\bm{g}\right> + \left<\bm{v}_{i(\bm{g})} - \bm{v}_{h(\bm{g})},\bm{g}\right> \geq \delta + \frac{3}{2}$.
In other words, $\Pr_{\bm{g} \sim N^m(0,1)}[F_{j,k+1}(\bm{g}) \geq \delta + \frac{3}{2}] \geq \frac{c^2}{16}$.
Again, the function $F_{j,k+1}$ is $\sqrt{(k+1)\Delta}$-Lipschitz and
\[
\Pr_{\bm{g} \sim N^m(0,1)}\Big[|F_{j,k+1}(\bm{g}) - \mu_{j,k+1}| \geq \frac{1}{2}\Big] \stackrel{\textrm{Lem.~\ref{lem:ConcentrationForLipschitzFunctions}}}{\leq} 2\exp\Big(-\frac{1}{8(k+1)\Delta}\Big) < \frac{c^2}{16}
\]
and hence $\mu_{j,k+1} \geq \delta+1$. This shows the claim.
\end{proof}
Now, suppose we run Lemma~\ref{lem:ExpansionOrExtensionLemma} iteratively, starting with $U := V$
and in each iteration we replace the current $U$ by the set $U'$.
We iterate this until the upper bound on $k$ is reached. Note that case (B) cannot happen more often than case (A)
as always $|\Gamma(U)| \leq n$.
Then after being $k = \frac{1}{100\Delta}\log(1/c)$ times in Case (A) and $\ell \in \{ 0,\ldots k\}$ times in Case (B),
we end up with a set $U \subseteq V$ with $|U| \geq n \cdot (\frac{c}{4})^{k-\ell} \geq n \cdot (\frac{c}{4})^{k}$ and
$\mu_{i,2k} \geq \mu_{i,k+\ell} \geq k$ for all $ i \in U$.
On the other hand, $\mu_{i,2k} \leq 10\sqrt{\log n} \cdot \sqrt{2k\Delta}$ by Lemma~\ref{lem:UpperBoundOnMuKI}.
Choosing $\Delta := \Theta_c(\frac{1}{\sqrt{\log n}})$ and $k := \Theta_c(\sqrt{\log n})$
with proper choice of constants, then gives a contradiction.
\paragraph{Acknowledgement.}
The author is very grateful to James R. Lee, Harishchandra Ramadas, Rebecca Hoberg and Alireza Rezaei for helpful discussion and comments.
\bibliographystyle{alpha}
| {
"timestamp": "2016-07-05T02:13:18",
"yymm": "1607",
"arxiv_id": "1607.00854",
"language": "en",
"url": "https://arxiv.org/abs/1607.00854",
"abstract": "One of the landmarks in approximation algorithms is the $O(\\sqrt{\\log n})$-approximation algorithm for the Uniform Sparsest Cut problem by Arora, Rao and Vazirani from 2004. The algorithm is based on a semidefinite program that finds an embedding of the nodes respecting the triangle inequality. Their core argument shows that a random hyperplane approach will find two large sets of $\\Theta(n)$ many nodes each that have a distance of $\\Theta(1/\\sqrt{\\log n})$ to each other if measured in terms of $\\|\\cdot \\|_2^2$.Here we give a detailed set of lecture notes describing the algorithm. For the proof of the Structure Theorem we use a cleaner argument based on expected maxima over $k$-neighborhoods that significantly simplifies the analysis.",
"subjects": "Data Structures and Algorithms (cs.DS); Computational Geometry (cs.CG)",
"title": "Lecture Notes on the ARV Algorithm for Sparsest Cut",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795095031688,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7092379470593639
} |
https://arxiv.org/abs/2012.07221 | Intersecting longest paths in chordal graphs | We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\omega(G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\lceil\frac{\omega(G)}{5}\rceil$. We also consider the analogous question for longest cycles, and show that if $G$ is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most $2\lceil\frac{\omega(G)}{3}\rceil$. | \section{Introduction}
In this paper we consider the paths of maximum length among all paths in a graph. As longest paths represent the most indirect way one could travel through the graph from some place to another, we will adopt the terminology of Kapoor et al.~\cite{KapoorEtal,JobsonEtal} and refer to them as \emph{detours}.
It is not too difficult to convince yourself that any two detours in a connected graph must intersect. In 1966 Gallai asked whether all detours in a connected graph share a common vertex~\cite{Erdosproc}.
While the answer turned out to be negative for graphs in general, this question has led to a large number of positive results for particular graph classes.
In the negative direction, Walther gave a counterexample with 25 vertices soon after the question was posed~\cite{Walther}. The smallest possible counterexample is obtained from the Petersen graph by breaking one vertex into three degree one vertices~\cite{Zamfirescu76,WaltherVoss}. Thomassen showed there are infinitely many planar counterexamples (the `hypotraceable' planar graphs)~\cite{Thomassen76}.
The known counterexamples contain sets of at least seven detours that do not have a common intersection. It remains an open question whether every $k$ detours in any connected graph have a vertex in common, for $k = 3,4,5,6$~\cite{Zamfirescu01,Voss}.
On the other hand, there are a number of classes of connected graphs for which it has been proven that all detours intersect in some vertex. These include trees, cacti~\cite{KlavzarPet}, split graphs~\cite{KlavzarPet}, circular arc graphs~\cite{BalisterEtal,Joos}, outerplanar graphs~\cite{RezendeEtal13,Axenovich}, 2-trees~\cite{RezendeEtal13}, graphs with all non-trivial blocks Hamiltonian~\cite{RezendeEtal13}, partial 2-trees~\cite{ChenEtal17}, connected dually chordal graphs~\cite{JobsonEtal}, cographs~\cite{JobsonEtal} and more~\cite{CerioliLima20,GolanShan18,ChenFu}. The motivating question for the present research is whether chordal graphs (and thus all $k$-trees) can be added to this list~\cite{West}. Recall that a graph is \emph{chordal} if every cycle of four or more vertices contains a chord, or in other words, the only induced cycles are triangles.
Even if there is no single vertex intersecting every detour, we may ask for the smallest \emph{transversal} of vertices that intersects all detours~\cite{RautenbachSereni}. This is the approach we take in our study of chordal graphs, and
the main result of this paper is the following, where $\omega(G)$ is the clique number.
\begin{theorem}
\label{theorem:chordal}
Given a connected chordal graph $G$, there exists a transversal of order at most $4\ceil{\tfrac{\omega(G)}{5}}$ that intersects every detour. Moreover, this transversal induces a clique in $G$.
\end{theorem}
The order of the smallest transversal intersecting all detours in a chordal graph was recently studied by Cerioli et al.~\cite{CerioliEtal20}.
They showed that there is a transversal of order at most $\max\{1,\omega(G)-2\}$.
Theorem~\ref{theorem:chordal} is an improvement on their result for $\omega = 15,19,20,23,24,25$ and $\omega \geq 27$.
The intersection properties of longest cycles in graphs have also been investigated, perhaps to a lesser extent. We also prove the following.
\begin{theorem}
\label{theorem:cycles}
Given a $2$-connected chordal graph $G$, there exists a transversal of order at most $2\ceil{\tfrac{\omega(G)}{3}}$ that intersects every longest cycle of $G$. Moreover, this transversal induces a clique in $G$.
\end{theorem}
For large $\omega$ this improves on recent results of Guti\'errez~\cite{Gutierrez}, who showed that a 2-connected chordal graph $G$ contains a transversal intersecting all longest cycles of order at most $\max\{1,\omega(G)-3\}$.
Thus Theorem~\ref{theorem:cycles} is an improvement for $\omega = 12$ and $\omega \geq 14$.
We also note the following result about the order of a transversal of detours in a planar graph. Cerioli et al.~\cite{CerioliEtal20} showed that, given a connected graph $G$ with treewidth $k$, there exists a transversal intersecting every detour of order at most $k$. Fomin and Thilikos~\cite{FominThil06} showed that a planar graph of order $n$ has treewidth at most $3.182\sqrt{n}$. Together these give the following result.
\begin{theorem}
\label{theorem:planar}
Given a connected planar graph $G$ with $n$ vertices, there exists a transversal intersecting every detour of order at most $3.182\sqrt{n}$.
\end{theorem}
This improves a previous result of Rautenbach and Sereni~\cite{RautenbachSereni}.
\section{Preliminaries}
Given a graph $G$, a \emph{tree decomposition} of $G$ consists of a tree $T$ and a collection of bags $\mathcal{B}$ indexed by the nodes of $T$ such that:
\begin{itemize}
\item the bags of $\mathcal{B}$ contain vertices of $V(G)$,
\item for each $v \in V(G)$, the set of bags containing $v$ correspond to a non-empty connected subtree of $T$,
\item for each edge $vw \in E(G)$ there is at least one bag containing both $v$ and $w$.
\end{itemize}
The width of a tree-decomposition is the order of the largest bag, minus 1. The treewidth of a graph $G$ is the minimum width over all tree decompositions of $G$. Technically, we should distinguish a node of the tree $T$ and the bag of $\mathcal{B}$ indexed by the node, but in practice there is rarely any need, and as such we conflate the two.
The following key result about tree decompositions of chordal graph we shall use extensively:
\begin{lemma}\label{lemma:tdcliques}
If $G$ is a chordal graph, then $G$ has a tree decomposition of minimum width in which every bag corresponds to a clique.
\end{lemma}
This result is well-known, and can be proven multiple ways. A result of Gavril showed that the chordal graphs are equivalent to the \emph{subtree graphs}: the intersection graph of subtrees of a tree. It is reasonably clear that the collection of subtrees corresponding to a chordal graph $G$ can be used to construct a tree decomposition for $G$ with the desired properties; we omit the full proof.
Now we consider the behaviour of the detours in a chordal graph. We start with an extremely straight-forward result which will be helpful later.
\begin{proposition}
\label{prop:extend}
Let $G$ be a connected chordal graph, $X$ a clique in $G$ and $P$ a detour in $G$. If $P$ has either an end vertex in $X$ or contains an edge of $X$, then $P$ must contain every vertex of $X$.
\end{proposition}
\begin{proof}
If $P$ has an end vertex in $X$ or contains an edge of $X$ but $v \in X$ is not in $P$, we could extend $P$ to contain $v$ either by adding an edge from the end vertex to $v$, or by replacing an edge of $P$ in $X$ with a 2-edge path containing $v$. Since $P$ is a detour, this is a contradiction.
\end{proof}
A \emph{clique-cut} in $G$ is a clique that is also a cut-set.
Suppose $P$ is a detour and $X$ is a clique such that $V(P) \cap X \neq \emptyset$.
\begin{itemize}
\item If all the vertices of $V(P)-X$ are in a single component of $G-X$ then we say $P$ \emph{touches} $X$ (or is \emph{touching} with respect to $X$). Alternatively, if $X$ is a clique-cut and there are vertices of $V(P)-X$ in at least two components of $G-X$ then we say that $P$ \emph{crosses} $X$ (or is \emph{crossing} with respect to $X$).
\item If $P$ has both end vertices in the same component of $G-X$ then we say $P$ is \emph{closed} with respect to $X$, and we call the component of $G-X$ containing both end vertices the \emph{home component} of $P$ with respect to $X$. Alternatively, if the end vertices of $P$ are in different components we say $P$ is \emph{open} with respect to $X$.
\item If $|V(P) \cap X| \leq \ceil{\tfrac{\omega(G)}{5}}$ then we say $P$ is \emph{small} with respect to $X$, otherwise it is \emph{large}.
\end{itemize}
Often the clique $X$ will be clear from context, in these cases we will often drop ``with respect to $X$". We call a clique \emph{total} if every detour intersects $X$, otherwise it is \emph{non-total}. When $X$ is total, every detour is either small or large, and either closed touching, closed crossing or open crossing. It is not possible from the definition for a detour to be open and touching.
Recall a \emph{separation} is a partition of the vertex set $(M,X,N)$ such that no edge exists with an end vertex in $M$ and in $N$. Note that if $B$ is a non-leaf bag in a tree decomposition, then $G-B$ is disconnected and $(M,B,N)$ is a separation where the components of $G-B$ are sorted into $M$ and $N$. Given a path $P$, let $f(P,M)$ denote the number of end vertices of the path in $M$ (and define $f(P,N)$ equivalently). Clearly $f(P,M),f(P,N) \in \{0,1,2\}$, and if $P$ has no end vertices in $X$, then $f(P,M)+f(P,N)=2$.
\begin{lemma}
\label{lemma:paste}
Let $(M,X,N)$ be a separation in $G$ where $X$ is a cut-clique. If there exist two detours $P,Q$ in $G$ such that
\begin{enumerate}
\item neither ends in $X$,
\item both intersect both of $M$ and $N$, and
\item $f(P,M)=f(Q,M)$
\end{enumerate}
then $(P \cap X) \cap (Q \cap X) \neq \emptyset$.
\end{lemma}
\begin{proof}
Suppose otherwise for the sake of a contradiction, that is, there exist detours $P$ and $Q$ satisfying the three properties but also $(P \cap X) \cap (Q \cap X) = \emptyset$. Note that since neither $P$ nor $Q$ end in $X$ and since every path has two ends, it follows $f(P,N)=f(Q,N)$ also. We construct a path $P_M$ as follows:
\begin{itemize}
\item Consider the graph $G-N$ and the path $P$ inside this graph. Since $P$ intersects $N$, $G-N$ will not contain all of $P$. It is possible that $P-N$ is a set of subpaths of $P$, and as such the number of end vertices of $P-N$ may not be two. However note that the number of end vertices of the subpaths in $P-N$ that are contained in $M$ is still $f(P,M)$; all other end vertices must be in $X$. Since $P$ enters $N$, there is at least one end vertex in $X$; if $f(P,M) \neq 1$ then there are at least two end vertices in $X$ (as exactly one end vertex only occurs when $P$ starts in $M$ and ends in $N$).
\item Since $X$ is a clique, we can join the subpaths of $P-N$ together using edges of $X$ to create one long path. Call this $P_M$. Note again that $f(P_M,M)=f(P,M)$. If $f(P,M)=0$ then $P_M$ has two end vertices in $X$, if $f(P,M)=1$ then $P_M$ has one end vertex in $X$, and if $f(P,M)=2$ then $P_M$ has no end vertices in $X$ but since $P$ enters $N$ at least one edge of $X$ is in $P_M$.
\end{itemize}
Construct $P_N,Q_M$ and $Q_N$ equivalently. Note that since $P \cap Q \cap X = \emptyset$, it follows $P_M \cap Q_N = \emptyset$ and $Q_M \cap P_N = \emptyset$. Also note $f(P_M,M) + f(Q_N,N) = f(P,M) + f(Q,N) = f(P,M) + f(P,N) = 2$, and also $f(Q_M,M)+f(P_N,N) =2$.
We now construct a new path $I$. If $f(P_M,M)=f(Q_N,N)=1$, then create $I$ by linking the endvertex of $P_M$ in $X$ and the endvertex of $Q_N$ in $X$ via an edge of $X$. Since $P_M \cap Q_N = \emptyset$, $I$ will be a path. Alternatively without loss of generality $f(P_M,M) = 0$ and $f(Q_N,N) = 2$. Now create $I$ by deleting an edge of $Q_N$ in $X$ and attaching these new end vertices to the two end vertices of $P_M$ in $X$. Again this will be a path. Also create a second path $I'$ in the same way using $Q_M$ and $P_N$. Now every vertex of $P$ and $Q$ is in $I \cup I'$. However, let $v$ be a vertex of $P \cap X$. The vertex $v$ is in $P_M$ and $P_N$ and so is in both $I$ and $I'$. Thus $|I| + |I'| > |P| + |Q|$. But since $P,Q$ have maximum length, $|I| + |I'| \leq |P|+|Q|$, a contradiction.
\end{proof}
\section{Proof of Theorem~\ref{theorem:chordal}}
We now provide a proof of Theorem~\ref{theorem:chordal}. This proof proceeds in several stages. First, we show that a connected chordal graph $G$ contains a special kind of clique, in which we shall find our traversal.
Note that by Lemma~\ref{lemma:tdcliques} and the Helly property of subtrees, there must exist at least one total clique in $G$.
We define a \emph{central clique} $X$ to be a total clique that satisfies at least one of the following properties:
\begin{enumerate}
\item $X$ is a total clique such that there is no detour $P$ which is small and touching with respect to $X$, but there is a small crossing detour,
\item $X$ is a total clique such that there exist two detours $P,Q$ that are both small and touching with respect to $X$, and have different home components,
\item $X$ is a total clique such that either $|X| \leq 4\ceil{\tfrac{\omega(G)}{5}}$ or there are no small detours at all with respect to $X$.
\end{enumerate}
We will use the above numbers to refer to the different types of central cliques.
\begin{lemma}
\label{lemma:central}
A connected chordal graph $G$ has a central clique.
\end{lemma}
\begin{proof}
Suppose no central clique exists.
Fix a tree decomposition $T$ of $G$ such that each bag is a clique and such that for any two adjacent bags $X$ and $Y$ neither $X \subseteq Y$ nor $Y \subseteq X$. (The first property follows from $G$ being a chordal graph, and the second is achieved by contracting an edge in the tree decomposition whenever the property is violated.)
We construct a set of directed edges $D$ with one going out of each bag $B$ as follows:
\begin{itemize}
\item If $B$ is a total clique, then since $B$ is not a central clique, there must be at least one small touching detour, and all small touching detours have the same home component, which we call $C$. Direct the arc from $B$ towards the subtree of $T-B$ containing $C$.
\item If $B$ is a non-total clique, there is some detour $P$ that does not intersect $B$, and so there is a component $C$ of $G-B$ such that $P \subseteq C$. The component $C$ is entirely within a single subtree of $T-B$, and so we construct an arc from $B$ towards its neighbour in said subtree. Note that even if several detours do not intersect $B$, they must be inside the same component since any two detours intersect, and so our arc is well-defined.
\end{itemize}
Using the bags of $T$ as nodes and the directed edges $D$ we construct a directed graph where the underlying graph is acyclic and every node has outdegree exactly one. Thus there exists at least one 2-cycle.
Let $e$ be the underlying edge of the tree-decomposition corresponding to a $2$-cycle, and label the bags at the end vertices $B$ and $B'$. The edge $e$ splits $T$ into two sides, which we label left and right such that $B$ is on the left and $B'$ is on the right.
There are now three possibilities to consider. Firstly, suppose that $B$ and $B'$ are both non-total cliques. So there exists a detour $P$ that does not intersect $B$ and is entirely on the right, and a detour $Q$ that does not intersect $B'$ and is entirely on the left. But then $P \cap Q = \emptyset$, which cannot occur as detours pairwise intersect.
Secondly, suppose that $B$ and $B'$ are both total cliques. Let $P$ be a small touching detour with respect to $B$, which will have its home component to the right of $e$. Label the home component of $P$ by $C$. Let $Q$ be a small touching detour with respect to $B'$ with its home component to the left of $e$. Define $X=B \cap B'$. Suppose $P$ contains a vertex $v$ in $B-X$. Now every vertex of $P$ is contained within $B$ or $C$, but $v$ has no neighbours in $C$ as $v$ and $C$ are separated by $X$. Thus, since $v$ is incident to some edge in $P$, the vertex $u$ at the other end of this edge must be in $B$, and so all of $B$ is in $P$ by Proposition~\ref{prop:extend}. Since $P$ is small with respect to $B$, it follows $B$ is a central clique of type 3. Hence $P$ does not visit $B-X$, and similarly $Q$ does not visit $B'-X$.
Thus both $P$ and $Q$ are small and touching with respect to the clique-cut $X$. The detours $P,Q$ still have different home components (to the right and left of $e$ respectively) and so $X$ is a central clique of type 2.
Thirdly, suppose (without loss of generality) that $B$ is total and $B'$ is non-total. Let $P$ be a small touching detour with respect to $B$, which thus has its home component to the right of $e$. Let $Q$ be a detour that does not intersect $B'$ (since it is non-total). Note $Q$ intersects $B$, and is entirely to the left of $e$. Again, let $X := B \cap B'$. As in the previous case, if $P$ contains a vertex $v$ in $B-X$, then all of $B$ is in $P$ and $B$ is a central clique of type 3. Hence we suppose that $P$ does not intersect $B-X$. However, $P$ and $Q$ must intersect, and $Q$ does not intersect $B'$ and therefore does not intersect $X$, which leaves nowhere for the two detours to intersect. This gives our final contradiction.
\end{proof}
We now let $X$ be a central clique as guaranteed by Lemma~\ref{lemma:central}. If $X$ is a central clique of type 3, then either $X$ itself or any subset of $X$ of size $4\ceil{\tfrac{\omega(G)}{5}}$ satisfies the requirements for Theorem~\ref{theorem:chordal}. So we assume that $X$ is a central clique of type 1 or type 2, and not type 3.
The next step is to construct a special set of detours $\mathcal{F}$ which will help us find our transversal. We do so via the following algorithm:
\begin{enumerate}
\item Initialise $\mathcal{F} = \emptyset$.
\item Consider the small, closed, crossing detours with respect to $X$.
\begin{enumerate}
\item If no such detours exist, go to Step 3.
\item Otherwise, if all such detours have the same home component, add one of them to $\mathcal{F}$, then go to Step 3.
\item Otherwise, there exist small closed crossing detours with different home components. If there exists two such detours with different home components that do not intersect in $X$, add them both to $\mathcal{F}$. Go to Step 3.
\item Finally, there exist small closed crossing detours with different home components, but they all pairwise intersect in $X$. Add one such pair $P,Q$ to $\mathcal{F}$, then go to Step 3.
\end{enumerate}
\item Now consider small touching detours with respect to $X$. If none exist, go to Step 4, otherwise go to Step 5.
\item If there exists a small open crossing detour, add it to $\mathcal{F}$ and finish.
\item Add two small touching detours $P,Q$ with different home components to $\mathcal{F}$ and finish.
\end{enumerate}
Note that if the algorithm reaches Step 5, $X$ had at least one small touching detour (from the condition in Step 3). Thus $X$ was a central clique of type 2, so it has two small touching detours with different home components. Hence Step 5 is a valid operation. All other steps of the algorithm are self-evidently valid.
From the construction of $\mathcal{F}$ it is clear that every detour in $\mathcal{F}$ is small, and $\mathcal{F}$ contains at most four detours (e.g.~if Steps 2(c) and 5 operate). Also, no detour $P \in \mathcal{F}$ has an end vertex in $X$, otherwise by Proposition~\ref{prop:extend} all of $X$ is in $P$ and so, since $P$ is small, $X$ is a central clique of type 3. Initially define $F := \bigcup_{P \in \mathcal{F}} P \cap X$. Since $\mathcal{F}$ contains at most four small detours, it follows $|F| \leq 4\ceil{\tfrac{\omega(G)}{5}}$ and since $F \subseteq X$, $F$ is itself a clique of $G$. If $|F| < 4\ceil{\tfrac{\omega(G)}{5}}$, then add other vertices of $X$ to $F$ in order to force $|F| = 4\ceil{\tfrac{\omega(G)}{5}}$ while maintaining the fact that $F$ is a clique. Our final step is to show that $F$ is the transversal we require.
\begin{claim}
$F$ is a transversal for the set of detours in $G$.
\end{claim}
\begin{proof}
Let $R$ be a detour in $G$. There are two cases to consider: either $R$ is large with respect to $X$ or $R$ is small with respect to $X$.
If $R$ is large with respect to $X$, then $|R \cap X| \geq \ceil{\tfrac{\omega(G)}{5}} + 1$. If $R \cap F = \emptyset$, then $$|X| \geq |F| + |R \cap X|\geq 4\ceil{\tfrac{\omega(G)}{5}} + \ceil{\tfrac{\omega(G)}{5}} + 1 \geq \omega(G) + 1,$$ which is a clear contradiction. Hence we may assume that $R$ is small with respect to $X$. We may also assume that $R \not\in \mathcal{F}$, since that case is clear.
Suppose $R$ is a touching detour. Thus by Step 5, $\mathcal{F}$ contains two small touching detours $P,Q$ with different home components. Without loss of generality, suppose that the home component for $P$ is different to the home component for $R$. Since $P$ and $R$ are touching detours and need to intersect while having different home components, it follows they intersect in $X$ itself, and thus $R$ intersects $F$.
Hence we may assume that $R$ is a crossing detour. The first subcase we will now consider is the case when $R$ is open and no touching detours exist. Thus by Step 4 and the existence of $R$ there exists a small open crossing detour $P \in \mathcal{F}$. We construct a separation $(M,X,N)$ where the detours $P$ and $R$ both have one endvertex in $M$ and one in $N$. Thus by Lemma~\ref{lemma:paste} it follows $(P \cap X) \cap (R \cap X) \neq \emptyset$, and hence $R$ intersects $F$.
The next subcase is when $R$ is open and crossing but some touching detours exist. Then by Step 5 there exist two small touching detours $P,Q \in \mathcal{F}$ with different home components. Partition the components of $G-X$ into $M$ and $N$ such that the home component of $P$ is in $M$, the home component of $Q$ is in $N$ and both $M$ and $N$ contain one endvertex of $R$; since $R$ is open this partition is achievable. Let $R_M$ be the subpath of $R$ created by deleting $N$ and pasting the remaining subpaths of $R$ together using the edges of $X$. Define $R_N$ analogously, and without loss of generality we say $|R_M| > \tfrac{1}{2}|R|$. Let $Q'$ be a subpath of $Q$ with one endvertex in $N$, one endvertex in $X$ such that $|Q'| > \tfrac{1}{2}|Q|$. If $R_M$ and $Q'$ do not intersect in $X$ then they can be concatinated to create a longer detour, if they do intersect in $X$ then so do $R$ and $Q$, and so $R$ intersects $F$. These subcases account for when $R$ is open and crossing.
Finally we need to account for when $R$ is closed and crossing. Since $R$ exists, Step 2 will add some detours to $\mathcal{F}$. First suppose that all of the small, closed crossing detours have the same home component, and let $P$ be the detour Step 2(b) added to $\mathcal{F}$. Now let $M$ be the home component of $P$ (and $R$), and let $N$ be all other components of $G-X$. Thus by Lemma~\ref{lemma:paste} applied to the separation $(M,X,N)$ it follows that $P$ and $R$ intersect in $X$, and thus $R$ intersects $F$.
Hence we may assume there exists a pair of small closed crossing detours $P,Q$ in $\mathcal{F}$ with different home components. Suppose $P,Q$ do not intersect in $X$ (that is, Step 2(c) occurred). Clearly we may also assume that $R$ does not intersect either $P$ or $Q$ in $X$. Label the home components of $P,Q,R$ by $C_P,C_Q$ and $C_R$ respectively, and note $C_P \neq C_Q$. If $C_P = C_R$, then let $M = C_P$ and $N = G - X - C_P$ and apply Lemma~\ref{lemma:paste} to $(M,X,N)$, so that $P$ intersects $R$ in $X$. So we assume $C_P \neq C_R$ and similarly that $C_Q \neq C_R$, that is, that $C_P,C_Q,C_R$ are three distinct components. Let $M = C_P \cup C_Q$ and $N = G-X-(C_P \cup C_Q)$. If both $P$ and $Q$ intersect $N$ then by Lemma~\ref{lemma:paste} the paths $P$ and $Q$ must intersect in $X$, which they do not. Hence without loss of generality we suppose $P$ does not intersect $N$, that is, $P \subset C_P \cup C_Q \cup X$. Since $P$ is crossing it must intersect both $C_P$ and $C_Q$. By a similar argument one of $P$ and $R$ cannot intersect $G-X-(C_P \cup C_R)$; since $P$ intersects $C_Q$ it must be that $R \subset C_P \cup C_R \cup X$ and that $R$ intersects $C_P$ and $C_R$. Again, by a similar argument one of $Q$ and $R$ cannot intersect $G-X-(C_Q \cup C_R)$, and given previous results it must be $Q \subset C_Q \cup C_R \cup X$ and that $Q$ intersects $C_Q$ and $C_R$.
We will now argue in a similar fashion to Lemma~\ref{lemma:paste} to construct a longer detour. Consider the subpaths of $P \cap (C_P \cup X)$. All of the end vertices of these subpaths are in $X$ itself except for two inside $C_P$, and since $P$ crosses $X$ there are at least two subpaths in $P \cap (C_P \cup X)$. Create $P'$ by connecting the subpaths of $P \cap (C_P \cup X)$ using the edges of $X$; because $P \cap (C_P \cup X)$ has at least two subpaths it follows that $P'$ will contain at least one edge of $X$. Similarly, consider the subpaths of $P \cap (C_Q \cup X)$ and connect them up using edges of $X$ to create a path $P''$. Note that $P''$ will have both end vertices in $X$.
Construct $Q',Q'',R',R''$ in the same way (substituting different components as appropriate). Finally, construct the path $P^*$ from $P'$ and $Q''$ by removing an edge $e$ of $P'$ in $X$ and adding the edges from the two endvertices of $e$ to the endvertices of $Q''$. This will be a path since $P' \subset C_P \cup X$ and $Q'' \subset C_R \cup X$ and $P,Q$ do not intersect in $X$. Construct $Q^*$ from $Q'$ and $R''$ and $R^*$ from $R'$ and $P''$ in the same fashion. Every vertex of $P,Q,R$ is used at least once in $P^*,Q^*,R^*$, but the vertices of $P \cap X$ are used twice (in both $P^*$ and $R^*$). Hence $|P^*| + |Q^*| + |R^*| > |P| + |Q| + |R|$, but since $P,Q,R$ are detours, it follows at least one of $P^*,Q^*,R^*$ is longer than a detour, a contradiction. This accounts for the case when there exists a pair of small closed crossing detours with different home components that do not intersect in $X$.
The final subcase we must consider is when every pair of small closed crossing detours with different home components has an intersection between the detours in $X$. (That is, Step 2(d) occurred.) Let $P,Q$ denote the pair we added to $\mathcal{F}$. Now at least one of these detours has a different home component to $R$; without loss of generality we suppose it is $P$. Then $P$ and $R$ would have been a legitimate choice of detours to add to $\mathcal{F}$ instead of $P$ and $Q$. But we know that for each valid choice the detours intersect in $X$, that is, $R$ must intersect $P$ in $X$ and is therefore intersects $F$. This completes all possible cases.
\end{proof}
\section{Longest cycles in chordal graphs}
In this section we discuss how to apply the same ideas as those used in the proof of Theorem~\ref{theorem:chordal} to prove the following result for longest cycles in chordal graphs.
\newtheorem*{repeatcycles}{Theorem \ref{theorem:cycles}}
\begin{repeatcycles}
Given a $2$-connected chordal graph $G$, there exists a transversal of order at most $2\ceil{\tfrac{\omega(G)}{3}}$ that intersects every longest cycle of $G$. Moreover, this transversal induces a clique in $G$.
\end{repeatcycles}
We will describe the modifications required to adapt the proof to longest cycles, rather than including a full proof of this result.
First note that longest cycles in $2$-connected graphs have the property that any two intersect~\cite{Gutierrez}, and that similarly to Proposition~\ref{prop:extend}, a longest cycle with an edge in a given clique must visit every vertex of that clique.
Cycles have no ends, so there is no need for the open/closed distinction used above. As before, given a clique-cut $X$ we say that a cycle $C$ \emph{touches} $X$ if $V(C) - X$ lies in a single component of $G-X$ (its home component), and otherwise it \emph{crosses} $X$.
We say a cycle $C$ is \emph{small} with respect to a clique $X$ if $|V(C) \cap X| \leq \ceil{\tfrac{\omega(G)}{3}}$.
Lemma~\ref{lemma:paste} can be replaced with a lemma that states that any two longest cycles crossing $X$ must intersect in $X$; otherwise the pieces of the cycles can be rearranged into two other cycles that also use some edges of $X$, contradicting the fact they were longest cycles.
For longest cycles, a \emph{central clique} is a total clique-cut that has either: a small crossing longest cycle but no small touching longest cycles; two small touching longest cycles with different home components; order at most $2\ceil{\tfrac{\omega(G)}{3}}$; or no small longest cycles at all.
To show there exists a central clique we argue as before. Assume there is none, and direct edges of the tree decomposition toward disjoint or small touching longest cycles. This means every bag has out degree one and so there is a directed $2$-cycle.
We consider the bags that constitute this 2-cycle, and distinguish cases by whether they are total clique-cuts or not. The argument proceeds exactly as in the proof of Lemma~\ref{lemma:central}.
Given a central clique $X$ we construct a transversal as follows.
Firstly, if there is a small crossing longest cycle but no small touching longest cycles, we only need to take the vertices from one such longest cycle in $X$.
Secondly, if there are small touching longest cycles with different home components then we take the vertices of two such cycles $P$ and $Q$ in $X$.
Now any other touching longest cycle must hit $P$ or $Q$ in $X$ since it avoids one of the home components.
If $R$ is a crossing longest cycle, at least half of $R$ is outside of one of the home components, say that of $P$.
If $P$ and $R$ don't intersect in $X$ it is possible to take at least half of each of $R$ and $P$ and combine them into a longer cycle using edges of $X$.
Thirdly, if $X$ has order at most $2\ceil{\tfrac{\omega(G)}{3}}$ we take all of $X$.
Finally, if $X$ is larger but has no small longest cycles at all, we take any $2\ceil{\tfrac{\omega(G)}{3}}$ vertices from $X$.
Thus we have a transversal of order at most $2\ceil{\tfrac{\omega(G)}{3}}$ as required.
| {
"timestamp": "2020-12-15T02:27:31",
"yymm": "2012",
"arxiv_id": "2012.07221",
"language": "en",
"url": "https://arxiv.org/abs/2012.07221",
"abstract": "We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\\omega(G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\\lceil\\frac{\\omega(G)}{5}\\rceil$. We also consider the analogous question for longest cycles, and show that if $G$ is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most $2\\lceil\\frac{\\omega(G)}{3}\\rceil$.",
"subjects": "Combinatorics (math.CO)",
"title": "Intersecting longest paths in chordal graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795125670755,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379433130173
} |
https://arxiv.org/abs/1011.6646 | Sparse random graphs: Eigenvalues and Eigenvectors | In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erdős-Rényi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial. | \section{Introduction}
\subsection{Overview}
In this paper, we consider two models of random graphs, the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and the random regular graph $G_{n,d}$.
Given a real number $p=p(n)$,$0\le p\le 1$, the Erd\H{o}s-R\'{e}nyi graph on a vertex set of size $n$ is obtained by drawing an edge between each pair of vertices, randomly and independently, with probability $p$.
On the other hand, $G_{n,d}$, where $d=d(n)$ denotes the degree, is a random graph chosen uniformly from the set of
all simple $d$-regular graphs on $n$ vertices. These are basic models in the theory of random graphs. For further information, we refer the readers
to the excellent monographs $ \cite{bollobas} ,\cite{janson}$ and
survey $ \cite{MRRG}$.
Given a graph $G$ on $n$ vertices, the adjacency matrix $A$ of $G$ is an $n\times n$ matrix whose entry $a_{ij}$ equals one if there
is an edge between the vertices $i$ and $j$ and zero otherwise. All diagonal entries $a_{ii}$ are defined to be zero.
The eigenvalues and eigenvectors of $A$ carry valuable information about the structure of the graph
and have been studied by many researchers for quite some time, with both theoretical and practical motivations (see, for example, $\cite{bauer},\cite{bhamidi},\cite{feige},\cite{semerjian}$ $\cite{FK1981},\cite{fried1991},\cite{fried2003}$, $\cite{fried1993},\cite{tvrandom},\cite{erdos2009semi}$, $\cite{shi2000},\cite{pothen1990}$).
The goal of this paper is to study the eigenvalues and eigenvectors of $G(n,p)$ and $G_{n,d}$.
We are going to consider:
\begin{itemize}
\item The global law for the limit of the empirical spectral distribution (ESD) of adjacency matrices of $G(n,p)$ and $G_{n,d}$.
For $p = \omega (1/n)$, it is well-known that eigenvalues of $G(n,p)$ (after a proper scaling) follows Wigner's semicircle law
(we include a short proof in the Appendix \ref{appendix:SCL} for completeness).
Our main new result shows that the same law holds for random regular graph with $d \rightarrow \infty$ with $n$.
This complements the well known result of McKay for the case when $d$ is an absolute constant (McKay's law) and extends recent results of
Dumitriu and Pal \cite{SRRG} (see Section \ref{section:SCL} for more discussion).
\item Bound on the infinity norm of the eigenvectors. We first prove that the infinity norm of any (unit) eigenvector $v$ of $G(n,p)$ is almost surely $o(1)$ for $p=\omega(\log n/n)$. This gives a positive answer to a question raised by Dekel, Lee and Linial
\cite{DLL}. Furthermore, we can show that $v$
satisfies the bound $\|v\|_{\infty} = O\left(\sqrt { \log^{2.2} g(n){\log n}/{np} } \right) $ for $p=\omega(\log n/n)=g(n)\log n/n$, as long as the corresponding eigenvalue
is bounded away from the (normalized) extremal values $-2$ and $2$.
\end{itemize}
We finish this section with some notation and conventions.
Given an $n\times n$ symmetric matrix $M$, we denote its $n$ eigenvalues as $${\lambda}_1 {(M)} \le {\lambda}_2 {(M)} \le \ldots \le {\lambda}_n {(M)},$$ and let $u_1(M),\ldots,u_n(M) \in \mathbb{R}^n$ be an orthonormal basis of eigenvectors of $M$ with $$M u_i(M)={\lambda}_i u_i(M).$$
The empirical spectral distribution (ESD) of the matrix $M$ is a one-dimensional function $$F^{\bf M}_n(x)=\frac{1}{n} |\{ 1\le j \le n: \lambda_j(M) \le x\}|,$$
where we use $|\mathbf{I}|$ to denote the cardinality of a set $\mathbf{I}$.
Let $A_n$ be the adjacency matrix of $G(n,p)$. Thus $A_n$ is a random symmetric $n\times n$ matrix whose upper triangular entries are iid copies of a real random variable $\xi$ and diagonal entries are $0$. $\xi$ is a Bernoulli random variable that takes values $1$ with probability $p$ and $0$ with probability $1-p$. $$\mathbb{E} \xi=p, \mathbb{V}ar{\xi}=p(1-p)={\sigma}^2.$$
Usually it is more convenient to study the normalized matrix
$$M_n=\frac{1}{\sigma}(A_n-pJ_n)$$
where $J_n$ is the $n\times n$ matrix all of whose entries are 1. $M_n$ has entries with mean zero and variance one. The global properties of the eigenvalues of $A_n$ and $M_n$ are essentially the same (after proper scaling), thanks to the following lemma
\begin{lem}
\emph{(Lemma 36, \cite{tvrandom})}
\label{EigenDiff}
Let $A,B$ be symmetric matrices of the same size where $B$ has rank one. Then for any interval $I$,
$$|N_I(A+B)-N_I(A)| \le 1,$$
where $N_I(M)$ is the number of eigenvalues of $M$ in $I$.
\end{lem}
\begin{defn}
Let $E$ be an event depending on $n$. Then $E$ holds with {\it overwhelming probability} if ${\hbox{\bf P}}(E)\geq 1- \exp(-\omega(\log n))$.
\end{defn}
The main advantage of this definition is that if we have a polynomial number of events, each of which holds with overwhelming probability, then their intersection also holds with overwhelming probability.
Asymptotic notation is used under the assumption that $n \rightarrow \infty$. For functions $f$ and $g$ of parameter $n$, we use the following notation as $n\rightarrow \infty$: $f=O(g)$ if $|f|/|g|$ is bounded from above; $f=o(g)$ if $f/g \rightarrow 0$; $f=\omega(g)$ if $|f|/|g|\rightarrow \infty$, or equivalently, $g=o(f)$; $f=\Omega(g)$ if $g=O(f)$; $f=\Theta(g)$ if $f=O(g)$ and $g=O(f)$.\\
\subsection{The semicircle law} \label{section:SCL}
In 1950s, Wigner \cite{wigner} discovered the famous semi-circle for the limiting distribution of the eigenvalues of random matrices.
His proof extends, without difficulty, to the adjacency matrix of $G(n,p)$, given that $np \rightarrow \infty$ with $n$. (See Figure \ref{fig:SCL1} for a numerical simulation)
\begin{thm}
\label{thm:SCL1}
For $p=\omega(\frac{1}{n})$, the empirical spectral distribution (ESD) of the matrix $\frac{1}{\sqrt{n}\sigma} A_n$ converges in distribution to the semicircle distribution which has a density ${{}\rho}_{sc}(x)$ with support on $[-2,2]$, $${{\rho}}_{sc}(x) := \frac{1}{2 \pi } \sqrt{4 -x^2}.$$
\end{thm}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.7]{SCL1.pdf}
\captionstyle{center}
\onelinecaptionsfalse
\caption{The probability density function of the ESD of
$G(2000,0.2)$ }
\label{fig:SCL1}
\end{figure}
If $np= O(1)$, the semicircle law no longer holds. In this case, the graph almost surely has $\Theta(n)$ isolated vertices, so
in the limiting distribution, the point $0$ will have positive constant mass.
The case of random regular graph, $G_{n,d}$, was
considered by McKay \cite{mckay} about 30 years ago. He proved that if $d$ is fixed, and $n \rightarrow \infty$, then the
limiting density function is
$$\displaystyle f_d(x)=\left\{ \begin{array}{ll}
\frac{d\sqrt{4(d-1)-x^2}}{2\pi (d^2-x^2)}, & \mbox{if $|x| \leq 2\sqrt{d-1}$};\\
\\
0 & \mbox{otherwise}.\end{array} \right.$$
This is usually referred to as McKay or Kesten-McKay law.
It is easy to verify that as $d\rightarrow \infty$, if we normalize the variable $x$ by $\sqrt{d-1}$, then the above density converges to the semicircle distribution on $[-2,2]$.
In fact, a numerical simulation shows the convergence is quite fast(see Figure \ref{fig:SCL-RRG}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.6]{SCL-RRG.pdf}
\captionstyle{center}
\onelinecaptionsfalse
\caption{The probability density function of the ESD of \protect\\
Random $d$-regular graphs with 1000 vertices}
\label{fig:SCL-RRG}
\end{figure}
It is thus natural to conjecture that Theorem \ref{thm:SCL1} holds for $G_{n,d}$ with $d \rightarrow \infty$.
Let $A'_n$ be the adjacency matrix of $G_{n,d}$, and set
$$M'_n=\frac{1}{\sqrt{\frac{d}{n}(1- \frac{d}{n} ) }}(A'_n-\frac{d}{n}J).$$
\begin{con} \label{conj:SCL2}
If $d \rightarrow \infty$ then the ESD of $\frac{1}{\sqrt{n}}M'_n$ converges to the standard semicircle distribution.
\end{con}
Nothing has been proved about this conjecture, until recently. In \cite{SRRG}, Dimitriu and Pal showed that the conjecture holds
for $d$ tending to infinity slowly, $d= n^{o(1)}$. Their method does not extend to larger $d$.
We are going to establish Conjecture \ref{conj:SCL2} in full generality. Our method is very different from that of \cite{SRRG}.
Without loss of generality we may assume $d\le n/2$, since the adjacency matrix of the complement graph of $G_{n,d}$ may be written as $J_n-A'_n$, thus by Lemma \ref{EigenDiff} will have the spectrum interlacing between the set $\{-\lambda_n(A'_n),\dots,-\lambda_1(A'_n)\}$. Since the semi-circular distribution is symmetric, the ESD of $G_{n,d}$ will converges to semi-circular law if and only if the ESD of its complement does.
\begin{thm}\label{thm:SCL-regular} If $d$ tends to infinity with $n$, then the empirical spectral distribution of $\frac{1}{\sqrt{n}}M'_n$ converges in distribution to the semicircle distribution.
\end{thm}
Theorem \ref{thm:SCL-regular} is a direct consequence of the following stronger result, which shows convergence at small scales. For an interval $I$ let $N'_I$ be the number of eigenvalues of $M'_n$ in $I$.
\begin{thm}\label{thm:ESD-regular}(Concentration for ESD of $G_{n,d}$). Let $\delta>0$ and consider the model $G_{n,d}$. If $d$ tends to $\infty$ as $n\rightarrow \infty$ then for any interval $I\subset[-2,2]$ with length at least $\delta^{-4/5}d^{-1/10}\log^{1/5} d$, we have
$$|N'_I-n\int_I\rho_{sc}(x)dx|< \delta n\int_I\rho_{sc}(x)dx$$
with probability at least $1-O(\exp(-cn\sqrt{d}\log d))$.
\end{thm}
\begin{rem}
Theorem \ref{thm:ESD-regular} implies that with probability $1-o(1)$, for $d=n^{\Theta(1)}$, the rank of $G_{n,d}$ is at least $n -n^{c}$ for some constant $ 0< c< 1$ (which can be computed
explicitly from the lemmas). This is a partial result toward the conjecture by the second author that $G_{n,d}$ almost surely has full rank (see \cite{Vusur}).
\end{rem}
\subsection{Infinity norm of the eigenvectors}
Relatively little is known for eigenvectors in both random graph models under study. In \cite{DLL}, Dekel, Lee and Linial, motivated by the study of nodal domains, raised the
following question.
\begin{question} \label{question:DLL}
Is it true that almost surely every eigenvector $u$ of $G(n, p)$ has $||u||_{\infty}=o(1)$? \end{question}
Later, in their journal paper \cite{DLL2}, the authors added one sharper question.
\begin{question} \label{question:DLL2}
Is it true that almost surely every eigenvector $u$ of $G(n, p)$ has $||u||_{\infty}= n^{-1/2+o(1)} $?
\end{question}
The bound $n^{-1/2 +o(1)}$ was also conjectured by the second author of this paper in an NSF proposal (submitted Oct 2008). He and Tao \cite{tvrandom} proved
this bound for eigenvectors corresponding to the eigenvalues in
the bulk of the spectrum for the case $p=1/2$. If one defines the adjacency matrix by writting $-1$ for non-edges, then this bound holds
for all eigenvectors \cite{tvrandom, tvrandom2}.
The above two questions were raised under the assumption that $p$ is a constant in the interval $(0,1)$. For $p$ depending on $n$, the statements may fail. If $p \le \frac{(1-\epsilon) \log n }{n} $, then the graph has (with high probability) isolated vertices and so one cannot expect that $\|u \| _{\infty} =o(1)$ for every eigenvector $u$.
We raise the following questions:
\begin{question} \label{question:TVW1} Assume $ p \ge \frac{(1+ \epsilon) \log n }{n} $ for some constant $\epsilon >0$.
Is it true that almost surely every eigenvector $u$ of $G(n, p)$ has $||u||_{\infty}=o(1)$? \end{question}
\begin{question} \label{question:TVW2}
Assume $ p \ge \frac{(1+ \epsilon) \log n }{n} $ for some constant $\epsilon >0$.
Is it true that almost surely every eigenvector $u$ of $G(n, p)$ has $||u||_{\infty}= n^{-1/2+o(1)} $?
\end{question}
Similarly, we can ask the above questions for $G_{n,d}$:
\begin{question} \label{question:TVW3} Assume $ d \ge (1+ \epsilon) \log n$ for some constant $\epsilon >0$.
Is it true that almost surely every eigenvector $u$ of $G_{n,d}$ has $||u||_{\infty}=o(1)$? \end{question}
\begin{question} \label{question:TVW4}
Assume $ d \ge (1+ \epsilon) \log n $ for some constant $\epsilon >0$.
Is it true that almost surely every eigenvector $u$ of $G_{n,d}$ has $||u||_{\infty}= n^{-1/2+o(1)} $?
\end{question}
As far as random regular graphs is concerned, Dumitriu and Pal \cite{SRRG} and Brook and Lindenstrauss \cite{BL} showed that for any normalized eigenvector of a
sparse random regular graph is delocalized in the sense that one can not have too much mass on a small set of coordinates. The readers may want to consult their papers for explicit statements.
We generalize our questions by the following conjectures:
\begin{con}
Assume $ p \ge \frac{(1+ \epsilon) \log n }{n} $ for some constant $\epsilon >0$. Let $v$ be a random unit vector whose distribution is uniform in the $(n-1)$-dimensional unit sphere. Let $u$ be a unit eigenvector of $G(n,p)$ and $w$ be any fixed $n$-dimensional vector. Then for any $\delta>0$
$${\hbox{\bf P}}(|w\cdot u-w\cdot v|>\delta)=o(1).$$
\end{con}
\begin{con}
Assume $ d \ge (1+ \epsilon) \log n $ for some constant $\epsilon >0$. Let $v$ be a random unit vector whose distribution is uniform in the $(n-1)$-dimensional unit sphere. Let $u$ be a unit eigenvector of $G_{n,d}$ and $w$ be any fixed $n$-dimensional vector. Then for any $\delta>0$
$${\hbox{\bf P}}(|w\cdot u-w\cdot v|>\delta)=o(1).$$
\end{con}
In this paper, we focus on $G(n,p)$. Our main result settles (positively) Question \ref{question:DLL} and almost Question \ref{question:TVW1} . This result follows from Corollary \ref{cor:ESDconvrate} obtained in Section 2.
\begin{thm}
\emph{(Infinity norm of eigenvectors)}
\label{Delocal}
Let $p =\omega(\log n /n)$ and let $A_n$ be the adjacency matrix of $G(n,p)$. Then there exists an orthonormal basis of eigenvectors of $A_n$, $\{ u_1,\ldots, u_n \}$, such that for every $1\le i \le n$, $||u_i||_{\infty}=o(1)$ almost surely.
\end{thm}
For Questions \ref{question:DLL2} and \ref{question:TVW2}, we obtain a good quantitative bound for those eigenvectors
which correspond to eigenvalues bounded away from the edge of the spectrum.
For convenience, in the case when $p=\omega(\log n/ n) \in (0,1)$, we write $$p=\frac{g(n) \log n}{n},$$ where $g(n)$ is a positive function such that $g(n)\rightarrow \infty$ as $n\rightarrow \infty$ ($g(n)$ can tend to $\infty$ arbitrarily slowly).
\begin{thm}
\label{DelocalBulk}
Assume $p={g(n)\log n}/{n} \in (0,1)$, where $g(n)$ is defined as above. Let $B_n=\frac{1}{\sqrt{n}\sigma} A_n$. For any $\kappa > 0$, and any $1 \le i \le n$ with $\lambda_i (B_n) \in [-2+\kappa, 2-\kappa]$, there exists a corresponding eigenvector $u_i$ such that $||u_i||_{\infty}=O_\kappa (\sqrt{ \frac{\log^{2.2} g(n)\log n}{np} } ) $with overwhelming probability.
\end{thm}
The proofs are adaptations of a recent approach developed in random matrix theory (as in \cite{tvrandom},\cite{tvrandom2},\cite{erdos2009semi}, \cite{erdos09local}).
The main technical lemma is a concentration theorem about the number of eigenvalues on a finer scale for $p=\omega(\log n/n) $.
\section{Semicircle law for regular random graphs}
\subsection{Proof of Theorem \ref{thm:ESD-regular}}
We use the method of comparison. An important lemma is the following
\begin{lem}\label{lem:np-reg} If $np\rightarrow \infty$ then $G(n,p)$ is $np$-regular with probability at least $\exp(-O(n(np)^{1/2})$.
\end{lem}
For the range $p\ge\log^2 n/n$, Lemma \ref{lem:np-reg} is a consequence of
a result of Shamir and Upfal \cite{SU} (see also \cite{KSVW}). For smaller values of $np$, McKay and Wormald \cite{MKW1} calculated
precisely the probability that $G(n,p)$ is $np$-regular, using the fact that the joint distribution of the degree sequence of $G(n,p)$ can be approximated by a simple model derived from independent random variables with binomial distribution. Alternatively, one may calculate the same probability directly using the asymptotic formula for the number of $d$-regular graphs on $n$ vertices (again by McKay and Wormald \cite{MKW2}). Either way, for $p=o(1/\sqrt{n})$, we know that
$${\hbox{\bf P}}(G(n,p)\text{ is }np\text{-regular})\ge \Theta(\exp(-n\log(\sqrt{np})).$$
which is better than claimed in Lemma \ref{lem:np-reg}.\\
Another key ingredient is the following concentration lemma, which may be of independent interest.
\begin{lem}\label{thm:ESD-con-general} Let $M$ be a $n\times n$ Hermitian random matrix whose off-diagonal entries $\xi_{ij}$ are i.i.d. random variables with mean zero, variance 1 and $|\xi_{ij}|< K$ for some common constant $K$. Fix $\delta>0$ and assume that the forth moment $M_4:=\sup_{i,j}{\hbox{\bf E}}(|\omega_{ij}|^4)=o(n)$. Then for any interval $I\subset [-2,2]$ whose length is at least $\Omega(\delta^{-2/3}(M_4/n)^{1/3})$, the number $N_I$ of the eigenvalues of $\frac{1}{\sqrt{n}}M$ which belong to $I$ satisfies the following concentration inequality
$${\hbox{\bf P}}(|N_I-n\int_I\rho_{sc}(t)dt|>\delta n\int_I\rho_{sc}(t)dt)\le 4\exp(-c\frac{\delta^4n^2|I|^5}{K^2}).$$
\end{lem}
Apply Lemma \ref{thm:ESD-con-general} for the normalized adjacency matrix $M_n$ of $G(n,p)$ with $K=1/\sqrt{p}$ we obtain
\begin{cor}\label{cor:ESDconvrate} Consider the model $G(n,p)$ with $np\rightarrow \infty$ as $n\rightarrow \infty$ and let $\delta>0$. Then for any interval $I\subset[-2,2]$ with length at least $\big(\frac{\log(np)}{\delta^4(np)^{1/2}}\big)^{1/5}$, we have
$$|N_I-n\int_I\rho_{sc}(x)dx|\ge \delta n\int_I\rho_{sc}(x)dx$$
with probability at most $\exp(-cn(np)^{1/2}\log(np))$.
\end{cor}
\begin{rem} If one only needs the result for the bulk case $I\subset [-2+\epsilon,2-\epsilon]$ for an absolute constant $\epsilon>0$ then the minimum length of $I$ can be improved to $\big(\frac{\log(np)}{\delta^4(np)^{1/2}}\big)^{1/4}$.
\end{rem}
By Corollary \ref{cor:ESDconvrate} and Lemma \ref{lem:np-reg}, the probability that $N_I$ fails to be close to the expected value in the model $G(n,p)$ is much smaller than the probability that $G(n,p)$ is $np$-regular. Thus the probability that $N_I$ fails to be close to the expected value in the model $G_{n,d}$ where $d=np$ is the ratio of the two former probabilities, which is $O(\exp(-cn\sqrt{np}\log np))$ for some small positive constant $c$. Thus, Theorem \ref{thm:ESD-regular} is proved, depending on Lemma \ref{thm:ESD-con-general} which we turn to next.
\subsection
{Proof of Lemma \ref{thm:ESD-con-general}}
Assume $I=[a,b]$ and $a-(-2)<2-b$.
We will use the approach of Guionnet and Zeitouni in \cite{GZ}. Consider a random Hermitian matrix $W_n$ with independent entries $w_{ij}$ with support in a compact region $S$. Let $f$ be a real convex $L$-Lipschitz function and define
$$Z:=\sum_{i=1}^nf(\lambda_i)$$
where $\lambda_i$'s are the eigenvalues of $\frac{1}{\sqrt{n}}W_n$. We are going to view $Z$ as the function of the atom variables $w_{ij}$. For our application we need $w_{ij}$ to be random variables with mean zero and variance 1, whose absolute values are bounded by a common constant $K$.
The following concentration inequality is from \cite{GZ}
\begin{lem}\label{GZconcentration} Let $W_n, f, Z$ be as above. Then there is a constant $c>0$ such that for any $T>0$
$${\hbox{\bf P}}(|Z-{\hbox{\bf E}}(Z)|\ge T)\le 4\exp(-c\frac{T^2}{K^2L^2}).$$
\end{lem}
In order to apply Lemma \ref{GZconcentration} for $N_I$ and $M$, it is natural to consider
$$Z:=N_I=\sum_{i=1}^n \chi_I(\lambda_i)$$
where $\chi_I$ is the indicator function of $I$ and $\lambda_i$ are the eigenvalues of $\frac{1}{\sqrt{n}}M_n$. However, this function is neither convex nor Lipschitz. As suggested in \cite{GZ}, one can overcome this problem by a proper approximation. Define $I_l=[a-\frac{|I|}{C},a]$, $I_r=[b,b+\frac{|I|}{C}]$ and construct two real functions $f_1, f_2$ as follows(see Figure \ref{fig:fg}):
\begin{equation*}
f_1(x)=\Bigg\{
\begin{array}{ll}
-\frac{C}{|I|}(x-a)-1& \text{if }x\in (-\infty, a-\frac{|I|}{C})\\
0&\text{if }x\in I\cup I_l\cup I_r\\
\frac{C}{|I|}(x-b)-1& \text{if }x\in (b+\frac{|I|}{C},\infty)
\end{array}
\end{equation*}
\begin{equation*}
f_2(x)=\Bigg\{
\begin{array}{ll}
-\frac{C}{|I|}(x-a)-1& \text{if }x\in (-\infty, a)\\
-1&\text{if }x\in I\\
\frac{C}{|I|}(x-b)-1& \text{if }x\in (b,\infty)
\end{array}
\end{equation*}
where $C$ is a constant to be chosen later. Note that $f_j$'s are convex and $\frac{C}{|I|}$-Lipschitz. Define $$X_1=\sum_{i=1}^n f_1(\lambda_i),\ X_2=\sum_{i=1}^n f_2(\lambda_i)$$ and apply Lemma \ref{GZconcentration} with $T=\frac{\delta}{8} n\int_I\rho_{sc}(t)dt$ for $X_1$ and $X_2$. Thus, we have
\begin{align*}
{\hbox{\bf P}}(|X_j-{\hbox{\bf E}}(X_j)|\ge \frac{\delta}{8} n\int_I\rho_{sc}(t)dt)&\le4\exp(-c\frac{\delta^2n^2|I|^2(\int_I\rho_{sc}(t)dt)^2}{K^2C^2}).\\
\end{align*}
At this point we need to estimate the value of $\int_I\rho_{sc}(t)dt$. There are two cases: if $I$ is in the ``bulk'' i.e. $I\subset[-2+\epsilon,2-\epsilon]$ for some positive absolute constant $\epsilon$, then $\int_I\rho_{sc}(t)dt=\alpha |I|$ where $\alpha$ is a constant depending on $\epsilon$. But if $I$ is very near the edge of $[-2,2]$ i.e. $a-(-2)<|I|=o(1)$, then $\int_I\rho_{sc}(t)dt=\alpha'|I|^{3/2}$ for some absolute constant $\alpha'$. Thus in both case we have
$${\hbox{\bf P}}(|X_j-{\hbox{\bf E}}(X_j)|\ge \frac{\delta}{8} n\int_I\rho_{sc}(t)dt)\le4\exp(-c_1\frac{\delta^2n^2|I|^5}{K^2C^2})$$
Let $X=X_1-X_2$, then
$${\hbox{\bf P}}(|X-{\hbox{\bf E}}(X)|\ge \frac{\delta}{4} n\int_I\rho_{sc}(t)dt)\le O(\exp(-c_1\frac{\delta^2n^2|I|^5}{K^2C^2})).$$
Now we compare $X$ to $Z$, making use of a result of
G\"otze and Tikhomirov \cite{GT}. We have ${\hbox{\bf E}}(X-Z)\le {\hbox{\bf E}}(N_{I_l}+N_{I_r})$. In \cite{GT}, G\"{o}tze and Tikhomirov obtained a convergence rate for ESD of Hermitian random matrices whose entries have mean zero and variance one, which implies that for any $I\subset[-2,2]$
$$|{\hbox{\bf E}}(N_I)-n\int_I\rho_{sc}(t)dt|<\beta n\sqrt{\frac{M_4}{n}},$$
where $\beta$ is an absolute constant, $M_4=\sup_{i,j}{\hbox{\bf E}}(|\omega_{ij}|^4)$.
Thus
$${\hbox{\bf E}}(X)\le{\hbox{\bf E}}(Z)+n\int_{I_l\cup I_r}\rho_{sc}(t)dt+\beta n\sqrt{\frac{M_4}{n}}.$$
In the ``edge" case we can choose $C=(4/\delta)^{2/3}$, then because $|I|\ge \Omega(\delta^{-2/3}(M_4/n)^{1/3})$, we have
$$n\int_{I_l\cup I_r}\rho_{sc}(t)dt=\Theta(n(\frac{|I|}{C})^{3/2})>\Omega( n\sqrt{\frac{M_4}{n}})$$
and
$$n\int_{I_l\cup I_r}\rho_{sc}(t)dt+\beta n\sqrt{\frac{M_4}{n}}=\Theta(n(\frac{|I|}{C})^{3/2})=\Theta(\frac{\delta}{4}n\int_I\rho_{sc}(t)dt).$$
In the ``bulk'' case we choose $C=4/\delta$, then
$$n\int_{I_l\cup I_r}\rho_{sc}(t)dt+\beta n\sqrt{\frac{M_4}{n}}=\Theta(n\frac{|I|}{C})=\Theta(\frac{\delta}{4}n\int_I\rho_{sc}(t)dt).$$
Therefore in both cases, with probability at least $1-O(\exp(-c_1\frac{\delta^4n^2|I|^5}{K^2}))$, we have
$$Z\le X\le {\hbox{\bf E}}(X)+ \frac{\delta}{4}n\int_I\rho_{sc}(t)dt < {\hbox{\bf E}}(Z)+\frac{\delta}{2} n\int_I\rho_{sc}(t)dt.$$
The convergence rate result of G\"{o}tze and Tikhomirov again gives
$${\hbox{\bf E}}(N_I)<n\int_I\rho_{sc}(t)dt+ \beta n\sqrt{\frac{M_4}{n}}<(1+\frac{\delta}{2})n\int_I\rho_{sc}(t)dt,$$
hence with probability at least $1-O(\exp(-c_1\frac{\delta^4n^2|I|^5}{K^2}))$
$$Z<(1+\delta) n\int_I\rho_{sc}(t)dt,$$
which is the desires upper bound.
The lower bound is proved using a similar argument. Let $I'=[a+\frac{|I|}{C}, b-\frac{|I|}{C}]$, $I'_l=[a,a+\frac{|I|}{C}]$, $I'_r=[b-\frac{|I|}{C},b]$ where $C$ is to be chosen later and define two functions $g_1$, $g_2$ as follows (see Figure \ref{fig:fg}):
\begin{equation*}
g_1(x)=\Bigg\{
\begin{array}{ll}
-\frac{C}{|I|}(x-a)& \text{if }x\in (-\infty, a)\\
0&\text{if }x\in I'\cup I'_l\cup I'_r\\
\frac{C}{|I|}(x-b)& \text{if }x\in (b,\infty)
\end{array}
\end{equation*}
\begin{equation*}
g_2(x)=\Bigg\{
\begin{array}{ll}
-\frac{C}{|I|}(x-a)& \text{if }x\in (-\infty, a+\frac{|I|}{C})\\
-1&\text{if }x\in I'\\
\frac{C}{|I|}(x-b)& \text{if }x\in (b-\frac{|I|}{C},\infty)
\end{array}
\end{equation*}
Define $$Y_1=\sum_{i=1}g_1(\lambda_i),\ Y_2=\sum_{i=1}g_2(\lambda_i).$$ Applying Lemma \ref{GZconcentration} with $T=\frac{\delta}{8} n\int_{I}\rho_{sc}(t)dt$ for $Y_j$ and using the estimation for $\int_I\rho(t)dt$ as above, we have
$${\hbox{\bf P}}(|Y_j-{\hbox{\bf E}}(Y_j)|\ge \frac{\delta}{8} n\int_{I}\rho_{sc}(t)dt)\le4\exp(-c_2\frac{\delta^2n^2|I|^5}{K^2C^2}).$$
Let $Y=Y_1-Y_2$, then
$${\hbox{\bf P}}(|Y-{\hbox{\bf E}}(Y)|\ge \frac{\delta}{4} n\int_{I}\rho_{sc}(t)dt)\le O(\exp(-c_2\frac{\delta^2n^2|I|^5}{K^2C^2})).$$
We have ${\hbox{\bf E}}(Z-Y)\le {\hbox{\bf E}}(N_{I'_l}+N_{I'_r})$. A similar argument as in the proof of the upper bound (using the convergence rate of G\"{o}tze and Tikhomirov) shows
$${\hbox{\bf E}}(Y)\ge{\hbox{\bf E}}(Z)-n\int_{I'_l\cup I'_r}\rho_{sc}(t)dt-\beta n\sqrt{\frac{M_4}{n}}>E(Z)- \frac{\delta}{4}n\int_I\rho_{sc}(t)dt.$$
Therefore with probability at least $1-O(\exp(-c_2\frac{\delta^2n^2|I|^5}{K^2C^2}))$, we have
$$Z\ge Y\ge {\hbox{\bf E}}(Y)- \frac{\delta}{4}n\int_I\rho_{sc}(t)dt > {\hbox{\bf E}}(Z)-\frac{\delta}{2} n\int_I\rho_{sc}(t)dt,$$
and by the convergence rate, with probability at least $1-O(\exp(-c2\frac{\delta^2n^2|I|^5}{K^2C^2}))$
$$Z>(1-\delta)n\int_I\rho_{sc}(t)dt.$$
Thus, Theorem \ref{thm:ESD-con-general} is proved.
\endproof
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.52]{f1f2.pdf
\hfill{
\includegraphics[scale=0.52]{g1g2.pdf}
\caption{Auxiliary functions used in the proof}
\label{fig:fg}
\end{figure}
\section{Infinity norm of the eigenvectors}
\subsection{Small perturbation lemma}
$A_n$ is the adjacency matrix of $G(n,p)$. In the proofs of Theorem \ref{Delocal} and Theorem \ref{DelocalBulk}, we actually work with the eigenvectors of a perturbed matrix $$A_n+\epsilon N_n,$$ where $\epsilon=\epsilon(n) >0$ can be arbitrarily small and $N_n$ is a symmetric random matrix whose upper triangular elements are independent with a standard Gaussian distribution.
The entries of $A_n+\epsilon N_n$ are continuous and thus with probability 1, the eigenvalues of $A_n+\epsilon N_n$ are simple. Let $$\mu_1 <\ldots <\mu_n$$ be the ordered eigenvalues of $A_n+\epsilon N_n$, which have a unique orthonormal system of eigenvectors $\{w_1,\ldots,w_n\}$. By the Cauchy interlacing principle, the eigenvalues of $A_n+\epsilon N_n$ are different from those of its principle minors, which satisfies a condition of Lemma \ref{EigenEntry}.
Let $\lambda_i$'s be the eigenvalue of $A_n$ with multiplicity $k_i$ defined as follows: $$\ldots \lambda_{i-1} < \lambda_{i}=\lambda_{i+1}=\ldots=\lambda_{i+k_i}<\lambda_{i+k_i+1}\ldots$$
By Weyl's theorem, one has for every $1\le j\le n$,
\begin{equation} \label{eq:weyl}
|\lambda_j - \mu_j | \le \epsilon || N_n ||_{\text{op}} = O(\epsilon \sqrt{n})
\end{equation}
Thus the behaviors of eigenvalues of $A_n$ and $A_n+\epsilon N_n$ are essentially the same by choosing $\epsilon$ sufficiently small. And everything (except Lemma \ref{EigenEntry}) we used in the proofs of Theorem \ref{Delocal} and Theorem \ref{DelocalBulk} for $A_n$ also applies for $A_n+\epsilon N_n$ by a continuity argument. We will not distinguish $A_n$ from $A_n+\epsilon N_n$ in the proofs.
The following lemma will allow us to transfer the eigenvector delocaliztion results of $A_n+\epsilon N_n$ to those of $A_n$ at some expense.
\begin{lem}\label{lem:perturb}
In the notations of above, there exists an orthonormal basis of eigenvectors of $A_n$, denoted by $\{u_1,\ldots,u_n \}$, such that for every $1\le j \le n$, $$||u_j||_{\infty} \le || w_j ||_{\infty} +\alpha(n),$$
where $\alpha(n)$ can be arbitrarily small provided $\epsilon(n)$ is small enough.
\end{lem}
\begin{proof}
First, since the coefficients of the characteristic polynomial of $A_n$ are integers, there exists a positive function $l(n)$ such that either $|\lambda_s- \lambda_t|=0$ or $|\lambda_s-\lambda_t| \ge l(n)$ for any $1\le s,t \le n$.
By (\ref{eq:weyl}) and choosing $\epsilon$ sufficiently small, one can get
$$|\mu_i- \lambda_{i-1}| > l(n) ~~ \text{and} ~~ | \mu_{i+k_i} -\lambda_{i+k_i+1}| > l(n)$$
For a fixed index $i$, let $E$ be the eigenspace corresponding to the eigenvalue $\lambda_i$ and $F$ be the subspace spanned by $\{w_i,\ldots,w_{i+k_i} \}$. Both of $E$ and $F$ have dimension $k_i$. Let $P_E$ and $P_F$ be the orthogonal projection matrices onto $E$ and $F$ separately.
Applying the well-known Davis-Kahan theorem (see \cite{SS} Section IV, Theorem 3.6) to $A_n$ and $A_n+\epsilon N_n$, one gets
$$|| P_E- P_F ||_{\text{op} } \le \frac{\epsilon || N_n ||_{\text{op} }} {l(n)} := \alpha(n),$$
where $\alpha(n)$ can be arbitrarily small depending on $\epsilon.$
Define $v_j=P_F w_j \in E$ for $i \le j \le i+k_i$, then we have $||v_j- w_j||_2 \le \alpha(n)$. It is clear that $\{v_i,\ldots,v_{k_i}\}$ are eigenvectors of $A_n$ and
$$||v_j||_{\infty} \le ||w_j||_{\infty} + ||v_j - w_j||_2 \le ||w_j||_{\infty} + \alpha(n).$$
By choosing $\epsilon$ small enough such that $n\alpha(n) < 1/2$, $\{v_i,\ldots,v_{k_i}\}$ are linearly independent. Indeed, if $\sum_{j=i}^{k_i} c_j v_j=0$, one has for every $i \le s \le i+k_i$, $\sum_{j=i}^{k_i} c_j \langle P_F w_j , w_s \rangle=0$, which implies $c_s = -\sum_{j=i}^{k_i} c_j \langle P_F w_j - w_j, w_s\rangle$. Thus
$|c_s| \le \alpha(n) \sum_{j=i}^{k_i} |c_j|,$ summing over all $s$, we can get $\sum_{j=i}^{k_i} |c_j| \le k\alpha(n) \sum_{j=i}^{k_i} |c_j|$ and therefore $ c_j=0$.
Furthermore the set $\{v_i,\ldots,v_{k_i}\}$ is 'almost' an orthonormal basis of $E$ in the sense that
\begin{equation*}
\begin{split}
|~ ||v_s||_2 -1 ~| & \le ||v_s- w_s||_2 \le \alpha(n) ~~~~~\text{for any $i \le s \le i+k_i$ }\\
\\
|\langle v_s, v_t \rangle| &=|\langle P_F w_s, P_F w_t\rangle| \\
&=|\langle P_F w_s -w_s, P_F w_t\rangle + \langle w_s, P_F w_t - w_t\rangle | \\
&= O( \alpha(n) ) ~~~~~~~~\text{for any $i \le s\neq t \le i+k_i$ }\\
\end{split}
\end{equation*}
We can perform a Gram-Schmidt process on $\{v_i,\ldots,v_{k_i}\}$ to get an orthonormal system of eigenvectors $\{u_i,\ldots,u_{k_i} \}$ on $E$ such that $$||u_j||_{\infty} \le || w_j ||_{\infty} +\alpha(n),$$
for every $i \le j \le i+k_i$.
We iterate the above argument for every distinct eigenvalue of $A_n$ to obtain an orthonormal basis of eigenvectors of $A_n$.
\end{proof}
\subsection{Auxiliary lemmas}
\begin{lem}
\emph{(Lemma 41, \cite{tvrandom})}
\label{EigenEntry}
Let $$B_n=
\left(
\begin{array}{cc}
a & X^* \\
X & B_{n-1}
\end{array}
\right)
$$
be a $n\times n$ symmetric matrix for some $a\in \mathbb{C}$ and $X \in \mathbb{C}^{n-1}$, and let
$\left(
\begin{array}{cc}
x\\
v
\end{array}
\right)$ be a eigenvector of $B_n$ with eigenvalue $\lambda_i(B_n)$, where $x\in \mathbb{C}$ and $v \in \mathbb{C}^{n-1}$. Suppose that none of the eigenvalues of $B_{n-1}$ are equal to $\lambda_i(B_n)$. Then
$$|x|^2= \frac{1}{1+\sum_{j=1}^{n-1} (\lambda_j (B_{n-1}) - \lambda_i(B_n))^{-2} |u_j (B_{n-1})^* X|^2},$$
where $u_j(B_{n-1})$ is a unit eigenvector corresponding to the eigenvalue $\lambda_j (B_{n-1}).$
\end{lem}
The \textit{Stieltjes transform} $s_n(z)$ of a symmetric matrix $W$ is defined for $z \in \mathbb{C}$ by the formula
$$s_n(z):=\frac{1}{n} \displaystyle\sum_{i=1}^{n} \frac{1}{\lambda_i(W)-z}.$$ It has the following alternate representation:
\begin{lem}
\emph{(Lemma 39, \cite{tvrandom})}
\label{StieTran}
Let $W=(\zeta_{ij})_{1\le i,j\le n}$ be a symmetrix matrix, and let $z$ be a complex number not in the spectrum of $W$. Then we have
$$s_n(z)=\frac{1}{n} \displaystyle\sum_{k=1}^{n} \frac{1}{\zeta_{kk}-z- a^*_k (W_k -zI)^{-1} a_k }$$
where $W_k$ is the $(n-1) \times (n-1)$ matrix with the $k^\text{th}$ row and column of $W$ removed, and $a_k \in \mathbb{C}^{n-1}$ is the $k^\text{th}$ column of $W$ with the $k^\text{th}$ entry removed.
\end{lem}
We begin with two lemmas that will be needed to prove the main results. The first lemma, following the paper \cite{tvrandom} in Appendix B, uses Talagrand's inequality. Its proof is presented in the Appendix \ref{appendix:projection}.
\begin{lem}
\label{ConcenLem}
Let $Y=(\zeta_1,\ldots,\zeta_n) \in \mathbb{C}^n$ be a random vector whose entries are i.i.d. copies of the random variable $\zeta=\xi-p$ (with mean $0$ and variance $\sigma^2$). Let $H$ be a subspace of dimension $d$ and $\pi_H$ the orthogonal projection onto H. Then
$$\textbf{P}(|\parallel \pi_H (Y) \parallel -\sigma \sqrt{d} |\ge t) \le 10 \exp(-\frac{t^2}{4}).$$
In particular,
\begin{equation} \label{eq:1.1}
\parallel \pi_H(Y) \parallel= \sigma \sqrt{d}+O( \omega(\sqrt{\log n}) )
\end{equation}
with overwhelming probability.
\end{lem}
The following concentration lemma for $G(n,p)$ will be a key input to prove Theorem \ref{DelocalBulk}.
Let $B_n=\frac{1}{\sqrt{n}\sigma} A_n$
\begin{lem}[Concentration for ESD in the bulk]
\emph{(Concentration for ESD in the bulk)}
\label{ConBulkAdj}
Assume $p={g(n)\log n}/{n}$. For any constants $\varepsilon, \delta > 0$ and any interval $I$ in $[-2+\varepsilon, 2-\varepsilon]$ of width $|I|=\Omega( {\log^{2.2} g(n) \log n}/{np} )$, the number of eigenvalues $N_I$ of $B_n$ in $I$ obeys the concentration estimate $$|N_I(B_n) - n \displaystyle\int_I {{}\rho}_{sc}(x)\,dx| \le {\delta} n |I|$$ with overwhelming probability.
\end{lem}
The above lemma is a variant of Corollary \ref{cor:ESDconvrate}. This lemma allows us to control the ESD on a smaller interval and the proof, relying on a projection lemma (Lemma \ref{ConcenLem}), is a different approach. The proof is presented in Appendix \ref{appendix:concentration}.
\subsection{Proof of Theorem \ref{Delocal}:}
Let $\lambda_n(A_n)$ be the largest eigenvalue of $A_n$ and $u=(u_1,\ldots,u_n)$ be the corresponding unit eigenvector. We have the lower bound $\lambda_n(A_n) \ge np$. And if $np=\omega(\log n)$, then the maximum degree $\Delta = (1+o(1))np$ almost surely (See Corollary 3.14, \cite{bollobas}).
For every $1\le i \le n$, $$\lambda_n(A_n) u_i = \sum_{j \in N(i)} u_j,$$
where $N(i)$ is the neighborhood of vertex $i$. Thus, by Cauchy-Schwarz inequality,
$$|| u ||_{\infty}=\text{max}_i \frac{| \sum_{j \in N(i)} u_j |}{\lambda_n(A_n)} \le \frac{\sqrt{ \Delta } }{\lambda_n(A_n) } = O(\frac{1}{\sqrt{np} }).$$
Let $B_n=\frac{1}{\sqrt{n} \sigma} A_n$. Since the eigenvalues of $W_n=\frac{1}{\sqrt{n} \sigma } (A_n - p J_n)$ are on the interval $[-2,2]$, by Lemma \ref{EigenDiff}, $\{\lambda_1(B_n), \ldots, \lambda_{n-1}(B_n) \} \subset [-2,2] $.
Recall that $np = g(n)\log n$. By Corollary \ref{cor:ESDconvrate}, for any interval $I$ with length at least $(\frac{\log (np)}{{\delta}^4 (np)^{1/2}})^{1/5}$(say $\delta=0.5$),with overwhelming probability, if $I \subset [-2+\kappa,2-\kappa]$ for some positive constant $\kappa$, one has $N_I(B_n)= \Theta(n \int_{I} \rho_{sc}(x) dx) = \Theta(n |I|)$; if $I$ is at the edge of $[-2,2]$, with length $o(1)$, one has $N_I(B_n)=\Theta(n\int_{I} \rho_{sc}(x) dx) = \Theta(n |I|^{3/2})$. Thus we can find a set $J \subset \{1,\ldots,n-1\}$ with $|J| =\Omega( n |I_0|$) or $|J| =\Omega(n |I_0|^{3/2})$ such that $|\lambda_j(B_{n-1})-\lambda_i(B_n)| \ll |I_0|$ for all $j \in J$, where $B_{n-1} $ is the bottom right $(n-1) \times (n-1)$ minor of $B_n$. Here we take $|I_0|=(1/g(n)^{1/20})^{2/3}$. It is easy to check that $|I_0| \ge (\frac{\log (np)}{{\delta}^4 (np)^{1/2}})^{1/5}$.
By the formula in Lemma \ref{EigenEntry}, the entry of the eigenvector of $B_n$ can be expressed as
\begin{equation} \label{eq:entry}
\begin{split}
|x|^2 &=\displaystyle\frac{1}{1+\sum_{j=1}^{n-1} (\lambda_j (B_{n-1}) - \lambda_i(B_n))^{-2} |u_j (B_{n-1})^* \frac{1}{\sqrt{n}\sigma}X|^2} \\
&\le \frac{1}{1+\sum_{j \in J} (\lambda_j (B_{n-1}) - \lambda_i(B_n))^{-2} |u_j (B_{n-1})^* \frac{1}{\sqrt{n}\sigma}X|^2} \\
&\le \frac{1}{1+\sum_{j \in J} n^{-1}|I_0|^{-2} |u_j (B_{n-1})^* \frac{1}{\sigma}X|^2} = \frac{1}{1+ n^{-1}|I_0|^{-2} ||\pi_{H}(\frac{X}{\sigma})||^2}\\
&\le \frac{1}{1+{n^{-1}|I_0|^{-2}}{|J|}}
\end{split}
\end{equation}
with overwhelming probability, where $H$ is the span of all the eigenvectors associated to $J$ with dimension $\text{dim}(H)=\Theta(|J|)$, $\pi_{H}$ is the orthogonal projection onto $H$ and $X \in \mathbb{C}^{n-1}$ has entries that are iid copies of $\xi$. The last inequality in (\ref{eq:entry}) follows from
Lemma \ref{ConcenLem} (by taking $t=g(n)^{1/10}\sqrt{\log n}$) and the relations $$||\pi_H(X)||=||\pi_H(Y+p \bold{1}_n)|| \ge ||\pi_{H_1} (Y+p\bold{1}_n)|| \ge ||\pi_{H_1} (Y)|| .$$ Here $Y=X-p\bold{1}_n$ and $H_1=H \cap H_2$, where $H_2$ is the space orthogonal to the all 1 vector $\bold{1}_n$. For the dimension of $H_1$, $\text{dim}(H_1)\ge \text{dim}(H)-1$ .
\medskip
Since either $|J|=\Omega (n |I_0|$) or $|J| =\Omega( n |I_0|^{3/2})$, we have ${n^{-1}|I_0|^{-2}}{|J|} =\Omega( {|I_0|}^{-1}$) or ${n^{-1}|I_0|^{-2}}{|J|} =\Omega( {|I_0|}^{-1/2}$). Thus $|x|^2 =O( |I_0|)$ or $|x|^2 =O( \sqrt{|I_0|})$. In both cases, since $|I_0| \rightarrow 0$, it follows that $|x|=o(1)$.
\hfill $\Box$
\subsection{Proof of Theorem \ref{DelocalBulk}}
With the formula in Lemma \ref{EigenEntry}, it suffices to show the following lower bound
\begin{equation} \label{eq:1.11}
\sum_{j=1}^{n-1} (\lambda_j (B_{n-1}) - \lambda_i(B_n))^{-2} |u_j (B_{n-1})^* \frac{1}{\sqrt{n}\sigma}X|^2 \gg \frac{np}{\log^{2.2} g(n) \log n}
\end{equation}
with overwhelming probability, where $B_{n-1} $ is the bottom right $n-1 \times n-1$ minor of $B_n$ and $X \in \mathbb{C}^{n-1}$ has entries that are iid copies of $\xi$. Recall that $\xi$ takes values $1$ with probability $p$ and $0$ with probability $1-p$, thus $\mathbb{E} \xi=p, \mathbb{V}ar{\xi}=p(1-p)={\sigma}^2$.
\medskip
By Theorem \ref{ConBulkAdj}, we can find a set $J \subset \{1,\ldots,n-1\}$ with $|J| \gg \frac{\log^{2.2} g(n)\log n}{p}$ such that $|\lambda_j(B_{n-1})-\lambda_i(B_n)| =O(\log^{2.2} g(n)\log n/{np})$ for all $j \in J$. Thus in (\ref{eq:1.11}), it is enough to prove
$$\displaystyle\sum_{j \in J} |u_j(B_{n-1})^T \frac{1}{\sigma}X|^2= ||\pi_{H}(\frac{X}{\sigma})||^2 \gg |J| $$
or equivalently
\begin{equation}
||\pi_{H}(X)||^2 \gg {\sigma}^2 |J|
\end{equation}
with overwhelming probability, where $H$ is the span of all the eigenvectors associated to $J$ with dimension $\text{dim}(H)=\Theta(|J|)$.
\medskip
Let $H_1=H\cap H_2$, where $H_2$ is the space orthogonal to $\bold{1}_n$. The dimension of $H_1$ is at least $\text{dim}(H)-1$.
Denote $Y=X- p \bold{1}_n$. Then the entries of $Y$ are iid copies of $\zeta$. By Lemma \ref{ConcenLem}, $$||\pi_{H_1}(Y)||^2 \gg {\sigma}^2 |J|$$ with overwhelming probability.
\medskip
Hence, our claim follows from the relations
$$||\pi_H(X)||=||\pi_H(Y+p \bold{1}_n)|| \ge ||\pi_{H_1} (Y+p\bold{1}_n)|| = ||\pi_{H_1} (Y)||.$$
\hfill $\Box$
\begin{appendices}
In this appendix, we complete the proofs of Theorem \ref{thm:SCL1}, Lemma \ref{ConcenLem} and Lemma \ref{ConBulkAdj}.
\section{Proof of Theorem \ref{thm:SCL1}}\label{appendix:SCL}
\medskip
We will show that the semicircle law holds for $M_n$. With Lemma \ref{EigenDiff}, it is clear that Theorem \ref{thm:SCL1} follows Lemma \ref{ESD} directly.
The claim actually follows as a special case discussed in the paper \cite{ch04spectra}. Our proof here uses a standard moment method.
\begin{lem}
\label{ESD}
For $p=\omega(\frac{1}{n})$, the empirical spectral distribution (ESD) of the matrix $W_n=\frac{1}{\sqrt{n}} M_n$ converges in distribution to the semicircle law which has a density ${{}\rho}_{sc}(x)$ with support on $[-2,2]$, $${{\rho}}_{sc}(x) := \frac{1}{2 \pi } \sqrt{4 -x^2}.$$
\end{lem}
Let ${\eta}_{ij} $ be the entries of $M_n={\sigma}^{-1} (A_n-p J_n)$. For $i = j$, $\eta_{ij}=-p/\sigma$; and for $i\not= j$, $\eta_{ij}$ are iid copies of random variable $\eta$, which takes value $(1-p)/ \sigma$ with probability $p$ and takes value $-p/ \sigma$ with probability $1-p$.
$$\textbf{E}\eta = 0, \textbf{E}\eta^2 =1, \textbf{E}\eta^s= O \left(\frac{1}{(\sqrt{p})^ {s-2}} \right)~\text{for}~ s\ge 2.$$
\medskip
For a positive integer $k$, the $k^{\text{th}}$ moment of ESD of the matrix $W_n$ is
$$\displaystyle\int x^k dF_n^{W}(x)= \frac{1}{n} \textbf{E}( \text{Trace}({W_n}^k)),$$
and the $k^{\text{th}}$ moment of the semicircle distribution is
$$\displaystyle\int_{-2}^{2} x^k \rho_{\text{sc}}(x) dx.$$
On a compact set, convergence in distribution is the same as convergence of moments. To prove the theorem, we need to show, for every fixed number $k$,
\begin{equation}\label{conver}
\frac{1}{n} \textbf{E}( \text{Trace}({W_n}^k)) \rightarrow \displaystyle\int_{-2}^{2} x^k \rho_{\text{sc}}(x) dx, \ \text{as}~ n \rightarrow \infty.
\end{equation}
For $k=2m+1$, by symmetry, $\displaystyle\int_{-2}^{2} x^k \rho_{\text{sc}}(x) dx=0$.
For $k=2m$,
\begin{equation*}
\begin{split}
\displaystyle\int_{-2}^{2} x^k \rho_{\text{sc}}(x) dx
&= \frac{1}{\pi}\int_{0}^{2} x^k \sqrt{4-x^2} dx
= \frac{2^{k+2}}{\pi}\int_{0}^{\pi/2} {\sin^k{\theta}} {\cos^2{\theta}} dx\\
&= \frac{2^{k+2}}{\pi} \frac{\Gamma(\frac{k+1}{2})\Gamma(\frac{3}{2})}{\Gamma(\frac{k+4}{2})}
=\frac{1}{m+1} \dbinom{2m}{m}
\end{split}
\end{equation*}
Thus our claim (\ref{conver}) follows by showing that
\begin{equation}\label{trace}
\displaystyle\frac{1}{n} \textbf{E}( \text{Trace}({W_n}^k)) =\left\{ \begin{array}{ll}
O(\frac{1}{\sqrt{np}}) & \mbox{if $k = 2m+1$};\\
\\
\frac{1}{m+1} {{2m}\choose{m}} + O(\frac{1}{np}) & \mbox{if $k = 2m$}.\end{array} \right.
\end{equation}
We have the expansion for the trace of ${W_n}^k$,
\begin{equation}\label{expan}
\begin{split}
\displaystyle\frac{1}{n} \textbf{E}( \text{Trace}({W_n}^k))
& =\frac{1}{n^{1+k/2}} \textbf{E}( \text{Trace}({\sigma}^{-1} M_n)^k )\\
& =\frac{1}{n^{1+k/2}} \sum_{1 \le i_1, \ldots, i_k \le n} \textbf{E} \eta_{i_1 i_2} \eta_{i_2 i_3} \cdots \eta_{i_k i_1}
\end{split}
\end{equation}
Each term in the above sum corresponds to a closed walk of length $k$ on the complete graph $K_n$ on $\{1,2, \ldots, n \}$. On the other hand, $\eta_{ij}$ are independent with mean 0. Thus the term is nonzero if and only if every edge in this closed walk appears at least twice. And we call such a walk a \emph{good} walk. Consider a \emph{good} walk that uses $l$ different edges $e_1, \ldots, e_l$ with corresponding multiplicities $m_1, \ldots, m_l$, where $l \le m$, each $m_h \ge 2$ and $m_1+\ldots+m_l = k$. Now the corresponding term to this \emph{good} walk has form $$\textbf{E} \eta_{e_1}^{m_1}\cdots \eta_{e_l}^{m_l}.$$
\medskip
Since such a walk uses at most $l+1$ vertices, a naive upper bound for the number of \emph{good} walks of this type is $n^{l+1} \times l^k$.
\medskip
When $k=2m+1$, recall $\textbf{E}\eta^s= \Theta \left({(\sqrt{p})^ {2-s}} \right)~\text{for}~ s\ge 2$, and so
\begin{equation*}
\begin{split}
\displaystyle\frac{1}{n} \textbf{E}( \text{Trace}({W_n}^k))
& = \frac{1}{n^{1+k/2}} \sum_{l=1}^{m} \sum_{\text{\emph{good} walk of l edges}} \textbf{E} \eta_{e_1}^{m_1}\cdots \eta_{e_l}^{m_l}\\
& \le \frac{1}{n^{m+3/2}} \sum_{l=1}^{m} n^{l+1} l^k (\frac{1}{ \sqrt{p}})^{m_1-2}\ldots (\frac{1}{ \sqrt{p}})^{m_l-2}\\
& = O(\frac{1}{\sqrt{np}}).
\end{split}
\end{equation*}
When $k=2m$, we classify the \emph{good} walks into two types. The first kind uses $l \le m-1$ different edges. The contribution of these terms will be
\begin{equation*}
\begin{split}
\frac{1}{n^{1+k/2}} \sum_{l=1}^{m-1} \sum_{\text{1st kind of \emph{good} walk of l edges}} \textbf{E} \eta_{e_1}^{m_1}\cdots \eta_{e_l}^{m_l}
& \le \frac{1}{n^{1+m}} \sum_{l=1}^{m} n^{l+1} l^k (\frac{1}{ \sqrt{p}})^{m_1-2}\ldots (\frac{1}{ \sqrt{p}})^{m_l-2}\\
& = O(\frac{1}{{np}}).
\end{split}
\end{equation*}
The second kind of \emph{good} walk uses exactly $l=m$ different edges and thus $m+1$ different vertices. And the corresponding term for each walk has form $$\textbf{E} \eta_{e_1}^{2}\cdots \eta_{e_l}^{2}=1.$$
The number of this kind of \emph{good} walk is given by the following result in the paper (\cite{bai08method}, Page 617--618):
\begin{lem}
The number of the second kind of \emph{good} walk is $$\displaystyle\frac{n^{m+1}(1+O(n^{-1}))}{m+1} \dbinom{2m}{m}.$$
\end{lem}
Then the second conclusion of (\ref{conver}) follows.
\section{ Proof of Lemma \ref{ConcenLem}:}\label{appendix:projection}
The coordinates of $Y$ are bounded in magnitude by $1$. Apply Talagrand's inequality to the map $Y \rightarrow ||\pi_H(Y)||$, which is convex and $1$-Lipschitz. We can conclude
\begin{equation} \label{eq:1.2}
\textbf{P}(|\parallel \pi_H (Y) \parallel -M(\parallel \pi_H (Y) \parallel)| \ge t) \le 4 \exp(-\frac{t^2}{16})
\end{equation}
where $M(\parallel \pi_H (Y) \parallel)$ is the median of $\parallel \pi_H (Y) \parallel$.
\medskip
Let $P=(p_{ij})_{1 \le i,j \le n}$ be the orthogonal projection matrix onto $H$. One has trace$P^2=$trace$P= \sum_i p_{ii}=d$ and $|p_{ii}| \le 1$, as well as,
$${\parallel \pi_H (Y) \parallel}^2 = \sum_{1 \le i,j \le n}^{} p_{ij} \zeta_i \zeta_j = \sum_{i=1}^{n} p_{ii} \zeta_{i}^2 + \sum_{i\neq j} p_{ij} \zeta_{i} \zeta_j$$
and
$$\mathbf{E}{\parallel \pi_H (Y) \parallel}^2=\mathbf{E}(\sum_{i=1}^{n} p_{ii} \zeta_{i}^2)+\mathbf{E}(\sum_{i\neq j} p_{ij} \zeta_{i} \zeta_j)= \sigma^2 d.$$
\medskip
Take $L=4/\sigma$. To complete the proof, it suffices to show
\begin{equation} \label{eq:1.3}
|M(\parallel \pi_H (Y) \parallel) - \sigma \sqrt{d}| \le L\sigma.
\end{equation}
Consider the event $\mathcal{E}_{+}$ that $\parallel \pi_H (Y) \parallel \ge \sigma L + \sigma \sqrt{d}$, which implies that ${\parallel \pi_H (Y) \parallel}^2 \ge \sigma^2(L^2 + 2L\sqrt{d} +d^2).$
\medskip
Let $S_1= \sum_{i=1}^{n}p_{ii} (\zeta_{i}^2-\sigma^2)$ and $S_2=\sum_{i \neq j}^{} p_{ij}\zeta_i \zeta_j$.
\medskip
Now we have
$$\textbf{P} (\mathcal{E}_{+}) \le \textbf{P} (\sum_{i=1}^{n} p_{ii}\zeta_{i}^2 \ge \sigma^2 d + L\sqrt{d}\sigma^2) + \textbf{P} (\sum_{i \neq j}^{} p_{ij} \zeta_i \zeta_j \ge \sigma^2 L\sqrt{d}).$$
By Chebyshev's inequality,
$$\textbf{P} (\sum_{i=1}^{n} p_{ii}\zeta_{i}^2 \ge \sigma^2 d + L\sqrt{d}\sigma^2) =\textbf{P} (S_1 \ge L\sqrt{d}\sigma^2)) \le \frac{\textbf{E}(|S_1|^2)}{L^2 d \sigma^4},$$
where $\textbf{E}(|S_1|^2) = \textbf{E} (\sum_{i} p_{ii} (\zeta_{i}^2 -\sigma^2) )^2 = \sum_{i} p_{ii}^2 \textbf{E} (\zeta_i^4- \sigma^4) \le d\sigma^2(1-2\sigma^2)$.
\medskip
Therefore, $\textbf{P}(S_1 \ge L\sqrt{d}\sigma^4) \le \displaystyle\frac{d\sigma^2(1-2\sigma^2)}{L^2 d \sigma^4} < \frac{1}{16}.$
\medskip
On the other hand, we have $\textbf{E}(|S_2|^2)=\textbf{E}(\sum_{i\neq j}^{} p_{ij}^2 \zeta_i^2 \zeta_j^2) \le \sigma^4 d$ and
$$\textbf{P} (\sum_{i \neq j}^{} p_{ij} \zeta_i \zeta_j \ge \sigma^2 L\sqrt{d}) = \textbf{P} (S_2 \ge L\sqrt{d} \sigma^2) \le \frac{\textbf{E}(|S_2|^2)}{L^2 d \sigma^4} < \frac{1}{10}$$
\medskip
It follows that $\textbf{E}(\mathcal{E}_{+}) < 1/4$ and hence $M(\parallel \pi_H (Y) \parallel) \le L\sigma +\sqrt{d} \sigma.$
\medskip
For the lower bound, consider the event $\mathcal{E}_{-} $ that $\parallel \pi_H (Y) \parallel \le \sqrt{d}\sigma-L\sigma$ and notice that
$$\textbf{P}(\mathcal{E}_{-}) \le \textbf{P} (S_1 \le - L\sqrt{d} \sigma^2) + \textbf{P} (S_2 \le - L\sqrt{d} \sigma^2).$$
\medskip
The same argument applies to get $M(\parallel \pi_H (Y) \parallel) \ge \sqrt{d}\sigma-L\sigma$. Now the relations (\ref{eq:1.2}) and (\ref{eq:1.3}) together imply (\ref{eq:1.1}).
\endproof
\section{Proof of Lemma \ref{ConBulkAdj}: } \label{appendix:concentration}
Recall the normalized adjacency matrix
$$M_n=\frac{1}{\sigma}(A_n-p J_n),$$
where $J_n=\bold{1}_n \bold{1}^T_n $ is the $n \times n$ matrix of all $1$'s, and let $W_n=\frac{1}{\sqrt{n}}M_n$.
\begin{lem}
\label{Lem1}
For all intervals $I \subset \mathbb{R}$ with $|I| = \omega{(\log n)}/{np}$, one has
$$N_I(W_n)=O(n|I|)$$ with overwhelming probability.
\end{lem}
The proof of Lemma \ref{Lem1} uses the same proof as in the paper \cite{tvrandom} with the relation (\ref{eq:1.1}).
Actually we will prove the following concentration theorem for $M_n$. By Lemma \ref{EigenDiff}, $| N_I (W_n)-N_I(B_n) | \le 1$, therefore Lemma \ref{ConBulk} implies Lemma \ref{ConBulkAdj}.
\begin{lem}
\label{ConBulk}
(Concentration for ESD in the bulk) Assume $p={g(n)\log n}/{n}$. For any constants $\varepsilon, \delta > 0$ and any interval $I$ in $[-2+\varepsilon, 2-\varepsilon]$ of width $|I|=\Omega( g(n)^{0.6}\log n/{np} )$, the number of eigenvalues $N_I$ of $W_n=\frac{1}{\sqrt{n}} M_n$ in $I$ obeys the concentration estimate $$|N_I(W_n) - n \displaystyle\int_I {{}\rho}_{sc}(x)\,dx| \le {\delta} n |I|$$ with overwhelming probability.
\end{lem}
To prove Theorem \ref{ConBulk}, following the proof in \cite{tvrandom}, we consider the \textit{Stieltjes transform}
$$s_n(z):=\frac{1}{n} \displaystyle\sum_{i=1}^{n} \frac{1}{\lambda_i(W_n)-z},$$ whose imaginary part
$$\text{Im} s_n(x+\sqrt{-1} \eta)=\frac{1}{n} \displaystyle\sum_{i=1}^{n} \frac{\eta}{\eta^2 + (\lambda_i(W_n)-x)^2}>0$$in the upper half-plane $\eta >0$.
\medskip
The semicircle counterpart
$$s(z):= \displaystyle\int_{-2}^{2} \frac{1}{x-z} \rho_{sc}(x)\,dx=\frac{1}{2\pi}\displaystyle\int_{-2}^{2} \frac{1}{x-z} \sqrt{4-x^2}\,dx,$$ is the unique solution to the equation $$s(z)+\frac{1}{s(z)+z}=0$$ with $\text{Im} s(z) >0$.
\medskip
The next proposition gives control of ESD through control of Stieltjes transform (we will take $L=2$ in the proof):
\begin{prop}
\emph{(Lemma 60, \cite{tvrandom})}
\label{ESDStie}
Let $L, \varepsilon, \delta >0$. Suppose that one has the bound $$|s_n(z)-s(z)| \le \delta$$ with (uniformly) overwhelming probability for all $z$ with $|\text{Re}(z)| \le L$ and $\text{Im}(z) \ge \eta$. Then for any interval $I$ in $[-L+\varepsilon, L-\varepsilon]$ with $|I| \ge \text{max}(2\eta, \frac{\eta}{\delta} \log \frac{1}{\delta})$, one has $$|N_I- n \displaystyle\int_I {\rho}_{sc}(x)\,dx| \le \delta n |I|$$ with overwhelming probability.
\end{prop}
By Proposition \ref{ESDStie}, our objective is to show
\begin{equation}
|s_n(z)-{s}(z)| \le {\delta}
\end{equation}
with (uniformly) overwhelming probability for all $z$ with $|\text{Re}(z)| \le 2$ and $\text{Im}(z) \ge {\eta}$, where $$\eta =\frac{\log^2 g(n) \log n}{np}.$$
In Lemma \ref{StieTran}, we write
\begin{equation} \label{eq:1.6}
s_n(z)= \frac{1}{n} \displaystyle{\sum_{k=1}^{n} \frac{1}{-\frac{{\zeta}_{kk}}{ \sqrt{n}\sigma}-z- Y_k }}
\end{equation}
where $$Y_k=a^*_k (W_{n,k} -zI)^{-1} a_k,$$
$W_{n,k}$ is the matrix $W_n$ with the $k^{\text{th}}$ row and column removed, and $a_k$ is the $k^{\text{th}}$ row of $W_n$ with the $k^{\text{th}}$ element removed.
\medskip
The entries of $a_k$ are independent of each other and of $W_{n,k}$, and have mean zero and variance $1/n$. By linearity of expectation we have $$\mathbf{E}(Y_k|W_{n,k})=\frac{1}{n}\text{Trace}(W_{n,k}-zI)^{-1}=(1-\frac{1}{n})s_{n,k}(z)$$ where $$s_{n,k}(z)= \frac{1}{n-1} \displaystyle{\sum_{i=1}^{n-1} \frac{1}{\lambda_i (W_{n,k}) -z}}$$ is the \textit{Stieltjes transform} of $W_{n,k}$. From the Cauchy interlacing law, we get$$\displaystyle{|{} s_n(z)- (1-\frac{1}{n}) {} s_{n,k}(z)|= O(\frac{1}{n} \int_{\mathbb{R}} \frac{1}{|x-z|^2}\,dx) =O(\frac{1}{n\eta})}=o(1),$$
and thus $$\mathbf{E}(Y_k|W_{n,k})=s_n(z)+o(1).$$
In fact a similar estimate holds for $Y_k$ itself:
\begin{prop}
\label{YProp}
For $1 \le k \le n$, $Y_k= \mathbf{E}(Y_k|W_{n,k}) +o(1)$ holds with (uniformly) overwhelming probability for all $z$ with $|\text{Re}(z)| \le 2$ and $\text{Im}(z) \ge {\eta}$.
\end{prop}
Assume this proposition for the moment. By hypothesis, $|\frac{{\zeta}_{kk}}{ \sqrt{n}\sigma}|=|\frac{-p}{\sqrt{n}\sigma}|=o(1)$. Thus in (\ref{eq:1.6}), we actually get
\begin{equation}
{} s_n(z) + \frac{1}{n} \displaystyle{\sum_{k=1}^{n} \frac{1}{s_n(z) +z + o(1)}}=0
\end{equation}
with overwhelming probability. This implies that with overwhelming probability either $s_n(z)=s(z)+o(1)$ or that $s_n(z)=-z+o(1)$. On the other hand, as Im$s_n(z)$ is necessarily positive, the second possibility can only occur when Im$z=o(1)$. A continuity argument (as in \cite{erdos09local}) then shows that the second possibility cannot occur at all and the claim follows.
\medskip
Now it remains to prove Proposition \ref{YProp}.
\medskip
{\bf Proof of Proposition \ref{YProp}.} Decompose $$Y_k=\displaystyle{\sum_{j=1}^{n-1} \frac{|u_j (W_{n,k})^*a_k|^2}{\lambda_{j}(W_{n,k})-z}}$$
and evaluate
\begin{equation}
\begin{split}
Y_k- \mathbf{E}(Y_k|W_{n,k})&= Y_k- \displaystyle{(1-\frac{1}{n}) {} s_{n,k}(z)}+o(1)\\
&= \displaystyle{\sum_{j=1}^{n-1} \frac{|u_j (W_{n,k})^*a_k|^2- \frac{1}{n}}{\lambda_{j}(W_{n,k})-z}}+o(1)\\
&= \displaystyle{\sum_{j=1}^{n-1} \frac{R_j}{\lambda_{j}(W_{n,k})-z}}+o(1),
\end{split}
\end{equation}
where we denote $R_j=\displaystyle |u_j (W_{n,k})^*a_k|^2- \frac{1}{n}$, $\{u_j (W_{n,k})\}$ are orthonormal eigenvectors of $W_{n,k}$.
\medskip
Let $J \subset \{1,\ldots, n-1\}$, then
$$\displaystyle\sum_{j\in J} R_j =||P_H(a_k)||^2- \frac{\text{dim}(H)}{n}$$
where $H$ is the space spanned by $\{u_j (W_{n,k})\}$ for $j \in J$ and $P_H$ is the orthogonal projection onto $H$.
In Lemma \ref{ConcenLem}, by taking $t=h(n)\sqrt{\log n}$, where $h(n)=\log^{0.001} g(n) $, one can conclude with overwhelming probability
\begin{equation} \label{eq:1.9}
|\displaystyle\sum_{j\in J} R_j| \ll \frac{1}{n}\left(\frac{h(n)\sqrt{|J|\log n}}{\sqrt{p}}+\frac{h(n)^2 \log n}{p}\right).
\end{equation}
Using the triangle inequality,
\begin{equation} \label{eq:1.10}
\displaystyle\sum_{j\in J} |R_j| \ll \frac{1}{n}\left(|J| +\frac{h(n)^2 \log n}{p} \right)
\end{equation}
with overwhelming probability.
\medskip
Let $z=x+\sqrt{-1} {} \eta$, where $\eta =\log^2 g(n) \log n/{np}$ and $|x| \le 2 -\varepsilon$, define two parameters
$$\alpha = \frac{1}{\log^{4/3} g(n)} ~~~~~~~\text{and}~~~~~~~ \beta=\frac{1}{\log^{1/3} g(n) }.$$
\medskip
First, for those $j \in J$ such that $|\lambda_j(W_{n,k})-x| \le \beta \eta $, the function $\frac{1}{\lambda_j(W_{n,k})-x -\sqrt{-1}{} \eta}$ has magnitude $O(\frac{1}{{} \eta})$. From Lemma \ref{Lem1}, $|J| \ll n\beta \eta$, and so the contribution for these $j \in J$ is,
$$\displaystyle{|\sum_{j \in J}^{} \frac{R_j}{\lambda_{j}(W_{n,k})-z}|} \ll \frac{1}{n\eta} \left( n\beta \eta +\frac{h(n)^2 }{\log^2 g(n)} \right) = O(\frac{1}{ \log^{1/3}g(n) })=o(1).$$
\medskip
For the contribution of the remaining $j\in J$, we subdivide the indices as
$$a \le |\lambda_j(W_{n,k})-x| \le (1+\alpha)a$$
where $a=(1+\alpha)^l \beta \eta$, for $0 \le l \le L$, and then sum over $l$.
\medskip
For each such interval, the function $\frac{1}{\lambda_j(W_{n,k})-x -\sqrt{-1}{} \eta}$ has magnitude $O(\frac{1}{a})$ and fluctuates by at most $O(\frac{\alpha}{a})$. Say $J$ is the set of all $j$'s in this interval, thus by Lemma \ref{Lem1}, $|J| =O( n\alpha a)$. Together with bounds (\ref{eq:1.9}), (\ref{eq:1.10}), the contribution for these $j$ on such an interval,
\begin{equation*}
\begin{split}
\displaystyle{|\sum_{j \in J}^{} \frac{R_j}{\lambda_{j}(W_{n,k})-z}|} &\ll \frac{1}{an} \left(\frac{h(n)\sqrt{|J|\log n}}{\sqrt{p}}+\frac{h(n)^2 \log n}{p} \right)+ \frac{\alpha}{an} \left(|J| +\frac{h(n)^2 \log n}{p} \right)\\
&=O\left( \frac{ \sqrt{\alpha} }{ \sqrt{(1+\alpha)^l} } \frac{ h(n) }{ \sqrt{\beta}\log g(n) } + \frac{h^2(n)}{ (1+\alpha)^l \beta \log^2 g(n) } +\alpha^2 \right)\\
&= O\left( \frac{1}{ \sqrt{\alpha\beta} } \frac{h(n)}{ \log g(n) } + \alpha \log \frac{1}{\beta \eta} \right)
\end{split}
\end{equation*}
\medskip
Summing over $l$ and noticing that $(1+\alpha)^{L} \eta/g(n)^{1/4} \le 3$, we get
\begin{equation*}
\begin{split}
\displaystyle{|\sum_{j \in J, \text{all} J}^{} \frac{R_j}{\lambda_{j}(W_{n,k})-z}|} &= O\left( \frac{1}{\sqrt{\alpha \beta}} \frac{h(n)}{\log g(n)} + \alpha \log\frac{1}{\beta \eta} \right)\\
&=O\left( \frac{h(n)}{\log^{1/6} g(n)} \right) =o(1).
\end{split}
\end{equation*}
\hfill $\Box$
\end{appendices}
{\bf\Large Acknowledgement.} The authors thank Terence Tao for useful conversations.
\bibliographystyle{plain}
| {
"timestamp": "2010-12-01T02:02:55",
"yymm": "1011",
"arxiv_id": "1011.6646",
"language": "en",
"url": "https://arxiv.org/abs/1011.6646",
"abstract": "In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\\rightarrow \\infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erdős-Rényi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial.",
"subjects": "Combinatorics (math.CO); Probability (math.PR)",
"title": "Sparse random graphs: Eigenvalues and Eigenvectors",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986979510652134,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7092379419369511
} |
https://arxiv.org/abs/1609.02181 | Geometry and a natural symplectic structure of phase tropical hypersurfaces | First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure. | \section{introduction}
In this paper we deal with smooth algebraic hypersurfaces in the complex projective space $\mathbb{CP}^n$. So, let $V$ be a smooth hypersurface in $\mathbb{CP}^n$ of degree $d$. Recall that for a fixed degree, generically a hypersurface in the projective space is smooth and transverse to all coordinate hyperplanes and all their intersections. Moreover, hypersurfaces in $\mathbb{CP}^n$ with the same degree are all diffeomorphic, and if we equip these hypersurfaces with the Fubini-Study symplectic form on $\mathbb{CP}^n$ then they are also symplectomorphic. We denote by $\mathring{V}$ the intersection $V\cap (\mathbb{C}^*)^n$ where $(\mathbb{C}^*)^n$ is the complement of the coordinate hyperplanes in $\mathbb{CP}^n$. In this case, $\mathring{V}$ is given by some polynomial equation. One can degenerate the complex standard structure of the complex algebraic torus to a worst possible degeneration, called ``maximal degeneration'' by M. Kontsevich and Y. Soibelman (see \cite{KS-00} and \cite{KS-04}), and see what kind of geometry can have a degeneration of our variety $\mathring{V}$.
After taking the logarithm, $(\mathbb{C}^*)^n$ degenerates or, in other words, collapse onto $\mathbb{R}^n$, and our hypersurface onto a balanced rational polyhedral complex $\Gamma$ called {\em{tropical variety}}. One can ask the following question:{\em{ What kind of geometry one can have on a nice lifting in $(\mathbb{C}^*)^n$ of this balanced rational polyhedral complex?}} This paper give an answer to this question using tools from tropical and phase tropical geometry.
\vspace{0.2cm}
Tropical geometry is a recent area of mathematics that can be seen as a limiting aspect (or ``degeneration'') of algebraic geometry. Where complex curves viewed as Riemann surfaces turn to metric graphs (one dimensional combinatorial object), and $n$-dimensional complex varieties turn to $n$-dimensional polyhedral complexes with some properties such as the balancing condition. In other words, tropical varieties are finite dimensional polyhedral complexes with some additional properties. As example, the tropical projective space $\mathbb{TP}^1$ is a smooth projective tropical variety homeomorphic to the segment. In general, the tropical projective space $\mathbb{TP}^n$ is a smooth projective tropical variety homeomorphic to the $n$-dimensional simplex. Moreover, as in the classical algebraic geometry, a projective tropical $n$-variety $V$ is a certain $n$-dimensional polyhedral complex in $\mathbb{TP}^N$. One of the most interesting projective tropical varieties are obtained by the tropical limit of a family of projective algebraic varieties $V_t$ with $1\leq t<\infty$ and $t$ tends to $\infty$. To be more precise, they are the limit of amoebas where amoebas of algebraic (or analytic) varieties are their image under the logarithm with base a real number $t$. For example, every tropical hypersurface is provided by such way. Tropical objects are some how, the image of a classical objects under the logarithm with base infinity, they are also called non-Archimedean amoebas.
\vspace{0.2cm}
Phase tropical varieties are some lifting of tropical varieties in the complex algebraic torus. More precisely, for
any strictly positive real number $t$ we define the self diffeomorphism $H_t$ of $(\mathbb{C}^*)^n$. This defines a new complex structure $J_t$ on $(\mathbb{C}^*)^n$ denoted by $J_t$ different from the standard complex structure if $t\ne e^{-1}$. One way to define phase tropical varieties, is to take the limit $\mathring{V}_\infty$ (with respect to the Hausdorff metric on compact sets in $(\mathbb{C}^*)^n$) of a family of $J_t$-holomorphic varieties $\{\mathring{V}_t\}_{t\in [e^{-1}, \infty)}$ when $t$ goes to $\infty$. First, in case of hypersurfaces, we prove that if the hypersurfaces $\mathring{V}_t$ are smooth with same degree (i.e. their defining polynomials have the same Newton polytope $\Delta$), then for a sufficently large $t$ the $\mathring{V}_t$'s are diffeomorphic to their degeneration $\mathring{V}_\infty$, and the compactification $M_\infty$ of $\mathring{V}_\infty$ in the toric variety $X_\Delta$ associated to $\Delta$ (see Subsection 3.3 for the precise definition of $X_\Delta$) have the same properties, and we have the following:
\begin{theorem}\label{A}
Let $\mathring{V}_t\subset (\mathbb{C}^*)^n$ be a family of smooth complex algebraic hypersurfaces with a fixed degree $\Delta$, and denote by $\mathring{V}_\infty$ the phase tropical hypersurface associated to the family $\{\mathring{V}_t\}_t$ (i.e., the limit of $H_t(\mathring{V}_t)$ when $t$ goes to $\infty$). Then for a sufficiently large $t\gg 0$ the following statements hold:
\begin{itemize}
\item[(i)]\, The hypersurface $\mathring{V}_t$ is diffeomorphic to $\mathring{V}_\infty$;
\item[(ii)]\, The compactification $M_\infty$ of $\mathring{V}_\infty$ in the toric variety $X_\Delta$ associated to $\Delta$ is diffeomorphic to $V_t$, where $V_t$ is the closure of $\mathring{V}_t$ in $X_\Delta$.
\end{itemize}
\end{theorem}
\vspace{0.2cm}
Moreover, using the fact that pairs-of-pants possess a natural symplectic structure which gives rise to the standard symplectic structure on the complex projective space $\mathbb{CP}^n$ after compactification (i.e. collapsing the pair-of-pants boundary), and the gluing of pairs-of-pants can be done in a natural way symplectically, we obtain a natural symplectic structure on all our phase tropical hypersurface.
Let $(\mathring{V}_t, \iota_t^*(\omega))\subset ((\mathbb{C}^*)^n, \omega)$ be a family of smooth symplectic hypersurfaces where $\iota_t$ is the inclusion map $\iota_t: \mathring{V}_t\hookrightarrow (\mathbb{C}^*)^n$,
and $\omega$ is the symplectic form on the complex algebraic torus $(\mathbb{C}^*)^n$ defined by:
\begin{equation}\label{(1)}
\omega = \frac{1}{2\sqrt{-1}}\sum_{i=1}^n\frac{dz_i}{z_i}\wedge\frac{d\overline{z}_i}{\overline{z}_i}.
\end{equation}
Moreover, assume that the phase tropical hypersurface $\mathring{V}_\infty$ which the limit (with respect to the Hausdorff metric on compact sets in $(\mathbb{C}^*)^n$) exists
and is equipped with the natural symplectic structure (i.e., with the natural symplectic form $\omega_{nat}$) constructed by Theorem \ref{B}. One can ask the following natural question: {\em{Are $(\mathring{V}_t, \iota_t^*(\omega))$ and $(\mathring{V}_\infty, \omega_{nat})$ symplectomorphic?}}
\vspace{0.2cm}
The following theorem gives an affirmative answer to this question:
\vspace{0.2cm}
\begin{theorem}\label{B}
Let $\mathring{V}_t\subset (\mathbb{C}^*)^n$ be a family of smooth complex algebraic hypersurfaces with a fixed degree $\Delta$, and denote by $\mathring{V}_\infty$ the phase tropical hypersurface associated to the family $\{\mathring{V}_t\}_t$ (i.e., the limit of $H_t(\mathring{V}_t)$ when $t$ goes to $\infty$). With notations as above, and for a sufficiently large $t\gg 0$ the following statements hold:
\begin{itemize}
\item[(i)]\, The hypersurface $\mathring{V}_\infty$ possesses a natural smooth symplectic structure;
\item[(ii)]\, the hypersurfaces $(\mathring{V}_t, \iota_t^*(\omega))$ and $(\mathring{V}_\infty, \omega_{nat})$ are symplectomorphic.
\end{itemize}
\end{theorem}
\vspace{0.2cm}
We will use the natural logarithm i.e. with base the Napier's constant $e$, so that the Archimedean amoeba of
a subvariety of the complex torus $(\mathbb C^*)^n$ is its image under the coordinatewise logarithm map. Recall that amoebas were introduced by Gelfand, Kapranov, and Zelevinsky in 1994 \cite{GKZ-94}. The coamoeba of a subvariety of $(\mathbb C^*)^n$ is its image under the coordinatewise argument map to the real torus $(S^1)^n$.
Coamoebas were introduced by Passare in a talk in 2004 (see \cite{NS-11} and \cite{NS-13} for more details about coamoebas).
\vspace{0.2cm}
This paper is organized as follows. In Section 2, we explain preliminary results in this area. In Section 3, we define phase tropical hypersurface and describe tropical localization. In Section 4, we describe examples of coamoebas and phase tropical hypersurfaces. In Section 5, we give the proof of Theorem \ref{A}. In Section 6, we construct in a natural way a symplectic structure on phase tropical varieties which proves Theorem \ref{B}.
\section{Preliminaries}
In this section we recall basic concepts of tropical hyperurfaces relevant for our paper. For the general case we can see \cite{MS-15} with more details. We consider algebraic hypersurfaces $\mathring{V}$ in the complex algebraic torus $(\mathbb{C}^*)^n$, where $\mathbb{C}^*=\mathbb{C}\setminus \{ 0\}$ and $n\geq 1$ an integer. This means that $\mathring{V}$ is the zero locus of a polynomial:
\begin{equation}\label{(1)}
f(z) =\sum_{\alpha\in \mathop{\rm supp}\nolimits (f)} a_{\alpha}z^{\alpha}, \quad
z^{\alpha}=z_1^{\alpha_1}z_2^{\alpha_2}\ldots z_n^{\alpha_n},
\end{equation}
where each $a_{\alpha}$ is a non-zero complex number and $\mathop{\rm supp}\nolimits (f)$ is a
finite subset of $\mathbb{Z}^n$, called the support of the polynomial
$f$, and its convex hull in $\mathbb{R}^n$ is called the Newton polytope
of $f$ that we denote by $\Delta_f$.
Moreover, we assume that $\mathop{\rm supp}\nolimits (f)\subset
\mathbb{N}^n$ and $f$ has no factor of the form $z^{\alpha}$.
\vspace{0.3cm}
The {\em amoeba} $\mathscr{A}_f$ of an algebraic variety $\mathring{V}\subset (\mathbb{C}^*)^n$ is by definition (see M. Gelfand, M.M. Kapranov and A.V. Zelevinsky \cite{GKZ-94}) the image of $\mathring{V}$ under the map :
\[
\begin{array}{ccccl}
\mathop{\rm Log}\nolimits&:&(\mathbb{C}^*)^n&\longrightarrow&\mathbb{R}^n\\
&&(z_1,\ldots ,z_n)&\longmapsto&(\log |z_1| ,\ldots ,\log |
z_n|).
\end{array}
\]
\vspace{0.2cm}
Let $\mathbb{K}$ be the field of Puiseux series with real powers, which is the field of series $\displaystyle{a(t) =\sum_{j\in A_a}\xi_jt^j}$ with $\xi_j\in \mathbb{C}^*$ and $A_a$ is a well-ordered subset of $\mathbb{R}$ (it means any of its subsets has a smallest element). It is well known that the field $\mathbb{K}$ is algebraically closed of characteristic zero. Moreover, it has a non-Archimedean valuation $\mathop{\rm val}\nolimits (a) = - \min A_a$:
\[
\left\{ \begin{array}{ccc}
\mathop{\rm val}\nolimits (ab)&=& \mathop{\rm val}\nolimits (a) + \mathop{\rm val}\nolimits (b) \\
\mathop{\rm val}\nolimits (a + b)& \leq& \max \{ \mathop{\rm val}\nolimits (a) ,\, \mathop{\rm val}\nolimits (b) \} ,
\end{array}
\right.
\]
and we set $\mathop{\rm val}\nolimits (0) = -\infty$. Let $g\in \mathbb{K}[z_1,\ldots ,z_n]$ be a polynomial as in \eqref{(1)}.
If $<,>$ denotes the scalar product in $\mathbb{R}^n$, then the following piecewise affine linear convex function $\displaystyle{g_{trop}(x) = \max_{\alpha\in\mathop{\rm supp}\nolimits (g) } \{ \mathop{\rm val}\nolimits (a_{\alpha}) + <\alpha , x> \}}$, which is in the same time the Legendre transform of the function $\nu :\mathop{\rm supp}\nolimits (g)\rightarrow \mathbb{R}$ defined by $\nu (\alpha ) = \min A_{a_{\alpha}}$, is called the {\em tropical polynomial} associated to $g$.
\begin{definition} The tropical hypersurface $\Gamma_g$ is the set of points in $\mathbb{R}^n$ where the tropical polynomial
$g_{trop}$ is not smooth (called the corner locus of $g_{trop}$).
\end{definition}
\noindent We have the following Kapranov's theorem (see \cite{K-00}):
\begin{theorem}[\cite{K-00}, Kapranov] The tropical hypersurface $\Gamma_g$ defined by the tropical polynomial $g_{trop}$ is the subset of $\mathbb{R}^n$ image under the valuation map of the algebraic hypersurface with defining polynomial $g$.
\end{theorem}
\noindent $\Gamma_g$ is also called the non-Archimedean amoeba of the zero locus of $g$ in $(\mathbb{K}^*)^n$.
\vspace{0.1cm}
Let $g$ be a polynomial as above, $\Delta$ its Newton polytope, and $\tilde{\Delta}$ its extending Newton polytope, i.e.,
$\tilde{\Delta} := \mbox{convexhull} \{ (\alpha , r )\in \mathop{\rm supp}\nolimits (g)\times \mathbb{R} \mid \, r\geq \min A_{a_{\alpha}} \}$. Let us extend the above function $\nu$ (defined on $\mathop{\rm supp}\nolimits (g)$) to all $\Delta$ as follow:
\[
\begin{array}{ccccl}
\nu&:&\Delta&\longrightarrow&\mathbb{R}\\
&&\alpha&\longmapsto&\min \{ r\mid\, (\alpha ,r)\in \tilde{\Delta} \}.
\end{array}
\]
By taking the linear subsets of the lower boundary of $\tilde{\Delta}$, it is clear that the linearity domains of $\nu$ define a convex subdivision $\tau = \{\Delta_1,\ldots ,\Delta_l\}$ of $\Delta$. Let $y= <x,v_i>+r_i$ be the equation of the hyperplane $Q_i\subset \mathbb{R}^n\times\mathbb{R}$ containing points of coordinates $(\alpha ,\nu (\alpha ))$ with $\alpha \in \mathop{\rm Vert}\nolimits (\Delta_i)$.
\noindent There is a duality between the subdivision $\tau$ and the subdivision of $\mathbb{R}^n$ induced by $\Gamma_g$,
where each connected component of
$\mathbb{R}^n\setminus \Gamma_g$ is dual to some vertex of $\tau_f$ and each $k$-cell of $\Gamma_g$ is dual to some
$(n-k)$-cell of $\tau$. In particular, each $(n-1)$-cell of $\Gamma_g$ is dual to some edge of $\tau$.
If $x\in E_{\alpha\beta}^*\subset \Gamma_g$, then $<\alpha , x> -\nu (\alpha ) = <\beta , x> -\nu (\beta )$,
so $<\alpha -\beta , x - v_i> = 0$. This means that $v_i$ is a vertex of $\Gamma_g$ dual to some $\Delta_i$ having
$E_{\alpha\beta}$ as edge.
\begin{definition}
A tropical hypersurface $\Gamma\subset \mathbb{R}^n$ is smooth if and only if its dual subdivision is a triangulation where the Euclidean volume of every triangle is equal to $\frac{1}{n!}$.
\end{definition}
\noindent
Let $\mathring{V}\subset (\mathbb{C}^*)^n$ be an algebraic hypersurface defined by a polynomial
${f (z) = \sum_{\alpha_i\in A}a_{\alpha_i}z^{\alpha_i}}$, with support $A = \{ \alpha_1,\ldots , \alpha_l,\alpha_{l+1},\ldots ,\alpha_r\} \subset \mathbb{Z}^n$, and $A' = \{ \alpha_{l+1},\ldots ,\alpha_r\} = \mathop{\rm Im}\nolimits (\mathop{\rm ord}\nolimits )$ where $\mathop{\rm ord}\nolimits$ is the order mapping from the set of complement components of the amoeba $\mathscr{A}$ of $\mathring{V}$ to $\Delta\cap\mathbb{Z}^n$ (see \cite{FPT-00}).
It was shown by Mikael Passare and Hans Rullg\aa (see \cite{PR1-04}) that the spine $\Gamma$ of the amoeba
$\mathscr{A}$ is a non-Archimedean amoeba defined by the
tropical polynomial
$$
f_{trop}(x) = \max_{\alpha\in A'}\{ c_{\alpha}+<\alpha , x>\} ,
$$
where $c_{\alpha}$ are a constants defined by:
\begin{equation}\label{(2)}
c_{\alpha} = {\mathbb R} \left( \frac{1}{(2\pi i)^n}\int_{\mathop{\rm Log}\nolimits^{-1}(x)}\log
\left|\frac{f(z)}{z^{\alpha}}\right| \frac{dz_1\wedge \ldots \wedge
dz_n}{z_1\ldots z_n}\right)
\end{equation}
where $x\in E_{\alpha}$,\, $z = (z_1,\cdots ,z_n)\in (\mathbb{C}^*)^n$. In other words, the spine of
$\mathscr{A}$ is defined as the set of points in $\mathbb{R}^n$
where the piecewise affine linear function $f_{trop}$ is not
differentiable.
Let us denote by $\tau$ the convex subdivision of $\Delta$ dual
to the tropical variety $\Gamma$. Then the set of vertices of $\tau$ is precisely the image of the order mapping (i.e., $A'$).
By duality, this means that the convex subdivision $\tau =\cup _{i=l+1}^r \Delta_{v_i}$ of $\Delta$ is determined by a piecewise affine linear map $\nu : \Delta\longrightarrow \mathbb{R}$ so that:
\begin{itemize}
\item[(i)]\, $\nu_{\mid \Delta_{v_i}}$ is affine linear for each $v_i$,
\item[(ii)]\, if $\nu_{\mid U}$ is affine linear for some open set $U\subset
\Delta$, then there exists $v_i$ such that $U\subset \Delta_{v_i}$.
\item[(iii)]\, $\nu (\alpha ) = - c_\alpha$ for any $\alpha \in \mathop{\rm Im}\nolimits (\mathop{\rm ord}\nolimits )$.
\end{itemize}
\noindent We define the {\em generalized $s$-Passare-Rullg\aa rd function} by the following:
\begin{definition}
Let $s = (s_1,\ldots ,s_l)\in \mathbb{R}_+^l$ and $\nu_{PR}^s : A
\longrightarrow \mathbb{R}$ be
the function, called the generalized $s$-Passare-Rullg\aa rd function, is defined by:
\[
\nu_{PR}^s(\alpha ) = \left\{ \begin{array}{ll}
-c_{\alpha}& \mbox{if\, $\alpha\in \mathop{\rm Im}\nolimits (\mathop{\rm ord}\nolimits )$}\\
<\alpha_j , a_{v}>+b_{v}+s_j &\mbox{if\, $\alpha = \alpha_j$ \,for\, $j=1,\ldots ,l$},
\end{array}
\right.
\]
where $\alpha_j\in\Delta_{v}$,\, $\Delta_{v}\in \tau$ and $y = <x , a_{v}>+b_{v}$
is the equation of the hyperplane in $\mathbb{R}^n\times\mathbb{R}$ containing the
points of coordinates $(\beta ; -c_{\beta})$ with $\beta\in \mathop{\rm Vert}\nolimits (\Delta_{v})$.
\end{definition}
\noindent Assume that we have a hypersurface $\mathring{V}\subset (\mathbb{C}^*)^n$ defined by the polynomial ${f(z)=\sum_{\alpha\in A}a_\alpha z^{\alpha}}$ with $a_\alpha\in \mathbb{C}^*$, $A$ a finite subset of $\mathbb{Z}^n$ and $z^{\alpha} = z_1^{\alpha_1}z_2^{\alpha_2}\ldots z_n^{\alpha_n}$. We denote by $\Delta$ the convex hull of $A$ in $\mathbb{R}^n$ which is the Newton polytope of $f$. We can consider the family of
hypersurfaces $\mathring{V}_{f_{(t;\, s)}}\subset (\mathbb{C}^*)^n $ defined by the following family of polynomials :
\begin{equation}\label{(3)}
f_{(t;\, s)}(z) =
\sum_{\alpha\in A}\xi_\alpha t^{\nu^s_{PR}(\alpha )} z^{\alpha}
\end{equation}
with $\xi_\alpha = a_\alpha e^{\nu^s_{PR}(\alpha )}$, and we view this family as a deformation of $f$.
Let us denote by $\mathscr{C}oh_A(\Delta )$ the set of coherent (i.e. convex) triangulations of $\Delta$ such that the set of vertices of all its elements is contained in $A$. For each $\tau\in \mathscr{C}oh_A(\Delta )$, assume $\nu : \Delta \rightarrow \mathbb{R}$ is a convex function defining $\tau$.
Let $f^{(\tau )}$ be the non-Archimedean polynomial defined by:
$$
f^{(\tau )}(z) = \sum_{\alpha\in A} a_{\alpha}t^{\nu (\alpha )}z^{\alpha}.
$$
We denote by $co\mathscr{A}_{\mathbb{C}}(f)$ (resp. $co\mathscr{A}_{\mathbb{K}}(f)$) the complex coamoeba (resp. non-Archimedean coamoeba) of the hypersurface with defining polynomial $f$.
\section{Phase tropical hypersurfaces}
\subsection{Phase tropical hypersurfaces} ${}$
\vspace{0.2cm}
For every strictly positive real number $t$ we define the self diffeomorphism $H_t$ of $(\mathbb{C}^*)^n$ by :
\[
\begin{array}{ccccl}
H_t&:&(\mathbb{C}^*)^n&\longrightarrow&(\mathbb{C}^*)^n\\
&&(z_1,\ldots ,z_n)&\longmapsto&\left(\mid z_1\mid^{-\frac{1}{\log t}}\dfrac{z_1}{\mid
z_1\mid},\ldots ,\mid z_n\mid^{-\frac{1}{\log t}}\dfrac{z_n}{\mid z_n\mid} \right).
\end{array}
\]
This defines a new complex structure on $(\mathbb{C}^*)^n$
denoted by $J_t = (dH_t)^{-1}\circ J\circ (dH_t)$ where $J$ is the
standard complex structure.
\noindent A $J_t$-holomorphic hypersurface $\mathring{V}_t$ is a
holomorphic hypersurface with respect to the $J_t$ complex structure on
$(\mathbb{C}^*)^n$. It is equivalent to say that $\mathring{V}_t = H_t(\mathring{V})$ where
$\mathring{V}\subset (\mathbb{C}^*)^n$ is an holomorphic hypersurface for the
standard complex structure $J$ on $(\mathbb{C}^*)^n$.
Recall that the Hausdorff distance between two closed subsets $A,
B$ of a metric space $(E, d)$ is defined by:
$$
d_H(A,B) = \max \{ \sup_{a\in A}d(a,B),\sup_{b\in B}d(A,b)\}.
$$
Here $E =\mathbb{R}^n\times (S^1)^n$ is equipped with the distance
defined as the product of the
Euclidean metric on $\mathbb{R}^n$ and the flat metric on $(S^1)^n$.
\begin{definition} A phase tropical hypersurface $\mathring{V}_{\infty}\subset
(\mathbb{C}^*)^n$ is the limit (with respect to the Hausdorff
metric on compact sets in $(\mathbb{C}^*)^n$) of a sequence of a
$J_t$-holomorphic hypersurfaces $\mathring{V}_t\subset (\mathbb{C}^*)^n$ when
$t$ tends to $\infty$.
\end{definition}
We have an algebraic definition of phase tropical hypersurfaces in case of curves (called complex tropical curves)(see \cite{M2-04}) as follows :
\noindent Let $ a \in \mathbb{K}^*$ be the Puiseux series
${a = \sum_{j\in A_a}\xi_jt^j}$ with $\xi\in \mathbb{C}^*$
and $A_a\subset \mathbb{R}$ is a well-ordered set with smallest element
Then we have a non-Archimedean valuation on $\mathbb{K}$ defined by
$\mathop{\rm val}\nolimits (a ) = - \min A_a$. We complexify the valuation
map as follows :
\[
\begin{array}{ccccl}
w&:&\mathbb{K}^*&\longrightarrow&\mathbb{C}^*\\
&&a&\longmapsto&w(a ) = e^{\mathop{\rm val}\nolimits (a )+i\arg (\xi_{-\mathop{\rm val}\nolimits
(a )})}.
\end{array}
\]
Let $\mathop{\rm Arg}\nolimits$ be the argument map $\mathbb{K}^*\rightarrow S^1$ defined by: for any Puiseux series ${a = \sum_{j\in A_a}\xi_jt^j}$, we set $\mathop{\rm Arg}\nolimits (a) = e^{i\arg (\xi_{-\mathop{\rm val}\nolimits (a)})}$ (this map extends the map $\mathbb{C}^*\rightarrow S^1$ defined by $\rho e^{i\theta} \mapsto e^{i\theta}$ which we denote by $\mathop{\rm Arg}\nolimits$).
\vspace{0.3cm}
\noindent Applying this map coordinatewise we obtain a map :
\[
\begin{array}{ccccl}
W:&(\mathbb{K}^*)^n&\longrightarrow&(\mathbb{C}^*)^n
\end{array}
\]
\begin{theorem}[Mikhalkin, 2002] The set $\mathring{V}_{\infty}\subset (\mathbb{C}^*)^n$
is a phase tropical hypersurface if and only if there
exists an algebraic hypersurface
$\mathring{V}_{\mathbb{K}}\subset(\mathbb{K}^*)^n$ over $\mathbb{K}$ such that
$\overline{W(\mathring{V}_{\mathbb{K}})} = \mathring{V}_{\infty}$, where $\overline{W(\mathring{V}_{\mathbb{K}})}$ is the closure of $W(\mathring{V}_{\mathbb{K}})$ in $(\mathbb{C}^*)^n \approx \mathbb{R}^n\times (S^1)^n$ as a Riemannian manifold with metric defined by the standard Euclidean metric of $\mathbb{R}^n$ and the standard flat metric of the real torus.
\end{theorem}
Let $f_t(x)=\sum_{\alpha}a_\alpha t^{-v(\alpha)}z^j$ be a polynomial with a parameter $t$, and $\mathring{V}_t=\{f_t=0\}\subset (\mathbb C^*)^n$. The family of $f_t$ can be viewed as a single polynomial in $\mathbb K[z_1^{\pm 1},\cdots,z_n^{\pm 1}]$. We have the following theorems (see \cite{M2-04}, \cite{M3-04}, and \cite{R1-01}):
\begin{theorem}[Mikhalkin, Rullg{\aa}rd (2001)]
The amoebas $\mathscr{A}_t$ of $\mathring{V}_t$ converge in the Hausdorff metric to the non-archimedean amoeba $\mathscr{A}_{\mathbb K}$ when $t\to \infty$.
\end{theorem}
\begin{theorem}[Mikhalkin]
The sets $H_t(\mathring{V}_t)$ converge in the Hausdorff metric to $W(\mathring{V}_{\mathbb K})$ when $t\to\infty$.
\end{theorem}
\subsection{Tropical localization} ${}$
\vspace{0.2cm}
\noindent Let $\nu$ be the piecewise affine linear map defined in Section 2, and
$\tilde{\Delta}$ be the extended polyhedron of $\Delta$ associated
to $\nu$, that is the convex hull of the set $\{ (\alpha ,u)\in
\Delta\times \mathbb{R} \,|\, u\geq \nu (\alpha ) \}$. For any
$\Delta_{v_i}\in \tau$, let $\lambda (x) = <x, a_{v_i}> + b_{v_i}$ be the affine linear
map defined on $\Delta$ such that $\lambda_{\mid \Delta_{v_i}} = \nu_{\mid
\Delta_{v_i}}$ where $<~,~>$ is the scalar product in $\mathbb{R}^n$, $a_{v_i}=(a_{v_i,\, 1},\ldots ,a_{v_i,\, n})\in\mathbb{R}^n$ (which is the coordinates of the vertex of the spine $\Gamma$, dual to $\Delta_{v_i}$), and $b_{v_i}$ is a real number. Let $s\in \mathbb{R}_+^l$ as above and put $\nu ' = \nu_{PR}^{(s)} -\lambda$ and we define the family of polynomials
$\{ {f'}_{(t;\, s)} \}_{t\in (0,\, \frac{1}{e}]}$ by:
$$
f_{(t,s)} '(z) =\sum_{\alpha\in A}\xi_{\alpha}t^{\nu '(\alpha )}z^{\alpha},
$$
where $\xi_{\alpha}\in \mathbb{C}$. Then we have:
\begin{align*}
&f_{(t,s)} '(z) =t^{-b_v}\sum_{\alpha\in A}\xi_{\alpha}t^{\nu_{PR}^{(s)} (\alpha
)}(z_1t^{-a_{v_i,\, 1}})^{\alpha_1}\ldots (z_nt^{-a_{v_i,\,
n}})^{\alpha_n}\nonumber \\
&\phantom{f_{(t,s)} '(z)}=t^{-b_v}f_{(t;\, s)}\circ \Phi^{-1}_{\Delta_{v_i},\, t}(z),\nonumber
\end{align*}
where $f_{(t;\, s)}$ is the polynomial defined in \eqref{(3)}, and $\Phi_{\Delta_{v_i},\, t}$ is the self diffeomorphism of
$(\mathbb{C}^*)^n$ defined by:
\[
\begin{array}{ccccl}
\Phi_{\Delta_{v_i},\, t}&:&(\mathbb{C}^*)^n&\longrightarrow&(\mathbb{C}^*)^n\\
&&(z_1,\ldots ,z_n)&\longmapsto&(z_1t^{a_{v_i,\, 1}},\ldots ,z_nt^{a_{v_i,\,
n}} ).
\end{array}
\]
This means that the polynomials ${f'}_{(t;\, s)}$ and $f_{(t;\, s)}\circ
\Phi^{-1}_{\Delta_{v_i},\, t}$ define the same hypersurface. So we have:
$$
\mathring{V}_{{f'}_{(t;\, s)}} = \mathring{V}_{f_{(t;\, s)}\circ
\Phi^{-1}_{\Delta_{v_i},\, t}} = \Phi_{\Delta_{v_i},\, t}(\mathring{V}_{f_{(t;\, s)}}),
$$
where $\mathring{V}_g$ denotes algebraic hypersurface in $(\mathbb{C}^*)^n$ with defining polynomial $g$.
Let $U_{v_i}$ be a small ball in $\mathbb{R}^n$ with center the
vertex of $\Gamma_{(t;\, s)}$ dual to $\Delta_{v_i}$ where $\Gamma_{(t;\, s)}$ is the
spine of the amoeba $\mathscr{A}_{H_t(\mathring{V}_{f_{(t;\, s)}})}$ where $H_t$
denotes the self diffeomorphism of $(\mathbb{C}^*)^n$ defined as in Subsection 3.1,
and $\mathop{\rm Log}\nolimits_t = \mathop{\rm Log}\nolimits \circ H_t$.
Let $f_{(t;\, s)}^{\Delta_{v_i}}$ be the truncation of $f_{(t;\, s)}$ to $\Delta_{v_i}$, and
$\mathring{V}_{\infty ,\,\Delta_{v_i}}$ be the complex tropical hypersurface with tropical coefficients of index $\alpha\in\Delta_{v_i}$ (i.e., $\mathring{V}_{\infty ,\,\Delta_{v_i}} = \lim_{t\rightarrow 0} H_t(\mathring{V}_{f_{(t;\, s)}^{\Delta_{v_i}}})$). Using Kapranov's theorem (see \cite{K-00}), we obtain the following Proposition (called a tropical localization by Mikhalkin, see \cite{M2-04}):
\begin{proposition} Let $s$ be in $\mathbb{R}_+^l$. For any $\varepsilon >0$ there exists $t_0$ such that if $t\geq t_0$ then the image under $\Phi_{\Delta_{v_i},\, t}\circ H_t^{-1}$ of $H_t(\mathring{V}_{f_{(t;\, s)}})\cap \mathop{\rm Log}\nolimits^{-1}(U_{v_i})$ is contained in the $\varepsilon$-neighborhood of the image under $\Phi_{\Delta_{v_i},\, t}\circ H_t^{-1}$ of the phase tropical hypersurface $\mathring{V}_{\infty ,\,\Delta_{v_i}}$ corresponding to the family $\{\mathring{V}_{f_{(t;\, s)}}\}_{t}$, with respect to the product metric in $(\mathbb{C}^*)^n\approx\mathbb{R}^n\times (S^1)^n$.
\end{proposition}
\begin{proof}
By decomposition of $f_{(t,s)} '$, we obtain:
\begin{equation}\label{(4)}
f_{(t,s)} '(z) =t^{-b_v}\sum_{\alpha\in \Delta_v\cap A}\xi_{\alpha}t^{\nu
(\alpha ) -<\alpha ,a_v>}z^{\alpha} \,\, +\,\, \sum_{\alpha\in
A\setminus \Delta_v} \xi_{\alpha}t^{\nu
(\alpha ) -<\alpha ,a_v>-b_v}z^{\alpha}.
\end{equation}
On the other hand, we have the following commutative diagram:
\begin{equation}\label{(5)}
\xymatrix{
(\mathbb{C}^*)^n\ar[rr]^{\Phi_{\Delta_v,t}}\ar[d]_{\mathop{\rm Log}\nolimits_t}&&
(\mathbb{C}^*)^n\ar[d]^{\mathop{\rm Log}\nolimits_t}\cr
\mathbb{R}^n\ar[rr]^{\phi_{\Delta_v}}&&\mathbb{R}^n,
\end{equation}
such that if $v=(a_{v,\, 1},\ldots ,a_{v,\, n})\in \mathbb{R}^n$ is
the vertex of the tropical hypersurface $\Gamma$ dual to the
element $\Delta_v$ of the subdivision $\tau$, then
$\phi_{\Delta_v}(x_1,\ldots , x_n)=(x_1-a_{v,\, 1},\ldots
,x_n-a_{v,\, n})$. Let $U_v$ be a small open ball in
$\mathbb{R}^n$ centered at $v$.
\noindent Assume that $\mathop{\rm Log}\nolimits_t(z)\in \phi_{\Delta_v}(U_v)$ and $z$ is not
singular in $\mathring{V}_{t}$. Then the second sum in \eqref{(4)} converges to zero
when $t$ goes to infinity, because by the choice of $z$ and $U_v$, the
tropical monomials in $f_{trop,\, (t,s)}'$, corresponding to lattice points
of $\Delta_v$, dominates the monomials corresponding to lattice points
of $A\setminus \Delta_v$. But the first sum in \eqref{(4)} is just a
polynomial defining the hypersurface $ \Phi_{\Delta_v,\, t} (V_{f_{(t,s)}^{\Delta_v}})$.
\noindent By the commutativity of diagram \eqref{(5)}, if
$z\in \mathring{V}_{f_t'}$ is such that $\mathop{\rm Log}\nolimits_t(z)\in \phi_{\Delta_v}(U_v)$
then $\mathop{\rm Log}\nolimits_t\circ \Phi_{\Delta_v,t}^{-1}(z)\in U_v$, and hence
$H_t( \Phi_{\Delta_v,t}^{-1}(z))\in \mathop{\rm Log}\nolimits^{-1}(U_v)$. So, the image under
$\Phi_{\Delta_v,\, t}\circ H_t^{-1}$ of
$H_t(\mathring{V}_{f_{(t,s)}})\cap \mathop{\rm Log}\nolimits^{-1}(U_v)$ is contained in an
$\varepsilon$-neighborhood of the image under $\Phi_{\Delta_v,\, t}\circ H_t^{-1}$ of
$H_t(\mathring{V}_{f_{(t,s)}^{\Delta_v}})$ for
sufficiently large $t$ and the proposition is proved because
$\mathring{V}_{\infty ,\,\Delta_v}$ is the limit when $t$ tends to $\infty$ of
the sequence of $J_t$-holomorphic hypersurfaces
$H_t(V_{f_{(t,s)}^{\Delta_v}})$ (by taking a discrete sequence $t_k$
converging to $\infty$ if necessary). In particular the set of arguments of $\mathring{V}_{\infty ,\, f}\cap \mathop{\rm Log}\nolimits^{-1}(v)$
is contained in the set of arguments of $V_{\infty ,\,\Delta_v}$ i.e., $\mathop{\rm Arg}\nolimits (\mathring{V}_{\infty ,\, f}\cap \mathop{\rm Log}\nolimits^{-1}(v))\subseteq \mathop{\rm Arg}\nolimits (\mathring{V}_{\infty ,\,\Delta_v})$. If it is not the case, we can get away too after applying $\Phi_{\Delta_v,\, t}\circ H_t^{-1}$ for sufficiently large $t$.
\end{proof}
\subsection{Toric varieties} ${}$
\vspace{0.2cm}
\noindent To every convex polyhedron $\Delta\subset \mathbb R^n$ with integer vertices, there is a complex toric variety ${X}_{\Delta}$ containing $(\mathbb C^*)^n$. Indeed, we can consider the Veronese embedding $\rho: (\mathbb C^*)^n \rightarrow \mathbb{CP}^{\# (\Delta\cap \mathbb Z^n)-1}$ defined by the monomial map associated to $\Delta\cap \mathbb Z^n$: $(z_1,\cdots,z_n)\mapsto z_1^{\alpha_1}z_2^{\alpha_2}\cdots z_n^{\alpha_n}$, for each $\alpha:=(\alpha_1,\cdots,\alpha_n)\in \Delta\cap \mathbb Z^n$; and $X_{\Delta}$ is defined as the closure of the image of $(\mathbb C^*)^n$. Then the Fubini-Study symplectic form on the projective spaces $\mathbb{CP}^{\# (\Delta\cap \mathbb Z^n)-1}$ defines a natural symplectic form on $X_{\Delta}$. In particular we obtain a symplectic form $\omega_{\Delta}$ on $(\mathbb C^*)^n$ invariant under the Hamiltonian action of the real torus $(S^1)^n$. This gives a moment map $\mu_{\Delta}$ with respect to $\omega_{\Delta}$:
\[
\begin{matrix}
\mu_{\Delta}&:&(\mathbb C^*)^n&\longrightarrow& \Delta\\
& &z &\mapsto &\dfrac{\displaystyle\sum_{\alpha\in \Delta\cap \mathbb Z^n} \sum_{i=1}^n \alpha_i|z_i^{2\alpha_i}|}{\displaystyle\sum_{\alpha\in \Delta\cap \mathbb Z^n}\sum_{i=1}^n|z_i^{2\alpha_i}|},
\end{matrix}
\]
which is an embedding with image the interior of $\Delta$.
\begin{equation}
\xymatrix{
(\mathbb{C}^*)^n\ar[rr]^{\mathop{\rm Log}\nolimits}\ar[dr]_{\mu_{\Delta}}&&\mathbb{R}^n\ar[dl]^{\Psi_\Delta}\cr
&\Delta.
}\nonumber
\end{equation}
The maps $\mathop{\rm Log}\nolimits$ and $\mu_{\Delta}$ both have orbits $(S^1)^n$ as fibers, and we obtain a reparametrization of $\mathbb{R}^n$ which we denote by $\Psi_{\Delta}$ (see \cite{GKZ-94}).
\begin{definition}
Let $\Gamma\subset \mathbb R^n$ be an $n$-dimensional balanced polyhedral complex, and $\Delta$ its dual convex lattice polyhedron. $\overline{\Gamma}\subset \Delta$ is the compactification of $\Gamma$ by taking $\Psi_{\Delta}(\Gamma)$ in $\Delta$. $\overline{\Gamma}\backslash \Psi_{\Delta}(\Gamma)$ is called the boundary of $\overline{\Gamma}$.
\end{definition}
Let $f$ be a Laurent polynomial in $\mathbb C[z_1^{\pm 1},\cdots , z_n^{\pm 1}]$, and $\Delta$ be its Newton polytope. Let $\mathring{V} :=\{z\in (\mathbb C^*)^n\,|\,f(z)=0\}$ be the hypersurface in $(\mathbb C^*)^n$ with defining polynomial $f$. Let $X_{\Delta}$ be the complex toric variety as defined before. We denote by $V$ the closure of the hypersurface $\mathring{V}$ in $X_{\Delta}$.
\vspace{0.2cm}
\noindent Let $\Delta$ be a compact convex lattice polyhedron such that the singularity of its corresponding toric variety $X_{\Delta}$ are on the vertices of $\Delta$. Let $(\mathbb C^*)^{\# (\Delta\cap \mathbb Z^n)}$ be the set of all polynomial $f(z)=\sum_{\alpha\in \Delta\cap\mathbb Z^n}a_{\alpha}z^{\alpha}$ such that $a_{\alpha}\ne 0$. Then for a generic choice of a polynomial, the closure $V$ in $X_{\Delta}$ of the zero set of $f$ is a smooth hypersurface transverse to all toric subvarieties $X_{\Delta'}$, corresponding to the faces $\Delta'\subset \Delta$. In particular, all such hypersurfaces $V$ are diffeomorphic, even symplectomorphic if they are equipped with the symplectic form coming from the one of $X_{\Delta}$.
\section{Examples of coamoebas and phase tropical hypersurfaces}
\begin{itemize}
\item[(a)] Let $\mathring{V}$ be the line in $(\mathbb{C}^*)^2$ defined by the polynomial
$f(z,w)=r_1e^{i\alpha_1}z+r_2e^{i\alpha_2}w+r_3e^{i\alpha_3}$ where $r_i$ are real positive numbers and $\alpha_1>\alpha_3>\alpha_2>0$. Then its coamoeba is as displayed in Figure 1. The equations of the external hyperplanes are given by $(1)$ \, $y = x + \alpha_1 - \alpha_2 + (2k+1)\pi$, \, $(2)$\, $x = \alpha_3 - \alpha_1 + (2l+1)\pi$, and $(3)$\, $y = \alpha_3 - \alpha_2 + (2m+1)\pi$ with $k,\, l$ and $m$ in $\mathbb{Z}$ (the external hyperplanes are seen in $\mathbb{R}^2$ the universal covering of the torus).
\begin{figure}[h!]
\begin{center}
\includegraphics[angle=0,width=0.5\textwidth]{Line-1.eps}\quad
\caption{The coamoeba of the line in $(\mathbb{C}^*)^2$ defined by the polynomial $f(z,w)=r_1e^{i\alpha_1}z+r_2e^{i\alpha_2}w+r_3e^{i\alpha_3}$ where $r_i$ are real positive numbers and $\alpha_1>\alpha_3>\alpha_2>0$.}
\label{}
\end{center}
\end{figure}
We can remark that in this case there are no extra-pieces, and all the boundary of the closure of this coamoeba in the torus is contained in three external hyperplanes.
\item[(b)] Consider now the example of a parabola.
Let $\mathring{V}_f\subset (\mathbb{C}^*)^2$ the curve defined by the polynomial $f(z,w) = w-z^2+2z-\lambda$ with $\lambda >1$. Consider the parametrization defined by :
$$
\left\lbrace
\begin{array}{l}
z(r, \alpha ) = r{e}^{i\alpha},\\
w(r, \alpha ) = r^2{e}^{2i\alpha}-2r{e}^{i\alpha}+\lambda ,
\end{array}
\right.
$$
with $r>0$ and $\alpha\in [0,2\pi ]$. We have to compute the argument
of $r^2{e}^{2i\alpha}-2r{e}^{i\alpha}+\lambda$, with $r\in
\mathbb{R}_+^*$. Let $a=\lambda -1$, so we have $w(r,\alpha ) =
(r{e}^{i\alpha} - 1)^2 + a$ and then $\beta = \arg (w(r,\alpha )) =
\arg \left[ \dfrac{(r{e}^{i\alpha} - 1)- i\sqrt{a}}{(r{e}^{-i\alpha}
- 1)- i\sqrt{a}}\right]$.
\begin{itemize}
\item[(i)] Let $0\leq\alpha\leq \arctan\sqrt{a}$ then
$0\leq\beta\leq 2\alpha$ if $1+\tan^2\alpha\leq r^2<\infty$ and
$g_\alpha (r)\leq \beta\leq 2\pi$ if $0<r^2<1+\tan^ 2\alpha$ where
for each $\alpha$,\, $g_\alpha$
is a differentiable function with one maximum in the interval
$0<r^2<1+\tan^2\alpha$ (see Figure 2);
\item[(ii)] If $\pi\geq\alpha\geq\arctan\sqrt{a}$ then
$2\alpha\leq\beta\leq 2\pi$;
\item[(iii)] For $\alpha > \pi$ we have the conjugate of the sets
in (i) and (ii).
\end{itemize}
\begin{figure}[h!]
\begin{center}
\includegraphics[angle=0,width=0.4\textwidth]{Parabola-1.eps}\quad
\caption{Coamoeba of a parabola.}
\label{}
\end{center}
\end{figure}
We can view a parabola as an algebraic curve
$\mathring{V}_{f_{\mathbb{K}}}$ over the field of the Puiseux series with
real powers $\mathbb{K}$, defined by the polynomial
$f_{\mathbb{K}}(z,w)=f_t(z,w) =t^0w-t^0z^2+2t^0z-t^{-\mathop{\rm Log}\nolimits \lambda}$ with $z,\, w\in \mathbb{K}^*$
and $t\in \mathbb{R}_+^*$.
It is clear that the limit of the coamoebas of the
curves $\mathring{V}_{f_t}$ converge to the coamoeba of the phase tropical
curve with tropical coefficients
$a_{01}=1,\, a_{00}=-\lambda$ and
$a_{20}=-1$, which are the coefficients with index in $\mathop{\rm Vert}\nolimits (\tau )$
where $\tau$ is the triangulation of the Newton polygon of $f$ dual to
$\Gamma$, with $\Gamma$ the tropical curve that is the spine of the amoeba of
$\mathring{V}_f$ (see Figure 3, the coamoeba of a phase tropical parabola).
\begin{figure}[h!]
\begin{center}
\includegraphics[angle=0,width=0.3\textwidth]{Parabola-2.eps}\quad
\caption{Coamoeba of a parabola with coefficients only in the vertices of the Newton polygon of its defining polynomial.}
\label{}
\end{center}
\end{figure}
We can see in Figure 2 extra-pieces in the coamoeba of our parabola.
\item[(c)]
Let $V_\lambda$ be the complex curve defined by the polynomial
$f(z,w) = \lambda + z + w + zw$ with $\lambda \in \mathbb{R}^*$ with Newton
polygon the standard square of vertices $(0,0), (1,0), (0,1)$ and
$(1,1)$.
\end{itemize}
\begin{itemize}
\item[$1^{st}$ {\it case}.]
Assume $0<\lambda < 1$, and we parametrize
$z=re^{i\alpha}$ with $\alpha \in [0, 2\pi ]$ and $r\in
\mathbb{R}^*_+$. So $\arg (w(r,\alpha )) = \theta (r,\alpha )$ with:
$$
\theta (r,\alpha ) = \arcsin \left( \frac{-r (1-\lambda )\sin
\alpha}{((\lambda + r(1+\lambda )\cos \alpha +r^2)^2+ r^2(1-\lambda
)^2\sin^2\alpha )^{\frac{1}{2}}}\right)
$$
and we have $\dfrac{\partial \theta}{\partial r}(r, \alpha ) = 0$ if
and only if $r=\pm \sqrt{\lambda}$, so
$r=\sqrt{\lambda}$ and the maximum of the argument of $w$ is
attained at $r=\sqrt{\lambda}$, this means that we have
$$
\theta_{\max}(\alpha )= \arcsin \left( \frac{-\sqrt{\lambda} (1-\lambda )\sin
\alpha}{((2\lambda +\sqrt{\lambda} (1+\lambda )\cos \alpha
)^2+ \lambda (1-\lambda
)^2\sin^2\alpha )^{\frac{1}{2}}}\right)
$$
If $0<\lambda < 1$ it can be viewed as a parameter, and hence as an
element of $\mathbb{K}^*$, which means that the curve $V_\lambda$ is viewed as
an algebraic curve over $\mathbb{K}$, i.e. $V_{\lambda}^{\mathbb{K}} =\{ (z,w)\in (\mathbb{K}^*)^2
\,|\,\lambda + z + w + zw =0 \}$ and $\mathop{\rm Log}\nolimits_{\mathbb{K}}(V_{\lambda}^{\mathbb{K}})$ is
the tropical curve with tropical polynomial
$f_{trop}(x,y) = \max \{ x,y,x+y, -1 \}$. We have $\mathop{\rm Log}\nolimits^{-1}(v_1)\cap
W(V_{\lambda}^{\mathbb{K}})$ is the union of the two sets of $S^1\times S^1$ with
boundary the two half of the cycles $\delta_1 =\{ \alpha =\pi \}$ and
$\delta_2 =\{ \beta =\pi \}$ and the half of the cycle defined by the
graph of the function $\theta_{\max}$, which is homotopic to the
product of $\delta_1$ and $\delta_2$. We have the same
result for the vertex $v_2$.
\begin{figure}[h!]
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{Trop-Hyperbola.eps}\quad
\caption{The spine of the amoeba of the hyperbola defined by the polynomial $f_{\lambda}$ with $0<\lambda<1$ and its coamoeba.}
\label{}
\end{center}
\end{figure}
\vspace{0.3cm}
\item[$2^{nd}$ {\it case}.]
Suppose $\lambda > 1$, and let $\tau
=\frac{1}{\lambda}$. So, $\lambda = \tau^{-1}$ and then $V_{\tau}^{\mathbb{K}} =\{ (z,w)\in ({\mathbb{K}}^*)^2
\,|\, \tau^{-1} + z + w + zw =0 \}$. Hence $\mathop{\rm Log}\nolimits_{\mathbb{K}}(V_{\tau}^{\mathbb{K}})$ is
the tropical curve with tropical polynomial
$f_{trop}(x,y) = \max \{ x,y,x+y, +1 \}$. Hence, we have:
\begin{align*}
\mathop{\rm Log}\nolimits^{-1}(v_1)\cap
&W(V_{\tau}^{\mathbb{K}})=\{ (\alpha ,\beta) \in S^1\times S^1 / 0\leq \alpha\leq
\pi ,\,\, \theta_{\max}(\alpha )\leq\beta\leq \pi \}\\
&\phantom{W(V_{\tau}^{\mathbb{K}})=}\cup \{ (\alpha ,\beta) \in S^1\times S^1 / \pi\leq \alpha\leq
2\pi ,\,\, \pi\leq\beta\leq \theta_{\max}(\alpha )\}
\end{align*}
\vspace{0.3cm}
\item[$3^{rd}$ {\it case}.] Assume $\lambda =1=t^0$, so we have $f_1(z,w) =
(1+z)(1+w)$, and the corresponding tropical curve is the union of
two axes, and $\mathop{\rm Log}\nolimits^{-1}(v_1)\cap
W(V_1^{\mathbb{K}})$ is the union of two circles (the valuation of the
constant coefficient is zero in this case).
\vspace{0.3cm}
\item[$4^{th}$ {\it case}.] Suppose $\lambda < 0$ and $\lambda \ne -1$. If
$\mid \lambda \mid <1$, then consider $\lambda$ as a parameter and we
have the tropical curve of the first case (it means that the
valuation of the constant coefficient is negative). So, if we put $z(t)
=t^{-x}e^{i\alpha}$ then $w(t)_{\alpha} = -\left(
\dfrac{t-t^{-x}e^{i\alpha}}{-1-t^{-x}e^{i\alpha}} \right)$ and then $\mathop{\rm Log}\nolimits^{-1}(v_1)\cap
W(V_{\lambda}^{\mathbb{K}})$ is the closure in $S^1\times S^1$ of the set
$$\left(\alpha , \lim_{t\rightarrow 0,\, x\rightarrow 0} \arg (w(t)_{\alpha}\right)$$
with $0\leq \alpha \leq 2\pi$. We then obtain the union of two
triangles. For the second vertex we have $\mathop{\rm Log}\nolimits^{-1}(v_2)\cap
W(V_{\lambda}^{\mathbb{K}})$ is the closure in $S^1\times S^1$ of the set
$$\left(\alpha , \lim_{t\rightarrow 0, x\rightarrow 1} \arg (w(t)_{\alpha}\right)$$
with $0\leq \alpha \leq 2\pi$, and we obtain the union of two
triangles.
\vspace{0.3cm}
\item[$5^{th}$ {\it case}.] Suppose $\lambda < -1$ and write $\lambda
= \tau^{-1}$ with $-1<\tau <0$. So we have the tropical curve of the
second case (this means that the valuation of the constant
coefficient is positive).
\end{itemize}
\section{A differential structure on phase tropical hypersurfaces}
\vspace{0.2cm}
\subsection{A differential structure on phase tropical hyperplanes} ${}$
\vspace{0.2cm}
In \cite{M2-04}, Mikhalkin gives the following definition of a generalized pair-of-pants:
\begin{definition}\label{definition-pants}
Let $\mathscr{H} \subset \mathbb{CP}^n$ be an arrangement of $n+2$ generic hyperplanes in $\mathbb{CP}^n$. Let $\mathscr{U}\subset \mathbb{CP}^n$ be the union of their tubular $\varepsilon$-neighborhood for a small $0<\varepsilon\ll1$. The complement $\overline{\mathscr{P}}_n=\mathbb{ CP}^n\backslash \mathscr{U}$ is called the $n$-dimensional pair-of-pants, and $\mathscr{P}_n=\mathbb{CP}^n\backslash \mathscr{H}$ is called the $n$-dimensional open pair-of-pants.
\end{definition}
As $\mathscr{H}\subset \mathbb{CP}^n$ is unique up to the action of the projective special linear group $PSL_{n+1}(\mathbb C)$, then $\mathscr{P}_n$ can be given a canonical complex structure. The one dimensional pair-of pants
$\mathscr{P}_1$ is diffeomorphic to the Riemann sphere punctured at 3 points.
Moreover, Mikhalkin constructs a foliation $\mathscr F$ of the complement in $\mathbb{R}^n$ of the complex defined by the standard tropical hyperplane $\Gamma_n$. As before, if $v\in \Gamma$ is a vertex, then there exists a neighborhood ${U}_v$ of $v$ in $\Gamma$ and an affine linear transformation $F$ with linear part $A_v$ in $SL_n(\mathbb Z)$ such that up to a translation in $\mathbb R^n$, ${}^tA^{-1}_v({U}_v)$ is a neighborhood of the origin in $\Gamma_n$. Let $W_v$ be a neighborhood of $\overline{F({U}_v)}$. According to Mikhalkin, a partition of unity gives a foliation $\mathscr{F}_{\Gamma}$ of a neighborhood $W$ of $\Gamma$.
\noindent Let $\pi_{\mathscr{F}_{\Gamma}}:W(\Gamma)\to \Gamma$ the projection along $\mathscr{F}_{\Gamma}$. By Theorem 5.4 of Mikhalkin and Rullgard, $\mathop{\rm Log}\nolimits_t(V_t)\subset W(\Gamma)$ for $t\gg 0$. Let $$\lambda_t:=\pi_{\mathscr{F}_{\Gamma}}\circ \mathop{\rm Log}\nolimits_t :V_t\to \Gamma .$$
\vspace{0.2cm}
The example of hyperplanes in the projective space is fundamental for our Theorem \ref{A}. So, let $H=\{ (z_1,\ldots,z_n)\in \mathbb{C}^n\,| \, z_1+\cdots + z_n +1=0\}\subset \mathbb{CP}^n$ be a hyperplane. Consider its toric part $\mathring{H}= H\cap (\mathbb{C}^*)^n$. Let us denote by $\mathscr{A}_n\subset\mathbb{R}^n$ the amoeba of $\mathring{H}$ and by
$\Gamma_n\subset \mathbb{R}^n$ the tropical hyperplane defined by the tropical polynomial:
$$
f_{trop}(x_1,\ldots,x_n) = \max \{ 0,x_1,\ldots, x_n\}.
$$
It is well known that $\Gamma_n\subset \mathscr{A}_n$ and it is called the spine of the amoeba $\mathscr{A}_n$. Moreover, $\Gamma_n$ is a strong deformation retract of $\mathscr{A}_n$ (see \cite{PR1-04}). The number of connected components of the complement of the amoeba $\mathscr{A}_n$ in $\mathbb{R}^n$ is equal to $n+1$. Each component $\mathscr{C}_i$ of $\mathbb{R}^n\setminus \mathscr{A}_n$ is equal to the subset of $\mathbb{R}^n$ where one the functions $\{ 0,x_1,\ldots, x_n\}$ is maximal.
\noindent Let us recall Mikhalkin's construction of the foliation mentioned above (\cite{M2-04}, Section 4.3) to obtain a singular foliation of the amoeba $\mathscr{A}_n$. More precisely, let $\mathscr{L}_i$ be the foliation of the complement component of $\Gamma$ corresponding to $x_i$ (i.e., the set of $\mathbb{R}^n$ where the tropical polynomial $f_{trop}$ achieved its maximum) into straight lines parallel to the gradient $v_i:=\frac{\partial}{\partial x_i}$ of $x_i$ for $i=1,\ldots, n$ and in the component corresponding to the constant function equal to $0$ we consider the foliation into straight lines parallel to the vector with coordinates $v_0=(1,\ldots,1)$. Consider $\pi_i: \mathscr{C}_i\rightarrow\Gamma_n$ the linear projection onto $\Gamma_n$ and parallel to the vector $v_i$. Let $\pi$ the following map:
$$
\pi: \mathscr{A}_n\setminus\Gamma_n \rightarrow \Gamma_n,
$$
where $\pi_{|\mathscr{C}_i\cap\mathscr{A}_n} = {\pi_i}_{|\mathscr{A}_n}$ for each $i=0, 1,\ldots n$. The foliations of the $\mathscr{C}_i$'s glue to a global foliation $\mathscr{L}$ of $\mathscr{A}_n$ which has singularities at $\Gamma_n$ and the leaves passing through a point $p$ in an open $(n-1-k)$-cell of $\Gamma_n$ is diffeomorphic to the union of $k+2$ segments having a common boundary point $p$ (in other word a cone over $k+2$ points). We can smooth the foliation $\mathscr{L}$ over all open $(n-1)$-cells of $\Gamma_n$, but not at the lower dimensional cells because their leaves are not even a topological manifolds. The only leaves diffeomorphic to a manifold are those passing through open $(n-1)$-cells which are diffeomorphic to the closed interval $[-1,+1]$. Let us denote the foliation obtained by this smoothing by $\mathscr{F}$.
\begin{proposition}
A phase tropical hyperplane ${H}_\infty\subset (\mathbb{C}^*)^n$ is diffeomorphic to a hyperplane in the projective space $\mathbb{CP}^n$ minus $n+1$ generic hyperplanes.
\end{proposition}
\begin{proof}
Since each phase tropical hyperplane is a translated in $(\mathbb{C}^*)^n$ of the following phase tropical hyperplane ${H}_\infty = W(\{(z_1,\ldots ,z_n)\in (\mathbb{K}^*)^n | \, z_1+\cdots+z_n+1=0\})$, then it suffices to consider this case.
Let us start by the case of phase plane tropical line in $(\mathbb{C}^*)^2$. In the case of lines the inverse image by the logarithmic map of the vertex of the tropical line $\Gamma := \mathop{\rm Log}\nolimits (\mathscr{H})$ is a union of two triangles whose vertices pairwise identified, and the inverse image by the logarithmic map of any point in the interior of its rays is a circle (see Example (a)). This means that the inverse image of each ray is a holomorphic annulus $\mathscr{R}_j$ for $j=1,2,3$. It is clear now that a phase tropical line in $(\mathbb{C}^*)^2$ is diffeomorphic to a sphere punctured in three points. In fact, if we denote $v_0$ the vertex of $\Gamma$ and $\mathcal{R}_j$ for $j=1, 2, 3$ are the three rays going to the infinity, then the phase tropical line in $(\mathbb{C}^*)^2$ is diffeomorphic the the gluing of
the closure $\overline{\mathop{\rm Log}\nolimits^{-1}(v_0)}$ in the real torus $(S^1)^2$ and the three semi-open holomorphic annulus $\mathscr{R}_j= \mathop{\rm Log}\nolimits^{-1}(\mathcal{R}_j)$ for $j=1, 2, 3$. A complete description of $\overline{\mathop{\rm Log}\nolimits^{-1}(v_0)}$ is given in \cite{NS-13}. For any dimension, it is the same as the complement in the real torus $(S^1)^n$ of an open zonotope (i.e. the coamoeba of a hyperplane). In case where $n>2$, using the description of the coamoeba of a hyperplane given in Theorem 3.3 \cite{NS-13} and the description of the $(n-1)$-dimensional pair-of-pants given in Proposition 2.24 \cite{M2-04}, one can check the phase tropical hyperplane $(\mathbb{C}^*)^n$ is diffeomorphic to the complex projective space $\mathbb{CP}^{n-1}$ minus a tubular neirghborhood of the union $\mathscr{H}$ of $n+1$ hyperplanes in $\mathbb{CP}^{n-1}$. Let us be more explicite.
\vspace{0.1cm}
The hyperplane $H_\mathbb{K} := \{ (z_1,\ldots ,z_n)\in (\mathbb{K}^*)^n | \, z_1+\cdots+z_n+1=0 \}$ can be parametrized as follows:
\begin{displaymath}
\left\{ \begin{array}{lll}
z_1(t) &=& t^{-x_1}e^{i\alpha_1}\\
z_2(t) &=& t^{-x_2}e^{i\alpha_2}\\
\vdots&\vdots&\vdots\\
z_{n-1}(t) &=& t^{-x_{n-1}}e^{i\alpha_{n-1}}\\
z_n(t)&=& 1-\sum_{j=1}^{j=n-1}t^{-x_j}e^{i\alpha_j}
\end{array} \right.
\end{displaymath}
with $x_j\in \mathbb{R}$ and $0\leq \alpha_j\leq 2\pi$ for $j=1,\ldots , n-1$. If we denote $\mathring{H}_t\subset (\mathbb{C}^*)^n$ the hyperplane given by the parametrization for a fixed $t$. Then all the family of hyperplanes $\{ \mathring{H}_t\}_{0<t\leq 1}$ is viewd as a single hyperplane in $(\mathbb{K}^*)^n$ and we have $\mathring{H}_\infty = W(\mathring{H}_\mathbb{K})$ where $W$ is the map from $(\mathbb{K}^*)^n$ to $(\mathbb{C}^*)^n$ defined in Section 3. Also, the tropical hyperplane $\Gamma_n$ is the image by the logarithmic map of $\mathring{H}_\infty$. The following lemma gives a complete topological description of $\mathring{H}_\infty$.
\end{proof}
\begin{lemma}\label{description-of-H}
Let $\mathring{H}_\infty \subset (\mathbb{C}^*)^n$ be a phase tropical hyperplane and $\Gamma_n$ its image by the logarithmic map.
Then the inverse image of a point in the interior of an $l$-cell $\sigma\subset \Gamma_n$ is the product of a real $l$-torus with the coamoeba of a hyperplane in $(\mathbb{C}^*)^{n-l}$, i.e. if $x = (x_1,\ldots,x_n)\in \mathring{\sigma}$ then we have:
$$
\mathop{\rm Log}\nolimits^{-1}(x) = (S^1)^l\times co\mathscr{A}(n-1-l),
$$
where $co\mathscr{A}(n-1-l)$ is the coamoeba a $(n-1-l)$-plane in $(\mathbb{C}^*)^{n-l}$.
\end{lemma}
\begin{proof}
Let $x$ be a point in the interior of an $l$-cell, then there exist $x_{j_1}, \ldots , x_{j_l}$ strictly negative and all the other $x_j$ are equal to zero. As $\mathring{H}_\infty$ is the limit when $t$ tends to zero (if we want t goes to infinity then we can make the change of variable in the parametrization, $t$ by $\frac{1}{\tau}$), then for any fixed $\alpha_{j_1}, \ldots, \alpha_{j_l}$ we obtain the coamoeba of a hyperplane in $(\mathbb{C}^*)^{n-l}$ (recall that $\lim_{t\rightarrow 0}t^{-x_{j_u}} =0$, because $x_{j_u}<0$ for any $u=1,\ldots, l$).
But $0\leq \alpha_{j_u}\leq 2\pi$, which means that the fiber over $x$
is the product of the torus $(S^1)^l$ with the coamoeba of a hyperplane in $(\mathbb{C}^*)^{n-l}$. In particular, the inverse image of a $0$-cell is the coamoeba of a hyperplane in $(\mathbb{C}^*)^{n}$ which is equal to its phase limit set, and its topological description is given in \cite{NS-13}.
\end{proof}
Lemma \ref{description-of-H} gives a complete description of the phase tropical hyperplane $\mathring{H}_\infty$, which coincide with the description of a hyperplane in the projective space $\mathbb{CP}^n$ minus $n+1$ generic hyperplanes.
\vspace{0.2cm}
\subsection{A differential structure on phase tropical hypersurfaces} ${}$
\vspace{0.2cm}
In the general case, let us denote by $\Gamma$ the tropical variety limit of the family of amoebas $\{\mathscr{A}_t\}$, where $\mathscr{A}_t$ is the amoeba of the variety $\mathring{V}_t$. Also,
we assume that the tropical hypersurface $\Gamma$ is smooth in the sense that every vertex of $\Gamma$ is dual to a simplex of Euclidean volume equal to $\frac{1}{n!}$. Therefore, locally for any vertex $v$ of $\Gamma$ there exists an open neighborhood ${U}_v$ diffeomorphic to the standard tropical hyperplane, in other words, tropical pair-of-pants. More precisely, there exists an affine linear transformation of $\mathbb R^n$ whose linear part ${A}_v$ belongs to $SL_n(\mathbb Z)$ such that ${U}_v$ is the image of the standard tropical hyperplane by ${}^tA_v^{-1}.$ Namely, $\overline{{U}}_v$ has $n+1$ boundary components isomorphic to an $(n-2)$-dimensional tropical hyperplane in $\mathbb R^{n-1}$ where $\mathbb R^{n-1}$ can be viewed as a boundary component of the tropical projective space $\mathbb{PT}^n$ represented by the standard simplex.
Let $v_1$ and $v_2$ be two adjacent vertices of $\Gamma$, in other words, there exists a compact edge $e$ with boundary $v_1$ and $v_2$. Then ${U}_{v_i}$ has a boundary component $\mathscr{B}_{ij}: = \partial_j{U}_{v_i}$ that can be viewed as a component of the boundary of a tubular neighborhood of a boundary component $\mathscr{B}_{ji}:=\partial_i{U}_{v_j}$. In other words, there exists an open neighborhood ${U}_{v_1v_2}$ of $v_1$ and $v_2$ containing ${U}_{v_1}$ and $\mathscr{U}_{v_2}$ such that ${U}_{v_1v_2}$ is the interior of the gluing of $\overline{{U}}_{v_1}$ and $\overline{{U}}_{v_2}$ along their boundaries $\mathscr{B}_{ij}$ and $\mathscr{B}_{ji}$ are joined by a vertical edge and all the other edges adjacent to $v_i ~(i=1,2)$ are horizontal (i.e., they are mutually parallel) such that the reversing orientation diffeomorphism is given by $(z_1,\cdots,z_{n-1},z_n)\mapsto (z_1,\cdots,z_{n-1},\overline{z}_n)$. After gluing all pieces, we obtain a manifold $W_\infty(\Gamma)$ with boundary coming from unbounded 1-cells of $\Gamma_f$ where each unbounded 1-cell will corresponds to $\mathscr{B}_{ij}$ for some vertex $v_i$. Each $\mathscr{B}_{ij}$ is a circle fibration over a union of lower dimensional pair-of-pants $\mathscr{P}_{n-2}$ (see Figure 5). We can remark that $W_\infty(\Gamma)$ is a topological description of the decomposition of $\mathring{H}_\infty= W(\mathring{V}_{\mathbb{K}})$, where $\mathring{V}_{\mathbb{K}}$ is the hypersurface of $(\mathbb{K}^*)^n$ representing the family $\{\mathring{V}_t\}$. In other words, the family $\{\mathring{V}_t\}$ is viewed as a single hypersurface in the algebraic torus $(\mathbb{K}^*)^n$.
\begin{figure}[h!]
\begin{center}
\includegraphics[angle=0,width=0.6\textwidth]{Pants-1.eps}\quad
\caption{Gluing of two $2$-dimensional pairs-of-pants in $(\mathbb{C}^*)^3$ along one component of their boundaries.}
\label{}
\end{center}
\end{figure}
Let us denote by $M_\infty(\Gamma)$ the result of collapsing all fibers of these fibrations on the boundary $\partial W_\infty(\Gamma)$ of $W_\infty(\Gamma)$. Then $M_\infty(\Gamma)$ is a smooth manifold. Indeed, this construction coincide locally with collapsing the boundary on $\overline{\mathscr{P}}_{n-1}$ which results in the projection space $\mathbb{CP}^{n-1}$ which is smooth.
\subsection{Proof of Theorem \ref{A}} ${}$
\vspace{0.2cm}
\noindent Since all smooth hypersurface with a fixed Newton polytope are isotopic, then we can choose any of them. More precisely, we will use for our subject the convenient one. Let $f_t(x)=\sum_{j\in \Delta\cap\mathbb{Z}^n}a_jt^{-v(j)}z^j$ be a polynomial with a parameter $t$, and $\mathring{V}_t=\{f_t=0\}\subset (\mathbb C^*)^n$. The family of $f_t$ can be viewed as a single polynomial in $\mathbb K[z_1^{\pm 1},\cdots,z_n^{\pm 1}]$. Therefore this family defines a hypersurface $\mathring{V}_{\mathbb K}\subset (\mathbb K^*)^n$. Let $\mathscr{A}_t:=\mathop{\rm Log}\nolimits_t(\mathring{V}_t)$ and $\mathscr{A}_{\mathbb K}:=\mathop{\rm Log}\nolimits_{\mathbb K}(\mathring{V}_{\mathbb K})$.
\noindent Let $\Gamma$ be a maximally dual $\Delta$-complex (i.e. all the element of its dual the subdivision are simplex of Euclidean volume $\frac{1}{n!}$) and $\nu:\Delta \cap \mathbb Z^n\to \mathbb R$ be the function such that $\Gamma=\Gamma_\nu$ i.e., $\Gamma_\nu$ is the tropical hypersurface defined by the tropical polynomial $\displaystyle{ \max_{\alpha\in \Delta \cap \mathbb Z^n } \{ \nu ({\alpha}) + <\alpha , x> \}}$.
Then we obtain a family of polynomial called a Viro-patchworking polynomial \cite{V-90}
$$
f_t(z)=\sum_{v\in \Delta\cap \mathbb Z^n}t^{-v(j)}z^j.
$$
Let us denote $\mathring{V}_t\subset (\mathbb C^*)^n$ the zero locus of the polynomial $f_t$. Using a foliation of the amoeba of $\mathring{V}_t$ Mikhalkin obtains a map $\lambda_t=\pi_{\mathcal F_{\Gamma}}\circ \mathop{\rm Log}\nolimits_t: V_t \to \Gamma$, and proves in Lemma 6.5, \cite{M2-04} that $\mathring{V}_t$ is smooth for a sufficiently large $t\gg 0$.
\noindent First of all, $\Gamma$ looks locally as a tropical hyperplane after a linear transformation with linear part $SL_n(\mathbb Z)$. It means that $\Gamma$ can be locally identified to a tropical hyperplane in $\mathbb R^n$ by a linear transformation $F$ of $\mathbb R^n$ with a linear part in $SL_n(\mathbb Z)$.
\noindent It was shown in Lemma 6.5 \cite{M2-04} that $V_t$ is also smooth, and $\lambda_t$ satisfies a nice properties. Indeed, for $t\gg 0$, $\mathring{V}_t$ is smooth, and $\mathring{V}_t$ is an union of finite number of open sets, where each set is the image of a small perturbation of a hyperplane. Hence, its compactification $V_t\subset X_{\Delta}$ is smooth and transverse to the coordinate hyperplanes. Also, for a large $t\gg 0$, $V_t$ is isotopic to the variety $M_\infty(\Gamma)$ constructed above (which is a compactification of the phase tropical variety $\mathring{V}_\infty = W_\infty(\Gamma)$ the lifting of $\Gamma$ in $(\mathbb{C}^*)^n$), this comes from Theorem 4 of Mikhalkin \cite{M2-04}, which proves the second statement of Theorem \ref{A}. This shows that $\mathring{V}_\infty$ is also diffeomorphic to $\mathring{V}_t$ for sufficently large $t\gg 0$ and the first statement of Theorem \ref{A} is proved.
\section{Construction of a natural symplectic structure on $\mathring{V}_{\infty}$}
Note that every pair-of-pants inherit a natural symplectic structure coming from the one of the projective space $\mathbb{CP}^n$. Namely, the projective space $\mathbb{CP}^n$ is obtained from a closed pair-of-pants after collapsing its boundary. Indeed, each component of the boundary of a pair-of pants $\mathscr{P}^n$ is a $S^1-$fibration over a lower dimensional pair-of-pants $\mathscr{P}^{n-1}$, and the result of collapsing all fibers of these $S^1-$fibrations is precisely the projective space $\mathbb{CP}^n$.
\vspace{0.2cm}
\subsection{Proof of Theorem \ref{B}} ${}$
\vspace{0.2cm}
Let $M_\infty(\Gamma)$ be the variety constructed in Section 5, which is a compactification of $\mathring{V}_\infty$ in the toric variety $X_\Delta$ where $\Delta$ is the degree of our original hypersurface $V$. The variety $M_\infty(\Gamma)$
is obtained by gluing pairs-of-pants along a part of their boundary $\mathscr{B}_j$ that is a product of a holomorphic cylinder (i.e. an annulus) in $\mathbb C^*$ with a lower dimensional pair-of-pants $\mathscr{P}^{n-2}$ (i.e. along $[0,1]\times \mathscr{B}_j$).
Moreover, each $\mathscr{B}_j$ is a circle fibration over $\mathscr{P}^{n-2}$, where the fibers are precisely the fibers of the annulus over the interval $[0,1]$:
$$
\mathscr{B}_j \longrightarrow \mathscr{P}^{n-2} \qquad \textrm{is an $S^1$-fibration},
$$
and
$$
\mathcal{A}\times\mathscr{P}^{n-2} = [0,1]\times \mathscr{B}_j \longrightarrow \mathscr{P}^{n-2} \qquad \textrm{is an annulus fibration},
$$
where $\mathcal{A}$ is the annulus $[0,1]\times S^1$.
Let us denote by $\omega_j^{(n-2)}$ the symplectic form on the pair-of-pants $\mathscr{P}^{n-2}$ coming from the projective space
$\mathbb{CP}^{n-2}$ and $ds\wedge dt$ the symplectic form on $S^1\times \mathbb{R}$. Hence, we obtain a symplectic form $\omega_j := ds\wedge dt + \omega_j^{(n-2)}$ on $[0,1]\times \mathscr{B}_j$. It means that we have a symplectic form on parts where the gluing was done.
Recall that $[0,1]\times \mathscr{B}_j$ can be seen as a neighborhood of a boundary component of the pair-of-pants $\mathscr{P}^{n-1}$. On the other part of $\mathscr{P}^{n-1}$ i.e., $\mathscr{P}^{n-1}\setminus \cup_j([0,1]\times\mathscr{B}_j)$, we already have the symplectic form of a pair-of-pants $\omega^{(n)}$ and the pull back of $\omega^{(n)}$ on the factor $\mathscr{P}^{n-2}$ of any boundary component is precisely $\omega_j^{(n-2)}$.
However, when we glue $[0,1]\times \mathscr{B}_j$ and $[0,1]\times \mathscr{B}_i$ where the first part is equipped with the form $ds\wedge dt + \omega_j^{n-2}$ then the second should be equipped with the form $-ds\wedge dt + \omega_i^{n-2}$ because the gluing was done with a reversing orientation (recall that the forms $\omega_j^{(n-2)}$ and $\omega_i^{(n-2)}$ are the same).
On the other hand, the symplectic forms outside of the gluing parts are well defined since each component is symplectically an open pair-of-pants which is a hyperplane in the complex algebraic torus $(\mathbb{C}^*)^n$. After taking the compactification of such hyperplanes in the projective space $\mathbb{CP}^{n-1}$, the restriction of these forms on the infinite parts (i.e. the $\mathbb{CP}^{n-2}$'s) are precisely the forms $\omega_j^{n-2}$'s. This gives rise to a global symplectic form $\mathring{\omega}_{nat}$ on $\mathring{V}_\infty$. This proves that $\mathring{V}_\infty$ has a natural symplectic structure because all the forms that we used are constructed naturally and the first part of Theorem \ref{B} is proved.
\vspace{0.1cm}
Let us denote by $\mathring{\omega}_t = \iota_t^*(\omega)$ the symplectic form on $\mathring{V}_t$ where $\iota_t$ is the inclusion of $\mathring{V}_t$ in the complex algebraic torus $(\mathbb{C}^*)^n$, $\omega$ is the symplectic form on $(\mathbb{C}^*)^n$ defined by (1). Using Moser's trick, Mikhalkin showed that $M_\infty(\Gamma)$ is symplectomorphic
to $V_t$ for a sufficiently large $t\gg0$. Let us denote this symplectomorphism by $\phi$. Hence we have the following commutative digram:
\begin{equation}\label{(2)}
\xymatrix{
(\mathring{V}_\infty, \mathring{\omega}_{nat})\ar[rr]^{\psi = \phi_{|\mathring{V}_\infty}}\ar@{^{(}->}[d]_{j}&&
(\mathring{V}_t, \mathring{\omega}_{t})\ar@{^{(}->}[d]^{i}\cr
(M_\infty(\Gamma), \omega_{nat})\ar[rr]^{\phi}&&(V_t, \omega_t).
\end{equation}
This means that for a sufficiently large $t\gg0$, \, $\mathring{V}_\infty$ is also symplectomorphic to $\mathring{V}_t$, and the second statement Theorem \ref{B} is proved. Recall that we can prove Theorem \ref{B} using a generalization of Moser's trick for non compact manifolds proved by R. E. Greene and K. Shiohama on 1979 in \cite{GS-79}.
| {
"timestamp": "2016-09-09T02:00:39",
"yymm": "1609",
"arxiv_id": "1609.02181",
"language": "en",
"url": "https://arxiv.org/abs/1609.02181",
"abstract": "First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.",
"subjects": "Algebraic Geometry (math.AG)",
"title": "Geometry and a natural symplectic structure of phase tropical hypersurfaces",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795102691455,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379416617377
} |
https://arxiv.org/abs/2209.09379 | Characterization of Graphs With Failed Skew Zero Forcing Number of 1 | Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Hence the failed zero forcing number of a graph was defined to be the size of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. The difference is that vertices that are not in $S$ can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by $F^{-}(G)$. In this paper we provide a complete characterization of all graphs with $F^{-}(G)=1$. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of $1$ are either: the union of two isolated vertices; $P_3$; $K_3$; or $K_4$. In this paper we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with $F^{-}(G)=1$. | \section{Introduction}
Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied \cite{Fallat}. Investigations of the largest size of a set $S$ that does not force all of the vertices in a graph to be in $S$. This quantity is known as the failed zero forcing number of a graph and is denoted by $F(G)$ \cite{Fetcie} and \cite{Adams}. Shitov \cite{Shitov}, proved that determining the failed zero forcing number of a graph is NP-complete. Independently, a closely related property called the zero blocking number of a graph was introduced in 2020 by Beaudouin-Lafona, Crawford, Chen, Karst, Nielsen, and Sakai Troxell \cite{Beaudouin} and Karst, Shen, and Vu \cite{KarstB}. The zero blocking number of a graph $G$ equals $|V(G)|-F(G)$.
In 2010, researchers from the IMA-ISU research group \cite{ISU} introduced skew zero forcing where any vertex that has all but one of its neighbors colored will force the last remaining vertex to be forced.
In 2016, Ansill, Jacob, Penzellna, and Saavedra \cite{Ansill} introduced the failed skew zero forcing number, which is the largest size of a set of vertices that does not skew force all of the vertices in the graph.
We will use $K_n$, $P_n$, $C_n$, to denote the complete graph, path, and cycle on $n$ vertices, respectively. A vertex $v$ is a cut-vertex of a graph $G$ if $G-v$ has more components than $G$.
Throughout the paper we will use colorings to describe failed zero forcing sets where the vertex in $S$ will be referred to as colored and the vertices not in $S$ will be referred to as uncolored.
In 2016 Ansill, Jacob, Penzellna, and Saavedra \cite{Ansill} characterized graphs with extreme values of $F^{-}(G)=0,n-2,n-1$ and $n$. The characterization of graphs with $F^{-}(G)=0$ was surprisingly complex. We review this in the next section.
In this paper we characterize all graphs with a failed skew zero forcing number of $1$. In these graphs, there is a vertex which is a stalled set (meaning that this set of vertices does not force any other vertices), and any pair of vertices forces all vertices in the graph. Within this class of graphs there is an interesting subclass of graphs where there is a unique vertex $v$ that is a failed skew zero forcing set of size 1 and all other vertices in the graph are zero forcing sets.
In 2015, Fetcie, Jacob, and Saavedra \cite{Fetcie}, characterized all graphs with a failed zero forcing number of $1$ - which turned out to be only four graphs: a pair of isolated vertices; $K_3$; $P_3$; and $P_4$. It is surprising to see that the change of the forcing rule where vertices not in $S$ can also force other vertices results in an infinite number of graphs with a failed skew zero forcing number of $1$.
\section{Failed skew zero forcing numbers of graphs}
Ansill, Jacob, Penzellna, and Saavedra \cite{Ansill} presented new results involving failed skew zero forcing numbers of graphs. In particular they provided a characterization for graphs with failed skew zero forcing numbers of $0$.
They described a family of graphs called doubly extended bouquet-dipoles which is defined below.
\begin{definition}
A graph $G$ is a doubly extended bouquet-dipole if it consists of vertices $u$ and $v$ that are each on a nonempty set of odd cycles, where all other vertices on the cycles have degree two, and $u$ and $v$ are joined by a path of even order that alternates between single even order paths whose internal vertices all have degree two, and multiple even order paths whose internal vertices all have degree two.
\end{definition}
We restate a theorem from Ansill, Jacob, Penzellna, and Savvedra \cite{Ansill} which gives a characterization of all graphs with a failed skew zero forcing number of $0$.
\begin{theorem}{\cite{Ansill}}
$F^{-}(G)=0$ if and only if $G$ is one of the following graphs.
(i) An odd cycle, or a nonempty set of odd cycles whose intersection is a single vertex or (ii) A doubly extended bouquet-dipole.
\end{theorem}
In this paper we provide a characterization of all graphs with a failed skew zero forcing number of $1$. To show a graph has $F^{-}(G)=0$ one has to show that the set where $S=\varnothing $ is a failed zero forcing set and if $S\neq \varnothing $ then all of the vertices in the graph are forced. However to show a graph has $F^{-}(G)=1$ it is more complex. We need to show that there is a vertex which is a stalled set, and that any pair of vertices forces all vertices in the graph.
\section{Results}
\begin{theorem}
The only disconnected graph with $F^-(G)=1$ is $2K_1$.
\end{theorem}
\begin{proof}
Suppose $G$ is disconnected and has more than one two components or a component with more than one vertex. Then let $S$ contain all but one component or the component with two or more vertices. Now $|S|>1$. Now let $G$ have two components that both have one vertex; $G=2K_1$. Let $|S|=1$. It is clear that this is the maximum stalling set of $2K_1$. Therefore, $2K_1$ is the only disconnected graph for which $F^-(G)=1$.
\end{proof}
From here we will assume that all graphs are connected. We first show two small graphs that have a failed skew zero forcing number of $1$.
\begin{lemma}
$F^-(P_3)=1$.
\end{lemma}
\begin{proof}
Let the external vertices be labeled $v_1$ and $v_2$ and the middle vertex be $w$. If $S=\{w\}$, then $S$ is skew stalled; $F^-(P_3)\geq 1$. If $|S|=2$, then without loss of generality there will be a vertex in $S$ with one neighbor outside of $S$, so the whole graph will be skew forced. Therefore, $F^-(P_3)\leq 1$, giving the desired conclusion that $F^-(P_3)=1$.
\end{proof}
\begin{lemma}
$F^-(K_4)=1$.
\end{lemma}
\begin{proof}
Without loss of generality, choose a vertex on $K_4$. Then that vertex has three neighbors outside of $S$ which implies it will not skew force. Then each neighbor outside of $S$ has two neighbors outside of $S$ which implies each neighbor will not skew force. Thus, $F^-(K_4) \geq 1$. Now suppose $|S|=2$. Then the vertices outside of $S$ only have one uncolored neighbor which will be skew forced into $S$, and then any vertex in $S$ will force the remaining vertex that is outside $S$. So, $F^-(K_4) \leq 1$. Therefore, $F^-(K_4)=1$.
\end{proof}
\indent Now we define a substructure and and a lemma that provides a lower bound for the failed skew zero forcing number of a graph.
\begin{definition}
An $n$-blocking is a $P_{2n+1}$ subgraph whose external vertices have degree higher than 2 and internal vertices have degree 2.
\end{definition}
\indent When $n=1$, we will call this substructure a 1-blocking.
\begin{lemma}
(n-Blocking Lemma) If $G$ has no vertices of degree 1 and contains $k$ disjoint blockings of size $n_1,...n_m$, $$F^-(G)\geq \Sigma_{i=1}^m n_i$$.
\end{lemma}
\begin{proof}
Let $P_{2n+1}$ be the largest $n$-blocking on a graph, $G$ with vertex indexing $v_1,...v_{2n+1}$, where $v_1$ and $v_{2n+1}$ are the end vertices with degree larger than 2. Then define $S=\{v_2,v_4,...,v_{2n}\}$ such that $|S|=n$. Then each vertex in $S$ has two neighbors outside $S$ and will not skew force, $v_1$ and $v_{2n+1}$ have degree higher than 2, so they have at least two neighbors outside $S$ and will not skew force, and each $v_{2i+1}$ inside the $n$-blocking has exactly 2 neighbors inside $S$. Since $G$ has no vertices of degree 1, no vertex outside of the $n$-blocking will skew force either. So, $S$ is skew stalled, and $F^-(G)\geq n$.
\end{proof}
\begin{corollary}
If $G$ is an odd cycle with a chord that creates an odd cycle of length $2n+1$, then $F^-(G)=n$
\end{corollary}
\indent Notice that when $n=1$, then this family of graphs is defined by a 1-blocking. This useful result begins our notions of how to characterize all graphs with $F^-(G)=1$. The following three lemmas provide useful descriptions of graphs with $F^-(G)=1$, specifically surrounding their failed skew stalling sets.
It is possible for a graph where $F^-(G)=1$ and there is more than one vertex that is a stalling set. An example of a graph with two different vertices that are both stalling sets is shown in Figure 1.
\begin{center}
\begin{tikzpicture}
\coordinate (1) at (0,0);
\coordinate (2) at (0,2);
\coordinate (3) at (1.5,1);
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\coordinate (8) at (5.5,1);
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\draw (1) -- (2) -- (3) -- (1);
\draw (3) -- (4) -- (5) -- (6) -- (7) -- (8);
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\foreach \point in {5} \fill (\point) circle (4pt)[blue];
\foreach \point in {7} \fill (\point) circle (4pt)[red];
\end{tikzpicture}
Figure 1. A graph with two different stalling sets of size 1.
\end{center}
In our next lemma we show that it is not possible to have more than 2 different vertices that are each stalling sets.
\begin{lemma}
Unless $G=K_4$, there cannot be more than $2$ vertices that are each failed skew zero forcing sets of size $1$ in a graph where $F^{-}(G)=1$.
\end{lemma}
\begin{proof}
We will assume that $G$ is not $K_4$. We then proceed by contradiction. Let $G$ be a graph and $k\geq 3$ where $F^{-}(G)=1$ and $a_{1},a_{2},...,a_{k}$ are vertices that are each failed skew zero forcing sets of size $1$. We consider two cases.
\begin{itemize}
\item Case 1. There does not exist more than two vertices from the set $\{a_{1},a_{2},..,a_{t}\}$ that share a common neighbor. Without loss of generality consider three vertices $a_{1},a_{2}$, and $a_{3}$ where $N(a_{1})\cap N(a_{2})=\emptyset $. Then $\{a_{1},a_{2}\}$ form a failed skew zero forcing set of size $2$, which contradicts the assumption that $F^{-}(G)=1$.
\item Case 2. There exist vertices $a_{1},a_{2},$ and $a_{3}$ that share a common neighbor $v$. Now since each of the vertices $a_{1},a_{2},$ and $a_{3}$ are failed skew zero forcing sets of size $1$, all of the neighbors of each $a_{i}$, $1\leq i\leq t$ have degree $3$ or more. If $N(a_{1})=N(a_{2})$, $N(a_{1})=N(a_{3})$, and $N(a_{2})=N(a_{3})$ then $F^{-}(G)=\left\vert V(G)\right\vert =n-2.$ Suppose that $N(a_{1})\neq N(a_{2})$, $N(a_{1})\neq N(a_{3})$, and $N(a_{2})\neq N(a_{3})$. Then there is a vertex $x$ where $x\in N(a_{1})$ but $x\notin N(a_{2})$, and there is a vertex $y$ where $y\in N(a_{3})$ but $x\notin N(a_{2})$. Hence each of the vertices $a_{1},a_{2},$ and $a_{3}$ have two uncolored neighbors each of which have degree at least $3$. Then $\{a_{1},a_{2},a_{3}\}$ forms a failed skew zero forcing set of size $3$, which contradicts the assumption that $F^{-}(G)=1$.
\end{itemize}
\end{proof}
\begin{lemma}
(Cut vertex lemma) Let $G$ be a graphs with a cut vertex $v$ with $\deg(v)>2$. Let $H_1,H_2,..,H_t$ be the components of $G-v$ where $\left\vert H_{1}\right\vert \geq \left\vert H_{2}\right\vert \geq \cdots \geq \left\vert H_{t}\right\vert $.
Then $F^{-1}(G)\geq 1+\sum_{i=1}^{t-2}\left\vert H_{i}\right\vert $
\end{lemma}
\begin{proof}
If $v$ is a cut vertex of degree $t>2$, we can take the vertices in the $t-2$ largest component(s) and add $v$. Since each vertex in this set either has 0 or 2 uncolored neighbors, this set is a failed zero forcing set. Hence $F^{-1}(G)\geq 1+\sum_{i=1}^{t-2}\left\vert H_{i}\right\vert $
\end{proof}
\begin{lemma}
If $F^-(G)=1$ and $\{v\}=S$, $v$ cannot have a neighbor of degree 2
\end{lemma}
\begin{proof}
If $v$ has a neighbor, $w$, of degree 2, then $w$ will skew force its neighbor, $u$, contradicting that $S$ is skew stalled. So, $v$ cannot have a neighbor of degree 2.
\end{proof}
\indent With these lemmas in mind, we can create a necessary condition for all graphs with $F^-(G)=1$.
\begin{lemma}
Excluding $P_3$ and $K_4$, if $F^-(G)=1$, then $G$ contains 1 disjoint 1-blocking and no other disjoint $n$-blocking.
\end{lemma}
\begin{proof}
By the $n$-blocking lemma, since $F^-(G)=1$, if $G$ contains an $n$-blocking, then $\Sigma_{i=0}^m n_i =1$, where $n_i$ is the size of each disjoint $n$-blocking. Hence $G$ contains at most 1 $n$-blocking where $n=1$.\\
\indent To show that $G$ contains at least one disjoint 1-blocking, recall that if $\{v\}$ is our maximum failed skew stalling set, that unless $v$ is on a $P_3$ the neighbors of $v$ must have degree greater than 2. It remains to confirm that $v$ necessarily has degree 2 which will imply the existence of the 1-blocking.\\
\indent Suppose $\deg(v)\geq 3$. We know its neighbors must have degree higher than 2 since if a neighbor $u$ has degree 1 then ${u,v}$ would be a stalled set, contradicting the assumption that $F^{-}(G)=1$. \\\indent Suppose $v$ is a cut vertex of degree $t\geq3$. By Lemma 3.5 the set $S$ consisting of $v$ and vertices in the $t-2$ largest subsets of $G-v$ is skew stalled. \\\indent Next suppose $v$ is not a cut vertex. Since $v$ has no neighbors of degree 2, $v$ must be adjacent to a vertex $u$, which prevents the creation of a new $n$-blocking. Recall that $\deg(u)\geq 3$. If $u$ has no neighbor of degree 2, let $S=\{v,u\}$. $S$ is skew stalled with $|S|=2$, contradicting the hypothesis that $F^-(G)=1$. If $u$ has a neighbor of degree 2, that neighbor must either be part of an even path or an odd cycle. If that neighbor is on an even path, let $S=\{v,u\}$ and skew force along the even path until it skew stalls at the next vertex with degree higher than 2. If it is part of an odd cycle and has a neighbor with degree higher than 2, then let $S$ contain only $u$ and the vertices of the cycle to which $u$ belongs. Since $u$ has two uncolored neighbors of degree higher than 2, $S$ is skew stalled with $|S|>1$, which contradicts the original hypothesis. Therefore, if $F^-(G)=1$, then it must be that $\deg(v)=2$.\\
\indent We still have to consider the case when $G$ has two distinct maximum skew stalling sets of cardinality 1, $\{v\}$ and $\{w\}$. By Lemma 3.6, neither vertex can have a neighbor of degree 2. Their shared neighbor, $u$ must have degree 3 or $\{v,w\}$ will be a skew stalling set.\\
\indent Suppose first that $\deg(v)$ and $\deg(w)>2$. If $u$ is a cut vertex, then by the above lemma, $F^-(G)>1$, so $u$ cannot be a cut vertex. If $u$ is not a cut vertex, then its neighbor, $x$ will have degree 2 or higher. If $\deg(x)=2$, then set $S=\{u\}$ and skew force along the even path until $S$ skew stalls upon reaching some vertex with degree 3 or higher. If $\deg(x)>2$ and the neighbor of $x$ has degree 3 or higher, then set $S=\{x,u\}$, which will stall and contradict the original statement of the theorem. So, let $x$ have at least one neighbor of degree 2. Then, since $u$ is not a cut vertex, this neighbor may be on an even path or an odd cycle. If $x$ only has a degree 2 neighbor on an odd cycle, then $x$ must have some other degree 3 neighbor that connects to other parts of the graph, since $u$ is not a cut vertex. In that case, let $S$ contain $x$ and the vertices of that odd cycle. Otherwise, let $S=\{x\}$ and skew force along optional odd cycle and then the even path until it reaches a degree 3 vertex and skew stalls. Then $|S|>1$, contradicting the original hypothesis.
\\\indent Therefore, $G$ contains at most and at least 1 disjoint 1-blocking; $G$ contains exactly 1 disjoint 1 blocking and no other $n$-blocking if $F^-(G)=1$.
\end{proof}
\indent As we close in on the necessary and sufficient conditions for $F^-(G)=1$, we consider the following important substructure that can exist within such graphs:
\begin{definition}
An even multiple path has vertices $u$ and $v$ with 2 or more even paths connecting them and are buffered on both sides by an even path.
\end{definition}
An example of an even multiple path can be seen in the middle of the first graph in Figure 2. Next we provide our main result, giving a full characterization of all graphs with $F^-(G)=1$.
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\newline
\newline
\medskip
Figure 2. Some graphs with $F^-(G)=1$
\end{center}
\begin{theorem}
$F^-(G)=1$ if and only if $G$ is $K_4$ or $P_3$ or $G$ has:
\begin{itemize}
\item Exactly 1 disjoint 1-blocking, no more than 2 non-disjoint 1-blockings that share a vertex of degree 3, and no other $n$-blocking
\item All degree 2 vertices are in a 1-blocking, an even path with external vertices with higher degree than 2, and even multiple path, or an odd cycle with exactly one vertex of degree higher than 2
\item There are no pendant even cycles on $G$ and no vertices of degree 1.
\item Outside of the 1-blocking, no vertex has more than one neighbor of degree 3.
\item All but one of the exterior vertices of the 1-blockings can have one neighbor that is in an odd cycle or adjacent to another exterior of a 1-blocking through an even path.
\end{itemize}
\end{theorem}
\begin{proof}
We know $F^-(G)=1$ for $K_4 $ and $P_3$ by Lemmas 3.1 and 3.2. Suppose $F^-(G)=1$. By Lemma 3.7, if $F^-(G)=1$, then $G$ contains 1 disjoint 1-blocking and no other disjoint $n$-blocking, satisfying the first criterion. Furthermore, suppose there is a degree 2 vertex $v$ in $V(G)$ that is not in one of the specified structures of the second criterion. Then $v$ would be in a separate $n$-blocking, contradicting the first criterion. $G$ cannot have a pendant even cycle since this contradicts the second criteria. If $G$ has a degree 1 vertex, it will skew force its neighbor. If that neighbor is not an element of a maximum failed skew forcing set, this contradicts that $F^-(G)=1$. If it is, then this vertex is not in a 1-blocking. For completeness, suppose that this vertex is a maximum skew stalling set. Therefore it cannot have a neighbor of degree 2. By Lemma 3.5, this would cause $F^-(G)>1$, contradicting that $F^-(G)=1$. Now suppose a vertex outside the 1-blocking has 2 neighbors of degree higher than 2. Then, this vertex will stall in $S$ but color any neighbors of degree 2. These neighbors are either in an odd cycle or an even path. If they are in an odd cycle, they are all skew forced, but will not be able to spread skew forcing to the rest of the graph, as they only have one vertex of degree higher than 2. If they are on an even path, then skew forcing will stall before the next vertex of degree higher than 2. So, the fourth criterion holds. Finally, if all exterior vertices of 1-blockings have a neighbor that is not in an odd cycle or on an even path connected to another exterior vertex of a 1-blocking. Then if each exterior vertex of the 1-blocking is in $S$, they will skew force along their neighbors of degree 2, but, since these vertices are in neither an odd cycle nor a 1-blocking, they must be on an even path where skew forcing will stall before the next vertex of degree higher than 3 that is not an exterior vertex of the 1-blocking. Therefore, the theorem holds that if $F^-(G)=1$, it must have all of the criteria.
Suppose $G$ satisfies all the conditions. Let $v$ be the first vertex we place in $S$. Begin with $v$ being the vertex of degree 2 in the 1-blocking. Each neighbor has degree higher than 2. Therefore, $S$ is skew stalled and $F^-(G)\geq 1$.
Now, there are 5 cases for the different vertices we can start with for $v$.
\begin{itemize}
\item (Case 1) Suppose $v$ is the vertex on a nonempty collection of odd cycles with its degree higher than 2. This will skew force the whole collection of odd cycles into $S$. Then it will follow one of the following cases:
\begin{itemize}
\item Case 1(a): Let $w$ be a degree 2 neighbor of $v$ in $S$.
\begin{itemize}
\item Case 1(a)i: $w$ is on a $P_{2n}$ where it will continue to skew force along the path until the next vertex with degree higher than 2 is skew forced into $S$. Then refer to case 2.
\item Case 1(a)ii: $w$ is the vertex of degree 2 in a 1-blocking. It will then skew force its neighbor with degree higher than 3, which is the intersection point of a nonempty collection of odd cycles which will be forced or the beginning of one or more even paths, which leads to case 1(a).
\end{itemize}
\item Case 1(b): Let $w$ be a degree 3 neighbor of $v$.
\begin{itemize}
\item Case 1(b)i: $w$ is the beginning of one or more even paths. Then skew forcing will continue along each path; this case naturally leads to case 1(a).
\item Case 1(b)ii: $w$ is a vertex in the 1-blocking with degree higher than 2. The vertex of degree 2 in the 1-blocking will skew force its neighbor which has degree higher than 2. Then, that vertex is the intersection of a nonempty collection of odd cycles which will now be skew forced, or the beginning of one or more even path(s), which leads to case 1(a).
\end{itemize}
\end{itemize}
\indent Since $G$ does not have any pendant even cycles and no vertex with degree 1, eventually the skew forcing will reach some vertex, $z$, that is the beginning of a pendant. $z$ cannot be cases 1(a) or 1(b) since it then would imply the existence of a degree 1 vertex. Furthermore, $z$ cannot be the vertex in case 2(a) since that would create one or more pendant even cycle(s). So, $z$ is the beginning of a pendant of a nonempty collection of odd cycles. Besides the degree 2 vertex of a 1-blocking, all neighbors of $z$ will be in $S$. Then all degree 2 neighbors of $z$ will skew force their neighbors and $z$ will skew force that vertex in a 1-blocking if they are neighbors. Now, each vertex in $S$ has at most one neighbor outside of $S$, so they will skew force their remaining neighbors. Finally, any degree 3 vertex with an edge incident to a nonempty set of odd cycles will skew force the connecting degree 3 vertex and thus skew force the entire set.
\item (Case 2) Now suppose $v$ is a vertex of degree 2 on a cycle. Skew forcing will continue along the cycle until you hit the only vertex with degree higher than 2 on the cycle. At this point, the reader can refer back to the case where that vertex with degree higher than 2 is $v$.
\item (Case 3) Suppose $v$ is the exterior point of a 1-blocking. Automatically, the other exterior vertex of the 1-blocking is skew forced. If $v$ shares a common exterior of a 1-blocking with that vertex, that exterior vertex will also be skew forced. So, all exterior vertices of 1-blockings are in $S$, causing each even path between them to be skew forced. Since there is one exterior vertex with no neighbors besides vertices in an odd cycle or vertices that are connected via an even path to another exterior of a 1-blocking, that exterior vertex will now skew force the degree 2 vertex of the 1-blocking. If any other exterior vertex is adjacent to an even path or multiple even paths that do not lead to another exterior of a 1-blocking, then it will skew force two consecutive vertices along this path, that will end in a nonempty collection of odd cycles or a multiple path, which would lead back to Cases 1 or 2.
\item (Case 4) Suppose $v$ has degree 2 on an even path. It will then skew force along the path until one of the degree 3 vertices mentioned above is skew forced.
\item (Case 5) Suppose there are two non-disjoint 1-blockings. Let $x$ and $y$ be the degree 2 vertices in these 1-blockings. Then if $S=\{x,y\}$, their common vertex has degree 3 and therefore will skew force another vertex, leading to one of the above cases.
\end{itemize}
Since selecting any vertex other than the degree 2 vertex in a 1-blocking will cause the whole graph to become skew forced, and selecting two degree 2 vertices from a 1-blocking causes the whole graph to become skew forced, we know that $F^-(G)\leq 1$. Therefore, $F^-(G)=1$.
\end{proof}
\indent From this result, we observe the following corollary:
\begin{corollary}
If $F^-(G)<2$, $G$ is planar.
\end{corollary}
\section{Conclusion}
Now that the characterization of graphs with $F^-(G)=0$ and $F^-(G)=1$ have been completed the next logical step is to look at graphs where $F^-(G)=k$ for $k\geq2$. We also proved that a graph with $F^-(G)=1$ can have at most two vertices that can be failed skew zero forcing sets of size 1. An analogous question can be asked. How many different failed skew zero forcing sets of size $t$ exist for a graph with $F^-(G)=t$?
\newline
\medskip
\textbf{Acknowledgements} This research was supported by the National Science Foundation Research for Undergraduates Award 1950189.
\printbibliography
\end{document}
| {
"timestamp": "2022-09-21T02:05:59",
"yymm": "2209",
"arxiv_id": "2209.09379",
"language": "en",
"url": "https://arxiv.org/abs/2209.09379",
"abstract": "Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Hence the failed zero forcing number of a graph was defined to be the size of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. The difference is that vertices that are not in $S$ can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by $F^{-}(G)$. In this paper we provide a complete characterization of all graphs with $F^{-}(G)=1$. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of $1$ are either: the union of two isolated vertices; $P_3$; $K_3$; or $K_4$. In this paper we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with $F^{-}(G)=1$.",
"subjects": "Combinatorics (math.CO)",
"title": "Characterization of Graphs With Failed Skew Zero Forcing Number of 1",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795102691455,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379416617377
} |
https://arxiv.org/abs/1109.2934 | On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime | Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of m(A) over F-free sets A in G, where m is the normalized Haar measure on G. Our main result is that, for any such collection F of forms in at least three variables, the sequence d_F(Z_p) converges to d_F(R/Z) as p tends to infinity over primes. This answers an analogue for Z_p of a question that Ruzsa raised about sets of integers. | \section{Introduction}
Much work in arithmetic combinatorics concerns the maximal density that a subset of a finite abelian group can have if it does not contain a non-trivial solution to a given linear equation. Examples include the study of sum-free sets, where the equation to be avoided is $x_1+x_2-x_3=0$, and the improvement of bounds for Roth's theorem, which concerns the equation $x_1-2x_2+x_3=0$.
A natural question about these maximal densities is whether they exhibit some particular asymptotic behaviour as the groups get larger in a family of groups, and a typical family to consider in this context is that of the groups $\Zmod{p}$ of residues modulo a prime. One may thus ask whether the maximal density of a subset of $\Zmod{p}$ avoiding solutions to some linear equation converges as $p\to \infty$. The analogous question for subsets of the integers $\{1, \ldots, N\}$ was raised by Ruzsa \cite[Problem 2.3]{ruzsa:linear-equationsII}, who conjectured that the corresponding limit exists for any linear equation. The main result in this paper implies that for the groups $\Zmod{p}$ the maximal density does indeed converge. The result itself is more general, and to state it precisely we shall use the following notation.
\begin{defn}\label{L-free}
Let $L(x) = c_1 x_1 + \cdots + c_t x_t$ be a linear form with non-zero integer coefficients. We say that a subset $A$ of an abelian group $G$ is \emph{$L$-free} if there is no $t$-tuple $x=(x_1,\ldots,x_t) \in A^t$ such that $L(x) = 0$. For a family $\mathcal{F}$ of linear forms we say that $A$ is \emph{$\mathcal{F}$-free} if $A$ is $L$-free for every $L \in \mathcal{F}$.
\end{defn}
For a given family $\mathcal{F}$ of linear forms, we define
\[ \md_\mathcal{F}(\Zmod{p}) = \max\{ |A|/p : \text{$A \subset \Zmod{p}$ and $A$ is $\mathcal{F}$-free} \}. \]
Our main result implies firstly that for any finite family $\mathcal{F}$ of forms in at least 3 variables, the maximal density $\md_\mathcal{F}(\Zmod{p})$ converges as $p \to \infty$ over primes. This is analogous to a result of Croot \cite{croot} establishing the convergence of the minimal normalised-count of three-term arithmetic progressions in subsets of $\Z_p$ of some fixed density,
and indeed a variant of Croot's method will be an essential part of our argument. Before stating the result in full, let us note that convergence can fail in our result if we do not restrict to prime moduli, as was also the case in \cite{croot}. Indeed, sum-free sets in $\Zmod{p}$ are easily shown to have maximal density converging to $1/3$ as $p \to \infty$ over primes (using the Cauchy-Davenport inequality \cite{T-V}), but in $\Zmod{2p}$ the odd residues form a sum-free set of density $1/2$.
The convergence of $\md_\mathcal{F}(\Zmod{p})$ leads to the problem of determining the limit. To address this it is potentially helpful to have a fixed group on which the limit can be analyzed. We show that the circle is one possible such group.
\begin{defn}
For a family $\mathcal{F}$ of linear forms and a compact abelian group $G$ with normalized Haar measure $\mu$, we define
\[\md_\mathcal{F}(G) = \sup \{ \mu(A) : \text{$A \subset G$ is measurable and $\mathcal{F}$-free}\}.\]
\end{defn}
We can now state our main result.
\begin{theorem}\label{mainres}
Let $\mathcal{F}$ be a finite family of linear forms, each in at least three variables.
Then $\md_{\mathcal{F}}(\Zmod{p})\to \md_{\mathcal{F}}(\T)\textrm{ as }p \to \infty\textrm{ over primes.}$
\end{theorem}
The methods in this paper do not allow to extend this result to families containing forms in two variables, essentially because such forms do not
fall under the purview of Fourier analysis; in the language of \cite{GTlin}, equations in two variables are of unbounded complexity, while equations in at least three variables are of complexity 1. (Note however that for a single non-trivial form $L$ in two variables it is easy to show that $d_L(\Zmod{p})\to 1/2$.)
It is an important fact that if $L$ is a translation-invariant form, that is a form $c_1 x_1 + \cdots + c_t x_t$ with $c_1 + \cdots + c_t = 0$, then $\md_L(\Zmod{p}) \to 0$ as $p \to \infty$, even if we allow an $L$-free set
to contain certain trivial solutions (constant solutions $(x,\ldots,x)$, for instance); this follows from Roth's method, as observed in a very similar context in \cite[Theorem 1.3]{ruzsa:linear-equationsI}. Therefore the limit in Theorem \ref{mainres} is also $0$ for any family $\mathcal{F}$ containing a translation-invariant form. On the other hand, if every member of $\mathcal{F}$ is not translation-invariant (to be concise let us say \emph{non-invariant}), then the limit will be positive.
\begin{proposition}\label{positivity}
Let $\mathcal{F}$ be a finite family of non-invariant linear forms, each in at least three variables. Then
$\md_{\mathcal{F}}(\T) = \lim_{p \to \infty} \md_{\mathcal{F}}(\Zmod{p}) > 0$.
\end{proposition}
This situation is thus analogous to that of the maximal edge-densities of graphs with forbidden subgraphs: if one of the forbidden subgraphs is bipartite then the maximal edge-density of a graph on $n$ vertices that does not contain the forbidden subgraphs tends to $0$ as $n \to \infty$; otherwise it tends to a positive limit determined by the chromatic numbers of the forbidden subgraphs.
It would be very interesting to have a simple asymptotic formula for $\md_\mathcal{F}(\Zmod{p})$ in terms of the coefficients of the linear forms, if indeed such a formula exists. While Theorem \ref{mainres} does not give such a formula, it does show that it suffices to analyze families of linear equations on $\T$ in order to find one.
The proof of Theorem \ref{mainres} and the remainder of this paper are laid out as follows. In Section \ref{TL_props} we gather tools that will enable us to use Fourier analysis to tackle the problem. In Section \ref{Transference} we review and extend a result from \cite{OlofTh}, itself based on work of Croot \cite{croot}, that allows us to transfer a solution-free set in $\Zmod{p}$ of density $\alpha$ to an almost-solution-free function on $\T$ with average $\alpha$. Section \ref{functions-to-sets} then shows how one may use this function to produce an almost-solution-free set of density close to $\alpha$; we then use a removal lemma on $\T$, proved in Section \ref{Removal}, to obtain a truly solution-free subset of $\T$ of density close to $\alpha$.
In Section \ref{Final} we combine these results to prove Theorem \ref{mainres}, and we also prove Proposition \ref{positivity}.
The transference result in Section \ref{Transference} concerns general compact abelian groups and we shall therefore work in
this general setting until the end of that section. From Section \ref{functions-to-sets} onwards we shall focus on the groups $\Zmod{p}$ and $\T$.
\section{Properties of $\Tsol_L$}\label{TL_props}
One of the main tools used in this paper is a multilinear operator associated with a given linear form $L$. This operator, which we denote $\Tsol_L$, is well known in arithmetic combinatorics in the setting of finite groups. In this section we define this operator and establish some of its main properties.
In a finite abelian group $G$, a set $A$ is $L$-free in the sense of Definition \ref{L-free} if and only if $|A^t\cap \ker L|=0$,
where $t$ is the number of variables in $L$ and $\ker L$ denotes the subgroup $\{x\in G^t:L(x)=0\}$ of the direct product $G^t$.
The quantity $|A^t\cap \ker L|/|\ker L|$ is the normalized count of solutions to $L(x)=0$ inside $A$. This notion has a natural generalization to any compact abelian group $G$, using the normalized Haar measure on $\ker L$.
\begin{defn}\label{TLs}
Let $G$ be a compact abelian group, let $L$ be a linear form in $t$ variables, and let $\mu_L$ denote the Haar measure on the closed subgroup $\ker L$ of $G^t$ satisfying $\mu_L(\ker L)=1$. Then for any $t$ measurable sets $A_1,A_2,\ldots,A_t$ we define the \emph{solution measure}
$\Tsol_L(A_1,\ldots,A_t) = \mu_L\big( (A_1\times \cdots\times A_t) \cap \ker L\big)$.
When $A_1=A_2=\cdots=A_t=A$ we write more concisely $\Tsol_L(A)$.
\end{defn}
The operator $\Tsol_L$ is the result of extending this definition from sets to functions.
\begin{defn}\label{TLf}
Let $G$ be a compact abelian group, let $L$ be a linear form in $t$ variables, and let $f_1,f_2,\ldots,f_t : G\rightarrow \C$ be measurable functions. We then define
\begin{equation}\label{TLfeq}
\Tsol_L(f_1,\ldots,f_t) = \int_{\ker L} f_1\otimes f_2\otimes \cdots \otimes f_t(x) \ud\mu_L(x),
\end{equation}
where $f_1\otimes f_2\otimes \cdots \otimes f_t(x) := f_1(x_1)f_2(x_2)\cdots f_t(x_t)$ for all $x=(x_1,\ldots,x_t) \in \ker L \subset G^t$. When $f_1=f_2=\cdots=f_t=f$ we write more concisely $\Tsol_L(f)$.
\end{defn}
Clearly Definitions \ref{TLs} and \ref{TLf} agree on measurable sets $A_1,\ldots,A_t$, that is we have $\Tsol_L(A_1,\ldots,A_t)=\Tsol_L(1_{A_1},\ldots,1_{A_t})$, where $1_X$ denotes the indicator function of a set $X$.
Let us recall the following standard fact, which will be used throughout the paper.
\begin{lemma}\label{cts-surj-hom}
Let $G$ and $H$ be compact abelian groups and let $\phi : G \to H$ be a surjective continuous homomorphism. Then $\phi$ preserves the normalized Haar measures, that is $\mu_H = \mu_G \circ \phi^{-1}$.
\end{lemma}
We shall also use the Gowers $U^2$ norm.
\begin{defn}
Let $\mathcal{L}_2(G)$ denote the Hilbert space of square-integrable complex-valued functions on $G$.
The $U^2$ norm $\norm{\cdot}_{U^2(G)}$ can be defined on $\mathcal{L}_2(G)$ by the formula
\begin{equation}\label{U2}
\norm{f}_{U^2(G)}^4=\int_{G^4} f(x+y)\,\overline{f(x+y')}\,\overline{f(x'+y)}\, f(x'+y') \ud x\ud x'\ud y\ud y'.
\end{equation}
When the group $G$ in question is clear we write more concisely $\norm{f}_{U^2}$.
\end{defn}
By a simple application of the Cauchy-Schwarz inequality to the right hand side of \eqref{U2}, we obtain the bound $\norm{f}_{U^2}\leq\norm{f}_{\mathcal{L}_2}$.
For an integer $n$ we let $n\cdot X$ denote the image of $X\subset G$ under the continuous map $x\mapsto n x$. We refer to this map as \emph{dilation} by $n$ or $n$-\emph{dilation}.
The following result is well known in the setting of finite groups and will be used below.
\begin{theorem}\label{U2control}
Let $L$ be a linear form in $t\geq 3$ variables, and let $G$ be a compact abelian group such that each coefficient of $L$ gives a surjective dilation on $G$. Then for any $f_1,\ldots,f_t\in \mathcal{L}_2(G)$ we have
\begin{equation}\label{U2C}
|\Tsol_L(f_1,\ldots,f_t)|\leq \min_{i\in [t]}\; \norm{f_i}_{U^2} \prod_{j\neq i} \norm{f_j}_{\mathcal{L}_2}.
\end{equation}
\end{theorem}
Here $[t]$ denotes the set of integers $\{1,2,...,t\}$.
\begin{proof}
Writing $L(x) = c_1 x_1 + \cdots + c_t x_t$, by assumption we have $c_i \cdot G = G$ for each $i\in [t]$. We shall prove the upper bound in
\eqref{U2C} just for $i=t$, which is sufficient by symmetry.
By the assumption on $c_j$-dilations, the map $G^{t-1}\to \ker L$, \[(y_1,...,y_{t-1})\mapsto (c_ty_1,c_ty_2,\ldots,c_ty_{t-1},-c_1y_1-c_2y_2-\cdots-c_{t-1}y_{t-1})\]
is surjective, so by Lemma \ref{cts-surj-hom} we have
\begin{equation}\label{param}
\Tsol_L(f_1,\ldots,f_t)=\int_{G^{t-1}} f_1(c_ty_1)\cdots f_{t-1}(c_ty_{t-1})f_t(-c_1y_1-\cdots-c_{t-1}y_{t-1})\ud y_1\cdots \ud y_{t-1}.
\end{equation}
We can use this expression to prove the desired upper bound by an argument which is standard in the setting of finite abelian groups, consisting
in two applications of the Cauchy-Schwarz inequality. We include the details in the present setting for completeness.
First we apply Fubini's theorem and the Cauchy-Schwarz inequality over variables $y_2,...,y_{t-1}$ to obtain
\begin{align*}
|\Tsol_L(f_1,\ldots,f_t)|^2 &\leq \int_{G^{t-2}} |f_2(c_t y_2) \cdots f_{t-1}(c_t y_{t-1})|^2 \ud y_2\cdots \ud y_{t-1} \\
&\quad \cdot \int_{G^{t-2}}\left|\int_G f_1(c_ty_1)f_t(-c_1y_1-\cdots-c_{t-1}y_{t-1}) \ud y_1 \right|^2\ud y_2\cdots \ud y_{t-1}.
\end{align*}
Applying Lemma \ref{cts-surj-hom} to $c_t$-dilation and $(y_2,\ldots,y_{t-1})\mapsto x=-c_2y_2-c_3y_3-\cdots-c_{t-1}y_{t-1}$, the right hand side above is found to equal
\[\norm{f_2}_{\mathcal{L}_2}^2\cdots \norm{f_{t-1}}_{\mathcal{L}_2}^2 \int_G\left|\int_G f_1(c_ty)f_t(x-c_1y) \ud y \right|^2\ud x.\]
By Fubini's theorem the integral here equals
\[\int_{G^2} f_1(c_ty)\overline{f_1(c_ty')} \left(\int_G f_t(x-c_1y)\overline{f_t(x-c_1y')} \ud x\right)\ud y\ud y'.\]
Applying Cauchy-Schwarz over $(y,y')$, Lemma \ref{cts-surj-hom} for dilations by $c_1$ and by $c_t$, and Fubini's theorem again, we find that this integral is at most $\norm{f_1}_{\mathcal{L}_2}^2$ times
\[
\left(\int_{G^4} f_t(x+y)\,\overline{f_t(x+y')}\, \overline{f_t(x'+y)}\, f_t(x'+y') \ud x\ud x'\ud y\ud y'\right)^{1/2}=\norm{f_t}_{U^2}^2.\qedhere
\]
\end{proof}
Note that, since the $\mathcal{L}_2$ norm dominates the $U^2$ norm, \eqref{U2C} implies immediately
\begin{equation}\label{L2control}
|\Tsol_L(f_1,\ldots,f_t)|\leq \prod_i \norm{f_i}_{\mathcal{L}_2}.
\end{equation}
We now turn to Fourier-analytic aspects of $\Tsol_L$. Let us first settle on some notation. For a compact abelian group $G$ and any real $r\geq 1$ we
denote by $\mathcal{L}_r(G)$ the Banach space of complex-valued functions on $G$ with integrable $r$th power. The implicit measure here
is the normalized Haar measure on $G$. The Pontryagin dual $\widehat{G}$ of $G$ is a discrete group, so by $\mathcal{L}_r(\widehat{G})$ we denote
the analogous Banach space but with the implicit measure being the counting measure on $\widehat{G}$, which assigns value 1 to each singleton.
We shall use the Fourier transform on $\mathcal{L}_2(G)$ (or Plancherel transform). By the compactness of $G$ we have $\mathcal{L}_2(G)\subset\mathcal{L}_1(G)$.
If $f\in \mathcal{L}_1(G)$ then the Fourier transform $\widehat{f}$ is defined for $\gamma\in \widehat{G}$ by
$\widehat{f}(\gamma)=\int_G f(x)\overline{\gamma(x)}\ud\mu_{G}(x)$.
If $\widehat{f}$ is also in $\mathcal{L}_1(\widehat{G})$ then we have the Fourier inversion formula
\begin{equation}\label{inversion}
f(x)=\int_{\widehat G} \widehat{f}(\gamma)\gamma(x) \ud\mu_{\widehat{G}}(\gamma).
\end{equation}
Plancherel's theorem gives us that for any $f\in \mathcal{L}_2(G)$, $\widehat{f}\in\mathcal{L}_2(\widehat{G})$ and $\norm{f}_{\mathcal{L}_2(G)}=\norm{\widehat{f}}_{\mathcal{L}_2(\widehat{G})}$.
The following expression of $\Tsol_L$ in terms of the Fourier transforms $\widehat{f_i}$ will be used in the next section.
\begin{proposition}\label{TLinversion}
Let $L = c_1 x_1 + \cdots + c_t x_t$ be a linear form and let $G$ be a compact abelian group such that
$c_i\cdot G=G$ for every $i$. Then for any $f_1,\ldots,f_t\in\mathcal{L}_2(G)$ we have
\begin{equation}\label{FI}
\Tsol_L(f_1,\ldots,f_t)= \int_{\widehat{G}} \widehat{f_1} (\gamma^{c_1}) \cdots \widehat{f_t} (\gamma^{c_t}) \ud\mu_{\widehat{G}}(\gamma).
\end{equation}
\end{proposition}
\begin{proof}
Let $H=\ker L$. First we prove \eqref{FI} for $f_1,\ldots,f_t$ having Fourier transforms in $\mathcal{L}_1(\widehat{G})$. In this case each $f_i$ is continuous (by \eqref{inversion}), and it follows that the function $F$ defined on $G^t/H$ by $F(x+H)= \int_H f(x+y)\ud\mu_H(y)$ is continuous on $G^t/H$. Note that $\Tsol_L(f_1,\ldots,f_t)=F(0)$. Let us consider the Fourier coefficients $\widehat{F}(\chi)$, $\chi\in \widehat{G^t/H}\cong H^\perp$. On one hand, it follows from $c_i\cdot G=G$ that $\gamma\mapsto \chi=\gamma\circ L$ is an isomorphism $\widehat{G}\to H^{\perp}$. On the other hand, using a standard formula for integration on quotient groups \cite[2.7.3 (2)]{RudLCA} one checks that $
\widehat{F}(\gamma\circ L)=\int_{G^t}f(x)\overline{\gamma\circ L(x)} \ud\mu_{G^t}(x)=\widehat{f_1}(\gamma^{c_1})\cdots \widehat{f_t}(\gamma^{c_t})$.
Therefore, provided $\widehat{F}$ is in $\mathcal{L}_1(H^\perp)$, we can apply Fourier inversion to conclude that
\[
\Tsol_L(f_1,\ldots,f_t)=F(0)=\int_{H^\perp} \widehat{F}(\chi) \ud\mu_{H^\perp}(\chi) =\int_{\widehat{G}} \widehat{f_1}(\gamma^{c_1})\cdots \widehat{f_t}(\gamma^{c_t})\ud\mu_{\widehat{G}}(\gamma).
\]
The function $\gamma\mapsto \widehat{f_1}(\gamma^{c_1})\cdots \widehat{f_t}(\gamma^{c_t})$ is shown to be indeed in $\mathcal{L}_1(\widehat{G})$ using Cauchy-Schwarz, Plancherel's theorem, and the fact that $\gamma\mapsto \gamma^{c_i}$ is injective (i.e. that $c_i\cdot G=G$).
Now let $f_i\in\mathcal{L}_2(G)$, so $\widehat{f_i}\in\mathcal{L}_2(\widehat{G})$ with $\norm{\widehat{f_i}}_{\mathcal{L}_2(\widehat{G})}=\norm{f_i}_{\mathcal{L}_2(G)}$. For each $i$ we have a sequence $\widehat{g}_{i,n}$ in
$\mathcal{L}_1(\widehat{G})$ with $\widehat{g}_{i,n}\to \widehat{f_i}$ in the $\mathcal{L}_2(\widehat{G})$ norm, whence also $g_{i,n}\to f_i$ in $\mathcal{L}_2(G)$. One then uses multilinearity of $\Tsol_L$, \eqref{L2control}, and Cauchy-Schwarz on $\mathcal{L}_2(\widehat{G})$ to show that
\begin{eqnarray*}
\Tsol_L(f_1,\ldots,f_t)&=&\lim_{n\to\infty} \Tsol_L(g_{1,n},g_{2,n},\ldots,g_{t,n})\\
&=&\lim_{n\to\infty}\int_{\widehat{G}} \widehat{g}_{1,n}(\gamma^{c_1})\cdots \widehat{g}_{t,n}(\gamma^{c_t}) \ud\mu_{\widehat{G}}(\gamma)
=\int_{\widehat{G}} \widehat{f}_1(\gamma^{c_1})\cdots \widehat{f}_t(\gamma^{c_t})\ud\mu_{\widehat{G}}(\gamma).
\end{eqnarray*}
\end{proof}
We close this section with the observation that for the proof of Theorem \ref{mainres} we can assume that each $L\in\mathcal{F}$ has coprime coefficients. This is justified by the following lemma.
\begin{lemma}
Let $\mathcal{F}$ be a finite family of linear forms, and let $\mathcal{F}'$ be obtained by multiplying each $L\in\mathcal{F}$ by some non-zero integer $n_L$. Let $G$ be a compact abelian group such that $n_L\cdot G=G$ for every $L$. Then $\md_\mathcal{F}(G)= \md_{\mathcal{F}'}(G)$.
\end{lemma}
\begin{proof}
It is clear that any $nL$-free set is also $L$-free, so $\md_{\mathcal{F}'}(G) \leq \md_\mathcal{F}(G)$. On the other hand, if $A$ is $L$-free then $n^{-1}A$ is $nL$-free for any $n$, for if $x_i \in n^{-1}A$ and $(n c_1) x_1 + \cdots + (n c_t) x_t = 0$ then $(n x_1, \ldots, n x_t) \in A^t$ is a solution. If in addition $n \cdot G = G$ then $\mu(n^{-1}A) = \mu(A)$. These properties imply easily that if $A$ is $\mathcal{F}$-free then $m^{-1} A$ is $\mathcal{F}'$-free, where $m = \prod_{L \in \mathcal{F}} n_L$, and so $\md_\mathcal{F}(G) \leq \md_{\mathcal{F}'}(G)$.
\end{proof}
\section{Transference}\label{Transference}
In proving Theorem \ref{mainres} we shall need to move sets between the groups $\Zmod{p}$ and $\T$. A result essentially allowing us to do so was established in Chapter 4 of \cite{OlofTh} by extending ideas of Croot \cite{croot}, though for simplicity it was assumed there that all linear forms considered had at least one coefficient equal to $1$. In this section we shall review this result and also show how to eliminate the assumption on the coefficients, thus obtaining Proposition \ref{transfer} and, as our main application, Corollary \ref{transfer-cor}.
Central to the results we are about to discuss is the notion of Freiman isomorphism.
\begin{defn} Let $k \geq 2$ and let $A \subset G$, $B \subset H$ be subsets of two abelian groups. We call a function $\varphi : A \to B$ a \emph{Freiman $k$-isomorphism} if it is a bijection and
\[ a_1 + \cdots + a_k = a_{k+1} + \cdots + a_{2k} \Longleftrightarrow \varphi(a_1) + \cdots + \varphi(a_k) = \varphi(a_{k+1}) + \cdots + \varphi(a_{2k}) \]
for all $a_i \in A$.
\end{defn}
Thus Freiman $k$-isomorphisms, or just $k$-isomorphisms for short, preserve additive relations of length at most $k$. The main result of \cite[Chapter 4]{OlofTh} was that if one can find $k$-isomorphisms between small subsets of the duals of $G$ and $H$, then one can model functions on one group by functions on the other, in a particular sense. That sense uses the following notion of \emph{admissibility} of a linear form.
\begin{defn}\label{GenAdm}
Let $G$ be a compact abelian group and let $L = c_1 x_1 + \cdots + c_t x_t$ be a linear form in at least three variables with coprime coefficients.
We say that $L$ is \emph{$G$-admissible} if $c_i \cdot G = G$ for each coefficient $c_i$. Since the $c_i$ are coprime there are integers $n_i$ such that $n_1 c_1 + \cdots + n_t c_t = 1$; we call the minimum value of $\abs{n_1} + \cdots + \abs{n_t}$ over such integers the \emph{multiplier-height} of $L$ and we denote this quantity by $h(L)$. We shall say that $L$ is \emph{$k$-admissible} if
\begin{equation}\label{klb}
k \geq \max\{ h(L), \abs{c_1}, \abs{c_2}, \ldots, \abs{c_t} \}.
\end{equation}
We give the obvious meaning to $(G,k)$-admissibility and, if $H$ is another compact abelian group, to $(G,H,k)$-admissibility.
\end{defn}
One more definition is needed in order to state the transference result.
\begin{defn}
Let $G$ and $H$ be abelian groups. We say that \emph{$H$ can Freiman $(n,k)$-model $G$} if for any subset $A$ of $G$ of size $|A|\leq n$ there exists a Freiman $k$-isomorphism from $A$ to a subset of $H$.
\end{defn}
\begin{proposition}\label{transfer}
Let $\epsilon>0$ and $h$ be a positive integer. Suppose $G$ and $H$ are two compact abelian groups such that,
for some $k \geq k_0(\epsilon)$ and all $n \leq C(\epsilon)^h$, $\widehat{H}$ can Freiman $(n,k)$-model $\widehat{G}$. Let $f : G \to [0,1]$ be a measurable function with $\int_G f = \alpha$. Then there is a continuous function $g : H \to [0,1]$ with $\int_H g = \alpha$ such that
\[ \abs{\Tsol_L(f) - \Tsol_L(g)} \leq t \epsilon \alpha^{t-2} \]
for any $(G,H,k)$-admissible linear form $L$ in $t \geq 3$ variables with multiplier-height at most $h$.
\end{proposition}
The main difference between this proposition and \cite[Proposition 4.2.8]{OlofTh} lies in the parameter $h$: since the latter result only dealt with forms where at least one coefficient was equal to $1$ one could always take $h = 1$. The purpose of the rest of this section is to indicate the proof of \cite[Proposition 4.2.8]{OlofTh} and show how one may adapt it to take the parameter $h$ into account.
\emph{Step 1: regularize $f$.} The proof begins with a modification of a standard procedure: one replaces $f$ by a function $f' : G \to [0,1]$ with Fourier support $R \subset \widehat{G}$ of bounded size, with $R$ containing the identity $1_{\widehat{G}}$ and with $R = R^{-1}$, so that
\[ f'(x) = \sum_{\gamma \in R} \widehat{f'}(\gamma)\gamma(x) \]
for all $x \in G$ and $\abs{R} \leq C(\epsilon)$. One can do this in such a way that $\int_G f' = \int_G f$ and $\abs{\Tsol_L(f) - \Tsol_L(f')} \leq t \epsilon \alpha^{t-2}/2$ for any $G$-admissible linear form $L$ in $t$ variables.
\emph{Step 2: transfer to $H$.} By the assumption that $\widehat{H}$ can $(n,k)$-model $\widehat{G}$ one can find a $k$-isomorphism $\varphi : R \to R' \subset \widehat{H}$. One then defines $g : H \to \R$ by
\[ g(x) := \sum_{\gamma \in R} \widehat{f'}(\gamma) \varphi(\gamma)(x). \]
The key properties of $g$ used in \cite{OlofTh} were that $\int_H g = \int_G f$ and that $\Tsol_L(g) = \Tsol_L(f')$ for any $(G,H,k)$-admissible form $L$ that has at least one coefficient equal to $1$; we shall therefore need to modify this part of the argument.
\emph{Step 3: control the range of $g$.} The function $g$ produced in the previous step does not a priori take values in $[0,1]$, as we need it to do. However, one can then produce a function $g' : H \to [0,1]$ such that $\int_H g' = \int_H g$ and $\abs{\Tsol_L(g) - \Tsol_L(g')} \leq t \epsilon \alpha^{t-2}/2$ for any $H$-admissible linear form $L$ (there are several ways to do this; see \cite{OlofTh}). This step completes the proof.
\begin{proof}[Proof of Proposition \ref{transfer}]
We shall only need to modify Step 2 above. We are thus given a function $f : G \to \R$ and a symmetric set $R \subset \widehat{G}$ containing the identity $1_{\Ghat}$ such that
\[ f(x) = \sum_{\gamma \in R} \widehat{f}(\gamma)\gamma(x) \]
for all $x \in G$, and we have that $R$ has bounded size: $\abs{R} \leq C(\epsilon)$. Let $Q \subset \widehat{G}$ be the set $Q := R^h = R \cdot R \cdots R $ where $h$ is the height-parameter supplied to Proposition \ref{transfer}. Certainly we have $\abs{Q} \leq C(\epsilon)^h$, and so the modelling hypothesis guarantees the existence of a Freiman $k$-isomorphism $\varphi : Q \to Q' \subset \widehat{H}$. By translating we may assume that $\varphi(1_{\Ghat}) = 1_{\Hhat}$. Note that this means that
\[ \gamma_1^{r_1} \cdots \gamma_n^{r_n} = \gamma_1^{s_1} \cdots \gamma_n^{s_n} \Longleftrightarrow \varphi(\gamma_1)^{r_1} \cdots \varphi(\gamma_n)^{r_n} = \varphi(\gamma_1)^{s_1} \cdots \varphi(\gamma_n)^{s_n} \]
whenever $\gamma_i \in Q$ and the $r_i, s_i$ are non-negative integers with $\sum r_i \leq k$ and $\sum s_i \leq k$. Note also that $\varphi(\gamma^{-1}) = \varphi(\gamma)^{-1}$ for all $\gamma \in Q$ since $\varphi(1_{\Ghat}) = 1_{\Hhat}$.
We now establish the following lemma, which replaces \cite[Lemma 4.6.3]{OlofTh}.
\begin{lemma}
Define $g : H \to \R$ by setting
\begin{equation} g(x) := \sum_{\gamma \in R} \widehat{f}(\gamma) \varphi(\gamma)(x) \label{g_formula} \end{equation}
for each $x \in H$. Then $\int_H g = \int_G f$, and if $L$ is a $(G,H,k)$-admissible linear form of multiplier-height at most $h$ then $\Tsol_L(g) = \Tsol_L(f)$.
\end{lemma}
\begin{proof}
The properties follow from the fact that \eqref{g_formula} is the Fourier expansion of $g$ and from the Fourier-inversion of $\Tsol_L$ provided by Proposition \ref{TLinversion}. Indeed,
\begin{align*} \widehat{g}(\chi) =
\begin{cases}
\widehat{f}(\gamma) & \text{if $\chi = \varphi(\gamma)$ for some $\gamma \in R$},\\
0 & \text{otherwise.}
\end{cases}
\end{align*}
Hence $\int_H g = \ghat(1_{\Hhat}) = \fhat(1_{\Ghat}) = \int_G f$. Now let $L = c_1 x_1 + \cdots + c_t x_t$ be a $(G,H,k)$-admissible form of multiplier-height at most $h$; thus we have integers $n_1, \ldots, n_t$ such that $n_1 c_1 + \cdots + n_t c_t = 1$ and $\abs{n_1} + \cdots + \abs{n_t} \leq h$. Proposition \ref{TLinversion} gives that
\[ \Tsol_L(f) = \sum_{\substack{\gamma \in \Ghat \\ \gamma^{c_i} \in R\,\, \forall i}} \fhat(\gamma^{c_1}) \cdots \fhat(\gamma^{c_t}) \quad \text{ and }\quad \Tsol_L(g) = \sum_{\substack{\chi \in \Hhat \\ \chi^{c_i} \in \varphi(R)\,\, \forall i}} \ghat(\chi^{c_1}) \cdots \ghat(\chi^{c_t}). \]
Let us write $\Gamma := \{ \gamma \in \Ghat : \gamma^{c_i} \in R \text{ for all $i$} \}$ and $\Psi := \{ \chi \in \Hhat : \chi^{c_i} \in \varphi(R) \text{ for all $i$} \}$ for the index sets occurring in these two sums. The result will follow if we can show that $\varphi$ is a bijection from $\Gamma$ to $\Psi$ such that $\varphi(\gamma)^{c_i} = \varphi(\gamma^{c_i})$ for all $i$ and $\gamma \in \Gamma$. We certainly have the second property, for if $\gamma \in \Gamma$ then
\[ \gamma = \gamma^{n_1 c_1 + \cdots + n_t c_t} = (\gamma^{c_1})^{n_1} \cdots (\gamma^{c_t})^{n_t} \in R^{n_1} \cdots R^{n_t} \subset Q, \]
and $\varphi$ is a $k$-isomorphism on $Q$, where $k \geq \abs{c_i}$ for all $i$. So we just need to establish that $\varphi(\Gamma) = \Psi$. Let us first deal with $\varphi(\Gamma) \subset \Psi$: let $\gamma \in \Gamma$. Then we need to show that $\varphi(\gamma)^{c_i} \in \varphi(R)$ for all $i$. But this is immediate since $\varphi(\gamma)^{c_i} = \varphi(\gamma^{c_i})$. The opposite inclusion follows in the same way using $\varphi^{-1}$ since $\Psi \subset \varphi(Q)$, which follows from the fact that if $\chi \in \Psi$ then
\[ \chi = (\chi^{c_1})^{n_1} \cdots (\chi^{c_t})^{n_t} \in \varphi(R)^{n_1} \cdots \varphi(R)^{n_t} = \varphi(R^{n_1} \cdots R^{n_t}) \subset \varphi(Q), \]
$\varphi$ being a Freiman $h$-isomorphism.
\end{proof}
The rest of the proof of Proposition \ref{transfer} is identical to that of the proof of Proposition 4.2.8 in \cite{OlofTh}, and so we are done.
\end{proof}
Proposition \ref{transfer} gives us a criterion for transferring functions between two compact abelian groups. The following lemmas show that this criterion allows us to work with the groups $\Zmod{p}$ and $\T$, since $\widehat{\Zmod{p}} \cong \Zmod{p}$ and $\widehat{\T} \cong \Z$.
\begin{lemma}
Let $n, k \in \N$ and let $p \geq (2k)^n$ be a prime. Then for any set $A \subseteq \Zmod{p}$ of size $n$ there is a set $B \subset \Z$ that is Freiman $k$-isomorphic to $A$.
In other words, $\Z$ can Freiman $(n,k)$-model $\Zmod{p}$ provided $p \geq (2k)^n$.
\end{lemma}
This result is standard and follows from an application of Dirichlet's box principle. The proof of the following lemma from \cite{OlofTh} is slightly more subtle but still elementary.
\begin{lemma}
Let $n, k$ be positive integers and let $N \geq (4k)^n$ be an integer. Then for any set $A \subseteq \Z$ of size $n$ there is a set $B \subset \Zmod{N}$ that is Freiman $k$-isomorphic to $A$.
In other words, $\Zmod{N}$ can Freiman $(n,k)$-model $\Z$ provided $N \geq (4k)^n$.
\end{lemma}
Thus we obtain the following immediate corollary of Proposition \ref{transfer}.
\begin{corollary}\label{transfer-cor}
Let $\epsilon>0$ and $h$ be a positive integer, and let each of $G$ and $H$ be $\Zmod{p}$ or $\T$, where $p \geq C(\epsilon, h)$ is a prime. Then for any measurable function $f : G \to [0,1]$ with $\int_G f = \alpha$ there is a continuous function $g : H \to [0,1]$ with $\int_H g = \alpha$ such that
\[ \abs{\Tsol_L(f) - \Tsol_L(g)} \leq t \epsilon \alpha^{t-2} \]
for any $(G,H,k)$-admissible linear form $L$ in $t \geq 3$ variables with multiplier-height at most $h$.
\end{corollary}
\section{From functions to sets}\label{functions-to-sets}
In our proof of Theorem \ref{mainres} we shall use Proposition \ref{transfer} to obtain a function $g$ with certain properties, and we shall then require a set with similar properties. The existence of such a set will be guaranteed by the following result.
\begin{lemma}\label{MainF2S}
Let $\epsilon > 0$, let $G$ be $\T$ or $\Zmod{p}$ for $p$ sufficiently large, and let $f : G \to [0,1]$ be measurable. Then there exists a measurable set $A\subset G$ such that $\left|\mu_G(A)-\int_G f\right|\leq \epsilon$ and
$|\Tsol_L(A)-\Tsol_L(f)|\leq t\epsilon$ for any $G$-admissible linear form $L$ in $t \geq 3$ variables.
\end{lemma}
Note that any linear form is $\T$-admissible and most forms are $\Zmod{p}$-admissible. To prove this lemma we shall use Theorem \ref{U2control} and a probabilistic construction that is familiar in the setting of finite abelian groups.
Given two sets $A,B$ let $A\Delta B$ denote their symmetric difference. The following notion of discretization will be used here and in the next section.
\begin{defn}
A set $A \subset \T$ is \emph{$(\delta,n)$-measurable} if there exists a set $B$ which is the union of dyadic intervals $I_{n,j} := [(j-1)/2^n, j/2^n)$, $j\in [2^n]$, such that $\mu(A\Delta B)<\delta$.
A function $f : \T \to [0,1]$ is \emph{$(\delta,n)$-measurable} if there is a function $g : \T \to [0,1]$ that is constant on the intervals $I_{n,j}$ such that $\norm{f-g}_{\mathcal{L}_1} < \delta$.
\end{defn}
We shall use both of the following equivalent instances of Littlewood's first principle, which are discussed in \cite[\S 2.4]{tao-year1}.
\begin{lemma}\label{LW}
Let $A \subset \T$ be measurable. Then for every $\delta > 0$ there exists $n$ such that $A$ is $(\delta, n)$-measurable.
\end{lemma}
\begin{lemma}\label{LW-fns}
Let $f : \T \to [0,1]$ be measurable. Then for every $\delta > 0$ there exists $n$ such that $f$ is $(\delta, n)$-measurable.
\end{lemma}
While one cannot in general approximate a $[0,1]$-valued function by a set in $\mathcal{L}_1(G)$, one can do so in $U^2(G)$ in the sense
of the following result. It is this that allows us to establish Lemma 4.1.
\begin{lemma}\label{F2S}
Let $\epsilon > 0$, let $G$ be $\T$ or $\Zmod{p}$ for $p$ sufficiently large, and let $f : G \to [0,1]$ be measurable.
Then there is a measurable set $A \subset G$ such that $\lVert f - 1_A \rVert_{U^2} \leq \epsilon$. Moreover, for $G = \T$ we may take $A$ to be a finite union of dyadic intervals.
\end{lemma}
\begin{proof}
For $G=\Zmod{p}$ this a standard result: define a subset $A$ of $G$ randomly by letting $x \in A$ with probability $f(x)$ independently for each $x \in G$. Then by independence the expectation of $\norm{f-1_A}_{U^2}^4$ equals 0, up to an error of magnitude $O(p^{-1})$ due to averaging over degenerate parallelograms (i.e. parallelograms $(x+y,x+y',x'+y,x'+y')$ with at least two vertices being equal). Thus there exists a choice of $A$ for which $\norm{f-1_A}_{U^2}=O(p^{-1/4})$.
Now we adapt this standard argument to deal with the group $\T$. First we approximate $f$ by a suitable step-function: by Lemma \ref{LW-fns} there is an integer $n$ and coefficients $\gamma_j\in [0,1]$ such that the function $g=\sum_{j\in [2^n]} \gamma_j 1_{I_{n,j}}$ satisfies $\norm{f-g}_{\mathcal{L}_1} < \epsilon^4$. We thus have $\norm{f-g}_{U^2} \leq \norm{f-g}_{\mathcal{L}_1}^{1/4}<\epsilon$.
Next we define an appropriate finite probability space. Consider the set $\Omega$ of functions of the form $1_A(x)=\sum_{j\in [2^n]} \alpha_j 1_{I_{n,j}}(x)$, where for all $j$ we have $\alpha_j\in\{0,1\}$. An element
in $\Omega$ can be identified with an element $\alpha=(\alpha_1,...,\alpha_{2^n})$ of $\{0,1\}^{2^n}$, and we can then
define a probability on $\Omega$ by declaring the events $\{\alpha_j=1\}$ to be independent and assigning to $\{\alpha_j=1\}$
the probability $\gamma_j$.
Now we compute the expectation of $\norm{g-1_A}_{U^2}^4$ for a randomly chosen $1_A \in \Omega$, using the following familiar expression for the $U^2$ norm:
\[\norm{f}_{U^2(G)}^4=\int_{G^3} f(x)\,\overline{f(x+h)}\,\overline{f(x+k)}\,f(x+h+k) \ud x \ud h \ud k.\]
This expression is seen to equal \eqref{U2} by Lemma \ref{cts-surj-hom}.
By Fubini's theorem we have
\[
\mathbb{E} \norm{g-1_A}_{U^2}^4=\int_{\T^3}
\mathbb{E} (g-1_A)(x)(g-1_A)(x+h)(g-1_A)(x+k)(g-1_A)(x+h+k)\; \ud x\ud h\ud k.
\]
Fix any values for $x,h,k$ such that no two vertices of the corresponding parallelogram $(x,x+h,x+k,x+h+k)$ lie in the same interval
$I_{n,j}$. Then the expectation of the product $(g-1_A)(x)(g-1_A)(x+h)(g-1_A)(x+k)(g-1_A)(x+h+k)$ is 0, by independence. Now consider the integral of this product over values of $x,h,k$ such that at least two vertices of the corresponding parallelogram lie in the same interval. This integral is at most the Haar measure of the set $D$ of these parallelograms,
\[
D=\{(x,h,k)\in \T^3: \exists \omega,\omega'\in\{0,1\}^2, \omega\neq\omega', \exists
j\in [2^n], I_{n,j}\ni x+\omega\cdot (h,k),x+\omega'\cdot (h,k)\},
\]
where $\omega\cdot (h,k)=\omega_1 h+\omega_2 k$. Note that $D$ is the support of the measurable function
\[
F_D:(x,h,k)\mapsto \sum_{\substack{\omega,\omega'\in \{0,1\}^2 \\\omega\neq\omega'}} \sum_{j\in[2^n]}
1_{I_{n,j}}\big(x+\omega\cdot (h,k)\big) 1_{I_{n,j}}\big(x+\omega'\cdot (h,k)\big),
\]
so $D$ is a measurable subset of $\T^3$. We want to show that the Haar measure of $D$ vanishes as $n\rightarrow \infty$, and for that it clearly suffices to show that the integral of $F_D$ vanishes. Fix any pair of distinct $\omega,\omega'$. Then $\omega-\omega'$ has at least one non-zero coordinate (where the subtraction here is coordinate-wise). For a fixed $x$, we have $x+\omega\cdot (h,k),x+\omega'\cdot (h,k)$ in the same interval $I_{n,j}$ only if $(\omega-\omega')\cdot (h,k)\in (-2^{-n},2^{-n})$. It follows that, for this pair $\omega,\omega'$, the integral
\[
\int_{\T^3} \sum_{j\in[2^n]}
1_{I_{n,j}}\big(x+\omega\cdot (h,k)\big) 1_{I_{n,j}}\big(x+\omega'\cdot (h,k)\big)\ud x\ud h \ud k
\]
is at most the Haar measure of the slab $S=\{(h,k)\in\T^2:(\omega-\omega')\cdot (h,k)\in (-2^{-n},2^{-n})\}$, which is $2^{-n+1}$. Applying this argument to each pair $\omega,\omega'$ shows that the Haar measure of $D$ vanishes as required.
\end{proof}
\begin{remark}
A minor modification of the above proof allows one to replace the $U^2$ norm by the $U^d$ norm for any $d \geq 2$; these norms are useful generalizations of the $U^2$ norm \cite{T-V}.
\end{remark}
\begin{proof}[Proof of Lemma \ref{MainF2S}]
Let $G$ be either $\T$ or $\Zmod{p}$, where $p$ is large. For a measurable function $f : G\to[0,1]$, let $A$ be the set given by Lemma \ref{F2S} applied with initial parameter $\epsilon$. Then by the Cauchy-Schwarz inequality we have $\abs{\int_G 1_A-\int_G f} \leq \norm{1_A-f}_{U^2}\leq\epsilon$. Now let $L$ be any $G$-admissible linear form in at least three variables. Then multilinearity of $S_L$ and Theorem \ref{U2control} imply $\abs{\Tsol_L(f)-\Tsol_L(A)} \leq t \norm{f-1_A}_{U^2}\leq t \epsilon$.
\end{proof}
\section{Removal lemmas}\label{Removal}
In this section, for $G=\T$ or $\Zmod{p}$, we are interested in measurable subsets $A\subset G$ of positive measure such that $\Tsol_L(A)<\delta$ for some small $\delta>0$. We shall prove the following result.
\begin{lemma}\label{Tremoval-t}
Let $\epsilon>0$ and let $L$ be a linear form in $t$ variables. There exists $\delta>0$ such that, for any measurable sets $A_1,A_2,\ldots,A_t\subset\T$ satisfying $\Tsol_L(A_1,A_2,\ldots,A_t)<\delta$, there are measurable sets $E_i\subset\T$ with $\mu(E_i)< \epsilon$ such that $(A_1\setminus E_1) \times (A_2\setminus E_2)\times\cdots\times (A_t\setminus E_t) \cap \ker L =\emptyset$.
\end{lemma}
This result has a well-known analogue for finite abelian groups, an analogue that now has several proofs. One such proof, given in \cite{KSVRL}, proceeds by turning the problem into one of removing small subgraphs from a certain auxiliary graph, and then applying a removal lemma for graphs. Unfortunately this argument does not seem to extend straightforwardly to the infinitary setting of the group $\T$. An earlier proof was given by Green \cite{GAR}, who established a Fourier-analytic regularity lemma from which the finitary removal lemma follows. This proof does generalize to the infinitary setting, allowing one to establish a removal lemma for arbitrary compact abelian groups, but checking this is somewhat technical. Instead, we shall prove Lemma \ref{Tremoval-t} using a finitary removal lemma as a black box. We first reduce to the special case where $L$ is the linear form $L(x)=x_1+\cdots+x_t$; this form will be denoted by $\mathbf{1}$ throughout this section.
\begin{lemma}\label{offdiag}
For any $\epsilon>0$ and any positive integer $t$ there exists $\delta>0$ such that the following holds. Let $A_1,A_2,\ldots,A_t$ be measurable subsets of $\T$ such that $\Tsol_{\mathbf{1}}(A_1,\ldots,A_t)<\delta$. Then there exist measurable sets $E_i\subset\T$ such that $(A_1\setminus E_1)\times\cdots\times(A_t\setminus E_t)\cap\ker\mathbf{1}=\emptyset$ and $\mu(E_i)<\epsilon$ for each $i$.
\end{lemma}
When $N$ is prime, the equivalence between the $\Zmod{N}$-analogues of Lemmas \ref{Tremoval-t} and \ref{offdiag} follows simply from inverting the dilations. For the circle we can still prove the desired equivalence at the cost of a slight worsening in the dependence between the parameters.
To show that Lemma \ref{offdiag} implies Lemma \ref{Tremoval-t} we use the following.
\begin{lemma}\label{T_L_L'}
Let $L(x) = c_1 x_1 + \cdots + c_t x_t$ and let $A_1,A_2,\ldots,A_t\subset \T$ be measurable. Then
\[\Tsol_L(A_1,\ldots,A_t) \leq \Tsol_\mathbf{1}(c_1 A_1, \ldots, c_t A_t) \leq |c_1 \cdots c_t|\, \Tsol_L(A_1,\ldots,A_t).\]
\end{lemma}
\begin{proof}
The map $\phi:(x_1,\ldots,x_t)\to (c_1x_1,\ldots,c_tx_t)$ is a continuous endomorphism on $\T^t$ which restricts to a continuous surjective homomorphism from $\ker L$ to $\ker \mathbf{1}$. In fact, since $L$ is the composition $\mathbf{1} \circ \phi$,
we have $\ker L = \phi^{-1} \ker \mathbf{1}$. By Lemma \ref{cts-surj-hom} we therefore have $\mu_\mathbf{1}=\mu_L\circ \phi^{-1}$. It follows that
$\Tsol_\mathbf{1}(A_1,\ldots,A_t)=\Tsol_L(c_1^{-1} A_1,\ldots,c_t^{-1} A_t)$, where $c_i^{-1}$ denotes taking the preimage under dilation by $c_i$.
Applying this to the sets $c_i A_i$ gives $\Tsol_\mathbf{1}(c_1 A_1,\ldots, c_t A_t)=\Tsol_L(c_1^{-1} c_1 A_1,\ldots,c_t^{-1} c_t A_t)$,
and it is easy to check that $c_i^{-1} c_i A_i = \bigcup_{j \in \Zmod{c_i}} \left( A_i + j/c_i \right)$.
The claim then follows from basic properties of $\Tsol_L$, namely multilinearity and invariance under translation by elements of $\ker L$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{Tremoval-t} using Lemma \ref{offdiag}]
Fix $\epsilon>0$ and let $\delta>0$ be as given by Lemma \ref{offdiag}. Now suppose $A_1,A_2,\ldots,A_t\subset \T$ are measurable subsets satisfying $\Tsol_L(A_1,\ldots,A_t)<\delta/|c_1\cdots c_t|$. By Lemma \ref{T_L_L'} we then have
$\Tsol_{\mathbf{1}}(c_1A_1,\ldots,c_tA_t) < \delta$. Lemma \ref{offdiag} therefore provides us with sets $F_i$ with $\mu(F_i) < \epsilon$ such that
$(c_1A_1\setminus F_1)\times\cdots\times(c_tA_t\setminus F_t) \cap \ker\mathbf{1}=\emptyset$.
Let $E_i = c_i^{-1} F_i$, and let $\phi$ be the map from Lemma \ref{T_L_L'}. Since
\[ \phi\big((A_1\setminus E_1)\times\cdots\times(A_t\setminus E_t) \cap\ker L \big) = (c_1A_1\setminus F_1)\times\cdots\times(c_tA_t\setminus F_t)\cap\ker\mathbf{1}=\emptyset,\]
the sets $E_i$ satisfy the conclusion of Lemma \ref{Tremoval-t}.
\end{proof}
We shall now prove Lemma \ref{offdiag}. For notational convenience we assume in the proof that $\mathbf{1}$ has last coefficient $c_t=-1$, without loss of generality. Note that as a special case of \eqref{param} we have
\begin{equation}\label{T_1}
\Tsol_{\mathbf{1}}(A_1,\ldots, A_t)= \int_{\T^{t-1}} 1_{A_1}(x_1)\cdots 1_{A_{t-1}}(x_{t-1}) 1_{A_t}(x_1+x_2+\cdots+x_{t-1}) \ud x_1\cdots \ud x_{t-1}.
\end{equation}
Using Lemma \ref{LW} one may reduce the proof of Lemma \ref{offdiag} to establishing the following variant.
\begin{lemma}\label{removal_lemma_discrete}
For any $\epsilon > 0$ and $t\in \N$ there exists $\delta > 0$ such that the following holds. Let $A_1, A_2,\ldots, A_t \subset \T$ be unions of intervals $[(j-1)/N, j/N)$, $j\in [N]$, for some $N \in \N$. Suppose further that $\Tsol_\mathbf{1}(A_1,\ldots, A_t)< \delta$.
Then there are measurable sets $E_i \subset A_i$ with $\mu(E_i)<\epsilon$ such that
$(A_1 \setminus E_1)\times\cdots\times(A_t \setminus E_t)\,\cap\ker\mathbf{1} = \emptyset$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{offdiag} using Lemma \ref{removal_lemma_discrete}]
Fix $\epsilon>0$ and let $\delta \in (0, \epsilon)$ be such that Lemma \ref{removal_lemma_discrete} holds with initial parameter $\epsilon/2$. Let
$A_1,\ldots,A_t\subset\T$ be measurable sets satisfying $\Tsol_\mathbf{1}(A_1,\ldots,A_t)< \delta/(t+1)$. Using \eqref{T_1} and a telescoping expansion (multilinearity of $\Tsol_\mathbf{1}$) we have $|\Tsol_\mathbf{1}(A_1,\ldots,A_t)-\Tsol_\mathbf{1}(B_1,\ldots,B_t)|\leq \mu(A_1\Delta B_1)+\cdots+\mu(A_t\Delta B_t)$
for any measurable sets $B_1,\ldots,B_t$. From Lemma \ref{LW} we obtain a positive integer $n$ and sets $B_i$ that are unions of intervals $[(j-1)2^{-n},j2^{-n})$ such that $\mu(A_i\Delta B_i)\leq \delta/(t+1)$ for each $i \in [t]$. It follows that $\Tsol_\mathbf{1}(B_1,\ldots,B_t)< \delta$. We apply Lemma \ref{removal_lemma_discrete} to these discretized sets, obtaining measurable sets $F_1,\ldots,F_t$ such that $\mu(F_i)< \epsilon/2$ and
\[ (B_1 \setminus F_1)\times\cdots\times(B_t \setminus F_t) \cap \ker\mathbf{1} = \emptyset. \]
Now letting $E_i=F_i\cup (A_i\setminus B_i)$, we have $\mu(E_i)<\epsilon$, and $A_i \setminus E_i\subset B_i\setminus F_i$, whence
$(A_1 \setminus E_1)\times\cdots\times(A_t \setminus E_t) \cap \ker\mathbf{1} = \emptyset$ as required.
\end{proof}
We shall deduce Lemma \ref{removal_lemma_discrete} from the following analogue for $\Zmod{N}$, proved in \cite{GAR}.
\begin{lemma}\label{removal_lemma_finite}
For any $\epsilon > 0$ and $t\in\N$ there exists $\delta > 0$ such that the following holds. Let $N \in \N$ and suppose that $A_1,\ldots, A_t \subset \Zmod{N}$ and that $\Tsol_\mathbf{1}(A_1,\ldots, A_t)< \delta$.
Then there are sets $E_i \subset A_i$ with $\abs{E_i} \leq \epsilon N$ such that
$(A_1 \setminus E_1)\times\cdots\times(A_t \setminus E_t) \cap \ker\mathbf{1}=\emptyset$.
\end{lemma}
To use this we need to express the solution measure $\Tsol_\mathbf{1}$ on $\T$ in terms of solution measures on $\Z_N$.
Recall that the Eulerian number $\eulerian{n}{k}$ is the number of permutations $a_1, \ldots, a_n$ of $[n]$ in which there are precisely $k$ values of $i$ such that $a_i < a_{i+1}$.
\begin{lemma}\label{discrete_solutions}
Let $t\geq 3$ and suppose $A_1,\ldots, A_t \subset \T$ are unions of intervals $[(j-1)/N, j/N)$, $j\in [N]$, for some positive integer $N$. Let $A_i'\subset \Zmod{N}$ be defined by $1_{A_i'}(x)=1_{A_i}(x/N)$. Then
\[ \Tsol_\mathbf{1}(A_1,\ldots, A_t) = \frac{1}{(t-1)!}\sum_{r=0}^{t-2} \eulerian{t-1}{r}\, \Tsol_\mathbf{1}(A_1',\ldots, A_t'-r). \]
In particular, $\Tsol_\mathbf{1}(A_1',\ldots, A_t'-r) \leq (t-1)!\, \Tsol_\mathbf{1}(A_1,\ldots, A_t)$ for any $r\in \{0,1, \ldots, t-2\}$.
\end{lemma}
\begin{proof}
For any $x \in \T$ we have $1_{A_i}(x) = 1_{A_i}( \lfloor N x \rfloor / N )$, where multiplication by $N$, floor function, and division by $N$ are all taken in $\R$.
Then $\Tsol_\mathbf{1}(A_1,\ldots, A_t)$ equals
\begin{eqnarray*}
&& \int_{\T^{t-1}} 1_{A_1}(\lfloor Nx_1 \rfloor /N)\cdots 1_{A_t}( \lfloor N(x_1+\cdots+x_{t-1}) \rfloor/N) \ud x_1\cdots \ud x_{t-1} \\
&=& \sum_{a_1,\ldots,a_{t-1}\in \Zmod{N}} 1_{A_1}(a_1/N)\cdots 1_{A_{t-1}}(a_{t-1}/N) \\
&&\hspace{1.5cm}\cdot\int_{[0,1/N)^{t-1}} 1_{A_t}( \lfloor N((a_1/N + y_1)+\cdots+ (a_t/N + y_{t-1})) \rfloor/N) \ud y_1\cdots \ud y_{t-1} \\
&=& \frac{1}{N^{t-1}} \sum_{a_1,\ldots,a_{t-1} \in \Zmod{N}} 1_{A_1'}(a_1)\cdots 1_{A_{t-1}'}(a_{t-1})\\
&&\hspace{1.5cm}\cdot\int_{[0,1)^{t-1}} 1_{A_t'}(a_1+\cdots+a_{t-1}+\lfloor y_1+\cdots+ y_{t-1} \rfloor) \ud y_1\cdots \ud y_{t-1}.
\end{eqnarray*}
For any real-valued function $f$ on the integers, \cite[6.65]{graham-knuth-patashnik} gives
\[ \int_0^1 \cdots \int_0^1 f(\lfloor x_1 + \cdots + x_n \rfloor) \ud x_1 \cdots \ud x_n = \sum_{k=0}^{n-1} \eulerian{n}{k} \frac{f(k)}{n!}. \]
The result now follows immediately.
\end{proof}
\begin{proof}[Proof of Lemma \ref{removal_lemma_discrete}]
We can assume that $t\geq 3$. Given $\epsilon > 0$, set $\epsilon' = \epsilon/(t-1)$ and let $\delta'$ be given by Lemma \ref{removal_lemma_finite} applied with initial parameter $\epsilon'$. Define $\delta = \delta'/(t-1)!$ and let $A_1,\ldots,A_t \subset \T$ be sets that are unions of intervals of the form $[j/N, (j+1)/N)$ for some positive integer $N$ and satisfy $\Tsol_\mathbf{1}(A_1,\ldots, A_t)< \delta$.
Let the sets $A_i' \subset \Zmod{N}$ be as in Lemma \ref{discrete_solutions}. We then have $\Tsol_\mathbf{1}(A_1',\ldots, A_t'-r)< \delta'$ for each $r = 0, \ldots, t-2$, and so Lemma \ref{removal_lemma_finite} gives us sets $E_{i,r}' \subset A_i'$ such that $|E_{i,r}'| \leq \epsilon' N$ and
\begin{equation}\label{removed'}
(A_1' \setminus E_{1,r}')\times\cdots\times\big((A_t'-r) \setminus (E_{t,r}'-r)\big) \cap \ker\mathbf{1}=\emptyset.
\end{equation}
Setting $E_i' = \bigcup_r E_{i,r}'$ we therefore see that there are no solutions to $y_1 + \cdots + y_{t-1} = y_t - r$ with $y_i \in A_i' \setminus E_i'$ for $r \in [0, t-2] \subset \Zmod{N}$.
We now define corresponding removal-sets in $\T$: for each $i\in [t]$ let
\[ E_i = \bigcup_{x \in E_i'} [x/N, (x+1)/N). \]
Since $\abs{E_i'} \leq \epsilon N$ we have $\mu(E_i) \leq \epsilon$ for each $i$. Now suppose that we had an element
\[ (x_1, \ldots, x_t)\; \in \;(A_1 \setminus E_1) \times \cdots \times (A_t \setminus E_t) \cap \ker \mathbf{1}. \]
Then, since each of the sets $A_i \setminus E_i$ is a union of intervals $[j/N, (j+1)/N)$, we have that the element $\lfloor N x_i \rfloor$ of $\Zmod{N}$ lies in $A_i' \setminus E_i'$. But we also have $x_t = x_1 + \cdots + x_{t-1}$, whence $\lfloor N x_t \rfloor = \lfloor N x_1 \rfloor + \cdots + \lfloor N x_{t-1} \rfloor + r$ for some $r \in [0, t-2] \subset \Zmod{N}$. This contradicts \eqref{removed'}, and therefore
$(A_1 \setminus E_1) \times \cdots \times (A_t \setminus E_t) \cap \ker \mathbf{1} = \emptyset$ as required.
\end{proof}
\section{Proofs of convergence and positivity}\label{Final}
We can now prove our main result.
\begin{proof}[Proof of Theorem \ref{mainres}]
We are given a finite family $\mathcal{F}$ of linear forms. Let $n$ be the cardinality of $\mathcal{F}$, and fix $\epsilon>0$.
Let $\delta \in (0, \epsilon/2)$ be such that Lemma \ref{Tremoval-t} and its analogue for $\Zmod{p}$ both work for each $L\in\mathcal{F}$, with initial parameter $\epsilon/2n$. Let $t$ be the maximum number of variables that occurs in a form in $\mathcal{F}$ and let $C=C(\epsilon,\mathcal{F})$ be such that Corollary \ref{transfer-cor} (transference) and Lemma \ref{MainF2S} (functions to sets) both work when applied with initial parameter $\delta/2t$ for any $p > C$ and $L\in\mathcal{F}$. We claim that $\abs{ \md_\mathcal{F}(\T) - \md_\mathcal{F}(\Zmod{p}) } \leq \epsilon$ for any prime $p > C$.
To see that $\md_\mathcal{F}(\T)\geq \md_\mathcal{F}(\Zmod{p})-\epsilon$, let $\alpha= \md_\mathcal{F}(\Zmod{p})$ and let $A\subset \Zmod{p}$ be $\mathcal{F}$-free with $\mu(A)=\alpha$. Corollary \ref{transfer-cor} then gives us a measurable function $g : \T\to [0,1]$ with $\int_\T g=\alpha$ such that $\Tsol_L(g) < \delta/2$ for every $L\in\mathcal{F}$. Applying Lemma \ref{MainF2S} to $g$ with initial parameter $\delta/2t$, we obtain a measurable subset $B$ of $\T$ of density at least $\alpha-\delta$ such that $\Tsol_L(B) < \delta$ for every $L\in\mathcal{F}$.
We now apply the removal lemma on $\T$. By our choice of $\delta$, Lemma \ref{Tremoval-t} gives us an $\mathcal{F}$-free subset $D$ of $B$ with $\mu(D)\geq \mu(B)- n \epsilon/2n\geq \alpha -\delta-\epsilon/2$. Therefore $\md_\mathcal{F}(\T)\geq \md_\mathcal{F}(\Zmod{p})-\epsilon$ as required.
The same argument, but with the roles of $\Zmod{p}$ and $\T$ swapped, shows that we also have $\md_\mathcal{F}(\Zmod{p})\geq \md_\mathcal{F}(\T)-\epsilon$, and so the result follows.
\end{proof}
\begin{remark}\label{rem}
One of our aims for the argument above was to treat the direction $d_\mathcal{F}(\Zmod{p})>d_\mathcal{F}(\T)-\epsilon$ and its opposite in a unified manner. It should be noted however that the first direction can also be treated more directly, without using Fourier analysis. In a nutshell, if $A\subset \T$ is $\mathcal{F}$-free with $\mu(A)>d_\mathcal{F}(\T)-\epsilon$ then the continuity of the forms in $\mathcal{F}$ implies that one can find an $\mathcal{F}$-free \emph{open} set $A'$ of measure at least $d_\mathcal{F}(\T)-2\epsilon$, and then for large $p$ the set $A_p=\{x\in \Zmod{p}: x/p\in A'\}$ is $\mathcal{F}$-free and of density at least $d_\mathcal{F}(\T)-3\epsilon$ in $\Zmod{p}$.
\end{remark}
We now prove Proposition \ref{positivity}, which said that $\md_\mathcal{F}(\T)$ is positive for any finite family $\mathcal{F}$ of non-invariant forms. We can do this by modifying an idea employed by Ruzsa \cite{ruzsa:linear-equationsII}. For a form $L(x) = c_1 x_1 + \cdots + c_t x_t$ we write $s_L=\abs{c_1} + \cdots + \abs{c_t}$.
\begin{proposition}
Let $\mathcal{F}$ be a finite family of non-invariant linear forms. Then
\[ \md_\mathcal{F}(\T) \geq \left(\sum_{L \in \mathcal{F}} s_L\right)^{-1}. \]
\end{proposition}
\begin{proof}
Let $s = \sum_{L \in \mathcal{F}} s_L$ and let $A$ be the interval $(-1/2s, 1/2s)$ embedded in $\T$. We claim that there is a translate $A-y$ of $A$ that is $\mathcal{F}$-free. This will be the case if $L(y,\ldots,y) \notin L(A \times \cdots \times A) = (-s_L/2s, s_L/2s)$ for each $L \in \mathcal{F}$. Since each $L$ is non-invariant, each such condition on $y$ excludes a finite union of open intervals, with total length $s_L/s$. Thus all the conditions together exclude a finite union of open intervals, the lengths of which sum to at most $1$. Hence there is some $y \in \T$ outside this union of intervals, and the result follows.
\end{proof}
\section{Concluding remarks}
Our proof of Theorem \ref{mainres} consists essentially of a combination of the transference result Proposition \ref{transfer} with the removal result Lemma \ref{Tremoval-t}. Two remarks should be made about this. The first is that the combination of Fourier-regularization (a key tool in the proof of the transference result) and removal results has been used successfully before; in particular, Green \cite[Theorem 9.3]{GAR} used it to relate the number of subsets of $[N]$ that are free from non-trivial solutions to $L(x)=0$ to the maximum size of an $L$-free subset of $[N]$. The second remark is that the argument in our proof of Theorem \ref{mainres} readily yields analogues of the theorem for many other families of groups, provided in particular that the appropriate removal lemmas are available (examples of such families include $(\Zmod{kp})$, where $k\in \N$ is fixed and $p$ ranges over the primes). We have not treated such generalizations here in order to avoid certain technicalities. Let us note, however, that if all one is interested in is convergence of maximal densities, rather than convergence to a particular quantity on a group, then all the theory one needs is finitary and the appropriate removal lemmas are well-known.
It would be interesting to generalize Theorem \ref{mainres} to allow the family $\mathcal{F}$ to consist not just of single linear equations but also of systems of linear equations. To this end it can be useful to classify systems according to a notion of \emph{complexity} related to the Gowers norms (see \cite{gowers-wolf}). The methods of this paper readily extend to give convergence of $d_\mathcal{F}(\Z_p)$ for a family of systems of complexity 1, but establishing $\T$ as a limit group along these lines requires an extension of Lemma \ref{Tremoval-t}. Convergence for systems of greater complexity requires other methods.
Finally, regarding the original question of Ruzsa mentioned in the introduction, we note that there is a simple transference result for functions on $[N]$, proved using an argument somewhat different from that employed for transference here, and that this can be used to answer Ruzsa's question affirmatively (we shall detail this elsewhere).\\
\textbf{Acknowledgements.} The authors are grateful to Tom Sanders for helpful comments and to an anonymous referee for pointing out the argument in Remark \ref{rem}. The second-named author would also like to thank Ben Green for his supervision and encouragement that led to the main results of \cite[Chapter 4]{OlofTh}.
| {
"timestamp": "2011-09-15T02:00:37",
"yymm": "1109",
"arxiv_id": "1109.2934",
"language": "en",
"url": "https://arxiv.org/abs/1109.2934",
"abstract": "Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of m(A) over F-free sets A in G, where m is the normalized Haar measure on G. Our main result is that, for any such collection F of forms in at least three variables, the sequence d_F(Z_p) converges to d_F(R/Z) as p tends to infinity over primes. This answers an analogue for Z_p of a question that Ruzsa raised about sets of integers.",
"subjects": "Combinatorics (math.CO); Classical Analysis and ODEs (math.CA)",
"title": "On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795091201804,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379408360978
} |
https://arxiv.org/abs/1001.4169 | Stolarsky's conjecture and the sum of digits of polynomial values | Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$ \liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0. $$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + ... + a_0 \in \Z[x]$ with $h\geq 2$ and $a_h>0$ and any base $q$, \[ \liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.\] For any $\epsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \epsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \epsilon$. | \section{Introduction}
Let $q\geq 2$ and denote by $s_q(n)$ the sum of digits in the $q$-ary representation of an integer $n$. In recent years, much effort has been made to get a better understanding of the distribution properties of $s_q$ regarding certain subsequences of the positive integers. We mention the ground-breaking work by C.~Mauduit and J.~Rivat on the distribution of $s_q$ of primes~\cite{MR09-1} and of squares~\cite{MR09-2}. In the case of general polynomials $p(n)$ of degree $h\geq 2$ very little is known. For the current state of knowledge, we refer to the work of C.~Dartyge and G.~Tenenbaum~\cite{DT06}, who provided some density estimates for the evaluation of $s_q(p(n))$ in arithmetic progressions. The authors \cite{HLS} recently examined the special case when $s_q(p(n)) \approx s_q(n)$.
A problem of a more elementary (though, non-trivial) nature is to study extremal properties of $s_q(p(n))$. Here we will always assume that
\begin{equation}\label{genpoly}
p(x)=a_h x^h+a_{h-1} x^{h-1} + \dots + a_0 \in \mbox{$\mathbb Z$}[x]
\end{equation}
is a polynomial of degree $h\geq 2$ with leading coefficient $a_h>0$.
In the binary case when $q = 2$, B.~Lindstr\"om~\cite{Li97} showed that
\begin{equation}\label{lind}
\limsup_{n \to \infty} \frac{s_2(p(n))}{\log_2 n}= h.
\end{equation}
In the proof of~(\ref{lind}), Lindstr\"om uses a sequence of integers $n$ with many $1$'s in their binary expansions such that $p(n)$ also has many $1$'s.
The special case $p(n)=n^2$ of~(\ref{lind}) has been reproved by M.~Drmota and J.~Rivat~\cite{DR05} with constructions due to J. Cassaigne and G. Baron.
On the other hand, it is an intriguing question whether it is possible to generate infinitely many integers $n$ such that $p(n)$ has few $1$'s compared to $n$. If this is possible, then this is indeed a rare event.
It is well-known~\cite{De75, Pe02} that the average order of magnitude of $s_q(n)$ and $s_q(n^h)$ is
\begin{equation}\label{avorder}
\sum_{n<N} s_q(n)\sim \frac{1}{h}\sum_{n<N} s_q(n^h)\sim \frac{q-1}{2\log q} \; N\log N.
\end{equation}
In particular, the average value of $s_q(n^h)$ is $h$ times larger than the
average value of $s_q(n)$.
In 1978, K. Stolarsky~\cite{St78} proved several results on the extremal values of $s_q(p(n))/s_q(n)$ for the special case when $q = 2$ and $p(n) = n^h$. He showed that the maximal order of magnitude is $$c(h) (\log_2 n)^{1-1/h},$$ where $c(h)$ only depends on $h$. This result is best possible, which follows from the Bose-Chowla theorem~\cite{BC62, HR83}. His proof can be generalized to base $q$ and to general polynomials $p(n)$.
Although this generalization is straightforward, we include it here for completeness. Recall that $p(n)$ may have negative coefficients as well.
\begin{theo}\label{limsupsto} Let $p(x) \in \mbox{$\mathbb Z$}[x]$ have degree at least $2$ and positive leading coefficient.
\begin{enumerate}
\item If $p(n)$ has only nonnegative coefficients then there exists $c_1$, dependent only on $p(x)$ and $q$, such that for all $n\geq 2$,
$$\frac{s_q(p(n))}{s_q(n)}\leq c_1 (\log_q n)^{1-1/h}.$$
This is best possible in that there is a constant $c_1'$, dependent only on $p(x)$, such that
$$\frac{s_q(p(n))}{s_q(n)}> c_1' (\log_q n)^{1-1/h}$$
infinitely often.
\item If $p(n)$ has at least one negative coefficient then there exists $c_2$ and $n_0$, dependent only on $p(x)$ and $q$,
such that for all $n\geq n_0$,
$$\frac{s_q(p(n))}{s_q(n)}\leq c_2 \log_q n.$$
This is best possible in that for all $\varepsilon>0$ we have
$$\frac{s_q(p(n))}{s_q(n)}> (q-1-\varepsilon) \log_q n$$
infinitely often.
\end{enumerate}
\end{theo}
The proof of this result along with some useful preliminary results are
given in Section \ref{sec:prelim}.
For the minimal order of $s_q(p(n))/s_q(n)$, Stolarsky treated the special case $q=2$ and $p(n)=n^2$. He
proved that there are infinitely many integers $n$ such that
\begin{equation}\label{stolthm}
\frac{s_2(n^2)}{s_2(n)}\leq \frac{4 (\log \log n)^2}{\log n}.
\end{equation}
He conjectured that an analogous result is true for every fixed $h\geq 2$ but he did ``not see how to prove this''.
\begin{conj}[Stolarsky~\cite{St78}, 1978]\label{conj}
For fixed $h\geq 2$,
$$ \liminf_{n\to \infty} \frac{s_2(n^h)}{s_2(n)}=0.$$
\end{conj}
By naive methods, it can be quite hard to find even a single value
$n$ such that $s_2(n^h)<s_2(n)$ for some $h$, let alone observe that the
limit infimum goes to $0$.
For example, an extremely brute force calculation shows that
the minimal $n$ such that $s_2(n^3) < s_2(n)$ is
$n=407182835067\approx 2^{39}$.
In Section \ref{sec:mtheo} we prove and generalize
Conjecture~\ref{conj}.
\begin{theo}\label{mtheo}
We have
$$ \liminf_{n\to \infty} \frac{s_q(p(n))}{s_q(n)}=0.$$
\end{theo}
In view of our generalization, it is natural to ask how quickly we can expect this ratio to go to zero. Recall that $h = \deg p$.
\begin{theo}\label{mtheo2}
There exist explicitly computable constants $B$ and $C$,
dependent only on $p(x)$ and $q$,
such that for all $\varepsilon$ with $0<\varepsilon<h(4h+1)$
there exists an $n < B \cdot C^{1/\varepsilon}$ with
\[ \frac{s_q(p(n))}{s_q(n)} < \varepsilon. \]
\end{theo}
The proof of this result along with an explicit construction for $B$ and $C$
is given in Section \ref{sec:mtheo2}.
As a nice Corollary to this result we have
\begin{cor}
There exists a constant $C_0$, dependent only on $p(x)$ and $q$,
such that there exists infinitely many $n$ with
\[ \frac{s_q(p(n))}{s_q(n)} \leq \frac{C_0}{\log n}. \]
\end{cor}
This is an improvement and generalization upon~\eqref{stolthm}.
\begin{proof}
By solving for $\varepsilon$ in $n < B \cdot C^{1/\varepsilon}$, one
easily sees that $\varepsilon < \frac{\log C }{\log n -\log B}$.
Without loss of generality we may assume that $B > 1$, hence we can take $C_0 = \log C $.
\end{proof}
One might expect that the ratio $s_q(p(n))/s_q(n)$ is small only rarely, with most of its time being spent near $h = \deg p$.
It turns out that this ratio is small somewhat more often than expected.
\begin{theo} \label{mtheo2.5}
For any $\varepsilon > 0$ there exists an explicitly computable $\alpha > 0$, dependent only on $\varepsilon$, $p(x)$ and $q$,
such that
\[ \# \left\{n < N : \quad \frac{s_q(p(n))}{s_q(n)} < \varepsilon
\right\} \gg N^\alpha \]
where the implied constant also only depends on $\varepsilon$, $p(x)$ and $q$.
\end{theo}
The proof of this result is given in Section \ref{sec:mtheo2.5}.
In Section \ref{sec:conc} we collect together questions raised in this paper
and pose some further lines of inquiry for this research.
\section{Preliminaries and Proof of Theorem~\ref{limsupsto}}\label{sec:prelim}
First we prove some preliminary results about $s_q$ which we need in the proofs.
Recall (cf.~\cite{Li97}) that terms are said to be \textit{noninterfering} if we can use the following splitting formul\ae:
\begin{prop}\label{propsplit}
For $1\leq b<q^k$ and $a,k\geq 1$,
\begin{align}
s_q(aq^k+b)&=s_q(a)+s_q(b),\label{splitpos}\\
s_q(aq^k-b)&=s_q(a-1)+(q-1)k-s_q(b-1).\label{splitneg}
\end{align}
\end{prop}
\begin{proof} Relation~(\ref{splitpos}) is a consequence of the (strong) $q$-additivity of $s_q$. For~(\ref{splitneg}) we write
$b-1=\sum^{k-1}_{i=0}b_iq^i$ with $0\leq b_i\leq q-1$. Then
\begin{align*}
s_q(aq^k-b) &=s_q((a-1)q^k+q^k-b)=s_q(a-1)+s_q(q^k-b)\\
&=s_q(a-1)+s_q\left(\sum_{i=0}^{k-1} (q-1-b_i)q^i\right)\\
&=s_q(a-1)+\sum_{i=0}^{k-1} (q-1-b_i)
\end{align*}
implying \eqref{splitneg}.
\end{proof}
\begin{prop}\label{propsub}
The function $s_q$ is subadditive and submultiplicative, i.e., for all $a,b\in \mathbb{N}$ we have
\begin{align}
s_q(a+b) &\leq s_q(a)+s_q(b),\label{subadd}\\
s_q(ab) &\leq s_q(a) s_q(b).\label{submult}
\end{align}
\end{prop}
\begin{proof}
The proof follows on the lines of~\cite[Section~2]{Ri08}.
As for~(\ref{subadd}), an even stronger result is true,
namely that
$s_q(a+b) = s_q(a) + s_q(b) - (q-1) \cdot r$
where $r$ is the number of ``carry'' operations needed when adding $a$
and $b$.
Writing $b=\sum_{i=0}^{k-1} b_i q^i$ we also have
\begin{align*}
s_q(ab)&=s_q\left(a\sum_{i=0}^{k-1} b_iq^i\right)\leq \sum_{i=0}^{k-1} s_q(ab_i)\\
&=\sum_{i=0}^{k-1} s_q(\underbrace{a+\dots+a}_{\mbox{$b_i$ times}})\leq s(a)\sum_{i=0}^{k-1} b_i,
\end{align*}
where we used twice the subadditivity of $s_q$ and we get~(\ref{submult}).
\end{proof}
\begin{proof}[Proof of Theorem \ref{limsupsto}]
This is an almost direct generalization of Stolarsky's proof (see~\cite[Section~2]{St78}) and Propositions~\ref{propsplit}
and~\ref{propsub}. First, suppose that $p(n)$ has only nonnegative coefficients. Then using
Proposition~\ref{propsub} we see that $s_q(p(n))\leq p(s_q(n))$. Therefore
\begin{align}
\frac{s_q(p(n))}{s_q(n)}
&\leq\frac{\min\{(q-1) \left(\log_q p(n)+1\right), p(s_q(n))\}}{s_q(n)}\nonumber\\
&\leq c_1\cdot \frac{\min\{\log_q n, s_q(n)^h\}}{s_q(n)} \label{stolcases}
\end{align}
where $c_1$ only depends on $p(x)$ and $q$.
If $\log_q n \leq s_q(n)^h$ then we have $(\log_q n)^{1/h} \leq s_q(n)$.
From this and~(\ref{stolcases}), we get that
$$ \frac{s_q(p(n))}{s_q(n)}
\leq c_1\cdot \frac{\log_q n}{(\log_q n)^{1/h}}
= c_1 (\log_q n)^{1-1/h}.
$$
Alternately, if $\log_q n > s_q(n)^h$ then we have $(\log_q
n)^{1/h}
> s_q(n)$ and
$$ \frac{s_q(p(n))}{s_q(n)}
\leq c_1\cdot s_q(n)^{h-1}
\leq c_1 (\log_q n)^{1-1/h}.
$$
For the lower bound, set
\begin{equation}\label{kdef}
k=\lfloor \log_q (\lambda (h+1)!)\rfloor +1,
\end{equation}
where $\lambda = \max \{a_i: 0\leq i\leq h\}$. By Stolarsky's use of the Bose-Chowla Theorem,
there are infinitely many integers $M\geq 3(k+1)$ such that there are integers $y_1, y_2, \dots, y_N$
with $N:=\lfloor (M+1)/(k+1)\rfloor-1,$ with the following three properties:
\begin{enumerate}
\item[(i)] $1\leq y_1<y_2<\dots<y_N\leq M^h$,
\item[(ii)] $y_i\equiv 0 \bmod (k+1)$,
\item[(iii)] all sums $y_{j_1}+\dots +y_{j_h}$ are distinct (\textit{distinct sum property});
here $j_1, j_2, \ldots, j_{h} \in \{1, 2, \ldots, N\}$ with possible
repetition.
\end{enumerate}
Note that (iii) implies the distinct sum property for all $y_{j_1}+\dots +y_{j_i}$ with $1\leq i\leq h$.
Now set
$$n=\sum_{i=1}^N q^{y_i},$$
such that
\begin{equation}\label{pnq}
p(n)=\sum_{i=0}^h a_i n^i = \sum_{i=0}^h \sum {}^{'} a_i \alpha(i; h_1, \dots, h_N) q^{y_1h_1+\dots +y_N h_N}
\end{equation}
where the summation $\sum {}^{'}$ is over all vectors $(h_1,\dots,h_N)$ satisfying $h_1+\dots+h_N=i$, and
$\alpha(i; h_1, \dots, h_N)$ denote the multinomial coefficients $i!/(h_1!\dots h_N!)$ bounded by $i!$.
Consider~(\ref{pnq}) as a polynomial in $q$. By the distinct sum property (iii) we have for all $0\leq i\leq h$ that
$$\#\{y_1h_1+\dots+y_N h_N: \; h_1+\dots +h_N=i\}= \binom{N+i-1}{N-1}.$$
Thus the coefficients of $q^{y_1h_1+\dots +y_N h_N}= q^{R}$ with $h_1+\dots+h_N=h$ in~(\ref{pnq}) are nonzero and bounded by
\begin{equation}\label{abound}
a_h h!+a_{h-1} (h-1)!+\dots +a_0 \leq \lambda (h+1) h!< q^k.
\end{equation}
By~(\ref{abound}) and (ii), the sums $y_1h_1+\dots +y_N h_N\equiv 0$ mod $(k+1)$ and hence the powers $q^R$ are
noninterfering and we get
$$\frac{s_q(p(n))}{s_q(n)}\geq \binom{N+h-1}{N-1}\cdot \frac{1}{N}\geq \frac{N^{h-1}}{h!}.$$
By construction,
$$\log_q n\leq y_N+1 \leq 2^{h+1} N^h (k+1)^h.$$
The claim now follows by observing that $k$ is largest for $q=2$.
Secondly suppose that $p(n)$ has at least one negative coefficient. Then the first claim follows by observing
that $s_q(p(n))\leq \lfloor \log_q p(n)\rfloor +1$ for sufficiently large $n$. For the lower bound, denote by
$a_j$ the negative coefficient with smallest index $j$, i.e., $a_j<0$ and $a_{j-l}\geq 0$ for $1\leq l\leq j $.
Then for all sufficiently large $k$ we have
\begin{align*}
s_q(p(q^k))&=s_q(a_h q^{hk}+\dots+a_{j+1} q^{(j+1)k}+a_j q^{jk}+a_{j-1} q^{(j-1)k}+\dots +a_0)\\
&=s_q(a_h q^{(h-j)k}+\dots+a_{j+1} q^{k}+a_j )+\sum_{l=0}^{j-1} s_q(a_l)\\
&\geq k(q-1)-s(-a_j-1)\\
&> k(q-1-\varepsilon).
\end{align*}
Here we have used Proposition~\ref{propsplit}.
As $s_q(q^k) = 1$ and $\log_q(q^k) = k$, the result follows.
This completes the proof of Theorem~\ref{limsupsto}.
\end{proof}
\section{Proof of Theorem~\ref{mtheo}}\label{sec:mtheo}
The proof of Theorem~\ref{mtheo} will use a construction of a sequence with noninterfering terms. First
assume that $p(x)=x^h$, $h\geq 2$ and define the polynomial
$$t_m(x)=m x^4+mx^3-x^2+mx+m$$
where $m\in \mbox{$\mathbb Z$}$ with $m\geq 3$.
By consecutively employing~(\ref{splitpos}) and~(\ref{splitneg}) we see that for all $k$ with $q^k>m$,
\begin{equation}\label{crx}
s_q(t_m(q^k))=(q-1)k+s_q(m-1)+3s_q (m).
\end{equation}
The appearance of $k$ in~(\ref{crx}) is crucial. The next lemma lies at the heart of the proofs. We
will use it to see that $s_q(t_m(q^k)^h)$, $h\geq 2$, is independent of $k$ whenever $k$ is sufficiently large.
Furthermore, we will exploit the fact that the coefficients of $[x^i]$ in $t_m(x)^h$ are polynomials in $m$ with
alternating signs.
\begin{lem}\label{lemuseful}
For fixed $h\geq 2$ and $m\geq 3$, we have
$$ t_m(x)^h =\sum_{i=0}^{4h} c_{i,h}(m) \; x^i$$
satisfying
\begin{equation}\label{cestim}
0<c_{i,h}(m) \leq (2mh)^h \qquad i=0,1,\dots, 4h.
\end{equation}
In fact, we have
\begin{align}\label{coc1}
c_{0,h}(m)=c_{4h,h}(m)=m^h, \ \ c_{1,h}(m)=c_{4h-1,h}(m)=hm^h.
\end{align}
\end{lem}
\begin{proof}A direct calculation shows that $t_m(x)^2$ and $t_m(x)^3$ have
property \eqref{cestim} provided $m\geq 3$. Set $h=2h_1+3h_2$ with
$\max(h_1,h_2)\geq 1$. Then
$$t_m(x)^h = \underbrace{t_m(x)^2\dots t_m(x)^2}_{\mbox{$h_1$ times}} \cdot \underbrace{t_m(x)^3 \dots t_m(x)^3}_{\mbox{$h_2$ times}}.$$
Since products of polynomials with all positive coefficients have all positive coefficients too, we get $c_{i,h}(m)>0$ for all
$i=0,1,\dots, 4h$. On the other hand, the coefficients of $t_m(x)^h$ are
clearly bounded by the corresponding coefficients of the polynomial
\begin{align*}
m^h(1+x+x^2+x^3+x^4)^h=m^h\sum_{0\leq l\leq k\leq j\leq i\leq h}\binom{h}{i}
\binom{i}{j}\binom{j}{k}\binom{k}{l}x^{i+j+k+l}.
\end{align*}
Therefore, for all $i$ with $0\leq i\leq 4h$, we have
\begin{align}\label{eq:c_i}
c_{i,h}(m)&\leq m^h \sum_{0\leq l\leq k\leq j\leq i\leq h} \frac{h!}{(h-i)!(i-j)!(j-k)!(k-l)!}\\
&\leq m^hh! \;{\rm exp}(h-i+i-j+j-k+k-l)\nonumber\\
&\leq m^hh!e^h\leq (2mh)^h.\nonumber
\end{align}
\end{proof}
\begin{proof}[Proof of Theorem \ref{mtheo}]
Now let $k$ be such that $q^k>(2mh)^h$. By~(\ref{cestim}) and~(\ref{splitpos}) we then have
$$s_q(t_m(q^k)^h)=s_q(c_{0,h}(m))+s_q(c_{1,h}(m))+\dots+s_q(c_{4h,h}(m))$$
where $s_q(c_{i,h}(m))$ is bounded by a function which only depends on $q$, $m$ and $h$.
Together with~(\ref{crx}) and letting $k\to \infty$ we thus conclude for fixed $m\geq 3$,
$$\lim_{k\to\infty} s_q(t_m(q^k)^h)/s_q(t_m(q^k))=0,$$
as wanted.
Finally we consider the case with a general polynomial instead of $x^h$. Write
\begin{equation}\label{gen}
p(t_m(x))=a_h t_m(x)^h+a_{h-1} t_m(x)^{h-1}+\dots+a_1 t_m(x) +a_0
\end{equation}
where $a_h>0$ and $h\geq 2$. First suppose that all the coefficients
are nonnegative. Lemma \ref{lemuseful} shows that for $i$ with
$2\leq i\leq h$ all the coefficients of $t_m(x)^i$ are positive. Also,
the coefficient
$[x^2]$ in $p(t_m(x))$ is nonnegative if we choose $m\geq 3$ sufficiently large. In fact, a sufficient
condition is $a_h \left( \binom{h}{2} m^h-h m^{h-1}\right)\geq a_1$ which is true whenever
\begin{equation}\label{mestim}
m\geq \left(\frac{2a_1}{h(3h-5) a_h}\right)^{1/(h-1)}.
\end{equation}
If the polynomial $p(x)$ has negative coefficients then there is a positive
integer $b$ such that the polynomial $p(x+b)$ has all positive coefficients.
A good choice for $b$ is
\begin{equation}\label{bestim}
b=\left\lceil 1+\frac{\lambda}{a_h}\right\rceil=
1+\left\lceil \frac{\lambda}{a_h}\right\rceil
,\qquad \lambda=\max \{|a_i|: 0\leq i\leq h\}.
\end{equation}
This is easy to see since both $p(x+b)-(a_h(x+b)^h-\lambda \sum^{h-1}_{i=0}
(x+b)^i)$ and
\begin{align*}
a_h(x+b)^h-&\lambda \sum^{h-1}_{i=0}(x+b)^i\\ &=
\left(a_h-\frac{\lambda}{x+b-1}\right)(x+b)^h+\frac{\lambda}{x+b-1}\\
&=\frac{1}{x+b-1}\left((a_hx+(b-1)a_h-\lambda)(x+b)^h+\lambda\right)
\end{align*}
have nonnegative coefficients when $b\geq 1+\frac{\lambda}{a_h}$. Thus
if $q^k>m+b$ then
$s_q(t_m(q^k)+b)=(q-1)k+s_q(m-1)+2s_q(m)+s_q(m+b)$ and one similarly obtains
for fixed $m$,
$$\lim_{k\to\infty} s_q(p(t_m(q^k)+b))/s_q(t_m(q^k)+b)=0.$$
This completes the proof of Theorem~\ref{mtheo}.
\end{proof}
\section{Proof of Theorem~\ref{mtheo2}}\label{sec:mtheo2}
The construction of an extremal sequence in the proof of Theorem~\ref{mtheo} gives a rough
bound on the minimal $n$ such that $s_q(n^h)<s_q(n).$ We first illustrate the method in the case $q=2$, $h=3$.
Set $m=3$. Then for all $k$ with $2^k>\max\limits_{0\leq i\leq 4h}{c_{i,h}(m)}=225$ we have
\begin{align*}
s_2(t_3(2^k)) &= k+1+6=k+7,\\
s_2(t_3(2^k)^3) &= 2\cdot(4+3+4+4+4+4)+4=50.
\end{align*}
Therefore, by setting $k=44$, we get
$$\min \{n: s_2(n^3)<s_2(n)\} < 2^{178}.$$
It is possible to show that the minimal such $n$ to be
$n=407182835067\approx 2^{39}$.
\begin{proof}[Proof of Theorem \ref{mtheo2}]
Consider the general polynomial
$$p(x)=a_hx^h+a_{h-1}x^{h-1}+\dots+a_0 \in \mbox{$\mathbb Z$}[x]$$
with $a_h>0$, $h\geq 2$.
Let $\lambda = \max |a_i|$.
Pick $b$ such that $p(x+b)$ has only nonnegative coefficients, as in \eqref{bestim}.
Pick $m \geq 3$ such that $p(t_m(x)+b)$ has only nonnegative coefficients,
as in \eqref{mestim}.
Our task is to bound the coefficients of of $p(t_m(x)+b)\in \mbox{$\mathbb Z$}[x]$.
To begin with, we estimate the coefficient of $x^i, 0\leq i\leq h$ of
$p(x+b)$,
\begin{equation}\label{pyestim}
\sum_{j=i}^h a_j b^{j-i} \binom{j}{i} \leq \sum_{j=i}^h \left\vert a_j b^{j-i} \binom{j}{i} \right\vert \leq \lambda (2b)^h.
\end{equation}
Combining \eqref{pyestim} with \eqref{coc1}, we find that the
constant term of $p(t_m(x)+b)$ is bounded by
\begin{equation*}
\lambda (2b)^h \sum^h_{i=0} m^i=\lambda (2b)^h\frac{m^{h+1}-1}{m-1}
\leq \lambda h(4mbh)^h
\end{equation*}
since $m\geq 3$ and $h\geq 2$. Again from \eqref{pyestim} and
\eqref{cestim}, we find that the other coefficients of
$p(t_m(x)+b)$ are bounded by
\begin{equation}\label{finalestim}
\lambda (2b)^h \sum_{i=1}^h (2mi)^i \leq \lambda h(4 mbh)^h.
\end{equation}
Therefore the coefficients of $p(t_m(x)+b)$ are bounded by
$\lambda h(4 mbh)^h$. Hence for $q^k>m+b$, we have
\begin{equation}\label{sqhestim}
s_q(p(t_m(q^k))+b)\leq (q-1)(4h+1)
\left(\frac{\log(\lambda h(4mbh)^h)}{\log q}+1\right).
\end{equation}
On the other hand, we clearly have $s_q(t_m(q^k)+b)> (q-1)k$ for $q^k>m+b$.
Let
\begin{align*}
k=\left\lfloor \frac{4h+1}{\varepsilon}
\left(\frac{\log(\lambda h(4mbh)^h)}{\log q}+1\right)\right\rfloor +1.
\end{align*}
Then for $0<\varepsilon<h(4h+1)$ we have $q^k>m+b$ and hence
$$\frac{s_q(p(t_m(q^k)+b))}{s_q(t_m(q^k)+b)} <\varepsilon.$$
Therefore,
\begin{align*}
\min \left\{n:\; \frac{s_q(p(n))}{s_q(n)}<\varepsilon\right\}
&\leq t_m(q^k)+b\\
&< m(q^{4k}+q^{3k}+q^k+1)\\
&<2m q^{4k}\\
&\leq 2mq^4 \left(q\lambda h(4mbh)^h\right)^{(16h+4)/\varepsilon}.
\end{align*}
Setting
$B := 2 m q^4$ and
$C := \left(q\lambda h(4mbh)^h\right)^{16h+4}$, it gives the desired
result.
\end{proof}
\section{Proof of Theorem~\ref{mtheo2.5}}\label{sec:mtheo2.5}
We start our analysis with the simple case of $p(n) = n^h$. Let $t_m(x) = m x^4 + m x^3 - x^2 + m x + m$
as in Section~\ref{sec:mtheo}.
Letting $n = n_{k,m} = t_m(q^k)$ we see from equation~(\ref{crx}) that, for $m<q^k$,
\[ s_q(n) = (q-1) k + s_q(m-1) + 3 s_q(m) \geq (q-1) k. \]
If $m$ has $i$ $q$-ary digits then $n$ will have
$4 k + i$ $q$-ary digits. We see that $t_m(q^k)^h$ is of length at most $h (4 k + i)$.
Let $t_m(q^k)^h = \sum_{j=0}^{4h} c_{j} q^{kj}$.
These $c_{j}$ are dependent upon $m$ and $h$, but are independent of $k$ for $k$ sufficiently large.
We see from equation (\ref{eq:c_i}) that
$c_j \leq (m h \cdot 2)^h$ and hence has
at most $h i + h \log_q h + h$ $q$-ary digits.
As there are $(4 h + 1)$ coefficients $c_j$ and
$s_q(c_j)\leq (q-1)(hi + h\log_q h + h)$, we get
$$s_q(n^h) \leq (q-1) (4h+1) \left(h i + h \log_q h + h \right).$$
Combining these together we have
\begin{align*}
\frac{s_q(n^h)}{s_q(n)} &\leq
\frac{(q-1) (4h+1) \left(hi + h\log_q h + h \right)}{(q-1) k}\\
& =
\frac{(4h+1) \left(h i + h\log_q h + h \right)}{k}.
\end{align*}
Without loss of generality suppose that $0<\varepsilon<h(4h+1)$.
Let $k_0$ be large enough so that $k_0>i$ and
\[\frac{(4h+1) \left(h i + h \log_q h + h\right)}{k_0} < \varepsilon. \]
For $i$ sufficiently large, we can take $k_0 =
\left \lfloor \frac{(4h+1)(hi+i)}{\varepsilon}\right \rfloor $.
Then this says that for every sufficiently large $m$ having $i$ $q-$ary digits,
there is an integer $n$ having $4 k_0 + i$ $q-$ary digits such that
\[ \frac{s_q(n^h)}{s_q(n)} < \varepsilon. \]
Moreover, by construction, each distinct $m$ will give rise to a distinct
$n$. Letting \[ \alpha = \frac{i}{4 k_0 + i} \geq
\frac{i}{4(4h+1) (h+1)i/\varepsilon + i} =
\frac{\varepsilon}{4 (4h+1) (h+1) + \varepsilon} \]
we get as $N\to \infty$ that
\[ \# \left\{n < N : \quad \frac{s_q(n^h)}{s_q(n)} < \varepsilon
\right\} \gg N^\alpha. \]
Now to extend this for general $p(x)$, we proceed as we did in the proof
of Theorem \ref{mtheo}.
First consider the case where $p(x)$ has only nonnegative coefficients.
There is a lower bound on $m$ such that $p(t_m(x))$ will have only
nonnegative coefficients and we proceed as before, after which the
result follows as before.
Second, if $p(x)$ has at least one negative coefficient, then consider instead the polynomial
$p(x+b)$ for sufficiently large $b$, which will have only nonnegative
coefficients, and the result follows.
\section{Conclusions and further work}
\label{sec:conc}
All results in this paper have explicitly computable constants for
existence or density results. Many times these constants are far
from the observed experimental values, and it is quite likely that
many of them may be strengthened. Examples include
Theorems~\ref{mtheo2} and~\ref{mtheo2.5}.
Some obvious generalizations of this problem are in looking at the ratios of
$\frac{s_q(p_1(n))}{s_q(p_2(n))}$, or even more generally
of $\frac{s_{q_1}(p_1(n))}{s_{q_2}(p_2(n))}$ with respect to two different
bases $q_1, q_2$. Alternately, instead of looking at polynomials $p(x) \in \mbox{$\mathbb Z$}[x]$,
we could look at quasi-polynomials $\lfloor p(n) \rfloor$ with $p(x) \in \R[x]$.
As another direction, we could consider expansions in other numeration
systems, e.g. the Zeckendorf expansion (or expansions with respect to linear
recurrences) or the balanced based $q$ representation. In the latter
case, for example, $11 = 1 \cdot 3^2 + 1 \cdot 3^1 - 1 \cdot 3^0$, and $s_3'(11) = 1 + 1 - 1 = 1$,
being the sum-of-digits function in this representation. This value will quite often be $0$, but its
extremal distribution could still have some interesting properties.
\section*{Acknowledgements.}
The authors thank J. Shallit for his remarks on a previous version of this paper.
| {
"timestamp": "2010-01-23T17:27:11",
"yymm": "1001",
"arxiv_id": "1001.4169",
"language": "en",
"url": "https://arxiv.org/abs/1001.4169",
"abstract": "Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$ \\liminf_{n\\to\\infty} \\frac{s_2(n^2)}{s_2(n)} = 0. $$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + ... + a_0 \\in \\Z[x]$ with $h\\geq 2$ and $a_h>0$ and any base $q$, \\[ \\liminf_{n\\to\\infty} \\frac{s_q(p(n))}{s_q(n)}=0.\\] For any $\\epsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \\epsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \\epsilon$.",
"subjects": "Number Theory (math.NT); Combinatorics (math.CO)",
"title": "Stolarsky's conjecture and the sum of digits of polynomial values",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986979508737192,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379405608844
} |
https://arxiv.org/abs/1205.4416 | On the Local-Global Conjecture for integral Apollonian gaskets | We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket. | \section{Introduction}
\begin{figure}
\includegraphics[width
2in]{pic.pdf}
\vskip-1.25in
\hskip-2.5in
{\Huge -11}
\vskip1in
\caption{The Apollonian gasket with root quadruple $v_{0}=(-11, 21,24,28)^{t}$.}
\label{fig1}
\end{figure}
\subsection{The
Local-Global
Conjecture}\
Let $\sG$ be an Apollonian
gasket, see Fig. \ref{fig1}. The number $b(C)$ shown inside a circle $C\in\sG$ is its curvature, that is, the reciprocal of its radius (the bounding circle has
negative
orientation). Soddy \cite{Soddy1937} first observed the existence of {\it integral}
gaskets $\sG$, meaning
ones for which $b(C)\in\Z$ for all $C\in\sG$. Let
$$
\sB=\sB_{\sG}:=\{b(C):C\in\sG\}
$$
be the set of all
curvatures in $\sG$. We call a
gasket {\it primitive} if $\gcd(\sB)=1$. From now on, we restrict our attention to a fixed primitive integral Apollonian gasket $\sG$.
Graham, Lagarias, Mallows, Wilks, and Yan \cite{LagariasMallowsWilks2002, GrahamLagarias2003} initiated a detailed study of Diophantine properties of $\sB$, with two separate families of problems (see also e.g. \cite{KontorovichOh2011, FuchsSanden2011, Sarnak2011}): studying $\sB$ with multiplicity (that is, studying circles), or without multiplicity (studying the integers which arise).
In the present paper, we are concerned with the latter
In particular,
the following striking local-to-global conjecture for $\sB$
is given in
\cite[p. 37]{GrahamLagarias2003}, \cite{FuchsSanden2011}.
Let $\sA=\sA_{\sG}$ denote the {\it admissible} integers, that is, those passing
all
local (congruence) obstructions:
$$
\sA:=\{n\in\Z:n\in\sB(\mod q),\text{ for all $q\ge1$}\}.
$$
\begin{conj}[Local-Global Conjecture]\label{conj}
Fix
a primitive, integral Apollonian
gasket $\sG$.
Then every sufficiently large admissible number is the curvature of a circle in $\sG$.
That is, if $n\in\sA$ and $n\g
1$, then $n\in\sB$
\end{conj}
The purpose of this paper is to prove the following
\begin{thm}\label{thm:Main}
Almost every admissible number is the curvature of a circle in
$\sG$. Quantitatively, the number of exceptions up to $N$ is bounded by
$O(N^{1-\eta})$, where $\eta>0$ is effectively computable.
\end{thm}
\begin{comment}
Tha
We restate the conjecture in the following way. For $N\ge1$, let
$$
\sB(N):
\sB\cap[1,N]
,\quad\text{and}\quad
\sA(N):
\sA\cap[1,N]
.
$$
Then the
XXXXXX Strong Density Conjecture is equivalent to the assertion that
\be\label{eq:Conj}
\#\sB(N)
\#\sA(N)
+
(1)
,
\quad
\text{
as $N\to\infty$.
}
\ee
In her thesis, Fuchs \cite{FuchsThesis} proved that $\sA$ is a union of arithmetic progressions with modulus dividing $24$,
that is,
\be\label{eq:locA}
n\in\sA\qquad \Longleftrightarrow\qquad
n(\mod 24)\in\sA(24),
\ee
cf. Lemma \ref{lem:GGq}.
Hence obviously
$$
\#\sA(N)=
\frac {\#\sA(24)}{24} \cdot N+O(1)
.
$$
\\
\subsection{Partial Progress and Statement of the Main Theorem}\
\end{comment}
Admissibility is completely explained in Fuchs's thesis \cite{FuchsThesis}, and is a condition restricting to certain residue classes modulo $24$, cf. Lemma \ref{lem:GGq}.
E.g. for the gasket in Fig. \ref{fig1}, $n\in\sA$ iff
\be\label{eq:locA}
n\equiv0, 4, 12, 13, 16,\text{ or }21(\mod 24).
\ee
Thus $\sA$ contains one of every four numbers (six admissible residue classes out of $24$), and Theorem \ref{thm:Main} can be restated in this case as
$$
\#(\sB\cap[1,N])={N\over4}\left(1+O(N^{-\eta})\right).
$$
In general, the local obstructions are easily determined (see Remark \ref{rmk:locA}) from the so-called {\it root quadruple}
\be\label{eq:v0Def}
v_{0}=v_{0}(\sG)
,
\ee
which is the column vector of the four smallest curvatures in $\sB$. For the gasket in Fig. \ref{fig1}, $v_{0}=(-11, 21, 24, 28)$.
\\
The history of this problem is as follows. The first progress
towards
the Conjecture
was already made in \cite{GrahamLagarias2003}, who showed
that
\be\label{eq:GLbnd}
\#(\sB\cap[1,N])\gg N^{1/2}.
\ee
Sarnak \cite{SarnakToLagarias}
improved this to
\be\label{eq:SarBnd}
\#(\sB\cap[1,N])\gg {N\over (\log N)^{1/2}},
\ee
and then Fuchs \cite{FuchsThesis}
showed
$$
\#(\sB\cap[1,N])\gg {N\over (\log N)^{0.150\dots}}.
$$
Finally Bourgain and Fuchs \cite{BourgainFuchs2010} settled the so-called ``Positive Density Conjecture,'' that
$$
\#(\sB\cap[1,N])\gg N.
$$
\subsection{Methods}\
Our main approach is through the
Hardy-Littlewood
circle method, combining two new ingredients. The first, applied to the major arcs, is effective bisector counting
in infinite volume
hyperbolic $3$
folds, recently achieved by I. Vinogradov \cite{Vinogradov2013}, as well as the uniform spectral gap over congruence towers of such,
see the Appendix by P\'eter Varj\'u.
The second ingredient is the minor arcs analysis,
inspired by
that given recently by the first-named author in \cite{Bourgain2012}, where it was proved that the prime curvatures in a
gasket
constitute a positive proportion of the primes.
(Obviously Theorem \ref{thm:Main} implies that
$100\%$ of the admissible prime curvatures appear.)
\subsection{Plan for the Paper}\
A more detailed outline of the proof, as well as the setup of some relevant exponential sums, is given in \S\ref{sec:outline}. Before we can do this, we need to recall the Apollonian group
and some of its subgroups in \S\ref{sec:prelim}. After the outline in \S\ref{sec:outline}, we use \S\ref{sec:preII} to collect some background
from the spectral and representation theory of infinite volume hyperbolic quotients.
Then some lemmata are reserved for \S\ref{sec:lems}, the major arcs are estimated in \S\ref{sec:Maj}, and the minor arcs are dealt with in \S\S\ref{sec:qQ0}-\ref{sec:QXT}.
The Appendix, by P\'eter Varj\'u, extracts the spectral gap property for the Apollonian group from that of its arithmetic subgroups.
\subsection{Notation}\
We use the following standard notation. Set $e(x)=e^{2\pi i x}$ and $e_{q}(x)=e(\frac xq)$. We use $f\ll g$ and $f=O(g)$ interchangeably; moreover $f\asymp g$ means $f\ll g\ll f$.
Unless otherwise specified, the implied constants may depend at most on the
gasket
$\sG$ (or equivalently on the root quadruple $v_{0}$), which is treated as fixed.
The symbol $\bo_{\{\cdot\}}$ is the indicator function of the event $\{\cdot\}$.
The greatest common divisor of $n$ and $m$ is written $(n,m)$, their least common multiple is $[n,m]$, and $\gw(n)$ denotes the number of distinct prime factors of $n$. The cardinality of a finite set $S$ is denoted $|S|$ or $\# S$.
The transpose of a matrix $g$ is written $g^{t}$. The prime symbol $'$ in $\underset{r(q)}{\gS} {}'$ means the range of $r(\mod q)$ is restricted to $(r,q)=1$. Finally, $p^{j}\| q$ denotes $p^{j}\mid q$ and $p^{j+1}\nmid q$.
\subsection*{Acknowledgements} The authors are grateful to Peter Sarnak for illuminating discussions, and many detailed comments improving the exposition of an earlier version of this paper. We
thank
Tim Browning, Sam Chow, Hee Oh, Xin Zhang, and
the referee for numerous
corrections and suggestions.
\newpage
\section{Preliminaries I: The Apollonian Group and Its Subgroups}\label{sec:prelim}
\subsection{Descartes Theorem and Consequences}\
Descartes' Circle Theorem states that a quadruple $v$
of (oriented) curvatures of four mutually tangent circles
lies on the cone
\be\label{eq:Fv}
F(v)=0,
\ee
where
$F$ is the Descartes quadratic form:
\be\label{eq:Fdef}
F(a,b,c,d)
=
2(a^{2}+b^{2}+c^{2}+d^{2})
-(a+b+c+d)^{2}
.
\ee
Note that $F$ has signature $(3,1)$ over $\R$,
and let
$$
G:=\SO_{F}(\R)=\{g\in\SL(4,\R):F(g v)=F(v),\text{ for all }v\in\R^{4}\}
$$
be the real special orthogonal group preserving $F$.
It follows immediately that for $b,c$ and $d$ fixed, there are two solutions $a,a'$ to \eqref{eq:Fv}, and
$$
a+a'=2(b+c+d).
$$
Hence we observe that $a$ can be changed into $a'$ by a reflection, that is,
$$
(a,b,c,d)^{t}=S_{1}\cdot (a',b,c,d)^{t},
$$
where the reflections
$$
S_{1}=
\bp
-1&2&2&2\\
&1&&\\
&&1&\\
&&&1
\ep
,\qquad
S_{2}=
\bp
1&&&\\
2&-1&2&2\\
&&1&\\
&&&1
\ep
,
$$
$$
S_{3}=
\bp
1&&&\\
&1&&\\
2&2&-1&2\\
&&&1
\ep
,\qquad
S_{4}=
\bp
1&&&\\
&1&&\\
&&1&\\
2&2&2&-1
\ep
,
$$
generate the so-called {\it Apollonian group}
\be\label{eq:GamDef}
\cA
=
\<S_{1},S_{2},S_{3},S_{4}
\>
.
\ee
It is a Coxeter group, free
except for
the relations $S_{j}^{2}=I$, $1\le j\le 4$. We immediately pass to the
index two
subgroup
$$
\G:=\cA\cap \SO_{F}
$$
of orientation preserving transformations, that is, even words in the generators.
Then $\G$ is freely generated by $S_{1}S_{2}$, $S_{2}S_{3}$ and $S_{3}S_{4}$.
It is known
that $\G$ is Zariski dense in $G$ but {\it thin}, that is, of infinite index in $G(\Z)$; equivalently, the Haar measure of $\G\bk G$ is infinite.
\subsection{Arithmetic Subgroups%
}\
Now we review the arguments from \cite{GrahamLagarias2003, SarnakToLagarias} which
lead to
\eqref{eq:GLbnd} and \eqref{eq:SarBnd}, as our setup depends critically on them.
Recall that for any fixed
gasket
$\sG$, there is a root quadruple $v_{0}$ of the four smallest
curvatures in $\sG$, cf. \eqref{eq:v0Def}.
It follows
from
\eqref{eq:Fv} and \eqref{eq:GamDef}
that
the set $\sB$ of all
curvatures can be realized as the orbit of the root quadruple $v_{0}$ under $\cA$.
Let
$$
\sO=\sO_{\sG}:=\G\cdot v_{0}
$$
be the orbit of $v_{0}$ under $\G$.
Then
the set of all
curvatures
certainly contains
\be\label{eq:setB}
\sB
\supset
\bigcup_{j=1}^{4}\<e_{j},\sO\>
=
\bigcup_{j=1}^{4}\<e_{j},\G\cdot v_{0}\>
,
\ee
where $e_{1}=(1,0,0,0)^{t},\dots, e_{4}=(0,0,0,1)^{t}$ constitute the standard basis for $\R^{4}$, and the inner product above is the standard one. Recall we are treating $\sB$ as a set, that is, without multiplicities.
\\
It was observed in \cite{GrahamLagarias2003} that $\G$ contains unipotent elements, and hence one can use these to furnish an injection of affine space in the otherwise intractable orbit $\sO$, as follows.
Note first that
\be\label{eq:C1def}
C_{1}:=
S_{4}S_{3}
=
\left(
\begin{array}{cccc}
1 & & & \\
& 1 & & \\
2 & 2 & -1 & 2 \\
6 & 6 & -2 & 3
\end{array}
\right)
\in\G
,
\ee
and
after conjugation by
$$
J:=
\left(
\begin{array}{cccc}
1 & & & \\
-1 & 1 & & \\
-1 & 1 & -2 & 1 \\
-1 & & & 1
\end{array}
\right)
,
$$
we have
$$
\tilde C_{1}:=
J^{-1}\cdot C_{1}\cdot J
=
\left(
\begin{array}{cccc}
1 & & & \\
& 1 & & \\
& 2 & 1 & \\
& 4 & 4 & 1
\end{array}
\right)
.
$$
Recall the spin homomorphism $\rho:
\SL_{2}\to\SO(2,1)$, embedded for our purposes in $\SL_{4}$, given explicitly by
\be\label{eq:spin}
\rho
:
\mattwo
\ga\gb
\g\gd
\mapsto
{1\over \ga\gd-\gb\g}
\bp
1&&&\\
&{\ga^{2}}&{2\ga\g}&{\g^{2}}\\
&{\ga\gb}&{\ga\gd+\gb\g}&{\g\gd}\\
&{\gb^{2}}&{2\gb\gd}&{\gd^{2}}
\ep
.
\ee
In fact $\SL_{2}$ is a double cover of $\SO(2,1)$ under $\rho$, with kernel $\pm I$.
It is clear from inspection that
$$
\rho:
\mattwo1201=:T_{1}
\mapsto
\tilde C_{1}
.
$$
Since
$T_{1}^{n}=\mattwo1{2n}01$,
for each $n\in\Z$, $\G$ contains the element
$$
C_{1}^{n}=
J\cdot \rho(T_{1}^{n})\cdot J^{-1}
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
4 n^2-2 n & 4 n^2-2 n & 1-2 n & 2 n \\
4 n^2+2 n & 4 n^2+2 n & -2 n & 2 n+1
\end{array}
\right).
$$
(Of course this can be seen directly from \eqref{eq:C1def}; these transformations will be more enlightening below.)
Thus if
$v=(a,b,c,d)^{t}\in\sO$ is
a
quadruple
in
the orbi
, then $\sO$ also contains
$
C_{1}^{n}\cdot v
$
for all $n$.
From \eqref{eq:setB}, we then have that the set $\sB$ of curvatures contains
\b
\label{eq:oneParab}
\sB\ni\<e_{4},C_{1}^{n}\cdot v\>
4( a
+ b) n^2
+2
( a
+ b
- c
+ d) n
+d
.
\e
The circles thus generated
are all tangent to two fixed circles, which explains the square
curvatures in Fig. \ref{fig2}.
Of course \eqref{eq:oneParab} immediately implies \eqref{eq:GLbnd}.
\\
\begin{figure}
\includegraphics[width=2.5in]{picInf.pdf}
\caption{Circles tangent to two fixed circles.}
\label{fig2}
\end{figure}
Observe further that
$$
C_{2}
:=
S_{2}S_{3}
=
\left(
\begin{array}{cccc}
1 & & & \\
6 & 3 & -2 & 6 \\
2 & 2 & -1 & 2 \\
& & & 1
\end{array}
\right)
$$
is another unipotent element, with
$$
\tilde C_{2}:=
J^{-1}\cdot C_{2}\cdot J
=
\left(
\begin{array}{cccc}
1 & & & \\
& 1 & 4 & 4 \\
& & 1 & 2 \\
& & & 1
\end{array}
\right)
,
$$
and
$$
\rho:
\mattwo1021
=:T_{2}
\mapsto
\tilde C_{2}
.
$$
Since
$T_{1}$ and $T_{2}$ generate $\gL(2)$, the principal $2$-congruence subgroup of $\PSL(2,\Z)$, we see that
the Apollonian group $\G$ contains the subgroup
\be\label{eq:G1
\Xi:=
\<C_{1},C_{2}\>
=
J\cdot
\rho\bigg(
\gL(2)
\bigg)
\cdot
J^{-1}
<
\G
.
\e
In particular,
whenever $(2x,y)=1$,
there is an element
$$
\mattwo*{2x}*{y}\in\gL(2)
,
$$
and thus $
\X
$ contains
the element
\bea
\label{eq:xiXYdef}
\xi_{x,y}
&:=&
J\cdot
\rho
\mattwo*{2x}*{y}
\cdot
J^{-1}
\\
\nonumber
&=&
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
*&*&*&*\\
*&*&*&*\\
4x ^2+2xy +y ^2-1 & 4x ^2+2xy &-2xy &
2xy+ y ^2
\end{array}
\right)
.
\eea
Write
\bea
\label{eq:wxyDef}
w_{x,y}
&=&\xi_{x,y}^{t}\cdot e_{4}
\\
\nonumber
&=&
(4x^{2}+2xy+y^{2}-1,4x^{2}+2xy,-2xy,2xy+y^{2})^{t}.
\eea
Then again by \eqref{eq:setB}, we have shown the following \pagebreak
\begin{lem}[\cite{SarnakToLagarias}]
Let $x,y\in\Z$ with $(2x,y)=1$,
and take any element $\g\in\G$ with corresponding quadruple
\be\label{eq:vgIs}
v_{\g}=(a_{\g},b_{\g},c_{\g},d_{\g})^{t}=\g\cdot v_{0}\in\sO.
\ee
Then the number
\b
\label{eq:binQuad}
\<e_{4},\xi_{x,y}\cdot \g\cdot v_{0}\>
=
\<w_{x,y},\g\cdot v_{0}\>
4A_{\g}x^{2}
+4B_{\g}xy
+C_{\g}y^{2}
-a _{\g}
\e
is the curvature of some circle in $\sG$,
where we have defined
\bea
\label{eq:ABCdef}
A_{\g}
&:=&
a_{\g}+b_{\g},
\\
\nonumber
B_{\g}
&:=&
{a _{\g}
+ b _{\g}
- c _{\g}
+ d _{\g}\over 2},
\\
\nonumber
C_{\g}
&:=&
a_{\g}
+ d_{\g}
.
\eea
Note from \eqref{eq:Fv} that $B_{\g}$ is integral.
\end{lem}
Observe that, by construction, the value of $a_{\g}$ is unchanged under the orbit of
the group
\eqref{eq:G1}, and
the circles whose
curvatures are generated by \eqref{eq:binQuad} are all tangent to the circle corresponding to $a_{\g}$.
\begin{comment}
For example, with the root quadruple $v_{0}=(-11,21,24,28)^{t}$ in Fig. \ref{fig1}, observe that the
curvatures
$$
28,40,\text{ and }96
$$
are all
tangent to the outermost circle, and all
values of the shifted binary quadratic
$$
40 x^{2} +28 xy+17 y^{2}+11.
$$
\end{comment}
It is classical (see \cite{Bernays1912})
that the number of distinct primitive values
up to $N$
assumed by a
positive-definite binary quadratic form is of order at least $ N(\log N)^{-1/2}$, proving \eqref{eq:SarBnd}.
To fix notation, we define the binary quadratic appearing in \eqref{eq:binQuad} and its shift by
\be\label{eq:ffVdef}
f_{\g}(x,y)
:=
A_{\g}x^{2}+2B_{\g} xy+C_{\g}y^{2}
,
\qquad
\ff_{\g}(x,y):=
f_{\g}(x,y)
-a_{\g}
,
\ee
so that
\be\label{eq:ffToInProd}
\<
w_{x,y},
\g\cdot v_{0}\>
=
\ff_{\g}(2x,y)
.
\ee
Note
from \eqref{eq:ABCdef} and \eqref{eq:Fv}
that the discriminant of $f_{\g}$ is
\be
\label{eq:disc}
\gD_{\g}
=
4(B_{\g}^{2}-A_{\g}C_{\g})
=
-4 a_{\g}^2
.
\ee
When convenient, we will drop the subscripts $\g$ in all the above.
\subsection{Congruence Subgroups}\
For each $q\ge1$, define the ``principal'' $q$-congruence subgroup
\be\label{eq:GqDef}
\G(q)
:=
\{
\g\in\G:
\g\equiv I(\mod q)
\}
.
\ee
These groups all have infinite index in $G(\Z)$, but finite index in $\G$. The quotients $\G/\G(q)$ have been determined
completely
by Fuchs \cite{FuchsThesis}
by proving an explicit
Strong Approximation theorem (see \cite{MatthewsVasersteinWeisfeiler1984}), Goursat's Lemma, and other ingredients, as we explain below.
Since
$G$ does not itself have
the
Strong Approximation
Property,
we pass to its connected spin double cover $\SL_{2}(\C)$. We will need the covering map explicitly later, so record it here.
First change variables from the Descartes form $F$ to
$$
\tilde F(x,y,z,w):=xw+y^{2}+z^{2}.
$$
Then there is a homomorphism
$\iota_{0}:\SL(2,\C)\to \SO_{\tilde F}(\R)$, sending
$$
g=
\mattwo{a+\ga i}{b+\gb i}{c+\g i}{d+\gd i} \in\SL(2,\C)
$$
to
$
{1\over |\det(g)|^{2}}
\left(
\begin{array}{cccc}
a^2+\alpha ^2 & 2 (a c+\alpha \gamma ) & 2 (c \alpha - a \gamma) & -c^2-\gamma ^2 \\
a b+\alpha \beta & b c+a d+\beta \gamma +\alpha \delta & d \alpha +c \beta -b \gamma -a \delta & -c d-\gamma \delta
\\
a \beta -b \alpha & -d \alpha +c \beta -b \gamma +a \delta & -b c+a d-\beta \gamma +\alpha \delta & d \gamma -c \delta
\\
-b^2-\beta ^2 & -2 (b d+\beta \delta ) & 2( b \delta -d \beta ) & d^2+\delta ^2
\end{array}
\right)
.
$
To map from $\SO_{\tilde F}$ to $\SO_{F}$, we apply a
conjugation, see \cite[(4.1)]{GrahamLagarias2003}. Let
\be\label{eq:iota}
\iota:\SL(2,\C)\to\SO_{F}(\R)
\ee
be
the composition of this conjugation with $\iota_{0}$. Let $\tilde\G$ be the preimage of $\G$ under $\iota$.
\begin{lem}[\cite{GrahamLagariasMallowsWilksYanI, FuchsThesis}]
The group $\tilde\G$ is generated by
$$
\pm
\mattwo1{4i}{}1,\quad
\pm
\mattwo{-2}ii{},\quad
\pm
\mattwo{2+2i}{4+3i}{-i}{-2i}.
$$
\end{lem}
With this explicit realization of $\tilde\G$ (and hence $\G$), Fuchs was able to
explicitly
determine
the images of $\tilde\G$ in $\SL(2,\Z[i]/(q))$, and hence understand the quotients $\G/\G(q)$ for all $q$.
\begin{lem}[\cite{FuchsThesis}]\label{lem:GGq}\
(1) The quotient groups $\G/\G(q)$ are multiplicative, that is, if $q$ factors as
$$
q=p_{1}^{\ell_{1}}\cdots p_{r}^{\ell_{r}},
$$
then
$$
\G/\G(q)\cong \G/\G(p_{1}^{\ell_{1}})\times\cdots\times \G/\G(p_{r}^{\ell_{r}}).
$$
(2) If $(q,6)=1$ then
\be\label{eq:Fuc}
\G/\G(q)\cong\SO_{F}(\Z/q\Z).
\ee
(3)
If $q=2^{\ell}$, $\ell\ge3$, then $\G/\G(q)$ is the full preimage of $\G/\G(8)$ under the projection $\SO_{F}(\Z/q\Z)\to \SO_{F}(\Z/8\Z)$. That is, the powers of $2$ stabilize at $8$.
Similarly, the powers of $3$ stabilize at $3$, meaning that for $q=3^{\ell}$, $\ell\ge1$, the quotient $\G/\G(q)$ is the preimage of $\G/\G(3)$ under the corresponding projection map.
\end{lem}
\begin{rmk}\label{rmk:locA}
This of course explains all local obstructions, cf. \eqref{eq:locA}.
The admissible numbers are precisely those
residue classes $(\mod 24)$ which appear as some entry in the orbit of $v_{0}$ under $\G/\G(24)$.
\end{rmk}
\newpage
\section{Setup and Outline of the Proof}\label{sec:outline}
In this section, we introduce the main exponential sum and give an outline of the rest of the argument.
Recall the fixed
gasket
$\sG$ having curvatures $\sB$ and root quadruple $v_{0}$.
Let $\G$ be the Apollonian subgroup with subgroup $\Xi$, see \eqref{eq:G1}. Let $\gd\approx1.3$ be the Hausdorff dimension of the gasket $\sG$; see \S\ref{sec:preII} for the important role played by this geometric invariant.
Recall also from \eqref{eq:binQuad} that for any $\g\in\G$ and $\xi\in\Xi$,
$$
\<e_{4},\xi\g v_{0}\>\in\sB.
$$
Our approach, mimicing \cite{BourgainKontorovich2010, BourgainKontorovich2011a}, is to exploit the bilinear (or multilinear) structure above.
We first give an informal description of the main ensemble from which we will form an exponential sum.
Let $N$ be our main growing parameter.
We construct our ensemble by decomposing a ball in $\G$ of norm $N$ into two balls,
a small
one in all of $\G$ of norm $T$, and a larger one of norm $X^{2}$ in $\Xi$, corresponding to $x,y\asymp X$. Specifically, we take
\be\label{eq:TXNis}
T=N^{1/100}\quad\text{and}\quad X=N^{99/200},\qquad\text{so that}\qquad TX^{2}=N.
\ee
See \eqref{eq:gS2p} and \eqref{eq:gS12p} where these numbers are used.
We further need the technical condition that in the $T$-ball, the value of $a_{\g}=\<e_{1},\g\, v_{0}\>$ (see \eqref{eq:vgIs}) is of order $T$. This is used crucially in
\eqref{eq:cJfbnd}
and
\eqref{eq:ACisT}.
Finally, for technical reasons (see Lemma \ref{lem:spec3} below), we need to further split the $T$-ball into two: a small ball of norm $T_{1}$, and a big ball of norm $T_{2}$. Write
\be\label{eq:TT1T2}
T=T_{1}T_{2},\qquad T_{2}=T_{1}^{\cC},
\ee
where $\cC$ is a large constant depending only on the spectral gap for $\G$; it is determined in \eqref{eq:cCis}. We now make formal the above discussion.
\subsection{Introducing the Main Exponential Sum}\
Let $N,X, T, T_{1}$, and $T_{2}$ be as in \eqref{eq:TXNis} and \eqref{eq:TT1T2}. Define the family
\be\label{eq:fFdef}
\fF=\fF_{T}:=
\left\{
\g=\g_{1}\g_{2}:
\begin{array}{c}
\g_{1},\g_{2}\in\G,\\
T_{1}<\|\g_{1}\|<2T_{1},\\
T_{2}<\|\g_{2}\|<2T_{2},\\
\<e_{1},\g_{1}\,\g_{2}\,v_{0}\>>T/100
\end{array}
\right\}
.
\ee
From Lax-Phillips \cite{LaxPhillips1982}
(or see \eqref{eq:Vin}),
we have the bound
\be\label{eq:fFTbnd}
\#\fF_{T}\ll T^{\gd}.
\ee
From \eqref{eq:ffToInProd}, we can identify $\g\in\fF$ with a shifted binary quadratic form $\ff_{\g}$ of discriminant $-4a_{\g}^{2}$ via
$$
\ff_{\g}(2x,y)=\<w_{x,y},\g\, v_{0}\>.
$$
Recall from \eqref{eq:binQuad} that whenever $(2x,y)=1$, the above is a curvature in the
gasket. We sometimes drop $\g$, writing simply $\ff\in\fF$; then the latter can also be thought of as a family of shifted quadratic forms. Note also that the decomposition $\g=\g_{1}\g_{2}$ in \eqref{eq:fFdef} need not be unique, so some forms may appear with multiplicity.
One final technicality is to smoothe the sum on $x,y\asymp X$. To this end, we fix a smooth, nonnegative function
$\gU$, supported in $[1,2]$ and having unit mass,
$
\int_{\R}\gU(x)dx=1.
$
\\
Our main object of study is then the representation number
\be\label{eq:cRNis}
\cR_{N}(n):=\sum_{\ff\in\fF_{T}}\sum_{(2x,y)=1}
\gU\left(\frac {2x}X\right)
\gU\left(\frac yX\right)
\bo_{\{n=\ff(2x,y)\}}
,
\ee
and the corresponding exponential sum, its Fourier transform
\be\label{eq:cRNhatIs}
\widehat{\cR_{N}}(\gt):=\sum_{\ff\in\fF}\sum_{(2x,y)=1}
\gU\left(\frac {2x}X\right)
\gU\left(\frac yX\right)
e(\gt\, \ff(2x,y))
.
\ee
Clearly $\cR_{N}(n)\neq0$ implies that $n\in\sB$. Note also from \eqref{eq:fFTbnd} that the total mass satisfies
\be\label{eq:totMass}
\widehat{\cR_{N}}(0)\ll T^{\gd}X^{2}.
\ee
The condition $(2x,y)=1$ will be a technical nuisance,
and can be freed by a standard use of the M\"obius inversion formula. To this end,
we introduce another parameter
\be\label{eq:Uis}
U=N^{\fu},
\ee
a small power of $N$, with $\fu>0$ depending only on the spectral gap of $\G$; it is determined in \eqref{eq:fuIs}. Then by truncating M\"obius inversion, define
\be\label{eq:cRNhatUis}
\widehat{\cR_{N}^{U}}(\gt)
:=
\sum_{\ff\in\fF}\sum_{x,y\in\Z}
\gU\left(\frac {2x}X\right)
\gU\left(\frac yX\right)
e(\gt\, \ff(2x,y))
\sum_{u\mid(2x,y)\atop u<U}\mu(u)
,
\ee
with corresponding ``representation
function'' $\cR_{N}^{U}$ (which could be negative).
\\
\subsection{Reduction to the Circle Method}\
We are now in position to outline the argument in the rest of the paper. Recall that $\sA$ is the set of admissible numbers. We first reduce our main Theorem \ref{thm:Main} to the following
\begin{thm}\label{thm:RNU}
There exists an $\eta>0$ and a function $\fS(n)$ with the following properties. For $\foh N<n<N$, the singular series $\fS(n)$ is nonnegative, vanishes only when $n\notin\sA$, and is otherwise
\gg_{\vep}N^{-\vep}
$
for any $\vep>0$. Moreover, for $\foh N<n<N$ and admissible,
\be\label{eq:RNUbnd}
\cR_{N}^{U}(n)\gg \fS(n) T^{\gd-1},
\ee
except for a set of cardinality $\ll N^{1-\eta}$.
\end{thm}
\pf[Proof of Theorem \ref{thm:Main} assuming Theorem \ref{thm:RNU}:]\
We first show that the difference between $\cR_{N}$ and $\cR_{N}^{U}$ is small in $\ell^{1}$. Using \eqref{eq:fFTbnd}
we have
\beann
\sum_{n<N}|\cR_{N}(n)-\cR_{N}^{U}(n)|
&=&
\sum_{n<N}
\left|
\sum_{\ff\in\fF}\sum_{x,y\in\Z}
\gU\left(\frac {2x}X\right)
\gU\left(\frac yX\right)
\bo_{\{n=\ff(2x,y)\}}
\sum_{u\mid(2x,y)\atop u\ge U}\mu(u)
\right|
\\
&\ll&
\sum_{\ff\in\fF}
\sum_{u
\ge U}
\sum_{y\ll X\atop y\equiv0(\mod u)}
\sum_{x\ll X\atop 2x\equiv0(\mod u)}
1
\\
&\l
&
T^{\gd
{X^{2}\over U}
,
\eeann
for any $\vep>0$. Recall from \eqref{eq:Uis} that $U$ is a fixed power of $N$, so the above saves a power from the total mass \eqref{eq:totMass}.
Now let $Z$ be the ``exceptional'' set of admissible $n<N$ for which $\cR_{N}(n)=0$. Futhermore, let $W$ be the set of admissible $n<N$ for which \eqref{eq:RNUbnd} is satisfied. Then
\beann
T^{\gd
{X^{2}\over U}
&\g
&
\sum_{n<N} |\cR_{N}^{U}(n)-\cR_{N}(n)|
\ge
\sum_{n\in Z\cap W} |\cR_{N}^{U}(n)-\cR_{N}(n)|
\\
&\gg_{\vep}&
| Z\cap W|
\cdot
T^{\gd-1}
N^{-\vep}
.
\eeann
Note also from Theorem \ref{thm:RNU} that $|Z\cap W^{c}|
\le |W^{c}|\ll N^{1-\eta}$.
Hence by \eqref{eq:TXNis} and \eqref{eq:Uis},
\be\label{eq:nZbnd}
|Z|
=
| Z\cap W^{c}|
+
| Z\cap W|
\ll_{\vep}
N^{1-\eta}
+
{N^{1+\vep}\over U}
,
\ee
which
is a power savings since $\vep>0$ is arbitrary. This completes the proof.
\epf
To establish \eqref{eq:RNUbnd}, we decompose $\cR_{N}^{U}$ into ``major'' and ``minor'' arcs, reducing Theorem \ref{thm:RNU} to the following
\begin{thm}\label{thm:CircMeth}
There exists an $\eta>0$ and a decomposition
\be\label{eq:RNcMcE}
\cR_{N}^{U}(n)=\cM_{N}^{U}(n)+\cE_{N}^{U}(n)
\ee
with the following properties. For $\foh N<n<N$ and admissible, $n\in\sA$, we have
\be\label{eq:cMNUbnd}
\cM_{N}^{U}(n)\gg \fS(n) T^{\gd-1},
\ee
except for a set of cardinality $\ll N^{1-\eta}$. The singular series $\fS(n)$ is the same as in Theorem \ref{thm:RNU}. Moreover,
\be\label{eq:cEN2bnd}
\sum_{n<N}|\cE_{N}^{U}(n)|^{2}
\ll
N\, T^{2(\gd-1)} N^{-\eta}
.
\ee
\end{thm}
\pf[Proof of Theorem \ref{thm:RNU} assuming Theorem \ref{thm:CircMeth}:]\
We restrict our attention to the set of admissible $n<N$ so that \eqref{eq:cMNUbnd} holds (the remainder having sufficiently small cardinality). Let $Z$ denote the subset of these $n$ for which $\cR_{N}^{U}(n)<\foh \cM_{N}^{U}(n)$; hence for $n\in Z$,
$$
1\ll
{|\cE_{N}^{U}(n)|\over N^{-\vep}T^{\gd-1}}
.
$$
Then by \eqref{eq:cEN2bnd},
\beann
|Z|
&\ll_{\vep}&
\sum
_{n<N}
{|\cE_{N}^{U}(n)|^{2}
\over
N^{-\vep}T^{2(\gd-1)}}
\ll
N^{1-\eta+\vep}
,
\eeann
whence the claim follows, since $\vep>0$ is arbitrary.
\epf
\subsection{Decomposition into Major and Minor Arcs}\
Next we
explain
the decomposition \eqref{eq:RNcMcE}. Let $M$ be a parameter controlling the depth of approximation in Dirichlet's theorem:
for any irrational $\gt\in[0,1]$, there exists some $q<M$ and $(r,q)=1$ so that $|\gt-r/q|<1/(qM)$. We will eventually set
\be\label{eq:Mis}
M=XT,
\ee
see \eqref{eq:nmRest}
where this value is used. (Note that $M$ is a bit bigger than $N^{1/2}=XT^{1/2}$.)
Writing $\gt=r/q+\gb$, we introduce parameters
\be\label{eq:Q0K0intro}
Q_{0}, K_{0},
\ee
small powers of $N$ as determined in \eqref{eq:Q0is
, so that the ``major arcs'' correspond to $q<Q_{0}$ and $|\gb|<K_{0}/N$.
In fact, we need a smooth version of this decomposition.
To this end, recall the ``hat'' function and its Fourier transform
\be\label{eq:hatFunc}
\ft(x):=\min(1+x,1-x)^{+},\qquad
\hat\ft(y)=\left({\sin(\pi y)\over \pi y}\right)^{2}.
\ee
Localize $\ft$ to the width $K_{0}/N$, periodize it to the circle, and put this spike on each fraction in the major arcs:
\be\label{eq:hatFuncN}
\fT(\gt)
=
\fT_{N,Q_{0},K_{0}}(\gt):=\sum_{q<Q_{0}}\sum_{(r,q)=1}\sum_{m\in\Z}\ft\left({N\over K_{0}}\left(\gt+m-\frac rq\right)\right)
.
\ee
By construction, $\fT$ lives on the circle $\R/\Z$ and is supported within $K_{0}/N$ of fractions $r/q$ with small denominator, $q<Q_{0}$, as desired.
Then define the ``main term''
\be\label{eq:cMNUdef}
\cM_{N}^{U}(n):=
\int_{0}^{1}
\fT(\gt)
\widehat{
\cR_{N}^{U}}
(\gt)
e(-n\gt)
d\gt
,
\ee
and ``error term''
\be\label{eq:cENUdef}
\cE_{N}^{U}(n):=
\int_{0}^{1}
(1-\fT(\gt))
\widehat{
\cR_{N}^{U}}
(\gt)
e(-n\gt)
d\gt
,
\ee
so that \eqref{eq:RNcMcE} obviously holds.
Since $\cR_{N}^{U}$ could be negative, the same holds for $\cM_{N}^{U}$. Hence we will establish \eqref{eq:cMNUbnd} by first proving a related result for
\be\label{eq:cMNdef}
\cM_{N}(n):=
\int_{0}^{1}
\fT(\gt)
\widehat{
\cR_{N}}
(\gt)
e(-n\gt)
d\gt
,
\ee
and then showing that $\cM_{N}$ and $\cM_{N}^{U}$ cannot differ by too much for too many values of $n$. This is the same (but in reverse) as the transfer from $\cR_{N}$ to $\cR_{N}^{U}$ in \eqref{eq:nZbnd}. See Theorem \ref{thm:cMNis} for the lower bound on $\cM_{N}$, and Theorem \ref{thm:cMNtoMNU} for the transfer.
To prove \eqref{eq:cEN2bnd}, we apply Parseval and decompose dyadically:
\beann
\sum_{n}|\cE_{N}^{U}(n)|^{2}
&=&
\int_{0}^{1}
|1-\fT(\gt)|^{2}
\left|
\widehat{\cR_{N}^{U}}(\gt)
\right|^{2}
d\gt
\\
&\ll&
\cI_{Q_{0},K_{0}}
+
\cI_{Q_{0}}
+
\sum_{Q_{0}\le Q<M\atop \text{dyadic}}
\cI_{Q}
,
\eeann
where we have dissected the circle into the following regions (using that $|1-\ft(x)|=|x|$ on $[-1,1]$):
\bea
\label{eq:IQ0K0}
\cI_{Q_{0},K_{0}}
&:=&
\int\limits_{\gt=\frac rq+\gb \atop q<Q_{0},(r,q)=1,|\gb|<K_{0}/N}
\left|\gb\frac N{K_{0}}\right|^{2}
\left|
\widehat{\cR_{N}^{U}}(\gt)
\right|^{2}
d\gt
,
\\
\label{eq:IQ0}
\cI_{Q_{0}}
&:=&
\int\limits_{\gt=\frac rq+\gb \atop q<Q_{0},(r,q)=1,K_{0}/N<|\gb|<1/(qM)}
\left|
\widehat{\cR_{N}^{U}}(\gt)
\right|^{2}
d\gt
,
\\
\label{eq:IQdef}
\cI_{Q}
&:=&
\int\limits_{\gt=\frac rq+\gb \atop Q\le q<2Q,(r,q)=1,|\gb|<1/(qM)}
\left|
\widehat{\cR_{N}^{U}}(\gt)
\right|^{2}
d\gt
.
\eea
Bounds of the quality \eqref{eq:cEN2bnd} are given for \eqref{eq:IQ0K0} and \eqref{eq:IQ0} in \S\ref{sec:qQ0}, see Theorem \ref{thm:IQ0}. Our estimation of \eqref{eq:IQdef} decomposes further into two cases, whether $Q<X$ or $X\le Q<M$, and are handled separately in \S\ref{sec:QX} and \S\ref{sec:QXT}; see Theorems \ref{thm:IQX} and \ref{thm:IQM}, respectively.
We point out again that our averaging on $n$ in the minor arcs makes this quite crude as far as individual $n$'s (the subject of Conjecture \ref{conj}) are concerned.
\subsection{The Rest of the Paper}\
The only section not yet described is \S\ref{sec:lems}, where we furnish some lemmata which are useful in the sequel. These decompose into two categories: one set of lemmata is related to
some infinite-volume counting problems, for which the background in \S\ref{sec:preII} is indispensable.
The other lemma is of a classical flavor, corresponding to a local analysis for the shifted binary form $\ff$; this studies a certain exponential sum which is dealt with via Gauss and Kloosterman/Sali\'e sums.
\\
This completes our outline of the rest of the paper.
\newpage
\section{Preliminaries II: Automorphic Forms and Representations}\label{sec:preII}
\begin{figure}
\begin{center}
\vskip-.3in
\includegraphics[width=3in]{Orbit11.pdf}
\end{center}
\vskip-.3in
\caption{The orbit of a point in hyperbolic space under the Apollonian group.}
\label{fig:lim}
\end{figure}
\subsection{Spectral Theory}\
Recall the general spectral theory in our present context.
We abuse notation (in this
section only), passing from $G=\SO_{F}(\R)$ to its spin double cover $G=\SL(2,\C)$. Let $\G<G$ be a geometrically finite discrete group. (The Apollonian group is such, being a Schottky group, see Fig. \ref{fig:lim}.) Then $\G$ acts discontinuously on the upper half space $\bH^{3}$, and any $\G$ orbit has a limit set $\gL_{\G}$ in the boundary $\dd\bH^{3}\cong
S^{2}
$ of some Hausdorff dimension $\gd=\gd(\G)\in[0,2]$. We assume that $\G$ is non-elementary (not virtually abelian), so $\gd>0$, and moreover that $\G$ is not a lattice,
that is, the quotient $\G\bk\bH^{3}$ has infinite hyperbolic volume;
then $\gd<2$.
The hyperbolic Laplacian $\gD$ acts on the space $L^{2}(\G\bk \bH^{3})$ of functions automorphic under $\G$ and square integrable on the quotient; we choose the Laplacian to be positive definite. The spectrum is controlled via the following, see \cite{Patterson1976, Sullivan1984, LaxPhillips1982}.
\begin{thm}[Patterson, Sullivan, Lax-Phillips]
The spectrum above $1$ is purely continuous, and the spectrum below $1$ is purely discrete. The latter is empty unless $\gd>1$, in which case, ordering the eigenvalues by
\be\label{eq:gls}
0<\gl_{0}<\gl_{1}
\le
\cdots\le
\gl_{max}<1,
\ee
the base eigenvalue $\gl_{0}$ is given by
$$
\gl_{0}=\gd(2-\gd).
$$
\end{thm}
\begin{rmk}
In our application to the Apollonian group, the limit set is precisely the underlying gasket, see Fig. \ref{fig:lim}. It has dimension
\be\label{eq:dim}
\gd\approx 1.3...>1.
\ee
\end{rmk}
Corresponding to $\gl_{0}$ is the Patterson-Sullivan base eigenfunction,
$\varphi_{0}$, which can be realized explicitly as the integral of a Poisson kernel against the so-called Patterson-Sullivan measure $\mu$.
Roughly speaking, $\mu$ is the weak$^{*}$ limit as $s\to\gd^{+}$ of the measures
\be\label{eq:muPS}
\mu_{s}(x):=
{
\sum_{\g\in\G}\exp({-s\,d(\fo,\g\cdot\fo)){\bf 1}_{x=\g\fo}}
\over
\sum_{\g\in\G}\exp({-s\,d(\fo,\g\cdot\fo))}
}
,
\ee
where $d(\cdot,\cdot)$ is the hyperbolic distance, and $\fo$ is any fixed point in $\bH^{3}$.
\begin{comment}
The measure $\mu$ is non-atomic \cite{Sullivan1984}; in particular if $\Phi$ is a sector in $S^{2}$ of radius $r$ which intersects the limit set $\gL_{\G}$ nontrivially and has boundary $\dd\Phi$ disjoint from $\gL_{\G}$, then
\be\label{eq:muBnd}
\mu(\Phi)\gg r^{\gd}.
\ee
\end{comment}
\\
\subsection{Spectral Gap}\label{sec:specGap}\
We assume henceforth that $\G$
moreover
satsifies
$\G<\SL(2,
\cO
)$, where $\cO=\Z[i]$. Then we have a tower of congruence subgroups: for any integer $q\ge1$, define $\G(q)$ to be the kernel of the projection map $\G\to \SL(2,
\cO
/\fq)$, with $\fq=(q)$ the principal ideal. As in \eqref{eq:gls}, write
\be\label{eq:glQs}
0<\gl_{0}(q)<\gl_{1}(q)
\le
\cdots\le
\gl_{max(q)}(q)<1,
\ee
for the discrete spectrum of $\G(q)\bk\bH^{3}$. The groups $\G(q)$, while of infinite covolume, have finite index in $\G$, and hence
\be\label{eq:gl0Qgl0}
\gl_{0}(q)=\gl_{0}=\gd(2-\gd).
\ee
But the second eigenvalues $\gl_{1}(q)$ could {\it a priori} encroach on the base. The fact that this does not happen is the spectral gap property for $\G$.
\begin{thm
\label{thm:specGap}
Given $\G$ as above, there exists some $\vep=\vep(\G)>0$ such that for all $q\ge1$,
\be\label{eq:specGap}
\gl_{1}(q)\ge \gl_{0}+\vep.
\ee
\end{thm}
This is proved in the Appendix by P\'eter Varj\'u.
\subsection{Representation Theory and Mixing Rates}\
By the Duality Theorem of Gelfand, Graev, and Piatetski-Shapiro \cite{GelfandGraevPS1966}, the spectral decomposition above is equivalent to the decomposition into irreducibles of the right regular representation
acting on $L^{2}(\G\bk G)$. That is, we identify $\bH^{3}\cong G/K$, with $K=\SU(2)$ a maximal compact subgroup, and lift functions from $\bH^{3}$ to (right $K$-invariant) functions on $G$. Corresponding to \eqref{eq:gls} is the decomposition
\be\label{eq:RepDecomp}
L^{2}(\G\bk G)=
V_{\gl_{0}}\oplus
V_{\gl_{1}}\oplus
\cdots
\oplus
V_{\gl_{max}}\oplus
V_{temp}
.
\ee
Here $V_{temp}$ contains the tempered spectrum (for $\SL_{2}(\C)$, every non-spherical irreducible representation is tempered), and each $V_{\gl_{j}}$ is an infinite dimensional vector space, isomorphic as a $G$-representation to a complementary series representation with parameter $s_{j}\in(1,2)$ determined by $\gl_{j}=s_{j}(2-s_{j})$.
Obviously, a similar decomposition holds for $L^{2}(\G(q)\bk G)$, corresponding to \eqref{eq:glQs}.
We also have the following well-known general fact about mixing rates of matrix coefficients, see e.g. \cite{CowlingHaagerupHowe1988}.
First we recall the relevant Sobolev norm. Let $(\pi,V)$ be a unitary $G$-representation, and
let $\{X_{j}\}$ denote an orthonormal basis of the Lie algebra $\fk$ of $K$ with respect to an $Ad$-invariant scalar product. For a smooth vector $v\in V^{\infty}$, define
the (second order) Sobolev norm $\cS$ of $v$ by
$$
\cS v
:=
\|v\|_{2}
+\sum_{j} \|d\pi(X_{j}).v\|_{2}
+\sum_{j} \sum_{j'} \|d\pi(X_{j})d\pi(X_{j'}).v\|_{2}
.
$$
\begin{thm}[{\cite[Prop. 5.3]{KontorovichOh2011}}]
Let $\gT>1$ and $(\pi,V)$ be a unitary representation of $G$ which does not weakly contain any complementary series representation with parameter $s>\gT$. Then for any smooth vectors $v,w\in V^{\infty}$,
\be\label{eq:decayMtrx}
\left|
\<\pi(g).v,w\>
\right|
\ll
\|g\|^{-2(2-\gT)}
\cdot
\cS v\cdot
\cS w
.
\ee
Here $\|\cdot\|$ is the standard Frobenius matrix norm.
\\
\end{thm}
\subsection{Effective Bisector Counting}\
The next ingredient which we require is the recent work by Vinogradov \cite{Vinogradov2013} on effective bisector counting for such infinite volume quotients. Recall the following su
(semi)groups
of $G$:
$$
A=\left\{a_{t}:=\mattwo{e^{t/2}}{}{}{e^{-t/2}}:t\in\R
\right\},
A^{+}=\left\{a_{t}:t\ge0
\right\},
$$
$$
M=\left\{\mattwo{e^{2\pi i\gt}}{}{}{e^{-2\pi i\gt}}:\gt\in\R/\Z
\right\},
K=\SU(2)
.
$$
We have the Cartan decomposition $G=KA^{+}K$, unique up to the normalizer $M$ of $A$ in $K$. We require it in the following more precise form. Identify $K/M$ with the sphere $S^{2}\cong\dd\bH^{3}.$ Then for every $g\in G$ not in $K$, there is a unique decomposition
\be\label{eq:gDecomp}
g=s_{1}(g)\cdot a(g)\cdot m(g) \cdot s_{2}(g)^{-1}.
\ee
with $s_{1},s_{2}\in K/M$, $a\in A^{+}$ and $m\in M$, corresponding to
$$
G=K/M\times A^{+}\times M\times M\bk K,
$$
see, e.g., \cite[(3.4)]{Vinogradov2013}. The following theorem follows easily
from \cite[Thm 2.2]{Vinogradov2013}.
\begin{thm}[{\cite{Vinogradov2013}}]\label{thm:Vin}
Let $\Phi,\Psi\subset S^{2}$ be
spherical caps
and let $\cI\subset\R/\Z$ be an interval. Then under the above hypotheses on $\G$ (in particular $\gd>1$), and using the decomposition
\eqref{eq:gDecomp},
we have
\be\label{eq:Vin}
\sum_{\g\in\G}
\bo
{
\left\{
\begin{array}{c}
s_{1}(\g)\in\Phi
\\
s_{2}(\g)\in \Psi
\\
\|a(\g)\|^{2}<T
\\
m(\g)\in\cI
\end{array}
\right\}
}
=
c_{\gd}\cdot
\mu(\Phi)
\mu(\Psi)
\ell(\cI)
T^{\gd}
+
O\big(
T^{
\gT
}
\big)
,
\ee
as $T\to\infty$.
Here $c_{\gd}>0$, $\|\cdot\|$ is the Frobenius norm, $\ell$ is Lebesgue measure, $\mu$ is Patterson-Sullivan measure (cf. \eqref{eq:muPS}),
and
\be\label{eq:ThIs}
\gT<\gd
\ee
depends only on the spectral gap for $\G$. The implied constant does not depend on $\Phi,\Psi,$ or $\cI$.
\end{thm}
This generalizes from $\SL(2,\R)$ to $\SL(2,\C)$ the main result of \cite{BourgainKontorovichSarnak2010}, which is itself a
generalization (with weaker exponents) to
our infinite volume setting
of \cite[Thm 4]{GoodBook}.
\newpage
\section{Some Lemmata}\label{sec:lems}
\subsection{Infinite Volume Counting Statements}\label{sec:count}\
Equipped with the tools of
\S\ref{sec:preII}, we isolate here some consequences
which
will be needed in the sequel.
We return to the notation $G=\SO_{F}$, with $F$ the Descartes
form
\eqref{eq:Fdef},
$\G=\cA\cap G$, the orientation preserving Apollonian subgroup,
and $\G(q)$ its principal congruence subgroups.
Moreover,
we import all the notation from the previous section.
First we use the spectral gap to see that summing over a coset of a congruence group can be reduced to summing over the original group.
\begin{lem}\label{lem:spec0}
Fix $\g_{1}\in\G$, $q\ge1$, and any ``congruence'' group $\tilde\G(q)$ satisfying
\be\label{eq:G1G}
\G(q)<\tilde\G(q)<\G.
\ee
Then as $Y\to\infty$,
\bea
\label{eq:lemLHS}
&&
\hskip-.5in
\#\{
\g\in\tilde\G(q)
:
\|\g_{1}\g\|<Y
\}
\\
\label{eq:lemRHS}
&&
=
{1\over [\G:\tilde\G(q)]}
\cdot
\#\{
\g\in\G
:
\|\g\|<Y
\}
+
O(Y^{\gT_{0}})
,
\eea
where $\gT_{0}<\gd$ depends only on the spectral gap for $\G$. The implied constant above
does not depend on $q$ or $\g_{1}$.
The same holds with $\g_{1}\g$ in \eqref{eq:lemLHS} replaced by $\g\g_{1}$.
\end{lem}
This simple lemma follows from a more-or-less standard argument. We give a sketch below, since a slightly more complicated result will be needed later, cf. Lemma \ref{lem:spec1}, but with essentially no new ideas. After proving the lemma below, we will use the argument as a template for the more complicated statement.
\pf[Sketch of Proof]\
Denote the left hand side \eqref{eq:lemLHS} by $\cN_{q}$, and let $\cN_{1}/[\G:\tilde\G(q)]$ be the first term of \eqref{eq:lemRHS}. For $g\in G$, let
\be\label{eq:fgIs}
f(g)=f_{Y}(g):=\bo_{\{\|g\|<Y\}},
\ee
and define
\be\label{eq:FqIs}
F_{q}(g,h):=
\sum_{\g\in\tilde\G(q)}
f(g^{-1}\g h),
\ee
so that
\be\label{eq:cNq}
\cN_{q}=F_{q}(\g_{1}^{-1},e).
\ee
By construction, $F_{q}$ is a function on $\tilde\G(q)\bk G\times \tilde\G(q)\bk G$, and we smooth $F_{q}$ in
both
copies of $\tilde\G(q)\bk G$, as follows.
Let $\psi\ge0$ be a smooth bump function supported in a ball of radius $\eta>0$ (to be chosen later) about the origin in $G$ with $\int_{G}\psi=1$, and
automorphize it to
$$
\Psi_{q}(g):=\sum_{\g\in\tilde\G(q)}\psi(\g g)
.
$$
Then clearly $\Psi_{q}$ is a bump function in $\tilde\G(q)\bk G$ with $\int_{\tilde\G(q)\bk G}\Psi_{q}=1$.
Let
$$
\Psi_{q,\g_{1}}(g):=\Psi_{q}(g\g_{1})
.
$$
Smooth the variables $g$ and $h$ in $F_{q}$ by
considering
\beann
\cH_{q}
&:=&
\<F_{q},\Psi_{q,\g_{1}}\otimes\Psi_{q}\>
=
\int_{\tilde\G(q)\bk G}
\int_{\tilde\G(q)\bk G}
F_{q}(g,h)\Psi_{q,\g_{1}}(g)\Psi_{q}(h) dg\, dh
\\
&=&
\sum_{\g\in\tilde\G(q)}
\int_{\tilde\G(q)\bk G}
\int_{\tilde\G(q)\bk G}
f(\g_{1}g^{-1}\g h)\Psi_{q}(g)\Psi_{q}(h) dg\, dh
.
\eeann
First we estimate the error from smoothing:
\beann
\cE
&=&
|\cN_{q}-\cH_{q}|
\\
&\le&
\sum_{\g\in\G}
\int_{\tilde\G(q)\bk G}
\int_{\tilde\G(q)\bk G}
|f(\g_{1}g^{-1}\g h)
-f(\g_{1}\g)
|
\Psi_{q}(g)\Psi_{q}(h) dg\, dh
,
\eeann
where we have increased $\g$ to run over all of $\G$. The analysis splits into three ranges.
\begin{enumerate}
\item
If $\g$ is such that
\be\label{eq:reg1Lem}
\|\g_{1}\
\|>Y(1+10\eta),
\ee
then both $f(\g_{1}g^{-1}\g h)$ and $f(\g_{1}\g)$ vanish.
\item
In the range
\be\label{eq:reg2Lem}
\|\g_{1}\g\|<Y(1-10\eta),
\ee
both $f(\g_{1}g^{-1}\g h)$ and $f(\g_{1}\g)$ are $1$, so their difference vanishes.
\item In the intermediate range, we
apply \cite{LaxPhillips1982}, bounding the count by
\be\label{eq:reg3Lem}
\ll Y^{\gd}\eta + Y^{\gd-\vep},
\ee
where $\vep>0$ depends on the spectral gap for $\G$.
\end{enumerate}
Thus it remains to analyze $\cH_{q}$.
Use a simple change of variables (see \cite[Lemma 3.7]{BourgainKontorovichSarnak2010}) to express $\cH_{q}$ via matrix coefficients:
$$
\cH_{q}
=
\int_{G}
f(g)
\<
\pi(g)
\Psi_{q},
\Psi_{q,\g_{1}}
\>_{\tilde\G(q)\bk G}
dg.
$$
Decompose the matrix coefficient into its projection onto the base irreducible $V_{\gl_{0}}$ in
\eqref{eq:RepDecomp} and an orthogonal term, and bound the remainder by the mixing rate \eqref{eq:decayMtrx} using the uniform spectral gap $\vep>0$ in \eqref{eq:specGap}. The functions $\psi$ are bump functions in six real dimensions, so can be chosen
to have second-order Sobolev norms bounded by $\ll \eta^{-5}$.
Of course the projection onto the base representation is just $[\G:\tilde\G(q)]^{-1}$ times the same projection at level one, cf. \eqref{eq:gl0Qgl0}.
Running the above argument in reverse at level one (see \cite[Prop. 4.18]{BourgainKontorovichSarnak2010}) gives:
\be\label{eq:NqErr}
\cN_{q}=
{1\over[\G:\tilde\G(q)]}\cdot \cN_{1}
+
O(\eta Y^{\gd}+Y^{\gd-\vep})
+
O(Y^{\gd-\vep} \eta^{-10})
.
\ee
Optimizing $\eta$ and renaming $\gT_{0}<\gd$ in terms of the spectral gap $\vep$ gives
the claim.
\epf
Next we exploit the previous lemma and the product structure of the family $\fF$
in \eqref{eq:fFdef}
to save a small power of $q$ in the following modular restriction. Such a bound is needed at several places in \S\ref{sec:QX}.
\begin{lem}\label{lem:spec3}
Let $\gT_{0}$ be as in \eqref{eq:lemRHS}. Define $\cC$ in \eqref{eq:TT1T2} by
\be\label{eq:cCis}
\cC:={10^{30}\over \gd-\gT_{0}},
\ee
hence determining $T_{1}$ and $T_{2}$.
There exists some $\eta_{0}>0$ depending only on the spectral gap of $\G$ so that
for any $1\le q<N$ and any $r(\mod q)$,
\be\label{eq:spec3}
\sum_
{\g\in\fF}
\bo_{\{
\<e_{1},\g v_{0}\>
\equiv r(\mod q)\}}
\ll
{1\over q^{\eta_{0}}}
T^{\gd}
.
\ee
The implied constant is independent of $r$.
\end{lem}
\pf
Dropping the condition $\<e_{1},\g_{1}\,\g_{2}v_{0}\>>T/100$ in \eqref{eq:fFdef}, bound the left hand side of \eqref{eq:spec3} by
\be\label{eq:1}
\sum_
{\g_{1}\in\G\atop\|\g_{1}\|\asymp T_{1}}
\sum_
{\g_{2}\in\G\atop\|\g_{2}\|\asymp T_{2}}
\bo_{\{
\<e_{1},\g_{1}\g_{2} v_{0}\>
\equiv r(\mod q)\}}
\ee
We decompose the argument into two ranges of $q$.
{\bf Case 1: $q$ small.}
In this range, we fix $\g_{1}$, and follow a standard argument for $\g_{2}$.
Let $\tilde\G(q)<\G$ denote the stabilizer of $v_{0} (\mod q)$, that is
\be\label{eq:G0q}
\tilde\G(q):=\{\g\in\G:\g v_{0}\equiv v_{0}(\mod q)\}.
\ee
Clearly \eqref{eq:G1G} is satisfied, and it is elementary that
\be\label{eq:G0qBnd}
[\G:\tilde\G(q)]\asymp q^{2},
\ee
cf. \eqref{eq:Fuc}.
Decompose $\g_{2}=\g_{2}'\g_{2}''$ with $\g_{2}''\in\tilde\G(q)$ and $\g_{2}'\in \G/\tilde\G(q)$.
Then by \eqref{eq:lemRHS} and \cite{LaxPhillips1982}, we have
\beann
\eqref{eq:1}
&=&
\sum_
{\g_{1}\in\G\atop\|\g_{1}\|\asymp T_{1}}
\sum_{\g_{2}'\in\G/\tilde\G(q)}
\bo_{\{
\<e_{1},\g_{1}\g_{2}' v_{0}\>
\equiv r(\mod q)\}}
\sum_
{\g_{2}''\in\tilde\G(q)\atop\|\g_{2}'\g_{2}''\|\asymp T_{2}}
1
\\
&\ll&
T_{1}^{\gd}\
q\
\left(
\frac1{q^{2}}\
T_{2}^{\gd}
+
T_{2}^{\gT_{0}}
\right)
.
\eeann
Hence we have saved a whole power of $q$, as long as
\be\label{eq:q1range}
q<T_{2}^{(\gd-\gT_{0})/2}
.
\ee
{\bf Case 2: $q\ge T_{2}^{{\gd-\gT_{0}\over 2}}$.}
Then by \eqref{eq:cCis} and \eqref{eq:TT1T2}, $q$ is actually a
very large
power of $T_{1}$,
\be\label{eq:qToT1}
q\ge T_{1}^{10^{29}}
.
\ee
In this range,
we exploit Hilbert's Nullstellensatz and effective versions of Bezout's theorem;
see a related argument in \cite[Proof of Prop. 4.1]{BourgainGamburd2009}.
Fixing $\g_{2}$ in \eqref{eq:1} (with $\ll T_{2}^{\gd}$ choices), we set
$$
v:=\g_{2}v_{0},
$$
and play now with $\g_{1}$. Let $S$ be the set of $\g_{1}$'s in question (and we now drop the subscript $1$):
$$
S=S_{v,q}(T_{1}):=\{\g\in\G:\|\g\|\asymp T_{1}, \<e_{1},\g v\>\equiv r(\mod q)\}.
$$
This congruence restriction is to a modulus much bigger than the parameter, so we
{\bf Claim:} There is an integer vector
v_{*}\neq0$ and an integer $z_{*}$ such that
\be\label{eq:wStar}
\<e_{1}
,
\g v_{*}\>=z_{*}
\ee
holds for all $\g\in S$. That is, the modular condition can be lifted to an exact equality.
First we assume the Claim and complete the proof of \eqref{eq:spec3}. Let $q_{0}$ be a
prime
of size $\asymp T_{1}^{(\gd-\gT_{0})/2}$, say, such that $v_{*}\not\equiv0(\mod q_{0})$; then
\beann
|S|
&\ll&
\#
\{
\|\g_{1}\|<T_{1}:\<e_{1
,\g v_{*}\>\equiv z_{*}(\mod q_{0})
\}
\\
&\ll&
q_{0}\left(
\frac1{q_{0}^{2}}
T_{1}^{\gd}
+
T_{1}^{\gT_{0}}
\right)
\ll
\frac1{q_{0}}
T_{1}^{\gd}
,
\eeann
by the argument in Case 1.
Recall we assumed that $q<N$. Since $q_{0}$ above is a small power of $N$,
the above saves a tiny power of $q$, as desired.
\\
It remains to establish the Claim.
For each $\g\in S$, consider the condition
$$
\<e_{1},\g\, v\>
=
\sum_{1\le j\le 4}\g_{1,j}\, v_{j}\equiv r(\mod q).
$$
First massage the equation into one with no trivial solutions.
Since $v$ is a primitive vector, after a linear change of variables we may assume that $(v_{1},q)=1$.
Then multiply through by $\bar v_{1}$, where $v_{1}\bar v_{1}\equiv 1(\mod q)$, getting
\be\label{eq:3}
\g_{1,1}
+
\sum_{2\le j\le 4}\g_{1,j}\, v_{j}\bar v_{1}\equiv r \bar v_{1}(\mod q).
\ee
Now, for variables $V=(V_{2},V_{3},V_{4})$ and $Z$, and each $\g\in S$,
consider the (linear) polynomials $P_{\g}\in\Z[V,Z]$:
$$
P_{\g}(V,Z):=
\g_{1,1}
+
\sum_{2\le j\le 4}\g_{1,j}\, V_{j}-Z,
$$
and the affine variety
$$
\cV:=\bigcap_{\g\in S}\{P_{\g}=0\}.
$$
If this variety
$\cV(\C)$
is non-empty,
then there is clearly a rational solution,
$(V^{*},Z^{*})\in\cV(\Q)$.
Hence
we have found a rational solution to \eqref{eq:wStar}, namely $v^{*}=(1,V_{2}^{*},V_{3}^{*},V_{4}^{*})\neq0$ and $z^{*}=Z^{*}$. Since \eqref{eq:wStar} is
homogeneous, we may clear denominators, getting an integral solution, $v_{*},z_{*}$.
Thus we henceforth assume by contradiction that the variety $\cV(\C)$ is empty.
Then by Hilbert's Nullstellensatz, there are polynomials $Q_{\g}\in\Z[V,Z]$ and an integer $\fd\ge1$ so that
\be\label{eq:2}
\sum_{\g\in S}P_{\g}(V,Z)\ Q_{\g}(V,Z)=\fd,
\ee
for all $(V,Z)\in\C^{4}$. Moreover,
Hermann's method \cite{Hermann1926} (see \cite[Theorem IV]{MasserWustholz1983})
gives effective bounds on the heights of $Q_{\g}$ and $\fd$ in the above Bezout equation. Recall the height
of a polynomial is the
logarithm of its largest coefficient (in absolute value); thus the polynomials $P_{\g}$ are linear in four variables with
height
\le\log T_{1}$.
Then $Q_{\g}$ and $\fd$ can be found so that
\be\label{eq:fdIs}
\fd\le e^{8^{4\cdot 2^{4-1}-1}(\log T_{1}+8\log 8)}
\ll
T_{1}^{10^{28}}
.
\ee
(Much better bounds are known, see e.g.
\cite[Theorem 5.1]{BerensteinYger1991}, but these suffice for our purposes.)
On the other hand, reducing \eqref{eq:2} modulo $q$ and evaluating at
$$
V_{0}=(
v_{2}\bar v_{1},
v_{3}\bar v_{1},
v_{4}\bar v_{1}),
\qquad
Z_{0}=r\bar v_{1},
$$
we have
$$
\sum_{\g\in S}P_{\g}(V_{0},Z_{0}) Q_{\g}(V_{0},Z_{0})\equiv0\equiv\fd(\mod q),
$$
by \eqref{eq:3}. But then since $\fd\ge1$, we in fact have $\fd\ge q$, which is incompatible with \eqref{eq:fdIs} and \eqref{eq:qToT1}.
This furnishes our desired contradiction, completing the proof.
\epf
Next we need a slight generalization of Lemma \ref{lem:spec0}, which will be used in the major arcs analysis, see \eqref{eq:fMdecomp}.
\begin{lem}\label{lem:spec1}
Let $1<K\le T_{2}^{1/10}$, fix $|\gb|<K/N$, and fix $x,y\asymp X$. Then for any $\g_{0}\in\G$,
any $q\ge1$, and any group $\tilde\G(q)$ satisfying \eqref{eq:G1G}, we have
\bea
\nonumber
\sum_
{\g\in\fF\cap\{ \g_{0}\tilde\G(q)\}}
e
\bigg(
\gb\,
\ff_{\g}(2x,y)
\bigg)
&=&
{1\over [\G:\tilde\G(q)]}
\sum_
{\g\in\fF}
e
\bigg(
\gb\,
\ff_{\g}(2x,y)
\bigg)
\\
\label{eq:spec1}
&&
\hskip1in
+
O(
T^{\gT}
K)
,
\eea
where $\gT<\gd$ depends only on the spectral gap for $\G$, and the implied constant
does
not
depend
on
$q$, $\g_{0}$, $\gb$, $x$ or $y$.
\end{lem}
\pf
The proof follows with minor changes that of Lemma \ref{lem:spec0}, so we give a sketch; see also \cite[\S4]{BourgainKontorovichSarnak2010}.
According to the construction \eqref{eq:fFdef} of $\fF$, the $\g$'s in question satisfy $\g=\g_{1}\g_{2}\in\g_{0}\tilde\G(q)$, and hence we can write
$$
\g_{2}=\g_{1}^{-1}\g_{0}\g_{2}',
$$
with $\g_{2}'\in\tilde\G(q)$. Then $\g_{2}'=\g_{0}^{-1}\g_{1}\g_{2}$, and using \eqref{eq:ffToInProd}, we can write the left hand side of \eqref{eq:spec1} as
$$
\sum_{\g_{1}\in\G\atop T_{1}<\|\g_{1}\|<2T_{1}}
\sum_{\g_{2}'\in\tilde\G(q)\atop T_{2}<\|\g_{1}^{-1}\g_{0}\g_{2}'\|<2T_{2}}
\bo_{\{\<e_{1},\g_{0}\g_{2}'\,v_{0}\>>T/100\}}\
e
\bigg(
\gb\,
\<w_{x,y},
\g_{0}\g_{2}'\,v_{0}\>
\bigg)
.
$$
Now we fix $\g_{1}$ and mimic the proof of Lemma \ref{lem:spec0} in $\g_{2}'$.
Replace \eqref{eq:fgIs} by
$$
f(g):=
\bo_{\{T_{2}<\|\g_{1}^{-1}g\|<2T_{2}\}}
\bo_{\{\<e_{1},g\,v_{0}\>>T/100\}}\
e
\bigg(
\gb\,
\<w_{x,y},
g\,v_{0}\>
\bigg)
.
$$
Then \eqref{eq:FqIs}-\eqref{eq:reg1Lem} remains essentially unchanged, save cosmetic changes such as replacing \eqref{eq:cNq} by
$F_{q}(\g_{1}\g_{0}^{-1},e)$. Then in the estimation of the difference $|\cN_{q}-\cH_{q}|$ by splitting the sum on $\g_{2}'$ into ranges, the argument now proceeds as follows.
\begin{enumerate}
\item
The range \eqref{eq:reg1Lem} should be replaced by
$$
\|\g_{1}\g_{0}^{-1}\g_{2}'\|<T_{2}(1-10\eta),\text{ or }
\|\g_{1}\g_{0}^{-1}\g_{2}'\|>2T_{2}(1+10\eta),
$$
$$
\text{ or }\<e_{1},\g_{1}\g_{0}^{-1}\g_{2}'\,v_{0}\><\frac T{100}(1-10\eta).
$$
\item
The range \eqref{eq:reg2Lem} should be replaced by the
range
$$
T_{2}(1+10\eta)<\|\g_{1}\g_{0}^{-1}\g_{2}'\|<2T_{2}(1-10\eta),
\text{ and }\<e_{1},\g_{1}\g_{0}^{-1}\g_{2}'\,v_{0}\>>\frac T{100}(1+10\eta),
$$
in which $f$ is differentiable.
Here instead of the difference $|f(\g_{1}\g_{0}^{-1}g\g_{2}'h)-f(\g_{1}\g_{0}^{-1}\g_{2}')|$ vanishing, it is now bounded by
$$
\ll \eta K,
$$
for a net contribution to the error of $\ll \eta K T^{\gd}$.
\item
In the remaining range, \eqref{eq:reg3Lem} remains unchanged, using $|f|\le1$.
\end{enumerate}
The error in \eqref{eq:NqErr} is then replaced by
$$
O(\eta\, K\, T_{2}^{\gd} + T_{2}^{\gd-\vep}\eta^{-10}).
$$
Optimizing $\eta$ and renaming $\gT$ gives the bound $O(T_{2}^{\gT}K^{10/11})$, which is better than claimed in the power of $K$. Rename $\gT$ once more using \eqref{eq:TT1T2} and \eqref{eq:cCis}, giving \eqref{eq:spec1}.
\epf
The following is our last counting lemma, showing a certain equidistribution among the values of $\ff_{\g}(2x,y)$ at the scale $N/K$. This bound is used in the major arcs, see
the proof of Theorem \ref{thm:cMNis}.
\begin{lem}
\label{lem:spec2}
Fix $N/2<n<N$, $1<K\le T_{2}^{1/10}$, and $x,y\asymp X$. Then
\be\label{eq:spec2}
\sum_
{\g\in
\f
}
\bo_{
\big\{
|
\ff_{\g}(2x,y)-
n|
<
\frac N{K}
\big\}
}
\gg
{T^{\gd}\over K}
+
T^{\gT}
,
\ee
where $\gT<\gd$ only depends on the spectral gap for $\G$. The implied constant is independent of $x,y,$ and $n$.
\end{lem}
\pf[Sketch]
The proof is an
explicit calculation nearly identical to the one given in \cite[\S5]{BourgainKontorovichSarnak2010};
we give only a sketch here.
Write the left hand side of \eqref{eq:spec2} as
$$
\sum_{\g_{1}\in\G\atop T_{1}<\|\g_{1}\|<2T_{1}}
\sum_{\g_{2}\in\G\atop T_{2}<\|\g_{2}\|<2T_{2}}
\bo_{\{
\<e_{1},\g_{1}\g_{2}v_{0}\>
>
T/100
\}}
\bo_{\{
|\<w_{x,y},\g_{1}\g_{2}v_{0}\>-n|<N/K
\}}
.
$$
Fix $\g_{1}$ and express the condition on $\g_{2}$ as $\g_{2}\in R\subset G$, where $R$ is the region
$$
R=
R_{\g_{1},x,y,n}:=
\left\{
g\in G
:
\begin{array}{c}
T_{2}<\|g\|<2T_{2}
\\
\<\g_{1}^{t}e_{1},g\, v_{0}\>>T/100
\\
|\<\g_{1}^{t}w_{x,y},
g\,
v_{0}
\>
-n|
<
\frac N{K}
\end{array}
\right\}
.
$$
Lift $G=\SO_{F}(\R)$ to its spin cover $\tilde G=\SL_{2}(\C)$ via the map $\iota$ of \eqref{eq:iota}. Let $\tilde R\subset\tilde G$ be the corresponding pullback region, and decompose $\tilde G$ into Cartan $KAK$ coordinates according to \eqref{eq:gDecomp}.
Note that $\iota$ is quadratic in the entries, so, e.g., the condition
\be\label{eq:gToIg}
\text{$\|g\|^{2}\asymp T$ gives $\|\iota(g)\|\asymp T$, }
\ee
explaining the factor $\|a(g)\|^{2}$ appearing in \eqref{eq:Vin}.
Then chop $\tilde R$
into
spherical caps
and apply Theorem \ref{thm:Vin}.
The same argument as in \cite[\S5]{BourgainKontorovichSarnak2010} then leads to \eqref{eq:spec2}, after renaming $\gT$; we suppress the details.
\epf
\newpage
\subsection{Local Analysis Statements}
\
In this subsection, we study a certain exponential sum which arises in a crucial way in our estimates.
Fix $\ff\in\fF$, and write $\ff=f-a$ with
$$
f(x,y)=Ax^{2}+2Bxy+Cy^{2}
$$
according to \eqref{eq:ffVdef}.
Let
$q_{0}\ge1$,
fix $r$ with $(r,q_{0})=1$, and fix $n,m\in\Z$.
(The notation is meant to be consistent with its later use; there will be another parameter $q$, and $q_{0}$ will be a divisor of $q$.)
Define the exponential sum
\be\label{eq:cSfDef}
\cS_{f}(q_{0},r;n,m)
:=
{1\over q_{0}^{2}}
\sum_{k(q_{0})}
\sum_{\ell( q_{0})}
e_{q_{0}}
\bigg(
r
f(k,\ell)
+nk+m\ell
\bigg)
.
\ee
This sum appears naturally in many places in the minor arcs analysis, see e.g. \eqref{eq:cRfuIs} and \eqref{eq:RfuNew}.
Our first lemma is completely standard, see, e.g. \cite[\S12.3]{IwaniecKowalski}.
\begin{lem}
With the above conditions,
\be\label{eq:cSfbnd1}
|
\cS_{f}(q_{0},r;n,m)
|
\le q_{0}^{-1/2}
.
\ee
\end{lem}
\begin{rmk}
Being a sum in two variables, one might expect square-root cancellation in each, giving a savings of $q_{0}^{-1}$; indeed this is what we obtain, modulo some coprimality conditions, see \eqref{eq:SfEval}. For some of our applications, saving just one square-root is plenty, and we can ignore the coprimality; hence the cleaner statement in \eqref{eq:cSfbnd1}.
\end{rmk}
\pf
Write $\cS_{f}$ for $\cS_{f}(q_{0},r;n,m)$.
Note first that
$\cS_{f}$
is multiplicative in $q_{0}$, so we study the case $q_{0}=p^{j}$ is a prime power.
Assume for simplicity $(q_{0},2)=1$; similar calculations are needed to handle the $2$-adic case.
First we re-express $\cS_{f}$ in a
more convenient form.
By
Descartes theorem \eqref{eq:Fv}, primitivity of the
gasket
$\sG$, and \eqref{eq:ABCdef}, we have that $(A,B,C)=1$;
assume henceforth that $(C,q_{0})=1$, say.
Write $\bar x$ for the multiplicative inverse of $x$ (the modulus will be clear from context). Recall throughout that $(r,q_{0})=1$.
Looking at the terms in the summand of $\cS_{f}$, we have
\beann
&&
\hskip-.5in
r
f(k,\ell)
+
nk+m\ell
\quad(\mod q_{0})
\\
&\equiv&
r
(Ak^{2}+2Bk\ell+C\ell^{2})
+
nk+m\ell
\\
&\equiv&
rC(\ell
+B\bar Ck
)^{2}
+
r\bar Ck^{2}
(AC
-
B^{2}
)
+
nk+m\ell
\\
&\equiv&
rC(\ell
+B\bar Ck
)^{2}
+
a^{2}
r\bar C
k^{2}
+
nk+m\ell
\\
&\equiv&
rC(\ell
+B\bar Ck
+
\overline{2rC}
m
)^{2}
-\overline{4rC} m^{2}
+
a^{2}
r\bar C
k^{2}
+
k(n
-
B\bar C m)
,
\eeann
where we used \eqref{eq:disc}. Hence we have
\beann
\cS_{f
&=&
{1\over q_{0}^{2}}
e_{q_{0}}
(
-\overline{4rC} m^{2}
)
\sum_{k( q_{0})}
e_{q_{0}}
\bigg(
a^{2}
r\bar C
k^{2}
+
k(n
-
B\bar C m)
\bigg)
\\
&&
\qquad
\times
\sum_{\ell( q_{0})}
e_{q_{0}}
\bigg(
rC(\ell
+B\bar Ck
+
\overline{2rC}
m
)^{2}
\bigg)
,
\eeann
and the $\ell$ sum is just a classical Gauss sum. It can be evaluated explicitly, see e.g. \cite[eq. (3.38)]{IwaniecKowalski}. Let
$$
\vep_{q_{0}}:=\twocase{}1{if $q_{0}\equiv 1(\mod 4)$}i{if $q_{0}\equiv3(\mod 4)$.}
$$
Then the Gauss sum on $\ell$ is $\vep_{q_{0}}\sqrt q_{0}\left({rC\over q_{0}}\right)$, where $({\cdot\over q_{0}})$ is the Legendre symbol. Thus we have
\beann
\cS_{f
&=&
{\vep_{q_{0}}\over q_{0}^{3/2}}
\left({rC\over q_{0}}\right)
e_{q_{0}}
(
-\overline{4rC} m^{2}
)
\sum_{k( q_{0})}
e_{q_{0}}
\bigg(
a^{2}
r\bar C
k^{2}
+
k(n
-
B\bar C m)
\bigg)
.
\eeann
Let
\be\label{eq:tildQdef}
\tilde q_{0}:=(a^{2},q_{0}), \qquad q_{1}:=q_{0}/\tilde q_{0},\qquad\text{and }\qquad a_{1}:=a^{2}/\tilde q_{0},
\ee
so that $a^{2}/q_{0}=a_{1}/q_{1}$ in lowest terms.
Break the sum on $0\le k< q_{0}$ according to $k= k_{1}+q_{1}\tilde k$,
with $0\le k_{1}< q_{1}$ and $0\le\tilde k< \tilde q_{0}$. Then
\beann
\cS_{f
&=&
{\vep_{q_{0}}\over q_{0}^{3/2}}
\left({rC\over q_{0}}\right)
e_{q_{0}}
(
-\overline{4rC} m^{2}
)
\\
&&
\qquad
\times
\sum_{k_{1}( q_{1})}
e_{q_{1}}
\bigg(
a_{1}
r\bar C
(k_{1})^{2}
\bigg)
e_{q_{0}}
\bigg(
{k_{1}
}
(n
-
B\bar C m)
\bigg)
\\
&&
\qquad
\times
\sum_{\tilde k( \tilde q_{0})}
e_{\tilde q_{0}}
\bigg(
{\tilde k}
(n
-
B\bar C m)
\bigg)
.
\eeann
The last sum vanishes unless $n-B\bar Cm\equiv0(\mod \tilde q_{0})$, in which case it is $\tilde q_{0}$. In the latter case, define $L$ by
\be\label{eq:Ldef}
L:=(Cn-Bm )/ \tilde q_{0} .
\ee
Then we have
\beann
\cS_{f
&=&
\bo_{
nC\equiv mB( \tilde q_{0})
}
{\vep_{q_{0}}\over q_{0}^{3/2}}
\left({rC\over q_{0}}\right)
e_{q_{0}}
(
-\overline{4rC} m^{2}
)
\\
&&
\qquad
\times
e_{q_{1}}
\bigg(
-
\overline{4a_{1}rC}
L^{2}
\bigg)
\left[
\sum_{k_{1}( q_{1})}
e_{q_{1}}
\bigg(
a_{1}
r\bar C
(
k_{1}
+
\overline{2a_{1}r}
L
)^{2}
\bigg)
\right]
\tilde q_{0}
.
\eeann
The Gauss sum in brackets is again evaluated as
$
\vep_{q_{1}}
q_{1}^{1/2}
\left(
{
a_{1}
r\bar C
\over q_{1}
}
\right)
,
$
so we have
\bea
\label{eq:SfEval}
\cS_{f}(q_{0},r;n,m)
&=&
\bo_{
nC\equiv mB(\tilde q_{0})
}
{
\vep_{q_{0}}\vep_{q_{1}}
\tilde q_{0}
^{1/2}
\over q_{0}
}
e_{q_{0}}
(
-\overline{4rC} m^{2}
)
\\
\nonumber
&&
\qquad
\times
e_{q_{1}}
\bigg(
-
\overline{4a_{1}rC}
L^{2}
\bigg)
\left({rC\over q_{0}}\right)
\left(
{
a_{1}
r\bar C
\over q_{1}
}
\right)
.
\eea
The claim then follows trivially.
\epf
\
Next we introduce a certain average of a pair of such sums. Let $f, q_{0}, r, n,$ and $m$ be as before, and fix $q\equiv0(\mod q_{0})$
and
$(u_{0},q_{0})=1$. Let $\ff'\in\fF$ be another
shifted form $\ff'=f'-a'$, with
$$
f'(x,y)=A'x^{2}+2B'xy+C'y^{2}.
$$
Also let $n',m'\in\Z$. Then define
\bea\label{eq:cSdef}
\hskip-.5in
\cS&=&
\cS(q,q_{0},f,f',n,m,n',m';u_{0})
\\
\nonumber
&:=&
\sideset{}{'}
\sum_{r(q)}
\cS_{f}(q_{0},r\d_{0};n,m)
\overline{\cS_{f'}(q_{0},r\d_{0};n',m')}
e_{q}(r (a'- a))
.
\eea
This sum also
appears naturally
in the minor arcs analysis, see \eqref{eq:IQbnd1} and \eqref{eq:IQbnd2}.
\begin{lem}
With the above notation,
we have the estimate
\be
\label{eq:modBndaNeqAp}
|\cS|
\ll
\left({q/ q_{0}}\right)^{2}
{
\{(a^{2},q_{0})
\cdot
((a')^{2},q_{0})\}
^{1/2}
\over q^{5/4}
}
(a-a',q)^{1/4}
.
\ee
\end{lem}
\begin{rmk}
Treating all $\gcd$'s above as $1$ and pretending $q=q_{0}$, the trivial bound here (after having saved essentially a whole $q$ from each of the two $\cS_{f}$ sums) is $1/q$, since the $r$ sum is unnormalized. So \eqref{eq:modBndaNeqAp} saves an extra $q^{1/4}$ in the $r$ sum. (In fact we could have saved the expected $q^{1/2}$, but this does not improve our final estimates.)
\end{rmk}
\pf
Observe that $\cS$ is multiplicative in $q$, so we again consider the prime power case $q=p^{j}$, $p\neq2$; then $q_{0}$ is also a prime power, since $q_{0}\mid q$. As before, we may assume $(C,q_{0})=(C',q_{0})=1$.
Recall $a_{1}$, $\tilde q_{0}$,
and $L$
given in \eqref{eq:tildQdef}
and \eqref{eq:Ldef},
and let $a_{1}'$, $\tilde q_{0}'$ and $L'$
be defined similarly.
Inputting the analysis from
\eqref{eq:SfEval} into both $\cS_{f}$ and $\cS_{f'}$,
we have
\bea
\nonumber
\c
&=&
\bo_{
nC\equiv 2mB(\tilde q_{0})
\atop
n'C'\equiv 2m'B'(\tilde q'_{0})
}
{
\vep_{q_{1}}
\bar\vep_{q_{1}'}
(\tilde q_{0}\tilde q'_{0})^{1/2}
\over q_{0}^{2}
}
\left(
CC'\over q_{0}}\right)
\left(
{
a_{1}
u_{0}\bar C
\over q_{1}
}
\right)
\left(
{
a_{1}'
u_{0}\bar C'
\over q_{1}'
}
\right)
\\
\label{eq:cS1}
&&
\times
\Bigg[
\sideset{}{'}
\sum_{r(q)}
\left(
{
r\over q_{1}
}
\right)
\left(
{
r\over q_{1}'
}
\right)
e_{q}(r \{a'- a\})
\\
\nonumber
&&
\qquad
\times
e_{q_{0}}
\Bigg(
\overline{ 4
r
u_{0}
}
\bigg\{
\overline{C'} (m')^{2}
-\overline{C} m^{2}
+
\overline{a_{1}'C'}
(L')^{2}
\tilde q'
-
\overline{a_{1}C}
L^{2}
\tilde q
\bigg\}
\Bigg)
\Bigg]
.
\eea
The term in brackets $\big[\cdot\big]$ is a Kloosterman- or Sali\'e-type sum, for which we have an elementary bound \cite{Kloosterman1927} to the power $3/4$:
\bea
\nonumber
|\cS
&\ll&
{
(\tilde q_{0}
\tilde q'_{0})
^{1/2}
\over q_{0}^{2}
}
q^{3/4}
(a-a',q)^{1/4}
,
\eea
giving
the claim. (There is no improvement in our use of this estimate from appealing to Weil's bound instead of Kloosterman's; any
power gain suffices).
\epf
In the case $a=a'$, \eqref{eq:modBndaNeqAp} only saves one power of $q$, and in \S\ref{sec:QXT} we will need slightly more; see the proof of \eqref{eq:cI42bnd}.
We get a bit more cancellation in the special case $f(m,-n)\neq f'(m',-n')$ below.
\begin{lem}
Assuming
$a=a'$ and $f(m,-n)\neq f'(m',-n')$,
we have the estimate
\be
\label{eq:modBndaEqAp}
|\cS|
\ll
(q/q_{0})^{5}\
{
(a^{2},q_{0})
\over q^{9/8}
}
\cdot
|f(m,-n)-f'(m',-n')|^{1/2}
.
\ee
\end{lem}
\pf
Assume first that $q$ (and hence $q_{0}$)
is
a prime power, continuing to omit the prime $2$.
Returning to the
definition of $\cS$ in \eqref{eq:cSdef},
it is clear in the case $a=a'$ that
$$
\sum_{r(q)}'=(q/q_{0})\sum_{r(q_{0})}'.
$$
Hence we again apply Kloosterman's $3/4$th bound to \eqref{eq:cS1}, getting
\bea
\label{eq:modBndaEqAp1}
|\c
|
&\ll&
\bo_{
nC\equiv 2mB(\tilde q_{0})
\atop
n'C'\equiv 2m'B'(\tilde q'_{0})
}
(q/q_{0})^{9/2}
{
(a^{2},q_{0})
\over q^{5/4}
}
\\
\nonumber
&&
\times
\prod_{p^{j}\| q_{0}}
\Bigg(
p^{j}
,
\bar 4
\left\{
\overline{C'} (m')^{2}
-\overline{C} m^{2}
+
\overline
a_{1}}
(a^{2},p^{j})
(
\overline{C'}
(L')^{2}
-
\overline{C}
L^{2}
)
\right\}
\Bigg)^{1/4}
,
\eea
which is valid now without the assumption
that $q_{0}$ is a prime power. (Here $a_{1}$ satisfies $a^{2}=a_{1}(a^{2},p^{j})$ as in \eqref{eq:tildQdef}, and $L$ is given in \eqref{eq:Ldef}, so both depend on $p^{j}$.)
Break the primes diving $q_{0}$ into two sets, $\cP_{1}$ and $\cP_{2}$, defining $\cP_{1}$ to be the set of those primes $p$ for which
\be\label{eq:pr2}
\overline
C} m^{2}
+
\overline
C}
L^{2}
\overline
a_{1}}
(a^{2},p^{j})
\equiv
\overline
C'} (m')^{2}
+
\overline
C'}
(L')^{2}
\overline
a_{1}}
(a^{2},p^{j})
\ \
(\mod p^{\lceil j/2\rceil})
,
\ee
and $\cP_{2}$ the rest.
For the latter, the $\gcd$ in $(p^{j},\cdots)$ of \eqref{eq:modBndaEqAp1} is at most $p^{j/2}$, so we clearly have
\be\label{eq:cP2}
\prod_{p^{j}\| q_{0}\atop p\in\cP_{2}}(p^{j},\cdots)^{1/4}
\le
\prod_{p^{j}\| q_{0}
}p^{j/8}
=
q_{0}^{1/8}
.
\ee
For $p\in\cP_{1}$, we multiply both sides of \eqref{eq:pr2} by
$$
a^{2}
AC-B^{2
A'C'-(B')^{2
a_{1}(a^{2},p^{j})
,
$$
giving
\bea
\label{eq:15}
&&
\hskip-.5in
(AC-B^{2})\overline{C} m^{2}
+
\overline{C}
L^{2}
(a^{2},p^{j})^{2}
\\
\nonumber
&
\equiv
&
(A'C'-(B')^{2})\overline{C'} (m')^{2}
+
\overline{C'}
(L')^{2}
(a^{2},p^{j})^{2}
\qquad(\mod p^{\lceil j/2\rceil})
.
\eea
Using \eqref{eq:Ldef} that
$$
nC-
mB = (a^{2},p^{j}) L
,
\qquad
n'C'-
m'B' = (a^{2},p^{j}) L'
$$
and subtracting $a$ from both sides of \eqref{eq:15},
we have shown that
\be
\label{eq:fmnfp}
f'(m',-n')
\equiv
f(m,-n)
\qquad(\mod p^{\lceil j/2\rceil})
.
\ee
Let
$$
Z=|f(m,-n)-f'(m',-n')|
.
$$
By assumption $Z\neq0$. Moreover \eqref{eq:fmnfp} implies that
$$
\left(
\prod_{p\in\cP_{1}}p^{\lceil j/2\rceil}
\right)
\mid
Z
,
$$
and hence
\be\label{eq:cP1}
\prod_{p^{j}\|q_{0}\atop p\in\cP_{1}}p^{j/4}
\le
Z^{1/2}
.
\ee
Combining \eqref{eq:cP1} and \eqref{eq:cP2} in \eqref{eq:modBndaEqAp1} gives the claim.
\epf
Finally we need some savings in the case $a=a'$ and $f(m,-n)=f'(m',-n')$. This will no longer come from $\cS$ itself, but from the following supplementary lemmata.
\begin{lem}
Fix an equivalence class $\cK$ of primitive binary quadratic forms of discriminant $-4a^{2}$.
We claim that the number of equivalent forms $f\in\cK$ with $\ff=f-a\in\fF$ is bounded, that is,
\be\label{eq:fincK}
\#\{\ff\in\fF:f\in\cK\}=O(1).
\ee
\end{lem}
\pf
From \eqref{eq:ABCdef}, \eqref{eq:fFdef}, and \eqref{eq:disc}, we have that
$f(m,n)=Am^{2}+2Bmn+Cn^{2}$ has coefficients of size
$$
A,B,C\ll T,
$$
and $AC-B^{2}=a^{2}$, with
a\asymp T.
$
It follows that $AC\asymp T^{2}$, and hence
\be\label{eq:ACisT}
A,C\asymp T.
\ee
Now suppose we have $\ff=f-a$ and $\ff'=f'-a$ with $f$ as above and $f'$ having coefficients $A',B',C'$. If $f$ and $f'$ are equivalent then there is an element $\mattwo ghij\in\GL(2,\Z)$ so that
\bea\label{eq:ApBpCpABC}
A' &=& g^{2} A + 2 gi B + i^{2} C,\\
\nonumber
B' &=& gh A + (gj+hi) B + ij C ,\\
\nonumber
C' &=& h^{2} A + 2 hj B + j^{2} C.
\eea
The first line can be rewritten as
$$
A'
=
C
( i
+ g B /C
)^{2}
+
g^{2} {4a^{2}\over C}
,
$$
so that
$$
g^{2}
\le
A'
{C\over4a^{2}}
\ll
1
.
$$
Similarly,
$$
( i
+ g B /C
)^{2}
\le
{A' \over C}
\ll
1
,
$$
and
hence
$|i|\ll 1$. In a similar fashion, we see that $|h|$ and $|j|$ are also bounded, thus the number of equivalent forms in $\cK$ is bounded, as claimed.
\epf
\begin{lem}
For a fixed large integer $z$, the number of inequivalent classes $\cK$ of primitive quadratic forms
of determinant $-4a^{2
$ which represent $z$ is
\be\label{eq:cKzBnd}
\ll_{\vep}\
z^{\vep}\cdot
(z,4a^{2})^{1/2}
,
\qquad
\text{for any $\vep>0$.}
\ee
\end{lem}
\pf
If $f\in\cK$ represents $z$, say $f(m,n)=z$, then, setting $w=(m,n)$, $f$ represents $z_{1}:=z/w^{2}$ primitively.
We see from \eqref{eq:ApBpCpABC} that $f$ is then in the same class as $f_{1}(m,n)=z_{1}m^{2}+2Bmn+Cn^{2}$, with
$$
-4a^{2}=z_{1}C-B^{2}
.
$$
Moreover, by a unipotent change of variables preserving $z_{1}$, we can force $B$ into the range $[0,z_{1})$, that is, $B$ is determined mod $z_{1}$. So the number of inequivalent such $f_{1}$ is equal to
\be\label{eq:BtoZ1}
\#\{B(\mod z_{1}):B^{2}\equiv-4a^{2}(z_{1})\}
=
\prod_{p^{e}\mid\mid z_{1}}
\#\{B^{2}\equiv-p^{2f}(p^{e})\}
,
\ee
where $p^{f}\mid\mid 2a$. If $2f\ge e$, then the number of local solutions is at most $p^{e/2}$. Otherwise, write $B=B_{1}p^{f}$; then there are at most $2$ solutions to $B_{1}^{2}\equiv-1(\mod p^{e-2f}),$
and there are $p^{f}$ values for $B$ once $B_{1}$ is determined. Hence the number of local solutions is at most $2\cdot\min(p^{e/2},p^{f})$, so the number of solutions to \eqref{eq:BtoZ1} is at most
$$
2^{\gw(z)}(z_{1},4a^{2})^{1/2}\ll_{\vep}z^{\vep}(z,4a^{2})^{1/2}.
$$
The number of divisors $z_{1}$ of $z$ is $\ll_{\vep} z^{\vep}$, completing the proof.
\epf
\begin{lem} \label{lem:kEllD}
Fix $(A,B,C)=1$ and $d\mid AC-B^{2}$. Then there are integers $k,\el
$ with $(k,\ell,d)=1$ so that, whenever $Am^{2}+2Bmn+Cn^{2}\equiv0(d)$, we have
\be\label{eq:nkmell}
(mk+n\ell)^{2}\equiv0(d).
\ee
\end{lem}
\pf
We will work locally, then lift to a global solution. Let $p^{e}\mid\mid d$.
Case 1: If $(p,A)=1$, then $Am^{2}+2Bmn+Cn^{2}\equiv0(p^{e})$ implies
$$
(m+\bar A Bn)^{2}-\bar A ^{2}B^{2}n^{2}+\bar ACn^{2}\equiv
(m+\bar A Bn)^{2}\equiv
0(p^{e}).
$$
In this case, we set $k_{p}:=1$, and $\ell_{p}:={\bar A B}$.
Case 2:
If $(p,A)>1$, then by primitivity, $(p,C)=1$. As before, we have
$
(n+\bar C Bm)^{2}\equiv
0(p^{e}),
$
and we choose $k_{p}={\bar C B}$, $\ell_{p}:=1$.
By the Chinese Remainder Theorem, there are integers $k$ and $\ell$ so that $k\equiv k_{p}(\mod p^{e})$, and similarly with $\ell$. By construction, we have $(k,\ell,d)=1$,
as claimed.
\epf
\begin{lem}
Given large $M$, $(A,B,C)=1$ and $d\mid AC-B^{2}$,
\be\label{eq:nmUpToM}
\#\{m,n<M:Am^{2}+2Bmn+Cn^{2}\equiv0(d)\}
\ee
$$
\ll_{\vep}
d^{\vep}
\left({M^{2}\over d^{1/2}}+M\right)
.
$$
\end{lem}
\pf
As in Lemma \ref{lem:kEllD}, $A,B,C$ and $d$ determine $k,\ell$ so that
$$
\sum_{m,n<M}\bo_{\{Am^{2}+2Bmn+Cn^{2}\equiv0(d)\}}
\le
\sum_{m,n<M}\bo_{\{(mk+n\ell)^{2}\equiv0(d)\}}
.
$$
But then there is a $d_{1}\mid d$, with $d\mid d_{1}^{2}$ so that $mk+n\ell\equiv0(d_{1})$.
Let $w=(\ell, d_{1})$; then $mk\equiv0(w)$ implies $m\equiv0(w)$ since $(k,\ell,d)=1$. There are at most $1+M/w$ such $m$ up to $M$. With $m$ fixed, $n$ is uniquely determined mod $d_{1}/w$.
Hence we get
the bound
\beann
\eqref{eq:nmUpToM}
&\le&
\sum_{d_{1}\mid d\atop d\mid d_{1}^{2}}
\sum_{w\mid d_{1}}
\sum_{m,n<M}\bo_{\{m\equiv0(\mod w)\}}\bo_{\{ n\equiv-\overline{\frac \ell w}\frac mw k(\mod \frac{d_{1}}w)\}}
\\
&\ll&
\sum_{d_{1}\mid d\atop d\mid d_{1}^{2}}
\sum_{w\mid d_{1}}
\left(
{M\over w}
+1
\right)
\left(
{wM\over d_{1}}
+1
\right)
\ll_{\vep}
d^{\vep}
\left(
{M^{2}\over d^{1/2}}
+M
\right)
,
\eeann
as claimed.
\epf
Finally we collect the above lemmata into our desired estimate, essential in the proof of \eqref{eq:cI41bnd}.
\begin{prop}
For large $M$ and $\ff=f-a\in\fF$ fixed,
\be\label{eq:eqeq}
\#
\left\{
\begin{array}{c}
\ff'\in\fF
\\
m,n,m',n'<M
\end{array}
\Bigg|
\begin{array}{c}
a'=a\\
\ff(m,-n)=\ff'(m',-n')
\end{array}
\right\}
\ll_{\vep}
(TM)^{\vep}
\left(
M^{2}
+
TM
\right)
,
\ee
for any $\vep>0$.
\end{prop}
\pf
Once $f,m,n$, and $\ff'=f'-a\in\fF$ are determined, it is elementary that there are $\ll_{\vep} M^{\vep}$ values of $m',n'$ with $f(m,-n)=f'(m',-n')$.
Decomposing $f'$ into classes and applying
\eqref{eq:fincK},
\eqref{eq:cKzBnd},
and
\eqref{eq:nmUpToM},
in succession, we have
\beann
&&
\hskip-1in
\sum_{m,n<M}
\sum_{\ff'\in\fF\atop a'=a}
\sum_{m',n'<M}
\bo_{\{f(m,-n)=f'(m',-n')\}}
\\
&\ll_{\vep}&
\sum_{m,n<M}
\sum_{\ff'\in\fF\atop a'=a}
\bo_{\{f'\text{ represents }f(m,-n)\}}
M^{\vep}
\\
&\ll&
M^{\vep}
\sum_{m,n<M}
\sum_{\text{classes }\cK\atop\text{representing }f(m,-n)}
\sum_{\ff'\in\fF\atop a'=a,f'\in\cK}
1
\\
&\ll_{\vep}&
(TM)^{\vep}
\sum_{m,n<M}
(f(m,-n),4a^{2})^{1/2}
\\
&\ll&
(TM)^{\vep}
\sum_{d\mid 4a^{2}}
d^{1/2}
\sum_{m,n<M}
\bo_{\{f(m,-n)\equiv0(d)\}}
\\
&\ll&
(TM)^{\vep}
\sum_{d\mid 4a^{2}}
d^{1/2}
\left(
{M^{2}\over d^{1/2}}
+
M
\right)
\\
&\ll&
(TM)^{\vep}
\left(
M^{2}
+
M
a
\right)
,
\eeann
from which the claim follows since $a\ll T$.
\epf
\newpage
\section{Major Arcs}\label{sec:Maj}
We return to the setting and notation of \S\ref{sec:outline} with the goal of establishing \eqref{eq:cMNUbnd}. Thanks to the counting lemmata in \S\ref{sec:count}, we can now define the major ars parameters $Q_{0}$ and $K_{0}$ from \eqref{eq:Q0K0intro}. First recall the two numbers $\gT<\gd$ appearing in \eqref{eq:spec1}, \eqref{eq:spec2}, and define
\be\label{eq:gT1def}
1<\gT_{1}<\gd
\ee
to be the larger of the two. Then set
\be\label{eq:Q0is}
Q_{0}= T ^{(\gd-\gT_{1})/20},
\qquad
K_{0}=Q_{0}^{2}.
\ee
We may now also set the parameter $U$ from \eqref{eq:Uis} to be
\be\label{eq:fuIs}
U=Q_{0}{}^{(\eta_{0})^{2}/100},
\ee
where $0<\eta_{0}<1$ is the number which appears in Lemma \ref{lem:spec3}.
Let $\cM_{N}^{(U)}(n)$ denote either $\cM_{N}(n)$ or $\cM_{N}^{U}(n)$ from \eqref{eq:cMNdef}, \eqref{eq:cMNUdef}, respectively.
Putting \eqref{eq:hatFuncN} and \eqref{eq:cRNhatIs} (resp. \eqref{eq:cRNhatUis}) into \eqref{eq:cMNdef} (resp. \eqref{eq:cMNUdef}),
making a change of variables $\gt=r/q+\gb$, and unfolding the integral from $\sum_{m}\int_{0}^{1}$ to $\int_{\R}$
gives
\be\label{eq:cMNUn}
\cM_{N}^{(U)}(n)
=
\sum_{x,y\in\Z}
\gU\left(\frac {2x}X\right)
\gU\left(\frac yX\right)
\cdot
\fM(n)
\cdot
\sum_{u}
\mu(u)
,
\ee
where in the last sum, $u$ ranges over $u\mid (2x,y)$ (resp. and $u<U$). Here we have defined
\bea
\label{eq:fMis}
&&
\hskip-.5in
\fM(n)
=
\fM_{x,y}(n)
\\
\nonumber
&:=&
\sum_{q<Q_{0}}
\sideset{}{'}\sum_{r(q)}
\sum_{\g\in\fF}
e_{q}(r(\<w_{x,y},\g v_{0}\>-n))
\int_{\R}
\ft
\left(
\frac N{K_{0}}
\gb
\right)
e(\gb (\ff_{\g}(2x,y)-n))
d\gb
,
\eea
using \eqref{eq:ffToInProd}.
As in \eqref{eq:G0q}, let $\tilde\G(q)$ be the stabilizer of $v_{0}(\mod q)$. Decompose the sum on $\g\in\fF$ in \eqref{eq:fMis} as
a sum on ${\g_{0}\in\G/\tilde\G(q)}$ and ${\g\in\fF\cap\g_{0}\tilde\G(q)}$.
Applying Lemma \ref{lem:spec1} to the latter sum, using the definition of $\gT_{1}$ in \eqref{eq:gT1def}, and recalling the estimate \eqref{eq:G0qBnd} gives
\be\label{eq:fMdecomp}
\fM(n)
=
\fS_{Q_{0}}(n)\cdot
\fW(n)
+
O\left({T^{\gT_{1}}\over N}K_{0}^{2}Q_{0}^{4}\right)
,
\ee
where
\beann
\fS_{Q_{0}}(n)
&:=&
\sum_{q<Q_{0}}
\sideset{}{'}\sum_{r(q)}
\sum_{\g_{0}\in\G/\tilde\G(q)}
{e_{q}(r(\<w_{x,y},\g_{0} v_{0}\>-n))
\over
[\G:\tilde\G(q)]
}
,
\\
\fW(n)
&:=&
{K_{0}\over N}\,
\sum_{\ff\in\fF}
\hat\ft
\left(
(\ff(2x,y)-n)
\frac {K_{0}}N
\right)
.
\eeann
Clearly we have thus split $\fM$ into ``modular'' and ``archimedean'' components. It is now a simple matter to prove the following
\begin{thm}\label{thm:cMNis}
For $\foh N<n<N$, there exists a function $\fS(n)$ as in Theorem \ref{thm:RNU} so that
\be\label{eq:cMNbnd}
\cM_{N}(n)\gg\fS(n)T^{\gd-1}
.
\ee
\end{thm}
\pf
First we discuss the modular component. Write $\fS_{Q_{0}}$ as
$$
\fS_{Q_{0}}(n)
=
\sum_{q<Q_{0}}
{1\over
[\G:\tilde\G(q)]
}
\sum_{\g_{0}\in\G/\tilde\G(q)}
c_{q}(\<w_{x,y},\g_{0} v_{0}\>-n)
,
$$
where $c_{q}$ is the Ramanujan sum, $c_{q}(m)=\sum_{r(q)}'e_{q}(rm)$.
By \eqref{eq:Fuc},
the analysis now
reduces to a classical estimate for the singular series. We may
use the transitivity of the $\g_{0}$ sum to replace $\<w_{x,y},\g_{0} v_{0}\>$ by $\<e_{4},\g_{0} v_{0}\>$,
extend the sum on $q$ to all natural numbers, and use multiplicativity to write the sum
as an Euler product. Then the resulting singular series
$$
\fS(n)
:=
\prod_{p}
\left[
1+
\sum_{k\ge1}
{1\over[\G:\G_{0}(p^{k})]}
\sum_{\g_{0}\in\G/\G_{0}(p^{k})}
c_{p^{k}}
\bigg(
\
e_{4},
\g_{0}\,
v_{0}
\>
-
\bigg)
\right]
$$
vanishes only on non-admissible numbers, and can easily be seen to satisfy
\be\label{eq:fSbnd}
N^{-\vep}\ll_{\vep} \fS(n)\ll_{\vep} N^{\vep},
\ee
for any $\vep>0$. See, e.g. \cite[\S4.3]{BourgainKontorovich2010}.
Next we handle the archimedean component. By our choice of $\ft$ in \eqref{eq:hatFunc}, specifically that $\hat\ft>0$ and $\hat\ft(y)>2/5$ for $|y|<1/2$, we have
$$
\fW(n)\gg {K_{0}\over N} \sum_{\ff\in\fF}
\bo_{\{
|\ff(2x,y)-n|
<
\frac N {2K_{0}}
\}}
\gg
{T^{\gd}\over N}+{T^{\gT_{1}}K_{0}\over N}
,
$$
using Lemma \ref{lem:spec2}.
Putting everything into \eqref{eq:fMdecomp} and then into \eqref{eq:cMNUn} gives \eqref{eq:cMNbnd}, using \eqref{eq:Q0is} and \eqref{eq:TXNis}.
\epf
Next we derive from the above that the same bound holds for $\cM_{N}^{U}$ (most of the time).
\begin{thm}\label{thm:cMNtoMNU}
There is an $\eta>0$ such that the bound \eqref{eq:cMNbnd} holds with $\cM_{N}$ replaced by $\cM_{N}^{U}$,
except on a set
of cardinality $\ll N^{1-\eta}$.
\end{thm}
\pf
Putting \eqref{eq:fMdecomp} into \eqref{eq:cMNUn} gives
\beann
&&
\hskip-.5in
\sum_{n<N}
|\cM_{N}(n)-\cM_{N}^{U}(n)|
\ll
\sum_{x,y\asymp X}
\sum_{n<N}|\fM(n)|
\sum_{u\mid(2x,y)\atop u\ge U}
1
\\
&\ll_{\vep}&
\sum_{y<X}
\sum_{u\mid y\atop u\ge U}
\sum_{x<X\atop 2x\equiv0(\mod u)}
\left\{
N^{\vep}
\sum_{\ff\in\fF}
{K_{0}\over N}
\left[
\sum_{n<N}
\hat\ft
\left(
(\ff(2x,y)-n)
\frac {K_{0}}N
\right)
\right]
+
K_{0}^{2}Q_{0}^{4}T^{\gT_{1}}
\right\}
\\
&\ll&
N^{\vep}
X\frac XU
T^{\gd}
,
\eeann
using \eqref{eq:fSbnd} and \eqref{eq:Q0is}.
The rest of the argument is identical to that leading to \eqref{eq:nZbnd}.
\epf
This establishes \eqref{eq:cMNUbnd}, and hence completes our Major Arcs analysis; the rest of the paper is devoted to proving \eqref{eq:cEN2bnd}.
\newpage
\section{Minor Arcs I: Case $q<Q_{0}$}\label{sec:qQ0}
We keep all the notation of \S\ref{sec:outline}, our goal in this section being to bound \eqref{eq:IQ0K0} and \eqref{eq:IQ0}. First we return to
\eqref{eq:cRNhatUis} and reverse orders of summation, writing
\be\label{eq:RNRf}
\widehat{\cR_{N}^{U}}
(\gt)
=
\sum_{\d< U}
\mu(\d)
\sum_{\ff\in\fF}
e(-a\gt)
\hat\cR_{f,\d}
(\gt)
,
\ee
where $\ff=f-a$ according to \eqref{eq:ffVdef}, and we have set
$$
\hat\cR_{f,\d}(\gt)
:=
\sum_{ 2x\equiv0( u)}
\sum_{ y\equiv0( u)}
\gU\left(\frac {2x}X\right)
\gU\left(\frac{y}X\right)
e
\bigg(
\gt
f(2 x, y)
\bigg)
.
$$
If $u$ is even, then we have
\be\label{eq:cRffuIs}
\hat\cR_{f,\d}(\gt)
=
\sum_{ x,y\in\Z}
\gU\left(\frac {x u}X\right)
\gU\left(\frac{y u}X\right)
e
\bigg(
\gt
f( x u , y u)
\bigg)
.
\ee
If $u$ is odd, we have
$$
\hat\cR_{f,\d}(\gt)
=
\sum_{ x,y\in\Z}
\gU\left(\frac {2x u}X\right)
\gU\left(\frac{y u}X\right)
e
\bigg(
\gt
f(2 x u , y u)
\bigg)
.
$$
From now on, we focus exclusively on the case $u$ is even, the other case being handled similarly.
We first massage $\hat \cR_{f,u}$ further.
Since $f$
is homogeneous quadratic,
we have
$$
f( x u , y u)
=
u^{2}
f(x,y)
.
$$
Hence expressing $\gt=\frac rq+\gb$, we will
need to write $u^{2}/q$ as a reduced fraction; to this end, introduce the notation
\be\label{eq:tilQis}
\tilde q:=(u^{2},q)\qquad
u_{0}:=u^{2}/\tilde q
,\qquad
q_{0}:=q/\tilde q
,
\ee
so that $u^{2}/q=u_{0}/q_{0}$ in lowest terms, $(u_{0},q_{0})=1$.
\begin{lem}
Recalling the notation \eqref{eq:cSfDef},
we have
\be\label{eq:cRfuIs
\hat\cR_{f,\d }\left(\frac rq+\gb\right)
=
{1\over \d ^{2}}
\sum_{n,m\in\Z}
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right)
\cS_{f}(q_{0},r\d _{0};n,m)
,
\e
where we have set
\be\label{eq:cJfIs}
\cJ_{f}\left(X,\gb;{n\over uq_{0}},{m\over uq_{0}}\right)
:=
\iint\limits_{x,y\in\R}
\gU\left({x\over X}\right)
\gU\left({y\over X}\right)
e
\bigg(
\gb
f(x,y)
-{n\over uq_{0}}x-{m\over uq_{0}}y
\bigg)
dx dy
.
\ee
\end{lem}
\pf
Returning to \eqref{eq:cRffuIs}, we have
\beann
\hat\cR_{f,\d}\left(\frac rq+\gb\right)
&=&
\sum_{x,y\in\Z}
\gU\left({\d x\over X}\right)
\gU\left({\d y\over X}\right)
e_{q_{0}}
\bigg(
r
\d_{0}
f( x,y)
\bigg)
e
\bigg(
\gb
\d^{2}
f(x,y)
\bigg)
\\
&=&
\sum_{k(q_{0})}
\sum_{\ell( q_{0})}
e_{q_{0}}
\bigg(
r
\d_{0}
f(k,\ell)
\bigg)
\\
&&
\times
\left[
\sum_{x\in\Z\atop x\equiv k(q_{0})}
\sum_{y\in\Z\atop y\equiv \ell(q_{0})}
\gU\left({\d x\over X}\right)
\gU\left({\d y\over X}\right)
e
\bigg(
\gb
\d^{2}
f(x,y)
\bigg)
\right]
.
\eeann
Apply Poisson summation to the bracketed term above:
\beann
\Bigg[
\cdot
\Bigg]
&=&
\sum_{x,y\in\Z}
\gU\left({\d (q_{0}x+k)\over X}\right)
\gU\left({\d (q_{0}y+\ell)\over X}\right)
e
\bigg(
\gb
\d^{2}
f(q_{0}x+k,q_{0}y+\ell)
\bigg)
\\
&=&
\sum_{n,m\in\Z}\
\iint\limits_{x,y\in\R}
\gU\left({\d (q_{0}x+k)\over X}\right)
\gU\left({\d (q_{0}y+\ell)\over X}\right)
e
\bigg(
\gb
\d^{2}
f(q_{0}x+k,q_{0}y+\ell)
\bigg)
\\
&&
\hskip3.5in
\times
e(-nx-my)
dx dy
\\
&=&
{1\over \d^{2}q_{0}^{2}}
\sum_{n,m\in\Z}
e_{q_{0}}(nk+m\ell)
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right),
\eeann
Inserting this in the above, the claim follows immediately.
\epf
We are now in position to prove the following
\begin{prop}
With the above notation,
\be\label{eq:hatRfuBnd}
\left|
\hat\cR_{f,\d }\left(\frac rq+\gb\right)
\right|
\ll
u
(\sqrt{q}|\gb|T)^{-1}
.
\ee
\end{prop}
\pf
By (non)stationary phase (see, e.g., \cite[\S8.3]{IwaniecKowalski}), the
integral
in
\eqref{eq:cJfIs}
has negligible contribution unless
$$
{|n|\over \d q_{0}},
{|m|\over \d q_{0}}
\ll
|\gb|\cdot |\nabla f|\ll |\gb|\cdot TX,
$$
so the $n,m$ sum can be restricted to
\be\label{eq:nmRest}
|n|,|m|\ll |\gb|\cdot TX\cdot \d q_{0}
\ll
\d
.
\ee
Here we used $|\gb|\ll (qM)^{-1}$ with $M$ given by \eqref{eq:Mis}.
In this range, stationary phase gives
\be\label{eq:cJfbnd}
\left|
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right)\right|
\ll\min\left(X^{2},{1\over |\gb|\cdot|\operatorname{discr}(f)|^{1/2}}\right)
\ll\min\left(X^{2},{1\over |\gb| T}\right)
,
\ee
using
\eqref{eq:disc} and \eqref{eq:fFTbnd}
that
$|\operatorname{discr}(f)|=4|B^{2}-AC|=4 a^{2}\gg T^{2}$.
Putting \eqref{eq:nmRest}, \eqref{eq:cJfbnd} and \eqref{eq:cSfbnd1} into \eqref{eq:cRfuIs}, we have
$$
\left|
\hat\cR_{f,\d }\left(\frac rq+\gb\right)
\right|
\ll
{1\over \d ^{2}}
\sum_{|n|,|m|\ll u}
{1\over |\gb|T
\cdot
{1\over \sqrt {q_{0}}
,
$$
from which the claim follows, using \eqref{eq:tilQis}.
\epf
Finally, we prove the desired estimates of the strength \eqref{eq:cEN2bnd}.
\begin{thm}\label{thm:IQ0}
Recall the integrals $\cI_{Q_{0},K_{0}},\ \cI_{Q_{0}}$ from \eqref{eq:IQ0K0}, \eqref{eq:IQ0}.
There is an $\eta>0$ so that
$$
\cI_{Q_{0},K_{0}}, \ \cI_{Q_{0}}\ll N\, T^{2(\gd-1)}\, N^{-\eta}
,
$$
as $N\to\infty$.
\end{thm}
\pf
We first handle $\cI_{Q_{0},K_{0}}$.
Returning to \eqref{eq:RNRf} and applying \eqref{eq:hatRfuBnd} gives
$$
\left|
\widehat{\cR_{N}^{U}}
\left(\frac rq+\gb\right)
\right|
\ll
\sum_{\d< U}
\sum_{\ff\in\fF}
u
(\sqrt{q}|\gb|T)^{-1}
\ll
U^{2}
T^{\gd-1}
(\sqrt{q}|\gb|)^{-1}
.
$$
Inserting this into \eqref{eq:IQ0K0} and using \eqref{eq:Q0is}, \eqref{eq:fuIs} gives
\beann
\cI_{Q_{0},K_{0}}
&\ll&
\sum_{q<Q_{0}}
\sideset{}{'}\sum_{r(q)}
\int_{|\gb|<K_{0}/N}
\left|
\gb
\frac N{K_{0}}
\right|^{2}
U^{4}
T^{2(\gd-1)}
{1\over q|\gb|^{2}}
d\gb
\\
&\ll&
Q_{0}
\frac N{K_{0}}
U^{4}
T^{2(\gd-1)}
\ll
N
T^{2(\gd-1)}
N^{-\eta}
.
\eeann
Next we handle
\beann
\cI_{Q_{0}}
&\ll&
\sum_{q<Q_{0}}
\sideset{}{'}
\sum_{r(q)}
\int\limits_{{K_{0}\over N}<|\gb|<\frac1{qM}}
U^{4}
T^{2(\gd-1)}
{1\over q|\gb|^{2}}
d\gb
\\
&\ll&
Q_{0}
U^{4}
T^{2(\gd-1)}
\left({
N\over K_{0}}+{Q_{0}M}\right)
\\
&\ll&
N
T^{2(\gd-1)}
{
Q_{0}
U^{4}
\over K_{0}}
,
\eeann
which is again a power savings.
\epf
\newpage
\section{Minor Arcs II: Case $Q_{0}\le Q<X$}\label{sec:QX}
Keeping all the notation from the last section, we now turn our attention to the integrals $\cI_{Q}$ in \eqref{eq:IQdef}.
It is no longer sufficient just to get cancellation in $\hat \cR_{f,u}$ alone, as in \eqref{eq:hatRfuBnd}; we must use the fact that $\cI_{Q}$ is an $L^{2}$-norm.
To this end, recall the notation \eqref{eq:tilQis}, and put \eqref{eq:cRfuIs} into \eqref{eq:RNRf}, applying Cauchy-Schwarz in the $u$-variable:
\bea
\label{eq:cRsq}
\left|\widehat{\cR_{N}^{U}}
\left(\frac rq+\gb\right)\right|^{2}
&\ll&
U
\sum_{\d< U}
\Bigg|
\sum_{\ff\in\fF}
e_{q}(-r a)
e(-a\gb)
\\
\nonumber
&&
\times
{1\over \d ^{2}}
\sum_{n,m\in\Z}
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right)
\cS_{f}(q_{0},r\d _{0};n,m)
\Bigg|^{2}
.
\eea
Recall from \eqref{eq:ffVdef} that
$\ff=f-a$.
Insert \eqref{eq:cRsq} into \eqref{eq:IQdef} and
open the square, setting $\ff'=f'-a'$. This gives
\bea
\nonumber
\cI_{Q}
&\ll&
U
\sum_{\d< U}
{1\over \d ^{4}}
\sum_{q\asymp Q}
\sideset{}{'}
\sum_{r(q)}
\int_{|\gb|<\frac1{qM}}
\Bigg|
\sum_{\ff\in\fF}
e_{q}(-r a)
e(-a\gb)
\\
\nonumber
&&
\times
\sum_{n,m\in\Z}
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right)
\cS_{f}(q_{0},r\d _{0};n,m)
\Bigg|^{2}
d\gb
\\
\label{eq:IQbnd1}
&
=
&
U
\sum_{\d< U}
{1\over \d ^{4}}
\sum_{n,m,n',m'\in\Z}\
\sum_{\ff,\ff'\in\fF}\
\sum_{q\asymp Q}
\\
\nonumber
&&
\times
\left[
\sideset{}{'}
\sum_{r(q)}
\cS_{f}(q_{0},r\d _{0};n,m)
\overline{\cS_{f'}(q_{0},r\d _{0};n',m')}
e_{q}(r (a'-a))
\right]
\\
\nonumber
&&
\times
\left[
\int_{|\gb|<\frac1{qM}}
\cJ_{f}\left(X,\gb;{n\over \d q_{0}},{m\over \d q_{0}}\right)
\overline{\cJ_{f'}\left(X,\gb;{n'\over \d q_{0}},{m'\over \d q_{0}}\right)}
e(\gb(a'-a))
d\gb
\right]
.
\eea
Note that again the sum has split into ``modular'' and ``archimedean'' pieces (collected in brackets, respectively), with the former being exactly equal to $\cS$ in \eqref{eq:cSdef}.
Decompose \eqref{eq:IQbnd1} as
\be\label{eq:cIQ12}
\cI_{Q}\ll \cI_{Q}^{(=)}+\cI_{Q}^{(\neq)},
\ee
where, once $\ff$ is fixed, we collect $\ff'$ according to whether $a'=a$ (the ``diagonal'' case) and the off-diagonal $a'\neq a$.
\begin{lem}
Assume $Q<X$. For $\square\in\{=,\neq\}$, we have
\be\label{eq:cIQ1def}
\cI_{Q}^{(\square)}
\ll
U^{6}
{X^{2}\over T}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\square a}
\sum_{q\asymp Q}
{
\{(a^{2},q)\cdot((a')^{2},q)\}^{1/2}(a-a',q)^{1/4}
\over
q^{5/4}
}
.
\ee
\end{lem}
\pf
Apply \eqref{eq:modBndaNeqAp} and \eqref{eq:nmRest}, \eqref{eq:cJfbnd} to \eqref{eq:IQbnd1}, giving
\beann
\cI_{Q}^{(\square)}
&\ll&
U
\sum_{u<U}
{1\over u^{4}}
\sum_{|n|,|m|,|n'|,|m'|\ll u}\
\sum_{\ff,\ff'\in\fF\atop a'\square a}\
\sum_{q\asymp Q}
\\
&&
\qquad
\times
{u^{4}
\{(a^{2},q)\cdot((a')^{2},q)\}^{1/2}(a-a',q)^{1/4}
\over
q^{5/4}
}
\\
&&
\qquad
\qquad
\times
\int_{|\gb|<1/(qM)}
\min\left(
X^{2},
{1\over |\gb|T}
\right)^{2}
d\gb
,
\eeann
where we used \eqref{eq:tilQis}. The claim then follows immediately from \eqref{eq:Mis} and $Q<X$.
\epf
We treat $\cI_{Q}^{(=)}, \ \cI_{Q}^{(\neq)}$ separately, starting with the former; we give bounds of the quality claimed in \eqref{eq:cEN2bnd}.
\begin{prop}
There is an $\eta>0$ such that
\be\label{eq:cIQ2bnd}
\cI_{Q}^{(=)}\ll N \, T^{2(\gd-1)}N^{-\eta},
\ee
as $N\to\infty$.
\end{prop}
\pf
From \eqref{eq:cIQ1def}, we have
\beann
\cI_{Q}^{(=)}
&\ll&
U^{6}
{X^{2}\over T}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'=a}
\sum_{q\asymp Q}
{(a^{2},q)\over q}
\\
&\ll&
{U^{6}
X^{2}\over QT}
\sum_{\ff\in\fF}
\sum_{\tilde q_{1}\mid a^{2}\atop\tilde q_{1}\ll Q}
\tilde q_{1}
\sum_{q\asymp Q\atop q\equiv0(\tilde q_{1})}
\sum_{\ff'\in\fF\atop a'=a}
1
\\
&\ll_{\vep}&
{U^{6}
X^{2}\over T}
\sum_{\ff\in\fF}
T^{\vep}
\sum_{\ff'\in\fF\atop a'=a}
1
.
\eeann
Recalling that $a=a_{\g}=\<e_{1},\g v_{0}\>$, replace the condition $a'=a$ with $a'\equiv a(\mod \lfloor Q_{0}\rfloor)$, and apply \eqref{eq:spec3}:
$$
\cI_{Q}^{(=)}
\ll_{\vep}
{U^{6}
X^{2}\over T}
T^{\gd}
T^{\vep}
{T^{\gd}\over Q_{0}^{\eta_{0}}}
.
$$
Then \eqref{eq:fuIs} and \eqref{eq:TXNis}
imply the claimed power savings.
\epf
Next we turn our attention to $\cI_{Q}^{(\neq)}$, the off-diagonal
contribution.
We decompose
this sum
further according to whether $\gcd(a,a')$ is large or not. To this end, introduce a parameter $H$, which we will eventually set to
\be\label{eq:His}
H=U^{10/\eta_{0}}=Q_{0}{}^{\eta_{0}/10},
\ee
where, as in \eqref{eq:fuIs}, the constant $\eta_{0}>0$ comes from Lemma \ref{lem:spec3}.
Write
\be\label{eq:cIQ1break}
\cI_{Q}^{(\neq)}=\cI_{Q}^{(\neq,>)}+\cI_{Q}^{(\neq,\le)},
\ee
corresponding to whether $(a,a')>H$ or $(a,a')\le H$, respectively. We deal first with the large $\gcd$.
\begin{prop}
There is an $\eta>0$ such that
\be\label{eq:cIQ11bnd}
\cI_{Q}^{(\neq,>)}\ll N \, T^{2(\gd-1)}N^{-\eta},
\ee
as $N\to\infty$.
\end{prop}
\pf
Writing $(a,a')=h>H$,
$\tilde q_{1}=(a^{2},q)$,
$\tilde q_{1}'=((a')^{2},q)$,
and
using
$(a-a',q)\le q$ in \eqref{eq:cIQ1def},
we have
\beann
\cI_{Q}^{(\neq,>)}
&\ll&
U^{6}
{X^{2}\over T}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\neq a,(a,a')>H}
\sum_{q\asymp Q}
{
\{(a^{2},q)\cdot((a')^{2},q)\}^{1/2}(a-a',q)^{1/4}
\over
q^{5/4}
}
\\
&\ll&
U^{6}
{X^{2}\over T}
\sum_{\ff\in\fF}
\sum_{h\mid a\atop h>H}
\sum_{\ff'\in\fF\atop a'\equiv 0(\mod h)}
\sum_{\tilde q_{1}\mid a^{2}\atop \tilde q_{1}\ll Q}
\sum_{\tilde q_{1}'\mid (a')^{2}\atop [\tilde q_{1},\tilde q_{1}']\ll Q}
(\tilde q_{1}
\tilde q_{1}')^{1/2}
\sum_{q\asymp Q\atop q\equiv 0([\tilde q_{1},\tilde q_{1}'])}
{1\over Q}
\\
&\ll_{\vep}&
U^{6}
{X^{2}\over T}
T^{\vep}
\sum_{\ff\in\fF}
\sum_{h\mid a\atop h>H}
\sum_{\ff'\in\fF\atop a'\equiv 0(\mod h)}
1
,
\eeann
where we used $[n,m]>(nm)^{1/2}$. Apply \eqref{eq:spec3} to the innermost sum, getting
\beann
\cI_{Q}^{(\neq,>)}
&\ll_{\vep}&
U^{6}
{X^{2}\over T}
T^{\vep}
T^{\gd}
{1\over H^{\eta_{0}}}
T^{\gd}
.
\eeann
By \eqref{eq:His} and \eqref{eq:fuIs}, this is a power savings, as claimed.
\epf
Finally, we handle small $\gcd$.
\begin{prop}
There is an $\eta>0$ such that
\be\label{eq:cIQ12bnd}
\cI_{Q}^{(\neq,\le)}\ll N \, T^{2(\gd-1)}N^{-\eta},
\ee
as $N\to\infty$.
\end{prop}
\pf
First note that
\beann
\cI_{Q}^{(\neq,\le)}
&=&
U^{6}
{X^{2}\over T}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\neq a,(a,a')\le H}
\sum_{q\asymp Q}
{
\{(a^{2},q)\cdot((a')^{2},q)\}^{1/2}(a-a',q)^{1/4}
\over
q^{5/4}
}
\\
&\ll&
U^{6}
{X^{2}\over T}
{1\over Q^{5/4}}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\neq a, (a,a')\le H}
\sum_{q\asymp Q}
(a,q)
(a',q)
(a-a',q)^{1/4}
.
\eeann
Write $g=(a,q)$ and $g'=(a',q)$, and let $h=(g,g')$; observe then that $h\mid (a,a')$ and $h\ll Q$. Hence we can write $g=hg_{1}$ and $g'=hg_{1}'$ so that $(g_{1},g_{1}')=1$.
Note also that $h\mid (a-a',q)$, so we can write $(a-a',q)=h\tilde g$; thus $g_{1},g_{1}',$ and $\tilde g$ are pairwise coprime, implying
$$
[hg_{1},hg_{1}',h\tilde g]
\ge
g_{1}g_{1}'\tilde g
.
$$
Then we have
\beann
\cI_{Q}^{(\neq,\le)}
&\ll&
U^{6}
{X^{2}\over T}
{1\over Q^{5/4}}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\neq a, (a,a')\le H}
\sum_{h\mid (a,a')\atop h\le H}
\sum_{g_{1}\mid a\atop g_{1}\ll Q}
\sum_{g_{1}'\mid a'\atop g_{1}'\ll Q}
\\
&&
\qquad\times
\sum_{\tilde g\mid (a-a')\atop [hg_{1},hg_{1}',h\tilde g]\ll Q}
(hg_{1})
(hg_{1}')
(h\tilde g)^{1/4}
\sum_{q\asymp Q\atop q\equiv0([hg_{1},hg_{1}',h\tilde g])}
1
\\
&\ll_{\vep}&
U^{6}
{X^{2}\over T}
{H^{9/4}\over Q^{5/4}}
\sum_{\ff,\ff'\in\fF}
T^{\vep}
\sum_{g_{1}\mid a\atop g_{1}\ll Q}
\sum_{g_{1}'\mid a'\atop g_{1}'\ll Q}
\sum_{\tilde g\mid (a-a')\atop \tilde g\ll Q
g_{1}\,
g_{1}'\,
\tilde g^{1/4}
{Q\over g_{1}g_{1}'\tilde g}
\\
&\ll&
U^{6}
{X^{2}\over T}
{H^{9/4}\over Q^{1/4}}
\sum_{\ff\in\fF}
\sum_{\tilde g\ll Q}
{1\over \tilde g^{3/4}}
T^{\vep}
\sum_{\ff'\in\fF\atop a'\equiv a(\mod \tilde g)
.
\eeann
To the last sum, we again apply Lemma \ref{lem:spec3}, giving
\beann
\cI_{Q}^{(\neq,\le)}
&\ll_{\vep}&
U^{6}
{X^{2}\over T}
{H^{9/4}\over Q^{1/4}}
T^{\gd}
\sum_{\tilde g\ll Q}
{1\over \tilde g^{3/4}}
T^{\vep}
{1\over \tilde g^{\eta_{0}}}
T^{\gd}
\ll
U^{6}
{X^{2}\over T}
{H^{9/4}\over Q_{0}^{\eta_{0}}}
T^{\gd}
T^{\vep}
T^{\gd}
,
\eeann
since $Q\ge Q_{0}$.
By \eqref{eq:His} and \eqref{eq:fuIs}, this is again a power savings, as claimed.
\epf
Putting together \eqref{eq:cIQ12}, \eqref{eq:cIQ2bnd}, \eqref{eq:cIQ1break}, \eqref{eq:cIQ11bnd}, and \eqref{eq:cIQ12bnd}, we have proved the following
\begin{thm}\label{thm:IQX}
For $Q_{0}\le Q<X$, there is some $\eta>0$ such that
$$
\cI_{Q}\ll N\, T^{2(\gd-1)}\, N^{-\eta},
$$
as $N\to\infty$.
\end{thm}
\newpage
\section{Minor Arcs III: Case $X\le Q<M$}\label{sec:QXT}
In this section, we continue our analysis of $\cI_{Q}$ from \eqref{eq:IQdef}, but now we need different methods to handle the very large $Q$ situation. In particular, the range of $x,y$ in \eqref{eq:cRffuIs} is now such that we have incomplete sums, so our first step is to complete them.
To this end, recall the notation \eqref{eq:tilQis} and introduce
\be\label{eq:glfDef}
\gl_{f}\left(X,\gb;\frac n{q_{0}},\frac m{q_{0}},u\right)
:=
\sum_{x,y\in\Z}
\gU\left(\frac {ux}X\right)
\gU\left(\frac {uy}X\right)
e
\left(
-{n\over q_{0}}x-{m\over q_{0}}y
\right)
e
\bigg(
\gb
u^{2}
f(x,y)
\bigg)
,
\ee
so that, using \eqref{eq:cSfDef}, an elementary calculation gives
\be\label{eq:RfuNew}
\hat\cR_{f,u}\left(\frac rq+\gb\right)
=
\sum_{n(q_{0})}
\sum_{m(q_{0})}
\gl_{f}\left(X,\gb;\frac n{q_{0}},\frac m{q_{0}},u\right)
\cS_{f}(q_{0},ru_{0};n,m)
.
\ee
Put \eqref{eq:RfuNew} into \eqref{eq:RNRf} and apply Cauchy-Schwarz in the $u$-variable:
\bea
\label{eq:cRsq2}
\left|\widehat{\cR_{N}^{U}}
\left(\frac rq+\gb\right)\right|^{2}
&\ll&
U
\sum_{\d< U}
\Bigg|
\sum_{\ff\in\fF}
e_{q}(-r a)
e(-a\gb)
\\
\nonumber
&&
\times
\sum_{0\le n,m<q_{0}}
\gl_{f}\left(X,\gb;\frac n{q_{0}},\frac m{q_{0}},u\right)
\cS_{f}(q_{0},ru_{0};n,m)
\Bigg|^{2}
.
\eea
As before, open the square, setting $\ff'=f'-a'$, and insert the result into \eqref{eq:IQdef}:
\bea
\label{eq:IQbnd2}
\cI_{Q}
&\ll&
U
\sum_{\d< U}
\sum_{q\asymp Q}
\sum_{n,m,n',m'<q_{0}}\
\sum_{\ff,\ff'\in\fF}\
\\
\nonumber
&&
\times
\left[
\sideset{}{'}
\sum_{r(q)}
\cS_{f}(q_{0},r\d _{0};n,m)
\overline{\cS_{f'}(q_{0},r\d _{0};n',m')}
e_{q}(r (a'-a))
\right]
\\
\nonumber
&&
\times
\left[
\int_{|\gb|<1/(qM)}
\gl_{f}\left(X,\gb;\frac n{q_{0}},\frac m{q_{0}},u\right)
\overline{\gl_{f'}\left(X,\gb;\frac {n'}{q_{0}},\frac {m'}{q_{0}},u\right)}
e(\gb (a-a'))
d\gb
\right]
.
\eea
Yet again the sum has split into modular and archimedean components with the former being exactly equal to $\cS$ in \eqref{eq:cSdef}. As before, decompose
$\cI_{Q}$ according to the diagonal ($a=a'$) and off-diagonal terms:
\be\label{eq:cIQ34}
\cI_{Q}\ll \cI_{Q}^{(=)}+\cI_{Q}^{(\neq)}.
\ee
\begin{lem}
Assume $Q\ge X$. For $\square\in\{=,\neq\}$, we have
\be\label{eq:cIQ3def}
\cI_{Q}^{(\square)}
\ll
{U
X^{3}
\over QT}
\sum_{u<U}
{1\over u^{4}}
\sum_{q\asymp Q}\
\sum_{n,m,n',m'\ll {U Q\over X}}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'\square a}
|\cS|
.
\ee
\end{lem}
\pf
Consider the sum $\gl_{f}$ in \eqref{eq:glfDef}. Since $x,y\asymp X/u$,
$|\gb|<1/(qM)$, $X\le Q$, and using \eqref{eq:Mis}, we have that
$$
|\gb u^{2} f(x,y)|\ll \frac1{QM} u^{2} T\left(\frac Xu\right)^{2}= \frac XQ\le 1.
$$
Hence
there is contribution only if $nx/q_{0}, my/q_{0} \ll 1$, that is, we may restrict to the range
$$
n,m\ll u q_{0}/X
.
$$
In this range, we give $\gl_{f}$ the trivial bound of $X^{2}/u^{2}$.
Putting this analysis into
\eqref{eq:IQbnd2},
the claim follows.
\epf
We handle the off-diagonal term first.
\begin{prop}
Assuming $X\le Q<M$, there is some $\eta>0$ such that
\be\label{eq:cI3bnd}
\cI_{Q}^{(\neq)}
\ll
N\, T^{2(\gd-1)}\, N^{-\eta}
,
\ee
as $N\to\infty$.
\end{prop}
\pf
Since \eqref{eq:modBndaNeqAp} is such a large savings in $q>X$, we can afford to lose in the much smaller variable $T$.
Hence put \eqref{eq:modBndaNeqAp} into \eqref{eq:cIQ3def}, estimating $(a-a',q)\le |a-a'|$ (since $a\neq a'$):
\beann
\cI_{Q}^{(\neq)}
&\ll&
{U
X^{3}
\over QT}
\sum_{u<U}
{1\over u^{4}}
\sum_{q\asymp Q}\
\sum_{n,m,n',m'\ll {UQ\over X}}
\sum_{\ff,\ff'\in\fF}
u^{4}
{
a
\cdot
a'
\over q^{5/4}
}
|a-a'|^{1/4}
\\
&\ll&
{U^{6}
X^{3}
\over
T}
\left( {Q\over X}\right)^{4}
T^{2\gd}
{
T^{2}\over Q^{5/4}}
T^{1/4}
\\
&\ll&
U^{6}
X^{7/4}
T^{2\gd}
T^{4}
=
X^{2}T\,
T^{2(\gd-1)}
\,
\left(
U^{6}
X^{-1/4}
T^{5}
\right)
,
\eeann
where we used \eqref{eq:tilQis}, $Q<M$, and \eqref{eq:Mis}.
Using \eqref{eq:TXNis} we have that
\be\label{eq:gS2p}
X^{-1/4}T^{5}=N^{-59/800},
\ee
so together with \eqref{eq:fuIs}, this is clearly a substantial power savings.
\epf
Lastly, we deal with the diagonal term. We no longer save enough from $a=a'$ alone. But recall that here more cancellation can be gotten from \eqref{eq:modBndaEqAp} in the special case that $\ff(m,-n)\neq\ff'(m',-n')$.
Hence we return to
\eqref{eq:cIQ3def} and, once $n,m,$ and $\ff$ are determined, separate $n',m'$, and $\ff'$ into cases corresponding to whether $\ff(m,-n)=\ff'(m',-n')$ or not.
Accordingly, write
\be\label{eq:cIQ312}
\cI_{Q}^{(=)}\
=\
\cI_{Q}^{(=,=)}
+
\cI_{Q}^{(=,\neq)}
.
\ee
We now estimate $\cI_{Q}^{(=,\neq)}$ using the extra cancellation in \eqref{eq:modBndaEqAp}.
\begin{prop}
Assuming $Q<XT$, there is some $\eta>0$ such that
\be\label{eq:cI42bnd}
\cI_{Q}^{(=,\neq)}
\ll
N\, T^{2(\gd-1)}\, N^{-\eta}
,
\ee
as $N\to\infty$.
\end{prop}
\pf
Returning to \eqref{eq:cIQ3def}, apply \eqref{eq:modBndaEqAp}:
\beann
\cI_{Q}^{(=,\neq)}
&\ll&
{U
X^{3}
\over QT}
\sum_{u<U}
{1\over u^{4}}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'= a}
\sum_{q\asymp Q}\
\sum_{n,m\ll {U Q\over X}}
\sum_{n',m'\ll {U Q\over X}\atop \ff(m,-n)\neq\ff'(m',-n')}
|\cS|
\\
&\ll&
{U
X^{3}
\over QT}
\sum_{u<U}
{1\over u^{4}}
\sum_{\ff,\ff'\in\fF}
\sum_{\tilde q_{1}\mid a^{2}\atop \tilde q_{1}\ll Q}
\sum_{q\asymp Q\atop q\equiv0(\tilde q_{1})}
\sum_{n,m,n',m'\ll {U Q\over X}}
u^{10}
{
\tilde q_{1}
\over Q^{9/8}
}
\left(T\left({UQ\over X}\right)^{2}\right)^{1/2}
\\
&\ll_{\vep}&
{U^{8}
X^{3}
\over T}
T^{2\gd}\,
T^{\vep}
\left( {U Q\over X}\right)^{4}
{
1\over Q^{9/8}
}
T^{1/2}
{UQ\over X}
\\
&\ll&
X^{2}
T
\
T^{2(\gd-1)}\,
\left(
X^{-1/8}
T^{35/8}
U^{13}
T^{\vep}
\right)
,
\eeann
where we used
that
$\ff(m,n)\ll T (UQ/X)^{2}$ and $Q<XT$.
From \eqref{eq:TXNis}, we have
\be
\label{eq:gS12p}
X^{-1/8}
T^{35/8}
=
N^{-29/1600}
,
\ee
so we have again a power savings, as claimed.
\epf
Lastly, we turn to the case $\cI_{Q}^{(=,=)}$, with $\ff(m,-n)=\ff'(m',-n')$. We exploit this condition to get savings using \eqref{eq:eqeq}.
\begin{prop}
Assuming $Q<XT$, there is some $\eta>0$ such that
\be\label{eq:cI41bnd}
\cI_{Q}^{(=,=)}
\ll
N\, T^{2(\gd-1)}\, N^{-\eta}
,
\ee
as $N\to\infty$.
\end{prop}
\pf
Returning to \eqref{eq:cIQ3def}, apply \eqref{eq:modBndaNeqAp}, and \eqref{eq:eqeq}:
\beann
\cI_{Q}^{(=,=)}
&\ll&
{U
X^{3}
\over QT}
\sum_{u<U}
{1\over u^{4}}
\sum_{q\asymp Q}\
\sum_{n,m\ll {UQ\over X}}
\sum_{\ff\in\fF}
\sum_{\ff'\in\fF\atop a'= a}
\sum_{n',m'\ll UQ/X\atop \ff(m,-n)=\ff'(m',-n')}
\hskip-.3in
u^{4}
{(a^{2},q)\over q^{5/4}}q^{1/4}
\\
&\ll&
{U
X^{3}
\over Q^{2}T}
\sum_{u<U}
\sum_{\ff\in\fF}
\sum_{\tilde q_{1}\mid a^{2}\atop\tilde q_{1}\ll Q}
\tilde q_{1}
\sum_{q\asymp Q\atop q\equiv0(\tilde q_{1})}\
\left[
\sum_{n,m\ll {UQ\over X}}
\sum_{\ff'\in\fF\atop a'= a}
\sum_{n',m'\ll UQ/X\atop \ff(m,-n)=\ff'(m',-n')}
1
\right]
\\
&\ll_{\vep}&
N^{\vep}
{U
X^{3}
\over Q^{2}T}
U
T^{\gd}
Q
\left[
\left( {UQ\over X}\right)^{2}
+
T
{UQ\over X}
\right]
\\
&\ll_{\vep}&
N^{\vep}
U^{4}
X^{2}
T^{\gd}
\ll
X^{2}T\,
T^{2(\gd-1)}\,
\left(
T^{1-\gd}
U^{4}
N^{\vep}
\right)
.
\eeann
From \eqref{eq:dim}, this is a power savings.
\epf
Combining
\eqref{eq:cIQ34}, \eqref{eq:cI3bnd}, \eqref{eq:cIQ312}, \eqref{eq:cI42bnd}, and \eqref{eq:cI41bnd}, we have the following
\begin{thm}
\label{thm:IQM}
If $X\le Q<M$, then there is some $\eta>0$ so that
$$
\cI_{Q}\ll N\, T^{2(\gd-1)} \, N^{-\eta},
$$
as $N\to\infty$.
\end{thm}
Finally, Theorems \ref{thm:IQ0}, \ref{thm:IQX}, and \ref{thm:IQM} together complete the proof of \eqref{eq:cEN2bnd}, and hence Theorem \ref{thm:Main} is proved.
\newpage
| {
"timestamp": "2013-05-15T02:00:46",
"yymm": "1205",
"arxiv_id": "1205.4416",
"language": "en",
"url": "https://arxiv.org/abs/1205.4416",
"abstract": "We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.",
"subjects": "Number Theory (math.NT)",
"title": "On the Local-Global Conjecture for integral Apollonian gaskets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.986979508737192,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379405608844
} |
https://arxiv.org/abs/1607.08865 | Law of Iterated Logarithm for random graphs | A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$$$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{\sigma_n \sqrt {2 \log \log n }} =1 \right] =1 . $$In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph $H$. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random $k$-uniform hypergraphs, we obtain the Central Limit Theorem (CLT) and LIL for the number of Hamilton cycles. | \section{Introduction}
Let $\{t_i\}_{i=1}^{\infty}$ be an infinite sequence of iid random variables with mean $0$ and variance $1$. Two key results in probability theory are the central limit theorem and the law of the iterated logarithm. The \emph{central limit theorem}
(CLT) states that for $X_n:= \sum_{i=1}^n t_i$, one has
$$ \frac{ X_n }{\sigma_n } \longrightarrow N(0,1),$$ where $\sigma_n := \sqrt {Var X_n } =\sqrt n $ and $N(0,1)$ denotes the standard gaussian distribution. The \emph{law of the iterated logarithm} (LIL), proved by Khinchin \cite{Khinchin} and Kolmogorov \cite{Kolmogorov}, asserts that
$$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ X_n }{\sigma_n \sqrt {2 \log \log n }} =1 \right] =1 . $$
\noindent The $\log \log n$ term reveals a subtle
correlation between the $X_i$'s, especially those with indices close to each other.
The theory of random graphs (hypergraphs) contains several central limit theorems, some of which are among the most well known
results in the field. It is natural to wonder if the LIL also holds.
The goal of this paper is to initiate this investigation and provide
the first few rigorous results. To our surprise, this natural problem has not been studied before and we hope this paper will motivate further activity.
Let $p$ be a fixed constant in $(0,1)$. We consider the infinite random hypergraph $H^k(\mathbb N , p)$ on the vertex set $\mathbb N $ where we add every $k$-subset $S\subseteq \mathbb N $ as an edge with probability $p$ independently. This gives rise to a nested sequence of
random hypergraphs where $H^k(n,p)$ is defined by restriction to the first $n$ vertices $[n]:=\{1, \dots, n \}$. The atom iid variables are $t_{S}$ which represent the edges ($t_{S} =1$ if $S$ forms an edge and $0$ otherwise). In the case of graphs (that is $k=2$) we denote $H^k(\mathbb N ,p)$ by $G(\mathbb N ,p)$ and $H^k(n,p)$ by $G(n,p)$. In this way, we obtain the usual binomial random graph model. We also consider the infinite random bipartite graph $B(\mathbb N ,p)$ on vertex set $A\cup B$, where $A$ and $B$ are two disjoint copies of $\mathbb N $, and every pair $ab\in A\times B$ forms an edge with probability $p$, independently. Let $B(n,p)$ be obtained from $B(\mathbb N ,p)$ by restricting $A$ and $B$ to their first $n$ elements.
Many CLT's in the theory of random graphs involve
some sort of counting functions. For instance, counting
the number of copies of a fixed graph (such as triangles or $C_4$'s)
is a classical problem; see \cite{KarR, Kar, Rucinski} and the references therein (the interested reader can also find a detailed discussion in \cite{JansonLuczakRucinski}, Chapter 6). In this case, the question of when the CLT holds is well understood.
\begin{theorem} Fix a nonempty graph $G$, and let $X_n$ count the number of copies of $G$ in $G(n,p)$. Let $m(G)=\max\{|E(H)|/|V(H)| : H\subset G\}$. If $p=p(n)$ is such that $np^{m(G)}\rightarrow \infty$ and $n^2(1-p)\rightarrow \infty$, then $(X_n-\mathbb{E}[X_n])/\sqrt{Var(X_n)}$ tends in distribution to $N(0,1)$.
\end{theorem}
It is more challenging to count global objects.
In \cite{Janson} Janson considered the numbers of spanning trees, perfect matchings and Hamilton cycles in random graphs. He showed these counting functions are
log-normal for $G(n,p)$ in certain ranges of density. Results of a similar flavor (and shorter proofs) were also obtained later by Gao \cite{Gao}.
\begin{theorem}Let $X_n$ be the random variable that counts number of spanning trees, perfect matchings, or Hamilton cycles in $G(n,p)$. Fix a constant $p<1$. Let $p(n)\rightarrow p$. If $\liminf n^{1/2} p(n)>0$, then
$$
p(n)^{1/2}\left(\log(X_n)-\log(\mathbb{E}[X_n])+\frac{1-p(n)}{c p(n)}\right)\rightarrow N\left(0,\frac{2(1-p)}{c}\right)
$$where $c=1$ in the case of spanning trees and Hamilton cycles, and $c=4$ in the case of perfect matchings.
\end{theorem}
Throughout this paper, we use $X_n$ to denote a statistic of the random model under consideration (that is, $H^k(n,p)$ or $B(n,p)$), with mean $\mu_n$ and variance $\sigma_n^2$, which may vary in each occasion.
First, we consider the case $X_n$ is the number of copies of a fixed graph $H$ in $G(n,p)$ and prove
\begin{theorem}\label{main1} For a fixed graph $H$, let $X_n$ denote the number of copies of $H$ in $G(n,p)$.
The sequence $X_n$ satisfies the LIL, namely
$$\Pr \left[\limsup_{n\rightarrow \infty}
\frac{X_n -\mu_n }{\sigma_n \sqrt{2\log\log n}}=1\right]=1.$$
\end{theorem}
The key ingredient in the
proof of Theorem \ref{main1} is to overcome the fact that the terms in $X_n$ are not completely independent.
Second, we consider the case where $X_n$ is the number of perfect matchings in $B(n,p)$. In this case, we obtain a LIL for the random variable $\log X_n$.
\begin{theorem}\label{thm:main}Let $X_n$ be the number of perfect matchings in $B(n,p)$ and set $Y_n := \log X_n$. Then the sequence $Y_n$ satisfies the LIL, namely
\begin{align}\label{eqtn main thm}
\Pr \left[\lim\sup_{n\rightarrow \infty}\frac{ Y_n- \log (n! p^n ) + \frac{1-p}{2p} }{ \sqrt{2\log \log n} \sqrt { \frac{1-p}{p}} }=1\right]=1
\end{align}
\end{theorem}
Third, we consider the number of Hamilton cycles in $G(n,p)$ and prove
\begin{theorem}
\label{thm:main2} Let $X_n$ be the number of Hamiltonian cycles in $G(n,p)$ and set $Y_n := \log X_n$.
The sequence $Y_n$ satisfies the LIL, namely
$$\Pr \left[\limsup_{n\rightarrow \infty}
\frac{Y_n - \log \left(\frac{(n-1)!}{2}p^n\right)+\frac{1-p}{p} }{\sqrt{\frac{2(1-p)}{p}}\sqrt{2\log\log {n}}}=1\right]=1.$$
\end{theorem}
The proofs of the last two theorems are more involved.
Our new key ingredient is a large deviation bound
on $X_n$ (the number of perfect matchings or Hamiltonian cycles, respectively), which appears to be new and could be of independent interest.
\begin{remark} Note that we did not write $(Y_n-\mathbb{E}[Y_n])/\sqrt{Var(Y_n)}$ in theorems \ref{thm:main} and \ref{thm:main2}, as the expected value and variance of $Y_n$ are unknown. We conjecture that
constants used in the theorem are good approximations of these quantities.
\end{remark}
Next, we consider the case of $k$-uniform random hypergraphs. In this setting, the CLT and the LIL for the number of copies of a fixed subhypergraph can be obtained in a similar way to the graph case. Therefore, we focus on global structures, Hamiltonian cycles in particular.
To start, there are many ways to define a cycle in a hypergraph. We work with the following: an $\ell$-overlapping Hamilton cycle is a cyclic ordering of the vertices $v_1,\ldots,v_n$
for which the edges consisting of $k$ consecutive vertices and two
consecutive edges overlap in exactly $\ell$ vertices. The case $\ell=1$ is known as a ``loose Hamilton cycle" and the case $\ell=k-1$ is known as ``tight Hamilton cycle" (note that the case $\ell=0$ corresponds to a perfect matchings).
Our next result works for all $\ell$, but for the sake of presentation we state it for loose Hamilton cycles (which from now on will be referred to as Hamilton cycles).
Let $X_{n}(k) $ denote the number of Hamilton cycles in $H^k(n,p)$ with mean $\mu_{n}(k)$ and variance $\sigma_{n}(k) ^2$.
We have found out, somewhat surprisingly, that for $k \ge 3$, $X_n(k) $ themselves satisfy
the CLT, as opposed to the case $k=2$ where $\log X_n(2)$ satisfies the CLT. The reason lies in the fact that unlike the case $k=2$, for $k\geq 3$, if we choose a few Hamilton cycles at random, it is very unlikely for them to have common edges and therefore the variance
of the counting function is much smaller compared to $\mu_n(k)^k$.
A similar observation has been used by Dudek and Frieze in \cite{DF} and \cite{DF1} where they determined the threshold behavior of $\ell$ Hamilton cycles.
\begin{theorem}\label{main:CLT}
For any $ k \ge 3$, the sequence $ X_{n}(k) $ satisfies the CLT, namely
$$\frac{ X_{n} (k) - \mu_{n} (k) } {\sigma_{n} (k) } \longrightarrow N(0,1). $$
\end{theorem}
Finally, we show that for $k \ge 4$, the sequence $X_{n} (k) $ satisfies a LIL.
\begin{theorem}\label{main:ILL}
For $k \ge 4$, the sequence $X_{n} (k) $ satisfies the LIL, namely
$$\Pr \left[\limsup_{n\rightarrow \infty} \frac{X_{n} (k) - \mu_{n} (k) }{\sigma_{n} (k) \sqrt {2\log\log n}}=1\right]=1.$$
\end{theorem}
We conclude this section with a few remarks. First, there are many other CLTs in the random graphs/hypergraphs literature, and
it is natural to raise the validity of the LIL in each situation. We hope that this paper will motivate further research in this direction.
As far as the new results are concerned, we prove them under the condition that $p$ is a fixed constant in $(0,1)$. Since we work with
a random infinite graph, letting $p$ depend on $n$ (as one usually does for $G(n,p)$) does not make sense.
However, one can still consider the sparse case by modifying the definition. For instance, one can say that the edge $ij\in \mathbb{N}^2$ appears with probability $p(\max\{i,j\})$, independently,
where $p(k)$ is a sequence of positive numbers tending to $0$ with $k$. It is an interesting question to determine those ranges of densities for which LIL holds.
For a technical reason, the proof of Theorem \ref{main:ILL} requires $k \ge 4$. We leave the case $k=3$ as an open problem.
\vskip2mm
\noindent {\bf Notation.} Throughout the paper, we assume that $n$ is sufficiently large, whenever needed. All asymptotic notation is used under the assumption that
$n \rightarrow \infty$. We will be using the following notation through the paper:
\begin{itemize}
\item
$K_n$ the complete graph on the vertex set $[n]=\{1,\ldots,n\}$.
\item $(t)_\ell:=t(t-1)\ldots(t-\ell+1)$.
\item $G(n,m)$ is the random graph chosen uniformly at random from the set of all graphs on vertex set $[n]$ with exactly $m$ edges.
\item $B(n,m)$ is the random graph chosen uniformly at random from the set of all bipartite graphs, with vertex sets of sizes $n$ with exactly $m$ edges.
\item For a random variable $X$, we write $X^*$ for its normalization: $X^*:= (X-\mathbb{E}[X])/(\sqrt{Var(X)})$.
\item For a graph $H$ we define $\mathcal H$ to be the set of all (labeled) copies of $H$ in
the infinite complete graph on vertex set $\mathbb{N}$. For each
$n\in \mathbb{N}$, we define $\mathcal H_n$ to be the subset of
$\mathcal H$, consisting of all copies of $H$ in $K_n$ (that is, all graphs in $\mathcal H$ which are contained in $[n]$).
\item Given a
copy $h\in \mathcal H$, we denote by $V(h)$ and $E(h)$ its vertex
set and edge set, respectively.
\item In the special case where $H$ is a
\emph{triangle} (that is, a graph on $3$ vertices $\{x,y,z\}$ where
all the three possible edges $\{xy,yz,zx\}$ appear), we replace
$\mathcal H$ with $\mathcal T$ in all of the previous notation.
\item We assume that an enumeration $\mathcal
H=\{h_1,h_2,\ldots\}$ is fixed so that for every $n\in \mathbb{N}$
we have $\mathcal H_n=\{h_1,\ldots,h_\ell\}$, where $\ell$ is the
number of labeled copies of $H$ in $K_n$. Note that such an
enumeration can be easily obtained by an induction on $n$.
\item Suppose $G$ is a random graph (taken from any arbitrary distribution). To each copy $h\in \mathcal H$, we associate an indicator random variable $\xi^G_h $. Whenever the model $G$ is clear from the context, we simply write $\xi_h$.
\item For a collection $\mathcal S$ of copies of $H $ we have $X_ {\mathcal S} :=\sum_{h\in
\mathcal S}\xi_h $.
\item Let $\Phi(x)$ denote the cumulative distribution function of the standard gaussian $N(0,1)$:
$$\Phi(x):= \Pr[ N(0,1) \le x] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-t^2/2}dt. $$
\item For an event $\mathcal E$, we denote its complement by $\neg \mathcal E$ (i.e., the event that $\mathcal E$ does not hold).
\end{itemize}
{\bf Organization of the paper.} The rest of the paper is organized
as follows. In Section \ref{sec:tools} we collect the tools which
are used for the proof of our main results. In Sections
\ref{sec:upper} and \ref{sec:lower} we prove the upper and lower
bounds for Theorem \ref{main1}, and in Section \ref{sec:calculations}
we explain some inequalities we use during the proof. In Section \ref{perfect matchings} we prove Theorem \ref{thm:main}, and in Section \ref{sec: main2} we prove Theorem \ref{thm:main2}. Both of these sections are split into two subsections containing the proof of the upper bound and the lower bound, respectively. Section \ref{sec:clt hypergraphs} contains the proof of \ref{main:CLT}, and Section \ref{sec: ill hypergraph} contains the proof of \ref{main:ILL}. Section \ref{upper tail estimates} contains the new large deviation estimates we need on perfect matchings and Hamilton cycles. The appendix contains some rather routine, but tedious, calculations and approximations that we use throughout the paper.
\section{Tools}\label{sec:tools}
In this section we introduce the main tools to be used in the proofs
of our results.
As a first tool, we present Janson's inequality (see e.g.\
\cite{JansonLuczakRucinski}, Theorem 2.14), which will be used in
order to get lower tail estimates for the number of copies of a
fixed graph $H$ in certain random graphs. We only use it in the model $G(\mathbb N,p)$ where $p$ is a fixed constant. For the convenience of the
reader, we state the inequality tailored for our use later (with respect to the
$\xi^G_h$'s which were previously defined). Before doing so, we need
some notation. Let $m\leq n$ be two positive integers, and let
$\mathcal S:=\mathcal H_n\setminus \mathcal H_m$. Consider the
random variable $\xi^G_{\mathcal S}=\sum_{h\in \mathcal S}\xi^G_h$, let $\mu_{\mathcal S}$ be its
expectation, and let $$\Delta:=\sum_{h,h'\in \mathcal S \text{ s.t }\\ h\cap h'\neq \emptyset}
\mathbb{E}[\xi_h\xi_{h'}].$$ With this notation in hand we are ready to state
the theorem.
\begin{theorem}\label{Janson}
For a fixed graph $H$ and for every $0\leq t\leq \mu_{\mathcal S}$
we have
$$\Pr\left[\xi^G_{\mathcal S}\leq \mu_{\mathcal S}-t\right]\leq
e^{-\frac{t^2}{2\Delta}}.$$
\end{theorem}
\begin{remark} \label{remJanson}
For the special case where $H$ is a triangle, it is easy to show (by
fixing the intersection edge) that $\Delta\leq \mu_{\mathcal
S}+(\binom{m}{2}(n-m)^2+\binom{n-m}{2}n^2+m(n-m)n^2)p^5$. We make
use of this later.
\end{remark}
Another tool to be used in our proofs is the following well known
lemma due to Borel and Cantelli.
\begin{lemma}[Borel-Cantelli Lemma]
Let $(A_i)_{i=1}^{\infty}$ be a sequence of events. Then
\begin{enumerate}[$(a)$]
\item If $\sum_{k}\Pr\left[A_k\right]<\infty$, then
$$\Pr\left[A_k \textrm{ for infinitely many }k\right]=0.$$
\item If $\sum_k \Pr\left[A_k\right]=\infty$ and in addition all
the $A_k$'s are independent, then
$$\Pr\left[A_k \textrm{ for infinitely many }k\right]=1.$$
\end{enumerate}
\end{lemma}
The following theorem due to Rinott \cite{Rinott} shows that, under some assumptions, the sum
of dependent random variables satisfies CLT, and measures the error
term based on the dependencies between the variables. Before stating
it explicitly, we need the following definition.
\begin{definition} \label{def:dependencygraph}
Let $(X_i)_{i\in I}$ be a collection of random variables. A graph
$D$ on a vertex set $I$ is called a \emph{dependency graph} for the
collection if for any pair of disjoint subsets $I_1,I_2\subseteq I$
for which there are no edges of $D$ between $I_1$ and $I_2$, the
random variables $(X_i)_{i\in I_1}$ and $(X_j)_{j\in I_2}$ are
independent.
\end{definition}
Now we state the result from
\cite{Rinott} which we are going to use.
\begin{theorem}[Theorem 2.2 in \cite{Rinott}]\label{thm:Rinott}
Let $(t_i)_{i=1}^n$ be a collection of random variables. Let
$X=\sum_{i=1}^n t_i$ and $\mu:=\mathbb{E}(X)$ and $\sigma^2:=Var(X)>0$.
Let $D$ be a dependency graph for the collection and suppose that
$|t_i-\mathbb{E}(t_i)|\leq B$ a.s.\ for every $i$ and that
$\Delta(D)\leq C$. Then
$$\left|\Pr\left[\frac{X-\mu}{\sigma}\leq x\right]-\Phi(x)\right|\leq \frac
{BC}\sigma \left(\sqrt{\frac 1{2\pi}}+16
\left(\frac{n}{\sigma^2}\right)^{1/2}C^{1/2}B+10\left(\frac{n}{\sigma^2}\right)
CB^2\right).$$
\end{theorem}
\begin{remark}\label{rem1}
Note that whenever $\sigma^2=\Omega(nCB^2)$ the expression in the
right hand side of the inequality in Theorem \ref{thm:Rinott} is
$O\left(\frac{BC}{\sigma}\right)$. Assuming this, since
$\lim_{x\rightarrow \infty}\frac{\int_x^{\infty}e^{-t^2/2}dt}{\frac
1x e^{-x^2/2}}=1$, for large enough $x$ it follows by Theorem
\ref{thm:Rinott} that
$$\left|\Pr\left[\frac{X-\mu}{\sigma}\geq x\right]-\frac
{1}{x\sqrt{2\pi}} e^{-x^2/2}\right|=
O\left(\frac{BC}{\sigma}\right).$$
\end{remark}
The key tools in the proofs of Theorems \ref{thm:main} and \ref{thm:main2} are the following concentration bounds, which may be of independent interest. We postpone their proofs to Section \ref{upper tail estimates}.
\begin{lemma}\label{conc0}Let $X_{n,m}$ be the number of perfect matchings in $B(n,m)$. Let $0<\delta<1/2$ be a constant. There is a constant $C$, depending on $\delta$, such that for any $\delta n^2\leq m\leq (1-\delta)n^2$, and $k=o(n^{1/3})$, we have
\[
\mathbb{E}[X_{n,m}^k]\leq C^k(\mathbb{E}[X_{n,m}])^k
\]
\end{lemma}Markov's bound implies that for $K\geq C$ one has:
\[
\Pr[X_{n,m}\geq K \mathbb{E}[X_{n,m}]]\leq (C/K)^k
\]by taking $\delta:=\min\{p/2,(1-p)/2\}$, $k=4\log n$, and $K=Ce$, we have the following corollary
\begin{corollary}\label{concentration0}Let $0<p<1$ be a constant. There is a constant $K$ (depending on $p$) such that for any $\frac{p}{2}n^2\leq m\leq \frac{1+p}{2}n^2$ one has
$$
\Pr[X_{n,m}\geq K\mathbb{E}[X_{n,m}]]\leq n^{-4}
$$
\end{corollary}
\noindent The concentration bounds for Hamilton cycles are as follows
\begin{lemma}\label{conc1} Let $X_{n,m}$ be the number of Hamilton cycles in $G(n,m)$. Let $0<\delta<1/2$ be a constant. There is a constant $C$, depending on $\delta$, such that for any $\delta {n \choose 2}\leq m\leq (1-\delta){n \choose 2}$, and $k\leq \frac{\log n}{8}$ we have:
\[
\mathbb{E}[X_{n,m}^k]\leq C^k(\mathbb{E}[X_{n,m}])^k
\]
\end{lemma}
Again, Markov's bound implies that for $K\geq C$ one has:
\[
\Pr[X_{n,m}]\geq K \mathbb{E}[X_{n,m}]]\leq (C/K)^k
\]by taking $\delta:=\min\{p/2,(1-p)/2\}$, $k=\frac{\log n}{8}$, and $K=Ce^{32}$, we have the following corollary
\begin{corollary}\label{concentration}Let $0<p<1$ be a constant. There is a constant $K$ (depending on $p$) such that for any $\frac{p}{2}{n\choose 2}\leq m\leq \frac{1+p}{2}{n\choose 2}$ one has
$$
\Pr[X_{n,m}\geq K\mathbb{E}[X_{n,m}]\leq n^{-4}
$$
\end{corollary}
The last lemma is an approximation to the lower factorial that we will use throughout.
\begin{lemma}\label{lower factorial}Let $t,\ell$ be integers such that $\ell=o(t^{2/3})$. Then,
\[
(t)_\ell=t^\ell\exp\left(-\frac{\ell(\ell-1)}{2t}+o(1)\right)
\]
\end{lemma}
In the proof of the upper-tail estimate for perfect matchings, we will need Bregman's theorem, which allows us to bound the number of perfect matchings given the degree sequence:
\begin{theorem}[Bregman-Minc inequality; \cite{Bregman}]\label{bregman}Let $G$ be a bipartite graph with two color classes $V=\{v_1,\ldots,v_n\}$ and $W=\{w_1,\ldots,w_n\}$. Denote by $M$ the number of perfect matchings and
$d_{v_i}$ the degree of $v_i$. Then \[
M\leq \prod_{i=1}^n(d(v_i)!)^{1/d(v_i)}
\]
\end{theorem}
\section{Proof of Theorem \ref{main1}} \label{sec:proofs}
\begin{proof}
Let $H$ be a graph on $\ell$ vertices, where $\ell$ is a fixed
constant. For the sake of simplicity of notation, throughout the
whole proof we omit the up-script $G$ from the random variables. In
order to prove Theorem \ref{main1} we aim to show that for every
$\varepsilon>0$ we have both the upper bound
$$\Pr\left[\frac{X_n-\mu_n}{\sigma_n} \geq (1+\varepsilon)\sqrt{2\log\log n} \textrm{ for infinitely
many }n\right]=0,$$ and the lower bound
$$\Pr\left[\frac{X_n-\mu_n}{\sigma_n} \geq (1-\varepsilon)\sqrt{2\log\log n} \textrm{ for infinitely
many }n\right]=1.$$
Since throughout the proof we make use of Theorem \ref{thm:Rinott}
for estimating the upper tails of random variables of the form
$X_n-X_m$, it will be convenient to introduce some notation. For
every $n\geq m$ let $\mathcal S_{n,m}=\mathcal H_n\setminus \mathcal
H_m$, where $\mathcal S_{n,0}=\mathcal H_n$. Let us define a
dependency graph for $\mathcal S_{n,m}$ in the following manner. The
vertex set of $D_{n,m}$ is $\mathcal S_{n,m}$, and the edge set
consists of all pairs $s,t\in \mathcal S_{n,m}$ for which $|E(s)\cap
E(t)|\geq 1$ (that is, pairs of copies of $H$ which share at least
one edge). Note that it trivially follows from the way we labeld
$\mathcal H$ that $V(D_{n,m})=\mathcal S_{n,m}$ is the number of
copies of $H$ with at least one vertex taken from
$\{m+1,\ldots,n\}$. In addition, it is easy to see that
$$\Delta(D_{n,m})\leq
c'_H|E(H)|n^{|V(H)|-2}=\Theta(n^{\ell-2}),$$ where $c'_H$ is the
maximum number of automorphisms of $H$ preserving some edge. Now,
let us denote by $X_{n,m}:=X_{\mathcal S_{n,m}}$ and let
$\mu_{n,m}$ and $\sigma^2_{n,m}$ be its expectation and variance,
respectively. Trivially, we have $\mu_{n,m}=\mu_n-\mu_m$ and
$|\xi_t-\mathbb{E}(\xi_t)|\leq 1$ for every $t\in V(D_{n,m})$.
Therefore, while applying Theorem \ref{thm:Rinott} for a large $x$
with $C=\Delta(D_{n,m})$ and $B=1$, using Remark \ref{rem1} we
obtain
\begin{equation} \label{eq:rinott}
\Pr\left[\frac{X_{n,m}-\mu_{n,m}}{\sigma_{n,m}}\geq x\right]=\frac 1{x\sqrt{2\pi}} e^{-x^2/2}+O\left(\frac{n^{\ell-2}}{\sigma_{n,m}}\right).
\end{equation}
Note that whenever we use \eqref{eq:rinott}, one should verify that the error
term is negligible compared to the first summand on the right hand
side. Most of the times it will be quite easy to check and therefore
we omit the calculations. For some relevant estimates on the variances that we use in the proof, the reader should consult Section \ref{sec:calculations}.
Let us start with proving the upper bound.
\subsection{Upper bound}\label{sec:upper}
Let $\varepsilon>0$ be some positive constant and let
$x=(1+\varepsilon/4)\sqrt{2\log\log n}$. Note that for a fixed $n$, by distinguishing between the two cases $|h\cap h'|=2$ and $|h\cap h'|>2$, we obtain $$\sigma^2_n:=\sigma^2_{n,0}=\Theta\left(n^{\ell}+\sum_{h,h'\in \mathcal S_n}Cov(\xi_h\xi_h')\right)=\Theta\left(n^{\ell}+n^{2\ell-2}+n^{\ell}n^{\ell-3}\right).$$
Therefore, by \eqref{eq:rinott} we have
\begin{align*}\Pr\left[\frac{X_n-\mu_n}{\sigma_n}\geq
x\right]&=\frac 1{x\sqrt{2\pi}}
e^{-x^2/2}+O\left(\frac{1}{n}\right)\\
&=O\left((\log n)^{-(1+\varepsilon/4)^2}\right).
\end{align*}
Using this estimate for every (large enough) $n$ of the form $a^k$ (where $a>1$),
we obtain that
$$\sum_{k=1}^\infty \Pr\left[\frac{X_{a^k}-\mu_{a^k}}{\sigma_{a^k}}\geq
(1+\varepsilon/4)\sqrt{2\log\log {a^k}}\right]=\sum_{k=1}^\infty
O\left(k^{-(1+\varepsilon/4)^2}\right)<\infty ,$$
and therefore, it follows from the Borel-Cantelli Lemma that for
some $k_0\in \mathbb{N}$ we have
$$\Pr\left[\frac{X_{a^k}-\mu_{a^k}}{\sigma_{a^k}}\leq
(1+\varepsilon/4)\sqrt{2\log\log {a^k}} \textrm{ for all }k\geq
k_0\right]=1.$$
Note that if $a$ is not an integer then we always assume that $k$ is sufficiently large and we set $n=\lfloor a^k\rfloor$. As it does not affect any of our asymptotic calculations, we will omit the flooring signs.
In order to complete the proof (of the upper bound), we need to
``close the gaps". That is, we need to show that there exists
$k_1\in \mathbb{N}$ such that
$$\Pr\left[\frac{X_{n}-\mu_{n}}{\sigma_{n}}\leq
(1+\varepsilon)\sqrt{2\log\log {n}}\textrm{ for every } n\geq
a^{k_1}\right]=1.$$
To this end we act in the following way. Fix $a>1$ which is close
enough to $1$ (to be determined later), and we show that
$\sum_{k}\delta_k<\infty$, where
$$\delta_k:=\Pr\left[X_{n,a^k}-\mu_{n,a^k}\geq
\frac{\varepsilon}2\sigma_n\sqrt{2\log\log {n}} \textrm{ for some
}a^k\leq n\leq a^{k+1}\right].$$
Therefore, using the Borel-Cantelli Lemma we conclude that there
exists $k_1$ for which
$$\Pr\left[X_{n,a^k}-\mu_{n,a^k}<
\frac{\varepsilon}2\sigma_n\sqrt{2\log\log {n}} \textrm{ for every }
k\geq k_1 \text{ and }a^k\leq n\leq a^{k+1}\right]=1.$$
Next, recall that
$$\Pr\left[X_{a^k}-\mu_{a^k}\leq
(1+\varepsilon/4)\sigma_{a^k}\sqrt{2\log\log {a^k}} \textrm{ for all
}k\geq k_0\right]=1,$$ and set $k_2:=\max\{k_0,k_1\}$.
All in all, we obtain that with probability 1, for every $k\geq k_2$
and for every $a^k\leq n\leq a^{k+1}$ we have
\begin{align*} X_n-\mu_n&=[(X_n-X_{a^k})-(\mu_n-\mu_{a^k})]+
(X_{a^k}-\mu_{a^k})\\&=(X_{n,a^k}-\mu_{n,a^k})+(X_{a^k}-\mu_{a^k})\\
&< \frac{\varepsilon}2\sigma_n\sqrt{2\log\log
{n}}+(1+\varepsilon/4)\sigma_{a^k}\sqrt{2\log\log n}\\
&< (1+\varepsilon)\sigma_n\sqrt{2\log\log n},
\end{align*}
as desired.
In order to complete our argument, we need to estimate $\delta_k$
and to show that indeed $\sum \delta_k<\infty$. This is done in the
following claim, which is a modification of Levy's inequality to our special case of dependent random variable.
\begin{claim} \label{claim1}
$\delta_k$ is such that $\sum \delta_k<\infty$.
\end{claim}
\begin{proof} Fix $k\in \mathbb{N}$. For each $m\leq n$ and for each $\tau>0$, let $\mathcal
E_{n,m,\tau}$ denote the event $\{X_{n,m}-\mu_{n,m}\leq \tau\}$.
Let $n=a^{k+1}$, $\tau=\frac{\varepsilon}2\sigma_n\sqrt{2\log\log
n}$, and for every $a^k\leq j\leq a^{k+1}$ define
$$A_j:=\begin{cases}\left(\bigcap_{i=a^k}^{j-1}\mathcal E_{i,a^k,\tau}\right)\cap \neg\mathcal E_{j,a^k,\tau} &\mbox{ for } j\geq a^k+1 \\
\neg\mathcal E_{a^k,a^k,\tau}& \mbox{ for } j=a^k.
\end{cases}$$
Note that we have $\tau=\sigma_n\cdot \omega(1)$ and that $\tau\leq \mu_{n,j}$, both will be used later in the proof. In order to see the latter, recall that
$\mu_{n,j}=\Theta((n-j)n^{\ell-1})$ and $\sigma^2_n\leq n^{\ell}+O(n^{2\ell-2})$. Therefore, $\sigma_n=O(n^{\ell-1})= o(\mu_{n,j}/\log n)$ for all $j\leq n-\omega(\log n)$.
Now, let $M_n:=\bigcup_{j=a^k}^n \neg\mathcal E_{j,a^k,\tau}$ and
note that $M_n=\bigcup_{j=a^k}^{n} A_j$ and that
$\delta_k=\Pr\left[M_{n}\right]$. We start with evaluating the
following probability:
\begin{align}\label{1} \Pr\left[M_{n}\cap \mathcal
E_{n,a^k,\tau/2}\right]=\sum_{j=a^k}^{n}\Pr\left[A_j\cap \mathcal
E_{n,a^k,\tau/2}\right].
\end{align}
Note that if $A_j\cap \mathcal E_{n,a^k,\tau/2}$, then in particular
we have
$$X_{j,a^k}-\mu_{j,a^k}> \tau \textrm{ and }
X_{n,a^k}-\mu_{n,a^k}\leq \tau/2.$$
Therefore, we conclude that
$$(X_{j,a^k}-\mu_{j,a^k})-(X_{n,a^k}-\mu_{n,a^k})>\tau/2,$$
which is equivalent to
\begin{align}\label{2}
X_{n,j}<\mu_{n,j}-\tau/2.
\end{align}
Moreover, a moment's thought reveals that for every $j$, the events
$\{X_{n,j}<\mu_{n,j}-\tau/2\}$ and $A_j$ are negatively
correlated, and therefore, one can upper bound \eqref{1} by
\begin{align}\label{3}
\sum_{j=a^k}^{n}\Pr\left[A_j \textrm{ and
}(X_{n,j}<\mu_{n,j}-\tau/2)\right]&\leq
\sum_{j=a^k}^{n}\Pr\left[A_j\right]\Pr\left[X_{n,j}<\mu_{n,j}-\tau/2\right].
\end{align}
Now, since clearly $\sigma_{n,j}\leq \sigma_n$, and since
$\tau=\sigma_n \cdot \omega(1)$, it follows by \eqref{eq:rinott}
that for every $a^k\leq j\leq n-\log^2n$ we have
\begin{align}\label{4}
\Pr\left[X_{n,j}<\mu_{n,j}-\tau/2\right]&=\Pr\left[\frac{X_{n,j}-\mu_{n,j}}{\sigma_{n,j}}<\frac{\tau}{2\sigma_{n,j}}\right]\nonumber \\
&=\exp\left(-\omega(1)\right)=o(1).
\end{align}
For larger values of $j$ we will simply observe that
$$\Pr\left[X_{j,a^k}>\mu_{j,a_k}+\tau\right]=o(1),$$
as desired.
Combining \eqref{4} with \eqref{1} and \eqref{3}, we obtain
\begin{align} \label{5}
\Pr\left[M_{n}\cap \mathcal E_{n,a^k,\tau/2}\right]= \delta_k \cdot
o(1).
\end{align}
As a penultimate step, we need to estimate $\Pr\left[\neg\mathcal
E_{n,a^k,\tau/2}\right]$. In order to do so we first observe that
since we choose $a>1$ to be very close to $1$, it is easy to verify
that in this case we have
$\sigma^2_{n,j}=\Theta\left(j(n-j)n^{2\ell-4}\right)$ (while
$\sigma^2_n=\Theta\left(n^{2\ell-2}\right)$). Now, together with
\eqref{eq:rinott}, these estimates imply that for some small constant
$C:=C(\varepsilon)>0$ we have
\begin{align}\label{6}
\Pr\left[\neg\mathcal
E_{n,a^k,\tau/2}\right]&=\Pr\left[\frac{X_{n,a^k}-\mu_{n,a^k}}{\sigma_{n,a^k}}\geq
\tau/(2\sigma_{n,a^k})\right] \nonumber \\
&\leq \exp\left( -\frac{\tau^2}{8\sigma^2_{n,a^k}}\right)
\nonumber\\
&=\exp \left(-\frac{Ca^{4k}\log\log a^{k+1}}{a^{4k}(a-1)}\right),
\end{align}
and by choosing $a-1<C/2$, one can upper bound \eqref{6} with
$k^{-2}$ (for large $k$).
All in all, we obtain
\begin{align*}
\delta_k=\Pr\left[M_n\right]&=\Pr\left[M_n\cap \mathcal
E_{n,a^k,\tau/2}\right]+\Pr\left[M_n\cap \neg\mathcal
E_{n,a^k,\tau/2}\right]\\
&\leq \delta_k\cdot o(1)+k^{-2},
\end{align*}
and therefore, $\delta_k=O(k^{-2})$ and $\sum \delta_k<\infty$ as
desired. This completes the proof of the claim, and therefore the
proof of the upper bound as well.
\end{proof}
Before we proceed to the lower bound, let us make a few observations
which can be obtained in a similar way as the above proof. We make
use of those in the next subsection.
\begin{enumerate}[$(O1)$]
\item For every $\varepsilon>0$ we have $\Pr\left[X_n-\mu_n \leq
-(1+\varepsilon)\sigma_n\sqrt{2\log\log n} \textrm{ for infinitely
many }n\right]=0$.
\item For $k\in \mathbb{N}$, let $\zeta_k$ be the random variable
counting the number of copies of $H$ with vertices from both
$\{a^{k}+1,\ldots, a^{k+1}\}$ and $[a^k]$. Let us also denote by
$\widetilde {\mu}_k$ and $\widetilde{\sigma}_k^2$ its expectation
and variance, respectively. Then, for every $\varepsilon>0$ we have
$$\Pr\left[\zeta_k-\widetilde{\mu}_{k}\leq
-(1+\varepsilon)\widetilde{\sigma}_k\sqrt{2\log\log a^{k+1}}
\textrm{ for infinitely many }k\right]=0.$$
\end{enumerate}
\subsection{Lower bound}\label{sec:lower}
Let $\varepsilon>0$ be some fixed positive constant, we aim to show
that
$$\Pr\left[\frac{X_n-\mu_n}{\sigma_n}\geq
(1-\varepsilon)\sqrt{2\log\log n} \textrm{ for infinitely many
}n\right]=1.$$
To this end, we focus on integers $n_k$ of the form $a^k$, where
$a>1$ is a large enough constant to be determined later.
For a fixed $k\in \mathbb{N}$, let $\eta_k$ be the random variable
that counts the number of copies of $H$ which are fully contained in
$\{a^k+1,\ldots,a^{k+1}\}$. Note that the set $\{\eta_k :
k\in\mathbb{N}\}$ is clearly independent, and that the random
variables $\eta_k$ are distributed the same as $X_{a^{k+1}-a^k}$
(and therefore, $\sigma^2_{\eta_k}=\sigma^2_{a^{k+1}-a^k}$ for every
$k$). Therefore, one can easily check that for large $a$ and $k$ we
have
\begin{align}\label{lowerbound:1}(1-\varepsilon/4)\sigma_{\eta_k}\sqrt{2\log\log
(a^{k+1}-a^{k})}\geq
(1-\varepsilon/2)\sigma_{a^{k+1}}\sqrt{2\log\log a^{k+1}}
\end{align}
(this can be verified using the simple observation that
$\lim_{a,k\rightarrow \infty}\frac{\log\log
(a^{k}-a^{k-1})}{\log\log a^k}=1$ and the estimate \eqref{varXin}
given in Section \ref{sec:calculations}).
Now, letting $x=(1-\varepsilon/4)\sqrt{2\log\log (a^{k+1}-a^k)}$ it
follows by $(*)$ that for some $\gamma>0$ we have
$$\Pr\left[\frac{\eta_k-\mu_{\eta_k}}{\sigma_{\eta_k}}\geq
x\right]=\Omega\left(k^{-1+\gamma}\right),$$
and therefore,
$$\sum_k\Pr\left[\frac{\eta_k-\mu_{\eta_k}}{\sigma_{\eta_k}}\geq
x\right]=\infty.$$
Using the Borel-Cantelli Lemma it thus follows that
$$\Pr\left[\frac{\eta_k-\mu_{\eta_k}}{\sigma_{\eta_k}}\geq
x \textrm{ for infinitely many } k\right]=1.$$
Now, let us choose $a>1$ to be a fixed large enough constant so
that for sufficiently large $k$ the following inequalities hold (the
existence of such $a$ for which all these inequalities hold follows
immediately from the relevant estimates in Section
\ref{sec:calculations}):
\begin{enumerate} [$(i)$]
\item $(1-\varepsilon/4)\sigma_{\eta_k}\sqrt{2\log\log (a^{k+1}-a^{k})}\geq (1-\varepsilon/2)\sigma_{a^{k+1}}\sqrt{2\log\log
a^{k+1}},$ and
\item $(1+\varepsilon)\sigma_{a^{k}}\sqrt{2\log\log a^{k}}\leq
(\varepsilon/4)\sigma_{a^{k+1}}\sqrt{2\log\log a^{k+1}},$ and
\item $(1+\varepsilon)\widetilde{\sigma}_{k}\sqrt{2\log\log a^{k+1}}\leq
(\varepsilon/4)\sigma_{a^{k+1}}\sqrt{2\log\log a^{k+1}}.$
\end{enumerate}
All in all, combining the above mentioned estimates and
$(i)$-$(iii)$ we conclude
\begin{align*}
X_{a^{k+1}}-\mu_{a^{k+1}}&=(\eta_k-\mu_{\eta_k})+(X_{a^{k}}-\mu_{a^{k}})+(\zeta_{k}-\widetilde{\mu}_{k})\\
&\geq
(1-\varepsilon/4)\sigma_{\eta_k}\sqrt{2\log\log(a^{k+1}-a^{k})}-(1+\varepsilon)\sigma_{a^{k}}\sqrt{2\log\log
a^{k}}-(1+\varepsilon)\widetilde{\sigma}_{k}\sqrt{2\log\log
a^{k+1}}\\
&\geq (1-\varepsilon/2)\sigma_{a^{k+1}}\sqrt{2\log\log
a^{k+1}}-(\varepsilon/2)\cdot\sigma_{a^{k+1}}\sqrt{2\log\log a^{k+1}}\\
&\geq (1-\varepsilon)\sigma_{a^{k+1}}\sqrt{2\log\log a^{k+1}},
\end{align*}
as desired. This completes the proof.
\end{proof}
\subsection{Relevant estimates for the variances appearing in the proof of theorem \ref{main1}}\label{sec:calculations} In this section we verify \eqref{lowerbound:1}, $(ii)$ and $(iii)$, by estimating the relevant variances.
Before doing so, recall that
\begin{align*}
Var(X_1+\ldots+X_n)=\sum_{i=1}^nVar(X_i)+\sum_{i\neq j}Cov(X_i,X_j),
\end{align*}
where $Cov(X,Y)=\mathbb{E}XY-\mathbb{E}X\mathbb{E}Y$. Moreover, note
that whenever $X$ and $Y$ are independent, then $Cov(X,Y)=0$.
Therefore, given a subset $\mathcal S\subseteq \mathcal H$, it
follows that
\begin{align}\label{sumVar}
Var(X_{\mathcal S})&=\sum_{t\in \mathcal S}Var(\xi_t)+\sum_{t\neq
s \textrm{ and } E(t)\cap E(s)\neq \emptyset}Cov(\xi_t,\xi_s)
\nonumber \\
&=\sum_{t\in \mathcal
S}Var(\xi_t)+\sum_{i=1}^{\ell-1}\sum_{|E(t)\cap
E(s)|=i}Cov(\xi_t,\xi_s).
\end{align}
In addition, recall that each of the $\xi_t$'s is an indicator
random variable for an appearance of a certain copy of $H$ (where
$|V(H)|=\ell$ and $|E(H)|=m$), and therefore we have
\begin{align*}
\mathbb{E}\xi_t=p^m \textrm{ and } Var(\xi_t)=p^m(1-p^m)=p^m-p^{2m}.
\end{align*}
Next, recall that $p$ and $\ell:=|V(H)|$ are fixed constants and
that we always assume $a$ and $k$ to be large enough. In particular,
it easy to see that the (asymptotically) largest element in the right
hand side of \eqref{sumVar} is the case $i=1$.
Now we can give some easy estimates.
{\bf Estimating $\sigma^2_n:=Var(X_n)$:} Recall that $X_n$ is a
sum of indicator random variables for all the (labeled) copies of
$H$ in $K_n$. Therefore, there exists a constant $C$ (which depend
of the number of automorphisms which preserve some edge) such that
the number of pairs $(s,t)$ of copies of $H$ which intersect in
exactly one edge is roughly $(1+o(1))Cn^{2\ell-2}$. Therefore,
running over all possible intersection edges we obtain that
\begin{align}\label{varXin}
\sigma^2_n&=(1+o(1))Cn^{2\ell}(p^{2m-1}-p^{2m}).
\end{align}
Now, note that since
\begin{align*}
\sigma^2_{a^{k+1}-a^k}&=(1+o(1))C(a^{k+1}-a^k)^{2\ell}(p^{2m-1}-p^{2m})\\
&=(1+o(1))C(a^k(a-1))^{2\ell}(p^{2m-1}-p^{2m}),
\end{align*}
by taking $a$ to be sufficiently large we obtain that
$$\sigma^2_{a^{k+1}-a^k}=(1+o(1))Ca^{2\ell(k+1)}(p^{2m-1}-p^m)$$
which is of the same order of magnitude as $\sigma^2_{a^{k+1}}$.
This verifies \eqref{lowerbound:1}.
In order to verify $(ii)$ all we need is to note that the quantity
$\sigma^2_{a^{k+1}}/\sigma^2_{a^k}$ is a function that tends to
infinity whenever $a$ does.
Finally, in order to verify $(iii)$ let us first estimate
$\widetilde{\sigma}^2_k$.
{\bf Estimating $\widetilde{\sigma}^2_{k}:=Var(\zeta_k)$:} Let $k\in
\mathbb{N}$ and $a>0$. Recall that $\zeta_k$ counts the number of
copies of $H$ with vertices from both $\{a^k+1,\ldots,a^{k+1}\}$ and
$[a^k]$. In this case, assuming $a$ goes to infinity, it is easy to
see that the largest summand in \ref{sumVar} is obtained whenever
the intersection edge is between $[a^k]$ and
$\{a^k+1,\ldots,a^{k+1}\}$. Therefore, for some constant $C'$ (which
does not depend on $a$) we obtain
\begin{align}\label{varZeta}
\widetilde{\sigma}^2_k&=(1+o(1))C'a^k(a^{k+1}-a^k)\left(a^{k+1}\right)^{2\ell-2}(p^{2m-1}-p^{2m})\nonumber\\
&=(1+o(1))C'\frac{1}{a}\left(a^{k+1}\right)^{2\ell}(p^{2m-1}-p^{2m}).
\end{align}
Note that by \eqref{varZeta} and \eqref{varXin} it follows that
$\widetilde{\sigma}^2_k=\Theta\left(\frac 1a
\sigma^2_{a^{k+1}}\right)$, and therefore, by taking $a$ to be
sufficiently large, $(iii)$ trivially holds.
\section{Proof of Theorem \ref{thm:main}}\label{perfect matchings}
Throughout the next section we are going to let $X_{n,m}$ be the number of perfect matchings in $B(n,m)$, $X_n$ the number of perfect matchings in $B(n,p)$ and $Y_n:=\log X_n$. We aim to prove:
\[
\Pr \left[\limsup_{n\rightarrow \infty}\frac{ Y_n- \log (n! p^n ) + \frac{1-p}{2p} }{ \sqrt{2\log \log n} \sqrt { \frac{1-p}{p}} }=1\right]=1
\]It will be enough to show that for
$\varepsilon>0$ we have both the upper bound
\begin{align}\Pr\left[\frac{ Y_n- \log (n! p^n ) + \frac{1-p}{2p} }{ \sqrt { \frac{1-p}{p}} } \geq (1+\varepsilon)\sqrt{2\log\log n^2} \textrm{ for infinitely
many }n\right]=0,\nonumber
\end{align}
and the lower bound
\begin{align}\Pr\left[\frac{ Y_n- \log (n! p^n ) + \frac{1-p}{2p} }{ \sqrt { \frac{1-p}{p}} } \geq (1-\varepsilon)\sqrt{2\log\log n^2} \textrm{ for infinitely
many }n\right]=1.\nonumber
\end{align}Note that in the equations above we have $\log\log{n^2}$, but those can be replaced by $\log\log n$ since the two quantities are asymptotically equal.
\subsection{Upper Bound}\label{upper bound section pm}
We need to prove that for any fixed $\varepsilon>0$
\begin{equation} \label{upper pm}
\Pr \left[\frac{\log X_n-\log (n! p^n)+\frac{1-p}{2p} }{ \sqrt { \frac{1-p}{p} }}\geq (1+\varepsilon)\sqrt{2 \log \log n^2}\text{ for infinite many }n\right]=0. \end{equation}
\noindent By Corollary \ref{concentration0}, there is a constant $K$ such that for all $ \frac{p}{2} n^2 \le m \le \frac{1+p}{2} n^2 $
$$ X_{n,m} \le K \mathbb{E}[X_{n,m}] $$ with probability at least $1 -n^{-4} $. Taking $\log$, we conclude that with the same probability
\begin{equation} \label{upper pm1}
\log X_{n,m} \le \log \mathbb{E}[X_{n,m}] + \log K. \end{equation}
\noindent We use the following approximation of the expected value,
\[
\mathbb{E}[X_{n,m}]=n!p_m^n\exp\left(-\frac{1-p_m}{2p_m}+O(1/n)\right)
\]
\noindent (where $p_m := \frac{m}{n^2}$). The calculation for which can be found in the Appendix. This yields,
$$\log \mathbb{E}[ X_{n,m}] =\log (n! p_m^n) - \frac{1-p_m}{2p_m } + o(1), $$The RHS can be written as
$$ \log(n!)+n\log\frac{m}{n^2}-\frac{n^2}{2}\left(\frac{1}{m}-\frac{1}{n^2}\right)+ o(1). $$
\noindent Let $E_n$ be the random variable that counts the number of edges in $B(n,p)$. By conditioning on $E_n=m$ and using the union bound
(over the range $\frac{p}{2}n^2 \le m \le \frac{1+p}{2} n^2$), we can conclude that with probability at least $1 -n^{-2} $
$$ \mathbb{I}_{\mathcal E} \log X_n \le \mathbb{I}_{\mathcal E} \Big( \log(n!)+n\log\frac{E_n}{n^2}-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{n^2} \right) + \log K + o(1) \Big), $$
\noindent where $X_n$ denotes the number of perfect matchings in $B(n,p)$, and $\mathbb{I}_{\mathcal E} $ is the indicator of the event $\mathcal E$ that
$B(n,p)$ has at least $\frac{p}{2} n^2$ and at most $\frac{1+ p}{2} n^2 $ edges. By Chernoff's bound, $\mathbb{I}_{\mathcal E} = 1$ with probability at least $1 -n^{-2} $. By the union bound
\begin{align} \label{upper0} \log X_n \le \Big( \log(n!)+n\log\frac{E_n}{n^2}-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{n^2}\right) + O(1) \Big),
\end{align} with probability at least $1 - 2n^{-2} $. Then,
\begin{align}
\log \frac{E_n}{n^2}&=\log \left(\frac{\sqrt{Var(E_n)}E_n^*}{n^2}+\frac{\mathbb{E}[ E_n]}{n^2}\right)\nonumber\\
&=\log\left(\left(\frac{p(1-p)}{n^2}\right)^{1/2}E_n^*+p\right)\nonumber\\
&=\log\left(p\left(\frac{1-p}{p}\right)^{1/2}\frac{E_n^*}{n}+p\right)\nonumber\\
&=\log p+\log \left(1+\left(\frac{1-p}{p}\right)^{1/2}\frac{E_n^*}{n}\right)\nonumber\\
&=\log p+\left(\frac{1-p}{p}\right)^{1/2}\frac{E_n^*}{n}+O(1/n^2). \nonumber
\end{align}
\noindent Plugging the last estimate into \eqref{upper0} we obtain, with the same probability
\
\[
\log X_n \le \log (n! p^n) +\left(\frac{1-p}{p}\right)^{1/2}E_n^*-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{n^2p}-\frac{p-1}{n^2p}\right) + O(1). \nonumber
\]
\noindent Note that with probability at least $1-n^{-2}$ we have $E_n=n^2p+O(n\log^2 n)$, in which case $\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{n^2p}\right)$ becomes $o(1)$. Thus, with probability at least $1-3n^{-2}$ we obtain
\begin{align}\label{comparison}
\frac{\log X_n-\log (n! p^n)+\frac{1-p}{2p} }{ \sqrt { \frac{1-p}{p} }}\leq E_n^*+ O(1).
\end{align}
Since $\sum _n n^{-2} < \infty$, we have, by the Borell-Cantelli lemma that the event in \eqref{comparison} holds with probability 1 for all sufficiently large $n$.
On the other hand, by the Kolmogorov-Khinchin theorem, $E_n^*$ satisfies LIL and thus
\[
E_n^*\leq (1+\varepsilon/2)\sqrt{2\log\log n^2}
\]happens with probability $1$ for all sufficiently large $n$. For all sufficiently large $n$, $ (\varepsilon/2)\sqrt{2\log\log n^2 }$ is larger than the error term
$O(1)$, and we have
\[
\frac{\log X_n-\log (n! p^n)+\frac{1-p}{2p} }{\sqrt {\frac{1-p}{p}}} \leq (1+\varepsilon)\sqrt{2\log\log n^2 },
\] proving equation (\ref{upper pm}).
\subsection{Proof of the Lower bound}
For the lower bound we need to show that there exists a sequence $n_k, k=1, 2 \dots$ of indices such that with probability 1,
\[
\frac{\log X_{n_k} -\log (n_k ! p^{n_k} )+\frac{1-p}{2p} }{\sqrt {\frac{1-p}{p}}} \ge (1- \varepsilon)\sqrt{2\log\log n_k^2 },
\]
\noindent holds for infinitely many $k$.
Let $C>0$ be a constant. By the proof of \cite[Theorem 15]{Janson}, we know
\begin{align}\label{lower bound eqtn}
E_n^*-\frac{\log X_{n} -\log (n ! p^{n} )+\frac{1-p}{2p} }{\sqrt {\frac{1-p}{p}}}>C
\end{align}happens with probability $O(1/n)$, and $E_n^*$ is as in the last section. From the standard proof of LIL for the sum of iid random variables \cite{Khinchin, Kolmogorov}, we see that there is a sequence $\{n_k\}:=\{c^k\}$ (where $c$ is an integer larger than 1) for which we have:
\[
E_{n_k}^*\geq (1-\varepsilon/2)\sqrt{2\log \log n_k^2}
\]
happens infinitely often with probability one. Restricting ourselves to this subsequence and denoting by $A_k$ the event that $\eqref{lower bound eqtn}$ holds for $n_k$, we have
\[
\Pr [A_k]=O(1/c^k)
\]so in particular we have
\[
\sum_{k} \Pr[A_k] <\infty
\] By Borel-Cantelli lemma, we have that with probability equal to $1$, for all large $k$:
\[
E_{n_k}^*-C\leq \frac{\log X_{n_k} -\log (n_k ! p^{n} )+\frac{1-p}{2p} }{\sqrt {\frac{1-p}{p}}}
\]
\noindent Let $k$ be large enough so that $C<(\varepsilon/2)\sqrt{2\log \log n_k^2}$. Then, with probability equal to $1$ we have that for infinite many $k$:
\[
(1-\varepsilon)\sqrt{2\log \log n_k}\leq \frac{\log X_{n_k} -\log (n_k ! p^{n_k} )+\frac{1-p}{2p} }{\sqrt {\frac{1-p}{p}}}
\]just as desired.
\section{Proof of Theorem \ref{thm:main2}}\label{sec: main2}
Throughout the next section we are going to let $X_{n,m}$ be the number of Hamilton cycles in $G(n,m)$, $X_n$ the number of Hamilton cycles in $G(n,p)$ and $Y_n:=\log X_n$. The structure of the proof is identical to the one done for theorem \eqref{thm:main}, so we omit some of the calculations. We aim to prove:
\[
\Pr\left[\limsup_{n\rightarrow\infty}\frac{Y_n-\log \mathbb{E}[X_n]+\frac{1-p}{p}}{\sqrt{\frac{2(1-p)}{p}}\sqrt{2\log \log {n}}}=1\right]=1
\]It will be enough to show that
$\varepsilon>0$ we have both the upper bound
\begin{align}\Pr\left[\frac{\log X_n-\log \mathbb{E}[X_n]+\frac{1-p}{p}}{\sqrt{\frac{2(1-p)}{p}}} \geq (1+\varepsilon)\sqrt{2\log\log {n\choose 2}} \textrm{ for infinitely
many }n\right]=0,\nonumber
\end{align}
and the lower bound
\begin{align}\Pr\left[\frac{\log X_n-\log \mathbb{E}[X_n]+\frac{1-p}{p}}{{\sqrt{\frac{2(1-p)}{p}}}} \geq (1-\varepsilon)\sqrt{2\log\log {n\choose 2}} \textrm{ for infinitely
many }n\right]=1.\nonumber
\end{align}Note that in the equations above we have $\log\log{n\choose 2}$, but those can be replaced by $\log\log n$ since the two quantities are asymptotically equal.
\subsection{Proof of upper bound}\label{sec:ub hamilton}
Let $\varepsilon>0$, and let $N:=(n-1)!/2$ be the number of Hamilton cycles in the complete graph $K_n$. With this notation one has,
\begin{align}\label{ey}
\mathbb{E}[X_{n,m}]=Np_m^n\exp\left(-\frac{n^2}{2m}(1-p_m)+o(1)\right).
\end{align}where in this section $p_m:=m/{n\choose 2}$. For a proof of \eqref{ey}, the reader can check the Appendix.
By using corollary \ref{concentration}, we have
\[
X_{n,m}\leq K \mathbb{E}[X_{n,m}]
\]
with probability at least $1-n^{-4}$.
Applying the log function and using estimate (\ref{ey}) we obtain
\begin{align}\label{logy}
\log X_{n,m}\leq \log K+\log N+n\log \frac{m}{{n\choose 2}}-\frac{n^2}{2}\left(\frac{1}{m}-\frac{1}{{n\choose 2}}\right)+o(1).
\end{align}Let $E_n$ be the random variable which counts the number of edges in $G\sim G(n,p)$, by conditioning on $E_n=m$ and using union bound (over the range $\frac{p}{2}{n\choose 2}\leq m\leq \frac{1+p}{2}{n\choose 2}$), with probability at least $1-n^{-2}$ we have
\begin{align}
\mathbb{I}_{\mathcal{E}}\log X_n\leq \mathbb{I}_{\mathcal{E}}\left(\log K+\log N+n\log \frac{E_n}{{n\choose 2}}-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{{n\choose 2}}\right)+o(1)\right)
\end{align}Where now we use $X_n$ (number of Hamilton cycles in $ G(n,p)$) and $\mathbb{I}_{\mathcal{E}}$ is the indicator random variable that the number of edges in $G(n,p)$ is in the range $[\frac{p}{2}{n\choose 2},\frac{1+p}{2}{n\choose 2}]$. By Chernoff's bound, $\mathbb{I}_{\mathcal{E}}=1$ with probability at least $1-n^{-2}$. Hence, by the union bound we have
\begin{align}\label{logx}
\log X_n\leq \log K+\log N+n\log \frac{E_n}{{n\choose 2}}-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{{n\choose 2}}\right)+o(1)
\end{align}with probability at least $1-2n^{-2}$. By a similar calculation to the one done in section \eqref{upper bound section pm}, we get
\begin{align}
\log\frac{E_n}{{n\choose 2}}&=\log p+\left(\frac{1-p}{{n\choose 2}p}\right)^{1/2}E_n^*+O(1/n^2).\nonumber
\end{align}
Plugging it into \eqref{logx}, we obtain that
\begin{align}
\log X_n&\leq \log K+\log N+n\left(\log p+\left(\frac{1-p}{{n\choose 2}p}\right)^{1/2}E_n^*\right)-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{{n\choose 2}}\right)+o(1)\nonumber \\
&=\log \mathbb{E}[X_n]+\left(\frac{2(1-p)}{p}\right)^{1/2}E_n^*-\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{{n\choose 2}p}-\frac{p-1}{{n\choose 2}p}\right)+O(1).\nonumber
\end{align}
Note that since with probability $1-o(1/n^2)$ we have that (say) $E_n=m+\Theta(n\log^2 n)$, it follows that $\frac{n^2}{2}\left(\frac{1}{E_n}-\frac{1}{{n\choose 2}p}\right)=o(1)$ without affecting the error probability.
All in all, with probability $1-O(1/n^2)$ we have
\begin{align}\label{ineq}
\frac{\log X_n-\log\mathbb{E}[X_n] +\frac{1-p}{p}}{\sqrt{\frac{2(1-p)}{p}}}\leq E_n^*+O(1).
\end{align}
By the Borel-Cantelli lemma, we see that for large $n$, with probability one, equation $\eqref{ineq}$ holds. Since $E_n^*$ satisfies LIL, we can upper bound the RHS of of $\eqref{ineq}$ by $(1+\varepsilon)\sqrt{2\log \log {n\choose 2}}$ for large $n$ with probability one. All in all,
\[
\frac{\log X_n-\log\mathbb{E}[X_n] +\frac{1-p}{p}}{\sqrt{\frac{2(1-p)}{p}}}\leq(1+\varepsilon)\sqrt{2\log \log {n\choose 2}}
\]holds for all large $n$ with probability one, which proves the upper bound.
\subsection{Proof of lower bound}\label{sec:lb hamilton}
Recall that in order to prove the lower bound one needs to show that for every $\varepsilon>0$ we have
\begin{align}\label{lb}
\Pr\left[\frac{\log X_n-\log \mathbb{E}[X_n]+\frac{1-p}{p}}{{\sqrt{\frac{2(1-p)}{p}}}}\geq (1-\varepsilon)\sqrt{2\log \log {n\choose 2}} \text{ for infinite many }n\right]=1
\end{align}By the proof of [\cite{Janson},Theorem 1] we have that for any fixed constant $C>0$:
\[
\Pr\left[E_n^*-\frac{\log X_{n} -\log \mathbb{E}[X_n]+\frac{1-p}{p} }{\sqrt{\frac{2(1-p)}{p}}}>C\right]=O(1/n)
\]
By repeating the idea of the lower bound on theorem \eqref{main1}, we obtain
\[
\frac{\log X_n-\log \mathbb{E}[X_n]+\frac{1-p}{p}}{\sqrt{\frac{2(1-p)}{p}}}\geq (1-\varepsilon)\sqrt{2\log \log {n\choose 2}}\]holds for infinite many $n$ with probability $1$, which proves \eqref{lb}.
\section{Proof of Theorem \ref{main:CLT}}\label{sec:clt hypergraphs}
\begin{proof}In this section we will be working with loose Hamilton cycles in random hypergraphs $H^k(n,p)$. Note that we require that $m:=n/(k-1)$ is an integer (which shall denote the number of edges of a Hamilton cycle). Thus, we will assume the divisibility condition $k-1\mid n$ throughout the rest of the section. Let $\mathcal H$ be the set of all Hamilton cycles in the complete $k$-uniform hypergraph on $n$ vertices. Then,
\begin{align}\label{size}
|\mathcal H|=\frac{n!}{2m((k-2)!)^m}
\end{align}Indeed, there are $n!$ ways to label the vertices consecutively (and the edges are determined trivially, including the one edge which goes back to the beginning of the labeling). In each of the $m$ edges, for the ``non-overlapping" vertices (there are $k-2$ such vertices), the order is not important. Therefore, one should divide by $(k-2)!^m$. Finally, note that each Hamilton cycle can be obtained in $2m$ ways ($m$ ``overlapping vertices" to be placed as vertex number $1$, and two isomorphic ways to label the vertices consecutively).
Now we are ready to prove Theorem \ref{main:CLT}. Let $E_n$ denote the number of edges of $H^k(n,p)$, and $X_n(k):=X_n$ be the number of Hamilton cycles of $H^k(n,p)$. The idea of the proof is to compare $X_n$ to $E_n$. Specifically, we want to show that
\begin{align}
\label{distance}
\mathbb{E}[|X_n^*-E_n^*|^2]
\end{align}goes to zero. Since clearly $E_n^*$ converges to $N(0,1)$, the theorem will follow.
To this end we will show that $X_n^*$ and $E_n^*$ are almost perfectly linearly correlated. Meaning that $Cov(X_n^*,E_n^*)\rightarrow 1$. Recall that
\begin{align}
\label{covariance}
Cov(X^*,E^*)=\frac{\mathbb{E}[X_nE_n]-\mathbb{E}[X_n]\mathbb{E}[E_n]}{\sqrt{Var(X_n)Var(E_n)}}.
\end{align}
Let $X_H$ be the event ``$H$ appears in $H^k(n,p)$". Hence,
\[
\mathbb{E}[X_H]=p^m
\]
Let $N:=\frac{n!}{2m((k-2)!)^m}$ (that is, $N=|\mathcal H|$), and by linearity of expectation, we have:
\[
\mathbb{E}[X_n]=Np^{m}
\] Also, since $E_n\sim Bi({n\choose k},p)$ we have $Var(E_n)={n\choose k} p(1-p)$ and $\mathbb{E}[E_n]={n\choose k} p$. We compute the missing quantities. Denote by $\mathcal E$ the set of edges in the complete $k$-uniform hypergraph, and denote by $E_e$ the event ``The edge $e$ appears in $H^k(n,p)$". Then,
\[
\mathbb{E}[X_nE_n]=\sum_{H\in\mathcal H,e\in\mathcal E} \mathbb{E}[X_H\cdot E_e]
\]By symmetry, by fixing one Hamilton cycle $H\in \mathcal H$, we have:
\[
\mathbb{E}[X_nE_n]=N\left(\sum_{e\in\mathcal E} p^{|H\cup e|}\right)=N\left(\left( {n\choose k} - m\right) p^{m+1}+mp^{m}\right)
\]Hence, $\mathbb{E}[X_nE_n]=\mathbb{E}[X_n](\mathbb{E}[E_n]+m(1-p))$, and we get $Cov(X_n,E_n)=\mathbb{E}[X_n](m(1-p))$. Lastly, we compute the variance of $X_n$.
\[
\mathbb{E}[X_n^2]=\sum_{H_1,H_2\in\mathcal H} p^{|H_1\cup H_2|}
\]Again, by fixing an arbitrary Hamilton cycle $H$, we get
\[
\mathbb{E}[X_n^2]=N\left(\sum_{H_1}p^{|H\cup H_1|}\right)
\]Let $N(a)$ be the number of Hamilton cycles that intersect $H$ in exactly $a$ edges. With this notation,
\[
\mathbb{E}[X_n^2]=N\left(\sum_{a= 0}^{m} N(a) p^{2m-a}\right)
\]Let $\alpha_a:=N(a)/N$. Then,
\[
\mathbb{E}[X_n^2]=N^2p^{2m}\left(\sum_{a=0}^{m} \alpha_a p^{-a}\right)
\]Hence,
\[
Var(X_n)=(\mathbb{E}[X_n])^2\left(-1+\left(\sum_{a= 0}^{m} \alpha_a p^{-a}\right)\right):=(\mathbb{E}[X_n])^2f(n)
\]Plugging back into \eqref{covariance}:
\begin{align}\label{cov}
Cov(X_n^*,E_n^*)=\frac{Cov(X_n,E_n)}{\sqrt{Var(X_n)Var(E_n)}}=\frac{\mathbb{E}[X_n](m)(1-p)}{\sqrt{(\mathbb{E}[X_n])^2f(n){n\choose k}p(1-p)}}=\frac{m(1-p)}{\sqrt{{n\choose k}p(1-p)f(n)}}
\end{align}Writing out $f(n)$:
\[
f(n)=(\alpha_0-1)+\frac{\alpha_1}{p}+\frac{\alpha_2}{p^2}+...=\alpha_1\left(\frac{1}{p}-1\right)+\alpha_2\left(\frac{1}{p^2}-1\right)+\ldots+\alpha_{m}\left(\frac{1}{p^{m}}-1\right)
\]Hence,
\[
f(n)\leq \alpha_1\left(\frac{1}{p}-1\right)+\sum_{t=2}^m\frac{\alpha_t}{p^t}
\]We are going to show that the sum is negligible compared to the first summand. First of all, note that $\alpha_1\leq m^2/{n\choose k}$ by a simple union bound. In general, to bound $\alpha_t$, we pick the $t$ edges from $H$ we are going to intersect. There are ${m\choose t}$ ways to do so. Next, collapse each one of those edges into a single vertex. Thus, we now have $n-t(k-1)$ vertices. Note that the number of vertices is still divisible by $k-1$, as it should be the case. Next, we form a Hamilton cycle on these vertices. There are
\[
\frac{(n-t(k-1))!}{2(m-t)((k-2)!)^{m-t}}
\]ways to do so. In order to see this, just note that we replace $n$ by $n-t(k-1)$ and $m$ by $m-t$ in equation (\ref{size}). Lastly, once the Hamilton cycle has been formed, we can uncollapse each one of the $t$ edges, so we obtain an extra factor of $(k!)^t$. Hence,
\begin{align}
\alpha_t&\leq \frac{1}{N}\cdot{m\choose t}\frac{(n-t(k-1))!(k!)^t}{2(m-t)((k-2)!)^{m-t}}\nonumber\\
&=\frac{(m)_t(k!)^tm((k-2)!)^t}{t!(m-t)(n)_{(k-1)t}}\nonumber\\
&=\frac{m(m)_tC^t}{(m-t)t!(n)_{(k-1)t}}
\end{align}for a constant $C$ depending on $k$. Plugging back on $f(n)$ we get:
\[
f(n)\leq \frac{m^2}{{n\choose k}}\left(\frac{1-p}{p}\right)+\sum_{t=2}^m\frac{m(m)_tC^t}{p^t(m-t)t!(n)_{(k-1)t}}
\]To handle the summation, we are going to split it into two sums:
\[
\sum_{t=2}^{\log n}\frac{m(m)_tC^t}{p^t(m-t)t!(n)_{(k-1)t}}+\sum_{t>\log n}^m\frac{m(m)_tC^t}{p^t(m-t)t!(n)_{(k-1)t}}:=S_1+S_2
\]Note that in the range $2\leq t\leq \log n$, we have by lemma \ref{lower factorial}:
\begin{itemize}
\item $(m)_t=m^t(1+o(1))$,
\item $(n)_{(k-1)t}=(1+o(1))n^{(k-1)t}$, and
\item $m/(m-t)\leq 2$.
\end{itemize}Hence,
\begin{align}
S_1&\leq(1+o(1))\sum_{t=2}^{\log n}\frac{2m^tC^t}{n^{(k-1)t}p^tt!}=O\left(\frac{m^2}{n^{2(k-1)}}\right)=O\left(\frac{1}{n^{2(k-2)}}\right)
\end{align}For $S_2$, we can upper bound $m/(m-t)\leq n$, and $(m)_t/(n)_{(k-1)t}\leq 1$ to obtain:
\[
S_2\leq \sum_{t>\log n}^m\frac{nC^t}{p^tt!}\leq \frac{n^2C^{\log n}}{p^{\log n}(\log n)!}=o\left(\frac{1}{n^{2(k-2)}}\right)
\]using this in the definition of $f$ we obtain:
\begin{align}
f(n)&\leq \frac{m^2}{{n\choose k}}\left(\frac{1-p}{p}\right)+O\left(\frac{1}{n^{2(k-2)}}\right)\nonumber\\
&=\frac{m^2}{{n\choose k}}\left(\frac{1-p}{p}\right)\left(1+O\left(n^{2-k}\right)\right)
\end{align}Thus,
\[
\frac{1}{\sqrt{1+O\left(n^{2-k}\right)}}\leq Cov(X_n^*,E_n^*)\leq 1
\]where the second inequality is just from Cauchy Schwarz. Then we have that the lower bound is:
\[
\frac{1}{\sqrt{1+O\left(n^{2-k}\right)}}
\]which we can re-write using a Taylor expansion as:
\begin{align}
\label{convergence}
1-O(n^{2-k})
\end{align}Hence, expanding \eqref{distance} and using \eqref{convergence} we have:
\[
\mathbb{E}[|X_n^*-E_n^*|^2]=\mathbb{E}[(X_n^*)^2]+\mathbb{E}[(E_n^*)^2]-2Cov(X_n^*,E_n^*)=2-2(1-O(n^{2-k})=O(n^{2-k})
\]
Hence, when $k\geq 3$, we have that the above tends to zero. This completes the proof of Theorem \ref{main:CLT}.
\end{proof}
\section{Proof of Theorem \ref{main:ILL}}\label{sec: ill hypergraph}
\begin{proof} Now we are going to use Theorem \ref{main:CLT} to derive LIL for $X_n^*$. First we note that since $E$ is the summation of ${n\choose k}$ i.i.d. random variables, then we have that $E_n^*$ obeys the LIL. That is,
\[
E_n^*\leq (1+\varepsilon/2)\sqrt{2\log\log {n}}
\]with probability $1$ for large enough $n$ and with probability 1 we also have
\[
E_n^*\geq (1-\varepsilon/2)\sqrt{2\log \log n}
\] infinitely often. Note that we write $\log\log n$ instead of $\log \log {n\choose k}$, which holds because they are asymptotically equal (as $k$ is fixed). Furthermore,
\[
\Pr\left(|X_n^*-E_n^*|\geq t)\leq \Pr(|X_n^*-E_n^*|^2\geq t^2\right)\leq \frac{\mathbb{E}[(X_n^*-E_n^*)^2]}{t^2}=O\left(\frac{1}{t^2n^{k-2}}\right)
\]let $t=(\varepsilon/2)\sqrt{2\log\log n}$. We obtain:
\begin{align}\label{errorterm}
\Pr\left(|X_n^*-E_n^*|\geq (\varepsilon/2)\sqrt{2\log\log n}\right)\leq O\left(\frac{1}{n^{k-2}\log\log n}\right)
\end{align}if $k\geq 4$, then we have:
\[
\sum_{n} \Pr(|X_n^*-E_n^*|\geq (\varepsilon/2)\sqrt{2\log\log n}) <\infty
\]and by the Borel-Cantelli Lemma we have that with probability 1, only finite many of those events can happen. That is, with probability 1 we have $|X_n^*-E_n^*|<(\varepsilon/2)\sqrt{2\log\log n}$ for all $n$ sufficiently large. Hence, with probability one, for infinitely many $n$ we have:
\[
(1-\varepsilon)\sqrt{2\log \log n}\leq X_n^*\leq (1+\varepsilon)\sqrt{2\log \log n}
\]Hence, we obtain the Law of Iterated Logarithm for Hamilton cycles provided that $k\geq 4$.
\end{proof}
\section{Upper-tail Estimates}\label{upper tail estimates}
In this section we present new upper-tail estimates needed in the proofs of Theorems \ref{thm:main} and \ref{thm:main2}.
\subsection{Proof of Lemma \ref{conc0}}
We denote by $K_{n,n}$ the complete bipartite graph and let $\mathcal P$ denote the set of all perfect matchings in $K_{n,n}$. Clearly, we have
$$| \mathcal P | = n! . $$
For each $P\in \mathcal P$, let $X_P$ to denote the indicator random variable for the event ``$P$ appears in $ B(n,m)$".
It is easy to see that
\begin{equation} \label{formula1} \mathbb{E}[ X_P] =\frac{(m)_n}{(n^2)_n}, \end{equation}and
\begin{equation} \label{Xexpectation} \mathbb{E}[ X_{n,m}] = n! \frac{ (m)_n} { (n^2) _n }= n!p_m^n\left( - \frac{1 -p_m}{p_m } +O(1/n) \right)
\end{equation}
\noindent where $p_m:=\frac{m}{n^2}$. For the calculation of equation \eqref{Xexpectation}, see the Appendix. In general, for any fixed bipartite graph
$H$ with $h$ edges, the probability that $B(n,m)$ contains $H$ is precisely
$$ \frac{(m)_{h}}{{(n^2)}_{h}}.$$
Thinking of $H$ as the (simple) graph formed by the union of perfect matchings $P_1, \dots, P_k$, observing that $X_H=X_{P_1}\cdots X_{P_k}$, we obtain that
\begin{equation} \label{rearranged}
\mathbb{E} [X_{n,m}^k] = \sum_{P_1,...,P_k\in \mathcal P} \mathbb E [X_{P_1} \dots X_{P_k}] = \sum_{a=0}^{(k-1)n}M(a)\frac{(m)_{kn-a}}{{(n^2)}_{kn-a}},
\end{equation}
where $M(a)$ is the number of (ordered) $k$-tuples
$(P_1,...,P_k)\in \mathcal P^k$, whose union contains exactly $kn-a$ edges. Our main task is to bound $M(a)$ from above.
Fix $a$ and let $\mathcal L:=\mathcal L(a)$ be the set of all sequences $L:= \ell_2,\ldots,\ell_k$ of non-negative integers where $$ \ell_2 + \dots + \ell_k = a. $$ For each sequence $L = \ell_2, \dots, \ell_k$, let $N_{L} $ be the number of $k$-tuples $(P_1,\ldots,P_k)$ such that
for every $2 \le t \le k$, we have $|P_t \cap (\cup_{j<t}P_j)|=\ell_t$. Clearly, we have
$$M(a) =\sum _{ L \in \mathcal L } N_L . $$
We construct a $k$-tuple in $N_L$ according to the following algorithm:
\begin{itemize}
\item Let $P_1$ be an arbitrary perfect matching.
\item Suppose that $P_1,\ldots,P_{t-1}$ are given, our aim is to construct $P_t$. Pick $\ell_t$ edges to be in $P_t \cap \cup_{j=1}^{t-1}P_j$ as follows: first, pick a subset $B_{1,t}$ of $\ell_t$ vertices from the first color class (say $V_1$). Next,
from each vertex pick an edge which appears in
$\cup_{j=1}^{t-1}P_j$ so that the chosen edges form a matching. Let us denote the obtained partial matching by $E_t$, and observe that $|E_t|=\ell_t$, and that $B_{2,t}:=\left(\cup E_t\right)\cap V_2$ is a set of size $\ell_t$ (where $V_2$ denotes the second color class).
\item Find a perfect matching $M_t$ between $V_1 \backslash B_{1,t} $ and $V_2 \backslash B_{2,t} $ which has an empty intersection with $\cup_{j=1}^{t-1} P_j$, and set $P_t:= E_t \cup M_t $.
\end{itemize}
Next, we wish to analyze the algorithm. There are $n!$ ways to choose $P_1$. Having chosen $P_1,\ldots,P_{t-1}$, there are ${n \choose {\ell_t} }$ ways to choose $B_{1,t}$. Each vertex in $B_{1,t}$ has at most $t-1$ different edges in $\cup_{j=1}^{t-1}P_j$. Thus, the number of ways to choose
$E_t$ is at most $(t-1)^{\ell_t} $. Moreover, once $B_{1,t}$ and $B_{2,t} $ are defined, the number of ways to choose $M_t$ is at most $(n-\ell_t) !$. This way, we obtain
$$N_L \le n ! \prod_ {t=2}^{k} {n \choose {\ell_t} } (t-1)^{\ell_t} (n-\ell_t)! = n ! \prod_{t=2}^k n! \frac{(t-1)^{\ell_t} } {\ell_t ! } = (n !)^k \prod_{t=2}^k \frac{(t-1)^{\ell_t} } {\ell_t ! }. $$
\noindent By the multinomial identity and the definition of the set $\mathcal L$,
$$\sum_{L \in \mathcal L} \prod_{t=2}^k \frac{(t-1)^{\ell_t} } {\ell_t ! } = \frac{1}{a!} (1+ \dots + (k-1)) ^a = \frac{ { k \choose 2}^a}{a!}. $$
\noindent Therefore
\begin{equation} \label{bound1} M(a) = \sum_{L \in \mathcal L} N_L \le (n!)^k \sum_{ L \in \mathcal L } \prod_{t=2}^k \frac{(t-1)^{\ell_t} } {\ell_t ! } = (n!)^k \frac{ {k \choose 2} ^a}{a!}. \end{equation}
This estimate is sufficient in the case when $a$ is relatively large. However, it is too generous in the case when $a$ is small (the main contribution in LHS of \eqref{rearranged} comes from this case). In order to sharpen the bound, we refine the estimate on the number of possible $M_t$'s that one can choose in the last step of the algorithm, call this number $\mathcal M_t$ (clearly, $\mathcal M_t$ also depends on the $B_{i,t}$s and we estimate a worse case scenario).
Let $G_t$ be the bipartite graph between $V_1 \backslash B_{1,t} $ and $V_2 \backslash B_{2,t}$ formed by the edges which are not in $\cup_{j=1}^{t-1}P_j$. For each $v \in V_1 \backslash B_{1,t} $, let $d_v$ be its degree in $G_t$. By the
Bregman-Minc inequality (see theorem \ref{bregman})
$$\mathcal M_t \le \prod_{ v \in V_1 \backslash B_{1,t} } (d_v !)^{1/d_v} . $$
\noindent It is clear from the definition that for each $v$
\[
d := n- \ell_t -(t-1) \le d_v \le n -\ell_t : = D
\]
Call a vertex $v$ {\it good} if $d_v = d $ and {\it bad} otherwise. It is easy to see that $v$ is good if and only if it has exactly $t-1$ different edges in $\cup_{j=1}^{t-1} P_j$ and none of these edges hits $B_{2,t} $. It follows that the number of good vertices is at least
\[
n- \ell_t (t-1) - \sum_{j=2}^{t-1} \ell_j \ge n -a(k-1) -a = n - ka .
\]
Since $(d!)^{1/d} $ is monotone increasing, it follows that
\[
\mathcal M_t \le (d !) ^{\frac{n-ka}{d} } (D! )^{ \frac{ ka -\ell_t }{D } }.
\]
Comparing to the previous bound of $(n-\ell_t) ! $, we gain a factor of
\begin{equation} \label{bound2} \frac { (d !) ^{\frac{n-ka}{d} } (D! )^{ \frac{ ka -\ell_t }{D } } }{ (n-\ell_t) ! } = \left[ \frac{ (d!)^{1/d} } { (D!)^{1/D } } \right] ^{n-ka}. \end{equation}
\noindent A routine calculation (see Appendix) shows that whenever $ka= o( n)$, the RHS is
\begin{equation} \label{bound3} (1 + o(1)) e^{- (t-1) }. \end{equation}
\noindent Thus, for such values of $a$, we have
\begin{equation} \label{bound4} M(a) \le (n!)^k \frac{ {k \choose 2} ^a}{a!} \prod_{t=2}^k (1+o(1)) e^{-(t-1)} < 2^k \exp\left(- \frac{k(k-1)}{2} \right) (n!)^k \frac{ {k \choose 2} ^a}{a!} , \end{equation} where the constant 2 can be replaced by any constant larger than 1.
\\
\\
Now we are ready to bound $\mathbb E X_{n,m} ^k$. Recall (\ref{rearranged})
$$ \mathbb E X_{n,m}^k =\sum_{a=0}^{(k-1)n}M(a)\frac{(m)_{kn-a}}{{(n^2)}_{kn-a}} . $$
\noindent We split the RHS as
$$ \sum_{a=0}^{T}M(a)\frac{(m)_{kn-a}}{{(n^2)}_{kn-a}}+\sum_{a=T+1 }^{(k-1)n}M(a)\frac{(m)_{kn-a}}{{(n^2)}_{kn-a}} = S_1 +S_2. $$
\noindent where $T= p_mek^2$. The assumption $k^3 =o(n)$ of the lemma guarantees that $kT =o(n)$. Let $p_m := \frac{m}{n^2} $. By \eqref{bound4} and lemma \ref{lower factorial} and a routine calculation, we have
$$ S_1= \sum_{a=0}^{T}M(a)\frac{(m)_{kn-a}}{(n^2)_{kn-a}}
\leq \frac{2^k(n!)^kp_m^{nk}}{e^{{k\choose 2}}}\exp\left(-\frac{k^2(1-p_m)}{2p_m}+o(1)\right)\sum_{a=0}^{T}\frac{({k\choose 2})^{a}}{a!}p_m^{-a}. $$
\noindent On the other hand,
$$ \sum_{a=0}^{T}\frac{({k\choose 2})^{a}}{a!}p_m^{-a} < \sum_{a=0}^{\infty }\frac{({k\choose 2})^{a}}{a!}p_m^{-a} = e^{ { k \choose2 } /p_m }, $$ so
$$S_1 \le \frac{2^k (n!)^kp_m ^{nk}}{e^{{k\choose 2}}}\exp\left(-\frac{k^2(1-p_m)}{2p_m}+o(1)\right)e^{{k\choose 2}/p_m}=
C_1^k(n!)^k p_m^{nk}, $$
\noindent where $C_1$ is a constant depending on $p_m$. (In fact we can replace the constant $2$ by any constant larger than 1 in the definition of $C_1$; see the remark following \eqref{bound4}).
To bound $S_2$, we use \eqref{bound1} and lemma \ref{lower factorial} to obtain
$$S_2 = \sum_{a >T} M(a)\frac{(m)_{kn-a}}{(n^2)_{kn-a}}
\leq (n!)^kp_m^{nk} \exp\left(-\frac{k^2(1-p_m)}{2p_m}+o(1)\right)\sum_{a >T } \frac{({k\choose 2})^{a}}{a!}p_m^{-a}. $$
Notice that we no longer have the term $\frac{2^k}{ e^{{ k \choose 2}}}$. However, as $a$ is large, there is a much better way to bound
$\sum_{a >T } \frac{({k\choose 2})^{a}}{a!}p_m^{-a}. $ Stirling's approximation yields
$$ \sum_{a >T } \frac{({k\choose 2})^{a}}{a!}p_m^{-a} \le \sum_{a >T} \left(\frac{ek^2}{ 2 p_m a } \right)^a < \sum_{a >T} \left(\frac{1}{2} \right)^a = O(1). $$
\noindent It follows that
$$S_2= o( (n!)^k p_m^{nk}) , $$ and thus is negligible for our needs. Therefore,
$$\mathbb E [X_{n,m} ^k]= S_1 + S_2 \le C_1^k (n!) p_m^{nk}. $$
\noindent Finally, note that \eqref{Xexpectation} implies
$$(\mathbb E [X_{n,m}])^k = (n!)^k p_m^{nk} \exp \left(\frac{k(1-p_m )}{p_m } +O(k/n)\right) \ge C_2^k(n!)^k p_m^{nk}, $$ for an appropiate constant $C_2$. Thus, we get $\mathbb{E}[X_{n,m}^k]/(\mathbb{E}[X_{n,m}]^k)\leq C^k$ by setting $C:=C_1/C_2$.
\subsection{Proof of Lemma \ref{conc1}}
\begin{proof}[Proof of lemma \ref{conc1}]Let $K_n$ be the complete graph of $n$ vertices and denote by $\mathcal H$ the set of Hamilton cycles in $K_n$. Clearly,
\[
|\mathcal H|=\frac{(n-1)!}{2}
\]
For each $H\in \mathcal H$, let $X_H$ denote the indicator random variable for the event ``$H$ appears in $G(n,m)$". It is easy to see that
\[
\mathbb{E}[X_H]=\frac{(m)_n}{{n\choose 2}_n}
\]Thus,
\begin{align}\label{exp x}
\mathbb{E}[X_{n,m}]=N\frac{(m)_n}{{n\choose 2}_n}
\end{align}where above and henceforth we let $N:=(n-1)!/2$. By lemma \ref{lower factorial},
\[
\frac{(m)_n}{{n\choose 2}_n}=p_m^n \exp\left(-\frac{1-p_m}{p_m}+o(1)\right)
\]Hence, calculating the $k$-th moment we obtain:
\begin{align}\label{k moment}
\mathbb{E}[X_{n,m}^k]=\sum_{H_1,\ldots,H_k\in\mathcal H}\mathbb{E}[X_{H_1}\ldots X_{H_k}]=\sum_{a=0}^{(k-1)n}M(a)\frac{(m)_{kn-a}}{{n\choose 2}_{kn-a}}
\end{align}where $M(a)$ is the number of (ordered) $k$-tuples $(H_1,\ldots,H_k)\in \mathcal H^k$. The following lemma gives us bounds for $M(a)$, and it is true for $k\leq \frac{\log n}{8}$.
\begin{lemma}\label{upper bound Ma}For $M(a)$ defined above, if $0\leq a\leq \log^3 n$ we have:
\[
M(a)\leq 3^k N^k\frac{(k(k-1))^a}{e^{k(k-1)}a!}
\]and for $\log^3 n< a\leq (k-1)n$ we have the following weaker bound:
\[
M(a)\leq 3^kN^k\frac{(k(k-1))^a}{a!}
\]
\end{lemma}
\noindent Splitting the sum in \eqref{k moment},
\begin{align}
\mathbb{E}[X_{n,m}^k]&=\sum_{a=0}^{\log^3 n}M(a)\frac{(m)_{kn-a}}{{n\choose 2}_{kn-a}}+\sum_{a=\log^3 n+1}^{(k-1)n}M(a)\frac{(m)_{kn-a}}{{n\choose 2}_{kn-a}}=S_1+S_2
\end{align}allows us to use lemma \eqref{upper bound Ma}. We bound the two sums separately:
\[
S_1=\sum_{a=0}^{\log^3n}M(a)\frac{(m)_{kn-a}}{{n\choose 2}_{kn-a}}\leq \frac{3^kN^kp_m^{nk}}{e^{k(k-1)}}\exp\left(-\frac{k^2(1-p_m)}{p_m}+o(1)\right)\sum_{a=0}^{\log^3n}\frac{(k(k-1))^a}{a!}p_m^{-a}
\]On the other hand,
\[
\sum_{a=0}^{\log^3n}\frac{(k(k-1))^a}{a!}p_m^{-a}\leq \sum_{a=0}^\infty\frac{(k(k-1))^a}{a!}p_m^{-a}=e^{k(k-1)/p_m}
\]so
\[
S_1\leq\frac{3^kN^kp_m^{nk}}{e^{k(k-1)}}\exp\left(-\frac{k^2(1-p_m)}{p_m}+o(1)\right)e^{k(k-1)/p_m}:=C_1^kN^kp_m^{nk}
\]for some appropriate constant $C_1$ (which depends on $k$). To bound $S_2$:
\[
S_2=\sum_{a>{\log^3n}}M(a)\frac{(m)_{kn-a}}{{n\choose 2}_{kn-a}}\leq 3^kN^kp_m^{nk}\exp\left(-\frac{k^2(1-p_m)}{p_m}+o(1)\right)\sum_{a>{\log^3n}}\frac{(k(k-1))^a}{a!}p_m^{-a}
\]However for this case, it is enough to bound the summation using Stirling's approximation, and use $k=O(\log n)$:
\[
\sum_{a>{\log^3n}}\frac{(k(k-1))^a}{a!}p_m^{-a}\leq \sum_{a>{\log^3n}}\left(\frac{k(k-1)e}{p_ma}\right)^a\leq\sum_{a>{\log^3n}}\left(\frac{1}{2}\right)^a=o(1)
\]It follows that
\[
S_2=o(3^kN^kp_m^{nk}),
\]and is thus totally negligible for our needs. Therefore,
\[
\mathbb{E}[X_{n,m}^k]=S_1+S_2\leq C_1^kN^kp_m^{nk}
\]Finally, raising equation (\ref{exp x}) to the $k$-th power yields:
\[
(\mathbb{E}[X_{n,m}])^k=N^kp_m^{nk}\exp\left(-\frac{(1-p_m)k}{p_m}+o(1)\right)\geq C_2^kN^kp_m^{nk}
\]for some constant $C_2$. Hence,
\[
\frac{\mathbb{E}[X_{n,m}^k]}{(\mathbb{E}[X_{n,m}])^k}\leq (C_1/C_2)^k
\]and setting $C:=C_1/C_2$ finishes the proof.
\end{proof}
\begin{proof}[Proof of lemma \ref{upper bound Ma}]Fix $a\leq \log^3 n$, and let $\mathcal L:=\mathcal L(a)$ be the set of all the sequences $L:=(\ell_1,\ldots,\ell_k)$ of non-negative integers where
\[
\ell_2+\ell_3+\ldots+\ell_k=a
\]For each $L=(\ell_2,\ldots,\ell_k)$, let $N_L$ be the number of $k$-tuples $(H_1,\ldots,H_k)$ such that for $2\leq t\leq k$ we have $|H_t\cap (\cup_{i<t}H_i)|=\ell_t$. Clearly we have,
\[
M(a)=\sum_{L\in \mathcal L}N_L
\]we know describe how to construct $k$-tuples in $N_L$.
\begin{enumerate}
\item Pick an arbitrary $H_1$.
\item Assume we are given $H_1,\ldots, H_{t-1}$. Construct a set $E_t$ of edges, of size $\ell_t$ such that $E_t\subset \cup_{i<t}H_i$.
\item Complete $E_t$ into a Hamilton cycle.
\end{enumerate}Next we analyze the algorithm. Clearly there are $N$ ways to perform the first step. For the moment, assume that the number of ways to perform step 2 and 3 (for a fixed $t$) is given by:
\[
3N\frac{(2(t-1))^{\ell_t}}{e^{2(t-1)}\ell_t!}
\]Then, for fixed $L$ we would have the following upper bound on $N_L$:
\[
N_L\leq 3^kN^k\prod_{t=2}\frac{(2(t-1))^{\ell_t}}{e^{2(t-1)}\ell_t!}
\]by the multinomial identity and the definition of the set $\mathcal L$ we have,
\[
\sum_{L\in \mathcal L}\prod_{t=2}\frac{(2(t-1))^{\ell_t}}{e^{2(t-1)}\ell_t!}=\frac{1}{e^{k(k-1)}a!}(2+4+\ldots+2(k-1))^a=\frac{(k(k-1))^a}{e^{k(k-1)}a!}
\]so we obtain the upper bound on $M(a)$,
\[
M(a)\leq 3^kN^k\frac{(k(k-1))^a}{e^{k(k-1)}a!}
\]as claimed. Hence to finish we need to upper bound steps 2-3 of the algorithm.
\\
\\
\textbf{Upper bound on steps 2 and 3.} Assume we are given $H_1,\ldots,H_{t-1}$. For each vertex $v$, consider the set $L(v)$ defined as follows:
\[
L(v):=\{w\mid vw\in (H_1\cup\ldots\cup H_{t-1})\}
\]which we shall refer to as the list of bad vertices of $v$. Note that for each $v$, we have $|L(v)|\leq 2(t-1)$. Pick a subset $V_t\subset V(K_n)$ of size $\ell_t$, say $V_t=\{u_1,\ldots,u_{\ell_t}\}$. We can do so in ${n\choose \ell_t}$ ways. Then, for each $u_i\in V_t$, we select an element, $w_i$, on its list $L(u_i)$. Perform this selection such that if $i\neq j$, then $w_i\neq w_j$. Note that this might not always be possible, in which case the number of ways to perform this step is zero (and we obtain the upper bound trivially). Having chosen the pairs $(u_i,w_i)$, we are going to match them through an edge. Hence, we have at most
\[
{n\choose \ell_t}(2(t-1))^{\ell_t}
\]number of ways to construct $E_t$. Now our task is to upper bound the number of ways we can complete $E_t$ into a Hamilton cycle without using any edges in $\cup_{i<t}H_i$.
\\
\\
First, we are going to collapse the edges in $E_t$ into vertices, and identify them by $w_i$. Hence, we now have $V(K_n)\backslash V_t$ as vertex set (that is, $n-\ell_t$ vertices). We are going to upper bound a bigger quantity: The number of \textbf{oriented} Hamilton cycles, such that for no vertex $v$, we have $v\rightarrow w$ for some $w\in L(v)$, which henceforth we shall refer to as ``$v$ is bad".
\\
\\
Let $N(t)$ be the quantity we wish to upper bound (that is, the number of oriented Hamilton cycles with no bad vertices). Hence,
\begin{align}
N(t)&=(n-\ell_t-1)!-\sum_{v_1}\#\{H\mid v_1 \text{ bad in }H\}+\sum_{v_1,v_2}\#\{H\mid v_1,v_2 \text{ bad in }H\}-\dots\nonumber\\
=&s_0-s_1+s_2-\cdots
\end{align}where $s_i=\sum_{v_1,\ldots,v_i}\#\{H\mid v_1\ldots,v_i \text{ bad in } H\}$. We now give upper and lower bounds on $s_i$, and we also argue why it is enough to consider the terms up to $i=\log^2 n$:
\\
\\
\textbf{Upper bound on $s_{t}$}: First we choose the $i$ vertices that will be bad. There are ${n-\ell_t\choose i}$ ways to do so. Say we chose $\{v_1,\ldots,v_i\}$. Then there are at most $2(t-1)$ many ways to make each vertex bad, hence a total of at most $(2(t-1))^i$ ways to make $v_r$ bad ($1\leq r\leq i$). Hence, we have $v_r\rightarrow x_r$ for some $x_r$ in its set $L(v_r)$. Collapse $v_r$ and $x_r$ onto a single vertex (for $1\leq r\leq i$), so now we have $n-\ell_t-i$ vertices. Then form any oriented Hamilton cycle on these vertices, so we have $(n-\ell_t-i-1)!$ ways to do so (then uncollapse them to obtain an oriented Hamilton cycles on $n-\ell_t$ vertices). Hence,
\begin{align}
s_i&\leq {n-\ell_t\choose i}(2(t-1))^i(n-\ell_t-i-1)!\nonumber\\
&=\frac{(n-\ell_t)!}{n-\ell_t-i}\cdot \frac{(2(t-1))^i}{i!}\nonumber\\
&=\frac{n-\ell_t}{n-\ell_t-i}\cdot (n-\ell_t-1)!\cdot \frac{(2(t-1))^i}{i!}\nonumber\\
&=\left(1+O\left(\frac{\ell_t+i}{n}\right)\right)(n-\ell_t-1)!\cdot \frac{(2(t-1))^i}{i!}
\end{align}but since we are considering $i\leq \log^2 n$ and $\ell_t\leq a\leq \log^3 n$ we have: $$s_i\leq (1+O(\log ^3 n/n))(n-\ell_t-1)!\cdot \frac{(2(t-1))^i}{i!}$$
\\
\\
\textbf{Truncation:} We show that $|\sum_{i=\log^2 n}^{n-1} (-1)^i s_i|$ is small. Indeed,
\begin{align}
\left|\sum_{i=\log^2 n}^{n-1} (-1)^i s_i\right|&\leq \sum_{i=\log^2 n}^{n-1}\frac{n-\ell_t}{n-\ell_t-i}(n-\ell_t-1)!\cdot \frac{(2(t-1))^i}{i!}\nonumber\\
&\leq (n-\ell_t-1)! \sum_{i=\log^2 n}^{n-1}n\frac{(2(t-1))^i}{i!}\nonumber\\
&\leq(n-\ell_t-1)!n^2\frac{(2(t-1))^{\log^2n}}{(\log^2 n)!}\nonumber\\
&=(n-\ell_t-1)!o(e^{-2(t-1)}/n)
\end{align}where the second to last inequality holds since the summands are in decreasing order (as $t$ is at most $\frac{\log n}{8}$).
\\
\\
\textbf{Lower bound on $s_t$}: For this bound, we are only going to consider $\{v_1,\ldots,v_i\}$ such that their lists are disjoint. Intuitively, almost all ${n-\ell_t\choose i}$ options are good since the sizes of the lists are of order $t$ (which will be logarithmic). Let $\alpha_i$ be the number of $\{v_1,\ldots,v_i\}$ such that $L(v_t)\cap L(v_r)=\emptyset$ for $t\neq r$ and $|L(v_r)|=2(t-1)$. Hence,
\begin{align}
s_i&\geq \alpha_i (2(t-1))^i(n-\ell_t-i-1)!\nonumber\\
&=\left(\frac{\alpha_i}{{n-\ell_t\choose i}}\right){n-\ell_t\choose i}(2(t-1))^i(n-\ell_t-i-1)!\nonumber\\
&=\left(\frac{\alpha_i}{{n-\ell_t\choose i}}\right)\left(1+O\left(\frac{i+\ell_t}{n}\right)\right)(n-\ell_t-1)!\cdot \frac{2(t-1))^i}{i!}
\end{align}Now, we compute $\alpha_i$: First we choose $v_1$ so that $|L(v)|=2(t-1)$. There are $n-\ell_t-O(\log^3 n)$ options for $v_1$. Then, choose $v_2$ so that $|L(v_2)|=2(t-1)$ and $L(v_2)\cap L(v_1)$ is empty. There are at most $(2(t-1))^2$ many vertices, $u$, such that $L(u)\cap L(v_1)$ is not empty (to see this note that $L(v_1)$ has size $(2(t-1))$ and each member of $L(v_1)$ is in at most $(2(t-1))$ many lists). Hence, the number of ways to pick $v_2$ is at least $n-\ell_t-O(\log^3 n)-(2(t-1))^2$. Continue in the manner to obtain (after dividing by the $i!$ that comes from double counting) the following lower bound:
\begin{align}
\alpha&\geq \frac{(n-\ell_t-O(\log^3 n))(n-\ell_t-O(\log^3 n)-(2(t-1))^2)\cdots (n-\ell_t-O(\log^3 n)-(i-1)(2(t-1))^2)}{i!}\nonumber\\
&\geq \frac{(n-\ell_t-O(\log^4 n ))^i}{i!}
\end{align}where the last inequality uses $i\leq \log^2n$ and $t\leq (\log n)/8$. We compare with ${n-\ell_t\choose i}$ as follows:
\begin{align}
\frac{\alpha_i}{{n-\ell_t\choose i}}&\geq\frac{\frac{(n-\ell_t-O(\log^4 n ))^i}{i!}}{\frac{(n-\ell_t)_i}{i!}}\nonumber\\
&=\frac{(n-\ell_t-O(\log^4 n ))^i}{(n-\ell_t)_i}\nonumber\\
&=\frac{(n-\ell_t)^i(1-O(\log^4 n/n))^i}{(n-\ell_t)^i(1+O(i^2/n))}\nonumber\\
&=(1-O(\log^{6}n/n))
\end{align}where above we use $(n-\ell_t)_i=(n-\ell_t)^i(1+O(i^2/n))$ which is valid for $i\leq \log^2 n$. Hence, putting everything together we arrive at the lower bound:
\[
s_i\geq (1-O(\log^{6}n /n))(n-\ell_t-1)!\cdot \frac{2(t-1))^i}{i!}
\]Hence, we have that for all $i\leq \log^2 n$ the following bounds on $s_i$:
\[
(1-O(\log^{6}n /n))(n-\ell_t-1)!\cdot \frac{2(t-1))^i}{i!}\leq s_i\leq (1+O(\log^{6}n /n))(n-\ell_t-1)!\cdot \frac{2(t-1))^i}{i!}
\]which implies:
\begin{align}
\sum_{i=0}^{\log^2 n}(-1)^is_i&\leq\sum_{i=0}^{\log^2n}(n-\ell_t-1)!\frac{(-2(t-1))^i}{i!}(1+(-1)^iO\left(\log^{6}n/n\right))\nonumber\\
&\leq \left(\sum_{i=0}^{\log^2 n}(n-\ell_t-1)!\frac{(-2(t-1))^i}{i!}\right)+\left(\sum_{i=0}^{\log^2 n}(n-\ell_t-1)!\frac{(2(t-1))^i}{i!}O\left(\frac{\log^{6}n}{n}\right)\right)\nonumber\\
&\leq (n-\ell_t-1)!\left(e^{-2(t-1)}(1+o(1))+e^{2(t-1)}\cdot O\left(\frac{\log^6n}{n}\right)\right)\nonumber\\
&=(n-\ell_t-1)!e^{-2(t-1)}\left((1+o(1)+O\left(\frac{e^{4(t-1)}\log^6 n}{n}\right)\right)\nonumber\\
&=(n-\ell_t-1)!e^{-2(t-1)}(1+o(1))\nonumber
\end{align}where the last equality uses the fact that $t\leq k\leq \frac{\log n}{8}$. Putting everything together we have:
\begin{align}
\sum_{i=0}^{n-1}(-1)^is_i&=\sum_{i=0}^{\log^2n}(-1)^is_i+\sum_{i=\log^2n}^{n-1}(-1)^is_i\nonumber\\
&\leq \sum_{i=0}^{\log^2n}(-1)^is_i+(n-\ell_t-1)!o(e^{-2(t-1)}/n)\nonumber\\
&\leq (n-\ell_t-1)!e^{2(t-1)}(1+o(1))\nonumber
\end{align}Thus, the number of ways to complete $E_t$ into a Hamilton cycles is upper bounded by:
\[
(n-1-\ell_t)!e^{-2(t-1)}(1+o(1))
\]Putting it together with the upper bound on the number of ways to construct $E_t$ we obtain that the upper bound on Steps 2 and 3 of our algorithm is given by:
\begin{align}
(1+o(1))(n-1-\ell_t)!e^{-2(t-1)}{n\choose \ell_t}(2(t-1))^{\ell_t}=(1+o(1))2N\frac{(2(t-1))^{\ell_t}}{e^{2(t-1)}}\leq 3N\frac{(2(t-1))^{\ell_t}}{e^{2(t-1)}}\nonumber
\end{align}
\end{proof}
\section{Appendix}
\textbf{Proof of lemma \ref{lower factorial}:}
Let $t, \ell$ be such that $\ell=o(t^{2/3})$. Then,
\begin{align}
(t)_\ell&=t(t-1)\cdots (t-\ell+1) \nonumber\\
&=t^{\ell}\prod_{i=0}^{\ell-1}(1-i/t)\nonumber\\
&=t^{\ell}\prod_{i=0}^{\ell-1}e^{-i/t+O(i^2/t^2)}\nonumber\\
&=t^{\ell}\exp\left(\sum_{i=0}^{\ell-1}-i/t+O(i^2/t^2)\right)\nonumber\\
&=t^{\ell}\exp\left(-\frac{\ell(\ell-1)}{2t}+O(\ell^3/t^2)\right)\nonumber\\
&=t^{\ell}\exp\left(-\frac{\ell(\ell-1)}{2t}+o(1)\right)\nonumber
\end{align}as claimed.\\
\\
\textbf{Approximation of expected value (Perfect matchings):} For a subgraph $H$ of $K_{n,n}$ with exactly $h$ edges, the probability that $H$ appears in $B(n,m)$ is exactly:
$$
\frac{{n^2-h \choose m-h}}{{n^2 \choose m}}=\frac{(m)_h}{(n^2)_h}
$$Let $H$ be a perfect matching on $K_{n,n}$, then $h=n$, so we can apply Lemma \ref{lower factorial} to obtain:
\begin{align*}
\frac{(m)_n}{(n^2)_n}&=\frac{ m^n \exp\left(-\frac{n(n-1)}{2m}+O(1/n)\right)}{(n^2)^n\exp\left(-\frac{n(n-1)}{2n^2}+O(1/n)\right)}\\
&=\frac{m^n}{(n^2)^n}\exp\left(-\frac{n^2}{2m}+\frac{1}{2}+O(1/n)\right)\\
&=p_m^n\exp\left(-\frac{1-p_m}{2p_m}+O(1/n)\right)
\end{align*}where in the last equality we used $p_m:=m/n^2$. Since there are a total of $n!$ perfect matchings, we obtain by linearity:
\[
\mathbb{E}[X_{n,m}]=n!p_m^n\exp\left(-\frac{1-p_m}{2p_m}+O(1/n)\right)
\]
\textbf{Approximation of expected value (Hamilton cycles):} Just like above, let $H$ be a hamilton cycle in $K_n$. Then the probability that $H$ appears in $G(n,m)$ is given by:
\begin{align*}
\frac{{{n\choose 2}-n \choose m-n} }{ {{n\choose 2}\choose m }}&=\frac{(m)_n}{{n\choose 2}_n}\\
&=\frac{m^n}{{n\choose 2}^n}\exp\left(-\frac{n^2}{2m}+\frac{n^2}{2{n\choose 2}}+O(1/n)\right)\\
&=p_m^n\exp\left(-\frac{1-p_m}{p_m}+O(1/n)\right)
\end{align*}by linearity, one obtains the desired approximation.
\\
\\
\textbf{Computation of equation \eqref{bound2}}: We are going to use the following upper and lower bounds for the factorial:
\[
\sqrt{2\pi s}(s/e)^s\leq s!\leq \sqrt{2\pi s}(s/e)^se^{1/12s}
\]Hence,
\begin{align}
\left[ \frac{ (d!)^{1/d} } { (D!)^{1/D } } \right] ^{n-ka}&\leq\left[\frac{(\sqrt{2\pi d}(d/e)^de^{1/12d})^{1/d}}{(\sqrt{2\pi D}(D/e)^D)^{1/D}}\right]^{n-ka}\nonumber\\
&=\left[(1+O(n^{-2}))\frac{d(2\pi d)^{1/2d}}{D(2\pi D)^{1/2D}}\right]^{n-ka}\nonumber\\
&=(1+O(n^{-1})\left[\frac{(2\pi d)^{1/2d}}{(2\pi D)^{1/2D}}\right]^{n-ka}\left[\frac{d}{D}\right]^{n-ka}\nonumber\\
&=(1+o(1))\left[1-\frac{t-1}{n-\ell_t}\right]^{n-ka}\nonumber\\
&=(1+o(1))e^{t-1}\nonumber
\end{align}as desired. (Here we use the assumption that $ka =o(n)$.)
\section{Acknowledgments}
We would like to thank Kyle Luh for his useful comments during the draft of this paper.
| {
"timestamp": "2017-10-12T02:02:51",
"yymm": "1607",
"arxiv_id": "1607.08865",
"language": "en",
"url": "https://arxiv.org/abs/1607.08865",
"abstract": "A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\\{t_i\\}_{i=1}^{\\infty}$ with mean $0$ and variance $1$$$ \\Pr \\left[ \\limsup_{n\\rightarrow \\infty} \\frac{ \\sum_{i=1}^n t_i }{\\sigma_n \\sqrt {2 \\log \\log n }} =1 \\right] =1 . $$In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph $H$. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random $k$-uniform hypergraphs, we obtain the Central Limit Theorem (CLT) and LIL for the number of Hamilton cycles.",
"subjects": "Combinatorics (math.CO)",
"title": "Law of Iterated Logarithm for random graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795083542037,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379402856712
} |
https://arxiv.org/abs/1009.4913 | Equivalence of concentration inequalities for linear and non-linear functions | We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\mathcal{X}$. We show that, with respect to an appropriate "normal distance" on $\mathcal{X}$, concentration inequalities for linear and non-linear functions of $X$ are equivalent. This normal distance corresponds naturally to the concentration rate in classical concentration results such as Gaussian concentration and concentration on the Euclidean and Hamming cubes. Under suitable assumptions on the roundness of the sets of interest, the concentration inequalities so obtained are asymptotically optimal in the high-dimensional limit. | \section{Introduction}
\label{sec:intro}
It is by now almost classical that smooth enough convex functions enjoy good concentration properties; see \emph{e.g.}\ \cite{Ledoux:2001} \cite{Lugosi:2009} \cite{McDiarmid:1998} \cite{MilmanSchechtman:1986} for surveys of the literature. It is also known that convexity can be neglected in the Gaussian case and that the smoothness assumptions are not essential and can be replaced, for instance, with bounded martingale differences; see \emph{e.g.}\ \cite{McDiarmid:1989} \cite{McDiarmid:1997} and also \cite{Vu:2002}.
A common feature of many concentration results is that an appropriate notion of distance is needed, \emph{e.g.}\ Talagrand's convex distance \cite{Talagrand:1995}. In this paper, a notion of ``normal distance'' on a topological vector space $\mathcal{X}$ is introduced through a technique commonly used in large deviations theory, Chernoff bounding, \emph{i.e.}\ estimating the measure of a set by using a containing half-space. Although simple, this method leads to a notion of distance that is in some sense ``natural'' with respect to the duality structure on $\mathcal{X}$. Remarkably, with respect to this distance, concentration inequalities on the tails of linear, convex, quasiconvex and non-linear functions on $\mathcal{X}$ are mutually equivalent.
Concentration of measure is based on a simple but non-trivial observation originally due to L{\'e}vy \cite{Levy:1951}: in a high-dimensional probability space, ``nearly all'' the probability mass lies close to any set with measure at least $\frac{1}{2}$; put another way, functions of many independent variables with small sensitivity to each individual input are very nearly constant. A typical concentration inequality is of the form
\begin{equation}
\label{eq:deviation_ineq}
\P[ | f(X) - m | \geq r ] \leq C_{1} \exp ( - C_{2} r^{2} ),
\end{equation}
where $f$ is a suitably well-behaved function, $X$ is a random variable such that the push-forward measure $(f \circ X)_{\ast} \P$ has some concentration property, and $m$ is either the mean value $\mathbb{E}[f(X)]$ or median value $\mathbb{M}[f(X)]$; sometimes the control is one-sided, and the absolute value in \eqref{eq:deviation_ineq} is omitted. A notable feature of this paper is that it provides concentration inequalities with $m = f(\mathbb{E}[X])$.
The key property of the normal distance of this paper is contained in the following \emph{portmanteau theorem} for the equivalence of various concentration inequalities with respect to normal distance:
\begin{thm}
\label{thm:portmanteau}
Let $\mathcal{X}$ be a real topological vector space and $\mathcal{X}^{\ast}$ its continuous dual space. Let $\Psi \colon \mathcal{X}^{\ast} \to [0, +\infty]$ be positively homogeneous of degree one. Define the \emph{$\Psi$-normal distance} from $x \in \mathcal{X}$ to $A \subseteq \mathcal{X}$ by
\[
d_{\perp, \Psi}(x, A) := \sup \left\{ \frac{\langle \nu, x - p \rangle_{+}}{\Psi(\nu)} \,\middle|\, \begin{matrix} p \in \mathcal{X} \text{ and } \nu \in \mathcal{X}^{\ast} \text{such that,} \\ \text{for all $a \in A$, } \langle \nu, a \rangle \leq \langle \nu, p \rangle \end{matrix} \right\},
\]
with the convention that $0 / 0 = 0$. Then the following statements about any random variable $X$ that takes values in $\mathcal{X}$ are equivalent:
\begin{compactenum}[(i)]
\item \label{port-1} for every closed half-space $\H_{p, \nu} := \{ x \in \mathcal{X} \mid \langle \nu, x - p \rangle \leq 0 \} \subseteq \mathcal{X}$, where $p \in \mathcal{X}$ and $\nu \in \mathcal{X}^{\ast}$,
\[
\P[X \in \H_{p, \nu}] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], \H_{p, \nu})^{2}}{2} \right);
\]
\item \label{port-2} for every convex set $K \subseteq \mathcal{X}$,
\[
\P[X \in K] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], K)^{2}}{2} \right);
\]
\item \label{port-3} for every measurable $A \subseteq \mathcal{X}$,
\[
\P[X \in A] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right);
\]
\item \label{port-4} for every measurable $f \colon \mathcal{X} \to \mathbb{R} \cup \{ \pm \infty \}$ and every $\theta \in \mathbb{R} \cup \{ \pm \infty \}$,
\[
\P[f(X) \leq \theta] \leq \exp \left( - \frac{d_{\perp, \Psi} (\mathbb{E}[X], f^{-1}([-\infty, \theta]))^{2}}{2} \right);
\]
\item \label{port-5} for every quasiconvex $f \colon \mathcal{X} \to \mathbb{R} \cup \{ \pm \infty \}$ and every $\theta \in \mathbb{R} \cup \{ \pm \infty \}$,
\[
\P[f(X) \leq \theta] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], f^{-1}([-\infty, \theta]))^{2}}{2} \right).
\]
\end{compactenum}
\end{thm}
Note that if $f$ is quasilinear (\emph{i.e.}\ both $f$ and $-f$ are quasiconvex), then formulation (\ref{port-5}) yields concentration inequalities for both the lower and upper tails of $f(X)$.
The notation and setting of the paper are covered in section \ref{sec:notation_and_background}, along with a review of some definitions and results from the concentration-of-measure literature. Normal distance is defined and its properties (including theorem \ref{thm:portmanteau}) are examined in section \ref{sec:normal_distance}. In section \ref{sec:main}, the normalizing function $\Psi$ is determined explicitly in several cases, thereby connecting theorem \ref{thm:portmanteau} with classical concentration results. In particular, proposition \ref{prop:emp_mean} identifies the normal distance that corresponds to the concentration of a vector, the entries of which are the empirical (sampled) means of functions of independent random variables. In section \ref{sec:asymptotics}, it is shown that the inequality in theorem \ref{thm:portmanteau}(\ref{port-3}) is asymptotically sharp (in the sense used in large deviations theory) in the high-dimensional limit, provided that $A$ is convex and ``sufficiently round'' at those points of $A$ that are closest to the center of mass $\mathbb{E}[X]$. Finally, for completeness, the method of Chernoff bounds and its consequences for convex sets are reviewed in an appendix (section \ref{sec:Chernoff}).
\section{Notation and Background}
\label{sec:notation_and_background}
Let $\mathcal{X}$ be a real topological vector space. Let $\mathcal{X}^{\ast}$ denote the continuous dual space of $\mathcal{X}$ and let $\langle \ell, x \rangle$ denote the dual pairing between $\ell \in \mathcal{X}^{\ast}$ and $x \in \mathcal{X}$; $\langle v, \ell \rangle$ will also denote the dual pairing between $v \in \mathcal{X}^{\ast \ast}$ and $\ell \in \mathcal{X}^{\ast}$. It is not strictly necessary to assume that $\mathcal{X}$ is locally convex, but the results of this paper may be trivially true if $\mathcal{X}^{\ast}$ does not contain enough linear functionals.
\subsection{Half-Spaces}
Given $p \in \mathcal{X}$ and $\nu \in \mathcal{X}^{\ast}$, $\H_{p, \nu}$ will denote the closed half-space of $\mathcal{X}$ that has $p$ in its frontier and outward-pointing normal $\nu$, \emph{i.e.}
\begin{equation}
\label{eq:halfspace}
\H_{p, \nu} := \left\{ x \in \mathcal{X} \,\middle|\, \langle \nu, x \rangle \leq \langle \nu, p \rangle \right\}.
\end{equation}
Note well the degenerate case $\H_{p, 0} = \mathcal{X}$. Every $(p, \nu) \in \mathcal{X} \times \mathcal{X}^{\ast}$ defines a unique closed half-space of $\mathcal{X}$, whereas a given closed half-space can have multiple distinct representations: $\H_{p, \nu} = \H_{p', \nu'}$ if, and only if, $\nu$ is a positive multiple of $\nu'$ and $\langle \nu, p - p' \rangle = \langle \nu', p - p' \rangle = 0$.
\subsection{Convex Sets and Cones}
The closed convex hull of $A \subseteq \mathcal{X}$ will be denoted by $\mathop{\overline{\mathrm{co}}}(A)$. Given a closed convex set $K \subseteq \mathcal{X}$ and $p \in K$, $\mathrm{N}_{p}^{\ast} K$ denotes the \emph{outward normal cone} to $K$ at $p$, and $\mathrm{N}^{\ast} K$ denotes the \emph{outward normal bundle} of $K$:
\begin{equation}
\label{eq:normal_cone}
\mathrm{N}_{p}^{\ast} K := \left\{ \nu \in \mathcal{X}^{\ast} \,\middle|\, K \subseteq \H_{p, \nu} \right\},
\end{equation}
\begin{equation}
\label{eq:normal_bundle}
\mathrm{N}^{\ast} K := \left\{ (p, \nu) \in \mathcal{X} \times \mathcal{X}^{\ast} \,\middle|\, p \in K, \nu \in \mathrm{N}_{p}^{\ast} K \right\}.
\end{equation}
The outward normal cone $\mathrm{N}_{p}^{\ast} K$ is a \emph{pointed convex cone}: it contains $0$, is convex, and $s_{1} \nu_{1} + s_{2} \nu_{2} \in \mathrm{N}_{p}^{\ast} K$ for all $s_{1}, s_{2} \geq 0$ and all $\nu_{1}, \nu_{2} \in \mathrm{N}_{p}^{\ast} K$. Also, $\mathrm{N}_{p}^{\ast} K = \{ 0 \}$ if $p$ is an interior point of $K$. Note that $\mathrm{N}^{\ast} K \subseteq \mathcal{X} \times \mathcal{X}^{\ast}$ is not necessarily a convex set. See figure \ref{fig:normal_cone} for an illustration.
\begin{figure}
\noindent\framebox[\linewidth]{\parbox{\linewidth-20pt}{
\begin{center}
\scalebox{1}
{
\input ./fig-normal_cones.tex
}
\end{center}
\caption{A convex set $K$ and its outward normal cones at points $p, q, r \in K$. $\partial K$ is smooth at $p \in \partial K$, so $\mathrm{N}_{p}^{\ast} K$ is a half-line; $\partial K$ has a vertex at $q$, so $\mathrm{N}_{q}^{\ast} K$ is a pointed convex cone with non-empty interior; at the interior point $r$, $\mathrm{N}_{r}^{\ast} K$ is the empty set.}
\label{fig:normal_cone}
}}
\end{figure}
\subsection{Quasiconvexity}
If $K \subseteq \mathcal{X}$ is a convex set, then a function $f \colon K \to \mathbb{R} \cup \{ \pm \infty \}$ is said to be \emph{quasiconvex} if, for every $\theta \in \mathbb{R} \cup \{ \pm \infty \}$, the sublevel set
\begin{equation}
\label{eq:sublevel}
f^{-1}([-\infty, \theta]) := \{ x \in K \mid - \infty \leq f(x) \leq \theta \}
\end{equation}
is a convex set; equivalently, $f$ is quasiconvex if, for all $x, y \in K$ and $t \in [0, 1]$,
\begin{equation}
\label{eq:equivalent_quasiconvex}
f((1 - t)x + t y) \leq \max \{ f(x), f(y) \}.
\end{equation}
$f$ is said to be \emph{quasiconcave} if $-f$ is quasiconvex, and $f$ is said to be \emph{quasilinear} if it is both quasiconvex and quasiconcave. Every convex (resp.\ concave, linear) function is quasiconvex (resp.\ quasiconcave, quasilinear), but not \emph{vice versa}. In particular, a function $f \colon \mathbb{R}^{N} \to \mathbb{R}$ is quasilinear if, and only if, it is the composition of a monotone function with a linear functional on $\mathbb{R}^{N}$ \cite[p.\ 122]{BoydVandenberghe:2004}.
\subsection{Indicator and Characteristic Functions.}
Given a set $A \subseteq \mathcal{X}$, $\mathbbm{1}_{A}$ and $\chi_{A}$ denote its \emph{indicator function} and \emph{characteristic function} respectively:
\begin{equation}
\label{eq:indicator_fn}
\mathbbm{1}_{A}(x) :=
\begin{cases}
1, & \text{if $x \in A$,} \\
0, & \text{if $x \notin A$;}
\end{cases}
\end{equation}
\begin{equation}
\label{eq:characteristic_fn}
\chi_{A}(x) :=
\begin{cases}
0, & \text{if $x \in A$,} \\
+\infty, & \text{if $x \notin A$.}
\end{cases}
\end{equation}
Note that, for any convex set $K \subseteq \mathcal{X}$, $\chi_{K}$ is a convex function.
\subsection{Probabilistic Notions}
Let $(\Omega, \mathscr{F}, \P)$ be a probability space and let $X \colon \Omega \to \mathcal{X}$ be an $\mathcal{X}$-valued random variable. $\mathbb{E}[\cdot]$ denotes the expectation operator with respect to the probability measure $\P$: $\mathbb{E}[X]$ is defined to be any $m \in \mathcal{X}$ such that
\begin{equation}
\label{eq:expectation_vector}
\mathbb{E}[\langle \ell, X - m \rangle] \equiv \int_{\Omega} \langle \ell, X(\omega) - m \rangle \, \d \P (\omega) = 0 \text{ for all } \ell \in \mathcal{X}^{\ast};
\end{equation}
if $\mathcal{X}^{\ast}$ separates the points of $\mathcal{X}$ (\emph{e.g.}\ if $\mathcal{X}$ is a Banach space), then $\mathbb{E}[X]$ is unique. For $Y \colon \Omega \to \mathbb{R}$, any $m \in \mathbb{R}$ that satisfies
\begin{equation}
\label{eq:median}
\sup \left\{ v \in \mathbb{R} \,\middle|\, \P[Y \leq v] \leq \frac{1}{2} \right\} \leq m \leq \inf \left\{ v \in \mathbb{R} \,\middle|\, \P[Y \leq v] \geq \frac{1}{2} \right\}
\end{equation}
will be called a \emph{median} of $Y$ and denoted $\mathbb{M}[Y]$. $M_{X} \colon \mathcal{X}^{\ast} \to [0, + \infty]$ denotes the \emph{moment-generating function} defined by
\begin{equation}
\label{eq:MGF}
M_{X}(\ell) := \mathbb{E} \left[ \exp \langle \ell, X \rangle \right] \text{ for all } \ell \in \mathcal{X}^{\ast}.
\end{equation}
$\Lambda_{X} \colon \mathcal{X}^{\ast} \to \mathbb{R} \cup \{ \pm \infty \}$ denotes the \emph{cumulant generating function} (or \emph{logarithmic moment-generating function}) defined by
\begin{equation}
\label{eq:logMGF}
\Lambda_{X}(\ell) := \log M_{X}(\ell) = \log \mathbb{E} \left[ \exp \langle \ell, X \rangle \right] \text{ for all } \ell \in \mathcal{X}^{\ast}.
\end{equation}
By H{\"o}lder's inequality, $\Lambda_{X}$ is a convex function.
\subsection{Talagrand's Inequalities}
It has been known for some time that convex sets and functions enjoy good concentration properties; moreover, to get good concentration results, it is necessary to measure distances in the right way.
For example, a theorem of Talagrand shows that if a convex set $K \subseteq \mathbb{R}^{N}$ occupies a ``significant'' portion of the Hamming cube $\{ -1, +1 \}^{N}$ and $t \gg 1$, then ``nearly all'' of the points of the Hamming cube lie within Euclidean distance $t$ of $K$. Define the \emph{Euclidean Hausdorff distance} from $x \in \mathbb{R}^{N}$ to $A \subseteq \mathbb{R}^{N}$ by
\begin{equation}
\label{eq:Hausdorff_distance}
d_{\mathrm{Haus}}(x, A) := \inf \{ \| x - a \|_{2} \mid a \in A \}.
\end{equation}
Talagrand \cite{Talagrand:1988} showed that if $X$ is uniformly distributed in $\{ -1, +1 \}^{N}$ then, for any $A \subseteq \mathbb{R}^{N}$, $\mathbb{E}[\exp(d_{\mathrm{Haus}}(X, \mathop{\overline{\mathrm{co}}}(A))^{2} / 8)] \leq \P[X \in A]^{-1}$; hence, Chebyshev's inequality implies that, for any $t \geq 0$,
\begin{equation}
\label{eq:Talagrand-1}
\P[X \in A] \P [d_{\mathrm{Haus}}(X, \mathop{\overline{\mathrm{co}}}(A)) \geq t] \leq \exp \left( - \frac{t^{2}}{8} \right).
\end{equation}
More interesting results can be obtained if one uses not the Euclidean distance but the Hamming distance --- or, more accurately, an infimum over weighted Hamming distances. For $w = (w_{1}, \dots, w_{N}) \in [0, + \infty)^{N}$, define the \emph{$w$-weighted Hamming distance} $d_{w}$ on a product of sets $\mathcal{X} = \prod_{n = 1}^{N} \mathcal{X}_{n}$ by
\begin{equation}
\label{eq:w_Hamming_distance}
d_{w}(x, y) := \sum_{n = 1}^{N} w_{n} (1 - \delta_{x_{n}, y_{n}});
\end{equation}
that is, $d_{w}(x, y)$ is the $w$-weighted sum of the number of components in which $x, y \in \mathcal{X}$ differ. For $x \in \mathcal{X}$ and $A \subseteq \mathcal{X}$, set $d_{w}(x, A) := \inf_{a \in A} d_{w}(x, a)$. Define \emph{Talagrand's convex distance} from $x \in \mathcal{X}$ to $A \subseteq \mathcal{X}$ by
\begin{equation}
\label{eq:Talagrand_distance}
d_{\mathrm{Tal}}(x, A) := \sup \left\{ d_{w}(x, A) \,\middle|\, w \in [0, + \infty)^{N}, \sum_{n = 1}^{N} w_{n}^{2} = 1 \right\},
\end{equation}
and, for $A, B \subseteq \mathcal{X}$, let $d_{\mathrm{Tal}}(A, B) := \inf_{a \in A} d_{\mathrm{Tal}}(a, B)$. Talagrand \cite[\S4.1]{Talagrand:1995} showed that if $X = (X_{1}, \dots, X_{N})$ is any $\mathcal{X}$-valued random variable with independent components, then
\begin{equation}
\label{eq:Talagrand-3}
\P[X \in A] \P[X \in B] \leq \exp \left( - \frac{d_{\mathrm{Tal}}(A, B)^{2}}{4} \right).
\end{equation}
These bounds on the probabilities of sets lead to deviation inequalities for convex Lipschitz functions. For example (\emph{cf.}\ \cite{JohnsonSchechtman:1991} \cite{Talagrand:1988}), let $X$ be any random variable in the unit cube in $\mathbb{R}^{N}$ with independent components, and let $f \colon [0, 1]^{N} \to \mathbb{R}$ be convex and Lipschitz with $\| f \|_{\mathrm{Lip}} \leq 1$; then, for any $t \geq 0$,
\begin{equation}
\label{eq:Talagrand-2}
\P[f(X) \geq \mathbb{M}[f(X)] + t] \leq 2 \exp \left( - \frac{t^{2}}{4} \right).
\end{equation}
Note, however, that these results use not only the convexity of the function of interest, but also require Lipschitz continuity. What concentration inequalities can be shown to hold without smoothness assumptions?
\subsection{McDiarmid's Inequality}
One smoothness-free concentration inequality is \emph{McDiarmid's inequality} \cite{McDiarmid:1989}, also known as the \emph{bounded differences inequality}, which itself generalizes an earlier inequality of Hoeffding \cite{Hoeffding:1963}. McDiarmid's inequality is by no means the strongest concentration-of-measure inequality in the literature, but is useful because of its simple hypotheses and proof.
Define the \emph{McDiarmid diameter} of $f$, denoted $\mathcal{D}[f]$, by
\begin{equation}
\label{eq:McD_diameter}
\mathcal{D}[f] := \left( \sum_{n = 1}^{N} \mathcal{D}_{n}[f]^{2} \right)^{1/2},
\end{equation}
where the $n^{\mathrm{th}}$ \emph{McDiarmid subdiameter} $\mathcal{D}_{n}[f]$ is defined by
\begin{equation}
\label{eq:McD_subdiameter}
\mathcal{D}_{n}[f] := \sup \{ | f(x) - f(y) | \mid x_{j} = y_{j} \text{ for } j \neq n \}.
\end{equation}
When $\mathbb{E}[|f(X)|]$ is finite and $X_{1}, \dots, X_{N}$ are independent, McDiarmid's inequality bounds the deviations of $f(X)$ from $\mathbb{E}[f(X)]$ in terms of the McDiarmid diameter of $f$: for any $r > 0$,
\begin{subequations}
\label{eq:McD}
\begin{align}
\label{eq:McD-leq}
\P[f(X) - \mathbb{E}[f(X)] \leq - r] & \leq \exp \left( - \frac{2 r^{2}}{\mathcal{D}[f]^{2}} \right), \\
\label{eq:McD-geq}
\P[f(X) - \mathbb{E}[f(X)] \geq r] & \leq \exp \left( - \frac{2 r^{2}}{\mathcal{D}[f]^{2}} \right).
\end{align}
\end{subequations}
McDiarmid's inequality implies that, for any $\theta \in \mathbb{R} \cup \{ \pm \infty \}$,
\begin{subequations}
\label{eq:McD2}
\begin{align}
\label{eq:McD2-leq}
\P[f(X) \leq \theta] & \leq \exp \left( - \frac{2 (\mathbb{E}[f(X)] - \theta)_{+}^{2}}{\mathcal{D}[f]^{2}} \right), \\
\label{eq:McD2-geq}
\P[f(X) \geq \theta] & \leq \exp \left( - \frac{2 (\theta - \mathbb{E}[f(X)])_{+}^{2}}{\mathcal{D}[f]^{2}} \right).
\end{align}
\end{subequations}
McDiarmid's inequality (and similar inequalities such as martingale inequalities) have the advantage that a bound on the tails of $f(X)$ is obtained solely in terms of the mean output $\mathbb{E}[f(X)]$ and the McDiarmid diameter $\mathcal{D}[f]$. However, McDiarmid's inequality cannot take advantage of any other properties of $f$ such as convexity or monotonicity; furthermore, if $f$ has infinite McDiarmid diameter on the essential range of $X$, then the trivial upper bound $1$ is obtained.
There are many other sources of concentration-of-measure inequalities: these include logarithmic Sobolev inequalities and the Herbst argument \cite{BakryEmery:1985} \cite{Gross:1975} \cite{HolleyStroock:1987}, the entropy method \cite{BobkovLedoux:1998} \cite{BoucheronLugosiMassart:2003} \cite{Ledoux:1996}, and information-theoretic methods \cite{Dembo:1997} \cite{Marton:1996}. Of particular interest are those concentration results that apply to infinite-dimensional settings \cite{LedouxTalagrand:1991}.
\section{Normal Distance}
\label{sec:normal_distance}
As noted above, efficient presentation of many concentration-of-measure inequalities relies on having an appropriate notion of function variation (\emph{e.g.}\ the Lipschitz norm or McDiarmid diameter) or distance (\emph{e.g.}\ Talagrand's convex distance). The inequalities that will be established in section \ref{sec:main} can be phrased in terms of transforms of moment-generating functions, but are more transparent if phrased in terms of a \emph{normal distance}, which will introduced in this section.
Fix a function $\Psi \colon \mathcal{X}^{\ast} \to [0, + \infty]$ that is positively homogeneous of degree one, \emph{i.e.}\ such that $\Psi(\alpha \ell) = \alpha \Psi(\ell)$ for all $\alpha \geq 0$ and all $\ell \in \mathcal{X}^{\ast}$. By analogy with the situation in finite-dimensional Euclidean space, in which $\Psi = \| \cdot \|_{2}$ on $(\mathbb{R}^{N})^{\ast}$, define the distance from a point $x \in \mathcal{X}$ to a half-space $\H_{p, \nu} \subseteq \mathcal{X}$ by
\begin{equation}
\label{eq:normal_distance_halfspace}
d_{\perp, \Psi}(x, \H_{p, \nu}) := \frac{\langle \nu, x - p \rangle_{+}}{\Psi(\nu)},
\end{equation}
with the convention that $0 / 0 = 0$, since the distance from $x \in \mathcal{X}$ to the trivial half-space $\H_{p, \nu} = \mathcal{X}$ ought to be zero. Note that $d_{\perp, \Psi}(x, \H_{p, \nu}) = 0$ whenever $x \in \H_{p, \nu}$; note also that the homogeneity assumption on $\Psi$ ensures that \eqref{eq:normal_distance_halfspace} is an unambiguous definition. We now generalize \eqref{eq:normal_distance_halfspace} to more general subsets of $\mathcal{X}$ than half-spaces. The heuristic is that the distance from $x$ to $A \subseteq \mathcal{X}$ should be the greatest possible distance (in the sense of \eqref{eq:normal_distance_halfspace}) from $x$ to any half-space that contains $A$; the existence of the degenerate half-space $\H_{p, 0}$ ensures that the normal distance is zero if there are no proper half-spaces that contain $A$.
\begin{defn}
\label{defn:normal_distance}
Let $x \in \mathcal{X}$ and $A \subseteq \mathcal{X}$. The \emph{$\Psi$-normal distance} from $x$ to $A$, denoted $d_{\perp, \Psi}(x, A)$, is defined (with the same convention that $0 / 0 = 0$) by
\begin{equation}
\label{eq:normal_distance}
d_{\perp, \Psi}(x, A) := \sup \left\{ \frac{\langle \nu, x - p \rangle_{+}}{\Psi(\nu)} \,\middle|\, \begin{matrix} p \in \mathcal{X} \text{ and } \nu \in \mathcal{X}^{\ast} \\ \text{such that } A \subseteq \H_{p, \nu} \end{matrix} \right\}.
\end{equation}
The $\Psi$-normal distance from $A \subseteq \mathcal{X}$ to $B \subseteq \mathcal{X}$ is defined by $d_{\perp, \Psi}(A, B) := \inf_{a \in A} d_{\perp, \Psi}(a, B)$. In the special case $\mathcal{X} = \mathbb{R}^{N}$ and $\Psi = \| \cdot \|_{2}$ on $(\mathbb{R}^{N})^{\ast}$, we shall simply write $d_{\perp}$ for $d_{\perp, \Psi}$, \emph{i.e.}
\begin{equation}
\label{eq:normal_distance_Euclidean}
d_{\perp}(x, A) := \sup \left\{ \frac{(\nu \cdot (x - p))_{+}}{\| \nu \|_{2}} \,\middle|\, \begin{matrix} p \in \mathbb{R}^{N} \text{ and } \nu \in (\mathbb{R}^{N})^{\ast} \\ \text{such that } A \subseteq \H_{p, \nu} \end{matrix} \right\}.
\end{equation}
\end{defn}
Note well that the definition of the normal distance $d_{\perp, \Psi}(x, A)$ does not require $\mathcal{X}$ to be normed; even when $\mathcal{X}$ is equipped with a norm $\| \cdot \|_{\mathcal{X}}$ and $\Psi$ is the corresponding operator norm, the normal distance $d_{\perp, \Psi}(x, A)$ is not the same as the Hausdorff distance from $x$ to $A$ defined by
\begin{equation}
\label{eq:Hausdorff_distance_X}
d_{\mathrm{Haus}}(x, A) := \inf \{ \| x - a \|_{\mathcal{X}} \mid a \in A \};
\end{equation}
see figure \ref{fig:normal_distance} for an illustration. Note also that it is not generally true that $d_{\perp, \Psi}(A, B) = d_{\perp, \Psi}(B, A)$: consider \emph{e.g.}\ $B := \{ (0, 1) \}$ and $A$ as in figure \ref{fig:normal_distance}, in which case
\[
d_{\perp, \Psi}(A, B) = \inf_{a \in A} d_{\perp, \Psi}(a, B) = 1 \neq 0 = d_{\perp, \Psi}(B, A).
\]
\begin{figure}
\noindent\framebox[\linewidth]{\parbox{\linewidth-20pt}{
\begin{center}
\scalebox{1}
{
\input ./fig-normal_distance.tex
}
\end{center}
\caption{An example of a subset $A$ of the Euclidean plane $\mathbb{R}^{2}$ for which the normal distance $d_{\perp}(0, A) = 1$ unit (\emph{cf.}\ the dashed line), as opposed to the Euclidean Hausdorff distance $d_{\mathrm{Haus}}(0, A) = 2$ units (\emph{cf.}\ the dotted arc).}
\label{fig:normal_distance}
}}
\end{figure}
For any $x \in \mathcal{X}$ and $A \subseteq B \subseteq \mathcal{X}$, it holds that $d_{\perp, \Psi}(x, B) \leq d_{\perp, \Psi}(x, A)$. Furthermore, since a closed half-space $\H_{p, \nu}$ contains $A$ if, and only if, it contains the closed convex hull $\mathop{\overline{\mathrm{co}}}(A)$ of $A$, the following equality holds:
\begin{equation}
\label{eq:normal_distance_convex_hull}
d_{\perp, \Psi}(x, A) = d_{\perp, \Psi}(x, \mathop{\overline{\mathrm{co}}}(A)) \text{ for all } x \in \mathcal{X} \text{ and all } A \subseteq \mathcal{X}.
\end{equation}
\begin{rmk}
It is natural to ask what, if any, relation there is between the normal distance and Talagrand's convex distance. The simplest answer is to say that the two distances should be compared only with great caution, since each belongs to a different setting: Talagrand's distance is defined on a product of sets, whereas the normal distance is defined on a topological vector space. Even on $\mathbb{R}^{N}$, the two distances measure different quantities: in some sense, $d_{\mathrm{Tal}}(x, A)$ measures how many of the coordinates of $x$ are covered by $A$, but does not measure the geometric distance between them; on the other hand, $d_{\perp, \Psi}(x, A)$ is a much more geometric measure of how far $x$ is from $A$ in terms of linear functionals on $\mathcal{X}$, and the ``size'' of those linear functionals is measured by $\Psi$. In particular, Talagrand's convex distance is positively homogeneous of degree zero, whereas the normal distance is positively homogeneous of degree one: for any $x \in \mathbb{R}^{N}$, $A \subseteq \mathbb{R}^{N}$, and $\alpha > 0$,
\[
d_{\mathrm{Tal}}(\alpha x, \alpha A) = d_{\mathrm{Tal}}(x, A),
\]
\[
d_{\perp, \Psi}(\alpha x, \alpha A) = \alpha d_{\perp, \Psi}(x, A).
\]
\end{rmk}
This section concludes with the proof of the portmanteau theorem (theorem \ref{thm:portmanteau}) and some final remarks on its applicability:
\begin{proof}[Proof of theorem \ref{thm:portmanteau}.]
The equivalence will be established by showing that
\[
\text{(\ref{port-1})} \implies \text{(\ref{port-2})} \implies \text{(\ref{port-3})} \implies \text{(\ref{port-4})} \implies \text{(\ref{port-5})} \implies \text{(\ref{port-1})}.
\]
Suppose that (\ref{port-1}) holds. Then
\begin{align*}
& \P[X \in K] \\
& \quad \leq \inf_{\H_{p, \nu} \supseteq K} \P[X \in \H_{p, \nu}] & & \text{by monotonicity of $\P$,} \\
& \quad \leq \inf_{\H_{p, \nu} \supseteq K} \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], \H_{p, \nu})^{2}}{2} \right) & & \text{by (\ref{port-1}),} \\
& \quad = \exp \left( - \frac{1}{2} \sup_{\H_{p, \nu} \supseteq K} d_{\perp, \Psi}(\mathbb{E}[X], \H_{p, \nu})^{2} \right) & & \\
& \quad = \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], K)^{2}}{2} \right) & & \text{by \eqref{eq:normal_distance}.}
\end{align*}
Hence, (\ref{port-1}) implies (\ref{port-2}).
Suppose that (\ref{port-2}) holds; then
\begin{align*}
\P[X \in A]
& \leq \P[X \in \mathop{\overline{\mathrm{co}}}(A)] & & \text{since $A \subseteq \mathop{\overline{\mathrm{co}}}(A)$,} \\
& \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], \mathop{\overline{\mathrm{co}}}(A))^{2}}{2} \right) & & \text{by (\ref{port-2}),} \\
& = \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right) & & \text{by \eqref{eq:normal_distance_convex_hull},}
\end{align*}
and so (\ref{port-2}) implies (\ref{port-3}). (\ref{port-4}) follows from (\ref{port-3}) upon setting $A := \{ x \in \mathcal{X} \mid f(x) \leq \theta \}$. (\ref{port-5}) is clearly a special case of (\ref{port-4}). (\ref{port-1}) follows from (\ref{port-5}) upon setting $f := \chi_{\H_{p, \nu}}$ and $\theta := 1$.
\end{proof}
\begin{rmk}
It is important to note that all the bounds in theorem \ref{thm:portmanteau} may be trivial if the dual space $\mathcal{X}^{\ast}$ is not rich enough. For example, given a measure space $(\mathcal{Z}, \mathscr{F}, \mu)$, for $0 < p < 1$, the space
\[
\mathcal{L}^{p}(\mathcal{Z}, \mathscr{F}, \mu; \mathbb{R}) := \left\{ f \colon \mathcal{Z} \to \mathbb{R} \,\middle|\, \| f \|_{p} := \left( \int_{\mathcal{Z}} | f(z) |^{p} \, \d \mu(z) \right)^{1/p} < + \infty \right\}
\]
is a topological vector space with respect to the quasinorm topology generated by $\| \cdot \|_{p}$. This space is not locally convex and has a trivial dual space: the only continuous linear functional on this space is the zero functional, and so the only closed half-space is the whole space. See \emph{e.g.}\ \cite[\S1.47]{Rudin:1991} for further discussion of spaces such as $\mathcal{L}^{p}([0, 1]; \mathbb{R})$ for $0 < p < 1$.
It is tempting to eliminate these pathologies by working with the algebraic, instead of the topological, dual of $\mathcal{X}$. This can be done, and most results go through \emph{mutatis mutandis}; in particular, it is necessary to replace all references to the closed convex hull $\mathop{\overline{\mathrm{co}}}(A)$ of $A \subseteq \mathcal{X}$ with the convex hull $\mathop{\mathrm{co}}(A)$; the analogue of \eqref{eq:normal_distance_convex_hull} (with $\Psi$ now defined on the algebraic dual of $\mathcal{X}$) is
\[
d_{\perp, \Psi}(x, A) = d_{\perp, \Psi}(x, \mathop{\mathrm{co}}(A)) \text{ for all } x \in \mathcal{X} \text{ and all } A \subseteq \mathcal{X}.
\]
The principal disadvantage of ignoring all topological structure on $\mathcal{X}$, of course, is that there are no longer notions of interior, closure and frontier --- although it still makes sense to discuss the extremal points of convex sets.
\end{rmk}
\section{Normal Distance as a Concentration Rate}
\label{sec:main}
The method of Chernoff bounding (reviewed in lemma \ref{lem:Chernoff}) gives bounds on $\P[X \in \H_{p, \nu}]$ in terms of the moment-generating function $M_{X}$. If these bounds can be formulated in terms of a suitable normal distance, then theorem \ref{thm:portmanteau} produces equivalent bounds for on $\P[X \in K]$ for convex $K$, on $\P[X \in A]$, \emph{\& c.}. As noted in \cite[\S2]{Lugosi:2009}, the best Chernoff bound on $\P[f(X) \geq \theta]$ is never better than the best bound using the all the moments of $f(X)$: if $f$ takes only non-negative values, then
\begin{equation}
\label{eq:moments_vs_Chernoff}
\inf_{k \in \mathbb{N}} \theta^{-k} \mathbb{E} \big[ f(X)^{k} \big] \leq \inf_{s \geq 0} e^{-s \theta} \mathbb{E} \big[ e^{s f(X)} \big].
\end{equation}
However, Chernoff bounds have the advantage that they are geometrically very easy to handle.
The next result provides the normal distance formulation for an $\mathcal{X}$-valued Gaussian random variable (in fact, for a family of such variables). In the special case of a single Gaussian random vector $X$ on $\mathcal{X} = \mathbb{R}^{N}$ with covariance operator $C_{X} = \sigma \mathbb{I}_{N}$, proposition \ref{prop:Gaussian} yields the classical Chernoff bound for a multivariate normal random variable.
\begin{prop}
\label{prop:Gaussian}
Let $\Gamma$ be a family of Gaussian random vectors in $\mathcal{X}$. For each $X \in \Gamma$, let $C_{X} \colon \mathcal{X}^{\ast} \to \mathcal{X}^{\ast \ast}$ be its covariance operator defined by
\begin{equation}
\label{eq:covariance}
\langle C_{X} \ell, \nu \rangle := \mathbb{E} \left[ \langle \ell, X \rangle \langle \nu, X \rangle \right].
\end{equation}
Let $E := \{ \mathbb{E}[X] \mid X \in \Gamma \}$, let
\begin{equation}
\label{eq:Gaussian_Psi}
\Psi(\nu) := \sup_{X \in \Gamma} \sqrt{\langle C_{X} \nu, \nu \rangle},
\end{equation}
and let $d_{\perp, \Psi}$ be the corresponding normal distance. Then, for any $A \subseteq \mathcal{X}$,
\begin{equation}
\label{eq:Gaussian_bound}
\sup_{X \in \Gamma} \P[X \in A] \leq \exp \left( - \frac{d_{\perp, \Psi}(E, A)^{2}}{2} \right).
\end{equation}
\end{prop}
\begin{proof}
For each $X \in \Gamma$, the moment-generating function for $X$ is given by
\begin{equation}
\label{eq:Gaussian_MGF_bound}
M_{X}(\ell) := \mathbb{E} \left[ e^{\langle \ell, X \rangle} \right] = \exp \left( \langle \ell, \mathbb{E}[X] \rangle + \frac{\langle C_{X} \ell, \ell \rangle}{2} \right).
\end{equation}
Therefore,
\begin{align*}
& \P[X \in \H_{p, \nu}] \\
& \quad \leq \inf_{s \geq 0} \exp \left( s \langle \nu, p - \mathbb{E}[X] \rangle + s^{2} \frac{\langle C_{X} \nu, \nu \rangle}{2} \right) & & \text{by \eqref{eq:Gaussian_MGF_bound} and lemma \ref{lem:Chernoff},} \\
& \quad = \exp \left( - \frac{\langle \nu, \mathbb{E}[X] - p \rangle_{+}^{2}}{2 \langle C_{X} \nu, \nu \rangle^{2}} \right) & & \\
& \quad \leq \exp \left( - \frac{\langle \nu, \mathbb{E}[X] - p \rangle_{+}^{2}}{2 \Psi(\nu)^{2}} \right) & & \text{by \eqref{eq:Gaussian_Psi},} \\
& \quad = \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], \H_{p, \nu})^{2}}{2} \right) & & \text{by \eqref{eq:normal_distance}.}
\end{align*}
Hence, by theorem \ref{thm:portmanteau},
\[
\P[X \in A] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right),
\]
and so
\begin{align*}
\sup_{X \in \Gamma} \P[X \in A]
& \leq \sup_{X \in \Gamma} \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right) \\
& = \exp \left( - \inf_{X \in \Gamma} \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right) \\
& = \exp \left( \frac{d_{\perp, \Psi}(E, A)^{2}}{2} \right). \qedhere
\end{align*}
\end{proof}
Lemma \ref{lem:Chernoff} also has the following consequences for random vectors supported in a cuboid in $\mathbb{R}^{N}$; this encompasses two standard situations in which concentration is often studied, namely concentration for functions on the Euclidean unit cube and on the Hamming cube.
\begin{prop}
\label{prop:cuboid}
Let $X$ be a random vector in $\mathbb{R}^{N}$ with independent components such that each component $X_{n}$ almost surely takes values in a fixed interval of length $L_{n}$. Let
\begin{equation}
\label{eq:cuboid_Psi}
\Psi(\nu) := \frac{1}{2} \sqrt{ \sum_{n = 1}^{N} L_{n}^{2} \nu_{n}^{2} }
\end{equation}
and let $d_{\perp, \Psi}$ be the corresponding normal distance. Then, for any $A \subseteq \mathbb{R}^{N}$,
\begin{equation}
\label{eq:cuboid_bound}
\P[X \in A] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[X], A)^{2}}{2} \right).
\end{equation}
\emph{A fortiori}, if $X$ takes values in (a translate of) the unit cube $[0, 1]^{N}$, then
\begin{equation}
\label{eq:unit_cube_bound}
\P[X \in A] \leq \exp \left( - 2 d_{\perp}(\mathbb{E}[X], A)^{2} \right),
\end{equation}
and if $X$ takes values in (a translate of) the Hamming cube $\{ -1, +1 \}^{N}$, then
\begin{equation}
\label{eq:hamming_cube_bound}
\P[X \in A] \leq \exp \left( - \frac{d_{\perp}(\mathbb{E}[X], A)^{2}}{2} \right).
\end{equation}
\end{prop}
\begin{proof}
The proof is similar to the Gaussian case: it is an application of lemma \ref{lem:Chernoff} and Hoeffding's lemma \cite[lemma 1 and (4.16)]{Hoeffding:1963}, which bounds the moment-generating function of $X_{n}$ as follows:
\[
M_{X_{n}}(\ell_{n}) := \mathbb{E} \left[ \exp(\ell_{n} X_{n}) \right] \leq \exp \left( \ell_{n} \mathbb{E}[X_{n}] + \frac{\ell_{n}^{2} L_{n}^{2}}{8} \right).
\]
Note that the claim can also be proved directly by applying McDiarmid's inequality to the function $\langle \nu, \cdot \rangle$, which has mean $\mathbb{E}[ \langle \nu, X \rangle] = \langle \nu, \mathbb{E}[X] \rangle$ and McDiarmid diameter $\sqrt{L_{1}^{2} + \dots + L_{N}^{2}}$.
\end{proof}
\begin{rmk}
Note the similarity between the normal distances of propositions \ref{prop:Gaussian} and \ref{prop:cuboid}. In the Gaussian case, the norm on $\mathcal{X}^{\ast}$ is the one induced by the ``largest'' covariance operator in the family of random variables $\Gamma$. In the bounded-range case, the norm on $\mathcal{X}^{\ast}$ is the one induced by the ``largest'' covariance operator for random variables satisfying the range constraint: if $X$ is a real-valued random variable taking values in an interval $[a, b]$, then $\Psi(\nu)^{2} = \frac{1}{4} (b - a)^{2} \nu^{2}$ and $\mathrm{Var}[X] \leq \frac{1}{4} (b - a)^{2}$; this upper bound on the variance is attained by a Bernoulli random variable with law $\frac{1}{2} \delta_{a} + \frac{1}{2} \delta_{b}$.
\end{rmk}
The next result identifies the normal distance that corresponds to the concentration of a vector, the entries of which are the empirical (sampled) means of functions of independent random variables.
\begin{prop}
\label{prop:emp_mean}
For $n = 1, \dots, N$, let $Z_{n} := f_{n}(Y_{n, 1}, \dots, Y_{n, K(n)})$ be a real-valued function of independent random variables $Y_{n, k}$, and suppose that $f_{n}$ has finite McDiarmid diameter $\mathcal{D}[f_{n}]$. Let $Z = (Z_{1}, \dots, Z_{N})$. Suppose that the random inputs of each $f_{n}$ are sampled independently $M(n)$ times according to the distribution $\P$ and that the empirical average
\begin{equation}
\label{eq:emp_mean}
\widehat{\mathbb{E}}[Z] = \left( \frac{1}{M(n)} \sum_{m = 1}^{M(n)} f_n \left( Y_{n, 1}^{(m)}, \dots, Y_{n, K(n)}^{(m)} \right) \right)_{n = 1}^{N} \in \mathbb{R}^{N}
\end{equation}
is formed. Then, for any $A \subseteq \mathbb{R}^{N}$,
\begin{equation}
\label{eq:emp_mean_bound}
\P \left[ \widehat{\mathbb{E}}[Z] \in A \right] \leq \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[Z], A)^{2}}{2} \right),
\end{equation}
where the distance $\Psi \colon (\mathbb{R}^{N})^{\ast} \to [0, +\infty)$ is given in terms of the McDiarmid diameters of the functions $f_{1}, \dots, f_{N}$ and the sample sizes $M(1), \dots, M(N)$:
\begin{equation}
\label{eq:emp_mean_Psi}
\Psi(\nu) := \frac{1}{2} \left( \sum_{n = 1}^{N} \frac{\nu_{n}^{2} \mathcal{D}[f_{n}]^{2}}{M(n)} \right)^{1/2}.
\end{equation}
\end{prop}
\begin{proof}
Let $\H_{p, \nu} \subsetneq \mathbb{R}^{N}$ be a half-space. Consider the real-valued random variable $\left\langle \nu, \widehat{\mathbb{E}}[Z] \right\rangle$ as a function of the sampled input random variables $Y_{n, k}^{(m)}$. Suppose that the McDiarmid subdiameter of $f_{n}$ with respect to $Y_{n, k}$ is $D_{n, k}$. Then the McDiarmid subdiameter of $\left\langle \nu, \widehat{\mathbb{E}}[Z] \right\rangle$ with respect to the $m^{\mathrm{th}}$ sample of $Y_{n, k}$ is $\nu_{n} D_{n, k} / M(n)$. Hence, the McDiarmid diameter of $\left\langle \nu, \widehat{\mathbb{E}}[Z] \right\rangle$ is
\[
\sqrt{ \sum_{k, n, m} \frac{\nu_{n}^{2} D_{n, k}^{2}}{M(n)^{2}} } = \sqrt{ \sum_{n, m} \frac{\nu_{n}^{2} \mathcal{D}[f_{n}]^{2}}{M(n)^{2}} } = \sqrt{ \sum_{n} \frac{\nu_{n}^{2} \mathcal{D}[f_{n}]^{2}}{M(n)} }
\]
Therefore, since $\widehat{\mathbb{E}}[Z]$ is an unbiased estimator for $\mathbb{E}[Z]$ (\emph{i.e.}\ $\mathbb{E} \left[ \widehat{\mathbb{E}}[Z] \right] = \widehat{\mathbb{E}}[Z]$), McDiarmid's inequality \eqref{eq:McD2-leq} implies that
\begin{align*}
\P \left[ \widehat{\mathbb{E}}[Z] \in \H_{p, \nu} \right]
& = \P \left[ \left\langle \nu, \widehat{\mathbb{E}}[Z] \right\rangle \leq \langle \nu, p \rangle \right] \\
& \leq \exp \left( - \frac{2 \left( \langle \nu, \mathbb{E}[Z] \rangle - \langle \nu, p \rangle \right)_{+}^{2}}{ \sum_{n = 1}^{N} \frac{\nu_{n}^{2} \mathcal{D}[f_{n}]^{2}}{M(n)} } \right) \\
& = \exp \left( - \frac{\langle \nu, \mathbb{E}[Z] - p \rangle_{+}^{2}}{2 \cdot \frac{1}{4} \cdot \sum_{n = 1}^{N} \frac{\nu_{n}^{2} \mathcal{D}[f_{n}]^{2}}{M(n)}} \right) \\
& = \exp \left( - \frac{d_{\perp, \Psi}(\mathbb{E}[Z], \H_{p, \nu})^{2}}{2} \right).
\end{align*}
The claim now follows from theorem \ref{thm:portmanteau}.
\end{proof}
An example of the application of proposition \ref{prop:emp_mean} is the following:
\begin{eg}[Functions of empirical means]
The Chernoff bounding method can be used to provide much-improved confidence levels for quantities derived from many empirical --- as opposed to exact --- means; see \emph{e.g.}\ \cite[\S5]{SullivanTopcuMcKernsOwhadi:2010}. Suppose that $H_{0} \colon \mathbb{R}^{N} \to \mathbb{R}$ is some function of interest: in particular, the quantity of interest is $H_{0} \left( \mathbb{E}[Z_{1}], \dots, \mathbb{E}[Z_{N}] \right)$ for some absolutely integrable real-valued random variables $Z_{1}, \dots, Z_{N}$. If, however, the exact means $\mathbb{E}[Z_{n}]$ are unknown, then empirical means $\widehat{\mathbb{E}}[Z_{n}]$ may be used in their place if appropriate confidence corrections are made. Suppose that ``error'' corresponds to concluding, based on the empirical means, that $H_{0}(\mathbb{E}[Z])$ is smaller than it actually is. Given $\alpha \in \mathbb{R}^{N}$, set
\begin{equation}
\label{eq:H_alpha}
H_{\alpha}(z_{1}, \dots, z_{N}) := H_{0}(z_{1} + \alpha_{1}, \dots, z_{N} + \alpha_{N}).
\end{equation}
Therefore, given any $\varepsilon > 0$, we seek an appropriate ``margin hit'' $\alpha = \alpha(\varepsilon) \in \mathbb{R}^{N}$ (typically, $\alpha_{n} \geq 0$ for each $n \in \{ 1, \dots, N \}$) such that
\[
\P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z_{1}], \dots, \widehat{\mathbb{E}}[Z_{N}] \right) \geq H_{0} \left( \mathbb{E}[Z_{1}], \dots, \mathbb{E}[Z_{N}] \right) \right] \geq 1 - \varepsilon.
\]
Dually, given $\alpha \in \mathbb{R}^{N}$, we seek a sharp upper bound on the probability of error, \emph{i.e.}\ on
\[
\P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z_{1}], \dots, \widehat{\mathbb{E}}[Z_{N}] \right) \leq H_{0} \left( \mathbb{E}[Z_{1}], \dots, \mathbb{E}[Z_{N}] \right) \right].
\]
If $H_{0}$ (and hence $H_{\alpha}$) is monotonic in each of its $N$ arguments and $Z_{1}, \dots, Z_{N}$ are independent, then the probability of non-error can be bounded from below as follows:
\begin{align*}
\P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z] \right) \leq H_{0}(\mathbb{E}[Z]) \right]
& = \P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z] \right) \leq H_{\alpha}(\mathbb{E}[Z] - \alpha) \right] \\
& \leq \prod_{n = 1}^{N} \P \left[ \widehat{\mathbb{E}}[Z_{n}] \leq \mathbb{E}[Z_{n}] - \alpha_{n} \right] \\
& \leq 1 - \prod_{n = 1}^{N} \left( 1 - \exp \left( - \frac{- 2 M(n) (\alpha_{n})_{+}^{2}}{\mathcal{D}[f_{n}]^{2}} \right) \right).
\end{align*}
Unfortunately, when $N$ is large, the last line of this inequality is typically close to zero unless the sample sizes are very large, and so this bound is of limited use. Geometrically, this is analogous to the fact that a high-dimensional orthant (product of half-lines) appears to be very narrow from the perspective of an observer at its vertex. In contrast, half-spaces always fill a half of the observer's field of view. To bound the probability of sublevel or superlevel sets using half-spaces requires $H_{\alpha}$ to have some convexity --- not monotonicity --- properties.
If $H_{\alpha}$ is quasiconvex, then the bounds using normal distances can be applied to good effect, and yield estimates that actually perform better the larger $N$ is. In particular, if $H_{\alpha}$ is both quasiconvex and differentiable, then the outward normal to its $t$-level set at some point $p$ is just any positive multiple of the derivative of $H_{\alpha}$ at $p$, and this yields the bound
\begin{equation}
\label{eq:pre_link_formula}
\P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z] \right) \leq \theta \right] \leq \inf_{p : H_{\alpha}(p) \leq \theta} \exp \left( - \frac{2 \left( \sum_{n = 1}^{N} \partial_{n} H_{\alpha}(p) (\mathbb{E}[Z_{n}] - p_{n}) \right)_{+}^{2}}{ \sum_{n = 1}^{N} \frac{(\partial_{n} H_{\alpha}(p))^{2} \mathcal{D}[f_{n}]^{2}}{M(n)} } \right).
\end{equation}
In particular, taking $\theta = H_{0}(\mathbb{E}[Z]) = H_{\alpha}(\mathbb{E}[Z] - \alpha)$ and evaluating the exponential in \eqref{eq:pre_link_formula} at $p = \mathbb{E}[Z] - \alpha \in \mathbb{R}^{N}$ yields that
\begin{equation}
\label{eq:link_formula}
\P \left[ H_{\alpha} \left( \widehat{\mathbb{E}}[Z] \right) \leq H_{0}(\mathbb{E}[Z]) \right] \leq \exp \left( - \frac{2 \left( \sum_{n = 1}^{N} \partial_{n} H_{\alpha}(p) \alpha_{n} \right)_{+}^{2}}{ \sum_{n = 1}^{N} \frac{(\partial_{n} H_{\alpha}(p))^{2} \mathcal{D}[f_{n}]^{2}}{M(n)} } \right).
\end{equation}
\eqref{eq:link_formula} is particularly useful since it links the margin hits $\alpha_{n}$, the sample sizes $M(n)$, and the maximum probability of error. For example, given a desired level of confidence, margin hits $\alpha_{n}$, and a total number of samples $M \in \mathbb{N}$, one can choose sample sizes $M(1), \dots, M(N)$ that sum to $M$ and minimize the right-hand side of \eqref{eq:link_formula}; this yields an optimal distribution of sampling resources so as to ensure that $H_{\alpha} \left( \widehat{\mathbb{E}}[Z] \right) \geq H_{0}(\mathbb{E}[Z])$ with the desired level of confidence.
\end{eg}
\section{High-Dimensional Asymptotics}
\label{sec:asymptotics}
The topic of this section is the asymptotic sharpness of the bounds introduced above as the dimension of the space $\mathcal{X}$ becomes large. We begin with a comparison of the McDiarmid and half-space bounds for a simple function: a quadratic form on $\mathbb{R}^{N}$.
\begin{eg}[Comparison with McDiarmid's inequality]
\label{eg:quadratic_form_comparison}
The following example serves to illustrate how the half-space method can produce upper bounds on the measure of suitable sublevel sets that are superior to those offered by McDiarmid's inequality; it also shows how this effect is more pronounced in higher-dimensional spaces. Consider the following quadratic form $Q_{N}$ on $\mathbb{R}^{N}$:
\begin{equation}
\label{eq:quadratic_form}
Q_{N}(x) := \tfrac{1}{2} \left\| x - \left( \tfrac{1}{2}, \dots, \tfrac{1}{2} \right) \right\|_{2}^{2}.
\end{equation}
For any $\theta > 0$, the sublevel set $Q_{N}^{-1}([-\infty, \theta])$ is simply a ball of radius $\sqrt{2 \theta}$ about the point $\left( \tfrac{1}{2}, \dots, \tfrac{1}{2} \right)$. Suppose that a random vector $X$ takes values in $\left[ - \tfrac{1}{2}, + \tfrac{1}{2} \right]^{N}$ with independent components. McDiarmid's inequality \eqref{eq:McD2-leq} implies that
\[
\P[Q_{N}(X) \leq \theta] \leq \exp \left( - 8 \left( \frac{\sqrt{N}}{6} - \frac{\theta}{\sqrt{N}} \right)_{+}^{2} \right),
\]
If also $\mathbb{E}[X] = 0$, then corollary \ref{prop:cuboid} implies that
\[
\P[Q_{N}(X) \leq \theta] \leq \exp \left( - \frac{( \sqrt{N} - \sqrt{8 \theta} )_{+}^{2}}{2} \right).
\]
For small $N$ and large $\theta$, McDiarmid's bound is the sharper of the two. However, for small $\theta$ (and, notably, as $N \to \infty$ for any fixed $\theta$), the half-space bound is the sharper bound. See figure \ref{fig:quadratic_form_comparison} for an illustration.
\begin{figure}
\noindent\framebox[\linewidth]{\parbox{\linewidth-20pt}{
\begin{center}
\scalebox{1}
{
\input ./fig-quadratic_form_comparison.tex
}
\end{center}
\caption{For the quadratic form $Q_{N}$ on $\mathbb{R}^{N}$ given in \eqref{eq:quadratic_form}, a comparison of the McDiarmid upper bound (squares) and half-space upper bound (triangles) on $\P[Q_{N}(X) \leq \theta]$ in the cases $\theta = \tfrac{1}{4}$ (dotted line and hollow polygons) and $\theta = \tfrac{1}{8}$ (solid line and filled polygons).}
\label{fig:quadratic_form_comparison}
}}
\end{figure}
\end{eg}
The previous example suggests that bounds constructed using the half-space method may perform very well in high dimension but also that the sharpness of the bound may depend on ``how round'' the set whose measure we wish to bound is. To fix ideas, suppose that $X = (X_{1}, \dots, X_{N}) \colon \Omega \to \mathbb{R}^{N}$ is a random vector with independent components, where $X_{n}$ is supported on an interval of length $L_{n}$. For $A \subseteq \mathbb{R}^{N}$, how sharp is the bound
\begin{equation}
\label{eq:bound_for_asymptotics}
\P[X \in A] \leq \exp \left( - \frac{d_{\perp}(\mathbb{E}[X], A)^{2}}{2} \right)?
\end{equation}
First, note that since $d_{\perp}(\mathbb{E}[X], A) = d_{\perp}(\mathbb{E}[X], \mathop{\overline{\mathrm{co}}}(A))$, the bound cannot be expected to be sharp if $A$ differs greatly from its closed convex hull, and so it makes sense to restrict investigation to the case that $A = K$, a closed and convex subset of $\mathbb{R}^{N}$. Secondly, it is not reasonable to expect the bound \eqref{eq:bound_for_asymptotics} on $\P[X \in K]$ to be sharp if $K$ is sharply pointed, \emph{e.g.}\ if $K$ is the narrow wedge $K_{\varepsilon}$ of angle $\varepsilon \ll 1$ based at $e_{1} := (1, 0, \dots, 0)$ in $\mathbb{R}^{N}$:
\begin{equation}
\label{eq:narrow_wedge}
K_{\varepsilon} := \left\{ x \in \mathbb{R}^{N} \,\middle|\, \frac{(x - e_{1}) \cdot e_{1}}{\| x - e_{1} \|_{2}} \leq \varepsilon \right\};
\end{equation}
see figure \ref{fig:narrow_wedge}. Therefore, we wish to consider the opposite situation in which $K$ has no sharp points, which will be made precise by requiring that $K$ satisfy an interior ball condition.
\begin{figure}
\noindent\framebox[\linewidth]{\parbox{\linewidth-20pt}{
\begin{center}
\scalebox{1}
{
\input ./fig-narrow_wedge.tex
}
\end{center}
\caption{It is not reasonable to expect that (an upper bound for) the measure of the half-space $\H_{e_{1}, -e_{1}}$ is a sharp upper bound for the measure of the narrow wedge $K_{\varepsilon}$ when $\varepsilon$ is small.}
\label{fig:narrow_wedge}
}}
\end{figure}
Suppose that $(p, \nu) \in \mathrm{N}^{\ast} K$ is such that $d_{\perp}(x, \H_{p, \nu}) = d_{\perp}(x, K)$. Suppose also that $\mathbb{B}_{r}(p - r \omega) \subseteq K$, with $r > 0$ and $\omega \in \mathbb{R}^{N}$ a unit vector, is an interior ball for $K$ at $p \in \partial K$; \emph{cf.}\ figure \ref{fig:interior_ball}. If the law of $X$ on $\mathbb{R}^{N}$ is highly singular, then it cannot be expected that the bound \eqref{eq:bound_for_asymptotics} is sharp, so suppose that the law of $X$ has a density with respect to Lebesgue measure that is bounded above by some constant $C > 0$. Then the bound \eqref{eq:bound_for_asymptotics} is
\[
\P[X \in K] \leq \exp \left( - \frac{2 \langle \nu, \mathbb{E}[X] - p \rangle_{+}^{2}}{\sum_{n = 1}^{N} \nu_{n}^{2} L_{n}^{2}} \right).
\]
In the extreme case, $K$ is precisely the closed ball $\overline{\mathbb{B}}_{r}(p - r \omega)$, the $\P$-measure of which is at most $C r^{N} \pi^{N / 2} / \Gamma (1 + N/2)$.
\begin{figure}
\noindent\framebox[\linewidth]{\parbox{\linewidth-20pt}{
\begin{center}
\scalebox{1}
{
\input ./fig-interior_ball.tex
}
\end{center}
\caption{An interior ball of radius $r$ for the closed convex set $K$ at the frontier point $p$. Necessarily, $p$ is a point at which $\partial K$ is smooth; $K$ admits no interior ball of positive radius at the vertex $q$. For convenience, the unit vector $\omega \in \mathbb{R}^{N}$ has been identified with $\nu \in \mathrm{N}_{p}^{\ast} K \subseteq (\mathbb{R}^{N})^{\ast}$.}
\label{fig:interior_ball}
}}
\end{figure}
In large deviations theory, the standard notion of asymptotic sharpness is logarithmic equivalence \cite[\S{}I.1]{denHollander:2000}; see also \emph{e.g.}\ \cite{DemboZeitouni:1998} \cite{Varadhan:2008} for surveys of the large deviations literature. Two sequences $(\alpha_{n})_{n \in \mathbb{N}}$ and $(\beta_{n})_{n \in \mathbb{N}}$ are said to be \emph{logarithmically equivalent}, denoted $\alpha_{n} \simeq \beta_{n}$, if
\begin{equation}
\label{eq:logarithmic_equivalence}
\frac{1}{n} \log \alpha_{n} - \frac{1}{n} \log \beta_{n} \equiv \log \left( \frac{\alpha_{n}}{\beta_{n}} \right)^{1/n} \to 0 \text{ as } n \to \infty.
\end{equation}
Are the half-space bound \eqref{eq:bound_for_asymptotics} and the measure of $\overline{\mathbb{B}}_{r}(p - r \omega)$ logarithmically equivalent? That is, does the conditional probability ${ \P \left[ X \in \overline{\mathbb{B}}_{r}(p - r \omega) \,\middle|\, X \in \H_{p, \nu} \right] }$, when raised to the power $\frac{1}{N}$, converge to $1$ as $N \to \infty$? To simplify the asymptotic expansions below, in all lines after the first two, we shall take $\mathbb{E}[X] = 0$ and $L_{1} = \dots = L_{N} = 1$. Then
\begin{align*}
& \frac{1}{N} \log \P \left[X \in \overline{\mathbb{B}}_{r}(p - r \omega) \right] - \frac{1}{N} \log \left( \text{r.h.s.\ of \eqref{eq:bound_for_asymptotics}} \right)\\
& \quad \leq \frac{1}{N} \left( \log \frac{C r^{N} \pi^{N / 2}}{\Gamma (1 + N/2)} + \frac{2 \langle \nu, \mathbb{E}[X] - p \rangle_{+}^{2}}{\sum_{n = 1}^{N} \nu_{n}^{2} L_{n}^{2}} \right) \\
& \quad = \frac{2 \langle \nu, p \rangle_{-}^{2}}{N \| \nu \|_{2}^{2}} + \frac{\log (C r^{N} \pi^{N/2})}{N} - \frac{\log \Gamma (1 + N/2)}{N} \\
\intertext{which, by Stirling's approximation for the Gamma function \cite[p.\ 256, eq.\ (6.1.37)]{AbramowitzStegun:1992}, is approximately}
& \quad \approx \frac{2 \langle \nu, p \rangle_{-}^{2}}{N \| \nu \|_{2}^{2}} + \frac{\log (C r^{N} \pi^{N/2})}{N} - \frac{1}{N} \log \left( \sqrt{\frac{2 \pi}{1 + N/2}} \left( \frac{1 + N/2}{e} \right)^{1 + N/2} \right) \\
& \quad \sim \frac{2 \langle \nu, p \rangle_{-}^{2}}{N \| \nu \|_{2}^{2}} + \frac{\log C}{N} - \frac{1}{2 N} \log \frac{4 \pi}{N} - \frac{1 + N/2}{N} \log \frac{N}{2 e} \\
& \quad \sim \frac{2 \langle \nu, p \rangle_{-}^{2}}{N \| \nu \|_{2}^{2}} + \log r - \log \sqrt{N}
\end{align*}
Note that $\langle \nu, p \rangle_{-} / \| \nu \|_{2} \leq \sqrt{N} d_{\mathbf{1}}(0, p)$, where $d_{\mathbf{1}}$ denotes the weighted Hamming distance with weight $w = (1, \dots, 1)$. Therefore, a necessary (but not sufficient) condition for the half-space bound to be asymptotically sharp in the sense of logarithmic equivalence is that $r$ is of the same order as $\sqrt{N}$. That is, it is necessary that $K$ is sufficiently round that it has an interior ball of radius comparable to $\sqrt{N}$ at those frontier points where the normal distance $d_{\perp}(\mathbb{E}[X], K)$ is attained.
Now suppose that $K = f^{-1}([- \infty, \theta])$ is a convex sublevel set for twice-differentiable function $f$. Let $\eta_{1}, \dots, \eta_{N - 1}, \nu$ be a basis of $\mathbb{R}^{N}$ such that
\[
\| \eta_{1} \|_{2} = \dots = \| \eta_{N - 1} \|_{2} = \| \nu \|_{2} = 1
\]
and, for each $n \in \{ 1, \dots, N - 1 \}$, $\eta_{n}$ is perpendicular to $\nu$. Suppose that, in this system of normal coordinates, near $p$, the frontier of $K$ can be approximated by a parabola:
\[
\partial K = \left\{ y_{1} \eta_{1} + \dots y_{N - 1} \eta_{N - 1} - y_{N} \nu \,\middle|\, y_{N} = \sum_{n = 1}^{N - 1} \lambda_{n} y_{n}^{2} \right\}
\]
with $\lambda_{1} \geq \lambda_{2} \geq \dots \geq \lambda_{N - 1} \geq 0$. Then the condition that $K$ admits an interior ball of radius $r$ at $p$ is the inequality
\[
r - \sqrt{r^{2} - \sum_{n = 1}^{N - 1} y_{n}^{2}} \geq \sum_{n = 1}^{N - 1} \lambda_{n} y_{n}^{2} \text{ whenever } \sum_{n = 1}^{N - 1} y_{n}^{2} \leq r^2.
\]
This, in turn, leads to the following condition on $\lambda_{1}$: it must hold that $\lambda_{1} \leq \frac{1}{2 r}$. Put another way, the half-space method cannot be expected to provide asymptotically sharp bounds for $\P[f(X) \leq \theta]$ if, when $f$ is approximated in normal coordinates near the closest point of $f^{-1}([- \infty, \theta])$ to $\mathbb{E}[X]$ by a non-negative quadratic form, that quadratic form has an eigenvalue greater than $(4 N)^{-1/2}$.
\section{Appendix: Chernoff Bounds}
\label{sec:Chernoff}
The method of Chernoff bounds \cite[\S7.4.3]{BoydVandenberghe:2004} \cite{Chernoff:1952} is a simple one in which the probability of a subset of $\mathcal{X}$ is bounded by that of a containing half-space, and the probability of that half-space is bounded using the moment-generating function of the probability measure.
\begin{lem}[Chernoff bounds]
\label{lem:Chernoff}
For any half-space $\H_{p, \nu} \subseteq \mathcal{X}$,
\begin{equation}
\label{eq:Chernoff_halfspace}
\P[X \in \H_{p, \nu}] \leq \inf_{s \geq 0} e^{s \langle \nu, p \rangle} M_{X} (- s \nu).
\end{equation}
For any convex set $K \subseteq \mathcal{X}$,
\begin{subequations}
\begin{align}
\P[X \in K]
\label{eq:Chernoff_convex-1}
& \leq \inf_{(p, \nu) \in \mathrm{N}^{\ast} K} e^{\langle \nu, p \rangle} M_{X} (- \nu) \\
\label{eq:Chernoff_convex-2}
& = \exp \left( - \sup_{p \in K} (\Lambda_{X} + \chi_{-\mathrm{N}_{p}^{\ast} K})^{\star}(p) \right).
\end{align}
\end{subequations}
In particular, for any $x \in \mathcal{X}$,
\begin{equation}
\label{eq:Chernoff_convex-3}
\P[X = x] \leq \exp ( - \Lambda_{X}^{\star}(x) ).
\end{equation}
\end{lem}
\begin{proof}
By the definition of the half-space $\H_{p, \nu}$,
\begin{align*}
\P \left[ X \in \H_{p, \nu} \right]
& = \P \left[ \langle \nu, X \rangle \leq \langle \nu, p \rangle \right] \\
& = \mathbb{E} \left[ \mathbbm{1}_{[ \langle \nu, p - X \rangle \geq 0 ]} \right] \\
& \leq \mathbb{E} \left[ e^{ s \langle \nu, p - X \rangle } \right] & & \text{for any $s \geq 0$,}\\
& = e^{s \langle \nu, p \rangle} \mathbb{E} \left[ e^{\langle - s \nu, X \rangle} \right] \\
& \leq e^{s \langle \nu, p \rangle} M_{X} (- s \nu).
\end{align*}
Since this inequality holds for any $s \geq 0$, taking the infimum over all such $s$ yields \eqref{eq:Chernoff_halfspace}. Recall that the outward normal cone to a convex set at any point is closed under multiplication by non-negative scalars; hence, for any convex set $K \subseteq \mathcal{X}$, taking the infimum of the right-hand side of \eqref{eq:Chernoff_halfspace} over half-spaces $\H_{p, \nu}$ that contain $K$ yields \eqref{eq:Chernoff_convex-1}. Now observe that
\begin{align*}
& \inf_{(p, \nu) \in \mathrm{N}^{\ast} K} e^{\langle \nu, p \rangle} M_{X} (- \nu) \\
& \quad = \inf_{(p, \nu) \in \mathrm{N}^{\ast} K} \exp ( \langle \nu, p \rangle + \Lambda_{X} (- \nu) ) \\
& \quad = \exp \left( \inf_{p \in K} \inf_{\nu \in \mathrm{N}_{p}^{\ast} K} \left( \langle \nu, p \rangle + \Lambda_{X} (- \nu) \right) \right) \\
& \quad = \exp \left( - \sup_{p \in K} \sup_{\nu \in - \mathrm{N}_{p}^{\ast} K} \left( \langle \nu, p \rangle - \Lambda_{X} (\nu) \right) \right) \\
& \quad = \exp \left( - \sup_{p \in K} (\Lambda_{X} + \chi_{\vphantom{\tfrac{\sim}{\sim}} -\mathrm{N}_{p}^{\ast} K})^{\star}(p) \right),
\end{align*}
which establishes \eqref{eq:Chernoff_convex-2}; \eqref{eq:Chernoff_convex-3} follows as a special case.
\end{proof}
\bibliographystyle{amsplain}
| {
"timestamp": "2010-09-27T02:02:24",
"yymm": "1009",
"arxiv_id": "1009.4913",
"language": "en",
"url": "https://arxiv.org/abs/1009.4913",
"abstract": "We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\\mathcal{X}$. We show that, with respect to an appropriate \"normal distance\" on $\\mathcal{X}$, concentration inequalities for linear and non-linear functions of $X$ are equivalent. This normal distance corresponds naturally to the concentration rate in classical concentration results such as Gaussian concentration and concentration on the Euclidean and Hamming cubes. Under suitable assumptions on the roundness of the sets of interest, the concentration inequalities so obtained are asymptotically optimal in the high-dimensional limit.",
"subjects": "Probability (math.PR)",
"title": "Equivalence of concentration inequalities for linear and non-linear functions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795079712153,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7092379400104578
} |
https://arxiv.org/abs/1905.06366 | Equivalence and invariance of the chi and Hoffman constants of a matrix | We show that the following two condition measures of a full column rank matrix $A \in \mathbb{R}^{m\times n}$ are identical: the chi constant and a signed Hoffman constant. This identity is naturally suggested by the evident invariance of the chi constant under sign changes of the rows of $A$. We also show that similar equivalence and invariance properties extend to variants of the chi and Hoffman constants that depend only on the linear subspace $A(\mathbb{R}^n):=\{Ax: x\in\mathbb{R}^n\} \subseteq \mathbb{R}^m$. Finally, we show similar identities between the chi constants and signed versions of Renegar's and Grassmannian condition measures. | \section{Introduction}
\label{sec.intro}
We show a novel equivalence between the following two condition measures of a matrix that play central roles in
numerical linear algebra and in convex optimization: the chi measure~\cite{BenTT90,Diki74,Fors96,Stew89,Todd90} and
the Hoffman constant~\cite{Hoff52,guler1995,KlatT95,WangL14}. We also show some similar equivalences for some
variants of these constants.
Let $A\in {\mathbb R}^{m\times n}$ be a full column rank matrix. The chi constant $\chi(A)$ and its variant $\overline
\chi(A)$ arise in the analysis of weighted least squares problems~\cite{BobrV01,Fors96,ForsS01,HougV97}. In
particular, $\overline \chi(A)$
plays a central role in the
analysis of Vavasis and Ye's interior-point algorithm for linear programming~\cite{MontT03,VavaY96}.
A remarkable feature of Vavasis and Ye's algorithm is its sole dependence on the matrix $A$ defining the primal and
dual constraints.
The Hoffman constant $H(A)$ is associated to Hoffman's Lemma~\cite{Hoff52,guler1995}, a fundamental {\em error bound}
for systems of linear constraints of the form $Ax \le b$. The Hoffman constant and other similar error bounds are
used to establish the convergence rate of a wide variety of optimization algorithms~\cite{WangL14,
BeckS15,Garb18,GutmP19,LacoJ15,LeveL10,LuoT93,NecoNG18,Pang97,PenaR16,WangL14}.
As we discuss in Section~\ref{sec.def}, the chi constant $\chi(A)$ and its variant $\overline\chi(A)$ can be seen as
measures of {\em worst} behavior of a canonical solution mapping for the following weighted least squares problems
\[
b \mapsto \argmin_{x\in {\mathbb R}^n} (Ax-b)^{\text{\sf T}} D(Ax - b)
\]
where $D$ is a diagonal matrix with positive diagonal entries.
Similarly, the Hoffman constant $H(A)$ and its variant $\overline H(A)$ can be seen as measures of {\em worst}
behavior of a canonical solution mapping for the following system of linear inequalities
\[
b\mapsto \{x\in{\mathbb R}^n: Ax \le b\}.
\]
It is not immediately obvious that there should be a relationship between the chi and Hoffman constants. Nonetheless,
it is known that $H(A)\le \chi(A)$ and that $\chi(A)$ can be arbitrarily larger~\cite{HoT02,PenaVZ19}. Thus an equivalence
between the constants $H(A)$ and $\chi(A)$ appears impossible. The main goal of this paper is to show that this
apparent impossibility can be attributed to and rectified via a canonical {\em sign invariance property} of $\chi(A)$
detailed in equation~\eqref{eq.sign.inv} below.
Namely, the constant $\chi(A)$ does not change when the signs of some of the rows of $A$ are flipped as the solution
mapping~\eqref{eq.wls} satisfies this sign invariance property. On the other hand, the constant $H(A)$ does
not satisfy this sign invariance property and thus $H(A)$ and $\chi(A)$ cannot be identical.
Our main result (Theorem~\ref{thm.main}) shows that $\chi(A)$ and
$H(A)$ become identical after properly tweaking $H(A)$ to ensure the sign invariance property.
A similar type of
invariance consideration yields identities between the
variants $\overline \chi(A)$ and $\overline H(A)$. Our developments can be further extended to obtain analogous
identities between the four measures $\chi(A), \overline \chi(A), H(A), \overline H(A)$ and the following two popular
condition measures for systems of linear inequalities: Renegar's distance to ill-posedness $\mathcal{R}(A)$~\cite{Rene95a}
and the Grassmannian condition measure $\mathcal{G}(A)$~\cite{AmelB12}.
The above developments are similar in spirit to results previously derived by Tun\c{c}el~\cite{Tunc99}, by Todd,
Tun\c{c}el, and Ye~\cite{ToddTY01}, and by
Ho and Tun\c{c}el~\cite{HoT02}. These articles compare various condition measures for linear programming including
the chi and Hoffman constants. However, there are two major differences between our developments and theirs. First,
most of the results in~\cite{Tunc99,HoT02,ToddTY01} provide only inequalities and hence are weaker than our identities
concerning the chi and Hoffman constants. Second, the articles~\cite{Tunc99,HoT02,ToddTY01} do not deal with
Renegar's and Grassmannian condition measures but instead relate the chi and Hoffman constants with Ye's condition
measure~\cite{Ye94} for polyhedra of the form $\{A^{\text{\sf T}} y: y\ge 0, \|y\|_1 = 1\}$. Hence we deliberately chose not
to discuss Ye's condition measure in this paper. However, we note that our results can be extended to
identities involving Ye's condition measure by drawing on the recent work by Pe\~na and Roshchina~\cite{PenaR20}.
To formally state the sign invariance property, we rely on the following convenient notation. Let $\S \subseteq
{\mathbb R}^{m\times m}$ denote the set of {\em signature} matrices defined as follows
\begin{equation}\label{eq.signature}
\S:=\{\operatorname{Diag}(s): s \in \{-1,1\}^m\}.
\end{equation}
The constant $\chi(A)$ satisfies the following {\em sign invariance property:}
\begin{equation}\label{eq.sign.inv}
\chi(A) = \chi(SA) \text{ for all } S\in \S.
\end{equation}
Our main result states that $\chi(A)$ and $H(A)$ become identical if we take a suitable {\em closure} of $H(A)$ to
ensure the sign invariance property.
\begin{theorem}\label{thm.main}
Let $A\in {\mathbb R}^{m\times n}$ be a full column-rank matrix. Then
\begin{equation}\label{eq.thm}
\chi(A) = \max_{S\in\S} H(SA).
\end{equation}
\end{theorem}
A similar type of invariance property relates the measures $\chi(A)$ and $\overline \chi(A)$. The construction of
$\overline \chi(A)$ depends only on the subspace $A({\mathbb R}^n)$. Thus $\overline \chi(A)$ readily
satisfies the following invariance under right multiplication by non-singular matrices
\[
\overline\chi(A) = \overline\chi(AR) \text{ for all } R\in {\mathbb R}^{n\times n} \text{ non-singular}.
\]
In analogy to Theorem~\ref{thm.main}, the measures $\chi(A)$ and $\overline\chi(A)$ become identical if we take a
suitable closure of $\chi(A)$ to ensure the same invariance under right multiplication by non-singular matrices (see
Proposition~\ref{prop.chi.invar}):
\[
\overline \chi(A) = \min_{R \in {\mathbb R}^{m\times m} \atop \text{ non-singular}} \|AR\| \cdot \chi(AR).
\]
Furthermore, the same kind of identity holds for the measures $H(A)$ and $\overline H(A)$ (see
Proposition~\ref{prop.H.invar}):
\[
\overline H(A) = \min_{R \in {\mathbb R}^{m\times m} \atop \text{ non-singular}} \|AR\| \cdot H(AR).
\]
In particular, identity~\eqref{eq.thm} in Theorem~\ref{thm.main} readily extends to the measures $\overline \chi(A)$
and $\overline H(A)$ as follows (see Corollary~\ref{cor}):
\begin{equation}\label{eq.thm.bar}
\overline \chi(A) = \max_{S\in\S} \overline H(SA).
\end{equation}
Our proof of Theorem~\ref{thm.main} will actually show the following stronger identity when all rows of $A$ are
non-zero (see Theorem~\ref{thm.main.strong}):
\[
\chi(A) = \max_{S\in\S\atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} H(SA).
\]
This stronger identity in turn yields some interesting connections with Renegar's distance to ill-posedness
$\mathcal{R}(A)$~\cite{Rene95a,Rene95b} and the Grassmannian condition number of $\mathcal{G}(A)$~\cite{AmelB12}. More precisely,
in Section~\ref{sec.renegar} we show the following identity analogous to~\eqref{eq.thm} (see
Proposition~\ref{prop.chi.ren}):
\begin{equation}\label{eq.chi.ren.0}
\chi(A) = \max_{S\in\S\atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} \frac{1}{\mathcal{R}(SA)}
\end{equation}
and the following identity analogous to~\eqref{eq.thm.bar} (see Corollary~\ref{corol.chi.grass}):
\begin{equation}\label{eq.chi.grass.0}
\overline\chi(A) = \max_{S\in\S\atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset}\mathcal{G}(SA).
\end{equation}
The main sections of the paper are organized as follows. Section~\ref{sec.def} recalls the construction of the chi
constants $\chi(A), \overline \chi(A)$ as well as the Hoffman constants $H(A), \overline H(A)$ and some of their main
properties. Our presentation deliberately follows separate but similar formats for $\chi(A), \overline \chi(A)$ and
for $H(A), \overline H(A)$. Section~\ref{sec.proof} presents the proof of Theorem~\ref{thm.main}. To do so, we state
and prove the stronger Theorem~\ref{thm.main.strong}. Finally, Section~\ref{sec.renegar} recalls the construction of
Renegar's condition measure $\mathcal{R}(A)$ and of the Grassmannian condition measure $\mathcal{G}(A)$. This section also
proves identities~\eqref{eq.chi.ren.0} and~\eqref{eq.chi.grass.0}.
Throughout the paper whenever we encounter an Euclidean space ${\mathbb R}^d$ we implicitly assume that it is endowed with the
Euclidean norm defined by the canonical inner product in ${\mathbb R}^d$, that is, $\|u\|:=\sqrt{u^{\text{\sf T}} u}$ for all $u\in
{\mathbb R}^d$. Likewise, whenever we encounter a space of matrices ${\mathbb R}^{p\times d}$ we implicitly assume that it is endowed
with the operator norm, that is,
\[
\|A\| = \max_{x\in {\mathbb R}^d\atop \|x\|\le 1} \|Ax\|
\]
for all $A\in {\mathbb R}^{p\times d}$.
\section{Definition and properties of the chi and Hoffman constants}
\label{sec.def}
This section recalls the construction and main properties of the constants $\chi(A), \overline \chi(A)$ and
$H(A),\overline H(A)$. These constants can be seen as condition measures for two fundamental problems in scientific
computing, namely {\em weighted least squares} and {\em linear inequalities.}
\subsection{Weighted least squares}
Let ${\mathscr D}\subseteq {\mathbb R}^{m\times m}$ denote the set of diagonal matrices in ${\mathbb R}^{m\times m}$ with positive diagonal
entries. That is,
\[
{\mathscr D}:=\{\operatorname{Diag}(d): d\in{\mathbb R}^m_{++}\},
\]
where ${\mathbb R}^m_{++} \subseteq {\mathbb R}^m$ denotes the set of vectors in ${\mathbb R}^m$ with positive entries.
Suppose $A\in {\mathbb R}^{m\times n}$. Given $D \in {\mathscr D}$, consider the weighted least squares problem
\begin{equation}\label{eq.wls}
\min_{x\in {\mathbb R}^n} \; (Ax - b)^{\text{\sf T}} D (Ax-b).
\end{equation}
When $A$ is full column-rank, it is easy to see that the solution to~\eqref{eq.wls} is precisely $A_D^{\dagger}b$
where $A_D^\dagger$ is the following weighted pseudo-inverse of $A$~\cite{Fors96,Stew89}:
\begin{equation}\label{eq.wpseudo}
A_D^{\dagger} = (A^{\text{\sf T}} D A)^{-1} A^{\text{\sf T}} D.
\end{equation}
\subsubsection{Condition measures $\chi(A)$ and $\overline \chi(A)$}
Suppose $A\in {\mathbb R}^{m\times n}$ is full column-rank. The condition measure $\chi(A)$ is defined as the following
worst-case characteristic of the family of solution mappings $A^{\dagger}_D: {\mathbb R}^m\rightarrow {\mathbb R}^n$ constructed
via~\eqref{eq.wpseudo}:
\begin{equation}\label{eq.def.chi}
\chi(A):=\max_{D\in{\mathscr D}} \|A^{\dagger}_D\|.
\end{equation}
Consider the following alternative reformulation of the weighted least-squares problem~\eqref{eq.wls} in the subspace
$A({\mathbb R}^n)$:
\begin{equation}\label{eq.wls.image}
\min_{y\in A({\mathbb R}^n)} \; (y - b)^{\text{\sf T}} D (y-b).
\end{equation}
The solution to~\eqref{eq.wls.image} is evidently the $D$-projection of $b$ onto $A({\mathbb R}^n)$. Once again, it is easy to
see that if $A$ is full column-rank then the $D$-projection onto $A({\mathbb R}^n)$ is
\[
A(A^{\text{\sf T}} D A)^{-1} A^{\text{\sf T}} D = AA_D^{\dagger}.
\]
The condition measure $\overline\chi(A)$ is defined as the
following worst-case characteristic of the family of solution mappings
$AA_D^{\dagger}: {\mathbb R}^m\rightarrow A({\mathbb R}^n)$:
\begin{equation}\label{eq.def.barchi}
\overline\chi(A):=\max_{D\in{\mathscr D}} \|AA_D^{\dagger}\| = \max_{D\in{\mathscr D}} \|A(A^{\text{\sf T}} D A)^{-1} A^{\text{\sf T}} D\|.
\end{equation}
Although it is not immediately evident, the constants $\chi(A)$ and $\overline \chi(A)$ are finite for any full-rank matrix $A\in
{\mathbb R}^{m\times n}$. This fact was independently shown by Ben-Tal and Teboulle~\cite{BenTT90},
Dikin~\cite{Diki74}, Stewart~\cite{Stew89}, and Todd~\cite{Todd90}. The constants $\chi(A)$ and $\overline \chi(A)$
arise in and
play a key role in weighted least-squares problems~\cite{Fors96,ForsS01,BobrV01} and in linear
programming~\cite{HoT02,ToddTY01,Tunc99,VavaY96}.
We record some alternative expressions for $\chi(A)$ and $\overline \chi(A)$ that are closely related to the
constructions of $H(A)$ and $\overline H(A)$ discussed below. First, observe that
\[
\chi(A)= \max_{D\in {\mathscr D}} \max_{b\in {\mathbb R}^m, x\in {\mathbb R}^n\atop Ax \ne b} \frac{\|x-A^{\dagger}_D(b)\|}{\|Ax-b\|}.
\]
Second, observe that
\[
\overline\chi(A)= \max_{D\in {\mathscr D}} \max_{b\in {\mathbb R}^m, y\in A({\mathbb R}^n)\atop y \ne b} \frac{\|y-AA_D^{\dagger}(b)\|}{\|y-b\|}.
\]
\subsubsection{Properties of $\chi(A)$ and $\overline \chi(A)$}
Suppose $A\in {\mathbb R}^{m\times n}$ is full column-rank and $D\in {\mathscr D}$. By construction, the solution mappings
$A_D^{\dagger}$ and $AA_D^{\dagger}$ satisfy the following property: For $S\in \S$ then
$(SA)_D^\dagger = A_D^\dagger S$. In particular $\|(SA)_D^\dagger\| =\|A_D^{\dagger}\|$ and $\|(SA)(SA)_D^\dagger\| = \|AA_D^\dagger\|$. Therefore~\eqref{eq.def.chi} and~\eqref{eq.def.barchi} imply that the
constants $\chi(A)$ and $\overline \chi(A)$ satisfy the following sign invariance property:
\[
\chi(A) = \chi(SA) \text{ and } \overline\chi(A) = \overline\chi(SA), \text{ for all }S \in \S.
\]
Furthermore, the quantity $\overline\chi(A)$ depends only on the subspace $A({\mathbb R}^n)$ which evidently satisfies
$A({\mathbb R}^n) = AR({\mathbb R}^n)$ for all non-singular $R\in {\mathbb R}^{n\times n}$.
Therefore, the constant $\overline \chi(A)$ is invariant under multiplication by non-singular matrices, that is,
\begin{equation}\label{eq.chi.right.invar}
\overline \chi(A) = \overline \chi(AR), \text{ for all non-singular }R\in {\mathbb R}^{n\times
n}.
\end{equation}
The constant $ \chi(A)$ is not invariant under multiplication by singular matrices. Proposition~\ref{prop.chi.invar} shows that $\overline\chi(A)$ is the closure of $ \chi(A)$ under this kind of invariance.
\begin{proposition}\label{prop.chi.invar}
Suppose $A\in {\mathbb R}^{m\times n}$ is full column-rank. Then $\overline \chi(A) \le \|A\| \cdot \chi(A)$ and $\overline
\chi(A) = \chi(A)$ when the columns of $A$ are orthonormal. In particular,
\begin{equation}\label{eq.chi.invar}
\overline \chi(A) = \min_{R \in {\mathbb R}^{n\times n} \atop \text{ non-singular}} \|AR\| \cdot \chi(AR).
\end{equation}
\end{proposition}
\begin{proof}
Since $\|AA_D^\dagger\| \le \|A\| \cdot \|A_D^\dagger\|$, the construction~\eqref{eq.def.chi}
and~\eqref{eq.def.barchi} of $\chi(A)$ and $\overline \chi(A)$
readily implies that
\begin{equation}\label{eq.chi.sub}
\overline \chi(A) \le \|A\| \cdot \chi(A).
\end{equation}
Next, we show that $\overline \chi(A) = \chi(A)$ when the columns of $A$ are orthonormal. To that end, observe that
if the columns of $A$ are orthonormal then $\|Ax\| = \|x\|$ for all $x\in {\mathbb R}^n$. In particular, if the columns of $A$
are orthonormal then $\|AA_D^\dagger\| = \|A_D^\dagger\|$ for all $D\in{\mathscr D}$. Thus~\eqref{eq.def.chi}
and~\eqref{eq.def.barchi} imply that $\overline \chi(A) = \chi(A)$.
Finally, from~\eqref{eq.chi.right.invar} and~\eqref{eq.chi.sub} it follows that
$
\overline \chi(A) = \overline \chi(AR) \le \|AR\|\cdot \chi(AR)
$
for all $R\in{\mathbb R}^{m\times m}$ non-singular. Thus~\eqref{eq.chi.invar} follows.
\end{proof}
In the special case when $m=n$ and $A\in {\mathbb R}^{n\times n}$ is non-singular it is easy to see that
\[
\chi(A) = \|A^{-1}\|.
\]
We will rely on the following related characterization of $\chi(A)$
from~\cite{Fors96}. The same characterization is also stated and proved in~\cite{Zhan00} by adapting a technique
from~\cite{ToddTY01}. In the statement below for $A\in {\mathbb R}^{m\times n}$ and $J\subseteq[m] := \{1,\dots,m\}$ the matrix $A_J\in {\mathbb R}^{J \times n}$ denotes the $|J|\times n$ submatrix of $A$ defined by the rows of $A$ indexed by $J$.
\begin{proposition}\label{prop.chi} Suppose $A\in {\mathbb R}^{m\times n}$ has full column-rank. Then
\begin{equation}\label{eq.chi}
\chi(A) = \max_{J\subseteq[m]\atop A_J \text{non-singular}} \|A_J^{-1}\| =
\max_{J\subseteq[m]\atop A_J \text{non-singular}} \max_{v \in {\mathbb R}^J\atop \|A_J^{\text{\sf T}} v \|= 1} \|v\|.
\end{equation}
\end{proposition}
\subsection{Linear inequalities}
Suppose $A\in {\mathbb R}^{m\times n}$. Consider the feasibility problem
\begin{equation}\label{eq.li}
Ax \le b.
\end{equation}
The solution of~\eqref{eq.li} is the set
\begin{equation}\label{eq.poly}
P_A(b):=\{x\in {\mathbb R}^n: Ax \le b\}.
\end{equation}
Observe that $P_A(b) \ne \emptyset$ if and only if $b \in A({\mathbb R}^n) + {\mathbb R}^m_+$.
\subsubsection{Condition measures $H(A)$ and $\overline H(A)$}
Suppose $A\in {\mathbb R}^{m\times n}$ is a nonzero matrix. The condition measure $H(A)$ is defined as the following worst-case
characteristic of the solution mapping $P_A:{\mathbb R}^m \rightrightarrows {\mathbb R}^n$ constructed via~\eqref{eq.poly}:
\begin{equation}\label{eq.def.H}
H(A) = \max_{b \in A({\mathbb R}^n) + {\mathbb R}^m_+\atop x \in {\mathbb R}^n\setminus P_A(b)} \frac{{\mathrm{dist}}(x,P_A(b))}{\|(Ax-b)_+\|}.
\end{equation}
Here and throughout the paper, ${\mathrm{dist}}(u,S)$ denotes the following point-to-set distance for all $u\in {\mathbb R}^d$ and
$S\subseteq {\mathbb R}^d$:
\[
{\mathrm{dist}}(u,S) = \inf_{v\in S} \|u-v\|.
\]
The constant $H(A)$ can be equivalently defined as the smallest constant depending only on $A$ such that the following
{\em error bound} holds for all $b \in A({\mathbb R}^n)+{\mathbb R}^m_+$ and all $x\in {\mathbb R}^n$:
\[
{\mathrm{dist}}(x,P_{A}(b)) \le H(A) \cdot \|(Ax-b)_+\|.
\]
Again, it is not immediately evident that $H(A)$ is finite. This fact was shown by Hoffman in his seminal
paper~\cite{Hoff52}. Other proofs of this fundamental result can be found in~\cite{guler1995,PenaVZ19,WangL14}.
After Hoffman's initial work, the literature in error bounds has developed
extensively~\cite{luo1994,LuoT93,ngai2015,Nguy17,Pang97,ZhouS17}. Error bounds play a key role in optimization and
variational analysis. In particular, error bounds are widely used to established the convergence rate of a variety of
algorithms~\cite{BeckS15,Garb18,GutmP19,LacoJ15,LeveL10,LuoT93,NecoNG18,Pang97,PenaR16,WangL14}.
Consider the following reformulation of~\eqref{eq.li} in the subspace $A({\mathbb R}^n)$:
\begin{equation}\label{eq.li.image}
y \le b, \; y \in A({\mathbb R}^n).
\end{equation}
The solution of~\eqref{eq.li.image} is the set
\[
(b-{\mathbb R}^m_+) \cap A({\mathbb R}^n) = AP_A(b).
\]
Define $\overline H(A)$ as the
following worst-case characteristic of the solution mapping $AP_A:{\mathbb R}^m\rightrightarrows A({\mathbb R}^n)$:
\begin{equation}\label{eq.def.barH}
\overline H(A) = \max_{b \in A({\mathbb R}^n)+{\mathbb R}^m_+ \atop y \in A({\mathbb R}^n) \setminus AP_A(b)}
\frac{{\mathrm{dist}}(y,AP_A(b))}{\|(y-b)_+\|}.
\end{equation}
The constant $\overline H(A)$ can be equivalently defined as the smallest constant depending only on the subspace
$A({\mathbb R}^n)$ such that the following error bound holds for all $b\in A({\mathbb R}^n) + {\mathbb R}^m_+$ and $v\in A({\mathbb R}^n) + b$
\[
{\mathrm{dist}}(v,(A({\mathbb R}^n) + b)\cap{\mathbb R}^m_+) \le \overline H(A) \cdot {\mathrm{dist}}(v,{\mathbb R}^m_+).
\]
\subsubsection{Properties of $H(A)$ and $\overline H(A)$}
By construction, $\overline H(A)$ depends only on $A({\mathbb R}^n)$ and thus is invariant under multiplication by non-singular matrices, i.e.,
\begin{equation}\label{eq.H.right.invar}
\overline H(A) = \overline H(AR), \text{ for all non-singular } R\in {\mathbb R}^{n\times n}.
\end{equation}
On the other hand, $H(A)$ is not invariant under multiplication by non-singular matrices. Proposition~\ref{prop.H.invar} shows that $\overline H(A)$ is the closure of $H(A)$ under this kind of invariance.
\begin{proposition}\label{prop.H.invar}
Suppose $A\in {\mathbb R}^{m\times n}$ is a nonzero matrix. Then $\overline H(A) \le \|A\| \cdot H(A)$ and $\overline H(A) =
H(A)$ when the \underline{nonzero} columns of $A$ are orthonormal. In particular,
\begin{equation}\label{eq.H.invar}
\overline H(A) = \min_{R \in {\mathbb R}^{n\times n} \atop \text{ non-singular}} \|AR\| \cdot H(AR).
\end{equation}
\end{proposition}
\begin{proof}
This proof is similar to the proof of Proposition~\ref{prop.chi.invar}. Observe that ${\mathrm{dist}}(Ax,AP_A(b)) \le
\|A\|\cdot {\mathrm{dist}}(x,P_A(b))$ for all $x\in {\mathbb R}^n$ because $\|Ax - Au\| \le \|A\|\cdot \|x-u\|$ for all $x,u\in {\mathbb R}^n$.
Hence~\eqref{eq.def.H} and~\eqref{eq.def.barH} imply that
\begin{equation}\label{eq.H.sub}
\overline H(A) \le \|A\| \cdot H(A).
\end{equation}
We next show that $\overline H(A) = H(A)$ when the nonzero columns of $A$ are orthonormal. For ease of exposition,
consider first the case when all columns of $A$ are nonzero and orthonormal. In this case it is easy to see that $y
\in A({\mathbb R}^n)$ if and only if $y = Ax$ for some unique $x\in {\mathbb R}^n$ with $\|y\| = \|x\|$.
Therefore ${\mathrm{dist}}(y,AP_A(b)) = {\mathrm{dist}}(x,P_A(b))$ for all $y = Ax\in A({\mathbb R}^n)$.
From~\eqref{eq.def.H} and~\eqref{eq.def.barH} it follows that $\overline H(A) = H(A)$.
Next consider the more general case when some columns of $A$ are zero. Without loss of generality assume that $A =
\matr{\tilde A & 0}$ for some $\tilde A\in {\mathbb R}^{m \times k}$ with nonzero orthonormal columns for some $k < n$. Since
the columns of $\tilde A$ are orthonormal, we have $\overline H(\tilde A) = H(\tilde A)$. To finish, it suffices to
show that $\overline H( A) = \overline H(\tilde A)$ and
$ H( A) = H(\tilde A)$. Indeed, $\overline H( A) = \overline H(\tilde A)$ holds because $A({\mathbb R}^n) = \tilde A({\mathbb R}^k)$
and $AP_A(b) = \tilde A P_{\tilde A}(b)$. On the other hand, for $x\in {\mathbb R}^n$ let $\tilde x\in {\mathbb R}^k$ denote the
subvector of first $k$ entries of $x$. Then $Ax = \tilde A \tilde x$ for all $x\in {\mathbb R}^n$ and thus $P_A(b) = P_{\tilde
A}(b) \times {\mathbb R}^{n-k}$. Hence
\[
H(A) = \max_{b \in A({\mathbb R}^n) + {\mathbb R}^m_+\atop x \in {\mathbb R}^n\setminus P_A(b)} \frac{{\mathrm{dist}}(x,P_A(b))}{\|(Ax-b)_+\|} =
\max_{b \in \tilde A({\mathbb R}^k) + {\mathbb R}^m_+\atop \tilde x \in {\mathbb R}^n\setminus P_A(b)} \frac{{\mathrm{dist}}(\tilde x,P_{\tilde
A}(b))}{\|(\tilde A\tilde x-b)_+\|} = H(\tilde A).
\]
Finally from~\eqref{eq.H.right.invar} and~\eqref{eq.H.sub} it follows that
$
\overline H(A) = \overline H(AR) \le \|AR\|\cdot H(AR)
$
for all $R\in{\mathbb R}^{m\times m}$ non-singular. Thus~\eqref{eq.H.invar} follows.
\end{proof}
We will also rely on the following two properties of $H(A)$. First, in the special case when $A({\mathbb R}^n) \cap {\mathbb R}^m_{++}
\ne \emptyset$ or equivalently $A({\mathbb R}^n) + {\mathbb R}^m_+ = {\mathbb R}^m$ we have~\cite[Corollary 1]{PenaVZ19}
\begin{equation}\label{eq.H.simple}
H(A) = \max_{v\in {\mathbb R}^m_+\atop \|A^{\text{\sf T}} v\| =1} \|v\|.
\end{equation}
Second, for general $A\in {\mathbb R}^{m\times n}$ we have the following related characterization of $H(A)$ discussed
in~\cite{PenaVZ19} but that can be traced back to~\cite{KlatT95,WangL14,Zhan00}.
\begin{proposition}\label{prop.H} Suppose $A\in {\mathbb R}^{m\times n}$ is full column-rank. Then
\begin{equation}\label{eq.H}
H(A) = \max_{J\subseteq[m]\atop A_J({\mathbb R}^n) \cap {\mathbb R}^J_{++} \ne \emptyset} \max_{v \in {\mathbb R}^J_+ \atop \|A_J^{\text{\sf T}} v\| = 1}
\|v\| = \max_{J\subseteq[m]\atop A_J \text{non-singular}} \max_{v \in {\mathbb R}^J_+ \atop \|A_J^{\text{\sf T}} v\| = 1} \|v\|.
\end{equation}
\end{proposition}
Observe both the similarity and subtle difference between the right-most expressions in the
characterization~\eqref{eq.chi} of $\chi(A)$ in Proposition~\ref{prop.chi} and the characterization~\eqref{eq.H} of
$H(A)$ in Proposition~\ref{prop.H}: the first maximum is taken over the same collection of sets $J$ in
both~\eqref{eq.chi} and~\eqref{eq.H} whereas the second maximum is taken over $v\in{\mathbb R}^J$ in~\eqref{eq.chi} and over
$v\in {\mathbb R}^J_+$ in~\eqref{eq.H}.
\section{Proof of Theorem~\ref{thm.main}}
\label{sec.proof}
We will prove the following stronger version of Theorem~\ref{thm.main}.
\begin{theorem}\label{thm.main.strong}
Let $A\in {\mathbb R}^{m\times n}$ be a full column-rank matrix. Then
\begin{equation}\label{eq.thm.full}
\chi(A) = \max_{S\in\S} H(SA) = H(\mathbf{A}),
\end{equation}
where
$\mathbf{A}\in{\mathbb R}^{2m \times n}$ is the column-wise concatenation of $A$ and $-A$, that is,
\begin{equation}\label{eq.concatenate}
\mathbf{A} = \matr{A \\ -A}.
\end{equation}
Furthermore, if all rows of $A$ are nonzero then~\eqref{eq.thm.full} can be sharpened to
\begin{equation}\label{eq.thm.sharper}
\chi(A) = \max_{S\in\S \atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} H(SA).
\end{equation}
\end{theorem}
\begin{proof}
From~\eqref{eq.chi} in Proposition~\ref{prop.chi} and~\eqref{eq.H} in Proposition~\ref{prop.H} it immediately follows
that $H(A) \le \chi(A)$. Thus the sign invariance of $\chi(A)$ readily yields
\[
\chi(A) = \max_{S\in\S} \chi(SA) \ge \max_{S\in\S} H(SA).
\]
To prove the reverse inequality we rely on~\eqref{eq.chi} and~\eqref{eq.H} again. Suppose $\hat J \subseteq [m]$ is
such that $A_{\hat J}$ is non-singular and
\[
\chi(A) = \|A_{\hat J}^{-1}\| = \max_{v\in {\mathbb R}^{\hat J}\atop \|A_{\hat J}^{\text{\sf T}} v\|=1} \|v\|.
\]
Thus $\chi(A) = \|\hat v\|$ for some $\hat v \in {\mathbb R}^{\hat J}$ such that $\|A_{\hat J}^{\text{\sf T}} \hat v\|=1$. Choose
$\hat S\in \S$ such that $\hat S_{ii} = \text{sign}(v_i)$ for each $i\in \hat J$ and let $u:=\hat S_{\hat J}\hat v \in
{\mathbb R}^{\hat J}_+$. Observe that $(\hat SA)_J = \hat S_{\hat J} A_{\hat J}$ is nonsingular and
\[
\|(\hat SA)_{\hat J}^{\text{\sf T}} u\| = \|A_{\hat J}^{\text{\sf T}} \hat S_{\hat J} u \|= \|A_{\hat J}^{\text{\sf T}} v\| = 1.
\]
Therefore
\[
\max_{S\in\S} H(SA) \ge H(\hat SA) \ge \max_{w \in {\mathbb R}^{\hat J}_+ \atop \|(\hat SA)_{\hat J}^{\text{\sf T}} w\|=1} \|w\| \ge
\|u\| = \|\hat v\| = \chi(A).
\]
Thus the first identity in~\eqref{eq.thm.full} is established. Next, Proposition~\ref{prop.chi} and
Proposition~\ref{prop.H} imply that for all $S\in \S$
\[
\chi(A) = \chi(\mathbf{A}) \ge H(\mathbf{A}) \ge H(SA).
\]
The second inequality follows because all rows of $SA$ are rows of $\mathbf{A}$ as well. Hence by taking the maximum over
$S\in\S$ and applying the first identity in~\eqref{eq.thm.full}, we obtain the second identity in~\eqref{eq.thm.full}.
When all rows of $A$ are non-zero, it follows that $A \tilde v$ has all nonzero entries for an arbitrarily small
perturbation $\tilde v$ of $\hat v$. Therefore the matrix $\hat S \in \S$ above can be chosen so that both $\hat
S_{\hat J}\hat v \in {\mathbb R}^{\hat J}_+$ and $\hat S A^{\text{\sf T}} \tilde v \in {\mathbb R}^m_{++}.$ Thus the sharper
identity~\eqref{eq.thm.sharper} follows.
\end{proof}
\begin{corollary}\label{cor}
Let $A\in {\mathbb R}^{m\times n}$ be a full column-rank matrix. Then
\[
\overline\chi(A) = \max_{S\in\S} \overline H(SA) = \overline H(\mathbf{A}),
\]
where $\mathbf{A}$ is as in~\eqref{eq.concatenate}.
Furthermore, if all rows of $A$ are nonzero then
\[
\overline\chi(A) = \max_{S\in\S \atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} \overline H(SA).
\]
\end{corollary}
\begin{proof}
This is an immediate consequence of Theorem~\ref{thm.main.strong}, Proposition~\ref{prop.chi.invar} and
Proposition~\ref{prop.H.invar}.
\end{proof}
We note that when $A\in {\mathbb R}^{m\times n}$ is full column-rank but some rows of $A\in {\mathbb R}^{m\times n}$ are zero, then the
following amended version of~\eqref{eq.thm.sharper} holds for the submatrix $\tilde A\in {\mathbb R}^{\ell \times n}$ obtained
after deleting the zero rows from $A$:
\[
\chi(\tilde A) = \max_{S\in\S \atop S\tilde A({\mathbb R}^n) \cap {\mathbb R}^\ell_{++} \ne \emptyset} H(S\tilde A).
\]
The construction of $\chi(A)$ and $H(A)$ enables us to rewrite the latter identity as follows
\[
\chi(A) = \max_{S\in\S \atop S\tilde A({\mathbb R}^n) \cap {\mathbb R}^\ell_{++} \ne \emptyset} H(SA).
\]
\section{Renegar's and Grassmannian condition numbers}
\label{sec.renegar}
Suppose $A \in {\mathbb R}^{m\times n}$ is such that $A({\mathbb R}^n)\cap {\mathbb R}^m_{++} \ne \emptyset$. This property can be equivalently
stated as $A({\mathbb R}^n) + {\mathbb R}^m_{+} = {\mathbb R}^m$, that is, for all $b\in {\mathbb R}^m$ the system of linear inequalities
\[
Ax \le b
\]
is feasible. In his seminal paper on condition measures for optimization~\cite{Rene95a}, Renegar
defined the {\em distance to infeasibility} of $A$ as the smallest perturbation that can be made on $A$ so that this
property is lost. That is
\[
\mathcal{R}(A):=\inf\{\|\Delta A\|: (A+\Delta A)({\mathbb R}^n)\cap {\mathbb R}^m_{++} = \emptyset\}.
\]
Renegar also defined $\|A\|/\mathcal{R}(A)$ as a condition number of $A$.
We have the following characterization of $\chi(A)$ in terms $\mathcal{R}(A)$ analogous to that in
Theorem~\ref{thm.main.strong}.
\begin{proposition}\label{prop.chi.ren}
Let $A\in {\mathbb R}^{m\times n}$ be a full column-rank matrix. If $A({\mathbb R}^n)\cap {\mathbb R}^m_{++}\ne \emptyset$ then $H(A) = 1/\mathcal{R}(A)$. Consequently, if all rows of full column-rank matrix $A\in{\mathbb R}^{m\times n}$ are nonzero then
\begin{equation}\label{eq.chi.ren}
\chi(A) = \max_{S\in\S \atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} \frac{1}{\mathcal{R}(SA)}.
\end{equation}
\end{proposition}
\begin{proof}
When $A({\mathbb R}^n)\cap {\mathbb R}^m_{++} \ne \emptyset$, the distance to ill-posedness $\mathcal{R}(A)$ has the following property similar
in spirit to Proposition~\ref{prop.chi} and Proposition~\ref{prop.H} (see\cite[Theorem 3.5]{Rene95b}):
\begin{equation}\label{eq.Ren}
\frac{1}{\mathcal{R}(A)} = \displaystyle\max_{v\in {\mathbb R}^m_+ \atop \|v\| =1} \|A^{\text{\sf T}} v\|.
\end{equation}
From~\eqref{eq.H.simple} and~\eqref{eq.Ren} it follows that $H(A) = 1/\mathcal{R}(A)$ when $A({\mathbb R}^n)\cap {\mathbb R}^m_{++} \ne
\emptyset$. The latter condition and~\eqref{eq.thm.sharper} in turn imply \eqref{eq.chi.ren} if all rows of $A$ are nonzero.
\end{proof}
Ameluxen and Burgisser~\cite{AmelB12} proposed a condition number via the Grassmannian manifold of linear subspaces of
${\mathbb R}^m$ of some fixed dimension. This condition number can be seen as a variant of Renegar's condition measure that
depends only on $A({\mathbb R}^n)$ akin to the variants $\overline \chi(A)$ and $\overline H(A)$ of $\chi(A)$ and $H(A)$
respectively. We next recall the description of the Grassmannian condition number proposed by Ameluxen and
Burgisser~\cite{AmelB12}. First, define the {\em Grassmannian} distance ${\mathrm{dist}}(L,L')$ between two linear subspaces
$L,L' \subseteq {\mathbb R}^m$ of the same dimension as
\[
{\mathrm{dist}}(L,L'):=\|\Pi_L - \Pi_{L'}\|,
\]
where $\Pi_L$ and $\Pi_{L'}$ denote the orthogonal projection matrices onto $L$ and $L'$ respectively.
Suppose $A\in {\mathbb R}^{m\times n}$ satisfies $A({\mathbb R}^n)\cap {\mathbb R}^m_{++} \ne\emptyset$. Let $L:=A({\mathbb R}^n)$ and define the {\em
Grassmannian condition number} of $A$ as follows
\[
\mathcal{G}(A):=\frac{1}{\min\{{\mathrm{dist}}(L,L'): \dim(L') = \dim(L) \text{ and } L\cap {\mathbb R}^m_{++} = \emptyset\}}.
\]
Since $\mathcal{G}(A)$ depends only on $A({\mathbb R}^n)$, it automatically satisfies the following invariance property just as
$\overline \chi(A)$ and $\overline H(A)$ do: For all non-singular $R\in {\mathbb R}^{m\times m}$
\begin{equation}\label{eq.grass.invar}
\mathcal{G}(AR) = \mathcal{G}(A).
\end{equation}
The pair of quantities $1/\mathcal{R}(A), \mathcal{G}(A)$ are related to each other in the same way the pairs of quantities $\chi(A),\overline\chi(A)$ and $H(A),\overline H(A)$ are. More precisely, we have the following analogue of Proposition~\ref{prop.chi.invar} and Proposition~\ref{prop.H.invar}.
\begin{proposition}\label{prop.ren.grass.invar}
Suppose $A\in {\mathbb R}^{m\times n}$ is a nonzero matrix and $A({\mathbb R}^n) \cap {\mathbb R}^m_{++}\ne \emptyset$. Then $\mathcal{G}(A) \le \|A\|/\mathcal{R}(A)$ and $\mathcal{G}(A) = 1/\mathcal{R}(A)$ when the non-zero columns of $A$ are orthonormal. Consequently, if $A\in {\mathbb R}^{m\times n}$ is a nonzero matrix
\begin{equation}\label{eq.ren.grass.invar}
\mathcal{G}(A) = \min_{R \in {\mathbb R}^{m\times m} \atop \text{non-singular}} \frac{\|AR\|}{\mathcal{R}(AR)}.
\end{equation}
\end{proposition}
\begin{proof}
Suppose $A\in {\mathbb R}^{m\times n}$ and $A({\mathbb R}^n) \cap {\mathbb R}^m_{++}\ne \emptyset$. Then
the inequality $\mathcal{G}(A) \le 1/\mathcal{R}(A)$ follows from~\cite[Theorem 1.4]{AmelB12} and the identity $\mathcal{G}(A) = 1/\mathcal{R}(A)$ when the nonzero columns of $A$ are orthonormal follows from~\cite[Theorem 1.3]{AmelB12}. The latter two facts and~\eqref{eq.grass.invar}
in turn imply~\eqref{eq.ren.grass.invar} when $A\in {\mathbb R}^{m\times n}$ is a nonzero matrix.
\end{proof}
We conclude with the following characterization of $\overline \chi(A)$ in terms $\mathcal{G}(A)$ analogous to that
in Corollary~\ref{cor}.
\begin{corollary}\label{corol.chi.grass}
Suppose $A\in {\mathbb R}^{m\times n}$ is a full column-rank matrix and all rows of $A$ are nonzero. Then
\begin{equation}\label{eq.chi.grass}
\overline\chi(A) = \max_{S\in\S \atop SA({\mathbb R}^n) \cap {\mathbb R}^m_{++} \ne \emptyset} \mathcal{G}(SA).
\end{equation}
\end{corollary}
\begin{proof}
This is an immediate consequence of Proposition~\ref{prop.chi.invar}, Proposition~\ref{prop.chi.ren}, and
Proposition~\ref{prop.ren.grass.invar}.
\end{proof}
\bibliographystyle{plain}
| {
"timestamp": "2020-05-19T02:33:29",
"yymm": "1905",
"arxiv_id": "1905.06366",
"language": "en",
"url": "https://arxiv.org/abs/1905.06366",
"abstract": "We show that the following two condition measures of a full column rank matrix $A \\in \\mathbb{R}^{m\\times n}$ are identical: the chi constant and a signed Hoffman constant. This identity is naturally suggested by the evident invariance of the chi constant under sign changes of the rows of $A$. We also show that similar equivalence and invariance properties extend to variants of the chi and Hoffman constants that depend only on the linear subspace $A(\\mathbb{R}^n):=\\{Ax: x\\in\\mathbb{R}^n\\} \\subseteq \\mathbb{R}^m$. Finally, we show similar identities between the chi constants and signed versions of Renegar's and Grassmannian condition measures.",
"subjects": "Optimization and Control (math.OC)",
"title": "Equivalence and invariance of the chi and Hoffman constants of a matrix",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795118010988,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7092379368145378
} |
https://arxiv.org/abs/1501.01685 | Spaces of regular abstract martingales | In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm. | \section{The space of regular martingales on a vector lattice}
Let $F$ be a vector lattice. A sequence $(E_n)$ of positive projections on $F$ such that $E_nE_m=E_{n\wedge m}$ is said to be a \term{filtration}. We will try to impose as few additional assumptions on the filtration as possible. A sequence $X=(x_n)$ in $F$ is a \term{martingale} with respect to the filtration $(E_n)$ if $E_nx_m=x_n$ whenever $m\ge n$. A sequence $X=(x_n)$ in $F$ is a \term{supermartingale} if $E_nx_m\le x_n$ whenever $m\ge n$ (note that in our definition we do not require that $E_nx_n=x_n$). We denote with $\mathcal{M}=\mathcal{M}\bigl(F,(E_n)\bigr)$ the space of all martingales on $F$ with respect to the filtration $(E_n)$. The space $\mathcal{M}$ equipped with the coordinate-wise order is an ordered vector space and we denote with $\mathcal{M}_+$ the positive cone of $\mathcal{M}$. There is an extensive literature on abstract martingales on vector and Banach lattices, see, e.g., \cite{DeMarr:66,Uhl:71,Kuo:05,Troitsky:05,Korostenski:08,Labuschagne:10,Gessesse:11,Grobler:14,Gao:14,Gao:14a}. For unexplained terminology on ordered vector and Banach spaces we refer the reader to \cite{Aliprantis:06,Aliprantis:07,Meyer-Nieberg:91}.
The space of \term{regular martingales} is defined as follows:
\begin{displaymath}
\mathcal{M}_r=\mathcal{M}_r\bigl(F,(E_n)\bigr)
=\bigl\{X_1-X_2 \mid X_1,X_2 \in \mathcal{M}_+\bigr\}.
\end{displaymath}
Equivalently, a martingale $X$ is regular iff $\pm X\le Y$ for some positive martingale $Y$. This definition is motivated by the definition of a regular operator. In this setting, it is a well known fact that the space of regular operators $\mathcal L_r(F)$ is itself a vector lattice when $F$ is order complete. So it has been a natural conjecture that $\mathcal M_r$ is a vector lattice whenever $F$ is order complete. We will prove that this is indeed the case. This result improves \cite[Theorem 2.3]{Korostenski:08}, which asserts that
$\mathcal{M}_r$ is a vector lattice (and is even order complete) provided that $F$ is order complete and for every $n$ the projection $E_n$ is order continuous and $\Range E_n$ is an order complete sublattice of $F$.
It has also been an open question whether $\mathcal M_r$ is \emph{always} a vector lattice. We will present an example to the contrary.
\begin{theorem}\label{Mr-VL}
Let $F$ be an order complete vector lattice and $(E_n)$ a filtration
on $F$. Then $\mathcal M_r$ is an order complete vector lattice.
\end{theorem}
\begin{proof}
Let $\mathcal A$ be a subset of $\mathcal{M}_r$ such that $\mathcal A$ is bounded from above in $\mathcal{M}_r$. We will show that $\sup\mathcal A$ exists in $\mathcal M_r$. Let $\mathcal S$ be the set of all supermartingales $Y$ that dominate $\mathcal A$, that is, $X\le Y$ for all $X\in\mathcal A$. By assumption, $\mathcal S$ is non-empty.
For every $n$, put
\begin{displaymath}
z_n=\inf\bigl\{y_n\mid Y=(y_k)\in\mathcal S\bigr\}.
\end{displaymath}
We claim that $Z=(z_n)$ is a martingale and $Z=\sup\mathcal A$.
First, observe that $X\le Z$ for each $X=(x_n)\in\mathcal A$. Indeed, for each $n$ and each $Y=(y_n)\in\mathcal S$ we have $x_n\le y_n$, so that $x_n\le z_n$.
Next, observe that $Z$ is a supermartingale. Let $m\ge n$, then for
every $Y\in\mathcal S$ we have $z_m\le y_m$, so that $E_nz_m\le
E_ny_m\le y_n$. It follows that $E_nz_m\le z_n$.
Next, we will show that $Z$ is, in fact, a martingale. Fix $k\in\mathbb
N$ and define $Y=(y_n)$ as follows:
\begin{displaymath}
Y=(E_1z_k,E_2z_k,\dots,E_{k-1}z_k,z_k,z_{k+1},z_{k+2},\dots).
\end{displaymath}
We claim that $Y$ is a supermartingale. Indeed, let $n\le m$.
\begin{eqnarray*}
\text{If}\quad k\le n&\text{ then }&E_ny_m=E_nz_m\le z_n=y_n;\\
\text{if}\quad m<k&\text{ then }&E_ny_m=E_nE_mz_k=E_nz_k=y_n;\\
\text{if}\quad n<k\le m&\text{ then }&E_ny_m=E_nz_m=E_nE_kz_m\le E_nz_k=y_n.\\
\end{eqnarray*}
Next, note that $X\le Y$ for each $X=(x_n)\in\mathcal A$. Indeed, if $n\ge k$ then
$y_n=z_n\ge x_n$. If $n<k$ then $x_k\le z_k$ implies $E_nx_k\le E_nz_k$, so that $x_n\le y_n$.
This yields that $Y\in\mathcal S$, so that $Z\le Y$. It follows that
for every $n<k$ we have $z_n\le y_n=E_nz_k$, so that $z_n\le
E_nz_k$. Therefore, $z_n=E_nz_k$ for all $n$ and $k$ with $n<k$. Also note that
\begin{math}
E_nz_n=E_nE_nz_{n+1}=E_nz_{n+1}=z_n
\end{math}
for every $n$. Thus, $Z$ is a martingale and $X\le Z$ for each
$X\in\mathcal A$. Clearly, every martingale $Y$ dominating $\mathcal
A$ is in $\mathcal S$ and, therefore, $Z\le Y$. Hence,
$Z=\sup\mathcal A$.
Let $X\in\mathcal M_r$. Applying the previous argument with $\mathcal A=\{\pm X\}$, we conclude that $\abs{X}$ exists. It follows that $\mathcal M_r$ is a vector lattice. By the preceding computation, it is order complete.
\end{proof}
\begin{example}
\emph{$\mathcal{M}_r$ need not be a vector lattice}.
Let $F=c$. For each $n$, define $E_n\colon F\to F$ as follows: if $x=(\alpha_i)$ we put
\begin{multline*}
E_nx=
\bigl(\alpha_1,\dots,\alpha_{3n},
\tfrac{\alpha_{3n+1}+\alpha_{3n+2}}{2},
\tfrac{\alpha_{3n+1}+\alpha_{3n+2}}{2},\alpha_{3n+3},\\
\tfrac{\alpha_{3n+4}+\alpha_{3n+5}}{2},
\tfrac{\alpha_{3n+4}+\alpha_{3n+5}}{2},\alpha_{3n+6},
\dots\bigr).
\end{multline*}
It is easy to see that $(E_n)$ is a filtration on $c$ and each $E_n$ is order continuous. It is a \term{dense} filtration in the sense that
\begin{math}
\bigcup_{n=1}^\infty\Range E_n
\end{math}
is dense in $F$. Define $(x_n)$ as follows:
\begin{displaymath}
x_n=(\underbrace{1,-1,0,1,-1,0,\dots,1,-1,0}_{3n},0,0,\dots).
\end{displaymath}
Note that $X=(x_n)$ is a martingale with respect to $(E_n)$; it is regular because $\pm x_n\le\one$ for every $n$, where $\one$ is the constant one sequence. We will write $x_{n,i}$ for the $i$-th coordinate of $x_n$. We claim that $X$ has no modulus in $\mathcal{M}_r$. Indeed, suppose that $\pm X\le Y$ for some martingale $Y$, $Y=(y_n)$. For each $n$ we have $y_n\ge\pm x_n$, so that
\begin{displaymath}
y_n\ge(\underbrace{1,1,0,1,1,0,\dots,1,1,0}_{3n},0,0,\dots)=u_n.
\end{displaymath}
It follows that
\begin{displaymath}
y_1=E_1y_n\ge E_1u_n
=u_n.
\end{displaymath}
Since $n$ is arbitrary, it follows that $y_{1,3k+1}\ge 1$ and $y_{1,3k+2}\ge 1$ for every $k$. Since $y_1$ is an element of $c$, there is $k_0$ such that $y_{1,3k_0}>0$. Define a martingale $Z=(z_n)$ as follows: for every $n$ and $i$, put $z_{n,i}=y_{n,i}$ except when $i=3k_0$, in this case put $z_{n,3k_0}=0$ (for every $n$). It is easy to see that $Z$ is a martingale and $\pm X\le Z<Y$. It follows that $X$ has no modulus.
\end{example}
\section{Krickeberg's formula}
Once again using the analogy with regular operators, we recall that if $F$ is an order complete vector lattice then $\mathcal L_r(F)$ is a vector lattice and the lattice operations on $\mathcal L_r(F)$ are given by the Riesz-Kantorovich formula. There is a similar formula for lattice operations on $\mathcal M_r$. Let $X=(x_n)$ be a martingale. It has been observed in the literature that that the modulus $\abs{X}$ of a regular martingale $X=(x_n)$ often satisfies the following identity:
\begin{displaymath}
\abs{X}_n=\sup_{m\ge n} E_n\abs{x_m}.
\end{displaymath}
For classical martingales, this identity goes back to Krickeberg's decomposition (see i.e., \cite[p.~32]{Meyer:72}); in the following we will refer to it as \term{Krickeberg's formula}. If the Krickeberg's formula is valid for $F$, that is, if the modulus of every martingale in $\mathcal M_r\bigl(F,(E_n)\bigr)$ is given by the Krickeberg's formula, then, clearly, $\mathcal M_r$ is a vector lattice and the other lattice operations are given by similar formulae; see, e.g., \cite[Theorem~7]{Troitsky:05}.
In the following proposition, we summarize several cases where $\mathcal{M}_r$ is a vector lattice and the lattice operations are given by Krickeberg's formula; it extends \cite[Theorem~2.3]{Korostenski:08}, \cite[Proposition 11]{Troitsky:05}, and \cite[Proposition 4]{Gessesse:11}.
\begin{proposition}\label{Krickeberg}
Let $F$ be a vector lattice and $(E_n)$ a filtration on $F$. Suppose that any of the following hold.
\begin{enumerate}
\item\label{oc-oc} $F$ is order complete and each $E_n$ is order continuous;
\item\label{fin-rank} $F$ is Archimedean and $E_n$ is of finite rank for each $n$;
\item\label{lat-hom} $E_n$ is a lattice homomorphism for each $n$.
\end{enumerate}
Then $\mathcal{M}_r$ is a vector lattice and the lattice operations are given by the Krickeberg's formula.
\end{proposition}
\begin{proof}
Let $X=(x_n)$ be a martingale such that $\pm X\le Y$ for some positive martingale $Y=(y_n)$. For a fixed $n$, the sequence $\bigl(E_n\abs{x_m}\bigr)_{m=n}^\infty$ is increasing in $m$, bounded below by $\abs{x_n}$ and above by $y_n$. Indeed, if $n\le m$ then
\begin{displaymath}
E_n\abs{x_m}=E_n\bigabs{E_mx_{m+1}}\le E_nE_m\abs{x_{m+1}}=E_n\abs{x_{m+1}},
\end{displaymath}
\begin{displaymath}
\abs{x_n}=\bigabs{E_nx_n}\le E_n\abs{x_n},\quad\text{and}\quad
E_n\abs{x_m}\le E_ny_m=y_n.
\end{displaymath}
\eqref{oc-oc} Since $F$ is order complete, $\sup\limits_{m\ge n}E_n\abs{x_m}$ exists for every $n$. Denote it $z_n$ and put $Z=(z_n)$. Clearly, $\pm X\le Z\le Y$. Note that $Z$ is a martingale: for every $k\le n$, since $E_k$ is order continuous, we have
\begin{displaymath}
E_kz_n=E_k\bigl(\sup_{m\ge n}E_n\abs{x_m}\bigr)
=\sup_{m\ge n}E_kE_n\abs{x_m}=\sup_{m\ge n}E_k\abs{x_m}=z_k.
\end{displaymath}
It follows that $Z=\abs{X}$. Note that $Z$ is given by the Krickeberg's formula.
\eqref{fin-rank} Let $H_n=\Range E_n$ and $H_n^+=F^+\cap H_n$. Since $H_n$ is finite-dimensional, we may view it as an ordered Banach space. Note that $\bigl(E_n\abs{x_m}\bigr)_{m=n}^\infty$ and $y_n$ are in $H_n^+$. Since $F$ is Archimedean and $H_n$ is finite-dimensional, $H_n^+$ is closed (in $H_n$) by \cite[Corollary~3.4]{Aliprantis:07} and normal by \cite[Lemma~3.1]{Aliprantis:07}. It follows from \cite[Theorem~2.45]{Aliprantis:07} that
\begin{math}
\lim_mE_n\abs{x_m}=z_n
\end{math}
in $H_n$. Since $H_n^+$ is closed, it follows from
\begin{math}
\abs{x_n}\le E_n\abs{x_m}\le y_n
\end{math}
that $\abs{x_n}\le z_n\le y_n$.
Repeating this process for every $n$, we produce a sequence $Z=(z_n)$ in $F_+$. To show that $Z$ is a martingale, let $k\le n$. Since $E_k$ is a continuous operator on $H_n$, we have
\begin{displaymath}
E_kz_n=E_k\bigl(\lim_{m\to\infty}E_n\abs{x_m}\bigr)=
\lim_{m\to\infty}E_kE_n\abs{x_m}=
\lim_{m\to\infty}E_k\abs{x_m}=z_k,
\end{displaymath}
where the limit is taken in $H_n$. It follows from $\pm X\le Z\le Y$ that $Z=\abs{X}$.
To verify Krickeberg's formula, it suffices to show that $z_n=\sup_mE_n\abs{x_m}$. It follows from \cite[Lemma~2.3(4)]{Aliprantis:07} that $z_n=\sup_mE_n\abs{x_m}$ in $H_n$. Let $a\in F$ such that $E_n\abs{x_m}\le a$ for all $m\ge n$. Let $G$ be the subspace of $F$ spanned by $H_n$ and $a$. Again, we may view it as an ordered Banach space with closed positive cone $G_+=F_+\cap G$; $H_n$ is a closed subspace of $G$. Hence, we still have
\begin{math}
\lim_mE_n\abs{x_m}=z_n
\end{math}
in $G$. Applying \cite[Lemma~2.3(4)]{Aliprantis:07} to $G$, we conclude that $z_n=\sup_mE_n\abs{x_m}$ in $G$, and, therefore, $z_n\le a$.
\eqref{lat-hom} It is easy to see that the sequence $\bigl(\abs{x_n}\bigr)$ is a martingale and is the modulus of $X$. Also, for every fixed $n$ and every $m\ge n$ we have $E_n\abs{x_m}=\bigabs{E_nx_m}=\abs{x_n}$, so that Krickeberg's formula s valid.
\end{proof}
It is an open problem whether the modulus of an operator is always given by the Riesz-Kantorovich formula, see~\cite[p.~59]{Aliprantis:07} for details. Similarly, it has been a natural conjecture that Krieckeberg's formula is always valid whenever $\mathcal M_r$ is a vector lattice. However, we will present a counterexample to the contrary. Our example will be based on \cite[Example~6]{Gessesse:11}, which we outline here for convenience of the reader.
\begin{example}(\cite{Gessesse:11})\label{halves}
Let $F=\mathbb{R}^\mathbb{N}$. For $n=0,1,2,\dots$, define $E_n$ via
\begin{multline*}
E_n\bigl((a_i)\bigr)=
\bigl(a_1,a_2,\dots,a_{2n},
\frac{a_{2n+1}+a_{2n+2}}{2},\frac{a_{2n+1}+a_{2n+2}}{2},\\
\frac{a_{2n+3}+a_{2n+4}}{2},\frac{a_{2n+3}+a_{2n+4}}{2},\dots\bigr)
\end{multline*}
Let $X=(x_n)_{n=0}^\infty$ where
\begin{displaymath}
x_n=\bigl(\underbrace{-1,1,\dots,-1,1}_{2n},0,0,\dots\bigr)
\end{displaymath}
(we take $x_0=0$).
It is easy to see that $(E_n)$ is a filtration on $F$ and $X$ is a martingale with respect to $(E_n)$.
\end{example}
\begin{example}\emph{$\mathcal{M}_r$ is a vector lattice, yet Krickeberg's formula fails.}
Let $F=\ell_\infty$; let $(E_n)$ and $X=(x_n)$ be as in Example~\ref{halves}, $n=0,1,\dots$. Let
$\varphi\colon F\to\mathbb R$ be a Banach limit. Put $P=\varphi\otimes\one$. It is easy to see that $P$ is a rank-one projection and that $E_nP=P$ for every $n$. It follows that the sequence $\bigl(PE_0,E_0,E_1,E_2,\dots\bigr)$ is a filtration on $F$ and the sequence $X=(x_0,x_0,x_1,x_2,\dots)$ is a martingale with respect to this filtration. Note that $\mathcal M_r$ is a vector lattice by Theorem~\ref{Mr-VL}.
We claim that $\abs{X}=(\one,\one,\one,\one,\dots)$. Indeed, it is easy to see that the sequence $(\one,\one,\one,\one,\dots)$ is a martingale which dominates $\pm X$. Now suppose $\pm X\le Y$ for some martingale $Y$. Put $Y=(z,y_0,y_1,\dots)$. For every $n\ge 0$ and $m\ge n$ we have $y_m\ge\abs{x_m}$, so that $y_n=E_ny_m\ge E_n\abs{x_m}$. Note that
\begin{displaymath}
E_n\abs{x_m}=\abs{x_m}=
\bigl(\underbrace{1,1,\dots,1,1}_{2m},0,0,\dots\bigr).
\end{displaymath}
This yields $y_n\ge\one$ for every $n\ge 0$. It follows from $y_0\ge\one$ that $z=PE_0y_0\ge\one$. Hence, $Y\ge(\one,\one,\one,\one,\dots)$. Therefore,
$\abs{X}=(\one,\one,\one,\one,\dots)$.
However, Krickeberg's formula for the initial term gives $\sup_{m}PE_0\abs{x_m}=0$ instead of $\one$.
\end{example}
\section{The space of regular bounded martingales on a~Banach lattice}
We say that $(E_n)$ is \term{uniformly bounded} if $\sup_n\norm{E_n}<+\infty$; we say that $(E_n)$ is \term{contractive} if $\norm{E_n}\le 1$ for every $n$. A martingale $X=(x_n)$ in $\mathcal{M}\bigl(F,(E_n)\bigr)$ is said to be \term{bounded} if its \term{martingale norm} defined by $\norm{X}=\sup_n\norm{x_n}$ is finite. We denote by $M=M\bigl(F,(E_n)\bigr)$ the space of all bounded martingales on $F$ with respect to the filtration $(E_n)$. It is easy to see that $M$ is a closed subspace of $\ell_\infty(F)$; hence $M$ is a Banach space. It can be easily verified that the martingale norm is \term{monotone}, i.e., $0\le X\le Y$ implies $\norm{X}\le\norm{Y}$. The space of regular bounded martingales is the following subspace of $M$:
\begin{displaymath}
M_r=M_r\bigl(F,(E_n)\bigr)=
\bigl\{X_1-X_2 \mid X_1,X_2 \in M_+\bigr\}.
\end{displaymath}
Again, one may expect similarities with the well-known theory of regular operators; see, e.g., \cite{Wickstead:07,Xanthos:15}. It is well known that every regular operator on a Banach lattice is bounded and the space of regular operators on an order complete Banach lattice is a Banach lattice under the \emph{regular norm}. We will prove that if $F$ is order complete then $M_r$ is a Banach lattice under the regular norm. We will show that, in contrast to the setting of regular operators, in general $M_r \neq \mathcal{M}_r$. Furthermore, $F$ is a KB-space iff $F$ is order continuous and every bounded martingale with respect to every uniformly bounded filtration is regular.
\begin{example}
\emph{Positive unbounded martingale on $\ell_1$.}
For any $0\le \alpha\le 1$, define
\begin{math}
P_{\alpha}=
\Bigl[
\begin{smallmatrix}
0 & 0 \\ \alpha & 1
\end{smallmatrix}
\Bigr].
\end{math}
It is easy to see that $P_{\alpha}$ is a positive projection onto $e_2$, and $P_\alpha$ is a contraction when viewed as an operator on $\ell_1^2$. Define a filtration on $\ell_1$ as follows.
\begin{multline*}
E_1=
\begin{bmatrix}
0 & 0 & & & & & \\
1 & 1 & & & && \\
& & 0 & 0 &&& \\
& & \frac12 & 1 &&& \\
& & & & 0 & 0 & \\
& & & & \frac14 & 1 & \\
& & & & & & \ddots \\
\end{bmatrix},
\quad
E_2=
\begin{bmatrix}
1 & 0 & & & & & & \\
0 & 1 & & & &&& \\
& & 0 & 0 &&& \\
& & \frac12 & 1 &&& \\
& & & & 0 & 0 & \\
& & & & \frac14 & 1 & \\
& & & & & & \ddots \\
\end{bmatrix},\\
E_3=
\begin{bmatrix}
1 & 0 & & & & & & \\
0 & 1 & & & &&& \\
& & 1 & 0 &&& \\
& & 0 & 1 &&& \\
& & & & 0 & 0 & \\
& & & & \frac14 & 1 & \\
& & & & & & \ddots \\
\end{bmatrix},
\text{ etc.}
\end{multline*}
It is easy to see that this is a filtration $\norm{E_n}=1$. Further, define
\begin{eqnarray*}
x_1&=&(0,1,\ 0,\tfrac12,\ 0,\tfrac14,\ 0,\tfrac18,\dots),\\
x_2&=&(1,0,\ 0,\tfrac12,\ 0,\tfrac14,\ 0,\tfrac18,\dots),\\
x_3&=&(1,0,\ 1,0,\ 0,\tfrac14,\ 0,\tfrac18,\dots),\\
x_4&=&(1,0,\ 1,0,\ 1,0,\ 0,\tfrac18,\dots),\\
\text{etc.}&&
\end{eqnarray*}
It can be easily verified that $(x_n)$ is a positive martingale with respect to the filtration $(E_n)$, but $\norm{x_n}>n-1$, so this martingale is unbounded.
\end{example}
On $M_r$, we define the following so called \term{regular norm}:
$$\norm{X}_r=\inf\bigl\{\norm{Y}\mid Y \in M_+,\ Y \ge\pm X\bigr\}.$$
We claim that the space $\bigl(M_r,\norm{\cdot}_r\bigr)$ is a Banach space. We will prove a more general result for ordered Banach spaces. In particular this result is known to be true for the space of regular operators and the space generated by positive compact operators (\cite{Chen:97}, Proposition 2.2).
\begin{theorem}\label{reg-norm}
Suppose that $\bigl(X,\norm{\cdot}\bigr)$ is an ordered normed space with a closed cone $X_+$ and a monotone norm. Then the following formula defines a norm on $X_r=X_+-X_+$:
$$\norm{x}_r=\inf\bigl\{\norm{y}\mid y \in X_+,\ y \ge\pm x\bigr\}.$$
For every $z \in X_r$, we have $\norm{z}\le 2\norm{z}_r$. Moreover, if $X$ is complete then $\bigl(X,\norm{\cdot}_r\bigr)$ is complete and if, in addition, $X=X_r$ then $\norm{\cdot}$ and $\norm{\cdot}_r$ are equivalent. If $X_r$ is a vector lattice then $\norm{x}_r=\bignorm{\abs{x}}$ for all $x\in X_r$.
\end{theorem}
\begin{proof}
It is easy to see that $\norm{\cdot}_r$ is positively
homogeneous. To verify the triangle inequality, let $u,v\in X_r$,
take any $\varepsilon>0$, and find $x,y\in X_+$ such that $-x\le
u\le x$ and $-y\le v\le y$, $\norm{x}\le\norm{u}_r+\varepsilon$, and
$\norm{y}\le\norm{v}_r+\varepsilon$. It follows that
\begin{math}
-(x+y)\le u+v\le x+y,
\end{math}
so that
\begin{displaymath}
\norm{u+v}_r\le\norm{x+y}\le\norm{x}+\norm{y}\le\norm{u}_r+\norm{v}_r+2\varepsilon.
\end{displaymath}
This yields $\norm{u+v}_r\le\norm{u}_r+\norm{v}_r$.
Fix $z\in X_r$. Let $x\in X_+$ be such that $-x\le z\le x$. Then $0\le x\pm z\le 2x$. It follows that
\begin{displaymath}
\norm{z}=\bignorm{\tfrac12(x+z)-\tfrac12(x-z)}
\le\tfrac12\norm{x+z}+\tfrac12\norm{x-z}
\le\tfrac12\norm{2x}+\tfrac12\norm{2x}
=2\norm{x}.
\end{displaymath}
Taking the infimum
over all such $x$, we get $\norm{z}\le2\norm{z}_r$. In particular,
if $\norm{z}_r=0$ then $\norm{z}=0$ and, therefore, $z=0$.
Now suppose that $\bigl(X,\norm{\cdot}\bigr)$ is complete and show
that $\bigl(X_r,\norm{\cdot}_r\bigr)$ is complete. Let $(z_n)$ be a
$\norm{\cdot}_r$-Cauchy sequence in $X_r$. Note that since $\norm{\cdot}\le
2\norm{\cdot}_r$, the sequence is also Cauchy in the original norm,
hence $z_n\xrightarrow{\norm{\cdot}}z$ for some $z\in X$. It suffices
to show that $z\in X_r$ and some subsequence of $(z_n)$ converges to
$z$ in $\norm{\cdot}_r$ because, in this case, the entire sequence
would still converge to $z$ in $\norm{\cdot}_r$.
Without loss of generality, passing to a subsequence, we may assume
that for every $n$ and every $k\ge n$ we have
$\norm{z_n-z_k}_r\le\frac{1}{3^n}$. For each $n$, $z_{n+1}-z_n\in
X_r$, so we can find $x_n\in X_+$ such that $-x_n\le z_{n+1}-z_n\le
x_n$ and $\norm{x_n}\le\frac{1}{2^n}$. Fix $m$. It follows from
$z_n\xrightarrow{\norm{\cdot}}z$ that
$z-z_m=\sum_{n=m}^\infty(z_{n+1}-z_n)$, where the series converges
in $\norm{\cdot}$. Note that
\begin{displaymath}
\sum_{n=m}^k(z_{n+1}-z_n)\le\sum_{n=m}^kx_n
\end{displaymath}
for every $k>m$ and $\sum_{n=m}^\infty x_n$ converges in $\norm{\cdot}$. Since $X_+$
is closed, $z-z_m\le\sum_{n=m}^\infty x_n$.
Similarly, $-(z-z_m)\le\sum_{n=m}^\infty x_n$.
It follows that
\begin{displaymath}
\norm{z-z_m}_r\le\Bignorm{\sum_{n=m}^\infty x_n}
\le\sum_{n=m}^\infty\norm{x_n}
\le\tfrac{1}{2^{m-1}}\to 0.
\end{displaymath}
If, moreover, $X=X_r$, we have that $\norm{\cdot}$ and $\norm{\cdot}_r$ are two complete norms on the same space with one of them dominating the other; it follows that the norms are equivalent.
Suppose that $X_r$ is a vector lattice and $x\in X_r$. It follows from $\abs{x}\ge\pm x$ that $\bignorm{\abs{x}}\ge\norm{x}_r$. On the other hand, if $\pm x\le y$ for some $y\in X_+$ then $\abs{x}\le y$ and the monotonicity of norm yields $\bignorm{\abs{x}}\le\norm{y}$, hence $\bignorm{\abs{x}}\le\norm{x}_r$.
\end{proof}
\begin{corollary}\label{Mr-BS}
Let $F$ be a Banach lattice and $(E_n)$ a filtration on $F$. The space $\bigl(M_r,\norm{\cdot}_r\bigr)$ is a Banach space and $\norm{X}\le\norm{X}_r$ for every $X\in M_r$. If $M_r$ is a vector lattice then $\norm{X}_r=\bignorm{\abs{X}}$ for all $X\in M_r$.
\end{corollary}
\begin{proof}
Applying Theorem \ref{reg-norm} to $M$, we conclude that $\bigl(M_r,\norm{\cdot}_r\bigr)$ is a Banach space and that if $M_r$ is a vector lattice then $\norm{X}_r=\bignorm{\abs{X}}$ for all $X\in M_r$. For $Y \in M_+$ such that $\pm X\le Y$, by the definition of the martingale norm we have $\norm{X}\le\norm{Y}$. It follows that $\norm{X}\le\norm{X}_r$.
\end{proof}
The regular norm may coincide with the martingale norm on $M_r$. We recall here that a Banach lattice $F$ is said to have the \term{Fatou property} if $0\le x_\alpha\uparrow x$ implies $\norm{x_\alpha}\to\norm{x}$. Dual and order continuous Banach lattices enjoy the Fatou property; see, e.g., \cite[p.~96]{Meyer-Nieberg:91} or \cite[p.~65]{Abramovich:02}.
\begin{proposition}\label{Fatou}
Let $F$ be a Banach lattice with the Fatou property and $(E_n)$ a contractive filtration on $F$. If $M_r$ is a vector lattice and Krickeberg's formula is valid on $M_r$ then $\norm{X}=\norm{X}_r$ for every $X \in M_r$.
\end{proposition}
\begin{proof}
Let $X=(x_n) \in M_r$ and $Z=(z_n)=\abs{X}$. By Corollary~\ref{Mr-BS},
$\norm{X}_r=\norm{Z}\ge\norm{X}$. We need to show that $\norm{X}\ge\norm{Z}$. By Krickeberg's formula, for each $n \in \mathbb{N}$ we have $E_n\abs{x_m}\uparrow z_n$ (in~$m$). The Fatou property yields $\norm{z_n}=\sup_m\bignorm{E_n\abs{x_m}}\le\sup_m\norm{x_m}=\norm{X}$, hence $\norm{Z}\le\norm{X}$.
\end{proof}
The following is the main result of our paper. Note that in view of Proposition \ref{Fatou} this result extends Theorem 13 in \cite{Troitsky:05}. Recall that if $F$ is order complete then $\mathcal M_r$ is a vector lattice by Theorem~\ref{Mr-VL}.
\begin{theorem}\label{Mr-BL}
Let $F$ be an order complete Banach lattice and $(E_n)$ a filtration on $F$. Then the space $M_r$ is an ideal of $\mathcal{M}_r$ and a Banach lattice under the regular norm.
\end{theorem}
\begin{proof}
Note first that if $X\in M_r$ and $Y\in \mathcal{M}_r$ such that $0\le Y\le X$ then $Y\in M_r$. Let $X\in M_r$. Then there exists $Y\in M_r$ such that $Y\ge\pm X$. By Theorem~\ref{Mr-VL}, $\abs{X}$ exists in $\mathcal{M}_r$. Clearly, $\abs{X}\le Y$ and, therefore, $\abs{X}\in M_r$. Hence, $M_r$ is a vector lattice and an ideal of $\mathcal{M}_r$.
By Corollary~\ref{Mr-BS}, $\bigl(M_r,\norm{\cdot}_r\bigr)$ is a Banach lattice.
\end{proof}
\begin{example} \emph{$M_r$ need not be a Banach lattice under the martingale norm.}
Let $F=\ell_\infty$, equipped with following equivalent norm:
$$\bignorm{(a_i)} =\bignorm{(a_i)}_\infty+\limsup\abs{a_i}$$
It can be easily verified that $\bigl(F,\norm{\cdot}\bigr)$ is a Banach lattice. Let $(E_n)$ and $X=(x_n)$ be as in Example~\ref{halves}. Note that each $E_n$ is order continuous. Clearly, $M=M_r$. By Theorem~\ref{Mr-BL}, $M_r$ is a Banach lattice under $\norm{\cdot}_r$ and an ideal in $\mathcal M_r$. For each $n$, we have
\begin{displaymath}
E_n\abs{x_m}=\abs{x_m}=(\underbrace{1,1,\dots,1,1}_{2m},0,\dots)\uparrow\one,
\end{displaymath}
By Proposition~\ref{Krickeberg}\eqref{oc-oc}, lattice operations are given by Krickeberg's formula. It follows that the modulus of $X$ is the constant martingale $\abs{X}_n=\one$. We have $\bignorm{\abs{X}}=2$ and $\norm{x_n}=1$ for each $n$, thus $1=\norm{X}<\bignorm{\abs{X}}$, so that $M_r$ fails to be a Banach lattice under the martingale norm.
\end{example}
Finally, we study under which conditions we have $M=M_r$. Recall that a vector lattice $F$ has a \term{strong unit} $e$ whenever $F=\bigcup_{n=1}^\infty [-ne,ne]$. A Banach lattice $F$ has an \term{order continuous norm} whenever $x_\alpha\downarrow 0$ implies $x_\alpha\to 0$; $F$ is a \term{KB-space} if it does not contain a sublattice isomorphic to $c_0$.
\begin{proposition}
Let $F$ be a Banach lattice with a strong unit $e$ and $(E_n)$ a uniformly bounded filtration on $F$. Then $M=M_r$. If, in addition, $F$ is order complete then $M$ is a vector lattice with a strong unit.
\end{proposition}
\begin{proof}
It is known that the original norm of $F$ is equivalent to the norm $\norm{\cdot}_\infty$ generated by $e$; see, e.g., \cite[p.~194]{Aliprantis:06}. In particular, there exists $C>0$ such that $\abs{x}\le C\norm{x}e$ for every $x\in F$.
For each $n$, put $y_n=E_ne$. Clearly, $Y=(y_n)$ is a bounded positive martingale. Let $X\in M$. Then for every $n$ we have $\pm x_n\le C\norm{x_n}e\le C\norm{X}e$. Applying $E_n$, we get $\pm x_n\le C\norm{X}y_n$. It follows that $\pm X\le C\norm{X}Y$, so that $X$ is regular. Hence, $M=M_r$. If, in addition, $F$ is order complete then $M$ is a vector lattice by Theorem~\ref{Mr-BL}. It follows from $\pm X\le C\norm{X}Y$ that $Y$ is a strong unit in $M$.
\end{proof}
\begin{theorem}
Let $F$ be a Banach lattice with an order continuous norm. Then the following are equivalent.
\begin{enumerate}
\item\label{KB-KB} $F$ is a KB-space;
\item\label{KB-MMr} $M=M_r$ for every uniformly bounded filtration $(E_n)$ on $F$;
\item\label{KB-M-VL} $M$ is a vector lattice for any uniformly bounded filtration $(E_n)$ on $F$.
\end{enumerate}
\end{theorem}
\begin{proof}
\eqref{KB-M-VL}$\Rightarrow$\eqref{KB-MMr} trivially. \eqref{KB-MMr}$\Rightarrow$\eqref{KB-M-VL} by Theorem~\ref{Mr-BL}
\eqref{KB-KB}$\Rightarrow$\eqref{KB-M-VL} The proof is similar to that of \cite[Theorem~7]{Troitsky:05}. Let $X=(x_n) \in M$. Fix some $n \in \mathbb{N}$. The sequence $\bigl(E_n\abs{x_m}\bigr)_{m=n}^\infty$ is increasing and norm bounded. Since $F$ is a KB-space, it follows that $z_n=\lim_m E_n\abs{x_m}$ exists; then $Z$ is a martingale and $Z=\abs{X}$.
\eqref{KB-MMr}$\Rightarrow$\eqref{KB-KB} Suppose that $F$ is not a KB-space. Then $c_0$ is lattice embeddable in $F$. Without lose of generality, we can assume that $c_0$ is a closed sublattice of $F$. Let $(E_n)$ and $X=(x_n)$ be as in Example~\ref{halves}; view $(E_n)$ as a (uniformly bounded) filtration on $c_0$ and $X$ as a martingale in $c_0$.
By \cite[Corollary~2.4.3]{Meyer-Nieberg:91}, there exist a positive projection $P:F \rightarrow c_0$. It is easy to see that $(PE_n)$ is again a uniformly bounded filtration on $F$ and $X$ is a bounded martingale with respect to it. We claim that $X$ is not regular. Indeed, suppose that $\pm X\le Z$ for some positive martingale $Z=(z_n)$ in $F$. Then $\pm x_n\le z_n$ for every $n$ yields $\abs{x_n}\le z_n$, so that
\begin{math}
E_0\abs{x_n}=PE_0\abs{x_n}\le PE_0z_n=z_0.
\end{math}
It follows that the increasing sequence $\bigl(E_0\abs{x_n}\bigr)$ is bounded above in $F$, hence it converges in $F$ because $F$ is order continuous. Therefore, $(E_0\abs{x_n})$ converges in $c_0$, which is clearly false.
\end{proof}
\medskip
\textbf{Acknowledgements.} We would like to thanks O.Blasco and N.Gao for helpful discussions.
| {
"timestamp": "2015-01-09T02:04:35",
"yymm": "1501",
"arxiv_id": "1501.01685",
"language": "en",
"url": "https://arxiv.org/abs/1501.01685",
"abstract": "In \\cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm.",
"subjects": "Functional Analysis (math.FA)",
"title": "Spaces of regular abstract martingales",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795114181106,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7092379365393247
} |
https://arxiv.org/abs/math/0601143 | L-functions and higher order modular forms | It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain cases one can associate a kind of ``higher order modular form'' to such Dirichlet series. This suggests a possible approach to a proof of the conjecture. | \section{Introduction}
We investigate the relationship between degree-2 $L$-functions and
modular forms.
We find that degree-2 $L$-functions can be associated to functions on the
upper half-plane which have similar properties to ``second order
modular forms.'' Since it is conjectured that degree-2 $L$-functions
can be associated to modular forms, this looks like a step in the right direction.
We review some
classical results on modular forms and then describe
the conjecture which motivates our work.
A good reference for this material is Iwaniec's book~\cite{Iw}.
Let
$$
\Gamma_0(N)=\left\{\mm abcd : a,b,c,d \hbox{ are integers, }
ad - bc = 1 \hbox{, and } c \equiv 0 \bmod N \right\}
$$
be the Hecke congruence group of level~$N$, and suppose
$\chi$ is a character mod~$N$. The group $\Gamma_0(N)$ acts
on functions $f:\mathcal H\to \mathbb C$ by $f\to f|\gamma$ where
\begin{equation}
f(z)\big| \mm abcd = \chi(d)^{-1} (cz+d)^{-k} f\(\frac{az+b}{cz+d}\) .
\end{equation}
Here $\mathcal H=\{x+i y\in \mathbb C\ :\ y>0\}$ is the upper half of the
complex plane.
The vector space of \emph{cusp forms of weight $k$ and character $\chi$ for
$\Gamma_0(N)$}, denoted $S_k(\Gamma_0(N),\chi)$, is the set of
holomorphic functions $f:\mathcal H\to \mathbb C$ which
satisfy $f|\gamma=f$ for all $\gamma\in \Gamma_0(N)$ and which
vanish at all cusps of~$\Gamma_0(N)$. Since
\begin{equation}
T := \mm 1101 \in \Gamma_0(N)
\end{equation}
we have $f(z)=f(z+1)$, so there is a Fourier expansion of the form
\begin{equation}\label{eqn:fourier}
f(z)=\sum_{n=1}^\infty a_n e^{2 \pi i n z} .
\end{equation}
In the case $\chi$ is the trivial character~$\chi_0$, the
newforms in $S_k(\Gamma_0(N),\chi_0)$ have a distinguished basis
of \emph{Hecke eigenforms} which satisfy
\begin{equation}\label{eqn:frickepm}
f|H_N=\pm f
\end{equation}
and
\begin{equation}\label{eqn:hecke}
f|T_p=a_p f
\end{equation}
for prime $p$.
Here
$$
H_N = \mm {}{-1}N{}
$$
is the Fricke involution. If $\ell$ is prime,
\begin{equation}\label{def:heckeop}
T_\ell = \chi(\ell) \mm \ell{}{}1 + \sum_{a=0}^{\ell-1} \mm 1a{}\ell,
\end{equation}
is the Hecke operator. If $\ell | N$ then $\chi(\ell)=0$ and $T_\ell$
is known as the Atkin-Lehner operator~$U_\ell$.
We will now state our motivating conjecture, and then explain its relevance
to the theory of $L$-functions.
\begin{conjecture}\label{theconjecture} If $f:\mathcal H\to \mathbb C$ is
analytic, is periodic with period~1~(\ref{eqn:fourier}), and satisfies
the Fricke~(\ref{eqn:frickepm}) and Hecke~(\ref{eqn:hecke})
relations with $\chi=\chi_0$, then $f\in S_k(\Gamma_0(N),\chi_0)$.
\end{conjecture}
Thus, the invariance property $f|\gamma=f$, which leads to the
Fricke and Hecke relations, would actually follow from
them.
We will rephrase the conjecture in terms of $L$-functions.
Associated to a cusp form with Fourier expansion~(\ref{eqn:fourier})
is an $L$-function
\begin{equation}
L(s,f)=\sum_{n=1}^\infty \frac{a_n}{n^s} .
\end{equation}
Using the Mellin transform and its inverse, it can be shown that
the Fricke relation~(\ref{eqn:frickepm}) is
equivalent to the functional equation
\begin{equation} \label{eqn:thefunctionalequation}
\xi_f(s):=\(\frac{2\pi}{\sqrt{N}}\)^{-s}\Gamma\(s\)L_f(s)=
\pm (-1)^{k/2}\xi_f(k-s).
\end{equation}
Also, the Hecke relations~(\ref{eqn:hecke}) are equivalent to
$L(s,f)$ having an Euler product of the form
\begin{equation} \label{eqn:theEulerproduct}
L(s,f)=\prod_p
\(1-a_p p^{-s} + \chi(p) p^{k-1-2s}\)^{-1},
\end{equation}
because both statements are equivalent to
$a_{p^n m}=a_{p^n} a_m$ for $p\nmid m$ and
\begin{equation}
a_{p^{n+1}} = a_p a_{p^n} - \chi(p)p^{k-1} a_{p^{n-1}}.
\end{equation}
Thus, Conjecture~\ref{theconjecture} is equivalent to
\begin{conjecture}\label{theconjectureL} If a Dirichlet series
continues to an entire function of order one which is bounded in
vertical strips and satisfies the functional
equation~(\ref{eqn:thefunctionalequation})
and the Euler product~(\ref{eqn:theEulerproduct}) with $\chi=\chi_0$,
then the Dirichlet series equals $L(s,f)$ for some
$f\in S_k(\Gamma_0(N),\chi_0)$.
\end{conjecture}
This conjecture should be viewed as part of the Langlands'
program. Note that one does not require functional equations for
twists of the $L$-function, as in Weil's converse theorem.
As a special case, the $L$-function of a rational elliptic curve
automatically has an Euler product of form~(\ref{eqn:theEulerproduct})
with $k=2$ and $\chi=\chi_0$,
so the modularity of a rational elliptic curve would
follow from the analytic continuation and functional equation
for one $L$-function.
Progress on the conjecture has been made only for small $N$,
for the trivial character~\cite{CF}, and (appropriately modified)
for almost the
same cases for nontrivial character~\cite{Har}.
For $N\le 4$, Hecke's original converse theorem establishes
the conjecture. This follows from the fact that the
group generated by $T$ and $H_N$ contains $\Gamma_0(N)$
exactly when $N\le 4$. Note that this only uses the
functional equation, not the Euler product.
For larger~$N$, one must use the Euler product in a
nontrivial way. This possibility was introduced in~\cite{CF},
and examples were given for certain~$N\le 23$.
In this paper we specialize to the case~$N=13$, for
the simple reason that this is the first case which has
not been solved. Our hope is to discover some
structure which can be used to attack the general case.
It turns out that the $N=13$ case leads to relations reminiscent of
``higher order modular forms,'' which are described in the next section.
In Section~\ref{sec:prior} we describe prior work and then
in Section~\ref{sec:13} we apply those methods to the case~$N=13$.
In recent work, Conrey, Odgers, Snaith, and the first author~\cite{CFOS}
have used some of the relations in this paper along with a new
generalization of Weil's lemma to complete the proof for~$N=13$.
\section{Higher order modular forms}
Our discussion here is imprecise and will only convey the general
flavor of this new subject. For details see~\cite{CDO, DKMO}.
We first introduce some slightly simpler notation. If
$f|\gamma=f$ then we have
\begin{equation}
\gamma \equiv 1 \bmod \Omega_f
\end{equation}
where $\Omega_f$ is the right ideal in the group ring $\mathbb C[GL(2,\mathbb R)]$
which annihilates~$f$, the action of matrices on $f$ being extended
linearly.
We will write $\gamma \equiv 1$ instead of $\gamma \equiv 1 \bmod \Omega_f$
throughout this paper.
Thus, if $f$ is a cusp form for the group $\Gamma$, then the invariance
properties of $f$ can be written as
$f|(1-\gamma)=0$ for all $\gamma\in \Gamma$, or equivalently,
$1-\gamma \equiv 0$.
This notation will make it easier to describe the properties of
higher order modular forms.
If $f$ is a \emph{second order} cusp form for the group $\Gamma$,
then $f$ satisfies the
relation
\begin{equation}\label{eqn:secondorder}
(1-\gamma_1)(1-\gamma_2)\equiv 0
\end{equation}
for all
$\gamma_1, \gamma_2 \in \Gamma$. Similarly, third order modular
forms satisfy
\begin{equation}\label{eqn:thirdorder}
(1-\gamma_1)(1-\gamma_2)(1-\gamma_3)\equiv 0,
\end{equation}
and so on. Roughly speaking, if $f$ is an $n$th order modular
form then $f|(1-\gamma)$ is an $(n-1)$st order modular form.
There are additional conditions involving the cusps and the parabolic
elements of $\Gamma$, but our goal here is just to introduce
the general idea. Indeed, it is nontrivial to determine
the proper technical conditions, see~\cite{CDO, DKMO}.
In connection with our exploration of Conjecture~\ref{theconjecture},
a condition of form~(\ref{eqn:secondorder})
will arise where $\gamma_1$ and $\gamma_2$ come from \emph{different} groups.
This first appeared in the original work of Weil on the converse theorem
involving functional equations for twists. Specifically, the
relation~(\ref{eqn:secondorder}) arose where
$\gamma_2$ was \emph{elliptic of infinite order}. The following
lemma applies:
\begin{lemma}\label{thm:weil} Suppose $f$ is holomorphic in $\H$ and
$\varepsilon\in GL_2(\mathbb R)^+$ is elliptic. If \hbox{$f|_k\varepsilon=f$},
then either $\varepsilon$ has finite order, or $f$ is constant.
\end{lemma}
This is known as ``Weil's lemma''~\cite{W}. See also the discussion in
Section~7.4 of Iwaniec's book~\cite{Iw}.
By the lemma, if $\gamma_2$ is elliptic of infinite order then
(\ref{eqn:secondorder}) implies that actually $1-\gamma_1\equiv 0$,
which is the conclusion Weil sought.
Denote by $S_k(\Gamma_1,\Gamma_2)$
the set of analytic functions (with appropriate technical
conditions) satisfying~(\ref{eqn:secondorder})
for all $\gamma_1\in\Gamma_1$ and $\gamma_2\in\Gamma_2$.
The above lemma says that if $\Gamma_2$ contains an elliptic
element of infinite order then
$S_k(\Gamma_1,\Gamma_2) = S_k(\Gamma_1)$. Note that the
analyticity of $f$ is necessary, and an analogue of Weil's
converse theorem for Maass form $L$-functions has not been
proven in classical language.
In Section~\ref{sec:13} we will see that our assumptions on the
Fricke involution and the Hecke operators lead to
condition~(\ref{eqn:secondorder})
with $\gamma_1\in\Gamma_0(13)$ and $\gamma_2$ in some other
discrete group. We also obtain higher order
conditions~(\ref{eqn:thirdorder})
where each $\gamma_j$ comes from a different group.
This suggests the following question:
\begin{question}What conditions on $\Gamma_1$ and $\Gamma_2$ ensure that
$S_k(\Gamma_1,\Gamma_2)$ is finite dimensional? What conditions
imply that $S_k(\Gamma_1,\Gamma_2) = S_k(\Gamma_1)$?
\end{question}
Part of the problem is determining the appropriate technical
conditions to incorporate into the definition of~$S_k(\Gamma_1,\Gamma_2)$.
Even when $\Gamma_1=\Gamma_2$ this is nontrivial. See~\cite{CDO, DKMO}.
\section{Manipulating the Hecke Operators}\label{sec:prior}
In \cite{CF} results were obtained for various $N$ up to $N=23$.
The idea is to manipulate the relations $T\equiv 1$, $H_N\equiv \pm 1$ and
$T_n\equiv a_n$ to obtain $\gamma\equiv 1 $ for all
$\gamma$ in a generating set for $\Gamma_0(N)$. We will describe
the cases of $N=5,7,9,11$ from~\cite{CF}, and then the remainder of the paper
will concern the interesting relationships that arose in our exploration
of the case~$N=13$.
We have the following generating sets:
\begin{eqnarray}
\Gamma_0(N)&=&\left\langle T,\ W_N,\ \mm{2}{-1}{-N}{\frac{N+1}{2}} \right\rangle,
\phantom{XXXXX} N=5,7,9 , \cr
\Gamma_0(11)&=&\left\langle T,\ W_{11},\ \mm{2}{-1}{{-11}}{6} ,
\mm{3}{-1}{-11}{4} \right\rangle, \cr
\Gamma_0(13)&=&\left\langle T,\ W_{13},\ \mm{2}{-1}{{-13}}{7} ,
\mm{-3}{-1}{13}{4} , \mm{3}{-1}{13}{-4} \right\rangle ,
\end{eqnarray}
where
\begin{equation}
T=\mm 11{}1
\ \ \ \ \ \ \ \ \ \ \
\mathrm{and}
\ \ \ \ \ \ \ \ \ \ \
W_N=\mm 1{}{N}1 .
\end{equation}
The generator $T$ is for free
because we have assumed a Fourier expansion. The generator
$W_N$ now follows from the Fricke relation, because $W_N=H_N T H_N$.
So for these groups we have two of the generators.
Note that this uses the functional equation, but not the Euler product.
In the next section we repeat the calculations from~\cite{CF} in
the cases $N=5,7,9,11$, and in the following sections we treat
the case $N=13$.
\subsection{Levels 5, 7, 9, and 11}
For every $N$ we obtain a new generator from $T_2$.
This will resolve the cases $N=5$, $7$, and~$9$.
\begin{lemma}[Lemma 2 of \cite{CF}] If $H_N \equiv \pm 1$ and
$T_2\equiv a_2$ then
$$
\mm 2{-1}{-N}{\frac{N+1}{2}}
\equiv 1 .
$$
\end{lemma}
\begin{proof}
Note that
$$
H_N^{-1} T_2 H_N
= \mm 1{}{}2 + \mm 2{}{}1 + \mm 2{}{-N}1.
$$
Since $H_N^{-1} T_2 H_N \equiv a_2 H_N^{-1} H_N \equiv a_2\equiv T_2$, we have:
$$
\mm 1{}{}2 + \mm 2{}{}1 + \mm 2{}{-N}1 \equiv
\mm 2{}{}1 + \mm 1{}{}2 + \mm 11{}2.
$$
Canceling common terms from both sides we are left with
$$
\mm 2{}{-N}1 \equiv \mm 11{}2.
$$
Right multiplying by $\mm 11{}2^{-1}$ we have
$$
M_2:=\mm 2{-1}{-N}{\frac{N+1}{2}} \equiv 1.
$$
\end{proof}
The lemma provides the final generator for
$\Gamma_0(5)$, $\Gamma_0(7)$, and $\Gamma_0(9)$.
To obtain the final generator for $\Gamma_0(11)$ we will combine
the Hecke operators $T_3$ and $T_4$
For $T_3$ we have
\begin{eqnarray}\label{eqn:T3manipulation}
0 &\equiv & \mathstrut H_{N}(T_3-a_3)H_{N} -(T_3-a_3) \cr
&=&
- \mm 1{1}{}3 - \mm 1{2}{}3 + \mm 3{}{-N}1 + \mm 3{}{-2N}1 \cr
&\equiv&
- \mm 1{1}{}3 - \mm 1{-1}{}3 + \mm 3{}{-N}1 + \mm 3{}{N}1 ,
\end{eqnarray}
where the second step used
\begin{equation}
\mm 1{-1}{}{1}\equiv 1
\ \ \ \ \ \ \
\text{ and }
\ \ \ \ \ \ \
\mm 1{}{N}{1}\equiv 1.
\end{equation}
We can combine the terms in pairs using
$$
\mm {1}{a}{}{p} - \mm {p}{}{Nb}{1} = \( 1- \mm {p}{-a}{Nb}{\frac{-Nab+1}{p}} \) \mm {1}{a}{}{p}
$$
to get
\begin{equation}\label{eqn:T3rel}
\left(1-\mm 3{-1}{-11}4 \right)\beta(1/3)
+\left(1- \mm 31{11}4 \right)\beta(-1/3)
\equiv 0 ,
\end{equation}
where $\beta(x)=\displaystyle{\mm{1}{x}{}{1}}$.
We will combine this with a relation obtained from $T_4$ .
Since $T_4$ and $T_2$ are not
independent, there is more than one way to proceed. The calculation
which seems most natural to us begins with
\begin{eqnarray}
0&\equiv &
H_{N}(T_{4} -a_4 )H_{N} - (T_{4} -a_4 ) \cr
&&\mathstrut - \left[H_{N}(T_{2} -a_2 )H_{N} -(T_{2} -a_2 )\right]\mm 2{}{}1 \cr
&&\mathstrut - \left[H_{N}(T_{2} -a_2 )H_{N}-(T_{2} -a_2 )\right]\mm 1{}{}2 \cr
&= &
- \mm {1}{1}{}{4} + \mm{4}{}{-3N}{1}
- \mm {1}{3}{}{4} + \mm {4}{}{-N}{1} .
\end{eqnarray}
Combining terms as in the $T_3$ case gives
\begin{equation}\label{eqn:T4rel}
\left(1- \mm 4{-1}{-11}3 \right)\beta(1/4)
+\left(1- \mm 41{11}3 \right)\beta(-1/4)
\equiv 0.
\end{equation}
Combining~(\ref{eqn:T3rel}) and~(\ref{eqn:T4rel}) we obtain
\begin{eqnarray}
1- \mm 3{-1}{-11}4
&\equiv&\mathstrut-\left(1- \mm 31{11}4 \right)\beta\left(-\frac 23 \right)\cr
&=&\mathstrut\left(1- \mm 4{-1}{-11}3 \right) \mm 31{11}4 \beta\left(-\frac 23 \right)\cr
&\equiv&\mathstrut -\left(1- \mm 41{11}3 \right)\beta\left(-\frac{2}{4} \right)
\mm 31{11}4 \beta\left(-\frac 23 \right)\cr
&=&\mathstrut\left(1-\mm 3{-1}{-11}4 \right)\mm 41{11}3 \beta\left(
-\frac 24 \right)\mm 31{11}4 \beta\left(-\frac 23 \right) .
\end{eqnarray}
However,
$$
\mm 41{11}3
\beta\left(-\frac{2}{4} \right)
\mm 31{11}4 \beta\left(-\frac 23 \right)
= \mm 1{-2/3}{11/2}{-8/3}
$$
is elliptic but not of finite order.
So by Lemma~\ref{thm:weil},
$$
\mm 3{-1}{-11}4 \equiv 1 .
$$
This is the final generator for~$\Gamma_0(11)$.
\section{Level $13$, mimic previous methods}\label{sec:13}
We will mimic the method used for $\Gamma_0(11)$ for
$\Gamma_0(13)$, but things will not work out as nicely.
What will arise is an expression of the form~(\ref{eqn:secondorder}) that
appears in the definition of second order modular form.
\subsection{The case of $T_3$. }\label{sec:T3}
From $ T_3$ we obtain the following expression, which is analogous
to~(\ref{eqn:T3rel}),
\begin{equation}
\left(1-\mm 3{-1}{13}{-4} \right)\beta(1/3)
+\left(1- \mm 3{1}{-13}{-4} \right)\beta(-1/3)
\equiv 0 .
\end{equation}
We manipulate this similarly to the example for $\Gamma_0(11)$:
\begin{eqnarray}
1-\mm 3{1}{-13}{-4}
&\equiv&\mathstrut-\left(1- \mm 3{-1}{13}{-4} \right)\beta\left(\frac 23 \right)\cr
&=&\mathstrut\left(1- \mm {-4}{1}{-13}3 \right) \mm 3{-1}{13}{-4}
\beta\left(\frac 23 \right)\cr
&\equiv&\mathstrut\left(1- H_{13} \mm {-4}{1}{-13}3 H_{13} \right)
H_{13} \mm 3{-1}{13}{-4}
\beta\left(\frac 23 \right)\cr
&=& \mathstrut\left(1- \mm 3{1}{-13}{-4}\right)
H_{13} \mm 3{-1}{13}{-4}
\beta\left(\frac 23 \right) .
\end{eqnarray}
So,
\begin{equation}
\left(1- \mm 3{1}{-13}{-4}\right) \left ( 1- \varepsilon_1\right) \equiv 0
\end{equation}
where
\begin{equation}
\varepsilon_1= H_{13} \mm 3{-1}{13}{-4}
\beta\left(\frac 23 \right)
= \mm {\sqrt{13}}{\frac{14}{3 \sqrt{13}}}{-3 \sqrt{13}}{- \sqrt{13}} .
\end{equation}
Since $\varepsilon_1$ is elliptic of order 2
we cannot obtain anything from Lemma~\ref{thm:weil}.
However, we do have an expression of the form~(\ref{eqn:secondorder}) which
looks like the definition of a second order modular form.
\subsection{The case of $T_4$. }\label{sec:T4}
From $T_4$, again proceeding as in the $\Gamma_0(11)$ example,
we first have
\begin{equation}
\left(1- \mm 4{-1}{13}{-3} \right)\beta(1/4)
+\left(1- \mm 41{-13}{-3} \right)\beta(-1/4)
\equiv 0.
\end{equation}
Continuing exactly as above, this leads to
\begin{equation}
\left(1- \mm 3{1}{-13}{-4}\right) \left ( 1- \varepsilon_2\right) \equiv 0
\end{equation}
where
\begin{equation}
\varepsilon_2
= \mm {-\sqrt{13}}{\frac{-4}{ \sqrt{13}}}{\frac{7 \sqrt{13}}{2}}{\sqrt{13}} .
\end{equation}
Again $\varepsilon_2$ is elliptic of order 2.
\subsection{Combining $T_3$ and $T_4$. }
We can combine the two relationships to obtain
\begin{equation}
0\equiv \left[1- \mm{3}{1}{-13}{-4} \right]
\left (1 -\varepsilon \right)
\end{equation}
for any $\varepsilon$ in the group generated by $\varepsilon_1$ and $\varepsilon_2$,
and perhaps one of those elements will be elliptic of infinite order?
Unfortunately, this is not the case. Note that
$$
\varepsilon_1 \varepsilon_2 = \mm{\frac{10}{3}}{\frac{2}{3}}{-\frac{13}{2}}{-1} ,
$$
which is hyperbolic. Since $\varepsilon_1$ and $\varepsilon_2$ have order~2,
the group they generate contains only the elements
$(\varepsilon_1 \varepsilon_2)^n$ and $\varepsilon_2 (\varepsilon_1 \varepsilon_2)^n$,
so that group is discrete.
Although $T_3$ and $T_4$ were not sufficient to obtain the missing
generator, there are an infinite number of other Hecke operators to try.
\subsection{The case of $T_6$. }
We now proceed with similar calculations with $T_6$. We have
\begin{eqnarray}
0&\equiv& H_{13}(T_{6} -a_6 )H_{13} -(T_{6} -a_6 ) \cr
&&- \left[H_{13}(T_{2} -a_2 )H_{13} -(T_{2} -a_2 )\right] \mm 3{}{}1
- \left[H_{13}(T_{2} -a_2 )H_{13} -(T_{2} -a_2 )\right] \mm 1{}{}3 \cr
&&- \left[H_{13}(T_{3} -a_3 )H_{13} -(T_{3} -a_3 )\right]\mm 2{}{}1
- \left[H_{13}(T_{3} -a_3 )H_{13} -(T_{3} -a_3 )\right]\mm 1{}{}2 \cr
&\equiv &- \mm {1}{1}{}{6} + \mm{6}{}{-65}{1}
- \mm {1}{5}{}{6} + \mm {6}{}{-13}{1} .
\end{eqnarray}
Using manipulations similar to those above gives
\begin{eqnarray}
0&\equiv& \left[-1+ \mm{6}{-1}{65}{11} \right] \mm{1}{1}{}{6} + \left[- 1 + \mm{6}{-5}{-13}{11} \right]
\mm{1}5{}6 \cr
&\equiv&\left[-1+ \mm{6}{-1}{65}{11} \right] \mm{1}{1}{}{6} , \cr
\end{eqnarray}
because $ \mm{6}{-5}{-13}{11} = M_2^{-1} H_{13}T^{-1} H_{13} T^{-1}$ so the
second term on the first line is~$\equiv 0$. So we have
$$
\mm{1}{1}{}{6} + \mm{6}{}{-65}{1} \equiv 0 ,
$$
so
$$
\mm 6{-1}{-65}{11} \equiv 1
$$
This is not a new matrix because $\mm 6{-1}{-65}{11} =H_{13}T H_{13}T H_{13} M_2H_{13}$.
That is, the above manipulations with $T_6$ produce results that can
be obtained from $T_2$.
\subsection{Computer manipulation of Hecke operators}\label{sec:132}
The explicit manipulation of Hecke operators described in this paper are quite
tedious to do by hand, so we decided to make use of a computer.
We modified Mathematica to do calculations in the group ring
$\mathbb C[SL(2,\mathbb R)]$, made functions for the Hecke operators,
automated manipulations that occur repeatedly (such as the first step
in every example in the previous section of this paper), and
implemented some crude simplifications procedures.
For the simplification procedures, we sought to automate the discovery,
for example, that if $T\equiv1$, $H_{13}\equiv \pm 1$, and $M_2\equiv 1$,
then
\begin{equation}
-1+ \mm{6}{-1}{65}{11} \equiv 0,
\end{equation}
as we saw at the end of the previous section.
Our approach was to put all of the matrices in each expression
in ``simplest form'' by considering all products (on the left)
with, for example, fewer than 6 matrices where are known
to be $\equiv 1$, and then keeping the representative which
has the smallest entries. This idea worked surprisingly well.
We also implemented a ``factorization'' function which would do the
(trivial) calculation to check such things as whether
$1-\gamma_1-\gamma_2+\gamma_3$ was of the form
$(1-\gamma_1)(1-\gamma_2)$ or $(1-\gamma_2)(1-\gamma_1)$.
\subsection{The case of $T_7$. }
Calculations with $T_7$ yield interesting results. We have
\begin{eqnarray}
0 &\equiv & H_{13}(T_7 -a_7 )H_{13} -(T_7 -a_7 ) \cr
&\equiv& - \mm {1}{2}{}{7} + \mm{7}{}{-52}{1} - \mm {1}{3}{}{7} + \mm {7}{}{-65}{1} \cr
&&\mathstrut - \mm {1}{4}{}{7} + \mm {7}{}{-26}{1} - \mm {1}{5}{}{7} + \mm {7}{}{-39}{1} . \cr
\end{eqnarray}
Note that the expression on the right consists of 4~pair of matrices, as
opposed to the 6~pair that one would expect to obtain from~$T_7$.
This is because
two pair
canceled during simplification.
It turns out that the right side of the above expression factors as
\begin{eqnarray}
&& \left[- 1+ \mm{-3}{1}{-13}{4} \right] \mm 12{}7
+\left[- 1+ \mm{7}{4}{-65}{-37} \right] \mm 1{-4}{}7 \cr
&&\hskip 0.3cm + \left[- 1+ \mm{7}{-4}{-26}{15} \right] \mm 1{4}{}7
+\left[- 1+ \mm{3}{1}{-13}{-4} \right]\mm 1{-2}{}7 \cr
&&\hskip 0.7cm= \left[ -\mm {3}{1}{-13}{-4} +1 \right] \mm {3}{1}{-13}{-4} ^{-1} H_{13} \mm {1}{2}{}{7} \cr
&&\hskip 1.2cm+ \left[ -\mm {3}{1}{-13}{-4} +1 \right] \mm {3}{1}{-13}{-4} ^{-1} H_{13}
\mm 7{-1}{-13}{2} ^{-1} \mm 1{-4}{}7 \cr
&&\hskip 1.2cm +\left[- 1+ \mm{3}{1}{-13}{-4} \right] \mm {7}{1}{13}{2} ^{-1} \mm 1{4}{}7 +
\left[- 1+ \mm{3}{1}{-13}{-4} \right]\mm 1{-2}{}7 \cr
&&\hskip 0.7cm= \left[- 1+ \mm{3}{1}{-13}{-4} \right]\cr
&&\hskip 1.2cm \left(-\mm{-\sqrt{13}}{\frac{2}{\sqrt{13}}}{3 \sqrt{13}}{-\sqrt{13}}
- \mm {2\sqrt{13}}{\frac{1}{\sqrt{13}}}{-7\sqrt{13}}{} + \mm 21{-13}{-3}
+\mm {1}{-2}{}{7} \right). \cr
\end{eqnarray}
We can right multiply by the inverse of any of the four matrices in the second
factor to rewrite this in the form $(1- \gamma) (1+ A-B -C)$.
For no good reason we choose the first term, giving
\begin{eqnarray}
0&\equiv& \left[ 1- \mm{3}{1}{-13}{-4} \right] \cr
&&\hskip 1cm \times \left( 1 +\mm {\frac{29}{7}}{\frac{5}{7}}{-13}{-2}
-\mm {\frac{5\sqrt{13}}{7}}{\frac{17}{7\sqrt{13}}}{\frac{-22\sqrt{13}}{7}}{\frac{-5\sqrt{13}}{7}}
-\mm {\frac{5\sqrt{13}}{7}}{\frac{24}{7\sqrt{13}}}{-3\sqrt{13}}{-\sqrt{13}} \right) \cr
&=& \left[ 1- \mm{3}{1}{-13}{-4} \right] \( 1+A-B-C\),
\end{eqnarray}
say.
This expression factors further. Specifically, one can check that
$A=C B$, so we have
\begin{equation}
0\equiv
\left[ 1- \mm{3}{1}{-13}{-4} \right] (1-C)(1-B)
\end{equation}
Unfortunately, $B^2=1$, so we cannot immediately
cancel the final factor to reduce to a second-order type expression.
It would be good if that happened, because we would have another matrix
to combine with the $\varepsilon_1$ and $\varepsilon_2$ from
Sections~\ref{sec:T3} and~\ref{sec:T4}.
However, there is a curious benefit to having $B^2=1$, for we also
have $A B = C$, so
\begin{equation}
0\equiv
\left[ 1- \mm{3}{1}{-13}{-4} \right] (1-A)(1-B).
\end{equation}
Note that if $B^2=1$, independent of any conditions on
$A$ and $C$, then
$( 1+A-B-C)(1+B)=(1-C A^{-1})(1+A B A^{-1}) A $, so
\begin{equation}\label{eqn:order2factorization}
0\equiv
\left[ 1- \mm{3}{1}{-13}{-4} \right] (1-C A^{-1})(1+A B A^{-1}),
\end{equation}
which is almost a third-order condition. Such expressions arise
whenever we have an order-2 matrix, so some types of factorization
are not a surprise. In the particular case at hand,
$C A^{-1}=A B A^{-1}$, which has order~2, so
$(1-C A^{-1})(1+A B A^{-1})=0$ and~(\ref{eqn:order2factorization})
contains absolutely no information. Perhaps one should think that
if $B^2=1$ then there always is some factorization, for
either~(\ref{eqn:order2factorization}) is
nontrivial, or the expression factors nontrivially in another way.
\subsection{A few other cases}
From $T_{10}$ we get
\begin{eqnarray}
0&\equiv&\left[ 1- \mm{3}{1}{-13}{-4} \right] \cr
&&\hskip 1cm \times \left(1+
\mm {\frac{21}{5}}{\frac{2}{5}}{-13}{-1}
-\mm {\frac{2\,{\sqrt{13}}}{5}}{\frac{7}{5\,{\sqrt{13}}}}{\frac{-11\,{\sqrt{13}}}{5}}{\frac{-2\,{\sqrt{13}}}{5}}
-\mm {\frac{4\,{\sqrt{13}}}{5}}{\frac{19}{5\,{\sqrt{13}}}}{-3\,{\sqrt{13}}}{-{\sqrt{13}}} \right) \cr
&=&\left[ 1- \mm{3}{1}{-13}{-4} \right] (1+A-B-C),
\end{eqnarray}
say. Again $A=CB$ and $B^2=1$, so we obtain two factorizations.
From $T_{15}$ we get
\begin{eqnarray}
0&\equiv&\left[ 1- \mm{3}{1}{-13}{-4} \right] \cr
&&\hskip 1cm \times \left(1
+\mm {\frac{16}{5}}{1}{- \frac{117}{5} }{-7}
-\mm {4\,{\sqrt{13}}}{\frac{15}{{\sqrt{13}}}}{\frac{-209\,{\sqrt{13}}}{15}}{-4\,{\sqrt{13}}}
-\mm {\frac{17\,{\sqrt{13}}}{15}}{\frac{4}{{\sqrt{13}}}}{\frac{-59\,{\sqrt{13}}}{15}}{-{\sqrt{13} }} \right), \cr
\end{eqnarray}
which again factors in the same two ways.
From $T_{9}$ we get
\begin{eqnarray}
0&\equiv&\left[ 1- \mm{3}{1}{-13}{-4} \right] \cr
&&\hskip 1cm \times \left(1
+\mm {\frac{10}{3}}{1}{-\frac{13}{3}}{-1}
-\mm {2{\sqrt{13}}}{\frac{9}{{\sqrt{13}}}}{\frac{-53\,{\sqrt{13}}}{9}}{-2\,{\sqrt{13}\ }}
-\mm {\frac{7\,{\sqrt{13}}}{9}} {\frac{4}{{\sqrt{13}}}} {\frac{-25\,{\sqrt{13}}}{9}} {-{\sqrt{13}}} \right). \cr
\end{eqnarray}
which again factors in the same two ways.
It would be helpful to understand the underlying reason why
these expressions factor.
More time on the computer should produce more relations, but it is
not clear how they will combine to produce the desired result.
It would be interesting if the relations could build to the point where
one could reduce higher order relations to lower order ones,
which could then combine with previously found relations to
cause additional cancellation, and so on, reducing down to
the one missing generator for~$\Gamma_0(13)$. It would be
more satisfying if one could find manipulations which produce any
specific matrix, as one does in the proof of Weil's converse
theorem.
Our approach here is to look for factorizations
$(1-\gamma)(1-\delta)(1-\varepsilon)\equiv 0$ in the
hopes of eliminating the last factor, perhaps because
$\varepsilon$ is elliptic of infinite order. In the case of expressions
that do not factor, it would be interesting to know if there are
cancellation laws beyond those implied by Weil's lemma. That
is, are there conditions on $A$, $B$, $C$ such that
$f|(1+A-B-C) =0$ implies some apparently stronger condition on~$f$,
beyond those cases where $1+A-B-C$ factors and Weil's lemma applies?
\subsection{A curiosity}
All the manipulations in this paper involve ``pairing up'' the terms
in a linear combination of matrices. Usually there is a natural
way to do this, for one is hoping to produce matrices in $\Gamma_0(N)$.
However, it is possible to pair the matrices in different ways,
and one would like some justification for the choices and to know the
consequences of making the right (or wrong) choices. This is
discussed extensively in~\cite{FW}.
We now give an example by repeating the analysis of Section~\ref{sec:prior}
making the wrong choices.
From~(\ref{eqn:T3manipulation}) with $N=11$ we have
\begin{equation}
\left( 1- \mm{3}{-1}{11}{-\frac{10}{3} } \right) \beta(1/3)
+
\left( 1- \mm{3}{1}{-11}{-\frac{10}{3} } \right) \beta(-1/3)
\equiv 0 ,
\end{equation}
where $\beta(x)=\displaystyle{\mm{1}{x}{}{1}}$.
Now doing manipulations exactly as in Section~\ref{sec:T3} we obtain
\begin{equation}
0\equiv \left( 1- \mm{3}{-1}{11}{-\frac{10}{3} } \right) (1-\varepsilon),
\end{equation}
where
\begin{equation}
\varepsilon = H_{11} \mm{3}{1}{-11}{-10/3} \beta(-2/3)=
\mm {\sqrt{11}}{-\frac{4}{\sqrt{11}}}{3\sqrt{11}}{-\sqrt{11}} ,
\end{equation}
which has order~2.
Note that the above manipulations cannot lead to
$\mm{3}{-1}{11}{-\frac{10}{3}} \equiv 1$.
Indeed,
if $p$ is prime, the group generated by $\Gamma_0(p)$ and $H_{p}$
is a maximal discrete subgroup of $SL(2,\mathbb R)$. So no manipulation
can lead to a new matrix which is~$\equiv 1$.
Yet, we do obtain additional
second order modular form type properties for newforms in
$S_k(\Gamma_0(11))$. It is not clear what mechanism
will lead to the production of new matrices for~$N=13$, yet not
produce a contradiction when~$N=11$.
Using $T_4$ in the same way gives
\begin{equation}
0\equiv \left( 1- \mm{4}{-1}{11}{-\frac{5}{2} } \right) \left(1-
\mm {\sqrt{11}}{-\frac{3}{\sqrt{11}}}{4\sqrt{11}}{-\sqrt{11}} \right),
\end{equation}
and from $T_6$ you get
\begin{equation}
0\equiv \left( 1- \mm{6}{-1}{11}{-\frac{5}{3} } \right) \left(1-
\mm {\sqrt{11}}{-\frac{2}{\sqrt{11}}}{6\sqrt{11}}{-\sqrt{11}} \right),
\end{equation}
where the inner matrix is hyperbolic.
This illustrates that $f|(1-\varepsilon)(1-\delta)=0$ need not
imply $f$ is constant, and even having multiple independent relations of that
form is not sufficient. In the case here, we have the above relations
in addition to $f|(1-\gamma)$ for all $\gamma\in\Gamma_0(11)$. This
suggest that these ``second order'' conditions may be weaker than they
appear.
| {
"timestamp": "2006-01-07T20:54:52",
"yymm": "0601",
"arxiv_id": "math/0601143",
"language": "en",
"url": "https://arxiv.org/abs/math/0601143",
"abstract": "It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain cases one can associate a kind of ``higher order modular form'' to such Dirichlet series. This suggests a possible approach to a proof of the conjecture.",
"subjects": "Number Theory (math.NT)",
"title": "L-functions and higher order modular forms",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795087371921,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7092379346128317
} |
https://arxiv.org/abs/2008.00584 | Optimal rates of convergence and error localization of Gegenbauer projections | Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and $\lambda\leq0$ or differentiable and $\lambda\leq1$, where $\lambda$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $\lambda>0$ or differentiable and $\lambda>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^{\lambda}$ and $n^{\lambda-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $\lambda>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods. | \section{Introduction}\label{sec:introduction}
Orthogonal polynomial approximations, such as Gegenbauer as well as
Legendre and Chebyshev approximations, play an important role in
many branches of numerical analysis, including function
approximations and quadrature
\cite{davis1984methods,trefethen2013atap}, the resolution of Gibbs
phenomenon \cite{gottlieb1992gibbs,gottlieb1997gibbs} and spectral
methods for the numerical solution of differential equations
\cite{guo1998gegenbauer,guo2000gegenbauer,hesthaven2007spectral,olver2013spectral,shen2011spectral}.
One of the most attractive features of them is that their
convergence rate depends strongly on the regularity of the
underlying function and can give highly accurate approximations for
smooth functions. Due to the important role that orthogonal
polynomial approximations play in many fields of applications, their
convergence analysis has attracted considerable interest, especially
in the spectral methods community.
Let $\mathrm{d}\mu$ be a positive Borel measure on the interval
$[a,b]$, for which all moments of $\mathrm{d}\mu$ are finite. We
introduce the inner product $\langle f,g \rangle_{\mathrm{d}\mu} =
\int_{a}^{b} f(x) g(x) \mathrm{d}\mu(x)$ and let
$\{\varphi_k\}_{k=0}^{\infty}$ be a set of orthogonal polynomials
with respect to $\mathrm{d}\mu$. Then, for any $f\in L^2(a,b)$, it
can be expanded in terms of $\{\varphi_k\}$ as
\begin{align}\label{eq:FourierSeries}
f(x) = \sum_{k=0}^{\infty} f_k \varphi_k(x), \quad f_k =
\frac{\langle f,\varphi_k \rangle_{\mathrm{d}\mu}}{\langle
\varphi_k,\varphi_k \rangle_{\mathrm{d}\mu}}.
\end{align}
Let $S_n(f)$ denote the truncation of the above series after $n+1$
terms, i.e., $S_n(f) = \sum_{k=0}^{n} f_k \varphi_k(x)$, it is well
known that $S_n(f)$ is the orthogonal projection of $f$ onto the
space $\mathcal{P}_{n}=\mathrm{span}\{1,x,\ldots,x^{n}\}$. Existing
approaches for error estimates of $S_n(f)$ in the maximum norm can
be roughly categorized into two types: (i) applying the classical
inequality $\|f - S_n(f) \|_{\infty} \leq (1+\Lambda)\| f -
\mathcal{B}_n(f) \|_{\infty}$, where $\Lambda =
\sup_{f\not\equiv0}\|S_n(f)\|_{\infty}/\|f\|_{\infty}$ is the
Lebesgue constant of $S_n(f)$ and $\mathcal{B}_n(f)$ is the best
approximation of degree $n$ to $f$. Hence, this approach transforms
the error estimate of $S_n(f)$ to the problem of finding the
asymptotic behavior of the corresponding Lebesgue constant; (ii)
using the inequality $\|f - S_n(f) \|_{\infty} \leq
\sum_{k=n+1}^{\infty} |f_k| \|\varphi_k\|_{\infty}$, and the
remaining task is to find some sharp estimates of the coefficients
$\{f_k\}$. The former approach plays a key role in the convergence
analysis for polynomial projections and nowadays the asymptotic
behavior of the Lebesgue constants associated with classical
orthogonal projections has been well-understood (see, e.g.,
\cite{frenzen1986Jacobi,levesley1999gegen,szego1975orthogonal}).
However, it is difficult to establish computable error bounds for
$S_n(f)$ with this approach. Moreover, to the best of the author's
knowledge, the sharpness of the derived error estimates has not been
addressed. For the latter approach, a remarkable advantage is that
some computable error bounds of $S_n(f)$ can be established (see,
e.g.,
\cite{bernstein1912cheb,liu2019optimal,liu2020legendre,trefethen2013atap,wang2012legendre,wang2018legendre,wang2020legendre,xiang2012error,xiang2020jacobi,zhao2013sharp}).
However, as shown in \cite{wang2018legendre,wang2020legendre}, the
convergence rate predicted by this approach may be overestimated for
differentiable functions.
In this work, we investigate optimal rates of convergence of
Gegenbauer projections in the maximum norm, i.e.,
$\mathrm{d}\mu(x)=(1-x^2)^{\lambda-1/2}\mathrm{d}x$, where
$\lambda>-1/2$ and $[a,b]=[-1,1]$. In order to exhibit the
dependence on the parameter $\lambda$, we denote by
$S_n^{\lambda}(f)$ the Gegenbauer projection of degree $n$. From the
preceding discussion we have
\begin{align}\label{eq:errorG}
\|f - S_n^{\lambda}(f) \|_{\infty} &\leq (1 + \Lambda_n(\lambda)) \|
f - \mathcal{B}_n(f) \|_{\infty},
\end{align}
where
$\Lambda_n(\lambda)=\sup_{f\not\equiv0}\|S_n^{\lambda}(f)\|_{\infty}/\|f\|_{\infty}$
is the Lebesgue constant of Gegenbauer projections. For
$\Lambda_n(\lambda)$ it is known (see, e.g.,
\cite{frenzen1986Jacobi,levesley1999gegen,lorch1959lebesgue}) that
\begin{align}\label{eq:Lebesgue}
\Lambda_n(\lambda) = \left\{
\begin{array}{ll}
{\displaystyle O(n^{\lambda}) }, & \hbox{$\lambda>0$,} \\[8pt]
{\displaystyle O(\log n) }, & \hbox{$\lambda=0$,} \\[8pt]
{\displaystyle O(1) }, & \hbox{$\lambda<0$.}
\end{array}
\right.
\end{align}
Regarding \eqref{eq:errorG} and \eqref{eq:Lebesgue}, it is natural
to ask: When using \eqref{eq:errorG} to predict the convergence rate
of $S_n^{\lambda}(f)$, how sharp the result is? If the predicted
rate is not sharp, what is the optimal rate? It is easily seen that
the predicted rate of convergence of $S_n^{\lambda}(f)$ is optimal
when $\lambda<0$ since it is the same as that of $\mathcal{B}_n(f)$,
and is near-optimal when $\lambda=0$ since $\Lambda_n(\lambda)$
grows very slowly. When $\lambda>0$, we can deduce that the rate of
convergence of $S_n^{\lambda}(f)$ is slower than that of
$\mathcal{B}_n(f)$ by at most a factor of $n^{\lambda}$. More
recently, the particular case of $\lambda=1/2$, which corresponds to
the case of Legendre projections, was examined in
\cite{wang2020legendre}. It is shown that the predicted rate of
convergence by \eqref{eq:errorG} is sharp when the underlying
function is analytic, but is optimistic for piecewise analytic
functions. For the latter, it has been shown that the convergence
rate of $S_n^{1/2}(f)$ is actually the same as than that of
$\mathcal{B}_n(f)$. This finding provides new insight into the
approximation power of Legendre projections and inspire us to
explore the case of Gegenbauer projections.
The aim of this paper is to present a comprehensive convergence rate
analysis of Gegenbauer projections and clarify the role of the
parameter $\lambda$. Our main contributions can be summarized as
follows:
\begin{itemize}
\item For analytic functions, we show that the inequality \eqref{eq:errorG} is sharp
in the sense that the optimal rate of convergence of
$S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$ by a
factor of $n^{\lambda}$ when $\lambda>0$. When $-1/2<\lambda\leq0$,
the rate of convergence of $S_n^{\lambda}(f)$ is the same as that of
$\mathcal{B}_n(f)$;
\item For piecewise analytic functions, we show that the optimal
rate of convergence of $S_n^{\lambda}(f)$ is the same as that of
$\mathcal{B}_n(f)$ when $-1/2<\lambda\leq1$. When $\lambda>1$,
however, we prove that the optimal rate of convergence of
$S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$ by a
factor of $n^{\lambda-1}$. Comparing this finding with the predicted
results by \eqref{eq:errorG} and \eqref{eq:Lebesgue}, we see that
the convergence rate of the Gegenabuer projection is better than the
predicted result by a factor of $n^{\lambda}$ when $\lambda\in(0,1]$
and by a factor of $n$ when $\lambda>1$;
\item We extend our discussion to functions of fractional smoothness,
including functions with endpoint singularities and functions with
an interior singularity of fractional order. The optimal rate of
convergence of $S_n^{\lambda}(f)$ for some model functions is also
analyzed.
\end{itemize}
The remainder of the paper is organized as follows. In the next
section, we present some preliminary results on Gegenbauer
polynomials and gamma functions. In section \ref{sec:experiment}, we
first carry out numerical experiments on the convergence rates of
$S_n^{\lambda}(f)$ and $\mathcal{B}_n(f)$ and then give some
observations. In section \ref{sec:Analytic} we analyze the
convergence behavior of $S_n^{\lambda}(f)$ for analytic functions.
We first establish some explicit bounds for the Gegenbauer
coefficients and then applied them to establish the optimal rate of
convergence of $S_n^{\lambda}(f)$. In section \ref{sec:Piecewise} we
establish optimal rates of convergence of $S_n^{\lambda}(f)$ for
piecewise analytic functions with the help of some refined estimates
of the Dirichlet kernel of Gegenbauer polynomials. We extend our
discussion to functions of fractional smoothness in section
\ref{sec:extension} and give some concluding remarks in section
\ref{sec:conclusion}.
\section{Preliminaries}
In this section, we introduce some basic properties of Gegenbauer
polynomials and the gamma function that will be used throughout the
paper. All these properties can be found in
\cite{olver2010nist,szego1975orthogonal}.
\subsection{Gamma function}
For $\Re(z)>0$, the gamma function is defined by
\begin{align}
\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} \mathrm{d}t.
\end{align}
When $\Re(z)\leq0$, $\Gamma(z)$ is defined by analytic continuation.
The gamma function satisfies the recursive property $ \Gamma(z+1) =
z \Gamma(z)$, and the classical reflection formula
\begin{align}\label{eq:reflection}
\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin(\pi z)}, \quad
z\neq0,\pm1,\ldots.
\end{align}
Moreover, the duplication formula of the gamma function reads
\begin{align}\label{eq:duplication}
\Gamma(2z) = \pi^{-1/2} 2^{2z-1} \Gamma(z) \Gamma\left(z +
\frac{1}{2}\right), \quad 2z\neq0,-1,-2,\ldots.
\end{align}
The ratio of two gamma functions will be crucial for the derivation
of explicit bounds for the Gegenbauer coefficients and the
asymptotic behavior of the Dirichlet kernel of Gegenbauer
projections. Let $a,b$ be some real or complex and bounded
constants, then we have
\begin{align}\label{eq:AsyGAMMA}
\frac{\Gamma(z+a)}{\Gamma(z+b)} = z^{a-b} \left[1 +
\frac{(a-b)(a+b-1)}{2z} + O(z^{-2}) \right], \quad
z\rightarrow\infty.
\end{align}
In the special case of either $a=1$ or $b=1$, the following simple
and sharp bounds will be useful.
\begin{lemma}\label{lem:gamma}
Let $k\in \mathbb{N}$ and $\gamma>-1$. Then
\begin{align}\label{eq:ratio1}
\frac{\Gamma(k+1)}{\Gamma(k+\gamma)} \leq k^{1-\gamma} \left\{
\begin{array}{ll}
{\displaystyle \frac{1}{\Gamma(1+\gamma)} }, & \hbox{$0\leq \gamma < 1$,} \\[8pt]
{\displaystyle 1}, &
\hbox{$-1<\gamma<0$~\text{or}~$\gamma\geq1$.}
\end{array}
\right.
\end{align}
and
\begin{align}\label{eq:ratio2}
\frac{\Gamma(k+\gamma)}{\Gamma(k+1)} \leq k^{\gamma-1} \left\{
\begin{array}{ll}
{\displaystyle 1}, & \hbox{$0\leq \gamma < 1$,} \\[8pt]
{\displaystyle \Gamma(1+\gamma)}, &
\hbox{$-1<\gamma<0$~or~$\gamma\geq1$.}
\end{array}
\right.
\end{align}
These upper bounds are sharp in the sense that they can be attained
when $k=1$ or $k=\infty$.
\end{lemma}
\begin{proof}
We only prove \eqref{eq:ratio1} and the proof of \eqref{eq:ratio2}
is completely analogous. In the cases $\gamma=0$ and $\gamma=1$,
\eqref{eq:ratio1} is trivial. Now consider the cases $-1<\gamma<0$
and $\gamma>0$ and $\gamma\neq1$. To this end, we introduce the
following sequence
\[
\psi(k) = \frac{\Gamma(k+1)}{\Gamma(k+\gamma)} k^{\gamma-1}.
\]
In view of the recursive property of $\Gamma(z)$, we obtain
\begin{align}
\frac{\psi(k+1)}{\psi(k)} = \frac{k+1}{k+\gamma} \left(\frac{k+1}{k}
\right)^{\gamma-1}. \nonumber
\end{align}
By differentiating the right-hand side of the above equation with
respect to $k$, one can easily check that the sequence
$\{\psi(k+1)/\psi(k)\}_{k=1}^{\infty}$ is strictly increasing when
$0<\gamma<1$ and is strictly decreasing when $-1<\gamma<0$ or
$\gamma>1$. Since $\lim_{k\rightarrow\infty} \psi(k+1)/\psi(k)=1$,
we deduce that $\{\psi(k)\}_{k=1}^{\infty}$ is strictly decreasing
when $0<\gamma<1$ and is strictly increasing when $-1<\gamma<0$ or
$\gamma>1$. Hence, if $0<\gamma<1$, we have
\begin{align}
\psi(k)\leq\psi(1) ~~ \Longrightarrow ~~
\frac{\Gamma(k+1)}{\Gamma(k+\gamma)} \leq
\frac{k^{1-\gamma}}{\Gamma(1+\gamma)}, \nonumber
\end{align}
and the upper bound can be attained when $k=1$. If $-1<\gamma<0$ or
$\gamma>1$, then
\begin{align}
\psi(k)\leq \lim_{k\rightarrow\infty} \psi(k) = 1 ~~ \Longrightarrow
~~ \frac{\Gamma(k+1)}{\Gamma(k+\gamma)} \leq k^{1-\gamma},
\nonumber
\end{align}
and the upper bound can be attained when $k=\infty$. This proves
\eqref{eq:ratio1} and the proof of Lemma \ref{lem:gamma} is
complete.
\end{proof}
\subsection{Gegenbauer polynomials}
Let $n\geq0$ be an integer, the Gegenbauer (or ultraspherical)
polynomial of degree $n$ is defined by
\begin{align}\label{def:GegenPoly}
C_{n}^{\lambda}(x) = \frac{(2\lambda)_n}{n!} {}_2\mathrm{
F}_1\left[\begin{matrix} -n, & n+2\lambda ;
\\ \lambda+\frac{1}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1-x}{2}
\right],
\end{align}
where ${}_2 \mathrm{F}_1(\cdot)$ is the Gauss hypergeometric
function defined by
\[
{}_2 \mathrm{F}_1 \left[\begin{matrix} a,~ b;& \\ c;
\end{matrix} \hspace{-.25cm} z \right] = \sum_{k = 0}^{\infty} \frac{ (a)_k
(b)_k }{ (c)_k } \frac{ z^k }{ k! },
\]
and where $(z)_k$ denotes the Pochhammer symbol defined by $(z)_{k}
= (z)_{k-1} (z + k - 1)$ for $k \geq 1$ and $(z)_0 = 1$. The
sequence of Gegenbauer polynomials
$\{C_k^{\lambda}(x)\}_{k=0}^{\infty}$ forms a system of polynomials
orthogonal over the interval $[-1,1]$ with respect to the weight
function $\omega_{\lambda}(x)=(1-x^2)^{\lambda-1/2}$ and
\begin{equation}\label{eq:GegenOrthog}
\int_{-1}^{1} \omega_{\lambda}(x) C_{m}^{\lambda}(x)
C_{n}^{\lambda}(x) \mathrm{d}x = h_n^{\lambda} \delta_{mn},
\end{equation}
where $\delta_{mn}$ is the Kronnecker delta and
\[
h_n^{\lambda} = \frac{\pi 2^{1-2\lambda}
\Gamma(n+2\lambda)}{\Gamma(\lambda)^2\Gamma(n+1) (n+\lambda)}, \quad
\lambda>-1/2,~~ \lambda\neq0.
\]
Since $\omega_{\lambda}(x)$ is even, it follows that
$C_{n}^{\lambda}(x)$ satisfy the following symmetry relation
\begin{align}\label{eq:symmetry}
C_{n}^{\lambda}(x) = (-1)^n C_{n}^{\lambda}(-x), \quad n\geq 0,
\end{align}
which implies that $C_{n}^{\lambda}(x)$ is an even function for even
$n$ and an odd function for odd $n$. For $\lambda>0$, Gegenbauer
polynomials satisfy the following inequality
\begin{align}\label{eq:Bound1}
|C_{n}^{\lambda}(x)| \leq C_{n}^{\lambda}(1) =
\frac{\Gamma(n+2\lambda)}{\Gamma(n+1) \Gamma(2\lambda)}, \quad
|x|\leq1,~~ n\geq 0,
\end{align}
and for $-1/2<\lambda<0$,
\begin{align}\label{eq:Bound2}
|C_{n}^{\lambda}(x)| \leq C_{\lambda} n^{\lambda-1}, \quad
|x|\leq1,~~ n\gg1,
\end{align}
where $C_{\lambda}$ is a positive constant independent of $n$. The
Rodrigues formula of Gegenbauer polynomials reads
\begin{align}\label{eq:Rodrigues}
\omega_{\lambda}(x) C_n^{\lambda}(x) =
\frac{-2\lambda}{n(n+2\lambda)} \frac{\mathrm{d}}{\mathrm{d}x}
\left\{ \omega_{\lambda+1}(x) C_{n-1}^{\lambda+1}(x) \right\},
\end{align}
which plays an important role in asymptotic analysis of the
Gegenbauer coefficients.
Gegebauer polynomials include some important polynomials such as
Legendre and Chebyshev polynomials as special cases. More
specifically, we have
\begin{align}\label{eq:GegChebU}
P_n(x) = C_n^{1/2}(x), \quad U_n(x) = C_n^{1}(x), \quad n\geq 0,
\end{align}
where $P_n(x)$ is the Legendre polynomial of degree $n$ and $U_n(x)$
is the Chebyshev polynomial of the second kind of degree $n$. When
$\lambda = 0$, the Gegenbauer polynomials reduce to the Chebyshev
polynomials of the first kind by the following definition
\begin{align}\label{eq:GegChebT}
\lim_{\lambda\rightarrow0^{+}} \lambda^{-1} C_{n}^{\lambda}(x) =
\frac{2}{n} T_n(x), \quad n \geq 1,
\end{align}
where $T_n(x)$ is the Chebyshev polynomial of the first kind of
degree $n$.
\section{Experimental observations}\label{sec:experiment}
In this section we present some experimental observations on the
convergence behavior of Gegenbauer projections. First, from the
orthogonality \eqref{eq:GegenOrthog} we have
\begin{align}\label{eq:GegenSeries}
S_n^{\lambda}(f) &= \sum_{k=0}^{n} a_k^{\lambda} C_k^{\lambda}(x),
\quad a_k^{\lambda} = \frac{1}{h_k^{\lambda} } \int_{-1}^{1}
\omega_{\lambda}(x) C_k^{\lambda}(x) f(x) \mathrm{d}x.
\end{align}
In order to quantify the difference between the rates of convergence
of $\mathcal{B}_n(f)$ and $S_n^{\lambda}(f)$, we introduce the
quantity
\begin{align}\label{eq:Indicator}
\mathcal{R}^{\lambda}(n) = \frac{\|f - S_n^{\lambda}(f) \|_{\infty}
}{\|f - \mathcal{B}_n(f) \|_{\infty}}.
\end{align}
Throughout the rest of the paper, we may use $S_n^{\lambda}(f,x)$
instead of $S_n^{\lambda}(f)$ when computing $S_n^{\lambda}(f)$ at
the point $x$. In addition, we compute $\mathcal{B}_n(f)$ using the
barycentric-Remez algorithm in \cite{pachon2009bary} and its
implementation is available in Chebfun with the \texttt{minimax}
command (see \cite{driscoll2014chebfun}).
\subsection{Analytic functions}
We consider the following three test functions
\begin{align}\label{eq:TestFun1}
f_1(x) = e^{2x^3}, \quad f_2(x) = \ln(1.2+x), \quad f_3(x) =
\frac{1}{1+9x^2},
\end{align}
It is clear that the first function is analytic in the whole complex
plane and the last two functions are analytic in a neighborhood of
the interval $[-1,1]$. We divide the choice of the parameter
$\lambda$ into two ranges: $\lambda\in(-1/2,0]$ and $\lambda>0$.
\begin{figure}[ht]
\centering
\includegraphics[trim=36mm 0mm 0mm 0mm,width=16cm,height=9cm]{AnalExam1}
\caption{Top row shows the log plot of the maximum errors of
$\mathcal{B}_n(f)$ ($\bullet$) and $S_n^{\lambda}(f)$ with
$\lambda=-2/5$ ($\circ$) and $\lambda=-1/10$ ($\Box$), for $f_1$
(left), $f_2$ (middle) and $f_3$ (right). Bottom row shows the plot
of the corresponding $\mathcal{R}^{\lambda}(n)$ for $\lambda=-2/5$
($\triangleleft$)
and $\lambda=-1/10$ ($\triangleright$).
} \label{fig:ExamI}
\end{figure}
Figure \ref{fig:ExamI} illustrates the maximum errors of
$\mathcal{B}_n(f)$ and $S_n^{\lambda}(f)$ for $\lambda=-2/5$ and
$\lambda=-1/10$ and the quantity $\mathcal{R}^{\lambda}(n)$ as a
function of $n$. From the top row of Figure \ref{fig:ExamI}, we see
that the accuracy of $\mathcal{B}_n(f)$ is indistinguishable with
that of $S_n^{\lambda}(f)$. From the bottom row of Figure
\ref{fig:ExamI}, we see that these two $\mathcal{R}^{\lambda}(n)$
tend, respectively, to some finite constants as $n$ grows, and thus
the rate of convergence of $S_n^{\lambda}(f)$ is the same as that of
$\mathcal{B}_n(f)$. Figure \ref{fig:ExamII} illustrates the maximum
errors of $\mathcal{B}_n(f)$ and $S_n^{\lambda}(f)$ for $\lambda=1$
and $\lambda=2$ and $n^{-\lambda}\mathcal{R}^{\lambda}(n)$ as a
function of $n$. From the top row of Figure \ref{fig:ExamII}, we see
clearly that the rate of convergence of $\mathcal{B}_n(f)$ is faster
than that of $S_n^{\lambda}(f)$. From the bottom row of Figure
\ref{fig:ExamII}, we see that these two
$n^{-\lambda}\mathcal{R}^{\lambda}(n)$ tend, respectively, to some
finite constants as $n$ grows, which implies that the convergence
rate of $S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$
by a factor of $n^{\lambda}$.
\begin{figure}[ht]
\centering
\includegraphics[trim = 36mm 0mm 0mm 0mm,width=16cm,height=9cm]{AnalExam2}
\caption{Top row shows the log plot of the maximum errors of
$\mathcal{B}_n(f)$ ($\bullet$) and $S_n^{\lambda}(f)$ with
$\lambda=1$ ($\circ$) and $\lambda=2$ ($\Box$), for $f_1$ (left),
$f_2$ (middle) and $f_3$ (right). Bottom row shows the log plot of
the corresponding $n^{-\lambda}\mathcal{R}^{\lambda}(n)$ for
$\lambda=1$ ($\triangleleft$) and $\lambda=2$ ($\triangleright$). }
\label{fig:ExamII}
\end{figure}
In summary, the above observations suggest the following
conclusions:
\begin{itemize}
\item For $\lambda\in(-1/2,0]$, the rate of convergence of
$S_n^{\lambda}(f)$ is the same as that of $\mathcal{B}_n(f)$;
\item For $\lambda>0$, however, the rate of convergence of
$S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$ by a
factor of $n^{\lambda}$.
\end{itemize}
\subsection{Differentiable functions}
We consider the following test functions
\begin{align}\label{eq:TestFun2}
f_4(x) = (x)_{+}^4, \quad f_5(x) = \left| \sin(4x) \right|^5, \quad
f_6(x) = \left\{
\begin{array}{ll}
{\displaystyle 2\cos(x) }, & \hbox{$x\in[-1,0)$,} \\[8pt]
{\displaystyle 2x^3 - x^2 + 2 }, & \hbox{$x\in[0,1]$,}
\end{array}
\right.
\end{align}
where $(x)_{+}^k$ is the {\it truncated power function} defined by
\begin{align}
(x)_{+}^k = \left\{
\begin{array}{ll}
{\displaystyle x^k }, & \hbox{$x\geq0$,} \\[8pt]
{\displaystyle 0 }, & \hbox{$x<0$,}
\end{array}
\right. ~~ k\geq1,~~ \mbox{and} ~~
(x)_{+}^0 = \left\{
\begin{array}{ll}
{\displaystyle 1 }, & \hbox{$x\geq0$,} \\[8pt]
{\displaystyle 0 }, & \hbox{$x<0$.}
\end{array}
\right.
\end{align}
It is clear that these three functions are all piecewise analytic
functions, whose definition will be given in section
\ref{sec:Piecewise}. In our numerical tests, we divide the choice of
the parameter $\lambda$ into ranges: $\lambda \in (-1/2,1]$ and
$\lambda>1$.
\begin{figure}[ht]
\centering
\includegraphics[trim = 36mm 0mm 0mm 0mm,width=16cm,height=9cm]{DiffExam1}
\caption{Top row shows the log-log plot of the maximum errors of
$\mathcal{B}_n(f)$ ($\bullet$), $S_n^{\lambda}(f)$ with
$\lambda=-1/5$ ($\circ$) and $\lambda=9/10$ ($\Box$), for $f_4$
(left), $f_5$ (middle) and $f_6$ (right). Bottom row shows the plot
of the corresponding $\mathcal{R}^{\lambda}(n)$ for $\lambda=-1/5$
($\triangleleft$) and $\lambda=9/10$ ($\triangleright$). }
\label{fig:ExamIII}
\end{figure}
Figure \ref{fig:ExamIII} illustrates the maximum errors of
$\mathcal{B}_n(f)$ and $S_n^{\lambda}(f)$ for $\lambda=-1/5$ and
$\lambda=9/10$ and the quantity $\mathcal{R}^{\lambda}(n)$ as a
function of $n$. From the top row of Figure \ref{fig:ExamIII}, we
see that the accuracy of $S_n^{\lambda}(f)$ is slightly worse than
that of $\mathcal{B}_n(f)$. From the bottom row of Figure
\ref{fig:ExamIII}, we see that these two $\mathcal{R}^{\lambda}(n)$
tend to or oscillate around some finite constants as $n$ grows,
which implies that the rate of convergence of $S_n^{\lambda}(f)$ is
the same as that of $\mathcal{B}_n(f)$. Figure \ref{fig:ExamIV}
illustrates the maximum errors of $\mathcal{B}_n(f)$ and
$S_n^{\lambda}(f)$ for $\lambda=3/2$ and $\lambda=3$ and
$n^{1-\lambda}\mathcal{R}^{\lambda}(n)$ as a function of $n$. From
the top row of Figure \ref{fig:ExamIV}, we see that the rate of
convergence of $S_n^{\lambda}(f)$ is significantly slower than that
of $\mathcal{B}_n(f)$. From the bottom row of Figure
\ref{fig:ExamIV}, we see that these two
$n^{1-\lambda}\mathcal{R}^{\lambda}(n)$ tend to or oscillate around
some finite constants as $n$ grows, which implies that the rate of
convergence of $S_n^{\lambda}(f)$ is slower than that of
$\mathcal{B}_n(f)$ by a factor of $n^{\lambda-1}$.
\begin{figure}[ht]
\centering
\includegraphics[trim = 36mm 0mm 0mm 0mm,width=16cm,height=9cm]{DiffExam2}
\caption{Top row shows the log-log plot of the maximum errors of
$\mathcal{B}_n(f)$ ($\bullet$), $S_n^{\lambda}(f)$ with
$\lambda=3/2$ ($\circ$) and $\lambda=3$ ($\Box$), for $f_4$ (left),
$f_5$ (middle) and $f_6$ (right). Bottom row shows the log plot of
the corresponding $n^{1-\lambda}\mathcal{R}^{\lambda}(n)$ for
$\lambda=3/2$ ($\triangleleft$) and $\lambda=3$ ($\triangleright$).}
\label{fig:ExamIV}
\end{figure}
In summary, the above observations suggest the following
conclusions:
\begin{itemize}
\item For $\lambda\in(-1/2,1]$, the rate of convergence of
$S_n^{\lambda}(f)$ is the same as that of $\mathcal{B}_n(f)$;
\item For $\lambda>1$, however, the rate of convergence of
$S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$ by a
factor of $n^{\lambda-1}$, which is one power of $n$ smaller than
the predicted result using \eqref{eq:errorG} and
\eqref{eq:Lebesgue}.
\end{itemize}
We remark that the convergence results of the particular case
$\lambda=0$ (that corresponds to Chebyshev projections) has been
included in the above two observations. We refer to
\cite{liu2019optimal,trefethen2013atap} for more details on the
convergence rate analysis of Chebyshev projections and to
\cite{wang2020legendre} for a comparison of Chebyshev projections
with $\mathcal{B}_n(f)$. Hereafter, we will omit discussion of this
case.
\section{Optimal rate of convergence of Gegenbauer projections for analytic functions}
\label{sec:Analytic} In this section we study the optimal rate of
convergence of Gegenbauer projections for analytic functions. Let
$\mathcal{E}_{\rho}$ denote the Bernstein ellipse
\begin{equation}\label{def:Bern}
\mathcal{E}_{\rho} = \left\{ z \in \mathbb{C} ~\bigg|~ z = \frac{u +
u^{-1}}{2},~~ |u| = \rho\geq1 \right\},
\end{equation}
and it has foci at $\pm 1$ and the major and minor semi-axes are
given by $(\rho+\rho^{-1})/2$ and $(\rho-\rho^{-1})/2$,
respectively.
The starting point of our analysis is the contour integral
expression of the Gegenbauer coefficients.
\begin{lemma}\label{lem:Contour}
Suppose that $f$ is analytic in the region bounded by the ellipse
$\mathcal{E}_{\rho}$ for some $\rho>1$, then for each $k\geq0$ and
$\lambda>-1/2$ and $\lambda\neq0$,
\begin{align}\label{eq:Contour}
a_k^{\lambda} & = \frac{ c_{k,\lambda} }{i \pi }
\oint_{\mathcal{E}_{\rho}} \frac{f(z)}{ (z \pm \sqrt{z^2 - 1})^{k+1}
} {}_2\mathrm{ F}_1\left[\begin{matrix} k + 1, & 1 - \lambda;
\\ k + \lambda + 1; \hspace{-1cm} &\end{matrix} \frac{1}{( z \pm \sqrt{z^2 - 1}
)^{2}} \right] \mathrm{d}z,
\end{align}
where the sign in $z \pm \sqrt{z^2 - 1}$ is chosen so that
$|z\pm\sqrt{z^2 - 1}|>1$ and the constant $c_{k,\lambda}$ is defined
by
\begin{align}\label{eq:c}
c_{k,\lambda} = \frac{\Gamma(\lambda)
\Gamma(k+1)}{\Gamma(k+\lambda)},
\end{align}
\end{lemma}
\begin{proof}
With a different normalization condition on $\{C_k^{\lambda}(x)\}$,
\eqref{eq:Contour} was first derived by Cantero and Iserles in
\cite{cantero2012rapid} for developing some fast algorithms.
The idea is to express $a_k^{\lambda}$ as a linear combination of
$\{f^{(j)}(0)\}$ and then as an integral transform with a Gauss
hypergeometric function as its kernel. Due to the slow convergence
of the Taylor series of the kernel, a hypergeometric transformation
was used to replace the original kernel with a new one that
converges much more rapidly, which gives \eqref{eq:Contour}. An
alternative and simpler approach for the derivation of
\eqref{eq:Contour} was proposed in \cite{wang2016gegenbauer} and the
idea is to rearrange the Chebyshev coefficients of the second kind.
We refer the interested readers to
\cite{cantero2012rapid,wang2016gegenbauer} for more details.
\end{proof}
We now state some new bounds on the Gegenauer coefficients
$\{a_k^{\lambda}\}$ for all $\lambda>-1/2$ and $\lambda\neq0$.
Compared to \cite[Thm.~4.3]{wang2016gegenbauer}, our new bounds are
more concise for $\lambda>0$ and are new for $-1/2<\lambda<0$.
\begin{theorem}\label{thm:GegBound}
Under the assumptions of Lemma \ref{lem:Contour}. Then, for
$\lambda\neq0$,
\begin{align}\label{eq:GenBound}
|a_0^{\lambda}| \leq D_{\lambda}(\rho) \left\{
\begin{array}{ll}
{\displaystyle \frac{1}{|\Gamma(\lambda)|} }, & \hbox{$-1/2<\lambda<0$,} \\[14pt]
{\displaystyle \lambda }, & \hbox{$0<\lambda\leq 1$,} \\[8pt]
{\displaystyle \frac{1}{\Gamma(\lambda)} }, & \hbox{$\lambda>1$,}
\end{array}
\right. \quad |a_k^{\lambda}| &\leq D_{\lambda}(\rho)
\frac{k^{1-\lambda}}{\rho^{k}}, \quad k\geq1,
\end{align}
where $D_{\lambda}(\rho)$ is defined by
\begin{align}\label{eq:D}
D_{\lambda}(\rho) &= \frac{\displaystyle M
L(\mathcal{E}_{\rho})}{\pi\rho} \left\{
\begin{array}{ll}
{\displaystyle \frac{|\Gamma(\lambda)|
\Gamma(1+\lambda)\Gamma(1-2\lambda)}{\Gamma(1-\lambda)}
\left(1 - \frac{1}{\rho^2} \right)^{2\lambda-1}}, & \hbox{$-1/2<\lambda<0$,} \\[15pt]
{\displaystyle \frac{1}{\lambda} \left(1 - \frac{1}{\rho^2} \right)^{\lambda-1}}, & \hbox{$0<\lambda\leq1$,} \\[15pt]
{\displaystyle \Gamma(\lambda) \left(1 + \frac{1}{\rho^2}
\right)^{\lambda-1}}, & \hbox{$\lambda>1$,}
\end{array}
\right.
\end{align}
and $M=\max_{z\in \mathcal{E}_{\rho}}|f(z)|$ and
$L(\mathcal{E}_{\rho})$ is the length of the circumference of
$\mathcal{E}_{\rho}$.
\end{theorem}
\begin{proof}
We follow the idea of the proof in \cite{wang2016gegenbauer}. From
Lemma \ref{lem:Contour} and \cite[Theorem 4.1]{wang2016gegenbauer}
we have that
\begin{align}\label{eq:GenS1}
|a_k^{\lambda}| &\leq \frac{|c_{k,\lambda}| M
L(\mathcal{E}_{\rho})}{\pi \rho^{k+1}} \left\{
\begin{array}{ll}
{\displaystyle {}_2\mathrm{ F}_1\left[\begin{matrix} k + 1, &
1-\lambda ;
\\ k+\lambda+1 ; \hspace{-1cm} &\end{matrix} ~ \frac{1}{\rho^2}
\right]}, & \hbox{$-1/2<\lambda\leq 1$ and $\lambda\neq0$,} \\[15pt]
{\displaystyle {}_2\mathrm{ F}_1\left[\begin{matrix} k + 1, &
1-\lambda ;
\\ k+\lambda+1 ; \hspace{-1cm} &\end{matrix} ~ -\frac{1}{\rho^2}
\right]}, & \hbox{$\lambda>1$.}
\end{array}
\right.
\end{align}
The remaining task is to bound the constant $c_{k,\lambda}$ and
these hypergeometric functions on the right hand side of
\eqref{eq:GenS1}. For the former, it is easy to see that
$|c_{k,\lambda}|=1$ when $k=0$. For $k\geq1$, using Lemma
\ref{lem:gamma} we obtain
\begin{align}
|c_{k,\lambda}| \leq k^{1-\lambda} \left\{
\begin{array}{ll}
{\displaystyle |\Gamma(\lambda)| }, & \hbox{$-1/2 <\lambda<0$,} \\[8pt]
{\displaystyle \lambda^{-1} }, & \hbox{$0<\lambda \leq 1$,} \\[8pt]
{\displaystyle \Gamma(\lambda) }, & \hbox{$\lambda>1$.}
\end{array}
\right.
\end{align}
We now aim to bound these hypergeometric functions on the right hand
side of \eqref{eq:GenS1}. For $\lambda>0$ and $|z|<1$, using the
Euler integral representation of the Gauss hypergeometric function
\cite[Equation~(15.6.1)]{olver2010nist}, we obtain
\begin{align}\label{eq:GenS2}
\left| {}_2\mathrm{ F}_1\left[\begin{matrix} k + 1, & 1-\lambda ;
\\ k+\lambda+1 ; \hspace{-1cm} &\end{matrix} ~ z
\right] \right| &=
\frac{\Gamma(k+\lambda+1)}{\Gamma(k+1)\Gamma(\lambda)}
\left| \int_{0}^{1} t^{k} (1-t)^{\lambda-1} \left(1 - zt\right)^{\lambda-1} \mathrm{d}t \right| \nonumber \\[8pt]
&\leq \left\{
\begin{array}{ll}
{\displaystyle \left(1 - |z| \right)^{\lambda-1}}, & \hbox{$0<\lambda\leq 1$,} \\[10pt]
{\displaystyle \left(1 + |z| \right)^{\lambda-1}}, &
\hbox{$\lambda>1$.}
\end{array}
\right.
\end{align}
When $-1/2<\lambda<0$, it is easy to verify that
\begin{align}
\frac{(k+1)_j}{(k+\lambda+1)_j} \leq \frac{(1)_j}{(\lambda+1)_j},
\quad \frac{(1-\lambda)_j}{(1+\lambda)_j} \leq
\frac{\Gamma(1+\lambda)\Gamma(1-2\lambda)}{\Gamma(1-\lambda)}
\frac{(1-2\lambda)_j}{(1)_j}. \nonumber
\end{align}
It follows that
\begin{align}\label{eq:GenS3}
\left| {}_2\mathrm{ F}_1\left[\begin{matrix} k + 1, & 1-\lambda ;
\\ k+\lambda+1 ; \hspace{-1cm} &\end{matrix} ~ z
\right] \right| &\leq \sum_{j=0}^{\infty} \frac{(k+1)_j
(1-\lambda)_j}{(k+\lambda+1)_j} \frac{|z|^{j}}{j!} \leq
\sum_{j=0}^{\infty} \frac{(1)_j (1-\lambda)_j}{(\lambda+1)_j}
\frac{|z|^{j}}{j!} \nonumber \\
&\leq \frac{\Gamma(1+\lambda)\Gamma(1-2\lambda)}{\Gamma(1-\lambda)}
\sum_{j=0}^{\infty} \frac{(1-2\lambda)_j}{j!} |z|^{j} \nonumber \\
&= \frac{\Gamma(1+\lambda)\Gamma(1-2\lambda)}{\Gamma(1-\lambda)} (1
- |z|)^{2\lambda-1}.
\end{align}
Combining \eqref{eq:GenS1}, \eqref{eq:GenS2} and \eqref{eq:GenS3},
the desired bounds follow immediately.
\end{proof}
With the above preparation, we now derive error estimates of
Gegenbauer projections in the maximum norm. Compared to the results
in \cite[Thm.~4.8]{wang2016gegenbauer}, our new results are more
concise and informative. Throughout the paper, $\lfloor x \rfloor$
denotes the integer part of $x$.
\begin{theorem}\label{thm:RateAnal}
Suppose that $f$ is analytic in the region bounded by the ellipse
$\mathcal{E}_{\rho}$ for some $\rho>1$.
\begin{enumerate}
\item[(i).] If $\lambda>0$, then for $n\geq \lfloor \lambda/\ln\rho \rfloor$ we have
\begin{align}\label{eq:AnalBound1}
\|f - S_n^{\lambda}(f) \|_{\infty} \leq \mathcal{K}
\frac{n^{\lambda}}{\rho^n},
\end{align}
where the quantity $\mathcal{K}$ is defined by
\begin{align}
\mathcal{K}= c \frac{D_{\lambda}(\rho)}{\ln\rho} \left\{
\begin{array}{ll}
{\displaystyle \frac{1}{\Gamma(2\lambda)} }, & \hbox{$0<\lambda\leq 1/2$,} \\[10pt]
{\displaystyle 2\lambda}, & \hbox{$\lambda>1/2$,}
\end{array}
\right. \nonumber
\end{align}
and $c\cong1$ is a generic positive constant.
\item[(ii).] If $-1/2<\lambda<0$, we have
\begin{align}\label{eq:AnalBound2}
\|f - S_n^{\lambda}(f) \|_{\infty} \leq \frac{D_{\lambda}(\rho)
C_{\lambda}}{(\rho-1) \rho^n},
\end{align}
where $C_{\lambda}$ is the positive constant defined in
\eqref{eq:Bound2}.
\end{enumerate}
Up to constant factors, these bounds on the right hand side are
optimal in the sense that they can not be improved in any negative
powers of $n$ further.
\end{theorem}
\begin{proof}
For (i), by Lemma \ref{lem:gamma} we have
\begin{align}
|C_k^{\lambda}(x)| \leq \frac{\Gamma(k+2\lambda)}{\Gamma(k+1)
\Gamma(2\lambda)} \leq \frac{k^{2\lambda-1}}{\Gamma(2\lambda)}
\left\{
\begin{array}{ll}
{\displaystyle 1}, & \hbox{$0<\lambda\leq 1/2$,} \\[10pt]
{\displaystyle \Gamma(1+2\lambda)}, & \hbox{$\lambda>1/2$.}
\end{array}
\right. \nonumber
\end{align}
Combining these bounds together with Theorem \ref{thm:GegBound}
gives
\begin{align}
\| f - S_n^{\lambda}(f) \|_{\infty} &\leq \sum_{k=n+1}^{\infty}
|a_k^{\lambda}| |C_k^{\lambda}(x)| \nonumber \\
&\leq D_{\lambda}(\rho) \left( \sum_{k=n+1}^{\infty}
\frac{k^{\lambda}}{\rho^{k}} \right) \left\{
\begin{array}{ll}
{\displaystyle \frac{1}{\Gamma(2\lambda)}}, & \hbox{$0<\lambda\leq 1/2$,} \\[10pt]
{\displaystyle 2\lambda}, & \hbox{$\lambda>1/2$.}
\end{array}
\right. \nonumber
\end{align}
For the sum inside the bracket, note that $k^{\lambda}/\rho^k$ is
strictly decreasing with respect to $k$ for $k\geq\lambda/\ln\rho$,
we obtain that
\begin{align}\label{eq:BoundS2}
\sum_{k=n+1}^{\infty} \frac{k^{\lambda}}{\rho^{k}} &\leq
\int_{n}^{\infty} \frac{x^{\lambda}}{\rho^{x}} \mathrm{d}x =
\frac{\Gamma(\lambda+1,n\ln\rho)}{(\ln\rho)^{1+\lambda}},
\end{align}
where $\Gamma(a,x)$ is the incomplete gamma function (see, e.g.,
\cite[p.174]{olver2010nist}). Finally, we note that
$|\Gamma(a,x)|\leq c x^{a-1} e^{-x}$, where the constant $c\cong1$,
the desired result \eqref{eq:AnalBound1} follows. The proof of (ii)
is similar and we omit the details.
We now turn to prove the optimality of \eqref{eq:AnalBound1} and
\eqref{eq:AnalBound2}. Here we only prove the former since the
latter can be proved by a similar argument. Suppose by contradiction
that there exist constants $\gamma,c>0$ independent of $n$ such that
\begin{align}\label{eq:LegProjBoundS3}
\| f - S_n^{\lambda}(f) \|_{\infty} \leq c
\frac{n^{\lambda-\gamma}}{\rho^n}.
\end{align}
We consider the function $f(x)=(x-\omega)^{-1}$ with
$\omega+\sqrt{\omega^2-1}>1+\lambda^{-1}$. It is easily seen that
this function has a simple pole at $x=\omega$ and therefore
$\rho\leq\omega+\sqrt{\omega^2-1}-\epsilon$, where $\epsilon>0$ may
be taken arbitrary small. Using Lemma \ref{lem:Contour} and the
residue theorem, we can write the Gegenbauer coefficients of $f(x)$
as
\begin{align}\label{eq:PoleCoeff}
a_k^{\lambda} & = \frac{(-2
c_{k,\lambda})}{(\omega+\sqrt{\omega^2-1})^{k+1}} {}_2\mathrm{
F}_1\left[\begin{matrix} k + 1, & 1-\lambda ;
\\ k + \lambda + 1 ; \hspace{-1cm} &\end{matrix} ~ \frac{1}{(\omega+\sqrt{\omega^2-1})^{2}}
\right].
\end{align}
Clearly, we see that $a_k^{\lambda}<0$ for all $k\geq0$. Moreover,
by considering the ratio $a_{k+1}^{\lambda}/a_k^{\lambda}$, it is
not difficult to verify that the sequence
$\{a_{k}^{\lambda}\}_{k=0}^{\infty}$ is strictly increasing. We now
consider the error of $S_n^{\lambda}(f)$ at the point $x=1$. In view
of \eqref{eq:Bound1}, we obtain that
\begin{align}
| f(1) - S_n^{\lambda}(f,1) | &= -\sum_{k=n+1}^{\infty}
a_{k}^{\lambda} C_{k}^{\lambda}(1) \geq -a_{n+1}^{\lambda}
C_{n+1}^{\lambda}(1). \nonumber
\end{align}
Combining this with \eqref{eq:LegProjBoundS3} we deduce that
\begin{align}
-a_{n+1}^{\lambda} C_{n+1}^{\lambda}(1) \leq \|f(x) -
S_n^{\lambda}(f) \|_{\infty} \leq c
\frac{n^{\lambda-\gamma}}{\rho^n}.
\end{align}
By using \eqref{eq:GenS2}, \eqref{eq:AsyGAMMA} and
\eqref{eq:PoleCoeff}, we obtain that $|a_{n+1}^{\lambda}
C_{n+1}^{\lambda}(1)|=O(n^{\lambda}
(\omega+\sqrt{\omega^2-1})^{-n})$. On the other hand, we know that
$n^{\lambda-\gamma}\rho^{-n}=O(n^{\lambda-\gamma}
(\omega+\sqrt{\omega^2-1}-\epsilon)^{-n})$. This leads to a
contradiction since the upper bound may be smaller than the lower
bound when $\epsilon$ is sufficiently small. Therefore, we can
conclude that the derived bound \eqref{eq:AnalBound1} is optimal and
can not be improved in any negative powers of $n$. This completes
the proof.
\end{proof}
\begin{remark}
From \cite[p.~131]{cheney1998approximation} we know that $\|f -
\mathcal{B}_n(f) \|_{\infty}=O(\rho^{-n})$. Comparing this with
\eqref{eq:AnalBound1} and \eqref{eq:AnalBound2}, it is easily seen
that the rate of convergence of $S_n^{\lambda}(f)$ is slower than
that of $\mathcal{B}_n(f)$ by a factor of $n^{\lambda}$ for
$\lambda>0$ and is the same as that of $S_n^{\lambda}(f)$ for $-1/2<
\lambda<0$. This explains the convergence behavior of
$S_n^{\lambda}(f)$ illustrated in Figures \ref{fig:ExamI} and
\ref{fig:ExamII}.
\end{remark}
\begin{remark}
Polynomial interpolation in the zeros of Gegenbauer polynomials is
also a powerful approach for analytic functions. When the
interpolation nodes are the zeros of $C_{n+1}^{\lambda}(x)$, it has
been shown in \cite[Thm.~4.1]{xie2013gegenbauer} that the rate of
convergence of Gegenbauer interpolation in the maximum norm is
$O(n^{\lambda} \rho^{-n})$ for $\lambda>0$ and is $O(\rho^{-n})$ if
$-1/2<\lambda<0$. Comparing this with Theorem \ref{thm:RateAnal}, we
see that Gegenbauer interpolation and projection of the same degree
enjoy the same convergence rate.
\end{remark}
\section{Optimal rate of convergence of Gegenbauer projections for
piecewise analytic functions}\label{sec:Piecewise} In this section
we study optimal rate of convergence of Gegenbauer projections for
piecewise analytic functions. Throughout this section, we denote by
$K$ a generic positive constant independent of $n$ which may take
different values at different places.
We first introduce the definition of piecewise analytic functions.
\begin{definition}\label{def:PiecewiseAnal}
Suppose that $f\in C^{k}[-1,1]$ where $k\geq0$. Let
$\ell\in\mathbb{N}$ and and assume that $\{\xi_j\}_{j=1}^{\ell}\in
(-1,1)$ is a set of distinct points such that $f$ is analytic on
each of the subinterval $[-1,\xi_1),~(\xi_1,\xi_2),\ldots,
(\xi_{\ell-1},\xi_{\ell}),~ (\xi_{\ell},1]$, but $f$ itself is not
analytic at these points $\xi_1,\ldots,\xi_{\ell}$. We then say that
$f$ a piecewise analytic function on $[-1,1]$.
\end{definition}
We now consider the convergence rate analysis of Gegenbauer
projections. First of all, using the integral expression of
Gegenbauer coefficients, we can rewrite the Gegenbauer projection as
\begin{align}
S_n^{\lambda}(f) & = \int_{-1}^{1} \omega_{\lambda}(t) f(t)
D_n^{\lambda}(x,t) \mathrm{d}t,
\end{align}
where $D_n^{\lambda}(\cdot,\cdot)$ is the Dirichlet kernel of
Gegenbauer projection defined by
\begin{align}\label{eq:Kn}
D_n^{\lambda}(x,t) &= \sum_{k=0}^{n} \frac{C_k^{\lambda}(x)
C_k^{\lambda}(t)}{h_k^{\lambda}} \nonumber
\\
&= \frac{\Gamma(\lambda)^2}{2^{2-2\lambda} \pi}
\frac{\Gamma(n+2)}{\Gamma(n+2\lambda)}
\frac{C_{n+1}^{\lambda}(x)C_{n}^{\lambda}(t) -
C_{n+1}^{\lambda}(t)C_{n}^{\lambda}(x) }{x-t},
\end{align}
and the last equation follows from the Christoffel-Darboux formula
of Gegenbauer polynomials.
The following refined estimates for the Dirichlet kernel will be
useful.
\begin{lemma}\label{lem:Kn}
Let $|x|\leq1$. Then, for $\lambda\neq0$ and large $n$,
\begin{enumerate}
\item[(i)] If $|t|\leq1$, it holds that $|D_n^{\lambda}(x,t)| \leq K
n^{2 \max\{\lambda,0\} +1}$.
\item[(ii)] If $|t|\leq 1-\varepsilon$ with $\varepsilon\in(0,1)$,
it holds that $|D_n^{\lambda}(x,t)| \leq K
n^{\max\left\{\lambda,1\right\}}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first consider (i). From \eqref{eq:AsyGAMMA}, \eqref{eq:Bound1}
and \eqref{eq:Bound2} we have
\begin{align}\label{eq:AsyMaxGen}
\max_{x\in[-1,1]}|C_n^{\lambda}(x)| = \left\{
\begin{array}{ll}
{\displaystyle O(n^{2\lambda-1}) }, & \hbox{$\lambda>0$,} \\[8pt]
{\displaystyle O(n^{\lambda-1}) }, & \hbox{$-1/2<\lambda<0$,}
\end{array}
\right.
\end{align}
and similarly $h_n^{\lambda}=O(n^{2\lambda-2})$. Combining these
estimates we find that
\begin{align}
|D_n^{\lambda}(x,t)| \leq \sum_{k=0}^{n} \frac{|C_k^{\lambda}(x)
C_k^{\lambda}(t)|}{h_k^{\lambda}} = \sum_{k=0}^{n} O(k^{2
\max\{\lambda,0\}}) = O(n^{2 \max\{\lambda,0\}+1}). \nonumber
\end{align}
This proves (i). To prove (ii), we split our discussion into two
cases: $|x-t|<\varepsilon/2$ and $|x-t|\geq\varepsilon/2$. For the
case $|x-t|<\varepsilon/2$, it is easily verified that $|x|\leq
1-\varepsilon/2$. Recall that $|C_n^{\lambda}(x)| =
O(n^{\lambda-1})$ for $x\in(-1,1)$, we obtain
\begin{align}
\max_{\substack{|x|\leq 1-\varepsilon/2\\ |t|\leq 1 - \varepsilon}}
\frac{|C_k^{\lambda}(x) C_k^{\lambda}(t)|}{h_k^{\lambda}} = O(1),
\nonumber
\end{align}
and thus
\begin{align}
|D_n^{\lambda}(x,t)| \leq \sum_{k=0}^{n} \frac{|C_k^{\lambda}(x)
C_k^{\lambda}(t)|}{h_k^{\lambda}} = \sum_{k=0}^{n} O(1) = O(n).
\nonumber
\end{align}
Next, we consider the case $|x-t|\geq \varepsilon/2$. Combining the
estimate $\max_{|t|\leq1-\varepsilon} |C_n^{\lambda}(t)| =
O(n^{\lambda-1})$ with \eqref{eq:AsyMaxGen}, and the last equality
in \eqref{eq:Kn}, we immediately infer that
\begin{align}
|D_n^{\lambda}(x,t)| = O(n^{\max\{\lambda,0\}}). \nonumber
\end{align}
A combination of the above two estimates gives (ii). This completes
the proof.
\end{proof}
Now, we prove the main result of this section.
\begin{theorem}\label{thm:PieceRate}
Let $m\in\mathbb{N}$ and let $f\in C^{m-1}[-1,1]$ be a piecewise
analytic function on $[-1,1]$. Then, for $\lambda<m+1$ and large
$n$, we have
\begin{align}\label{eq:DiffRate}
\|f - S_n^{\lambda}(f)\|_{\infty} \leq K \left\{
\begin{array}{ll}
{\displaystyle n^{-m} }, & \hbox{if $\lambda\leq1$,} \\[8pt]
{\displaystyle n^{-m-1+\lambda} }, & \hbox{if $\lambda>1$.}
\end{array}
\right.
\end{align}
Moreover, the convergence rates on the right-hand side in
\eqref{eq:DiffRate} are optimal in the sense that they can not be
improved further.
\end{theorem}
\begin{proof}
We assume that $\{\xi_j\}_{j=1}^{\ell}\in (-1,1)$, where
$\ell\geq1$, are the points of singularity of $f$. According to
\cite[Theorem~3]{saff1989poly}, there exists a polynomial $q_n$ of
degree $n$ such that for all $x\in[-1,1]$
\begin{align}\label{eq:SaffBound}
|f(x) - q_n(x) | \leq \frac{C}{n^{m}} e^{-c n^{\zeta} d(x)^{\eta} },
\end{align}
where $\zeta\in(0,1)$ and $\eta\geq\zeta$ or $\zeta=1$ and $\eta>1$,
$d(x)=\min_{1\leq k \leq \ell}|x-\xi_k|$ and $C,c$ are some positive
constants. Taking $\zeta=\eta\in(0,1)$ in \eqref{eq:SaffBound} and
using the fact that $S_n^{\lambda}(f) \equiv f$ whenever $f$ is a
polynomial of degree up to $n$, we immediately obtain
\begin{align}\label{eq:ModFunS1}
|f - S_n^{\lambda}(f)| &= |f - q_n + S_n^{\lambda}(q_n-f) | \leq |f
- q_n| + |S_n^{\lambda}(f-q_n)|
\nonumber \\
&\leq \frac{C}{n^{m}} e^{-c (n d(x))^\eta} + \frac{C}{n^{m}}
\int_{-1}^{1} e^{-c (nd(t))^{\eta}} \omega_{\lambda}(t) |
D_n^{\lambda}(x,t) | \mathrm{d}t.
\end{align}
We now consider the asymptotic behavior of the last integral in
\eqref{eq:ModFunS1}. For simplicity of notation we denote it by $I$.
Moreover, let
$I_1=[\xi_1-\epsilon,\xi_1+\epsilon],\ldots,I_{\ell}=[\xi_{\ell}-\epsilon,\xi_{\ell}+\epsilon]$,
where $\epsilon>0$ is chosen to be small enough such that these
subintervals $I_1,\ldots,I_{\ell}$ are pairwise disjoint and are
contained in the interior of $[-1,1]$, i.e.,
$I_1,\ldots,I_{\ell}\subset[-1,1]$. Therefore,
\begin{align}\label{eq:ModFunS2}
I &= \sum_{k=1}^{\ell} \int_{I_k} e^{-c (nd(t))^{\eta}}
\omega_{\lambda}(t) | D_n^{\lambda}(x,t) | \mathrm{d}t
\nonumber \\
&~~~~~~~~ + \int_{[-1,1]\backslash\bigcup_{k=1}^{\ell} I_k} e^{-c
(nd(t))^{\eta}} \omega_{\lambda}(t) | D_n^{\lambda}(x,t) |
\mathrm{d}t.
\end{align}
For the sum in \eqref{eq:ModFunS2}, notice that $d(t)=|t-\xi_k|$
when $t\in I_{k}$, and thus we get
\begin{align}
\sum_{k=1}^{\ell} \int_{I_k} e^{-c(nd(t))^{\eta}}
\omega_{\lambda}(t) | D_n^{\lambda}(x,t) | \mathrm{d}t &=
\sum_{k=1}^{\ell} \int_{\xi_k-\epsilon}^{\xi_k+\epsilon} e^{-c
(n|t-\xi_k|)^{\eta}} \omega_{\lambda}(t) |
D_n^{\lambda}(x,t) | \mathrm{d}t \nonumber \\
&= \sum_{k=1}^{\ell} \int_{-\epsilon}^{\epsilon} e^{-c
(n|y|)^{\eta}} \omega_{\lambda}(y+\xi_k) | D_n^{\lambda}(x,y+\xi_k)
| \mathrm{d}y, \nonumber
\end{align}
where we have applied the change of variable $t=y+\xi_k$ in the last
step. Furthermore, using Lemma \ref{lem:Kn} and a change of variable
$s=ny$, we obtain
\begin{align}\label{eq:ModFunS3}
\sum_{k=1}^{\ell} \int_{I_k} e^{-c (nd(t))^{\eta}}
\omega_{\lambda}(t) | D_n^{\lambda}(x,t) | \mathrm{d}t &\leq \ell
\max_{t\in\cup_{k=1}^{\ell}I_{k}} |\omega_{\lambda}(t)| K
n^{\max\{\lambda,1\}} \int_{-\epsilon}^{\epsilon}
e^{-c(n|y|)^{\eta}}
\mathrm{d}y \nonumber \\
&\leq \ell \max_{t\in\cup_{k=1}^{\ell}I_{k}} |\omega_{\lambda}(t)| K
n^{\max\{\lambda,1\} - 1} \int_{0}^{\infty} e^{-cs^{\eta}}
\mathrm{d}s
\nonumber \\
&= O(n^{\max\{\lambda,1\}-1}).
\end{align}
For the second term in \eqref{eq:ModFunS2}, notice that $d(t)\geq
\epsilon$ when $t\in[-1,1]\backslash\bigcup_{k=1}^{\ell} I_k$, we
obtain
\begin{align}\label{eq:ModFunS4}
\int_{[-1,1]\backslash\bigcup_{k=1}^{\ell} I_k} e^{-c
(nd(t))^{\eta}} \omega_{\lambda}(t) | D_n^{\lambda}(x,t) |
\mathrm{d}t &\leq e^{-c (n\epsilon)^{\eta}} \int_{-1}^{1}
\omega_{\lambda}(t) |
D_n^{\lambda}(x,t) | \mathrm{d}t \nonumber \\
&\leq K n^{2\max\{\lambda,0\}+1} e^{-c(n\epsilon)^{\eta}}
\int_{-1}^{1} \omega_{\lambda}(t) \mathrm{d}t \nonumber \\
&= O(n^{2\max\{\lambda,0\}+1} e^{-c(n\epsilon)^{\eta}}),
\end{align}
where we have used Lemma \ref{lem:Kn} in the second step. Combining
\eqref{eq:ModFunS1}, \eqref{eq:ModFunS3} and \eqref{eq:ModFunS4}
gives the desired result and thus completes the proof.
We now turn to prove the optimality of the convergence rates on the
right-hand side of \eqref{eq:DiffRate}. Recall that $\|f -
\mathcal{B}_n(f)\|_{\infty}=O(n^{-m})$ (see, e.g., \cite[Chapter
7]{timan1963approximation}). In the case $\lambda\leq1$, the rate of
convergence of $S_n^{\lambda}(f)$ is obviously optimal since it is
the same as that of $\mathcal{B}_n(f)$. In the case $\lambda>1$, the
predicted convergence rate is $\|f -
S_n^{\lambda}(f)\|_{\infty}=O(n^{-m-1+\lambda})$. To show the
optimality of this rate, we consider a specific example
$f(x)=(x)_{+}^5$, which corresponds to $m=5$. According to
\cite[Eqn.~(18.17.37)]{olver2010nist}, the Gegenbauer coefficients
of $f$ can be expressed as
\begin{align}
a_k^{\lambda} &= \frac{15}{8} \frac{\Gamma(\lambda) (k +
\lambda)}{\Gamma(\lambda + \frac{k+7}{2}) \Gamma(\frac{7-k}{2})},
\quad k\geq0.
\end{align}
From this, we can see that $a_{2k+1}^{\lambda}=0$ for $k\geq3$. For
$k\geq6$ is even and large, we have, using \eqref{eq:reflection} and
\eqref{eq:AsyGAMMA},
\begin{align}\label{eq:diffS1}
a_{k}^{\lambda} &= (-1)^{\frac{k}{2}+1}
\frac{15\Gamma(\lambda)}{8\pi} \frac{(k+\lambda)
\Gamma(\frac{k-5}{2})}{\Gamma(\frac{k+7}{2}+\lambda)} \nonumber \\
&= (-1)^{\frac{k}{2}+1} \frac{15\Gamma(\lambda)}{4\pi}
\left(\frac{k}{2} \right)^{-\lambda-5} + O(k^{-\lambda-6}).
\end{align}
In addition, employing \eqref{eq:Bound1} and \eqref{eq:AsyGAMMA}, we
deduce that $C_{k}^{\lambda}(1) = k^{2\lambda-1}/\Gamma(2\lambda)+
O(k^{2\lambda-2})$ for large $k$. Now we consider the error estimate
of $S_n^{\lambda}(f)$ at $x=1$. Suppose $n\geq6$ is even and large,
using \eqref{eq:diffS1} and the asymptotic estimate of
$C_{k}^{\lambda}(1)$, we obtain that
\begin{align}
f(1) - S_n^{\lambda}(f,1) &= \sum_{k=1}^{\infty}
a_{n+2k}^{\lambda} C_{n+2k}^{\lambda}(1) \nonumber \\
&\sim (-1)^{\frac{n}{2}+1} \frac{15\Gamma(\lambda)
2^{\lambda+3}}{\pi \Gamma(2\lambda)} n^{\lambda-6}
\sum_{k=1}^{\infty} (-1)^k \left(1 +
\frac{2k}{n} \right)^{\lambda-6} \nonumber \\
&= O(n^{\lambda-6}), \nonumber
\end{align}
where in the last step we have used the fact that the alternating
series is always bounded for $\lambda<6$. Similarly, it is not
difficult to show that $f(1) - S_n^{\lambda}(f,1)=O(n^{\lambda-6})$
if $n\geq6$ is odd and large. Finally, note that $\|f -
S_n^{\lambda}(f)\|_{\infty} \geq |f(1) - S_n^{\lambda}(f,1)|$, we
can conclude that the predicted rate $\|f -
S_n^{\lambda}(f)\|_{\infty} = O(n^{\lambda-6})$ is optimal. This
completes the proof.
\end{proof}
In order to verify the convergence rates predicted by Theorem
\ref{thm:PieceRate}, we consider the three test functions defined in
\eqref{eq:TestFun2}, which corresponds to $m=4,5,3$, respectively.
From Theorem \ref{thm:PieceRate} we know that the predicted rate of
$S_n^{\lambda}(f_4)$ is $O(n^{-4})$ if $\lambda\leq1$ and is
$O(n^{\lambda-5})$ if $\lambda>1$, and the predicted rate of
$S_n^{\lambda}(f_5)$ is $O(n^{-5})$ if $\lambda\leq1$ and is
$O(n^{\lambda-6})$ if $\lambda>1$, and the predicted rate of
$S_n^{\lambda}(f_6)$ is $O(n^{-3})$ if $\lambda\leq1$ and is
$O(n^{\lambda-4})$ if $\lambda>1$. For each $f_j$, where $j=4,5,6$,
we test the convergence rate of $S_n^{\lambda}(f_j)$ with four
values of $\lambda$. The results are plotted in Figure
\ref{fig:ExamV}. Clearly, we see that their convergence rates
coincide quite well with the predicted rates. Moreover, these
results also explain the observations in Figures \ref{fig:ExamIII}
and \ref{fig:ExamIV} since the convergence rates of
$\mathcal{B}_n(f)$ for $f_4,f_5$ and $f_6$ are $O(n^{-4}),O(n^{-5})$
and $O(n^{-3})$, respectively.
\begin{figure}[ht]
\centering
\includegraphics[width=.5\textwidth,height=6cm]{DiffRate1.eps}~
\includegraphics[width=.5\textwidth,height=6cm]{DiffRate2.eps}
\includegraphics[width=.5\textwidth,height=6cm]{DiffRate3.eps}
\caption{Top row shows convergence rates of $S_n^{\lambda}(f_4)$
(left) and $S_n^{\lambda}(f_5)$ (right) and bottom row shows
convergence rate of $S_n^{\lambda}(f_6)$. Dashed lines indicate the
convergence rates predicted by Theorem \ref{thm:PieceRate}. Here $n$
ranges from 2 to 250.} \label{fig:ExamV}
\end{figure}
Before closing this section, we make some further comments regarding
Theorem \ref{thm:PieceRate}:
\begin{itemize}
\item The convergence rate predicted by Theorem \ref{thm:PieceRate}
is optimal for functions with interior singularities of integer
order and this can be seen from Figure \ref{fig:ExamV}. For
functions with singularities of fractional order, however, the
predicted rate may be overestimated. In this case, a new analysis
should be carried out and we will consider it in the next section.
\item
Let $m\geq1$ be an integer and define the space
\begin{align}\label{def:FunSpace}
H_m = \left\{f~|~ f,f',\ldots,f^{(m-1)}\in \mathrm{AC}[-1,1], ~~
f^{(m)}\in \mathrm{BV}[-1,1] \right\},
\end{align}
where $\mathrm{AC}[-1,1]$ and $\mathrm{BV}[-1,1]$ denote the space
of absolutely continuous functions and the space of bounded
variation functions on $[-1,1]$, respectively. It is clear that the
test functions in \eqref{eq:TestFun2} also belong to this space and
$f_4\in H_4$, $f_5\in H_5$ and $f_6\in H_3$. This space is
preferable when developing error estimates for various orthogonal
polynomial approximations to differentiable function (see, e.g.,
\cite{liu2019optimal,liu2020legendre,trefethen2013atap,wang2018legendre,xiang2020jacobi}).
Comparing the space $H_m$ and the assumption of piecewise analytic
functions, which one is better in the sense that optimal rate of
convergence of $S_n^{\lambda}(f)$ can be predicted? To gain some
insight, we consider an example
\[
f(x) = \left\{
\begin{array}{ll}
{\displaystyle x^2\sin(\pi/x) }, & \hbox{if $x\neq0$,} \\[8pt]
{\displaystyle 0 }, & \hbox{if $x=0$,}
\end{array}
\right.
\]
which is piecewise analytic on $[-1,1]$. Moreover, it is known that
$f\in\mathrm{ AC}[-1,1]$, but $f{'}\notin \mathrm{BV}[-1,1]$.
Clearly, we can apply Theorem \ref{thm:PieceRate} to predict the
convergence rate of $S_n^{\lambda}(f)$, e.g., $\|f -
S_n^{\lambda}(f) \| = O(n^{-1})$ for $\lambda\leq1$. Note that
$f\notin H_1$, and thus the latest result on the error estimate of
$S_n^{\lambda}(f)$ in \cite[Thm.~9]{xiang2020jacobi} cannot be used.
\end{itemize}
\section{Extensions}\label{sec:extension}
In this section we extend our discussion
in two directions, including the convergence rate of Gegenbauer
projections for functions of fractional smoothness and error
estimates of Gegenbauer spectral differentiation.
\subsection{Functions of fractional smoothness}
In this subsection we consider the convergence rate of Gegenbauer
projections for functions of fractional smoothness. For the sake of
simplicity, we only consider some model functions and their
convergence results will shed light on the study of more complicated
functions.
\subsubsection{Functions with endpoint singularities}
We consider the model function $f_{\alpha}(x)=(1+x)^{\alpha}$, where
$\alpha>0$ is not an integer. Using
\cite[Eqn.~(7.311.3)]{gradshteyn2007table} and
\eqref{eq:duplication} and \eqref{eq:reflection}, we can write the
Gegenbauer coefficients of $f_{\alpha}$ as
\begin{align}\label{eq:akEnd}
a_k^{\lambda} &= (-1)^{k+1} \frac{2^{2\lambda+\alpha}
\sin(\alpha\pi) \Gamma(\lambda) \Gamma(\alpha+\lambda +\frac{1}{2})
\Gamma(\alpha+1) (k+\lambda) \Gamma(k-\alpha) }{\pi^{3/2}
\Gamma(k+\alpha+2\lambda+1)}.
\end{align}
For $k\geq \lfloor \alpha \rfloor + 1$, it is easily seen that the
sequence $\{a_k^{\lambda}\}$ is an alternating sequence. Thus, for
$\lambda>0$ and $n\geq \lfloor \alpha \rfloor$, we can deduce from
\eqref{eq:symmetry} and \eqref{eq:Bound1} that
\begin{align}
\|f_{\alpha} - S_n^{\lambda}(f_{\alpha}) \|_{\infty} &\leq
\sum_{k=n+1}^{\infty} | a_k^{\lambda} | C_k^{\lambda}(1) = \left|
\sum_{k=n+1}^{\infty} a_k^{\lambda} C_k^{\lambda}(-1) \right| =
|f_{\alpha}(-1) - S_n^{\lambda}(f_{\alpha},-1)|, \nonumber
\end{align}
and this implies that the maximum error of
$S_n^{\lambda}(f_{\alpha})$ is attained at $x=-1$. Combining
\eqref{eq:akEnd}, \eqref{eq:symmetry}, \eqref{eq:Bound1} and
\eqref{eq:AsyGAMMA} we have
\begin{align}
\|f_{\alpha} - S_n^{\lambda}(f_{\alpha}) \|_{\infty} &=
\frac{2^{\alpha+1} |\sin(\alpha\pi)| \Gamma(\alpha+\lambda
+\frac{1}{2}) \Gamma(\alpha+1) }{\pi\Gamma(\lambda+\frac{1}{2})}
\sum_{k=n+1}^{\infty} \frac{(k+\lambda) \Gamma(k-\alpha)
\Gamma(k+2\lambda) }{\Gamma(k+\alpha+2\lambda+1)
\Gamma(k+1)} \nonumber \\
&= \frac{2^{\alpha+1} |\sin(\alpha\pi)| \Gamma(\alpha+\lambda
+\frac{1}{2}) \Gamma(\alpha+1) }{\pi\Gamma(\lambda+\frac{1}{2})}
\sum_{k=n+1}^{\infty} \left[ \frac{1}{k^{2\alpha+1}} + O(k^{-2\alpha-2}) \right] \nonumber \\
&= \frac{2^{\alpha} |\sin(\alpha\pi)| \Gamma(\alpha+\lambda
+\frac{1}{2}) \Gamma(\alpha) }{\pi\Gamma(\lambda+\frac{1}{2})
n^{2\alpha}} + O(n^{-2\alpha-1}).
\end{align}
Hence we see that the convergence rate of
$S_n^{\lambda}(f_{\alpha})$ is $O(n^{-2\alpha})$. After recalling
from \cite[p.~411]{timan1963approximation} that the rate of
convergence of $\mathcal{B}_n(f_{\alpha})$ is also
$O(n^{-2\alpha})$, we can therefore conclude that both
$S_n^{\lambda}(f_{\alpha})$ and $\mathcal{B}_n(f_{\alpha})$ converge
at the same rate.
Some remarks are in order.
\begin{remark}
Observe that the constant in the leading term of $\|f_{\alpha} -
S_n^{\lambda}(f_{\alpha}) \|_{\infty}$ behaves like
$O(\lambda^{\alpha})$ as $\lambda\rightarrow\infty$, we can deduce
that the accuracy of $S_n^{\lambda}(f)$ will deteriorate as
$\lambda$ increases; see Figure \ref{fig:ExamVI} for an
illustration.
\end{remark}
\begin{remark}
For more general class of functions with endpoint singularities, the
conclusion that the rate of convergence of $S_n^{\lambda}(f)$ is the
same as that of $\mathcal{B}_n(f)$ still holds since the error at
the endpoint singularities will dominate the maximum error.
\end{remark}
\begin{figure}[ht]
\centering
\includegraphics[width=7.5cm,height=6cm]{EndExam1}~
\includegraphics[width=7.5cm,height=6cm]{EndExam2}
\caption{Convergence rates of $\mathcal{B}_n(f)$ (dots) and
$S_n^{\lambda}(f)$ with four values of $\lambda$ for
$f(x)=(1+x)^{3/2}$ (left) and $f(x)=\arccos(x)$ (right). The dashed
line in the left panel is $O(n^{-3})$ and in the right panel is
$O(n^{-1})$.} \label{fig:ExamVI}
\end{figure}
In Figure \ref{fig:ExamVI} we test convergence rates of
$\mathcal{B}_n(f)$ and $S_n^{\lambda}(f)$ for $f(x)=(1+x)^{3/2}$ and
$f(x)=\arccos(x)$. It is easily verified that $\alpha=3/2$ for the
former and $\alpha=1/2$ for the latter. As expected, we see that the
rate of convergence of $\mathcal{B}_n(f)$ is better than that of
$S_n^{\lambda}(f)$ by only a constant factor. Moreover, we also see
that the accuracy of $S_n^{\lambda}(f)$ indeed deteriorates as
$\lambda$ increases.
\subsubsection{Functions with an interior singularity of fractional
order} As a canonical example, we consider the function
\begin{align}\label{def:Model}
f(x)=|x-\theta|^{\alpha},
\end{align}
where $\theta\in(-1,1)$ and $\alpha>0$ is not an integer. When
$\alpha=1,3,5,\ldots$, it follows from Theorem \ref{thm:PieceRate}
that
\begin{align}\label{eq:RateInterior}
\|f - S_n^{\lambda}(f)\|_{\infty} = \left\{
\begin{array}{ll}
{\displaystyle O(n^{-\alpha}) }, & \hbox{if $\lambda\leq1$,} \\[8pt]
{\displaystyle O(n^{-\alpha-1+\lambda}) }, & \hbox{if $\lambda>1$.}
\end{array}
\right.
\end{align}
When $\alpha$ is not an integer, we conjecture that this result
still holds. In the particular case of Legendre projections (i.e.,
$\lambda=1/2$), it has been shown in \cite{wang2020legendre} that
\eqref{eq:RateInterior} indeed holds based on the observation that
the maximum error of Legendre projections is attained at $x=\theta$
for large $n$. For the Gegenbauer case, however, the situation may
be different and it is highly interesting to clarify the dependence
of the location of the maximum error on $\lambda$. To get some
insight, we plot in Figure \ref{fig:ExamVII} the pointwise error of
$S_n^{\lambda}(f)$ for three values of $\lambda$. Clearly, we
observe that, for $\lambda$ greater than a critical value, the
location of the maximum error of $S_n^{\lambda}(f)$ will jump from
the interior singularity $x=\theta$ to one of the endpoints $x=1$ or
$x=-1$. Following this key observation, we shall restrict our
attention to convergence rates of $S_n^{\lambda}(f)$ at these three
critical points, from which we will clarify the dependence of the
location of the maximum error on $\lambda$.
\begin{figure}[ht]
\centering
\includegraphics[width=4.9cm,height=4.3cm]{Pointwise1}~
\includegraphics[width=4.9cm,height=4.3cm]{Pointwise2}~
\includegraphics[width=4.9cm,height=4.3cm]{Pointwise3}
\caption{Pointwise error of $S_n^{\lambda}(f)$ for $\lambda=-2/5$
(left), $\lambda=3/4$ (middle) and $\lambda=2$ (right). Here
$f(x)=|x-1/4|^{3/2}$ and $n=30$. These red points are the errors of
$S_n^{\lambda}(f)$ at the critical points.} \label{fig:ExamVII}
\end{figure}
We start with the following result.
\begin{lemma}\label{lem:IntSing}
Let $f$ be the function defined in \eqref{def:Model}. Then, for each
$k\geq\alpha+1$\footnote{This condition is imposed here due to the
definition of generalized Gegenbauer functions proposed in
\cite[Definition~2.1]{liu2019optimal}. However, numerical tests show
that the formula \eqref{eq:ExpFormula} is valid for all $k\geq0$. To
keep the proof concise, we will not pursue this here.}, we have
\begin{align}\label{eq:ExpFormula}
a_k^{\lambda} &= \omega_{\lambda+\alpha+1}(\theta)
\frac{\Gamma(\lambda) \Gamma(\alpha+1) (k+\lambda)}{2^{1+\alpha}
\Gamma(\lambda+\alpha+\frac{3}{2}) \sqrt{\pi}} \left( {}_2\mathrm{
F}_1\left[\begin{matrix} \alpha+1-k, & k+2\lambda+\alpha+1 ;
\\ \alpha + \lambda + \frac{3}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1-\theta}{2} \right] \right. \nonumber \\
&~~~~~~~~~~ \left. + (-1)^k {}_2\mathrm{ F}_1\left[\begin{matrix}
\alpha+1-k, & k+2\lambda+\alpha+1 ;
\\ \alpha + \lambda + \frac{3}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1+\theta}{2} \right]
\right).
\end{align}
As $k\rightarrow\infty$ the coefficient $a_k^{\lambda}$ has the
following asymptotic behavior:
\begin{align}\label{eq:AsymInt}
a_k^{\lambda} &=
-\omega_{\frac{\lambda+\alpha+1}{2}}(\theta)\sin\left(\frac{\alpha\pi}{2}\right)
\frac{2^{1+\lambda}\Gamma(\lambda)\Gamma(\alpha+1)}{\pi
k^{\alpha+\lambda}} \cos\left(2(k+\lambda)\phi(\theta) -
\frac{\lambda\pi}{2} \right) \nonumber \\
&~~~~~ + O(k^{-\alpha-\lambda-1}).
\end{align}
where $\phi(\theta)=\arccos(\sqrt{(1+\theta)/2})$.
\end{lemma}
\begin{proof}
To show \eqref{eq:ExpFormula}, we follow the idea of Theorem 4.3 in
\cite{liu2019optimal} for Chebyshev coefficients. Let $m=\lfloor
\alpha \rfloor$ and $s=\alpha-m\in(0,1)$. Invoking the Rodrigues
formula \eqref{eq:Rodrigues} and using integration by parts $m+1$
times, we have for $k\geq m+1$ that
\begin{align}\label{eq:IntS1}
a_k^{\lambda} &= \frac{1}{h_k^{\lambda}} \prod_{j=0}^{m}
\frac{2(\lambda+j)}{(k-j)(k+2\lambda+j)} \int_{-1}^{1} f^{(m+1)}(x)
\omega_{\lambda+m+1}(x) C_{k-m-1}^{\lambda+m+1}(x) \mathrm{d}x \nonumber \\
&= \frac{1}{h_k^{\lambda}} \prod_{j=0}^{m}
\frac{2(\lambda+j)}{(k-j)(k+2\lambda+j)} \left[ \int_{-1}^{\theta}
f^{(m+1)}(x) \omega_{\lambda+m+1}(x)
C_{k-m-1}^{\lambda+m+1}(x) \mathrm{d}x \right. \nonumber \\
&~~~~~~~~~~ \left. + \int_{\theta}^{1} f^{(m+1)}(x)
\omega_{\lambda+m+1}(x) C_{k-m-1}^{\lambda+m+1}(x) \mathrm{d}x
\right].
\end{align}
In the following, we consider to derive explicit forms of these two
integrals inside the bracket. For simplicity of notation, we denote
the former one by $J_1$ and the latter one by $J_2$. From
\cite[Eqn.~(3.12b)]{liu2019optimal}, we know that
\begin{align}\label{eq:IntS2}
\omega_{\lambda+m+1}(x) C_{k-m-1}^{\lambda+m+1}(x) &=
\frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})}{\Gamma(k-m)\Gamma(2m+2\lambda+2)
2^{s-1} \Gamma(\lambda+\alpha+\frac{1}{2})} \nonumber \\
&~~~~~ \times {}_{-1}\mathcal{I}_{x}^{1-s} \left\{
\omega_{\lambda+\alpha}(x) {}^{l}G_{k-\alpha}^{(\lambda+\alpha)}(x)
\right\},
\end{align}
where ${}_{a}\mathcal{I}_{x}^{\nu}(\cdot)$ is the left fractional
integral of order $\nu$ and ${}^{l}G_{\nu}^{(\lambda)}(x)$ is the
left generalized Gegenbauer function of fractional degree $\nu$
defined by
\begin{align}
{}_{a}\mathcal{I}_{x}^{\nu}(f) = \frac{1}{\Gamma(\nu)} \int_{a}^{x}
\frac{f(t)}{(x-t)^{1-\nu}} \mathrm{d}t, \quad
{}^{l}G_{\nu}^{(\lambda)}(x) = (-1)^{\lfloor \nu \rfloor}
{}_2\mathrm{ F}_1\left[\begin{matrix} -\nu, & \nu+2\lambda ;
\\ \lambda + \frac{1}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1+x}{2}
\right]. \nonumber
\end{align}
For $J_1$, using \eqref{eq:IntS2} and fractional integration by
part, we obtain
\begin{align}\label{eq:IntS3}
J_1 &=
\frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})}{\Gamma(k-m)\Gamma(2m+2\lambda+2)
2^{s-1} \Gamma(\lambda+\alpha+\frac{1}{2})} \nonumber \\
&~~~~~~~~~~ \times \int_{-1}^{\theta} f^{(m+1)}(x)
{}_{-1}\mathcal{I}_{x}^{1-s} \left\{ \omega_{\lambda+\alpha}(x)
{}^{l}G_{k-\alpha}^{(\lambda+\alpha)}(x) \right\} \mathrm{d}x
\nonumber \\
&=
\frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})}{\Gamma(k-m)\Gamma(2m+2\lambda+2)
2^{s-1} \Gamma(\lambda+\alpha+\frac{1}{2})} \nonumber \\
&~~~~~~~~~~ \times \int_{-1}^{\theta} \omega_{\lambda+\alpha}(x)
{}^{l}G_{k-\alpha}^{(\lambda+\alpha)}(x)
{}_{x}\mathcal{I}_{\theta}^{1-s} \left\{ f^{(m+1)}(x) \right\}
\mathrm{d}x,
\end{align}
where ${}_{x}\mathcal{I}_{\theta}^{\nu}(\cdot)$ is the right
fractional Riemann-Liouville integral of order $\nu$. For
$x\in(-1,\theta)$, a direction calculation shows that
${}_{x}\mathcal{I}_{\theta}^{1-s}\{ f^{(m+1)}\} = (-1)^{m+1}
\Gamma(\alpha+1)$. Moreover, using
\cite[Eqn.~(3.13b)]{liu2019optimal}, we have
\begin{align}
\omega_{\lambda+\alpha}(x) {}^{l}G_{k-\alpha}^{(\lambda+\alpha)}(x)
=
-\frac{\Gamma(\lambda+\alpha+\frac{1}{2})}{2\Gamma(\lambda+\alpha+\frac{3}{2})}
\frac{\mathrm{d}}{\mathrm{d}x} \left\{ \omega_{\lambda+\alpha+1}(x)
{}^{l}G_{k-\alpha-1}^{(\lambda+\alpha+1)}(x) \right\}, \nonumber
\end{align}
and therefore, we arrive at
\begin{align}\label{eq:IntS4}
J_1 &= (-1)^m
\frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})
\Gamma(\alpha+1)}{\Gamma(k-m)\Gamma(2m+2\lambda+2) 2^{s}
\Gamma(\lambda+\alpha+\frac{3}{2})}
\omega_{\lambda+\alpha+1}(\theta)
{}^{l}G_{k-\alpha-1}^{(\lambda+\alpha+1)}(\theta) \nonumber\\
&= (-1)^k \frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})
\Gamma(\alpha+1)}{\Gamma(k-m)\Gamma(2m+2\lambda+2) 2^{s}
\Gamma(\lambda+\alpha+\frac{3}{2})}
\omega_{\lambda+\alpha+1}(\theta) \nonumber \\
&~~~~~~ \times {}_2\mathrm{ F}_1\left[\begin{matrix} \alpha+1-k, &
k+2\lambda+\alpha+1 ;
\\ \alpha + \lambda + \frac{3}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1+\theta}{2}
\right].
\end{align}
Using similar arguments, we can obtain
\begin{align}\label{eq:IntS5}
J_2 &= \frac{\Gamma(k+m+2\lambda+1)\Gamma(\lambda+m+\frac{3}{2})
\Gamma(\alpha+1)}{\Gamma(k-m)\Gamma(2m+2\lambda+2) 2^{s}
\Gamma(\lambda+\alpha+\frac{3}{2})}
\omega_{\lambda+\alpha+1}(\theta) \nonumber \\
&~~~~~~ \times {}_2\mathrm{ F}_1\left[\begin{matrix} \alpha+1-k, &
k+2\lambda+\alpha+1 ;
\\ \alpha + \lambda + \frac{3}{2} ; \hspace{-1cm} &\end{matrix} ~ \frac{1-\theta}{2}
\right].
\end{align}
Inserting \eqref{eq:IntS4} and \eqref{eq:IntS5} into
\eqref{eq:IntS1}, we obtain \eqref{eq:ExpFormula}.
As for \eqref{eq:AsymInt}, it follows from applying the asymptotic
expansion of Gauss hypergeometric function in
\cite[Eqn.~(4.7)]{paris2013asym} (with $\varepsilon=1$) to
\eqref{eq:ExpFormula}. This ends the proof.
\end{proof}
An immediate corollary of Lemma \ref{lem:IntSing} is the comparison
of Chebyshev and Legendre coefficients, which was studied in
\cite{boyd2014cheb,wang2016gegenbauer}. More specifically, let
$k\geq1$ and let $a_k^L$ and $a_k^C$, respectively, denote the $k$th
Legendre and Chebyshev coefficients of $f$ defined in
\eqref{def:Model}, i.e.,
\begin{align}\label{eq:ChebLeg}
a_k^L = \frac{2k+1}{2} \int_{-1}^{1} f(x) P_k(x) \mathrm{d}x, \quad
a_k^C = \frac{2}{\pi} \int_{-1}^{1} \frac{f(x) T_k(x)}{\sqrt{1-x^2}}
\mathrm{d}x.
\end{align}
It has been observed in the right panel of
\cite[Fig.~7]{wang2016gegenbauer} that $a_k^C$ decays faster than
$a_k^L$ by a factor of $O(k^{1/2})$ and the sequence $\{a_k^L/a_k^C
k^{-1/2}\}$ oscillates around a finite value as
$k\rightarrow\infty$. However, a result to explain these
observations is still lacking. Here, we provide a precise result to
this problem.
\begin{corollary}
Let $f$ be defined by \eqref{def:Model} and let $a_k^L$ and $a_k^C$
be defined by \eqref{eq:ChebLeg}. As $k\rightarrow\infty$, we have
\begin{align}
\frac{a_k^{L}}{a_k^{C}} &= \omega_{\frac{3}{4}}(\theta)
\frac{\cos\left((2k+1)\phi(\theta) -
\frac{\pi}{4}\right)}{\cos\left(2k\phi(\theta) \right)}
\sqrt{\frac{\pi k}{2}} + O(k^{-1/2}).
\end{align}
where $\phi(\theta) $ is defined in \eqref{eq:AsymInt}.
\end{corollary}
\begin{proof}
First, setting $\lambda=1/2$ in \eqref{eq:AsymInt} gives
\begin{align}
a_k^{L} &=
-\omega_{\frac{2\alpha+3}{4}}(\theta)\sin\left(\frac{\alpha\pi}{2}\right)
\frac{2^{3/2}\Gamma(\alpha+1)}{\pi^{1/2} k^{\alpha+1/2}}
\cos\left((2k+1)\phi(\theta) - \frac{\pi}{4} \right) +
O(k^{-\alpha-3/2}). \nonumber
\end{align}
Moreover, in view of \eqref{eq:GegChebT} we find that
\begin{align}
a_k^{C} = \frac{2}{k} \lim_{\lambda\rightarrow0} \lambda
a_k^{\lambda} &= -\omega_{\frac{\alpha+1}{2}}(\theta)
\sin\left(\frac{\alpha\pi}{2}\right) \frac{4\Gamma(\alpha+1)}{\pi
k^{\alpha+1}} \cos\left(2k\phi(\theta)\right) + O(k^{-\alpha-2}).
\nonumber
\end{align}
Combining these two results, we immediately obtain the desired
result.
\end{proof}
We now give convergence rates of $S_n^{\lambda}(f)$ at the critical
points.
\begin{theorem}\label{thm:RateCritical}
Let $f$ be defined by \eqref{def:Model}. Then, for large $n$, it
holds that
\begin{itemize}
\item[(i).] $|f(\pm1) - S_n^{\lambda}(f,\pm1)| =
O(n^{-\alpha-1+\lambda})$ .
\item[(ii).] $|f(\theta) - S_n^{\lambda}(f,\theta)| =
O(n^{-\alpha})$.
\end{itemize}
\end{theorem}
\begin{proof}
For (i), combining the fact
$C_k^{\lambda}(1)=k^{2\lambda-1}\Gamma(2\lambda)^{-1} +
O(k^{2\lambda-2})$ for $k\gg1$ and \eqref{eq:AsymInt} we have
\begin{align}
f(1) - S_n^{\lambda}(f,1) &= \sum_{k=n+1}^{\infty} a_k^{\lambda}
C_k^{\lambda}(1) \nonumber \\
&= -\omega_{\frac{\lambda+\alpha+1}{2}}(\theta)
\sin\left(\frac{\alpha\pi}{2}\right)
\frac{2^{1+\lambda}\Gamma(\lambda)\Gamma(\alpha+1)}{\pi\Gamma(2\lambda)}
\nonumber \\
&~~~~~ \times \sum_{k=n+1}^{\infty} \left[
\frac{\cos\left(2(k+\lambda)\phi(\theta) - \frac{\lambda\pi}{2}
\right)}{k^{\alpha+1-\lambda}} + O(k^{-\alpha+\lambda-2}) \right].
\nonumber
\end{align}
For the leading term of the sum on the right-hand side of above, we
have
\begin{align}
\sum_{k=n+1}^{\infty} \frac{\cos\left(2(k+\lambda)\phi(\theta) -
\frac{\lambda\pi}{2} \right)}{k^{\alpha+1-\lambda}} &=
\cos\left(2\lambda\phi(\theta)-\frac{\lambda\pi}{2}\right)
\sum_{k=n+1}^{\infty} \frac{\cos\left(2k\phi(\theta)
\right)}{k^{\alpha+1-\lambda}} \nonumber \\
&~ - \sin\left(2\lambda\phi(\theta)-\frac{\lambda\pi}{2}\right)
\sum_{k=n+1}^{\infty} \frac{\sin\left(2k\phi(\theta)
\right)}{k^{\alpha+1-\lambda}}. \nonumber
\end{align}
For sums of the form $\sum_{k=n}^{\infty} e^{ik\beta}k^{\alpha}$,
where $\beta\in\mathbb{R}$ is not a multiple of $2\pi$ and
$\alpha\in\mathbb{R}$, we can deduce from
\cite[Eqn.~(5.09)]{olver1974asymp} and
\cite[Eqn.~(5.10)]{olver1974asymp} that they behave like
$O(n^{\alpha})$ as $n\rightarrow\infty$. This proves the convergence
rate of $S_n^{\lambda}(f)$ at $x=1$. The convergence rate of
$S_n^{\lambda}(f)$ at $x=-1$ can be proved in a similar manner and
we omit the details.
To prove (ii). On the one hand, it follows from \eqref{eq:AsymInt}
that $a_k^{\lambda}=O(k^{-\alpha-\lambda})$. On the other hand, from
\eqref{eq:Bound2} we see that $|C_k^{\lambda}(x)| =
O(k^{\lambda-1})$ for $x\in(-1,1)$. Combining these two estimates we
get
\begin{align}
f(\theta) - S_n^{\lambda}(f,\theta) = \sum_{k=n+1}^{\infty}
a_k^{\lambda} C_k^{\lambda}(\theta) = \sum_{k=n+1}^{\infty}
O(k^{-\alpha-1}) = O(n^{-\alpha}). \nonumber
\end{align}
This completes the proof.
\end{proof}
By combining the key observation in Figure \ref{fig:ExamVII} and the
convergence results in Theorem \ref{thm:RateCritical}, we can deduce
that the convergence results in \eqref{eq:RateInterior} still hold
when $\alpha>0$ is not an integer. Moreover, we recall from
\cite{timan1963approximation} that the convergence rate of
$\mathcal{B}_n(f)$ is $O(n^{-\alpha})$. Therefore, we conclude that
the rate of convergence of $S_n^{\lambda}(f)$ is the same as that of
$\mathcal{B}_n(f)$ if $-1/2<\lambda\leq1$. For $\lambda>1$, however,
the rate of convergence of $S_n^{\lambda}(f)$ is slower than that of
$\mathcal{B}_n(f)$ by a factor of $n^{\lambda-1}$, which is one
power of $n$ better than the predicted result by \eqref{eq:errorG}.
In Figure \ref{fig:ExamVIII} we test a numerical example with
$\theta=-0.4$ and $\alpha=5/2$. As expected, we see that the
predicted convergence rates by \eqref{eq:RateInterior} agree quite
well with the observed convergence rates.
\begin{figure}[ht]
\centering
\includegraphics[width=7.5cm,height=6cm]{InteriorExam1}~
\includegraphics[width=7.5cm,height=6cm]{InteriorExam2}
\caption{The left panel shows the convergence of $\mathcal{B}_n(f)$
(circles) and $S_n^{\lambda}(f)$ for $\lambda=1/6,1/3,2/3,1$. The
right panel shows the convergence of $S_n^{\lambda}(f)$ for
$\lambda=3/2,2,5/2,3$. The dashed line in the left panel is
$O(n^{-5/2})$ and these dashed lines in the right panel indicate the
convergence rates predicted by \eqref{eq:RateInterior}. Here
$f(x)=|x+0.4|^{5/2}$.} \label{fig:ExamVIII}
\end{figure}
\subsection{Error estimates of spectral differentiation}
In practical applications, it is a powerful approach to compute
derivatives of the underlying functions using spectral
approximations. As a consequence of Theorem \ref{thm:GegBound} we
have the following error estimates.
\begin{theorem}
Suppose that $f$ is analytic in the region bounded by the ellipse
$\mathcal{E}_{\rho}$ for some $\rho>1$. Then, for $\lambda>-1/2$ and
$\lambda\neq0$ and $j\in \mathbb{N}$, we have
\begin{align}
\|f^{(j)} - (S_n^{\lambda}(f))^{(j)} \|_{\infty} \leq c \left[
\frac{2^{1-j-2\lambda} \sqrt{\pi} D_{\lambda}(\rho)}{
\Gamma(\lambda)\Gamma(\lambda+j+\frac{1}{2})\ln\rho} \right]
\frac{n^{\lambda+2j}}{\rho^n},
\end{align}
where $c\simeq1$ is a generic positive constant.
\end{theorem}
\begin{proof}
According to \cite[Eqn.~(18.9.19)]{olver2010nist} we obtain
\[
\frac{\mathrm{d}^j}{\mathrm{d}x^j} C_{n}^{\lambda}(x) = 2^j
\frac{\Gamma(\lambda+j)}{\Gamma(\lambda)} C_{n-j}^{\lambda+j}(x),
\]
and thus
\begin{align}\label{eq:DerivS1}
f^{(j)}(x) - \frac{\mathrm{d}^j}{\mathrm{d}x^j} S_n^{\lambda}(f,x)
&= 2^j \frac{\Gamma(\lambda+j)}{\Gamma(\lambda)}
\sum_{k=n+1}^{\infty} a_k^{\lambda} C_{k-j}^{\lambda+j}(x).
\end{align}
We note that
\[
|C_{k-j}^{\lambda+j}(x)| \leq C_{k-j}^{\lambda+j}(1) \simeq
\frac{k^{2\lambda+2j-1}}{\Gamma(2\lambda+2j)}, \quad k\gg1.
\]
Hence, combining this with Theorem \ref{thm:GegBound} and
\eqref{eq:DerivS1}, we get
\begin{align}
\| f^{(j)} - S_n^{\lambda}(f)^{(j)} \|_{\infty} &\leq 2^j
\frac{D_{\lambda}(\rho)
\Gamma(\lambda+j)}{\Gamma(\lambda)\Gamma(2\lambda+2j)}
\sum_{k=n+1}^{\infty} \frac{k^{\lambda+2j}}{\rho^k}. \nonumber
\end{align}
The remaining proof is similar to that of Theorem
\ref{thm:RateAnal}, we omit the details.
\end{proof}
\section{Concluding remarks}\label{sec:conclusion}
In this work, we have generalized our earlier work on the
convergence rate analysis of Legendre projections in
\cite{wang2020legendre} to the case of Gegenbauer projections. We
have shown that the rates of convergence of $S_n^{\lambda}(f)$ and
$\mathcal{B}_n(f)$ are the same in the context of either $f$ is
analytic and $\lambda\leq0$ or $f$ is piecewise analytic and
$\lambda\leq1$. Otherwise, the rate of convergence of
$S_n^{\lambda}(f)$ is slower than that of $\mathcal{B}_n(f)$ by a
factor of $n^{\lambda}$ when $f$ is analytic and $\lambda>0$ and by
a factor of $n^{\lambda-1}$ when $f$ is piecewise analytic and
$\lambda>1$. We also extended our discussion to functions with
endpoint singularities and functions with an interior singularity of
fractional order.
Finally, we note that Jacobi projections are also widely used for
the numerical solution of differential equations. Sharp error
estimates of Jacobi projections for analytic functions have been
analyzed in \cite{zhao2013sharp} and for differentiable functions in
the space $H_m$ were recently analyzed in \cite{xiang2020jacobi}. In
the latter case, however, the convergence rate in the maximum norm
derived in \cite[Thm.~8]{xiang2020jacobi} is suboptimal when
$\max\{\alpha,\beta\}>-1/2$, where $\alpha,\beta$ are parameters in
Jacobi polynomials. In future work, it would be interesting to
explore the optimal rate of convergence of Jacobi projections under
the assumption of piecewise analytic functions.
| {
"timestamp": "2020-08-04T02:26:43",
"yymm": "2008",
"arxiv_id": "2008.00584",
"language": "en",
"url": "https://arxiv.org/abs/2008.00584",
"abstract": "Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and $\\lambda\\leq0$ or differentiable and $\\lambda\\leq1$, where $\\lambda$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $\\lambda>0$ or differentiable and $\\lambda>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^{\\lambda}$ and $n^{\\lambda-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $\\lambda>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods.",
"subjects": "Numerical Analysis (math.NA); Classical Analysis and ODEs (math.CA)",
"title": "Optimal rates of convergence and error localization of Gegenbauer projections",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795075882267,
"lm_q2_score": 0.7185943865443349,
"lm_q1q2_score": 0.7092379337871916
} |
https://arxiv.org/abs/2212.01211 | Sometimes Two Irrational Guards are Needed | In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur. | \section{Introduction}
In the art gallery problem, we are given a closed polygon $P$, on
$n$ vertices, with rational coordinates and
an integer $k$.
We are asked whether it is possible to find a set (of guards) $G$ of size $k$
such that any point $p\in P$ is seen by a point in $G$.
We say two points $p$, $q$ see each other if the line segment $pq$
is contained inside $P$.
We show that an optimal solution of two guards might need to have irrational coordinates.
In such a case, we say a polygon has \textit{irrational guards}.
The art gallery problem was formulated in 1973 by Victor Klee. See, for example, the book by O'Rourke \cite[page 2]{o1987art}.
One of the earliest results states that every simple polygon on $n$ vertices can always be guarded with $\lfloor n/3 \rfloor$ guards~\cite{chvatal1975combinatorial,Fisk78a}.
\begin{figure}[tbp]
\centering
\includegraphics{figures/Fisk.pdf}
\caption{Any triangulation of a simple polygon can be three-colored.
At least one of the color classes has at most $\lfloor n/3\rfloor$ vertices. This color class also guards the entire polygon, as every triangle is incident to all three colors~\cite{Fisk78a}.}
\label{fig:Fisk}
\end{figure}
Interestingly, it is actually very tough to find any positive algorithmic results on the art gallery problem. It seems like the art gallery problem is almost impenetrable.
For instance, only in 2002, Micha Sharir pointed out that the problem was even decidable~\cite[see acknowledgments]{EfratH02,EfratH06}.
The decidability of the art gallery problem is actually easy once you know methods from real algebraic geometry~\cite{Basu2006_RealAlgebraicGeometry}.
The idea is to reduce the problem to the first-order theory of the reals.
We encode guard positions by variables, and then we check if every point in the polygon is seen by at least one guard.
Note that this is easy to encode in the first-order theory of the reals, as we are allowed to use existential ($\exists g_1,g_2,\ldots$) and universal quantifiers ($\forall p =(x,y)$).
Since then, despite much research on the art gallery problem, no better algorithm appeared, as far as worst-case complexity is concerned.
The underlying reason for the difficulty to find better algorithms
can be explained by the fact that the art gallery problem is \ensuremath{\exists\mathbb{R}}\xspace-complete~\cite{stade2022complexity, Abrahamsen2022_artGallery}.
In a nutshell, \ensuremath{\exists\mathbb{R}}\xspace-completeness precisely entails that there is no better method for the worst-case complexity of the problem.
(\ensuremath{\exists\mathbb{R}}\xspace can be defined as the class of problems that are equivalent to finding a real root to a multivariate polynomial with integer coordinates. See \Cref{sub:ETR} for an introduction.)
More specifically, it was shown that arbitrary algebraic numbers may be needed to describe an optimal solution to the art gallery problem.
This may come as a surprise to some readers, and was clearly a surprise back then.
Specifically, ``in practice'', it seems very rare that irrational guards are ever needed.
The reason is that a typical situation is one of the following two.
Either the guards have some freedom to move around and still see the entire polygon.
Or if a guard has no freedom, it is forced to be on a line defined by vertices of the polygon.
As the vertices of the polygon are at rational coordinates, the guards will be at rational coordinates in that case as well.
Indeed, only in 2017, the first polygon requiring irrational guards was found~\cite{abrahamsen2017irrational}.
Even though \ensuremath{\exists\mathbb{R}}\xspace-reductions exhibit an infinite number of polygons that require irrational guards, those polygons are not ``concrete'' in the naive sense of the word.
And up to this day, this is the only ``concrete'' polygon~\cite{abrahamsen2017irrational} that we know does require irrational guards.
In this work, we find a second polygon.
It is superior to the first one in the sense that it shows that
two guards are already enough to enforce irrational guards.
As a single guard can always be chosen to have rational coordinates,
we settle the question of the minimum number of guards required to
have irrational guards.
We summarize our results in the following theorem.
\begin{theorem}\label{thm:main}
There exists a polygon with rational coordinates, such that there is only one way of guarding this polygon optimally with two guards.
Those two guards have irrational coordinates.
\end{theorem}
\paragraph{Organization.}
We first discuss our results from different angles (\Cref{sub:discussion}).
Then we give a selected overview of related research on the art gallery problem (\Cref{sub:art}).
We finish this introduction with some background on the existential theory of the reals (\Cref{sub:ETR}).
In \Cref{sec:preparation}, we give an overview of how we constructed the polygon and what is the intuition behind the different parts.
In \Cref{sec:Polygon}, we give the polygon with coordinates of all vertices and we provide a formal proof of correctness.
In \Cref{sec:Challenges}, we explain how we constructed the polygon and what technical challenges we had to overcome.
\subsection{Discussion}
\label{sub:discussion}
In this section, we discuss different aspects of our findings.
\paragraph{Minimum number of irrational guards.}
It is known that one guard can always be chosen to be rational~\cite{LeePreparataoptimal}.
The polygon by Abrahamsen, Adamaszek, and Miltzow~\cite{abrahamsen2017irrational} requires three irrational guards.
The main strength of our finding is to determine the minimum number of guards required to have irrational guards.
\paragraph{Grid Approximation.}
One way to circumvent worst-case complexity is to discretize the polygon and restrict oneself to a dense grid~\cite{BonnetM17Approx,EfratH02,EfratH06}.
The polygon by Abrahamsen, Adamaszek, and Miltzow showed that a grid cannot have a better approximation factor than $4/3 = 1.333\ldots$.
We improve this lower bound to $3/2 = 1.5$.
\begin{figure}[tbp]
\centering
\includegraphics[]{figures/Grid-Approx.pdf}
\caption{We may restrict the guards to lie on a dense grid.
This may make the optimal solution worse.}
\label{fig:Grid-Approx}
\end{figure}
Note that Bonnet and Miltzow showed that under some mild assumptions, the grid contains a constant factor approximation~\cite{BonnetM17Approx}.
\paragraph{Intuitive Understanding.}
It is good to have multiple different concrete polygons that require irrational guards.
This result complements the \ensuremath{\exists\mathbb{R}}\xspace-completeness of the art gallery problem nicely.
While \ensuremath{\exists\mathbb{R}}\xspace-completeness is clearly stronger from a theoretical perspective, concrete polygons may be useful to get a better intuitive understanding of the difficulty.
\paragraph{Test Cases.}
From a practical perspective, our polygon can serve as a test case on which we can compare the performance of different algorithms.
Usually, we would like to have a host of difficult and diverse instances
that can be automatically generated.
With difficult instances, we mean polygons that require irrational guards.
We leave this as a future research question.
\paragraph{Technical Depth.}
The principal methods that we used for the construction of our polygon are in spirit similar to the methods used by Abrahamsen, Adamaszek, and Miltzow~\cite{abrahamsen2017irrational}.
However, it turned out that it was considerably more difficult to find the polygon.
On the one hand, our construction is smaller and thus there were fewer parameters that we had to manipulate to find a solution.
On the other hand, the two guards interact in more intertwined ways.
Thus making it much harder to find a correct placement of all the polygon vertices.
To be concrete, the construction by \cite{abrahamsen2017irrational} has the guards $a$, $m$, and $t$.
Guards $a$ and $m$ cover together two pockets.
Similarly, guards $m$ and $t$ cover together two separate pockets.
Thus there is no direct interaction between guards $a$ and $t$.
This makes it easier to construct the different parts independently.
In our case the two guards \ensuremath{l}\xspace and \ensuremath{t}\xspace together guard three pockets.
And this implies that the interaction between \ensuremath{l}\xspace and \ensuremath{t}\xspace is much more integrated.
This leads to a construction with some vertices being extremely close.
Furthermore, our construction has pockets that were inside other pockets.
\paragraph{Irrational Boundary Guards.}
Both our polygon and the polygon by~\cite{abrahamsen2017irrational} have their
guards in the interior.
It is an interesting open problem if one can enforce irrational guards, in case all guards are restricted to lying on the boundary and are only required to guard the boundary.
\paragraph{Integer Coordinates.}
One may wonder whether there is also a polygon with integer coordinates that exposes irrational guards.
The answer is yes, and this can be achieved by scaling all coordinates by all the appearing denominators.
\paragraph{Special Polygons.}
One may wonder whether it is possible to enforce irrational guards on
polygons with some extra properties like, being monotone or being rectilinear.
Both of those questions have been positively resolved by~\cite{abrahamsen2017irrational}.
\subsection{Art Gallery Problem}
\label{sub:art}
The literature on the art gallery is vast.
Therefore, we decide to focus here on algorithmic results.
\paragraph{Exact Algorithmic Results.}
In 1979, the first algorithm for guarding a polygon in linear time with one guard appeared~\cite{LeePreparataoptimal}.
It took until 1992 until there was an algorithm that could determine if a polygon could be guarded by two guards in $O(n^4)$ time~\cite{BellevilleCCCG,belleville1991computing}.
As mentioned already above it took until 2002 to find the first correct algorithm that solves the art gallery problem~\cite{EfratH02, EfratH06}.
There is still no other algorithm known.
On the lower bound side, we know \text{NP}\xspace-hardness~\cite{LeeLin86}, APX-hardness~\cite{eidenbenz2001inapproximability} and W[1]-hardness~\cite{BonnetW1HARD}.
One may argue that \text{NP}\xspace-hardness is enough evidence that there are
no efficient algorithms for the art gallery problem and that this
may fully explain the lack of algorithmic results.
However, for other \text{NP}\xspace-complete problems like Clique, Subset-sum, Dominating Set, and TSP, we do know a myriad of algorithms.
Although many of them run in exponential time in the worst case they give huge improvements in many different situations.
We believe that the lack of algorithmic results may stem from the fact that
we do not know how to discretize the art gallery problem efficiently.
Note that all the mentioned problems are already discrete.
We believe that the \ensuremath{\exists\mathbb{R}}\xspace-completeness of the art gallery problem may give the most compelling explanation of why a concise discretization of the art gallery problem is unlikely~\cite{Abrahamsen2022_artGallery,stade2022complexity}.
Specifically, many discretization schemes would imply that the art gallery problem lies in \text{NP}\xspace and thus imply $\text{NP}\xspace = \ensuremath{\exists\mathbb{R}}\xspace$.
The first proof that the art gallery problem is \ensuremath{\exists\mathbb{R}}\xspace-comlete was given by Abrahamsen, Adamaszek, and Miltzow in 2017~\cite{Abrahamsen2022_artGallery}.
It was recently improved by Stade, who showed that it is even \ensuremath{\exists\mathbb{R}}\xspace-complete if we only require the boundary to be guarded~\cite{stade2022complexity}.
It remains open whether guarding the boundary from the boundary is \ensuremath{\exists\mathbb{R}}\xspace-complete.
\paragraph{Practical Difficulty.}
There is a series of papers that studied the art gallery problem
from a practical perspective.
In other words, they implemented algorithms and tested them on benchmark instances~\cite{PracticalBorrmann, PracticalARTMasterFriedrich, Simon-hengeveld2021practical, tozoni, Hengeveld2021_artGallery}.
The practical experiences from those papers suggest that irrational coordinates do not play any role in the pursuit to find an optimal solution.
\begin{figure}[tbp]
\centering
\includegraphics[page = 4]{figures/RayArrangement}
\caption{Left: The dark green region is added to the visibility polygon. Right: The orange region is removed from the visibility polygon.}
\label{fig:SmallChanges}
\end{figure}
\paragraph{Explaining the Discrepancy.}
We are aware of two theoretical explanations for the discrepancy between the theoretical and the practical results.
One such finding uses smoothed analysis and argues that there is with high probability an optimal solution after a small random perturbation of the polygon~\cite{Hengeveld2021_artGallery, Erickson2022_SmoothingGap}.
A second explanation comes from Hengeveld and Miltzow.
They introduced the notion of vision-stability.
To explain this concept, we consider guards that can either see by some small angle $\delta$ around reflex vertices or are blocked by an angle $\delta$ by reflex vertices.
See the green visibility regions in \Cref{fig:SmallChanges}.
Intuitively, if $\delta$ is small enough then the optimal number of guards will not change.
Vision-stability states that there indeed exists such a $\delta>0$.
Using this assumption Hengeveld and Miltzow could find a polynomially-sized discretization scheme for the art gallery problem.
It remains an open question to improve their discretization scheme.
\paragraph{Topology.}
Given a polygon, we can consider the set $\mathcal{G}(P)$ of all possible guard sets of minimum size.
Together with the Hausdorff distance, $\mathcal{G}(P)$ forms a topological space.
From the algebraic encoding by Sharir, we know that the topological space must be compact and semi-algebraic.
The question arises: given a compact semi-algebraic $S$, is there a polygon $P$ such that $\mathcal{G}(P)$ is topologically equivalent to $S$?
This question was positively answered by Bertschinger, El Maalouly, Miltzow, Schnider, and Weber for homotopy-equivalence~\cite{TOPOLOGY-SIMON} and shortly improved by Stade and Tucker-Foltz to homeomorphic-equivalence~\cite{stade2022topological}.
\subsection{Existential theory of the Reals}
\label{sub:ETR}
The complexity class \ensuremath{\exists\mathbb{R}}\xspace (often pronounced as ``ER'') has gained a lot of interest in recent years.
It is defined via its canonical complete problem \problemname{ETR} (short for \emph{Existential Theory of the Reals}) and contains all problems that polynomial-time many-one reduce to it.
In an \problemname{ETR} instance, we are given an integer~$n$ and a sentence of the form
\[
\exists X_1, \ldots, X_n \in \ensuremath{\mathbb{R}}\xspace :
\varphi(X_1, \ldots, X_n),
\]
where~$\varphi$ is a well-formed and quantifier-free formula consisting of polynomial equations and inequalities in the variables and the logical connectives $\{\land, \lor, \lnot\}$.
The goal is to decide whether this sentence is true.
As an example, consider the formula $\varphi(X,Y) :\equiv X^2 + Y^2 \leq 1 \land Y^2 \geq 2X^2 - 1$;
among (infinitely many) other solutions, $\varphi(0,0)$ evaluates to true, witnessing that this is a yes-instance of \problemname{ETR}.
It is known that
\[
\text{NP}\xspace \subseteq \ensuremath{\exists\mathbb{R}}\xspace \subseteq \text{PSPACE}\xspace
\text{.}
\]
Here the first inclusion follows because a \problemname{SAT} instance can trivially be written as an equivalent \problemname{ETR} instance.
The second inclusion is highly non-trivial and was first proven by Canny in his seminal paper~\cite{Canny1988_PSPACE}.
Note that the complexity of working with continuous numbers was studied in various contexts.
To avoid confusion, let us make some remarks on the underlying machine model.
The underlying machine model for \ensuremath{\exists\mathbb{R}}\xspace (over which sentences need to be decided and where reductions are performed) is the \textnormal{word RAM}\xspace (or equivalently, a Turing machine) and not the \textnormal{real RAM}\xspace~\cite{Erickson2022_SmoothingGap} or the Blum-Shub-Smale model~\cite{Blum1989_ComputationOverTheReals}.
The complexity class \ensuremath{\exists\mathbb{R}}\xspace gains its importance by numerous important algorithmic problems that have been shown to be complete for this class in recent years.
The name \ensuremath{\exists\mathbb{R}}\xspace was introduced by Schaefer in~\cite{Schaefer2010_GeometryTopology} who also pointed out that several \text{NP}\xspace-hardness reductions from the literature actually implied \ensuremath{\exists\mathbb{R}}\xspace-hardness.
For this reason, several important \ensuremath{\exists\mathbb{R}}\xspace-completeness results were obtained before the need for a dedicated complexity class became apparent.
Common features of \ensuremath{\exists\mathbb{R}}\xspace-complete problems are their continuous solution space and the nonlinear relations between their variables.
Important \ensuremath{\exists\mathbb{R}}\xspace-completeness results include the realizability of abstract order types~\cite{Mnev1988_UniversalityTheorem,Shor1991_Stretchability} and geometric linkages~\cite{Schaefer2013_Realizability}, as well as the recognition of geometric segment graphs~\cite{Kratochvil1994_IntersectionGraphs,Matousek2014_IntersectionGraphsER}, unit-disk graphs~\cite{Kang2012_Sphere,McDiarmid2013_DiskSegmentGraphs}, and ray intersection graphs~\cite{Cardinal2018_Intersection}.
More results appeared in the graph drawing community~\cite{Dobbins2022_areaUniversality,Erickson2019_CurveStraightening,Lubiw2022_DrawingInPolygonialRegion,Schaefer2021_FixedK}, regarding polytopes~\cite{Dobbins2019_NestedPolytopes,Richter1995_Polytopes}, the study of Nash-equilibria~\cite{Berthelsen2019_MultiPlayerNash,Bilo2016_Nash, Bilo2017_SymmetricNash,Garg2018_MultiPlayer,Schaefer2017_FixedPointsNash}, training neural networks~\cite{Abrahamsen2021_NeuralNetworks, train-fully-neural-networks}, matrix factorization~\cite{Chistikov2016_Matrix,Schaefer2018_tensorRank,Shitov2016_MatrixFactorizations,Shitov2017_PSMatrixFactorization}, or continuous constraint satisfaction problems~\cite{Miltzow2022_ContinuousCSP}.
In computational geometry, we would like to mention the art gallery problem~\cite{Abrahamsen2022_artGallery, stade2022complexity} and covering polygons with convex polygons~\cite{Abrahamsen2022_Covering}.
Recently, the community started to pay more attention to higher levels of the ``real polynomial hierarchy'', which surprisingly captures some interesting algorithmic problems~\cite{Blanc2021_ESS,DCosta2021_EscapeProblem,Dobbins2022_areaUniversality,Jungeblut2022_Hausdorff,Real-Poly-Hierarchy,Burgisser2009_ExoticQuantifiers}.
\section{Preparation}
\label{sec:preparation}
We aim to construct a polygon.
This polygon should be guarded by two guards at irrational coordinates but requires three guards at rational coordinates.
We must restrict the possible coordinates the guards can be positioned.
In this section, we will explore the tools to restrict the possible positions of the two guards within the polygon.
\begin{figure}[tbp]
\centering
\includegraphics[page=5]{figures/Galleries.pdf}
\caption{Our final polygon: it has a core (gray), three quadrilateral pockets (blue), and four narrow triangular pockets (yellow).}
\label{fig:examplepolygon}
\end{figure}
\subsection{Basic Definitions}
Each guard $g$ will be able to guard some region of the polygon:
we call this region its \textit{visibility polygon} $\ensuremath{\text{vis}}\xspace(g)$.
The visibility polygon includes all points for which the line segment between the guard and the point is included in the polygon $P$.
Notably, the union of the visibility polygons of the two guards must be the art gallery. Otherwise, the art gallery is not completely guarded.
A \textit{window} is an edge of the visibility polygon $\ensuremath{\text{vis}}\xspace(g)$ that is not part of the boundary of $P$.
We can find windows in a guard $g$'s visibility polygon, by shooting rays from $g$ to reflex vertices (the vertices of the polygon, with an interior angle larger than $\pi$).
If these rays do not leave the polygon at the reflex vertex,
a window will exist between the reflex vertex and the position where the ray does intersect the boundary of the polygon.
Let the \textit{window's end} be the intersection of the ray with an edge of the polygon.
Our final polygon consists of the core and a number of pockets, as shown in \Cref{fig:examplepolygon}.
The \textit{core} of the polygon is the square in the center.
We will enforce that both guards are located in the core.
As a square is a convex shape, this implies that both guards will guard the core.
The \textit{pockets} are all regions outside the core.
We will use pockets that are either quadrilateral or triangular.
Pockets are \textit{attached} to either the core or another pocket: they have one edge that lies on the boundary of the core or on the boundary of another pocket.
Quadrilateral pockets will always be attached to the core.
Each quadrilateral pocket has one edge that is not on the boundary of the core, nor adjacent to it.
We will call this edge the \textit{wall} of a
quadrilateral pocket.
Similarly, triangular pockets will be attached to either the core or a quadrilateral pocket.
We will use pockets as a tool to limit the locations of the two guards.
\subsection{Guard Segments}
\begin{figure}[tbp]
\centering
\includegraphics[page=7]{figures/GalleriesLucas.pdf}
\caption{A small polygon that can only be guarded by two guards, because each guard segment (yellow dashed line) must contain a guard. The region where a guard could guard at least one pocket is shaded in light yellow.}
\label{fig:guardsegment}
\end{figure}
We can force a guard to be positioned on a line segment within the polygon.
Such a line segment is called a \textit{guard segment}.
Guard segments are commonly used in the context of the art gallery problem~\cite{abrahamsen2017irrational, stade2022complexity}.
In this section, we will describe how we construct a guard segment.
Let us denote by $s$ the segment and by $\ell$ its supporting line.
To make $s$ a guard segment, we add two triangular pockets
where $\ell$ intersects $\partial P$.
Each of the triangular pockets has an edge on $\ell$.
Besides this one edge, the pockets lay on different sides of $\ell$.
Only a guard on the line segment between the two pockets can guard both triangular pockets at the same time.
We have two guards in our polygon and both will be on a guard segment.
If the two guard segments are not intersecting, we can enforce that there must be one guard on each of them as follows.
First, note that there are in total four triangular pockets.
Second, we make the triangular pockets sufficiently narrow.
In this way, it is impossible to guard two of the triangular pockets outside of a guard segment.
Thus at least one guard must be on each guard segment.
A simple construction with two non-intersecting guard segments is shown in \Cref{fig:guardsegment}.
\subsection{Guarding Quadrilateral Pockets}
We will now describe how given the position of guard \ensuremath{l}\xspace and a quadrilateral pocket $Q$ will limit the position of guard \ensuremath{t}\xspace.
See \Cref{fig:simplepocket} for an illustration of the following description.
First, note that if \ensuremath{l}\xspace will not guard $Q$ completely then there will remain some unguarded region (orange) in $Q$.
The part of the guard segment of \ensuremath{t}\xspace where the unguarded region is visible is denoted the \textit{\textnormal{feasible segment}\xspace}.
It is bounded from the \textit{\textnormal{back ray}\xspace} and the \textit{\textnormal{front ray}\xspace{}}.
It is clear that \ensuremath{t}\xspace must be on the feasible segment.
We can compute the \textnormal{front ray}\xspace{} by first computing the window end's $s$ from \ensuremath{l}\xspace to the wall of $Q$ and then shooting a ray from $s$ in the direction of the second reflex vertex of $Q$.
\begin{figure}[tbp]
\centering
\includegraphics[page=8]{figures/Galleries.pdf}
\caption{A polygon with guard \ensuremath{l}\xspace. The guard \ensuremath{l}\xspace defines an unguarded region in the quadrilateral pocket, a \textnormal{front ray}\xspace{} and a \textnormal{back ray}\xspace{}, and a feasible segment.}
\label{fig:simplepocket}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[page=2]{figures/GalleriesLucas.pdf}
\caption{Our complete polygon. The art gallery is shaded according to the function of each region: white is the core, yellow is the pockets used to create guard segments, and turquoise are other pockets. The yellow dashed lines represent the guard segments. The coordinates of important vertices are given.}
\label{fig:fullpolygon1}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[page=3, trim={0 0 0 3cm}, clip=true]{figures/GalleriesLucas.pdf}
\caption{Our complete polygon. The optimal solution has two guards at irrational coordinates is shown. The blue regions are guarded by the upper left guard; the red regions are guarded by the bottom right guard; the purple regions are guarded by both. The dashed lines are rays shot from the guards through reflex vertices. For each pocket, these windows meet at a point on the art gallery's wall, of which the coordinates are also given.}
\label{fig:fullpolygon2}
\end{figure}
\begin{table}[H]
\centering
\caption{Coordinates of the vertices of the polygon ($v_1, \dots, v_{28}$), the guards (\ensuremath{l}\xspace and \ensuremath{t}\xspace), and the window's ends ($w_1, w_2, w_3$).}
\begin{tabular}{|ll|ll|ll|}
\hline
$v_1$ & $(0, 10)$ & $v_{12}$ & $(12.7, 7)$ & $v_{23}$ & $(4, -1.7)$ \\
$v_2$ & $(2, 10)$ & $v_{13}$ & $(11.7, 6)$ & $v_{24}$ & $(4, 0)$ \\
$v_3$ & $(3, 11)$ & $v_{14}$ & $(\frac{1230422}{101007}, 6)$ & $v_{25}$ & $(0, 0)$ \\
$v_4$ & $(2.3, 10)$ & $v_{15}$ & $(\frac{1016072}{101007}, 4)$ & $v_{26}$ & $(0, 8)$ \\
$v_5$ & $(4, 10)$ & $v_{16}$ & $(10, 4)$ & $v_{27}$ & $(-1, 7)$ \\
$v_6$ & $(4, \frac{465522}{29357})$ & $v_{17}$ & $(10, 0)$ & $v_{28}$ & $(0, 8.3)$ \\
$v_7$ & $(6, \frac{312388}{29357})$ & $v_{18}$ & $(6, 0)$ & $\ensuremath{l}\xspace^*$ & $(3.7-2.2\cdot\sqrt{2}, 11.7-2.2\cdot\sqrt{2})$ \\
$v_8$ & $(6, 10)$ & $v_{19}$ & $(6, \frac{-25442}{34407})$ & $\ensuremath{t}\xspace^*$ & $(7.4-0.5\cdot\sqrt{2}, 1.7-0.5\cdot\sqrt{2})$ \\
$v_9$ & $(10, 10)$ & $v_{20}$ & $(4, \frac{-84128}{34407})$ & $w_1$ & $(\frac{293570\cdot\sqrt{2}+8052346}{1425913}, \frac{-765670\cdot\sqrt{2}+16485384}{1425913})$ \\
$v_{10}$ & $(10, 6)$ & $v_{21}$ & $(4, 2)$ & $w_2$ & $(\frac{1071750\cdot\sqrt{2}+29733818}{2673483}, \frac{1010070\cdot\sqrt{2}+13370606}{2673483})$ \\
$v_{11}$ & $(11.4, 6)$ & $v_{22}$ & $(3, -2.7)$ & $w_3$ & $(\frac{344070\cdot\sqrt{2}+3108526}{760803}, \frac{293430\cdot\sqrt{2}+1804526}{760803})$ \\ \hline
\end{tabular}
\label{tab:my_label}
\end{table}
\section{Complete Polygon}
\label{sec:Polygon}
In this section, we will present our complete polygon: a polygon that can be guarded by two guards if and only if both guards are situated at irrational points.
\subsection{The Polygon}
As we described in \Cref{sec:preparation} and displayed in \Cref{fig:fullpolygon1}, the polygon consists of a core and some pockets.
The polygon has four triangular pockets defining two guard segments.
The two guard segments lie on the lines $y=x+8$ and $y=x-5.7$.
Furthermore, the polygon has three quadrilateral pockets.
In \Cref{tab:my_label}, the coordinates of the vertices of the polygon, the coordinates of the two guards, and the coordinates of the window's ends are given.
The walls of the three quadrilateral pockets have the supporting lines:
\begin{enumerate}
\item Top pocket: $y=\frac{-76567\cdot x+771790}{29357}$.
\item Right pocket: $y=\frac{101007\cdot x-587372}{107175}$.
\item Bottom pocket: $y=\frac{29343\cdot x-201500}{34407}$.
\end{enumerate}
\subsection{Proof}
We prove that our polygon can be guarded with two irrational guards, but cannot be guarded with two rational guards.
We state that the polygon can be guarded by two guards placed at
\[\ensuremath{l}\xspace^\star= (3.7-2.2\cdot\sqrt{2}, 11.7-2.2\cdot\sqrt{2})\] and \[ \ensuremath{t}\xspace^\star = (7.4-0.5\cdot\sqrt{2}, 1.7-0.5\cdot\sqrt{2}).\]
\Cref{fig:fullpolygon2} displays the visibility polygons of these two guards.
It can be checked using simple calculations that the two visibility polygons cover the complete polygon.
First, both guard segments will contain a guard.
Furthermore, the window's ends are in the same location (so no unseen region between them) and both vertices on the wall are guarded.
Now, we prove that no two rational guards can guard our polygon.
Specifically, we show that no two guards, except the ones mentioned, will guard the entire polygon.
Clearly, both guard segments must contain a guard.
Let \ensuremath{l}\xspace be on the guard segment in the top left of the polygon, and \ensuremath{t}\xspace be on the guard segment in the bottom right of the polygon.
Given the position of \ensuremath{l}\xspace, we calculate where \ensuremath{t}\xspace can guard all regions not guarded by \ensuremath{l}\xspace.
For each of the three pockets, we will bound the position of \ensuremath{t}\xspace given the position of \ensuremath{l}\xspace.
To represent their positions, we will use their x-coordinates.
As both \ensuremath{l}\xspace and \ensuremath{t}\xspace lie on a non-vertical guard segment, their x-coordinates will uniquely describe their position.
First, we calculate which part of the pocket \ensuremath{l}\xspace guards and which part it does not.
Due to the guard segment of \ensuremath{l}\xspace, we will have exactly one window along with its corresponding window's end.
As described in \Cref{sec:preparation}, we can use this construction to determine the region where \ensuremath{t}\xspace can guard the region of the pocket \ensuremath{l}\xspace does not cover.
Notably, \ensuremath{t}\xspace will always be on the correct side of the \textnormal{back ray}\xspace.
Indeed, the entire guard segment of \ensuremath{t}\xspace lies on one side of the \textnormal{back ray}\xspace.
As such, we will bound the \textnormal{feasible segment}\xspace by ensuring \ensuremath{t}\xspace lies on the correct side of the \textnormal{front ray}\xspace.
We calculate the intersection of this ray and the guard segment.
Then, the x-coordinate of \ensuremath{t}\xspace may either not be smaller than, or be greater than the x-coordinate of the intersection between the \textnormal{front ray}\xspace and the guard segment.
\begin{figure}[tbp]
\centering
\includegraphics[page=10]{figures/GalleriesLucas.pdf}
\caption{The solution to the system of equations (\Cref{eq:1}, \Cref{eq:2}, \Cref{eq:3}). Here, $\bullet$ denotes a closed interval, while $\circ$ denotes an open interval.}
\label{fig:proof}
\end{figure}
It depends on the pocket whether the x-coordinate of \ensuremath{t}\xspace can not be smaller or greater than the x-coordinate of the intersection.
Guard \ensuremath{t}\xspace must lie on the same side of the \textnormal{front ray}\xspace as the unguarded region of the pocket.
As can be verified in \Cref{fig:fullpolygon2}, the x-coordinate of \ensuremath{t}\xspace interacts with the intersections we find for the pockets in the following way:
\begin{enumerate}
\item Top pocket: the x-coordinate of \ensuremath{t}\xspace must be smaller or equal to the intersection.
\item Right pocket: the x-coordinate of \ensuremath{t}\xspace must be greater or equal to the intersection.
\item Bottom pocket: the x-coordinate of \ensuremath{t}\xspace must be smaller or equal to the intersection.
\end{enumerate}
It is important to note that the x-coordinate of \ensuremath{t}\xspace does not lie on the same side of all three intersections.
If it did lie on the same side of all three, then the position of \ensuremath{t}\xspace could trivially be at any coordinate greater or smaller than all three intersections.
We can use the x-coordinate of \ensuremath{l}\xspace to determine its position.
So, we use \ensuremath{l}\xspace's x-coordinate ($x_\ensuremath{l}\xspace$) to calculate inequalities that limit the x-coordinate of \ensuremath{t}\xspace ($x_\ensuremath{t}\xspace$):
\begin{align}
x_\ensuremath{t}\xspace\leq \frac{42734\cdot x_\ensuremath{l}\xspace-70239}{5460\cdot x_\ensuremath{l}\xspace-9950} \label{eq:1} \\
x_\ensuremath{t}\xspace\geq \frac{11928\cdot x_\ensuremath{l}\xspace-269330}{2570\cdot x_\ensuremath{l}\xspace-40705} \label{eq:2} \\
x_\ensuremath{t}\xspace\leq \frac{13538\cdot x_\ensuremath{l}\xspace-61793}{4220\cdot x_\ensuremath{l}\xspace-93450} \label{eq:3}
\end{align}
We will use \Cref{eq:1}, \Cref{eq:2}, \Cref{eq:3} as a system of equations.
A solution to the system of equations will have a corresponding pair of guards.
We use an algebraic computer program to calculate the solution to this system of equations.
\Cref{fig:proof} shows the values for $x_\ensuremath{l}\xspace$ for which the system of equations has a valid solution.
Then, any value for $x_\ensuremath{t}\xspace$ is chosen between the bounds imposed by $x_\ensuremath{l}\xspace$.
However, not all solutions for $x_\ensuremath{l}\xspace$ correspond to valid positions for guard \ensuremath{l}\xspace.
Specifically, notice in \Cref{fig:proof} that the only possible value for $x_\ensuremath{l}\xspace < \frac{10539}{9974} < \frac{995}{546}$ is irrational.
We will argue that $x_\ensuremath{l}\xspace$ must be smaller than $\frac{10539}{9974}$.
For the following description, we refer to \Cref{fig:Left-Bounding-Guard}
\begin{figure}
\centering
\includegraphics{figures/Left-Bounding-Guard.pdf}
\caption{The guard \ensuremath{l}\xspace must be to the left of the line $x = \frac{10539}{9974}$.}
\label{fig:Left-Bounding-Guard}
\end{figure}
Suppose for the purpose of contradiction that there is a valid guard placement with $x_\ensuremath{l}\xspace\geq\frac{10539}{9974}$.
In this case, guard \ensuremath{l}\xspace fails to guard any part of the wall of the top pocket.
Guard $x_\ensuremath{t}\xspace$ must be less than or equal to $6$ to guard the entire wall.
However, for $x_\ensuremath{t}\xspace$ less than or equal to $6$, guard \ensuremath{t}\xspace fails to guard the wall of the right pocket.
Guard \ensuremath{l}\xspace can never guard the entire wall of the right pocket.
This gives a contradiction and implies that
$x_\ensuremath{l}\xspace < \frac{10539}{9974}$.
As such, we can limit the possible locations for guard \ensuremath{l}\xspace as $x_\ensuremath{l}\xspace < \frac{10539}{9974}$.
Evidently, in this range, the only valid x-coordinate for $x_\ensuremath{l}\xspace$ is $3.7-2.2\cdot\sqrt{2}$, see \Cref{fig:proof}.
For this x-coordinate of $x_\ensuremath{l}\xspace$, the only possible position for \ensuremath{t}\xspace is at $x_\ensuremath{t}\xspace = 7.4-0.5\cdot\sqrt{2}$.
Finally, this shows that the only possible configuration of two guards in this polygon is at $(3.7-2.2\cdot\sqrt{2}, 11.7-2.2\cdot\sqrt{2})$, and at $(7.4-0.5\cdot\sqrt{2}, 1.7-0.5\cdot\sqrt{2})$: both guards must be at irrational coordinates.
\section{Challenges}
\label{sec:Challenges}
We encountered new challenges while searching for our polygon, compared to Abrahamsen, Adamaszek, and Miltzow~\cite{abrahamsen2017irrational}'s polygon that requires three irrational guards.
\begin{figure}[tbp]
\centering
\includegraphics[page=9]{figures/GalleriesLucas.pdf}
\hspace{2cm}
\includegraphics[page=11]{figures/GalleriesLucas.pdf}
\caption{Left: Two guards and two reflex vertices, where the corresponding wall leads to an invalid polygon. Right: Three \textnormal{front ray}\xspace{}s intersecting the guard segment.
Guard \ensuremath{t}\xspace must be on the dotted side of each ray.
If the intersection points were reversed, there does not exist a valid solution either.}
\label{fig:properites}
\end{figure}
\paragraph{Irrational Lines.}
To describe how we found the polygon described in the previous section, we introduce the concept of a rational and an irrational line.
Give a line $\ell = \{mx + n : x\in \ensuremath{\mathbb{R}}\xspace \}$, we say that $\ell$ is \textit{rational} if and only if $m,n\in \ensuremath{\mathbb{Q}}\xspace$, and \textit{irrational} otherwise.
Note that a rational line might contain arbitrarily many irrational points. (For example, the line $y=0$ contains all the irrational points of the form $(a,0)$, with \ensuremath{l}\xspace being irrational.)
Now, if we have two distinct rational points, then they span a rational line.
And by point-line duality, if we have two rational lines, then they intersect at a rational point.
This implies that any irrational point is contained in at most one rational line.
And reversely any irrational line contains at most one rational point.
(For instance, the point $(\pi,\pi)$ is on the rational line $x=y$.)
This is relevant to us, as our guards are irrational points.
Our guard segments are rational lines, as they are defined by triangular pockets.
And the triangular pockets consist of rational vertices.
Specifically, any window's end must be irrational, for the following reason.
The line $\ell$ defined by the guard and the reflex vertex must be irrational, as it is not the guard segment.
Furthermore, $\ell$ already contains a rational point, namely the reflex vertex.
Thus, the window's end must be irrational as well.
\paragraph{Constructing the Polygon.}
To start, like in the construction with three guards, we parameterize the position of the (irrational) positions of the two guards and the reflex vertices before creating the polygon.
In other words, $g = (g_1\cdot\sqrt{2} + g_2, g_3\cdot\sqrt{2} + g_4)$, with $g_i \in \ensuremath{\mathbb{Q}}\xspace$.
To define the guard segments, we compute the unique rational line through each guard.
Then, we define the quadrilateral pockets.
To do this, we note that the window's ends of the two guards must meet on each pocket's wall.
For a pocket, its window's end comes from two rays shot from the two guards through the reflex vertices of the pockets.
Let $w$ be the point where the two rays intersect.
Recall that $w$ will always be an irrational point.
One unique rational line will contain $w$, so this must be the supporting line of the wall.
Now, changing the coordinates of the two parameterized guards and the reflex vertices will change the position of the guard segments and the walls of all the quadrilateral pockets.
Note that not all walls and guard segments lead to a valid polygon, or to a polygon at all.
We have to check a few properties that we will describe next.
\paragraph{Properties.}
Here, we list some properties that we need to ensure.
We start with the properties of the guard segments.
As already mentioned, the two guard segments should not intersect.
Furthermore, if the guard segments go across the whole polygon,
it might be possible to guard all quadrilateral pockets with a single guard.
Thirdly, if a guard segment is aligned such that a larger region of each pocket is observed when the guard is moved in one direction, then the interaction with the other guard is not meaningful.
As such, many positions of guards cause invalid guard segments.
Now, let's move to the properties of the quadrilateral pockets.
For each pocket, we ensured (by construction) that the window's ends of the two guards are at the exact same point.
To create the walls of the pockets, we cast a ray from each guard to each pocket, to illustrate where the window in its visibility polygon would be.
For each pocket, we found the (irrational) intersection of these rays of the two guards.
Then, we define the wall of the pocket as a line segment on the rational line through the intersection.
This construction of the walls also poses a problem.
As mentioned above, there only exists a single rational line through an irrational point.
Thus, there will be only one possible wall for the given guards and reflex vertices.
However, not all walls will be valid.
An example of an invalid wall is shown in the left picture in \Cref{fig:properites}.
\begin{figure}[tbp]
\centering
\includegraphics{figures/FeasibleRays.pdf}
\caption{We draw the two guard segments without the surrounding polygon.
On the top, we move the guard \ensuremath{l}\xspace from left to right.
On the bottom, we see the corresponding feasible segment for guard \ensuremath{t}\xspace.
Recall that we denote by $\ensuremath{l}\xspace^{\star}$ and
$\ensuremath{t}\xspace^\star$ the coordinates of the predetermined guard positions.}
\label{fig:Bad-frontRAYs}
\end{figure}
Even a valid polygon does not guarantee that the construction will work.
For the following description, refer to \Cref{fig:Bad-frontRAYs}.
Our construction guarantees that the feasible segment for guard \ensuremath{t}\xspace is limited to a single point, given the predetermined position of $\ensuremath{l}\xspace = \ensuremath{l}\xspace^*$.
(Recall that the feasible segment is an interval on the guard segment of \ensuremath{t}\xspace).
However, it does not guarantee that there does not exist a feasible segment for \ensuremath{t}\xspace when \ensuremath{l}\xspace is arbitrarily close to its predetermined position.
Recall that we determine the feasible segment for \ensuremath{t}\xspace based on the \textnormal{front ray}\xspace{}s.
Guard \ensuremath{t}\xspace must lie on the correct side of all three of these \textnormal{front ray}\xspace{}s.
So, when $x_\ensuremath{l}\xspace$ is less than its predetermined value, there must not exist an interval on the guard segment of \ensuremath{t}\xspace that lies on the correct side of all these rays.
(Recall from \Cref{sec:Polygon} that we note $\ensuremath{l}\xspace = (x_\ensuremath{l}\xspace,y_\ensuremath{l}\xspace)$.)
Otherwise, there must exist a rational point for \ensuremath{t}\xspace on the interval.
For this rational point of \ensuremath{t}\xspace, we can find a corresponding rational point for \ensuremath{l}\xspace.
However, when $x_\ensuremath{l}\xspace$ is greater than its predetermined position, the order of the intersection of these rays with the guard segment will reverse.
So there must not exist a segment on the guard segment that lies on the correct side of all rays when the intersection points are reversed, either.
To prevent this, we arrange the \textnormal{front ray}\xspace{}s' intersection pattern as shown in the right picture of \Cref{fig:properites}.
This implies that the middle intersection requires the guard to be on one side, and the first and the last intersection require the guard to be on the other side.
\paragraph{Summary.}
In comparison to the polygon construction with three guards~\cite{abrahamsen2017irrational}, our polygon has fewer parameters that we can adjust, as we have one guard less.
As everything depends on those few parameters, it was difficult to find a configuration that satisfies all the desired properties simultaneously.
Furthermore, we need to check some additional properties that didn't play a role in the previous construction, as the middle guard was surrounded by the other guards from two different sides.
At last, we could not avoid the supporting line of the guard segment intersecting two quadrilateral pockets.
\vfill
\paragraph{Acknowledgments.}
We would like to thank Thekla Hamm and Ivan Bliznets for their helpful comments on the presentation.
Lucas Meijer is generously supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.
Tillmann Miltzow is generously supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 016.Veni.192.250 and no. VI.Vidi.213.150.
\bibliographystyle{plain}
| {
"timestamp": "2022-12-05T02:13:36",
"yymm": "2212",
"arxiv_id": "2212.01211",
"language": "en",
"url": "https://arxiv.org/abs/2212.01211",
"abstract": "In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.",
"subjects": "Computational Geometry (cs.CG)",
"title": "Sometimes Two Irrational Guards are Needed",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795102691455,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7092379297656319
} |
https://arxiv.org/abs/1812.00114 | The Duality between Ideals of Multilinear Operators and Tensor Norms | We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $\Sigma$-operators. To this end, we introduce and develop the notions of $\Sigma$-ideals of multilinear operators and $\Sigma$-tensor norms. We establish the foundations of this theory by proving a representation theorem for maximal $\Sigma$-ideals of multilinear operators by finitely generated $\Sigma$-tensor norms and a duality theorem for $\Sigma$-tensor norms. For these notions we also develop basic theory and present concrete examples. | \section{Introduction}
The theory of operator ideals, mainly developed by Pietsch in \cite{pietsch78}, is a branch of Functional Analysis that provides a systematic framework to study linear operators grouping them according to the so called ideal property. Many examples of operator ideals find applications in a wide variety of topics related to Functional Analysis, for instance, Operator, Probability and Measure theories as well as Harmonic Analysis and geometry of Banach spaces. Some of these applications are shown throughout the monographs \cite{defant93, diestel95, pietsch78, ryan02, tomczak89,wojtaszczyk91}.
The theory of operator ideals is closely related with the theory of tensor norms developed by Grothendieck \cite{grothendieck53}. Indeed, if we restrict our attention to finite dimensional normed spaces, there exists a one to one relation between operator ideals and tensor norms; this is summarized by $E^*\tens_{\nu} F=\mathcal{A}(E;F)$ where the equality stands for an isometric isomorphism. This relation does not hold for general Banach spaces; however, it is possible to extend it under certain assumptions. To be specific, maximal ideals of linear operators are represented by finitely generated tensor norms through the Representation Theorem for Maximal Ideals. This result is a keystone in the matter since it provides a bridge between operator ideals and tensor norms phenomena in the general case of Banach spaces. This time, the interplay between this two fields has the form $(X\tens_\alpha Y)^*=\mathcal{A}(X;Y^*)$. For a detailed exposition of this result we strongly recommend \cite[Section 17.5]{defant93}.
The purpose of this article is to develop the duality theory for ideals of multilinear operators and tensor norms based on the geometric approach of $\Sigma$-operators. Our two main results are a representation and a duality theorem. These results are situated in the research developed in recent years and devoted to extend the theory of linear operators to other nonlinear situations. Among other perspectives, this research consists of extending ideals of linear operators to the multilinear setting. It is noteworthy the nonexistence of a canonical path to be followed to extend the theory of linear operators to the multilinear setting. For this reason, some classes of linear operators have more than one multilinear version, for instance, the cases of absolutely summing and nuclear operators. A recent approach, detailed in \cite{fernandez-unzueta18}, proposes to study multilinear operators based on the idea of considering multilinear mappings as homogeneous functions. This approach gave rise to the notion of $\sos$. In \cite{angulo18, fernandez-unzueta18, fernandez-unzueta18a, fernandez-unzueta18b} the reader can appreciate the results of this way of thinking. These references develop the classes of bounded multilinear operators, multilinear operators factoring through Hilbert spaces, Lipschitz $p$-summing and $(p,q)$-dominated multilinear operators under the perspective of $\Sigma$-operators. Each of these classes seems to work properly since they verify analogues of the fundamental features of their linear counterparts. Even more, these classes are closed under certain compositions and have a peculiar maximal ideal behavior. Furthermore, they are in duality relation with particular tensor norms which enjoy of a peculiar finitely generated behavior. These duality relations along with the relation between operator ideals and tensor norms lead, naturally, to investigate a proper notion of ideals of multilinear operators and tensor norms following the geometric approach of $\sos$. For achieving our goals we introduce a proper notion of ideals of multilinear operators and tensor norms. We also specify local properties for these concepts, that is, properties that ensure a good behavior in general Banach spaces regarding finite dimensional subspaces. Specifically, we introduce $\Sigma$-ideals of multilinear operators and $\Sigma$-tensor norms. For these concepts we prove a Representation Theorem for Maximal $\Sigma$-ideals and a Duality Theorem for $\Sigma$-tensor norms.
Let us we describe our results. Fix a positive integer $n$ and Banach spaces $\xxx$ and $Y$. Every bounded multilinear operator $\Txxy$ has associated a homogeneous function $f_T:\sxx\into Y$ where $\sxx\subset \xtx$ is the so called Segre cone of the Banach spaces $X_i$. The projective tensor norm $\pi$ defines a metric structure on $\sxx$. The continuity of the mapping $T$ is equivalent to the Lipschitz condition of $f_T$. A function $f_T$ as above is referred to as the bounded $\Sigma$-operator associated to $T$. For a better development of the theory, particularly on the local features, we have to consider not just the norm $\pi$ but all reasonable crossnorms $\beta$ on $\xtx$.
The notion of $\Sigma$-ideal is intended to group multilinear operators taking into account their associated $\Sigma$-operators (see \eqref{idealproperty}). A component in a $\Sigma$-ideal $\ideal$ is denoted by $\Abxxy$ and is determined by an arrangement of the form $(n,\xxx,Y,\beta)$ where $n$ is a positive integer, $\xxx$ and $Y$ are Banach spaces and $\beta$ is a reasonable crossnorm on $\xtx$. By definition, the component $\Abxxy$ is a linear subspace of $\Lxxy$ and is normed with its ideal norm denoted by $\Ab$. Roughly speaking, $\mathcal{A}$ is the union of all the possible components and $A$ is an application whose restriction to each component agrees with the norm $\Ab$ (see Definition \ref{ideal}).
The idea of representing $\Sigma$-ideals in tensorial terms gives rise to the notion of $\Sigma$-tensor norms. These norms appear in two different versions: \emph{on spaces} and \emph{on duals}. Roughly speaking, a $\Sigma$-tensor norm on spaces $\alpha$ assigns a norm $\ab$ on $\xtxty$ such that the dual space $(\xtxty,\ab)^*$ to be isometrically isomorphic to a component $\Abxxyd$ for a certain $\Sigma$-ideal $\ideal$. For a better development of the theory we are in the necessity of consider tensor products of the form $\Lbxxy\tens Y$, where $\Lbxxy$ stand for the dual space of $\xtxb$. A $\Sigma$-tensor norm on duals assigns a norm $\vb$ on $\Lbxxy\tens Y$. As we will see, these two versions are dual to each other (see Definition \ref{tns}, Definition \ref{tnd} and Theorem \ref{tnstotnd}).
Each of the concepts we are presenting defines the two others, uniquely, in the class of finite dimensional normed spaces. This fact is summarized by the mappings
\begin{eqnarray}\label{relationfinitedimensions}
(\Lbee\tens F^*,\vb)=(\etetf,\ab)^*=\Abeefd\\
\fhi\tens y^*\hspace{1cm}\mapsto\hspace{1.4cm}\fhi\tens y^*\hspace{1.3cm}\mapsto\hspace{1.1cm} \fhi\cdot y^*\hspace{1cm}\nonumber
\end{eqnarray}
where $E_i$ and $F$ are finite dimensional normed spaces, $\nu$ is a $\Sigma$-tensor norm on duals, $\alpha$ is a $\Sigma$-tensor norm on spaces, $\ideal$ is a $\Sigma$-ideal, the tensor $\fhi\tens y^*$ acts by evaluation on $\etetf$ and the operator $\fhi\cdot y^*$ is defined by $\fhi\cdot y^*(z)=\fhi(z)y^*$ for all $z$ in $\epe$. Here, the equality symbol stands for an isometric isomorphism (see Theorems \ref{tnstotnd} and \ref{tndsideal}).
In order to extend \eqref{relationfinitedimensions} to general Banach spaces we are in the necessity of defining maximal $\Sigma$-ideals, finitely generated $\Sigma$-tensor norms on spaces and cofinitely generated $\Sigma$-tensor norms on duals. Our main results are expressed in these terms. Theorem \ref{rt} is The Representation Theorem for Maximal $\Sigma$-ideals and affirms that
\begin{align*}
\left(\xtxty,\ab\right)^*&=\Abxxyd\\
\left(\xtxtyd,\ab\right)^*\cap\mathcal{L}(\xxx;Y) &=\Abxxy
\end{align*}
hold linearly and isometrically if $\alpha$ is a finitely generated $\Sigma$-tensor norm on spaces related with the maximal $\Sigma$-ideal $\ideal$ by \eqref{relationfinitedimensions}. Theorem \ref{dt} is The Duality Theorem for $\Sigma$-tensor norms and affirms that the linear inclusion
\begin{eqnarray*}
\left(\Lbxx\tens Y,\vb\right)\hookrightarrow\left(\xtxtyd,\ab\right)^*,
\end{eqnarray*}
given by evaluation, is isometric for all Banach spaces if $\nu$ is cofinitely generated, $\alpha$ is finitely generated, and they are related in finite dimensions as in \eqref{relationfinitedimensions}.
Throughout this article we present, as examples of $\Sigma$-ideals, multilinear versions of the most classical examples of operator ideals. On the side of tensor norms, we exhibit the greatest and least $\Sigma$-tensor norms in both versions. We also recall from \cite{fernandez-unzueta18a} the Laprest\'e-type $\Sigma$-tensor norms on spaces.
It is noteworthy the consistency of the results presented here with those of \cite{floret_hunfeld02} and \cite{achour16, cabrera-padilla15, cabrera-padilla16a, cabrera-padilla17} where other relations of ideals and tensor norms are developed for multilinear and Lipschitz mappings.
The distribution of the material is the next. In Section 1.1 we fix notation. In Section 2 we recall basic theory of $\Sigma$-operators. In Section 3 we introduce and develop the notion of $\Sigma$-ideals. In Section 4 we introduce the notion of $\Sigma$-tensor norms. In this section we also specify the relation between $\Sigma$-tensor norms and $\Sigma$-ideals in finite dimensions. In Section 5 we relate $\Sigma$-tensor norms and $\Sigma$-ideals in general Banach spaces. In this section we prove the representation and duality theorems.
\subsection{Notation}
The letter $\kk$ denotes the filed of real or complex numbers. The unit ball of the normed space $X$ is denoted by $B_X$. The operator $K_X:X\into X^{**} $ denote the canonical embedding given by evaluation. The set of finite dimensional subspaces of $X$ is denoted by $\mathcal{F}(X)$ and the set of finite codimensional subspaces is denoted by $\mathcal{CF}(X)$. The class of all Banach spaces is denoted by $\ban$ while $\finn$ denote the class of all finite dimensional normed spaces. If $f:A\into B$ is a function, then we use the notation $\lev f , a \rev:=f(a)$.
Throughout this article $n$ denotes a positive integer and the letters $X_1,\dots, X_n$, $Z_1,\dots, Z_n$, $Y$, $W$ denote Banach spaces. The symbol $\Lxxy$ denotes the Banach space of all bounded multilinear operators $T:\xpx\into Y$ with the usual norm $\|T\|=\sup\{ \|T\xxp\| \;|\; x_i\in B_{X_i}, 1\leq i\leq n\}$. If $Y=\kk$ we simply write $\mathcal{L}(\xxx)$.
\section{$\Sigma$-operators and Multilinear Operators}
The set of decomposable tensors of the algebraic tensor product $\xtx$ is denoted by $\sxx$. That is, $\sxx:=\left\{\; \xxt \;|\; x^i\in X_i, 1 \leq i \leq n \;\right\}$.
From now on $\beta$ denote a reasonable crossnorm on $\xtx$. That is, a norm such that $\varepsilon (u)\leq \beta(u) \leq \pi(u)$ for all $u$ in $\xtx$, where $\varepsilon$ and $\pi$ are the injective and projective tensor norms given by
\begin{align*}
\pi(u)&= \inf \left\{\; \sumim \|x_i^1\|\cdots\|x_i^n\| \;\Big|\; u=\sumim x_i^1\tens\cdots\tens x_i^n \;\right\},\\
\varepsilon(u)&= \sup \left\{\; |x_1^*\tens\cdots\tens x_n^*(u)| \;\Big|\; x_i^*\in B_{X_i^*}, 1\leq i\leq n \;\right\}
\end{align*}
for all $u$ in $\xtx$.
Every norm $\beta$ defines a metric on the set $\sxx$ but according to \cite[Theorem 2.1]{fernandez-unzueta18} all these metrics are equivalent. Explicitly,
\begin{equation}\label{lipschitzequivalent}
\pi(\pmq) \leq 2^{n-1} \beta(\pmq) \qquad \forall\, p,q\in \sxx.
\end{equation}
We also use the symbol $\sxx$ to denote the metric space resulting of restricting the norm $\pi$ to the set of decomposable tensors of $\xtx$. This metric space is the so called Segre cone of $\xxx$.
\begin{remark}\label{betarestricted}
It is straightforward that if $E_i$ is a closed subspace of $X_i$, $1 \leq i \leq n$, the restriction of the norm $\beta$ to the tensor product $\ete$ is a reasonable crossnorm. Such restrictions are denoted by $\br$. The consideration of these restrictions will be crucial for defining the local behavior of $\Sigma$-ideals and $\Sigma$-tensor norms (see Definition \ref{maximalhull}, Definition \ref{finitelygenerated} and Definition \ref{cofinitelygenerated}).
\end{remark}
The symbol $\Lbxx$ denote the Banach space of all multilinear forms $\fhi:\xpx\into \kk$ whose linearization $\tilde{\fhi}:\xtxb\into \kk$ is bounded provided with the norm $\|\fhi\|_\beta:=\|\tilde{\fhi}\|$. Plainly, the mapping $\fhi\mapsto\tilde{\fhi}$ defines a linear isometry between $\Lbxx$ and $\xtxb^*$.
The universal property of the projective tensor product establishes that for every bounded multilinear operator $T:\xpx\into Y$ there exists a unique bounded linear operator $\tlin:\xtxp\into Y$ such that $T\xxp=\tlin(\xxt)$ for all $x^i\in X_i$, $1 \leq i \leq n$. In particular, the restriction $\tlin|_{\sxx}:\sxx\into Y$ is a homogeneous Lipschitz function. In this situation, the operator $\tlin$ is called the linearization of $T$ and the function $f_T:=\tlin|_{\sxx}$ is named the bounded $\so$ associated to $T$. Under these circumstances \eqref{lipschitzequivalent} implies
\begin{equation}\label{lipnorm}
\|T\|=Lip^\pi(f_T)\leq\Lb(f_T)\leq 2^{n-1}Lip^\pi(f_T)
\end{equation}
for all bounded multilinear operators $T$ and reasonable crossnorms $\beta$.
\section{ $\Sigma$-Ideals of Multilinear Operators}
We begin our study extending the definition of linear operator ideals (see \cite[Section 9]{defant93}) to the multilinear case following the approach of $\Sigma$-operators. Specifically, we introduce the notion of $\Sigma$-ideal of multilinear operators. In a few words, this new concept is intended to group multilinear operators taking into account their associated $\Sigma$-operators. For this reason, all the properties defining $\Sigma$-ideals require the language of $\Sigma$-operators. For a better exposition, before presenting each new definition we expose necessary details and notation.
An arrangement of the form $\xxyb$ where $n$ is a positive integer, $X_i$, $1 \leq i \leq n$, $Y$ are Banach spaces and $\beta$ is a reasonable crossnorm on $\xtx$ is named an \emph{election in the class} $\ban$. Obvious definition follows for elections in the class of finite dimensional normed spaces $\finn$.
The essential feature of a linear operator ideal is the ideal property. That is, the property which ensures that a composition $RTS$ of linear operators is in a given ideal whenever $T$ is. The first step in our development is to specify the corresponding ideal property. To this end, consider two elections $\xxyb$ and $(m,\zzzm,W)$ in the class $\ban$, a bounded multilinear operator $\Txxy$, a bounded linear operator $S:Y\into W$ and a bounded multilinear operator $R:\zpzm\into \xtxb$ such that $\rlin:(\ztzm,\theta)\into \xtxb$ is bounded and has image contained in $\sxx$. With the help of the diagram
\begin{equation}\label{idealproperty}
\begin{array}{c}
\xymatrix{
\zpzm\ar[d]\ar[drr]^-R &&\xpx\ar[d]\ar[drr]^-T && &\\
\szzm\ar[d]\ar[rr]^{\fr} &&\sxx\ar[d]\ar[rr]^{f_T} &&Y\ar[r]^S &W\\
(\ztzm,\theta)\ar[rr]^-{\rlin} &&\xtxb &&&
}
\end{array}
\end{equation}
it is plain that $Sf_TR=Sf_Tf_R\tens$, where $\tens:\zpzm\into \ztzm$ is the natural multilinear mapping. Thus, instead of studying a composition from $\zpzm$ to $W$ we study $Sf_Tf_R$ taking advantage of the geometry that the normed spaces $(\ztzm,\theta)$ and $\xtxb$ has to offer. A $\Sigma$-operator $f_R$ as above is named $\mathbf{\Sigma}$-$\mathbf{\beta}$-$\mathbf{\theta}$-{\bf operator}.
\begin{definition}\label{ideal}
A {\bf $\mathbf{\Sigma}$-ideal of multilinear operators} $\ideal$ defined on $\ban$ assigns to each election $\xxyb$ in the class of Banach spaces $\ban$ a linear subspace, named component, $\Abxxy$ of $\Lxxy$ and a norm $\Ab$ on $\Abxxy$ which makes it a Banach space and satisfy the following properties:
\begin{itemize}
\item [I1] Every rank-1 multilinear operator $\fhi\cdot y$, where $\fhi\in\Lbxx$ and $y\in Y$, is in the component $\Abxxy$ and $\Ab(\fhi\cdot y)\leq\|\fhi\|_\beta\;\|y\|$.
\item [I2] $\Lb(f_T) \leq\Ab(T)$ for all $T$ in $\Abxxy$.
\item [I3] If in the composition
\[\begin{array}{c}
\xymatrix{
\szzm\ar[r]^-\fr & \sxx\ar[r]^-{f_T} & Y\ar[r]^-S & W \\
}
\end{array}\]
$\fr$ is a $\Sigma$-$\beta$-$\theta$-operator, $T$ is in $\Abxxy$ and $S:Y\into W$ is a bounded linear operator, then the multilinear operator $Sf_TR:\zpzm\into W$ belongs to $\Atzzmw$ and $\At(Sf_TR)\leq \|\rlin\| \Ab(T) \|S\|$.
\end{itemize}
\end{definition}
As the reader knows, there exist other approaches of ideals of multilinear operators highly influenced by ideas of Pietsch \cite{pietsch84}. This is the case of hyper-ideals developed by Botelho and Torres \cite{botelho15} and multi-ideals developed by Floret and Hunfeld \cite{floret_hunfeld02}. This late is the most popular notion and many of known examples of classes of multilinear operators fit in this framework. Next, we will see that a $\Sigma$-ideal $\ideal$ enjoys the benefits of multi-ideals although, strictly speaking, they are not comparable (since for defining a component in a $\Sigma$-ideal we have to provide a reasonable crossnorm $\beta$).
\begin{proposition}\label{multiideal}
Let $\ideal$ be a $\Sigma$-ideal in the class $\ban$. Let $(n, \xxx, Y,\beta)$ be an election in the class of Banach spaces. Then
\begin{itemize}
\item [(i)] Every rank-1 multilinear operator of the form
\begin{eqnarray*}
x_1^*\tens \dots \tens x_n^*\tens y,:\xpx &\into & Y\\
\xxp &\mapsto & x_1^*(x^1)\dots x_n^*(x^n)y,
\end{eqnarray*}
where $x_i^*\in X_i^*$, $1 \leq i \leq n$ and $y\in Y$, is in $\Abxxy$ and $\Ab(x_1^*\tens \dots \tens x_n^*\tens y)\leq \|x_1^*\|\dots \|x_n^*\|\| y\|$.
\item [(ii)] The inclusion $\Abxxy\subset \Lxxy$ is bounded and has norm at most 1.
\item [(iii)] $\Ab(\kk\times\dots\times\kk \ni (\lambda_1,\dots, \lambda_n)\mapsto \lambda_1\cdots\lambda_n\in\kk)=1$.
\item [(iv)] Let $T_i:Z_i\into X_i$, $1\leq i\leq n$ and $S:Y\into W$ be bounded linear operators and let $T$ in $\Abxxy$. Then $ST(T_1\times\dots\times T_n)$ is in $\mathcal{A}^\pi(\zzzn;W)$ and $A^\pi(STR)\leq \|T_1\|\dots\|T_n\|\Ab(T)\|S\|$.
\end{itemize}
\end{proposition}
\begin{proof}
The proof of (i) is immediate from Property I1; (ii) is a direct consequence of \eqref{lipnorm} and I2; (i) and (ii) imply (iii). For the proof of (iv) take $R=T_1\times\dots\times T_n$. The composition $STR$ can be factored as $Sf_T(\tens\circ (T_1\times\dots\times T_n))$, where $\tens:\xpx\into \sxx\subset\xtx$ is the natural multilinear map. It is clear that the composition $\tens\circ (T_1\times\dots\times T_n)$ is a $\Sigma$-$\beta$-$\pi$-operator. Then, I3 completes the proof.
\end{proof}
Among the benefits of the ideal property for $\Sigma$-ideals we have that it enable us to relate components of $n$-linear operators with components of $m$-linear operators not just $n$-linear operators with itself as is the case of multi-ideals.
An interesting situation occurs when considering those multilinear operators whose rank is contained in finite dimensional subspaces. Explicitly, let $\Fbxxy$ be the linear space of all bounded multilinear operators $T:\xpx\into Y$ such that their linearization $\tlin:\xtxb\into Y$ is bounded and $T(\xpx)$ is contained in a finite dimensional subspace of $Y$. According to Proposition \ref{multiideal} (i), $\Fbxxy$ is contained in the component $\Abxxy$ for every $\Sigma$-ideal $\ideal$. Recall that this does not occur, in general, for multi-ideals (see \cite[Lemma 5.4]{jarchow07}).
The next step in our development is to specify the maximality for $\Sigma$-ideals. This property is intended to study multilinear operators by means of their behavior on finite dimensional spaces. For specifying maximality, fix a $\Sigma$-ideal $\ideal$ and an election $\xxyb$ in $\ban$ and let $T\in\Abxxy$. Let $E_i$ be a finite dimensional subspace of $X_i$ for $1\leq i \leq n$ and let $L$ be a cofinite dimensional subspace of $Y$. Consider the multilinear operator between finite dimensional normed spaces $Q_L T (I_{E_1}\times\dots\times I_{E_n}):\epe\into Y/L$ where $I_{E_i}$ is the natural inclusion of $E_i$ into $X_i$ and $Q_L$ is the canonical quotient map from $Y$ onto $Y/L$. In a maximal $\Sigma$-ideal we are able to calculate $\Ab(T)$ taking into account the values $A^\br(Q_L T (I_{E_1}\times\dots\times I_{E_n}))$ (see Remark \ref{betarestricted}). For shorten notation let $I_{\eee}:\epe\into \xtxb$ be the multilinear map defined by $I_{\eee}\xxp=\xxt$. Plainly, $I_{\eee}$ is a $\Sigma$-$\beta$-$\br$-operator.
\begin{definition}\label{maximality}
A $\Sigma$-ideal $\ideal$ is named maximal if
\[\Ab(T)=\sup A^{\br}(Q_L f_T I_{\eee}:\epe\into Y/L)\]
for all $T$ in $\Abxxy$ where the suprema is taken over all $E_i\in\mathcal{F}(X_i)$, $1\leq i\leq n$, and $L\in\mathcal{CF}(Y)$.
\end{definition}
Most of the classical maximal ideals of linear operators admit a multilinear version following the approach of $\Sigma$-operators.
\begin{example}\label{boundedmultilinearoperators} \textbf{The $\mathbf{\Sigma}$-ideal of bounded multilinear operators.} Let $\sxxb$ be the metric space resulting by restricting the norm $\beta$ to $\sxx$. Recall that a multilinear operator $\Txxy$ is bounded if and only if $f_T:\sxxb\into Y$ is Lipschitz (see Section 2). Let $\mathcal{L}_{Lip}^\beta(\xxx;Y)$ be the Banach space $\Lxxy$ endowed with the norm $\Lb$. The $\Sigma$-ideal whose components are $\mathcal{L}_{Lip}^\beta(\xxx;Y)$ is a maximal $\Sigma$-ideal.
\end{example}
\begin{example}\textbf{The $\mathbf{\Sigma}$-ideal of Lipschitz $\mathbf{p}$-summing multilinear operators.} Following \cite{angulo18}, a multilinear operator $\Txxy$ is Lipschitz $\beta$-$p$-summing if there exists a positive constant $C$ such that
\[\sumim \|T(a_i)-T(b_i)\|^p\leq C^p \sup \left\{ \sumim |\fhi(a_i)-\fhi(b_i)|^p : \fhi\in B_{\Lbxx} \right\}\]
for all finite sequences $(a_i)_{i=1}^m$ and $(b_i)_{i=1}^m$ in $\xpx$. The smallest constant $C$ as above is denoted by $\pi_p^{Lip,\beta}(T)$ and is named the Lipschitz $\beta$-$p$-summing norm of $T$. Let $\Pi_{p}^{Lip,\beta}(\xxx;Y)$ denote the Banach space of all Lipschitz $\beta$-$p$-summing multilinear operators from $\xpx$ to $Y$ endowed with the norm $\pi_p^{Lip,\beta}$. Then the collection defined as $\Pi_p^{Lip}=\bigcup \Pi_{p}^{Lip,\beta}(\xxx;Y),$ where the union is taken over all elections $(n,\xxx,Y,\beta)$ in $\ban$, is a maximal $\Sigma$-ideal. For the case $n=1$ see \cite[Chapter 2]{diestel95}.
\end{example}
\begin{example}\label{factorhilbert}\textbf{The $\mathbf{\Sigma}$-ideal of multilinear operators factoring through Hilbert spaces.} In \cite{fernandez-unzueta18a} it is specified the notion of these operators for the case $\pi$. An easy adaptation produced the next definition. Fix and election $(n,\xxx, Y, \beta)$ in $\ban$. The multilinear operator $T:\xpx\into Y$ factors through a Hilbert space with respect to $\beta$ if there exists a Hilbert space $H$, a subset $M$ of $H$, a bounded multilinear operator $A:\xpx\into H$ whose image is contained in $M$, and a Lipschitz function $B:M\into Y$ such that the next diagram commutes
\begin{small}
\begin{equation}\label{factorization}
\begin{array}{c}
\xymatrix{
\xpx\ar[dr]_{A}\ar[rr]^-T & &Y\\
&M\ar[ur]_B\ar[d] &\\
&H &
}
\end{array}.
\end{equation}
\end{small}Define $\Gamma^\beta(T)$ as $\inf \Lb(A)\, Lip(B)$ where the infimum is taken over all possible factorizations as above. Let $\Gamma^\beta(\xxx;Y)$ denote the Banach space of all these operators endowed with the norm $\Gamma^\beta$. Then the collection defined as $\Gamma=\bigcup \Gamma^{\beta}(\xxx;Y)$, where the union is taken over all elections $(n,\xxx,Y,\beta)$ in $\ban$ is a maximal $\Sigma$-ideal. For the case $n=1$ see \cite[Section 2.b]{pisier86}
\end{example}
\begin{example}\label{dominated}\textbf{The $\mathbf{\Sigma}$-ideal of $\mathbf{(p,q)}$-dominated multilinear operators.} In \cite[Definition 4.1]{fernandez-unzueta18b} we may find: Let $1\leq p,q \leq \infty$ such that $1/p+1/q \leq 1$. Take the unique $r\in[1,\infty]$ such that $1=1/r+1/p+1/q$. The multilinear operator $T:\xpx\into Y$ is called $\beta$-$(p,q)$-dominated if there exists a constant $C>0$ such that
\[\|(\lev y_i^* , T(x_i^1,\dots, x_i^n)- T(z_i^1,\dots, z_i^n) \rev)_{i=1}^m\|_{r^*}\leq C\|(x_i^1\tens\cdots\tens x_i^n-z_i^1\tens\cdots\tens z_i^n)_{i=1}^m\|_{p}^{w,\beta} \|(y_i^*)_{i=1}^m\|_{q}^w\]
holds for all finite sequences $(x_i^1,\dots, x_i^n)_{i=1}^m$, $(z_i^1,\dots, z_i^n)_{i=1}^m$ in $\xpx$ and $(y_i^*)_{i=1}^m$ in $Y$. Here
\[ \|(a_i-b_i )_{i=1}^m\|_{p}^{w,\beta} := \sup \left\{ \left(\sum_{i=1}^m |f_\fhi(a_i)-f_\fhi(b_i)|^{p}\right)^{1/p} : \fhi\in B_{\Lbxx} \right\}. \]
Define $D_{p,q}^\beta(T)$ as the infimum of the constants $C$ as above. Let $\mathcal{D}_{p,q}^\beta(\xxx;Y)$ denote the Banach space of all $\beta$-$(p,q)$-dominated multilinear operators endowed with the norm $D_{p,q}^\beta$. Then the collection defined as $\mathcal{D}_{p,q}=\bigcup \mathcal{D}_{p,q}^{\beta}(\xxx;Y)$, where the union is taken over all elections $(n,\xxx,Y,\beta)$ in $\ban$ is a maximal $\Sigma$-ideal. See \cite[Section 19]{defant93} for the case $n=1$.
\end{example}
As we can see, in some cases the ideals has a simple demeanor. For instance, in the case of the $\Sigma$-ideal of bounded multilinear operators $[\mathcal{L};Lip]$ (see Example \ref{boundedmultilinearoperators}) all the components are isomorphic once we fix a positive integer $n$ and Banach spaces $\xxx$ and $Y$ (see \eqref{lipnorm}). Concretely, we have that
\[ \|Id: \mathcal{L}_{Lip}^\pi(\xxx; Y)\into \mathcal{L}_{Lip}^\beta(\xxx; Y) \|\leq 1 \]
\[ \|Id: \mathcal{L}_{Lip}^\beta(\xxx; Y)\into \mathcal{L}_{Lip}^\pi(\xxx; Y) \|\leq 2^{n-1}. \]
A similar result is valid for the $\Sigma$-ideal of multilinear operators which can be factored through a Hilbert space $[\Gamma,\gamma]$ (see Example \ref{factorhilbert}). In other situation it is not clear if the components are related in general, for instance we do not know if a Lipschitz $\pi$-$p$-summing multilinear operator is Lipschitz $\beta$-$p$-summing for some other reasonable crossnorm $\beta$. In Section 5.2.1 we explore more about this phenomena.
\section{$\Sigma$-Tensor Norms}
Throughout this section we extend the concept of tensor norm for two factors (see \cite[Section 12.1]{defant93}) to the case of several factors having in mind that we are interested in studying multilinear operators by means of their associated $\Sigma$-operators. The outcome of this approach is the concept of $\Sigma$-tensor norm. As we will see, these norms appear in two different version: \emph{on spaces} and \emph{on duals}.
\subsection{$\mathbf{\Sigma}$-Tensor Norms on Spaces}
This type of tensor norms is a procedure that assign a norm to each tensor product of the form $\xtxty$ such that their topological dual identifies a component of a $\Sigma$-ideal, that is, multilinear operators from $\xpx$ to $Y^*$ (see Theorem \ref{rt} and \eqref{conclusion}).
\begin{definition}\label{tns}
A {\bf $\Sigma$-tensor norm on spaces} $\alpha$ on the class of $\ban$ assigns, to each election $\xxyb$ in $\ban$, a norm $\alpha^\beta$ on the algebraic tensor product $\xtxty$ such that:
\begin{itemize}
\item [S1] $\alpha^\beta((p-q)\tens y)\leq\beta(p-q)\,\|y\|$ for every $p,q\in\sxx$ and $y\in Y$.
\item [S2] For every $\fhi\in\Lbxx$ and $y^*\in Y$ the linear functional
\begin{eqnarray*}
\fhi\tens y^*:\xtxty &\into & \kk\\
\xxyt &\mapsto & f_\fhi(\xxt)y^*(y)
\end{eqnarray*}
is bounded and $\|\fhi\tens y^*\|\leq\|\fhi\|_\beta\,\|y^*\|$.
\item [S3] If $\fr:\szzm\into\sxx$ and $S:W\into Y$ denote a $\Sigma$-$\beta$-$\theta$-operator and a bounded linear operator respectively, then the tensor product operator
\begin{eqnarray*}
\fr\tens S:\left(\ztzm\tens W,\alpha^\theta\right) &\into & \left(\xtxty,\alpha^\beta\right)\\
z^1\tens\dots\tens z^m\tens w &\mapsto & \fr(z^1\tens\dots\tens z^m)\tens S(w)
\end{eqnarray*}
is bounded and $\|f_R\tens S\|\leq\|\rlin\|\,\|S\|$.
\end{itemize}
\end{definition}
The authors of \cite{floret_hunfeld02} also develop the notion of tensor norms for tensor products of several factors. Next, we show that $\Sigma$-tensor norms on spaces enjoy the same benefits.
\begin{proposition}
Let $\alpha$ be a $\Sigma$-tensor norm on spaces in the class $\ban$. Let $(n,\xxx,Y,\beta)$ be an election in the class of Banach spaces. Then
\begin{itemize}
\item [(i)] $\ab(\xxt\tens y)\leq \|x^1\|\dots \|x^n\|\|y\|$ for all $x^i\in X_i$ and $y\in Y$.
\item [(ii)] The functionals $x_1^*\tens\dots\tens x_n^*\tens y^*:(\xtxty,\ab)\into \kk$ are bounded and $\|x_1^*\tens\dots\tens x_n^*\tens y^* \|\leq \|x_1^*\|\dots \|x_n^*\| \|y^*\|$ for all $x_i^*\in X_i^*$ and $y\in Y^*$.
\item [(iii)] If $T_i:Z_i\into X_i$, $1\leq i\leq n$ and $S:Y\into W$ be bounded linear operators, then
\begin{eqnarray*}
T_1\tens\dots\tens T_n\tens S:\left(\ztz\tens W,\alpha^\pi\right) &\into & \left(\xtxty,\alpha^\beta\right)\\
z^1\tens\dots\tens z^n\tens w &\mapsto & T_1(z^1)\tens\dots\tens T_n(z^n)\tens S(w)
\end{eqnarray*}
is bounded and $\|T_1\tens\dots\tens T_n\tens S\|\leq \|T_1\|\dots\|T_n\|\|S\|$.
\end{itemize}
\end{proposition}
\begin{proof}
Items (i) and (ii) are immediate from definition since $\beta$ is a reasonable crossnorm. Item (iii) follows from the fact that the composition $R=\tens\circ (T_1\times\dots\times T_n)$ is a $\Sigma$-$\beta$-$\pi$-operator, where $\tens:\xpx\into \sxx\subset\xtxb$ is the natural multilinear map. The proof is complete by applying Property S3.
\end{proof}
The next step in our development is to specify a proper notion of finitely generation for $\Sigma$-tensor norms on spaces (see \cite[Section 12.4]{defant93}). Roughly speaking, a finitely generated $\Sigma$-tensor norm on spaces $\alpha$ allows to calculate $\ab(u)$ of any $u$ in $\xtxty$ by considering those finite dimensional tensor products $\etetf$, where $E_i\subset X_i$ and $F\subset Y$, which contain $u$. As well as maximality for $\Sigma$-ideals we have to consider the reasonable crossnorm $\br$ and take into account the values $\alpha^\br(u;\eee, F)$ (see Remark \ref{betarestricted}).
\begin{definition}\label{finitelygenerated}
A $\Sigma$-tensor norm on spaces $\alpha$ is named finitely generated if
\[\ab(u;\xxx,Y):=\inf \abr(u;\eee,F) \]
where the infimum is taken over all $E_i\in\mathcal{F}(X_i)$, $1\leq i\leq n$ and $F\in\mathcal{F}(Y)$ such that $u$ is contained in $\etetf$.
\end{definition}
\subsubsection*{The Greatest and Least $\Sigma$-Tensor Norms on Spaces}
Examining properties $S1$ and $S_2$ of Definition \ref{tns} and motivated by the injective and projective tensor norms for two factors, we define for each election $\xxyb$ in the class $\ban$
\begin{align*}
\pi^\beta(u) &:=\inf \left\{ \sumim \beta(\pimqi)\|y_i\| \;|\; u=\sumim (\pimqi)\tens y_i \; ; \; p_i,\,q_i\in\sxx, y_i\in Y \right\}\\
\varepsilon^\beta(u)&:=\sup \left\{\; |\lev \fhi\tens y^* , u \rev| \;|\; \|\fhi\|_\beta\leq1,\; \|y^*\|\leq 1 \;\right\}.
\end{align*}
for all $u\in \xtxty$.
Let $\pi$ denote the assignment defined as follows: For each election $(n,\xxx, Y, \beta)$ in the class $\ban$, $\pi$ assigns the norm $\pi^\beta$. Similarly define $\varepsilon$. Direct from definition we have that $\pi^\beta$ and $\varepsilon^\beta$ give rise to $\Sigma$-tensor norms on spaces. Moreover, in the next proposition we prove that $\pi$ and $\varepsilon$ are the greatest and least $\Sigma$-tensor norms on spaces, respectively.
\begin{proposition}
Let $\xxyb$ be an election in the class $\ban$. A norm $\ab$ on $\xtxty$ verifies S1 and S2 if and only if
\[\varepsilon^\beta(u)\leq \ab(u)\leq\pi^\beta(u)\qquad \forall\; u\in\xtxty.\]
\end{proposition}
The proof of this proposition just requires elementary tensor products techniques and are easily adapted from the case of two factors, so we omit it.
\begin{example} Let $\xxyb$ be an election in $\ban$. Define
\[\gamma^\beta(u)=\inf \|(a_i-b_i)\|_2^\beta\,\|(y_i)\|_2\]
where the infimum is taken over all representation of the form $u=\sum_{i=1}^m(\pimqi)\tens y_i$, such that $p_i,q_i\in\sxx, y_i\in Y$ and $(p_i,q_i)\leq_\beta (a_i,b_i)$. Here, $\|(a_i-b_i)\|_2^\beta=\left(\sum_{i=1}^m \beta(p_i-q_i)^2\right)^{1/2}$ and $(p_i,q_i)\leq_\beta (a_i,b_i)$ means that $\sum_{i=1}^m |f_\fhi(p_i)-f_\fhi(q_i)|^2\leq \sum_{i=1}^m |f_\fhi(a_i)-f_\fhi(b_i)|^2 $ for all $\fhi$ in $\Lbxx$. The case $n=1$ can be consulted in \cite[Section 2.b]{pisier86}
\end{example}
\begin{example}{\bf The Laprest\'e $\Sigma$-Tensor Norms on Spaces.} We recall the next definition from \cite[Definition 4.1]{fernandez-unzueta18a}; the case $n=1$ is detailed in \cite[Section 12.5]{defant93}. Let $1\leq p,q\leq \infty$ such that $1/p+1/q\geq1$. Take the unique $r\in[1,\infty]$ with the property $1=1/r+1/q^*+1/p^*$. The Laprest\'e $\Sigma$-tensor norm on spaces $\alpha_{p,q}^\beta$ on $\xtxty$ is defined by
\[\alpha_{p,q}^\beta(u):=\inf \|(\lambda_i)_{i=1}^m\|_r \|(\pimqi)_{i=1}^m\|_{q^*}^{w,\beta} \|(y_i)_{i=1}^m\|_{p^*}^w\]
where the infimum is taken over all representations of the form $u=\sum_{i=1}^m \lambda_i(p_i-q_i)\tens y_i$ with $\lambda_i\in \kk$, $p_i,q_i\in\sxx$, $y_i\in Y$ (see Example \ref{dominated}).
As it is well known, the Chevet-Saphar tensor norms are particular cases of the Laprest\'e tensor norm. The corresponding cases acquire the form
\begin{align*}
d_p^\beta(u):=\alpha_{1,p}^\beta(u)&=\inf\left\{ \|(\pimqi)_i \|_{p}^{w,\beta} \|(y_i)_i\|_{p^*} \,\Big|\, u=\sumim (\pimqi)\tens y_i \right\}\\
w_p^\beta(u):=\alpha_{p,p^*}^\beta(u)&=\inf\left\{ \|(\pimqi)_i \|_{p}^{w,\beta} \|(y_i)_i\|_{p^*}^w \,\Big|\, u=\sumim (\pimqi)\tens y_i \right\}\\
g_p^\beta(u):=\alpha_{p,1}^\beta(u)&=\inf\left\{ \|(\pimqi)_i \|_{p}^\beta \|(y_i)_i\|_{p^*}^w \,\Big|\, u=\sumim (\pimqi)\tens y_i \right\}\\
\alpha_{1,1}^\beta(u)=d_{1}^\beta(u)&=g_1^\beta(u)=\pi^\beta(u)
\end{align*}
\end{example}
The assignments $\pi$, $\varepsilon$, $\gamma_2$ and $\alpha_{p,q}$ are examples of finitely generated $\Sigma$-tensor norms on spaces. The cases $\pi$ and $\varepsilon$ follows from definition, the case $\gamma_2$ is \cite[Proposition 4.3]{fernandez-unzueta18b} and the Laprest\'e case is \cite[Proposition 4.3]{fernandez-unzueta18a}.
\subsection{$\mathbf{\Sigma}$-Tensor Norms on Duals}
In ideals of operators, not necessarily linear or multilinear, operators which have range contained in finite dimensional subspaces are of fundamental importance. As we saw in Section 3, $\Fbxxy\subset \Abxxy$ holds linearly for every $\Sigma$-ideal $\ideal$. It is not difficult to see that $\Fbxxy$ is linearly isomorphic to $\Lbxx\tens Y$. In this section we give a precise definition of those norms on $\Lbxx\tens Y$ which identify the space $\Fbxxy$ endowed with the norm $\Ab$ (see Theorem \ref{tndsideal} and \eqref{conclusion}).
The most technical requirement of the definition of $\Sigma$-tensor norms on duals is the uniform property. In Section 4.2 we saw that the uniform property for $\Sigma$-tensor norms on spaces requires the notion of the so called $\Sigma$-$\beta$-$\theta$-operators. Vaguely, for defining $\Sigma$-tensor norms on duals we need a dual concept of $\Sigma$-$\beta$-$\theta$-operators.
Let $n, m$ positive integers, $\xxx$, $\zzzm$ be Banach spaces and $\beta$ and $\theta$ be two reasonable crossnorms on $\xtx$ and $\ztzm$, respectively. If the linear operator $A:\Lbxx\into\Ltzzm$ is such that its adjoint linear operator $A^*:\ztzmt^{**}\into \xtxb^{**}$ verifies
\[A^*K_{\ztzt}(\szzm)\subset K_{\xtxb}(\sxx),\]
we say that $A$ {\bf preserves $\mathbf{\Sigma}$}.
\begin{definition}\label{tnd}
A {\bf $\mathbf{\Sigma}$-tensor norm on duals} $\nu$ on the class of Banach spaces assigns, to each election $\xxyb$ in $\ban$, a norm $\vb$ on the tensor product $\Lbxx\tens Y$ with the following properties:
\begin{itemize}
\item [D1] $\vb(\fhi\tens y)\leq\|\fhi\|_\beta \|y\|$ for every $\fhi\in\Lbxx,\; y\in Y.$
\item [D2] For every $p,q\in\sxx$ and $y^*\in Y^*$ the linear functional
\begin{eqnarray*}
(\pmq)\tens y^*:\Lbxx\tens Y &\into & \kk\\
\fhi\tens y &\mapsto & (f_\fhi(p)-f_\fhi(q))y^*(y)
\end{eqnarray*}
is bounded and $\|(\pmq)\tens y^*\|\leq\beta(\pmq)\|y^*\|$.
\item [D3] If $A:\Lbxx\into\Ltzzm$ and $B:Y\into W$ denote bounded linear operators where $A$ preserves $\Sigma$, then the linear operator
\begin{eqnarray*}
A\tens B:\left(\Lbxx\tens Y,\vb\right) & \into &\left(\Ltzzm\tens W,\vt\right)\\
\fhi\tens y& \mapsto & A(\fhi)\tens B(y)
\end{eqnarray*}
is bounded and $\|A\tens B\|\leq\|A\|\,\|B\|$.
\end{itemize}
\end{definition}
The next step int our development is to define a well behavior of $\Sigma$-tensor norm on duals in the general case of Banach spaces regarding finite dimensional spaces. To be explicit we define when a $\Sigma$-tensor norm on duals is cofinitely generated.
Consider Banach spaces $\xxx$ and finite dimensional subspaces $E_i$ of $X_i$ for $1 \leq i \leq n$. Let $R_{\eee}:\Lbxx\into \mathcal{L}^{\br}(\eee)$ be the linear operator which maps each $\fhi$ to its restriction to $\epe$ (see Remark \ref{betarestricted}). The analogous case of this property for the case $n=1$ is defined in \cite[Section 12.4]{defant93}.
\begin{definition}\label{cofinitelygenerated}
A $\stn$ on duals $\nu$ on the class $\ban$ is named cofinitely generated if
\[\vb(v;\Lbxx,Y):=\sup \vbr\left( R_{\eee}\tens Q_L (v); \Lbree, Y/L\right)\]
where the suprema is taken over all $E_i\in \mathcal{F}(X_i)$ and $L\in\mathcal{CF}(Y)$.
\end{definition}
\subsubsection*{The Greatest and the Least $\Sigma$-Tensor Norms on Duals}
For each election $\xxyb$ in $\ban$ define
\begin{align*}
\pi_\beta(v)&:= \inf \left\{\;\sumim \|\fhi\|_\beta \|y_i\| \;\Big|\; v=\sumim\fhi_i\tens y_i \;\right\}\\
\varepsilon_\beta(v)&:=\sup\left\{\; |\lev (\pmq)\tens y^* , v \rev|\;\Big|\; \beta(\pmq)\leq 1,\, \|y^*\|\leq 1 \;\right\}
\end{align*}
for all $v\in\Lbxx\tens Y$. As it is expected, these norms are the greatest and least $\Sigma$-tensor norms on duals, respectively.
\begin{proposition}
Let $\xxyb$ be an election in the class $\ban$. A norm $\vb$ on the tensor product $\Lbxx\tens Y$ verifies D1 and D2 if and only if
\[\varepsilon_\beta(v) \leq\vb (v) \leq\pi_\beta(v)\quad \forall v\in \Lbxx\tens Y.\]
\end{proposition}
The proof of this proposition is easily adapted from the case of two factors, so we omit it.
\subsection{The Duality Relation between $\Sigma$-Tensor Norms in Finite Dimensions}
In this section we exhibit the relation between the two types of $\Sigma$-tensor norms in the class of finite dimensional normed spaces. As we will see, this relation is in a dual fashion.
In general, the linear spaces $\Lbxx\tens Y$ and $\xtxtyd$ have not the same dimension. For this reason, we do not expect to relate the two types of tensor norms in general Banach spaces in an isomorphically fashion (see Theorem \ref{dt}). Nevertheless, the finite dimensional case is simpler and more illustrative.
\begin{theorem}\label{tnstotnd}
Every $\Sigma$-tensor norm on spaces $\alpha$ on the class $\finn$ defines a $\Sigma$-tensor norm on duals $\nu$ on the class $\finn$ by
\begin{equation}\label{linearduality}
\left(\Lbee\tens F,\vb\right):=\left(\etetfd,\ab\right)^*.
\end{equation}
Every $\Sigma$-tensor norm on duals $\nu$ on the class $\finn$ defines a $\Sigma$-tensor norm on spaces $\alpha$ on the class $\finn$ by
\[\left(\etetf,\ab\right):=\left(\Lbee\tens F^*,\vb\right)^*\]
\end{theorem}
Before proving this theorem let us make an observation: If we restrict \eqref{linearduality} to the case $n=1$ we obtain $(E^*\tens F,v)=(E\tens F^*,\alpha)^*$, or equivalently, $(E\tens F,v)=(E^*\tens F^*,\alpha)^*$ for all finite dimensional normed spaces. This last equation is nothing more than the construction of the dual tensor norm $\nu=\alpha'$ of $\alpha$ for the case of two factors. Under this viewpoint, we may consider \eqref{linearduality} as an extension of the procedure to define the dual tensor norm of a given tensor norm. In our setting we have that the dual norm of a $\Sigma$-tensor norm on spaces is a $\Sigma$-tensor norm on duals (and reciprocally).
\begin{proof}
We only proof that \eqref{linearduality} gives rise to a $\Sigma$-tensor norm on duals since the opposite direction is analogous. Let $\alpha$ be a $\Sigma$-tensor norm on spaces and let $\eefb$ be an election in $\finn$.
{\bf $\vb$ verifies D1}: S2 is equivalent to D1 since for every $\fhi\in\Lbee$ and $y\in F$ we have $\vb(\fhi\tens y):= \|\fhi\tens y : \left(\etetfd,\ab\right) \into \kk \|\leq \|\fhi\|_\beta\, \|y\|$.
{\bf$\vb$ verifies D2}: Since the involved spaces are finite dimensional we have that the functionals defined by $(\pmq)\tens y^*$ are bounded. The definition of $\vb$ implies
\[\sup\limits_{\vb(v)\leq 1}|\lev (\pmq)\tens y^* \,,\, v \rev|=\ab((\pmq)\tens y^*)\leq\beta(\pmq)\ \|y^*\|.\]
{\bf$\vb$ verifies D3}: Let $(m,M_1, \dots, M_m, N, \theta)$ be an election in the class $\finn$. Consider a bounded linear operator that preserves $\Sigma$ $A:\Lbee\into\mathcal{L}^\theta(M_1, \dots, M_m)$ and a bounded linear operator $B:N\into F$. The finite dimensional assumption lets us consider $A^*:(M_1\tens \cdots\tens M_m,\theta)\into (E_1\tens \cdots\tens E_n,\beta)$. The linearity of $A^*$ lets us define the multilinear operator
\begin{eqnarray*}
T:M_1\times\dots\times M_m &\into & (E_1\tens \cdots\tens E_n,\beta)\\
(z_1,\dots, z_m) &\mapsto & A^*(z_1\tens\cdots\tens z_m).
\end{eqnarray*}
The universal property of the tensor product implies $\tlin=A^*$. Hence, $\tlin:(M_1\tens\cdots\tens M_m,\theta)\into (\ete,\beta)$ is bounded and $\|\tlin\|=\|A\|$. Even more, the set $\{ A^*(p) \,|\, p\in\Sigma_{M_1,\dots, M_m} \}$ is contained in $\Sigma_{\eee}$ since $A$ preserves $\Sigma$. In other words, the $\Sigma$-operator $\ft:\Sigma_{M_1,\dots, M_m}^\theta \into (E_1\tens \cdots\tens E_n,\beta)$ is a $\Sigma$-$\beta$-$\theta$-operator. The uniform property of $\alpha$ implies that $\ft\tens B^* :\left(M_1,\tens\cdots\tens, M_m\tens N,\alpha^\theta\right) \into \left(\etetfd,\ab\right)$ is bounded and $\|\ft\tens B^*\|\leq\|A\|\, \|B^*\|$.
The boundedness of $A\tens B:\left(\Lbee\tens F,\vb\right)\into \left(\mathcal{L}^\theta(M_1,\dots, M_n)\tens N,\vt\right)$ is deduced from
\begin{align*}
|\lev A\tens B (v)\,,\, u\rev\big| =\Big|\lev v\,,\, \ft\tens B^*(u)\rev\big|&\leq \vb(v)\,\ab\left(\ft\tens B^*(u) ; \eee, F^*\right)\\
&\leq \vb(v)\,\|A\|\,\|B\|\,\alpha^\theta(u ; \mmm, N^*).
\end{align*}
\end{proof}
\begin{example}
Adapting the case of two factors leads us to the fact that $\varepsilon_\beta$ is dual to $\pi^\beta$. In other words they verify a relation as in \eqref{linearduality}. In this particular case we can say more: the space $ (\Lbxx\tens Y ,\varepsilon_{\beta})$ is isometrically contained in the dual space $(\xtxtyd,\pi^\beta)^*$ via the morphism given by evaluation for all election $(n,\xxx,Y,\beta)$ in $\ban$ (see Sections 4.1.1, 4.2.1, Theorem \ref{dt} and \eqref{conclusion}).
\end{example}
\subsection{$\mathbf{\Sigma}$-Tensor Norms and $\mathbf{\Sigma}$-Ideals in the Finite Dimensions}
Next, we show that each $\Sigma$-tensor norm (on duals) gives rise to a unique $\Sigma$-ideal in finite dimensions. For the linear case of this relation the reader may check \cite[Sections 17.1 and 17.2]{defant93}
\begin{theorem}\label{tndsideal}
Every $\Sigma$-tensor norm on duals $\nu$ on $\finn$ defines a $\Sigma$-ideal $\ideal$ on $\finn$ by
\[\Abeef:=\left(\Lbee\tens F,\vb\right).\]
Conversely, every $\Sigma$-ideal $\ideal$ on $\finn$ defines a $\stn$ on duals $\nu$ on $\finn$, by
\[\left(\Lbee\tens F,\vb\right):=\Abeef.\]
\end{theorem}
\begin{proof}
We only prove that a $\Sigma$-tensor norm on duals defines a $\Sigma$-ideal since the converse is completely analogous. Let $\nu$ be a $\stn$ on duals. Plainly, the assignment
$\fhi\tens y\mapsto \fhi\cdot y$ defines a linear isomorphism between $\Lbee\tens F$ and $\Abeef$.
\textbf{$\ideal$ verifies I1}: Immediate from D1.
\textbf{$\ideal$ verifies I2}: Let $T=\sum_{i=1}^l\fhi_i\cdot y_i$ in $\Abeef$ and take $p,q\in\Sigma_{\eee}$ and $y^*\in Y^*$. Algebraic manipulations lead to $\lev y^* , f_T(p)-f_T(q) \rev = \lev (\pmq)\tens y^* \,,\, \sum_{i=1}^l\fhi_i\tens y_i \rev$. Hence, Property D2 of $\nu$ and $\vb(v_T)=\Ab(T)$ assert that
\[|\lev y^* , f_T(p)-f_T(q) \rev|\leq \beta(\pmq)\, \|y^*\|\, \Ab(T).\]
\textbf{$\ideal$ verifies I3}: Consider the composition
\[\begin{array}{c}
\xymatrix{
\Sigma_{M_1\dots M_m}\ar[r]^-\fr & \Sigma_{\eee} \ar[r]^-{f_T} & F\ar[r]^-S & N \\
}
\end{array},\]
where $\fr$ is a $\Sigma$-$\beta$-$\theta$-operator, $T$ is in $\Abeef$ and $S:F\into N$ is a bounded linear operator. According to the isomorphism between $\mathcal{A}^\theta(M_1\dots M_m;N)$ and $\mathcal{L}^\theta(M_1\dots M_m)\tens N$ we have that the multilinear operator $Sf_T R$ is associated with $(\rlin^*\tens S) (v_T)$. The linear operator $\rlin^*: (\ete,\beta)^* \into (M_1\tens\cdots\tens M_m,\theta)^*$ preserves $\Sigma$ since $f_R$ is a $\Sigma$-$\beta$-$\theta$-operator. Hence, applying D3 we obtain $\At(S f_T R) = \vt\left( \rlin^*\tens S (v_f) \right) \leq \|\rlin\|\, \|S\|\, \vb(v_f) = \|\rlin\|\, \|S\|\, \Ab(T)$.
\end{proof}
\section{Duality and Representation Theorems}
In sections 4.3 and 4.4 we established the relation between $\Sigma$-tensor norms and $\Sigma$-ideals in the class $\finn$. Next we extend these relations for Banach spaces. The outcomes are the duality theorem for $\Sigma$-tensor norms and the representation theorem for maximal $\Sigma$-ideals.
\subsection{The Duality Theorem for $\Sigma$-Tensor Norms}
As we saw in Theorem \ref{tnstotnd} the two types of $\Sigma$-tensor norms are in duality relation on the class of finite dimensional normed spaces. For extending this relation to general Banach spaces we need to extend a given $\Sigma$-tensor norms on spaces on finite dimensional normed spaces to Banach spaces.
Let $\alpha$ be a $\Sigma$-tensor norm on spaces defined on the class $\finn$. Let $(n,\xxx,Y)$ be an election in $\ban$. For every $u$ in $\xtxty$ define
\[\ab(u;\xxx,Y):=\inf \abr(u;\eee,F) \]
where the infimum is taken over all $E_i\in\mathcal{F}(X_i)$, $1\leq i\leq n$ and $F\in\mathcal{F}(Y)$ such that $u$ is contained in $\etetf$. Following the definition and using elementary tensor product techniques it is possible to prove that $\alpha$ is a finitely generated $\Sigma$-tensor norm on spaces on the class $\ban$.
According to previous definition and Theorem \ref{tnstotnd}, every $\Sigma$-tensor norm on duals $\nu$ in the class $\ban$ gives rise to a unique finitely generated $\Sigma$-tensor norm on spaces $\alpha$ on the class $\ban$ such that $\left(\etetf,\ab\right)$ is isometrically isomorphic to the dual space $\left(\Lbee\tens F^*,\vb\right)^*$ for all election $(n,\eee, F)$ in $\finn$.
\begin{theorem}\label{dt}{\bf (Duality Theorem of $\Sigma$-Tensor Norms)}
Let $\nu$ be a cofinitely generated $\Sigma$-tensor norm on duals and let $\alpha$ be the finitely generated $\Sigma$-tensor norm on spaces defined by $\nu$. Then $\left(\Lbxx\tens Y,\vb\right)$ is isometrically contained in $\left(\xtxtyd,\ab\right)^*$ for all elections $\xxyb$ in $\ban$.
\end{theorem}
\begin{proof}
Let $\xxyb$ be an election in $\ban$. Let $v=\sum_{j=1}^l \fhi_j\tens y_j$ in $\Lbxx\tens Y$. If $\ab(u; \xxx, Y)<1$ then there exist $E_i\in\mathcal{F}(X_i)$, $1\leq i \leq n$ and $F\in\mathcal{F}(Y^*)$ such that $u\in\etetf$ and $\abr(u; \eee, F)<1$. The space $L=\{y \,|\, y^*(y)=0 \mbox{ for all } y^*\in F\}$ is an element of $\mathcal{CF}(Y)$ such that $F$ is isometric to $(Y/L)^*$. Take any representation of $u$ in $\etetf$ of the form $\sum_{i=1}^m(\pimqi)\tens y_i^*$. Then, algebraic manipulations show that $\lev v , u \rev = \lev R_{\eee}\tens Q_L(v) , \sum_{i=1}^m(\pimqi)\tens z_i^* \rev$ for some $z_i^*\in (Y/L)^*$, $1\leq i\leq m$. By assumption, the space $(\Lbree\tens Y/L,\vbr)$ is isometric to $( \ete\tens (Y/L)^*,\abr)^*$. Hence, $|\lev v , u \rev|\leq \vbr(R_{\eee}\tens Q_L(v)) \, \abr\left(\sum_{i=1}^m (\pimqi)\tens z_i^*\right).$ Since $\abr\left(\sum_{i=1}^m (\pimqi)\tens z_i^*\right)<1$, we have that $|\lev v , u \rev|\leq \vb\left(v;\, \Lbxx, Y\right)$. After taking suprema over all $\ab(u)<1$ in last inequality we obtain
\[\|v:\left(\xtxtyd,\ab\right)\into \kk\|\leq\vb\left(v;\, \Lbxx Y\right).\]
For the converse inequality, let $E_i\in\mathcal{F}(X_i)$, $1 \leq i \leq n$, $L\in\mathcal{CF}(Y)$ and $\eta>0$. There exist $u$ in the space $ E_1\tens\cdots E_n\tens (Y/L)^*$ such that $\abr(u; \eee, (Y/L)^*)<1$ and $\vbr(R_{\eee}\tens Q_L(v),\Lbree, Y/L)\, (1-\eta)\leq |\lev R_{\eee}\tens Q_L(v) \,,\, u\rev|$. The space $(Y/L)^*$ is isometrically isomorphic to a finite dimensional subspace of $F$ of $Y^*$. Then, after algebraic manipulations, we have
\[|\lev R_{\eee}\tens Q_L(v) \,,\, u\rev| = |\lev v , u \rev| \leq \|v:\left(\xtxtyd,\ab\right)\into \kk\|.\]
After taking suprema over all $E_i$, $1 \leq i \leq n$ and $L$ as above we obtain that
\[\vb(u;\ \Lbxx, Y)(1-\eta)\leq\|v:\left(\xtxtyd,\ab\right)\into \kk\|\]
holds for all $\eta>0$.
\end{proof}
The linear version of this result can be consulted in \cite[Section 15.5]{defant93}.
\subsection{The Representation Theorem for Maximal $\Sigma$-ideals of Multilinear Operators}
For proving this result, we need two preliminary results about maximal $\Sigma$-ideals and finitely generated $\Sigma$-tensor norms. A detailed exposition of this result for the case of linear operators can be found in \cite[Section 17]{defant93}.
\begin{proposition}\label{regular}
Let $\ideal$ be a maximal $\Sigma$-ideal. Then $T\in\Abxxy$ if and only if $K_Y T\in \Abxxydd$.
\end{proposition}
\begin{proof}
The ideal property asserts that $K_Y T\in\Abxxydd$ and $\Ab(K_Y T)\leq \Ab(T)$ whenever $T\in\Abxxy$.
Conversely, suppose $K_Y T\in\Abxxydd$. Let $E_i\in\mathcal{F}(X_i)$, $1 \leq i \leq n$ $L\in\mathcal{CF}(Y)$ and $\eta>0$. There exist a finite dimensional subspace $F$ of $Y^*$ isometric to $(Y/L)^*$ via $Q_L^*$. Consider the subspace $H$ of $Y^{**}$ defined as the span of the set $K_Y f_T(\Sigma_{\eee})$. By the Principle of Local Reflexivity there exist a finite dimensional subspace $G$ of $Y$ and an isomorphism $\psi:H\into G$ with $\|\psi\|\leq 1+\varepsilon$ such that $ \lev y^* , \psi(y^{**}) \rev = \lev y^{**} , y^* \rev$ for all $y^*\in F$ and $y^{**}\in H$. Then, algebraic manipulations lead to $\lev \varphi \,,\, Q_L f_T I_{\eee}(p) \rev = \lev \varphi \,,\, Q_L\psi K_Y f_T(p) \rev$ for all $\varphi \in (Y/L)^*$ and $p\in\Sigma_{\eee}$. In other words $Q_L f_T I_{\eee}=Q_L\psi K_Y T$. Finally, the ideal property implies
\[\Ab(Q_L f_T I_{\eee})=\Ab(Q_L\psi K_Y T)\leq\left(1+\eta \right)\, A^\beta(K_YT)\]
which ensures $\Ab(Q_L f_T I_{\eee})\leq \Ab(K_YT)$. The proof is complete after taking suprema, in previous inequality, over all $E_i$, $1\leq i \leq n$ and $L$ as above.
\end{proof}
As usual, every bounded functional $\fhi:X_1\tens_\pi\dots\tens_\pi X_n\tens_\pi Y\into\kk$ defines a bounded multilinear operator $T_\fhi:\xpx\into Y^*$ given by $\lev T_\fhi\xxp , y\rev:= \fhi (\xxt\tens y)$ for all $x^i\in X_i$, $1\leq i\leq n$ and $y\in Y$. The canonical extension of $\fhi$ is denoted by $\varfhi$ and defined by
\begin{eqnarray*}
\overline{\fhi}:\xtx\tens Y^{**} &\into & \kk\\
x^1\tens\dots\tens x^n\tens y^{**} &\mapsto & \lev y^{**} , T_\fhi(\xxt) \rev.
\end{eqnarray*}
If we consider $\xtxty$ as a linear subspace of $\xtx\tens Y^{**}$, the functional $\overline{\fhi}$ is actually an extension of $\fhi$ since for every $x^i\in X_i$, $1\leq i\leq n$ and $y\in Y$ we have $\overline{\fhi}(\xxt\tens y)=\lev T_\fhi(\xxt) , y \rev =\fhi(\xxt\tens y)$.
On the other hand, every bounded multilinear operator $T:\xpx\into Y$ defines a bounded functional $\fhi_T:X_1\tens_\pi\dots\tens_\pi X_n\tens_\pi Y^*\into\kk$ given by $\fhi_T(\xxt\tens y^*)=\lev y^*, T\xxp \rev$ for all $x^i\in X_i$, $1\leq i\leq n$ and $y^*\in Y^*$. It is not difficult to see that $\fhi_{(T_\fhi)}=\varfhi$ for all bounded functional $\fhi:X_1\tens_\pi\dots\tens_\pi X_n\tens_\pi Y\into\kk$ and $T_{(\fhi_T)}=K_YT$ for all bounded multilinear operator $\Txxy$.
\begin{proposition}\label{aronbernerextension}
Let $\alpha$ be a finitely generated $\Sigma$-tensor norm on spaces on the class $\ban$ and let $\xxyb$ be an election in $\ban$. Then:
\begin{itemize}
\item [(i)] $\left(\xtxty,\ab\right)$ is a closed subspace of $\left(\xtxtydd,\ab\right)$.
\item [(ii)] $\fhi\in\left(\xtxty,\ab\right)^*$ if and only if $\overline{\fhi}\in\left(\xtxtydd,\ab\right)^*$.
In this situation $\|\fhi\|=\|\overline{\fhi}\|$.
\end{itemize}
\end{proposition}
\begin{proof}
(i): In this proof we have identified the spaces $Y$ and $K_Y(Y)$. By uniformity of $\alpha$ it is clear that $\ab(u; \xxx ,Y^{**})\leq\ab(u; \xxx, Y)$. For the converse inequality let $u\in\xtxty$, $\eta>0$ and fix a representation of $u$ of the form $\sum_{i=1}^m x_i^1\tens\dots\tens x_i^n\tens y_i$. There exist finite dimensional subspaces $E_i\subset X_i$ and $F\subset Y^{**}$ such that $u\in\etetf$ and
\[\abr(u; \eee, F)\leq (1+\eta)\, \ab(u ; \xxx,Y^{**}).\]
We may assume that $y_i\in F$, $1\leq i \leq n$. The Principle of Local Reflexivity ensures the existence of a finite dimensional subspace $G$ of $Y$ and an isomorphism $\psi:F\into G$ such that $\psi(y_i)=y_i$, $1\leq i \leq n$, and $\|\psi\|\leq(1+\eta)$. Consider the $\Sigma$-$\br$-$\br$-operator given by the identity $I:\seebr\into \seebr$. Again, uniformity of $\alpha$ implies that
\[\abr(I\tens \psi (u); \eee, G) \leq (1+\eta)\,\abr(u; \eee, F).\]
Finally,
\begin{align*}
\ab( u; \xxx, Y) &\leq \abr(u; \eee, G)\\
&= \abr(I\tens \psi (u); \eee, G)\\
&\leq (1+\eta)\, \abr(u; \eee, F)\\
&\leq (1+\eta)^2\, \ab(u ; \xxx, Y^{**})\\
\end{align*}
holds for all $\eta>0$.
(ii): First, suppose $\overline{\fhi}$ is bounded. Let $u\in\left(\xtxty,\ab\right)$. Then $\fhi(u)=\overline{\fhi}(u)$ and $\ab\left(u; \xxx, Y\right)=\ab\left(u; \xxx Y^{**}\right)$. Therefore, $\fhi$ is bounded and $\|\fhi\|\leq\|\overline{\fhi}\|$.
Conversely, suppose $\fhi$ is bounded. Fix $u$ in $\left(\xtxtydd,\ab\right)$ and let $\eta>0$. Since $\alpha$ is finitely generated there exist finite dimensional subspaces
$E_i$ and $F$ of $X_i$, $1\leq i \leq n$, and $Y^{**}$ respectively such that $\etetf$ contains $u$ and
\[\abr\left(u; \eee, F\right)\leq(1+\eta)\, \ab\left(u; \xxx, Y^{**}\right).\]
Let $\sum_{i=1}^m x_i^1\tens\dots\tens x_i^n\tens y_i^{**}$ be a fixed representation of $u$ in $\etetf$. Now, we may apply the Principle of Local Reflexivity to $F\subset Y^{**}$ and $span\{f_\fhi(x_i^1\tens\dots\tens x_i^n)\}\subset Y^*$ to find a finite dimensional subspace $G\subset Y$ and an isomorphism $\psi:F\into G$ with $\|\psi\|\leq 1+\eta$ and $\lev f_\fhi(x_i^1\tens\dots\tens x_i^n) \,,\, \psi(y_i^{**}) \rev=\lev y_i^{**}\,,\,
f_\fhi(x_i^1\tens\dots\tens x_i^n)\rev$ for all $1\leq i \leq n$.
Then $\fhi(x_i^1\tens\dots\tens x_i^n\tens \psi(y_i^{**}))=\overline{\fhi}(x_i^1\tens\dots\tens x_i^n\tens y_i^{**})$, hence $\fhi\circ (I\tens\psi)(u)=\overline{\fhi}(u)$. Finally,
\begin{align*}
|\overline{\fhi}(u)|&=|\fhi\circ (I\tens\psi)(u)|\\
&\leq\|\fhi\|\;\ab\left(I\tens\psi(u); \xxx, Y\right)\\
&\leq\|\fhi\|\;\abr\left(I\tens\psi(u); \eee, G\right)\\
&\leq\|\fhi\|\;(1+\eta)\;\abr\left(u; \eee, F\right)\\
&\leq\|\fhi\|\;(1+\eta)^2\;\ab\left(u; \xxx, Y^{**}\right)
\end{align*}
completes the proof.
\end{proof}
If in Theorem \ref{tndsideal} we take $n=1$, then we recover the original relation between tensor norms and ideals of linear operators. For this reason we say that the $\Sigma$-ideal $\ideal$ and the $\Sigma$-tensor norm on duals $\nu$ are \emph{associated} if $\Abeef$ is isometrically isomorphic to $\left(\Lbee\tens F,\vb\right)$ for all elections $(n,\eee, F, \beta)$ in $\finn$.
\begin{theorem}\label{rt}{\bf (Representation Theorem for Maximal $\Sigma$-ideals)} Let $\nu$ be a $\Sigma$-tensor norm on duals and $\ideal$ be the maximal $\Sigma$-ideal associated to $\nu$. Then
\begin{align*}
\left(\xtxty,\ab\right)^*&=\Abxxyd\\
\left(\xtxtyd,\ab\right)^*\cap\Lxxy&=\Abxxy
\end{align*}
for every election $\xxyb$ in $\ban$, where $\alpha$ is the finitely generated $\stn$ on spaces on the class $\ban$ defined by $\nu$.
\end{theorem}
\begin{proof}
First, we will prove the second equality. Given finite dimensional subspaces $E_i$ of $X_i$, $1\leq i \leq n$ and a finite codimensional subspace $L$ of $Y$ we have, by hypothesis, that
\begin{equation}\label{rt1}
\left(\ete\tens (Y/L),^*\abr\right)^*=\mathcal{A}^{\br}\left(\eee;Y/L\right)
\end{equation}
is a linear isometric isomorphism.
Let $T\in\Abxxy$ and $\eta>0$ fixed. Let $\fhi_T$ be the associated functional of $T$. For $u\in\left(\xtxtyd,\ab\right)$ there exist $E_i\in \mathcal{F}(X_i)$, $1\leq i \leq n$ and $F\in \mathcal{F}(Y^*)$ such that $u\in\left(\etetf,\abr\right)$ and $\abr(u;\eee,F)\leq(1+\eta)\, \ab(u;\xxx, Y)$. The space $F$ defines a finite codimensional subspace $L$ of $Y$ such that $(Y/L)^*=F$ holds linearly and isometrically via $Q_L^*$. Then, \eqref{rt1} ensures
\begin{align*}
|\fhi_T(u)| &= |\fhi_T\circ (I_{\eee}\tens Q_L^*)(u)|\\
&\leq \|\fhi_T\circ (I_{\eee}\tens Q_L^*):\left(\ete\tens (Y/L)^*,\abr\right)\into \kk\| \abr(u)\\
&= A^{\br}(Q_L f_T I_{\eee}) (1+\eta) \ab(u;\xxx,Y)\\
&\leq \Ab(T) (1+\eta) \ab(u;\xxx, Y).
\end{align*}
Hence, $\fhi_T$ is bounded and $\|\fhi_T\|\leq A(T)$.
For the converse inequality let $\fhi\in\left(\xtxtyd, \ab\right)^*$ such that its associated multilinear operator has range contained in $Y$. First, notice that
\[\sup\left\{\; A(Q_L f_{T_\fhi} I_{\eee}) \;|\; E_i\in\mathcal{F}(X_i)\;L\in\mathcal{CF}(Y) \;\right\}<\infty.\]
This is easy to see since
\begin{equation}\label{rt2}
A(Q_L f_{T_\fhi} I_{\eee})=\|\fhi\circ (I_{\eee}\tens Q_L^*)\|\leq\|\fhi\|
\end{equation}
holds for all $E_i\in\mathcal{F}(X_i)$, $1\leq i \leq n$ and $L\in\mathcal{CF}(Y)$. This means that $T_\fhi\in\Abxxy$. Actually, if we take suprema in \eqref{rt2} over all $E_i$ and $L$ we obtain, by maximality, that $A(T_\fhi)\leq\|\fhi\|$.
For the first equality let $T\in\Abxxyd$. We will prove that
\begin{eqnarray*}
\zeta_T:(\xtxty,\ab) &\into &\kk\\
\xxt\tens y &\mapsto & \lev f_T(\xxt), y \rev
\end{eqnarray*}
is bounded. By Proposition \ref{aronbernerextension} this occurs exactly when its canonical extension $\overline{\zeta_T}:(\xtxtydd,\ab)\into \kk$ is bounded. We just have proved that the functional $\fhi_T:(\xtxtydd,\ab)\into \kk$ is bounded and $\|\fhi_T\|=\Ab(T)$. But
\[\fhi_T(\xxt\tens y^{**}) =\lev y^{**} \,,\, f_T(\xxt)\rev =\overline{\zeta_T}(\xxt\tens y^{**})\]
asserts that $\fhi_T=\overline{\zeta_T}$. Hence $\zeta_T$ is bounded and $\|\zeta_T\|=\Ab(T)$.
Conversely, let $\fhi\in\left(\xtxty,\ab\right)^*$. Consider the associated multilinear operators $T_\fhi:\xpx\into Y^*$ and $T_{\overline{\fhi}}:\xpx\into Y^{***}$ of $\fhi$ and $\overline{\fhi}$. The definition of the canonical extension implies that $\lev T_{\overline{\fhi}}(\xxt) \,,\, y^{**} \rev = \lev K_{Y^*}T_{\fhi}(\xxt)\,,\,y^{**}\rev$ for all $x_i^*\in X_i$, $1 \leq i \leq n$ and $y^{**}\in Y^{**}$. This means that $T_{\overline{\fhi}}$ has range contained in $Y^*$ and $T_{\overline{\fhi}}=K_{Y^*}T_\fhi$. Now, Proposition~\ref{aronbernerextension} implies
\[\overline{\fhi}\in\left(\xtxtydd,\ab\right)^*\cap\mathcal{L}\left(\xxx;Y^*\right) =\Abxxyd.\]
Finally, $K_{Y^*} T_\fhi\in\Abxxyddd$ asserts that $T_\fhi\in\Abxxyd$ and
\[\Ab(T_\fhi)=\Ab(K_{Y^*}T_{\fhi})=\|\overline{\fhi}\|=\|\fhi\|.\]
\end{proof}
In the theory of ideals of linear operators (and even of multilinear operators) it is common to have an isometry of the form $(X\tens_\alpha Y)^*=\mathcal{A}(X;Y^*)$ for all Banach spaces $X$ and $Y$. From this, it is easy to prove that the ideal is maximal if $\alpha$ is finitely generated, a condition which is easier to check than maximality. We finish this section by presenting a criterion for maximality of $\Sigma$-ideals. First, we prove a preliminary result.
\begin{proposition}\label{maximalhull}
Let $\ideal$ be a $\Sigma$-ideal on the class $\finn$. Define for every multilinear operator $\Txxy$
\[A^{max,\beta}(T):=\sup A^{\br}(Q_L f_T I_{\eee}:\epe\into Y/L)\leq\infty,\]
where the suprema is taken over all $E_i\in\mathcal{F}(X_i)$, $1\leq i\leq n$, and $L\in\mathcal{CF}(Y)$.
Also define
\[\mathcal{A}^{max,\beta}(\xxx,Y):=\{\; T:\xpx\into Y \;|\; A^{max,\beta}(T)<\infty \;\}.\]
Then the pair $[\mathcal{A}^{max},A^{max}]$ is a maximal $\Sigma$-ideal on the class $\ban$.
\end{proposition}
\begin{proof}
For shorten the article we only prove the more delicate issues of this proposition. That is we only prove the ideal property and that $A^{max,\beta}$ give place to a Banach space.
Let $(m,\zzzm, W, \theta)$ be an election in $\ban$. Consider the composition
\[\begin{array}{c}
\xymatrix{
\szzm\ar[r]^-\fr &\sxx\ar[r]^-{f_T} &Y\ar[r]^S & W\\
}
\end{array}\]
where $S$ is a bounded linear operator, $T$ is in $\mathcal{A}^{max,\beta}(\xxx;Y)$ and $\fr$ is a $\Sigma$-$\beta$-$\theta$-operator associated with $R$. Let $M_i$ be a finite dimensional subspace of $Z_i$, $1\leq i \leq m$ and $G$ be a finite codimensional subspace of $W$. Since $\fr(\szzm)$ is contained in $\sxx$ and $M_1\tens\cdots\tens M_m$ is a finite dimensional space then there exist finite dimensional subspaces $E_i$ of $X_i$, $1\leq i \leq n$ such that $\fr I_{M_1,\dots, M_m}(\Sigma_{M_1,\dots, M_m})\subset\Sigma_{\eee}$. Set $L=Ker(Q_GS)\in\mathcal{CF}(Y)$. Consider the commutative diagram
\[\begin{array}{c}
\xymatrix{
\szzm\ar[drr]_{\fr I_{M_1,\dots, M_m}}\ar[rr]^{\fr} &&\sxx\ar[rr]^{f_T} && Y\ar[d]_{Q_L}\ar[r]^S &W\ar[d]^{Q_G}\\
\Sigma_{M_1,\dots, M_m}\ar[u]^{I_{M_1,\dots, M_m}} &&\Sigma_{\eee}\ar[u]_{I_{\eee}} &&Y/L\ar[r]_B &W/G\\
}
\end{array}.\]
Then
\begin{align*}
A^{\theta|}\left(Q_G(Sf_T\fr)I_{\mmm}\right) &= A^{\theta|}(B(Q_L f_T I_{\eee})(\fr I_{\mmm}))\\
&\leq \|S\| A^{\br}(Q_L f_T I_{\eee}) \|\rlin\|\\
&\leq \|S\| A^{max,\beta}(T) \|\rlin\|.
\end{align*}
After taking suprema over all $M_i$, $1\leq i \leq m$ and $G$ as above we have
\[A^{max}(Sf_T R)\leq\|S\| A^{max,\beta}(T) \|\rlin\|.\]
To see that $A^{max,\beta}$ is a complete norm let $(T^k)_{k=1}^\infty$ be a sequence in $\mathcal{A}^{max,\beta}(\xxx;Y)$ such that $\sum_{k=1}^\infty A^{max,\beta}(T^k)<\infty$. Define $T=\sum_{k=1}^\infty T^k:\xpx\into Y$. Notice that
\[ \sum\limits_{k=1}^\infty \|f_{T^k}(p)\| \leq \sum\limits_{k=1}^\infty \Lb(f_{T^k}) \leq \sum\limits_{k=1}^\infty A^{max,\beta}(T^k).\]
Hence, $\sum_{k=1}^\infty f_{T^k}(p)$ converges in $Y$ for every $p\in\sxx$ since $Y$ is a Banach space. Moreover, $\sum_{k=1}^\infty \Lb(f_{T^k}) \leq \sum_{k=1}^\infty A^{max,\beta}(T^k)$ also implies that $f_T$ is a Lipschitz function on $\sxx$. That is, $T$ is a bounded multilinear operator. Finally, by definition of $f_T$, the triangle inequality and definition of $A^{max,\beta}$ we have that
\[A^\br(Q_Lf_T I_{\eee}) \leq \sum\limits_{k=1}^\infty A^\br( Q_L f_{T^k} I_{\eee}) \leq \sum\limits_{k=1}^\infty A^{max,\beta}(T^k)\]
holds for all $E_i$,$1\leq i \leq n$ and $L$.
\end{proof}
Given a $\Sigma$-ideal $\ideal$ on the class $\ban$ then $[\mathcal{A}^{max},A^{max}]$ is called the \emph{maximal hull} of $\ideal$. By definition, $\ideal$ and its maximal hull coincide on the class $\finn$ and $A^{max,\beta}(T)\leq \Ab(T)$ for all multilinear operator $T$. In this terms, a $\Sigma$-ideal is maximal if it coincides with its maximal hull.
\begin{proposition}\label{criterionmaximal}
Let $\ideal$ be a $\Sigma$-ideal. Suppose there exist a finitely generated $\Sigma$-tensor norm on spaces $\alpha$ such that for any election of Banach spaces $\xxyb$ we have that
\begin{equation}\label{criterion}
\left(\xtxty,\ab\right)^*=\Abxxyd
\end{equation}
holds linearly and isometrically. Then, $\ideal$ is maximal.
\end{proposition}
\begin{proof}
Theorem~\ref{tndsideal} ensures that $\ideal$ is a $\Sigma$-ideal on the class $\finn$. Proposition~\ref{maximalhull} tells us that its maximal hull is a $\Sigma$-ideal on $\ban$. The RT combined with (\ref{criterion}) asserts that
\begin{equation}\label{criterion1}
\Abxxyd=\mathcal{A}^{max,\beta}\left(\xxx;Y^*\right)
\end{equation}
holds linearly and isometrically for all elections $\xxyb$ in $\ban$. Another application of the RT combined with (\ref{criterion1}) leads us to
\[\mathcal{A}^{max,\beta}\left(\xxx;Y\right)=\mathcal{A}^\beta\left(\xxx;Y^{**}\right)\cap \Lxxy.\]
Finally, for $T$ in $\mathcal{A}^{max,\beta}\left(\xxx;Y\right)$ we have
\[\Ab(T)=\Ab(K_Y^{-1}K_YT)\leq \Ab(K_YT)= A^{max,\beta}(T).\]
The proof is complete since the converse inequality $A^{max,\beta}(T)\leq \Ab(T)$ is always true.
\end{proof}
\begin{example}
The following duality relation holds
\begin{align*}
(\xtxty,\pi^\beta)^*&=\mathcal{L}^\beta_{Lip}(\xxx;Y^*)\\
(\xtxty,\gamma_2^\beta)^*&=\Gamma^\beta(\xxx;Y^*)\\
(\xtxty,\alpha_{p^*,q^*}^\beta)^*&=\mathcal{D}_{p,q}^\beta(\xxx;Y^*)\\
(\xtxty,d_p^\beta)^*&=\Pi_p^{Lip,\beta}(\xxx;Y^*)\\
\end{align*}
The first equality follows easily adapting the linear case; the second is an adaptation of \cite[Theorem 4.4]{fernandez-unzueta18b}; the third is \cite[Theorem 4.7]{fernandez-unzueta18a}; and the fourth is \cite[Theorem 2.26]{angulo10}.
\end{example}
According to Proposition \ref{regular}, in the presence of maximality we always may assume that multilinear operators under study have range contained in a dual space. Combining the Representation and Duality Theorems we have that if a maximal $\Sigma$-ideal $\ideal$ is associated with the cofinitely generated $\Sigma$-tensor norm on duals $\nu$ then
\begin{equation}\label{conclusion}
\begin{array}{c}
\xymatrix{
(\xtxty,\ab)^* \ar[r] & \Abxxyd \\
(\Lbxx\tens Y^*,\vb) \ar[r]\ar@{^(->}[u] & (\mathcal{F}^\beta(\xxx;Y^*),\Ab)\ar@{^(->}[u] \\
}
\end{array}
\end{equation}
commutes, where the horizontal arrows are isomorphic isometries, the vertical ones are linear isometries and $\alpha$ is the finitely generated $\Sigma$-tensor norm on spaces defined by $\nu$ (see comment after Definition \ref{ideal}).
\subsubsection{Basic Results}
As can be seen, the most technical difficulty of the notion of $\Sigma$-ideals of multilinear operators is the ideal property since this involves all reasonable crossnorms $\beta$. Next, we explore some consequences of this consideration.
\begin{proposition}
Let $\alpha$ be a $\Sigma$-tensor norms on spaces and $\ideal$ be a $\Sigma$-ideal. Let $n$ be a positive integer, $\xxx$ and $Y$ be Banach spaces. Let $\beta$ and $\theta$ two reasonable crossnorms on $\xtx$ such that there exists $\lambda$ in $\kk$ such that $\beta\leq \lambda\theta$. Then
\begin{itemize}
\item [(i)] $\alpha^\beta (u)\leq \lambda \alpha^\theta(u)$ for all $u\in\xtxty$.
\item [(ii)] $A^\theta(T)\leq \lambda A^\beta(T)$ for all $T\in\Abxxy$.
\item [(iii)] The inclusions $\mathcal{A}^\varepsilon (\xxx; Y)\subset \mathcal{A}^\beta (\xxx; Y) \subset \mathcal{A}^\pi (\xxx; Y)$ are continuous.
\item [(iv)] The inclusions $(\xtxty; \alpha^\pi)\subset (\xtxty; \alpha^\beta) \subset (\xtxty; \alpha^\varepsilon)$ are continuous.
\end{itemize}
\end{proposition}
\begin{corollary}
Let $\alpha$ be a $\Sigma$-tensor norm on spaces and $\ideal$ be a $\Sigma$-ideal. Let $\xxx$ and $Y$ be Banach spaces. If $\beta$ and $\theta$ are equivalent reasonable crossnorms on $\xtx$, then
\begin{itemize}
\item[(i)] The components $\Abxxy$ and $\mathcal{A}^\theta(\xxx;Y)$ are isomorphic.
\item[(ii)] The products $(\xtxty,\ab)$ and $(\xtxty,\alpha^\theta)$ are isomorphic.
\end{itemize}
\end{corollary}
In some cases it is not necessary to consider the restriction of the reasonable crossnorm $\beta$.
\begin{corollary}
Let $\alpha$ be a finitely generated $\Sigma$-tensor norm on spaces and $\ideal$ be a maximal $\Sigma$-ideal. Let $(n, \xxx, Y, \beta)$ be and election in $\ban$ where $\beta$ is and injective tensor norm. Then
\begin{itemize}
\item [(i)] $\Ab(T)=\sup \{\Ab(Q_L f_T I_{\eee}) | E_i\in \mathcal{F}(X_i), F\in \mathcal{CF}(Y)\}$.
\item [(ii)] $\ab(u; \xxx,Y)=\inf\{ \ab(u; \eee,F) | E_i\in \mathcal{F}(X_i), F\in \mathcal{F}(Y)\}$.
\end{itemize}
\end{corollary}
The previous corollary is also true if we replace the Banach spaces $X_i$ by Hilbert spaces $H_i$ and $\beta$ by the reasonable crossnorm $\|\cdot\|_2$ (the norm which makes $H_1\tens\dots\tens H_n$ a (pre) Hilbert space, see \cite[Section 2.6]{kadison83} for details).
In every $\Sigma$-ideal $\ideal$ there exist cases where components coincide. Recall from \cite{pisier83} that Pisier prove the existence of Banach spaces $X$ such that $(X\hat{\tens} X,\varepsilon)$ is isomorphic to $(X\hat{\tens} X,\pi)$. For these spaces $X$ we have that every component $\mathcal{A}^\beta(X,X;Y)$ is isomorphic to $\mathcal{A}^\pi(X,X;Y)$ for all Banach space $Y$.
We finish the article presenting some consequences for the case $n=2$. According to Grothendieck there exist fourteen natural tensor norms, this is, tensor norms obtained from $\pi$ after the application of a sequence, in any order, of the operations: taking dual norm, injective associated, projective associated and transposing (see \cite[pp. 184]{ryan02}). In the following diagram we exhibit the consequence of this observation. In it, $\mathcal{A}^\beta$ stands for $\mathcal{A}^\beta(X_1,X_2;Y)$ and an arrow from $\mathcal{A}^\beta$ to $\mathcal{A}^\theta$ indicates the existence of a constant $\lambda$ such that $ A^\theta (T)\leq \lambda \Ab(T)$
\begin{equation*}
\begin{array}{c}
\xymatrix{
& \mathcal{A}^\pi & \\
\mathcal{A}^{\backslash(/\pi)}\ar[ur] & & \mathcal{A}^{(\pi\backslash)/}\ar[ul] \\
\mathcal{A}^{/\pi}\ar[u] & \mathcal{A}^{\backslash\varepsilon/}\ar[uu]\ar[ul]\ar[ur] & \mathcal{A}^{\pi\backslash}\ar[u] \\
\mathcal{A}^{/(\backslash\varepsilon/)}\ar[u]\ar[ur]& & \mathcal{A}^{(\backslash\varepsilon/)\backslash}\ar[u]\ar[ul]\\
\mathcal{A}^{\varepsilon/}\ar[u]& \mathcal{A}^{/\pi\backslash}\ar[uu]\ar[ul]\ar[ur] & \mathcal{A}^{\backslash\varepsilon}\ar[u] \\
\mathcal{A}^{(\varepsilon/)\backslash}\ar[u]\ar[ur]& & \mathcal{A}^{/(\backslash\varepsilon)}\ar[u]\ar[ul] \\
& \mathcal{A}^\varepsilon\ar[ul]\ar[uu]\ar[ur] &
}
\end{array}.
\end{equation*}
\bibliographystyle{plainurl}
| {
"timestamp": "2018-12-04T02:05:50",
"yymm": "1812",
"arxiv_id": "1812.00114",
"language": "en",
"url": "https://arxiv.org/abs/1812.00114",
"abstract": "We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $\\Sigma$-operators. To this end, we introduce and develop the notions of $\\Sigma$-ideals of multilinear operators and $\\Sigma$-tensor norms. We establish the foundations of this theory by proving a representation theorem for maximal $\\Sigma$-ideals of multilinear operators by finitely generated $\\Sigma$-tensor norms and a duality theorem for $\\Sigma$-tensor norms. For these notions we also develop basic theory and present concrete examples.",
"subjects": "Functional Analysis (math.FA)",
"title": "The Duality between Ideals of Multilinear Operators and Tensor Norms",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795095031687,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7092379292152053
} |
https://arxiv.org/abs/2105.05732 | Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control | In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-\sigma I$ ($\sigma\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control.We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed. | \section{Introduction}
In a separable Hilbert space $X$ consider the nonlinear control system
\begin{equation}\label{u}
\left\{
\begin{array}{ll}
u'(t)+Au(t)+p(t)Bu(t)=0,& t>0\\\\
u(0)=u_0.
\end{array}\right.
\end{equation}
where $A:D(A)\subset X\to X$ is a linear self-adjoint operator on $X$ such that $A\geq-\sigma I$, with $\sigma\geq0$, $B$ belongs to $\mathcal{L}(X)$, the space of all bounded linear operators on $X$, and $p(t)$ is a scalar function representing a bilinear control. We suppose that the spectrum of $A$ consists of a sequence of real numbers $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ which can be ordered, whithout loss of generality, as $-\sigma\leq\lambda_k\leq\lambda_{k+1}\to\infty$ as $k\to\infty$. We denote by $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the corresponding eigenfunctions, $A\varphi_k=\lambda_k\varphi_k,$ with $\norm{\varphi_k}=1$, $\forall\,k\in{\mathbb{N}}^*$.
In the recent paper \cite{acu}, we studied the stabilizability of \eqref{u} to the $j$-th eigensolution of the free equation ($p\equiv0$), $\psi_j(t)=e^{-\lambda_j t}\varphi_j$, for every $j\in{\mathbb{N}}^*$.
For this purpose, we introduced the notion of \emph{$j$-null controllability in time $T>0$} for the pair $\{A,B\}$:
denoting by $y(\cdot;v_0,p)$ the solution of the linear system
\begin{equation*}
\left\{\begin{array}{ll}
y'(t)+Ay(t)+p(t)B\varphi_j=0,&t\in[0,T]\\\\
y(0)=y_0,
\end{array}\right.
\end{equation*}
we say that $\{A,B\}$ is $j$-null controllable in time $T>0$ if for any initial condition $y_0\in X$ there exists a control $p\in L^2(0,T)$ such that
\begin{equation*}
y(T;v_0,p)=0
\quad\mbox{and}\quad \norm{p}_{L^2(0,T)}\leq N_T\norm{y_0} ,
\end{equation*}
where $N_T$ is a positive constant depending only on $T$.
Then, the \emph{control cost} is given by
\begin{equation*}
N(T)=\sup_{\norm{y_0}=1}\inf\left\{\norm{p}_{L^2(0,T)}\,:\, y(T;y_0,p)=0\right\}.\end{equation*}
In \cite[Theorem 3.7]{acu} we have shown that, if $\{A,B\}$ is $j$-null controllable, then \eqref{u} is locally superexponentially stabilizable to $\psi_j$: for all $u_0$ in some neighborhood of $\varphi_j$ there exists a control $p\in L^2_{loc}([0,+\infty))$ such that the corresponding solution $u$ of \eqref{u} satisfies
\begin{equation}\label{superex}
\norm{u(t)-\psi_j(t)} \leq Me^{-e^{\omega t}},\qquad \forall\,t\geq0
\end{equation}
for suitable constants $\omega,M>0$ independent of $u_0$. Notice that such a result holds only under the condition of $j$-null controllability for the pair $\{A,B\}$. In particular, no assumptions are required on the behavior of the control cost.
Moreover, in \cite[Theorem 3.8]{acu} we gave sufficient conditions to ensure the $j$-null controllability of $\{A,B\}$: a gap condition for the eigenvalues of $A$ and a rank condition on $B$.
In this paper, we address the related, more delicate, issue of the exact controllability of \eqref{u} to the eigensolutions $\psi_j$ via bilinear controls. The main differences between the results of this paper and \cite[Theorem 3.7]{acu} can be summarized as follows:
\begin{itemize}
\item in addition to assuming the pair $\{A,B\}$ to be $j$-null controllable, we further require that the control cost $N(\cdot)$ satisfies $N(\tau)\leq e^{\nu/\tau}$ for any $0<\tau\leq T_0$, with $\nu,T_0>0$,
\item under the above stronger assumptions, not only we prove local exact controllability in any time, but also global exact controllablity in large time for a wide set of initial data.
\end{itemize}
The following result ensures local exact controllability for problem \eqref{u} assuming a precise behavior of the control cost for small time. In the last section of this paper, we show that such a behavior of the control cost is typical of parabolic problems in one space dimension.
\begin{thm}\label{teo1}
Let $A:D(A)\subset X\to X$ be a densely defined linear operator such that
\begin{equation}\label{ipA}
\begin{array}{ll}
(a) & A \mbox{ is self-adjoint},\\
(b) &\exists\,\sigma\geq0\,:\,\langle Ax,x\rangle \geq -\sigma\norm{x} ^2,\,\, \forall\, x\in D(A),\\
(c) &\exists\,\lambda>-\sigma \mbox{ such that }(\lambda I+A)^{-1}:X\to X \mbox{ is compact},
\end{array}
\end{equation}
and let $B:X\to X$ be a bounded linear operator. Assume that $\{A,B\}$ is $j$-null controllable in any time $T>0$ for some $j\in{\mathbb{N}}^*$ and suppose that
\begin{equation}\label{bound-control-cost}
N(\tau)\leq e^{\nu/\tau},\quad\forall\,0<\tau\leq T_0,
\end{equation}
for some constants $\nu,T_0>0$.
Then, for any $T>0$, there exists a constant $R_{T}>0$ such that, for any $u_0\in B_{R_{T}}(\varphi_j)$, there exists a control $p\in L^2(0,T)$ such that the solution $u$ of \eqref{u} satisfies $u(T)=\psi_j(T)$.
Moreover, the following estimate holds
\begin{equation}\label{intro-estim-p}
\norm{p}_{L^2(0,T)}\leq \frac{e^{-\pi^2\Gamma_0/T}}{e^{2\pi^2\Gamma_0/(3T)}-1},
\end{equation}
where $\Gamma_0$ and $R_T$ can be computed as follows
\begin{equation}\label{Gamma_0}
\Gamma_0:=2\nu+\max\left\{\ln(D),0\right\},
\end{equation}
\begin{equation}\label{RT}
R_T:=e^{-6\Gamma_0/T_1},
\end{equation}
with
\begin{equation}\label{T_11}
T_1:=\min\left\{\frac{6}{\pi^2}T,1,T_0\right\},
\end{equation}
\begin{equation}\label{D}
D:=2\norm{B}e^{2\sigma+(3\norm{B})/2+1/2}\max\left\{1,\norm{B}\right\}.
\end{equation}
\end{thm}
The main idea of the proof consists of applying the stability estimates of \cite{acu} on a suitable sequence of time intervals of decreasing length $T_j$, such that $\sum_{j=1}^\infty T_j<\infty$. Such a sequence, which can be constructed only thanks to \eqref{bound-control-cost}, has to be carefully chosen in order to fit the error estimates that we take from \cite{acu}. We point out that our method is fully constructive, being based on an algorithm that allows to compute all relevant constants. In particular, we make no use of inverse mapping theorems.
In \cite{acu}, we gave sufficient conditions for $j$-null controllability. However, the hypotheses of \cite[Theorem 3.8]{acu} do not guarantee the validity of condition \eqref{bound-control-cost} for the control cost. In the result that follows, we provide sufficient conditions for $N(T)$ to satisfy \eqref{bound-control-cost}. It would be interesting to understand if \eqref{bound-control-cost} is also necessary for the local exact controllability of \eqref{u}.
\begin{thm}\label{Thm-suff-cond}
Let $A:D(A)\subset X\to X$ be such that \eqref{ipA} holds and suppose that there exists a constant $\alpha>0$ for which the eigenvalues of $A$ fulfill the gap condition
\begin{equation}\label{gap}
\sqrt{\lambda_{k+1}-\lambda_1}-\sqrt{\lambda_k-\lambda_1}\geq \alpha,\quad\forall\, k\in {\mathbb{N}}^*.
\end{equation}
Let $B: X\to X$ be a bounded linear operator such that there exist $b,q>0$ for which
\begin{equation}\label{ipB}
\begin{array}{l}
\langle B\varphi_j,\varphi_j\rangle\neq0\quad\mbox{and}\quad\left|\lambda_k-\lambda_1\right|^q|\langle B\varphi_j,\varphi_k\rangle|\geq b,\quad\forall\,k\neq j.
\end{array}
\end{equation}
Then, the pair $\{A,B\}$ is $j$-null controllable in any time $T>0$, and the control cost $N(T)$ satisfies \eqref{bound-control-cost} with
\begin{equation}\label{T_0}
T_0:=\min\left\{1,1/\alpha^2\right\},
\end{equation}
and $\nu=\Gamma_j$, where
\begin{eqnarray}\label{Gamma_jFatiha}
\lefteqn{2\Gamma_j(M,b,q,\alpha)}
\\
\nonumber
&=&M+ \frac{M^2}4 + (2q+5) e + \max\left\{ \ln\left(\dfrac{3M}{|\langle B\varphi_j,\varphi_j\rangle|^2}\right), \ln\left(\dfrac{3MC_q}{b^2}\right), \ln\left(\dfrac{3M C_{q,\alpha} }{b^2}\right),0\right\}
\end{eqnarray}
and
\begin{equation}\label{M}
M:=C^2\left(1+\frac{1}{\alpha^2}\right)^2+2|\lambda_1|,
\end{equation}
\begin{equation}\label{Cq-Cqa}
C_q=2\left(\frac{2q}{e}\right)^{2q},\quad C_{q,\alpha}=\frac{2\Gamma(2q+1)}{\alpha\sqrt{\lambda_2-\lambda_1}}.
\end{equation}
Here $\Gamma(\cdot)$ is the Gamma function and $C$ is a positive constant independent of $T$ and $\alpha$.
\end{thm}
Observe that assumption \eqref{ipB} is stronger that \cite[hypothesis (16)]{acu}. Nevertheless, it is satisfied by all the examples of parabolic problems that we presented in \cite{acu}.
From Theorems \ref{teo1} and \ref{Thm-suff-cond} we deduce the following Corollary.
\begin{cor}
Let $A:D(A)\subset X\to X$ be such that \eqref{ipA} holds and suppose that there exists a constant $\alpha>0$ for which \eqref{gap} is satisfied. Let $B: X\to X$ be a bounded linear operator that verifies \eqref{ipB} for some $b,q>0$. Then, problem \eqref{u} is locally controllable to the $j$th eigensolution $\psi_j$ in any time $T>0$.
\end{cor}
Furthermore, from Theorem \ref{teo1} we deduce two semi-global controllability results in the case of an accretive operator $A$. In the first one, Theorem \ref{teoglobal} below, we prove that all initial states lying in a suitable strip can be steered in finite time to the first eigensolution $\psi_1$ (see Figure~\ref{fig1}). Moreover, we give a uniform estimate for the controllability time depending on the size of the projection of the initial datum $u_0$ on $\varphi_1^\perp$.
\begin{thm}\label{teoglobal}
Let $A:D(A)\subset X\to X$ be a densely defined linear operator such that \eqref{ipA} holds with $\sigma=0$ and let $B:X\to X$ be a bounded linear operator. Let \{A,B\} be a $1$-null controllable pair which satisfies \eqref{bound-control-cost}. Then, there exists a constant $r_1>0$ such that for any $R>0$ there exists $T_{R}>0$ such that for all $u_0\in X$ with
\begin{equation}\label{ipu0}
\left|\langle u_0,\varphi_1\rangle-1\right|< r_1,\qquad
\norm{u_0-\langle u_0,\varphi_1\rangle\varphi_1}\leq R,
\end{equation}
problem \eqref{u} is exactly controllable to the first eigensolution $\psi_1(t)=e^{-\lambda_1 t}\varphi_1$ in time $T_{R}$.
\end{thm}
\begin{figure}[ht!]
\centering\begin{tikzpicture}
\fill[green(ryb)!30](4,-3)--(6,-3)--(6,3)--(4,3)--cycle;
\draw[] (0,0) -- (10,0);
\draw[] (2,4) -- (2,-4);
\fill(5,0) node[below]{\footnotesize{$\varphi_1$}} circle (.05);
\fill(3,0) node[below]{\footnotesize{$\psi_1(T_R)$}} circle (.05);
\draw[] (4,-4) -- (4,4);
\draw[] (6,-4) -- (6,4);
\draw[] (2,3) node[left]{\footnotesize{$R$}} -- (9,3);
\draw[] (2,-3) node[left]{\footnotesize{$-R$}} -- (9,-3);
\draw[<->] (4,3.5) --node[above]{\footnotesize{$r_1$}} (6,3.5);
\fill(5.5,2) node[above]{\footnotesize{$u_0$}} circle (.05);
\draw[ultra thick,->] (5.5,2) -- (3,0);
\end{tikzpicture}
\caption{the colored region represents the set of initial conditions that can be steered to the first eigensolution in time $T_R$.}\label{fig1}
\end{figure}
Our second semi-global result, Theorem \ref{teoglobal0} below, ensures the exact controllability of all initial states $u_0\in X\setminus \varphi_1^\perp$ to the evolution of their orthogonal projection along the first eigensolution. Such a function is defined by
\begin{equation}\label{exactphi1}
\phi_1(t)=\langle u_0,\varphi_1\rangle \psi_1(t), \quad\forall\, t \geq 0,
\end{equation}
where $\psi_1$ is the first eigensolution.
\begin{thm}\label{teoglobal0}
Let $A:D(A)\subset X\to X$ be a densely defined linear operator such that \eqref{ipA} holds with $\sigma=0$ and let $B:X\to X$ be a bounded linear operator. Let \{A,B\} be a 1-null controllable pair which satisfies \eqref{bound-control-cost}.
Then, for any $R>0$ there exists $T_R>0$ such that for all $u_0\in X$ with
\begin{equation}\label{cone}
\norm{u_0-\langle u_0,\varphi_1\rangle\varphi_1} \leq R |\langle u_0,\varphi_1\rangle|,
\end{equation}
system \eqref{u} is exactly controllable to $\phi_1$, defined in \eqref{exactphi1}, in time $T_R$.
\end{thm}
Notice that, denoting by $\theta$ the angle between the half-lines ${\mathbb{R}}_+\varphi_1$ and ${\mathbb{R}}_+ u_0$, condition \eqref{cone} is equivalent to
\begin{equation*}
|\tan\theta|\leq R,
\end{equation*}
which defines a closed cone, say $Q_R$, with vertex at $0$ and axis equal to ${\mathbb{R}}\varphi_1$ (see Figure~\ref{fig2}). Therefore, Theorem \ref{teoglobal0} ensures a uniform controllability time for all initial conditions lying in $Q_R$.
We observe that, since $R$ is any arbitrary positive constant, all initial conditions $u_0\in X\setminus \varphi_1^\perp$ can be steered to the corresponding projection to the first eigensolution. Indeed, for any $u_0\in X\setminus \varphi_1^\perp$, we define
\begin{equation*}
R_0:=\norm{\frac{u_0}{\langle u_0,\varphi_1\rangle}-\varphi_1}.
\end{equation*}
Then, for any $R\geq R_0$ condition \eqref{cone} is fulfilled:
\begin{equation*}
\frac{1}{|\langle u_0,\varphi_1\rangle|}\norm{u_0-\langle u_0,\varphi_1\rangle\varphi_1}=R_0\leq R.
\end{equation*}
\begin{figure}[ht!]
\centering\begin{tikzpicture}
\fill[green(ryb)!30](0,-4.5)--(7,0)--(0,4.5)--cycle;
\fill[green(ryb)!30](7,0)--(14,-4.5)--(14,4.5)--cycle;
\coordinate (v2) at (7,0);
\coordinate(v4) at (11.5,2);
\coordinate(v5) at (11.5,0);
\coordinate(v8) at (2,1);
\coordinate(v9) at (2,0);
\tkzMarkAngle[size=1cm](v5,v2,v4);
\tkzLabelAngle[pos=1.25](v5,v2,v4){\footnotesize{$\theta$}};
\tkzMarkAngle[size=1.75cm](v8,v2,v9);
\tkzLabelAngle[pos=2](v8,v2,v9){\footnotesize{$\hat{\theta}$}};
\draw[] (0,0) -- (14,0);
\draw[] (7,4.5) -- (7,-4.5);
\draw[] (7,0) -- (13,3.86);
\draw[dashed] (13,3.86) -- (14,4.5);
\draw[] (7,0) -- (13,-3.86);
\draw[dashed] (13,-3.86) -- (14,-4.5);
\draw[] (7,0) -- (1,3.86);
\draw[dashed] (1,3.86) -- (0,4.5);
\draw[] (7,0) -- (1,-3.86);
\draw[dashed] (1,-3.86) -- (0,-4.5);
\fill(11,0) node[below]{\footnotesize{$\varphi_1$}} circle (.05);
\fill(3,0) node[below]{\footnotesize{$\varphi_1$}} circle (.05);
\fill(10,0) node[below]{\footnotesize{$\phi_1(T_R)$}} circle (.05);
\fill(3.5,0) node[below]{\footnotesize{$\qquad\hat{\phi}_1(T_R)$}} circle (.05);
\fill(11.5,2) node[above]{\footnotesize{$u_0$}} circle (.05);
\fill(2,1) node[above]{\footnotesize{$\hat{u}_0$}} circle (.05);
\draw[] (11.5,2) -- (11.5,0);
\draw[] (11.5,2) -- (7,0);
\draw[] (2,1) -- (2,0);
\draw[] (2,1) -- (7,0);
\fill(11.5,0) circle (.05);
\fill(2,0) circle (.05);
\draw[ultra thick,->] (11.5,2) -- (10,0);
\draw[ultra thick,->] (2,1) -- (3.5,0);
\fill(13,-2.5) node{$Q_R$};
\end{tikzpicture}
\caption{fixed any $R>0$, the set of initial conditions exactly controllable in time $T_R$ to their projection along the first eigensolution is indicated by the colored cone $Q_R$.}\label{fig2}
\end{figure}
Finally, we would like to recall part of the huge literature on bilinear control of evolution equations, referring the reader to the references in \cite{acu} for more details. A seminal paper in this field is certainly the one by Ball, Marsden, Slemrod \cite{bms}, which establishes that system \eqref{u} is not controllable. More precisely, denoting by $u(t;u_0,p)$ the unique solution of \eqref{u}, the attainable set from $u_0$ defined by
\begin{equation*}
S(u_0)=\{ u(t;u_0,p);t\geq 0, p\in L^r_{loc}([0,+\infty),{\mathbb{R}}),r>1\}
\end{equation*}
is shown in \cite{bms} to have a dense complement.
As for positive results, we would like to mention Beauchard \cite{b}, on bilinear control of the wave equation, and Beauchard, Laurent \cite{bl} on bilinear control of the Schr{\"o}dinger equation (see also \cite{beau} for a first result on this topic). The results obtained in these papers rely on linearization around the ground state, the use of the inverse mapping theorem, and a regularizing effect which takes place in both problems. Local controllability is proved for any positive time for the Schr{\"o}dinger equation and for a sufficiently (optimal) large time for the wave equation. Both papers require the condition
\begin{equation}
\label{B}
\langle B\varphi_1,\varphi_k\rangle \neq 0,\quad \forall \, k \geq 1
\end{equation}
to be satisfied, together with a suitable asymptotic behavior with respect to the eigenvalues. Notice that the structure of the second order operator and the fact that the space dimension equals one allow the authors of \cite{b} and \cite{bl} to apply Ingham's theory (\cite{kl}) which requires a gap condition on the eigenvalues. We further observe that even if the genericity of assumption \eqref{B} is proved in both papers \cite{b,bl}, only few explicit examples of operators $B$ of multiplication type are available in the literature. We refer to \cite{au} where a general constructive algorithm for building potentials which satisfy the infinite non-vanishing conditions \eqref{B}, and further the asymptotic condition \eqref{ipB}, is established.
If \eqref{B} is violated then it has been first shown by Coron \cite{cor}, for a model describing a particle in a moving box, that there exists a minimal time for local exact controllability to hold. This model couples the Schr\"odinger equation with two ordinary differential equations modeling the speed and acceleration of the box (see also Beauchard, Coron \cite{beaucor} for local exact controllability for large time). A further paper by Beauchard and Morancey \cite{bm} for the Schr\"odinger equation extends \cite{bl} to cases for which the above condition is violated, that is, when there exist integers $k$ such that
$\langle B\varphi_1,\varphi_k\rangle =0$.
An example of controllability to trajectories for nonlinear parabolic systems is studied in \cite{fgip}, where, however, additive controls are considered. In such an example, one can obtain controllability to free trajectories by Carleman estimates and inverse mapping arguments. Such a strategy seems hard to adapt to the current setting.
The paper which has the strongest connection with our work is the one by Beauchard and Marbach in \cite{beau-mar}, where the authors study small-time null controllability for a scalar-input heat equation in one space dimension, with nonlinear lower order terms. Among the results of such paper, we mention null-controllability to the first eigenstate of a heat equation with bilinear control. From this result it would be possible to deduce local controllability only to the first eigenstate of the heat equation subject to Neumann boundary conditions. It is worth noting that \cite{beau-mar} addresses a specific parabolic equation. Moreover, the methods developed therein, relying on the so-called source term procedure, are totally different from ours.
We observe that the bilinear controls we use in this paper are just scalar functions of time. This fact explains why applications mainly concern problems in low space dimension, like the results in \cite{beau,b,beaucor,bl,beau-mar,bm,cor}.
A stronger control action could be obtained by letting controls depend on time and space. We refer the reader to \cite{cfk,ck} for more on this subject.
This paper is organized as follows. In section 2, we have collected some preliminaries as well as results from \cite{acu} that we need in order to prove Theorem \ref{teo1}. Section 3 contains such a proof, while section 4 is devoted to demonstrate Theorem \ref{Thm-suff-cond}. In section 5, we give the proof of our semi-global results (Theorems \ref{teoglobal} and \ref{teoglobal0}). Finally, applications of Theorem \ref{teo1} to parabolic problems are analyzed is section 6.
\section{Preliminaries}
In this section, we recall a well-known result for the well-posedness of our control problem and the regularity of the solution as well as some results from \cite{acu} that are necessary for the proof of Theorem \ref{teo1}. Moreover, we will remind the fundamental definition of $j$-null controllable pair.
We recall our general functional frame. Let $(X,\langle\cdot,\cdot\rangle,\norm{\cdot})$ be a separable Hilbert space, let $A:D(A)\subset X\to X$ be a densely defined linear operator with the following properties
\begin{equation}\label{ipAA}
\begin{array}{ll}
(a) & A \mbox{ is self-adjoint},\\
(b) &\exists\,\sigma\geq0\,:\,\langle Ax,x\rangle \geq -\sigma\norm{x} ^2,\,\, \forall\, x\in D(A),\\
(c) &\exists\,\lambda>-\sigma \mbox{ such that }(\lambda I+A)^{-1}:X\to X \mbox{ is compact}.
\end{array}
\end{equation}
We denote by $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ the eigenvalues of $A$, which can be ordered, whithout loss of generality, as $-\sigma\leq\lambda_k\leq\lambda_{k+1}\to\infty$ as $k\to\infty$, and by $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the corresponding eigenfunctions, $A\varphi_k=\lambda_k\varphi_k,$ with $\norm{\varphi_k}=1$, $\forall\,k\in{\mathbb{N}}^*$.
Let $B:X\to X$ be a bounded linear operator. Fixed $T>0$, consider the following bilinear control problem
\begin{equation}\label{a1f}\left\{
\begin{array}{ll}
u'(t)+A u(t)+p(t)Bu(t)+f(t)=0,& t\in [0,T]\\\\
u(0)=u_0.
\end{array}\right.
\end{equation}
If $u_0\in X$, $p\in L^2(0,T)$ and $f\in L^2(0,T;X)$, a function $u\in C^0([0,T],X)$ is called a \emph{mild solution} of \eqref{a1f} if it satisfies
\begin{equation*}
u(t)=e^{-tA }u_0-\int_0^t e^{-(t-s)A}[p(s)Bu(s)+f(s)]ds, \quad\forall t\in[0,T].
\end{equation*}
We introduce the following notation:
\begin{equation*}\begin{array}{l}
\norm{f}_{2}:=\norm{f}_{L^2(0,T;X)},\qquad\forall\,f\in L^2(0,T;X)\\\\
\norm{f}_{\infty}:=\norm{f}_{C([0,T];X)}=\sup_{t\in [0,T]}\norm{f(t)} ,\qquad\forall\, f\in C([0,T];X).
\end{array}
\end{equation*}
The well-posedness of \eqref{a1f} is ensured by the following proposition (see \cite{bms} for a proof).
\begin{prop}\label{propa24}
Let $T>0$. For any $u_0\in X$, $p\in L^2(0,T)$ and $f\in L^2(0,T;X)$ there exists a unique mild solution of \eqref{a1f}.
Furthermore, $u(\cdot)$ satisfies
\begin{equation}\label{a5}
\norm{u}_{\infty}\leq C(T) (\norm{u_0} +\norm{f}_{2}),
\end{equation}
for a suitable positive constant $C(T)$.
\end{prop}
\begin{oss}\label{rmk-regularity}
\emph{Under the hypotheses of Theorem \ref{propa24} it is possible to prove that the solution is more regular. Indeed, for every $\varepsilon\in(0,T)$ it holds that $u\in H^1(\varepsilon,T;X)\cap L^2(\varepsilon,T;D(A))$ and the following identity is satisfied
\begin{equation*}
u'(t)+A u(t)+p(t)Bu(t)+f(t)=0,\quad \text{for a.e. }t\in[\varepsilon,T].
\end{equation*}
Furthermore, if $u_0=0$ then $u\in H^1(0,T;X)\cap L^2(0,T;D(A))$ (it can be deduced by applying, for instance, \cite[Proposition 3.1, p.130]{bd}).}
\end{oss}
Let us now consider the following nonlinear control problem
\begin{equation}\label{v}
\left\{\begin{array}{ll}
v'(t)+A v(t)+p(t)Bv(t)+p(t)B\varphi_j=0,&t\in[0,T]\\\\
v(0)=v_0,
\end{array}\right.
\end{equation}
where $\varphi_j$ is the $j$th eigenfunction of $A$. We denote by $v(\cdot;v_0,p)$ the solution of \eqref{v} associated with initial condition $v_0$ and control $p$.
The following result establishes a bound for the solution of \eqref{v} in terms of the initial condition. We give its proof in \ref{appendix} for the sake of clarity and completeness. This proof follows that of \cite[Proposition 4.3]{acu}, with a different presentation, in particular with respect to the assumptions in the statement.
\begin{prop}\label{prop38}
Let $T>0$. Let $A:D(A)\subset X\to X$ be a densely defined linear operator that satisfies \eqref{ipAA} and let $B:X\to X$ be a bounded linear operator. Let $v_0\in X$ and let $p\in L^2(0,T)$ be such that
\begin{equation}\label{prelim-p-bound}
\norm{p}_{L^2(0,T)}\leq N_T\norm{v_0},
\end{equation}
with $N_T$ a positive constant.
Then, $v(\cdot;v_0,p)$ verifies
\begin{equation}\label{unifv}
\sup_{t\in[0,T]}\norm{v(t;v_0,p)}^2\leq C_1(T,\norm{v_0})\norm{v_0}^2,
\end{equation}
where $C_1(T,\norm{v_0}):=e^{(2\sigma+\norm{B})T+2\norm{B}N_T\sqrt{T}\norm{v_0}}(1+\norm{B}N_T^2)$ and $\sigma$ is defined in \eqref{ipAA}.
\end{prop}
For any $0\leq s_0\leq s_1$, we now introduce the linear problem
\begin{equation}\label{newlin}
\begin{cases}
y'(t)+Ay(t)+p(t)B\varphi_j=0,&t\in[s_0,s_1]\\\\
y(s_0)=y_0
\end{cases}
\end{equation}
and we denote by $y(\cdot;y_0,s_0,p)$ the solution associated with initial condition $y_0$ at time $s_0$ and control $p$. Let us recall that for any fixed $T>0$ and $j\in{\mathbb{N}}^*$, we say that the pair $\{A,B\}$ is \emph{$j$-null controllable in time $T$} if there exists a constant $N_T$ such that for every $y_0\in X$ there exists a control $p\in L^2(0,T)$ with
\begin{equation}\label{estimpnew}
\norm{p}_{L^2(0,T)}\leq N_T\norm{y_0},
\end{equation}
for which the solution of \eqref{newlin} with $s_0=0$ and $s_1=T$ satisfies $y(T;y_0,0,p)=0$. In this case, we define the \emph{control cost} as
\begin{equation}\label{def-control-cost}
N(T)=\sup_{\norm{y_0}=1}\inf\left\{\norm{p}_{L^2(0,T)}\,:\, y(T;y_0,0,p)=0\right\}.\end{equation}
With an approximation argument one realizes that \eqref{estimpnew} holds with $N_T=N(T)$, that is, for every $y_0\in X$ there exists $p\in L^2(0,T)$ with $\norm{p}_{L^2(0,T)}\leq N(T)\norm{y_0}$ such that $y(T;y_0,0,p)=0$.
Now, consider the following control problem
\begin{equation}\label{w}
\left\{\begin{array}{ll}
w'(t)+Aw(t)+p(t)Bv(t)=0,&t\in[0,T]\\\\
w(0)=0,
\end{array}\right.
\end{equation}
with $v$ the solution of \eqref{v}. We denote by $w(\cdot;0,p)$ the solution of \eqref{w} associated with control $p$.
In the following proposition we give a quadratic estimate of the solution of \eqref{w} in terms of the initial condition of the Cauchy problem solved by $v$. We give its proof in \ref{appendix} for the sake of clarity and completeness. This proof follows that of \cite[Proposition 4.4]{acu}, with a different presentation and a different hypothesis \eqref{v0} compared to the corresponding ones in the statement of \cite[Proposition 4.4]{acu}.
\begin{prop}\label{prop39}
Let $T>0$, $A:D(A)\subset X\to X$ be a densely defined linear operator that satisfies \eqref{ipAA} and $B:X\to X$ be a bounded linear operator. Let $p\in L^2(0,T)$ verify \eqref{prelim-p-bound} with $N_T=N(T)$ and $v_0\in X$ be such that
\begin{equation}\label{v0}
N(T)\norm{v_0}\leq 1.
\end{equation}
Then, $w(\cdot;0,p)$ satisfies
\begin{equation}\label{wT}
\norm{w(T;0,p)}\leq K(T)\norm{v_0}^2,
\end{equation}
where
\begin{equation}\label{KT}
K^2(T):=\norm{B}^2N(T)^2e^{(4\sigma+\norm{B}+1)T+2\norm{B}\sqrt{T}}\left(1+\norm{B}N(T)^2\right).
\end{equation}
\end{prop}
\section{Proof of Theorem \ref{teo1}}
Fixed any $j\in{\mathbb{N}}^*$ and any $T>0$, our aim is to prove local exact controllability in time $T$ for the following problem
\begin{equation}\label{sys}\left\{\begin{array}{ll}
u'(t)+A u(t)+p(t)Bu(t)=0,& t\in[0,T]\\\\
u(0)=u_0,
\end{array}\right.
\end{equation}
to the $j$th eigensolution $\psi_j=e^{-\lambda_j t}\varphi_j$ of $A$, that is the solution of \eqref{sys} when $p=0$ and $u_0=\varphi_j$. Hereafter, we will denote by $u(\cdot;u_0,p)$ the solution of \eqref{sys} associated with initial condition $u_0$ and control $p$.
We recall that $A:D(A)\subset X\to X$ is a densely defined linear operator that satisfies
\begin{equation}\label{ipAAA}
\begin{array}{ll}
(a) & A \mbox{ is self-adjoint},\\
(b) &\exists\,\sigma\geq0\,:\,\langle Ax,x\rangle \geq -\sigma\norm{x} ^2,\,\, \forall\, x\in D(A),\\
(c) &\exists\,\lambda>-\sigma \mbox{ such that }(\lambda I+A)^{-1}:X\to X \mbox{ is compact},
\end{array}
\end{equation}
and we denote by $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ and $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the eigenvalues and the eigenfunctions of $A$, respectively. $B:X\to X$ is a bounded linear operator. The pair $\{A,B\}$ is assumed to be $j$-null controllable in any time, with control cost that satisfies
\begin{equation}\label{bound-control-costt}
N(\tau)\leq e^{\nu/\tau},\quad\forall\,0<\tau\leq T_0,
\end{equation}
for some constants $\nu,T_0>0$.
The proof of Theorem \ref{teo1} is divided into two main parts: the case $\lambda_j=0$, that we build by a series of steps, and the case $\lambda_j\neq0$.
\subsection{Case $\lambda_j=0$}
If $\lambda_j=0$ our reference trajectory will be the stationary function $\psi_j\equiv\varphi_j$. Given $T>0$, we define $T_f$ as
\begin{equation}\label{T_f}
T_f:=\min\left\{T,\frac{\pi^2}{6},\frac{\pi^2}{6}T_0\right\},
\end{equation}
where $T_0$ is the constant in \eqref{bound-control-costt}. We will actually build a control $p\in L^2(0,T_f)$ such that $u(T_f;u_0,p)=\psi_j$, and then, by taking $p(t)\equiv 0$ for $t>T_f$, the solution $u$ of \eqref{sys} will remain forever on the target trajectory $\psi_j$.
Now, we define
\begin{equation}\label{T_1}
T_1:=\frac{6}{\pi^2}T_f,
\end{equation}
and we observe that $0<T_1\leq 1$. Then, we introduce the sequence $\{T_j\}_{j\in{\mathbb{N}}^*}$ as
\begin{equation}\label{Tj}
T_j:=T_1/j^2,
\end{equation}
and the time steps
\begin{equation}\label{taun}
\tau_n=\sum_{j=1}^n T_j,\qquad\forall\, n\in{\mathbb{N}},
\end{equation}
with the convention that $\sum_{j=1}^0T_j=0$. Notice that $\sum_{j=1}^\infty T_j=\frac{\pi^2}{6}T_1=T_f$.
\begin{oss}\label{ossF32}
Note that the sequence of times $(T_j)_{j \in NN^*}$ is strictly decaying towards $0$, whereas the sequence of times
$(\tau_j)_{j \in NN^*}$ is strictly increasing and converges to $T_f$.
\end{oss}
Set $v:=u-\varphi_j$. We will consider the equation satisfied by $v$ on suitable intervals of time $[s_0,s_1]$ and suitable initial data $v^0$ at the initial time $s_0$, as follows. Given any $0\leq s_0\leq s_1\leq T$, and any $v^0$ in $X$, $v$ is the solution of the following Cauchy problem
\begin{equation}\label{vv}
\left\{\begin{array}{ll}
v'(t)+A v(t)+p(t)Bv(t)+p(t)B\varphi_j=0,&t\in[s_0,s_1]\\\\
v(s_0)=v^0.
\end{array}\right.
\end{equation}
We denote by $v(\cdot;v^0,s_0,p)$ the solution of \eqref{vv} associated with initial condition $v^0$ at time $s_0$ and control $p$. Observe that proving the controllability of $u$ to $\psi_j=\varphi_j$ in time $T_f$ is equivalent to show the null controllability of $v$, that is, $v(T_f;v_0,0,p)=0$, where $v_0=u_0-\varphi_j$.
The strategy of the proof consists first of building a control $p_1\in L^2(0,T_1)$ such that at time $T_1$ the solution of \eqref{vv} can be estimated by the square of the initial condition. We then iterate the procedure on consecutive time intervals of the form $[\tau_{n-1},\tau_n]$: each time we construct a control $p_n\in L^2(\tau_{n-1},\tau_n)$ such that the solution of \eqref{vv} on $[\tau_{n-1},\tau_n]$ at time $\tau_n$ is estimated by the square of the initial condition on such interval. Hence, combining all those estimates and letting $n$ go to infinity, we finally deduce that there exists a control $p\in L^2_{loc}(0,+\infty)$ such that $v(T_f;v_0,0,p)=0$ and so $u(T_f;u_0,p)=\varphi_j$.
In practice, we shall build, by induction, controls $p_n\in L^2(\tau_{n-1},\tau_n)$ for $n\geq 1$ such that, setting
\begin{equation}\label{def-q-v}
\begin{array}{l}
\displaystyle q_{n}(t):=\sum_{j=1}^n p_j(t)\chi_{[\tau_{j-1},\tau_j]}(t),\\
v_n:=v(\tau_n;v_0,0,q_n),
\end{array}
\end{equation}
it holds that
\begin{equation}\label{iterative-stepp}
\begin{array}{ll}
1.&\norm{p_{n}}_{L^2(\tau_{n-1},\tau_n)}\leq N(T_n)\norm{v_{n-1}},\\
2.& y(\tau_{n};v_{n-1},\tau_{n-1},p_n)=0,\\
3.&\norm{v(\tau_n;v_{n-1},\tau_{n-1},p_n)}\leq e^{\left(\sum_{j=1}^n 2^{n-j}j^2-2^n6\right)\Gamma_0/T_1},\\
4.&\norm{v(\tau_n;v_{n-1},\tau_{n-1},p_n)}\leq \prod_{j=1}^{n}K(T_j)^{2^{n-j}}\norm{v_0}^{2^{n}}.
\end{array}
\end{equation}
Observe that, by construction,
\begin{equation*}
v_{n}=v(\tau_n;v_0,0,q_n)=v(\tau_n;v_{n-1},\tau_{n-1},p_n),\quad\forall\,n\geq1.
\end{equation*}
\subsubsection{First iteration}
Let us start by studying control problem \eqref{vv} in the first time interval $[s_0,s_1]=[\tau_0,\tau_1]=[0,T_1]$. Recalling that $\{A,B\}$ is $j$-null controllable in any time, given $v_0\in X$ there exists a control $p_1\in L^2(0,T_1)$ such that
\begin{equation}\label{p_0}
\norm{p_1}_{L^2(0,T_1)}\leq N(T_1)\norm{v_0},\quad\text{and}\quad y(T_1;v_0,0,p_1)=0,
\end{equation}
where $N(T_1)$ is the control cost and $y(\cdot;v_0,0,p_1)$ is the solution of the linear problem \eqref{newlin}. So, the first two items of \eqref{iterative-stepp} for $n=1$ are fulfilled. We now apply Proposition \ref{prop38} deducing that
\begin{equation}
\sup_{t\in[0,T_1]}\norm{v(t;v_0,0,p_1)}^2\leq C_1(T_1,\norm{v_0})\norm{v_0}^2,
\end{equation}
where $C_1(T_1,\norm{v_0})=e^{(2\sigma+\norm{B})T_1+2\norm{B}N(T_1)\sqrt{T_1}\norm{v_0}}(1+\norm{B}N(T_1)^2)$.
We measure how close from $0$ the solution of \eqref{vv} is steered at time $T_1$ by control $p_1$. For this purpose, we introduce the function $w(\cdot):=v(\cdot;v_0,0,p_1)-y(\cdot;v_0,0,p_1)$ which satisfies the following Cauchy problem
\begin{equation}\label{ww}
\left\{\begin{array}{ll}
w'(t)+Aw(t)+p_1(t)Bv(t)=0,&t\in[0,T_1]\\\\
w(0)=0.
\end{array}\right.
\end{equation}
Thanks to Proposition \ref{prop39}, if
\begin{equation}\label{v00}
N(T_1)\norm{v_0}\leq 1,
\end{equation}
then, the solution of \eqref{ww} satisfies
\begin{equation}\label{wTT}
\norm{w(T_1;0,p_1)}=\norm{v(T_1;v_0,0,p_1)}\leq K(T_1)\norm{v_0}^2,
\end{equation}
where $K(\cdot)$ is defined on $(0,\infty)$ as
\begin{equation}\label{KTT}
K^2(\tau):=\norm{B}^2N(\tau)^2e^{(4\sigma+\norm{B}+1)\tau+2\norm{B}\sqrt{\tau}}\left(1+\norm{B}N(\tau)^2\right).
\end{equation}
Notice that, the first equality in \eqref{wTT} holds true because control $p_1$ steers to $0$ the solution of the linear problem (see \eqref{p_0}).
\begin{oss}\label{oss32}
Observe that function $K(\cdot)$ satisfies
$$K^2(\tau)\leq \norm{B}^2N^2(\tau)e^{(4\sigma+3\norm{B}+1)}\left(1+\norm{B}N^2(\tau)\right),\quad \forall\,0<\tau\leq1.$$
Therefore, since $T_1=\min\{6T/\pi^2, 1,T_0\}$, combining the above inequality with \eqref{bound-control-costt}, we deduce that there exists a constant $\Gamma_0>\nu$ such that
\begin{equation}\label{estK}
K(\tau)\leq e^{\Gamma_0/\tau},\quad \forall\,0<\tau\leq T_1.
\end{equation}
where $T_0$ is defined in \eqref{bound-control-costt}.
Note that a suitable choice of constant $\Gamma_0$ such that \eqref{estK} holds is \eqref{Gamma_0}.
\end{oss}
We now define the radius of the neighborhood of $\varphi_j$ where we take the initial condition $u_0$ as in \eqref{RT}. Let $u_0\in B_{R_T}(\varphi_j)$, or equivalently $v_0=u_0 - \varphi_j \in B_{R_T}(0)$, be chosen arbitrarily. With this choice we have that
\begin{equation*}
N(T_1)\norm{v_0}\leq e^{\nu/T_1}e^{-6\Gamma_0/T_1}\leq e^{-5\Gamma_0/T_1}\leq1,
\end{equation*}
and \eqref{v00} is satisfied. Therefore, we get that
\begin{equation}\label{first-iteration-induction-for-v_n}
\norm{v(T_1;v_0,0,p_1)}\leq K(T_1)\norm{v_0}^2\leq e^{-11\Gamma_0/T_1},
\end{equation}
which proves 3. and 4. of \eqref{iterative-stepp} for $n=1$.
\subsubsection{Iterative step}
Now, suppose that we have built controls $p_j\in L^2(\tau_{j-1},\tau_j)$ such that \eqref{iterative-stepp} holds for each $j=1,\dots,n-1$. In particular, for $j=n-1$, there exists $p_{n-1}\in L^2(\tau_{n-1},\tau_n)$ which verifies
\begin{equation}\label{inductive-hyp-iter-step}
\begin{array}{ll}
1.&\norm{p_{n-1}}_{L^2(\tau_{n-2},\tau_{n-1})}\leq N(T_{n-1})\norm{v_{n-2}},\\
2.& y(\tau_{n-1};v_{n-2},\tau_{n-2},p_{n-1})=0,\\
3.&\norm{v(\tau_{n-1};v_{n-2},\tau_{n-2},p_{n-1})}\leq e^{\left(\sum_{j=1}^{n-1} 2^{n-1-j}j^2-2^{n-1}6\right)\Gamma_0/T_1},\\
4.&\norm{v(\tau_{n-1};v_{n-2},\tau_{n-2},p_{n-1})}\leq \prod_{j=1}^{n-1}K(T_j)^{2^{n-1-j}}\norm{v_0}^{2^{n-1}}.
\end{array}
\end{equation}
We shall now prove that there exists $p_n\in L^2(\tau_{n-1},\tau_n)$ such that every item of \eqref{iterative-stepp} is fulfilled. We defined $q_{n-1}$ and $v_{n-1}$ as in \eqref{def-q-v} and we consider the following problem
\begin{equation}\label{v-iterative-step}
\begin{cases}
v'(t)+Av(t)+p(t)Bv(t)+p(t)B\varphi_j=0,&[\tau_{n-1},\tau_n]\\
v(\tau_{n-1})=v_{n-1},
\end{cases}
\end{equation}
where the control $p$ has still to be suitably chosen. By the change of variables $s=t-\tau_{n-1}$ and the definition \eqref{taun}, we shift the problem from $[\tau_{n-1},\tau_n]$ into the interval $[0,T_n]$. We introduce the functions $\tilde{v}(s)=v(s+\tau_{n-1})$ and $\tilde{p}(s)=p\left(s+\tau_{n-1}\right)$ and we rewrite \eqref{v-iterative-step} as
\begin{equation}\label{tildevtn-1tn}
\left\{\begin{array}{ll}
\tilde{v}'(s)+A\tilde{v}(s)+\tilde{p}(s)B\tilde{v}(s)+\tilde{p}(s)B\varphi_j=0,&s\in \left[0,T_n\right]\\\\
\tilde{v}(0)=v_{n-1}.
\end{array}
\right.
\end{equation}
Recalling that $\{A,B\}$ is $j$-null controllable in any time, there exists a control $\tilde{p}_n\in L^2(0,T_n)$ such that
\begin{equation*}
\norm{\tilde{p}_n}_{L^2(0,T_n)}\leq N(T_n)\norm{v_{n-1}}\quad\text{and}\quad \tilde{y}(T_n,v_{n-1},0,\tilde{p}_n)=0,
\end{equation*}
where $\tilde{y}(\cdot;v_{n-1},0,\tilde{p}_n)$ is the solution of the linear problem \eqref{newlin} on $[0,T_n]$. Furthermore, since $v_{n-1}=v(\tau_{n-1};v_0,0,q_{n-1})=v(\tau_{n-1};v_{n-2},\tau_{n-2},p_{n-1})$, from 3. of \eqref{inductive-hyp-iter-step} we obtain that
\begin{equation}\label{induc}
\begin{split}
N(T_n)\norm{v_{n-1}}&\leq e^{\nu n^2/T_1}e^{\left(\sum_{j=1}^{n-1} 2^{n-1-j}j^2-2^{n-1}6\right)\Gamma_0/T_1}\\
&\leq e^{(n^2+(-(n-1)^2-4(n-1)+2^{n-1}6-6-2^{n-1}6)\Gamma_0/T_1}\\
&=e^{-(2n+3)\Gamma_0/T_1}\leq1,
\end{split}
\end{equation}
where we have used that the constant of the control cost $\nu$ is less than or equal to $\Gamma_0$ (see Remark \ref{oss32}), and the identity
\begin{equation}
\sum_{j=0}^n\frac{j^2}{2^j}=2^{-n}(-n^2-4n+6(2^n-1)), \qquad n\geq0,
\end{equation}
which can be easily checked by induction.
We now choose the control $\tilde{p}=\tilde{p}_n$ in \eqref{tildevtn-1tn} and still denote by $\tilde{v}$ the corresponding solution. We set $w=\tilde{v}-\tilde{y}$. Then, $w$ solves \eqref{w} with $T=T_n$ and $p=\tilde{p}_n$. So, we can apply Proposition \ref{prop39} with $T=T_n$ to problem \eqref{tildevtn-1tn} and since $w(T_n;0,\tilde{p}_n)=\tilde{v}(T_n;v_{n-1},0,\tilde{p}_n)$, we obtain that
\begin{equation*}
\norm{\tilde{v}(T_n;v_{n-1},0,\tilde{p}_n)}\leq K(T_n)\norm{v_{n-1}}^2.
\end{equation*}
We shift back the problem into the original interval $\left[\tau_{n-1},\tau_{n}\right]$, we define $p_n(t):=\tilde{p}_n( t-\tau_{n-1})$, and we get
\begin{equation*}
\norm{p_n}_{L^2(\tau_{n-1},\tau_n)}\leq N(T_n)\norm{v_{n-1}},\quad\text{and}\quad y(\tau_n,v_{n-1},\tau_{n-1},p_n)=0,
\end{equation*}
and
\begin{equation}\label{iterative-step-bound-v-taun}
\norm{v(\tau_{n};v_{n-1},\tau_{n-1},p_n)}\leq K(T_n)\norm{v_{n-1}}^2.
\end{equation}
So, we have proved the first two items of \eqref{iterative-stepp}. Moreover, thanks to 3. of \eqref{inductive-hyp-iter-step}, we deduce that
\begin{equation}
\norm{v(\tau_{n};v_{n-1},\tau_{n-1},p_n)}\leq e^{\Gamma_0 n^2/T_1}\left[e^{\left(\sum_{j=1}^{n-1} 2^{n-1-j}j^2-2^{n-1}6\right)\Gamma_0/T_1}\right]^2=e^{\left(\sum_{j=1}^n 2^{n-j}j^2-2^n6\right)\Gamma_0/T_1},
\end{equation}
that is the third item of \eqref{iterative-stepp}. Finally, using again \eqref{iterative-step-bound-v-taun} and 4. of \eqref{inductive-hyp-iter-step} we obtain that
\begin{equation*}
\norm{v(\tau_{n};v_{n-1},\tau_{n-1},p_n)}\leq K(T_n)\left[\prod_{j=1}^{n-1}K(T_j)^{2^{n-1-j}}\norm{v_0}^{2^{n-1}}\right]^2=\prod_{j=1}^{n}K(T_j)^{2^{n-j}}\norm{v_0}^{2^{n}}.
\end{equation*}
This concludes the induction argument and the proof of \eqref{iterative-stepp}.
We are now ready to complete the proof of Theorem \ref{teo1} for the case $\lambda_j=0$. We observe that for all $n\in{\mathbb{N}}^*$
\begin{equation}\label{final}
\begin{split}
\norm{v(\tau_{n};v_{n-1},\tau_{n-1},p_n)}&\leq\prod_{j=1}^nK(T_j)^{2^{n-j}}\norm{v_0}^{2^n}\\
&\leq \prod_{j=1}^n\left(e^{\Gamma_0 j^2/T_1}\right)^{2^{n-j}}\norm{v_0}^{2^n}=e^{\Gamma_0 2^n/T_1\sum_{j=1}^nj^2/2^j}\norm{v_0}^{2^n}\\
&\leq e^{\Gamma_0 2^n/T_1\sum_{j=1}^\infty j^2/2^j}\norm{v_0}^{2^n}\leq \left(e^{6\Gamma_0/T_1}\norm{v_0}\right)^{2^n}
\end{split}
\end{equation}
where we have used that $\sum_{j=1}^\infty j^2/2^j=6$. Notice that \eqref{final} is equivalent to
\begin{equation}\label{final2}
\norm{v(\tau_{n};v_0,0,q_n)}\leq \left(e^{6\Gamma_0/T_1}\norm{v_0}\right)^{2^n},
\end{equation}
where $q_n(t)=\sum_{j=1}^{n}p_j(t)\chi_{[\tau_{j-1},\tau_j]}(t)$. We now take the limit as $n\to \infty$ in \eqref{final2} and we get
\begin{equation}
\norm{u\left(\frac{\pi^2}{6}T_1;u_0,q_{\infty}\right)-\varphi_j}=\norm{v\left(\frac{\pi^2}{6}T_1;v_0,0,q_{\infty}\right)}=\norm{v(T_f;v_0,0,q_{\infty})}\leq 0
\end{equation}
because $\norm{v_0}<e^{-6\Gamma_0/T_1}$. This means that, we have built a control $p\in L^2_{loc}([0,\infty))$, defined by
\begin{equation}
p(t)=\left\{\begin{array}{ll}
\sum_{n=1}^\infty p_{n}(t)\chi_{\left[\tau_{n-1} ,\tau_{n}\right]}(t),& t\in \left(0,T_f\right]\\\\
0,&t\in(T_f,+\infty)
\end{array}\right.
\end{equation}
for which the solution $u$ of \eqref{sys} reaches the $j$th eigensolution $\psi_j=\varphi_j$ in time $T_f$, less than or equal to $T$, and stays on it forever.
Observe that, thanks to the first item of \eqref{iterative-stepp} and to \eqref{induc}, we are able to yield a bound for the $L^2$-norm of such a control:
\begin{equation}\label{pestimate}
\begin{split}
\norm{p}^2_{L^2\left(0,T\right)}&=\sum_{n=1}^\infty \norm{p_{n}}^2_{L^2\left(\tau_{n-1},\tau_{n}\right)}\leq \sum_{n=1}^\infty \left(N(T_n)\norm{v\left(\tau_{n-1} \right)}\right)^2\leq \sum_{n=1}^\infty e^{-2(2n+3)C_K/T_1}\\
&\leq\frac{e^{-6C_K/T_1}}{e^{4C_K/T_1}-1}=\frac{e^{-\pi^2C_K/T_f}}{e^{2\pi^2C_K/(3T_f)}-1}.
\end{split}
\end{equation}
Notice that since \eqref{T_f} holds, \eqref{pestimate} implies \eqref{intro-estim-p}.
\subsection{Case $\lambda_j\neq0$}
Now, we face the case $\lambda_j\neq0$. We define the operator
\begin{equation*}
A_j:=A-\lambda_jI.
\end{equation*}
We proved in \cite[Lemma 4.7]{acu} that if $\{A,B\}$ is $j$-null controllable, then the same holds for the pair $\{A_j,B\}$. Furthermore, it is easy to check that also condition \eqref{bound-control-costt} is verified by the control cost associated with $\{A_j,B\}$, if the same property holds for the control cost associated with the pair $\{A,B\}$.
It is possible to check that $A_j$ satisfies \eqref{ipAAA} and moreover it has the same eigenfuctions, $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$, of $A$, while the eigenvalues are given by
\begin{equation*}
\mu_k=\lambda_k-\lambda_j, \qquad\forall\, k\in{\mathbb{N}}^*.
\end{equation*}
In particular, $\mu_j=0$.
We define the function $z(t)=e^{\lambda_j t}u(t)$, where $u$ is the solution of \eqref{sys}. Then, $z$ solves the following problem
\begin{equation}\label{z}
\left\{\begin{array}{ll}
z'(t)+A_jz(t)+p(t)Bz(t)=0,&[0,T],\\\\
z(0)=u_0.
\end{array}\right.
\end{equation}
We define $T_f$ as in \eqref{T_f} (where $T_0$ is now the constant associated with the control cost relative to the pair $\{A_j,B\}$) and $R_{T}$ as in \eqref{RT}. We deduce from the previous analysis that, if $u_0\in B_{R_{T}}(\varphi_j)$, then there exists a control $p\in L^2([0,+\infty))$ that steers the solution $z$ to the eigenstate $\varphi_j$ in time $T_f\leq T$. This implies the exact controllability of $u$ to the eigensolution $\psi_j(t)=e^{-\lambda_j t}\varphi_j$: indeed,
\begin{equation*}
\norm{u\left(T_f;u_0,p\right)-\psi_j\left(T_f\right)}=\norm{e^{-\lambda_jT_f}z\left(T_f\right)-e^{-\lambda_jT_f}\varphi_j}=e^{-\lambda_jT_f}\norm{z\left(T_f\right)-\varphi_j}=0.
\end{equation*}
\begin{oss}
We observe that, from \eqref{pestimate}, it follows that $\norm{p}_{L^2(0,T_{f})}\to 0$ as $T_f\to 0$. This fact is not surprising since as $T_f$ approaches $0$, also the size of the neighborhood where the initial condition can be chosen goes to zero.
\end{oss}
\section{Proof of Theorem \ref{Thm-suff-cond}}
Before showing the proof of Theorem \ref{Thm-suff-cond}, we define formally the following function
\begin{equation}\label{G}
G_M(T):=\frac{M}{T^4}e^{M/T}\sum_{k=1}^{\infty}\frac{e^{-2\omega_kT+M\sqrt{\omega_k}}}{|\langle B\varphi_j,\varphi_k\rangle|^2},
\end{equation}
where $M$ is a positive constant, $\omega_k:=\lambda_k-\lambda_1$, for all $k\in{\mathbb{N}}^*$, $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ are the eigenvalues of $A$. In Lemma \ref{lem1} below, we investigate the behavior of $G_M(T)$ for small values of $T$. Such a result will be crucial for the analysis of the control cost $N(T)$ in Theorem \ref{Thm-suff-cond}.
\begin{lem}\label{lem1}
Let $A:D(A)\subset X\to X$ be such that \eqref{ipA} and \eqref{gap} hold and $B:X\to X$ be such that \eqref{ipB} holds. Then, for any $M,T>0$ the series in \eqref{G} is convergent and there exists a positive constant $\Gamma_j$, such that
\begin{equation}
G_M(T)\leq e^{2\Gamma_j/T},\quad\forall\,0<T\leq 1.
\end{equation}
Moreover, a suitable choice of $\Gamma_j=\Gamma_j(M,b,q, \alpha)$ is \eqref{Gamma_jFatiha}.
\end{lem}
\begin{proof}
Thanks to assumption \eqref{ipB}, we have that
\begin{equation}
\begin{split}
G_M(T)&=\frac{M}{T^4}e^{M/T}\sum_{k=1}^{\infty}\frac{e^{-2\omega_kT+M\sqrt{\omega_k}}}{|\langle B\varphi_j,\varphi_k\rangle|^2}\\
&\leq \frac{M}{T^4}e^{M/T}\left[\frac{e^{M^2/(8T)}}{|\langle B\varphi_j,\varphi_j\rangle|^2}+\frac{1}{b^2}\sum_{k\neq j,\,k=1}^{\infty}\left(\omega_k^{2q}e^{-\omega_kT}\right)e^{-\omega_kT+M\sqrt{\omega_k}}\right].
\end{split}
\end{equation}
For any $\omega\geq0$ we set $f(\omega)=e^{-\omega T+M\sqrt{\omega}}$. The maximum value of $f$ is attained at $\omega=\left(\frac{M}{2T}\right)^2$. So, we can bound $G_M(T)$ as follows
\begin{equation}\label{gamma4}
G_M(T)\leq\frac{M}{T^4}e^{M/T}\left[\frac{e^{M^2/(8T)}}{|\langle B\varphi_j,\varphi_j\rangle|^2}+\frac{e^{M^2/(4T)}}{b^2}\sum_{k=1}^{\infty}\omega_k^{2q}e^{-\omega_kT}\right].
\end{equation}
Now, for any $\omega\geq0$ we define the function $g(\omega)=\omega^{2q}e^{-\omega T}$. Its derivative is given by
\begin{equation*}
g'(\omega)=(2q-\omega T)\omega^{2q-1}e^{-\omega T}
\end{equation*}
and therefore we deduce that
\begin{equation*}
g(\omega)\mbox{ is }\left\{\begin{array}{ll}\mbox{increasing } & \mbox{if }0\leq \omega<(2q)/T\\\\
\mbox{decreasing}&\mbox{if } \omega\geq (2q)/T \end{array}\right.
\end{equation*}
and $g$ has a maximum at $\omega=(2q)/T$. We define the following index:
\begin{equation*}
k_1:=k_1(T)=\sup\left\{k\in{\mathbb{N}}^*\,:\,\omega_k\leq \frac{2q}{T}\right\}
\end{equation*}
Note that $k_1(T)$ goes to $\infty$ as $T$ converges to $0$.
We can rewrite the sum in \eqref{gamma4} as follows
\begin{equation}\label{s123}
\sum_{k=1}^{\infty}\omega_k^{2q}e^{-\omega_kT}=\sum_{k\leq k_1-1}\omega_k^{2q}e^{-\omega_kT}+\sum_{k_1\leq k\leq k_1+1}\omega_k^{2q}e^{-\omega_kT}+\sum_{k\geq k_1+2}\omega_k^{2q}e^{-\omega_kT}.
\end{equation}
For any $k\leq k_1-1$, we have
\begin{equation}\label{s1}
\int_{\omega_k}^{\omega_{k+1}}\omega^{2q}e^{-\omega T}d\omega\geq (\omega_{k+1}-\omega_k)\omega_k^{2q}e^{-\omega_k T}\geq \alpha\sqrt{\omega_2}\,\omega_k^{2q}e^{-\omega_k T}
\end{equation}
and for any $k\geq k_1+2$
\begin{equation}\label{s3}
\int_{\omega_{k-1}}^{\omega_k}\omega^{2q}e^{-\omega T}d\omega\geq (\omega_k-\omega_{k-1})\omega_k^{2q}e^{-\omega_k T}\geq \alpha\sqrt{\omega_2}\,\omega_k^{2q}e^{-\omega_k T}.
\end{equation}
So, by using estimates \eqref{s1} and \eqref{s3}, \eqref{s123} becomes
\begin{equation}\label{s2i}
\sum_{k=1}^{\infty}\omega_k^{2q}e^{-\omega_kT}\leq\frac{2}{\alpha\sqrt{\omega_2}}\int_0^\infty \omega^{2q}e^{-\omega T}d\omega+\sum_{k_1\leq k\leq k_1+1}\omega_k^{2q}e^{-\omega_kT}.
\end{equation}
Furthermore, recalling that $g$ has a maximum for $\omega=2q/T$, it holds that
\begin{equation}\label{s2}
k=k_1,k_1+1\quad\Rightarrow\quad \omega_k^{2q}e^{-\omega_k T}\leq\left(2q/T\right)^{2q}e^{-2q}.
\end{equation}
Finally, the integral term of \eqref{s2i} can be rewritten as
\begin{equation}\label{s13}
\int_0^\infty\omega^{2q}e^{-\omega T}d\omega=\frac{1}{T}\int_0^{\infty}\left(\frac{s}{T}\right)^{2q}e^{-s}ds=\frac{1}{T^{1+2q}}\int_0^{\infty}s^{2q}e^{-s}ds=\frac{\Gamma(2q+1)}{T^{1+2q}},
\end{equation}
where by $\Gamma(\cdot)$ we indicate the Euler integral of the second kind.
Therefore, we conclude from \eqref{s2} and \eqref{s13} that there exist two constants $C_q,C_{q,\alpha}>0$ such that
\begin{equation}
\sum_{k=1}^\infty \omega_k^{2q}e^{-\omega_k T}\leq \frac{C_q}{T^{2q}}+\frac{C_{\alpha,q}}{T^{1+2q}}.
\end{equation}
We use this last bound to prove that there exists $\Gamma_j>0$ such that
\begin{equation}\label{ineqCM}
G_M(T)\leq\frac{M}{T^4}e^{M/T}\left[\frac{e^{M^2/(8T)}}{|\langle B\varphi_j,\varphi_j\rangle|^2}+\frac{e^{M^2/(4T)}}{b^2}\left(\frac{C_q}{T^{2q}}+\frac{C_{\alpha,q}}{T^{1+2q}}\right)\right]\leq e^{2\Gamma_j/T} \quad \forall \ T \in (0,1],
\end{equation}
as claimed.
\end{proof}
Now we proceed with the proof of Theorem \ref{Thm-suff-cond}.
\begin{proof}[Proof (of Theorem \ref{Thm-suff-cond}).]
Let $T>0$ and consider problem \eqref{newlin}. For any $y_0\in X$ and $p\in L^2(0,T)$ there exists a unique strong solution $y\in C^0([0,T],X)$ of \eqref{newlin} that can be written as
\begin{equation}
y(t)=e^{-tA}y_0-\int_0^t e^{-(t-s)A}p(s)B\varphi_jds,
\end{equation}
(see, for instance, \cite[Proposition 3.1, p. 130]{bd}).
Our aim is to find a control $p\in L^2(0,T)$ for which $y(T;y_0,0,p)=0$, that is equivalent to the following identity
\begin{equation*}
\sum_{k\in{\mathbb{N}}^*}\langle y_0,\varphi_k\rangle e^{-\lambda_k T}\varphi_k=\int_0^T p(s)\sum_{k\in{\mathbb{N}}^*}\langle B\varphi_j,\varphi_k\rangle e^{-\lambda_k(T-s)}\varphi_kds.
\end{equation*}
Since, by hypothesis, the eigenfunctions of $A$ form an orthonormal basis of $X$, the above formula reads as
\begin{equation*}
\langle y_0,\varphi_k\rangle=\int_0^T e^{\lambda_ks}p(s)\langle B\varphi_j,\varphi_k\rangle ds,\quad\forall\,k\in{\mathbb{N}}^*,
\end{equation*}
or, equivalently,
\begin{equation}\label{momp}
\int_0^T e^{\lambda_ks}p(s)ds=\frac{\langle y_0,\varphi_k\rangle}{\langle B\varphi_j,\varphi_k\rangle},\quad\forall\,k\in{\mathbb{N}}^*.
\end{equation}
By defining $q(s):=e^{\lambda_1 s}p(s)$ and $\omega_k:=\lambda_k-\lambda_1\geq0$, the family of equations \eqref{momp} can be rewritten as
\begin{equation}\label{momp-q}
\int_0^T e^{\omega_k s}q(s)ds=\frac{\langle y_0,\varphi_k\rangle}{\langle B\varphi_j,\varphi_k\rangle},\quad\forall\,k\in{\mathbb{N}}^*.
\end{equation}
Thanks to hypothesis \eqref{gap}, we can apply \cite[Theorem 2.4]{cmv} that ensures the existence of a biorthogonal family $\{\sigma_k\}_{k\in{\mathbb{N}}^*}$ to the family of exponentials $\{\zeta_k\}_{k\in{\mathbb{N}}^*}$, $\zeta_k(s)=e^{\omega_ks}$, $s\in[0,T]$.
We claim that the series
\begin{equation}\label{p}
\sum_{k\in{\mathbb{N}}^*}\frac{\langle y_0,\varphi_k\rangle}{\langle B\varphi_j,\varphi_k\rangle}\sigma_k(s),
\end{equation}
is convergent in $L^2(0,T)$. Indeed, thanks to the following estimate, from \cite[Theorem 2.4]{cmv}, for the biorthogonal family $\{\sigma_k\}_{k\in{\mathbb{N}}^*}$
\begin{equation*}
\norm{\sigma_k}^2_{L^2(0,T)}\leq C_\alpha(T) e^{-2\omega_kT}e^{C \sqrt{\omega_k}/\alpha},\quad\forall\,k\in{\mathbb{N}}^*,
\end{equation*}
with $C>0$ independent of $T$ and $\alpha$, and
\begin{equation*}
C_\alpha(T)=\left\{\begin{array}{ll}
C\left(\frac{1}{T}+\frac{1}{T^2\alpha^2}\right)e^{\frac{C}{\alpha^2 T}}&\text{if }T<\frac{1}{\alpha^2},\\\\
C^2\alpha^2&\text{if }T\geq\frac{1}{\alpha^2},
\end{array}\right.
\end{equation*}
we obtain
\begin{equation*}
\begin{split}
\sum_{k\in{\mathbb{N}}^*}\left|\frac{\langle y_0,\varphi_k\rangle}{\langle B\varphi_j,\varphi_k\rangle}\right|\norm{\sigma_k}_{L^2(0,T)}&\leq \norm{y_0}\left(\sum_{k\in{\mathbb{N}}^*}\frac{\norm{\sigma_k}^2_{L^2(0,T)}}{|\langle B\varphi_j,\varphi_k\rangle|^2}\right)^{1/2}\\
&\leq\norm{y_0}\left( C^2_\alpha(T)\sum_{k\in{\mathbb{N}}^*}\frac{e^{-2\omega_kT}e^{C\sqrt{\omega_k}/\alpha}}{|\langle B\varphi_j,\varphi_k\rangle|^2})\right)^{1/2}.
\end{split}
\end{equation*}
Observe that, by Lemma \ref{lem1}, the right-hand side of the above estimate is finite for any $T>0$.
Therefore, we define the control $q$ as
\begin{equation*}
q(s):=\sum_{k\in{\mathbb{N}}^*}\frac{\langle y_0,\varphi_k\rangle}{\langle B\varphi_j,\varphi_k\rangle}\sigma_k(s),
\end{equation*}
and we deduce that $q\in L^2(0,T)$ satisfies \eqref{momp-q} and furthermore
\begin{equation*}
\norm{q}_{L^2(0,T)}\leq C_\alpha(T)\Lambda_T\norm{y_0},
\end{equation*}
where $\Lambda_T:=\left( \sum_{k\in{\mathbb{N}}^*}\frac{e^{-2\omega_kT}e^{C\sqrt{\omega_k}/\alpha}}{|\langle B\varphi_j,\varphi_k\rangle|^2}\right)^{1/2}$.
Finally, returning to $p$, we obtain that
\begin{equation}
\norm{p}^2_{L^2(0,T)}=\int_0^Te^{-2\lambda_1s}|q(s)|^2ds\leq \max\left\{1,e^{-2\lambda_1 T}\right\}\norm{q}^2_{L^2(0,T)}.
\end{equation}
By taking
\begin{equation}\label{def-control-cost-p}
N(T):=\max\left\{1,e^{-\lambda_1 T}\right\}C_{\alpha}(T)\Lambda_T,
\end{equation}
we deduce that $\{A,B\}$ is $j$-null controllable in any time $T>0$ with associated control cost \eqref{def-control-cost-p}.
What remains to prove is estimate \eqref{bound-control-cost} for the control cost $N(T)$ defined in \eqref{def-control-cost-p}, for $T$ small.
Let us define $T_0$ as in \eqref{T_0}. Then for any $0<T< T_0$, it holds that
\begin{equation*}
C_\alpha(T)=C\left(\frac{1}{T}+\frac{1}{T^2\alpha^2}\right)e^{\frac{C}{\alpha^2 T}}.
\end{equation*}
We can assume without loss of generality that the constant $C \geq 1$, since we can replace it by $\max\left\{1, C\right\}$. We assume for all the sequel that $C \geq 1$.
Since $0<T< T_0 \leq 1$, we claim that there exists $\widetilde{M}>0$ such that
\begin{equation}\label{aimM}
C_\alpha^2(T) \leq \frac{\widetilde{M}}{T^4}e^{\widetilde{M}/T} \quad \forall \ T \in (0,T_0).
\end{equation}
Indeed, we have
\begin{equation*}
C_\alpha^2(T)\leq C^2\left(1+\frac{1}{\alpha^2}\right)^2\frac{1}{T^4}e^{\frac{2C}{\alpha^2T}} \quad \forall \ T \in (0,T_0).
\end{equation*}
We set
\begin{equation*}
\widetilde{M}:=C^2\left(1+\frac{1}{\alpha^2}\right)^2.
\end{equation*}
We note that since $C\geq 1$, we have
$$
\dfrac{2C}{\alpha^2} \leq \widetilde{M}.
$$
Hence from the two above estimates, we deduce \eqref{aimM}.
Moreover, we easily prove that
$$
\max\left\{1,e^{-\lambda_1 T}\right\} \leq e^{|\lambda_1|} \quad \forall \ T \in (0,T_0).
$$
Therefore, the control cost $N(T)$ given by \eqref{def-control-cost-p} can be bounded from above as follows
\begin{equation*}
N(T)\leq \sqrt{G_{M}(T)},
\end{equation*}
where $M$ is defined as in \eqref{M} and the function $G_M(\cdot)$ is defined in \eqref{G}. Finally, thanks to Lemma \ref{lem1}, we deduce that $N(T)$ fulfills property \eqref{bound-control-cost} with $\nu=\Gamma_j$.
\end{proof}
\section{Proof of Theorems \ref{teoglobal} and \ref{teoglobal0}}
Before proving Theorem \ref{teoglobal}, let us show a preliminary result that demonstrates the statement in the case of a strictly accretive operator.
\begin{lem}\label{lemglobal}
Let $A:D(A)\subset X\to X$ be a densely defined linear operator such that \eqref{ipA} holds with $\sigma=0$ and let $B:X\to X$ be a bounded linear operator. Let \{A,B\} be a 1-null controllable pair which satisfies \eqref{bound-control-cost}. Furthermore, we assume $\lambda_1=0$. Then, there exists a constant $r_1>0$ such that for any $R>0$ there exists $T_{R}>0$ such that for all $v_0\in X$ that satisfy
\begin{equation}\label{ipv0}
\left|\langle v_0,\varphi_1\rangle\right| < r_1,\qquad\norm{v_0-\langle v_0,\varphi_1\rangle\varphi_1}\leq R,
\end{equation}
problem \eqref{v} is null controllable in time $T_{R}$.
\end{lem}
\begin{proof}
{\bf First step.} We fix $T=1$. Thanks to Theorem \ref{teo1}, there exists a constant $r_1>0$ such that, denoting by $u_1$ the solution of \eqref{u} on $[0,1]$, if $\norm{u_1(0)- \varphi_1}< \sqrt{2}r_1$ then there exists a control $p_1\in L^2(0,1)$ for which the solution of \eqref{u} of with $p$ replaced by $p_1$, satisfies $u_1(1)=\varphi_1$. We set $v_1=u_1- \varphi_1$ on $[0,1]$. We deduce that
if $\norm{v_1(0)} < \sqrt{2}r_1$ then there exists a control $p_1\in L^2(0,1)$ for which the solution $v_1$ of \eqref{v} on $[0,1]$ with $p$ replaced by $p_1$, satisfies $v_1(1)=0$.
\smallskip\noindent
{\bf Second step.} Let $v_0\in X$ be the initial condition of \eqref{v}. We decompose $v_0$ as follows
\begin{equation*}
v_0=\langle v_0,\varphi_1\rangle\varphi_1+v_{0,1},
\end{equation*}
where $v_{0,1}\in \varphi_1^\perp$ and we suppose that $\left|\langle v_0,\varphi_1\rangle\right| < r_1$. If $R\leq r_1$, then $ \norm{v_0}^2 \leq r^2_1+R^2\leq 2r^2_1$ and we can directly apply the first step of the proof
with $T_R=1$. Otherwise, we define $t_{R}$ as
\begin{equation}\label{trR}
t_{R}:=\frac{1}{2\lambda_2}\log{\left(\frac{R^2}{r_1^2}\right)},
\end{equation}
and in the time interval $[0,t_{R}]$ we take the control $p\equiv0$. Then, for all $t\in [0,t_{R}]$, we have that
\begin{equation*}
\norm{v(t)}^2\leq \norm{e^{-tA}\left(\langle v_0,\varphi_1\rangle\varphi_1+v_{0,1}\right)}^2\leq \left|\langle v_0,\varphi_1\rangle\right|^2+e^{-2\lambda_2 t}\norm{v_{0,1}}^2 < r_1^2+e^{-2\lambda_2 t}R^2.
\end{equation*}
In particular, for $t=t_{R}$, it holds that $\norm{v(t_{R})}^2 < 2 r^2_1$.
Now, we define $T_{R}:=t_{R}+1$ and set $v_1(0)=v(t_R)$. Thanks to the first step of the proof, there exists a control $p_1\in L^2(0,1)$, such that $v_1(1)=0$, where $v_1$ is the solution of \eqref{v} on $[0,1]$ with $p$ replaced by $p_1$.
Then $v(t)=v_1(t-t_R)$ solves \eqref{v} in the time interval $(t_{R},T_{R}]$ with the control $p_1(t-t_{R})$ that steers the solution $v$ to $0$ at $T_{R}$.
\end{proof}
\begin{proof}[Proof (of Theorem \ref{teoglobal})]
We start with the case $\lambda_1=0$. Let $u_0\in X$ satisfy \eqref{ipu0}. Set $v(t):=u(t)-\varphi_1$, then $v$ satisfies \eqref{v} and moreover $v_0:=v(0)=u_0-\varphi_1$ fulfills \eqref{ipv0}. Thus, by Lemma \ref{lemglobal}, problem \eqref{u} is exactly controllable to the first eigensolution $\psi_1
\equiv\varphi_1$ in time $T_{R}$.
Now, we consider the case $\lambda_1>0$. As in the proof of Theorem \ref{teo1}, we introduce the variable $z(t)=e^{\lambda_1t}u(t)$ that solves problem \eqref{z}. For such a system, since the first eigenvalue of $A_1$ is equal $0$, we have the exact controllability to $\varphi_1$ in time $T_{R}$. Namely $z(T_{R})=\varphi_1$, that is equivalent to the exact controllability of $u$ to $\psi_1$:
\begin{equation}
z(T_{R})=\varphi_1\
\quad\Longleftrightarrow\quad
e^{\lambda_1T_{R}}u(T_{R})=\varphi_1
\quad\Longleftrightarrow\quad
u(T_{R})=\psi_1(T_{R}).
\end{equation}
The proof is thus complete.
\end{proof}
The proof of Theorem \ref{teoglobal0} easily follows from Theorem \eqref{teoglobal}.
\begin{proof}[Proof (of Theorem \ref{teoglobal0})]
We assume \eqref{cone}. Suppose that $\gamma:=\langle u_0,\varphi_1\rangle\neq0$. We decompose $u_0$ as $u_0=\gamma\varphi_1+\zeta_1$, with $\zeta_1:=u_0-\langle u_0,\varphi_1\rangle\varphi_1\in\varphi_1^\perp$ and define $\tilde{u}(t):=u(t)/\gamma$. Hence, $\tilde{u}$ solves
\begin{equation}\label{utildeglobal}
\left\{
\begin{array}{ll}
\tilde{u}'(t)+A\tilde{u}(t)+p(t)B\tilde{u}(t)=0,& t>0\\
\tilde{u}(0)=\varphi_1+\tilde{\zeta_1},
\end{array}\right.
\end{equation}
where $\tilde{\zeta_1}:=\zeta_1/\gamma$.
We apply Theorem \ref{teoglobal} to \eqref{utildeglobal} to deduce the existence of $T_R>0$ such that $\tilde{u}(T_R)=\psi_1(T_R)$. Therefore, the solution of \eqref{u} with initial condition $u_0\in X$ that do not vanish along the direction $\varphi_1$ can be exactly controlled in time $T_R$ to the trajectory $\phi_1(\cdot)=\langle u_0,\varphi_1\rangle\psi_1(\cdot)$, where
$\phi_1$ is defined in \eqref{exactphi1}.
Note that if $u_0\in X$ satisfies both $u_0\in\varphi_1^\perp$ and \eqref{cone}, then we have trivially that $u_0\equiv 0$. We then choose $p\equiv 0$, so that the solution of \eqref{u} remains constantly equal to $\phi_1\equiv 0$.
\end{proof}
\section{Applications}
In this section we present some examples of parabolic equations for which Theorem \ref{teo1} can be applied. The hypotheses \eqref{ipA},\eqref{gap} and \eqref{ipB} have been verified in \cite{acu} and \cite{cu}, to which we refer for more details. We observe that, thanks to \cite[Remark 6.1]{acu}, since the second order operators considered in the examples are accretive ($\langle Ax,x\rangle\geq0$, for all $x\in D(A)$), it suffices to prove the following gap condition
\begin{equation}\label{gap-no-lambda1}
\exists\,\alpha>0\,:\,\sqrt{\lambda_{k+1}}-\sqrt{\lambda_k}\geq\alpha,\quad\forall\,k\geq1,
\end{equation}
which implies \eqref{gap}.
Moreover, in the case of an accretive operator it suffices to show that there exist $b,q>0$ such that
\begin{equation}\label{ipB-accr}
\begin{array}{l}
\langle B\varphi_j,\varphi_j\rangle\neq0\quad\mbox{and}\quad\left|\lambda_k-\lambda_j\right|^q|\langle B\varphi_j,\varphi_k\rangle|\geq b,\quad\forall\,k\neq j.
\end{array}
\end{equation}
to have \eqref{ipB}.
Furthermore, we note that the global results Theorem \ref{teoglobal} and Theorem \ref{teoglobal0} can be applied to any example below. Note also, that the given examples below, are non-exhaustive.
\subsection{Diffusion equation with Dirichlet boundary conditions}\label{ex1}
Let $I=(0,1)$ and $X=L^2(0,1)$. Consider the following problem
\begin{equation}\label{eqex1}\left\{\begin{array}{ll}
u_t(t,x)-u_{xx}(t,x)+p(t)\mu(x)u(t,x)=0 & x\in I,t>0 \\\\
u(t,0)=0,\,\,u(t,1)=0, & t>0\\\\
u(0,x)=u_0(x) & x\in I.
\end{array}\right.
\end{equation}
We denote by $A$ the operator defined by
\begin{equation*}
D(A)=H^2\cap H^1_0(I),\quad A\varphi=-\frac{d^2\varphi}{dx^2}.
\end{equation*}
and it can be checked that $A$ satisfies \eqref{ipA}. We indicate by $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ and $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the families of eigenvalues and eigenfunctions of $A$, respectively:
\begin{equation*}
\lambda_k=(k\pi)^2,\quad \varphi_k(x)=\sqrt{2}\sin(k\pi x),\quad \forall\, k\in{\mathbb{N}}^*.
\end{equation*}
It is easy to see that \eqref{gap-no-lambda1} holds true (and so \eqref{gap}):
\begin{equation*}
\sqrt{\lambda_{k+1}}-\sqrt{\lambda_k}=\pi,\qquad \forall\, k\in {\mathbb{N}}^*.
\end{equation*}
Let $B:X\to X$ be the operator
\begin{equation*}
B\varphi=\mu\varphi
\end{equation*}
with $\mu\in H^3(I)$ such that
\begin{equation}\label{mu}
\mu'(1)\pm\mu'(0)\neq 0\quad\mbox{ and }\quad\langle\mu\varphi_j,\varphi_k\rangle\neq0\quad\forall\, k \in {\mathbb{N}}^*.
\end{equation}
Then, there exists $b>0$ such that
\begin{equation*}
\left|\lambda_k-\lambda_j\right|^{3/2}|\langle \mu\varphi_j,\varphi_k\rangle|\geq b,\qquad\forall\, k\in{\mathbb{N}}^*.
\end{equation*}
For instance, a suitable function that satisfies \eqref{mu} is $\mu(x)=x^2$: indeed, in this case
\begin{equation*}
\langle \mu\varphi_j,\varphi_k\rangle=\left\{\begin{array}{ll}
\frac{4kj(-1)^{k+j}}{(k^2-j^2)^2},& k\neq j,\\\\
\frac{2j^2\pi^2-3}{6j^2\pi^2},&k=j.
\end{array}\right.
\end{equation*}
Therefore, problem \eqref{eqex1} is controllable to the $j$-th eigensolution $\psi_j$ in any time $T>0$ as long as $u_0\in B_{R_T}(\varphi_j)$, with $R_T>0$ a suitable constant, where $\psi_j(t,x)=\sqrt{2}\sin(j\pi x)e^{-j^2\pi^2t}$.
\subsection{Diffusion equation with Neumann boundary conditions}\label{ex2}
Let $I=(0,1)$, $X=L^2(I)$ and consider the Cauchy problem
\begin{equation}\label{eqex2}
\left\{\begin{array}{ll}
u_t(t,x)-u_{xx}(t,x)+p(t)\mu(x)u(t,x)=0 & x\in I,t>0 \\\\
u_x(t,0)=0,\,\,u_x(t,1)=0, &t>0\\\\
u(0,x)=u_0(x). & x\in I.
\end{array}\right.
\end{equation}
The operator $A$, defined by
\begin{equation*}
D(A)=\{ \varphi\in H^2(0,1): \varphi'(0)=0,\,\,\varphi'(1)=0\},\quad A\varphi=-\frac{d^2\varphi}{dx^2}
\end{equation*}
satisfies \eqref{ipA} and has the following eigenvalues and eigenfunctions
\begin{equation*}
\begin{array}{lll}
\lambda_0=0,&\varphi_0=1\\
\lambda_k=(k\pi)^2,& \varphi_k(x)=\sqrt{2}\cos(k\pi x),& \forall\, k\geq1.
\end{array}
\end{equation*}
Thus, the gap condition \eqref{gap-no-lambda1} is fulfilled with $\alpha=\pi$. Fixed $j\in{\mathbb{N}}$, the $j$-th eigensolution is the function $\psi_j(x)=e^{-\lambda_j t}\varphi_j(x)$.
We define $B:X\to X$ as the multiplication operator by a function $\mu\in H^2(I)$, $B\varphi=\mu\varphi$, such that
\begin{equation}
\mu'(1)\pm\mu'(0)\neq 0\quad\mbox{ and }\quad\langle\mu\varphi_j,\varphi_k\rangle\neq0\quad\forall\, k \in {\mathbb{N}}.
\end{equation}
It can be proved that, there exists $b>0$ such that
\begin{equation}\label{mu2}
\left|\lambda_k-\lambda_j\right||\langle \mu\varphi_j,\varphi_k\rangle|\geq b,\qquad\forall\, k\in{\mathbb{N}}^*.
\end{equation}
For example, $\mu(x)=x^2$ satisfies \eqref{mu2}. Indeed, it can be shown that
\begin{equation*}
\langle \mu\varphi_0,\varphi_k\rangle=\left\{\begin{array}{ll}
\frac{2\sqrt{2}(-1)^{k}}{(k\pi)^2},&k\geq1,\\\\
\frac{1}{3},&k=0,
\end{array}\right.
\end{equation*}
and for $j\neq0$
\begin{equation*}
\langle \mu\varphi_j,\varphi_k\rangle=\left\{\begin{array}{ll}
\frac{4(-1)^{k+j}(k^2+j^2)}{(k^2-j^2)^2\pi^2},&k\neq j,\\\\
\frac{1}{3}+\frac{1}{2j^2\pi^2},&k=j.
\end{array}\right.
\end{equation*}
Therefore, problem \eqref{eqex2} is controllable to the $j$-th eigensolution $\psi_j$ in any time $T>0$ as long as $u_0\in B_{R_T}(\varphi_j)$, with $R_T>0$ a suitable constant.
\subsection{Variable coefficient parabolic equation}\label{ex3}
Let $I=(0,1)$, $X=L^2(I)$ and consider the problem
\begin{equation}\label{eqex3}
\left\{
\begin{array}{ll}
u_t(t,x)-((1+x)^2u_x(t,x))_x+p(t)\mu(x)u(t,x)=0&x\in I,t>0\\\\
u(t,0)=0,\quad u(t,1)=0,&t>0\\\\
u(0,x)=u_0(x)&x\in I.
\end{array}
\right.
\end{equation}
We denote by $A:D(A)\subset X\to X$ the following operator
\begin{equation*}
D(A)=H^2\cap H^1_0(I),\qquad A\varphi=-((1+x)^2\varphi_x)_x.
\end{equation*}
It can be checked that $A$ satisfies \eqref{ipA} and that the eigenvalues and eigenfunctions have the following expression
\begin{equation*}
\lambda_k=\frac{1}{4}+\left(\frac{k\pi}{\ln2}\right)^2,\qquad\varphi_k=\sqrt{\frac{2}{\ln 2}}(1+x)^{-1/2}\sin\left(\frac{k\pi}{\ln2 }\ln(1+x)\right).
\end{equation*}
Furthermore, $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ verifies the gap condition \eqref{gap-no-lambda1} with $\alpha=\pi/\ln{2}$.
We fix $j\in{\mathbb{N}}^*$ and define the operator $B:X\to X$ by $B\varphi=\mu\varphi$, where $\mu\in H^2(I)$ is such that
\begin{equation}\label{mu3}
2\mu'(1)\pm\mu'(0)\neq0,\quad\mbox{and}\quad \langle \mu\varphi_j,\varphi_k\rangle\neq0\quad\forall k \in{\mathbb{N}}^*.
\end{equation}
Hence, thanks to \eqref{mu3}, it is possible to show that \eqref{ipB-accr} is fulfilled with $q=3/2$ (see \cite[Section 6.3]{acu}). For instance, when $j=1$, an example of a suitable function $\mu$ that satisfies \eqref{mu3} is $\mu(x)=x$, see \cite{acu} for the verification.
Thus, from Theorem \ref{teo1}, we deduce that, for any $T>0$, system \eqref{eqex3} is controllable to the $j$-th eigensolution if the initial condition $u_0$ is close enough to $\varphi_j$.
\subsection{Diffusion equation in a $3D$ ball with radial data}\label{ex4}
In this example, we study the controllability of an evolution equation in the three dimensional unit ball $B^3$ for radial data. The bilinear control problem is the following
\begin{equation}\label{eqex4}
\left\{\begin{array}{ll}
u_t(t,r)-\Delta u(t,r)+p(t)\mu(r)u(t,r)=0 & r\in[0,1], t>0 \\\\
u(t,1)=0,&t>0\\\\
u(0,r)=u_0(r) & r\in[0,1]
\end{array}\right.
\end{equation}
where the Laplacian in polar coordinates for radial data is given by the following expression
$$\Delta\varphi(r)=\partial^2_r \varphi(r)+\frac{2}{r}\partial_r\varphi(r).$$
The function $\mu$ is a radial function as well in the space $H^3_r(B^3)$, where the spaces $H^k_r(B^3)$ are defined as follows
$$X:=L^2_{r}(B^3)=\left\{\varphi\in L^2(B^3)\,|\, \exists \psi:{\mathbb{R}}\to{\mathbb{R}}, \varphi(x)=\psi(|x|)\right\}$$
$$H^k_r(B^3):=H^k(B^3)\cap L^2_{r}(B^3) .$$
The domain of the Dirichlet Laplacian $A:=-\Delta$ in $X$ is $D(A)=H^2_{r}\cap H^1_0(B^3)$. We observe that $A$ satisfies hypothesis \eqref{ipA}. We denote by $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ and $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the families of eigenvalues and eigenvectors of $A$, $A\varphi_k=\lambda_k\varphi_k$, namely
\begin{equation}\label{ee}\varphi_k=\frac{\sin(k\pi r)}{\sqrt{2\pi}r},\qquad\lambda_k=(k\pi)^2
\end{equation}
$\forall\, k\in{\mathbb{N}}^*$, see \cite[Section 8.14]{leb}. Since the eigenvalues of $A$ are actually the same of the Dirichlet $1D$ Laplacian, \eqref{gap-no-lambda1} is satisfied, as we have seen in Example \ref{ex1}.
Fixed $j\in{\mathbb{N}}^*$, let $B:X\to X$ be the multiplication operator $Bu(t,r)=\mu(r)u(t,r)$, with $\mu$ be such that
\begin{equation}\label{mu4}
\mu'(1)\pm\mu'(0)\neq 0,\quad\mbox{and}\quad \langle \mu\varphi_j,\varphi_k\rangle\neq0\quad\forall\, k\in{\mathbb{N}}^*.
\end{equation}
Then, it can be proved that
\begin{equation}\label{mu4p}
\left|\lambda_k-\lambda_j\right|^{3/2}|\langle \mu\varphi_j,\varphi_k\rangle|\geq b,\qquad \forall\, k\in{\mathbb{N}}^*,
\end{equation}
with $b$ a positive constant. For instance, $\mu(x)=x^2$ verifies \eqref{mu4} and \eqref{mu4p}:
\begin{equation*}
\langle B\varphi_j,\varphi_k\rangle=\left\{\begin{array}{ll}
\frac{4(-1)^{k+j}kj}{(k^2-j^2)^2\pi^2},&k\neq j,\\\\
\frac{2j^2\pi^2-3}{6j^2\pi^2},&k=j.
\end{array}\right.
\end{equation*}
Therefore, by applying Theorem \ref{teo1}, we conclude that for any $T>0$, the exists a suitable constant $R_T>0$ such that, if $u_0\in B_{R_T}(\varphi_j)$, problem \eqref{eqex4} is exactly controllable to the $j$-th eigensolution $\psi_j$ in time $T$.
\subsection{Degenerate parabolic equation}\label{ex5}
In this last section we want to address an example of a control problem for a degenerate evolution equation of the form
\begin{equation}\label{eqex5}
\left\{
\begin{array}{ll}
u_t-\left(x^{\gamma} u_x\right)_x+p(t)x^{2-\gamma}u=0,& (t,x)\in (0,+\infty)\times(0,1)\\\\
u(t,1)=0,\quad\left\{\begin{array}{ll} u(t,0)=0,& \mbox{ if }\gamma\in[0,1),\\\\ \left(x^{\gamma}u_x\right)(t,0)=0,& \mbox{ if }\gamma\in[1,3/2),\end{array}\right.\\\\
u(0,x)=u_0(x).
\end{array}
\right.
\end{equation}
where $\gamma\in[0,3/2)$ describes the degeneracy magnitude, for which Theorem \ref{teo1} applies.
If $\gamma\in[0,1)$ problem \eqref{eqex5} is called weakly degenerate and the natural spaces for the well-posedness are the following weighted Sobolev spaces. Let $I=(0,1)$ and $X=L^2(I)$, we define
\begin{equation*}
\begin{array}{l}
H^1_{\gamma}(I)=\left\{u\in X: u \mbox{ is absolutely continuous on } [0,1], x^{\gamma/2}u_x\in X\right\}\\\\
H^1_{\gamma,0}(I)=\left\{u\in H^1_\gamma(I):\,\, u(0)=0,\,\,u(1)=0\right\}\\\\
H^2_\gamma(I)=\left\{u\in H^1_\gamma(I): x^{\gamma}u_x\in H^1(I)\right\}.
\end{array}
\end{equation*}
We denote by $A:D(A)\subset X\to X$ the linear degenerate second order operator
\begin{equation}
\left\{\begin{array}{l}
\forall u\in D(A),\quad Au:=-(x^{\gamma}u_x)_x,\\\\
D(A):=\{u\in H^1_{\gamma,0}(I),\,\, x^{\gamma}u_x\in H^1(I)\}.
\end{array}\right.
\end{equation}
It is possible to prove that $A$ satisfies \eqref{ipA} (see, for instance \cite{cmp}) and furthermore, if we denote by $\{\lambda_k\}_{k\in{\mathbb{N}}^*}$ the eigenvalues and by $\{\varphi_k\}_{k\in{\mathbb{N}}^*}$ the corresponding eigenfunctions, it turns out that the gap condition \eqref{gap-no-lambda1} is fulfilled with $\alpha=\frac{7}{16}\pi$ (see \cite{kl}, page 135).
If $\gamma\in[1,2)$, problem \eqref{eqex5} is called strong degenerate and the corresponding weighted Sobolev space are described as follows: given $I=(0,1)$ and $X=L^2(I)$, we define
\begin{equation*}
\begin{array}{l}
H^1_{\gamma}(I)=\left\{u\in X: u \mbox{ is absolutely continuous on } (0,1],\,\, x^{\gamma/2}u_x\in X\right\}\vspace{.1cm}\\\\
H^1_{\gamma,0}(I):=\left\{u\in H^1_{\gamma}(I):\,\,u(1)=0\right\},\vspace{.1cm}\\\\
H^2_\gamma(I)=\left\{u\in H^1_\gamma(I):\,\, x^{\gamma}u_x\in H^1(I)\right\}.
\end{array}
\end{equation*}
In this case the operator $A:D(A)\subset X\to X$ is defined by
\begin{equation*}
\left\{\begin{array}{l}
\forall u\in D(A),\quad Au:=-(x^{\gamma}u_x)_x,\vspace{.1cm}\\\\
D(A):=\left\{u\in H^1_{\gamma,0}(I):\,\, x^{\gamma}u_x\in H^1(I)\right\}\vspace{.1cm}\\
\qquad\,\,\,\,\,=\left\{u\in X:\,\,u \mbox{ is absolutely continuous in (0,1] },\,\, x^{\gamma}u\in H^1_0(I),\right.\vspace{.1cm}\\
\qquad\qquad\,\,\,\left.x^{\gamma}u_x\in H^1(I)\mbox{ and } (x^{\gamma}u_x)(0)=0\right\}
\end{array}\right.
\end{equation*}
and it has been proved that \eqref{ipA} holds true (see, for instance \cite{cmvn}) and that \eqref{gap-no-lambda1} is satisfied for $\alpha=\frac{\pi}{2}$ (see \cite{kl}).
We fix $j=1$ and for all $\gamma\in[0,3/2)$, we define the linear operator $B:X\to X$ by $Bu(t,x)=x^{2-\gamma}u(t,x)$ and in \cite{cu} we have proved that there exists a constant $b>0$ such that
\begin{equation*}
\left|\lambda_k-\lambda_1\right|^{3/2}|\langle B\varphi_1,\varphi_k\rangle|\geq b\quad\forall k\in{\mathbb{N}}^*.
\end{equation*}
Finally, by applying Theorem \ref{teo1}, we ensure the exact controllability of problem \eqref{eqex5} to the first eigensolution, for both weakly and strongly degenerate problems.
\section*{Acknowledgments}
We are grateful to J. M. Coron and P. Martinez for their precious comments and suggestions.
| {
"timestamp": "2021-05-13T02:20:59",
"yymm": "2105",
"arxiv_id": "2105.05732",
"language": "en",
"url": "https://arxiv.org/abs/2105.05732",
"abstract": "In a separable Hilbert space $X$, we study the controlled evolution equation \\begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \\end{equation*} where $A\\geq-\\sigma I$ ($\\sigma\\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\\in L^2_{loc}(0,+\\infty)$ is a bilinear control.We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.",
"subjects": "Optimization and Control (math.OC); Analysis of PDEs (math.AP)",
"title": "Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9869795072052383,
"lm_q2_score": 0.7185943805178139,
"lm_q1q2_score": 0.7092379275639255
} |
https://arxiv.org/abs/0809.0918 | Intersecting random graphs and networks with multiple adjacency constraints: A simple example | When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.A simple idea to account for multiple constraints consists in taking the intersection of random graphs. In this paper we initiate the study of random graphs so obtained through a simple example. We examine the intersection of an Erdos-Renyi graph and of one-dimensional geometric random graphs. We investigate the zero-one laws for the property that there are no isolated nodes. When the geometric component is defined on the unit circle, a full zero-one law is established and we determine its critical scaling. When the geometric component lies in the unit interval, there is a gap in that the obtained zero and one laws are found to express deviations from different critical scalings. In particular, the first moment method requires a larger critical scaling than in the unit circle case in order to obtain the one law. This discrepancy is somewhat surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on both the unit interval and unit circle. | \section{Introduction}
\label{sec:Introduction}
Graphs provide simple and useful representations
for networks with the presence of an edge between a pair of nodes
marking their ability to communicate with each other.
Thus, for some set $V$ of nodes, an undirected
graph $G \equiv (V,E)$ with edge set $E$ is defined such that an edge
exists between nodes $i$ and $j$
if and only if these nodes can establish a communication link.
This adjacency between nodes in the graph
representation may depend on various constraints, both physical
and logical. In typical settings, only a {\em single} adjacency constraint
is considered. Here are some examples.
\begin{enumerate}
\item[(i)] In wireline networks, an edge between two nodes signifies the
existence of a physical point-to-point communication link
(e.g., fiber link) between the two nodes;
\item[(ii)] Imagine a wireless network serving a set of
users distributed over a region $\mathbb{D}$ of the plane.
A popular model, known as the disk model, postulates that
nodes $i$ and $j$ located at $\myvec{x}_i$ and $\myvec{x}_j$
in $\mathbb{D}$ are able to communicate if
$\| \myvec{x}_i - \myvec{x}_j \| \leq r$ where $r$ is the
transmission range;
\item[(iii)]
Eschenauer and Gligor \cite{EschenauerGligor} have recently proposed
a key pre-distribution scheme for use in wireless sensor networks:
Each node is randomly assigned a small set of distinct keys
from a large key pool. These keys form the key ring of the node,
and are inserted into its memory. Nodes
can establish a secure link between them when they have at least one key
in common in their key rings.
\end{enumerate}
Sometimes in applications there is a need to account
for {\em multiple} adjacency constraints to reflect the several citeria
that must be satisfied before communication can take place between
two users. For instance, consider the situation where
the Eschenauer-Gligor scheme is used in a wireless sensor network whose
nodes have only a finite transmission range (as is the case in practice).
Then, in order for a pair of nodes to establish a secure link,
it is not enough that the distance between them
does not exceed the transmission range.\footnote{This of course
assumes that the adopted communication model is compatible with
the disk model.}
They must also have at least one key in common.
Such situations can be naturally formalized in the following setting:
Suppose we have two adjacency constraints, say as in the example above,
modeled by the undirected graphs
$G_1 \equiv (V,E_1)$ and $G_2 \equiv (V,E_2)$.
The {\em intersection} of these graphs is the graph
$(V,E)$ with edge set $E$ given by
\[
E := E_1 \cap E_2 ,
\]
and we write $G_1 \cap G_2 := (V, E_1 \cap E_2 )$.
Through the intersection graph $G_1 \cap G_2$, we are
able to simultaneously capture two different adjacency constraints.
Of course the same approach can be extended to an arbitrary
number of constraints, but in the interest of concreteness
we shall restrict the discussion to the case of two constraints.
In an increasing number of contexts,
{\em random} graph models\footnote{Here we shall
understand random graphs in the broad sense to mean graph-valued random
variables. To avoid any ambiguity with current usage,
the random graphs introduced by Erd\H{o}s and R\'enyi
in their groundbreaking paper \cite{ER1960} will be called
Erd\H{o}s-R\'enyi graphs.}
have been found to be more appropriate.
For instance, in wireless networking
several classes of random graphs have been proposed
to model the effects of geometry, mobility and user interference
on the wireless communication link, e.g.,
geometric random graphs (also known as disk models)
\cite{GuptaKumar1998, HanMakowski-CommLetters, PenroseBook}
and signal-to-interference-plus-noise-ratio (SINR) graphs
\cite{DousseBaccelliThiran+Infocom2003,
DousseFranceschettiMacrisMeesterThiran}.
See Sections \ref{subsec:GRGs} and \ref{subsec:ERGs}
for a description of the two classes of random graphs considered here.
When random graphs are used, we can also define their
intersection in an obvious manner:
Given two random graphs with vertex set $V$, say
$\mathbb{G}_1 \equiv (V,\mathbb{E}_1)$
and
$\mathbb{G}_2 \equiv (V,\mathbb{E}_2)$,
their intersection is the random graph
$(V,\mathbb{E})$ where
\[
\mathbb{E} := \mathbb{E}_1 \cap \mathbb{E}_2.
\]
For simplicity assume
the random graphs $\mathbb{G}_1$ and $\mathbb{G}_2$ to be independent.
A natural question to ask is the following:
How are the structural properties
of the random graph $\mathbb{G}_1 \cap \mathbb{G}_2$
shaped by those of the random graphs $\mathbb{G}_1$ and $\mathbb{G}_2$?
Here we are particularly interested in zero-one laws for
certain graph properties -- More on that later.
Intersecting graphs represents a modular approach
to building more complex models. It could be argued
that this approach is of interest only if the known
structural properties for the component random graphs can be leveraged
to gain a better understanding of the resulting
intersection graphs.
As we shall see shortly through the simple example developed here,
successfully completing this program
is not as straightforward as might have been expected.
In this paper we consider exclusively the random graph
obtained by intersecting Erd\H{o}s-R\'enyi graphs
with certain geometric random graphs in one dimension.
We were motivated to consider this simple model
for the following reasons:
(i) The disk model popularized by the work of Gupta and Kumar
\cite{GuptaKumar1998}
assumes simplified pathloss, no user interference and no fading,
and the transmission range is a proxy for transmit
power to be used by the users. One crude way to include fading
is to think of it as link outage.
Thus an edge is present between a pair
of nodes if and only if they are within communication range
(so that there is a communication link in the sense of
the usual disk model) {\em and}
that link between them is indeed active (i.e., not in outage).
This simple model is simply obtained by
taking the intersection of the disk model with
an Erd\H{o}s-R\'enyi graph.
(ii) Both Erd\H{o}s-R\'enyi graphs and geometric random graphs
are well understood classes of random graphs with an extensive literature
devoted to them; see the monographs
\cite{Bollobas, JansonLuczakRucinski, PenroseBook} for
Erd\H{o}s-R\'enyi graphs and the text \cite{PenroseBook} for
geometric random graphs. Additional information concerning
one-dimensional graphs can be found in the references
\cite{GodehardtBook}
\cite{GodehardtJaworski}
\cite{HanThesis}
\cite{HanMakowski-TransactionsIT+Uniform}
\cite{Maehara}.
It is hoped that this wealth of results
will prove helpful in successfully carrying out the program outlined earlier.
(iii) Furthermore, this simple model is a trial balloon
for the study of more complicated situations. In particular, we have in mind
the study of wireless sensor networks employing the Eschenauer-Gligor
scheme to establish secure links.
In that case the resulting random intersection graphs,
the so-called random key graphs under partial visibility
\cite{DiPietroManciniMeiPanconesiRadhakrishnan2004}
\cite{DiPietroManciniMeiPanconesiRadhakrishnan2006}
\cite{YaganMakowskiISIT2008},
share some similarity with the models discussed here, but have far greater
complexity due to lack of independence in the link assignments in the
non-geometric component;
see comments in Section \ref{ConcludingRemarks}.
Our ability to successfully complete the study of the models
considered in this paper would provide
some measure of comfort that the more complicated cases are indeed
amenable to analysis, with pointers to possible results.
We would like to draw attention to a similar problem which
has been studied recently. In \cite{YiWanLiFrieder}, the authors
consider a geometric random graph where the {\em nodes} become inactive
independently with a certain probability. In contrast, we are interested
in the situation where the {\em edges} in the geometric random graph
can become inactive.
In the context of our simple model we investigate the zero-one laws
for the property that there are no isolated nodes;
particular emphasis is put on identifying the corresponding
critical scalings. This is done with the help of
the method of first and second moments.
Even this simple and well-structured situation gives rise
to some surprising results:
When the geometric component is defined on the unit circle,
a full zero-one law is established and we determine its critical scaling.
When the geometric component lies in the unit interval, there is a gap in
the results in that the obtained zero and one laws are
found to express deviations from different critical scalings.
In particular, we encounter a
situation where the first moment method requires a larger critical
scaling than in the unit circle case in order to obtain the one law.
This discrepancy is somewhat surprising given that the zero-one laws
for the absence of isolated nodes are identical in the geometric
random graphs on both the unit interval and unit circle.
Thus one is led to the (perhaps naive)
expectation that the boundary
effects of the geometric component play no role in shaping
the zero-one laws in the random intersection graphs.
Therefore, it appears that this discrepancy between the zero and
one laws is an artifact of the method of first moment, and a
different approach is needed to bridge this gap.
The analysis given here provides some insight into classical results.
This is done by developing a new interpretation
of the critical scalings (for the absence of isolated nodes)
in terms of the probability of an edge existing
between a pair of nodes. This interpretation seems to hold quite generally.
In fact, it is this observation which enabled us to guess the form
of the zero-one law for the random intersection graphs and
may find use in similar problems.
It is natural to wonder here what form take the zero-one laws for
the property of graph connectivity. We remark that this is now
a more delicate problem for contrary to what occurs
with one-dimensional graphs \cite{GodehardtBook} \cite{GodehardtJaworski}
\cite{HanThesis} \cite{HanMakowski-TransactionsIT+Uniform} \cite{Maehara},
the total ordering of the line cannot be used to advantage, and new
approaches are needed. But not all is lost:
In some sense the property that there are no isolated nodes can
be viewed as a \lq\lq first-order approximation" to the
property of graph connectivity -- This is borne out
by the fact that for many
classes of random graphs these two properties are asymptotically
equivalent under the appropriate scaling; see
the monographs \cite{Bollobas} and \cite{PenroseBook} for
Erd\H{o}s-R\'enyi graphs and geometric random graphs, respectively.
In that sense the preliminary results obtained here
constitute a first step on the road to
establish zero-one laws for the property of graph connectivity.
A word on the notation and conventions in use:
Throughout $n$ will denote the number of nodes in the random graph and
all limiting statements, including asymptotic equivalences, are understood
with $n$ going to infinity.
The random variables (rvs) under consideration are all
defined on the same probability triple $(\Omega, {\cal F}, \mathbb{P})$.
Probabilistic statements are made with respect to this probability
measure $\mathbb{P}$, and we denote the corresponding expectation
operator by $\mathbb{E}$.
Also, we use the notation $=_{st}$ to indicate distributional equality.
The indicator function of an event $E$ is denoted by $\1{E}$.
\section{Model and assumptions}
\label{sec:ModelAssumptions}
In this paper we are only concerned with undirected graphs.
As usual, a graph $G \equiv (V,E)$ is said to be {\em connected} if every
pair of nodes in $V$ can be linked by at least one path over the edges
(in $E$) of the graph. We say a node is {\em isolated} if no edge exists
between the node and any of the remaining nodes.
Also, let $\mathcal{E}(G)$ refer to the set of edges of $G$, namely
$\mathcal{E}(G) = E$. We begin by recalling the classical random graph models
used in the definition of the model analyzed here.
\subsection{The geometric random graphs}
\label{subsec:GRGs}
Two related geometric random graphs are introduced.
Fix $n =2, 3, \ldots $ and $r > 0$,
and consider a collection $X_1, \ldots , X_n$ of i.i.d. rvs
which are distributed uniformly in the interval $[0,1]$ (referred
to as the {\em unit interval}).
We think of $r$ as the transmission range and $X_1, \ldots , X_n$
as the locations of $n$ nodes (or users), labelled $1, \ldots , n$,
in the interval $[0,1]$.
Nodes $i$ and $j$ are said to be adjacent if
$|X_i - X_j | \leq r $,
in which case an undirected edge exists between them.
The indicator rv $\chi^{(L)}_{ij} (r)$ that
nodes $i$ and $j$ are adjacent is given by
\[
\chi^{(L)}_{ij} (r)
:= \1{ |X_i -X_j| \leq r } .
\]
This notion of edge connectivity gives rise to
an undirected geometric random graph on the unit interval,
thereafter denoted $\mathbb{G}^{(L)} (n; r)$.
For each $i=1, \ldots , n$, node $i$ is
an isolated node in $\mathbb{G}^{(L)} (n; r)$ if
$|X_i - X_j| > r$ for all $j=1, \ldots , n$ with $j\neq i$.
The indicator rv $\chi^{(L)}_{n,i}(r)$
that node $i$ is an isolated node in $\mathbb{G}^{(L)} (n; r)$ is
given by
\[
\chi^{(L)}_{n,i}(r)
:= \prod_{j=1, j \neq i}^n
\left ( 1 - \chi^{(L)}_{ij} (r) \right ).
\]
The number of isolated nodes in $\mathbb{G}^{(L)} (n; r)$
is then given by
\[
I^{(L)}_n (r) := \sum_{i=1}^n \chi^{(L)}_{n,i}(r).
\]
We also consider the geometric random graph obtained by
locating the nodes uniformly on the circle with unit circumference
(thereafter referred to as the {\em unit circle}) -- This corresponds
to identifying the end points of the unit interval.
In this formulation, we fix some reference point on the circle
and the node locations $X_1,\dots,X_n$ are given by the length
of the clockwise arc with respect to this reference point.
We measure the distance between any two
nodes by the length of the smallest arc between the nodes,
i.e., the distance between nodes $i$ and $j$ is given by
\[
\| X_i-X_j \|
:=\min(|X_i-X_j|,1-|X_i-X_j|).
\]
As we still think of $r$ as the transmission range,
nodes $i$ and $j$ are now said to be adjacent if
$ \| X_i-X_j \| \leq r.$
The indicator rv $\chi^{(C)}_{ij} (r)$ that
nodes $i$ and $j$ are adjacent is given by
\[
\chi^{(C)}_{ij} (r)
:= \1{ \|X_i-X_j \| \leq r }.
\]
This notion of adjacency leads to
an undirected geometric random graph on the unit circle,
thereafter denoted $\mathbb{G}^{(C)} (n; r)$.
This model is simpler to
analyze as the boundary effects have been removed.
The indicator rv $\chi^{(C)}_{n,i}(r)$ that node $i$ is an isolated node
in $\mathbb{G}^{(C)} (n; r)$ is again defined by
\[
\chi^{(C)}_{n,i}(r)
:= \prod_{j=1, j \neq i}^n
\left ( 1 - \chi^{(C)}_{ij} (r) \right ).
\]
The number of isolated nodes in $\mathbb{G}^{(C)} (n; r)$
is then given by
\[
I^{(C)}_n (r) := \sum_{i=1}^n \chi^{(C)}_{n,i}(r).
\]
Throughout, it will be convenient to view the graphs
$\mathbb{G}^{(L)} (n; r)$ and $\mathbb{G}^{(C)} (n; r)$
as {\em coupled} in that they are constructed from the {\em same}
rvs $X_1, \ldots , X_n$ defined on the {\em same}
probability space $(\Omega, {\cal F}, \mathbb{P})$.
Note that the two models differ only in the manner in which the
distance between two users is defined. To take advantage
of this observation, we shall write
\[
d(x,y) :=
\left \{
\begin{array}{ll}
|x-y| & \mbox{on~the~unit~interval} \\
& \\
\| x-y \|
& \mbox{on~the~unit~circle}
\end{array}
\right .
\]
for all $x,y$ in $[0,1]$ as a compact way to capture the appropriate
notion of \lq\lq distance".
Also, in the same spirit, as a way to lighten the notation,
we omit the superscripts $(L)$
and $(C)$ from the notation when the discussion
applies equally well to both cases.
\subsection{The Erd\H{o}s-R\'enyi graphs}
\label{subsec:ERGs}
Fix $n=2,3, \ldots $ and $p$ in $[0,1]$. In this case, $p$ corresponds
to the probability that an (undirected) edge exists between any pair of nodes.
We start with rvs $\{ B_{ij}(p), \ 1 \leq i < j \leq n \},$ which are i.i.d.
$\{0,1 \}$-valued rvs with success probability $p$.
Nodes $i$ and $j$ are said to be adjacent if $B_{ij} (p) = 1$.
This notion of edge connectivity defines
the undirected Erd\H{o}s-R\'enyi (ER) random graph,
thereafter denoted $\mathbb{G} (n; p)$.
For each $i=1, \ldots , n$, node $i$ is isolated in
$\mathbb{G} (n; p)$ if
$B_{ij}(p) = 0$ for $i < j \leq n $ and
$B_{ji}(p) = 0$ for $1\leq j < i $. The indicator $\chi_{n,i}(p)$
that node $i$ is an isolated node in $\mathbb{G} (n;p)$ is
then given by
\[
\chi_{n,i}(p)
:= \prod_{i < j \leq n } \left ( 1 - B_{ij}(p) \right )
\cdot \prod_{1\leq j < i } \left ( 1 - B_{ji} (p) \right ).
\]
The number of isolated nodes in $\mathbb{G} (n;p)$ is the rv $I_n(p)$
given by
\[
I_n (p) := \sum_{i=1}^n \chi_{n,i}(p).
\]
\subsection{Intersecting the geometric and Erd\H{o}s-R\'enyi graphs}
\label{subsec:Intersection}
The random graph model studied in this paper
is parametrized by the number $n$ of nodes, the transmission
range $r > 0$ and the probability $p$ ($ 0 \leq p \leq 1 $)
that a link is active (i.e., not in outage).
To lighten the notation we often group the parameters $r$ and $p$
into the ordered pair $\myvec{\theta} \equiv (r,p)$.
Throughout we always assume that the collections of rvs
$\{ X_i, \ i=1, \ldots , n \}$ and
$\{ B_{ij}(p), \ 1 \leq i < j \leq n \}$ are {\em mutually independent}.
With the convention introduced earlier,
the intersection of the two graphs
$\mathbb{G} (n;r)$ and $\mathbb{G} (n;p)$ is the graph
\[
\mathbb{G}(n;\myvec{\myvec{\theta}})
:=
\mathbb{G} (n; r)\cap \mathbb{G} (n;p)
\]
defined on the vertex set $\{ 1, \ldots , n\}$ with edge set
given by
\[
\mathcal{E} \left ( \mathbb{G}(n;\myvec{\theta}) \right )
=
\mathcal{E} \left ( \mathbb{G} (n; r) \right )
\cap \mathcal{E} \left ( \mathbb{G} (n;p) \right ).
\]
We refer to $\mathbb{G}(n;\myvec{\theta})$ as the intersection graph
on the unit interval (resp. unit circle)
when in this definition, $\mathbb{G} (n; r)$ is taken to be
$\mathbb{G}^{(L)}(n; r)$ (resp. $\mathbb{G}^{(C)}(n; r)$).
The nodes $i$ and $j$ are adjacent in
$\mathbb{G}(n;\myvec{\theta})$ if and only if
they are adjacent in {\em both}
$\mathbb{G} (n; r)$ {\rm and} $\mathbb{G} (n;p)$.
The indicator rv $\chi_{ij} (\myvec{\theta})$ that
nodes $i$ and $j$ are adjacent in
$\mathbb{G}(n;\myvec{\theta})$ is given by
\[
\chi_{ij} (\myvec{\theta})
=
\left \{
\begin{array}{ll}
\chi_{ij} (r) B_{ij}(p) & \mbox{if~ $ i < j $} \\
& \\
\chi_{ij} (r) B_{ji}(p) & \mbox{if~ $ j < i $.}
\end{array}
\right .
\]
For each $i=1, \ldots , n$, node $i$ is isolated
in $\mathbb{G}(n;\myvec{\theta})$
if either it is not within transmission range from
each of the $(n-1)$ remaining nodes, or being within range
from some nodes, the corresponding links all are inactive.
The indicator rv $\chi_{n,i}(\myvec{\theta})$ that node $i$ is an
isolated node in $\mathbb{G} (n;\myvec{\theta})$ can be expressed as
\[
\chi_{n,i}(\myvec{\theta})
:= \prod_{i=1,\ j \neq i}^n \left ( 1 - \chi_{ij} (\myvec{\theta}) \right ) .
\]
As expected, the number of isolated nodes in $\mathbb{G} (n;\myvec{\theta})$
is similarly defined as
\[
I_n (\myvec{\theta}) := \sum_{i=1}^n \chi_{n,i}(\myvec{\theta}).
\]
\subsection{Scalings}
Some terminology:
A scaling for either of the geometric graphs is a mapping
$r: \mathbb{N}_0 \rightarrow \mathbb{R}_+$, while
a scaling for ER graphs is simply a mapping
$p: \mathbb{N}_0 \rightarrow [0,1]$.
A scaling for the intersection graphs
combines scalings for each of the component graphs, and
is defined as a mapping
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
The main objective of this paper can be stated as follows:
Given that
\[
\bP{ \mathbb{G}(n;\myvec{\theta}_n) ~\mbox{\rm has~no~isolated~nodes} }
= \bP{ I_n(\myvec{\theta}_n)=0 }
\]
for all $n=2,3, \ldots $, what conditions are needed on the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$
to ensure that
\[
\lim_{n \rightarrow \infty}
\bP{ I_n(\myvec{\theta}_n)=0 } = 1
\quad (\mbox{\rm resp.} ~ 0).
\]
In the literature such results are known as
zero-one laws. Interest in them stems from their ability to
capture the threshold behavior of the underlying random graphs.
\section{Classical results}
\label{sec:ClassicalResultsER+GRG}
\subsection{Erd\H{o}s-R\'enyi graphs}
\label{subsec:ErdosrenyiGraphs}
There is no loss of generality in writing
a scaling $p: \mathbb{N}_0 \rightarrow [0,1]$ in the form
\begin{equation}
p_n = \frac{ \log n + \alpha_n }{n},
\quad n=1,2, \ldots
\label{eq:DeviationCondition+ER}
\end{equation}
for some deviation function
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}$.
The following result is well known \cite{Bollobas, JansonLuczakRucinski}.
\begin{theorem}
{\sl For any scaling $p: \mathbb{N}_0 \rightarrow [0,1]$ in the form
(\ref{eq:DeviationCondition+ER}), we have the zero-one law
\begin{equation*}
\lim_{n \rightarrow \infty }
\bP{ I_n(p_n)=0 }
= \left \{
\begin{array}{ll}
0 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = - \infty $} \\
& \\
1 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = + \infty $}
\end{array}
\right .
\label{eq:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}
\end{equation*}
where the deviation function $\alpha : \mathbb{N}_0 \rightarrow \mathbb{R}$
is determined through (\ref{eq:DeviationCondition+ER}).
}
\label{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}
\end{theorem}
This result identifies the scaling
$p^\star: \mathbb{N}_0 \rightarrow [0,1]$ given by
\begin{equation*}
p^\star_n = \frac{\log n}{n},
\quad n=1,2, \ldots
\label{eq:CriticalScaling+ER}
\end{equation*}
as the critical scaling for the absence of isolated nodes in ER graphs.
\subsection{Geometric random graphs}
\label{subsec:GeometricRandomGraphs}
Any scaling $r: \mathbb{N}_0 \rightarrow \mathbb{R}_+$
can be written in the form
\begin{equation}
r_n = \frac{\log n + \alpha_n}{2n},
\quad n=1,2, \ldots
\label{eq:TauScaling}
\end{equation}
for some deviation function
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}$.
The following result can be found in
\cite{AppelRusso}, \cite{PenroseBook}.
\begin{theorem}
{\sl For any scaling $r: \mathbb{N}_0 \rightarrow \mathbb{R}_+$
written in the form (\ref{eq:TauScaling}) for some deviation function
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}$,
we have the zero-one law
\begin{equation*}
\lim_{n \rightarrow \infty }
\bP{ I_n(r_n)=0 }
= \left \{
\begin{array}{ll}
0 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = - \infty $} \\
& \\
1 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = + \infty $}.
\end{array}
\right .
\label{eq:VeryStrongZeroOneLaw+Iso+Geo}
\end{equation*}
}
\label{thm:VeryStrongZeroOneLaw+Iso+Geo}
\end{theorem}
Theorem \ref{thm:VeryStrongZeroOneLaw+Iso+Geo}
identifies the scaling
$r^\star: \mathbb{N}_0 \rightarrow [0,1]$ given by
\begin{equation}
r^\star_n = \frac{\log n }{2n},
\quad n=1,2, \ldots
\label{eq:CriticalScaling+GRG}
\end{equation}
as the critical scaling for the absence of isolated nodes
in geometric random graphs.
For reasons that will become apparent shortly, we now develop
an equivalent version of Theorem \ref{thm:VeryStrongZeroOneLaw+Iso+Geo}
that bears a striking resemblance with the zero-one law of
Theorem \ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER} for ER graphs.
To that end, define
\[
\ell (r):=\min(1,2r),
\quad r \geq 0.
\]
Intuitively, $\ell(r)$ is akin to the probability that an edge exists
between any pair of nodes in $\mathbb{G}(n;r)$ -- In fact it has exactly that
meaning for $\mathbb{G}^{(C)}(n;r)$ while it is true approximately (when
boundary conditions are ignored) for $\mathbb{G}^{(L)}(n;r)$.
With this in mind, for any scaling
$r: \mathbb{N}_0 \rightarrow \mathbb{R}_+$ write
\begin{equation}
\ell(r_n)
= \frac{ \log n + \beta_n}{n},
\quad n=1,2, \ldots
\label{eq:TauScaling+Alternative}
\end{equation}
for some deviation function
$\beta: \mathbb{N}_0 \rightarrow \mathbb{R}$.
The representations (\ref{eq:TauScaling})
and (\ref{eq:TauScaling+Alternative}) together require
\[
\beta_n = \min(\alpha_n, n-\log n), \quad n=1,2, \ldots
\]
It is easily verified that
$\lim_{n \rightarrow \infty} \beta_n = -\infty$
(resp. $\lim_{n \rightarrow \infty} \beta_n = +\infty$)
if and only if $\lim_{n \rightarrow \infty} \alpha_n = -\infty$
(resp. $\lim_{n \rightarrow \infty} \alpha_n = +\infty$).
This implies the following equivalent rephrasing of
Theorem \ref{thm:VeryStrongZeroOneLaw+Iso+Geo}.
\begin{theorem}
{\sl For any scaling $r: \mathbb{N}_0 \rightarrow \mathbb{R}_+$
written in the form (\ref{eq:TauScaling+Alternative})
for some deviation function $\beta: \mathbb{N}_0 \rightarrow \mathbb{R}$,
we have the zero-one law
\begin{equation*}
\lim_{n \rightarrow \infty }
\bP{ I_n(r_n)=0 }
= \left \{
\begin{array}{ll}
0 & \mbox{if~ $\lim_{ n\rightarrow \infty }\beta_n = - \infty $} \\
& \\
1 & \mbox{if~ $\lim_{ n\rightarrow \infty }\beta_n = + \infty $}.
\end{array}
\right .
\label{eq:VeryStrongZeroOneLaw+Iso+Geo+Alternative}
\end{equation*}
}
\label{thm:VeryStrongZeroOneLaw+Iso+Geo+Alternative}
\end{theorem}
By Theorem \ref{thm:VeryStrongZeroOneLaw+Iso+Geo+Alternative}
any scaling $r^\star: \mathbb{N}_0 \rightarrow \mathbb{R}_+$
such that
\begin{equation}
\ell(r^\star_n) = \frac{\log n }{n},
\quad n=1,2,\ldots
\label{eq:CriticalScaling+GRG+Alternative}
\end{equation}
is a critical scaling for the absence of isolated nodes
in geometric random graphs.
As expected, it is easy to see that any scaling is critical under
the definition (\ref{eq:CriticalScaling+GRG+Alternative})
if and only if it is under
the definition (\ref{eq:CriticalScaling+GRG}).
\section{The basic difficulty}
\subsection{Intersecting Erd\H{o}s-R\'enyi graphs}
\label{subsec:IntersectingER}
As a d\'etour consider intersecting two {\em independent} ER graphs.
This results in another ER graph, i.e.,
\[
\mathbb{G}(n;p) \cap \mathbb{G}(n;p^\prime )
=_{st}
\mathbb{G}(n;pp^\prime ) ,
\quad 0 \leq p,p^\prime \leq 1 .
\]
It is therefore a simple matter to select scalings
$p,p^\prime: \mathbb{N}_0 \rightarrow [0,1]$ such that the
intersection graph
$\mathbb{G}(n;p) \cap \mathbb{G}(n;p^\prime )$ exhibits a zero-one law
for the absence of isolated nodes. By Theorem
\ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER},
it suffices to take these scalings such that
\begin{equation*}
p_n p^\prime_n
= \frac{ \log n + \alpha_n }{n},
\quad n=1,2, \ldots
\end{equation*}
for some appropriate deviation function
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}$.
Despite its simplicity, this result has some interesting implications:
For instance, select the two scalings such that
\[
p_n p^\prime_n
= \frac{1}{2} \frac{ \log n }{n},
\quad n=1,2, \ldots
\]
with
\[
p_n = p^\prime_n
= \sqrt{ \frac{1}{2} \frac{ \log n }{n}},
\quad n=1,2, \ldots
\]
In that case, upon writing
\[
\frac{1}{2} \frac{ \log n }{n}
= \frac{ \log n + \left ( -\frac{1}{2} \log n \right ) }{n},
\quad n=1,2, \ldots ,
\]
we conclude
\[
\lim_{n \rightarrow \infty}
\bP{ \mathbb{G}(n;p_n) \cap \mathbb{G}(n;p^\prime_n )
~\mbox{\rm has~no~isolated~nodes} } = 0
\]
by the zero law of Theorem
\ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}.
Yet, we also have
\[
\lim_{n \rightarrow \infty}
\bP{ \mathbb{G}(n;p_n) ~\mbox{\rm has~no~isolated~nodes} }
=1
\]
and
\[
\lim_{n \rightarrow \infty}
\bP{ \mathbb{G}(n;p^\prime_n) ~\mbox{\rm has~no~isolated~nodes} }
= 1
\]
by the one law of Theorem
\ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}
as we note that
\[
\sqrt{ \frac{1}{2} \frac{ \log n }{n}}
= \frac{ \log n + \alpha_n }{n}
\]
for all $n=1,2, \ldots$ with the choice
\[
\alpha_n := \sqrt{ \frac{n \log n}{2} } - \log n .
\]
Thus, even when the individual graphs
$\mathbb{G}(n;p_n )$ and $\mathbb{G}(n;p^\prime_n )$
contain {\em no} isolated nodes with a probability close to one,
it is possible for the
intersection graph $\mathbb{G}(n;p_n) \cap \mathbb{G}(n;p^\prime_n )$
to contain isolated nodes with a probability very close to one.
The reason for this is quite simple: A node that is isolated
in $\mathbb{G}(n;p_np_n^\prime )$
may not be isolated in either of the component graphs
$\mathbb{G}(n;p_n)$ and $\mathbb{G}(n;p_n^\prime )$.
For ER graphs, the answer, although very simple,
fails to give much insight into how the individual graphs interact
with each other and how this affects
the overall behavior of the intersection graph.
\subsection{Intersecting an Erd\H{o}s-R\'enyi graph
with a geometric random graph}
With this in mind, note that with $0 < r < 1$ for the unit interval
(resp. $0 < r < 0.5$ for the unit circle) and $0 < p < 1$,
the intersection graph
$\mathbb{G}(n;r) \cap \mathbb{G}(n;p)$ is {\em not}
stochastically equivalent
to either a geometric random graph or an Erd\H{o}s-R\'enyi graph, i.e.,
it is {\em not} possible to find parameters
$r^\prime = r^\prime(n;r,p)$ and $p^\prime = p^\prime(n;r,p)$
in $\mathbb{R}_+$ and $[0,1]$, respectively, such that
\[
\mathbb{G}(n;r) \cap \mathbb{G}(n;p)
=_{st}
\mathbb{G}(n;r^\prime)
\]
and
\[
\mathbb{G}(n;r) \cap \mathbb{G}(n;p)
=_{st}
\mathbb{G}(n;p^\prime ) .
\]
Consequently, results for either ER or geometric random graphs
(as given in Section \ref{sec:ClassicalResultsER+GRG}) cannot be used
in a straightforward manner to determine the zero-one laws for
the intersection graphs.
On the other hand, it is obvious that
if either $\mathbb{G}(n;r)$ or $\mathbb{G}(n;p)$ contains isolated nodes,
then
$\mathbb{G}(n;r) \cap \mathbb{G}(n;p)$ must contain isolated nodes.
Therefore, a zero law for the intersection graph should
follow by combining the zero laws for the ER and
geometric random graphs. However, as will become apparent from our main results,
such arguments are too loose to provide the best possible zero law.
A direct approach is therefore required with the difficulty
mentioned earlier remaining, namely that a node isolated
in $\mathbb{G}(n;r) \cap \mathbb{G}(n;p)$
may not be isolated in either $\mathbb{G}(n;r)$ or $\mathbb{G}(n;p)$.
Nevertheless the corresponding zero-one laws do provide a basis
for guessing the form of the zero-one law for the intersection graphs.
This is taken on in the next section.
\section{The main results}
\label{sec:MainResults}
\subsection{Guessing the form of the results}
\label{subsec:Guessing}
Upon comparing the zero-one laws of
Theorems \ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}
and \ref{thm:VeryStrongZeroOneLaw+Iso+Geo+Alternative},
the following shared structure suggests itself:
For the random graphs of interest here (as well as for others,
e.g., random key graphs \cite{YaganMakowskiISIT2008}),
it is possible to identify a quantity which gives
\[
\bP{\textrm{Edge exists between two nodes}} ,
\]
either exactly (e.g., $p$ in ER graphs or $\ell(r)$ in geometric random
graphs on the circle) or approximately (e.g., $\ell(r)$ in geometric
random graphs on the interval).
Critical scalings for the absence of isolated nodes are then determined
through the requirement
\begin{equation}
\bP{\textrm{Edge exists between two nodes}}
= \frac{\log n}{n}.
\label{eq:CriticalRequirement}
\end{equation}
In particular the zero-one law requires scalings satisfying
\[
\begin{array}{c}
\bP{\textrm{Edge exists between two nodes}} \\
\mbox{\rm \lq\lq much~smaller\rq\rq ~than~} \frac{\log n}{n},
\end{array}
\]
while the one-one law deals with scalings satisfying
\[
\begin{array}{c}
\bP{\textrm{Edge exists between two nodes}} \\
\mbox{\rm \lq\lq much~larger\rq\rq ~than~} \frac{\log n}{n}.
\end{array}
\]
The exact technical meaning of \lq\lq much smaller" and
\lq\lq much larger" forms the content of results such as
Theorems \ref{thm:VeryStrongZeroOneLaw+NoIsolatedNodes+ER}
and \ref{thm:VeryStrongZeroOneLaw+Iso+Geo+Alternative}.
With this in mind, for the random intersection graphs studied here
it is natural to take
\begin{equation}
\bP{\textrm{Edge exists between two nodes}} := p \ell(r)
\label{eq:CriticalRequirement2}
\end{equation}
under the enforced independence assumptions.
We expect that a critical scaling
$\myvec{\theta}^\star : \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$
for the random intersection graphs should be determined by
\begin{equation*}
p^\star_n \ell(r^\star_n) = \frac{\log n}{n},
\quad n=1,2, \ldots
\label{eq:CriticalScaling+Intersection}
\end{equation*}
The exact form taken by the results is discussed in
Sections \ref{subsec:IntersectionGraphsUnitCircle}
and \ref{subsec:IntersectionGraphsUnitInterval}.
We start with the model on the circle
for which we have obtained the most complete results.
\subsection{Intersection graphs on the unit circle}
\label{subsec:IntersectionGraphsUnitCircle}
With a scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we associate the sequence $\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}$
through
\begin{equation}
p_n \ell (r_n) =\frac{\log n + \alpha_n}{n},
\quad n=1,2, \ldots
\label{eq:DeviationUnitCircle}
\end{equation}
In the case of the intersection graph on the unit circle
we get a full zero-one law.
\begin{theorem}[Unit circle]
{\sl For any
scaling $\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have the zero-one law
\begin{equation*}
\lim_{n \rightarrow \infty }
\bP{ I^{(C)}_n(\myvec{\theta}_n)=0 }
= \left \{
\begin{array}{ll}
0 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = - \infty $} \\
& \\
1 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = + \infty $}
\end{array}
\right .
\label{eq:VeryStrongZeroOneLaw+Iso+Int1}
\end{equation*}
where the sequence $\alpha : \mathbb{N}_0 \rightarrow \mathbb{R}$
is determined through (\ref{eq:DeviationUnitCircle}).
}
\label{thm:MainThmCir}
\end{theorem}
\subsection{Intersection graphs on the unit interval}
\label{subsec:IntersectionGraphsUnitInterval}
With a scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we also associate the sequence
$\alpha^\prime: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
through
\begin{equation}
p_n \ell (r_n) =\frac{2(\log n - \log \log n ) + \alpha^\prime_n}{n},
\quad n=1,2, \ldots
\label{eq:DeviationUnitInterval}
\end{equation}
For the intersection graph on the unit interval
there is a gap between the zero and one laws.
\begin{theorem}[Unit interval]
{\sl For any scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have the zero-one law
\begin{equation*}
\lim_{n \rightarrow \infty }
\bP{ I^{(L)}_n(\myvec{\theta}_n)=0 }
= \left \{
\begin{array}{ll}
0 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha_n = - \infty $} \\
& \\
1 & \mbox{if~ $\lim_{ n\rightarrow \infty }\alpha^\prime_n = + \infty $}
\end{array}
\right .
\label{eq:VeryStrongZeroOneLaw+Iso+Int2}
\end{equation*}
where the sequences $\alpha, \alpha^\prime :
\mathbb{N}_0 \rightarrow \mathbb{R}$
are determined through (\ref{eq:DeviationUnitCircle})
and (\ref{eq:DeviationUnitInterval}), respectively.
}
\label{thm:MainThmInt}
\end{theorem}
An elementary coupling argument shows that
for any particular realization of the rvs $\{ X_i, \ i=1, \ldots , n \}$
and $\{ B_{ij}(p), \ 1 \leq i < j \leq n \}$,
the graph on the circle contains
more edges than the graph on the interval.
As a result, the zero law for the unit circle automatically implies
the zero law for the unit interval, and only the former needs to be
established.
\section{Method of first and second moments}
\label{sec:FirstAndSecond}
The proofs rely on the method of first and second moments
\cite[p. 55]{JansonLuczakRucinski},
an approach widely used in the theory of Erd\H{o}s-R\'enyi graphs:
Let $Z$ denote an $\mathbb{N}$-valued rv with finite second moment.
The method of first moment
\cite[Eqn. (3.10), p. 55]{JansonLuczakRucinski}
relies on the inequality
\begin{equation}
1 - \bE{ Z } \leq \bP{ Z = 0 },
\label{eq:LowerBoundFunction}
\end{equation}
while the method of second moment
\cite[Remark 3.1, p. 55]{JansonLuczakRucinski}
uses the bound
\begin{equation}
\bP{ Z = 0 }
\leq
1 - \frac{ \bE{Z}^2}{\bE{Z ^2} } .
\label{eq:UpperBoundFunction}
\end{equation}
Now, pick a scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
From (\ref{eq:LowerBoundFunction}) we see that the one law
\begin{equation*}
\lim_{n \rightarrow \infty} \bP{ I_n(\myvec{\theta}_n) = 0 } = 1
\label{eq:GenericOneLaw}
\end{equation*}
is established if we show that
\begin{equation}
\lim_{n \rightarrow \infty} \bE{ I_n(\myvec{\theta}_n) } = 0.
\label{eq:GenericOneLaw+Condition}
\end{equation}
On the other hand, it is plain from (\ref{eq:UpperBoundFunction}) that
\begin{equation*}
\lim_{n \rightarrow \infty} \bP{ I_n(\myvec{\theta}_n) = 0 } = 0
\label{eq:GenericZeroLaw}
\end{equation*}
if
\begin{equation}
\liminf_{n \rightarrow \infty}
\left ( \frac{ \bE{ I_n (\myvec{\theta}_n) }^2 }{ \bE{ I_n (\myvec{\theta}_n)^2 } }
\right )
\geq 1 .
\label{eq:GenericZeroLaw+Condition}
\end{equation}
Upon using the exchangeability and the binary nature of the rvs involved
in the count variables of interest,
we can obtain simpler characterizations of the convergence statements
(\ref{eq:GenericOneLaw+Condition}) and (\ref{eq:GenericZeroLaw+Condition}).
Indeed, for all $n=2,3, \ldots $ and every $\myvec{\theta}$ in
$\mathbb{R}_+ \times [0,1]$, the calculations
\[
\bE{ I_n (\myvec{\theta}) }
= \sum_{i=1}^n \bE{ \chi_{n,i}(\myvec{\theta}) }
= n \bE{ \chi_{n,1}(\myvec{\theta}) }
\]
and
\begin{eqnarray}
& & \bE{ I_n(\myvec{\theta})^2 }
\nonumber \\
&=&
\bE{\left (\sum_{i=1}^n \chi_{n,i}(\myvec{\theta}) \right )^2 }
\nonumber \\
&=& \sum_{i=1}^n \bE{ \chi_{n,i}(\myvec{\theta}) }
+ \sum_{i,j=1, \ i\neq j }^n \bE{ \chi_{n,i}(\myvec{\theta}) \chi_{n,j} (\myvec{\theta}) }
\nonumber \\
&=& n \bE{ \chi_{n,1}(\myvec{\theta}) }
+ n(n-1) \bE{ \chi_{n,1}(\myvec{\theta}) \chi_{n,2} (\myvec{\theta}) }
\nonumber
\end{eqnarray}
are straightforward, so that
\begin{eqnarray}
& & \frac{ \bE{ I_n(\myvec{\theta})^2 } }{ \bE{ I_n(\myvec{\theta}) }^2 }
\nonumber \\
&=&
\frac{ 1 }{ n \bE{ \chi_{n,1}(\myvec{\theta}) } }
+ \frac{n-1}{n}
\cdot
\frac{ \bE{ \chi_{n,1}(\myvec{\theta}) \chi_{n,2} (\myvec{\theta}) } }
{ \bE{ \chi_{n,1} (\myvec{\theta}) }^2 } .
\nonumber
\end{eqnarray}
Thus, for the given scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we obtain the one law by showing that
\begin{equation}
\lim_{n \rightarrow \infty}
n \bE{ \chi_{n,1}(\myvec{\theta}_n) } = 0,
\label{eq:ToShow1}
\end{equation}
while the zero law will follow if we show that
\begin{equation}
\lim_{n \rightarrow \infty} n \bE{ \chi_{n,1}(\myvec{\theta}_n) }
= \infty
\label{eq:ToShow2}
\end{equation}
and
\begin{equation}
\limsup_{n \rightarrow \infty}
\left (
\frac{ \bE{ \chi_{n,1}(\myvec{\theta}_n) \chi_{n,2} (\myvec{\theta}_n) } }
{ \bE{ \chi_{n,1} (\myvec{\theta}_n) }^2 }
\right )
\leq 1.
\label{eq:ToShow3}
\end{equation}
The bulk of the technical discussion therefore amounts to establishing
(\ref{eq:ToShow1}), (\ref{eq:ToShow2}) and (\ref{eq:ToShow3})
under the appropriate conditions on the scaling
$\myvec{\theta} : \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
To that end, in the next two sections
we derive expressions for the quantities
entering (\ref{eq:ToShow1}), (\ref{eq:ToShow2}) and (\ref{eq:ToShow3}).
Throughout we denote by $X$, $Y$ and $Z$ three mutually independent
rvs which are uniformly distributed on $[0,1]$, and by $B$, $B^\prime$
and $B^{\prime\prime}$ three mutually independent $\{0,1\}$-valued rvs
with success probability $p$. The two groups of rvs
are assumed to be independent.
\section{First moments} \label{sec:FirstMom}
Fix $n=2,3, \ldots $ and $\myvec{\theta}$ in $\mathbb{R}_+ \times [0,1]$.
For both the unit circle and unit interval, the enforced independence
assumptions readily imply
\begin{eqnarray}
\bE{ \chi_{n,1} (\theta) }
&=&
\bE{
\prod_{i=1,\ j \neq i}^n \left ( 1 - \chi_{ij} (\theta) \right )
}
\nonumber \\
&=&
\bE{ \left ( 1 - p a(X;r) \right )^{n-1} }
\nonumber \\
&=&
\int_0^1 \left ( 1 - p a(x;r) \right )^{n-1} dx
\label{eq:ExpressionsForFirstMoment}
\end{eqnarray}
where we have set
\begin{equation}
a(x;r) := \bP{ d( x , Y ) \leq r },
\quad
\begin{array}{c}
0 \leq x \leq 1 ,\\
r > 0 .
\end{array}
\label{eq:ExpressionsForA}
\end{equation}
Closed-form expressions for (\ref{eq:ExpressionsForA}) depend on the
geometric random graph being considered.
\subsection{The unit circle}
As there are no border effects, we get
\begin{equation}
a^{(C)}(x;r) = \ell(r),
\quad
\begin{array}{c}
0 \leq x \leq 1 ,\\
r > 0
\end{array}
\label{eq:AforCircle}
\end{equation}
and with the help of (\ref{eq:ExpressionsForFirstMoment})
this yields
\begin{equation}
\bE{ \chi^{(C)}_{n,1} (\myvec{\theta}) }
= \left ( 1 - p \ell(r) \right)^{n-1},
\quad
\begin{array}{c}
r > 0,\\
p \in [0,1].
\end{array}
\label{eq:FirstMomCircle}
\end{equation}
\subsection{The unit interval}
For $r \ge 1$, it is plain that
\begin{equation*}
a^{(L)} (x;r) = 1,
\quad 0 \leq x \leq 1 .
\end{equation*}
On the other hand, when $0 < r < 1$,
elementary calculations show that
\remove{
\begin{eqnarray*}
& & a^{(L)} (x;r) \\
& = & \left \{
\begin{array}{ll}
x + r & \mbox{if~ $0 \leq x \leq \min(r,1-r)$} \\
& \\
\ell(r) & \mbox{if~ $ \min(r,1-r) < x < \max(r,1-r) $} \\
& \\
1-x + r & \mbox{if~ $\max(r,1-r) \leq x < 1$.}
\end{array}
\right .
\end{eqnarray*}}
\begin{eqnarray*}
& & a^{(L)} (x;r) \\
& = & \left \{
\begin{array}{ll}
x + r & \begin{array}{l}
\text{if } 0 < r \le 0.5, 0 \leq x \leq r \\
\text{or } 0.5 < r < 1, 0 \leq x \leq 1-r
\end{array} \\
& \\
\ell(r) & \begin{array}{l}
\text{if } 0 < r \le 0.5, r \leq x \leq 1-r \\
\text{or } 0.5 < r < 1, 1-r \leq x \leq r
\end{array} \\
& \\
1-x + r & \begin{array}{l}
\text{if } 0 < r \le 0.5, 1-r \leq x \leq 1 \\
\text{or } 0.5 < r < 1, r \leq x \leq 1.
\end{array} \\
\end{array}
\right .
\end{eqnarray*}
Reporting this information into (\ref{eq:ExpressionsForFirstMoment}),
we obtain the following expressions in a straightforward manner:
\begin{itemize}
\item[(i)] For $0 < r \leq 0.5$ and $0 < p \leq 1$,
\begin{eqnarray}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) }
&=& (1-2r)(1-2pr)^{n-1}
\label{eq:Expression(i)} \\
& & + ~ \frac{2}{np} \left ( (1-pr)^n-(1-2pr)^n \right ).
\nonumber
\end{eqnarray}
\item[(ii)] For $0.5 < r < 1$ and $0 < p \leq 1$,
\begin{eqnarray}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) }
&=& (2r-1)(1-p)^{n-1}
\label{eq:Expression(ii)} \\
& & +~ \frac{2}{np} \left ( (1-pr)^n-(1-p)^n \right ).
\nonumber
\end{eqnarray}
\item[(iii)] For $r \geq 1$ and $0 < p \leq 1$,
\begin{equation}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) } = (1-p)^{n-1}.
\label{eq:Expression(iii)}
\end{equation}
\item[(iv)] For $r > 0$ and $p=0$,
\begin{equation}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) } = 1.
\label{eq:Expression(iv)}
\end{equation}
\end{itemize}
The expressions (\ref{eq:Expression(i)}) and (\ref{eq:Expression(ii)})
can be combined into the single expression
\begin{eqnarray}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) }
&=& \left | 2r-1 \right | (1-p\ell(r))^{n-1}
\label{eq:Expression(i+ii)} \\
& & +~ \frac{2}{np}
\left ( (1-pr)^n-(1-p\ell(r))^n \right )
\nonumber
\end{eqnarray}
on the range $0 < r < 1$ and $0 < p \leq 1$.
Collecting (\ref{eq:Expression(iii)}), (\ref{eq:Expression(iv)})
and (\ref{eq:Expression(i+ii)})
we get the upper bound
\begin{equation}
\bE{ \chi^{(L)}_{n,1} (\myvec{\theta}) }
\leq
(1-p \ell(r))^{n-1}
+
\frac{2}{np}\left(1-\frac{1}{2}p \ell(r)\right)^n
\label{eq:IntervalFirstMomUB}
\end{equation}
for any fixed $n=2,3,\dots,$
and $\myvec{\theta}$ in $\mathbb{R}_+ \times [0,1]$.
\section{Second moments}
Again fix $n=2,3, \ldots $
and $\myvec{\theta}$ in $\mathbb{R}_+ \times [0,1]$.
The same arguments apply for both the unit circle and unit interval:
For $x,y$ in $[0,1]$, write
\begin{eqnarray*}
& & b(x,y;\myvec{\theta})
\nonumber \\
&:=&
\bE{
\left ( 1 - B^\prime \1{ d(x,Z) \leq r } \right )
\left ( 1 - B^{\prime\prime} \1{ d(y,Z) \leq r } \right )
}
\nonumber \\
&=& 1 - p a(x;r) - pa(y;r) + p^2 u(x,y;r)
\label{eq:ExpressionsForB}
\end{eqnarray*}
with
\[
u(x,y;r)
:=
\bP{ d(x,Z) \leq r, d(y,Z) \leq r}.
\]
We then proceed with the decomposition
\begin{eqnarray}
& & \chi_{n,1}(\myvec{\theta}) \chi_{n,2} (\myvec{\theta})
\nonumber \\
&=& \prod_{j=2}^n \left ( 1 - \chi_{1j} (\myvec{\theta}) \right ).
\prod_{k=1, k \neq 2}^n \left ( 1 - \chi_{2k} (\myvec{\theta}) \right )
\nonumber \\
&=& \left ( 1 - \chi_{12} (\myvec{\theta}) \right )
\prod_{j=3}^n \left ( 1 - \chi_{1j} (\myvec{\theta}) \right )
\left ( 1 - \chi_{2j} (\myvec{\theta}) \right ) .
\nonumber
\end{eqnarray}
Under the enforced independence assumptions,
an easy conditioning argument
(with respect to the triple $X_1$, $X_2$ and $B_{12}$)
based on this decomposition now gives
\begin{eqnarray}
& & \bE{ \chi_{n,1}(\myvec{\theta}) \chi_{n,2} (\myvec{\theta}) }
\label{eq:SecondMom+DistMatters} \\
&=&
\bE{ \left ( 1 - B \1{ d(X,Y) \leq r } \right )
b( X,Y; \myvec{\theta} )^{n-2} } .
\nonumber
\end{eqnarray}
As mentioned earlier we need only consider the unit circle as we do
from now on: From (\ref{eq:AforCircle}) it is plain that
\[
b^{(C)} (x,y;\myvec{\theta})
= 1 - 2p \ell (r) + p^2 u^{(C)}(x,y;r)
\]
for all $x,y$ in $[0,1]$, where we note that
\begin{eqnarray*}
u^{(C)}(x,y;r) & = & \bP{ \| x - Z \| \leq r, \| y - Z \| \leq r } \\
& = & u^{(C)} ( 0, \| x-y \|; r )
\end{eqnarray*}
by translation invariance.
Thus, writing
\begin{equation}
\tilde b^{(C)} (z; \myvec{\theta} )
:= 1 - 2p \ell (r) + p^2 \tilde u^{(C)}(z;r),
\quad z \in [0,0.5]
\label{eq:bTildeuTilde}
\end{equation}
with
\[
\tilde u^{(C)} (z;r)
:= u^{(C)} ( 0, z; r ) ,
\]
we get
\[
b^{(C)} (x,y;\myvec{\theta})
= \tilde b^{(C)} (\| x-y \|;\myvec{\theta}),
\quad x,y \in [0,1] .
\]
Taking advantage of these facts we now find
\begin{eqnarray*}
& & \bE{ \chi_{n,1}^{(C)}(\myvec{\theta}) \chi_{n,2}^{(C)} (\myvec{\theta}) }
\nonumber \\
&=&
\bE{ \left ( 1 - B \1{ \| X - Y \| \leq r } \right )
\tilde b^{(C)} ( \| X - Y \| ; \myvec{\theta} )^{n-2} }
\nonumber \\
&=&
\bE{ \left ( 1 - p \1{ \| X - Y \| \leq r } \right )
\tilde b^{(C)} ( \| X - Y \| ; \myvec{\theta} )^{n-2} }
\nonumber \\
&=& 2 \int_0^{0.5}
\left ( 1 - p \1{ z \leq r } \right )
\tilde b^{(C)} (z ; \myvec{\theta} )^{n-2} dz
\label{eq:chi1chi2IntegralA}
\end{eqnarray*}
by a straightforward evaluation of the double integral
\[
\int_0^1 dx \int_0^1 dy
\left ( 1 - p \1{ \| x - y \| \leq r } \right )
\tilde b^{(C)}(\| x-y \|;\myvec{\theta} )^{n-2} .
\]
Consequently,
\begin{equation}
\bE{ \chi_{n,1}^{(C)}(\myvec{\theta}) \chi_{n,2}^{(C)} (\myvec{\theta}) }
\leq
2 \int_0^{0.5}
\tilde b^{(C)} (z ; \myvec{\theta} )^{n-2} dz .
\label{eq:chi1chi2IntegralB}
\end{equation}
It is possible to compute the value of $\tilde u^{(C)} (z;r)$
for various values for $z,r$:
For $0 < r < 0.5$, we find
\begin{align*}
& \tilde u^{(C)} (z;r) \\
= & \left \{
\begin{array}{ll}
2r-z & \mbox{if~ $0 < r < 0.25, 0 \leq z \le 2r$} \\
& \\
0 & \mbox{if~ $0 < r < 0.25, 2r < z \le 0.5$} \\
& \\
2r-z & \mbox{if~ $0.25 \le r < 0.5, 0 \leq z \le 1-2r$} \\
& \\
4r-1 & \mbox{if~ $0.25 \le r < 0.5, 1-2r < z \le 0.5$}.
\end{array}
\right .
\end{align*}
\remove{
\begin{align*}
& \tilde u^{(C)} (z;r) \\
= & \left \{
\begin{array}{ll}
2r-z & \mbox{if~ $0 \leq z \le \min(2r,1-2r)$} \\
& \\
(4r-1)^+ & \mbox{if~ $\min(2r,1-2r) < z < 0.5$}.
\end{array}
\right .
\end{align*}}
Details are outlined in Appendix \ref{sec:utilde}.
Obviously, if $r \geq 0.5$, then
$\tilde u^{(C)} (z;r)=1$ for every $z$ in $[0,0.5]$.
Thus, for $0 \le p \le 1$, through (\ref{eq:bTildeuTilde})
we obtain
\begin{align*}
& \tilde b^{(C)} (z; \myvec{\theta} ) \\
= & \left \{
\begin{array}{ll}
1-4pr + p^2 (2r-z) & \mbox{if~ $0 < r < 0.25, 0 \leq z \le 2r$} \\
& \\
1-4pr & \mbox{if~ $0 < r < 0.25, 2r < z \le 0.5$} \\
& \\
1-4pr + p^2(2r-z) & \mbox{if~ $\begin{array}{l}
0.25 \le r < 0.5, \\
0 \leq z \le 1-2r
\end{array}$ } \\
& \\
1-4pr + p^2(4r-1) & \mbox{if~ $\begin{array}{l}
0.25 \le r < 0.5, \\
1-2r < z \le 0.5.
\end{array}$ }
\end{array}
\right .
\end{align*}
Using this fact in (\ref{eq:chi1chi2IntegralB})
and evaluating the integral,
we obtain the following upper bounds,
see Appendix \ref{sec:chi1chi2UB} for details:
\begin{itemize}
\item[(i)] For $0 < r < 0.25$ and $0 < p \leq 1$,
\begin{align*}
& \bE{ \chi^{(C)}_{n,1}(\myvec{\theta}) \chi^{(C)}_{n,2} (\myvec{\theta}) } \\
\leq & (1-4r)(1-4pr)^{n-2} \\
& \hspace{1em} + \frac{2(1-4pr)^{n-1}}{(n-1)p^2} \left ( \left ( 1 + \frac{2p^2r}{1-4pr} \right )^{n-1} - 1
\right ) .
\end{align*}
\item[(ii)] For $0.25 \leq r < 0.5$ and $0 < p \leq 1$,
\begin{align*}
&
\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}) \chi^{(C)}_{n,2} (\myvec{\theta}) }
\\
\leq & (4r-1)(1-2pr)^{2(n-2)} \\
& \qquad + (2-4r)(1-4pr+2p^2r)^{n-2} .
\end{align*}
\item[(iii)] For $r \ge 0.5$ and $0 < p \leq 1$,
\[
\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}) \chi^{(C)}_{n,2} (\myvec{\theta}) }
= (1-p)^{2n-3}.
\]
\item[(iv)] For $r > 0$ and $p=0$,
\[
\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}) \chi^{(C)}_{n,2} (\myvec{\theta}) } = 1.
\]
\end{itemize}
Furthermore, combining these bounds with (\ref{eq:FirstMomCircle}),
we obtain the following upper bound on
\[
R_n(\myvec{\theta})
:=\frac{ \bE{ \chi^{(C)}_{n,1}(\myvec{\theta})
\chi^{(C)}_{n,2} (\myvec{\theta}) } }
{ \bE{ \chi^{(C)}_{n,1} (\myvec{\theta}) }^2 }
\]
in the various cases listed below.
\begin{itemize}
\item[(i)] For $0 < r < 0.25$ and $0 < p \leq 1$,
\begin{eqnarray}
R_n(\myvec{\theta}) & \leq & \frac{1-4r}{1-4pr}
\label{eqn:bnd1} \\
& & +~ \frac{2}{(n-1)p^2} \left(
\left(1+\frac{2p^2r}{1-4pr}\right)^{n-1} - 1
\right).
\nonumber
\end{eqnarray}
\item[(ii)] For $0.25 \le r < 0.5$ and $0 < p \le 1$,
\begin{eqnarray}
R_n(\myvec{\theta}) & \leq
& \frac{4r-1}{(1-2pr)^2}
\label{eqn:bnd2} \\
& & +~
(2-4r) \frac{(1-4pr+2p^2r)^{n-2}}{(1-2pr)^{2(n-1)}}.
\nonumber
\end{eqnarray}
\item[(iii)] For $r \geq 0.5$ and $0 < p \leq 1$,
\begin{equation}
R_n(\myvec{\theta}) = \frac{1}{1-p}.
\label{eqn:bnd3}
\end{equation}
\item[(iv)] For $r>0$ and $p=0$,
\begin{equation} R_n(\myvec{\theta}) = 1.
\label{eqn:bnd4}
\end{equation}
\end{itemize}
\section{Proof of the one laws}
As discussed in Section \ref{sec:FirstAndSecond}, the one law will be
established if we show that (\ref{eq:ToShow1}) holds.
Below we consider separately the unit circle and the unit interval.
In that discussion we repeatedly use the elementary bound
\begin{equation}
1 - x \leq e^{-x},
\quad x \geq 0.
\label{eq:B}
\end{equation}
\subsection{One law over the unit circle}
The one law over the unit circle reduces to showing
the following convergence.
\begin{lemma}
{\sl
For any scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have
\begin{equation*}
\lim_{n \rightarrow \infty} n\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}_n) } = 0
\quad \text{if} \quad \liminfty{n} \alpha_n = +\infty
\end{equation*}
where the sequence $\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
is determined through (\ref{eq:DeviationUnitCircle}).
}
\label{lem:OneLawCircle}
\end{lemma}
\myproof
Fix $n=1,2, \ldots $ and in
the expression (\ref{eq:FirstMomCircle}) substitute
$(r,p)$ by $(r_n,p_n)$ according to the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
We get
\begin{eqnarray*}
n\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}_n) }
&=&
n \left ( 1 - p_n \ell(r_n) \right)^{n-1}
\nonumber \\
&=& n \left(1-\frac{\log n +\alpha_n}{n}\right)^{n-1}
\nonumber \\
&\leq&
n e^{-\frac{n-1}{n}(\log n + \alpha_n)}
\nonumber \\
&=& n^{\frac{1}{n}} e^{ - \frac{n-1}{n} \alpha_n}
\end{eqnarray*}
where the bound (\ref{eq:B}) was used.
Letting $n$ go to infinity we get the desired conclusion since
$\lim_{n \rightarrow \infty} \alpha_n = \infty$.
\myendpf
\subsection{One law over the unit interval}
A similar step is taken for the random intersection graph
over the unit interval.
\begin{lemma}
{\sl
For any scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have
\begin{equation*}
\lim_{n \rightarrow \infty} n\bE{ \chi^{(L)}_{n,1}(\myvec{\theta}_n) } = 0
\quad \text{if} \quad \liminfty{n} \alpha^\prime_n = +\infty
\end{equation*}
where the sequence $\alpha^\prime: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
is determined through (\ref{eq:DeviationUnitInterval}).
}
\label{lem:OneLawInterval}
\end{lemma}
\myproof
Fix $n=1,2, \ldots $ and in
the upper bound (\ref{eq:IntervalFirstMomUB}) substitute
$(r,p)$ by $(r_n,p_n)$ according to the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
We get
\begin{eqnarray}
n\bE{ \chi^{(L)}_{n,1}(\myvec{\theta}_n) }
&\leq &
n \left ( 1 - p_n \ell(r_n) \right )^{n-1}
\nonumber \\
& & +~
\frac{2}{p_n}\left(1-\frac{1}{2}p_n \ell(r_n)\right)^n .
\nonumber
\end{eqnarray}
As in the proof of Lemma \ref{lem:OneLawCircle}, we can show that
\[
\lim_{n \rightarrow \infty}
n \left ( 1 - p_n \ell(r_n) \right )^{n-1}
= 0
\]
under the condition
$\lim_{n \rightarrow \infty} \alpha^\prime_n = \infty$; details are
left to the interested reader.
The desired conclusion will be established as soon
as we show that
\begin{equation}
\lim_{n \rightarrow \infty}
\frac{2}{p_n}\left(1-\frac{1}{2}p_n \ell(r_n)\right)^n
= 0
\label{eq:NeededLimit}
\end{equation}
under the same condition
$\lim_{n \rightarrow \infty} \alpha^\prime_n = \infty$.
To do so, fix $n=1,2,\ldots $ sufficiently large so that
$\alpha^\prime_n \geq 0$ -- This is always possible under
the condition
$\lim_{n \rightarrow \infty} \alpha^\prime_n = \infty$.
On that range we note that
\begin{eqnarray*}
& & \frac{1}{p_n}\left(1-\frac{1}{2}p_n \ell(r_n)\right)^n
\nonumber \\
&\leq&
\frac{1}{p_n \ell(r_n)}
\cdot \left(1-\frac{1}{2}p_n \ell(r_n)\right)^n
\nonumber \\
&\leq&
\left (
\frac{2 (\log n - \log \log n) + \alpha^\prime_n}{n}
\right )^{-1}
e^{-\log n + \log \log n - \frac{1}{2} \alpha^\prime_n }
\nonumber \\
&=&
\frac{ \log n }
{ 2 (\log n - \log \log n) + \alpha^\prime_n }
e^{ - \frac{1}{2} \alpha^\prime_n }
\nonumber \\
&\leq&
\frac{ \log n }
{ 2 (\log n - \log \log n) }
e^{ - \frac{1}{2} \alpha^\prime_n }
\end{eqnarray*}
upon using the fact $\ell(r_n) \leq 1$
and the bound (\ref{eq:B}).
Letting $n$ go to infinity we obtain (\ref{eq:NeededLimit}).
\myendpf
\section{Proof of the zero laws}
As observed earlier, when dealing with the zero law we need only concern
ourselves with the unit circle case. Throughout this section, we take
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$
and associate with it the sequence
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
through (\ref{eq:DeviationUnitCircle}).
We now show (\ref{eq:ToShow2}) and (\ref{eq:ToShow3}) under the condition
$\lim_{n \rightarrow \infty} \alpha_n = - \infty$.
This will complete the proof of the zero laws.
In the discussion we shall make use of the following elementary fact:
For any sequence $a: \mathbb{N}_0 \rightarrow \mathbb{R}_+$,
the asymptotic equivalence
\begin{equation}
(1-a_n)^n \sim e^{-na_n}
\label{eq:C}
\end{equation}
holds provided
$\lim_{n \to \infty} a_n = \lim_{n \to \infty} na^2_n = 0$.
\subsection{Establishing (\ref{eq:ToShow2})}
The first step is contained in the following zero-law complement
of Lemma \ref{lem:OneLawCircle}.
\begin{lemma}
{\sl For any scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have
\begin{equation*}
\lim_{n \rightarrow \infty} n\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}_n) }
= \infty
\quad \text{if} \quad \liminfty{n} \alpha_n = -\infty
\end{equation*}
where the sequence $\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
is determined through (\ref{eq:DeviationUnitCircle}).
}
\label{lem:ZeroLawCircle}
\end{lemma}
\myproof
Fix $n=1,2, \ldots $ and in
the expression (\ref{eq:FirstMomCircle}) substitute
$(r,p)$ by $(r_n,p_n)$ according to the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
As in the proof of Lemma \ref{lem:OneLawCircle} we start with the
expression
\begin{equation}
n\bE{ \chi^{(C)}_{n,1}(\myvec{\theta}_n) }
=
n \left ( 1 - p_n \ell(r_n) \right)^{n-1}.
\label{eq:ZeroLawCircleExpression}
\end{equation}
Under the condition $\lim_{n \rightarrow \infty } \alpha_n = -\infty$
we note that $\alpha_n = - |\alpha_n |$ for all $n$ sufficiently large,
say for all $n \geq n^\star$ for some finite integer $n^\star$.
Using (\ref{eq:DeviationUnitCircle}) we get
$|\alpha_n| \leq \log n $ on that range by
the non-negativity condition $p_n \ell(r_n) \geq 0$.
Therefore,
\begin{equation}
p_n \ell (r_n) \leq \frac{\log n}{n}
\quad \mbox{\rm and} \quad
n \left ( p_n \ell (r_n) \right )^2
\leq \frac{ (\log n )^2}{n}
\label{eq:pTimeslIneq}
\end{equation}
for all $n \geq n^\star$, and the equivalence (\ref{eq:C})
(with $a_n = p_n \ell (r_n)$) now yields
\begin{equation}
n \left ( 1 - p_n \ell(r_n) \right)^{n-1}
\sim n e^{- n p_n \ell (r_n) }
\label{eq:a}
\end{equation}
with
\begin{equation}
n e^{- n p_n \ell (r_n) }
=
n e^{- (\log n + \alpha _n ) } = e^{-\alpha_n} ,
\quad n=1,2, \ldots
\label{eq:b}
\end{equation}
Finally, letting $n$ go to infinity in (\ref{eq:ZeroLawCircleExpression})
and using (\ref{eq:a})-(\ref{eq:b}), we find
\[
\lim_{n \rightarrow \infty}
n \left ( 1 - p_n \ell(r_n) \right)^{n-1}
= \lim_{n \rightarrow \infty} e^{ - \alpha _n }
= \infty
\]
as desired under the condition
$\lim_{n \rightarrow \infty } \alpha_n = -\infty$.
\myendpf
\subsection{Establishing (\ref{eq:ToShow3})}
The proof of the one-law will be
completed if we establish the next result.
\begin{proposition}
{\sl For any scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$,
we have
\begin{equation*}
\limsup_{n \rightarrow \infty} R_n(\myvec{\theta}_n) \leq 1
\quad \text{if} \quad \liminfty{n} \alpha_n = -\infty
\end{equation*}
where the sequence
$\alpha: \mathbb{N}_0 \rightarrow \mathbb{R}_+ $
is determined through (\ref{eq:DeviationUnitCircle}).
}
\label{prop:CovarianceLimit}
\end{proposition}
The proof of Proposition \ref{prop:CovarianceLimit}
is organized around the following simple observation:
Consider a sequence $a: \mathbb{N}_0 \rightarrow \mathbb{R}$
and let $N_1, \ldots , N_K$ constitute a partition of $\mathbb{N}_0$
into $K$ subsets, i.e., $N_k \cap N_\ell = \emptyset$ for distinct
$k,\ell=1, \ldots , K$, and $\cup_{k=1}^K N_k = \mathbb{N}_0$.
In principle, some of the subsets $N_1, \ldots , N_K$ may be either
empty or finite. For each $k=1, \ldots , K$ such that $N_k$
is {\em non}-empty, we set
\[
\alpha_k :=
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_k } }
a_n
= \inf_{ n \in N_k}
\left ( \sup_{m \in N_k:~ m \geq n } a_m
\right )
\]
with the natural convention that $\alpha_k = -\infty $ when $N_k$ is finite.
In other words, $\alpha_k$ is the limsup for the subsequence
$\{ a_n , \ n \in N_k \}$.
It is a simple matter to check that
\begin{equation*}
\limsup_{n \rightarrow \infty} a_n
= {\max}^\star \left ( \alpha_k, \ k=1, \ldots , K \right )
\end{equation*}
with $\max^\star$ denoting the maximum operation over
all indices $k$ such that $N_k$ is non-empty.
\myproof
As we plan to make use of this fact with $K=4$, we write
\[
R_k := \limsup_{ \substack{n \rightarrow \infty\\
n \in N_k } }
R_n(\myvec{\theta}_n),
\quad k=1,\ldots , 4
\]
with
\begin{align*}
N_1 := & \{ n \in \mathbb{N}_0: \ 0 < r_n < 0.25, \ 0 < p_n \leq 1 \},
\\
N_2 := & \{ n \in \mathbb{N}_0: \ 0.25 \leq r_n < 0.5, \ 0 < p_n \leq 1 \},
\\
N_3 := & \{ n \in \mathbb{N}_0: \ 0.5 \leq r_n , \ 0 < p_n \leq 1 \}
\\
\intertext{and}
N_4 := & \{ n \in \mathbb{N}_0: \ r_n > 0, \ p_n = 0 \}.
\end{align*}
Therefore, we have
\begin{equation*}
\limsup_{n \rightarrow \infty}
R_n(\myvec{\theta}_n) = {\max}^\star (R_k, \ k=1, \ldots , 4 )
\label{eqn:limsup=max}
\end{equation*}
and the result will be established if we show that
\begin{equation*}
R_k \leq 1, \quad k=1, \ldots , 4 .
\label{eq:TOSHOW}
\end{equation*}
In view of the convention made earlier, we need only discuss
for each $k=1, \ldots , 4$,
the case when $N_k$ is countably infinite, as we do from now on.
The easy cases are handled first:
From (\ref{eqn:bnd4}) it is obvious that $R_4 = 1$.
Next as observed before,
(\ref{eq:pTimeslIneq}) holds for all $n$ sufficiently large
under the condition $\lim_{n \to \infty} \alpha_n = -\infty$.
Since $\ell(r_n)=1$ for all $n$ in $N_3$, we conclude that
\[
\lim_{ \substack{n \rightarrow \infty\\
n \in N_3 } }
p_n = 0
\]
and the conclusion $R_3 =1 $ is now immediate from (\ref{eqn:bnd3}).
We complete the proof by invoking
Lemmas \ref{lem:N1} and \ref{lem:N2} given next
which establish $R_1 \leq 1$ and $R_2 \leq 1$, respectively.
\myendpf
\begin{lemma}
{\sl Under the assumptions of Proposition \ref{prop:CovarianceLimit},
with $N_1$ countably infinite, we have $R_1 \leq 1$.
}
\label{lem:N1}
\end{lemma}
\myproof
Fix $n=2,3, \ldots $ and pick $(r,p)$ such that
$0 < r < 0.25$ and $0< p \leq 1$.
With $(\ref{eqn:bnd1})$ in mind, we note that
\begin{eqnarray}
& & \frac{2}{(n-1)p^2}
\left(\left(1+\frac{2p^2 r}{1-4pr}\right)^{n-1} - 1 \right)
\nonumber \\
&=&
\frac{2}{(n-1)p^2}
\left( \sum_{k=0}^{n-1} {n-1 \choose k} \left(\frac{2p^2 r}{1-4pr}\right)^k
- 1 \right)
\nonumber \\
&=&
\frac{2}{(n-1)p^2}
\sum_{k=1}^{n-1} {n-1 \choose k} \left(\frac{2p^2 r}{1-4pr}\right)^k
\nonumber \\
&=&
\frac{4r}{1-4pr}
+
\frac{2}{(n-1)p^2}
\sum_{k=2}^{n-1} {n-1 \choose k} \left(\frac{2p^2 r}{1-4pr}\right)^k
\nonumber
\end{eqnarray}
and
we can rewrite the right handside of (\ref{eqn:bnd1}) as
\begin{eqnarray}
& &
\frac{1-4r}{1-4pr}
+ \frac{2}{(n-1)p^2}
\left(\left(1+\frac{2p^2 r}{1-4pr}\right)^{n-1} - 1 \right)
\nonumber \\
&=&
\frac{1}{1-4pr}
+ \frac{2}{(n-1)p^2}
\sum_{k=2}^{n-1} {n-1 \choose k} \left(\frac{2p^2 r}{1-4pr}\right)^k
\nonumber \\
&\leq&
\frac{1}{1-4pr}
+ \frac{2}{(n-1)}
\sum_{k=2}^{n-1} {n-1 \choose k} \left(\frac{2pr}{1-4pr}\right)^k
\nonumber
\end{eqnarray}
since $p^k \leq p^2$ for $k=2, \ldots , n-1$.
Therefore,
\begin{equation*}
R_n (\myvec{\theta})
\leq
\frac{1}{1-4pr}
+ \frac{2}{(n-1)}
\left ( 1 + \frac{ 2pr }{1 - 4 pr} \right )^{n-1} .
\end{equation*}
In this last bound, fix $n$ in $N_1$ and substitute $(r,p)$
by $(r_n, p_n)$ according to the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$.
Standard properties of the limsup operation yield
\begin{eqnarray*}
R_1
&\leq&
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
\left ( \frac{1}{1-4p_n r_n} \right )
\\
& &
+~ \limsup_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
\left ( \frac{2}{(n-1)}
\left ( 1 + \frac{ 2p_n r_n}{1 - 4 p_n r_n} \right )^{n-1}
\right )
\nonumber
\end{eqnarray*}
and the desired result $R_1 \leq 1$ will follow if we show that
\begin{equation}
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
\left ( \frac{1}{1-4p_n r_n} \right )
= 1
\label{eq:N1+A}
\end{equation}
and
\begin{equation}
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
\left ( \frac{2}{(n-1)}
\left ( 1 + \frac{ 2p_n r_n}{1 - 4 p_n r_n} \right )^{n-1}
\right )
= 0 .
\label{eq:N1+B}
\end{equation}
To do so,
under the condition $\lim_{n \to \infty} \alpha_n = -\infty$ we
once again use the fact that (\ref{eq:pTimeslIneq}) holds
for large $n$ with $p_n \ell (r_n) = 2 p_nr_n$ for all $n$ in $N_1$.
Thus,
\[
\lim_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
p_n r_n = 0
\]
and the convergence (\ref{eq:N1+A}) follows.
Next, since $1+x \leq e^x$ for all $x$ in $\mathbb{R}$, we note
for all $n$ in $N_1$ that
\begin{eqnarray*}
& & \frac{2}{n-1} \left(1+\frac{2p_n r_n}{1-4p_n r_n} \right)^{n-1}
\nonumber \\
&=& \frac{2}{n-1}
\left(
1+\frac{p_n \ell(r_n)}{1-2 p_n \ell(r_n)}
\right)^{n-1}
\nonumber \\
&\leq& \frac{2}{n-1}
\left ( e^{ \frac{p_n \ell(r_n)}{ 1-2 p_n \ell(r_n)} } \right )^{n-1}
= 2 e^{\beta_n}
\end{eqnarray*}
with
\[
\beta_n
:= (n-1) \frac{p_n \ell(r_n)}{ 1-2 p_n \ell(r_n)} - \log (n-1).
\]
Thus, (\ref{eq:N1+B}) follows if we show that
\begin{equation}
\lim_{ \substack{n \rightarrow \infty\\
n \in N_1 } }
\beta_n = -\infty .
\label{eq:N1+C}
\end{equation}
From (\ref{eq:pTimeslIneq}) we get
\[
\beta_n
\leq \left ( \frac{n-1}{n} \right )
\frac{\log n + \alpha_n}{ 1-2 \frac{\log n}{n} }
- \log (n-1)
\]
for large $n$.
It is now a simple exercise to check that
\[
\lim_{n \rightarrow \infty}
\left ( \frac{n-1}{n} \right ) \frac{\log n }{ 1-2 \frac{\log n}{n} }
- \log (n-1)
= 0
\]
and the conclusion (\ref{eq:N1+C}) is obtained under the assumption
$\lim_{n \to \infty} \alpha_n = -\infty$.
\myendpf
\begin{lemma}
{\sl Under the assumptions of Proposition \ref{prop:CovarianceLimit},
with $N_2$ countably infinite, we have $R_2 \leq 1$.
}
\label{lem:N2}
\end{lemma}
\myproof
Fix $n=2,3, \ldots $ and pick $(r,p)$ such that
$0.25 < r \leq 0.5$ and $0< p \leq 1$.
From $(\ref{eqn:bnd2})$ we get
\begin{eqnarray}
R_n(\myvec{\theta})
&\leq&
\frac{4r-1}{(1-2pr)^2}
\nonumber \\
& & +~
\frac{2-4r}{ (1-2pr)^2} \frac{(1-4pr+2p^2 r)^{n-2}}{(1-2pr)^{2(n-2)}}
\nonumber \\
&=& \frac{4r}{ (1-2pr)^2}
\left ( 1 - \frac{(1-4pr+2p^2 r)^{n-2}}{(1-2pr)^{2(n-2)}} \right )
\nonumber \\
& & +~
\frac{1}{ (1-2pr)^2}
\left ( 2 \frac{(1-4pr+2p^2 r)^{n-2}}{(1-2pr)^{2(n-2)}}
-1 \right ) .
\nonumber
\end{eqnarray}
Now fix $n$ in $N_2$ and substitute $(r,p)$ by $(r_n, p_n)$ according to
the scaling
$\myvec{\theta}: \mathbb{N}_0 \rightarrow \mathbb{R}_+ \times [0,1]$
in (\ref{eqn:bnd2}).
As before, properties of the limsup operation yield
\begin{equation}
R_2 \leq R_{2c}
\left ( R_{2a} + R_{2b} \right )
\label{Inequality1}
\end{equation}
with
\[
R_{2a}
:=
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\left ( 4r_n
\left (
1 - \frac{(1-4p_n r_n+2p_n^2 r_n)^{n-2}}{(1-2p_n r_n)^{2(n-2)}}
\right )
\right ),
\]
\[
R_{2b}
:=
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\left ( 2 \frac{(1-4p_n r_n+2p_n^2 r_n)^{n-2}}{(1-2p_n r_n)^{2(n-2)}}
-1 \right )
\]
and
\[
R_{2c}
:=
\limsup_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\frac{1}{(1-2p_n r_n)^2} .
\]
As in the proof of Lemma \ref{lem:N1}, it is also the case here that
$R_{2c}$ exists as a limit and is given by
\[
R_{2c}
=
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\frac{1}{(1-2p_n r_n)^2} = 1 ;
\]
details are omitted in the interest of brevity.
Next, we show that
\begin{equation}
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\frac{(1-4p_n r_n+2p_n^2 r_n)^{n-2}}{(1-2p_n r_n)^{2(n-2)}}
= 1.
\label{eq:Intermediary1}
\end{equation}
Once this is done, we see from their definitions that
$R_{2a} = 0$ and $R_{2b} = 1$, and the conclusion
$R_2 \leq 1$ follows from (\ref{Inequality1}).
To establish (\ref{eq:Intermediary1}) we note that
\[
4p_n r_n - 2p_n^2 r_n
= p_n \ell(r_n) (2-p_n) \leq 2 p_n \ell(r_n)
\]
and
\[
2p_n r_n = p_n \ell (r_n)
\]
for all $n$ in $N_2$. Now making use of (\ref{eq:pTimeslIneq})
we conclude that
\[
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
\left ( 4p_n r_n - 2p_n^2 r_n \right )
=
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
(n-2) \left ( 4p_n r_n - 2p_n^2 r_n \right )^2
= 0
\]
while
\[
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
2 p_n r_n
= \lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
(n-2) \left ( 2p_n r_n \right )^2
= 0.
\]
By the equivalence (\ref{eq:C}) used
with $a_n = 4p_n r_n - 2p_n^2 r_n$
and $a_n = 2 p_n r_n$, respectively, we now conclude that
\begin{eqnarray}
& & \frac{(1-4p_n r_n+2p_n^2 r_n)^{n-2}}{(1-2p_n r_n)^{2(n-2)}}
\nonumber \\
&\sim&
\frac{
e^{- (n-2) \left ( 4p_n r_n - 2p_n^2 r_n \right ) }
}{
\left ( e^{ - (n-2) \left ( 2 p_n r_n \right ) } \right )^2
}
\nonumber \\
&=& e^{ 2 (n-2) \left ( p_n^2 r_n \right ) }
\label{eq:Intermediary2}
\end{eqnarray}
as $n$ goes to infinity in $N_2$.
Finally, for $n$ in $N_2$, because $\ell(r_n) = 2 r_n \geq 0.5$,
we get
\begin{eqnarray*}
2 (n-2) \left ( p_n^2 r_n \right )
&=& (n-2) \frac{ \left ( p_n \ell(r_n) \right )^2 }
{ \ell(r_n) }
\nonumber \\
&\leq& 2 (n-2) \cdot \left ( p_n \ell(r_n) \right )^2
\nonumber \\
&=& \frac{2(n-2)}{n} \cdot n \left ( p_n \ell(r_n) \right )^2
\end{eqnarray*}
so that
\[
\lim_{ \substack{n \rightarrow \infty\\
n \in N_2 } }
2 (n-2) \left ( p_n^2 r_n \right ) = 0
\]
with the help of (\ref{eq:pTimeslIneq}).
The conclusion (\ref{eq:Intermediary1})
now follows from (\ref{eq:Intermediary2}), and the proof of
Lemma \ref{lem:N2} is complete.
\myendpf
\section{Simulation results}
\begin{figure*}[htp]
\centerline{
\subfigure[Fix $n=100$, $p=0.25$ and vary $r$]{\includegraphics[width=3.25in]{sim1.eps}
\label{fig:sim-r}}
\hfil
\subfigure[Fix $n=100$, $r=0.1$ and vary $p$]{\includegraphics[width=3.25in]{sim2.eps}
\label{fig:sim-p}}}
\caption{Simulation results}
\label{fig:sim}
\end{figure*}
In this section, we present some plots from simulations in Matlab
which confirm the results in Theorem \ref{thm:MainThmCir}
and Theorem \ref{thm:MainThmInt}.
For given $n$, $p$ and $r$, we estimate
the probability that there are no isolated
nodes by averaging over $1,000$ instances
of the random graphs $\mathbb{G}^{(C)}(n;\myvec{\myvec{\theta}})$
and $\mathbb{G}^{(L)}(n;\myvec{\myvec{\theta}})$.
In Figure \ref{fig:sim-r}, we have taken $n=100$ and $p=0.25$,
and examine the threshold behavior of the probability
that there are no isolated nodes by varying $r$.
Theorem \ref{thm:MainThmCir} suggests that the critical range
for the graph over the unit circle when $n=100$ and $p=0.25$ should be
$r^\star=0.09$. This is confirmed by the simulation results.
In the case of the unit interval, we expect from
Theorem \ref{thm:MainThmInt} that the critical range will be
between $r^\star=0.09$ and $r^{\star \star} =0.12$; this is in
agreement with the plot.
In Figure \ref{fig:sim-p}, we have taken $n=100$ and $r=0.1$,
and repeat the analysis by choosing various values for $p$.
As expected from Theorem \ref{thm:MainThmCir},
the critical edge probability for the unit circle is found to occur
at $p^{\star} = 0.23$. It is also clear that for the unit interval,
the critical edge probability is between
$p^{\star} = 0.23$ and $p^{\star \star} = 0.31$ as predicted by
Theorem \ref{thm:MainThmInt}.
\section{Concluding remarks}
\label{ConcludingRemarks}
Theorem \ref{thm:MainThmInt} shows
a gap between the zero and one laws in the case of
the intersection graph on the unit interval:
The zero law expresses deviations
with respect to the scaling
$\myvec{\theta}^{\star}: \mathbb{N}_0 \rightarrow \mathbb{R}_+\times [0,1]$
determined through
\[
p^{\star}_n \ell(r^{\star}_n )
= \frac{ \log n }{n},
\quad n=1,2,\ldots
\]
as guessed.
On the other hand,
the one law reflects sensitivity with respect to the \lq\lq {\em larger}"
scaling $\myvec{\theta}^{\star\star}: \mathbb{N}_0 \rightarrow
\mathbb{R}_+\times [0,1]$
determined through
\[
p^{\star\star}_n \ell(r^{\star\star}_n )
= \frac{ 2(\log n - \log \log n )}{n},
\quad n=1,2,\ldots
\]
Inspection of the proof readily shows that the
method of first moment is not powerful enough to close the gap --
To the best of our knowledge we are not aware of any other instance
in the literature where this occurs.
While we still believe that this gap can be bridged,
it is clear that a different method of analysis will be needed.
The analysis given here also suggests the form of the zero-one law to expect
when the geometric component lives in higher dimensions. Specifically,
consider the case where the nodes are located in a region
$\mathbb{D} \subseteq \mathbb{R}^d$,
without boundary, e.g., a torus or a spherical surface.
Then it is easy to compute the probability of an edge between two nodes as
\[
p \ell(r)= p \bP{ d(\myvec{x},\myvec{Y}) \leq r }
\]
where $\myvec{x}$ is an arbitrary point in $\mathbb{D}$,
the rv $\myvec{Y}$ is uniformly distributed over $\mathbb{D}$
and $d(\cdot,\cdot)$ is the appropriate notion of distance.
As before, if we define the sequence $\alpha: \mathbb{N}_0 \to \mathbb{R}$
through
\[
p_n \ell(r_n) = \frac{\log n + \alpha_n}{n}, \quad n=1,2,\dots
\]
then the required dichotomy in the first moment
(cf. Lemma \ref{lem:OneLawCircle} and Lemma \ref{lem:ZeroLawCircle})
cleary holds even in higher dimensions. As a result, we expect the
critical scaling for the absence of isolated nodes to be given through
\[
p_n^\star \ell(r_n^\star) = \frac{\log n}{n}, \quad n=1,2, \dots.
\]
Finally, similar inferences can be made for modeling
wireless sensor networks which rely on the Eschenauer-Gligor scheme
to securize their communication links:
Power constraints restrict nodes to have a finite transmission range,
a physical communication constraint which is captured by the disk model,
the Eschenauer-Gligor scheme
introduces a logical constraint which is well modeled by the random
key graph \cite{YaganMakowskiISIT2008}.
Combining these two constraints amounts to taking the intersection
of a geometric random graph with a random key graph
\cite{DiPietroManciniMeiPanconesiRadhakrishnan2004}
\cite{DiPietroManciniMeiPanconesiRadhakrishnan2006}.\footnote{The case
when the transmission range is inifinite is the so-called full visibility
case \cite{YaganMakowskiISIT2008}.}
However, unlike Erd\H{o}s-R\'enyi
graphs, random key graphs exhibit dependencies between edges,
and this renders
the problem more complex. Nevertheless, we expect the determination
of critical scalings through the probability of an edge between two nodes
to take place here as well; see (\ref{eq:CriticalRequirement}).
This time, in (\ref{eq:CriticalRequirement2})
the probability $p$ is replaced by the probability that two
nodes share a common key in the Eschenauer-Gligor scheme.
| {
"timestamp": "2008-09-04T23:13:14",
"yymm": "0809",
"arxiv_id": "0809.0918",
"language": "en",
"url": "https://arxiv.org/abs/0809.0918",
"abstract": "When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.A simple idea to account for multiple constraints consists in taking the intersection of random graphs. In this paper we initiate the study of random graphs so obtained through a simple example. We examine the intersection of an Erdos-Renyi graph and of one-dimensional geometric random graphs. We investigate the zero-one laws for the property that there are no isolated nodes. When the geometric component is defined on the unit circle, a full zero-one law is established and we determine its critical scaling. When the geometric component lies in the unit interval, there is a gap in that the obtained zero and one laws are found to express deviations from different critical scalings. In particular, the first moment method requires a larger critical scaling than in the unit circle case in order to obtain the one law. This discrepancy is somewhat surprising given that the zero-one laws for the absence of isolated nodes are identical in the geometric random graphs on both the unit interval and unit circle.",
"subjects": "Information Theory (cs.IT); Probability (math.PR)",
"title": "Intersecting random graphs and networks with multiple adjacency constraints: A simple example",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075733703927,
"lm_q2_score": 0.7371581741774411,
"lm_q1q2_score": 0.7092254621480071
} |
https://arxiv.org/abs/2107.14054 | Detecting and diagnosing prior and likelihood sensitivity with power-scaling | Determining the sensitivity of the posterior to perturbations of the prior and likelihood is an important part of the Bayesian workflow. We introduce a practical and computationally efficient sensitivity analysis approach using importance sampling to estimate properties of posteriors resulting from power-scaling the prior or likelihood. On this basis, we suggest a diagnostic that can indicate the presence of prior-data conflict or likelihood noninformativity and discuss limitations to this power-scaling approach. The approach can be easily included in Bayesian workflows with minimal effort by the model builder and we present an implementation in our new R package priorsense. We further demonstrate the workflow on case studies of real data using models varying in complexity from simple linear models to Gaussian process models. |
\section{Introduction}
\label{sec:intro}
Bayesian inference is characterised by the derivation of a posterior
from a prior and a likelihood. As the posterior is dependent on the
specification of these two components, investigating its sensitivity
to perturbations of the prior and likelihood is a critical step in the Bayesian
workflow~\citep{gelmanBayesianWorkflow2020,depaoliImportancePriorSensitivity2020,lopesConfrontingPriorConvictions2011}. Along
with indicating the robustness of an inference in general, such sensitivity
is related to issues of \textit{prior-data
conflict}~\citep{evansCheckingPriordataConflict2006,allabadiOptimalRobustnessResults2017,reimherrPriorSampleSize2020}
and \textit{likelihood noninformativity}~\citep{gelmanPriorCanOften2017,poirierRevisingBeliefsNonidentified1998}.
Historically, sensitivity analysis has been an important topic in
Bayesian methods
research~\citep[e.g.][]{canavosBayesianEstimationSensitivity1975,skeneBayesianModellingSensitivity1986,bergerRobustBayesianAnalysis1990,bergerOverviewRobustBayesian1994,hillSensitivityBayesianAnalysis1994}.
However, the amount of research on the topic has
diminished~\citep{watsonApproximateModelsRobust2016,bergerBayesianRobustness2000}
and results from sensitivity analyses are seldom reported in empirical
studies employing Bayesian
methods~\citep{vandeschootSystematicReviewBayesian2017}. We
suggest that a primary reason for this is the lack of
sensitivity analysis approaches that are easily incorporated into existing
modelling workflows.
Common modelling workflows~\citep[e.g.\ those described in][]{grinsztajnBayesianWorkflowDisease2021,gelmanBayesianWorkflow2020,schadPrincipledBayesianWorkflow2021} typically involve
specifying a model in a probabilistic programming
language, such as Stan~\citep{standevelopmentteamStanModellingLanguage2021} or PyMC~\citep{pymc3} and using Markov
chain Monte Carlo algorithms to approximate the
posterior via posterior draws. In many cases, modellers begin with a base model with template priors and likelihood, which is then iteratively built on~\citep{gelmanBayesianWorkflow2020}. To check for prior and likelihood sensitivity, one can fit models with different
perturbations to the prior or likelihood, but this can require substantial amounts of both user and computing time~\citep{perezMCMCbasedLocalParametric2006,jacobiAutomatedSensitivityAnalysis2018}. Using more computationally efficient methods can reduce the computation time, but existing methods, while useful in many circumstances, have limitations; they are focused on particular types of
models~\citep{Roos2021,hunanyan2021quantification}
or inference
mechanisms~\citep{roosSensitivityAnalysisBayesian2015}, rely on
manual specification of
perturbations~\citep{mccartanAdjustrStanModel2021},
require substantial or technically complex changes to the model code that hinder
widespread
use~\citep{giordanoCovariancesRobustnessVariational2018,jacobiAutomatedSensitivityAnalysis2018}, or may still require substantial amounts of computation time~\citep{hoGlobalRobustBayesian2020,bornnEfficientComputationalApproach2010}.
In this work, we present an approach for sensitivity analysis that
addresses these limitations and aims to:
\begin{itemize}
\item be computationally efficient;
\item be applicable to a wide range of models;
\item provide automated diagnostics;
\item require minimal changes to existing model code and workflow.
\end{itemize}
We emphasize that the approach should not be used for repeated tuning of the priors until diagnostic warnings no longer appear. Instead, the approach should be considered as a diagnostic to detect accidentally misspecified priors and unexpected sensitivities or conflicts, and the reaction to diagnostic warnings should always involve careful consideration about domain expertise, priors, and model specification.
\begin{figure}[h]
\centering
\input{./figs/scaling-example.tex}
\caption{Example of our power-scaling sensitivity approach. Here, the prior is power-scaled, and the effect on the posterior is shown. In this case the prior
is \(\normal(0, 2.5)\) and the likelihood is equivalent to
\(\normal(10, 1)\). Power-scaling the prior by different
\(\alpha\) values (in this case 0.5 and 2.0) shifts the posterior
(shaded for emphasis), indicating prior sensitivity.}%
\label{fig:scaling-example}
\end{figure}
\newpage
The approach uses importance sampling to estimate properties of perturbed
posteriors that result from power-scaling (exponentiating by some \(\alpha > 0\)) the prior or likelihood (see Figure~\ref{fig:scaling-example}). We
propose a diagnostic, based on the the change to the posterior induced by this perturbation, that can indicate the
presence of prior-data conflict or likelihood noninformativity. Importantly, as long as the changes to the priors or likelihood induced by power-scaling are not too substantial, the procedure does not require refitting the model, which drastically increases its efficiency. The
envisioned workflow is as follows (also see
Figure~\ref{fig:workflow}):
\begin{enumerate}
\item[(1)] Fit a base model (either a template
model or a manually specified model) to data, resulting in a base
posterior distribution.
\item[(2, 3)] Estimate properties of perturbed posteriors that result from separately
power-scaling the prior and likelihood.
\item[(4, 5)] Evaluate the extent the perturbed posteriors differ from the
base posterior numerically and visually.
\item[(6)] Diagnose based on the pattern of prior and likelihood sensitivity.
\item[(7)] Reevaluate the assumptions implied by the base model and
potentially modify it (and repeat (1)--(6)).
\item[(8)] Continue with use of the model for its intended purpose.
\end{enumerate}
\begin{figure}[bt]
\centering
\small
\begin{tikzpicture}[
>=stealth,
node distance = 0.5cm
]
\node[draw,
align = center,
] (basefit) {(1) Run posterior inference};
\node[draw,
align = center,
below left = 1cm,
] (perturbprior) {(2) Estimate properties of\\ posteriors with\\perturbed \textit{prior}};
\node[draw,
align = center,
below right = 1cm,
] (perturblik) {(3) Estimate properties of\\ posteriors with\\perturbed \textit{likelihood}};
\node[draw,
align = center,
below = of perturbprior,
] (checkprior) {(4) Check \textit{prior} \\sensitivity};
\node[draw,
align = center,
below = of perturblik,
] (checklik) {(5) Check \textit{likelihood} \\sensitivity};
\node[draw,
align = center,
below = 4cm,
] (diagnose) {(6) Diagnose sensitivity};
\node[draw,
align = center,
below left = of diagnose,
] (adjust) {(7) Adjust model};
\node[draw,
align = center,
below = of diagnose,
] (use) {(8) Use model};
\draw[-stealth] (basefit) -- (perturbprior);
\draw[-stealth] (basefit) -- (perturblik);
\draw[-stealth] (perturbprior) -- (checkprior);
\draw[-stealth] (perturblik) -- (checklik);
\draw[-stealth] (checkprior) -- (diagnose);
\draw[-stealth] (checklik) -- (diagnose);
\draw[-stealth] (diagnose) -- (use);
\draw[-stealth] (diagnose) -- (adjust);
\draw[-stealth] (adjust) -| +(-2,0) |- (basefit);
\end{tikzpicture}
\caption{Workflow of our proposed sensitivity analysis approach.}
\label{fig:workflow}
\end{figure}
\section{Details of the approach}
\subsection{Power-scaling perturbations}
\label{sec:method}
The proposed sensitivity analysis approach relies on separately perturbing the prior or likelihood through
power-scaling (exponentiating by some \(\alpha > 0\)). This power-scaling is
a controlled, distribution-agnostic method of modifying a probability
distribution. Intuitively, it can be considered to weaken (when \(\alpha < 1\)) or strengthen (when \(\alpha > 1\)) the component being power-scaled in relation to the other. Although power-scaling changes the normalising constant, this is not a concern when using Monte Carlo approaches for estimating posteriors via posterior draws.
For all non-uniform distributions, as \(\alpha\)
diverges from \(1\), the shape of the distribution
changes. However, it retains the support of the base distribution
(if the density at a point in the base distribution is zero,
raising it to any power will still result in zero; likewise any
nonzero density will remain nonzero). In the context of prior perturbations, these properties are desirable as slight
perturbations from power-scaling
result in distributions that likely represent similar implied
assumptions to those of the base distribution. A set of slightly perturbed priors can thus be considered a reasonable class of distributions for prior sensitivity
analysis~\citep[see][]{bergerRobustBayesianAnalysis1990,bergerOverviewRobustBayesian1994}. For the likelihood, power-scaling acts as an approximation for decreasing or increasing the number of (conditionally independent) observations, akin to data cloning~\citep{lele2007DataCloning}.
For some common distributions, the effect of power-scaling
on the parameters can be expressed analytically and in a way that the resulting
distribution is of the same form. For instance, a normal distribution,
\(\normal(\theta \mid \mu, \sigma) \propto \exp{(-\frac{1}{2}
(\frac{\theta - \mu}{\sigma})^2)}\), when power-scaled by some
\(\alpha > 0\) simply scales the \(\sigma\) parameter by \(\alpha^{-1/2}\), thus
\(\normal(\theta \mid \mu, \sigma)^\alpha \propto \normal(\theta \mid
\mu, \alpha^{-1/2}\sigma )\). Other exponential family distributions commonly used as priors,
such as beta, gamma and exponential, behave similarly, such that the unnormalised
density can be expressed in the same form with modified parameters
(see Table~\ref{tab:dists} and Figure~\ref{fig:example-dists}). This
characteristic is particularly useful when investigating the prior
sensitivity as such power-scaled priors are intuitively understandable
in relation to the base prior, but power-scaling also works for distributions from other families, providing that the distribution is non-uniform (distributions with parameters controlling the support will only be power-scaled with respect to the base support). However, power-scaling, while intuitive and effective, is only able to perturb a distribution in a particular manner. For example, it is not possible to directly shift the location of a distribution via power-scaling, without also changing other aspects. Thus, like most diagnostics, when power-scaling sensitivity analysis does not indicate sensitivity, this only means that it could not detect sensitivity to power-scaling, not that the model is certainly well-behaved or insensitive to other types of perturbations.
\begin{table}[tb]
\centering
\caption{Forms of power-scaled distributions for common
distributions.}%
\label{tab:dists}
\begin{tabular}{ll}
\toprule
Base & Power-scaled \\
\midrule
\(\expdist(\theta \mid \lambda)\) & \( \propto \expdist(\theta \mid \alpha \lambda)\)\\
\(\normal(\theta \mid \mu, \sigma)\) & \( \propto \normal(\theta \mid \mu, \alpha ^{-1/2} \sigma)\) \\
\(\betadist(\theta \mid s_1, s_2)\) & \( \propto \betadist(\theta \mid \alpha s_1 -
\alpha + 1, \alpha s_2 - \alpha + 1)\) \\
\(\gammadist(\theta \mid s_1, s_2)\) & \( \propto \gammadist(\theta \mid \alpha s_1 -
\alpha + 1, \alpha s_2)\) \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[tb]
\centering
\input{./figs/example_dists.tex}
\caption{The effect of power-scaling on common
distributions. In each case, the resulting distributions can be
expressed in the same form as the base distribution with modified
parameters.}%
\label{fig:example-dists}
\end{figure}
\FloatBarrier
\subsection{Power-scaling priors}
\label{sec:prior}
In order for the sensitivity analysis approach to be independent of the number of parameters in the model, all priors are power-scaled simultaneously. However, in some cases, certain priors should be excluded from this set. For example, in hierarchical models, power-scaling both top- and intermediate-level priors can lead to unintended results. To illustrate this, consider two forms of prior, a non-hierarchical prior with two independent parameters \(p(\theta) \, p(\phi)\) and a hierarchical prior of the form \(p(\theta\mid\phi) \, p(\phi)\). In the first case, the appropriate power-scaling for the prior is \(p(\phi)^\alpha \, p(\theta)^\alpha\), while in the second, only the top level prior should be power-scaled, that is, \(p(\theta\mid\phi) \, p(\phi)^\alpha \). If the prior \(p(\theta\mid\phi)\) is also power-scaled, \(\theta\) will be affected by the power-scaling twice, directly and indirectly, perhaps even in opposite directions depending on the parameterization.
\subsection{Estimating properties of perturbed posteriors}
As the normalizing constant for the posterior distribution can rarely
be computed analytically in real-world analyses, our approach assumes
that the base posterior is approximated using Monte Carlo draws
(workflow step 1, see Figure~\ref{fig:workflow}). These draws are used
to estimate properties of the perturbed posteriors via importance sampling (workflow steps 2 and 3, see
Figure~\ref{fig:workflow}). Importance
sampling is a method to
estimate expectations of a target distribution by weighting draws from a proposal distribution~\citep{robert2004}. After computing these weights, there are several possibilities
for evaluating sensitivity.
For example, different summaries of perturbed posteriors can be computed
directly, or resampled draws can be generated
using importance resampling~\citep{rubin1988SIR}.
Importance sampling as a method for efficient sensitivity analysis has
been previously described by
\citet{bergerOverviewRobustBayesian1994,besagBayesianComputationStochastic1995,oneillImportanceSamplingBayesian2009,tsaiInfluenceMeasuresRobust2011}. However, one
limitation of importance sampling is that it can be unreliable when
the variance of importance weights is large or infinite. Hence,
as described by \citet{bergerOverviewRobustBayesian1994}, relying
on importance sampling to estimate a posterior resulting from a
perturbed prior or likelihood, without controlling the width of the
perturbation class (e.g. through a continuous parameter to control the
amount of perturbation, \(\alpha\) in our case) is likely to lead to unstable estimates.
To further alleviate issues with importance sampling, we
use Pareto smoothed importance sampling~\citep[PSIS;][]{vehtariParetoSmoothedImportance2019}, which stabilises the importance weights in an efficient,
self-diagnosing and trustworthy manner by modelling the upper tail of the importance weights with a generalised Pareto distribution. In cases where PSIS does not perform adequately, weights are adapted with
importance weighted moment matching~\citep[IWMM;][]{paananenImplicitlyAdaptiveImportance2021}, which is
a generic adaptive importance sampling algorithm that improves
the implicit proposal distribution by iterative weighted moment matching. The combination of using a continuous parameter to control the amount of perturbation, along with PSIS and IWMM, allows for a reliable and self-diagnosing method of estimating properties of perturbed posteriors.
\subsubsection{Calculating importance weights for power-scaling perturbations}
Consider an expectation
of a function \(h\) of parameters \({\theta}\), which come from a
target distribution \(f({\theta})\):
\begin{equation}
\mathbb{E}_f[h({\theta})] = \int h({\theta})f({\theta}) d{\theta}.
\end{equation}
In cases when draws can be generated from the target distribution, the
simple Monte Carlo estimate can be calculated from a sequence of \(S\)
draws from \(f({\theta})\):
\begin{equation}
\mathbb{E}_f[h({\theta})] \approx \frac{1}{S} \sum_{s=1}^Sh({\theta}^{(s)}), \text{where }
{\theta}^{(s)} \sim f({\theta}).
\end{equation}
As an alternative to calculating the expectation directly with draws from \(f({\theta})\), the
importance sampling estimate instead uses draws from a proposal
distribution \(g({\theta})\) and the ratio between the target and
proposal densities, known as the importance weights \(w\). The self-normalised importance sampling estimate does not require normalising constants of the target or proposal to be known and is thus well suited for use in the context of probabilistic programming languages, which do not calculate these:
\begin{equation} \label{eq:snis}
\mathbb{E}_f[h({\theta})] \approx \frac{\sum_{s=1}^S
h(\theta^{(s)}) \frac{f({\theta}^{(s)})}{g({\theta}^{(s)})}}{\sum_{s=1}^S \frac{f({\theta}^{(s)})}{g({\theta}^{(s)})}} =
\frac{\sum_{s=1}^S h(\theta^{(s)}) w^{(s)}}{\sum_{s=1}^S w^{(s)}}, \text{where } \theta^{(s)} \sim g(\theta).
\end{equation}
In the context of power-scaling perturbations, the proposal
distribution is the base posterior and the target distribution is
a perturbed posterior resulting from power-scaling. If the proposal and target
distributions are expressed as the products of the prior \(p(\theta)\) and
likelihood \(p(y \mid {\theta})\), with one of these components raised to the power of
\(\alpha\), the importance sampling weights only
depend on the density of the component being power-scaled. For prior power-scaling, the importance weights are
\begin{equation} \label{eq:wprior}
w_{\alpha_\text{pr}}^{(s)} = \frac{p({\theta}^{(s)})^\alpha p(y \mid {\theta}^{(s)})}{p({\theta}^{(s)})p(y \mid {\theta}^{(s)})}\\
= p({\theta}^{(s)})^{\alpha - 1}.
\end{equation}
Analogously, the importance weights for likelihood power-scaling
are
\begin{equation} \label{eq:wlik}
w_{\alpha_\text{lik}}^{(s)} = p(y \mid {\theta}^{(s)})^{\alpha - 1}.
\end{equation}
As the importance weights are only dependent on the density of the
power-scaled component at the location of the proposal draws, they are easy to compute for a range of \(\alpha\) values.
See Section~\ref{sec:implementation} for practical implementation details about
computing the weights.
\subsection{Measuring sensitivity}
There are different ways to evaluate the effect of power-scaling perturbations on
a posterior (workflow steps 4 and 5, see
Figure~\ref{fig:workflow}). Here we present two options: first, a method
that investigates changes in specific posterior quantities of interest
(e.g.\ mean and standard deviation)
(Section~\ref{sec:quantity}), and second, a
method based on the distances between the base marginal posteriors and the
perturbed marginal posteriors (Section~\ref{sec:distance}). These methods should not be considered competing, but rather allow for different levels of sensitivity
analysis, and depending on the context and what the modeller is interested in, one may be more useful than the other. Importantly, the proposed power-scaling approach is not tied to any
particular method of evaluating sensitivity. These methods are our
suggestions, but once quantities or weighted draws from perturbed posteriors
are computed, a multitude of comparisons to the base posterior and other posteriors
can be performed.
\subsubsection{Quantity-based sensitivity}
\label{sec:quantity}
In some cases it can be most useful to investigate
sensitivity of particular quantities of interest. Expectations of interest for a perturbed
posterior can be calculated from the base draws and the importance
weights using Equation~(\ref{eq:snis}). Other quantities that are not expectations (such as the median and quantiles) can be derived from the weighted empirical cumulative distribution function~(ECDF). Computed quantities can then be compared based on the specific interests of the modeller, or local sensitivity can be quantified by derivatives with respect to the perturbation parameter \(\alpha\) (see Section~\ref{sec:local}).
\subsubsection{Distance-based sensitivity}
\label{sec:distance}
We can investigate the sensitivity of marginal posteriors using a distance-based approach. Here, we follow previous work which has quantified sensitivity based on the distance between the base and perturbed posteriors~\citep{ohaganHSSS,allabadiMeasuringBayesianRobustness2021,kurtekBayesianSensitivityAnalysis2015}.
In principle, many different divergence or distance measures can be
used, although there may be slight
differences in interpretation~\citep[see, for example][]{lekHowChoiceDistance2019,chaComprehensiveSurveyDistance2007}, however, the cumulative Jensen-Shannon divergence~\citep[CJS;][]{nguyenNonparametricJensenShannonDivergence2015} has two properties that make it appropriate for our use case. First, its symmetrised form is upper-bounded, like the standard Jensen-Shannon divergence~\citep{linDivergenceMeasuresBased1991}, which aids interpretation. Second, instead of comparing probability density functions (PDFs) as the standard Jensen-Shannon divergence does, it compares cumulative distribution functions (CDFs), which can be easily and efficiently estimated from Monte Carlo draws. Although probability density functions could be estimated using kernel density estimates and then the standard Jensen-Shannon distance used, this may require substantially more draws to be accurate and lead to artefacts otherwise~\citep[for further discussion of the benefits of CDFs, see, for example][]{sailynoja2021graphical}.
Given two CDFs \(P(\theta)\) and \(Q(\theta)\),
\begin{equation}
\label{eq:cjs}
\cjs(P(\theta) \| Q(\theta)) = \int P(\theta)\log_2 \left(\frac{2 P(\theta)}{P(\theta) + Q(\theta)}\right) \mathop{d\theta} + \frac{1}{2\ln(2)} \int Q(\theta) - P(\theta) \mathop{d\theta}.
\end{equation}
As a distance measure, we use the symmetrised and metric (square root) version of CJS, normalised with respect to its upper bound, such that it is bounded on the 0 to 1 interval~\citep[for further details see][]{nguyenNonparametricJensenShannonDivergence2015}:
\begin{equation}
\label{eq:cjsdist}
\cjs_{\text{dist}}(P(\theta) \| Q(\theta)) = \sqrt{\frac{\cjs(P(\theta) \| Q(\theta) + \cjs(Q(\theta) \| P(\theta))}{\int P(\theta) + Q(\theta) \mathop{d\theta}}}.
\end{equation}
As \(\cjs\) is not invariant to the sign of the parameter values, \(\cjs(P(\theta)\|Q(\theta)) \neq \cjs(P(-\theta)\|Q(-\theta))\), we use \(\max(\cjs_{\text{dist}}(P(\theta)\|Q(\theta)),\cjs_{\text{dist}}(P(-\theta)\|Q(-\theta)))\) to account for this and ensure applicability regardless of possible transformations applied to posterior draws that may change the sign.
In our approach, we compare the ECDFs of the base posterior to the perturbed posteriors. The ECDF of the base posterior is estimated from the base posterior draws, whereas the ECDFs of the perturbed posteriors are estimated by first weighting the base draws with the importance weights. As described in \citet{nguyenNonparametricJensenShannonDivergence2015}, when using ECDFs, the integrals in Equations~(\ref{eq:cjs}) and (\ref{eq:cjsdist}) reduce to sums, which allows for efficient computation.
\subsubsection{Local sensitivity}
\label{sec:local}
Both distance-based and quantity-based sensitivity can be evaluated
for any \(\alpha\) value.
It is also possible to obtain an overall estimate of sensitivity
at \(\alpha = 1\) by differentiation.
This follows previous work which defines
the local sensitivity as the derivative with respect to the
perturbation parameter~\citep{gustafsonLocalRobustnessBayesian2000,maroufyLocalGlobalRobustness2015,sivaganesanRobustBayesianDiagnostics1993,giordanoCovariancesRobustnessVariational2018}. For power-scaling, we suggest considering the derivative with respect to \(\log_2(\alpha)\) as it captures the symmetry of power-scaling around \(\alpha = 1\) and provides values on a natural scale in relation to halving or doubling the log density of the component.
Because of the simplicity of the power-scaling procedure, local sensitivity at \(\alpha = 1\) can be computed analytically with importance sampling for certain quantities, such as the mean and variance, without knowing the analytical form of the posterior.
This allows for a highly computationally efficient method to probe for sensitivity
in common quantities before performing further sensitivity diagnostics. For quantities that are computed as an expectation of some function \(h\), the derivative at \(\alpha = 1\)
can be computed as follows. We denote the power-scaling importance weights as \(p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}\), where \(p_{\text{ps}}(\theta^{(s)})\) is the density of the power-scaled component, which can be either the prior or likelihood depending on the type of scaling. Then
\begin{align*}
\left . \frac{\sum_{s=1}^S h(\theta^{(s)}) p_{\text{ps}}(\theta^{(s)})^{\alpha - 1} }{\sum_{s=1}^S p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}}
\frac{\partial}{\partial \log_2(\alpha)}\right|_{\alpha = 1}\\
= \left . \frac{\left(\sum_{s=1}^S \alpha \ln(2) h(\theta^{(s)}) p_{\text{ps}}(\theta^{(s)})^{\alpha - 1} \ln(p_{\text{ps}}(\theta^{(s)}))\right)\left(\sum_{s=1}^S p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}\right)}{\left(\sum_{s=1}^S p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}\right)^2} \right|_{\alpha = 1}\\
- \left . \frac{\left(\sum_{s=1}^S h(\theta^{(s)}) p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}\right)\left(\sum_{s=1}^S \alpha \ln(2) p_{\text{ps}}(\theta^{(s)})^{\alpha - 1} \ln(p_{\text{ps}}(\theta^{(s)}))\right)}{\left(\sum_{s=1}^S p_{\text{ps}}(\theta^{(s)})^{\alpha - 1}\right)^2} \right|_{\alpha = 1}\\
= \ln(2) \left(\frac{1}{S} \sum_{s=1}^S \ln(p_{\text{ps}}(\theta^{(s)})) h( \theta^{(s)}) - \left (\frac{1}{S} \sum_{s=1}^S h(\theta^{(s)}) \right) \left (\frac{1}{S} \sum_{s=1}^S \ln(p_{\text{ps}}(\theta^{(s)})) \right )\right).
\end{align*}
Consider for example that we are interested in the sensitivity of the posterior
mean of the parameters \(\theta\). For prior scaling,
the derivative of the mean with respect to \(\log_2(\alpha)\) at \(\alpha = 1\) is then
\begin{align}
\ln(2) \left( \frac{1}{S} \sum_{s=1}^S \ln (p(\theta^{(s)})) \theta^{(s)} - \left (\frac{1}{S} \sum_{s=1}^S \theta^{(s)} \right) \left (\frac{1}{S} \sum_{s=1}^S \ln (p(\theta^{(s)})) \right ) \right).
\end{align}
As with quantity-based sensitivity, distance-based sensitivity can also be quantified
by taking the corresponding derivative.
\(\cjs_{\text{dist}}\) increases from 0 as \(\alpha\) diverges from 1, so we take the second derivative with respect to \(\log_2(\alpha)\) as an indication of local power-scaling sensitivity. We approximate this numerically from the ECDFs with finite differences (for \(\delta = 0.01\)):
\begin{align*}
D_{\cjs} = \frac{\cjs_{\text{dist}}(\hat{P}_1(\theta) \| \hat{P}_{1/(1 + \delta)}(\theta)) + \cjs_{\text{dist}}(\hat{P}_1(\theta) \| \hat{P}_{1 + \delta}(\theta))}{2\log_2(\delta)},
\end{align*}
where \(\hat{P}_1(\theta)\) is the ECDF of the base posterior (when \(\alpha = 1\)), \(\hat{P}_{1/(1+\delta)}(\theta)\) is the weighted ECDF when \(\alpha = 1/(1+\delta)\) and \(\hat{P}_{1+\delta}(\theta)\) is the weighted ECDF when \(\alpha = 1 + \delta\).
\subsubsection{Diagnostic threshold}
\label{sec:threshold}
We consider \(D_{\cjs} \geq 0.05\) to be a reasonable indication of sensitivity. For a normal distribution, this corresponds to the mean differing by more than approximately 0.3 standard deviations, or the standard deviation differing by a factor greater than approximately 0.3, when the power-scaling \(\alpha\) is changed by a factor of two. This distance is shown in Figure~\ref{fig:threshold-example}. However, depending on how concerned a modeller is with sensitivity, this threshold can be adapted to reflect what constitutes a meaningful change in the specific model.
\begin{figure}[tb]
\centering
\input{./figs/cdf_example.tex}
\caption{PDFs (top) and CDFs (bottom) of \(\normal(0, 1)\) and perturbed distributions differing by \(\cjs_{\text{dist}} \approx 0.05\) from the \(\normal(0, 1)\). Note that \(\cjs\) quantifies the difference between two CDFs; the corresponding PDFs are shown here to aid interpretation.}
\label{fig:threshold-example}
\end{figure}
\FloatBarrier
\subsection{Diagnosing sensitivity}
\label{sec:examples}
Sensitivity can be diagnosed by comparing the amount of exhibited prior and likelihood sensitivity (workflow step 6, see Table~\ref{tab:sensitivity-types}).
When a model is well-behaved, it is expected that there will be likelihood sensitivity, as power-scaling the likelihood is analogous to changing the number of (conditionally independent) observations. However, in hierarchical models, it is important to recognise that this is analogous to changing the number of observations within each group, rather than the number of groups. As such, in hierarchical models, lack of likelihood sensitivity based on power-scaling does not necessarily indicate that the likelihood is weak overall.
\begin{table}[tb]
\centering
\caption{The interplay between prior sensitivity and
likelihood sensitivity can be used to diagnose the cause. As there can be relations between parameter, the pattern of sensitivity for a single parameter should be considered in the context of others. Cases in which the posterior is insensitive to both
prior and likelihood power-scaling (i.e.\ uninformative
prior with likelihood noninformativity) will likely be detectable from model
fitting issues, and are not further addressed by our approach.}%
\label{tab:sensitivity-types}
\begin{tabular}{cccc}
\toprule
& & \multicolumn{2}{c}{\makecell{Prior\\ sensitivity}} \\
& & No & Yes \\
\midrule
\multirowcell{2}[-0.5em]{Likelihood\\ sensitivity} & No & & \makecell[l]{Likelihood\\ noninformativity} \\
\cmidrule{2-4}
& Yes & \makecell[l]{Likelihood\\ domination} & \makecell{Prior-data conflict} \\
\bottomrule
\end{tabular}
\end{table}
Likelihood domination (the combination of a weakly informative
or diffuse prior combined with a well-behaving and informative likelihood) will result
in likelihood sensitivity but no prior sensitivity. This indicates that the posterior is mostly reliant on the data and likelihood rather than the prior (see Figure~\ref{fig:weaklyinf}).
\begin{figure}[tb]
\centering
\input{./figs/weak.tex}
\caption{A weakly informative \(\normal(0, 10)\) prior and a
well-behaving \(\normal(10, 1)\) likelihood lead to likelihood domination. This is indicated by little to no
prior sensitivity and expected likelihood sensitivity. Top row: the prior is power-scaled; bottom row: the likelihood is power-scaled. Note that
in the figure the likelihood and posterior densities are almost
completely overlapping.}%
\label{fig:weaklyinf}
\end{figure}
In contrast, prior sensitivity can result from two primary causes, both of which are indications that the model may have an issue: 1)
\textit{prior-data conflict} and 2) \textit{likelihood noninformativity}. In the case of prior-data conflict, the posterior will exhibit both prior and likelihood sensitivity, whereas in the case of likelihood noninformativity (in relation to the prior) there
will be some marginal posteriors which are not as sensitive to
likelihood power-scaling as they are to prior power-scaling (or not at all sensitive to
likelihood power-scaling).
Prior-data conflict~\citep{walter2009,evansCheckingPriordataConflict2006,nottCheckingPriorDataConflict2020} can
arise due to intentionally or unintentionally informative priors
disagreeing with, but not being dominated by, the likelihood. When this is the case, the posterior will
be sensitive to power-scaling both the prior and the likelihood, as
illustrated in Figure~\ref{fig:conflict-1}. When prior-data conflict has been detected, the modeller may wish to modify the model by using a less informative prior~\citep[see][]{evansWeakInformativityInformation2011,nottUsingPriorExpansions2020} or using heavy-tailed distributions~\citep{gagnon2021robustness,ohaganBayesianHeavytailedModels2012}.
\begin{figure}[tb]
\centering
\input{./figs/conflict-example-1.tex}
\caption{Conflict between \(\normal(0, 2.5)\) prior and \(\normal(10, 1)\) likelihood,
results in the posterior (shaded for emphasis)
being sensitive to both prior and likelihood power-scaling. Top row: the prior is power-scaled; bottom row: the likelihood is power-scaled.}%
\label{fig:conflict-1}
\end{figure}
The presence of prior sensitivity but relatively low (or no) likelihood sensitivity is an indication that the likelihood is weakly informative (or noninformative) in relation to the prior. This can occur, for example, when there is complete
separation in a logistic regression. The simplest case of complete
separation occurs when there are observations of only one class. For
example, suppose a researcher is attempting to identify the occurrence
rate of a rare event in a new population. Based on previous research,
it is believed that the rate is close to 1 out of 1000. The researcher
has since collected 100 observations from the new population, all of
which are negative. As the data are only of one class, the posterior
will then exhibit prior sensitivity as the likelihood is relatively weak. In the case of weakly informative or noninformative likelihood, the choice of prior will have a direct impact on the posterior and is therefore of a greater importance and should be considered carefully.
In some cases, the likelihood (or the data) may not be problematic in
and of itself, but if the chosen prior dominates the likelihood, the
posterior will be relatively insensitive to power-scaling the
likelihood. This can occur, for example, when the likelihood is a
Student's \(t\) distribution and the prior is a normal distribution,
as shown in Figure~\ref{fig:conflict-2}.
\begin{figure}[tb]
\centering
\input{./figs/conflict-example-2.tex}
\caption{The posterior (shaded for emphasis) exhibits prior sensitivity, but relatively
low likelihood sensitivity. This is due to the \(\normal(0, 1)\) prior dominating the \(t_4(10,1)\) likelihood because of the forms of the distributions. Top row: the prior is power-scaled; bottom row: the likelihood is power-scaled.}%
\label{fig:conflict-2}
\end{figure}
\FloatBarrier
\subsubsection{Sensitivity for parameter combinations and other quantities}
As discussed, sensitivity can be evaluated for each marginal distribution separately in a relatively automated manner. However, this approach may lead to interpretation issues when individual parameters are by definition not informed by the likelihood, or are not readily interpretable. In the case when the likelihood may be informative for a combination of parameters, but not any of the parameters individually, it can be useful to perform a whitening transformation (such as principle components analysis)~\citep{kessy2018Sphere} on the posterior draws and then investigate sensitivity in the compressed parameter space. This can indicate which parameter combinations are sensitive to likelihood perturbations, indicating that they are jointly informed by the likelihood, and which are not.
This whitening approach works when there are few parameters, but as the number of parameters grows, the compressed components can be more difficult to interpret. Instead, in more complex cases, we suggest the modeller focus on target quantities of interest. For example, in the case of Gaussian process regression, it can be more useful to investigate the sensitivity of predictive distributions~\citep{paananen2021RSense,paananen2019VariableSelection} than posterior distributions of model parameters.
\section{Software implementation}
\label{sec:implementation}
Our approach for power-scaling sensitivity analysis is implemented in
\texttt{priorsense} (\url{https://github.com/n-kall/priorsense}), our new
R~\citep{rcoreteamLanguageEnvironmentStatistical2020} package for
prior sensitivity diagnostics. The implementation focuses on models
fitted with
Stan~\citep{standevelopmentteamStanModellingLanguage2021}, but it can
be extended to work with other probabilistic programming frameworks
that provide similar functionality. The package includes numerical
diagnostics and graphical representations of changes in
posteriors. These are available for both distance- and quantity-based
sensitivity.
\subsection{Usage}
Conducting a power-scaling sensitivity analysis with \texttt{priorsense} can be done as follows: given a fitted model object, \texttt{powerscale\_sensitivity} will automatically perform workflow steps 2--5 (Figure~\ref{fig:workflow}) and return the local sensitivity of each parameter in a model fit (based on numerical derivatives of \(\cjs_\text{dist}\) by default). Follow-up analysis for diagnosing the sensitivity can be performed with \texttt{powerscale\_sequence}, which returns an object containing the base posterior draws along with weights corresponding to each perturbed posterior (or optionally resampled posterior draws). This can be plotted to visualise the change in ECDFs (Figure~\ref{fig:bodyfat-both-sens}), kernel density etimates, or estimated quantities (Figure~\ref{fig:bodyfat-summary-sens}), with respect to the degree of power-scaling. Sensitivity of posterior quantities such as the mean, median or standard deviation can be assessed with the \texttt{powerscale\_derivative} (for analytical derivatives) and \texttt{powerscale\_gradients} (for numerical derivatives) functions. All functions will provide warnings when estimates derived from PSIS or IWMM may not be trustworthy due to too large differences between the perturbed and base posteriors.
\subsection{Practical implementation details}
In this section, we provide more details for a practical implementation of the approach.
As shown in Equations~\ref{eq:wprior} and \ref{eq:wlik}, the importance weights for power-scaling the prior or likelihood rely on density evaluations of the power-scaled component. Thus, the following are required for estimating properties of the perturbed posteriors:
\begin{itemize}
\item posterior draws from the base posterior
\item (log of) likelihood evaluations at the locations of the posterior draws
\item (log of) joint prior evaluations (for the priors to be power-scaled) at the locations of the posterior draws
\end{itemize}
In R, posterior draws can be accessed from the model fit object
directly, while the \texttt{posterior}
package~\citep{burknerPosteriorToolsWorking2020} provides
convenient functions for working with them. Existing R packages interfacing with Stan already make use of the log likelihood values~\citep[e.g.\ the \texttt{loo} package;][]{vehtariLooEfficientLeaveoneout2020}, and the log prior values can be specified in the model code, for example in the
\texttt{generated quantities} block of the Stan code (as shown
in Listing~\ref{lst:example-stan}). In cases where some
priors should be excluded from the power-scaling (such as intermediate
priors in hierarchical models, see Section~\ref{sec:prior}), only the priors to be power-scaled should be included here.
\begin{lstlisting}[
float,
basicstyle = \ttfamily,
caption = Example Stan code with log prior and log likelihood
specified such that the resulting fitted model can be used with \texttt{priorsense}.,
label = lst:example-stan
]
data {
int<lower=1> N;
vector[N] y;
}
parameters {
real mu;
real<lower=0> sigma;
}
model {
// priors
target += normal_lpdf(mu | 0, 10);
target += normal_lpdf(sigma | 0, 5);
// likelihood
target += normal_lpdf(y | mu, sigma);
}
generated quantities {
vector[N] log_lik; // log likelihood
real log_prior; // joint log prior
// log likelihood
for (n in 1:N) {
log_lik[n] = normal_lpdf(y[n] | mu, sigma);
}
// log prior
log_prior = normal_lpdf(mu | 0, 10)
+ normal_lpdf(sigma | 0, 5);
}
\end{lstlisting}
\texttt{priorsense} uses \texttt{loo} package for PSIS, while IWMM is
currently implemented directly. \(\cjs_\text{dist}\) is also implemented directly, while other divergence measures are imported from
\texttt{philentropy}~\citep{drostPhilentropyInformationTheory2018}. Functions from \texttt{matrixStats}~\citep{bengtssonMatrixStatsFunctionsThat2020} and \texttt{spatstat}~\citep{spatstat} are used for calculating weighted quantities and weighted ECDFs, respectively. Diagnostics graphics are created using \texttt{ggplot2}~\citep{wickhamGgplot2ElegantGraphics2016}.
\section{Case studies}
In this section, we show how \texttt{priorsense} can be used in a Bayesian model building workflow to detect and diagnose prior sensitivity
in realistic models fit to real data (corresponding data and code are available at \url{https://github.com/n-kall/powerscaling-sensitivity}). We use the \texttt{brms} package ~\citep{burknerBrmsPackageBayesian2017}, which is a high-level R interface to Stan, to specify and fit the simpler regression
models (Stan is used directly for the Gaussian process regression model) and \texttt{priorsense} to evaluate the prior and likelihood sensitivity. Unless further specified, we
use Stan to sample the posteriors using the default settings (4
chains, 2000 iterations per chain, half discarded as warmup). Convergence diagnostics for Hamiltonian Monte Carlo and effective sample sizes are checked for all models, and sampling parameters are adjusted to relieve any identified issues before proceeding with sensitivity analysis. As the primary indication of local sensitivity, we use the gradient of \(\cjs_{\text{dist}}\) with
respect to \(\log_2(\alpha)\). We consider \(D \geq 0.05\) to be indicative of sensitivity (see Section \ref{sec:threshold}).
\subsection{Body fat}
This case study shows a situation in which \textit{prior-data conflict}
can be detected by power-scaling sensitivity analysis. This
conflict results from choosing priors that are not of appropriate
scales for some predictors. For this case study, we use the
\texttt{bodyfat} data
set~\citep{johnsonFittingPercentageBody1996}, which has
previously been the focus of variable selection
experiments~\citep{pavoneUsingReferenceModels2020,heinzeVariableSelectionReview2018}. The
aim of the analysis is to predict an expensive and cumbersome water immersion measurement of body
fat percentage from a set of thirteen easier to measure
characteristics, including age, height, weight, and circumferences of
various body parts.
We begin with a linear regression model to predict body fat percentage
from the aforementioned variables. By default, in \texttt{brms} the
\(\beta_0\) (intercept) and \(\sigma\) parameters are given data-derived weakly
informative priors, and the regression coefficients are given improper
flat priors. Power-scaling will not affect flat priors, so we specify proper priors for the regression coefficients. We specify the
same prior for all coefficients, \(\normal(0, 1)\), which does not
seem unreasonable based on preliminary prior-predictive checks. We arrive at the following model:
\begin{align*}
y_i &\sim \normal(\mu_i, \sigma) \\
\mu_i &= \beta^0 + \sum_{k=1}^{13}x_{i}^k\beta_k \\
\beta^0 &\sim t_3(0, 9.2) \\
\sigma &\sim t_3^+(0, 9.2) \\
\beta^k &\sim \normal(0, 1) \\
\end{align*}
From the marginal posterior plot (Figure~\ref{fig:bodyfat-posterior}), there do not appear to be issues, and all estimates are in reasonable ranges. Power-scaling sensitivity
analysis, performed with the \texttt{powerscale\_sensitivity} function, however, shows that there is both prior sensitivity and likelihood sensitivity for one of the parameters, \(\beta^{\text{wrist}}\)
(Table~\ref{tab:bodyfat-sense}). This
indicates that there may be prior-data conflict.
\begin{figure}[tb]
\centering
\input{./figs/bodyfat_posterior_base.tex}
\caption{Marginal posteriors for bodyfat case study. Points show means, intervals correspond to 50\% and 95\% credible intervals.}%
\label{fig:bodyfat-posterior}
\end{figure}
\begin{table}[tb]
\centering
\caption{Sensitivity diagnostic values for the body fat case study using the original model. Values are based on second derivatives of \(\cjs_\text{dist}\) with respect to \(\log_2(\alpha)\) around \(\alpha = 1\). Higher sensitivity values indicate greater sensitivity. Prior sensitivity values above the threshold \((\geq 0.05)\) indicate possible informative prior (bold). Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate possible weak or noninformative likelihood (bold). The combined sensitivity information indicates that there may be prior-data conflict for \(\beta^\text{wrist}\).}
\label{tab:bodyfat-sense}
\begin{tabular}{llll}
\toprule
Parameter & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity} & Comment\\
\midrule
\(\beta^{\text{wrist}}\) & \textbf{0.12} & 0.09 & prior-data conflict\\
\(\beta^{\text{weight}}\) & 0.02 & 0.12 \\
\(\beta^{\text{thigh}}\) & 0.01 & 0.08 \\
\(\beta^{\text{neck}}\) & 0.01 & 0.11 \\
\(\beta^{\text{knee}}\) & 0.01 & 0.1 \\
\(\beta^{\text{hip}}\) & 0.01 & 0.11 \\
\(\beta^{\text{height}}\) & 0.00 & 0.09 \\
\(\beta^{\text{forearm}}\) & 0.02 & 0.12 \\
\(\beta^{\text{chest}}\) & 0.01 & 0.08 \\
\(\beta^{\text{biceps}}\) & 0.01 & 0.09 \\
\(\beta^{\text{ankle}}\) & 0.02 & 0.1 \\
\(\beta^{\text{age}}\) & 0.03 & 0.12 \\
\(\beta^{\text{abdomen}}\) & 0.00 & 0.09 \\
\bottomrule
\end{tabular}
\end{table}
We then check how the ECDF of the posterior is affected by power-scaling of the prior and likelihood. In \texttt{priorsense}, this is done creating a sequence of weighted draws (for a sequence of \(\alpha\) values) using \texttt{powerscale\_sequence}, and then plotting with \texttt{powerscale\_plot\_ecdf} (Figure~\ref{fig:bodyfat-both-sens}, left). We see that the posterior is sensitive to both prior and likelihood power-scaling, and that it shifts right (towards zero) as the prior is strengthened and left (away from zero) as the likelihood is strengthened. This is an indication of prior-data conflict,
which can be further seen by plotting the change in quantities using \texttt{powerscale\_plot\_quantities} (Figure~\ref{fig:bodyfat-summary-sens}). Prior-data conflict is evident by the `X' shape of the mean plot, as the mean is shifted in opposite directions.
\begin{figure}[tb]
\centering
\includegraphics{./figs/bodyfat_sens_joint.pdf}
\caption{Power-scaling diagnostic plot of maginal ECDFs
for posterior \(\beta^{\text{wrist}}\) in the body fat case
study. (Left) Original prior; There is both prior and likelihood sensitivity, as the ECDFs are not overlapping. (Right) Adjusted prior; There is now no prior sensitivity, as the ECDFs are overlapping, whereas there is still likelihood sensitivity.}%
\label{fig:bodyfat-both-sens}
\end{figure}
\begin{figure}[tb]
\centering
\input{./figs/bodyfat_quantities_joint.tex}
\caption{Posterior quantities of \(\beta^{\text{wrist}}\) as a function of power-scaling for the body fat case study. With this plot, we can compare the effect of prior and likelihood power-scaling on specific quantities. Shown as dashed lines are \(\pm2\) Monte Carlo standard errors (MCSE) of the base posterior quantity, as guides to whether an observed change is meaningful. Top: original prior; The pattern of the change in the mean indicates prior-data conflict, as power-scaling the prior and likelihood have opposite directional effects on the posterior mean. Bottom: adjusted prior; there is no longer prior or likelihood sensitivity for the mean, indicating no prior-data conflict. Likelihood sensitivity for the posterior standard deviation remains, indicating that the likelihood is informative.}
\label{fig:bodyfat-summary-sens}
\end{figure}
As there is prior sensitivity arising from prior-data conflict, which
is unexpected and unintentional as our priors were chosen to be weakly informative, we consider modifying the priors. On inspecting the raw
data, we see that although the predictor variables are all measured on
similar scales, the variance of the variables differs
substantially. For example, the variance of wrist circumference is
0.83, while the variance of abdomen is 102.65. This leads to our
chosen prior to be unintentionally informative for some of the
regression coefficients, including wrist, while being weakly
informative for others. To account for this, we refit the model with
priors empirically scaled to the data, \(\beta^k \sim \normal(0, 2.5 s_y/s_{x^k})\), where \(s_y\) is the standard deviation of \(y\) and \(s_{x^k}\) is the standard deviation of predictor variable \(x^k\). This corresponds to the default priors used by for regression models in the \texttt{rstanarm} package~\citep{rstanarm}, as described in \citet{gelmanRegressionOtherStories2020} and \citet{gabry2020PriorDistributions}. We refit the model and see that the
posterior mean for \(\beta^{\text{wrist}}\) changes from -1.45 to -1.86,
indicating that the base prior was indeed unintentionally
informative and in conflict with the data, pulling the estimate towards zero. Running power-scaling sensitivity
analysis on the adjusted model fit shows that there is no longer prior sensitivity but there is appropriate likelihood sensitivity (Table~\ref{tab:bodyfat-refit}, Figure~\ref{fig:bodyfat-both-sens} right).
\begin{table}[tb]
\centering
\caption{Sensitivity diagnostic values for the body fat case study using the adjusted model. Values are based on derivatives of \(\cjs_\text{dist}\) with respect to \(\log_2(\alpha)\) around \(\alpha = 1\). Higher sensitivity values indicate greater sensitivity. Prior sensitivity values above the threshold \((\geq 0.05)\) indicate possible informative prior (bold). Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate possible weak or noninformative likelihood (bold). The pattern of sensitivity indicates that no parameters exhibit prior sensitivity, and the likelihood is informative for all parameters.}%
\label{tab:bodyfat-refit}
\begin{tabular}{lll}
\toprule
Parameter & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity} \\
\midrule
\(\beta^{\text{wrist}}\) & 0.00 & 0.08 \\
\(\beta^{\text{weight}}\) & 0.00 & 0.09 \\
\(\beta^{\text{thigh}}\) & 0.00 & 0.10 \\
\(\beta^{\text{neck}}\) & 0.00 & 0.09 \\
\(\beta^{\text{knee}}\) & 0.00 & 0.08 \\
\(\beta^{\text{hip}}\) & 0.00 & 0.09 \\
\(\beta^{\text{height}}\) & 0.00 & 0.08 \\
\(\beta^{\text{forearm}}\) & 0.00 & 0.09 \\
\(\beta^{\text{chest}}\) & 0.00 & 0.09 \\
\(\beta^{\text{biceps}}\) & 0.00 & 0.08 \\
\(\beta^{\text{ankle}}\) & 0.00 & 0.09 \\
\(\beta^{\text{age}}\) & 0.00 & 0.08 \\
\(\beta^{\text{abdomen}}\) & 0.00 & 0.10 \\
\bottomrule
\end{tabular}
\end{table}
This is a clear example of how power-scaling sensitivity analysis
can detect and diagnose prior-data conflict. Unintentionally informative priors resulted in the conflict, which
could not be detected by only inspecting the posterior estimates of the base model. Once detected and diagnosed, the model could be adjusted and analysis could proceed. It is important to emphasise that the model was modified as the original priors were \emph{unintentionally} informative. If the original priors had been manually specified based on prior knowledge, it may not have been appropriate to modify the priors after observing the sensitivity.
\FloatBarrier
\subsection{Banknotes}
This case study is an example of using power-scaling sensitivity analysis to detect and diagnose \textit{likelihood noninformativity}. We use
the \texttt{banknote} data set~\citep{Flury1988} available from the \texttt{mclust} package~\citep{mclust},
which contains measurements of six properties of 100 genuine (\(Y = 0\)) and 100 counterfeit (\(Y = 1\)) Swiss banknotes. We fit a logistic regression on the status of a note based on these measurements. For priors, we
use the template priors
\(\normal(0, 10)\) for the intercept and \(\normal(0, 2.5/s_{x^k})\), where \(s_{x^k}\) is the standard deviation of predictor \(k\). The model is then
\begin{align*}
Y_{i} &\sim \Bernoulli(p_{i})\\
\log\left(\frac{p_{i}}{1-p_{i}}\right) &= \beta^0 + \sum_{k=1}^6{\beta^k x_{i}^k}\\
\beta^0 &\sim \normal(0, 10)\\
\beta^k &\sim \normal(0, 2.5/s_{x^k})
\end{align*}
Power-scaling sensitivity analysis indicates prior sensitivity for all predictor coefficients (Table~\ref{tab:bank-sens}). Furthermore, most exhibit low likelihood sensitivity, indicating a weak likelihood. In a Bernoulli model, this may arise if the binary outcome is completely separable by the predictors. This can be confirmed using the \texttt{detectseparation} package~\citep{kosmidis2021}, which detects infinite maximum likelihood estimates (caused by separation) in binary outcome regression models without fitting the model. Indeed, the data set is completely separable and the prior sensitivity will remain, regardless of choice of prior.
\begin{table}[t]
\centering
\caption{Sensitivity diagnostic values for the bank notes case study. Values are based on \(\cjs_\text{dist}\). Prior sensitivity values above the threshold \((\geq 0.05)\) indicate possible informative prior (bold). Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate possible weak or noninformative likelihood (bold). The pattern of sensitivity indicates that there is weak likelihood for most parameters.}
\label{tab:bank-sens}
\begin{tabular}{llll}
\toprule
Parameter & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity} & Comment\\
\midrule
\(\beta^{\text{length}}\) & \textbf{0.07} & \textbf{0.02} & weak likelihood \\
\(\beta^{\text{left}}\) & \textbf{0.10} & \textbf{0.01} & weak likelihood \\
\(\beta^{\text{right}}\) & \textbf{0.08} & \textbf{0.02} & weak likelihood \\
\(\beta^{\text{bottom}}\) & \textbf{0.25} & 0.11 & prior-data conflict \\
\(\beta^{\text{top}}\) & \textbf{0.18} & \textbf{0.04} & weak likelihood \\
\(\beta^{\text{diagonal}}\) & \textbf{0.13} & 0.05 & prior-data conflict \\
\bottomrule
\end{tabular}
\end{table}
\FloatBarrier
\subsection{Bacteria treatment}
Here, we use the \texttt{bacteria} data set, available from the \texttt{MASS} package~\citep{mass} to demonstrate power-scaling sensitivity analysis in hierarchical models as a way to compare priors. This data has previously been used by
\citet{kurtekBayesianSensitivityAnalysis2015} in a sensitivity analysis directly comparing posteriors resulting from different priors. We use the same model structure and similar priors and arrive at matching conclusions. Importantly, however, we show that the problematic prior can be detected from the resulting posterior, without the need to compare to other posteriors (and without the need for multiple fits).
The data set contains 220 observations of the effect of a treatment (placebo, drug with low compliance, drug with high compliance) on 50 children with middle ear infection over 5 time points (week). The outcome variable is the presence (\(Y = 1\)) or absence (\(Y = 0\)) of the bacteria targeted by the drug. We fit the same generalised linear multilevel model on the data as \citet{kurtekBayesianSensitivityAnalysis2015}, based on an example from \citet{brownMCMCGeneralizedLinear2010}:
\begin{align*}
Y_{ij} &\sim \Bernoulli(p_{ij})\\
\log\left(\frac{p_{ij}}{1-p_{ij}}\right) &= \mu + \sum_{k=1}^3{x_{ij}^k\beta^k + V_i} \\
\mu &\sim \normal(0, 10) \\
\beta^k &\sim \normal(0, 10) \\
V_i &\sim \normal(0, \sigma) \\
\tau = \frac{1}{\sigma^2} &\sim \gammadist(0.01, 0.01).
\end{align*}
We try different priors for the precision
hyperparameter \(\tau\).
We compare the sensitivity of the base model, with prior \(\tau \sim \gammadist(0.01, 0.01)\), to the comparison
priors. Three of which are considered reasonable, \(\tau \sim \normal^+(0, 10), \Cauchy^+(0, 100), \gammadist(1, 2)\), and one is considered unreasonable, \(\tau \sim \gammadist(9, 0.5)\). These priors are shown in Figure~\ref{fig:bacteria-priors}. We fit each model with four chains of 10000 iterations (2000 discarded as warmup) and perform power-scaling sensitivity analysis on each.
As discussed in Section~\ref{sec:prior},
only the top-level parameters in the hierarchical prior are power-scaled (i.e.\ the prior on \(V_i\) is not power-scaled). Posterior quantities and sensitivity diagnostics are shown in Table~\ref{tab:bacteria}. It is
apparent that the \(\tau\) parameter is sensitive to the prior when using the \(\gammadist(9, 0.5)\) prior. This is an indication that such a prior may be inappropriately informative. Although there is no indication of power-scaling sensitivity for the \(\mu\) and \(\beta\) parameters, comparing the posteriors for the models indicates differences in these parameters for the unreasonable \(\tau\) prior compared to the other priors (Table~\ref{tab:bacteria}). This is an important observation, and highlights that power-scaling is a local perturbation and may not influence the model strongly enough to change all quantities, yet can indicate the presence of potential issues.
\begin{figure}[tb]
\centering
\input{./figs/bacteria_priors.tex}
\caption{Visual comparison of priors for the hyperparameter \(\tau\)
in the bacteria case study. Priors considered reasonable for this application are shown on the left while priors considered unreasonable are shown on the right.}%
\label{fig:bacteria-priors}
\end{figure}
\begin{table}[b]
\centering
\caption{Sensitivity diagnostic values for the bacteria case study. Values derived from numerical derivatives of \(\cjs_\text{dist}\) with respect to \(\log_2(\alpha)\) around \(\alpha = 1\). Higher sensitivity values indicate greater sensitivity. Prior sensitivity values above the threshold \((\geq 0.05)\) indicate potential issues. Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate weak or noninformative likelihood. There appears to be prior-data conflict for the unreasonable prior (\(\tau \sim \text{gamma}(9, 0.5)\)), which leads to drastically differing posterior estimates for all parameters.}
\label{tab:bacteria}
\begin{tabular}{llllll}
\toprule
\makecell[l]{Prior\\(comment)} & Parameter & \makecell[l]{Post.\\mean} & \makecell[l]{Post.\\SD} & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity} \\
\midrule
\(\tau \sim \gammadist(0.01, 0.01)\) \\
& \(\tau\) & 0.36 & 0.3 & 0.02 & 0.10 \\
& \(\mu\) & 3.8 & 0.76 & 0.03 & 0.18 \\
& \(\beta^{\text{week}}\) & -0.17 & 0.06 & 0.02 & 0.10 \\
& \(\beta^{\text{trtDrugP}}\) & -1.01 & 0.92 & 0.02 & 0.10 \\
& \(\beta^{\text{trtDrug}}\) & -1.55 & 0.92 & 0.02 & 0.10 \\
\midrule
\(\tau \sim \normal^+(0, 1)\) \\
& \(\tau\) & 0.44 & 0.27 & 0.01 & 0.20 \\
& \(\mu\) & 3.6 & 0.76 & 0.00 & 0.16 \\
& \(\beta^{\text{week}}\) & -0.16 & 0.05 & 0.00 & 0.11 \\
& \(\beta^{\text{trtDrugP}}\) & -0.94 & 0.85 & 0.00 & 0.10 \\
& \(\beta^{\text{trtDrug}}\) & -1.48 & 0.84 & 0.01 & 0.10 \\
\midrule
\(\tau \sim \Cauchy^+(0, 1)\) \\
& \(\tau\) & 0.44 & 0.30 & 0.01 & 0.12 \\
& \(\mu\) & 3.62 & 0.76 & 0.01 & 0.17 \\
& \(\beta^{\text{week}}\) & -0.16 & 0.06 & 0.00 & 0.11 \\
& \(\beta^{\text{trtDrugP}}\) & -0.95 & 0.85 & 0.00 & 0.11 \\
& \(\beta^{\text{trtDrug}}\) & -1.47 & 0.85 & 0.00 & 0.09 \\
\midrule
\(\tau \sim \gammadist(1, 2)\) \\
& \(\tau\) & 0.37 & 0.21 & 0.02 & 0.17 \\
& \(\mu\) & 3.73 & 0.8 & 0.02 & 0.17 \\
& \(\beta^{\text{week}}\) & -0.17 & 0.06 & 0.01 & 0.09 \\
& \(\beta^{\text{trtDrugP}}\) & -0.99 & 0.90 & 0.01 & 0.09 \\
& \(\beta^{\text{trtDrug}}\) & -1.53 & 0.88& 0.01 & 0.09 \\
\midrule
\(\tau \sim \gammadist(9, 0.5)\) \\
&\\
prior-data conflict & \(\tau\) & 13.8 & 5.4 & \textbf{0.10} & 0.13 \\
posterior differs & \(\mu\) & 2.63 & 0.42 & 0.01 & 0.06 \\
posterior differs & \(\beta^{\text{week}}\) & -0.12 & 0.05 & 0.00 & 0.08 \\
posterior differs & \(\beta^{\text{trtDrugP}}\) & -0.66 & 0.46 & 0.01 & 0.07 \\
posterior differs & \(\beta^{\text{trtDrug}}\) & -1.14 & 0.45 & 0.01 & 0.08\\
\bottomrule
\end{tabular}
\end{table}
\FloatBarrier
\subsection{Motorcycle crash case study}
Here, we demonstrate power-scaling sensitivity analysis on a more complex model, without easily interpretable model parameters. We use the \texttt{mcycle} data set, also available in the \texttt{MASS} package and show the sensitivity of predictions to perturbations of the prior and likelihood.
The data set contains 133 measurements of head acceleration at different time points during a simulated motorcycle crash. It is further described by \citet{Silverman1985}. We fit a Gaussian process regression to the data, predicting the head acceleration (\(y\)) from the time (\(x\)). We use two Gaussian processes; one for the mean and one for the standard deviation of the residuals. The model is
\begin{align*}
y &\sim \normal(f(x), \exp(g(x))) \\
f &\sim \GP(0, K_1(x, x^\prime, \rho_f, \sigma_f))\\
g &\sim \GP(0, K_2(x, x^\prime, \rho_g, \sigma_g))\\
\rho_f &\sim \normal^+(0, 1)\\
\rho_g &\sim \normal^+(0, 1)\\
\sigma_f &\sim \normal^+(0, 0.05)\\
\sigma_g &\sim \normal^+(0, 0.5).
\end{align*}
For \(K_1\) and \(K_2\) we use Mat\'ern covariance functions with \(\nu = 3/2\),
\begin{align*}
K_1(x - x^\prime) &= \sigma_f\left(1 + \sqrt{\frac{3(x - x^\prime)^2}{\rho_f^2}} \right)\exp\left(-\sqrt{\frac{3(x - x^\prime)^2}{\rho_f^2}}\right)\\
K_2(x - x^\prime) &= \sigma_g\left(1 + \sqrt{\frac{3(x - x^\prime)^2}{\rho_g^2}} \right)\exp\left(-\sqrt{\frac{3(x - x^\prime)^2}{\rho_g^2}}\right).
\end{align*}
For efficient sampling with Stan, we use Hilbert space approximate Gaussian processes~\citep{solin2020HilbertSpace,ruitortmayol2022PracticalHilbert}.
The number of basis functions (\(m_f = m_g = 40\)) and the proportional extension factor (\(c_f = c_g = 1.5\)) are adapted such that the posterior length-scale estimates \(\hat\rho_f\) and \(\hat\rho_g\) are above the threshold of that which can be accurately approximated~\citep[see][]{ruitortmayol2022PracticalHilbert}.
We can then focus on the choice of priors for the length-scale parameters (\(\rho_f, \rho_g\)) and the marginal residual parameters (\(\sigma_f, \sigma_g\)). It is known that for a Gaussian process, the \(\rho\) and \(\sigma\) parameters are not well informed independently~\citep{diggle2007Geostatistics}, so the sensitivity of the marginals may not be properly representative as there may be prior sensitivity no matter the choice of prior. We first demonstrate the sensitivity of the marginals before proceeding with a focus on the sensitivity of the model predictions, in accordance with \citet{paananen2021RSense}.
\begin{table}[t]
\caption{Prior and likelihood sensitivity in the motorcycle crash case study using the original prior. Prior sensitivity values above the threshold \((\geq 0.05)\) indicate possible informative prior (bold). Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate possible weak or noninformative likelihood (bold). There is clear sensitivity in the posterior parameter marginals, however, as the parameters are difficult to interpret, it is unclear to what extent this poses an issue.}
\label{tab:motorcycle-base-sens}
\centering
\begin{tabular}{lll}
\toprule
Parameter & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity} \\
\midrule
\(\rho_f\) & \textbf{0.52} & 1.62\\
\(\rho_g\) & \textbf{0.18} & 0.06\\
\(\sigma_f\) & \textbf{0.92} & 2.09\\
\(\sigma_g\) & \textbf{0.14} & 0.18\\
\bottomrule
\end{tabular}
\end{table}
As expected, there is prior sensitivity in the marginals (Table~\ref{tab:motorcycle-base-sens}). The prior and likelihood sensitivity for \(\sigma_f\) and \(\sigma_g\) is high, which may be an indication of an issue, however it is difficult to determine based on the parameter marginals alone. Instead we follow up by plotting how the predictions are affected by power-scaling.
\begin{figure}[tb]
\centering
\input{./figs/motorcycle_sense_plot.tex}
\caption{Sensitivity of posterior predictions to prior and likelihood power-scaling in the motorcycle case study. Shown in the plots are the mean, 50\% and 95\% credible intervals for the posterior predictions. There is clear prior and likelihood sensitivity in the predictions around 20~ms after the crash.}%
\label{fig:motorcycle-sense}
\end{figure}
As shown in Figure~\ref{fig:motorcycle-sense}, the predictions around 20~ms exhibit sensitivity to both prior and likelihood power-scaling. The prediction interval widens as the prior is strengthened (\(\alpha > 1\)), and narrows as it is weakened (\(\alpha < 1\)). Likelihood power-scaling has the opposite effect. This indicates potential prior-data conflict from an unintentionally informative prior.
If we widen the prior on \(\sigma_f\) from \(\normal(0, 0.05)\) to \(\normal(0, 0.1)\), this alleviates the conflict such that it is no longer apparent in the predictions~(Figure~\ref{fig:motorcycle-adjust}) and plotting against the raw data indicates a good fit~(Figure~\ref{fig:motorcycle-data}). However, there remains sensitivity in the parameters, although it is lessened (Table~\ref{tab:motorcycle-adjust-sens}). This further demonstrates that depending on the model, prior sensitivity may be present, but is not necessarily an issue. We advise modellers to pay attention to specific quantities and properties of interest, particularly when performing sensitivity analyses on complex models, rather than focusing on parameters without clear interpretations.
\begin{table}[tb]
\centering
\caption{Prior and likelihood sensitivity in the motorcycle crash case study using the adjusted prior. Prior sensitivity values above the threshold \((\geq 0.05)\) indicate possible informative prior (bold). Likelihood sensitivity values below the threshold (\(< 0.05\)) indicate possible weak or noninformative likelihood (bold). Although alleviated, sensitivity remains in the posterior parameter marginals.}
\label{tab:motorcycle-adjust-sens}
\begin{tabular}{lll}
\toprule
Parameter & \makecell[l]{Prior\\ sensitivity} & \makecell[l]{Likelihood\\ sensitivity}\\
\midrule
\(\rho_f\) & \textbf{0.12} & 0.13 \\
\(\rho_g\) & \textbf{0.15} & 0.25 \\
\(\sigma_f\) & \textbf{0.35} & 0.20 \\
\(\sigma_g\) & \textbf{0.26} & 0.09 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[tb]
\centering
\input{./figs/motorcycle_sense_adjust_plot.tex}
\caption{Power-scaling sensitivity for the model with adjusted prior. Shown in the plots are the mean, 50\% and 95\% credible intervals for the posterior predictions. There is now no prior sensitivity and minimal likelihood sensitivity for the predictions.}%
\label{fig:motorcycle-adjust}
\end{figure}
\begin{figure}[tb]
\centering
\input{./figs/motorcycle_adjust_plot.tex}
\caption{Prediction plot for the adjusted model with the data superimposed. Shown in the plot are the mean, 50\% and 95\% credible intervals for the posterior predictions. The predictions capture the raw data well, indicating that we have arrived at a reasonable model.}%
\label{fig:motorcycle-data}
\end{figure}
\section{Conclusion}
\label{sec:discussion}
We have introduced an approach and corresponding workflow for prior and likelihood sensitivity
analysis using power-scaling perturbations of the prior and
likelihood. The proposed approach is computationally efficient and applicable to
a wide range of models with minor changes to existing model code. This will allow automated prior sensitivity diagnostics for packages like \texttt{brms} and \texttt{rstanarm}, and make the use of default priors safer as potential problems can be detected and warnings presented to users. The approach can also be used to identify which priors may need more careful specification. The use
of PSIS and IWMM ensures that the approach is reliable
while being computationally efficient. These
properties were demonstrated in several case studies of real data and our sensitivity analysis
workflow easily fits into a larger Bayesian workflow
involving model checking and model iteration.
We have demonstrated checking the
presence of sensitivity based on the second derivative of the cumulative Jensen-Shannon
distance between the base and perturbed priors with respect to the
power-scaling factor. While this is a useful diagnostic,
power-scaling sensitivity analysis is a general approach with multiple
valid variants. Future work could include further developing quantity-based sensitivity to identify meaningful changes in quantities and predictions
with respect to power-scaling, and working towards automated guidance on safe model adjustment after sensitivity has been detected and diagnosed.
Finally, it is important to recognise that the presence of prior sensitivity or the absence of likelihood sensitivity are not issues in and of themselves. Rather, context and intention of the model builder need to be taken into account. We suggest that the model builder pay particular attention when the pattern of sensitivity is unexpected or surprising, as this may indicate the model is not behaving as anticipated. We again emphasise that the approach should be coupled with thoughtful consideration of the model specification and not be used for repeated tuning of the priors until diagnostic warnings disappear.
\section{Acknowledgements}
We thank Osvaldo Martin, Andrew Manderson and Sona Hunanyan for insightful comments on a previous draft, and Cory McCartan for a helpful discussion on implementation details. We also acknowledge the computational resources provided by the Aalto Science-IT project and support by the Academy of Finland Flagship programme: Finnish Center for Artificial Intelligence, FCAI. This work was partially funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2075 -- 390740016.
| {
"timestamp": "2022-05-06T02:25:01",
"yymm": "2107",
"arxiv_id": "2107.14054",
"language": "en",
"url": "https://arxiv.org/abs/2107.14054",
"abstract": "Determining the sensitivity of the posterior to perturbations of the prior and likelihood is an important part of the Bayesian workflow. We introduce a practical and computationally efficient sensitivity analysis approach using importance sampling to estimate properties of posteriors resulting from power-scaling the prior or likelihood. On this basis, we suggest a diagnostic that can indicate the presence of prior-data conflict or likelihood noninformativity and discuss limitations to this power-scaling approach. The approach can be easily included in Bayesian workflows with minimal effort by the model builder and we present an implementation in our new R package priorsense. We further demonstrate the workflow on case studies of real data using models varying in complexity from simple linear models to Gaussian process models.",
"subjects": "Methodology (stat.ME)",
"title": "Detecting and diagnosing prior and likelihood sensitivity with power-scaling",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075722839015,
"lm_q2_score": 0.7371581741774411,
"lm_q1q2_score": 0.7092254613470913
} |
https://arxiv.org/abs/1908.10684 | Downlink Analysis for the Typical Cell in Poisson Cellular Networks | Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering a stationary point process of users, which is independent of the base station (BS) point process, it is well known that the typical user does not lie in the typical cell and thus it may not truly represent the typical cell performance. Inspired by this observation, we present a construction that allows a direct characterization of the downlink performance of the typical cell. For this, we present an exact downlink analysis for the 1-D case and a remarkably accurate approximation for the 2-D case. Several useful insights about the differences and similarities in the two viewpoints (typical user vs. typical cell) are also provided. |
\section{Introduction}
\label{sec:Introduction}
The previous decade has witnessed a significant growth in research efforts related to the modeling and analysis of cellular networks using stochastic geometry. A vast majority of these works, e.g., \cite{AndBacJ2011,DhiGanJ2012}, rely on the homogeneous PPP model for the BS locations. The user locations are then modeled as a stationary point process that is assumed to be independent of the BS process. Given the stationarity and independence of the user point process, the concept of coverage of the typical user and coverage of an arbitrary fixed location are identical. As a result, one does not need to explicitly consider Palm conditioning on the user point process and the analysis can just focus on the origin as a location of the typical user. However, it is well known that the origin falls in a Poisson-Voronoi (PV) cell that is bigger on an average than the typical cell \cite{BacBla2009}, called the {\em Crofton cell}. Therefore, this approach does not characterize the performance of the typical cell, which is the main focus of this letter.
One way of characterizing the typical cell performance is to consider a user distribution model that places a single user distributed uniformly at random in each cell independently of the other cells. This user process can be interpreted as the locations of the users scheduled in a given resource block. One can also argue that this point process is at least as meaningful as the one discussed above because practical cellular networks are dimensioned to ensure that the load of each cell is almost the same. Similar to~\cite{Haenggi2017}, we refer this user process as {\em Type I user process} and the aforementioned independent user process as {\em Type II user process}. Given that the Crofton cell is statistically larger than the typical cell, it is easy to establish that both the desired signal power and the interference power observed at the typical user of the Type I process will (stochastically) dominate the corresponding quantities observed by the typical user of the Type II process.
While the downlink analysis of the Type II user process is well understood, this letter deals with the downlink coverage analysis for the Type I user process.
\indent{\em Related Works:} The downlink analysis of cellular networks with the Type II user process involves using the {\em contact distribution} of the PPP to characterize the link distance, and using {\em Slivnyak's theorem} to argue that conditioned on the link distance, the point process of interferers remains a PPP~\cite{AndBacJ2011}. While the idea of using the Type I user process is relatively recent, there are two noteworthy works in this direction. First and foremost is~\cite{Haenggi2017}, which defined this user process and used it for the uplink analysis. This idea was extended to the downlink case in \cite{Martin2017_Meta}, where the meta distribution of signal-to-interference ratio ($\mathtt{SIR}$) is derived using an empirically obtained link distance distribution (for the Type I process) and approximating the point process of interferers as a homogeneous PPP beyond the link distance from the location of the typical user. In \cite{PraPriHar}, we derived the exact integral expression and a closed-form approximation for the serving link distance distribution for the Type I user process. Building on the insights obtained from \cite{Martin2017_Meta} and \cite{PraPriHar}, we provide an accurate downlink analysis for the Type I user process in this letter.
\indent{\em Contributions:} The most important contribution of this letter is to demonstrate that the well-accepted way of defining the typical user by considering an independent and stationary point process of users is not the only way of analyzing cellular networks modeled as point processes. More importantly, this construction does not result in the typical cell performance. In order to highlight the finer differences between the two viewpoints, we first present the exact analysis of the Type I process for the 1-D case. Leveraging the qualitative insights obtained from the 1-D case, we perform an approximate yet accurate analysis for the Type I user process in a 2-D cellular network.
In particular, for the Type I process, we empirically show that the point process of interfering BSs given a distance $R_o$ between user and serving BS exhibits a clustering effect at distances slightly larger than $R_o$ that is not captured by a homogeneous PPP approximation (beyond $R_o$) as used in \cite{Martin2017_Meta}. Using this insight, we propose a dominant-interferer based approach in order to accurately approximate the point process of interferers. This approximation allows us to accurately evaluate the interference received by the user conditioned on the link distance, which subsequently provides a remarkably tight approximation for the 2-D case.
\section{System Model and Preliminaries}
\label{sec:System_Model}
We assume that the locations of BSs form a homogeneous PPP
$\Psi\equiv\{\textbf{x}_1,\textbf{x}_2,\dots\}$ of density $\lambda$ on $\mathbb{R}^d$ for $d\in\{1,2\}$.
The PV cell with the nucleus at $\mathbf{x}\in\Psi$ can be defined as
\begin{equation}
V_\mathbf{x}=\{\mathbf{y}\in\mathbb{R}^d: \|\mathbf{y}-\mathbf{x}\|\leq\|\mathbf{x'}-\mathbf{y}\|, ~\forall \mathbf{x'}\in\Psi\}.
\label{eq:PV_Cell_Definition}
\end{equation}
Since by Slivnyak's theorem \cite{Haenggi2013}, conditioning on a point is the same as adding a point to a PPP, we focus on the typical cell of the point process $\Psi\cup\{o\}$ at $o$, which is given by
\begin{equation}
V_o=\{\mathbf{y}\in\mathbb{R}^d: \|\mathbf{y}\|\leq\|\mathbf{x}-\mathbf{y}\|, ~\forall \mathbf{x}\in\Psi\}.
\end{equation}
Henceforth, we consider $\Phi=\Psi\cup\{o\}$. Further, let $\tilde{V}_o$ be the cell of the PV tessellation of $\Psi$ containing the origin, called the Crofton cell.
Without loss of generality, the typical user from the Type II user process can be assumed to be located at the origin (see \cite{AndBacJ2011}) which means it resides in the Crofton cell $\tilde{V}_o$.
Now, we define Type I user point process as
\begin{equation}
\Omega\triangleq\{U(V_\mathbf{x}):\mathbf{x}\in\Phi\},
\label{eq:UserProcessTypeI}
\end{equation}
where $U(A)$ is the point chosen uniformly at random from the set $A$ independently for different $A$. Note that the typical user from the Type I user point process $\Omega$ represents a uniformly random point in the typical cell.
By the above construction, the location of the typical user in the typical cell becomes $\mathbf{y}\sim U(V_o)$ and $\Psi$ becomes the point process of interfering BSs to the typical user at $\mathbf{y}\in V_o$.
Let $R_o$ denote the {\em link distance}, i.e., the distance from the BS of $V_o$ (i.e., the origin) to the user at $\mathbf{y}$.
We consider the standard power law path loss model with exponent $\alpha>2$ for signal propagation. Further, assuming independent Rayleigh fading, we model the small-scale fading gains $h_{\mathbf{x}}$ associated with the typical user and the BS at $\mathbf{x}\in\Phi$ as exponentially distributed random variables with unit
mean. We assume $\{h_{\mathbf{x}}\}$ are independent for all $\mathbf{x}\in\Phi$.
Thus, $\mathtt{SIR}$ at the typical user located at $\mathbf{y}\in V_o$ in an interference-limited system is
\begin{equation}
\mathtt{SIR}=\frac{h_{o}R_o^{-\alpha}}{\sum\limits_{\mathbf{x}\in\Psi}h_{\mathbf{x}}\|\mathbf{x}-\mathbf{y}\|^{-\alpha}}.
\end{equation}\vspace{-.3cm}
\begin{definition}
The coverage probability is the probability that the $\mathtt{SIR}$ at the typical user is greater than a threshold $\tau$.
\end{definition}\vspace{-.2cm}
In the rest of this section, we briefly discuss the coverage probability of the Type II process
By definition, the link distance of the typical user of the Type II user process is $\hat{R}_o=\|\mathbf{x}\|$ where $\mathbf{x}\in\Psi$ is the closest point to the origin. The cumulative distribution function ($\mathtt{CDF}$) of $\hat{R}_o$ (i.e., the {\em contact distribution}) is $1-\exp(-\lambda\kappa_dr^d)$ \cite{Haenggi2013}, where $\kappa_d=1$ and $\kappa_d=\pi$ for $d=1$ and $d=2$, respectively.
The coverage probability of the Type II process in the $d$-dimensional Poisson cellular network is given by \cite{AndBacJ2011}
\begin{align}
\mathtt{P_{II}^d}(\tau)\triangleq{\mathbb{P}}[\mathtt{SIR}>\tau]= &\left[{1+\tau^{\frac{d}{\alpha}}\int_{\tau^{-\frac{d}{\alpha}}}^{\infty} \frac{1 }{1 + u^{\frac{\alpha}{d}}} {\rm d}u}\right]^{-1}. \numberthis
\label{eq:CovP_Crf_d12}
\end{align}
Note that \cite{AndBacJ2011} is focused on the case of $d=2$ but the extension to the general $d$-dimensional case is straightforward. $\mathtt{P_{II}^d}(\tau)$ can also be interpreted as the fraction of the covered area.
\section{Coverage Analysis of Type I User Process}
\label{sec:CovP_TypCell}
In this section, we present the exact and an approximate (yet accurate) coverage analysis of the Type I user process for $d=1$ and $d=2$.
\vspace{-.45cm}
\subsection{Exact Coverage Analysis for $d=1$}
\label{sec:CovP_Analysis_1d}
We begin our discussion with the distribution of the serving link distance conditioned on the distances from the typical BS at the origin to the neighboring BSs (one from each side). Let $R_1$ and $R_2$ be the distances from the typical BS to these two neighboring BSs. Since $\Phi$ is a Poisson process on $\mathbb{R}$, $R_1$ and $R_2$ are i.i.d. exponential with mean $\lambda^{-1}$. The joint distribution of $R_1$ and $R_2$ conditioned on $R_1<R_2$ is
\begin{align}\
\hspace{-.1cm}f_{R_1,R_2}(r_1, r_2) = 2 \lambda^2 \exp(-\lambda (r_1+r_2)), ~r_2\geq r_1 \geq 0. \numberthis
\label{eq:DIST_R1R2}
\end{align}
The serving link distance distribution for the user at $y\sim U(V_o)$, where $|y|=R_o$, conditioned on $R_1$ and $R_2$ becomes
\begin{equation}
F_{R_o}(R_o\leq r\mid R_1, R_2) = \begin{cases} \frac{4r}{R_1 + R_2}, &\text{if}~ \frac{R_1}{2}\geq r \geq 0, \\
\frac{2r + R_1}{R_1 + R_2}, &\text{if}~ \frac{R_2}{2} \geq r > \frac{R_1}{2}, \\
1, &\text{if}~ r>\frac{R_2}{2}. \\
\end{cases}
\label{eq:Cond_CDF_1D}
\end{equation}
Now, we present the exact coverage probability of the Type I process in the following theorem.
\vspace{-.1cm}
\begin{thm}
The coverage probability of the Type I process in a 1-D Poisson cellular network is
\begin{align}
\hspace{-.3cm}\mathtt{P_{I}^1}(\tau)=\int_{0}^{\infty} \hspace{-.25cm}\int_{0}^{r_2}\hspace{-.15cm}\dP{\mathtt{SIR}>\tau\mid r_1, r_2}f_{R_1,R_2}(r_1, r_2) {\rm d}r_1 {\rm d}r_2,
\label{eq:CovP_1d}
\end{align}
where $f_{R_1,R_2}(r_1,r_2)$ is given by \eqref{eq:DIST_R1R2},
\begin{align}
&\dP{\mathtt{SIR} > \tau\mid r_1, r_2} = \int_{0}^{\frac{r_1}{2}} {\cal L}_{\mathcal{I}}(\tau r^{\alpha}\mid r_1-r, r_2+r)\frac{2}{r_1 + r_2} {\rm d}r\nonumber\\
&~~~~~~~~~~~+ \int_{0}^{\frac{r_2}{2}} {\cal L}_{\mathcal{I}}(\tau r^{\alpha}\mid r_1+r, r_2-r)\frac{2}{r_1 + r_2} {\rm d}r,
\label{eq:CovP_Cond_1d}
\end{align}
\begin{align}
\hspace{-.25cm}~\text{and}~{\cal L}_{\mathcal{I}}(s\mid u,v)=\frac{\exp\left(-\lambda \int_{u}^{\infty} \frac{s {\rm d}r}{r^{\alpha} + s} - \lambda \int_{v}^{\infty} \frac{s {\rm d}r}{r^{\alpha} + s} \right)}{(1 + s u^{-\alpha})(1 + s v^{-\alpha})}.
\label{eq:LT_Int_1D}
\end{align}
\end{thm}
\begin{IEEEproof}
Let ${x}_{l}$ and ${x}_{r}$ be the neighboring interfering BSs to the typical user at $y\in V_o$ in $\Psi_1$ and $\Psi_2$, respectively, where $\Psi_1=\Psi\cap\mathbb{R}^-$ and $\Psi_2=\Psi\cap\mathbb{R}^+$. Let $\tilde{R}_1=|{x}_{l}-y|$ and $\tilde{R}_2=|{x}_{r}-y|$.
Thus, the aggregate interference can be written as $\mathcal{I}=\mathcal{I}_1+\mathcal{I}_2$ where $\mathcal{I}_1=h_{x_{l}}\tilde{R}_1^{-\alpha}+\sum_{x\in\Psi_1\setminus\{x_{l}\}} h_{x}|x-y|^{-\alpha}$ and $\mathcal{I}_2=h_{x_{r}}\tilde{R}_2^{-\alpha}+\sum_{x\in\Psi_2\setminus\{x_{r}\}} h_{x}|x-y|^{-\alpha}$. Now, the Laplace transform (LT) of $\mathcal{I}_1$ conditioned on $\tilde{R}_1$ is
\begin{align}
\mathcal{L}_{\mathcal{I}_1}(s\mid \tilde{R}_1)&=\mathbb{E}\left[e^{-sh_{x_{l}}\tilde{R}_1^{-\alpha}}\prod_{x\in\Psi_1\setminus\{x_{l}\}}e^{-sh_{x} |x-y|^{-\alpha}}\mid \tilde{R}_1\right]\nonumber\\
&\stackrel{(a)}{=}\mathbb{E}_{h}[e^{sh \tilde{R}_1^{-\alpha}}]\mathbb{E}\left[\prod_{x\in\Psi_1\setminus\{{x}_{l}\}}\mathbb{E}_{h}[e^{-sh|x-y|^{-\alpha}}]\mid\tilde{R}_1\right]\nonumber\\
&\stackrel{(b)}{=}\frac{1}{1+s \tilde{R}_1^{-\alpha}}\exp\left(-\lambda\int_{\tilde{R}_1}^\infty\frac{s{\rm d}r}{r^\alpha+s}\right),\nonumber
\end{align}
where (a) follows from the independence of the fading gains and (b) follows from the LT of an exponential r.v. and the probability generating functional ($\mathtt{PGFL}$) of the PPP \cite{Haenggi2013}. Similarly, we obtained LT of $\mathcal{I}_2$ condition on $\tilde{R}_2$. Thus, the LT of aggregate interference conditioned on $\tilde{R}_1=u$ and $\tilde{R}_2=v$ is given by \eqref{eq:LT_Int_1D}.
Now, conditioned on $R_1$ and $R_2$, the coverage probability becomes
$$\dP{\mathtt{SIR} > \tau\mid R_1, R_2}=\mathbb{E}_{R_o}\left[\mathcal{L}_{\mathcal{I}}(\tau r^\alpha\mid \tilde{R}_1,\tilde{R}_2)\mid R_1,R_2\right].$$
Finally, by deconditioning the above equation over the joint distribution of $R_1$ and $R_2$ given in \eqref{eq:DIST_R1R2}, we obtain \eqref{eq:CovP_1d}.
\end{IEEEproof}
From the above analysis, it is evident that the exact analysis of the Type I process requires conditioning on the locations of {\em all the neighboring} BSs around $V_o$. While this was manageable in 1-D, it becomes significantly more complicated for $d>1$, which prevents an exact analysis. In the next subsection, we present a new approximation that leads to a tight characterization of the Type I user performance for $d=2$.
\vspace{-.2cm}
\subsection{Approximate Coverage Analysis for $d=2$}
\label{sec:CovP_Analysis_2d}
The coverage analysis requires the joint distribution of the distances $\|\mathbf{x}-\mathbf{y}\|$, $\mathbf{x}\in\Psi$, and the link distance $R_o=\|\mathbf{y}\|$ of the typical user at $\mathbf{y}\sim U(V_o)$. Thus, we first discuss the distribution of $R_o$ and then approximate the point process of interferers $\Psi$ conditioned on $R_o$. Finally, using these distributions, we present the approximate coverage analysis.
\subsubsection{Approximation of the link distance distribution}
\label{subsec:CDF_Ro}
In \cite{PraPriHar}, we derived an exact expression for the distribution of $R_o$ which involves multiple integrals. Therein, we also derived a closed-form expression to approximate the $\mathtt{CDF}$ of $R_o$ which is
\begin{equation}
F_{R_o}(r)\approx 1-\exp\left(-\pi\rho_o\lambda r^2\right),~\text{for}~r\geq 0,
\label{eq:CDF_Ro}
\end{equation}
where $\rho_o=\frac{9}{7}$ is the correction factor (CF), which corresponds to the ratio of the mean volumes of $\tilde{V}_o$ and $V_o$.
\begin{figure*}[h]
\centering\vspace{-.2cm}
\includegraphics[width=.32\textwidth]{pcfCondRo_3by10_36by10_032919.eps}
\includegraphics[width=.32\textwidth]{Distance_UniformPointInTypicalCell_to_SecondNearestSeedPoint_New_New.eps}
\includegraphics[width=.32\textwidth]{CDF_RecPow_Int.eps}
\label{fig:CovP_TypCell}\vspace{-.25cm}
\caption{{\em Left:} The $\mathtt{pcf}$ of the point process of interferers observed by a user, from the Type I process, having link distance $R_o\in\{0.3,0.6\}$. {\em Middle:} Distributions of distances to the serving and dominant interfering BSs from the typical user of the Type I for $\lambda=1$. {\em Right:} CDF of desired signal and interference powers received by the typical users of the Type I and II processes for $\lambda=10^{-5}$ and $P=30$ dBm.}
\label{fig:CDF_Ro_R1_pcf}\vspace{-.3cm}
\end{figure*}
\subsubsection{Approximation of the point process of interferers $\Psi$}
\label{subsec:PP_PihI}
To understand the statistics of the point process of interferers observed by the typical user at $\mathbf{y}\in V_o$, we analyze the pair correlation function ($\mathtt{pcf}$) of $\Psi=\Phi\setminus\{o\}$ with reference to $\mathbf{y}\in V_o$ which is \cite{Haenggi2013}
\begin{equation}
g(r\mid R_o) = \frac{1}{2\pi r}\frac{{\rm d} K(r\mid R_o)}{{\rm d}r},~\text{for}~r>R_o,\nonumber
\end{equation}
where $R_o=\|\mathbf{y}\|$, $K(r\mid R_o)=\mathbb{E}[\Psi(\mathcal{B}_\mathbf{y}(r))\mid R_o]$ is Ripley's $K$ function given $R_o$ and $\mathcal{B}_\mathbf{y}(r)$ is the disk of radius $r$ centered at $\mathbf{y}$. Fig. \ref{fig:CDF_Ro_R1_pcf} (Left) shows the simulated user-interfering BS $\mathtt{pcf}$ conditioned on $R_o$. From the figure, it is easy to interpret that the point process of interferers exhibits a {\em clustering effect} at distances slightly larger than $R_o$ and complete spatial randomness for $r\gg R_o$. The exact characterization of such point process is complex because of the correlation in the points (in $\Psi$) that form the boundaries of $V_o$ (as seen by the typical user at $\mathbf{y}\in V_o$). Therefore, in order to accurately evaluate the interference received by the typical user, we need to carefully approximate the point process of interferers as seen by the typical user.
A natural candidate for the approximation is homogeneous PPP of density $\lambda$ outside of $\mathcal{B}_\mathbf{y}(R_o)$ \cite{Martin2017_Meta}. Henceforth, we refer to this approximation as $\mathtt{App1}$. $\mathtt{App1}$ ignores the clustering effect (see Fig. \ref{fig:CDF_Ro_R1_pcf} (Left)) and thus underestimates the interference. Therefore, in order to capture the effect of clustering to some degree, we explicitly consider the interference from the {\em dominant interferer} at distance $R_1=\arg\min_{\mathbf{x}\in\Psi}\|\mathbf{x}-\mathbf{y}\|$ and approximate the point process of interferers with homogeneous PPP of density $\lambda$ outside $\mathcal{B}_{\mathbf{y}}(R_1)$. We call this approximation $\mathtt{App2}$. Now, the crucial part is to obtain the distribution of $R_1$.
Given the complexity of the analysis of r.v. $R_o$ \cite{PraPriHar}, it is reasonable to deduce that the exact characterization of the distribution of $R_1$ is equally, if not more, challenging. Thus, we obtain an approximate distribution of $R_1$ as follows.
The $\mathtt{CDF}$ of $R_o$, given in \eqref{eq:CDF_Ro}, is the same as the contact distribution of PPP with density $\rho_o\lambda$. Therefore, using this and the argument of clustering discussed above, the $\mathtt{CDF}$ of $R_i$ (distance to $i$-th closed point in $\Psi$ from the user at $\mathbf{y}\in V_o$) can be approximated by inserting an appropriate CF $\rho_i$ to the $\mathtt{CDF}$ of $(i+1)$-th closest point to the origin in the PPP. From $g(r\mid R_o)\downarrow 1$ as $r\to\infty$, we have $\rho_i\to 1$ as $i\to \infty$. While $\rho_1=1.31$ gives the best fit for the empirical $\mathtt{CDF}$ of $R_1$, we approximate $\rho_1$ by $\rho_o$ for simplicity. Now, the $\mathtt{CDF}$ of $R_1$ conditioned on $R_o$ can be approximated as \cite{Moltchanov2012}
\begin{equation}
F_{R_1}(v\mid R_o)=1-\exp\left(-\pi\lambda\rho_o(v^2-R_o^2)\right)~\text{for}~v\geq R_o,
\label{eq:CDF_R1Ro}
\end{equation}
and thus the approximated marginal $\mathtt{CDF}$ of $R_1$ becomes
\begin{align}
\hspace{-.2cm}F_{R_1}(v)=1-(\pi\lambda\rho_o v^2 + 1)\exp\left(-\pi\lambda\rho_o v^2 \right) ~\text{for}~ v\geq 0.
\label{eq:CDF_R1}
\end{align}
Fig. \ref{fig:CDF_Ro_R1_pcf} (Middle) shows the accuracy of the approximated CDFs of $R_o$ and $R_1$ given in \eqref{eq:CDF_Ro} and \eqref{eq:CDF_R1}.
Fig. \ref{fig:CDF_Ro_R1_pcf}(left) shows that App2 provides a slightly pessimistic estimate for the $\mathtt{pcf}$ because of which it will slightly underestimate the interference power.
\subsubsection{Coverage Probability}
\label{subsec:CoVP}
Now, we derive the coverage probability of the Type I process using the distribution of link distance $R_o$, given in \eqref{eq:CDF_Ro}, and the approximated point process of interferers $\mathtt{App2}$, discussed in Subsection \ref{subsec:PP_PihI}, in the following theorem.
\begin{thm}
The coverage probability of the Type I process in a 2-D Poisson cellular network can be approximated as
\begin{align}
\mathtt{P_{I}^2}(\tau)=\rho_o^2\tau^{-\delta}\int_{0}^{\tau^\delta}\frac{(\tilde\beta(t)+\rho_o)^{-2}}{1+t^{\frac{1}{\delta}}}{\rm d}t,
\label{eq:CovP}
\end{align}
where $\tilde\beta(t)=t\int_{t^{-1}}^{\infty}\frac{1}{1+u^{\frac{1}{\delta}}}{\rm d}u$ and $\delta=\frac{2}{\alpha}$.
\end{thm}
\begin{IEEEproof}
Let $\tilde{\mathbf{x}}$ be the dominant interfering BS such that $\|\tilde{\mathbf{x}}-\mathbf{y}\|=R_1$. We write the interference received by the user with link distance $R_o$ as $\mathcal{I}(R_o)=h_{\tilde{\mathbf{x}}}R_1^{-\alpha} + \mathcal{I}(\tilde{\Psi})$ where $\tilde{\Psi}=\Psi\setminus\{\tilde{\mathbf{x}}\}$ and $\mathcal{I}(\tilde{\Psi})=\sum_{\mathbf{x}\in\tilde{\Psi}}h_{\mathbf{x}}\|\mathbf{x}-\mathbf{y}\|^{-\alpha}$.
Thus, the coverage probability conditioned on $R_o$ and $R_1$ becomes
\begin{align}
\mathtt{P_{I}^2}&(\tau\mid R_o,R_1) = \mathbb{P}\left[h_o>\tau R_o^\alpha\mathcal{I}(R_o)\right]\nonumber\\
&=\mathcal{L}_{h_{\tilde{\mathbf{x}}}}\left(\tau(R_o/R_1)^\alpha\right)\mathcal{L}_{\mathcal{I}(\tilde{\Psi})}\left(\tau R_o^\alpha\mid R_1\right).
\label{eq:CovP_Cond}
\end{align}
Now, the LT of $\mathcal{I}(\tilde{\Psi})$ at $\tau R_o^{\alpha}$ for given $R_1$ can be obtained as
\begin{align}
&\mathcal{L}_{\mathcal{I}(\tilde{\Psi})}(\tau R_o^\alpha\mid R_1)\stackrel{(a)}{=}\mathbb{E}_{\tilde{\Psi}}\prod_{\mathbf{x}\in\tilde{\Psi}}\frac{1}{1+\tau R_o^{\alpha}\|\mathbf{x}-\mathbf{y}\|^{-\alpha}}\nonumber\\
&\stackrel{(b)}{=}\exp\left(-\lambda\int_{\mathbb{R}^2\setminus\mathcal{B}_\mathbf{y}(R_1)}\frac{1}{1+\tau^{-1} R_o^{-\alpha}\|\mathbf{x}-\mathbf{y}\|^{\alpha}}{\rm d}\mathbf{x}\right)\nonumber\\
&\stackrel{(c)}{=}\exp\left(-2\pi\lambda R_o^2\tau^{\delta}\int_{\tau^{-\delta}(R_1/R_o)^2}^{\infty}\frac{1}{1+u^{\frac{1}{\delta}}}{\rm d}u\right),
\label{eq:LT}
\end{align}
where (a) follows due to the independence of the power fading gains and the LT of an exponential r.v., (b) follows using the $\mathtt{App2}$ and the $\mathtt{PGFL}$ of the PPP \cite{Haenggi2013}, and (c) follows through the substitution of $\mathbf{x}-\mathbf{y}=\mathbf{z}$ and then using Cartesian-to-polar coordinate conversion.
Now, substituting \eqref{eq:LT} along with $\mathcal{L}_{h_{\tilde{\mathbf{x}}}}\left(\tau(R_o/R_1)^\alpha\right)=\frac{1}{1+\tau(R_o/R_1)^\alpha}$ in \eqref{eq:CovP_Cond} and futher taking expectation over $R_1$ and $R_o$ yields the coverage probability as $\mathtt{P_{I}^2}(\tau)=$
\begin{align}
\left(2\pi\lambda\rho_o\right)^2\int_{0}^{\infty}\int_{r}^{\infty}\frac{\exp\left(-\pi\lambda r^2\beta(\tau,r,v)-\pi\lambda\rho_o v^2\right)}{1+\tau(r/v)^\alpha}v{\rm d}v r{\rm d}r,\nonumber
\end{align}
where $\beta(\tau,r,v)=\tau^{\delta}\int_{\tau^{-\delta}(\frac{v}{r})^2}^{\infty}\frac{1}{1+u^{1/{\delta}}}{\rm d}u$. Now, by interchanging the order of the integrals and further simplification, we obtain \eqref{eq:CovP}. This completes the proof.
\end{IEEEproof}\vspace{-.15cm}
\section{Numerical Results and Discussion}
Fig. \ref{fig:CDF_Ro_R1_pcf} (Right) shows that the desired signal power and interference power received by the typical users of the Type I and the Type II processes are significantly different (by 2-3 dB). Given the fundamental differences in the constructions of these two processes, this observation is not surprising. Besides, in Fig. \ref{fig:CovP_TypCell} we note that the coverage probabilities for the two processes are fairly similar, especially for higher values of $\alpha$. This is mainly because the desired signal power and the interference from a few dominant interfering BSs scale up by almost the same factors in the two cases (note that $\rho_1 \approx \rho_o$) and a few neighboring interfering BSs dominate the aggregate interference for higher values of $\alpha$. A key point to note here is that the fact that the coverage probabilities are similar in the two models does not imply that the other performance measures will also be close. Finally, note that $\mathtt{App2}$ results in a slightly higher coverage probability since it slightly underestimates the pcf of the point process of interferers (refer Fig. \ref{fig:CDF_Ro_R1_pcf}(left)).
\vspace{-.35cm}
\section{Conclusion}
In this letter, we have revisited the downlink analysis of cellular networks by arguing that the typical user analysis in the popular approach of considering a stationary and independent user point process results in the analysis of a Crofton cell, which is bigger on average than the typical cell. In order to characterize the performance of the typical cell, we consider a recent construction in which each cell is assumed to contain a single user distributed uniformly at random independently of the other cells. After highlighting the key analytical challenges in characterizing the typical cell performance in this case, we provide a remarkably accurate approximation that facilitates the general analysis of the typical cell in Poisson cellular networks. Even though the downlink coverage for the two cases is similar, we show that the other metrics, such as the received desired power and interference power, may exhibit significant differences, thus necessitating the need for a careful analysis of the typical cell. Although this letter was focused on the downlink coverage, the underlying characterization of the interference field can be used for the analysis of other key performance metrics as well.
\begin{figure}[h]
\centering\vspace{-.35cm}
\includegraphics[width=.45\textwidth]{CoverageProbability_d12_New_New_New_New.eps}\vspace{-.35cm}
\caption{Coverage probability of the Type I and the Type II processes.}
\label{fig:CovP_TypCell}
\end{figure}
\vspace{-.4cm}
| {
"timestamp": "2019-08-29T02:12:15",
"yymm": "1908",
"arxiv_id": "1908.10684",
"language": "en",
"url": "https://arxiv.org/abs/1908.10684",
"abstract": "Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering a stationary point process of users, which is independent of the base station (BS) point process, it is well known that the typical user does not lie in the typical cell and thus it may not truly represent the typical cell performance. Inspired by this observation, we present a construction that allows a direct characterization of the downlink performance of the typical cell. For this, we present an exact downlink analysis for the 1-D case and a remarkably accurate approximation for the 2-D case. Several useful insights about the differences and similarities in the two viewpoints (typical user vs. typical cell) are also provided.",
"subjects": "Information Theory (cs.IT)",
"title": "Downlink Analysis for the Typical Cell in Poisson Cellular Networks",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075733703925,
"lm_q2_score": 0.7371581684030623,
"lm_q1q2_score": 0.7092254565924334
} |
https://arxiv.org/abs/1912.11063 | Does scrambling equal chaos? | Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in a classically integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos. | \section{Saddle-dominated scrambling in Dicke model}\label{sec:dicke}
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{dicke.pdf}
\caption{(a) The OTOC in the quantum Dicke model ($N = 40$) with $\hat{O} = \hat{p}$, for a microcanonical ensemble of 40 eigenstates around $H = -1$, and for two representative values of $\gamma$. The dashed slopes show $\lambda_\text{sad}=\omega_1 = \sqrt{\gamma - 1} $. (b) Classical Lyapunov exponent in the Dicke model with $\gamma=2$ (computed as $(\partial q(t) / \partial q(0))^2 \sim e^{2 \lambda_{\text{chaos}} t}$, with $t = 2000$) for $600$ randomly sampled trajectories in the energy shell $[-1.3, 0.7]$. For all of them, $\lambda_{\text{chaos}}$ is smaller than $\lambda_{\text{saddle}}=\omega_1 = 1$, marked by a star. }
\label{fig:dicke}
\end{figure}
In this appendix, we show that saddle-dominated scrambling also takes place in the \textit{Dicke model}~\cite{dicke0,dickereview}. Originally, it was conceived to describe $N$ two-level atoms coupled to a single optical cavity mode. Several recent works~\cite{dicke,pilatowsky2019positive,rey} considered the OTOCs in this model. In the classical limit, its Hamiltonian \begin{equation}
H = \frac12 (q^2 + p^2) + x + \gamma z q
\end{equation}
describes an $SU(2)$ (pseudo)-spin ($x, y, z$) and a harmonic oscillator ($p, q$), interacting with a coupling constant $\gamma > 0$. A quantum phase transition occurs at $\gamma = 1$, and the super-radiant $\gamma > 1$ phase is characterized by a degenerate pair of ground states, and a saddle at $(q=p=0, x=1)$ with $H=-1$, associated with an excited state transition~\cite{wang19}. The saddle has a single unstable exponent $\omega_1 = \sqrt{\gamma - 1}$, which turns out to be larger than the Lyapunov exponent of a typical classical trajectory with comparable energy, see Fig.~\ref{fig:dicke}(b) and also Refs.~\cite{dicke,pilatowsky2019positive}. This suggests that $\lambda_{\text{OTOC}} \ge \omega_1$ is a tight bound, if the OTOC is computed in an ensemble that overlaps with the saddle point. Our numerics, following the method of~\cite{dicke,pilatowsky2019positive}, confirms this prediction, see Fig.~\ref{fig:dicke}(a).
\end{document}
| {
"timestamp": "2020-04-09T02:03:31",
"yymm": "1912",
"arxiv_id": "1912.11063",
"language": "en",
"url": "https://arxiv.org/abs/1912.11063",
"abstract": "Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in a classically integrable model. We derive a lower bound on the OTOC Lyapunov exponent which depends only on local properties of such fixed points. We present several models for which this bound is tight, i.e. for which scrambling is dominated by the local dynamics around the fixed points. We propose that the notion of scrambling be distinguished from that of chaos.",
"subjects": "Statistical Mechanics (cond-mat.stat-mech); Disordered Systems and Neural Networks (cond-mat.dis-nn); High Energy Physics - Theory (hep-th); Chaotic Dynamics (nlin.CD)",
"title": "Does scrambling equal chaos?",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075701109193,
"lm_q2_score": 0.7371581684030623,
"lm_q1q2_score": 0.7092254541896861
} |
https://arxiv.org/abs/2004.05177 | Dissipation-assisted operator evolution method for capturing hydrodynamic transport | We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision. |
\section{Additional data for the Ising chain and XX ladder models}
\subsection{Convergence with bond dimension}
In the main text, we showed that the dissipation leads to a decay of the operator entanglement at long times. Crucially, this makes the maximal operator entanglement encountered during the evolution independent of system size, depending only on the parameters of the dissipation. As we argued, we can therefore capture the diffusive spreading of correlations up to arbitrarily long times, without significant finite-size or truncation effects. Here we show explicitly how the curves for $D(t)$ converge as we increase the bond dimension $\chi$.
The results are shown in Fig.~\ref{fig:ChiConv} for the tilted field Ising model. We fix parameters $\ell_* = 4$, $\Delta t =1$, $\gamma = 0.2$ (same as in Fig.~\ref{fig:Test}(b)) and compare results for different bond dimensions $\chi=32,64,128,256,512$. As the operator entanglement peaks and decreases (see Fig.~\ref{fig:Test}(a)), the truncation error of the unitary TEBD time step also starts decreasing. While for small $\chi$, the truncation errors encountered around the peak time are already significant, they decrease (roughly linearly) with $\chi$. This also shows up in the results for the time-dependent diffusion constant, $D(t)$. While at small $\chi$ the truncation effects are clearly visible, the curves quickly converge as $\chi$ is increased.
Another way of testing the effects of truncation is by looking at whether the conservation law (in this case, of energy) is satisfied. We consider the correlations $C_j(t)$ and normalize them such that $\sum_j C_j(0)=1$. The exact dissipative dynamics would maintain this normalization at all subsequent times due to energy conservation (assuming $\ell_*$ is larger than the support of the terms in the Hamiltonian, in this case $\ell_* \geq 2)$. This is crucial for correctly capturing transport properties. We find that the errors in the conservation law, as measured by $\left| 1 - \sum_j C_j(t) \right|$ quickly decrease as $\chi$ becomes larger. We conclude that it is possible to simulate the dissipative dynamics~\eqref{eq:DissipativeEvol} up to long times, with a bond dimension that is independent of total system size.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{SM1.pdf}
\caption{Convergence of results with bond dimension $\chi$ in the Ising chain~\eqref{eq:IsingDef} for dissipation parameters $\ell_* = 4$, $\Delta t =1$, $\gamma = 0.2$. (a): Truncation error per TEBD step, summed over all bonds in the chain ($L=51$ sites). (b) Convergence of results for $D(t)$ (see main text for definition). (c) Errors in the energy conservation, as measured by the sum of the coefficients of local energy density terms $C_j(t)$. }
\label{fig:ChiConv}
\end{figure}
\subsection{Scaling collapse as a function of $\gamma / \Delta t$}
Here, we justify our claim in the main text that when $\gamma$ is sufficiently small, the results (in particular, estimates of $D$) are functions of the ratio $\gamma / \Delta t$ only. This can be seen by utilizing the Baker-Campbell-Hausdorff formula to rewrite the evolution operator~\eqref{eq:DissipativeEvol} as
\begin{align}\label{eq:BCH}
\left( \op{D}_{\ell_*,\gamma} e^{i\mathcal{L} \Delta t}\right)^N \equiv \left( e^{-\op{K}_{\ell_*}\gamma} e^{i\mathcal{L} \Delta t}\right)^N = \left( e^{-\op{K}_{\ell_*}\gamma + i\mathcal{L} \Delta t + O(\gamma \Delta t)} \right)^N = e^{-\op{K}_{\ell_*} N \gamma + i\mathcal{L} N \Delta t + O(\gamma N \Delta t)} = e^{t \left( i \op{L} - \op{K}_{\ell_*} \frac{\gamma}{\Delta t} \right) + O(\gamma t)},
\end{align}
where $t = N\Delta t$ and we have introduced the logarithm of the dissipator, acting on a Pauli string as
\begin{align}
\op{K}_{\ell_*}[\op{S}] =
\begin{cases}
0 &\text{ if } \ell_\op{S} \leq \ell_* \\
(\ell_\op{S} - \ell_*) \op{S} &\text{ otherwise}.
\end{cases}
\end{align}
In the second equality of Eq.~\eqref{eq:BCH} we assumed $\gamma \ll 1$ to drop higher order terms that scale as $\gamma^2 \Delta t$. We also assume that $\Delta t$ is at most an $O(1)$ quantity, so that terms that scale as $\gamma \Delta t^2$ are of the same order as $\gamma \Delta t$.
Eq.~\eqref{eq:BCH} shows that the dynamics only depends on the ratio $\gamma/\Delta t$, and not on the individual value of $\gamma$ and $\Delta t$, up to times $t \approx 1/\gamma$. As such, it does not directly constrain the diffusion constant, which is extracted from the long-time limit. However, in practice we find that $D(t)$ saturates to a constant at a finite time $t_\text{sat}$. While $t_\text{sat}$ itself depends on $\gamma$ and $\Delta t$ (as well as on the Hamiltonian), we find that this dependence is relatively weak; in particular, $t_\text{sat}$ should converge to a finite, $O(1)$ value as $\gamma\to 0$. Therefore, estimate of $D$ should also depend only on the ratio $\gamma/\Delta t$, provided that we are in the regime where $\gamma t_\text{sat} \lesssim 1$.
Testing this expectation on the Ising chain~\eqref{eq:IsingDef}, we find that it works remarkably well, even for $\gamma \approx 1$ (we also find that it works increasingly well as $\ell_*$ gets larger). This is shown in Fig.~\ref{fig:scaling}. Figs.~\ref{fig:scaling}(a,b) show that curves with identical ratio $\gamma/\Delta t$ are the same at early times, and, moreover, their late time saturation values are also close to one another, provided that we are in a regime with sufficiently small $\gamma$. Consequently, the estimates for $D$ show a scaling collapse when data for the same $\ell_*$ but different $\Delta t$, are plotted as a function of $\gamma / \Delta t$, see Fig.~\ref{fig:scaling}(c).
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{SM2.pdf}
\caption{Scaling collapse as a function of $\gamma / \Delta t$. (a,b): comparison of time-dependent diffusion constants for two curves with different $\Delta t$ but the same ratio $\gamma / \Delta t$. When $\gamma$ is sufficiently small, the results remain close to each other even at long times. (c): Estimates of $D \equiv
lim_{t\to\infty} D(t)$, comparing $\Delta t = 1$ and $\Delta t = 1/4$. In the small $\gamma$ regime, relevant for extrapolation, the curves with the same $\ell_*$ collapse when plotted as function of $\gamma / \Delta t$.}
\label{fig:scaling}
\end{figure}
\subsection{Operator weights}
In the main text, we noted that the operator von Neumann entropy of the dissipatively evolving local density approaches $1$ (in units of $\ln{2}$) at long times. Our interpretation was that this points to a long-time behavior where the evolving operator is increasingly dominated by its diffusive, `conserved' part, $\tilde q_0(t) \approx \sum_j C_j(t) q_j$. We now further support this by calculating the weight of various operators in $\tilde q_0$ (in this section we use a different notation from the main text, with $0$ denoting the center site).
To define what we mean by the weight of an operator, let us expand $\tilde q_0$ in the basis of Pauli strings, $\tilde q_0 = \sum_{\op{S}} c_\op{S}(t) \op{S}$; the weight of the Pauli string is then the squared coefficient, $|c_\op{S}|^2$. The total weight on operators with length $\ell$ is given by the following quantity:
\begin{equation}
W_\ell(t) \equiv \sum_{\substack{\op{S} \\ \ell_\op{S} = \ell}} |c_\op{S}(t)|^2.
\end{equation}
For unitary evolution one would have a conserved total weight, $\sum_\op{S} |c_\op{S}(t)|^2 = \sum_\ell W_\ell(t) = 1$. During evolution, the weight gets redistributed from short operators to an essentially random superposition of long ones, such that at time $t$ the operator is dominated by strings of length $\ell \sim v_\text{B} t$, with $v_\text{B}$ the butterfly velocity. This leads to the linear growth of operator entanglement with time.
The dissipator fundamentally changes this picture, as it \emph{removes} operator weight from long strings. This reverses the effect of the unitary dynamics, making the contribution of short operators dominant at long times, which leads to the observed decay in the entanglement. While short operators, with $\ell \leq \ell_*$, are not affected directly by the dissipator, their weight also decreases as they get converted into longer strings which are subsequently dissipated. However, due to the hydrodynamic nature of transport, we find that the weight associated to local densities, $|C_j|^2 \equiv |c_{q_j}|^2$ decreases parametrically more slowly than those of non-conserved operators, so that they dominate at long times.
To show this, we consider the XX ladder~\eqref{eq:XXladder} and consider the evolution of the spin density, $\tilde Z_0(t)$. Calculating operator weights for this object, we find that the weight on local densities decays as $W_{\ell=1} \sim t^{-1/2}$, as expected from the diffusive nature of spin transport~\cite{RvK2018,Khemani2018}. Considering larger $\ell$, we find two things. First, for $\ell > \ell_*$, the weight decreases exponentially with $\ell$, as expected from the form of the dissipator. More importantly, however, we find that the weights for $\ell > 1$ decay parametrically faster in time, $W_{\ell>1} \sim t^{-3/2}$ (even when $1 < \ell \leq \ell_*$); this is shown in Fig.~{}. This is consistent with our earlier prediction, that $\tilde Z_0(t)$ is dominated by the local densities at long times.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{SM3.pdf}
\caption{Total weight on strings of size $\ell$ as a function of time. The majority of the remaining (not yet dissipated) weight is on 1-site strings as decays as $t^{-1/2}$. The weight of longer strings decays as $t^{-3/2}$. Data shown for $\Delta t = 1$, $\ell_* = 5$ with $\gamma = 0.05$ (left) and $\gamma = 0.25$ (right).}
\label{fig:weights}
\end{figure}
\iffalse
The $t^{-3/2}$ power law can be explained using the operator spreading picture developed in Refs. \onlinecite{RvK2018,Khemani2018} in connection with random unitary circuits with a U$(1)$ conserved quantity $\sum_{x}Z_{x}$. The nature of operator spreading in these models becomes clearer through both Haar averaging and employing a large-$q$ limit, which we assume hereafter. Under time evolution, one can rewrite the time evolved $Z_x$ operator as
$$
Z_{x}(t)=Z_{x}^{\text{D}}(t)+Z_{x}^{\text{B}}(t)
$$
where $Z_{x}^{\text{D}}(t)\equiv\sum_{y}K(x,y,t) Z_{y}$ is the diffusive part of the operator and $K(x,y,t)$ is the lattice random walk/unbiased diffusion kernel. Applying a unitary gate to $Z_x^\text{D}(t)$ on bond $y,y+1$ at time $t$ both evolves $K(x,y,t)$ but also sources a new ballistically growing operator with weight $|\partial_y K(x,y,t)|^2$. In this way, at each time step $Z_x^\text{D}$ sources new ballistically growing operators which thereafter form part of $Z_{x}^{\text{B}}$. This picture can be used to deduce the behaviour of $W_\ell$ as a function of time, i.e., what is the weight on operators with support $\ell>1$ at time $t$. According to the above picture, operators of support $\ell$ would correspond to terms in $Z^{\text{B}}$ which have been ballistically growing for time interval $t-\tau=\ell/(2v_{\text{B}})$, where $v_\text{B}$ is the operator butterfly velocity. The weight of such terms is therefore expected to be
$$
\int dy\left(\partial_{y}K_{xy}(\tau)\right)^{2} \sim\left(D(t-\frac{\ell}{2v_{\text{B}}})\right)^{-3/2}.
$$
This shows that the weight on length $\ell$ operators at time $t\gg\frac{\ell}{2v_{\text{B}}} $ goes as $(Dt)^{-3/2}$.
\fi
The $t^{-3/2}$ power law can be explained using the operator spreading picture developed in Refs. \onlinecite{RvK2018,Khemani2018}. In this picture, we rewrite the time-evolved local density at position $x$ as $q_x(t)$
\begin{equation*}
q_{x}(t)=q_{x}^{\text{D}}(t)+q_{x}^{\text{B}}(t)
\end{equation*}
where $q_{x}^{\text{D}}(t)\equiv\sum_{y}C(x,y,t) Z_{y}$ is the diffusive part of the operator and we assume that $C(x,y,t) \equiv \braket{q_y|q_x(t)}$ is well approximated by an unbiased diffusion kernel. $q_{x}^\text{B}(t)$ contains the contributions from all remaining Pauli strings, and is dominated by those with length $\ell = 2 v_\text{B}t$, with $v_\text{B}$ the operator butterfly velocity~\cite{RvK2017,Nahum2017}. The unitary dynamics leads to a conversion of weight from the diffusive to the ballistic part, whose local rate is given by `current' squared, $|\partial_y C(x,y,t)|^2$. In this way, at each time step $q_x^\text{D}$ sources new ballistically growing operators which thereafter form part of $q_{x}^{\text{B}}$. This picture can be used to deduce the behaviour of $W_\ell$ as a function of time. According to the above picture, operators of support $\ell$ would correspond to terms in $q^{\text{B}}$ which have been ballistically growing for a time interval $t-\tau=\ell/(2v_{\text{B}})$. The weight of such terms is therefore expected to be
\begin{equation*}
\int \text{d}y\left(\partial_{y}C(x,y,\tau)\right)^{2} \sim\left[D\left(t-\frac{\ell}{2v_{\text{B}}}\right)\right]^{-3/2}.
\end{equation*}
This shows that the weight on length $\ell$ operators at time $t\gg\frac{\ell}{2v_{\text{B}}} $ goes as $(Dt)^{-3/2}$.
\section{Spin diffusion in Floquet circuits}
We now complement the results shown for energy-conserving, Hamiltonian dynamics in the main text, with data on time-periodic models. We construct these as circuits of local unitary gates, with a `brick-wall' structure and consequently, a strict light cone. This structure is illustrated in Fig.~\ref{fig:circuits}(a). We use the same two-site unitary $u$ in each gate, such that the system has translation invariance in space (with unit cells composed of two sites) and in time (by two layers of the circuit).
We want our circuit to conserve the total spin-$z$ component. For a spin-$1/2$ chain, such a circuit is fully parametrized by three numbers, and it corresponds to a Trotterized version of an XXZ chain with a staggered magnetic field
\begin{equation}\label{eq:CircuitDef}
u = e^{-i\left( J_{xy} (S^x_1S^x_2+S^y_1S^y_2) + J_{zz}S^z_1S^z_2 + g(S^z_1-S^z_2)\right)},
\end{equation}
where we have now used spin operator $S^\alpha$ instead of Pauli matrices (the two differ by a factor of $2$), and the subscripts refer to the two sites on which the gate acts. We choose irrational values of the three couplings, $J_{xy} = 2\sqrt{7}$, $J_{zz} = 2\sqrt{5}$, $g=2\sqrt{3}$.
We apply our dissipative evolution method for this circuit model, applying the dissipator after every second layer of the circuit (i.e., one Floquet period). We extract spin diffusion constant in the same way as in the main text. The results for the spin-$1/2$ circuit are plotted in Fig.~\ref{fig:circuits}(b). We find that the convergence to $\gamma=0$ is less clear than in the Hamiltonian models we studied in the main text. In particular, for $\ell_* =1,2$ we observe a strong non-monotonicity with $\gamma$, while $\ell_*=3,4$ do appear to converge linearly to compatible values of $D$. Nevertheless, we note that the variations in $D$ are all relatively small.
\begin{figure}
\centering
\includegraphics[width=1.\columnwidth]{SM4.pdf}
\caption{Diffusion constants in Floquet curcits. (a) The circuits have a brick-wall structure, updating even/odd bonds in turn. Every gate is given by the same $S_z$-conserving two-site unitary $u$. (b,c) Estimates of the spin diffusion constant for the circuit defined by Eq.~\eqref{eq:CircuitDef}, for spin-$1/2$ and spin-$1$ chains.}
\label{fig:circuits}
\end{figure}
Our interpretation is that the apparent lack of convergence in Fig.~\ref{fig:circuits}(b) is not related to the Floquet circuit nature of our model; rather, it has to do with the fact that it is close to an integrable point. It was recently shown that for $g=0$, the model in Eq.~\eqref{eq:CircuitDef} is integrable; this is closely related to the integrability of the XXZ Hamiltonian. In the latter case, a staggered field is known to break integrability~\cite{Huang2013,Steinigeweg2015,Mendoza2015}, so we expect that for generic $g$ our circuit is also non-integrable. However, we believe that the nearby integrable point is responsible for the non-trivial behavior we observe (for example some almost-conserved operator of length $\ell=3$ could explain why the $\ell_* \leq 2$ curves have a qualitatively different behavior from $\ell_* \geq 3$).
To test this intuition, we also consider the spin-$1$ version of the same model. That is, we use the same definition of the two-site gate as in Eq.~\eqref{eq:CircuitDef}, but with $S_{1,2}^\alpha$ standing for spin-$1$ operators. The results for this case are shown in Fig.~\ref{fig:circuits}(c). While we find that getting to smaller $\gamma$ becomes quite difficult in this case, due to a quick initial growth of operator entanglement, so that our results are not as precisely converged as for the models presented in the main text, we find no evidence of strong non-monotonicities in the regime we can simulate. This reinforces our belief that the peculiar behavior exhibited by the spin-$1/2$ model is tied to the presence of nearby integrable points. | {
"timestamp": "2020-04-14T02:00:16",
"yymm": "2004",
"arxiv_id": "2004.05177",
"language": "en",
"url": "https://arxiv.org/abs/2004.05177",
"abstract": "We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators. We represent the observable as a matrix product operator, and show that the dissipation leads to a decay of operator entanglement, allowing us to capture the dynamics to long times. We test this scheme by calculating spin and energy diffusion constants in a variety of physical models. By gradually weakening the dissipation, we are able to consistently extrapolate our results to the case of zero dissipation, thus estimating the physical diffusion constant with high precision.",
"subjects": "Strongly Correlated Electrons (cond-mat.str-el); Statistical Mechanics (cond-mat.stat-mech)",
"title": "Dissipation-assisted operator evolution method for capturing hydrodynamic transport",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075755433747,
"lm_q2_score": 0.7371581626286834,
"lm_q1q2_score": 0.7092254526386913
} |
https://arxiv.org/abs/1703.04973 | Real interpolation with variable exponent | We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable Besov and Lorentz spaces as introduced recently in Almeida and Hästö (J. Funct. Anal. 258 (5) 1628--2655, 2010) and L. Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407--420). | \section*{{\normalsize \bf #2}}\list
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\input{tcilatex}
\begin{document}
\title{Real interpolation with variable exponent}
\author{Douadi Drihem}
\date{\today }
\maketitle
\begin{abstract}
We present the real interpolation with variable exponent and we prove the
basic properties in analogy to the classical real interpolation. More
precisely, we prove that under some additional conditions, this method\ can
be reduced to the case of fixed exponent. An application, we give the real
interpolation of variable Besov and Lorentz spaces as introduced recently in
Almeida and H\"{a}st\"{o} (J. Funct. Anal. 258 (5) 1628--2655, 2010) and L.
Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407--420).\vski
5pt
\textit{MSC 2010\/}: 46B70, 46E30, 46E35.
\textit{Key Words and Phrases}: real interpolation, embeddings, Besov space,
Lorentz space, variable exponent.
\end{abstract}
\section{Introduction}
It is well known that real interpolation play an important role in several
different areas, especially for modern analysis and its theory started early
in 1960's by J-L. Lions and J. Peetre. There are two ways for introducing
the real interpolation method. The first is the $K$-method and the second is
the $J$-method. But the spaces generated by the $K$-and $J$-methods are the
same. For general literature on real interpolation we refer to \cite{BS88},
\cite{BL76}, \cite{T78} and references therein.
In recent years, there has been growing interest in generalizing classical
spaces such as Lebesgue spaces, Sobolev spaces, Besov spaces,
Triebel-Lizorkin spaces to the case with either variable integrability or
variable smoothness. The motivation for the increasing interest in such
spaces comes not only from theoretical purposes, but also from applications
to fluid dynamics, image restoration and PDE\ with non-standard growth
conditions.
From these in this paper we present a variable version of real
interpolation. First we study the variable version of $K$-method, where we
present some equivalent norms for the space generated by this method and we
prove their basic properties in analogy to the fixed exponent. Secondly, we
present the same analysis for the variable version of $J$-method and we
prove the first main statement of this paper. That is, under some additional
conditions the spaces generated by the $K$-and $J$-methods are the same.
Since the reiteration theorem is one of the most important general results
in interpolation theory, we will give its proof. Finally, we study the real
interpolation of variable exponent Besov and Lorentz spaces. Almost all of
the material we present is due to \cite{BS88}, and \cite{BL76}. Allowing the
exponent is vary from point to point will raise extra difficulties which, in
general, are overcome by imposing regularity assumptions on this exponent.
\section{Preliminaries}
As usual, we denote by $\mathbb{R}^{n}$ the $n$-dimensional real Euclidean
space, $\mathbb{N}$ the collection of all natural numbers and $\mathbb{N
_{0}=\mathbb{N}\cup \{0\}$. The letter $\mathbb{Z}$ stands for the set of
all integer numbers.\ The expression $f\lesssim g$ means that $f\leq c\,g$
for some independent constant $c$ (and non-negative functions $f$ and $g$),
and $f\approx g$ means $f\lesssim g\lesssim f$. \vskip5pt
By $c$ we denote generic positive constants, which may have different values
at different occurrences. Although the exact values of the constants are
usually irrelevant for our purposes, sometimes we emphasize their dependence
on certain parameters (e.g. $c(p)$ means that $c$ depends on $p$, etc.).
Further notation will be properly introduced whenever needed.
The variable exponents that we consider are always measurable functions $p$
on $\mathbb{R}$ with range in $[1,\infty \lbrack $. We denote the set of
such functions by $\mathcal{P}({\mathbb{R}})$. We use the standard notation
p^{-}:=\underset{x\in \mathbb{R}^{n}}{\text{ess-inf}}$ $p(x)$,$\quad p^{+}:
\underset{x\in \mathbb{R}^{n}}{\text{ess-sup }}p(x)$.
The variable exponent modular is defined by $\varrho _{p(\cdot )}(f):=\int_
\mathbb{R}^{n}}\varrho _{p(x)}(\left\vert f(x)\right\vert )dx$, where
\varrho _{p}(t)=t^{p}$. The variable exponent Lebesgue space $L^{p(\cdot )}
\ consists of measurable functions $f$ on $\mathbb{R}^{n}$ such that
\varrho _{p(\cdot )}(\lambda f)<\infty $ for some $\lambda >0$. We define
the Luxemburg (quasi)-norm on this space by the formula $\left\Vert
f\right\Vert _{p(\cdot )}:=\inf \Big\{\lambda >0:\varrho _{p(\cdot )}\Big
\frac{f}{\lambda }\Big)\leq 1\Big\}$. A useful property is that $\left\Vert
f\right\Vert _{p(\cdot )}\leq 1$ if and only if $\varrho _{p(\cdot )}(f)\leq
1$, see \cite{DHHR}, Lemma 3.2.4.
We say that $g:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is \textit{locally }lo
\textit{-H\"{o}lder continuous}, abbreviated $g\in C_{\text{loc}}^{\log }
\mathbb{R}^{n})$, if there exists $c_{\log }(g)>0$ such tha
\begin{equation}
\left\vert g(x)-g(y)\right\vert \leq \frac{c_{\log }(g)}{\log
(e+1/\left\vert x-y\right\vert )} \label{lo-log-Holder}
\end{equation
for all $x,y\in \mathbb{R}^{n}$. If
\begin{equation*}
|g(x)-g(0)|\leq \frac{c_{\log }}{\ln (e+1/|x|)}
\end{equation*
for all $x\in \mathbb{R}^{n}$, then we say that $g$ is \emph{$\log $-H\"{o
lder continuous at the origin} (or has a \emph{$\log $ decay at the origin
). We say that $g$ satisfies the log\textit{-H\"{o}lder decay condition}, if
there exists $g_{\infty }\in \mathbb{R}$ and a constant $c_{\log }>0$ such
tha
\begin{equation*}
\left\vert g(x)-g_{\infty }\right\vert \leq \frac{c_{\log }}{\log
(e+\left\vert x\right\vert )}
\end{equation*
for all $x\in \mathbb{R}^{n}$. We say that $g$ is \textit{globally}-lo
\textit{-H\"{o}lder continuous}, abbreviated $g\in C^{\log }$, if it i
\textit{\ }locally log-H\"{o}lder continuous and satisfies the log-H\"{o
lder decay\textit{\ }condition.\textit{\ }The constants $c_{\log }(g)$ and
c_{\log }$ are called the \textit{locally }log\textit{-H\"{o}lder constant
and the log\textit{-H\"{o}lder decay constant}, respectively\textit{.} We
note that all functions $g\in C_{\text{loc}}^{\log }(\mathbb{R}^{n})$ always
belong to $L^{\infty }$.\vskip5pt
We refer to the recent monograph $\mathrm{\cite{CF13}}$ for further
properties, historical remarks and references on variable exponent spaces.
\subsection{Technical lemmas}
In this subsection we present some results which are useful for us. The
following lemma is from \cite{DHHR}.
\begin{lemma}
\label{estimate -modular}Let$\ A\subset \mathbb{R}^{n}\ $and$\ p\in \mathcal
P}(\mathbb{R}^{n})$\ with$\ p^{-}<\infty $. If $\varrho _{p\left( \cdot
\right) }(f\chi _{A})>0$ or $p^{+}<\infty $, the
\begin{equation*}
\min \left\{ \left( \varrho _{p\left( \cdot \right) }(f\chi _{A})\right) ^
\frac{1}{p^{-}}},\left( \varrho _{p\left( \cdot \right) }(f\chi _{A})\right)
^{\frac{1}{p^{+}}}\right\} \leq \left\Vert f\right\Vert _{p\left( \cdot
\right) }\leq \max \left\{ \left( \varrho _{p\left( \cdot \right) }(f\chi
_{A})\right) ^{\frac{1}{p^{-}}},\left( \varrho _{p\left( \cdot \right)
}(f\chi _{A})\right) ^{\frac{1}{p^{+}}}\right\} .
\end{equation*}
\end{lemma}
The next lemma is a Hardy-type inequality which is easy to prove.
\begin{lemma}
\label{Hardy-inequality}\textit{Let }$0<a<1$ \textit{and }$0<q\leq \infty
\textit{. Let }$\left\{ \varepsilon _{k}\right\} _{k}$\textit{\ be a
sequences of positive real numbers and denote} $\delta _{k}=\sum_{j=-\infty
}^{\infty }a^{\left\vert k-j\right\vert }\varepsilon _{j}$.\textit{\ }Then
there exists constant $c>0\ $\textit{depending only on }$a$\textit{\ and }$q$
such tha
\begin{equation*}
\Big(\sum\limits_{k=-\infty }^{\infty }\delta _{k}^{q}\Big)^{1/q}\leq c\text{
}\Big(\sum\limits_{k=-\infty }^{\infty }\varepsilon _{k}^{q}\Big)^{1/q}.
\end{equation*}
\end{lemma}
We will make use of the following statement, see \cite{DHHMS}, Lemma 3.3 for
$w:=1$.
\begin{lemma}
\label{DHHR-estimate}Let $Q=(a,b)\subset \mathbb{R}$ with $0<a<b<\infty $.
Let $p\in \mathcal{P}(\mathbb{R})$ \textit{be log-H\"{o}lder continuous at
the origin }and $w:\mathbb{R}\rightarrow \mathbb{R}^{+}$ be a weight
function. Then for every $m>0$ there exists $\gamma =e^{-4mc_{\log
}(1/p)}\in \left( 0,1\right) $ such tha
\begin{eqnarray*}
&&\Big(\frac{\gamma }{w(Q)}\int_{Q}\left\vert f(y)\right\vert w(y)dy\Big
^{p\left( x\right) } \\
&\leq &\max \big(1,\left( w(Q)\right) ^{1-\frac{p\left( x\right) }{p^{-}}
\big)\frac{1}{w(Q)}\int_{Q}\left\vert f(y)\right\vert ^{p\left( y,0\right)
}w(y)dy \\
&&+\omega (m,b)\Big(\frac{1}{w(Q)}\int_{Q}g(x,y)w(y)dy\Big)
\end{eqnarray*
hold if $0<w(Q)<\infty $, all $x\in Q\subset \mathbb{R}$ and all $f\in
L^{p\left( \cdot \right) }(w)+L^{\infty }$\ with $\left\Vert fw^{1/p\left(
\cdot \right) }\right\Vert _{p\left( \cdot \right) }+\left\Vert f\right\Vert
_{\infty }\leq 1$, wher
\begin{equation*}
\omega (m,b)=\min \left( b^{m},1\right) \text{, }p\left( y,0\right) =p\left(
y\right) \text{ \ \ and \ \ }g(x,y)=(e+\frac{1}{x})^{-m}+(e+\frac{1}{y})^{-m}
\end{equation*
o
\begin{equation*}
\omega (m,b)=\min \left( b^{m},1\right) \text{, }p\left( y,0\right) =p\left(
0\right) \text{ \ \ and \ \ }g(x,y)=(e+\frac{1}{x})^{-m}\chi _{\{x:p(x){<
p(0)\}}(x).
\end{equation*
\textit{In addition we have the same estimate}, wher
\begin{equation*}
\omega (m,b)=1\text{, }p\left( y,0\right) =p_{\infty }\text{ \ \ and \ \
g(x,y)=(e+x)^{-m}\chi _{\{x:p(x){<}p_{\infty }\}}(x),
\end{equation*
if $p\in \mathcal{P}(\mathbb{R})$ satisfies the log\textit{-H\"{o}lder decay
condition, where }$\gamma =e^{-4mc_{\log }}$\textit{.}
\end{lemma}
The proof of this lemma\ is given in \cite{D4}. Notice that in the proof of
this theorem we need only that
\begin{equation*}
\int_{Q}\left\vert f(y)\right\vert ^{p\left( y\right) }w(y)dy\leq 1
\end{equation*
and/or $\left\Vert f\right\Vert _{\infty }\leq 1$.
The next lemma is the continuous version of Hardy-type inequality, see \cit
{D.S2007}.
\begin{lemma}
\label{lq-inequality}\textit{Let }$s>0$\textit{. Let }$q\in \mathcal{P}
\mathbb{R})$\textit{\ }be \emph{$\log $}-H\"{o}lder continuous both at the
origin\ and at infinity with $1\leq q^{-}\leq q^{+}<\infty $\textit{. Let }
\left\{ \varepsilon _{t}\right\} _{t}$\textit{\ be a sequence of positive
measurable functions.} Le
\begin{equation*}
\eta _{t}=t^{s}\int_{t}^{\infty }\tau ^{-s}\varepsilon _{\tau }\frac{d\tau }
\tau }\quad \text{and\quad }\delta _{t}=t^{-s}\int_{0}^{t}\tau
^{s}\varepsilon _{\tau }\frac{d\tau }{\tau }.
\end{equation*
Then there exists constant $c>0\ $\textit{depending only on }$s$, $q^{-}
\textit{, c}$_{\log }(q)$ \textit{and }$q^{+}$ such tha
\begin{equation*}
\left\Vert \eta _{t}\right\Vert _{L^{q(\cdot )}((0,\infty ),\frac{dt}{t
)}+\left\Vert \delta _{t}\right\Vert _{L^{q(\cdot )}((0,\infty ),\frac{dt}{t
)}\lesssim \left\Vert \varepsilon _{t}\right\Vert _{L^{q(\cdot )}((0,\infty
),\frac{dt}{t})}.
\end{equation*}
\end{lemma}
\section{The K-Method}
The fundamental notion of real interpolation is the $K$-functional, where it
is due to J. Peetre.
\begin{definition}
\label{Compatible}Let $A_{0}$ and $A_{1}$\ be Banach spaces over $\mathbb{K=
}$ or $\mathbb{C}$. We shall say that $A_{0}$ and $A_{1}$ are compatible if
there is a Hausdorff topological vector space $Z$ such that
\begin{equation*}
A_{0},A_{1}\hookrightarrow Z,
\end{equation*
with continuous embeddings.
\end{definition}
Let $A_{0}$ and $A_{1}$ be compatible. We will say that $(A_{0},A_{1})$ is a
compatible couple. Then we can form their sum $A_{0}+A_{1}$ and their
intersection $A_{0}\cap A_{1}$. The sum consists of all $f\in Z$ such that
we can write
\begin{equation*}
f=f_{0}+f_{1}
\end{equation*
\ for some $f_{0}\in A_{0}$\ and $f_{1}\in A_{1}$. Then $A_{0}+A_{1}$ is a
Banach space with norm defined by\
\begin{equation*}
\left\Vert f\right\Vert _{A_{0}+A_{1}}=\inf_{f=f_{0}+f_{1}}\left( \left\Vert
f_{0}\right\Vert _{A_{0}}+\left\Vert f_{1}\right\Vert _{A_{1}}\right) .
\end{equation*
$A_{0}\cap A_{1}$ is a Banach space with norm defined by\
\begin{equation*}
\left\Vert f\right\Vert _{A_{0}\cap A_{1}}=\max_{f=f_{0}+f_{1}}\left(
\left\Vert f_{0}\right\Vert _{A_{0}},\left\Vert f_{1}\right\Vert
_{A_{1}}\right) .
\end{equation*}
Let $(A_{0},A_{1})$ be a compatible couple. With $t>0$\ fixed, pu
\begin{equation*}
K(t,f,A_{0},A_{1})=\inf_{f=f_{0}+f_{1}}\left( \left\Vert f_{0}\right\Vert
_{A_{0}}+t\left\Vert f_{1}\right\Vert _{A_{1}}\right) ,\quad f\in
A_{0}+A_{1},
\end{equation*}
is the $K$-functional. For any $f\in A_{0}+A_{1}$, $K(t,f,A_{0},A_{1})$ is a
positive, increasing and concave function of $t$. In particula
\begin{equation}
K(t,f,A_{0},A_{1})\leq \max (1,\frac{t}{s})K(s,f,A_{0},A_{1}).
\label{K-property}
\end{equation
If there is no danger of confusion, we shall write
K(t,f)=K(t,f,A_{0},A_{1}) $.
\begin{definition}
Let $\theta \in (0,1)$ and $q\in \mathcal{P}(\mathbb{R})$. Let
(A_{0},A_{1}) $\ be a compatible couple. The space $(A_{0},A_{1})_{\theta
,q(\cdot )}$ consists of all $f$ in $A_{0}+A_{1}$ for which the functiona
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}=\left\Vert
t^{-\theta }K(t,f)\right\Vert _{L^{q(\cdot )}((0,\infty ),\frac{dt}{t})}
\end{equation*
is finite.
\end{definition}
\begin{definition}
Let $\theta \in \lbrack 0,1]$. Let $(A_{0},A_{1})$\ be a compatible couple.
The space $(A_{0},A_{1})_{\theta ,\infty }$ consists\ of\ all\ $f$\ in
A_{0}+A_{1}$ for which the functiona
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,\infty
}}=\sup_{t>0}t^{-\theta }K(t,f)
\end{equation*
is finite.
\end{definition}
In the next lemma we prove that the first definition can be given in
discrete\ version, where we need additional assumptions on $q$.
\begin{lemma}
\label{K-property2}Let $(A_{0},A_{1})$\ be a compatible couple and $f\in
A_{0}+A_{1}$.\ Let $\theta \in (0,1)$, $f\in (A_{0},A_{1})_{\theta ,q(\cdot
)}$\ and we put $\alpha _{v}=K(2^{v},f)$, $v\in \mathbb{Z}$\textit{. Let }
q\in \mathcal{P}(\mathbb{R})$ \textit{be log-H\"{o}lder continuous both at
the origin and at infinity. Then
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\approx \Big
\sum_{v=-\infty }^{0}2^{-v\theta q(0)}\alpha _{v}^{q(0)}\Big)^{\frac{1}{q(0)
}+\Big(\sum_{v=1}^{\infty }2^{-v\theta q_{\infty }}\alpha _{v}^{q_{\infty }
\Big)^{\frac{1}{q_{\infty }}}.
\end{equation*
Moreover
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\approx \Big
\int_{0}^{1}t^{-\theta q(0)}K(t,f)^{q(0)}\frac{dt}{t}\Big)^{\frac{1}{q(0)}}
\Big(\int_{1}^{\infty }t^{-\theta q_{\infty }}K(t,f)^{q_{\infty }}\frac{dt}{
}\Big)^{\frac{1}{q_{\infty }}}.
\end{equation*}
\end{lemma}
\textbf{Proof. }We will do the proof in two steps and we need only to prove
the first statement.
\textit{Step 1.} Let us prove tha
\begin{equation}
S=\Big(\sum_{v=-\infty }^{0}2^{-v\theta q(0)}\alpha _{v}^{q(0)}\Big)^{\frac{
}{q(0)}}+\Big(\sum_{v=1}^{\infty }2^{-v\theta q_{\infty }}\alpha
_{v}^{q_{\infty }}\Big)^{\frac{1}{q_{\infty }}}\lesssim \left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}. \label{est-step1}
\end{equation
By scaling argument, we need only to prove tha
\begin{equation*}
\sum_{v=1}^{\infty }2^{-v\theta q_{\infty }}\big(\alpha _{v}\big)^{q_{\infty
}}\lesssim 1\quad \text{and}\quad \sum_{v=-\infty }^{0}2^{-v\theta q(0)}\big
\alpha _{v}\big)^{q(0)}\lesssim 1
\end{equation*
for any $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$ with $\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\leq 1$. To prove the first
estimate we need to prove that
\begin{equation*}
2^{-v\theta q_{\infty }}\big(\alpha _{v}\big)^{q_{\infty }}\leq
\int_{2^{v-1}}^{2^{v}}\big(t^{-\theta }K(t,f)\big)^{q\left( t\right) }\frac
dt}{t}+2^{-v}=\delta
\end{equation*
for any $v\in \mathbb{N}$. This claim can be reformulated as showing tha
\begin{equation*}
\big(\delta ^{-\frac{1}{q_{\infty }}}2^{-v\theta }\alpha _{v}\big
^{q_{\infty }}=\Big(\frac{1}{\log 2}\int_{2^{v-1}}^{2^{v}}\delta ^{-\frac{1}
q_{\infty }}}2^{-v\theta }\alpha _{v}\frac{d\tau }{\tau }\Big)^{q_{\infty
}}\lesssim 1.
\end{equation*
Using the property $\mathrm{\eqref{K-property}}$, we find tha
\begin{equation*}
\int_{2^{v-1}}^{2^{v}}\delta ^{-\frac{1}{q_{\infty }}}2^{-v\theta }\alpha
_{v}\frac{d\tau }{\tau }\lesssim \int_{2^{v-1}}^{2^{v}}\delta ^{-\frac{1}
q_{\infty }}}\tau ^{-\theta }K(\tau ,f)\frac{d\tau }{\tau }.
\end{equation*
By Lemma \ref{DHHR-estimate} the last expression with power $q\left(
t\right) $ is bounded b
\begin{equation*}
c\int_{2^{v-1}}^{2^{v}}\delta ^{-\frac{q\left( \tau \right) }{q_{\infty }}
\big(\tau ^{-\theta }K(\tau ,f)\big)^{q\left( \tau \right) }\frac{d\tau }
\tau }+1
\end{equation*
for any $t\in \lbrack 2^{v-1},2^{v}]$. Since $q$ is \emph{$\log $-}H\"{o
lder continuous at infinity, we find tha
\begin{equation}
\delta ^{-\frac{q\left( \tau \right) }{q_{\infty }}}\approx \delta ^{-1}
\text{ \ \ }\tau \in \lbrack 2^{v-1},2^{v}],\quad v\in \mathbb{N}.
\label{log-at- zero}
\end{equation
Therefore, from the definition of $\delta $, we find tha
\begin{equation*}
\int_{2^{v-1}}^{2^{v}}\delta ^{-\frac{q\left( \tau \right) }{q_{\infty }}
\big(\tau ^{-\theta }K(\tau ,f)\big)^{q\left( \tau \right) }\frac{d\tau }
\tau }\lesssim 1.
\end{equation*
Now, let us prove the second estimate. We need to show that
\begin{equation*}
2^{-v\theta q(0)}\big(\alpha _{v}\big)^{q(0)}\lesssim \int_{2^{v}}^{2^{v+1}
\big(t^{-\theta }K(t,f)\big)^{q\left( t\right) }\frac{dt}{t}+2^{v}=\delta
\end{equation*
for any $v\leq 0$. This claim can be reformulated as showing tha
\begin{equation*}
\big(\delta ^{-\frac{1}{q(0)}}2^{-v\theta }\alpha _{v}\big)^{q(0)}=\Big
\frac{1}{\log 2}\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}2^{-v\theta
}\alpha _{v}\frac{d\tau }{\tau }\Big)^{q(0)}\lesssim 1.
\end{equation*
The property $\mathrm{\eqref{K-property}}$, gives\ tha
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}2^{-v\theta }\alpha _{v}\frac
d\tau }{\tau }\lesssim \int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}\tau
^{-\theta }K(\tau ,f)\frac{d\tau }{\tau }.
\end{equation*
Again by Lemma \ref{DHHR-estimate}
\begin{equation*}
\Big(\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}\tau ^{-\theta }K(\tau
,f)\frac{d\tau }{\tau }\Big)^{q\left( t\right) }\lesssim
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{q\left( \tau \right) }{q(0)}}\big(\tau
^{-\theta }K(\tau ,f)\big)^{q\left( \tau \right) }\frac{d\tau }{\tau }+1.
\end{equation*
for any $t\in \lbrack 2^{v},2^{v+1}]$\ and any $v\leq 0$. We use the
logarithmic decay\ condition at origin\ of $q$ to show tha
\begin{equation*}
\delta ^{-\frac{q\left( \tau \right) }{q(0)}}\approx \delta ^{-1},\text{ \ \
}\tau \in \lbrack 2^{v},2^{v+1}],\quad v\leq 0.
\end{equation*
Therefore, from the definition of $\delta $, we find tha
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{q\left( \tau \right) }{q(0)}}\big(\tau
^{-\theta }K(\tau ,f)\big)^{q\left( \tau \right) }\frac{d\tau }{\tau
\lesssim 1
\end{equation*
for any $v\leq 0$. Hence, we proved $\mathrm{\eqref{est-step1}}$.
\textit{Step 2.} Let us prove tha
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\lesssim S.
\end{equation*
This claim can be reformulated as showing tha
\begin{equation*}
\dint_{0}^{\infty }\big(t^{-\theta }K(t,\frac{f}{S})\big)^{q\left( t\right)
\frac{dt}{t}\lesssim 1.
\end{equation*
Now our estimate clearly follows from the inequalitie
\begin{equation*}
\int_{2^{v-1}}^{2^{v}}\big(t^{-\theta }K(t,\frac{f}{S})\big)^{q\left(
t\right) }\frac{dt}{t}\lesssim 2^{-v\theta q_{\infty }}\big(\frac{\alpha _{v
}{S}\big)^{q_{\infty }}+2^{-v}=\delta
\end{equation*
for any $v\in \mathbb{N}$ and
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\big(t^{-\theta }K(t,\frac{f}{S})\big)^{q\left(
t\right) }\frac{dt}{t}\lesssim 2^{-v\theta q(0)}\big(\frac{\alpha _{v}}{S
\big)^{q(0)}+2^{v}
\end{equation*
for any $v\leq 0$. The first claim can be reformulated as showing\ tha
\begin{equation*}
\int_{2^{v-1}}^{2^{v}}\big(\delta ^{-\frac{1}{q(t)}}t^{-\theta }K(t,\frac{f}
S})\big)^{q\left( t\right) }\frac{dt}{t}\lesssim 1.
\end{equation*
We need only to show tha
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}t^{-\theta }K(t,\frac{f}{S})\lesssim 1
\end{equation*
for any $v\in \mathbb{N}$ and any $t\in \lbrack 2^{v-1},2^{v}]$. From
\mathrm{\eqref{K-property}}$, the left-hand side is bounded b
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}2^{-\theta v}K(2^{v},\frac{f}{S}),
\end{equation*
and from $\mathrm{\eqref{log-at- zero}}$ we find that
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}2^{-\theta v}K(2^{v},\frac{f}{S})\lesssim \delta ^{
\frac{1}{q_{\infty }}}2^{-\theta v}K(2^{v},\frac{f}{S})\leq 1.
\end{equation*
for any $v\in \mathbb{N}$. Similarly we estimate the second claim. Hence the
lemma is proved.\quad $\square $
Let $(A_{0},A_{1})$\ be a compatible couple and\ $f\in A_{0}+A_{1}$.\ Let
\theta \in \lbrack 0,1]$, $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$\ and we
put $\alpha _{v}=K(2^{v},f)$, $v\in \mathbb{Z}$\textit{. }Then we hav
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,\infty }}\approx \sup_{v\in
\mathbb{Z}}2^{v\theta }\alpha _{v}.
\end{equation*}
We present\ some\ important properties of the spaces $(A_{0},A_{1})_{\theta
,q(\cdot )}$.
\begin{theorem}
\label{embedding}Let\ $\theta \in (0,1)$\ and\ $q\in \mathcal{P}(\mathbb{R})
. Let $(A_{0},A_{1})$\ be a compatible couple of Banach spaces. Then
(A_{0},A_{1})_{\theta ,q(\cdot )}$ is Banach space and
\begin{equation*}
K(s,f,A_{0},A_{1})\leq \gamma _{\theta ,q^{+}}s^{\theta }\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}
\end{equation*
for any $s>0$. Moreover we hav
\begin{equation*}
A_{0}\cap A_{1}\hookrightarrow \left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}\hookrightarrow A_{0}+A_{1}.
\end{equation*}
\end{theorem}
\textbf{Proof.}\ Let $\{f_{n}\}_{n}$ be a sequence in $A_{0}+A_{1}$ such tha
\begin{equation*}
\sum_{n=1}^{\infty }\left\Vert f_{n}\right\Vert _{(A_{0},A_{1})_{\theta
,q(\cdot )}}<\infty .
\end{equation*
Since $L^{q(\cdot )}((0,\infty ),\frac{dt}{t})$ is a Banach space, the
series $\sum_{n=1}^{\infty }t^{-\theta }K(t,f_{n})$\ converges in
L^{q(\cdot )}((0,\infty ),\frac{dt}{t})$ then we ge
\begin{equation*}
\sum_{n=1}^{\infty }t^{-\theta }K(t,f_{n})<\infty
\end{equation*
for all $t>0$. Since $A_{0}+A_{1}$\ is a Banach space, the
\begin{equation*}
t^{-\theta }K(t,\sum_{n=1}^{\infty }f_{n})\leq t^{-\theta
}\sum_{n=1}^{\infty }K(t,f_{n})
\end{equation*
for all $t>0$. Applying the $L^{q(\cdot )}((0,\infty ),\frac{dt}{t})$-norm
to each side, we obtain
\begin{equation*}
\big\|\sum_{n=1}^{\infty }f_{n}\big\|_{(A_{0},A_{1})_{\theta ,q(\cdot
)}}\leq \sum_{n=1}^{\infty }\left\Vert f_{n}\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}<\infty ,
\end{equation*
which ensure that $(A_{0},A_{1})_{\theta ,q(\cdot )}$ is Banach space. By
the property $\mathrm{\eqref{K-property}}$ we find tha
\begin{equation*}
\min (1,\frac{t}{s})K(s,f)\leq K(t,f),\quad s,t>0.
\end{equation*
Therefore
\begin{equation*}
\big\|t^{-\theta }\min (1,\frac{t}{s})\big\|_{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}K(s,f)\leq \left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta
,q(\cdot )}}.
\end{equation*
Let us prove tha
\begin{equation}
\big\|t^{-\theta }\min (1,\frac{t}{s})\big\|_{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}\gtrsim s^{-\theta }. \label{est-t-theta}
\end{equation
We hav
\begin{equation*}
\big\|t^{-\theta }\min (1,\frac{t}{s})\big\|_{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}\geq s^{-\theta }\big\|\big(\frac{t}{s}\big)^{1-\theta }\big\
_{L^{q(\cdot )}((0,s),\frac{dt}{t})},
\end{equation*
an
\begin{equation*}
\int_{0}^{s}\big(\frac{t}{s}\big)^{(1-\theta )q(t)}\frac{dt}{t}\geq
\int_{0}^{s}\big(\frac{t}{s}\big)^{(1-\theta )q^{+}}\frac{dt}{t}=\frac{1}
(1-\theta )q^{+}}.
\end{equation*
From Lemma \ref{estimate -modular}, we find our claim $\mathrm
\eqref{est-t-theta}}$. Therefore
\begin{equation*}
K(s,f)\leq \gamma _{\theta ,q^{+}}s^{\theta }\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}
\end{equation*
for any $s>0$. Taking $s=1$, we obtain
\begin{equation*}
\left\Vert f\right\Vert _{A_{0}+A_{1}}\lesssim \left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}.
\end{equation*
Now sinc
\begin{equation*}
K(t,f)\leq \min (1,t)\left\Vert f\right\Vert _{A_{0}\cap A_{1}},
\end{equation*
we find tha
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\lesssim
\left\Vert f\right\Vert _{A_{0}\cap A_{1}}.
\end{equation*
\quad $\square $
\begin{definition}
Let $(A_{0},A_{1})$\ and $(B_{0},B_{1})$ be two compatible couples of Banach
spaces and let $T$ be a linear operator defined on $A_{0}+A_{1}$ and taking
values in $B_{0}+B_{1}$. $T$ is said be admissible\ with respect to the
couples\ $(A_{0},A_{1})$\ and $(B_{0},B_{1})$ if, for each $i=1,0$ the
restriction of $T$ to $A_{i}$ maps $A_{i}$ into $B_{i}$ and furthermore is a
bounded operator from $A_{i}$ into $B_{i}:
\begin{equation*}
\left\Vert Tf\right\Vert _{B_{i}}\leq \left\Vert T\right\Vert
_{L(A_{i},B_{i})}\left\Vert f\right\Vert _{A_{i}},\quad f\in A_{i}.
\end{equation*}
\end{definition}
Notice that every admissible operator $T$\ with respect to the couples\
(A_{0},A_{1})$\ and $(B_{0},B_{1})$ is bounded from $A_{0}+A_{1}$ into
B_{0}+B_{1}$.
\begin{theorem}
\label{b-operators}Let\ $\theta \in (0,1)$\ and\ $q\in \mathcal{P}(\mathbb{R
)$. Let $(A_{0},A_{1})$\ and $(B_{0},B_{1})$ be two compatible couples of
Banach spaces and let $T$ be admissible\ with respect to the couples\
(A_{0},A_{1})$\ and $(B_{0},B_{1})$. The
\begin{equation*}
T:(A_{0},A_{1})_{\theta ,q(\cdot )}\longrightarrow (B_{0},B_{1})_{\theta
,q(\cdot )}
\end{equation*
an
\begin{equation*}
\left\Vert Tf\right\Vert _{(B_{0},B_{1})_{\theta ,q(\cdot )}}\leq \max
\left( \left\Vert T\right\Vert _{L(A_{0},B_{0})},\left\Vert T\right\Vert
_{L(A_{1},B_{1})}\right) \left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta
,q(\cdot )}}
\end{equation*
for all $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$.
\end{theorem}
\textbf{Proof.} Suppose that $T:(A_{0},A_{1})\longrightarrow (B_{0},B_{1})$.
The
\begin{eqnarray*}
K(t,Tf,B_{0},B_{1}) &\leq &\left\Vert T\right\Vert _{L(A_{0},B_{0})}K\Big
\frac{\left\Vert T\right\Vert _{L(A_{1},B_{1})}t}{\left\Vert T\right\Vert
_{L(A_{0},B_{0})}},f,A_{0},A_{1}\Big) \\
&\leq &\max \big(\left\Vert T\right\Vert _{L(A_{0},B_{0})},\left\Vert
T\right\Vert _{L(A_{1},B_{1})}\big)K(t,f,A_{0},A_{1}),
\end{eqnarray*
by the property $\mathrm{\eqref{K-property}}$. Multiplying by $t^{-\theta }$
and then applying the $L^{q(\cdot )}((0,\infty ),\frac{dt}{t})$-norm to each
side we obtain the desired estimate.\quad $\square $
\begin{proposition}
Let $\theta \in (0,1)$. Let $(A_{0},A_{1})$\ be a compatible couples of
Banach spaces. \newline
(i) Let $q,r\in \mathcal{P}(\mathbb{R})$ with $1\leq q(\cdot )\leq r(\cdot
)<\infty $. The
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}\hookrightarrow (A_{0},A_{1})_{\theta
,r(\cdot )}.
\end{equation*
an
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}\hookrightarrow (A_{0},A_{1})_{\theta
,\infty }
\end{equation*
(ii) \textit{Let }$q\in \mathcal{P}(\mathbb{R})$ \textit{be log-H\"{o}lder
continuous both at the origin and at infinity with $q(0)=q_{\infty }$. Then
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}=(A_{1},A_{0})_{1-\theta ,q(\cdot )}.
\end{equation*
(iii) \textit{Let }$q,r\in \mathcal{P}(\mathbb{R})$ \textit{be log-H\"{o
lder continuous both at the origin and at infinity with }$q(0)=r(0)$ and
q_{\infty }=r_{\infty }$\textit{. Then
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}=(A_{0},A_{1})_{\theta ,r(\cdot )}.
\end{equation*
(iv) If $A_{1}\hookrightarrow A_{0}$, the
\begin{equation*}
(A_{0},A_{1})_{\theta _{1},q(\cdot )}\hookrightarrow (A_{0},A_{1})_{\theta
,q(\cdot )}\text{\quad if }0<\theta \leq \theta _{1}<1.
\end{equation*
(v) If $A_{0}=A_{1}$, with equal norm, the
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}=A_{0}.
\end{equation*}
\end{proposition}
\textbf{Proof.} We prove (i). From Theorem \ref{embedding}, we obtai
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}\hookrightarrow (A_{0},A_{1})_{\theta
,\infty }\quad \text{and}\quad K(s,f)\lesssim s^{\theta }\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}
\end{equation*
for any $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$, any $s>0$\ with
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\neq 0$\ and
this implies tha
\begin{eqnarray*}
&&\int_{0}^{\infty }\big(t^{-\theta }K(t,\frac{f}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}})\big)^{r\left( t\right) }\frac{dt}{t}
\\
&\leq &\int_{0}^{\infty }\big(t^{-\theta }K(t,\frac{f}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}})\big)^{q\left( t\right)
\big(\sup_{t>0}t^{-\theta }K(t,\frac{f}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}})\big)^{r\left( t\right) -q\left(
t\right) }\frac{dt}{t} \\
&\lesssim &\int_{0}^{\infty }\big(t^{-\theta }K(t,\frac{f}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}})\big)^{q\left( t\right)
\frac{dt}{t}.
\end{eqnarray*
The last term is bounded since
\begin{equation*}
\Big\|t^{-\theta }K(t,\frac{f}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}})\Big\|_{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}=1,
\end{equation*
and henc
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,r(\cdot )}}\lesssim
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}.
\end{equation*
Hence the property (i) is proved. To prove (ii) we use Lemma \re
{K-property2} and the fact that
\begin{equation*}
K(t,f,A_{0},A_{1})=tK(t^{-1},f,A_{1},A_{0}),\quad v>0,
\end{equation*
and $q(0)=q_{\infty }$. The property (iii) follows by Lemma \ref{K-property2
. Now if $A_{1}\hookrightarrow A_{0}$ we have $\left\Vert f\right\Vert
_{A_{0}}\leq c\left\Vert f\right\Vert _{A_{1}}$ for any $f\in A_{0}$ and
\begin{equation*}
K(t,f)=\left\Vert f\right\Vert _{A_{0}},
\end{equation*
if $t>c$. The
\begin{equation*}
\left\Vert t^{-\theta }K(t,f)\right\Vert _{L^{q(\cdot )}((c,\infty ),\frac{d
}{t})}\lesssim \left\Vert f\right\Vert _{A_{0}},
\end{equation*
and
\begin{equation*}
\left\Vert t^{-\theta }K(t,f)\right\Vert _{L^{q(\cdot )}((0,\infty ),\frac{d
}{t})}\lesssim \left\Vert t^{-\theta }K(t,f)\right\Vert _{L^{q(\cdot
)}((0,c),\frac{dt}{t})}+\left\Vert f\right\Vert _{A_{0}}.
\end{equation*
Using the fact tha
\begin{equation*}
\left\Vert f\right\Vert _{A_{0}}\lesssim \left\Vert t^{-\theta
_{1}}K(t,f)\right\Vert _{L^{q(\cdot )}((c,\infty ),\frac{dt}{t})},
\end{equation*
and $0<\theta \leq \theta _{1}<1$, we obtai
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\lesssim
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta _{1},q(\cdot )}}.
\end{equation*
So, the property (iv) is proved. Now the property (v) is immediate. The
proof is complete.\quad $\square $
\section{The J-Method}
Let $(A_{0},A_{1})$ be a compatible couple. With $t>0$\ fixed, pu
\begin{equation*}
J(t,f,A_{0},A_{1})=\inf_{f=f_{0}+f_{1}}\left( \left\Vert f_{0}\right\Vert
_{A_{0}},t\left\Vert f_{1}\right\Vert _{A_{1}}\right) ,\quad f\in A_{0}\cap
A_{1}.
\end{equation*
Notice that $J(t,f,A_{0},A_{1})$\ is an equivalent norm on $A_{0}\cap A_{1}$
for a given $t>0$. If there is no danger of confusion, we shall write
J(t,f)=J(t,f,A_{0},A_{1})$. For any $f\in A_{0}\cap A_{1}$, $J(t,f)$ is a
positive, increasing and convex \ function of $t$, such tha
\begin{equation}
J(t,f)\leq \max (1,\frac{t}{s})J(s,f), \label{Est-J}
\end{equation
an
\begin{equation}
K(t,f)\leq \min (1,\frac{t}{s})J(s,f). \label{Est-J1}
\end{equation
Now we define the interpolation space constructed by the $J$-method.
\begin{definition}
Let $\theta \in (0,1)$ and $q\in \mathcal{P}(\mathbb{R})$. Let
(A_{0},A_{1}) $ be a compatible couple. The space $(A_{0},A_{1})_{\theta
,q(\cdot ),J}$ consists of all $f$ in $A_{0}+A_{1}$ that are representable
in the for
\begin{equation}
f=\int_{0}^{\infty }u(t)\frac{dt}{t} \label{rep}
\end{equation
where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$\ an
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}=\inf
\left\Vert t^{-\theta }J(t,u(t))\right\Vert _{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}<\infty ,
\end{equation*
where the infimum is taken over all $u$ such that $\mathrm{\eqref{rep}}$
holds.
\end{definition}
\begin{definition}
Let $\theta \in (0,1)$. The space $(A_{0},A_{1})_{\theta ,\infty ,J}$
consists of all $f$ in $A_{0}+A_{1}$ that are representable in the form
\mathrm{\eqref{rep}}$, where $u(t)$ is measurable with values in $A_{0}\cap
A_{1}$\ an
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,\infty ,J}}=\inf
\sup_{t>0}t^{-\theta }J(t,u(t))<\infty ,
\end{equation*
where the infimum is taken over all $u$\ such that\ $\mathrm{\eqref{rep}}$
holds.
\end{definition}
\begin{lemma}
\label{J-property2}Let $(A_{0},A_{1})$\ be a compatible couple and $f\in
A_{0}+A_{1}$.\ Let $\theta \in (0,1)$\textit{\ and }$q\in \mathcal{P}
\mathbb{R})$ \textit{be log-H\"{o}lder continuous both at the origin and at
infinity. Then }$f\in (A_{0},A_{1})_{\theta ,q(\cdot ),J}$ if and only if
there exist $u_{v}\in A_{0}\cap A_{1}$, $v\in \mathbb{Z}$, with
\begin{equation}
f=\sum_{v=-\infty }^{\infty }u_{v}\quad \text{convergence in }A_{0}\cap
A_{1}, \label{f-representation}
\end{equation
and such that
\begin{equation*}
\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert _{\lambda ^{\theta
,q(0),q_{\infty }}}=\Big(\sum_{v=-\infty }^{0}2^{-v\theta
q(0)}J(2^{v},u_{v})^{q(0)}\Big)^{\frac{1}{q(0)}}+\Big(\sum_{v=1}^{\infty
}2^{-v\theta q_{\infty }}J(2^{v},u_{v})^{q_{\infty }}\Big)^{\frac{1}
q_{\infty }}}<\infty .
\end{equation*
Moreover
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}\approx
\inf_{u_{v}}\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert
_{\lambda ^{\theta ,q(0),q_{\infty }}},
\end{equation*
where the infimum is extended over all sequences $(u_{v})_{v}$ satisfying\
\mathrm{\eqref{f-representation}}$.
\end{lemma}
\textbf{Proof.} Let $f\in (A_{0},A_{1})_{\theta ,q(\cdot ),J}$. Then we have
a representatio
\begin{equation*}
f=\int_{0}^{\infty }u(t)\frac{dt}{t},
\end{equation*
where $u(t)$ is measurable with values in $A_{0}\cap A_{1}$\ an
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}=\inf
\left\Vert t^{-\theta }J(t,u(t))\right\Vert _{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}<\infty .
\end{equation*
We se
\begin{equation*}
u_{v}=\int_{2^{v}}^{2^{v+1}}u(t)\frac{dt}{t},\quad v\in \mathbb{Z}.
\end{equation*
Then we hav
\begin{equation*}
f=\sum_{v=-\infty }^{\infty }u_{v}.
\end{equation*
Let us prove tha
\begin{equation}
S(\{u_{v}\})=\Big(\sum_{v=-\infty }^{0}2^{-v\theta q(0)}\alpha _{v}^{q(0)
\Big)^{\frac{1}{q(0)}}+\Big(\sum_{v=1}^{\infty }2^{-v\theta q_{\infty
}}\alpha _{v}^{q_{\infty }}\Big)^{\frac{1}{q_{\infty }}}\lesssim \left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}, \label{est-J}
\end{equation
with $\alpha _{v}=J(2^{v},u_{v})$, $v\in \mathbb{Z}$. We need only to prove
tha
\begin{equation*}
\sum_{v=1}^{\infty }2^{-v\theta q_{\infty }}\big(\frac{\alpha _{v}}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\big
^{q_{\infty }}\lesssim 1,
\end{equation*
an
\begin{equation*}
\sum_{v=-\infty }^{0}2^{-v\theta q(0)}\big(\frac{\alpha _{v}}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\big)^{q(0)}\lesssim 1.
\end{equation*
First let us prove that
\begin{equation*}
2^{-v\theta q_{\infty }}\Big(\frac{\alpha _{v}}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\Big)^{q_{\infty }}\leq
\int_{2^{v}}^{2^{v+1}}\big(t^{-\theta }J(t,\frac{u(t)}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\big)^{q\left( t\right)
}\frac{dt}{t}+2^{-v}=\delta
\end{equation*
for any $v\in \mathbb{N}$. This claim can be reformulated as showing tha
\begin{equation*}
\Big(\delta ^{-\frac{1}{q_{\infty }}}2^{-v\theta }\frac{\alpha _{v}}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\Big
^{q_{\infty }}\lesssim 1.
\end{equation*
Using the property $\mathrm{\eqref{Est-J}}$, we find tha
\begin{equation*}
\delta ^{-\frac{1}{q_{\infty }}}2^{-v\theta }\frac{\alpha _{v}}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\lesssim
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q_{\infty }}}\tau ^{-\theta }J(\tau
,\frac{u(t)}{\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}
\frac{d\tau }{\tau }.
\end{equation*
By Lemma \ref{DHHR-estimate}
\begin{eqnarray*}
&&\Big(\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q_{\infty }}}\tau ^{-\theta
}J(\tau ,\frac{u(t)}{\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta
,q(\cdot ),J}}}\frac{d\tau }{\tau }\Big)^{q\left( t\right) } \\
&\lesssim &\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{q\left( \tau \right) }
q_{\infty }}}\big(\tau ^{-\theta }J(\tau ,\frac{u(t)}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\big)^{q\left( \tau
\right) }\frac{d\tau }{\tau }+1
\end{eqnarray*
for any $t\in \lbrack 2^{v},2^{v+1}]$. Since, $q$ is \emph{$\log $-}H\"{o
lder continuous at the infinity we find tha
\begin{equation}
\delta ^{-\frac{q\left( \tau \right) }{q_{\infty }}}\approx \delta ^{-1}
\text{ \ \ }\tau \in \lbrack 2^{v},2^{v+1}],\quad v\in \mathbb{N}.
\label{Est-delta}
\end{equation
Therefore, from the definition of $\delta $, we find that the last integral
is dominated by a constant independent on $v\in \mathbb{N}$. Now, let us
prove that
\begin{equation*}
2^{-v\theta q(0)}\Big(\frac{\alpha _{v}}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot )}}}\Big)^{q(0)}\lesssim
\int_{2^{v}}^{2^{v+1}}\big(t^{-\theta }J(t,\frac{f}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\big)^{q\left( t\right) }\frac{dt}{t
+2^{v}=\delta
\end{equation*
for any $v\leq 0$. This claim can be reformulated as showing tha
\begin{equation*}
\Big(\delta ^{-\frac{1}{q(0)}}2^{-v\theta }\frac{\alpha _{v}}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\Big)^{q(0)}\lesssim 1.
\end{equation*
The property $\mathrm{\eqref{Est-J}}$, gives\ tha
\begin{equation*}
\delta ^{-\frac{1}{q(0)}}2^{-v\theta }\frac{\alpha _{v}}{\left\Vert
f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}\leq
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}\tau ^{-\theta }J(\tau ,\frac
u(t)}{\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\frac
d\tau }{\tau }.
\end{equation*
Again by Lemma \ref{DHHR-estimate}
\begin{eqnarray*}
&&\Big(\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{1}{q(0)}}\tau ^{-\theta }J(\tau
,\frac{u(t)}{\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}}
\frac{d\tau }{\tau }\Big)^{q\left( t\right) } \\
&\lesssim &\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{q\left( \tau \right) }{q(0)
}\big(\tau ^{-\theta }J(\tau ,\frac{u(t)}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\frac{d\tau }{\tau }\big)^{q\left(
\tau \right) }\frac{d\tau }{\tau }+1
\end{eqnarray*
for any $t\in \lbrack 2^{v},2^{v+1}]$\ and any $v\leq 0$. We use the
logarithmic decay\ condition at origin\ of $q$ to show tha
\begin{equation*}
\delta ^{-\frac{q\left( \tau \right) }{q(0)}}\approx \delta ^{-1},\text{ \ \
}\tau \in \lbrack 2^{v},2^{v+1}],\quad v\leq 0.
\end{equation*
Therefore and from the definition of $\delta $, we find tha
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\delta ^{-\frac{q\left( \tau \right) }{q(0)}}\big(\tau
^{-\theta }J(\tau ,\frac{u(t)}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\big)^{q\left( \tau \right) }\frac
d\tau }{\tau }\lesssim 1.
\end{equation*
Hence the left-hand side of $\mathrm{\eqref{est-J}}$ can be estimated b
\begin{equation*}
c\int_{0}^{\infty }\big(t^{-\theta }J(t,\frac{u(t)}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\big)^{q\left( t\right) }\frac{dt}{t
+1.
\end{equation*
The first term is bounded since
\begin{equation*}
\Big\|t^{-\theta }J(t,\frac{u(t)}{\left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}})\Big\|_{L^{q(\cdot )}((0,\infty )
\frac{dt}{t})}\leq 1.
\end{equation*
Now in $\mathrm{\eqref{est-J}}$ taking the infimum, we conclude tha
\begin{equation*}
\inf_{u_{v}}\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert
_{\lambda ^{\theta ,q(0),q_{\infty }}}\lesssim \left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}.
\end{equation*
Conversely, assume that
\begin{equation*}
f=\sum_{v=-\infty }^{\infty }u_{v},
\end{equation*
and
\begin{equation*}
S(\{u_{v}\})<\infty .
\end{equation*
Let us prove tha
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}\lesssim
\inf_{w_{v}}S(w_{v})=S,
\end{equation*
where the infimum is taking over all sequences $\{w_{v}\}$ satisfying
\mathrm{\eqref{f-representation}}$. Choos
\begin{equation*}
u(t)=\frac{u_{v}}{\log 2},\quad t\in \lbrack 2^{v},2^{v+1}].
\end{equation*
Then $f=\int_{0}^{\infty }u(t)\frac{dt}{t}$. This claim can be reformulated
as showing tha
\begin{equation*}
\int_{0}^{\infty }\big(t^{-\theta }J(t,\frac{u(t)}{S})\big)^{q\left(
t\right) }\frac{dt}{t}\lesssim 1.
\end{equation*
Now our estimate clearly follows from the inequalitie
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\big(t^{-\theta }J(t,\frac{u(t)}{S})\big)^{q\left(
t\right) }\frac{dt}{t}\lesssim 2^{-v\theta q_{\infty }}\big(\frac{\alpha _{v
}{S}\big)^{q_{\infty }}+2^{-v}=\delta
\end{equation*
for any $v\in \mathbb{N}$ and
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\big(t^{-\theta }J(t,\frac{u(t)}{S})\big)^{q\left(
t\right) }\frac{dt}{t}\lesssim 2^{-v\theta q(0)}\big(\frac{\alpha _{v}}{S
\big)^{q(0)}+2^{v}=\delta
\end{equation*
for any $v\leq 0$. The first claim can be reformulated as showing\ tha
\begin{equation*}
\int_{2^{v}}^{2^{v+1}}\big(\delta ^{-\frac{1}{q(t)}}t^{-\theta }J(t,\frac
u(t)}{S})\big)^{q\left( t\right) }\frac{dt}{t}\lesssim 1.
\end{equation*
We need only to show tha
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}t^{-\theta }J(t,\frac{u(t)}{S})\lesssim 1
\end{equation*
for any $v\in \mathbb{N}$ and any $t\in \lbrack 2^{v},2^{1+v}]$. The
left-hand side is bounded b
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}2^{-\theta v}J(2^{v},\frac{u(t)}{S}).
\end{equation*
From $\mathrm{\eqref{Est-delta}}$ we find that
\begin{equation*}
\delta ^{-\frac{1}{q(t)}}2^{-\theta v}J(2^{v},\frac{u(t)}{S
,A_{0},A_{1})\lesssim \delta ^{-\frac{1}{q_{\infty }}}2^{-\theta v}J(2^{v}
\frac{u_{v}}{S})\leq 1
\end{equation*
for any $v\in \mathbb{N}$. Similarly we estimate the second claim.
We conclude tha
\begin{equation*}
\inf_{u_{v}}\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert
_{\lambda ^{\theta ,q(0),q_{\infty }}}\lesssim \left\Vert f\right\Vert
_{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}.
\end{equation*
\quad $\square $
We shall prove that the spaces generated by the $K$-and $J$-methods are the
same.
\begin{theorem}
\label{Equi-J-and-K}Let $(A_{0},A_{1})$\ be a compatible couple.\ Let
\theta \in (0,1)$\textit{\ and }$q\in \mathcal{P}(\mathbb{R})$ \textit{be
log-H\"{o}lder continuous both at the origin and at infinity. Then
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot ),J}=(A_{0},A_{1})_{\theta ,q(\cdot )},
\end{equation*
with equivalence of norms.
\end{theorem}
\textbf{Proof.} Let $f\in (A_{0},A_{1})_{\theta ,q(\cdot ),J}$ with
f=\int_{0}^{\infty }u(s)\frac{ds}{s}$, where $u(t)$ is measurable with
values in $A_{0}\cap A_{1}$. By $\mathrm{\eqref{Est-J1}}$ we have
\begin{equation*}
K(t,f)\leq \int_{0}^{\infty }K(t,u(s))\frac{ds}{s}\leq \int_{0}^{\infty
}\min (1,\frac{t}{s})J(s,u(s))\frac{ds}{s}.
\end{equation*
Applying Hardy inequality, Lemma \ref{lq-inequality}, we ge
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\lesssim
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}.
\end{equation*
For the converse inequality, Lemma 3.3.2 of \cite{BL76}, and using Theorem
\ref{embedding}, implies the existence of a representation
\begin{equation*}
f=\sum_{v=-\infty }^{\infty }u_{v},
\end{equation*
such that
\begin{equation*}
J(2^{v},u_{v})\leq (\gamma +\varepsilon )K(2^{v},f)
\end{equation*
for any $v\in \mathbb{Z}$, $\varepsilon >0$ and $\gamma $ is a universal
constant less than or equal $3$. By Lemmas \ref{K-property2} and \re
{J-property2} we ge
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot ),J}}\lesssim
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}.
\end{equation*
This completes the proof of this theorem.$\quad \square $
\begin{theorem}
\label{Density}Let $(A_{0},A_{1})$\ be a compatible couple.\ Let $\theta \in
(0,1)$\textit{\ and }$q\in \mathcal{P}(\mathbb{R})$ \textit{be log-H\"{o
lder continuous both at the origin and at infinity. Then }$A_{0}\cap A_{1}$
is dense in $(A_{0},A_{1})_{\theta ,q(\cdot )}$.
\end{theorem}
\textbf{Proof.} Let $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$. From Theorem
\ref{Equi-J-and-K} we hav
\begin{equation*}
f=\sum_{v=-\infty }^{\infty }u_{v},
\end{equation*
where $u_{v}$, $v\in \mathbb{Z}$ is measurable with values in $A_{0}\cap
A_{1}$ an
\begin{equation*}
\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert _{\lambda ^{\theta
,q(0),q_{\infty }}}<\infty .
\end{equation*
The
\begin{eqnarray*}
&&\Big\|f-\sum_{|v|\leq N}u_{v}\Big\|_{(A_{0},A_{1})_{\theta ,q(\cdot )}} \\
&\leq &\Big(\sum_{v=N}^{\infty }2^{-v\theta q_{\infty
}}J(2^{v},u_{v})^{q_{\infty }}\Big)^{\frac{1}{q_{\infty }}}+\Big
\sum_{v=-\infty }^{-N}2^{-v\theta q(0)}J(2^{v},u_{v})^{q(0)}\Big)^{\frac{1}
q(0)}}.
\end{eqnarray*
Therefore
\begin{equation*}
\Big\|f-\sum_{|v|\leq N}u_{v}\Big\|_{(A_{0},A_{1})_{\theta ,q(\cdot )}},
\end{equation*
which tends to zero if $N\longrightarrow \infty $.$\quad \square $
\begin{definition}
Let $\theta \in \lbrack 0,1]$. Let $(A_{0},A_{1})$ be a compatible couple of
normed vector spaces. Suppose that $X$ is an intermediate space with respect
to $(A_{0},A_{1})$. Then we say that \newline
(i) $X$ is of class $\mathscr{C}_{K}(\theta ;A_{0},A_{1})$ if
K(t,f;A_{0},A_{1})\leq Ct^{\theta }\left\Vert f\right\Vert _{X},\quad f\in
X; $\newline
(ii) $X$ is of class $\mathscr{C}_{J}(\theta ;A_{0},A_{1})$ if $\left\Vert
f\right\Vert _{X}\leq Ct^{-\theta }J(t,f;A_{0},A_{1}),\quad f\in A_{0}\cap
A_{1}.$\newline
(iii) We say that $X$ is of class $\mathscr{C}(\theta ;A_{0},A_{1})$ if $X$
is of class $\mathscr{C}_{K}(\theta ;A_{0},A_{1})$ and of class\ $\mathscr{C
_{J}(\theta ;A_{0},A_{1})$.
\end{definition}
Let $q\in \mathcal{P}(\mathbb{R})$\textit{. }From \cite[Theorem 3.5.2]{BL76}
and Proposition 3 we see that $(A_{0},A_{1})_{\theta ,q(\cdot )}$ is of
class $\mathscr{C}(\theta ;A_{0},A_{1})$ if $\theta \in (0,1)$.
We are now ready to prove the reiteration theorem, which is one of the most
important general results in interpolation theory.
\begin{theorem}
\label{The reiteration theorem}Let $q\in \mathcal{P}(\mathbb{R})$ \textit{be
log-H\"{o}lder continuous both at the origin and at infinity. }Let
(A_{0},A_{1})$ and $(X_{0},X_{1})$ be two compatible couples of normed
linear spaces, and assume that $X_{i}$ ($i=0,1$) are complete and of class
\mathscr{C}(\theta _{i};A_{0},A_{1})$, where $\theta _{0},\theta _{1}\in
\lbrack 0,1]$ and $\theta _{0}\neq \theta _{1}$. Pu
\begin{equation*}
\theta =(1-\eta )\theta _{0}+\eta \theta _{1},\quad \eta \in (0,1).
\end{equation*
The
\begin{equation*}
(A_{0},A_{1})_{\theta ,q(\cdot )}=(X_{0},X_{1})_{\eta ,q(\cdot )}
\end{equation*
with equivalence of norms. In particular, if $\theta _{0},\theta _{1}\in
(0,1)$, $q_{0},q_{1}\in \mathcal{P}(\mathbb{R})$ \textit{are log-H\"{o}lder
continuous both at the origin and at infinity} and $(A_{0},A_{1})_{\theta
_{i},q_{i}(\cdot )}$ are complete then
\begin{equation*}
\left( (A_{0},A_{1})_{\theta _{0},q_{0}(\cdot )},(A_{0},A_{1})_{\theta
_{1},q_{1}(\cdot )}\right) _{\eta ,q(\cdot )}=(A_{0},A_{1})_{\theta ,q(\cdot
)}
\end{equation*
wher
\begin{equation*}
\frac{1}{q(\cdot )}=\frac{\theta _{0}}{q_{0}(\cdot )}+\frac{\theta _{1}}
q_{1}(\cdot )}.
\end{equation*}
\end{theorem}
\textbf{Proof. }We will do the proof in two steps.
\textit{Step 1.} Let us prove tha
\begin{equation}
(X_{0},X_{1})_{\eta ,q(\cdot )}\hookrightarrow (A_{0},A_{1})_{\theta
,q(\cdot )}. \label{First-emb}
\end{equation
Let $f\in (X_{0},X_{1})_{\eta ,q(\cdot )}$. The
\begin{equation*}
f=f_{0}+f_{1},\quad f_{0}\in X_{0},f_{1}\in X_{1}.
\end{equation*
Since $X_{i}$ ($i=0,1$) are of class $\mathscr{C}(\theta _{i};A_{0},A_{1})$
we hav
\begin{eqnarray*}
K(t,f;A_{0},A_{1}) &\leq &K(t,f_{0};A_{0},A_{1})+K(t,f_{1};A_{0},A_{1})\leq
c(t^{\theta _{0}}\left\Vert f_{0}\right\Vert _{X_{0}}+t^{\theta
_{1}}\left\Vert f_{1}\right\Vert _{X_{1}}) \\
&\leq &ct^{\theta _{0}}K(t^{\theta _{1}-\theta _{0}},f;X_{0},X_{1}).
\end{eqnarray*
Therefore, from Theorem \ref{K-property2}, we get
\begin{eqnarray*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}} &\lesssim &\Big
\int_{0}^{1}t^{(\theta _{0}-\theta )q(0)}K(t^{\theta _{1}-\theta
_{0}},f;X_{0},X_{1})^{q(0)}\frac{dt}{t}\Big)^{\frac{1}{q(0)}} \\
&&+\Big(\int_{1}^{\infty }t^{(\theta _{0}-\theta )q_{\infty }}K(t^{\theta
_{1}-\theta _{0}},f;X_{0},X_{1})^{q_{\infty }}\frac{dt}{t}\Big)^{\frac{1}
q_{\infty }}}.
\end{eqnarray*
Putting\ $s=t^{\theta _{1}-\theta _{0}}$ and observing that $\eta =\frac
\theta -\theta _{0}}{\theta _{1}-\theta _{0}}$ we find tha
\begin{equation*}
\left\Vert f\right\Vert _{(A_{0},A_{1})_{\theta ,q(\cdot )}}\lesssim
\left\Vert f\right\Vert _{(X_{0},X_{1})_{\eta ,q(\cdot )}},
\end{equation*
which gives $\mathrm{\eqref{First-emb}}$.
\textit{Step 2.} Let us prove tha
\begin{equation}
(A_{0},A_{1})_{\theta ,q(\cdot )}\hookrightarrow (X_{0},X_{1})_{\eta
,q(\cdot )}. \label{Second-emb}
\end{equation
Assume that $f\in (A_{0},A_{1})_{\theta ,q(\cdot )}$. We choose a
representation
\begin{equation*}
f=\sum_{v=-\infty }^{\infty }u_{v},
\end{equation*
where $u_{v}$, $v\in \mathbb{Z}$ is measurable with values in $A_{0}\cap
A_{1}$ an
\begin{equation*}
\left\Vert \left( J(2^{v},u_{v})\right) _{v}\right\Vert _{\lambda ^{\theta
,q(0),q_{\infty }}}<\infty .
\end{equation*
Applying $\mathrm{\eqref{Est-J1}}$, and that $X_{i}$ ($i=0,1$) are of class
\mathscr{C}(\theta _{i};A_{0},A_{1})$ we get for any $j\in \mathbb{Z}$
\begin{eqnarray*}
&&2^{(\theta _{0}-\theta )j}K(2^{(\theta _{1}-\theta _{0})j},f;X_{0},X_{1})
\\
&\leq &2^{(\theta _{0}-\theta )j}\sum_{v=-\infty }^{\infty }K(2^{(\theta
_{1}-\theta _{0})j},u_{v};X_{0},X_{1}) \\
&\leq &2^{(\theta _{0}-\theta )j}\sum_{v=-\infty }^{\infty }\min \left(
1,2^{(j-v)(\theta _{1}-\theta _{0})}\right) J(2^{v(\theta _{1}-\theta
_{0})},u_{v};X_{0},X_{1}) \\
&\leq &2^{-\theta j}\sum_{v=-\infty }^{\infty }\min \left( 2^{(j-v)\theta
_{0}},2^{(j-v)\theta _{1}}\right) J(2^{v},u_{v};A_{0},A_{1}).
\end{eqnarray*
The last term can be rewritten u
\begin{equation}
\sum_{v=-\infty }^{j}2^{(j-v)(\theta _{0}-\theta )}2^{-v\theta
}J(2^{v},u_{v};A_{0},A_{1})+\sum_{v=j+1}^{\infty }2^{(j-v)(\theta
_{1}-\theta )}2^{-v\theta }J(2^{v},u_{v};A_{0},A_{1}) \label{sum-equi}
\end{equation
for any $j\in \mathbb{Z}$. We treat the case where\ $j\geq 0$. The first sum
can be rewritten u
\begin{eqnarray*}
&&\sum_{v=-\infty }^{0}2^{(j-v)(\theta _{0}-\theta )}2^{-v\theta
}J(2^{v},u_{v};A_{0},A_{1})+\sum_{v=1}^{j}2^{(j-v)(\theta _{0}-\theta
)}2^{-v\theta }J(2^{v},u_{v};A_{0},A_{1}) \\
&\lesssim &2^{j(\theta _{0}-\theta )}\sup_{v\leq 0}(2^{-v\theta
}J(2^{v},u_{v};A_{0},A_{1}))+\sum_{v=1}^{j}2^{(j-v)(\theta _{0}-\theta
)}2^{-v\theta }J(2^{v},u_{v};A_{0},A_{1}).
\end{eqnarray*
Applying Lemma \ref{Hardy-inequality} we ge
\begin{equation*}
\left\Vert \left( 2^{(\theta _{0}-\theta )j}K(2^{(\theta _{1}-\theta
_{0})j},f;X_{0},X_{1})\right) _{j\geq 1}\right\Vert _{\lambda ^{\theta
,q(0),q_{\infty }}}\leq \left\Vert \left( J(2^{v},u_{v})\right)
_{v}\right\Vert _{\lambda ^{\theta ,q(0),q_{\infty }}}.
\end{equation*
Now if $j\leq 0$, the second sum of $\mathrm{\eqref{sum-equi}}$ can be
rewritten u
\begin{eqnarray*}
&&\sum_{v=j+1}^{0}2^{(j-v)(\theta _{1}-\theta )}2^{-v\theta
}J(2^{v},u_{v};A_{0},A_{1})+\sum_{v=1}^{\infty }2^{(j-v)(\theta _{1}-\theta
)}2^{-v\theta }J(2^{v},u_{v};A_{0},A_{1}) \\
&\leq &\sum_{v=j+1}^{0}2^{(j-v)(\theta _{1}-\theta )}2^{-v\theta
}J(2^{v},u_{v};A_{0},A_{1})+2^{j(\theta _{1}-\theta )}\sup_{v\geq 1}\left(
2^{-v\theta }J(2^{v},u_{v};A_{0},A_{1})\right) .
\end{eqnarray*
Applying again Lemma \ref{Hardy-inequality} we ge
\begin{equation*}
\left\Vert \left( 2^{(\theta _{0}-\theta )j}K(2^{(\theta _{1}-\theta
_{0})j},f;X_{0},X_{1})\right) _{j\leq 0}\right\Vert _{\lambda ^{\theta
,q(0),q_{\infty }}}\leq \left\Vert \left( J(2^{v},u_{v})\right)
_{v}\right\Vert _{\lambda ^{\theta ,q(0),q_{\infty }}}.
\end{equation*
This prove the embedding $\mathrm{\eqref{Second-emb}}$ by taking the infimum
in view of the Theorem \ref{J-property2} and the fact\ tha
\begin{equation*}
\left\Vert f\right\Vert _{(X_{0},X_{1})_{\eta ,q(\cdot )}}\approx \Big
\sum_{j=-\infty }^{0}2^{(\theta _{0}-\theta )q(0)}\alpha _{j}^{q(0)}\Big)^
\frac{1}{q(0)}}+\Big(\sum_{j=1}^{\infty }2^{(\theta _{0}-\theta )q_{\infty
}}\alpha _{j}^{q_{\infty }}\Big)^{\frac{1}{q_{\infty }}},
\end{equation*
wher
\begin{equation*}
\alpha _{j}=K(2^{(\theta _{1}-\theta _{0})j},f;X_{0},X_{1}),\quad j\in
\mathbb{Z}.
\end{equation*
This completes the proof of Theorem \ref{The reiteration theorem}.$\quad
\square $
\section{Application}
In this section, we give a simple application of the results of the previous
sections. We will present various real interpolation formulas in Besov
spaces with variable indices. The symbol $\mathcal{S}(\mathbb{R}^{n})$ is
used in place of the set of all Schwartz functions on $\mathbb{R}^{n}$. We
denote by $\mathcal{S}^{\prime }(\mathbb{R}^{n})$ the dual space of all
tempered distributions on $\mathbb{R}^{n}$. The Fourier transform of a
Schwartz function $f$ is denoted by $\mathcal{F}f$. To define the variable
Besov spaces, we first need the concept of a smooth dyadic resolution of
unity. Let $\Psi $\ be a function\ in $\mathcal{S}
\mathbb{R}
^{n})$\ satisfying $\Psi (x)=1$\ for\ $\left\vert x\right\vert \leq 1$\ and\
$\Psi (x)=0$\ for\ $\left\vert x\right\vert \geq 2$.\ We define $\varphi
_{0} $ and $\varphi _{1}$ by $\mathcal{F}\varphi _{0}(x)=\Psi (x)$,
\mathcal{F}\varphi _{1}(x)=\Psi (x)-\Psi (2x)$\ and
\begin{equation*}
\mathcal{F}\varphi _{j}(x)=\mathcal{F}\varphi _{1}(2^{-j}x)\quad \text
\textit{for}}\quad j=2,3,....
\end{equation*
Then $\{\mathcal{F}\varphi _{j}\}_{j\in \mathbb{N}_{0}}$\ is a smooth dyadic
resolution of unity, $\sum_{j=0}^{\infty }\mathcal{F}\varphi _{j}(x)=1$ for
all $x\in \mathbb{R}^{n}$.\ Thus we obtain the Littlewood-Paley
decomposition $f=\sum_{j=0}^{\infty }\varphi _{j}\ast f$ of all $f\in
\mathcal{S}^{\prime }
\mathbb{R}
^{n})$ $($convergence in $\mathcal{S}^{\prime }
\mathbb{R}
^{n}))$.
Let $p,q\in \mathcal{P}(\mathbb{R}^{n})$. The mixed Lebesgue-sequence space
\ell _{>}^{q(\cdot )}(L^{p(\cdot )})$ is defined on sequences of $L^{p(\cdot
)}$-functions by the modula
\begin{equation*}
\varrho _{\ell _{>}^{q(\cdot )}(L^{p\left( \cdot \right)
})}((f_{v})_{v}):=\sum\limits_{v=1}^{\infty }\inf \Big\{\lambda
_{v}>0:\varrho _{p(\cdot )}\Big(\frac{f_{v}}{\lambda _{v}^{1/q(\cdot )}}\Big
\leq 1\Big\}.
\end{equation*
The (quasi)-norm is defined from this as usual
\begin{equation}
\left\Vert \left( f_{v}\right) _{v}\right\Vert _{\ell _{>}^{q(\cdot
)}(L^{p\left( \cdot \right) })}:=\inf \Big\{\mu >0:\varrho _{\ell ^{q(\cdot
)}(L^{p(\cdot )})}\Big(\frac{1}{\mu }(f_{v})_{v}\Big)\leq 1\Big\}.
\label{mixed-norm}
\end{equation
If $q^{+}<\infty $, then we can replace $\mathrm{\eqref{mixed-norm}}$ by the
simpler expression $\varrho _{\ell _{>}^{q(\cdot )}(L^{p(\cdot
)})}((f_{v})_{v}):=\sum\limits_{v=1}^{\infty }\left\Vert |f_{v}|^{q(\cdot
)}\right\Vert _{\frac{p(\cdot )}{q(\cdot )}}$. The case $p:=\infty $ can be
included by replacing the last modular by $\varrho _{\ell _{>}^{q(\cdot
)}(L^{\infty })}((f_{v})_{v}):=\sum\limits_{v=1}^{\infty }\big\|\left\vert
f_{v}\right\vert ^{q(\cdot )}\big\|_{\infty }$.
We define the following class of variable exponents $\mathcal{P}^{\mathrm{lo
}}(\mathbb{R}^{n}):=\big\{p\in \mathcal{P}:\frac{1}{p}\in C^{\log }\big\}$,
were introduced in $\mathrm{\cite[Section \ 2]{DHHMS}}$. We define
1/p_{\infty }:=\lim_{|x|\rightarrow \infty }1/p(x)$\ and we use the
convention $\frac{1}{\infty }=0$. Note that although $\frac{1}{p}$ is
bounded, the variable exponent $p$ itself can be unbounded.
We state the definition of the spaces $B_{p(\cdot ),q(\cdot )}^{s(\cdot )}$,
which introduced and investigated in \cite{AH}.
\begin{definition}
\label{11}\textit{Let }$\left\{ \mathcal{F}\varphi _{j}\right\}
_{j=0}^{\infty }$\textit{\ be a resolution of unity}, $s:\mathbb{R
^{n}\rightarrow \mathbb{R}$ and $p,q\in \mathcal{P}(\mathbb{R}^{n})$.\textit
\ The Besov space }$B_{p(\cdot ),q(\cdot )}^{s(\cdot )}$\textit{\ consists
of all distributions }$f\in \mathcal{S}^{\prime }
\mathbb{R}
^{n})$\textit{\ such that
\begin{equation*}
\left\Vert f\right\Vert _{B_{p(\cdot ),q(\cdot )}^{s(\cdot )}}:=\left\Vert
(2^{js(\cdot )}\varphi _{j}\ast f)_{j}\right\Vert _{\ell _{>}^{q(\cdot
)}(L^{p\left( \cdot \right) })}<\infty .
\end{equation*}
\end{definition}
Taking $s\in \mathbb{R}$ and $q\in (0,\infty ]$ as constants we derive the
spaces $B_{p(\cdot ),q}^{s}$ studied by Xu in \cite{Xu08}. We refer the
reader to the recent papers \cite{D3}, \cite{KV121},\cite{KV122} and\ \cit
{IN14} for further details, historical remarks and more references on these
function spaces. For any $p,q\in \mathcal{P}_{0}^{\log }(\mathbb{R}^{n})$
and $s\in C_{\text{loc}}^{\log }$, the space $B_{p(\cdot ),q(\cdot
)}^{s(\cdot )}$ does not depend on the chosen smooth dyadic resolution o
\textit{\ }unity $\{\mathcal{F}\varphi _{j}\}_{j\in \mathbb{N}_{0}}$ (in the
sense of\ equivalent quasi-norms) and
\begin{equation*}
\mathcal{S}(\mathbb{R}^{n})\hookrightarrow B_{p(\cdot ),q(\cdot )}^{s(\cdot
)}\hookrightarrow \mathcal{S}^{\prime }(\mathbb{R}^{n}).
\end{equation*
Moreover, if $p,q,s$ are constants, we re-obtain the usual Besov spaces
B_{p,q}^{s}$, studied in detail in \cite{Pe76}, \cite{SiRu96}, \cite{T1},
\cite{T2} and \cite{T31}.
Applying Lemma \ref{K-property2} and using the same arguments of \cite
Theorem 3.1]{AH14} we obtain.
\begin{theorem}
\label{real-int-Besov}Let $\theta \in (0,1)$\textit{\ and }$q\in \mathcal{P}
\mathbb{R})$ \textit{be log-H\"{o}lder continuous both at the origin and at
infinity\ with }$q(0)=q_{\infty }$. Let $p,q_{0},q_{1}\in \mathcal{P}^{\log
}(\mathbb{R}^{n})$ and $\alpha _{0},\alpha _{1}\in C_{\mathrm{loc}}^{\log }
. If $0\neq \alpha _{0}-\alpha _{1}$ is constant, the
\begin{equation*}
(B_{p(\cdot ),q_{0}(\cdot )}^{\alpha _{0}(\cdot )},B_{p(\cdot ),q_{1}(\cdot
)}^{\alpha _{1}(\cdot )})_{\theta ,q(\cdot )}=B_{p(\cdot ),q(0)}^{\alpha
(\cdot )}
\end{equation*
with $\alpha (\cdot )=(1-\theta )\alpha _{0}(\cdot )+\theta \alpha
_{1}(\cdot )$. Moerove
\begin{equation*}
(B_{p(\cdot ),r_{0}}^{\alpha _{0}(\cdot )},B_{p(\cdot ),r_{1}}^{\alpha
_{1}(\cdot )})_{\theta ,q(\cdot )}=B_{p(\cdot ),q(0)}^{\alpha (\cdot )},
\end{equation*
with $r_{0},r_{1}\in \lbrack 1,\infty ]$\ and
\begin{equation*}
\frac{1}{q(0)}=\frac{1-\theta }{r_{0}}+\frac{\theta }{r_{1}}.
\end{equation*}
\end{theorem}
Now we present some interpolation results in variable exponent Lorentz
spaces $\mathcal{L}^{p(\cdot ),q(\cdot )}(\mathbb{R}^{n})$ introduced by
\cite{EKS08}.
\begin{definition}
If $f$ is a measurable function on $\mathbb{R}^{n}$, we define the
non-increasing rearrangement of $f$ throug
\begin{equation*}
f^{\ast }(t)=\sup \{\lambda >0:m_{f}(\lambda )>t\}
\end{equation*
where $m_{f}$ is the distribution function of $f$.
\end{definition}
\begin{definition}
Let $p,q\in \mathcal{P}(\mathbb{R})$. By $\mathcal{L}^{p(\cdot ),q(\cdot )}
\mathbb{R}^{n})$\ we denote the space of functions f on $\mathbb{R}^{n}$
such that
\begin{equation*}
\left\Vert f\right\Vert _{\mathcal{L}^{p(\cdot ),q(\cdot )}(\mathbb{R}^{n})}
\big\|t^{\frac{1}{p(t)}-\frac{1}{q(t)}}f^{\ast }(t)\big\|_{L^{q(\cdot )}
\mathbb{[}0,\infty ))}<\infty .
\end{equation*}
\end{definition}
We refer to the recent\ paper \cite{EKS08} for further details on these
scales of spaces. We present an equivalent quasi-norm for the space
\mathcal{L}^{p(\cdot ),q(\cdot )}(\mathbb{R}^{n})$, where the proof is quite
similar to that for Lemma \ref{K-property2}.
\begin{lemma}
\label{Lorentz-property}\textit{Let }$p,q\in \mathcal{P}(\mathbb{R})$
\textit{be log-H\"{o}lder continuous both at the origin and at infinity. The
\begin{equation*}
\left\Vert f\right\Vert _{\mathcal{L}^{p(\cdot ),q(\cdot )}(\mathbb{R
^{n})}\approx \Big(\sum_{v=-\infty }^{0}2^{-v\frac{q(0)}{p(0)}}\left(
f^{\ast }(2^{v})\right) ^{q(0)}\Big)^{\frac{1}{q(0)}}+\Big
\sum_{v=1}^{\infty }2^{-v\frac{q_{\infty }}{p_{\infty }}}\left( f^{\ast
}(2^{v})\right) ^{q_{\infty }}\Big)^{\frac{1}{q_{\infty }}}.
\end{equation*
Moreover
\begin{equation*}
\left\Vert f\right\Vert _{\mathcal{L}^{p(\cdot ),q(\cdot )}(\mathbb{R
^{n})}\approx \Big(\int_{0}^{1}t^{-\frac{q(0)}{p(0)}}\left( f^{\ast
}(t)\right) ^{q(0)}\frac{dt}{t}\Big)^{\frac{1}{q(0)}}+\Big(\int_{1}^{\infty
}t^{-\frac{q_{\infty }}{p_{\infty }}}\left( f^{\ast }(t)\right) ^{q_{\infty
}}\frac{dt}{t}\Big)^{\frac{1}{q_{\infty }}}.
\end{equation*}
\end{lemma}
Applying this lemma and \cite[Theorem 5.2.1]{BL76} we obtain.
\begin{theorem}
\label{real-int-Lorentz}Let $\theta \in (0,1)$\textit{\ and }$q\in \mathcal{
}(\mathbb{R})$ \textit{be log-H\"{o}lder continuous both at the origin and
at infinity\ with }$q(0)=q_{\infty }$. The
\begin{equation*}
(L^{1},L_{\infty })_{\theta ,q(\cdot )}=\mathcal{L}^{p,q(\cdot )}(\mathbb{R
^{n}),
\end{equation*
with $p=\frac{1}{1-\theta }$.
\end{theorem}
| {
"timestamp": "2017-03-16T01:07:59",
"yymm": "1703",
"arxiv_id": "1703.04973",
"language": "en",
"url": "https://arxiv.org/abs/1703.04973",
"abstract": "We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable Besov and Lorentz spaces as introduced recently in Almeida and Hästö (J. Funct. Anal. 258 (5) 1628--2655, 2010) and L. Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407--420).",
"subjects": "Functional Analysis (math.FA)",
"title": "Real interpolation with variable exponent",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075701109192,
"lm_q2_score": 0.7371581626286833,
"lm_q1q2_score": 0.7092254486341123
} |
https://arxiv.org/abs/2004.11160 | Calculating permutation entropy without permutations | A method for analyzing sequential data sets, similar to the permutation entropy one, is discussed. The characteristic features of this method are as follows: it preserves information about equal values, if any, in the embedding vectors; it is exempt of combinatorics; it delivers the same entropy value as does the permutation method, provided the embedding vectors do not have equal components. In the latter case this method can be used instead of the permutation one. If embedding vectors have equal components this method could be more precise in discriminating between similar data sets. | \section{Introduction}
Due to technical
progress in the areas of sensors and storage devices a huge amount of raw data about time course of different processes, such as ECG, EEG, climate data recordings, stock market data have become available.
These data are redundant. The data processing and classification, aimed at
extracting meaningful for nonspecialist characteristics, is based on reducing the excess of redundancy.
As a result, a new data is obtained, small in size and digestible by a human being.
Examples of those reduced data for time series can be mean value, variance,
Liapunov exponents, correlation dimension, attractor dimension and others.
A remarkable method suitable for reducing the excess of redundancy in time series
has been proposed by Ch.Bandt and B.Pompe in
\cite{Bandt2002}, known as permutation entropy. This method is simple and transparent, is robust with respect to monotonic distortions of the raw data, and is suitable for estimating the dynamical complexity
of the underlying dynamical process. Many interesting results, e.g.
\cite{Porta2001,Zanin2012,Bariviera2015,Tylov2018}, have been obtained with
straightforward application of the permutation entropy methodology in its initial form, as it is
described in \cite{Bandt2002}.
Nevertheless, this method is subjected to a critique for not taking into account absolute values of
the raw data and for not treating properly a possibility of having equal values in the embedding vector (ties),
\cite{Zunino2017,Cuesta-Frau2018}. In this connection, it should be taken into account that any
redundancy
reduction method leaves out some type of information, which may be useless for one process/task and may carry
useful information for another one. In the latter case, the bare idea of \cite{Bandt2002}
about how to treat equal values can/should be
modified in order to meet a purpose of concrete situation. Examples of such a modification can be found
in \cite{Azami2016,Chen2019} for taking into account absolute values, or in \cite{Bian2012,Haruna2013} for treating equal values. Interesting modification of the permutation entropy method has been proposed in
\cite{Berger2017} for 3‑tuple EEG data.
In the standard permutation entropy methodology, it is preferable that embedding vectors
have all their components different. Otherwise, they cannot be plainly symbolized by a permutation
without using additional rules, which actually treat equal values as not being such.
Situation with equal values in the embedding vector may arise for high embedding dimension,
for crude quantization of measured data, for very long data sequences
and when observed dynamical system has intrinsically
only a small number of possible outputs.
This note is aimed at discussing a slightly different symbolization technique of embedding vectors,
which does not refer to combinatorics, and which is capable of preserving information about equal values
in embedding vectors. Instead of permutation, an embedding vector is emblematized with a single integer number of base $D$, where $D$ is the embedding dimension. In the case of no ties (no equal components in the embedding vectors) the technique is equivalent to the standard permutation entropy methodology.
In the opposite case, it may discriminate between similar data sets better than the permutation
entropy method does.
\section{Permutation entropy}\label{P}
Consider a finite sequence
\begin{equation}\label{raw}
\mathsf{X} = (x_0,x_1,\dots, x_{N-1}),\quad x_i\in \mathbb{R}^1, \quad i=0,1,2,\dots N-1,
\end{equation}
of measurements. By choosing the embedding dimension $D<N$ the data
(\ref{raw}) can be embedded into a $D$-dimensional space by picking out consecutive
$D$-tuples from $\mathsf{X}$. As a result, a set of $D$-dimensional
embedding vectors is obtained:
\begin{equation}\label{Draw}
\mathbf{V} = \{V_0,V_1,\dots, V_{N-D}\},\quad V_i\in \mathbb{R}^D, \quad i=0,1,2,\dots N-D,
\end{equation}
where each vector has the following form:
\begin{align}\nonumber
V_0 =& \,(x_0,x_1,x_2\dots, x_{D-1}),\quad\dots\quad,
\\\label{Dvec}
V_i=& \,(x_i,x_{i+1},x_{i+2},\dots, x_{i+D-1}),\quad\dots\quad ,
\\\nonumber
V_{N-D}=& \,(x_{N-D},x_{N+1-D},\dots, x_{N-1}).
\end{align}
An additional parameter of the embedding procedure is delay $\tau =1,2,\dots$. In the above definition,
we put $\tau=1$ for simplicity. With $\tau\ne1$ one would have
$V_i= (x_i,x_{i+\tau},x_{i+2\tau},\dots, x_{i+(D-1)\tau})$ instead of (\ref{Dvec}).
The data represented in (\ref{Draw}) and/or (\ref{Dvec}) is even more redundant than that
represented in (\ref{raw}) since, for $D\ll N$, most data values from (\ref{raw}) are represented in (\ref{Dvec})
$D$ times. In the permutation entropy technique \cite{Bandt2002}, each embedding vector from
(\ref{Draw}) and/or (\ref{Dvec}) is replaced with a permutation $\pi$ of $D$ integers \{0,1,2,\dots,$D-1$\},
which is defined by the order pattern of values composing the vector. For any embedding vector
$V=(x_0,x_1,\dots,x_{D-1})$ the permutation $\pi$, which symbolizes it,
is calculated as follows. Arrange all components
of $V$ either in the descending, \cite{Keller2014}:
\begin{equation}\label{desc}
V=(x_0,x_1,\dots,x_{D-1}) \rightarrow V_\pi=(x_{r_0},x_{r_1},\dots,x_{r_{D-1}}),\,
x_{r_0}>x_{r_1}>\dots>x_{r_{D-1}},
\end{equation}
or in the ascending, \cite{Haruna2013,Gutjahr2020}:
\begin{equation}\label{asc}
V=(x_0,x_1,\dots,x_{D-1}) \rightarrow V_\pi=(x_{r_0},x_{r_1},\dots,x_{r_{D-1}}),\,
x_{r_0}<x_{r_1}<\dots<x_{r_{D-1}}
\end{equation}
order\footnote{Actually, in (\ref{desc}), (\ref{asc}) equal values (ties) are as well admitted.
Here, we exclude such a possibility for the sake of clarity. The equal values are discussed in
the next section.} keeping their subscripts unchanged.
The permutation $\pi$ which corresponds to $V$ is obtained as the row of the subscripts in the
rearranged vector $V_\pi$ from either (\ref{desc}), or (\ref{asc}):
\begin{equation}\label{pi}
\pi\equiv\pi(V)=(r_0,r_1,\dots,r_{D-1}).
\end{equation}
From the set of embedding vectors $\mathbf{V}$, calculate a new set $\Pi$ of order
patterns by replacing each vector in (\ref{Draw}) by the corresponding permutation:
\begin{equation}\nonumber
\Pi=\,\{\pi_0,\pi_1,\dots,\pi_{N-D}\}.
\end{equation}
Now, empirical probability of each permutation, $p(\pi_i)$, can be obtained by
dividing the number of occurrences of $\pi_i$ in the $\Pi$ by the total number of elements
in the $\Pi$. The permutation entropy of $\mathbf{V}$ is the Shannon entropy of the probability
distribution $p(\pi_i)$:
\begin{equation}\nonumber
H(\mathbf{V})\equiv H(\Pi)=-\sum\limits_{i=0}^{K-1}p(\pi_i)\log(p(\pi_i)),
\end{equation}
where $K$ is the number of different permutations in the $\Pi$.
\subsection{Treatment of equal values}\label{equal0}
Equal values in an embedding vector are, to an extent, inconvenient.
Indeed, if $x_{r}=x_{s}$ for some $0\le r,s< D$ in a vector $V=(x_0,x_1,\dots,x_{D-1})$,
then $r$ and $s$ should be placed side by side in the permutation (\ref{pi}),
but which one should go first?
Due to sameness of values it is impossible to uniquely determine a corresponding permutation
without introducing additional rules.
In some cases the possibility of equal values can be ignored due to their low probability.
This is reasonable when the embedding dimension is low, and/or a chaotic process data are
recorded with high precision, see \cite{Bandt2002,Bandt2005,Aziz2005}.
If equal values are inevitable, the following rule is applied\footnote{In some cases, e.g. \cite{Bian2012,Haruna2013}, the opposite inequality sign is used here.}
\begin{equation}\label{rule}
\text{\bf if }x_s=x_r \text{ \bf and } s>r \text{ \bf than } s \text{ goes first.}
\end{equation}
The rule (\ref{rule}) has different meaning depending of whether (\ref{desc}) or (\ref{asc})
convention is used. Namely, in the case of (\ref{desc}), an embedding vector with all
components equal will be equivalent to a vector with monotonically ascending components.
If (\ref{asc}) is adopted, then that same vector will be equivalent to a vector with
monotonically descending components, see Fig. \ref{asdes}.
\begin{figure}[t]
\includegraphics[width=0.3\textwidth]{Fig1a.pdf}\hfill
\includegraphics[width=0.3\textwidth]{Fig1b.pdf}\hfill
\includegraphics[width=0.3\textwidth]{Fig1c.pdf}
\caption{\label{asdes}In the standard permutation entropy symbolization,
a sequence of same values ({\tt a}) is
equivalent either to ascending ({\tt b}), or descending ({\tt c}) sequence, if either (\ref{desc}),
or (\ref{asc}) convention is used. }
\end{figure}
Without knowing a real system, it is not clear which case is better and whether it is
good or bad to label a sequence of same values as being decreasing or increasing.
Actually, the permutation symbolization technique aims at reducing redundancy.
Discrimination between constant and either increasing, or decreasing
sequences of data may appear to be excessive in some cases.
On the other hand, when a system
generating data has a few possible outputs, or the data was subjected to a crude
quantization, or embedding dimension is large,
it may happen to be useful if presence of
equal values in the embedding vector results in order pattern preserving this fact.
One possible approach to do this is discussed in the next section.
\section{Arithmetic entropy}
\subsection{Symbolization}\label{Sym}
The following symbolization is aimed to keep information about equal values
in embedding vectors. Having a vector $V=(x_0,x_1,\dots,x_{D-1})$ construct a
sequence of integers $\alpha$:
\begin{equation}\label{aldef}
V=(x_0,x_1,\dots,x_{D-1}) \rightarrow \alpha=\alpha(V)=(a_0,a_1,\dots,a_{D-1})
\end{equation}
by the following
rule. Find the smallest component, $c_0$, in the $V$. If $c_0$ is found at places
$r_1,r_2,\dots$, put number 0 at those places in the $\alpha(V)$. Find the next smallest
component $c_1$, $c_1>c_0$ in the $V$. If $c_1$ is found at places
$s_1,s_2,\dots$, put number 1 at those places in the $\alpha$. Proceed this way until
components of $V$ are exhausted. At this stage, all $D$ components of $\alpha$ will be
determined. The $\alpha$ obtained this way is used as a symbol of embedding vector $V$.
For example, consider $V=(7, 15, 7, 25, 15)$. The corresponding symbol, or the order pattern
is $\alpha=(0,1,0,2,1)$. Here, information about equal values and their positions is
preserved.
If $V$ has no equal components, it can be proven (see Appendix \ref{pi-1}) that $\alpha=\pi^{-1}$.
This means that $\alpha$ is the
inverse permutation of the one obtained for $V$ if convention (\ref{asc}) is used.
Since correspondence between permutations and their inverse is one-to-one, it does not matter
which one, $\pi$ or $\alpha$, is used for calculating entropy. This further means that for a data set
and embedding method, which does not deliver equal values
in the embedding vectors, symbolization used here is equivalent
to the permutation one\footnote{It seems, that in paper \cite{Tylov2018} symbolization method described
here is used. But, as it may be concluded
from \cite[Eq. (6)]{Tylov2018}, the issue of equal values is not addressed. Similar
approach is used in \cite{Kulp2014,Berger2017}, again without considering equal values.} while calculating
entropy.
\subsection{Arithmetization}\label{Ari}
Expect that embedding vector $V$ in (\ref{aldef}) has exactly $d$ unique components,
where $d\le D$. In this case, corresponding symbol $\alpha(V)$ will be a sequence
of $D$ numbers chosen from the set $\mathbbmss{d}=\{0,1,\dots,d-1\}$ in such a way that
not any element from $\mathbbmss{d}$ is missed. The latter can be formulated as the following condition:
\begin{equation}\label{cond}
\bigwedge_{b\in \mathbbmss{d}}\,\,b\in \alpha(V).
\end{equation}
The sequence $\alpha(V)$ can be considered as a single integer $A(V)$,
in a base-$D$ positional numeral system\footnote{For a single embedding vector, $d$ might be chosen
as radix instead of $D$. But $d$ may be different for different vectors. And a same integer may have different representation for different bases with (\ref{cond}) satisfied. E.g. 0112$_3$ = 1110$_2$.}, with digits
$a_{D-1}a_{D-2}\dots a_0$:
\begin{equation}\label{A}
A(V)\equiv A =a_0 + a_1\,D + a_2\,D^2 +\dots + a_{D-1}D^{D-1}.
\end{equation}
It is clear that there is one-to-one correspondence between order patterns $\alpha$ and
integers obtained as shown in (\ref{A}).
Therefore, a set of order patterns, constructed as described in Sec. \ref{Sym},
can be replaced with a set $\mathcal{A}$ of integers obtained as shown in (\ref{A}):
\begin{equation}\label{AA}
\mathcal{A}=\{A_0, A_1,\dots, A_{N-D}\}, \text{ where } A_i\equiv A(V_i).
\end{equation}
The empirical probabilities $p(A_i)$ to find an integer $A_i$ among those in $\mathcal{A}$ can be calculated
as usual, and we have for the arithmetic entropy:
\begin{equation}\nonumber
H_a(\mathbf{V})\equiv H_a(\mathcal{A})=-\sum\limits_{i=0}^{L-1}p(A_i)\log(p(A_i)),
\end{equation}
where $L$ is the number of different integers in the $\mathcal{A}$.
For a data sequence and embedding method which does not deliver equal values in the embedding vectors,
all $d_i=D$ and the integers $A_i$ will represent corresponding permutation order patterns
unambiguously. In this case, $A_{min}\le A_i \le A_{max}$, where $A_{min}$ corresponds to
pattern $\alpha_{min}=(D-1,D-2,\dots,1,0)$:
$$
A_{min}=D-1 + (D-2)\,D + (D-3)\,D^2 +\dots+D^{D-2},
$$
and $A_{max}$ corresponds to pattern $\alpha_{max}=(0,1,\dots,D-1)$:
$$
A_{max}=D + 2\,D^2 +3\,D^3+\dots+(D-1)\,D^{D-1}.
$$
In this case, only $D!$ integers will be used from $[A_{min}; A_{max}]$ due to condition (\ref{cond}).
\subsection{How many new possible order patterns are got?}\label{got}
If it is decided to treat order patterns generated from embedding $D$-vectors with some
components equal as not equivalent to those from vectors with all components different,
then the number of all possible patterns will be greater than $D!$. Here we attempt to
estimate how many new patterns can be obtained.
Any new pattern appears from embedding $D$-vector with $d$ different com\-po\-n\-ents, where
$d\in\{1,2,\dots,D-1\}$. So, having $d$ fixed, the number of corresponding new patterns is equal to the number
$N(D,d)$
of base-$D$ $D$-digit integers constructed from digits $\{0,1,\dots,d-1\}$ in such a way that
each of the $d$ digits is used at least once. This number can be calculated as
\newcommand{\Stirling}[2]{\genfrac{\{}{\}}{0pt}{}{#1}{#2}}
\newcommand{\stirling}[2]{\genfrac{\{}{\}}{0pt}{1}{#1}{#2}}
$
N(D,d)=d!\,\Stirling{D}{d},
$
where $\stirling{D}{d}$ --- is the Stirling numbers of the second kind,
\cite[Part 5, \S 2]{Riordan1958}. Considering all possible values for $d$, we have for the
total number of possible new patterns%
:
\begin{equation}\label{ND}
N(D)=\sum\limits_{0<d<D}d!\,\Stirling{D}{d} = b(D) - D!\,,
\end{equation}
where $b(D)$ are known as ordered Bell numbers, see \cite[p.337]{Pippenger2010}
for naming discussion.
Calculating\footnote{The Stirling numbers were calculated with {\tt stirling2(D,d)}
function in the ``maxima'' computer algebra system ({\tt http://maxima.sourceforge.net/}).}
$N(D)$ for $D\in\{2,3,4,5,6,7\}$ we see that the number of new patterns is
normally greater than $D!$, see Fig. \ref{comp} and also Table \ref{T0}. Of course, the possible new patterns may only be
significant when they can be observed (see discussion about this in \cite{Cuesta-Frau2018}).
This depends on the process under study and embedding method.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{Fig2.pdf}
\end{center}
\caption{\label{comp}Comparison of possible number of patterns. ({\small $\bullet$}) --- equal values are treated as described in Sec. \ref{equal0}, $D!$; ($\ast$) --- additional possible
patterns due to equal values, $N(D)$ graph, Eq. (\ref{ND}).}
\end{figure}
\begin{table}
\begin{center}
\begin{tabular}{c c c c c c c}
\hline
$D$ & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$N(D)+D!$ & 3 & 13 & 75 & 541 & 4683 & 47293 \\
\hline
\end{tabular}
\end{center}
\caption{\label{T0} Total number of possible patterns in the AE symbolization.}
\end{table}
\subsection{Coding}\label{cod}
Certainly, there are several possible implementations of the algorithm discussed
in Secs. \ref{Sym}, \ref{Ari}. Here, the one used for the examples in Sec. \ref{Exa},
and Appendix \ref{more},
below, is shown. It is a {\tt C++} program.
It is expected that the sequence (\ref{raw}) is organized into a
one-dimensional array {\tt X[N]}. For calculating arithmetic order pattern
of vector $V_i$ shown in (\ref{Dvec}) it is necessary to pass
a pointer to the {\tt X[i]} to the function
{\tt get\_numerical\_pattern}, below, as its third argument:
{ \tt data\_point = X + i}.
In the below example, {\tt X[i]} is declared as {\tt double}, but it can be of any
type with appropriate sorting defined.
The returning value is declared as {\tt mpz\_class}, which is a
GNU multiple precision integer ({\tt https://gmplib.org/}). This is used because
for embedding dimensions $D>15$ the returned number representing an order pattern
may exceed 64 bits in size\footnote{It makes sense to use large embedding dimensions only for very long sequences of data. Otherwise, any observed pattern appears only once, which is unfavorable for estimating
probabilities.}. For smaller $D$, {\tt mpz\_class} can be replaced with
{\tt int}, or {\tt long} everywhere in the code.
\begin{Verbatim}[numbers=left]
#include <gmp.h>
#include <gmpxx.h>
#include <forward_list>
/**
Function calculates numerical representation of order pattern
of an embedding vector V_i = {x_i, x_{i + tau}, ...}.
Here D is the embedding dimension, tau is the delay.
The data_point points to the first component of the Vi in the
array of raw data.
*/
mpz_class get_numerical_pattern(int D,int tau,double * data_point)
{
int k;
std::forward_list<double> FL;
auto it = FL.before_begin();
for (k=0;k<D;k++) it = FL.emplace_after(it, data_point[k*tau]);
FL.sort();
FL.unique();
int * pDpnm = new int [D]; // order pattern will be here
int tag = 0;
for (auto it = FL.begin(); it != FL.end(); ++it)
{
for (k=0;k<D;k++)
if (*it == data_point[k*tau]) pDpnm[k] = tag;
tag++;
}
mpz_class pnum = 0; // arithmetic order pattern (initial value)
mpz_class digval = 1; // initial value of a single digit
for (k=0;k<D;k++)
{
pnum += pDpnm[k]*digval;
digval *= D;
}
return pnum;
}
\end{Verbatim}
This code is transparent and does not refer to combinatorics.
At the same time, provided an embedding vector does not have equal components,
when loop at lines {\footnotesize 23-28} above is complete,
we obtain in the array \verb-pDpnm[D]- a permutation $\pi^{-1}$, where
$\pi$ is the permutation for that vector, obtained in accordance with the
standard rules of \cite{Bandt2002} reproduced in Sec. \ref{P} with (\ref{asc}) adopted.
\section{Example}\label{Exa}
The discussed methodology has been tested at two surrogate sequences.
The purpose was to demonstrate that for a pair of sequences the standard permutation
entropy method gives roughly the same entropy, whereas the arithmetic
entropy may be considerably different.
For calculating standard permutation entropy in situation when equal components in embedding
vectors are possible we replace the following fragment:
\begin{verbatim}
for (auto it = FL.begin(); it != FL.end(); ++it)
{
for (k=0;k<D;k++)
if (*it == data_point[k*tau]) pDpnm[k] = tag;
tag++;
}
\end{verbatim}
in the code of Sec. \ref{cod}, above, with the following one:\bigskip
\begin{verbatim}
for (auto it = FL.begin(); it != FL.end(); ++it)
{
for (k=D-1;k>=0;k--)
if (*it == data_point[k*tau]) pDpnm[k] = tag++;
}
\end{verbatim}
With such a replacement we get in the array {\tt pDpnm[D]}
above, the permutation, which is inverse to
one obtained for $V_i$ in the standard permutation entropy symbolization with
rules (\ref{asc}) and (\ref{rule}) adopted.
As it was mentioned above, usage of inverse permutations instead of the initial ones
delivers the same value for the standard permutation entropy.
The two sequences, S1 and S2 are obtained as follows.
By means of function \verb-gsl_rng_uniform_int- from the GNU Scientific Library,
\cite{Galassi2009},
we generate random numbers from the set $\{0,1,\dots,4\}$, which are equally probable.
Each obtained random number \verb-val- is written into the S1. The same number is
written into the S2, provided it is not equal to the number written to S2 at the
previous step. If it does, then the number (\verb-val- + 1) ({\bf mod} 5) is written
instead. This introduces a non-zero correlation between consecutive values in S2.
E.g., in the S2 any two consecutive values are always different. Examples of S1, S2
are as follows
$$
\text{S1} = (2,2,0,3,4,0,1,3,4,4,0,3,3,2,2,4,4,2,0,1,\dots),
$$$$
\text{S2} = (2,3,0,3,4,0,1,3,4,0,1,3,4,2,3,4,0,2,0,1,\dots).
$$
1\,000\,000 long S1 and S2 were produced and both permutation and arithmetic entropy
have been calculated. The results are shown in Tables \ref{T1} and \ref{T2}.
\begin{table}
\begin{center}
\begin{tabular}{c c c |c c c}
\hline
$D=3$ & PE & AE &$D=4$ & PE & AE \\
\hline
S1 & 2.497 & 3.684 & S1 & 4.390 & 6.165 \\
\hline
S2 & 2.368 & 2.919 & S2 & 4.187 & 5.238 \\
\hline
\end{tabular}
\end{center}
\caption{\label{T1}Comparison of permutation entropy (PE) and arithmetic entropy (AE) for embedding delay
$\tau=1$. Entropy is given in bits.}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{ccc|ccc}
\hline
$D=3$ & PE & AE &$D=4$ & PE & AE \\
\hline
S1 & 2.498 & 3.684 & S1 & 4.393 & 6.166 \\
\hline
S2 & 2.407 & 3.676 & S2 & 4.224 & 6.098 \\
\hline
\end{tabular}
\end{center}
\caption{\label{T2}Same as Table \ref{T1} for embedding delay $\tau=2$.}
\end{table}
Notice that arithmetic entropy is considerably greater than the permutation one.
This is due to high frequency of embedding vectors with equal components.
Also, from Table \ref{T1} with $\tau=1$ it can be seen that arithmetic entropy discriminates
better between S1 and S2.
Although, case with delay $\tau=2$ shown in Table \ref{T2} is not similarly conclusive.
This might be due to construction method of the S2 sequence.
Namely, by pulling from S2 embedding vectors with delay 2, we may get vectors with equal adjacent
components, similarly to S1 case. This alleviates difference between S1 and S2.
For $\tau=1$, embedding vectors for S2 do not have equal adjacent components. More
examples are in the Appendix \ref{more}, below.
\section{Conclusions and discussion}
In this note, we have discussed a method for calculating entropy in a sequence of data,
which is similar to the permutation entropy method. The characteristic features of this
method are as follows:
\begin{itemize}
\item[(i)] it treats equal components in the embedding vectors as being
equal instead of ordering them artificially;
\item[(ii)] it is entirely exempt of combinatorics, labeling order patterns by integers instead of permutations;
\item[(iii)] if embedding vectors do not have equal components, this method delivers exactly the same
value for the entropy as does the standard permutation entropy one.
\end{itemize}
In the symbolization procedure discussed in Sec. \ref{Sym}, new order patterns may
appear as compared to the standard permutation method, see Sec. \ref{got}, above.
Those new patterns arise from embedding vectors with some components being equal to each other.
In the standard permutation entropy method,
the embedding vectors characterized by those new patterns, if any,
are labeled by permutations as if there were no equal components.
This is made possible through ordering equal values in accordance with the rule (\ref{rule}).
Mathematically, replacing embedding vectors with their
order patterns means constructing a quotient set from
the set of all embedding vectors with respect to some equivalence relation,
\cite{Keller2007,Bian2012,Piek2019}. In the case of permutation entropy, the corresponding equivalence relation
is defined by (\ref{rule}) and either (\ref{desc}), or (\ref{asc}).
Denote it by $\sim_P$.
For arithmetic entropy, the corresponding equivalence relation
is defined by the algorithm described in the first paragraph of Sec. \ref{Sym}.
Denote it by $\sim_A$. It is clear that for two embedding vectors $U$, $V$,
if
$U\, \sim_A\,V$, then $ U\, \sim_P\,V$.
Namely, if $U$, $V$ have the same arithmetic order pattern
then they do have the same permutation order pattern.
That means that $\sim_P$ is coarser relation than $\sim_A$.
Other equivalence relations could be offered, which are courser than $\sim_P$,
or finer than $\sim_A$, or lying in between, or incomparable with the both,
see e.g. \cite{Berger2017}.
A symbolization which still uses permutations, but is equivalent to discussed here,
as regards the treatment of equal values in embedding vectors,
has been proposed in \cite{Haruna2013}, see discussion in the Appendix B, below.
Which one is better depends on the data sequence and which kind of redundancy
is intended to strip.\bigskip\medskip
{
\small
| {
"timestamp": "2020-08-14T02:11:58",
"yymm": "2004",
"arxiv_id": "2004.11160",
"language": "en",
"url": "https://arxiv.org/abs/2004.11160",
"abstract": "A method for analyzing sequential data sets, similar to the permutation entropy one, is discussed. The characteristic features of this method are as follows: it preserves information about equal values, if any, in the embedding vectors; it is exempt of combinatorics; it delivers the same entropy value as does the permutation method, provided the embedding vectors do not have equal components. In the latter case this method can be used instead of the permutation one. If embedding vectors have equal components this method could be more precise in discriminating between similar data sets.",
"subjects": "Data Analysis, Statistics and Probability (physics.data-an)",
"title": "Calculating permutation entropy without permutations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9621075733703925,
"lm_q2_score": 0.7371581510799252,
"lm_q1q2_score": 0.7092254399257121
} |
https://arxiv.org/abs/1002.3934 | Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics | We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture. | \section{Introduction}
\subsection{Definitions and the statement of the problem}
Consider a pseudo-Riemmanian metric $g=(g_{ij})$ on a surface $M^2$. A function $F:{T^*}M\to \mathbb{R}$ is called {\it an integral} of the geodesic flow of $g$, if $\{ H, F\}=0$, where $H:= \tfrac{1}{2} \sum_{i,j} g^{ij} p_ip_j:T^*M\to \mathbb{R}$ is the kinetic energy corresponding to the metric. Geometrically, the condition $\{ H, F\}=0$ means
that the function $F$ is constant on the trajectories of the Hamiltonian system with the Hamiltonian $H$. We say that the integral $F$ is {\it quadratic in momenta,} if in every local coordinate system $(x,y)$ on $M^2$ it has the form
\begin{equation} \label{integral}
a(x,y)p_x^2+ b(x,y)p_xp_y+ c(x,y)p_y^2
\end{equation} in the canonical coordinates $(x,y,p_x, p_y)$ on $T^*M^2$.
Geometrically, the formula \eqref{integral}
means that the restriction of the integral to every cotangent space $T^*_{(x,y)}M^2\equiv \mathbb{R}^2$ is a homogeneous quadratic function. As {\it trivial} examples of quadratic in momenta integrals
we consider those proportional to the Hamiltonian $H$.
Similarly, we say that the integral is {\it linear} in momenta, if for every local coordinate system $(x,y)$ on $M^2$ it has the form $\alpha(x,y) p_x + \beta(x,y) p_y$ in the canonical coordinates $(x,y,p_x, p_y)$ on $T^*M^2$; an integral linear in momenta is {\it trivial}, if it is identically zero.
The importance of integrals quadratic in momenta other than the Hamiltonian
for studying the metric was recognized long ago. Indeed,
it was Jacobi's realization that the geodesic flow of the ellipsoid
admitted such an `extra' quadratic integral that allowed him to
integrate the geodesics on the ellipsoid.
In the present paper we solve (see Model Examples 1, 2, 3 and Theorems \ref{main1}, \ref{main3}, \ref{main4} below) the following problem:
{\bf Problem.} {\it Find all metrics of signature $(+,-)$ on closed 2-dimensional manifolds whose geodesic flows admit nontrivial
integrals quadratic in momenta. }
Riemannian metrics whose geodesic flows admit integrals quadratic in momenta are quite good studied. Indeed, local description of such a metric in a neighborhood of almost every point is known since Liouville.
Moreover, the Riemannian version (and, therefore, if the signature of $g$ is (--,--)) of the problem above was solved. There exist two different approaches that lead to a solution: one, which is based on the ideas of Kolokoltsov \cite{Kol}, was realized in \cite{Kol,1,klein}, see also \cite{BMF,BF}.
Alternative approach to the description of metrics whose geodesic flows admit nontrivial integrals quadratic in momenta
is due to Kiyohara \cite{Kio}, see also \cite{Igarashi,memo}. Our solution uses main ideas from both approaches.
Metrics whose geodesic flows admit integrals quadratic in momenta were studied in the framework of differential geometry (at least since Darboux \cite{Darboux}) and mathematical physics (at least since Birkhoff \cite{3} and Whittaker \cite{Whi}). We give two
applications of our results in differential geometry and one application in mathematical physics. In differential geometry, we use the connection between integrals quadratic in momenta and geodesically equivalent metrics (we give the necessary definition in \S \ref{beltrami}) to solve the natural generalization of the Beltrami problem for closed manifolds, and to prove the two-dimensional pseudo-Riemannian
version of the projective Obata conjecture. In mathematical physics, we prove that all quadratically-superintegrable metrics on closed surfaces (the necessary definition is in \S \ref{superi}) have constant curvature. This generalizes the result of \cite{Kio,klein} to the pseudo-Riemannian metrics.
\subsection{Metrics on the torus whose geodesic flows admit nontrivial integrals quadratic in momenta}
Locally, pseudo-Riemannian metrics admitting integrals quadratic in momenta were described\footnote{As it mentioned in \cite{pucacco,BMP}, the essential part of the result appeared already in Darboux \cite[\S\S592--594,600--608]{Darboux}} in \cite[Theorem 1]{pucacco} and \cite[Theorem 1]{BMP}:
\begin{Th}[\cite{pucacco,BMP}] \label{main} Suppose a Riemannian or pseudo-Riemannian
metric $g$ on a connected surface ${M}^2$ admits an integral $F$ quadratic in momenta such that $F\ne \mbox{\rm const} \cdot H$ for all $\mbox{\rm const} \in \mathbb{R}$. Then, in a neighbourhood of almost every point there exist coordinates $x,y$ such that the metric and the integral are as in the following table:
\begin{center}\begin{tabular}{|c||c|c|c|}\hline & \textrm{Liouville case} & \textrm{Complex-Liouville case} & \textrm{Jordan-block case}\\ \hline \hline
$g$ & $(X(x)-Y(y))(dx^2 + \varepsilon dy^2)$ & $\Im(h)dxdy$ & $\left(\widehat Y(y)+\tfrac{x}{2} Y'(y)\right)dxdy $
\\ \hline $F$&$\tfrac{X(x)p_y^2 +\varepsilon Y(y)p_{x}^2}{X(x)-Y(y)} $&
$p_x^2 - p_y^2 + 2\tfrac{\Re(h)}{\Im(h)}p_xp_y $ & $\varepsilon \left(p_x^2 - \frac{Y(y)}{\widehat Y(y)+ \tfrac{x}{2} Y'(y) }p_xp_y\right)$ \\ \hline
\end{tabular}
\end{center}
where $\varepsilon = \pm1$, and $\Re (h)$ and $\Im (h)$ are the real and imaginary parts of a
holomorphic function $h$ of the variable $z:= x+ i\cdot y$.
\end{Th}
\begin{Rem} Within our paper, we understand ``{\it almost every}'' in the topological sense:
a condition is fulfilled at almost every point, if the set of the points where it is fulfilled is everywhere dense.
\end{Rem}
We see that the metric $g$ in the Jordan-block and Complex-Liouville cases always has indefinite signature
(+,--), and the metric $g$ in the Liouville case has signature (+,--) if and only if $\varepsilon =-1$.
The Liouville case with $\varepsilon =1$ was known to classics.
In Section \ref{puc}, we repeat the proof of Theorem \ref{main}, because we will need most techical details from it in the proof of our main result, which is Theorem \ref{main1} below.
Let us now discuss the case when $M^2$ is closed.
First of all, because of Euler characteristic, a closed surface admitting a pseudo-Riemannian metric of indefinite signature is homeomorphic to the torus or to the Klein bottle. Since a double cover of the Klein bottle is the torus, and the geodesic flow of the lift of a metric whose geodesic flow admits an integral quadratic in momenta also admits an integral quadratic in momenta, the most important case is when the surface is the torus. In Model Example 1 below we describe a class of pseudo-Riemannian metrics on the torus such that their geodesic flows admit nontrivial integrals quadratic in momenta. Theorem \ref{main1} claims that every metric such that its geodesic flow admits a nontrivial integral quadratic in momenta is isometric to one from Model Example 1.
{\bf Model Example 1.} We consider $\mathbb{R}^2 $ with the standard coordinates $(x,y)$, two linearly independent vectors $\xi=(\xi_1,\xi_2)$ and $\nu=(\nu_1, \nu_2)$, and two nonconstant functions $X$ and $Y$ of one variable (it is convenient to think that the variable of $X$ is $x$ and the variable of $Y$ is $y$)
such that \begin{itemize}
\item[(a)] for all $(x,y)\in \mathbb{R}^2$ we have $X(x)\ne Y(y)$, and
\item[(b)] for every $(x,y)\in \mathbb{R}^2$, $ X(x+\xi_1)= X(x+ \nu_1)= X(x)$ and $ Y(y+\xi_2)= Y(y+ \nu_2)= Y(y)$. \end{itemize}
Next, consider the metrics $(X(x)- Y(y))(dx^2 +\varepsilon dy^2)$ on $\mathbb{R}^2$, where $\varepsilon =\pm1$, and
the action of the lattice $G:=\{k \cdot \xi + m \cdot \nu \mid k, m\in \mathbb{Z}\}$ on $\mathbb{R}^2$. The action is free, discrete and
preserves the metric and the quadratic integral $ \tfrac{X(x)p_y^2 +\varepsilon Y(y)p_{x}^2}{X(x)-Y(y)}$.
Then, the geodesic flow of the
induced metric on the quotient space $ \mathbb{R}^2/G$ (homeomorphic to the torus) admits an integral quadratic in momenta. We will call such metrics \emph{ globally--Liouville}.
\begin{figure}[ht!]
{\includegraphics[width=0.7\textwidth]{liouville.eps}}
\caption{Vectors $\xi$ and $\nu$ and a fundamental region (gray parallelogram) of the action of $G$ from Model Example 1. The torus $ \mathbb{R}^2/G$ can be identified with this parallelogram with glued opposite sides. Since the action of $G$ preserves $X(x)$ and $Y(y)$, the metric $g$
induces a metric on $ \mathbb{R}^2/G$, and the integral $F$ induces an integral quadratic in momenta }\label{liouville}
\end{figure}
\begin{Th} \label{main1} Suppose a metric $g$ on the two-torus
${T}^2$ admits an integral $F$ quadratic in momenta. Assume the integral is not a linear combination of
the square of an integral linear in momenta and the Hamiltonian. Then, $(T^2, g)$ is globally Liouville, i.e., there exist $X$, $Y$, $\xi$, $\nu$ satisfying the conditions in the Model Example 1 above and a diffeomorphism $\phi:T^2 \to \mathbb{R}^2/G$ that takes $g$ to the globally-Liouville metric $(X(x)- Y(y))(dx^2 +\varepsilon dy^2)$ on $\mathbb{R}^2/G$ and the integral $F$ to the integral $\pm \left(\tfrac{X(x)p_y^2 +\varepsilon Y(y)p_{x}^2}{X(x)-Y(y)}\right)$.
\end{Th}
In the Riemannian case, Theorem \ref{main1} follows from \cite{1,Kio}, see also \cite{BMF,BF}.
We see that the answer in the pseudo-Riemannian case is essentially the same ( = no new phenomena appear) as the answer in the Riemannian case. This similarity with the Riemannian case was unexpected: indeed, by Theorem \ref{main}, in the pseudo-Riemannian case (different from the Riemannian case) there are three different types of metrics admitting quadratic integrals. Moreover, the examples from papers \cite{chanu,rastelli,smirnov,PR} show that, locally, the pair (metric,integral) can change the type, i.e., the pair (metric,integral) can be, for example, as in Liouville case from one side of a line, and as in Complex-Liouville case from another side of the line.
But it appears that only one type, namely the Liouville,
can exist on closed manifolds.
Moreover, as we show in Example \ref{Ex2}, if the integral is the square of an integral linear in momenta, then the Jordan-block case is possible (even if the surface is closed). Moreover, the pair
(metric,integral) can change the type: be of Jordan-block type in a neighborhood of one point, and of Liouville type in a neighborhood of another point. Moreover, one can modify Example \ref{Ex2}(c) such that the set of the points such that the pair (metric,integral) changes the type is the direct product of the Cantor set and a circle.
\subsection{Metrics on the Klein bottle whose geodesic flows admit integrals quadratic in momenta }
The scheme of the description is the same as for the torus: in Model Example 2 we describe a big family of metrics on the Klein bottle whose geodesic flows admit integrals quadratic in momenta. Theorem \ref{main3} claims that every metric such that its geodesic flow admits an integral quadratic in momenta and such that the geodesic flow of the lift of the metric to the oriented cover admits no integral linear in momenta is as in Model Example 2.
\\[.1cm]
{\bf Model Example 2.} We consider $\mathbb{R}^2 $ with the standard coordinates $(x,y)$, constants $c\ne 0, \ d\ne 0$, two vectors $\xi=(c,0)$ and $\nu=(0, d) $, and two nonconstant functions $X$ and $Y$ of one variable (it is convenient to think that the variable of $X$ is $x$ and the variable of $Y$ is $y$)
such that \begin{itemize}
\item[(a)] for all $(x,y)\in \mathbb{R}^2$ we have $X(x)\ne Y(y)$, and
\item[(b)] for every $(x,y)\in \mathbb{R}^2$, $ X(x+c)= X(x)$ and $ Y(y+d)= Y(-y)= Y(y)$. \end{itemize}
Next, consider the metrics $(X(x)- Y(y))(dx^2 +\varepsilon dy^2)$ on $\mathbb{R}^2$ and
the action of the group $G$
generated by the transformations $(x,y)\mapsto (x+ c, -y)$ and $(x,y)\mapsto (x, y+ d)$. The action is free, discrete and preserves the metric and the quadratic integral $ \tfrac{X(x)p_y^2 +\varepsilon Y(y)p_{x}^2}{X(x)-Y(y)}$.
Then, the geodesic flow of the
induced metric on the quotient space $ \mathbb{R}^2/G$ (homeomorphic to the Klein bottle) admits an integral quadratic in momenta. We will call such metrics \emph{ globally-(Klein)-Liouville}.
\begin{figure}[ht!]
{\includegraphics[width=0.7\textwidth]{kleinliouville.eps}}
\caption{Vectors $\xi$ and $\nu$ and a fundamental region (gray rectangle) of the action of $G$ from Model Example 2. The Klein bottle $ \mathbb{R}^2/G$ can be identified with this rectangle with glued opposite sides: the horizontal sides are glued with preserving the orientation, and the vertical sides are glued with inverting the orientation. The action of $G$ preserves the metric $g$
and the integral $F$; hence, the geodesic flow of the induced metric on $ \mathbb{R}^2/G$ admits an integral quadratic in momenta}\label{kleinliouville}
\end{figure}
\begin{Th} \label{main3} Suppose a metric $g$ on the Klein bottle
${K}^2$ admits an integral $F$ quadratic in momenta. Assume the lift of the integral to the oriented cover is not a linear combination of the lift of the Hamiltonian and
the square of a function linear in momenta. Then, $(K^2, g, F)$ is globally-(Klein)-Liouville, i.e., there exist $X$, $Y$, $c,d$ satisfying the conditions in the Model Example 2 above and a diffeomorphism $\phi:K^2 \to \mathbb{R}^2/G$ that takes $g$ to the globally-(Klein)-Liouville metric $(X(x)- Y(y))(dx^2 +\varepsilon dy^2)$ on $\mathbb{R}^2/G$
and $F$ to the integral $\pm \left(\tfrac{X(x)p_y^2 +\varepsilon Y(y)p_{x}^2}{X(x)-Y(y)}\right)$.
\end{Th}
In the Riemannian case, Theorem \ref{main3} was proved in \cite[Theorem 3]{klein}. We see that the answer in the pseudo-Riemannian case is essentially the same as the answer in the Riemannian case (similar to the torus).
The following example explains why we require that the LIFT of the integral (to the oriented cover) is not a linear combination of the lift of the Hamiltonian and the square of a function linear in momenta:
\begin{Ex} As in the Main Example 2, we consider $\mathbb{R}^2 $ with the standard coordinates $(x,y)$, constants $c\ne 0, \ d\ne 0$, two vectors $\xi=(c,0)$ and $\nu=(0, d)$, the function $X$ of the variable $x$ such that $ X(x+c)= X(x)$. Different from the Main Example 2, by $Y$ we denote a CONSTANT
such that $X(x)\ne Y$ for all $x\in \mathbb{R}$.
Under this assumptions, the metric $(X(x)- Y)(dx^2 +\varepsilon dy^2)$ and the integral $ \tfrac{X(x)p_y^2 +\varepsilon Yp_{x}^2}{X(x)-Y}$ induce a metric on the $K^2:= \mathbb{R}^2/G$, where $G$ is the group generated by the mappings $(x,y)\mapsto (x+ c, -y)$ and $(x,y)\mapsto (x, y+ d)$, and an integral quadratic in momenta for the geodesic flow of this metric.
The lift of the integral to the oriented cover $T^2:= \mathbb{R}^2/G'$, where $G':= \{ 2k\cdot \xi + m\cdot \nu \mid k, m\in \mathbb{Z}\}$, is a linear combination of the Hamiltonian $\tfrac{1}{2} \frac{p_x^2 + \varepsilon p_y^2}{X(x)- Y}$ and the square of the (linear in momenta) function $p_y$. Indeed, $
F= p_y^2 + {2} \varepsilon Y \cdot H.
$
But, on ${K}^2$, the integral in NOT a linear combination of the Hamiltonian and of the square of a function linear in momenta. The formal proof of this observation in the Riemannian case can be found in \cite[\S\S3,4]{klein}, the Riemannian proof can be easily generalized (using Theorem \ref{main1} of our paper) to the pseudo-Riemannian metrics. The main idea of the proof is that the function $p_y$ does not generate a function on the Klein bottle, since the mapping $(x,y)\mapsto (x+ c, -y)$
changes the sign of this function.
\end{Ex}
\subsection{Metrics on the torus whose geodesic flows admit integrals linear in momenta}
In order to complete the description of the metrics of signature (+,--) whose geodesic flows admit nontrivial integrals quadratic in momenta, we need to describe the metrics of signature (+,--)
on the torus such that their geodesic flows admit nontrivial integrals linear in momenta.
In the Riemannian case, metrics with geodesic flows admitting integrals linear in momenta can be considered as a partial case of the metrics whose geodesic flows admit integrals quadratic in momenta. Indeed, up to an isometry, any such metric is essentially as in Model Examples 1,2 (see \cite{BMF,BF}), the only difference is that the function $ X$ is constant.
In particular, it implies that one can always slightly perturb a metric whose geodesic flow admits an integral linear in momenta such that the geodesic flow of the result admits an integral quadratic in momenta, but admits no integral linear in momenta.
It appears that in the pseudo-Riemannian case the situation is different.
Below, we construct a family of metrics on the torus whose geodesic flows admit integrals linear in momenta. In Examples \ref{Ex2}, \ref{Ex3}, we use the construction to show that in the pseudo-Riemannian case the following new (compared with the Riemannian case) phenomena appear:
\begin{itemize}
\item Example \ref{Ex2}(a) shows that metric and the integral can be as in the Jordan-block case.
\item Example \ref{Ex2}(c) shows that the metric and the integral can be as in the Jordan-block case
in one neighborhood and as in the Liouville case in another neighborhood.
\item Example \ref{Ex3} shows the existence of a metric whose geodesic flow admits an integral linear in momenta, such that no small perturbation of this metric admits an integral quadratic in momenta which is not a linear combination of the square of an integral linear in momenta and the Hamiltonian.
\end{itemize}
{\bf Construction.}
We consider $\mathbb{R}^2$ with the standard coordinates $x,y$ and the standard orientation, the vector fields $\xi:= (1,0)$, $ \eta:=(0,1)$, and
a smooth foliation on $\mathbb{R}^2$ invariant with respect to the flow of the vector field $\xi$ and with respect to the mapping $(x,y)\mapsto (x,y+1)$.
With the help of these data, we construct a metric of signature $(+,-)$ on $\mathbb{R}^2$ such that $\xi$ is a Killing vector field for this metric.
At every point $p$, we consider two vectors $U_1(p)$ and $U_2(p)$
satisfying the following conditions:
\begin{itemize}
\item $U_1$ at every point is tangent to the leaf of the foliation containing this point,
\item $(U_1(p), U_2(p))$ is an orthonormal positive basis for the flat metric $\hat g =dx^2 + dy^2$, that is
\begin{itemize} \item[$\bullet$] $|U_1|_{\hat g}= |U_2|_{\hat g}=1$, $\hat g(U_1, U_2)=0$,
\item[$\bullet$] the orientation given by the basis coincides with the standard orientation, see Figure \ref{u1u2}.\end{itemize}
\end{itemize}
Clearly, ar every point there exist precisely two possibilities for such vector fields $U_1$, $U_2$ (the second possibility is $(-U_1, -U_2)$).
\begin{figure}[ht!]
{\includegraphics[width=0.7\textwidth]{u1u2.eps}}
\caption{A leaf of the foliation and two possibilities for the vectors $U_1$, $U_2$}\label{u1u2}
\end{figure}
Now, consider the metric $g$ such that in the basis $(U_1,U_2)$ it has the matrix
$
\begin{pmatrix}
0 & 1\\ 1& 0
\end{pmatrix}.$
The metric clearly
does not depend on the choice of vectors $U_1, U_2$ at every point, and is smooth. Since all objects we used to construct the metric are invariant with respect to the flow of $\xi$, the vector field $\xi$ is Killing for the metric. Then, the geodesic flow of the metric admits an integral $p_x$ linear in momenta.
Since all objects are invariant with respect to the lattice $G= \{ k\cdot \xi + m\cdot \eta\mid k,m\in \mathbb{Z}\}$, the metric induces a metric on the torus $\mathbb{R}^2/G$ whose geodesic flow admits an integral linear in momenta.
\begin{Rem}
By construction, the leaves of the foliation are light-line geodesics.
\end{Rem}
\begin{Ex}\label{Ex2} If the foliation is as on Figure \ref{fig1}(a), the square of the integral is as in the Jordan block case. If the foliation is as on Figure \ref{fig1}(b), the square of the integral is as in the Liouville case. If the foliation is as on Figure \ref{fig1}(c), the square of the integral is as the Jordan block case in an annulus $\{(x,y)\in \mathbb{R}^2 \mid y-[y]> \tfrac{1}{2}\}$ and as in Liouville case in the annulus $\{(x,y)\in \mathbb{R}^2 \mid y-[y]<\tfrac{1}{2}\}$, where $[y]$ denotes the integer part of $y$.
\end{Ex}
\begin{figure}[ht!]
{\includegraphics[width=1.1\textwidth]{foliations.eps}}
\caption{The foliations from Example \ref{Ex2}}\label{fig1}
\end{figure}
\begin{Ex}\label{Ex3} Let the foliation is as on Figure \ref{fig2} (the restriction of the foliation to the annulus $\{(x,y)\mid x-[x]<\tfrac{1}{2}\}$ is the so-called Reeb component). Then, the geodesic flow of
no small perturbation of this metric admits an integral quadratic in momenta that is not a linear combination of the Hamiltonian and the square of an integral linear in momenta. Indeed, the Reeb component is stable with respect to small perturbations, and the light line geodesics of the
metrics from Model Example 1 are winding on the torus and form no Reeb component.
\end{Ex}
\begin{center}
\begin{figure}[ht!]
{\includegraphics[width=.5\textwidth]{foliations1.eps}}
\caption{The foliation from Example \ref{Ex3}}\label{fig2}
\end{figure}
\end{center}
Let us now describe all metrics on closed manifolds whose geodesic flows admit nontrivial integrals linear in momenta.
{\bf Model Example 3.}
We consider $\mathbb{R}^2 $ with the standard coordinates $(x,y)$, the vectors $\xi:=(1,0)$ and $\nu:=(0, 1)$, and three functions $K(y), L(y),M(y)$ of the variable $y$ periodic with period $1$ such that at every point $\det\begin{pmatrix}K & L \\ L& M \end{pmatrix}= KM-L^2<0$.
Next, consider the metric $g= K(y)dx^2 +2L(y) dxdy + M(y) dy^2$ on $\mathbb{R}^2$, and
the action of the lattice $G:=\{k \cdot \xi + m \cdot \nu \mid k, m\in \mathbb{Z}\}$ on $\mathbb{R}^2$. The action is free, discrete and preserves the metric and the integral $ p_{x}$ linear in momenta.
Then, the geodesic flow of the
induced metric on the quotient space $ \mathbb{R}^2/G$ (homeomorphic to the torus) admits an integral linear in momenta.
\begin{Th} \label{main4} Let $g $ be a metric of signature $(+,-)$ on the torus $T^2$ such that it is not flat.
If the geodesic flow of $g$ admits an integral linear in momenta, then the metric is
as in Model Example 3, i.e., there exist functions $K(y), M(y), L(y)$ periodic with period
$1$ and a diffeomorphism $\phi:T^2\to \mathbb{R}^2/G$ that takes the metric $g$ to the metric $ K(y)dx^2 +2L(y) dxdy + M(y) dy^2$, and the integral to $\mbox{\rm const} \cdot p_x$.
\end{Th}
In Theorem \ref{main4}, we assume that the metric $g$ is not flat. For flat metrics, Theorem \ref{main4} is wrong, since the integral curves of the Killing vector field corresponding to the linear integral are non necessary closed curves for the flat metrics, but are closed curves in Model Example 3.
We need therefore to describe separately flat metrics of signature (+,--) on the torus.
By the { \it standard flat torus } we will consider $(\mathbb{R}^2/G, dxdy)$, where $(x,y)$ are the standard coordinates on $\mathbb{R}^2$, and $G$ is a lattice generated by two linearly independent vectors.
In \S \ref{flat} we will recall why every torus $(T^2, g)$ such that the metric $g$ is flat and has signature (+,--) is isometric to a standard one.
\section{Applications}
\subsection{Application I: Betrami problem on closed pseudo-Riemannian manifolds} \label{beltrami}
Two metrics $g$ and $\bar g$ on one manifold are {\it geodesically equivalent,} if every (unparametrized) geodesic of the first metric is a geodesic of the second metrics. Investigation of geodesically equivalent metrics is a classical topic in differential geometry, see the surveys \cite{Aminova2,mikes} or/and the introductions to \cite{threemanifolds,hyperbolic,diffgeo}.
In particular, in 1865 Beltrami \cite{Beltrami} asked\footnote{ Italian original from \cite{Beltrami}:
La seconda $\dots$ generalizzazione $\dots$ del nostro problema, vale a dire: riportare i punti di una superficie sopra un'altra superficie in modo che alle linee geodetiche della prima corrispondano linee geodetiche della seconda.} {\it to describe all pairs of geodesically equivalent Riemannian metrics on surfaces. }
From the context it is clear that he considered this problem locally, in a neighbourhood of almost every point, but the problem has sense, and is even more interesting globally.
Geodesically equivalent metrics and quadratic integrals are closely related:
\begin{Th} \label{main2}
Two metrics $g$ and $\bar g$ on $M^2$ are geodesically equivalent, if and only if the following (quadratic in momenta) function \begin{equation}\label{Integral} F:TM^2\to \mathbb{R}, \
\
F(x_1, x_2, p_1, p_2):= \left(\frac{\det(g)}{\det(\bar g)}\right)^{2/3}\cdot \sum_{i,j} \bar g^{ij}p_ip_j,
\end{equation}
where we raised the indexes of $\bar g$ with the help of $g$, i.e., $\bar g^{ij} = g^{ki} \bar g_{km} g^{mj}$,
is an integral of the geodesic flow of $g$. Moreover, $F=\mbox{\rm const}\cdot H $ for a certain $\mbox{\rm const} \in \mathbb{R}$ if and only if $g$ and $\bar g$ are proportional with a constant coefficient of proportionality.
\end{Th}
Theorem \ref{main2} above was essentially known to Darboux~\cite[\S\S600--608]{Darboux}; for recent proofs see \cite[Corollary 1]{pucacco}. See also the discussion in \cite[Section 2.4]{bryant}.
Combining Theorems \ref{main1}, \ref{main3}, \ref{main4} with Theorem \ref{main2}, we obtain a complete description of geodesically equivalent pseudo-Riemannian metrics on closed surfaces.
\subsection{ Application II: every quadratically-superintegrable metric on a closed surface has constant curvature } \label{superi}
Recall that a metric on $M^2$
is called quadratically-superintegrable, if the geodesic flow of the metric admits three linearly independent
integrals quadratic in momenta. Quadratically-superintegrable metrics were first considered by Koenigs \cite{koenigs}. Nowdays, investigation of quadratically-superintegrable metrics is a hot topic in mathematical physics due to various applications and deep mathematical structures behind it, see e.g. \cite{kress}.
For example, the standard flat metric $dxdy$ on the 2-torus $\mathbb{R}^2/G$, where $G$ is a lattice generated by two linearly independent vectors, is quadratically-superintegrable. Indeed, the Hamiltonian $H= 2p_xp_y$ and the quadratic in momenta functions $F_1:= p_x^2, \ F_2:= p_y^2$ are linearly independent integrals, and are invariant with respect to any lattice.
\begin{Cor} \label{super}
Let a metric $g$ on a closed surface be quadratically-superintegrable. Then, it has constant curvature.
If in addition the metric has signature {\rm (+,--)}, then it is flat.
\end{Cor}
In the proof of Corollary \ref{super} we will need the following
\begin{Lemma} \label{-5} Let the metric $g$ of signature $(+, -)$ on the two-torus $T^2$ admit an integral quadratic in momenta that is not a linear combination of the Hamiltonian and of the square of an integral linear in momenta. Then, there exists a Riemannian
metric $\bar g$ geodesically equivalent to $g$.
\end{Lemma}
{\bf Proof.} By Theorem \ref{main1}, without loss of generality we can assume that the metric $g$ and the integral $F$ are as in Model Example 1. Without loss of generality we can think that $X(x)>Y(y)$ for all $(x,y)\in \mathbb{R}^2$.
Let us cook with the help of $H, F_1$ a Riemannian metric $\bar g$ geodesically equivalent to $g$.
We put $X_{min}= \min_{x\in\mathbb{R} }X(x)$ and $Y_{max}= \max_{y\in\mathbb{R} }Y(y)$. Clearly, $X_{min}>Y_{max}$.
We consider
$$\bar F:= H + \frac{1}{X_{min} + Y_{max}} F_1=
\frac{\tfrac{1}{2}- \tfrac{Y}{X_{min} + Y_{max}}}{X-Y} p_x^2 + \tfrac{\frac{X}{X_{min} + Y_{max}}-\tfrac{1}{2}}{X-Y} p_y^2.
$$
Since $X > \tfrac{X_{min} +Y_{max }}{2} >Y$, the integral $\bar F$ is positively defined (considered as a quadratic form on $T^*M^2$). Consider the metric $\bar g$ constructed by $\bar F$ with the help of Theorem \ref{main2}. The metric is positively defined (i.e., is Riemannian), and is geodesically equivalent to $g$. Lemma \ref{-5} is proved.
{\bf Proof of Corollary \ref{super}.} The Riemannian version of Corollary \ref{super} is known (see \cite[Theorem 5.1]{Kio} and \cite[Lemma 3]{klein}, see also \cite[Theorem 6]{CMH}).
Then, without loss of generality we can assume that the metric has signature (+,--).
Let $H, F_1, F_2$ be the linearly independent integrals quadratic in momenta. If both $F_1$ and $F_2$ are linear combinations of the square of integrals linear in momenta and the Hamiltonian, the metric admits two Killing vector fields implying that it has constant curvature.
Assume now that there exists
an integral quadratic in momenta that is not a linear combination of the Hamiltonian and of the square
of an integral linear in momenta. By Lemma \ref{-5}, there exists a Riemannian
metric $\bar g$ geodesically equivalent to $g$.
The metric $\bar g$
is also quadratically-superintegrable. Indeed, as it was proved in \cite[Lemma 1]{dim2} (see also \cite[\S 2.8]{bryant} and \cite[Lemma 3]{kruglikov}), every metric geodesically equivalent to a quadratically-superintegrable metric is also quadratically-superintegable. Then, by the Riemannian version of Corollary \ref{super} (which is known, as we recalled above), the metric $\bar g$ has constant curvature. Then, by the Beltrami Theorem (see \cite{Beltrami,beltrami-small}), the metric $g$ also has constant curvature. The first part of Corollary \ref{super} is proved.
If the metric has signature (+,--), then the surface if the torus or the Klein bottle.
By the Gauss-Bonnet Theorem, a metric of constant curvature on the torus or on the Klein bottle
is flat. Corollary \ref{super} is proved.
\subsection{Application III: Proof of projective Obata conjecture for two-dimensional pseudo-Riemannian metrics}
Let $(M^n,g)$ be a pseudo-Riemannian manifold of dimension $n\ge 2$.
Recall that a \emph{projective transformation} of $M^n$ is a diffeomorphism of the manifold that takes unparameterized geodesics to geodesics.
The goal of this paper is to prove the two-dimensional pseudo-Riemannian version of the following
{\bf Projective Obata conjecture.} { \it Let a connected Lie group $G$ act on a closed
connected $(M^n, g)$ of dimension
$n\ge 2$ by projective
transformations. Then, it acts by isometries, or for some $c\in \mathbb{R}\setminus \{0\}$ the metric $c\cdot g$ is the Riemannian metric of constant positive sectional curvature $+1$.}
\begin{Rem} The attribution of conjecture to Obata is in folklore (in the sense we did not find a paper of Obata where he states this conjecture). Certain papers, for example \cite{hasegawa,nagano,Yamauchi1}, refer to this statement as to a classical conjecture. If we replace ``closedness" by
``completeness", the obtained conjecture is attributed in folklore to Lichnerowicz, see also the discussion in \cite{diffgeo}. \end{Rem}
For Riemannian metrics,
projective Obata conjecture was proved in \cite{obata,CMH,diffgeo}. Then, in dimension two we may assume that the signature of the metric is $(+, -)$, and that the manifold is covered by the torus $T^2$.
Thus, the two-dimensional version of the projective Obata conjecture follows from
\begin{Th} \label{obatath} Let $(T^2, g)$ be the two-dimensional torus $T^2$ equipped with a metric $g$ of signature $(+, -)$. Assume
a connected Lie group $G$ acts on
$(T^2, g)$ by projective
transformations.
Then, $G$ acts by isometries.
\end{Th}
Note that in the theory of geodesically equivalent metrics and projective transformations, dimension 2 is a special dimension: many methods that work in dimensions $n \ge 3$ do not work in dimension 2. In particular, the proof of the projective Obata conjecture in the Riemannian case was separately done for dimension 2 in \cite{obata, CMH} and for dimensions greater than 2 in \cite{diffgeo}.
Moreover, recently an essential progress was achived in the proof of the projective Obata conjecture in the pseudo-Riemannian case in dimensions $n\ge 3$, see \cite{kiosak2,mounoud}. This progress allows us to hope that it is possible to mimic (see \cite[\S 1.2]{kiosak2}) the Riemannian proof in the pseudo-Riemannian situation (assuming the dimension is $n\ge 3$). Thus, Theorem \ref{obatath} closes an important partial case in the proof of projective Obata conjecture.
{\bf Proof of Theorem \ref{obatath}.} Let $g$ be a pseudo-Riemannian metric of signature $(+, -)$ of nonconstant curvature on $T^2$. We denote by $\Proj_0(T^2, g)$ the connected component of the group of projective transformations of $(T^2, g)$, and by $\Iso_0(T^2, g)$ the connected component of the group of isometries. Clearly, $\Proj_0(T^2, g)\supseteq \Iso_0(T^2, g)$; our goal is to prove $\Proj_0(T^2, g)=\Iso_0(T^2, g)$.
We assume that $\Proj_0(T^2, g)\ne \Iso_0(T^2, g)$.
Then, there exists a vector field $v$ such that
it is a projective vector field, but is not Killing vector field. (Recall that
a vector field $v$ is {\em projective}, if its local flow takes geodesics considered as unparameterized curved
to geodesics).
Then, by \cite[Korollar 1]{obata}, \cite[Corollary 1]{CMH}, or \cite{Topalov},
the quadratic in velocities function
$$
I:TM\to \mathbb{R}, \ \ \ I(\xi):=({\cal L}_vg)(\xi, \xi)-\tfrac{2}{3} \mathbb{\rm
trace}(g^{-1}{\cal L}_vg)\, g(\xi, \xi), $$ where $\mathbb{\rm
trace}(g^{-1}{\cal L}_vg):= g^{ij} (\mathcal{L}_vg)_{ij}$
is a nontrivial (i.e., $\ne 0$) integral
for the geodesic flow of $g$.
Suppose first
$I$ is not a linear combination of the energy integral $g(\xi, \xi)$ and of the square of an integral linear in velocities. Since closed manifolds do not allow vector fields $v$ such that $\mathcal{L}_vg = \mbox{\rm const}\cdot g $ for $\mbox{\rm const} \ne 0$, $I$ is not proportional to the energy integral $g(\xi, \xi)$.
Then, by Lemma \ref{-5}, there exists a RIEMANNIAN metric $\bar g$ geodesically equivalent to $g$.
Every projective vector field for $g$ is also a projective vector field for $\bar g$ and vice versa, so that $\Proj_0(M, g) = \Proj_0(M, \bar g)$.
By the (already proved) Riemannian version of projective Obata conjecture we obtain that $\Iso_0(M, \bar g) = \Proj_0(M, \bar g)$. Thus, $\Proj_0(M, g)= \Iso_0(M,\bar g).$
By \cite[Corollary 1]{beltrami-small}, see also \cite{knebelman}, the dimensions of the Lie group of isometries of geodesically equivalent metrics coincide. Indeed, for every Killing vector field $\bar K$ for $\bar g$ the vector field
$K^i:= \left(\tfrac{\det g }{\det \bar g}\right)^{\tfrac{1}{n+1}}\bar g^{ik}g_{kj} \bar K^j $
is a Killing vector field for $g$. Then, $\dim(\Iso_0(M, g)) = \dim(\Iso_0(M, \bar g))$
implying that $\Iso_0(M, g) =\Proj_0(M, g)$. Hence,
the assumption that $I$ is not a linear combination of the energy integral $g(\xi, \xi)$ and of the square of an integral linear in velocities leads to a contradiction. Thus, there exists a nontrivial integral linear in velocities. Finally, there exists a nontrivial Killing vector field that we denote by $K$.
Then, the group $\Proj_0$ is at least two-dimensional (because it algebra contains $K$ and $v$).
The structures of possible Lie groups of projective transformations was understood already by S. Lie \cite{Lie}. He proved that the for a $2-$dimensional
metric of nonconstant curvature the Lie algebra of $\Proj_0$ is the noncommutative
two dimensional algebra, or is $\mathfrak{sl}(3,\mathbb{R})$. In both cases there exists a projective vector field
$u$ such that the linear span $span(u, K)$ is a two-dimensional noncommutative Lie algebra.
Then, without loss of generality we can assume that $[K, u]= u$ or $[K, u]= K$.
Now, by Theorem \ref{main4}, there exists a global coordinate system $\bigl(x\in \ (\mathbb{R}, \ \textrm{mod} \ 1), y\in \ (\mathbb{R}, \ \textrm{mod} \ 1)\bigr)$ such that in this coordinate system
$K= \alpha \cdot \tfrac{\partial }{\partial x}$, where $\alpha\ne 0$. Assume $u(x,y)=u_1(x,y) \tfrac{\partial }{\partial x} +u_2(x,y) \tfrac{\partial }{\partial y} $. Without loss of generality we assume that $(u_1(0,0), u_2(0,0))\ne (0,0)$.
Let $\phi_t$ be the flow of $K$. Since $K= \alpha \cdot \tfrac{\partial }{\partial x}$, $\phi_t(x,y)=(x+ \alpha t, y)$. Let us calculate the vector $d\phi_t(u(0,0))$ for $t=1/\alpha$ by two methods (and obtain two different results which gives us a contradiction).
First of all, since $\phi_{1/\alpha}$ is the identity diffeomorohism, $d\phi_t(u(0,0)) =u(0,0)$ for $t=1/\alpha$.
The other method of calculating $d\phi_t(u(0,0))$ is based on the commutative relation $[K, u]= u$ or $[K, u]= K$.
Let us first assume that $K,u$ satisfy
$[K, u]= u$. In the coordinates, this condition
reads $ \alpha \tfrac{\partial }{\partial x}u_1 = u_1$ and $ \alpha \tfrac{\partial }{\partial x}u_2 = u_2$ implying
$u_1(x,0) = u_1(0,0) \cdot e^{x/\alpha }$ and $u_2(x,0) = u_2(0,0) \cdot e^{x/\alpha }$. Then, $$d\phi_{1/\alpha}(u(0,0))= u_1(0,0) \cdot e^{1/\alpha^2 } \tfrac{\partial }{\partial x} + u_2(0,0) \cdot e^{1/\alpha^2}
\tfrac{\partial }{\partial y} = u(0,0) \cdot e^{1/\alpha^2} . $$ Since $(u_1(0,0),u_2(0,0))\ne (0,0)$ we obtain that $d\phi_{1/ \alpha}(u(0,0)) \ne u(0,0)$ which gives a contradiction.
Thus, the commutative relation $[K, u]= u$ is not possible.
Let us now consider the second possible commutative relation $[K, u]= K$.
In coordinates this relation reads
$ \alpha \tfrac{\partial }{\partial x}u_1 = \alpha$ and $ \alpha \tfrac{\partial }{\partial x}u_2 = 0$ implying
$u_1(x,0) = u_1(0,0) + {x}$. We again obtain that $d\phi_{1/\alpha}(u(0,0)) \ne u(0,0)$, which gives a contradiction. Thus, the commutative relation $[K, u]= K$ is also not possible. Finally, in all cases the existence of a nontrivial projective vector field on the torus
$T^2$ equipped with a metric of nonconstant curvature
leads to a contradiction.
Let us now consider the remaining case: we assume that $g$ has constant curvature.
By Gauss-Bonnet Theorem, a metrics of constant curvature on $T^2$ is flat. Then, as we show in \S \ref{flat}, $(T^2, g)$ is isometric
to the { \it standard flat torus } $(\mathbb{R}^2/L, dxdy)$, where $(x,y)$ are the standard coordinates on $\mathbb{R}^2$, and $L$ is a lattice generated by two linearly independent vectors. In particular, all geodesics of the lift of the metric to $\mathbb{R}^2$ are the standard straight lines. Clearly, any projective
transformation of $(\mathbb{R}^2/L, dxdy)$ generates a bijection $\phi:\mathbb{R}^2 \to \mathbb{R}^2 $ that commute with the lattice $L$ and maps straight lines to straight lines. It is easy to see that
the connected component of the group of such bijections consists of parallel translations, i.e., acts by isometries. Finally, $\Proj_0(\mathbb{R}^2/L, dxdy) = \Iso_0(\mathbb{R}^2/L, dxdy)$. Theorem \ref{obatath} is proved.
\weg{
{\bf Acknowledgement.} The author thanks
Deutsche Forschungsgemeinschaft
(Priority Program 1154 --- Global Differential Geometry and Research Training Group 1523 --- Quantum and Gravitational Fields) and FSU Jena for partial financial support, and D. Alekseevsky, O. Bauer, A. Bolsinov, G. Manno, P. Mounoud, G. Pucacco, and A. Zeghib for useful discussions. }
\section{Local theory and the proof of Theorem~1 } \label{puc}
\subsection{Admissible coordinate systems and
Birkhoff-Kolokoltsov forms} \label{admissible}
Let $g$ be a pseudo-Riemannian metric of signature (+,--) on connected oriented $M^2$.
Consider (and fix) two vector fields $V_1, V_2$ on $M^2$ such that
\begin{itemize}
\item[(A)] $g(V_1, V_1) =g(V_2, V_2)=0$ and
\item[(B)] $g(V_1, V_2)>0$,
\item[(C)] the basis $(V_1, V_2)$ is positive (i.e., induces the positive orientation).
\end{itemize}
Such vector fields always exist locally. Since locally there is precisely two possibilities in choosing the directions of such vector fields, the vector fields exist
on a finite (at most, double-) cover of $M^2$.
We will say that a local coordinate system $(x,y)$
is {\it admissible}, if the vector fields $\frac{\partial }{\partial x}$ and $\frac{\partial }{\partial y} $ are proportional to $V_1, V_2$ with positive coefficient of proportionality: $$\frac{\partial }{\partial x}= \lambda_1(x,y) V_1(x,y), \ \ \ \frac{\partial }{\partial y}= \lambda_2(x,y) V_2(x,y), \ \ \ \textrm{where $\lambda_i>0$}.$$
Obviously,
\begin{itemize}
\item admissible coordinates exist in a sufficiently small neighbourhood of every point, \item the metric $g$ in admissible coordinates has the form
\begin{equation}\label{metric}
g =f(x,y)dxdy , \ \ \ \textrm{where $f>0$}, \end{equation}
\item two admissible coordinate systems in
one neighbourhood are connected by \begin{equation} \label{coordinatechange} \begin{pmatrix} x_{new}\\
y_{new}\end{pmatrix}
= \begin{pmatrix} x_{new}(x_{old}) \\
y_{new}(y_{old})\end{pmatrix} , \ \ \textrm{where $\frac{dx_{ new}}{dx_{old}}>0$, $\frac{dy_{ new}}{dy_{old}}>0$}. \end{equation}
\end{itemize}
\begin{Rem} \label{admi} For further use let us note that smooth local functions $x,y$ form an admissible coordinate system, if and only if $V_1(x)>0$, $V_2(y)>0$, and $V_1(y)= V_2(x)=0$ (where $V(h)$ denotes the derivative of the function $h$ in the direction of the vector $V$).
\end{Rem}
\begin{Lemma}[\cite{pucacco}] \label{BK}
Let $(x,y)$ be an admissible coordinate system for $g$.
Let $F$ given by \eqref{integral} be an integral for $g$.
Then,
$$
B_1:= \frac{1}{\sqrt{|a(x,y)|}}dx, \;\; \left({\rm respectively },
B_2:= \frac{1}{\sqrt{|c(x,y)|}}dy \right)
$$
is a 1-form, which is defined at points such that $a\ne 0$ (respectively, $c\ne 0$). Moreover, the coefficient
$a$ (respectively, $c$) depends only on $x$ (respectively, $y$), which in particular implies that the forms $B_1$, $B_2$ are closed. \end{Lemma}
\begin{Rem} The forms $B_1, B_2$ are not the direct analog of the ``Birkhoff" 2-form introduced by
Kolokoltsov in \cite{Kol}. In a certain sense, they are real
analogs of the two branches of the square root of the Birkhoff form.
\end{Rem}
\noindent{\bf Proof of Lemma~\ref{BK}.} The first part of the statement, namely that $$
\frac{1}{\sqrt{|a(x,y)|}}dx, \;\; \left({\rm respectively },
\frac{1}{\sqrt{|c(x,y)|}}dy \right)
$$
transforms as a $1$-form under admissible coordinate changes is evident: indeed, after the coordinate change
\eqref{coordinatechange}, the momenta transform as follows:
$p_{x_{old}}= p_{x_{new}}\frac{d{x_{new}}}{d{x_{old}}}$, $p_{x_{old}}= p_{x_{new}}\frac{d{x_{new}}}{d{x_{old}}}$. Then, the integral $F$ in the new coordinates has
the form
$$ \underbrace{\left(\frac{d{x_{new}}}{d{x_{old}}}\right)^2{a}}_{a_{new} } {p_{x_{new}}^2} + \underbrace{\frac{d{x_{new}}}{d{x_{old}}}\frac{d{y_{new}}} {d{y_{old}}}{b}}_{b_{new}} {p_{x_{new}}} {p_{y_{new}}} + \underbrace{\left(\frac{d{y_{new}}}{d_{y_{old}}} \right)^2{c}}_{c_{new}} {p_{y_{new}}^2}.$$
Then, the formal expression $\frac{1}{\sqrt{|a|}}dx_{old}
$ ({\rm respectively}, $ \frac{1}{\sqrt{|c|}}dy_{old}$) transforms into
$$
\frac{1}{\sqrt{|a|}} \frac{d{x_{old}}}{d{x_{new}}} dx_{new} \ \ \ \ \ \left(\textrm{respectively, $ \frac{1}{\sqrt{|c|}}\frac{d{y_{old}}}{d{y_{new}}}dy_{new}$}\right), $$
which is precisely the transformation law of 1-forms.
Let us prove that the coefficient
$a$ (respectively, $c$) depends only on $x$ (respectively, $y$), which in particular implies that the forms $B_1$, $B_2$ are closed.
If $g$ is given by \eqref{metric}, its Hamiltonian is
$$H=\frac{2p_xp_y}{f} \, ,$$
and the condition $\{H, F\}=0$ reads \\
\begin{eqnarray*}
0&=& \left\{\frac{2p_xp_y}{f}, ap_x^2+ bp_xp_y+ cp_y^2\right\} \\
&=& \frac{2}{f^2}\left(p_x^3(fa_y) + p_x^2 p_y (fa_x + fb_y + 2 f_x a + f_y b)+ p_yp_x^2 (fb_x + fc_y+ f_x b + 2 f_y c)+ p_y^3 (c_xf)\right) \, ,
\end{eqnarray*}
i.e., is equivalent to the following system of PDE:
\begin{equation}\label{sys}
\left\{\begin{array}{rcc} a_y&=&0 \, ,\\
fa_x + fb_y + 2 f_x a + f_y b&=&0\, ,\\
fb_x + fc_y+ f_x b + 2 f_yc &=&0\, ,\\
c_x&=&0 \, .\end{array}
\right.\end{equation}
Thus, $a=a(x)$, $c=c(y)$ implying that
$B_1:= \frac{1}{\sqrt{|a|}} dx$ and $B_2:= \frac{1}{\sqrt{|c|}}dy$ are closed forms (assuming $a\ne 0$ and $c\ne 0$). Lemma~\ref{BK} is proved.
\begin{Rem} \label{rem3} For further use let us formulate one more consequence of equations \eqref{sys}: if $a\equiv c \equiv 0$ in a neighbourhood of a point, then $bf = \mbox{\rm const}$, implying $F- \tfrac{\mbox{\rm const}}{2} \cdot H=0$ in the neighborhood. If we consider \eqref{sys} as a system of PDE on the unknown functions $a,b,c$, we see that the system is linear and of finite type. Than, vanishing of the solution corresponding to the integral $\hat F:= \left(F- \tfrac{\mbox{\rm const}}{2} \cdot H\right)$ in the neighborhood implies vanishing of the solution on the whole connected manifold. Thus, if $a\equiv c \equiv 0$ in a neighborhood of a point, then for a certain $\mbox{\rm const} \in \mathbb{R}$ we have $F\equiv \mbox{\rm const} \cdot H$ on the whole manifold.
\end{Rem}
\begin{Rem} \label{rem4} For further use let us note that the set of the points where the form $B_1$ ($B_2$, resp.) is not defined coincides with the set of the points such that $a=0$ ($c=0$, resp.) and is invariant with respect to the (local) flow of the vector field $V_2$ ($V_1$, resp.)
\end{Rem}
A local coordinate system $(x,y)$ will be called \emph{perfect}, if it is admissible, and
if in this coordinates system the coefficients $a, c$ take values in the set $\{-1,0,1\}$ only.
\begin{Lemma} \label{adm} Let $F$ given by \eqref{integral} be an integral
for the geodesic flow of $ g =f(x,y) dxdy$
such that $F\ne \mbox{\rm const} \cdot H$ for all $\mbox{\rm const} \in \mathbb{R}$.
Then, almost every point $p$ has a neighborhood $U$ such that precisely one of the following conditions is fulfilled: \begin{itemize} \item[(i)] $ac>0$ at all points of $U$,
\item[(ii)] $ac<0$ at all points of $U$,
\item[(iii)(a)] $a=0$ and $c\ne 0$ at all points of $U$, or
\item[(iii)(b)] $a \ne 0$ and $c=0$ at all points of $U$.
\end{itemize}
Moreover, there exists a perfect coordinate system $\tilde x, \tilde y$ in a (possibly, smaller) neighborhood $U'(p)\subseteq U(p)$ of $p$.
In the perfect coordinate system, the metric and the integral
are given by
$$
g= \tilde f(\tilde x,\tilde y)dxdy \ \ \textrm{and} \ \ F= \mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(a(x,y)) p_{\tilde x}^2 + \tilde b(\tilde x,\tilde y) p_{\tilde x}p_{\tilde y} + \mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(c( x,y)) p_{\tilde y}^2,
$$
where $\mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(\tau) = \left\{\begin{array}{ccc} 1 &\textrm{if} & \tau >0 \\ -1 &\textrm{if} & \tau <0 \\ 0 &\textrm{if} & \tau =0. \end{array}\right.$
\end{Lemma}
\noindent {\bf Proof of Lemma \ref{adm}.} It is sufficient to prove the lemma assuming that $M^2$ is a small neighborhood $W$. We consider and fix admissible coordinates in this neighborhood. In this coordinates the coefficients $a,b,c$ of the integral \eqref{integral} are smooth functions.
We conisder the following subsets of $W:$
\begin{itemize}
\item $W_{ac\ne 0}:= \{ q\in W \mid a(q)c(q) \ne 0\}$,
\item $W_{a\ne 0,c = 0}:= \{q\in W \mid a(q) \ne 0, c(q)= 0\}$,
\item $W_{a= 0,c \ne 0}:= \{q\in W \mid a(q) = 0, c(q)\ne 0\}$,
\item $W_{a= 0,c = 0}:= \{q\in W \mid a(q) = 0, c(q)= 0\}$.
\end{itemize}
The sets are clearly disjunkt, there union coincides with the whole $W$.
We consider the set $W_{\textrm{perfect}}:= W_{ac\ne 0} \cup int(W_{a=0,c \ne 0}) \cup int(W_{a\ne 0,c = 0}), $ where ``$int$" denotes the set of inner points. The set $W_{\textrm{perfect}}$
is open, and is everywhere dense in $W$. Indeed, it is open, since $W_{ac\ne 0} $, $int(W_{a=0,c \ne 0})$, and $ int(W_{a\ne 0,c = 0}) $ are open. It is everywhere dense,
since it is everywhere dense in the set $W_{ac\ne 0} \cup W_{a=0,c \ne 0}\cup W_{a\ne 0,c = 0}, $ and the remaining set $W_{a= 0,c = 0}$ is nowhere dense by Remark \ref{rem3}.
Now, by definition, every point of $W_{\textrm{perfect}}$ has a neighborhood such that in this neighborhood one of the conditions (i)--(iii) is fulfilled. The first statement of the proposition is proved.
Let us now
prove the second statement.
Let $p_0\in int(W_{a\ne 0,c = 0})$. In a simply-connected
neighborhood $U(p_0)\subset W_{a\ne 0,c = 0} $, we consider the function
\begin{equation} \label{normalx}
x_{new}(p) :=\int\limits_{p_0}^p
B_1. \end{equation} Since the form $B_1$ is closed, and $U(p_0)$ is simply-connected, the function $x_{new} $ does not depend on the choice of the curve connecting the points $p_0,p$,
and is therefore well defined. The differential of the function $x_{new}$ is precisely the 1-form $B_1$, and does not vanish at $p_0$. We have $V_1(x_{new})= B_1(V_1)>0$,
$V_2(x_{new})= B_1(V_2)= 0$. Since the coordinates $(x,y)$ are admissible, $V_2(y)>0$ and
$V_1(y)= 0$. Then, by Remark \ref{admi}, $(x_{new}, y)$ is a local admissible
coordinate system in a possibly smaller neighborhood $U'\subseteq U$ containing $p_0$.
\begin{Rem} Let us note that, in the admissible coordinates the formula \eqref{normalx} looks \begin{equation}\label{loc:normalx} x_{new}(x{})=\int_{x_0}^{x_{}} \frac{1}{\sqrt{|a(t)|} }\, dt
\end{equation} implying that $x_{new}$ is independent of $y$, i.e., $x_{new}=x_{new}(x)$. \end{Rem}
In this coordinate system,
the integral $F$ is equal to $$ {\left(\frac{d{x_{new}}}{d{x}}\right)^2{a}} {p_{x_{new}}^2} + {\frac{d{x_{new}}}{d{x_{old}}}{b}} {p_{x_{new}}} {p_{y}} = \frac{{a}}{(\sqrt{|a|})^2} {p_{x_{new}}^2} + {\frac{b}{\sqrt{|a|}}} {p_{x_{new}}} {p_{y}} =
\mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(a) {p_{x_{new}}^2} + b_{new} {p_{x_{new}}} {p_{y}} .$$
The cases $p_0\in int(W_{a= 0,c \ne 0})$, $p_0\in W_{a\ne 0,c \ne 0}$ are similar: in the case $p_0\in int(W_{a =0,c \ne 0})$, in the coordinate system $(x, y_{new})$ in a possibly smaller neighborhood of $p_0$, where
\begin{equation} \label{normaly}
y_{new} :=\int\limits_{p_0}^pB_2,\end{equation}
the integral $F$ is given by $b_{new}p_{x}p_{y_{new}} + \mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(c) p_{y_{new}}^2$. In the case $p_0\in W_{a\ne 0,c \ne 0}$,
in the coordinate system $(x_{new}, y_{new})$, where $x_{new}$ is given by \eqref{normalx} and $y_{new}$ is given by \eqref{normaly}, the integral $F$ is given by $\mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(a) p_{x_{new}}^2+ b_{new}p_{x_{new}}p_{y_{new}} + \mbox{\rm sign}}\newcommand{\Proj}{\mbox{\rm Proj}}\newcommand{\Iso}{\mbox{\rm Iso}(c) p_{y_{new}}^2$. Lemma \ref{adm}
is proved.
\begin{Rem} \label{pp} If $a=0$ ($c=0$, resp.), the coordinate
transformation of the form $(x_{new}(x),y)$ ($(x,y_{new}(y))$, resp.) does not change the property of coordinates to be perfect.
If $ac\ne 0$,
the perfect coordinates are unique up to transformation $(x,y)\mapsto (x+ \mbox{\rm const}_1, y+\mbox{\rm const}_2)$. In particular, if $ac\ne 0$, the vector fields $\tfrac{\partial }{\partial x}$ and $\tfrac{\partial }{\partial y}$, where $x,y$ are local perfect coordinates,
do not depend on the choice of local perfect coordinates, and therefore are well-defined globally, at all points such that $ac\ne 0$ (provided that $V_1, V_2$ satisfying (A,B,C) are globally defined). \end{Rem}
\subsection{Proof of Theorem~2}
By Lemma \ref{adm}, almost every point of $M^2$ has a neighborhood such that
in perfect coordinates the metrics and the integral are as in one of the following cases:
\begin{itemize} \item[] {\bf Case 1: $ac >0$:} The metric is $f(x,y) dx dy $, the integral is $\pm (p_x^2 + b(x,y) p_xp_y + p_y^2)$.
\item[] {\bf Case 2: $ac<0$:} The metric is $f(x,y) dx dy $, the integral is $\pm(p_x^2 + b(x,y) p_xp_y - p_y^2)$.
\item[] {\bf Case 3a: $c\equiv 0$: } The metric is $f(x,y) dx dy $, the integral is $\pm(p_x^2 + b(x,y) p_xp_y)$.
\item[] {\bf Case 3b: $a\equiv 0$: } The metric is $f(x,y) dx dy $, the integral is $\pm( b(x,y) p_xp_y+ p_y^2)$. \end{itemize}
We will carefully consider all four cases.
\subsubsection{Case 1} \label{case1}
\begin{Prop} \label{c1} Let the geodesic flow of a metric $g=f(x,y)dxdy$ admits an integral \eqref{integral}. Assume $ac>0$ at the point $p$.
Then, in the coordinates $(u,v)= (\tfrac{x +y}{2}, \tfrac{x-y}{2})$, where $(x,y)$ are perfect coordinates in a neighborhood of $p$,
\begin{equation}\label{answer:case1}
g=(U(u)-V(v))(du^2 -dv^2) \ \textrm{and} \ \
F= \pm \left(\frac{p_v^2 U(u) - p_u^2V(v)}{U(u)-V(v)} \right)\, ,
\end{equation}
where $U,V$ are certain functions of one variable.
\end{Prop}
\noindent{\bf Proof. } Without loss of generality $a$ and $c$ are positive in a neighborhood of $p$.
Then, by Lemma \ref{adm}, in perfect coordinates in a neighborhood of $p$
the metric and the integral are
$g=f(x,y) dx dy $, $F=p_x^2 + b(x,y) p_xp_y + p_y^2$. Then, the system (\ref{sys}) has the following simple form:
$$
\left\{\begin{array}{rcc} (fb)_y+ 2 f_x &=&0 \,,\\
(fb)_x + 2 f_y&=&0 \, , \end{array}
\right. \ \textrm{which is equivalent to} \
\left\{\begin{array}{rcc} (fb+ 2 f)_x + (fb + 2 f)_y&=&0 \,,\\
(fb -2 f)_x -(fb- 2 f)_y&=&0 \,. \end{array}
\right.$$
After the (non-admissible) change of coordinates $u = \tfrac{x+y}{2}$, $v= \tfrac{x-y}{2}$, the system has the form
$$
\left\{\begin{array}{rcc} (fb+ 2 f)_u&=&0 \,,\\
(fb -2 f)_v&=&0 \,, \end{array}
\right.\ \textrm{which is equivalent to} \ \left\{\begin{array}{rcc} fb+ 2f & =& -4{V(v)}\, ,\\ fb-2f& =&-4{U(u)}\end{array} \right.
$$ for certain functions
$U(u) $ and $V(v)$.
Thus,
$$
f= {U(u)-V(v)} \, , \;\; b=-2 \frac{U(u)+V(v)}{U(u)-V(v)}\, .
$$
Let us now calculate the metric in the integral in the new coordinates: substituting
$dx= du + dv, dy= du- dv$ in the formula $g = f(x,y) dxdy = {(U(u)-V(v))} dxdy$, we obtain that in the new coordinates the metric is $(U(u)-V(v))(du^2 - dv^2)$. Substituting $p_x =\left(\tfrac{\partial u}{\partial x}p_u + \tfrac{\partial v}{\partial x}p_v \right) = \tfrac{1}{2} (p_u + p_v) $ and
$p_y =\left(\tfrac{\partial u}{\partial y}p_u + \tfrac{\partial v}{\partial y}p_v \right)=\tfrac{1}{2} (p_u - p_v) $ in the formula $F= p_x^2 + b p_xp_y +p_y^2 = p_x^2 -2 \frac{U(u)+V(v)}{U(u)-V(v)}p_xp_y +p_y^2$, we obtain that in the new coordinates $(u,v)$
$$
F= \tfrac{1}{2}\left( p_u^2 + p_v^2 - \frac{U(u)+V(v)}{U(u)-V(v)}(p_u^2 - p_v^2) \right)=
\frac{U(u) p_v^2 - V(v) p_u^2 }{U(u)-V(v)}.
$$
We see that, in the new coordinates, the metric and the integral are as in \eqref{answer:case1}. Proposition \ref{c1} is proved.
\subsubsection{ Case 2 }
\begin{Prop}
\label{c2} Let the geodesic flow of a metric $g=f(x,y)dxdy$ admits an integral \eqref{integral}. Assume $ac<0$ at the point $p$. Then, in perfect coordinates in a neighborhood of $p$,
\begin{equation}\label{answer:case2}
g= \Im(h)dxdy \ \ \ \textrm{and} \ \ \ F=\pm \left( p_x^2 - p_y^2 + 2\frac{\Re(h)}{\Im(h)}p_xp_y\right),
\end{equation}
where $\Re(h)$ and $\Im(h)$ are the real and the imaginary parts of a holomorphic function $h$ of the variable $z = x+ i\cdot y$.
\end{Prop}
\noindent{\bf Proof. }
Without loss of generality $a(p)>0, $ $c(p)<0$. By Lemma \ref{adm}, in perfect coordiantes
the metric and the integral are
$g=f(x,y) dx dy $, $F=p_x^2 + b(x,y) p_xp_y - p_y^2$. Then, the system (\ref{sys}) has the following simple form:
\begin{equation}\label{sys:case2}
\left\{\begin{array}{rcc} (fb)_y+ 2 f_x &=&0 \,,\\
(fb)_x -2 f_y&=&0 \,. \end{array}
\right.\end{equation}
We see that these equations are the Cauchy-Riemann conditions for the complex-valued
function $fb+ 2i f$. Thus, for an appropriate holomorphic
function $h= h(x+ iy)$ we have $fb=\tfrac{1}{2}\Re(h)$, $f =\Im(h)$.
Finally, the metric and the integral have the form \eqref{answer:case2}. Proposition \ref{c2} is proved.
\subsubsection{Case 3 } \label{3t}
In this case we prove two propositions: the first one is more general, and is the final step in the proof of Theorem \ref{main}. The second one requires additional assumptions, and will be used in the proof of Theorem \ref{main1}.
\begin{Prop}
\label{c3} Let the geodesic flow of a metric $g=f(x,y)dxdy$
admits an integral \eqref{integral}. Then, the following two statements are true:
\begin{itemize}
\item[(a)] If $a(p)\ne 0$, and $c(q)=0$ at every point $q$ of a small neighborhood of $p$, in perfect coordinates in a (possibly, smaller) neighborhood of $p$,
\begin{equation}\label{answer:case3a}
g= \left( \widehat{ Y}(y)+\frac{x}{2} Y'(y)\right)dxdy \ \ \textrm{and} \ \ F= \pm\left(p_x^2 - \frac{Y(y)}{\widehat{ Y}(y)+\frac{x}{2} Y'(y) }p_xp_y \right)\, ,
\end{equation} where $Y$ and $\widehat Y$ are functions of one variable.
\item[(b)] If $c(p)\ne 0$, and $a(q)=0$ at every point $q$ of a small neighborhood of $p$, in perfect coordinates in a (possibly, smaller) neighborhood of $p$,
\begin{equation}\label{answer:case3b}
g= \left( \widehat{ X}(x)+\frac{y}{2} X'(x)\right)dxdy \ \ \textrm{and} \ \ F=\pm \left( p_y^2 - \frac{X(x)}{\widehat{ X}(x)+\frac{y}{2} X'(x) }p_xp_y \right) \, ,
\end{equation} where $X$ and $\widehat X$ are functions of one variable. \end{itemize}
\end{Prop}
\noindent{\bf Proof. }
The cases (a) and (b) are clearly analogous;
without loss of generality we can assume $a(p)>0, $ $c\equiv 0$.
By Lemma \ref{adm}, in perfect coordinates
the metric and the integral are
$g=f(x,y) dx dy $, $ F=p_x^2 + b(x,y) p_xp_y $.
Then, the equation (\ref{sys}) has the following simple form:
\begin{equation}\label{sys:case3}
\left\{\begin{array}{rcc} (fb)_y+ 2 f_x &=&0 \, ,\\
(fb)_x &=&0 \, .\end{array}
\right.\end{equation}
This system can be solved. Indeed, the second equation implies $fb= -Y(y)$. Substituting this in the first equation we obtain
$Y'(y)= 2f_x$ implying
$$f= \frac{x}{2} Y'(y)+ \widehat{ Y}(y) \textrm{ \ \ and} \ \ \ b= - \frac{Y(y)}{\frac{x}{2} Y'(y)+ \widehat{ Y}(y)}\, .
$$
Finally, the metric and the integral are as in \eqref{answer:case3a}. Proposition \ref{c3}(a) is proved. The proof of Proposition \ref{c3}(b) is essentially the same.
{\bf Proof of Theorem 1. }
Theorem~\ref{main} follows directly from Lemma \ref{adm} and Propositions \ref{c1}, \ref{c2}, \ref{c3}. Indeed,
by Lemma \ref{adm}, almost every point has a neighborhood such that in this neighborhood the assumptions of one of Propositions \ref{c1}, \ref{c2}, \ref{c3} are fulfilled. Then, by Propositions \ref{c1}, \ref{c2}, \ref{c3} the metric and the integral are as in the table in
Theorem \ref{main}.
We will also need a slightly less general version of normal form of metrics satisfying the assumption of Case 3.
Let us observe that the function
$Y$ from \eqref{answer:case3a}, or the function $X$ from \eqref{answer:case3b}, can be given in invariant terms (i.e., they does not depend on the choice of a perfect coordinate system, and
can be smoothly prolonged to the whole manifold). Indeed, consider the symmetric
$(2,0)-$tensor $\tilde F^{ij}$ such that $F= \sum_{i,j}\tilde F^{ij}p_ip_j$
(if $F $ is given by \eqref{integral}, the matrix of $\tilde F$ is $\begin{pmatrix} a & b/2 \\ b/2 & c\end{pmatrix}$). Transvecting $\tilde F^{ij}$ with $g_{ij}$ we obtain the globally
defined smooth function $L:= \mbox{\rm trace}(\tilde F_j^i):= \sum_{i,j} \tilde F^{ij}g_{ij}$. Under assumptions of Case 3a, in the perfect coordinates, the function $L$ is given by
\begin{equation}\label{L}
L= \sum_{i,j} \tilde F^{ij} g_{ij} =
\mbox{\rm trace}\left( \begin{pmatrix} a & b/2 \\ b/2 & c\end{pmatrix} \begin{pmatrix} 0& f/2 \\ f/2 & 0\end{pmatrix} \right) = \mbox{\rm trace}\left( \begin{pmatrix} -Y/4& \ast \\ 0 & -Y/4\end{pmatrix} \right)=-Y/2.
\end{equation}
\weg{ Now, let us observe that if (under assumptions of Case 3a) $Y\equiv \mbox{\rm const}$ in a neighborhood $U$, then in this neighborhood the integral $F$ is a linear combination of a square of an integral linear in momenta and the Hamiltonian. Indeed, in this case $Y'=0$, and the formulas \eqref{answer:case3a} look
$$g= \widehat{ Y}(y)dxdy \ \ \textrm{and} \ \ F= \pm\left(p_x^2 - \frac{\mbox{\rm const} }{\widehat{ Y}(y)} p_xp_y \right).
$$
Since the components of the metric do not depend on $x$, $p_x$ is an integral (linear in momenta)
of the geodesic flow of $g$, and
$F$ is a linear combination of $p_x^2$ and $H$.
Since in Theorem \ref{main1} we require that the restriction of the integral to no neighborhood is a linear combination of a square of an integral linear in momenta and the Hamiltonian, we can
take $L$ as a local coordinate in the neighborhood of almost every point $p$ such that $c\equiv 0$ in $U(p)$, or $a\equiv 0$ in $U(p)$. }
\begin{Prop}
\label{c4} Let the geodesic flow of a metric $g=f(x,y)dxdy$
admits an integral \eqref{integral}. Then, the following two statements are true:
\begin{itemize}
\item[(a)] Suppose $a(p)\ne 0$, and $c(q)=0$ at every point $q$ of a small neighborhood of $p$.
Assume $dL_{|p}\ne 0$, where $L$ is given by \eqref{L}. Then, in a (possibly, smaller) neighborhood of $p$, in perfect coordinates $(x,y) $ such that $y(q)= -{2}L(q)$ for all $q$, the metric and the integral are given by
\begin{equation}\label{answer:case3abis}
g= \left( { Y}(y)+\frac{x}{2} \right)dxdy \ \ \textrm{and} \ \ F= \pm\left(p_x^2 - \frac{y}{{ Y}(y)+\frac{x}{2} }p_xp_y \right)\, ,
\end{equation}
where $Y$ is a function of one variable.
\item[(b)] Suppose $c(p)\ne 0$, and $a(q)=0$ at every point $q$ of a small neighborhood of $p$.
Assume $dL_{|p}\ne 0$, where $L$ is given by \eqref{L}. Then, in a (possibly, smaller) neighborhood of $p$,
in perfect coordinates $(x,y) $ such that $x(q)= -{2}L(q)$ for all $q$, the metric and the integral are given by
\begin{equation}\label{answer:case3bbis}
g= \left( { X}(x)+\frac{y}{2} \right)dxdy \ \ \textrm{and} \ \ F= \pm\left(p_y^2 - \frac{x}{{ X}(x)+\frac{y}{2} }p_xp_y \right)\, ,
\end{equation}
where $X$ is a function of one variable. \end{itemize}
\end{Prop}
\noindent{\bf Proof. } The cases (a) and (b) are clearly analogous;
without loss of generality we can assume $a(p)>0, $ $c\equiv 0$. In the perfect coordinates such that
$y= -{2}L$, we have $g=f(x,y)dxdy$ and $F= p_x^2 -\tfrac{y}{f} p_x p_{y} $. Then, the system
\eqref{sys} is equivalent to the equation
$
2 f_x =1.$
Thus, $f=Y(y) +\tfrac{x}{2}$. Proposition \ref{c4}(a) is proved. The proof of Proposition \ref{c4}(b) is similar.
\section{ Global theory and the main step in the proof of Theorem~2} \label{4}
\subsection{Notation, conventions, and the plan of the proof} \label{notation}
Within the whole section we assume that \begin{itemize} \item
the surface is the torus $T^2$,
\item $g$ is a pseudo-Riemannian metric of signature (+,--) on $T^2$.
\item The vector fields $V_1$, $V_2$ satisfying conditions (A,B,C) from \S \ref{admissible} are globally defined (the case when it is not possible will be considered in \S \ref{laststep}).
\item $F$ is a nontrivial integral of the geodesic flow of
$g$. We will reserve notation $x,y$ for admissible coordinates, or for perfect coordinates, and will denote the coefficients of the integral as in \eqref{integral}. As in \S \ref{admissible}, we will denote by $B_1$, $B_2$ the $1-$forms
$\tfrac{1}{\sqrt{|a|}} dx$ and $\tfrac{1}{\sqrt{|c|}} dy$. \end{itemize}
As in \S \ref{3t}, we denote by $\tilde F^{ij}$ the
symmetric $(2,0)-$tensor corresponding to the integral $F$, and by $\tilde F_j^i$ the $(1,1)-$tensor
$\tilde F_j^i:= \sum_k \tilde F^{ik}g_{k j }.$
We will proceed according to the following plan:
\begin{enumerate}
\item In \S \ref{inco} we show that there exists no point such that $ac<0$. This will imply that $\tilde F_j^i$ has real eigenvalues at every point of $T^2$.
\item By Remark \ref{invar}, $\tilde F_j^i$ has only one eigenvalue
(of algebraic multiplicity 2) at the points such that $B_1$ or $B_2$ is not defined. In \S \ref{injo}, we show that
this eigenvalue is
constant on each connected component of the set such that $B_1$ or $B_2$ is not defined.
\item In \S \ref{liou} we show that the existence a point such that $B_1$ or $B_2$ is not defined implies that
one of the eigenvalues of $\tilde F_j^i$ is constant on the whole manifold.
\item In \S \ref{eigenvalue}, we show that if one of the eigenvalues of $\tilde F_j^i$ is constant, the quadratic integral $F$, or the lift of the quadratic integral to the appropriate double cover is a linear combination the square of a function linear in momenta and the Hamiltonian. Later, in Corollary \ref{cover},
we show that if the lift of the quadratic integral to a double cover is a linear combination of the lift of the Hamiltonian and the square of a integral linear in momenta,
then the integral is a linear combination of the Hamiltonian and the square of an integral linear in momenta.
\item In \S\ref{endof} we show that if at every point $B_1$ and $B_2$ are defined, then the torus, the
metric $g$, and the integral $F$ are as in the Model Example 1.
\end{enumerate}
These will prove Theorem \ref{main1} under the additional assumption that the vector fields $V_1, V_2 $ exist on $T^2$. The case when this vector fields do not exist on $T^2$ will be considered later, in \S \ref{laststep}: we will prove that this case can not happen (if there exists an integral quadratic in momenta that is not a linear combination of the Hamiltonian and the square of an integral linear in momenta).
\subsection{At every point, the eigenvalues of $\tilde F_j^i$ are real } \label{inco}
\begin{Lemma} \label{impcomplex}
There is no point $p \in T^2$ such that at this point $ac<0$.
\end{Lemma}
\noindent{\bf Proof. }
Suppose at $p\in T^2$ we have $ac<0$. Let $W_0$ be the connected
component of the set
$$W:= \{ q \in T^2 \mid \textrm{$B_1$ and $B_2$ are defined} \}$$
containing the point $p$. At every $q\in W_0$ we have $ac<0$.
We consider the function $K:W_0\to \mathbb{R}$, $K= \frac{1}{g^*(B_1, B_2)}$, where $g^*$ is the scalar product on $T^*T^2$ induced by $g$.
In any perfect coordinates $(x,y)$ we have $B_1= dx$, $B_2=dy$, and $g = \Im(h) dx dy $ by Proposition \ref{c2}. Then, $K= \Im(h)$ for a holomorphic function $h$ implying it is harmonic function. When we approach the boundary $\overline W_0\setminus W_0$, the function $K$ converges to $0$. Indeed, in the admissible coordinates near a boundary point
the function $K$ is $ f \sqrt{|ac|}$, and $ac\stackrel{\textrm{converges}}{\longrightarrow} 0$ (because at least one of coefficients $a,c$ is zero at the points of boundary).
Finally, by the maximum principle (for harmonic functions), the function $h$ is identically zero, which clearly contradicts the assumptions. Lemma \ref{impcomplex} is proved.
\begin{Cor} \label{jjj}
At every point of $T^2$, the eigenvalues of $\tilde F_j^i$ are real.
\end{Cor}
{\bf Proof.} The eigenvalues are the roots of the characteristic polynomial
\begin{equation} \label{2-1}\chi(t)= \det(\tilde F_j^i- t\cdot \delta_j^i)= \det\left(\begin{pmatrix} fb/4 & af/2 \\ cf/2 & fb/4 \end{pmatrix} - t\cdot \begin{pmatrix} 1& 0 \\ 0 & 1 \end{pmatrix}\right)= t^2 - \tfrac{fb}{2}t+ \tfrac{(fb)^2}{16} - \tfrac{acf^2}{4}.\end{equation}
The discriminant of $\chi(t)$ is ${\cal D} = \tfrac{1}{4}\left(\tfrac{fb}{2}\right)^2 - \left( \tfrac{(fb)^2}{16} - \tfrac{acf^2}{4}\right) = \tfrac{acf^2}{4}$. We see that if $ac\ge 0$ (which is fulfilled by Proposition \ref{c2}) the discriminant is nonnegative implying the eigenvalues of $\tilde F_j^i$ are real. Corollary \ref{jjj} is proved.
\begin{Rem} \label{invar}
For further use let us note that at the points such that $ac=0$ the discriminant $\mathcal{D}$ of $\chi(t)$ given by \eqref{2-1} vanishes implying the tensor $\tilde F_j^i$ has only one eigenvalue (of algebraic multiplicity two), namely $\tfrac{fb}{4}$. At the points such that $ac>0$ the discriminant $\mathcal{D}>0 $ implying the tensor $\tilde F_j^i$ has two different real eigenvalues.
\end{Rem}
\subsection{The function $L:= \sum_i \tilde F_i^i (:= \mbox{\rm trace}(\tilde F_j^i))$ is constant on each connected component of the set of the points such that $B_1$ or $B_2$ is not defined. } \label{injo}
\begin{Lemma} \label{imp}
The function $L= \sum_i \tilde F_i^i (:= \mbox{\rm trace}(\tilde F_j^i))$ is constant on each connected component of the set of the points such that $B_1$ is not defined.
\end{Lemma}
{\bf Proof.} Let at the point $p$ the form $B_1$ is not defined. We consider a small neighborhood $U(p)$ of $p$. Lemma \ref{imp} is a direct corollary of the following
{ \bf Statement. } {\it $L$ is constant on each connected component of the set $\{q\in U(p)\mid \textrm{ $B_1$ is not defined } \} $.}
Now, the above statement follows from the following two propositions:
\begin{Prop} \label{C1} Assume $B_1$ is not defined at every point of a neighborhood of $p$. Then, $L$ is constant in this neighborhood.
\end{Prop}
\begin{Prop} \label{C2} Assume every neighborhood of $p$ has a point such that $B_1$ is defined.
Then, for a certain neighborhood $U(p)$ the function $L$ is constant on the connected component of the set $\{ q \in U(p) \mid \textrm{ $B_1$ is not defined} \}$
containing the point $p$. \end{Prop}
We will proceed as follows: we will first prove Proposition \ref{C1}. Then, we prove a technical Proposition \ref{ppp}. Finally, we will use Propositions \ref{C1}, \ref{ppp} in the proof of Proposition \ref{C2}.
{\bf Proof of Proposition \ref{C1}.} Our goal is to prove that $dL= 0$ at $p$. Without loss of generality, by Remark \ref{rem3}, we can assume $c\ne 0$ at the point $p$.
We assume that $dL\ne 0$ at $p$, and find a contradiction.
We denote by $W_0$ the connected component of the set
$W:= \{ q \in T^2 \mid \textrm{ $B_1$ is not defined } \}$ containing $p$.
We denote by $\alpha :(-\infty, +\infty)\to T^2$
the integral curve of $V_2$ such that $\alpha (0)=p$.
Since $W$ is invariant with respect to the flow of $V_2$, the curve $\alpha$ is a curve on $W_0$.
Let us show that the curve $\alpha$ is periodic.
In a small neighborhood of every point of the curve, we have $a\equiv 0$ implying
$$L= \mbox{\rm trace}{(\tilde F^{i}_j)} = \mbox{\rm trace}\left(\begin{pmatrix}0 & b/2 \\ b/2 & c \end{pmatrix}\begin{pmatrix}0 & f/2 \\ f/2 & 0 \end{pmatrix} \right) =fb/2. $$
Then, the second equation of \eqref{sys} implies that, on $W$, the function $L$ is invariant with respect to the flow of
$V_2$. Hence, at every point of the curve $\alpha$ we have $dL\ne 0$.
Then, the connected component of the set $\{q \in T^2\mid L(q)= L(p)\}$ containing $p$ coincides with the image of $\alpha$. Since $\{q \in T^2\mid L(q)= L(p)\}$ is compact, the image of $\alpha $ is compact implying the image of the curve is a closed circle.
The following cases are possible:
\begin{itemize}
\item[] {\bf Case (a):} For every $t\in \mathbb{R} $, the form $B_2$ is defined at the point $\alpha(t)$,
\item[] {\bf Case (b):} There exists $t\in \mathbb{R} $ such that at the point $\alpha(t)$ the form $B_2$ is not defined.
\end{itemize}
Under assumptions of Case (a),
let us construct a perfect coordinate system in a neighborhood $U(\alpha(t))$ of every point $\alpha(t)$.
We assume that every neighborhood $U(\alpha(t))$ is sufficiently small and is homeomorphic to the disk.
As the first coordinate $x$ we take the function $-{2}L$ (where $L= \sum_{i,j}\tilde F^{ij}g_{ij}$ as above). Since $L$ is preserved by flow of $V_2$, its differential is not zero in a small neighborhood of every point $\alpha(t)$. Since
$dL(V_2)=0$, the coordinate $x$ can be taken as the first admissible coordinate.
In order to construct the second
coordinate $y$, we consider the curve $\gamma:[0,t+1]\to W$ connecting the points $p= \gamma(0) $
and $q\in U(c(t))$ the such that $\gamma_{|[0,t]}= \alpha_{|[0,t]}$, $\gamma(t+1)=q$, and such that $\gamma_{|[t, t+1]}$ lies in $U(\alpha(t))$. We put $y(q):= \int_{\gamma}B_2$.
The function $y$ is well-defined, its differential is $B_2$ and is not zero at $\alpha(t)$.
The local coordinates $x,y$ are as in Proposition \ref{c4}(b). Then, in this coordinates, the metric $g$ is equal to $ \left(X(x) {+\tfrac{1}{2}y}\right)dxdy$. Since $V_2(X)= 0$ locally, and since the functions $X(\alpha(t))$ coincides on the intersection of the
neighborhoods $U(\alpha(t_0))$ and $U(\alpha(t_0+ \varepsilon))$ (for small $\varepsilon$),
for every point of the curve $\alpha$ we have $X(\alpha(t))= X(\alpha(0))= X(p)$.
When $t$ ranges from $-\infty$ to $+\infty$, the coordinate
$y$ also ranges from $-\infty $ to $+\infty$. Indeed, $\int_{\alpha_{|{[0,t]}}}B_2(\alpha'(t))= \int_0^t {B_2}_{|\alpha(s)}({V_2}_{| \alpha(s)}) ds $, and $B_2(V_2) $ is positive and is therefore separated from zero on the compact set $\textrm{image}(\alpha)$.
Then, there exists $t$ such that the value of $y$ corresponding to $\alpha(t)$ is $-2X(p)$ . At the point $\alpha(t)$, the metric $g = \left(X(x) {+\tfrac{1}{2}y}\right)dxdy$ is degenerate which contradicts the assumptions. Proposition \ref{C1} is proved under the additional assumptions of Case (a).
Let us now prove Proposition \ref{C1} under assumtions of Case (b): we
assume that there exists $t$ such $B_2=0$ at $\alpha(t)$ this point .
Let $(t_{min}, t_{max})$, where $ t_{min} < 0 < t_{max} \in \mathbb{R}$, be the (open) interval such that
\begin{itemize} \item $B_2$ is defined at $\alpha(t)$ for every $t\in (t_{min}, t_{max})$,
\item $B_2$ is not defined at
$\alpha(t_{min})$, and at $\alpha(t_{max})$. \end{itemize}
As in the proof for Case (a),
we construct a perfect local coordinate system $x,y$ in a neighborhood $U(\alpha(t))$ of every point $\alpha(t)$, where $t\in (t_{min}, t_{max})$. We put $x(q):= -{2}L(q) $ and
$y(q):= \int_{\gamma}B_2$, where $\gamma:[0,t+1]\to W$, \ $\gamma_{|[0,t]}= \alpha_{|[0,t]},$ \ $\gamma(t+1)=q$, and such that $\gamma_{|[t, t+1]}$ lies in $U(\alpha(t))$. We assume that the neighborhood $U(\alpha(t))$ is sufficiently small implying $B_2$ is defined at every point of $U(\alpha(t))$, and is homeomorphic to the disk.
By Proposition \ref{c4}, in this coordinates, the metric is $(X(x) + \tfrac{1}{2}y)dxdy$.
Let us show that the coordinate $y$ converges to $-2X(p)$ when $t$ converges to $t_{max}$.
In order to do this, we consider the scalar product on $T^*T^2$ induced by $g$ (we will denote this scalar product by $g^*$). We consider the
function $h:= g^*(-{2}dL , B_2)$. This is indeed a function (i.e., $h$ does not depend on the choice of an admissible coordinate system) which is defined at the points such that $B_2$ is defined. In admissible coordinates $(\tilde x, \tilde y)$
in the neighborhood of the point $\alpha(t_{max})$, the function is given by
$ h= -{2}\tfrac{1}{\tilde f} \cdot \tfrac{\partial L }{\partial \tilde x} \cdot \tfrac{1}{\sqrt{|\tilde c|}}$. Since
$\tilde c(\alpha(t_{max}))=0$, we have $h(\alpha(t))\stackrel{t \to t_{max}}{\longrightarrow} \pm \infty $.
In the constructed above coordinates $(x,y)$, we have $h(\alpha(t))= \tfrac{1}{X(p)+ \tfrac{y(\alpha(t))}{2} }\cdot 1 \cdot 1$. Then, ${X(p)}+ \tfrac{y(\alpha(t))}{2} \stackrel{t \to t_{max}}{\longrightarrow} 0$. Thus, $y(\alpha(t)) \stackrel{t \to t_{max}}{\longrightarrow} -2X(p)$.
Similarly one can show that the same is true for $t_{min}$, namely $y(\alpha(t)) \stackrel{t \to t_{min}}{\longrightarrow} -2X(p)$. .
Since $y(\alpha(t)) = \int_0^t {B_2}_{|\alpha(s)}({V_2}_{| \alpha(s)}) ds $, and ${B_2}_{|\alpha(s)}({V_2}_{| \alpha(s)}) $ is positive for all $s\in (t_{min}, t_{max})$, the values of $y(\alpha(t)) $ can not converge to the same number for $t\to t_{max}$ and for $t\to t_{min}$. The obtained contradiction proves Proposition \ref{C1}.
\begin{Prop} \label{ppp} The set $\{q\in T^2 \mid \textrm{ $B_1$ or $B_2$
is defined in $q $ } \} $ is connected.
\end{Prop}
{\bf Proof.} It is sufficiently to prove that every point $p $ has a neighborhood $U(p)$
such that the set $S(p):= \{q\in U(p) \mid \textrm{ $B_1$ or $B_2$
is defined in $q $ } \} $ is connected. We take a sufficiently small $U(p)$, and consider admissible coordinates $x,y$
in $U(p)$. We assume that the neighborhood is small enough so we can connect every two points of this neighborhood by a geodesic.
If the set $S(p)$ is not connected, at every point $q\in U(p)$ we have
$a(q)=0$, or $c(q)= 0$. Without loss of generality we can assume that at every point of $U(p) $ we have $a=0$. Then, the point $p$ satisfies the assumptions of Proposition \ref{C1} above implying $L= fb/2= \mbox{\rm const}$ on $U(p)$.
Let us now consider the points $U(p)\setminus S(p)$. At every such point, $a=c=0$ implying
$$ \tilde F^{ij} = \begin{pmatrix}0 & b/2 \\ b/2 & 0 \end{pmatrix} = \begin{pmatrix}0 & L/f \\ L/f & 0 \end{pmatrix} = \tfrac{{L}}{4} \cdot g^{ij}.$$ Thus, at such points, $F = \tfrac{L}{2} H= \mbox{\rm const} \cdot H$. Without loss of generality we can assume that $\mbox{\rm const} =0$, otherwise we can replace $F$ by $(F- \mbox{\rm const} \cdot H)$.
We take 5 points $p_1,...,p_5\in U(p)\setminus S(p)$ such that $F_{|T^*_{p_i}T^2}=0 $ at these points.
Since $F$ is an integral, it vanishes on every geodesic passing through any of the points $p_1,..., p_5$. Take a point $q\in U(p)$ in a small neighborhood of $S$, and connect this point with the points $p_1,...,p_5$ by geodesics, see Figure \ref{fig}.
Let $ \xi_1\in T^*_{p_1}T^2, \dots , \xi_5\in T^*_{p_5}T^2$ be the vector-momenta of these geodesics at $q$. At almost every $q$, the tangent vectors of the geodesics are mutually nonproportional implying the vector-momenta $\xi_i$ and $\xi_j$
are not proportional for $i\ne j$.
\begin{figure}[ht!]
{\includegraphics[width=.3\textwidth]{fivepoints.eps}}
\caption{The geodesic connecting the points $p_i$ with the point $q$, and their tangent vectors at the point $q$. For almost every $q$, the tangent vectors are mutually nonproportional}\label{fig}
\end{figure}
Since $F$ is an integral and $F_{|T^*_{p_i}T^2}\equiv 0$, we have $ F(\xi_i)= 0$. Thus, the quadratic function $F_{|T_qT^2 }$ vanishes in 5 mutually nonproportional points $\xi_i$. Hence, $ F_{|T^*qT^2}\equiv 0$. Thus, the restriction of $F$ to a small neighborhood of $p$ vanishes, which clearly contradicts the assumptions. The contradiction proves Proposition \ref{ppp}.
Combining Proposition \ref{ppp}, Remark \ref{4}, and Lemma \ref{impcomplex}, we obtain
\begin{Cor} \label{cc3}
Let $a>0$ at a point. Then, at every point of $T^2$ we have $a\ge 0$, $c\ge 0$.
\end{Cor}
{\bf Proof of Proposition \ref{C2}.} We consider admissible coordinates $x,y$ in a small neighborhood $U(p)$. We think that the point $p$ has the coordinates $(x(p), y(p))= (0,0)$.
In this coordinates, by Remark \ref{4}, the connected component of the set $\{q \in U(p) \mid \textrm{$B_1$ is not defined at $q$}\} $ containing $p$ is one of the following sets (for a certain $\varepsilon>0$):
$$
W_{+\varepsilon}:= \{q \in U(p) \mid 0\le x(q)\le \varepsilon \} \,, \ W_{-\varepsilon}:= \{q \in U(p) \mid 0\ge x(q)\ge -\varepsilon \} \, , \ \textrm{or} \ W_{0}:= \{q \in U(p) \mid x(q)=0 \}.
$$ If the connected component of the set $\{q \in U(p) \mid \textrm{$B_1$ is not defined at $q$}\} $ containing $p$ is $W_{+\varepsilon}$ or $W_{-\varepsilon}$, we are done by Proposition \ref{C1}.
We assume that the connected component of the set $\{q \in U(p) \mid \textrm{$B_1$ is not defined at $q$}\} $ containing $p$ is $W_0$. Our goal is to prove that $\frac{\partial L}{\partial y}=0 $ for the points of this set.
Let us first observe that $da_{|q} = 0$ for every $q\in W_0$.
Indeed, by Corollary \ref{cc3}, the function $a$ accepts an extremum (minimum or maximum) at $q$.
Then, the second equation of \eqref{sys} tells us that $\frac{\partial L}{\partial y}=0$, i.e., $L$ is constant on the set $\{q\in U(p)\mid x(q)= x(p)\}$. Proposition \ref{C2} and Lemma \ref{imp} are proved.
\begin{Rem} \label{thesame} Since there is no essential difference between $B_1$ and $B_2$,
the function $L$ is constant on every connected component of the set $\{ q\in {T}^2 \mid {B_1} \textrm{ or $B_2$ is not defined at $q$} \} $, as we claimed in the title of this section \end{Rem}
\subsection{ At a neighborhood of every point the metrics are Liouville, or one eigenvalue of $\tilde F_j^i$ is constant on the manifold } \label{liou}
Recall that integrals linear in momenta and Killing vector fields are closely related:
the function $I=\alpha(x,y) p_x + \beta(x,y) p_y$ is an integral of the geodesic flow of $g$, if and only if
the vector field $v= (\alpha, \beta) $ is a Killing vector field. Moreover, the mapping
$I=\alpha(x,y) p_x + \beta(x,y) p_y \mapsto v= (\alpha, \beta) $ is coordinate-independent.
By Lemma \ref{impcomplex}, at every point of $T^2$ we have $ac\ge 0$.
\begin{Lemma} \label{twodiff}
If there exists a point $q$ such that at this point at least one of the forms $B_1$, $B_2$ is not defined, then one of the eigenvalues of $\tilde F_j^i$ is constant on the manifold.
\end{Lemma}
\noindent{\bf Proof. } We consider two sets:
$$
W:= \{ p\in T^2 \mid \textrm{ $B_1$ and $B_2$ are defined at $p$ }\} \textrm{ \ and \ } T^2 \setminus W.
$$
Assume $T^2 \setminus W \ne \varnothing$. At every point $s \in T^2$, we denote by $E_1(s)\le E_2(s)$ the roots of the characteristic polynomial
$$
\chi(t):= \det(\tilde F_j^i - t \cdot \delta_j^i)
$$
at the point $s$ counted with multiplicities. (By Corollary \ref{jjj}, the roots of the polynomials $\chi(t)$ are real).
The functions $E_1$ and $E_2$ are at least continuous.
At the points of $T^2 \setminus W$, by Remark \ref{invar}, we have $E_1=E_2= L/2$, where $L= \mbox{\rm trace}(\tilde F_j^i)$.
Then, by Remark \ref{thesame}, both functions $E_1, E_2$ are constant on each connected component of $T^2 \setminus W$.
Since $W$ is open, and since $W\cup \left(T^2 \setminus W\right)= T^2$, in order to prove Lemma \ref{twodiff},
it is sufficient to show that at least one of the functions $E_1, E_2$ is constant on every connected connected component of $W$.
We consider a point $p$ such that at this point $ac>0$, and denote by $W_0$ the connected
component of $W$
containing $p$.
At every point $p_0$ of $W_0$, we consider the vector fields $\tfrac{\partial }{\partial x}$, $\tfrac{\partial }{\partial y}$, where $x,y$ are perfect coordinates is a neighborhood of $p_0$. Though the perfect coordinates are local coordinates, these vector fields are well defined at all points of $W_0$, see Remark \ref{pp}. Moreover, at every point $p_0$ the vectors $\tfrac{\partial }{\partial x}$, $\tfrac{\partial }{\partial y}$ form a dual basis to the basis $(B_1, B_2)$ in $T_{p_0}^*T^2$.
Let us show that the vector fields $\tfrac{\partial }{\partial x}$, $\tfrac{\partial }{\partial y}$ are complete on $W_0$. Since the basis $(B_1, B_2)$ is dual to the basis $\left(\tfrac{\partial }{\partial x}, \tfrac{\partial }{\partial y}\right)$, it is sufficient to show that
for every point $q$ of the boundary $\partial W_0 := \overline W_0 \setminus W_0$
the integral $\int_{p}^q B_1=\pm \infty$, or $\int_{p}^q B_2=\pm \infty$.
We consider admissible coordinates $\tilde x, \tilde y$ in a neighborhood of $q$.
Without loss of
generality, $\tilde x(q)= 0$ and $\tilde a(0)= 0$.
As we explained in the proof of Lemma \ref{imp}, the differential $d\tilde a_{|q}=0$ implying
$\tilde a(\tilde x)= \tilde x^2 \alpha(x)$, where
$\alpha(x)$ is a smooth function in a neigborhood of $0$. Then, $\int_{p}^q B_1 = \mbox{\rm const} + \int_{\tilde x_0}^0 \tfrac{1}{\sqrt{|\tilde a(s)|}} ds = \mbox{\rm const} \pm \int_{\tilde x_0}^0\left(\tfrac{1}{{|s| \sqrt{|\alpha(s)|}}} \right) ds = \pm \infty$.
Thus, the vector fields $\tfrac{\partial }{\partial x}$, $\tfrac{\partial }{\partial y}$ are complete on $W_0$.
We consider the local coordinates $u= \tfrac{1}{2}(x +y)$ and $v = \tfrac{1}{2}(x-y)$, and the corresponding vector fields
$\tfrac{\partial }{\partial u} = \tfrac{1}{2}\left( \tfrac{\partial }{\partial x} + \tfrac{\partial }{\partial y}\right)$ and $\tfrac{\partial }{\partial v} = \tfrac{1}{2}\left( \tfrac{\partial }{\partial x} - \tfrac{\partial }{\partial y}\right)$. Since $\tfrac{ \partial }{\partial x}$, $\tfrac{ \partial }{\partial y}$ are complete, the vector fields $\tfrac{\partial }{\partial u}$, $\tfrac{\partial }{\partial v} $
are also complete.
The coordinates $u,v$ are as in Proposition \ref{c1}. Then, by Proposition \ref{c1}, in the coordinates $(u,v)$, the metric and the integral have the form $(U(u)- V(v))(du^2 - dv^2)$ and $ \tfrac{U(u)p_v^2 - V(v)p_{u}^2}{U(u)-V(v)}$.
Since $f= U(u)-V(v)>0$, we have $U(u)>V(v)$.
Let us note that at every point of $W_0$,
the local functions $U$ and $V$ have a clear geometric sense, and, therefore, are globally given at all points of $W_0$, and can be continuously prolonged up to the boundary.
Indeed, in the coordinates $(u,v)$ the matrix of $\tilde F_j^i$ is
$$\begin{pmatrix}-V(v) & 0 \\ 0 & -U(u) \end{pmatrix}. $$
Thus, $U= -E_1$ and $V= -E_2$.
Consider the action of the group
$(\mathbb{R}^2, +)$ on $W_0$ generated by the vector fields $\tfrac{\partial }{\partial u} $ and $\tfrac{\partial }{\partial v}$. The action is well defined, since the vector fields commute and are complete. The action is transitive and locally-free. Then, $W_0$ is diffeomorphic to the torus, to the cylinder, or to $\mathbb{R}^2$.
Since $T^2\setminus W_0 \ne \varnothing$, $W_0$ can not be the torus.
Now suppose $W_0$ is a cylinder. Then, its boundary has at most two connected components.
Each integral curve of
at least one of the vector fields $\tfrac{\partial }{\partial u} $ and $\tfrac{\partial }{\partial v}$ is not closed. Without loss of generality, we
assume that for every $p\in W_0$ the integral curve of the vector field $\tfrac{\partial }{\partial v}$ is not closed (i.e., it is the generator of the cylinder, or a standard winding on the cylinder. In the case the boundary of $W_0$ has two boundary components, the integral curve of $\tfrac{\partial }{\partial v}$ attracts to one component of the boundary for $t\to +\infty$, and to another component of the boundary for $t\to -\infty$).
For every boundary component, there exists a sequence of the points of any
integral curve of $\tfrac{\partial }{\partial v}$ converging to a point of the boundary component. Indeed, the closure of $W_0$ is compact, so every sequence of points has a converging subsequence. We consider a converging subsequence of the sequence
$\phi(0,p)=p, \phi(1,p), \phi(2,p),\phi(3,p),...$ where $\phi:\mathbb{R}\times W_0\to W_0$
denotes the flow of the vector field $\tfrac{\partial }{\partial v}$.
Clearly, this sequence can not
converge to a point of $W_0$. Then, it converges to a point of a boundary component.
Since the function $E_1=-U$ is constant along the integral curve, the value of $E_1$ on the boundary coincides with the value of $E_1$ at the point $p$. Similarly,
the sequence
points $\phi(0,p)=p, \phi(-1,p), \phi(-2,p),\phi(-3,p),...$ has a subsequence converging to another component of the boundary. Then, the value of $E_1$ on both components of the boundary coincides and is equal to the value of $E_1$ at every point of $W_0$. Then, the function $E_1$ is constant on $W_0$.
Let us use the same idea to show that $W_0$ can not be diffeomorphic to $\mathbb{ R}^2$. Indeed, in this case $\partial W_0$ has one connected component, and the orbits of both vector fields $\tfrac{\partial }{\partial u}$, $\tfrac{\partial }{\partial v}$ are not closed implying $U(u)= V(v)$ at every point, which clearly contradicts the assumptions.
Finally, one of the eigenvalues of $\tilde F_j^i$ is constant on $W_0$. Lemma \ref{twodiff} is proved.
\subsection{ If one eigenvalue of $\tilde F_j^i$ is constant, then there exists an integral linear in momenta} \label{eigenvalue}
By Lemma \ref{twodiff}, we have the following two possibilities (not disjunkt):
\begin{itemize}
\item[{\bf (1) \ }] one of the eigenvalues of $\tilde F_j^i$ is constant,
\item[{\bf (2) \ }] at every point $ac>0$.
\end{itemize}
The goal of this section is to show that in the first case there exists an integral linear in momenta (at least on an appropriate double cover of the torus; later (in \S \ref{4.2})
we show that the integral exists already on the torus, see Corollary \ref{cover}).
\begin{Lemma} \label{existlinear} Let one of the eigenvalues of $\tilde F_j^i $ is constant. Then, for a certain (at most, double) cover of the torus, the lift of the integral is a linear combination of the square of an integral linear in momenta and the lift of the Hamiltonian. Moreover, there exists no point $q$ such that $F_{|T^*_qT^2}\equiv \mbox{\rm const}\cdot H_{|T^*_qT^2}$.
\end{Lemma}
{\bf Proof.} Without loss of generality we can assume that one of the eigenvalues of $\tilde F_j^i$ is identically $0$, otherwise we replace $F$ by $F- \mbox{\rm const}\cdot H$ for the appropriate $\mbox{\rm const}\in \mathbb{R}$. Then,
$
\tilde F_j^i
$ has rank at most 1.
Let $\tilde F_j^i \ne 0 $ at a point $q$.
We consider local coordinate $(u,v)$ in $U(q)$ such that $\tfrac{\partial }{\partial u} $ lies in the kernel of $\tilde F_j^i$. In this coordinates, the (symmetric) matrix of $\tilde F^{ij}$ satisfies the equation
$$
\begin{pmatrix} \tilde F^{11} & \tilde F^{12} \\ \tilde F^{21} & \tilde F^{22} \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = 0
$$
implying $\tilde F^{11} = \tilde F^{12}= \tilde F^{21} =0$.
Then, in this coordinates $F= \tilde F^{22} p_v^2 $ implying that the integral is locally the square of the function
$\sqrt{F^{22}} p_v$ , if $\tilde F^{22}>0$, or $\sqrt{-F^{22}} p_v$, if $\tilde F^{22}<0$. Then, the (linear in momenta) function $\sqrt{F^{22}} p_v$ (if $\tilde F^{22}>0$) or $\sqrt{-F^{22}} p_v$ (if $\tilde F^{22}<0$) in a local integral linear in momenta, and $\sqrt{F^{22}}\tfrac{\partial }{\partial v}$ (if $\tilde F^{22}>0$) or $\sqrt{-F^{22}}\tfrac{\partial }{\partial v}$ (if $\tilde F^{22}<0$) is a Killing vector field.
Let us show that the points such that $\tilde F_j^i=0$ are isolated. Indeed, otherwise there exist two such points, say $p_1$ and $p_2$, in a sufficiently small neigborhood $U$.
For every point $q$ of this neighborhood we consider the geodesics connecting $p_i$ with q.
For almost every $q$, the geodesics intersect transversally at the point $q$, see Figure \ref{fig3}.
\begin{center}
\begin{figure}[ht!]
{\includegraphics[width=.3\textwidth]{twopoints.eps}}
\caption{The geodesic connecting the points $p_i$ with the point $q$, and their tangent vectors at the point $q$. At almost every $q$, the tangent vectors of the geodesics at the point $q$ are linearly independent}\label{fig3}
\end{figure}
\end{center}
We denote by $\xi_1, \xi_2$ vector-momenta of these geodesics at the point $q$.
Since $F_{|T^*_{p_i}T^2}\equiv 0$, we have $F(\xi_1)=F(\xi_2)=0$ implyling
$F_{|T^*_{p_i}T^2}\equiv 0$. Since this is fulfilled for almost every point $q$ of a small neighborhood, the integral $F$ vanishes identically on two linearly independent vector-momenta, which impossible for the integral $F= \tilde F^{22} p_v^2 $ (for $\tilde F^{22}\ne 0$).
Thus, the points $q$ such that ${ F}_{|T^*qT^2 }\equiv 0$ are isolated.
Then, the set $N:= \{ q\in T^2 \mid { F}_{|T^*qT^2 }\equiv 0\}$ is discrete.
Hence, the set
$T^2\setminus N = \{ q\in T^2 \mid F_{|T^*qT^2 }\not \equiv 0\}$ is connected implying that $\tilde F^{ij}$ is nonpositive definite everywhere, or nonnegative definite everywhere. Without loss of generality we can think that $\tilde F^{ij}$ is nonnegative definite everywhere, otherwise we replace $F$ by $-F$.
Let us show that in {\it a small neighborhood $U(p)$ of every point $p$ there exists precisely two integrals linear in momenta such
that \begin{itemize} \item[(a)] they are smooth at every points $q \not\in N$, and \item[(b)] the square of each of these integrals is equal to $F$. \end{itemize}}
If $p\not\in N$, the statement is evident: in the constructed above local coordianates $u,v$ the integrals are $\pm \sqrt{F}= \pm \sqrt{ \tilde F^{22} p_v^2 }= \pm p_v \sqrt{ \tilde F^{22}} .$ Since every neighborhood has a point from $T^2 \setminus N$, in a neighborhood of every point there exist at
most two such integrals. Thus, in order to prove the statement
above we need to prove that in a neighborhood of every point from $N$ there exists at least one such integral (the second one will be minus the first).
Let $p\in N$. We take a small neighborhood $U(p)$ homeomorphic to the disk, and consider $U(p) \setminus \gamma$, where $\gamma$ is a geodesics starting at the point $p$, see Figure \ref{fig4}. Since $U(p)\setminus \gamma$ is simply-connected and contains no point from $N$, on $U(p)\setminus \gamma$ there exists an integral $I= \alpha(x,y) p_x + \beta(x,y) p_y$ linear in momenta such that $I^2=F$.
We consider the Killing vector field $v:= (\alpha, \beta)$ corresponding to this integral. Since the value of this integral on each geodesic passing through $p$ is zero, the Killing vector field $(\alpha, \beta)$ is orthogonal to geodesics containing $p$. Then, the qualitative behaviour of the vector field at the points of a small circle around $p$ is as on Figure \ref{fig4}.
Indeed, they are tangent to the level curves of the geodesic distance function to the point $p$, which are hyperbolas (one of them is on Figure \ref{fig4}) and light-line geodesics though $p$.
\begin{center}
\begin{figure}[ht!]
{\includegraphics[width=.7\textwidth]{killing.eps}}
\caption{Qualitative behaviour of the vector field $v$ at the points of a small circle around $p$}\label{fig4}
\end{figure}
\end{center}
We see that the vector field $v$ is oriented in the same direction on the different sides of $\gamma$ , implying that one can prolong the vector field to $U(p)\setminus \{p\}$. Then, there exists the integral $I$ linear in momenta such that $I^2= F$ in $U(p)\setminus p$ as we claimed.
Since in
a small neighborhood $U(p)$ of every point $p$ there exists precisely two integrals linear in momenta
satisfying the conditions (a), (b) above, an integral linear in momenta
satisfying the conditions (a), (b) above exists on $T^2$, or on the double cover of $T^2$. The first statement of Lemma \ref{existlinear} is proved.
Let us prove the second statement of Lemma \ref{existlinear}: let us show that the set $N$ is actually empty. Indeed, the index of the vector field
$v$ is negative at the points of $N$, see Figure \ref{fig4}, and is zero at all other points. But the sum of the indexes of any vector field on the torus must be zero.
Thus, there exists an integral linear in momenta
satisfying the condition (b) above on the torus, or on the double cover of the torus.
Lemma \ref{existlinear} is proved.
\begin{Cor} \label{zeros}
Let $v$ be a nontrivial Killing vector field of a pseudo-Riemannian metric $g$ on the torus $T^2$. Then, there is no point $p\in T^2$ such that $v=0$ at $p$.
\end{Cor}
{\bf Proof.} In the Riemannian case (and, therefore, if $g$ has signature (--,--)), Corollary \ref{zeros} is evident. Indeed, the Killing vector field preserves the complex structure corresponding to the metric, and is therefore holomorphic (with respect to the complex structure). By the Abel Lemma, it has no zeros.
Let now the signature of the metric be (+,--). We consider the integral linear in momenta corresponding to the Killing vector field. It vanishes at the points where the Killing vector field vanishes.
The square of this integral is an integral quadratic in momenta. If the linear integral is
$ \alpha(x,y) p_x + \beta(x,y) p_y$, its square is $F= \alpha^2 p_x^2 + 2 \alpha \beta p_xp_y + \beta^2 p_y^2$, and the matrix $\tilde F^{ij}$ (such that $F= \sum_{i,j}\tilde F^{ij} p_ip_j$) is
$$
\begin{pmatrix}
\alpha^2 & \alpha\beta \\ \alpha \beta & \beta^2
\end{pmatrix}.
$$
We see that its rang is $\le 1$ implying that $0$ is a (constant) eigenvalue of $\tilde F_j^i$.
Then, by Lemma \ref{existlinear}, there exists no point such that $\alpha= \beta=0$ implying there exists no point such that $v= 0$. Corollary \ref{zeros} is proved.
\begin{Rem} Actually, our final goal is to prove that a nontrivial integral linear in momenta exists already on the torus (and not on the double cover of the torus). We will do it later, in Section \ref{proof4}. By Corollary \ref{cover} (whose proof does not use Theorem \ref{main1}, so no logical loop appears), the integral linear in momenta
satisfying the condition (b) above exists already on the torus. \end{Rem}
\subsection{Proof of Theorem 2 under the assumption that the vector fields $V_1$, $V_2$ exist on the whole torus} \label{endof}
Let the geodesic flow of $g$ of signature (+,--) on the torus admits an integral quadratic in momenta; assume the integral is not a linear combination of the square of an integral linear in momenta and the Hamiltonian.
As everywhere in Section \ref{4}, we assume that the vector fields $V_1$, $V_2$ satisfying conditions (A,B,C) from \S \ref{admissible} exist on the whole torus. By Lemmas \ref{twodiff}, \ref{existlinear}, at every point of the manifold $ac >0$.
We consider the vector fields $\tfrac{\partial }{\partial u}$, $\tfrac{\partial }{\partial v}$ from the proof of Lemma \ref{twodiff}. These vector fields commute and never vanish. Then, they generate a locally free action of $(\mathbb{R}^2, +)$ on $T^2$. The stabilizer $G$ of this actions is a subgroup of $(\mathbb{R}^2, +)$ with the following properties: it is
\begin{itemize}
\item discrete, and
\item the quotient space is compact.
\end{itemize}
Then, it is a lattice, i.e., $G= \{ k\cdot \xi + m\cdot \eta \mid (k,m)\in \mathbb{R}\}$ for certain linearly independent vectors $\xi, \eta$. Then, there exists a natural diffeomorphism $\phi: \mathbb{R}^2/G \to T^2$. We identify $\mathbb{R}^2/G$ and
$ T^2$ by this diffeomorphism and consider the lift of the metric and
the integral to $\mathbb{R}^2$. By Proposition \ref{c1}, in the coordinate system $(u,v)$
on $\mathbb{R}^2$, the metric and the integral are $(U(u)- V(v))(du^2 - dv^2)$ and $\pm \tfrac{U(u)p_v^2 - V(v)p_{u}^2}{U(u)-V(v)}$, i.e., are as in Model Example 1.
Since the metric and the integral are preserved by the lattice, the functions $U$ and $V$ are preserved by the lattice as well. Thus, the metric on $\mathbb{R}^2/G $ are as in Model Example 1. Theorem \ref{main1} is proved (under the additional assumption that the vector fields $V_1$, $V_2$ exist on the whole torus).
\section{ Proof of Theorem 4, final step of the proof of Theorem 2,
and proof of Theorem 3} \label{proof4}
\subsection{ Flat metrics of signature (+,--) on $T^2$, and their Killing vector fields} \label{flat}
By the Gauss-Bonnet Theorem, a metric of constant curvature on the torus is flat (= has zero curvature).
Recall that by the { \it standard flat torus } we consider $(\mathbb{R}^2/G, dxdy)$, where $(x,y)$ are the standard coordinates on $\mathbb{R}^2$, and $G$ is a lattice generated by two linearly independent vectors.
It is well-known that every torus $(T^2, g)$ such that the metric $g$ is flat and has signature (+,--) is isometric to a standard one. Indeed, by \cite{carriere}, the flat torus is geodesically complete implying its universal cover is
isometric to $(\mathbb{R}^2, dxdy)$. The fundamental group of the torus,
$(\mathbb{Z}^2,+)$, acts on $(\mathbb{R}^2, dxdy)$. The action is isometric, free, and discrete. It is easy to
see that every orientation-preserving isometry of $(\mathbb{R}^2, dxdy) $ without fixed points is a translation. Then, $\mathbb{Z}^2$ acts as a lattice generated by two linearly independent vectors, and $(T^2, g)$ is isometric to a certain $(\mathbb{R}^2/G, dxdy)$.
The space of Killing vector fields of $(\mathbb{R}^2, dxdy)$ is a 3-dimensional linear vector space generated by two translations $(1,0) = \tfrac{\partial }{\partial x} $ and $(0,1) = \tfrac{\partial }{\partial y} $, and the pseudo-rotation $(y,x) =y\tfrac{\partial }{\partial x}+y\tfrac{\partial }{\partial y}$. Then, the space of Killing vector fields on the flat torus $(\mathbb{R}^2/G, dxdy)$ is two-dimensional and is generated by the Killing vector fields $(1,0) = \tfrac{\partial }{\partial x} $ and $(0,1) = \tfrac{\partial }{\partial y} $. Note that, depending on the values of the constants $(\mbox{\rm const}_1, \mbox{\rm const}_2)\ne (0,0)$, every integral curve of the Killing vector field $ \mbox{\rm const}_1\cdot \tfrac{\partial }{\partial x} + \mbox{\rm const}_2\cdot \tfrac{\partial }{\partial y} $ is either a closed curve, or
an everywhere dense winding on the torus.
\subsection{ Killing vector fields on the torus of nonconstant curvature } \label{4.2}
\begin{Prop} \label{kil}
Let the metric $g$ of nonconstant curvature on the torus $T^2$ admits a nonzero Killing vector field $v$.
Then,
there exists a free action of the group $(\mathbb{R}/\mathbb{Z}, +)$ on the torus such that the infinitesimal generator of this action is proportional to the Killing vector $v$ with a constant coefficient of proportionality.
\end{Prop}
{\bf Proof.} We denote by $R$ the scalar curvature of $g$.
By Corollary \ref{zeros}, the vector field $v$ has no zeros on $T^2$. Then, the Killing vector field generates a locally-free action of the group $(\mathbb{R},+)$.
Let us prove that the Killing vector field (after an appropriate scaling) actually generates the action of the group
$SO_1= \mathbb{R}/\mathbb{Z}$ without fixed points.
Indeed, take a point $p$ such that $dR\ne 0$, and consider the orbit of the Killing vector field containing the point. Since the flow of a Killing vector field preserves the curvature, at every point $q$ of the orbit we have $R(q)= R(p)$ and $dR\ne 0$. Then, the orbit coincides with the connected component of the set $\{q \in T^2 \mid R(q)=R(p)\}$ containing the point $p$ implying it is a circle.
We consider the action $\rho: \mathbb{R}\times T^2\to T^2$ of the group $(\mathbb{R},+)$ generated by the flow of the vector field.
Since the orbit through $p$ is a circle, for certain $t_0>0$ we have $\rho(t_0, p)= p$ and
for no $t\in (0, t_0)$ $\rho(t, p)= p$. Without loss of generality we can think that $t_0=1$, otherwise we replace $v$ by $t_0 \cdot v$.
Since the action $\rho$ is isometric and orientation-preserving, it
commutes with the exponential mapping $\exp:TT^2\to T^2$. Then, for every point $q\in T^2$ we have $\rho(1,q)= q$ and $\rho(t,q)\ne q$ for $t\in (0, 1)$. Thus, the action of the group $(\mathbb{R}/\mathbb{Z}, +)$ is well-defined, and has no fixed points. Proposition \ref{kil} is proved.
\begin{Cor} \label{cover1} Let $v$ be a nonzero Killing vector field on the torus $(T^2,g)$, where $g$ has signature $(+,-)$. Then, there exists
no involution $\sigma:T^2\to T^2$ without fixed point that preserves the orientation and the metric, and sends the vector field $v$ to $-v$.
\end{Cor}
{\bf Proof.} If the metric $g$ has constant curvature, as we have recalled in \S \ref{flat}, the torus is isometric to $(\mathbb{R}^2/G, dxdy)$ for a lattice $G$ generated by two linearly independent vectors $\xi$ and $\eta$, and the Killing vector field is
$\mbox{\rm const}_1 \cdot \xi + \mbox{\rm const}_2 \cdot \eta$ for $(\mbox{\rm const}_1, \mbox{\rm const}_2)\ne (0,0)$. The involution $\sigma$ without
fixed points that preserves the orientation and the metric induces an isometry of $(\mathbb{R}^2, dxdy)$ without fixed points that preserves the orientation and the metric. Such isometry is a translation and can not send the Killing vector field $\mbox{\rm const}_1 \cdot \xi + \mbox{\rm const}_2 \cdot \eta$ to $-\left(\mbox{\rm const}_1 \cdot \xi + \mbox{\rm const}_2 \cdot \eta\right)$. Corollary \ref{cover1} is proved under the
assumption that $g$ has constant curvature.
Assume now that the curvature of $g$ is not constant. Then, by Proposition \ref{kil}, the Killing vector field (after the appropriate scaling) generates a free action of $(\mathbb{R}/\mathbb{Z}, +)$ on $T^2$.
We consider the quotient space $T^2/_{(\mathbb{R}/\mathbb{Z})}.$ Since the action of $\mathbb{R}/\mathbb{Z}$ on $T^2$ is free,
the quotient space is a 1-dimensional closed manifold, i.e., is diffeomorphic to $S^1$. The orientation of the torus induces the orientation on $S^1$.
The involution $\sigma$ of the torus preserves the action, the orientation, and sends $v$ to $-v$.
Then, it inverses the orientation of $S^1 = T^2/_{(\mathbb{R}/\mathbb{Z})}.$ Then, it has a fix point. We consider the orbit of $\mathbb{R}/\mathbb{Z}$ corresponding to this point. The involution $\sigma$ preserves this orbits and
changes the direction of the vector field $v$ on this orbit. Then, it has a fixed point which contradicts the assumptions. The contradiction proves Corollary \ref{cover1}.
\begin{Cor} \label{cover} Let $F$ be a nontrivial integral quadratic in momenta for the geodesic flow of the metric $g$ on the torus $T^2$ and $\pi:\widetilde T^2 \to T^2 $ be a double cover of $T^2$.
Assume the lift of the integral to $\widetilde T^2$ is a linear combination of the square of a function linear in momenta and the lift of the Hamiltonian. Then, the integral $F$
is a linear combination of the square of an integral linear in momenta and the Hamiltonian
\end{Cor}
{\bf Proof.} We consider the involution $\sigma:\widetilde T^2 \to \widetilde T^2$ corresponding to the cover:
$\sigma(\tilde p)= \tilde q$ if $\pi(\tilde p)=\pi(\tilde q)$ and $\tilde p\ne \tilde q$. The involution preserves the lift of the Hamiltonian and of the integral.
We consider the function $I:T^*\widetilde T^2\to \mathbb{R}$ linear in momenta such that
$ F= \mbox{\rm const}_1\cdot H + \mbox{\rm const}_2\cdot I^2$, where $H$ and $F$ denote the lift of the Hamiltonian and the integral. Since the integral $F$ is nontrivial, $\mbox{\rm const}_2\ne 0$ implying $I$ is a nontrivial integral (linear in momenta). We consider the Killing vector field $v $ corresponding to the integral. Since the involution $ \sigma$ preserves $H$ and $F$, it preserves $I^2= \tfrac{1}{\mbox{\rm const}_2}(F- \mbox{\rm const}_1\cdot H)$. Since by Proposition \ref{kil} the vector field $v$ vanishes at no point,
either $d\sigma(v)= v $ for all points, or $d\sigma(v)= -v$ for all points. The second possibility is forbidden by Corollary \ref{cover1}. Then, $d\sigma(v)= v $ implying the integral $I$ on $\widetilde T^2$ induces an integral $I$ (linear in momenta) on $T^2= \widetilde T^2/\sigma$ such that, on $T^2$, $ F= \mbox{\rm const}_1\cdot H + \mbox{\rm const}_2\cdot I^2$. Corollary \ref{cover} is proved.
\subsection{ Proof of Theorem \ref{main4} } Let $F$ be an integral linear in momenta of the geodesic flow of a metric $g$ on the torus $T^2$. We denote by $v$ the corresponding Killing vector field.
We consider the action $\rho$ of
$(\mathbb{R}/\mathbb{Z}, +)$ on $T^2$ from Proposition \ref{kil}, the quotient space $T^2/_{(\mathbb{R}/\mathbb{Z})}$ diffeomorphic to the circle, and the tautological projection $\pi:T^2 \to T^2/_{(\mathbb{R}/\mathbb{Z})}=S^1 $.
Let us construct a coordinate system $(x \in \mathbb{R } \textrm{\ mod} \ 1, \ y \in \mathbb{R } \textrm{\ mod } \ 1 )$ on $T^2$. We parametrize $S^1$ by $(Y\in \mathbb{R} \ \textrm{ \ mod \ } \ 1 )$, and
put $y(q):= Y(\pi(q))\in \mathbb{R}/\mathbb{Z})$. In order to construct the coordinate $x$, we consider a smooth section $c: S^1 \to T^2$ of the bundle. By definition of the section, for every $q\in T^2$ there exists a unique $t\in (\mathbb{R } \textrm{\ mod\ } \ 1)$ such that $\rho(t,q) \in \textrm{image}(c)$. We put $x(q)= -t$.
By construction, in this coordinates, the vector field $v$ is $\tfrac{\partial }{\partial x}$, and the corresponding integral linear in momenta is $p_x$. Let in this coordiantes the metric $g$ is given by
$g= K(x,y) dx^2+ 2 L(x,y) dxdy + M(x,y) dy^2$. Since the metric has signature (+,--), we have $KM-L^2 = \det\begin{pmatrix} K &L\\ L & M\end{pmatrix} <0$. Thus, in order to prove Theorem \ref{main4}, it is sufficient to show that the functions $K, L, M$ are functions of the variable $y$ only, i.e., $ \tfrac{\partial K}{\partial x} = \tfrac{\partial L}{\partial x}= \tfrac{\partial M}{\partial x}=0.$
We denote by $k(x,y), l(x,y), m(x,y) $ the components of the inverse matrix to $g$:
$$
\begin{pmatrix} k &l\\ l &m\end{pmatrix} = \begin{pmatrix} K &L\\ L & M\end{pmatrix} ^{-1}.
$$
Evidently, $2H= k(x,y) p_x^2 + 2 l(x,y) p_xp_y + m(x,y) p_y^2$, and
the condition $\{F,2H \}=0$ reads \\
\begin{eqnarray*}
0&=& \left\{ p_x, k(x,y) p_x^2 + 2 l(x,y) p_xp_y + m(x,y) p_y^2\right\} \\
&=& \tfrac{\partial k}{\partial x} p_x^2 + 2 \tfrac{\partial l}{\partial x} p_xp_y + \tfrac{\partial m}{\partial x} p_y^2 \, ,
\end{eqnarray*}
i.e., is equivalent to the condition $ \tfrac{\partial k}{\partial x} = \tfrac{\partial l}{\partial x}= \tfrac{\partial m}{\partial x}=0.$ Then, the coefficients $k,l,m$ depend on the variable $y$ only, implying that the coefficients $K,L, M$ also depend on the variable $y$ only. Theorem \ref{main4} is proved.
\subsection{ Proof of Theorem \ref{main1} under the assumption that the vector fields $V_1, V_2$ do not exist on the torus } \label{laststep}
We assume that the geodesic flow of the metric $g$ on $T^2$ admits a nontrivial integral $F$ quadratic in momenta that is not a linear combination of the Hamiltonian and an integral linear in momenta. Assume the vector fields $V_1,V_2$ satisfying assumptions (A,B,C) from \S \ref{admissible} do not exist. We consider the double cover $\pi: \widetilde T^2 \to T^2$ such that $V_1,V_2$ satisfying (A,B,C) exist on $\widetilde T^2$. Then, by the proved
part of Theorem \ref{main1}, the lift of the metric to $\widetilde T^2$ is as in Model Example 1
(we idientify
$\widetilde T^2$ with $\mathbb{R}^2/G$ and the lift $\tilde g$ of the
metric with the metric from Model Example 1). On the torus $\widetilde T^2$, the only possibility for the vector fields $V_1,V_2$ are (we consider the standard orientation on $\mathbb{R}^2$):
$$ V_2 = \lambda \left(\tfrac{\partial }{\partial x} + \tfrac{\partial }{\partial y}\right), \ V_1 = \mu\left(\tfrac{\partial }{\partial x} - \tfrac{\partial }{\partial y}\right), $$
where $\lambda$ and $\mu$ are smooth functions on $\widetilde T^2$ such that for every $\tilde p\in \widetilde T^2$ we have $\lambda(\tilde p) \mu(\tilde p)>0$, and $x,y$ are the standard coordinates on $\mathbb{R}^2$.
We consider the involution $\sigma$
corresponding to the cover $\pi$, that it $\sigma(\tilde p)= \tilde q$ if and only if $\pi(\tilde p)= \pi(\tilde q)$ and $\tilde p\ne \tilde q$. Since by assumptions the vector fields $V_1, V_2$
do not exist on $T^2$, and the involution preserves the orientation, the metric $\tilde g$, and the lift of the integral, we have
$$
d\sigma\left(\tfrac{\partial }{\partial x} + \tfrac{\partial }{\partial y}\right) = -\left(\tfrac{\partial }{\partial x} + \tfrac{\partial }{\partial y}\right) \ \ \textrm{and} \ \ d\sigma\left(\tfrac{\partial }{\partial x} - \tfrac{\partial }{\partial y}\right) = -\left(\tfrac{\partial }{\partial x} - \tfrac{\partial }{\partial y}\right) $$
implying \begin{equation}\label{inversion}
d\sigma\left(\tfrac{\partial }{\partial x} \right) = -\tfrac{\partial }{\partial x} \ \ \textrm{and} \ \ d\sigma\left( \tfrac{\partial }{\partial y}\right) = -\tfrac{\partial }{\partial y}. \end{equation}
But on the torus $\mathbb{R}^2/G$ there is no involution with no fixed point with the property \eqref{inversion}. The contradiction shows that the situation assumed in this section, namely that the vector fields $V_1, V_2$ do not exist on $T^2$, is impossible. Theorem \ref{main1}
is proved.
\subsection{ Proof of Theorem \ref{main3}}
We assume that $g$ is a metric of signature (+,--) on the Klein bottle $K^2$ whose geodesic flow admits an integral quadratic in momenta. We also assume that the lift of the integral
to the oriented cover is not a linear combination of the lift of the Hamiltonian and the square of a function linear in momenta. Our goal is to prove that $(K^2, g)$ is as in Model Example 2.
We consider the oriented cover $\pi:T^2 \to K^2$, and the lift of the metric and the integral to $T^2$. They satisfy the assumptions in Theorem \ref{main1}. Hence we can think that $T^2$, the lift of the metric, and the lift of the integral are as Model Example 1:
$$T^2= \mathbb{R}^2/G\, , \ g=(X(x)- Y(y))(dx^2 - dy^2) \, , \ \textrm{and} \
F= \tfrac{X(x)p_y^2 - Y(y)p_{x}^2}{X(x)-Y(y)}, $$
where $G= \{k\cdot \xi + m \cdot \eta\mid k,m\in \mathbb{Z} \}$.
Next, consider the universal cover $\tilde \pi := \pi\circ P: \mathbb{R}^2 \to K^2$, where $P$ is the canonical projection from $\mathbb{R}^2$ to $\mathbb{R}^2/G$.
We conisder the action of the fundamental group of the Klein bottle on $\mathbb{R}^2$ corresponding to $\tilde \pi$. Recall that the fundamental group of $K^2$ is generated by two elements, say $A$ and $B$, satisfying the relation $ABA^{-1}B={\bf 1}:$
\begin{equation} \label{pi}
\pi_1(K^2)= \<A,B | ABA^{-1}B={\bf 1}\> .
\end{equation}
This action has the following properties:
\begin{itemize}
\item[(a)] It preserves the metric and the integral,
\item[(b)] It is free and discrete.
\end{itemize}
Let us show that the condition (a) implies the condition
\begin{itemize}
\item[($\textrm{a}'$)] For every element $\alpha \in \pi_1(K^2)$ we have \begin{equation} d\alpha(\tfrac{\partial}{\partial y}) = \pm \tfrac{\partial}{\partial y} \, , \ \ d\alpha (\tfrac{\partial}{\partial x}) = \tfrac{\partial}{\partial x}. \label{nn} \end{equation}
\end{itemize}
Indeed,
since at every point $(x,y)\in \mathbb{R}^2$ the factor $X(x)-Y(y)\ne 0$, and since every nonempty level $\{X= \mbox{\rm const}_1\}$ intersects with every nonempty level $\{Y= \mbox{\rm const}_2\}$, without loss of generality we can think that $X(x)>Y(y)$ for all $(x,y) \in \mathbb{R}^2$.
Now, in the coordinates $x,y$, the matrix of $F^i_j$ is $\begin{pmatrix} - Y(y) & \\ & - X(x) \end{pmatrix}, $ so $\tilde F_{j}^i$ has eigenvalues
$-X(x)$, $-Y(y)$.
Since the action preserves the metric and the integral, it preserves the eigenvalues $X,Y$ and the eigenspaces $\textrm{span}\left(\tfrac{\partial}{\partial y}\right)$ and $\textrm{span}\left(\tfrac{\partial}{\partial x}\right)$ of this eigenspaces.
Since $g\left(\tfrac{\partial}{\partial x}, \tfrac{\partial}{\partial x}\right) = X(x)-Y(y), $ and $\alpha$ preserves $X$ and $Y$, we have that
$g\left(d\alpha\left(\tfrac{\partial}{\partial x}\right), d\alpha\left(\tfrac{\partial}{\partial x}\right)\right) = X(x)-Y(y)$ implying $ d\alpha\left(\tfrac{\partial}{\partial x}\right)= \pm \tfrac{\partial}{\partial x}$. The proof that $ d\alpha\left(\tfrac{\partial}{\partial y}\right)= \pm \tfrac{\partial}{\partial y}$ is similar.
Thus, the action preserves the standard flat metric $dx^2 + dy^2$ on $\mathbb{R}^2$. Then, the fundamental group of $K^2$ as a crystallographic group. From the classification of crystallographic groups \cite[\S1.7]{berger}, it follows that every action of the group \eqref{pi} on $\mathbb{R}^2$ satisfying ($\textrm{a}'$, b) is
generated by $A$, $A(x,y)= (x+ c, -y)$ and $B$, $B(x,y)=(x, y + d)$ for certain $c\ne 0 \ne d$, i.e., is an the Model Example 2. Theorem \ref{main3} is proved.
\weg{
\section{Conclusion}
We gave a complete description of pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit integrals quadratic in momenta, and gave first applications in differential geometry (completed the solution of the Beltrami problem) and mathematical physics (proved that on closed manifolds, every quadratically-superintegrable metric has constant curvature). Both applications were obtained by simple combining known results with the results of our paper.
The goal of this section is to suggest two possible applications of our paper. These possible applications
were one of our motivations to study this problem. They are
more involving than simple combining the results of the present paper with the known results and are extremely interesting.
As an interesting (and potentially solvable problem) in differential geometry we suggest to prove the two-dimensional version of
{\bf Projective Obata Conjecture.} { \it Let a connected Lie group $G$ act on a closed
connected manifold $(M^n, g)$ of dimension
$n\ge 2$ by projective
transformations. Then, it acts by isometries, or for some $c\in \mathbb{R}\setminus \{0\}$ the metric $c\cdot g$
is the Riemannian metric of constant positive sectional curvature $+1$.}
Recall that a \emph{projective transformation} of a Riemannian or pseudo-Riemannian manifold is a diffeomorphism of the manifold that takes unparameterized geodesics to geodesics.
\begin{Rem} The attribution of conjecture to Obata is in folklore (in the sense we did not find a paper of Obata where he states this conjecture). Certain papers, for example \cite{hasegawa,nagano,Yamauchi1}, refer to this statement as to a classical conjecture. If we replace ``closedness" by
``completeness", the obtained conjecture is attributed in folklore to Lichnerowicz, see also the discussion in \cite{diffgeo}. \end{Rem}
Note that in the theory of geodesically equivalent metrics and projective transformations, dimension 2 is a special dimension: many methods that work in bigger dimensions do not work in dimension 2. In particular, the proof of the projective Obata Conjecture in the Riemannian case was separately done for dimension 2 in \cite{obata, CMH} and for dimensions greater than 2 in \cite{diffgeo}.
Moreover, recently an essential progress was done in the proof of the projective Obata Conjecture in the pseudo-Riemannian case in dimension $n\ge 3$, see \cite{kiosak2,mounoud}.
We expect that it is possible to combine the results of
\cite{bryant,alone} (where a local description of 2-dimensional metrics admitting projective vector fields, i.e., infinitesumal generators of projective transformations were constructed) and the results of our paper, and prove the
projective Obata Conjecture, though it will require a lot of work.
An interesting possible application in mathematical physics is related to the Schr\"odingen equations on closed (pseudo-Riemannian) surfaces: as it was proved in \cite[Theorem 5]{pucacco}, the existence of an integral
quadratic in momenta implies the existence of a differential operator of the second order that commute with the natural Schr\"odiger operator (i.e., essentially with the Beltrami-Laplace operator).
This observation was applied with success in the Riemannian case (see, for example \cite{sinai,dobrokhotov}), and brought deep insight in the behavior of the
quantum states of 2-dimensional Riemannian metrics. Global description of Riemannian metrics whose geodesic flows
admit integrals quadratic in momenta played an important role in this result. In view of our results, one can try now to do the same in the signature (+,--). }
{\bf Acknowledgement.} The author thanks
Deutsche Forschungsgemeinschaft
(Priority Program 1154 --- Global Differential Geometry and Research Training Group 1523 --- Quantum and Gravitational Fields) and FSU Jena for partial financial support, and D. Alekseevsky, O. Bauer, A. Bolsinov, G. Manno, P. Mounoud, G. Pucacco, and A. Zeghib for useful discussions.
| {
"timestamp": "2010-02-20T20:34:24",
"yymm": "1002",
"arxiv_id": "1002.3934",
"language": "en",
"url": "https://arxiv.org/abs/1002.3934",
"abstract": "We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.",
"subjects": "Differential Geometry (math.DG); Mathematical Physics (math-ph)",
"title": "Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846716491914,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.709201978775731
} |
https://arxiv.org/abs/1805.08756 | Analysis of Sequential Quadratic Programming through the Lens of Riemannian Optimization | We prove that a "first-order" Sequential Quadratic Programming (SQP) algorithm for equality constrained optimization has local linear convergence with rate $(1-1/\kappa_R)^k$, where $\kappa_R$ is the condition number of the Riemannian Hessian, and global convergence with rate $k^{-1/4}$. Our analysis builds on insights from Riemannian optimization -- we show that the SQP and Riemannian gradient methods have nearly identical behavior near the constraint manifold, which could be of broader interest for understanding constrained optimization. | \section{Introduction}
In this paper, we consider the equality-constrained optimization problem
\begin{equation}
\label{problem:manopt}
\begin{aligned}
\minimize_{x \in \R^n} & ~~ f(x), \\
\subjectto & ~~ x \in {\mathcal M} = \{x : F(x)=0 \},
\end{aligned}
\end{equation}
where we assume $f : \R^n \rightarrow \R$ and $F: \R^n \rightarrow
\R^m$ are $C^2$ smooth functions with $m\le n$.
The focus of this paper is on the local and global convergence rate of
``first-order'' methods for~\eqref{problem:manopt}: methods that only
query $\grad f(x)$ at each iteration (but can do whatever they want
with the constraint), e.g. projected gradient descent.
The iteration complexity of first-order unconstrained
optimization has been a foundational result in theoretical machine
learning~\cite{NemirovskiYu83,Bubeck15}, and it would be of interest
to improve our understanding in the constrained case too.
While numerous ``first-order'' methods can solve
problem~\eqref{problem:manopt}~\cite{Bertsekas99,nocedal2006sequential},
we will restrict attention to two types of methods: Riemannian
first-order methods and Sequential Quadratic Programming, which we now
briefly review. When $\mc{M}$ has a manifold structure near $x_\star$,
one could use Riemannian optimization algorithms~\cite{AbsilMaSe09},
whose iterates are maintained on the constraint set $\mc{M}$.
Classical Riemannian algorithms proceed by computing the
Riemannian gradient and then taking a descent step along the geodesics
based on this gradient~\cite{luenberger1972gradient,gabay1982minimizing}.
Later, Riemannian algorithms are simplified
by making use of \textit{retraction}, a mapping from the tangent space
to the manifold that can replace the necessity of computing the exact
geodesics while still maintaining the same convergence
rate. Intuitively, first-order Riemannian methods can be viewed as
variants of projected gradient descent that utilize the manifold
structure more carefully. Analyses of many such Riemannian algorithms
are given in \citep[Section 4]{AbsilMaSe09}.
An alternative approach for solving problem~\eqref{problem:manopt} is
Sequential Quadratic Programming (SQP)~\citep[Section
18]{nocedal2006sequential}. Each iteration of SQP
solves a quadratic programming problem which minimizes a quadratic
approximation of $f$ on the linearized constraint set $\set{x:F(x_k) +
\grad F(x_k)(x-x_k)=0}$.
When the quadratic approximation uses the Hessian of the objective
function, the SQP is equivalent to Newton method solving nonlinear
equations. When the full Hessian is intractable, one can either
approximate the Hessian with BFGS-type updates, or just use some raw
estimate such as a big PSD matrix~\cite{boggs1995sequential}. The
iterates need not be (and are often not) feasible, which makes SQP
particularly appealing when it is intractable to obtain a feasible
start or projection onto the constraint set.
\subsection{Contribution and related work}
In this paper, we consider the following ``first-order'' SQP
algorithm,
which sequentially solves the quadratic program
\begin{equation}
\label{algorithm:constrained}
\begin{aligned}
x_{k+1} =
\argmin & ~~ f(x_k) +\<\nabla f(x_k), x-x_k\> +
\frac{1}{2\eta}\dl x-x_k \dl_2^2 \\
\subjectto & ~~ F(x_k) + \nabla F(x_k)(x-x_k) = 0,
\end{aligned}
\end{equation}
where $\eta>0$ is the stepsize. Each iterate only requires
$\set{\grad f, \grad F}$ (hence first-order). This algorithm can be
seen as a cheap prototype SQP (compared with BFGS-type) and is more
suitable than Riemannian methods when the retraction onto the
constrain set is intractable.
We prove that the SQP~\eqref{algorithm:constrained} has local linear
convergence with rate $(1-1/\kappa_R)^k$ where $\kappa_R$ is the condition
number of the Riemannian Hessian at $x_\star$
(Theorem~\ref{theorem:convergence}), and global convergence with rate
$k^{-1/4}$ (Theorem~\ref{thm:global_closeness_and_convergence}).
Our work differs from the existing literature in the following ways.
\begin{enumerate}
\item We provide explicit convergence rates which is lacking in prior
work on SQP. Existing local convergence analysis has focused more on
the local quadratic convergence of more expensive BFGS-type
SQPs~\cite{boggs1995sequential,nocedal2006sequential}, whereas
global convergence results are mostly
asymptotic~\cite{boggs1995sequential,solodov2009global}.
\item We observe and make explicit the fact that the SQP iterates stay
``quadratically close'' to the manifold when initialized near it
(though potentially far from $x_\star$) -- see
Figure~\ref{fig:quadratic} for an illustration.
Such an observation
connects first-order SQP to Riemannian gradient methods and allows
us to borrow insights from Riemannian optimization to analyze the SQP.
\item We provide new analysis plans for SQP, based on the fact that
SQP iterates quickly becomes nearly identical to Riemannian gradient
steps once it gets near the constraint set.
Our local analysis builds on a new potential function
\begin{equation*}
\| {\mathsf P}_{x_\star}(x_k - x_\star) \|_2^2 + \sigma \|
{\mathsf P}_{x_\star}^\perp (x_k - x_\star) \|_2
\end{equation*}
for some $\sigma > 0$ (see Section~\ref{sec:geometry} for definition
of the projections), as opposed to the traditionally used exact
penalty functions. Our global analysis
constructs descent lemmas similar to those in Riemannian gradient
methods with additional second-order error terms. These results can
be of broader interest for understanding constrained optimization.
\end{enumerate}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.4\textwidth]{quadratic.pdf}
\end{center}
\caption{Illustration for SQP iterates}\label{fig:quadratic}
\end{figure}
\paragraph{Related work}
The convergence rate of many first-order algorithms on
problem~\eqref{problem:manopt}
are shown to achieve
local linear convergence with rate $(1 - 1/\kappa_R)$, which is termed as
the \textit{canonical rate} in~\cite{luenberger2008linear}.
Riemannian algorithms that achieve the canonical rate include geodesic
gradient projection~\citep[Theorem 2]{luenberger1972gradient},
geodesic steepest descent~\citep[Theorem 4.4]{gabay1982minimizing},
Riemannian gradient descent~\citep[Theorem 4.5.6]{AbsilMaSe09}. The
canonical rate is also achieved by the modified Newton method on the
quadratic penalty function~\citep[Section 15.7]{luenberger2008linear},
which resembles an SQP method in spirit.
Though analyses are well-established for both the Riemannian and the
SQP approach (even by the same author in~\cite{gabay1982minimizing}
for the Riemannian approach and~\cite{gabay1982reduced} for the SQP
approach), the connection between them did not receive much
attention until the past 10 years. Such connection
is re-emphasized in~\cite{absil2009all}: the authors
pointed out that the feasibly-projected sequential quadratic
programming (FP-SQP) method in \cite{wright2004feasible} gives the
same update as the Riemannian Newton update. \citet{MishraSe16} used
this connection to provide a framework for
selecting a preconditioning metric for Riemannian optimization, in particular
when the Riemannian structure is sought on a quotient manifold.
However, these connections
are established for second-order methods between the Riemannian and the
SQP approaches, and such connection for first-order methods are
not---we believe---explicitly pointed out yet.
\paragraph{Paper organization}
The rest of this paper is organized as the following. In
Section~\ref{sec:preliminaries}, we state our assumptions and give
preliminaries on the Riemannian geometry on and off the manifold
$\mc{M}$. We present our local and global convergence result in
Section~\ref{sec:main} and~\ref{sec:global}, and prove them in
Section~\ref{sec:proof} and~\ref{sec:proof-global}.
We give an example in Section~\ref{section:example} and perform
numerical experiments in Section~\ref{sec:exp}.
\subsection{Notation}
For a matrix $A \in \R^{m \times n}$, we denote
$A^{\dagger} \in \R^{n \times m}$ as its Moore-Penrose inverse,
$A^{\mathsf T} \in \R^{n \times m}$ as its transpose, and $A^{\dagger {\mathsf T}}$ as
the transpose of its Moore-Penrose inverse. As $n \ge m$ and $\rank(A) = m$, we have
$A^\dagger = A^{\mathsf T}(A A^{\mathsf T})^{-1}$. We denote $\sigma_{\min}(A)$ to be
the least singular value of matrix $A$. For a $k$'th order tensor
$T \in \R^{n_1 \times n_2 \times \cdots \times n_k}$, and $k$ vectors
$u_1 \in \R^{n_1}, \ldots, u_k \in \R^{n_k}$, we denote
$T[u_1, \ldots, u_k] = \sum_{i_1, \ldots, i_k} T_{i_1\cdots i_k} u_{1,
i_1} \cdots u_{k, i_k}$ as tensor-vectors multiplication. The
operator norm of tensor $T$ is defined as
$\dl T \dl_{\op} = \sup_{\dl u_1 \dl_2 = 1, \ldots, \dl u_k \dl_2 = 1}
T[u_1, \ldots, u_k]$.
For a scaler-valued function $f: \R^n \rightarrow \R$, we write its
gradient at $x \in \R^n$ as a column vector $\nabla f(x) \in \R^n$. We
write its Hessian at $x \in \R^n$ as a matrix
$\nabla^2 f(x) \in \R^{n \times n}$, and its third order derivative at
$x$ as a third order tensor
$\nabla^3 f(x) \in \R^{n \times n \times n}$. For a vector-valued
function $F: \R^n \rightarrow \R^m$, its Jacobian matrix at
$x \in \R^n$ is an $m \times n$ matrix
$\nabla F(x) \in \R^{m \times n}$, and its Hessian matrix at $x$ as a
third order tensor $\nabla^2 F(x) \in \R^{m \times n \times n}$.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Assumptions}\label{sec:assumptions}
Let $x_\star$ be a local minimizer of
problem~\eqref{problem:manopt}.
Throughout the rest of this paper, we make the following assumptions
on problem~\eqref{problem:manopt}. In particular, all these
assumptions are local, meaning that they only depend on the properties
of $f$ and $F$ in $\ball(x_\star,\delta)$ for some $\delta>0$.
\begin{assumption}[Smoothness]
\label{assumption:smoothness}
Within $\ball(x_\star, \delta)$, the functions $f$ and $F$ are $C^2$
with local Lipschitz constants
${L_f}, {L_F}$, Lipschitz gradients with constants ${\beta_f}$,
${\beta_F}$, and Lipschitz Hessians with constants ${\rho_f}$, ${\rho_F}$.
\end{assumption}
\begin{assumption}[Manifold structure and constraint qualification]
\label{assumption:cq}
The set $\mc{M}$ is a $m$-dimensional smooth submanifold of $\R^n$.
Further, $\inf_{x \in \ball(x_\star, \delta)}
\sigma_{\min}(\nabla F(x))\ge\gamma_F$ for some constant
$\gamma_F>0$.
\end{assumption}
Smoothness and constraint qualification together implies that the
constraints $F(x)$ are well-conditioned and
problem~\eqref{problem:manopt} is $C^2$ near $x_\star$. In particular,
we can define a matrix ${\rm Hess} f(x_\star)$ via the formula
\begin{equation}
\begin{aligned}
& \quad {\rm Hess} f(x_\star) = {\mathsf P}_{x_\star} \nabla^2 f(x_\star)
{\mathsf P}_{x_\star} - \sum_{i=1}^m [ \nabla F(x_\star)^{\dagger {\mathsf T}} \nabla
f(x_\star)]_i \cdot {\mathsf P}_{x_\star} \nabla^2 F_i(x_\star)
{\mathsf P}_{x_\star},
\end{aligned}
\end{equation}
where
\begin{equation}
\begin{aligned}
{\mathsf P}_{x_\star} = {\mathsf I}_n - \nabla F(x_\star)^\dagger \nabla F(x_\star).
\end{aligned}
\end{equation}
We can see later that ${\rm Hess} f(x_\star)$ is the matrix representation of the Riemannian Hessian of function $f$ on ${\mathcal M}$ at $x_\star$.
\begin{assumption}[Eigenvalues of the Riemannian Hessian]
\label{assumption:quadratic-growth}
Define
\begin{equation*}
\begin{aligned}
\lambda_{\max} = \sup\{ \< u, {\rm Hess} f(x_\star) u\> : \dl u
\dl_2 = 1, \nabla F(x_\star) u = 0 \},\\
\lambda_{\min} = \inf\{ \< u, {\rm Hess} f(x_\star) u\>: \dl u \dl_2
= 1, \nabla F(x_\star) u = 0 \}.\\
\end{aligned}
\end{equation*}
We assume $0 < {\lambda_{\min}} \le {\lambda_{\max}} < \infty$. We call $\kappa_R =
{\lambda_{\max}}/{\lambda_{\min}}$ the condition number of ${\rm Hess} f(x_\star)$.
\end{assumption}
\subsection{Geometry on the manifold ${\mathcal M}$}
\label{sec:geometry}
Since we assumed that the set ${\mathcal M}$ is a smooth submanifold of $
\R^{n}$, we endow ${\mathcal M}$ with the Riemannian geometry induced by the
Euclidean space $\R^{n}$. At any point $x \in {\mathcal M}$, the tangent space
(viewed as a subspace of $\R^n$) is obtained by taking the
differential of the equality constraints
\begin{equation}
T_{x} {\mathcal M}= \{ u \in \R^n : \nabla F(x) u = 0 \}.
\end{equation}
Let ${\mathsf P}_x$ be the orthogonal projection operator from $\R^n$ onto $T_x {\mathcal M}$. For any $u \in \R^n$, we have
\begin{equation}
\begin{aligned}
{\mathsf P}_x (u) &= [{\mathsf I}_n - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla F(x)^{\mathsf T})^{-1} \nabla F(x) ] u.
\end{aligned}
\end{equation}
Let ${\mathsf P}_x^{\perp}$ be the orthogonal projection operator from $\R^n$ onto the complement subspace of $T_x {\mathcal M}$. For any $u \in \R^n$, we have
\begin{equation}
\begin{aligned}
{\mathsf P}_x^{\perp} (u) &= \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla F(x)^{\mathsf T})^{-1} \nabla F(x) u.
\end{aligned}
\end{equation}
With a little abuse of notations, we will not distinguish ${\mathsf P}_x$ and ${\mathsf P}_x^{\perp}$ with their matrix representations. That is, we also think of ${\mathsf P}_x, {\mathsf P}_x^{\perp} \in \R^{n \times n}$ as two matrices.
We denote $\nabla f(x)$ and ${\rm grad} f(x)$
respectively the Euclidean gradient and the Riemannian gradient of $f$ at $x \in {\mathcal M}$.
The Riemannian gradient of $f$ is the projection of the Euclidean gradient onto the tangent space
\begin{equation}
\begin{aligned}
{\rm grad} f(x) = {\mathsf P}_x (\nabla f(x)) = [{\mathsf I}_n - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla F(x)^{\mathsf T})^{-1}
\nabla F(x) ] \nabla f(x).
\end{aligned}
\end{equation}
Since $x_\star$ is a local minimizer of $f$ on the manifold ${\mathcal M}$, we have ${\rm grad} f(x_\star) = 0$.
At $x \in {\mathcal M}$, let $\nabla^2 f(x)$ and ${\rm Hess} f(x)$ be respectively the Euclidean and the Riemannian Hessian of $f$. The Riemannian Hessian is a symmetric operator on the tangent space and is given by projecting the directional derivative of the gradient vector field. That is, for any $u, v \in T_x {\mathcal M}$, we have (we use ${\mathsf D}$ to denote the directional derivative)
\begin{equation}
\begin{aligned}
& {\rm Hess} f(x) [u, v] = \< v, {\mathsf P}_x ( {\mathsf D} {\rm grad} f(x) [u] )\> \\
= & \<v, {\mathsf P}_x \cdot \nabla^2 f(x) u - {\mathsf P}_x \cdot ({\mathsf D}
{\mathsf P}_x^{\perp} [u])\cdot \nabla f\> \\
= & v^{\mathsf T} {\mathsf P}_x \nabla^2 f(x) {\mathsf P}_x u - \nabla^2 F(x)[\nabla
F(x)^{\dagger {\mathsf T}} \nabla f(x), u, v].
\end{aligned}
\end{equation}
With a little abuse of notation, we will not distinguish the Hessian operator with its matrix representation. That is
\begin{equation}
\begin{aligned}
{\rm Hess} f(x) = {\mathsf P}_x \nabla^2 f(x) {\mathsf P}_x - \sum_{i=1}^m [ \nabla
F(x)^{\dagger {\mathsf T}} \nabla f(x)]_i \cdot {\mathsf P}_x \nabla^2 F_i(x)
{\mathsf P}_x.
\end{aligned}
\end{equation}
\subsection{Geometry off the manifold ${\mathcal M}$}
We can extend the definition of the matrix representations of the above Riemannian quantities outside the manifold ${\mathcal M}$. For any $x \in \R^n$, we denote
\begin{equation}
\begin{aligned}
& \quad {\mathsf P}_x = {\mathsf I}_n - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla F(x)^{\mathsf T})^{-1} \nabla F(x),\\
& \quad {\mathsf P}_x^{\perp} = \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla F(x)^{\mathsf T})^{-1} \nabla F(x),\\
& \quad {\rm grad} f(x) = {\mathsf P}_x \grad f(x).
\end{aligned}
\end{equation}
By the constraint qualification assumption (Assumption
\ref{assumption:cq}), $\nabla F(x)\nabla F(x)^{\mathsf T}$ is invertible and
the quantities above are well defined in $\ball(x_\star, \delta)$. We
call ${\rm grad} f(x)$ the extended Riemannian gradient of $f$ at $x$,
which extends the Riemannian gradient outside the manifold ${\mathcal M}$ as
$(f,F)$ are still well-defined there.
\subsection{Closed-form expression of the SQP iterate}
The above definitions makes it possible to have a concise closed-form
expression for the SQP iterate~\eqref{algorithm:constrained}. Indeed,
as each iterate solves a standard QP, the expression can be obtained
explicitly by writing out the optimality condition. Letting $x_k = x$,
the next iteration $x_{k+1} = x_+$ is given by
\begin{equation}
\label{equation:delta}
\begin{aligned}
& x_+ = x -\eta [{\mathsf I}_n - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla
F(x)^{\mathsf T} )^{-1} \nabla F(x)]\nabla f(x) \\
& \quad\quad\quad - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla
F(x)^{\mathsf T} )^{-1} F(x) \\
& = x - \eta{\mathsf P}_x \grad f(x) - \nabla F(x)^{\mathsf T} (\nabla F(x) \nabla
F(x)^{\mathsf T} )^{-1} F(x) \\
& = x - \eta{\rm grad} f(x) - \grad F(x)^\dagger F(x).
\end{aligned}
\end{equation}
We will frequently refer to this expression in our proof.
\section{Main results}
\subsection{Local convergence theorem}
\label{sec:main}
Under Assumptions~\ref{assumption:smoothness},~\ref{assumption:cq}
and~\ref{assumption:quadratic-growth}, we show that the SQP
algorithm~\eqref{algorithm:constrained} converges locally linearly
with rate $1-1/\kappa_R$. The proof can be found in
Section~\ref{sec:proof}.
\begin{theorem}[Local linear convergence of SQP with canonical rate]
\label{theorem:convergence}
There exists $\eps > 0$ and a constant $\sigma >0$ such that the
following holds. Let $x_0 \in \ball(x_\star, \eps)$ and $x_k$ be
the iterates of Equation (\ref{algorithm:constrained}) with stepsize
$\eta=1/\lambda_{\max}({\rm Hess} f(x_\star))$. Letting
\begin{equation*}
\begin{aligned}
a_k =& \dl {\mathsf P}_{x_\star} (x_k-x_\star) \dl_2,\\
b_k =& \dl {\mathsf P}_{x_\star}^\perp (x_k-x_\star) \dl_2,
\end{aligned}
\end{equation*}
we have
\begin{equation*}
a_{k+1}^2 + \sigma b_{k+1} \le \Big(1 -
\frac{1}{2\kappa_R}\Big)^2(a_k^2 + \sigma b_k),
\end{equation*}
where
$\kappa_R = \lambda_{\max}({\rm Hess} f(x_\star))/\lambda_{\min}({\rm Hess}
f(x_\star))$ is the condition number of the Riemannian Hessian of
function $f$ on the manifold ${\mathcal M}$ at $x_\star$. Consequently, the
distance $\dl x_k-x_\star \dl_2^2=a_k^2+b_k^2$ also converges
linearly: $$\ltwo{x_k-x_\star}^2\le O\left((1 -
1/(2\kappa_R))^k\right).$$
\end{theorem}
{\bf Remark}
Theorem~\ref{theorem:convergence} requires choosing the stepsize
$\eta$ according to the maximum eigenvalue of ${\rm Hess} f(x_\star)$,
which might not be known in advance. In practice, one could
implement a line-search (for example as in ~\citep[Section
4.2]{AbsilMaSe09} for Riemannian gradient methods) which would
hopefully achieve the same optimal rate.
\subsection{Global convergence}
\label{sec:global}
Let ${\mathcal M}_\eps = \{ x : \| F(x) \|_2 \le \eps \}$ denote an
$\eps$-neighborhood of the manifold ${\mathcal M}$. To show global properties
of SQP algorithm, we make the following additional assumptions:
\begin{assumption}[Global assumptions]\label{ass:global}~
\begin{enumerate}
\item There exists $\eps_0 \ge 0$, such that $\sup_{x \in
{\mathcal M}_{\eps_0}} \| \nabla f(x) \|_2 \le {G_f}$, and $\inf_{x \in
{\mathcal M}_{\eps_0}} f(x) \ge {\underline{f}}$.
\item The condition in Assumption \ref{assumption:cq} holds in this
neighborhood ${\mathcal M}_{\eps_0}$: $\inf_{x \in
{\mathcal M}_{\eps_0}}\sigma_{\min}(\nabla F(x))\ge\gamma_F$.
\item Some conditions in Assumption \ref{assumption:smoothness} holds
globally in $\R^n$: the functions $f$ and $F$ are $C^2$, $f$ has
Lipschitz constant ${L_f}$, and $f$ and $F$ have Lipschitz
gradients with constants ${\beta_f}$, ${\beta_F}$.
\end{enumerate}
\end{assumption}
The following theorem establishes the global convergence of SQP
algorithm with a small constant stepsize.
Our convergence guarantee is provided in terms of the norm of the
extended Riemannian gradient. The proof can be found in
Section~\ref{sec:proof-global}.
\begin{theorem}\label{thm:global_closeness_and_convergence}
There exists constants $K_1, K_2 > 0, \eps_\star > 0$, such that for any $\eps \le \eps_\star$, we initialize $x_0 \in {\mathcal M}_\eps$, and letting step size to be $\eta = \sqrt{\eps/K_1}$, then each iterates will be close to the manifold, i.e., $\{ x_i \}_{i \in \N} \subseteq {\mathcal M}_\eps$. Moreover, for any $k \in \N$, we have
\[
\min_{i \in [k]} \| {\rm grad} f(x_i) \|_2^2 \le K_2 \{ [f(x_0) - {\underline{f}}]/ (k \sqrt \eps) + \sqrt{\eps} \}.
\]
\end{theorem}
To minimize the bound in the right hand side, one can choose $\eps = O(1/ k)$, and we get the following corollary.
\begin{corollary}
There exists constants $\{ K_i \}_{i=1}^4$, such that for any $k \ge K_1$, if we take $\eps = K_2/k$, and initialize on the manifold ${\mathcal M}$ with step size $\eta = K_3/k^{1/2}$, we have
\[
\min_{i \in [k]} \| {\rm grad} f(x_i) \|_2 \le K_4/k^{1/4}.
\]
\end{corollary} {\bf Remark on the stationarity measure} While our
global convergence is measured the extended Riemannian gradient
$\ltwo{{\rm grad} f(x_i)}$ on the infeasible iterate $x_i$'s, one could
construct nearby feasible points with small Riemannian gradient via a
straightforward perturbation argument.
\section{Proof of Theorem~\ref{theorem:convergence}}\label{sec:proof}
\subsection{Riemannian Taylor expansion}
\label{sec:RTE}
The proof of Theorem~\ref{theorem:convergence} relies on a particular
expansion of the Riemannian gradient off the manifold, which we state
as follows.
\begin{lemma}[First-order expansion of Riemannian gradient]
\label{lemma:riemannian-taylor-expansion}
There exist constants $\eps_0>0$ and $C_{r,1}, C_{r,2}>0$ such that for all
$x\in\ball(x_\star,\eps_0)$,
\begin{equation}
{\rm grad} f(x) = {\rm Hess} f(x_\star) (x - x_\star) +
r(x),
\end{equation}
where the remainder term $r(x)$ satisfies the error bound
\begin{equation}
\| r(x) \|_2 \le C_{r,1} \| x-x_\star \|_2^2 +
C_{r,2} \| {\mathsf P}_{x_\star}^\perp(x-x_\star) \|_2.
\end{equation}
\end{lemma}
Lemma~\ref{lemma:riemannian-taylor-expansion} extends
classical Riemannian Taylor expansion~\citep[Section
7.1]{AbsilMaSe09} to points off the manifold, where the remainder
term contains an additional first-order error. While the error term
is linear in $\| {\mathsf P}_{x_\star}^\perp(x-x_\star) \|_2$, this
expansion is particularly suitable when $\| {\mathsf P}_{x_\star}^\perp(x-x_\star) \|_2$ is on the order
of $ \|{\mathsf P}_{x_\star}(x-x_\star) \|_2^2$, which results in a
quadratic bound on $\ltwo{r(x)}$.
In the proof of Theorem~\ref{theorem:convergence}, ${\rm grad} f(x)$ mostly
appears through its squared norm and inner product with
$x-x_\star$. We summarize the expansion of these terms in the
following corollary.
\begin{corollary}
\label{corollary:rgrad-expansions}
There exist constants $\eps_0>0$ and $C_{r,1},C_{r,2}>0$ such that the
following hold. For any $\eps\le \eps_0$, $x\in\ball(x,\eps)$,
\begin{align*}
& \<{\rm grad} f(x), x-x_\star\> = \<x-x_\star, {\rm Hess}
f(x_\star)(x-x_\star)\> + R_1, \\
& \| {\rm grad} f(x)\|_2^2 = \<x-x_\star, [{\rm Hess}
f(x_\star) ]^2(x-x_\star)\> + R_2,
\end{align*}
where
\begin{align*}
\max\set{|R_1|, |R_2|}
\le \eps (C_{r,1}\| {\mathsf P}_{x_\star}(x-x_\star)\|_2^2 +
C_{r,2} \| {\mathsf P}_{x_\star}^\perp(x-x_\star)\|_2 ).
\end{align*}
\end{corollary}
\subsection{Proof of Theorem~\ref{theorem:convergence}}
The following perturbation bound (see, e.g.~\cite{Sun01}) for
projections is useful in the proof.
\begin{lemma}
\label{lemma:projection-perturbation-bound}
For $x_1,x_2 \in \ball(x_\star, \delta)$, we have
\begin{equation*}
\begin{aligned}
&\opnorm{{\mathsf P}_{x_1} - {\mathsf P}_{x_2}} \le
{\beta_P} \dl x_1-x_2 \dl_2 \\
& \dl {\mathsf P}_{x_1} {\mathsf P}_{x_2}^\perp \dl_{\op} \le {\beta_P} \dl
x_1-x_2 \dl_2.
\end{aligned}
\end{equation*}
where ${\beta_P} = 2{\beta_F}/\gamma_F$.
\end{lemma}
We now prove the main theorem.
\paragraph{Step 1: A trivial bound on $\| x_+-x_\star\|_2^2$. }
Consider one iterate of the algorithm $x\to x_+$, whose closed-form
expression is given in~\eqref{equation:delta}. We have $x_+ =x
+\Delta$, where
\begin{equation}
\Delta = -\eta \cdot {\rm grad} f(x) - \nabla F(x)^\dagger F(x).
\end{equation}
We would like to relate $\dl x_+ - x_\star \dl_2$ with $\dl x - x_\star \dl_2$.
Observe that ${\rm grad} f(x_\star)=0$, we have
\begin{equation*}
\dl {\rm grad} f(x) \dl_2 = \dl {\rm grad} f(x) - {\rm grad} f(x_\star) \dl_2 \le
{\beta_E} \dl x-x_\star \dl_2.
\end{equation*}
Observe that $ F(x_\star) = 0$, we have
\begin{equation*}
\begin{aligned}
&\quad \dl \nabla F(x)^\dagger F(x) \dl_2 = \dl \nabla F(x)^\dagger
(F(x) - F(x_\star)) \dl_2 \\
& \le \dl \nabla F(x)^\dagger \dl_{\op} \dl F(x) - F(x_\star) \dl_2
\le ({\beta_F}/\gamma_F) \dl x - x_\star \dl_2.
\end{aligned}
\end{equation*}
Accordingly, we have
\begin{align*}
& \quad \dl x_+-x_\star \dl_2 \\
& \le \dl x-x_\star \dl_2 + \eta \dl
{\rm grad} f(x) \dl_2 + \dl \nabla F(x)^\dagger F(x) \dl_2 \\
& \le [1 + \eta {\beta_E} + ({\beta_F}/\gamma_F)] \dl x-x_\star \dl_2.
\end{align*}
Hence, for any stepsize $\eta$, letting ${C_d}= [1 + \eta {\beta_E} + ({\beta_F}/\gamma_F)]^2$, for $x \in \ball(x_\star, \delta)$, we have
\begin{equation} \label{equation:distance-growth}
\dl x_+-x_\star \dl_2^2 \le {C_d} \dl x-x_\star\dl_2^2.
\end{equation}
\paragraph{Step 2: Analyze normal and tangent distances. }
This is the key step of the proof.
We look into the \emph{normal} direction and the \emph{tangent} direction separately. The intuition
for this process is that the normal part of $x-x_\star$ is a measure of feasibility, and as we will see, converges much more quickly.
Now we look at equation $x_+-x_\star=x-x_\star+\Delta$. Multiplying it by ${\mathsf P}_x$ and
${\mathsf P}_x^\perp$ gives
\begin{align}
& {\mathsf P}_x^\perp (x_+-x_\star) = {\mathsf P}_x^\perp (x-x_\star) + {\mathsf P}_x^\perp \Delta =
{\mathsf P}_x^\perp (x-x_\star) - \nabla F(x)^\dagger
F(x), \label{equation:px-perp} \\
& {\mathsf P}_x (x_+-x_\star) = {\mathsf P}_x (x-x_\star) + {\mathsf P}_x \Delta = {\mathsf P}_x (x-x_\star) - \eta
\cdot {\rm grad} f(x). \label{equation:px}
\end{align}
We now take squared norms on both equalities and bound the
growth. Define the normal and tangent distances as
\begin{equation}
a = \dl {\mathsf P}_{x_\star} (x-x_\star)\dl_2,~~~
b = \dl {\mathsf P}_{x_\star}^\perp (x-x_\star) \dl_2 ,
\end{equation}
and $(a_+,b_+)$ similarly for $x_+$. Note that the definitions of $a$,
$b$ use the projection at $x_\star$, so they are slightly different
from quantities~\eqref{equation:px-perp} and~\eqref{equation:px}.
From now on, we assume
that $\| x-x_\star\|_2\le\eps_0$, and $\eps_0$ is sufficiently small
such that~\eqref{equation:distance-growth} holds. The requirements on
$\eps_0$ will later be tightened when necessary.
\subparagraph{The normal direction. }
We have
\[
\begin{aligned}
& \quad \dl {\mathsf P}_x^\perp (x_+ - x_\star)\dl_2^2 \dl {\mathsf P}_x^\perp (x - x_\star) \dl_2^2 + 2 \< x - x_\star,
{\mathsf P}_x^\perp \Delta \> + \dl {\mathsf P}_x^\perp \Delta \dl_2^2 \\
&= \dl {\mathsf P}_x^\perp (x - x_\star) \dl_2^2 - 2 \< x - x_\star,
\nabla F(x)^\dagger F(x) \> + \<F(x), (\nabla F(x) \nabla
F(x)^{\mathsf T})^{-1} F(x) \> \\
&= \dl {\mathsf P}_x^\perp (x - x_\star) \dl_2^2 - 2\big\<\nabla
F(x)(x-x_\star), (\nabla F(x)\nabla F(x)^{\mathsf T})^{-1}(\nabla
F(x)(x-x_\star)+r(x))\big\> \\
& \quad\quad+ \big\<\nabla
F(x)(x-x_\star)+r(x), (\nabla F(x)\nabla F(x)^{\mathsf T})^{-1}(\nabla
F(x)(x-x_\star)+r(x))\big\> \\
&= \<r(x), (\nabla F(x)\nabla F(x)^{\mathsf T})^{-1}r(x)\>,
\end{aligned}
\]
where $r(x)=F(x) - \nabla F(x)(x-x_\star)$. By the smoothness of function $F$, we have
\[
\dl r(x) \dl_2 \le {\beta_F}/2 \cdot \dl x-x_\star \dl_2^2.
\]
Accordingly, we get
\begin{equation*}
\begin{aligned}
& \quad \dl {\mathsf P}_x^\perp(x_+-x_\star) \dl_2^2 \\
& \le
{\beta_F}^2/(4\gamma_F^2) \cdot \dl x-x_\star \dl_2^4 =
{\beta_F}^2/(4\gamma_F^2) \cdot [\dl
{\mathsf P}_{x_\star}^\perp(x-x_\star) \dl_2^2 + \dl
{\mathsf P}_{x_\star}(x-x_\star)\dl_2^2]^2\\
& = {\beta_F}^2/(4\gamma_F^2) \cdot (a^2+b^2)^2.
\end{aligned}
\end{equation*}
Applying the perturbation bound on projections
(Lemma~\ref{lemma:projection-perturbation-bound}), we get
\begin{equation}
\label{equation:bound-normal}
\begin{aligned}
&\quad b_+ = \dl {\mathsf P}_{x_\star}^\perp (x_+ - x_\star) \dl_2 \\
& \le
\dl {\mathsf P}_x^\perp(x_+-x_\star) \dl_2 +
\dl {\mathsf P}_{x}-{\mathsf P}_{x_\star} \dl_{\op} \dl x_+-x_\star \dl_2 \\
&\le {\beta_F}/(2\gamma_F) \cdot (a^2+b^2) +
{\beta_P} \dl x-x_\star \dl_2 \dl x_+-x_\star\dl_2 \\
&\le {\beta_F}/(2\gamma_F) \cdot (a^2+b^2) +
{\beta_P}{C_d} \cdot \dl x-x_\star\dl_2^2 \\
&= \underbrace{[ {\beta_F}/(2\gamma_F) + {\beta_P}{C_d} ]}_{{C_b}} \cdot
(a^2+b^2) = {C_b}(a^2+b^2).
\end{aligned}
\end{equation}
\subparagraph{The tangent direction. }
We have
\begin{equation*}
\begin{aligned}
&\quad \dl{\mathsf P}_x (x_+-x_\star) \dl_2^2 \\
&= \dl {\mathsf P}_x (x - x_\star) \dl_2^2 + 2 \< x - x_\star, {\mathsf P}_x
\Delta \> + \dl {\mathsf P}_x \Delta \dl_2^2\\
&= \dl{\mathsf P}_x (x-x_\star)\dl_2^2 - 2\eta \cdot \<{\rm grad} f(x),
x - x_\star\> + \eta^2 \cdot \dl {\rm grad} f(x)\dl_2^2.
\end{aligned}
\end{equation*}
Applying Lemma~\ref{lemma:projection-perturbation-bound}, we get that
for any vector $v$,
\begin{equation*}
\begin{aligned}
\vert \dl {\mathsf P}_x v \dl_2^2 - \dl {\mathsf P}_{x_\star} v \dl_2^2 \vert
= \vert \<v,({\mathsf P}_{x_\star}-{\mathsf P}_x)v\> \vert \le {\beta_P} \dl
x-x_\star \dl_2 \cdot \dl v\dl_2^2.
\end{aligned}
\end{equation*}
Applying this to vectors $x_+-x_\star$ and $x-x_\star$ gives
\begin{equation}
\label{equation:aplus-bound}
\begin{aligned}
a_+^2
& \le a^2 - 2\eta\<{\rm grad} f(x),
x -x_\star\> + \eta^2\dl {\rm grad} f(x) \dl_2^2 \\
& \quad + {\beta_P}(\dl x-x_\star\dl_2^3 +
\dl x-x_\star\dl_2 \dl x_+ -x_\star\dl_2^2) \\
& \le a^2 - 2\eta\<{\rm grad} f(x),
x -x_\star\> + \eta^2\dl {\rm grad} f(x) \dl_2^2 \\
& \quad + {\beta_P}(1+{C_d})\dl x-x_\star
\dl_2^3.
\end{aligned}
\end{equation}
Applying Corollary~\ref{corollary:rgrad-expansions}, and
note that ${\rm Hess} f(x_\star)={\mathsf P}_{x_\star}{\rm Hess}
f(x_\star){\mathsf P}_{x_\star}$ by the property of the Riemannian Hessian,
we get
\begin{align*}
& \quad a_+^2 \le a^2 - 2\eta\<x-x_\star, {\rm Hess}
f(x_\star)(x-x_\star)\> +
\eta^2\<x-x_\star, ({\rm Hess} f(x_\star) )^2(x-x_\star)\> \\
& \quad\quad\quad + (-2\eta R_1 + \eta^2 R_2) + \eps_0 C(a^2+b^2) \\
& \quad\quad = \<{\mathsf P}_{x_\star}(x-x_\star), ({\mathsf I} -\eta{\rm Hess}
f(x_\star))^2{\mathsf P}_{x_\star}(x-x_\star)\> + (-2\eta R_1 +
\eta^2 R_2) + \eps_0 C(a^2+b^2),
\end{align*}
where the remainders are bounded as
\begin{equation*}
\begin{aligned}
\max\set{|R_1|, |R_2|} &\le
\eps_0\big(C_{r,1}\| {\mathsf P}_{x_\star}(x-x_\star)\|_2^2 +
C_{r,2}\| {\mathsf P}_{x_\star}^\perp(x-x_\star)\|_2 \big) \\
& = \eps_0(C_{r,1} a^2 + C_{r,2} b).
\end{aligned}
\end{equation*}
Choosing the stepsize as $\eta=1/\lambda_{\max}({\rm Hess} f(x_\star))$,
we have ${\mathsf I} -\eta{\rm Hess} f(x_\star) \preceq (1-1/\kappa_R) {\mathsf I}$. For this choice of
$\eta$, using the above bound for $R_1,R_2$, we get that there exists
some constant $C_1,C_2$ such that
\begin{equation}
\label{equation:bound-tangent}
a_+^2 \le
\Big(1-\frac{1}{\kappa_R}\Big)^2a^2 + \eps_0(C_1 a^2 + C_2 b).
\end{equation}
\subparagraph{Putting together. }
Let $\sigma > 0$ be a constant to be
determined. Looking at the quantity $a_+^2+ \sigma b_+$, by the
bounds~\eqref{equation:bound-normal}
and~\eqref{equation:bound-tangent} we have
\begin{align*}
& \quad a_+^2 + \sigma b_+ \\
& \le \Big(1-\frac{1}{\kappa_R}\Big)^2a^2
+ \eps_0(C_1a^2+C_2b) + \sigma{C_b}(a^2+b^2) \\
& \le \Big( \Big(1-\frac{1}{\kappa_R}\Big)^2 + \eps_0 C_1 +
\sigma{C_b} \Big)a^2 + \eps_0(C_2+ \sigma{C_b})b.
\end{align*}
The last inequality is by rearranging the terms and by the assumption that $b = \dl {\mathsf P}_{x_\star}^\perp (x-x_\star) \dl_2 \le \eps_0$.
Then, we would like to choose $\sigma$ and $\eps_0$ sufficiently small such that
\begin{align}
\Big(1-\frac{1}{\kappa_R}\Big)^2 + \eps_0 C_1 +
\sigma{C_b} \le& \Big( 1 - \frac{1}{2\kappa_R} \Big)^2, \label{eqn:thm1_margin1}\\
\eps_0(C_2+ \sigma{C_b}) \le& \sigma.\label{eqn:thm1_margin2}
\end{align}
This can be obtained by first choosing $\sigma$ small to satisfy Eq. (\ref{eqn:thm1_margin1}) leaving a small margin for the choice of $\eps_0$, then choosing $\eps_0$ small to satisfy both Eq. (\ref{eqn:thm1_margin1}) and (\ref{eqn:thm1_margin2}). With this choice of $\sigma$ and $\eps_0$, we have
\begin{equation*}
a_+^2 + \sigma b_+ \le \Big( 1 - \frac{1}{2\kappa_R} \Big)^2(a^2 +
\sigma b).
\end{equation*}
\paragraph{Step 3: Connect the entire iteration path. }
Consider iterates $x_k$ of the first-order
algorithm~\eqref{algorithm:constrained}. Initializing $x^0$
sufficiently close to $x_\star$, the descent on $a_k^2+\sigma b_k$
ensures $a_k^2+b_k^2\le\eps_0^2$. Thus we chain this analysis on
$(x, x_+)=(x_k,x_{k+1})$ and get that $a_k^2+\sigma b_k$ converges
linearly with rate $1-1/(2\kappa_R)$. This in turn implies the bound
$a_k^2+b_k^2\le O((1-1/(2\kappa_R))^k)$ by observing that
$a_k^2+b_k^2\le C(a_k^2+\sigma b_k)$ for some constant $C$.
\section{Proof of Theorem~\ref{thm:global_closeness_and_convergence}}
\label{sec:proof-global}
The proof of Theorem~\ref{thm:global_closeness_and_convergence} is
directly implied by the following two lemmas.
Lemma~\ref{lem:global_closeness} ensures that, as we initialize close
to the manifold and choosing step size reasonably small, the iterates
will be automatically close to the
manifold. Lemma~\ref{lem:global_convergence} shows that, as the
iterates are close to the manifold at each step, the value of function
$f$ can decrease by a reasonable amount, so that there will be a
iterates with small extended Riemannian gradient.
\begin{lemma}\label{lem:global_closeness}
Let $0 < \eps_1 \le [\gamma_F^2 / (2{\beta_F})] \wedge \eps_0$, $x_0 \in {\mathcal M}_{\eps_1}$ and $\eta \le [\eps_1/(2 {\beta_F} {G_f}^2)]^{1/2}$. Then $\{ x_k \}_{k \ge 0} \subseteq {\mathcal M}_{\eps_1}$.
\end{lemma}
\begin{proof}
We use proof by induction. We assume $x_k \in {\mathcal M}_{\eps_1}$, i.e., we have $\| F(x_k) \|_2 \le \eps_1$. We would like to show that $\| F(x_{k+1}) \|_2 \le \eps_1$, where $x_{k+1}$ gives
\begin{equation}\label{eqn:global_iterates}
\begin{aligned}
x_{k+1} = x_k - \eta{\rm grad} f(x_k) - \grad F(x_k)^\dagger F(x_k).
\end{aligned}
\end{equation}
Performing Taylor's expansion of $F(x_{k+1})$ at $x_k$, we have
\[
\begin{aligned}
&\| F(x_{k+1}) \|_2 = \| F(x_k + (x_{k+1 }- x_k)) \|_2 \\
\le ~& \| F(x_k) - \nabla F(x_k) [ \eta {\rm grad} f(x_k) + \grad F(x_k)^\dagger F(x_k)] \|_2 + {\beta_F} \| x_{k+1} - x_k \|_2^2.\\
\end{aligned}
\]
Note we have
\[
\begin{aligned}
\grad F(x_k) {\rm grad} f(x_k) =& \grad F(x_k) {\mathsf P}_{x_k} \grad f(x_k) = 0, \\
\nabla F(x_k) \nabla F(x_k)^\dagger =& {\mathsf I}_n,
\end{aligned}
\]
which gives
\[
\begin{aligned}
&F(x_k) - \nabla F(x_k) [ \eta {\rm grad} f(x_k) + \grad F(x_k)^\dagger
F(x_k)] = F(x_k) - F(x_k) = 0.
\end{aligned}
\]
As a result, we have
\[
\begin{aligned}
&\| F(x_{k+1}) \|_2 = {\beta_F} \| x_{k+1} - x_k \|_2^2\\
=& {\beta_F} [ \eta^2 \| {\rm grad} f(x_k) \|_2^2 + F(x_k)^{\mathsf T} \grad F(x_k)^{\dagger {\mathsf T}} \grad F(x_k)^\dagger F(x_k)]\\
\le& {\beta_F} \eta^2 {G_f}^2 + ({\beta_F}/\gamma_F^2) \| F(x_k) \|_2^2.
\end{aligned}
\]
Note by induction assumption, we have $\| F(x_k) \|_2 \le \eps_1$. Hence as long as $\eps_1 \le \gamma_F^2 / (2{\beta_F})$ and $\eta^2 \le \eps_1/(2 {\beta_F} {G_f}^2) $, we have
\[
\begin{aligned}
&\| F(x_{k+1}) \|_2 \le {\beta_F} \eta^2 {G_f}^2 + ({\beta_F}/\gamma_F^2) \| F(x_k) \|_2^2
\le \eps_1 / 2 + \eps_1 / 2 = \eps_1.
\end{aligned}
\]
The lemma holds by noting that the initialization of the induction holds since $x_0 \in {\mathcal M}_{\eps_1}$.
\end{proof}
\begin{lemma}\label{lem:global_convergence}
There exists constant $K < \infty$, such that for any $\eps$ satisfying $0 < \eps \le \eps_\star = [\gamma_F^2 / (2{\beta_F})] \wedge [{\beta_F} {G_f}^2/(2{\beta_f}^2)] \wedge \eps_0$, letting $\eta = [\eps/(2 {\beta_F} {G_f}^2)]^{1/2}$ and initializing $x_0 \in {\mathcal M}_\eps$, we have for any $k \in \N$:
\[
\min_{i \in [k]} \| {\rm grad} f(x_i) \|_2^2 \le K \{ [f(x_0) - {\underline{f}}]/ (k \sqrt \eps) + \sqrt{\eps} \}.
\]
\end{lemma}
\begin{proof}
Performing Taylor's expansion of $f(x_{k+1})$ at $x_k$, we have
\[
f(x_{k+1} ) - f(x_k) - \< \nabla f(x_k), x_{k+1} - x_k \> \le {\beta_f} \| x_{k+1} - x_k \|_2^2.
\]
Using Eq. (\ref{eqn:global_iterates}) gives
\[
\begin{aligned}
&f(x_{k+1}) - f(x_k) + \eta \| {\rm grad} f(x_k) \|_2^2 + \nabla f(x_k)^{\mathsf T} \nabla F(x_k)^\dagger F(x_k) \\
\le& {\beta_f}[ \eta^2 \| {\rm grad} f(x_k)\|_2^2 + F(x_k)^{\mathsf T} \nabla F(x_k)^{\dagger {\mathsf T}} \nabla F(x_k)^\dagger F(x_k)]\\
\le& {\beta_f}[ \eta^2 \| {\rm grad} f(x_k)\|_2^2 + (1 / \gamma_F^2) \| F(x_k) \|_2^2].
\end{aligned}
\]
By the choice of $\eps$ and $\eta$, we have $\eta \le 1/ (2{\beta_f})$, rearranging the terms gives
\[
\begin{aligned}
&f(x_{k+1}) - f(x_k) + (\eta/2) \| {\rm grad} f(x_k) \|_2^2 \\
\le& ({L_f} / \gamma_F) \| F(x_k) \|_2 + ({\beta_f} / \gamma_F^2) \| F(x_k) \|_2^2\\
\le& [({L_f} / \gamma_F) + ({\beta_f} / \gamma_F^2) \eps_\star] \eps \equiv K_0 \eps.
\end{aligned}
\]
Performing telescope summation and rearranging the terms,
\[
\begin{aligned}
&\frac{1}{k}\sum_{i=1}^k \| {\rm grad} f(x_i) \|_2^2
\le 2 [f(x_0) - f(x_k)] / (k\eta) + 2 K_0 \eps / \eta
\le K\{ [f(x_0) - f(x_k)] / (k \sqrt \eps) + \sqrt{\eps}\},
\end{aligned}
\]
for some constant $K$. Since we know $x_k \in {\mathcal M}_{\eps}$, this concludes the proof of this lemma.
\end{proof}
\begin{figure*}[ht!]
\centering
\subfigure[Vary $\kappa$]{
\includegraphics[width=0.2\textwidth]{Figures/d=1000_eps=1e-2.pdf}
}\label{figure:compare-kappa}
\subfigure[Vary $\eps$]{
\includegraphics[width=0.2\textwidth]{Figures/d=1000_kappa=1e2.pdf}
}\label{figure:compare-eps}
\subfigure[Compare SQP and Riemannian]{
\includegraphics[width=0.4\textwidth]{Figures/ropt.pdf}
}\label{figure:riemannian}
\caption{\small Convergence of SQP. (a)(b) Effect of condition
number and initialization radius on the convergence. Thins lines
show all the 20 instances and bold lines indicate the median
performance. (c) Comparing SQP and Riemannian gradient descent
over 5 instances.}
\label{figure:figure}
\end{figure*}
\section{Example}
\label{section:example}
We provide the eigenvalue problem as a simple example illustrating our
convergence result. We emphasize that this is a simple and
well-studied problem; our goal here is only to illustrate the
connection between SQP and the Riemannian algorithms.
\begin{example}[SQP for eigenvalue problems]
Consider the eigenvalue problem for a symmetric matrix
$A\in\R^{n\times n}$:
\begin{equation}
\label{problem:eigenval}
\begin{aligned}
\minimize & ~~ \frac{1}{2}x^\top Ax \\
\subjectto & ~~ \ltwo{x}^2 - 1 = 0.
\end{aligned}
\end{equation}
This is an instance of problem~\eqref{problem:manopt} with
$f(x)=(1/2)x^\top Ax$ and $F(x)=\ltwo{x}^2 - 1$. Let
$\lambda_1<\lambda_2\le \dots\le \lambda_n$ be the eigenvalues of
$A$, and $v_i(A)$ be the corresponding eigenvectors.
The SQP iterate for this problem is $x\mapsto x_+=x+\Delta$, where
$\Delta$ solves the subproblem
\begin{align*}
\minimize & ~~ \<Ax, \Delta\> + \frac{1}{2\eta}\ltwo{\Delta}^2 \\
\subjectto & ~~ \ltwo{x}^2 + 2\<x, \Delta\> - 1 = 0.
\end{align*}
Applying~\eqref{equation:delta}, we obtain the explicit formula
\begin{equation}
\label{equation:sqp-eigenvalue}
x_+ = \frac{\ltwo{x}^2 + 1}{2\ltwo{x}^2} x - \eta\Big({\mathsf I}_n -
\frac{xx^\top}{\ltwo{x}^2}\Big)Ax.
\end{equation}
By Theorem~\ref{theorem:convergence}, the local convergence
rate is $1-1/\kappa_R$, which we now examine. At $x_\star=v_1(A)$,
we have ${\rm Hess} f(x_\star) = A - \lambda_1 {\mathsf I}_n$, and the tangent
space $\mc{T}_{x_\star}\mc{M}={\rm
span}(v_2(A),\dots,v_n(A))$. Therefore, the condition number of
the Riemannian Hessian is
\begin{equation*}
\kappa_R = \frac{\sup_{v\in\mc{T}_{x_\star}\mc{M},\ltwo{v}=1}\<v,
{\rm Hess}
f(x_\star)v\>}{\inf_{v\in\mc{T}_{x_\star}\mc{M},\ltwo{v}=1} \<v,
{\rm Hess}
f(x_\star)v\>} = \frac{\lambda_n - \lambda_1}{\lambda_2 -
\lambda_1}.
\end{equation*}
Hence, choosing the right stepsize $\eta$, the convergence rate of
the SQP method is
\begin{equation*}
1 - \frac{1}{\kappa_R} = \frac{\lambda_n - \lambda_2}{\lambda_n -
\lambda_1},
\end{equation*}
matching the rate of power iteration and Riemannian gradient
descent.
The keen reader might find that the
iterate~\eqref{equation:sqp-eigenvalue} converges globally to
$x_\star$ as long as $\<x_0,x_\star\>\neq 0$. This is more
optimistic than our
Theorem~\ref{thm:global_closeness_and_convergence} (which only
guarantees global convergence to stationary point). Whether such
global convergence holds more generally would be an interesting
direction for future study.
\end{example}
\section{Numerical experiments}
\label{sec:exp}
\paragraph{Setup}
We experiment with the SQP algorithm on random instances of the
eigenvalue problem~\eqref{problem:eigenval} with $d=1000$. Each
instance $A$ was generated randomly with a controlled condition
number $\kappa_R=\frac{\lambda_n - \lambda_1}{\lambda_2 - \lambda_1}$,
and the stepsize was chosen as $\eta=\frac{1}{2(\lambda_n-\lambda_1)}$
to optimize for the local linear rate. The initialization $x_0$ is
sampled randomly around the solution $x_\star$ with average distance
$\eps$ (recall $\ltwo{x_\star}=1$).
We run the following three sets of comparisons and plot the results in
Figure~\ref{figure:figure}.
\begin{enumerate}[(a)]
\item Test the effect of $\kappa_R$ on the local linear rate, with
$\kappa_R\in\set{25, 50, 100}$ and fixed $\eps=0.01$.
\item Test the effect of initialization radius $\eps$ (i.e. localness)
on the convergence, with $\eps\in\set{0.01, 1, 100}$ and fixed
$\kappa=100$.
\item Test whether SQP is close to Riemannian gradient descent when
initialized at a same feasible start $x_0\in\mc{M}$.
\end{enumerate}
\paragraph{Results} In experiment (a), we see indeed that the local
linear rate of SQP scales as $1-C/\kappa_R$: doubling the condition
number will double the number of iterations required for halving the
sub-optimality. Experiment (b) shows that the linear convergence is
indeed more robust locally than globally; when
initialized very far away ($\eps=100$), linear convergence with the
same rate happens on most of the instances after a while, but there
does exist bad instances on which the convergence is slow. This
corroborates our theory that the global convergence of SQP is more
sensitive to the stepsize choice as the SQP is not guaranteed to
approach the manifold when initialized far away. Experiment (c)
verifies our intuition that SQP is approximately equal to Riemannian
gradient descent: with a feasible start, their iterates stay almost
exactly the same.
\section{Conclusion}
We established local and global convergence of a cheap SQP algorithm
building on intuitions from Riemannian optimization. Potential future
directions include generalizing our global result to ``far from the
manifold'', as well as identifying problem structures under which we
obtain global convergence to local minimum.
\section*{Acknowledgement}
We thank Nicolas Boumal and John Duchi for a number of helpful
discussions. YB was partially supported by John Duchi's National
Science Foundation award CAREER-1553086. SM was supported by an Office
of Technology Licensing Stanford Graduate Fellowship.
\bibliographystyle{amsalpha}
| {
"timestamp": "2019-02-01T02:08:29",
"yymm": "1805",
"arxiv_id": "1805.08756",
"language": "en",
"url": "https://arxiv.org/abs/1805.08756",
"abstract": "We prove that a \"first-order\" Sequential Quadratic Programming (SQP) algorithm for equality constrained optimization has local linear convergence with rate $(1-1/\\kappa_R)^k$, where $\\kappa_R$ is the condition number of the Riemannian Hessian, and global convergence with rate $k^{-1/4}$. Our analysis builds on insights from Riemannian optimization -- we show that the SQP and Riemannian gradient methods have nearly identical behavior near the constraint manifold, which could be of broader interest for understanding constrained optimization.",
"subjects": "Optimization and Control (math.OC)",
"title": "Analysis of Sequential Quadratic Programming through the Lens of Riemannian Optimization",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846697584034,
"lm_q2_score": 0.7248702880639792,
"lm_q1q2_score": 0.7092019774051551
} |
https://arxiv.org/abs/math/0605771 | Generalized Jacobian for Functions with Infinite Dimensional Range and Domain | In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon--Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the $\beta$-compactness, the $\beta$-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set. | \section{\bf Introduction}
The subject of nonsmooth analysis focuses on the study of a
derivative-like object for nonsmooth functions. When the function is a
convex real-valued, the notion of subgradient was introduced in the
late fifties by Rockafellar in \cite{Roc70}, and in
the references therein. Since then, the focus shifted to finding
derivative-like objects for nonconvex, in particular, locally Lipschitz
functions acting between two normed spaces $X$ and $Y$.
When $X$ and $Y$ are {\it both} finite dimensional normed spaces and
$f:\mathscr{D}\to Y$ is a {\it vector-valued} locally Lipschitz function,
Clarke introduced in \cite{Cla76c}, \cite{Cla83} the notion of the
{\it generalized Jacobian} based on Rademacher's theorem which gives
the almost everywhere differentiability of locally Lipschitz
functions. This generalized Jacobian is
\Eq{CJ}{
\partial f(p)
:=\mathop{\rm co}\nolimits\big\{A\in\mathscr{L}(X,Y)\mid\exists\,(x_i)_{i\in\mathbb{N}}\mbox{ in }\Omega(f)
: \lim_{i\to\infty} x_i= p\, \mbox{ and }
\lim_{i\to\infty} Df(x_i)= A\big\},
}
where $\Omega(f)$ denotes the set of the points of $\mathscr{D}$ where $f$ is
differentiable, it is of full Lebesgue measure.
Another but related generalized Jacobian based also on the use of
Rademacher's theorem was proposed for the same setting by Pourciau in
\cite{Pou77} and was defined as
\Eq{PJ}{
\partial^P f(p):=\bigcap_{\delta>0} \mathop{\overline{\mbox{\rm co}}}\nolimits
\big\{D f(x):x\in (p+\delta B_X)\cap\Omega(f)\big\}.
}
One can show that Clarke's and Pourciau's generalized Jacobians are
equivalent. These two objects are nonempty,
due to Rademacher's
theorem. Furthermore, in terms of these Jacobians, results have been
derived pertaining optimality conditions, implicit functions theorems,
metric regularity, and calculus rules including the sum rule and the
chain rule. Thereby, it has already been shown that these generalized
Jacobians are successful approximations of $f$ by linear operators.
When establishing calculus rules such as the sum and/or the chain
rules, a fundamental property, namely, the ``blindness'' of the
generalized Jacobian with respect to sets of Lebesgue measure zero, or
null sets is needed. The blindness of Clarke generalized gradient was
established by Clarke in \cite{Cla83} and that of Clarke's generalized
Jacobian was shown by Warga \cite{War81a} and by Fabian \& Preiss
\cite{FP87a}.
Thibault in \cite{Thi82a} extended Clarke's notion of generalized
Jacobian, equation \eq{CJ}, to the case where $X$ and $Y$ are
infinite dimensional separable Banach spaces such that $Y$ is
{\it reflexive}. This extension was based on the dense differentiability
of locally Lipschitz functions (cf.\ Aronszajn \cite{Aro76},
Christensen \cite{Chr73}, and Phelps \cite{Phe78}). Thibault's
definition is
\Eq{TJ}{
\partial_{H} f(p)
:=\mathop{\overline{\mbox{\rm co}}}\nolimits\big\{A\in\mathscr{L}(X,Y)\mid\exists\,(x_i)_{i\in\mathbb{N}}\mbox{ in }H
: \lim_{i\to\infty} x_i= p\, \mbox{ and }
\lim_{i\to\infty} Df(x_i)= A\big\},
}
where $H$ is a subset of $\mathscr{D}$ on which $f$ is G\^ateaux-differentiable
and such that $\mathscr{D}\setminus H$ is a Haar-null set in $\mathscr{D}$. The notion in
\eq{TJ} depends on the choice of the set $H$, and hence, unlike Clarke's
generalized Jacobian, is not known to be blind with respect to the
Haar-null sets. In other words, the notion in \eq{TJ}, assigns to
every locally Lipschitz function, not a single object but rather a
family of generalized Jacobians that is parametrized by certain null
sets. Thus, based on this approach all the chain rules derived in
\cite{Thi82a} are in terms of the Haar null set $H$.
Other notions are known in the infinite dimensional setting, such as
the notion of derivate containers in \cite{War76b}, \cite{War81a};
the concepts of screens and ``fans'' in \cite{Hal76a}, \cite{Hal76b};
the concept of shields \cite{Swe77}; Ioffe's fan derivative
\cite{Iof81c}, and the notion of coderivatives developed in
\cite{MS96c}. Most of these notions are not given in terms of relevant
sets of {\it linear operators}. A relatively recent survey on the
different subdifferentials and their properties is given in
\cite{BZ99a} where also an extended list of references could be found.
In recent papers \cite{PZ*a}and \cite{PZ*b} Clarke's generalized
Jacobian \eq{CJ} was extended to the case when $X$ was any {\it normed}
space and $Y$ was a finite dimensional space. In these references the
generalized Jacobian was defined to be a {\it set of linear operators}
from $X$ to $Y$. When the domain is infinite dimensional and the image
space is $\mathbb{R}$, the notion introduced in \cite{PZ*a} and \cite{PZ*b}
coincides with Clarke's generalized gradient which is defined as
\Eq{Cg}{
\partial^c f(p)
:= \big\{\zeta\in X^*\mid\langle\zeta, h\rangle\leq f^{\circ}(p, h),
\,\, \forall h\in X\big\},
}
where
\Eq{Cdd}{
f^{\circ}(p,h):=\limsup_{\over{x&\to p\\[-2mm]t&\to 0^+}}
\frac{f(x+th)-f(x)}{t}
}
is Clarke's {\it generalized directional derivative}.
In \cite{PZ*a}and \cite{PZ*b}, the nonemptiness, the $w^*$-compactness, the
convexity, and the upper semicontinuity property of the extended generalized
Jacobian were derived. Furthermore, a chain rule for
the composition of nonsmooth locally Lipschitz maps with finite
dimensional ranges was established.
The difficulty caused by the infinite dimensionality of the domain was
handled in \cite{PZ*a}and \cite{PZ*b} by introducing an intermediate
Jacobian $\partial_L f$ defined on finite dimensional spaces $L$ so
that Rademacher theorem remains applicable.
In this paper we are interested in extending the definition of Clarke's
generalized
Jacobian to the case when in addition to the domain also the range is
infinite dimensional. In this case, two extra difficulties manifest.
The first is the differentiability issue related to the Rademacher
theorem in infinite dimension. This issue will be handled by taking
image spaces satisfying the Radon--Nikod\'ym property. This implies that
the restriction of a Lipschitz function $f:\mathscr{D}\to Y$ to a
{\it finite dimensional} domain is almost everywhere differentiable
(cf.\ \cite{BL00}[Prop. 6.41]).
The second difficulty is pertaining finding a topology in the space of
linear operators $\mathscr{L}(X,Y)$, where the generalized Jacobian lives, so that
bounded sequences would have cluster points in this topology. To
overcome this difficulty, we also assume that the image space $Y$ is a
dual of a normed space.
The goal of this paper is to provide a generalized Jacobian for
locally Lipschitz functions defined between infinite dimensional
normed spaces with the range $Y$ is a dual space and satisfies the
Radon--Nikod\'ym property. We shall show that our generalized Jacobian
enjoys all the fundamental properties desired from a derivative set.
In Section 2 we introduce the $\beta$-topology on the space of linear
operators $\mathscr{L}(X,Y)$. This is a $w^*$-operator topology induced by the
predual of $Y$. We prove an analog of the Banach--Alaoglu theorem
as well as an extension theorem which will be
crucial for the proof of the nonemptiness of the generalized Jacobian.
In this section we also derive results related to various upper
semicontinuity properties which will be repeatedly used in the
subsequent section. In Section 3 the $L$-Jacobian, $\partial _Lf(p)$,
and the generalized Jacobian are defined as an extension of Pourciau's
notion, equation \eq{PJ}, to infinite dimensional spaces. We also show
that our generalized Jacobian could be equivalently defined in terms
of cluster points, which is a definition that corresponds to Clarke's
approach. Basic properties and a characterization of the generalized
Jacobian are established. A main tool named the ``restriction and the
vectorization'' theorem is developed which is central for
deriving many results in the rest of the paper.
Relationships to Thibault's limit set, to a Ioffe type fan derivative and
to Mordukhovich coderivative are given in Section 4. A generalization of
Lebourg mean-value theorem is obtained as well as that the
generalized Jacobian is a $w^*$-Hadamard prederivative. We also
characterize the cases when the generalized Jacobian is a strict
norm-G\^ateaux or a strict $w^*$-Fr\'echet prederivative. In Section 5 we
derive two chain rules: a nonsmooth-smooth, and a nonsmooth-nonsmooth
one. Their proofs evoke most of the results and properties
established in the previous sections. As a consequence, a sum rule
follows. Finally, in Section 6 we develop results for the generalized
Jacobian of continuous selections.
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| {
"timestamp": "2006-05-31T01:58:05",
"yymm": "0605",
"arxiv_id": "math/0605771",
"language": "en",
"url": "https://arxiv.org/abs/math/0605771",
"abstract": "In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon--Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the $\\beta$-compactness, the $\\beta$-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.",
"subjects": "Functional Analysis (math.FA)",
"title": "Generalized Jacobian for Functions with Infinite Dimensional Range and Domain",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846684978779,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7092019764914375
} |
https://arxiv.org/abs/2009.07305 | The general position number of the Cartesian product of two trees | The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees. | \section{Introduction}
\label{sec:intro}
Let $d_G(x,y)$ denote, as usual, the number of edges on a shortest $x,y$-path in $G$. A set $S$ of vertices of a connected graph $G$ is a {\em general position set} if $d_G(x,y) \ne d_G(x,z) + d_G(z,y)$ holds for every $\{x,y,z\}\in \binom{S}{3}$. The {\em general position number} $\gp(G)$ of $G$ is the cardinality of a largest general position set in $G$. Such a set is briefly called a {\em gp-set} of $G$.
Before the general position number was introduced in~\cite{PM}, an equivalent concept was proposed in~\cite{ullas-2016}. Much earlier, however, the general position problem has been studied by K\"orner~\cite{korner-1995} in the special case of hypercubes. Following~\cite{PM}, the graph theory general position problem has been investigated in~\cite{BSA, MGS, Sk, SKB, PMS, patkos-2019+, JT}.
The {\em Cartesian product} $G\,\square\, H$ of vertex-disjoint graphs $G$ and $H$ is the graph with vertex set $V(G) \times V(H)$, vertices $(g,h)$ and $(g',h')$ being adjacent if either $g=g'$ and $hh'\in E(H)$, or $h=h'$ and $gg'\in E(G)$. In this paper we are interested in $\gp(G\,\square\, H)$, a problem earlier studied in~\cite{MGS, SKB, PMS, JT}. More precisely, we are interested in Cartesian products of two (finite) trees. (For some of the other investigations of the Cartesian product of trees see~\cite{balak-2016, shiu-2018, wood-2011}.) An important reason for this interest is the fact that the general position number of products of paths is far from being trivial. First, denoting with $P_\infty$ the two-way infinite path, one of the main results from~\cite{PMS} asserts that $\gp(P_\infty \,\square\, P_\infty) = 4$. Denoting further with $G^n$ the $n$-fold Cartesian product of $G$, it was demonstrated in the same paper that $10\le \gp(P_\infty^3) \le 16$. The lower bound $10$ was improved to $14$ in~\cite{SKB}. Very recently, these results were superseded in~\cite{KR} by proving that if $n$ is an arbitrary positive integer, then $\gp(P_\infty^n) = 2^{2^{n-1}}$. Denoting with $n(G)$ the order of a graph $G$, in this paper we prove:
\begin{theorem}
\label{thm:main}
If $T$ and $T^{*}$ are trees with $\min\{n(T),n(T^*)\}\geq 3$, then
$$\gp(T\,\square\, T^{*}) = \gp(T) + \gp(T^{*})\,.$$
\end{theorem}
\noindent
Theorem~\ref{thm:main} widely extends the above mentioned result $\gp(P_\infty \,\square\, P_\infty) = 4$. Further, the equality $\gp(P_\infty^n) = 2^{2^{n-1}}$ shows that Theorem~\ref{thm:main} has no obvious (inductive) extension to Cartesian products of more than two trees. Hence, to determine the general position number of such products remains a challenging problem.
In the next section we give further definitions, recall known results needed, and prove several auxiliary new results. Then, in Section~\ref{sec:proof}, we prove Theorem~\ref{thm:main}.
\section{Preliminaries}
\label{sec:prelim}
Let $T$ be a tree. The set of leaves of $T$ will be denoted by $L(T)$, and let $\ell(T) = |L(T)|$. If $u$ and $v$ are vertices of $T$ with $\deg(u) \ge 2$ and $\deg(v) = 1$, then the unique $u,v$-path is a \textit{branching path} of $T$. If $u$ is not a leaf of $T$, then there are exactly $\ell(T)$ branching paths starting from $u$; we say that the $u$ is the \textit{root} of these branching paths and that the degree $1$ vertex of a branching path $P$ is the \textit{leaf of} $P$.
\begin{lemma}\label{tree}{\rm(\cite{PM})}
If $T$ is a tree, then $\gp(T)=\ell(T)$.
\end{lemma}
We next describe which vertices of a tree lie in some gp-set of the tree.
\begin{lemma}\label{T-non-leaf}
A non-leaf vertex $u$ in a tree $T$ belongs to a gp-set of $T$ if and only if $T-u$ has exactly two components and at least one of them is a path.
\end{lemma}
\begin{proof}
First, let $R$ be a gp-set of $T$ containing the non-leaf vertex $u$. Suppose that $T-u$ has at least three components, say $T_1,T_2$ and $T_3$. Since $R$ is a gp-set containing $u$, $R$ intersects with at most one of $T_1$, $T_2$ and $T_3$. Assume without loss of generality that $R\cap V(T_2)=\emptyset$ and $R\cap V(T_3)=\emptyset$. Choose vertices $v$ and $w$ in $T$ such that $v\in V(T_2)$ and $w\in V(T_3)$. Then $(R-\{u\})\cup\{v,w\}$ is a larger gp-set than $R$ in $T$, a contradiction. Hence $T-u$ has exactly two components, say $T_1$ and $T_2$. Now suppose that neither $T_1$ nor $T_2$ is a path. Then as above, we have $R\cap V(T_1)=\emptyset$ or $R\cap V(T_2)=\emptyset$. By symmetry, we assume that $R\cap V(T_2)=\emptyset$. Since $T_2$ is not a path, there are at least two leaves $x_1$ and $x_2$ in $T_2$.
Then the set $(R-\{u\})\cup\{x_1,x_2\}$ is a larger gp-set than $R$, again, in $T$. Therefore, at least one of $T_1$ and $T_2$ is a path.
Conversely, we observe that $u$ is a non-leaf vertex on a pendant path in $T$. Then $u$ belongs to a gp-set in $T$.
\end{proof}
In $G\,\square\, H$, if $h\in V(H)$, then the subgraph of $G\,\square\, H$ induced by the vertices $(g,h)$, $g\in V(G)$, is a {\em $G$-layer}, denoted with $G^h$. Analogously $H$-layers $\prescript{g}{}H$ are defined. $G$-layers and $H$-layers are isomorphic to $G$ and to $H$, respectively. The distance function in Cartesian products is additive, that is, if $(g_{1},h_{1}), (g_{2},h_{2})\in V(G\,\square\, H)$, then
\begin{equation}
\label{eq1}
d_{G\,\square\, H}((g_{1},h_{1}), (g_{2},h_{2})) = d_{G}(g_{1},g_{2})+d_{H}(h_{1},h_{2}).
\end{equation}
If $u,v\in V(G)$, then the {\em interval} $I_G(u,v)$ between $u$ and $v$ in $G$ is the set of all vertices lying on shortest $u,v$-paths, that is,
$$I_G(u,v) = \{w:\ d_G(u,v) = d_G(u,w) + d_G(w,u)\}\,.$$
In what follows, the notations $d_{G}(u,v)$ and $I_G(u,v)$ may be simplified to $d(u,v)$ and $I(u,v)$ if $G$ will be clear from the context.
Equality~\eqref{eq1} implies that intervals in Cartesian products have the following nice structure, cf.~\cite[Proposition 12.4]{WI}.
\begin{lemma}\label{geo-interval}
If $G$ and $H$ are connected graphs and $(g_{1},h_{1}), (g_{2},h_{2})\in V(G\,\square\, H)$, then
$$I_{G\,\square\, H}((g_{1},h_{1}), (g_{2},h_{2})) = I_{G}(g_{1},g_{2}) \times I_{H}(h_{1},h_{2})\,.$$
\end{lemma}
Equality (\ref{eq1}) also easily implies the following fact (also proved in \cite{JT}).
\begin{lemma}
\label{lem:lay-v}
Let $G$ and $H$ be connected graphs and $R$ a general position set of $G\,\square\, H$. If $u=(g,h)\in R$, then $V(\prescript{g}{}{H})\cap R=\{u\}$ or $V(G^{h})\cap R=\{u\}$.
\end{lemma}
For finite paths the already mentioned result $\gp(P_\infty \,\square\, P_\infty) = 4$ reduces to:
\begin{lemma}
\label{p-path}{\rm(\cite{PMS})}
If $n_1, n_2\ge 2$, then
$$\gp(P_{n_1}\,\square\, P_{n_2})= \left\{
\begin{array}{ll}
4; & \min\{n_1,n_2\}\geq 3, \\
\\
3;& {\rm otherwise}\,.
\end{array}
\right.$$
\end{lemma}
To conclude the preliminaries we construct special maximal (with respect to inclusion) general position sets in products of trees.
\begin{lemma}\label{lower-gp} Let $T$ and $T^*$ be two trees with $\min\{n(T),n(T^*)\}\geq 3$, $v_{i}\in V(T)\setminus L(T)$, and $v_{j}^{*}\in V(T^*)\setminus L(T^*)$. Then $(L(T)\times\{v_{j}^{*}\})\cup(\{v_{i}\}\times L(T^{*}))$ is a maximal general position set of $T\,\square\, T^{*}$.
\end{lemma}
\begin{proof}
Set $R=(L(T)\times\{v_{j}^{*}\})\cup(\{v_{i}\}\times L(T^{*}))$ and let $V_0=\{u,v,w\}\subseteq R$. We first consider the case when $V_0\subseteq L(T)\times\{v_{j}^{*}\}$ or $V_0\subseteq \{v_{i}\}\times L(T^{*})$. By symmetry, assume that $V_0\subseteq L(T)\times\{v_{j}^{*}\}$. Then each vertex of $V_0$ is corresponding to a leaf of $L(T)$ in the layer $T^{v_j^*}\cong T$. Therefore $u,v,w$ do not lie on a common geodesic in $T\,\square\, T^{*}$.
In the following, without loss of generality, we can assume that $u,w\in L(T)\times\{v_{j}^{*}\}$ with $u=(v_{k},v_{j}^{*})$, $w=(v_s,v_{j}^{*})$ and $v=(v_{i},v_{\ell}^{*})\in\{v_{i}\}\times L(T^{*})$. By Equality (\ref{eq1}), we have $d(u,v)=d_{T}(v_{k},v_{i})+d_{T^{*}}(v_{j}^{*},v_{\ell}^{*})$ and $d(u,w)=d_{T}(v_{k},v_s)$, $d(w,v)=d_{T}(v_s,v_{i})+d_{T^{*}}(v_{j}^{*},v_{\ell}^{*})$. Note that $v_k$, $v_s$ are two distinct vertices in $L(T)$ of $T$ and $v_i\in V(T)\setminus L(T)$. Then $d_T(v_k,v_i)<d_T(v_k,v_s)+d_T(v_s,v_i)$ whenever $v_i$ lies on the $v_k,v_s$-geodesic or outside $v_k,v_s$-geodesic of $T$. This implies that $d(u,v)<d(u,w)+d(w,v)$ in $T\,\square\, T^{*}$. Therefore $w$ does not lie on the $u,v$-geodesic in $T\,\square\, T^{*}$. Analogously, neither $u$ lies on the $v,w$-geodesic nor $v$ lies on the $u,w$-geodesic of $T\,\square\, T^{*}$. Thus $u,v,w$ do not lie on a common geodesic in $T\,\square\, T^{*}$, which implies that $R$ is a general position set in $T\,\square\, T^{*}$.
Next we prove the maximality of $(L(T)\times\{v_{j}^{*}\})\cup(\{v_{i}\}\times L(T^{*}))$ as a general position set in $T\,\square\, T^{*}$. Otherwise, there is a general position set $R^{\prime}$ in $T\,\square\, T^*$ of order greater than $\ell(T)+\ell(T^*)$ such that $R\subset R^{\prime}$. Then there exists a vertex $z\in R^{\prime}\backslash R$, say $z=(v_{p},v_{q}^{*})$. If $p=i$, then there exist two vertices $(v_i,v_s^*),(v_i,v_t^*)\in R$ such that $z\in I_{T\,\square\, T^{*}}((v_i,v_s^*),(v_i,v_t^*))$ (since $\prescript{v_i}{}{T^{*}}\cong T^{*}$). This is a contradiction showing that $p\neq i$. Similarly, we have $q\neq j$. Now we consider the positions of $v_p$ in $T$ and $v_q^{*}$ in $T^*$. Suppose first that $v_{p}\in L(T)$, $v_{q}^{*}\in L(T^{*})$. Then there are two vertices $(v_{p},v_{j}^{*}),(v_{i},v_{q}^{*})$ in $R$ such that $z\in I_{T\,\square\, T^{*}}((v_{p},v_{j}^{*}),(v_{i},v_{q}^{*}))$, contracting that $R\cup \{z\}$ is a general position set of $T\,\square\, T^{*}$. If $v_{p}\in L(T)$ and $v_{q}^{*}\notin L(T^{*})$, then we select a vertex $v_{q^{\prime}}^*\in L(T^*)$ such that $v_{q^{\prime}}^*$ is closer to the leaf of the corresponding branching path than $v_q^*$ in $T^*$. Then $z\in I_{T\,\square\, T^{*}}((v_{p},v_{j}^{*}),(v_{i},v_{q^{\prime}}^{*}))$, a contradiction. Similarly, $v_{p}\notin L(T)$ and $v_{q}^{*}\in L(T^{*})$ cannot occur. Finally we assume that $v_{p}\notin L(T)$, $v_{q}^{*}\notin L(T^{*})$. Now we select two vertices $v_{p^{\prime}}\in L(T)$ and $v_{q^{\prime}}^{*}\in L(T^{*})$ such that $v_{p^{\prime}}$ is closer to the leaf of the branching path than $v_p$ in $T$ and $v_{q^{\prime}}^{*}$ is closer to the leaf of the branching path than $v_q^*$ in $T^*$. But then $(v_{p},v_{q}^{*})\in I_{T\,\square\, T^{*}}((v_{p^{\prime}},v_{j}^{*}),(v_{i},v_{q^{\prime}}^{*}))$, a final contradiction.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:proof}
If $T$ and $T^*$ are both paths, then Theorem~\ref{thm:main} holds by Lemma~\ref{p-path}. In the following we may thus without loss of generality assume that $T^*$ is not a path.
Lemma~\ref{lower-gp} implies that $\gp(T\,\square\, T^{*}) \geq \gp(T) + \gp(T^{*})$, hence it remains to prove that $\gp(T\,\square\, T^{*})\leq \gp(T) + \gp(T^{*})$. Set $n = n(T)$, $n^{*} = n(T^*)$, $V(T)=\{v_{1},\ldots, v_{n}\}$, and $V(T^{*}) = \{v_{1}^{*}, \ldots,v_{n^{*}}^{*}\}$.
Assume on the contrary that there exists a general position set $R$ of $T$ such that $|R| > \gp(T) + \gp(T^{*})$. Since the restriction of $R$ to a $T$-layer of $T\,\square\, T^{*}$ is a general position set of the layer (which is in turn isomorphic to $T$), the restriction contains at most $\gp(T) = \ell(T)$ elements. Similarly, the restriction of $R$ to a $T^*$-layer contains at most $\gp(T^*) = \ell(T^*)$ elements. We now distinguish the following cases.
\medskip\noindent
\textbf{Case 1.} There exists a $T$-layer $T^{v_{j}^{*}}$ with $|V(T^{v_{j}^{*}}) \cap R| = \gp(T)$, or a $T^*$-layer $\prescript{v_{i}}{}T^{*}$ with $|V(\prescript{v_{i}}{}T^{*})\cap R| = \gp(T^*)$.
By the commutativity of the Cartesian product, we may without loss of generality assume that there is a layer $\prescript{v_{i}}{}{T^{*}}$ with $|R\cap V(\prescript{v_{i}}{}{T^{*}})| = \gp(T^*)$. Let $R = R_1\cup R_2$, where $R_1 = R\cap V(\prescript{v_{i}}{}{T^{*}})$ and $R_2 = R\setminus R_1$, that is, $R_2 = \bigcup\limits_{t\in [n]\setminus\{i\}}\Big(V(\prescript{v_{t}}{}T^{*})\cap R\Big)$. Let further $S^{*}$ be the projection of $R\cap V(\prescript{v_{i}}{}{T^{*}})$ on $T^{*}$, that is, $S^{*}=\{v_{j}^{*}:\ (v_{i},v_{j}^{*})\in R_1\}$. Since $|R_1| = \gp(T^*)$, our assumption implies $|R_2| \geq \gp(T)+1$. Then, as $\gp(T) = \ell(T)$, there exist two different vertices $w=(v_p,v_q^*)$ and $w^{\prime}=(v_{p^{\prime}},v_{q^{\prime}}^*)$ from $R_2$ such that $v_p$ and $v_{p^{\prime}}$ lie on a same branching path $P$ of $T$. (Note that it is possible that $v_p = v_{p^{\prime}}$.) We may assume that $d_{T}(v_{p^{\prime}}, x) \le d_{T}(v_p,x)$, where $x$ is the leaf of $P$. We proceed by distinguishing two subcases based on the position of $v_q^*$ and $v_{q^{\prime}}^*$ in $T^*$.
\medskip\noindent
\textbf{Case 1.1.} There exists a branching path $P^*$ of $T^*$ that contains both $v_q^*$ and $v_{q^{\prime}}^*$. \\
Recall that $T^*$ is not a path. Lemma~\ref{T-non-leaf} implies that a vertex of a tree belongs to a gp-set if and only if it lies on a pendant path and has degree $1$ or $2$. Therefore, we can select $P^*$ with the root of degree at least $3$. Assume that $d_{T^*}(v_{q^{\prime}}^*, y) \le d_{T^*}(v_q^*,y)$, where $y$ is the leaf of $P^*$. (The reverse case can be treated analogously.)
Since $S^*$ is a gp-set of $T^*$ which is not isomorphic to a path, there is a vertex $v_k^*\in S^*$ lying on $P^*$. So we may consider that $P^*$ is a branching path that contains $v_q^*$, $v_{q^{\prime}}^*$ and a vertex $v_{k}^*\in S^*$. (It is possible that some of these vertices are the same.) Let $z=(v_i,v_k^*)$. Then $z\in R_1$. We proceed by distinguishing the following subcases based on the position of $v_{p}$, $v_{p^{\prime}}$ and $v_{i}$ in $T$.
\medskip\noindent
\textbf{Subcase 1.1.1.} $v_{p^{\prime}}\in I(v_i,v_p)$. \\
In this subcase, if $v_{k}^*$ is closer than $v_q^*$, $v_{q^{\prime}}^*$ to the leaf $y$ of $P^*$, then, by Lemma \ref{geo-interval}, $w^{\prime} \in I_{T\,\square\, T^{*}}(w,z)$, a contradiction.
If $v_{k}^*\in I(v_{q}^*,v_{q^{\prime}}^{*})$, then since $\ell(T^*)\geq 3$, there exists $z^{\prime}=(v_{i},v_{k^{\prime}}^{*})\in \{v_{i}\}\times S^{*}$ such that $v_{k}^{*}$,$v_{q}^{*}\in I(v_{q^{\prime}}^*,v_{k^{\prime}}^{*})$ in $T^*$. Then we have \begin{eqnarray*}
d(w^{\prime},z^{\prime})&=&d_T(v_{p^{\prime}},v_i)+d_{T^*}(v_{q^{\prime}}^*,v_{k^{\prime}}^*)\\
&=&d_T(v_{p^{\prime}},v_i)+d_{T^*}(v_{q^{\prime}}^*,v_{k}^*)+d_{T^*}(v_{k}^*,v_{k^{\prime}}^*)\\
&=&d(w^{\prime},z)+d(z,z^{\prime}),
\end{eqnarray*}
which implies that $z\in I_{T\,\square\, T^{*}}(w^{\prime},z^{\prime})$, a contradiction.
\medskip\noindent
\textbf{Subcase 1.1.2.} $v_{i}\in I(v_{p},v_{p^{\prime}})$.\\
In this subcase, if $v_{k}^*\in I(v_q^*,v_{q^{\prime}}^*)$ in $P^*$, then $z\in I_{T\,\square\, T^{*}}(w,w^{\prime})$ by Lemma \ref{geo-interval}, a contradiction.
Assume that $v_{k}^*$ is closer than $v_q^*$, $v_{q^{\prime}}^*$ to the leaf of $P^*$. Since $|S^{*}|=\ell(T^*)\geq 3$, there is a vertex $z^{\prime}=(v_i,v_{k^{\prime}}^*)\in \{v_i\}\times S^{*}$ such that $v_q^*$, $v_{q^{\prime}}^*\in I(v_{k}^*,v_{k^{\prime}}^*)$ in $T^{*}$. Let $v_{k^{\prime}}^*$ be on a branching path ${P^{\prime}}^*$ in $T^{*}$ where ${P^{\prime}}^*\neq P^*$. Note that $\ell(T) + 1\geq3$. There exists at least one vertex $a=(v_{x},v_{y}^{*})\in R_2 \setminus\{w,w^{\prime}\}$. Next we consider the positions of $v_x,v_{y}^{*}$ in $T,T^*$, respectively.
Suppose first that $v_{y}^{*}\in V(P^*\cup {P^{\prime}}^*)$. If $v_x$, $v_p$, $v_{p^{\prime}}$ and $v_i$ lie on a path in $T$, then there are five vertices $w$, $w^{\prime}$, $z$, $z^{\prime}$ and $a$ in $R_2$, three of which lie on a common geodesic in $T\,\square\, T^*$, a contradiction. Note that if $T$ is a path, then we are done as above. Therefore, assume that $T$ is not isomorphic to a path in the following and the root of $P$ has degree at least $3$. Otherwise, $v_x\notin P$ and $v_x,v_p$ lie on a common branching path in $T$. Let $V_s$ be the set of vertices of $T$ but not contained in $T_{ip^{\prime}}$ where $T_{ip^{\prime}}$ is the subtree of $T-v_p$ containing $v_i$ and $v_{p^{\prime}}$.
If there is a vertex $a^{\prime}=(v_s,v_l^{*})\in R_2$ with $v_s\in V_s$, then $R_2$ contains $w$, $w^{\prime}$, $z$, $z^{\prime}$ and $a^{\prime}$, three of which are on a common geodesic, a contradiction. Therefore, the first coordinate of any vertex in $R_2$ cannot be in $V_s$. Assume that $P^{\prime}\neq P$ is any branching path containing $v_p$ and a leaf both in $T_{ip^{\prime}}$ and $T$. Then, besides $w$, $P^{\prime}\,\square\, T^*$ contains at most one vertex in $R_2$ of $T\,\square\, T^{*}$. Otherwise, $P^{\prime}\,\square\, T^*$ contain two vertices $h$, $h^{\prime}$ in $R_2$. Then there exist two vertices $h_0,h_{0}^{\prime} \in\{v_i\}\times S^*$ such that three vertices from $\{h,h^{\prime},h_0,h_{0}^{\prime},w\}$ lie on some geodesic in $T\,\square\, T^{*}$, a contradiction. (Here $h_0$ may be equal to $h_{0}^{\prime}$.) Note that $V_s$ contains at least two leaves of $T$ since the root of $P$ (just in $V_s$) has degree at least $3$. Then $T_{ip^{\prime}}$ has at most $\ell(T)-2$ leaves in $T$. Since $P\,\square\, T^*$ contains two vertices $w$ and $w^{\prime}$ in $R_2$, we have $|R_2|\leq \ell(T)-2+1< \ell(T)=\gp(T)$, a contradiction with the assumption.
Assume now that $v_{y}^{*}\notin V(P^*\cup {P^{\prime}}^*)$. Then there exists a vertex $z^{\prime\prime}=(v_i,v_{k^{\prime\prime}}^{*})\in\{v_i\}\times S^{*}$ such that $v_{y}^{*},v_{k^{\prime\prime}}^{*}$ lie on a common branching path in $T^*$. If $v_{y}^{*}$ is closer to the leaf of the branching path than $v_{k^{\prime\prime}}^{*}$ in $T^*$, then $v_i\in I(v_x,v_i)$ and $v_{k^{\prime\prime}}^{*}\in I(v_{y}^{*},v_{k}^{*})$. Therefore, by Lemma \ref{geo-interval}, we get $z^{\prime\prime}\in I_{T\,\square\, T^{*}}(a,z)$, a contradiction.
In the case that $v_{k^{\prime\prime}}^{*}$ is closer to the leaf of the branching path than $v_{y}^{*}$ in $T^*$, we consider the positions of $v_x$, $v_p$, $v_{p^{\prime}}$ and $v_i$ in $T$. Let $V_{1}=\{z,z^{\prime},w,w^{\prime},a,z^{\prime\prime}\}$. Then $V_1\subseteq R_2$. If $v_x$, $v_p$, $v_{p^{\prime}}$ and $v_i$ lie on a path in $T$, then there exist three vertices in $V_1$ lying on a common geodesic in $T\,\square\, T^*$, a contradiction again. Otherwise, $v_x\notin P$ and $v_x,v_p$ lie on a common branching path in $T$. Similarly as above, a contradiction occurs.
\medskip\noindent
\textbf{Subcase 1.1.3.} $v_{p}\in I(v_{i},v_{p^{\prime}})$.\\
In this subcase, since $\ell(T^*)\geq 3$, there exists a vertex $z^{\prime}=(v_{i},v_{k^{\prime}}^{*})\in \{v_{i}\}\times S^{*}$ such that $v_{k^{\prime}}^{*}\notin P^*$ and $v_{q}^{*}\in I(v_{k^{\prime}}^{*},v_{q^{\prime}}^{*})$ in $T^{*}$. Since \begin{eqnarray*}
d(z^{\prime},w^{\prime})&=&d_T(v_i,v_{p^{\prime}})+d_{T^*}(v_{k^{\prime}}^*,v_{q^{\prime}}^*)\\
&=&d_T(v_i,v_{p})+d_{T^*}(v_{k^{\prime}}^*,v_{q}^{*})+d_T(v_p,v_{p^{\prime}})+d_{T^*}(v_{q}^*,v_{q^{\prime}}^*)\\
&=&d(z^{\prime},w)+d(w,w^{\prime}),
\end{eqnarray*}
we have $w\in I_{T\,\square\, T^{*}}(z^{\prime},w^{\prime})$, a contradiction.
\medskip\noindent
\textbf{Subcase 1.1.4.} $v_{i}\notin V(P)$ such that $v_{i}$, $v_{p}$ lie on a same branching path in $T$.\\
In this subcase, since $\ell(T^*)\geq 3$, there is a vertex $z^{\prime}=(v_i,v_{k^{\prime}}^*)\in \{v_i\}\times S^{*}$ such that $v_{q}^{*}\in I(v_{k^{\prime}}^{*},v_{k}^{*})$ in $T^{*}$.
If $v_{k}^{*}\in I(v_{q}^{*},v_{q^{\prime}}^*)$ , then obviously $v_{k}^{*}\in I(v_{q}^{*},v_{k^{\prime}}^*)$ and therefore,
\begin{eqnarray*}
d(w^{\prime},z^{\prime})&=&d_T(v_{p^{\prime}},v_i)+d_{T^*}(v_{q^{\prime}}^*,v_{k^{\prime}}^*)\\
&=&d_T(v_{p^{\prime}},v_i)+d_{T^*}(v_{q^{\prime}}^*,v_{k}^*)+d_{T^*}(v_{k}^*,v_{k^{\prime}}^*)\\
&=&d(w^{\prime},z)+d(z,z^{\prime})\,.
\end{eqnarray*}
We conclude that $z\in I_{T \,\square\, T^{*}}(w^{\prime},z^{\prime})$, a contradiction.
If $v_{k}^{*}$ is closer to the leaf of $P^*$ than $v_{q}^{*},v_{q^{\prime}}^*$, then we get a contradiction similarly as in Subcase 1.1.2.
\medskip\noindent
\textbf{Case 1.2.} $v_q^*$ and $v_{q^{\prime}}^*$ do not lie on a same branching path in $T^*$.\\
In this subcase, we may assume that $v_q^*$ and $v_{q^{\prime}}^*$ lie on distinct branching paths $P^{*}$ and $P^{\prime*}$ in $T^{*}$, respectively. Since $\ell(T^*)\geq 3$ and $T^*$ is not isomorphic to a path, there exist two vertices $z=(v_i,v_{k}^*)$ and $z^{\prime}=(v_{i},v_{k^{\prime}}^{*})$ from $ \{v_i\}\times S^{*}$, such that $v_{k}^*\in P^*$ and $v_{k^{\prime}}^*\in P^{\prime*}$. We consider the following subcases based on the positions of $v_{p}$, $v_{p^{\prime}}$ and $v_{i}$ in $T$.
\medskip\noindent
\textbf{Subcase 1.2.1.} $v_{p^{\prime}}\in I(v_i,v_p)$.\\
In this subcase, if $v_{k^{\prime}}^*$ is closer than $v_{q^{\prime}}^*$ to the leaf of $P^{\prime*}$, then $v_{p^{\prime}}\in I(v_p,v_i)$ and $v_{q^{\prime}}^*\in I(v_q^*,v_{k^{\prime}}^*)$. Lemma \ref{geo-interval} gives $w^{\prime}\in I_{T\,\square\, T^{*}}(w,z^{\prime})$, a contradiction. On the other hand, if $v_{q^{\prime}}^*$ is closer than $v_{k^{\prime}}^*$ to the leaf of $P^{\prime*}$, then $v_i\in I(v_i,v_{p^\prime})$ and $v_{k^\prime}^*\in I(v_k^*,v_{q^\prime}^*)$, hence Lemma~\ref{geo-interval} gives $z^{\prime}\in I_{T\,\square\, T^{*}}(w^{\prime},z)$, a contradiction again.
\medskip\noindent
\textbf{Subcase 1.2.2.} $v_{i}\in I(v_{p},v_{p^{\prime}})$.\\
In this subcase, we first assume that $v_{q^{\prime}}^*$ is closer than $v_{k^\prime}^*$ to the leaf of $P^{\prime*}$. Then $v_i\in I(v_i,v_{p^{\prime}})$ and $v_{k^\prime}^*\in I(v_k^*,v_{q^\prime}^*)$. Therefore, by Lemma \ref{geo-interval}, we get $z^{\prime}\in I_{T\,\square\, T^{*}}(z,w^{\prime})$ as a contradiction.
Otherwise we suppose that $v_{k^\prime}^*$ is closer than $v_{q^{\prime}}^*$ to the leaf of $P^{\prime*}$. If $v_{q}^{*}$ is closer than $v_{k}^{*}$ to the leaf of $P^*$, then $v_i\in I(v_p,v_{i})$ and $v_{k}^*\in I(v_{q}^*,v_{k^\prime}^*)$. Therefore, by Lemma \ref{geo-interval}, we get $z\in I_{T\,\square\, T^{*}}(w,z^{\prime})$, a contradiction. In the case that $v_{k}^*$ is closer than $v_{q}^*$ to the leaf of $P^*$, we find a contradiction similarly as the proof of Subcase 1.1.2.
\medskip\noindent
\textbf{Subcase 1.2.3.} $v_{p}\in I(v_{i},v_{p^{\prime}})$.\\
In this subcase, if $v_{k}^*$ is closer than $v_{q}^*$ to the leaf of $P^{*}$, then $v_{p}\in I(v_{i},v_{p^{\prime}})$ and $v_{q}^*\in I(v_{k}^*,v_{q^{\prime}}^*)$. So Lemma~\ref{geo-interval} gives $w\in I_{T\,\square\, T^{*}}(z,w^{\prime})$, a contradiction. And if $v_{q}^*$ is closer than $v_{k}^*$ to the leaf of $P^{*}$, then $v_{i}\in I(v_{i},v_{p})$ and $v_{k}^*\in I(v_{k^{\prime}}^*,v_{q}^*)$, hence we get $z\in I_{T\,\square\, T^{*}}(z^{\prime},w)$.
\medskip\noindent
\textbf{Subcase 1.2.4.} $v_{i}\notin V(P)$ such that $v_{i}$, $v_{p}$ lie on a same branching path in $T$.\\
First suppose that $v_{q}^{*}$ is closer to the leaf than $v_{k}^{*}$ in $P^{*}$, then $v_{i}\in I(v_{i},v_{p})$ and $v_{k}^*\in I(v_{q}^*,v_{k^{\prime}}^*)$. Thus, by Lemma \ref{geo-interval}, we get $z\in I_{T\,\square\, T^{*}}(w,z^{\prime})$.
Assume that $v_{k}^{*}$ is closer than $v_{q}^{*}$ to the leaf of $P^{*}$. If $v_{q^{\prime}}^*$ is closer to the leaf than $v_{k^\prime}^*$, then $v_i\in I(v_i,v_{p^{\prime}})$ and $v_{k^\prime}^*\in I(v_k^*,v_{q^\prime}^*)$, which gives $z^{\prime}\in I_{T\,\square\, T^{*}}(z,w^{\prime})$. If $v_{{k}^{\prime}}^{*}$ is closer than $v_{{q}^{\prime}}^{*}$ to the leaf of ${P^{\prime}}^*$, we can proceed similarly as in Subcase 1.1.4.
Now we turn to the second case.
\medskip\noindent
\textbf{Case 2.} $|R\cap V(\prescript{v_{k}}{}{T^{*}})| < \ell(T^*)$ for any $k\in [n]$, and $|R\cap V(T^{v_t^*})| < \ell(T)$ for any $t\in [n^*]$.\\
In this case, let $\prescript{v_{i}}{}{T^{*}}$ be a layer with $|R\cap V(\prescript{v_{i}}{}{T^{*}})| = \max\{|R\cap V(\prescript{v_{k}}{}{T^{*}})|:k\in[n]\}$. Let $R = R_1\cup R_2$ where $R_1 = R\cap V(\prescript{v_{i}}{}{T^{*}})$ and $R_2 = R\setminus R_1$, that is, $R_2 = \bigcup\limits_{k\in [n]\setminus\{i\}}\Big(V(\prescript{v_{k}}{}T^{*})\cap R\Big)$. Set further $S^{*}=\{v_{j}^{*}:\ (v_{i},v_{j}^{*})\in R_1\}$. Then $1\leq|S^{*}|\leq \ell(T^*)-1$.
Assume first $|S^*|=1$. Therefore $|R\cap V(\prescript{v_{k}}{}{T^{*}})|\leq1$ for any $k\in[n]$. Next we only need to consider $|R\cap V(T^{v_j^*})|\leq1$ for any $j\in[n^*]$. (If $|R\cap V(T^{v_j^*})|\geq2$ for some $j\in[n^*]$, by commutativity of $T\,\square\, T^*$, the proof is similar to the subcase in which $2\leq|S^{*}|\leq\ell(T^*)-1$.) Therefore, suppose that $|R\cap V(T^{v_j^*})|\leq1$ for any $j\in[n^*]$. Then $|R|\leq\min\{n,n^{*}\}$. We now claim that $|R|\leq \ell(T)+\ell(T^*)$. If not, then since $|R|\geq \ell(T)+\ell(T^*)+1\geq 6$, there exist three vertices $u=(v_{p},v_{j}^{*})$, $v=(v_{p^{\prime}},v_{q}^{*})$ and $w=(v_{s},v_{\ell}^{*})$ from $R$ such that $v_p,v_{p^{\prime}}$ lie on a same branching path in $T$, and $v_{j}^{*},v_{\ell}^{*}$ lie on a common branching path in $T^*$. Note that there may be $p^{\prime} = s, q=\ell$. But we can always select a vertex $h\in R \setminus \{u,v,w\}$ such that $u,v,h$ or $u,w,h$ lie on a same geodesic in $T\,\square\, T^{*}$, which is a contradiction. So our result holds when $|S^*|=1$.
Suppose second that $2\leq|S^{*}|\leq\ell(T^*)-1$. As $|R_1| = |S^*|$, we need to prove that $|R_2|\leq \ell(T)+\ell(T^*)-|S^*|$. Assume on the contrary that $|R_2|\geq \ell(T)+\ell(T^*)-|S^*| + 1$.
Since $|S^*|\geq 2$, there are two distinct vertices $w=(v_{i},v_{j}^{*})$ and $w^{\prime}=(v_{i},v_{j^{\prime}}^{*})$ from $\{v_{i}\}\times S^{*}$. We distinguish the following cases based on the positions of $v_j^*$, $v_{j^{\prime}}^*$ in $T^*$.
\medskip\noindent
\textbf{Case 2.1.} $v_j^*$ and $v_{j^{\prime}}^*$ lie on a same branching path $P^*$ of $T^*$.\\
In this subcase, we may without loss of generality assume that $v_{j^{\prime}}^*$ is closer than $v_j^*$ to the leaf of $P^*$. Let $T^*_{v_{j^{\prime}}^*}$ be the maximal subtree of $T^*-v_j^*$ containing $v_{j^{\prime}}^*$ and let $V_{s^*} = V(T^*)\setminus V(T^*_{v_{j^{\prime}}^*})$. Let further $S_{1}^{*} = \{v_{q}^{*}:\ v_{q}^{*}\in I(v_j^*,v_{\ell}^*), v_\ell^*\in S^{*}\cap V(T^*_{v_{j^{\prime}}^*})\}$. Now we prove the following claim.
\medskip\noindent
\textbf{Claim 1.} If $z=(v_{p},v_{t}^{*})\in R_2$, then $v_{t}^{*}\in S_{1}^{*}$ .
\medskip\noindent
\textbf{Proof of Claim 1.}
If not, suppose first that $v_{t}^{*}\in V(P^*)$ is closer than $v_{j^{\prime}}^*$ to the leaf of $P^*$. Then $v_i\in I(v_i,v_p)$ and $v_{j^{\prime}}^*\in I(v_{t}^{*},v_{j}^*)$. Hence, $w^{\prime}\in I_{T\,\square\, T^{*}}(w,z)$. And if
$v_{t}^{*}\in V_{s^*}$, then $v_j^*\in I(v_{t}^{*},v_{j^{\prime}}^*)$. Combining this fact with $v_{i}\in I(v_{i},v_{p})$, we have $w\in I_{T\,\square\, T^{*}}({w^\prime},z)$. This proves Claim 1.
By Claim 1,
we have $|\bigcup\limits_{v_{t}^{*}\in S_{1}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)| \geq \ell(T)+\ell(T^{*})-|S^{*}| + 1 \geq \ell(T)+1$.
Then there exist two vertices $z=(v_{p},v_{\ell}^{*})$ and $z^{\prime}=(v_{p^{\prime}},v_{\ell^{\prime}}^{*})$ from $\cup_{v_{t}^{*}\in S_{1}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)$ such that $v_{\ell}^{*},v_{\ell^{\prime}}^{*}\in S_{1}^{*}$ and $v_{p},v_{p^{\prime}}$ lie on a same branching path $P$ in $T$. Without loss of generality, let $v_{p^{\prime}}$ be closer than $v_{p}$ to the leaf of $P$, and let $v_{\ell}^{*},v_{\ell^{\prime}}^{*}\in I(v_j^*,v_{j^{\prime}}^*)$ (by the definition of $S_1^*$). We consider the following subcases according to the positions of $v_{i},v_{p},v_{p^{\prime}}$ in $T$.
\medskip\noindent
\textbf{Subcase 2.1.1.} $v_{p^{\prime}}\in I(v_i,v_p)$.\\
If $v_{\ell^{\prime}}^{*}$ is closer than $v_{\ell}^{*}$ to $v_{j^{\prime}}^*$ in $P^{*}$, then we have $v_{p^{\prime}}\in I(v_{i},v_{p})$ and $v_{\ell^{\prime}}^*\in I(v_{\ell}^*,v_{{j^\prime}}^*)$. Therefore, $z^{\prime}\in I_{T\,\square\, T^{*}}(z,{w^\prime})$. And if $v_{\ell}^{*}$ is closer than $v_{\ell^{\prime}}^{*}$ to $v_{j^{\prime}}^*$ in $P^{*}$, then we have $v_{p^{\prime}}\in I(v_{i},v_{p})$ and $v_{\ell^{\prime}}^*\in I(v_{\ell}^*,v_{j}^*)$ and so $z^{\prime}\in I_{T\,\square\, T^{*}}(z,{w})$.
\medskip\noindent
\textbf{Subcase 2.1.2.} $v_{i}\in I(v_{p},v_{p^{\prime}})$.\\
Note that $\ell(T)+\ell(T^{*})-|S^{*}| + 1 \geq 4$. Then there exists at least a vertex $a=(v_{x},v_{y}^{*})\in \cup_{v_{t}^{*}\in S_{1}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)$ different from $z$ and $z^{\prime}$. Based on the position of $v_y^*$ ($v_{y}^{*}\in P^*$ or $v_{y}^{*}\notin P^*$) in $T^*$, and the positions of $v_x$, $v_{i}$, $v_{p}$ and $v_{p^{\prime}}$ in $T$, we get contradictions using a similar proof as in Subcase 1.1.2.
\medskip\noindent
\textbf{Subcase 2.1.3.} $v_{p}\in I(v_{i},v_{p^{\prime}})$.\\
If $v_{\ell^{\prime}}^{*}$ is closer than $v_{\ell}^{*}$ to $v_{j^{\prime}}^*$ in $T^{*}$, then $v_{p}\in I(v_{i},v_{p^{\prime}})$ and $v_{\ell}^*\in I(v_{j}^*,v_{\ell^\prime}^*)$, therefore $z\in I_{T\,\square\, T^{*}}(w,{z^\prime})$. And if $v_{\ell}^{*}$ is closer than $v_{\ell^{\prime}}^{*}$ to $v_{j^{\prime}}^*$ in $T^{*}$, then $v_{p}\in I(v_{i},v_{p^{\prime}})$ and $v_{\ell}^*\in I(v_{j^{\prime}}^*,v_{\ell^\prime}^*)$, hence $z\in I_{T\,\square\, T^{*}}(w,{z^\prime})$.
\medskip\noindent
\textbf{Subcase 2.1.4.} $v_{i}\notin V(P)$ such that $v_{i}$, $v_{p}$ lie on a same branching path in $T$.\\
Since $\ell(T)+\ell(T^{*})-|S^{*}| + 1 \geq 4$, there exists a vertex $(v_{x},v_{y}^{*})\in \cup_{v_{t}^{*}\in S_{1}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)$. Proceeding similarly as in Subcase 1.1.4, we get required contradictions. But then $|\cup_{v_{t}^{*}\in S_{1}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)| \leq \ell(T)+\ell(T^{*})-|S^{*}|$, a contradiction with the assumption.
\medskip\noindent
\textbf{Case 2.2.} $v_j^*$,$v_{j^{\prime}}^*$ lie on different branching paths $P^*$, $P^{\prime*}$ in $T^*$, respectively.\\
In this subcase, let $S_2^*$ be a set of vertices of $\prescript{v_{i}}{}T^{*}$ closer to the leaf of a branching path than $v_{g}^{*}$ for any $v_{g}^{*}\in S^*$. Note that $S^*\cap S_{2}^{*} = \emptyset$. We prove the following claim.
\medskip\noindent
\textbf{Claim 2.} If $(v_{p},v_t^{*})$ in $R_2$, then $v_{t}^{*}\in V(T^{*})\setminus(S^*\cup S_{2}^{*})$.
\medskip\noindent
\textbf{Proof of Claim 2.}
Lemma~\ref{lem:lay-v} implies $v_{t}^{*}\notin S^*$.
Assume that $v_{t}^{*}\in S_{2}^*$ lies on a same branching path for some $v_{g}^{*}$ in $T^{*}$. Note that $|S^*| \geq 2$. Then there exists another vertex $v_{g^{\prime}}^*$ such that $v_g^*\in I(v_t^*,v_{g^{\prime}}^*)$. Combining this fact with $v_{i}\in I(v_{i},v_{p})$, we arrive at a contradiction $w\in I_{T\,\square\, T^{*}}(z,{w^\prime})$. This proves Claim 2.
Let now $S_{1^\prime}^* = \{v_q^*:\ v_q^*\in I(v_g^*,v_{g^{\prime}}^*),v_g^*,v_{g^{\prime}}^*\in S^*\}$. By a parallel reasoning as in Subcase 2.1 and with Claim~2 in hands we infer that $|\cup_{v_{t}^{*}\in S_{1^{\prime}}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)|\leq \ell(T)$.
Let $S = \{v_{k}:\ (v_{k},v_{t}^{*})\in\cup_{v_{t}^{*}\in S_{1^{\prime}}^{*}}\big(V(T^{v_{t}^{*}})\cap R\big)\}$ and set $S^{**} = V(T^{*})\setminus (S^{*}\cup S_{1^\prime}^{*})$. From the assumption we have
$|\cup_{v_{t}^{*}\in S^{**}}\big(V(T^{v_{t}^{*}})\cap R\big)| \geq \ell(T)+\ell(T^*)-|S|-|S^*| +1$.
So there exists a vertex $z=(v_p,v_{\ell}^{*})\in\cup_{v_{t}^{*}\in S^{**}}\big(V(T^{v_{t}^{*}})\cap R\big)$, and we can always select two distinct vertices $u=(v_h,v_{g}^*)$ and $v=(v_{h^\prime},v_{g^\prime}^*)$ from $R$ such that $v_p$ and $v_h$ lie on a same branching path in $T$, while $v_{\ell}^{*}$ and $v_{g^\prime}^*$ lie on a common branching path in $T^*$. But we can choose another vertex $w\in R$ such that either $u,w,z$ or $u,v,z$ lie on a same geodesic in $T\,\square\, T^{*}$ as a contradiction. Therefore,
$$|\bigcup\limits_{v_{t}^{*}\in S^{**}}\Big(V(T^{v_{t}^{*}})\cap R\Big)| \leq \ell(T)+\ell(T^*)-|S|-|S^*|.$$
and we are done.
\section*{Acknowledgements}
Kexiang Xu is supported by NNSF of China (grant No.\ 11671202, and the China-Slovene bilateral grant 12-9). Sandi Klav\v{z}ar acknowledges the financial support from the Slovenian Research Agency (research core funding P1-0297, projects J1-9109, J1-1693, N1-0095, and the bilateral grant BI-CN-18-20-008).
| {
"timestamp": "2020-09-17T02:00:42",
"yymm": "2009",
"arxiv_id": "2009.07305",
"language": "en",
"url": "https://arxiv.org/abs/2009.07305",
"abstract": "The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.",
"subjects": "Combinatorics (math.CO)",
"title": "The general position number of the Cartesian product of two trees",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846666070896,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7092019751208611
} |
https://arxiv.org/abs/2108.05411 | Cohomology and deformations of weighted Rota-Baxter operators | Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in {\bf k}$, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by $\lambda$-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a $\lambda$-weighted relative Rota-Baxter operator $T$, and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal and finite order deformations of $T$ from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation which is the obstruction to extend the deformation. In the end, we also consider the cohomology of $\lambda$-weighted relative Rota-Baxter operators in the Lie case and find a connection with the case of associative algebras. | \section{Introduction}
Rota-Baxter operators (of weight $0$) was first appeared in the work of G. Baxter \cite{baxter} in his study of the fluctuation theory in probability. Such operators can be seen as an algebraic abstraction of the integral operator. Rota-Baxter operators on associative algebras were further studied by G.-C. Rota \cite{rota} and others \cite{atkinson,cartier,miller}. There is a close connection between Rota-Baxter operators and Yang-Baxter equation \cite{aguiar}. In last twenty years, Rota-Baxter operators have found important applications in renormalizations of quantum field theory \cite{connes}, pre-algebras \cite{bai-spl}, infinitesimal bialgebras \cite{aguiar} and double algebras \cite{gon}. See \cite{guo-book} for more on Rota-Baxter operators. The notion of relative Rota-Baxter operators (also called generalized Rota-Baxter operators or $\mathcal{O}$-operators) are a generalization of Rota-Baxter operators in the presence of a bimodule \cite{uchino}. They can be seen as a noncommutative analogue of Poisson structures. Rota-Baxter operators and relative Rota-Baxter operators also appeared in the context of Lie algebras \cite{kuper,guo-lax}. They are related to Rota-Baxter operators on Lie groups via global-infinitesimal correspondence \cite{sheng-adv,sheng-w1}.
\medskip
Weighted (relative) Rota-Baxter operators are generalization of (relative) Rota-Baxter operators. They are related to post-algebras \cite{guo-lax}, weighted infinitesimal bialgebras \cite{aybe}, weighted associative Yang-Baxter equations \cite{aybe}, combinatorics of planar rooted forests \cite{forest}, and play an important role in mathematical physics \cite{guo-lax}. Some classification result of weighted Rota-Baxter operators on matrix algebras are given in \cite{guba}. See \cite{brz,das-jmp} for some other generalizations of Rota-Baxter operators.
\medskip
On the other hand, the algebraic deformation theory began with the seminal work of M. Gerstenhaber \cite{gers} for associative algebras, followed by its extension to Lie algebras by A. Nijenhuis and R. Richardson \cite{nij-ric}. These deformations are governed by suitable cohomologies (e.g. Hochschild cohomology for associative algebras and Chevalley-Eilenberg cohomology for Lie algebras). See \cite{bala} for deformations of algebras over any binary quadratic operad. Deformations of (relative) Rota-Baxter operators on Lie algebras have been developed in \cite{tang} by introducing a new cohomology theory. It has been extended to associative algebras in \cite{das-rota}.
\medskip
Recently, the authors in \cite{sheng-w1} introduced a cohomology theory for relative Rota-Baxter operators of weight $1$ on Lie algebras and studied their linear deformations. Their cohomology is given by the Chevalley-Eilenberg cohomology of a suitable Lie algebra with coefficients in an appropriate representation. This description of cohomology has lacks of information to know the structure of the cohomology ring and to study formal and finite order deformations. Our aim in this paper is to define the cohomology of a relative Rota-Baxter operator of arbitrary weight (not necessary of weight $1$) using Maurer-Cartan element in a suitable differential graded Lie algebra. This will fulfil the gaps of \cite{sheng-w1}. However, we will mainly focus on associative algebras (the Lie case is also described at the end).
\medskip
Given a fixed scalar $\lambda \in {\bf k}$, we first construct a differential graded Lie algebra whose Maurer-Cartan elements are precisely $\lambda$-weighted relative Rota-Baxter operators on associative algebras. This characterization of $\lambda$-weighted relative Rota-Baxter operators allows us to define cohomology theory for such operators. We also interpret this cohomology as the Hochschild cohomology of a suitable associative algebra with coefficients in an appropriate bimodule. Next, we study various aspects of deformations (linear, formal and finite order deformations) of $\lambda$-weighted relative Rota-Baxter operators. We introduce Nijenhuis elements associated with a $\lambda$-weighted relative Rota-Baxter operator $T$ that generate trivial linear deformations of $T$. We find a sufficient condition for the rigidity of the operator $T$ in terms of Nijenhuis elements. For a finite order deformation of $T$, we also associate a second cohomology class, which is the obstruction to extend the given deformation to a next order deformation.
\medskip
We end this paper by considering $\lambda$-weighted relative Rota-Baxter operators on Lie algebras. We define the cohomology of such operators using Maurer-Cartan characterizations in a suitable differential graded Lie algebra. When $\lambda = 1$, our cohomology coincides with the one given in \cite{sheng-w1}. Finally, we relate it with the cohomology of $\lambda$-weighted relative Rota-Baxter operators on associative algebras by suitable skew-symmetrizations.
\medskip
\noindent {\bf Organization of the paper.} In Section \ref{sec-2}, we first recall weighted relative Rota-Baxter operators on associative algebras and prove some basic results about such operators. In Section \ref{sec-3}, we define cohomology of weighted relative Rota-Baxter operators using Maurer-Cartan characterizations. We also show that such cohomology can be interpreted as the Hochschild cohomology. Deformations of weighted relative Rota-Baxter operators are considered in Section \ref{sec-4}. Finally, in Section \ref{sec-5}, we focus on weighted relative Rota-Baxter operators on Lie algebras and define their cohomology. Relations with the case of associative algebras are also described.
\medskip
\noindent {\bf Notations.} Let $\big( \mathfrak{g} = \oplus \mathfrak{g}^n, [~,~]_\mathfrak{g}, d \big)$ be a differential graded Lie algebra. An element $\theta \in \mathfrak{g}^1$ is said to be a Maurer-Cartan element if $\theta$ satisfies
\begin{align*}
d \theta + \frac{1}{2} [\theta, \theta]_\mathfrak{g} = 0.
\end{align*}
We denote by $\mathbb{S}_n$ the set of all permutations on the set $\{1, \ldots, n \}$. A permutation $\sigma \in \mathbb{S}_n$ is called a $(p,q)$-shuffle (with $p+q= n$) if $\sigma (1) < \cdots < \sigma (p)$ and $\sigma (p+1) < \cdots < \sigma (p+q)$. The set of all $(p,q)$-shuffles are denoted by $\mathbb{S}_{(p,q)}$. All vector spaces, (multi)linear maps, tensor products, wedge products are over a field ${\bf k}$ of characteristic zero.
\section{Weighted relative Rota-Baxter operators}\label{sec-2}
In this section, we recall weighted relative Rota-Baxter operators \cite{aybe,forest,uchino} and study some basic properties.
Let $A$ and $B$ be two associative algebras. We denote the elements of $A$ by $a, b, a_1, a_2, \ldots $ and the elements of $B$ by $u, v, u_1, u_2, \ldots$. We also denote the multiplication in $A$ by $\cdot_A$ and the multiplication in $B$ by $\cdot_B$. Let the associative algebra $A$ acts on $B$. That is, $B$ is an $A$-bimodule \big(that consist of left and right $A$-actions $l: A \otimes B \rightarrow B,~(a, u) \mapsto l_a (u) = a \cdot u$ and $r: B \otimes A \rightarrow B,~ (u,a) \mapsto r_a (u) = u \cdot a$ with the followings $(a \cdot_A b) \cdot u = a \cdot ( b \cdot u)$, $(a \cdot u) \cdot b = a \cdot (u \cdot b)$ and $(u \cdot a) \cdot b = u \cdot ( a\cdot_A b)$ \big) satisfying additionally
\begin{align*}
(a \cdot u) \cdot_B v = a\cdot (u \cdot_B v), \qquad (u \cdot a) \cdot_B v = u \cdot_B (a \cdot v) ~~~ \text{ and } ~~~ (u \cdot_B v) \cdot a = u \cdot_B (v\cdot a),
\end{align*}
for $u, v \in B$ and $a, b \in A.$ In this case, we often say that $B$ is an associative $A$-bimodule. Note that any associative algebra $A$ is an associative $A$-bimodule with left and right $A$-actions are given by the algebra multiplication.
The following result is standard and we omit the details.
\begin{prop}
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Then for any $\lambda \in {\bf k}$, the direct sum $A \oplus B$ carries an associative structure given by
\begin{align*}
(a,u) \bullet_\lambda (b,v) = (a \cdot_A b ,~ a \cdot v + u \cdot b + \lambda~ u \cdot_B v), ~ \text{ for } a, b \in A \text{ and } u, v \in B.
\end{align*}
This is called the $\lambda$-weighted semidirect product and denoted by $A \ltimes_\lambda B$.
\end{prop}
\begin{defn}
\begin{itemize}
\item[(i)] Let $A$ be an associative algebra. A linear map $T: A \rightarrow A$ is said to be $\lambda$-weighted Rota-Baxter operator on $A$ if $T$ satisfies
\begin{align}
T(a) \cdot_A T (b) = T \big( T(a) \cdot_A b + a \cdot_A T(b) + \lambda~ a \cdot_A b \big), ~ \text{ for } a, b \in A.
\end{align}
\item[(ii)] Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. A linear map $T : B \rightarrow A $ is said to be a $\lambda$-weighted relative Rota-Baxter operator on $B$ over the algebra $A$ if
\begin{align}\label{rel-rb-id}
T(u) \cdot_A T (v) = T \big( T(u) \cdot v + u \cdot T(v) + \lambda~ u \cdot_A v \big), ~ \text{ for } u, v \in B.
\end{align}
We simply call $T$ as a $\lambda$-weighted relative Rota-Baxter operator when the domain and codomain of $T$ are clear from the context.
\end{itemize}
\end{defn}
\begin{remark}\label{rel-not}
It follows from the above definitions that a $\lambda$-weighted Rota-Baxter operator on $A$ is a particular case of $\lambda$-weighted relative Rota-Baxter operator.
\end{remark}
In the following, we characterize $\lambda$-weighted relative Rota-Baxter operators in terms of their graph.
\begin{prop}
A linear map $T : B \rightarrow A$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if the graph $Gr (T) = \{ (T(u), u) | ~u \in B \}$ is a subalgebra of the $\lambda$-weighted semidirect product $A \ltimes_\lambda B$.
\end{prop}
\begin{proof}
For any $u, v \in B$, we have
\begin{align*}
(T(u), u) \bullet_\lambda (T(v), v) = \big( T(u) \cdot_A T(v),~ T(u) \cdot v + u \cdot T(v) + \lambda~ u \cdot_B v \big).
\end{align*}
This is in $Gr (T)$ if and only if (\ref{rel-rb-id}) holds. Hence the result follows.
\end{proof}
As a consequence of the above proposition, we get the following.
\begin{prop}
Let $T : B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. Then $B$ carries a new associative algebra structure given by
\begin{align*}
u \cdot_T v := T(u) \cdot v + u \cdot T(v) + \lambda~ u \cdot_B v, ~ \text{ for } u, v \in B.
\end{align*}
\end{prop}
\medskip
Next, we consider $\lambda$-weighted modified associative Yang-Baxter equation (modified AYBE$_\lambda$) as an associative analogue of the modified Yang-Baxter equation considered in \cite{guo-lax}. We find its connection with $\lambda$-weighted Rota-Baxter operators.
\begin{defn}
Let $A$ be an associative algebra. For a linear map $R : A \rightarrow A$, the equation
\begin{align*}
R(a) \cdot_A R (b) = R \big( R(a) \cdot_A b + a \cdot_A R(b) \big) - \lambda^2 ~ a \cdot_A b, \text{ for } a, b \in A
\end{align*}
is called the $\lambda$-weighted modified associative Yang-Baxter equation (modified AYBE$_\lambda$).
\end{defn}
\begin{prop}\label{rb-myb}
Let $A$ be an associative algebra. Then there is a one-to-one correspondence between solutions of modified AYBE$_\lambda$ and $\lambda$-weighted Rota-Baxter operators on $A$.
\end{prop}
\begin{proof}
Let $T$ be a $\lambda$-weighted Rota-Baxter operator on $A$. Take $R = \lambda \mathrm{id}_A + 2 T$. We observe that
\begin{align}\label{modified1}
R(a) \cdot_A R(b) = \lambda^2~ a \cdot_A b + 2 \lambda \big( T(a) \cdot_A b + a \cdot_A T(b) \big) + 4 ~T(a) \cdot_A T(b).
\end{align}
On the other hand, by a direct calculation
\begin{align}\label{modified2}
&R \big( R(a) \cdot_A b + a \cdot_A R(b) \big) - \lambda^2 ~ a \cdot_A b \nonumber \\
&= \lambda^2~ a \cdot_A b + 2 \lambda \big( T(a) \cdot_A b + a \cdot_A T(b) \big) + 4 ~T \big( T(a) \cdot_A b + a \cdot_A T(b) + \lambda~ a \cdot_A b \big).
\end{align}
Since $T$ is a $\lambda$-weighted Rota-Baxter operator on $A$, it follows from (\ref{modified1}) and (\ref{modified2}) that $R$ is a solution of modified AYBE$_\lambda$. Conversely, if $R$ is a solution of modified AYBE$_\lambda$, then it is easy to see that $T = \frac{1}{2} (R - \lambda \mathrm{id}_A)$ is a $\lambda$-weighted Rota-Baxter operator on $A$. This completes the proof.
\end{proof}
Let $A$ be an unital associative algebra with unit $1 \in A$. Let $r = r_{(1)} \otimes r_{(2)}$ be an element in $A\otimes A$. Here we use the Sweedler notation. We define three elements of $A \otimes A \otimes A$, namely,
\begin{align*}
r_{12} = r_{(1)} \otimes r_{(2)} \otimes 1, \qquad r_{13} = r_{(1)} \otimes 1 \otimes r_{(2)} ~~~~ \text{ and } ~~~~ r_{23} = 1 \otimes r_{(1)} \otimes r_{(2)}.
\end{align*}
\begin{defn}
An element $r = r_{(1)} \otimes r_{(2)} \in A \otimes A$ is said to be a $\lambda$-weighted associative Yang-Baxter solution if the following holds
\begin{align}\label{w-aybe}
r_{13} r_{12} - r_{12} r_{23} + r_{23} r_{13} = \lambda r_{13}.
\end{align}
\end{defn}
We have the following relation between weighted associative Yang-Baxter solutions and weighted Rota-Baxter operators.
\begin{prop}
Let $A$ be an unital associative algebra and $r = r_{(1)} \otimes r_{(2)} \in A \otimes A$ be a $\lambda$-weighted associative Yang-Baxter solution. Then the map $T : A \rightarrow A$ defined by $T(a) = r_{(1)} \cdot_A a \cdot_A r_{(2)}$, is a $(-\lambda)$-weighted Rota-Baxter operator on $A$.
\end{prop}
\begin{proof}
Note that the identity (\ref{w-aybe}) can be written as
\begin{align*}
r_{(1)} \cdot_A \widetilde{r}_{(1)} \otimes \widetilde{r}_{(2)} \otimes r_{(2)} ~-~ r_{(1)} \otimes r_{(2)} \cdot_A \widetilde{r}_{(1)} \otimes \widetilde{r}_{(2)}~ +~ \widetilde{r}_{(1)} \otimes r_{(1)} \otimes r_{(2)} \cdot_A \widetilde{r}_{(2)} = \lambda~ r_{(1)} \otimes 1 \otimes r_{(2)},
\end{align*}
where $r_{(1)} \otimes r_{(2)}$ and $\widetilde{r}_{(1)} \otimes \widetilde{r}_{(2)}$ denote two copies of $r$. In the above identity, replacing the first tensor product by multiplication of $a$ and the second tensor product by multiplication of $b$, and using the definition of $R$, we get
\begin{align*}
T ( T(a) \cdot_A b ) - T(a) \cdot_A T(b) + T ( a \cdot_A T(b)) = \lambda T( a \cdot_A b).
\end{align*}
This shows that $T$ is a $(-\lambda)$-weighted Rota-Baxter operator on $A$.
\end{proof}
\medskip
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Let $T, T' : B \rightarrow A$ be two $\lambda$-weighted relative Rota-Baxter operators.
\begin{defn}
A morphism from $T$ to $T'$ consists of a pair $(\phi, \psi)$ of associative algebra morphisms $\phi : A \rightarrow A$ and $\psi : B \rightarrow B$ satisfying
\begin{align*}
\phi \circ T = T' \circ \psi , \quad \psi ( a \cdot u) = \phi (a) \cdot \psi (u) ~~~ \text{ and } ~~~ \psi ( u \cdot a) = \psi (u) \cdot \phi (a), ~ \text{ for } a \in A, u \in B.
\end{align*}
\end{defn}
The set of all $\lambda$-weighted relative Rota-Baxter operators and morphisms between them forms a category, denoted by $\mathsf{rRB}_\lambda (B,A)$.
\begin{prop}
Let $(\phi, \psi)$ be a morphism of $\lambda$-weighted relative Rota-Baxter operators from $T$ to $T'$. Then $\psi : B \rightarrow B$ is a morphism of induced associative algebras from $(B, \cdot_T)$ to $(B, \cdot_{T'})$.
\end{prop}
\begin{proof}
For any $u, v \in B$, we have
\begin{align*}
\psi ( u \cdot_T v) =~& \psi \big( T(u) \cdot v + u \cdot T(v) + \lambda ~ u \cdot_B v \big) \\
=~& \phi (T(u)) \cdot \psi (v) + \psi (u) \cdot \phi (T(v)) + \lambda ~ \psi(u) \cdot \psi (v) \\
=~& T' (\psi (u)) \cdot \psi (v) + \psi (u ) \cdot T' (\psi (v)) + \lambda ~ \psi (u) \cdot_B \psi (v) = \psi (u) \cdot_{T'} \psi (v).
\end{align*}
Hence the result follows.
\end{proof}
\section{Cohomology of $\lambda$-weighted relative Rota-Baxter operators}\label{sec-3}
The aim of this section is to provide Maurer-Cartan characterization of $\lambda$-weighted relative Rota-Baxter operators and define the cohomology of such operators.
\subsection{Maurer-Cartan characterization and Cohomology}\label{subsection31}
In this subsection, we construct a differential graded Lie algebra whose Maurer-Cartan elements are precisely $\lambda$-weighted relative Rota-Baxter operators. Using this characterization, we define the cohomology of a $\lambda$-weighted relative Rota-Baxter operator.
We start by recalling the Gerstenhaber bracket \cite{gers2}. Let $V$ be a vector space and consider the graded space $\oplus_{n \geq 1} \mathrm{Hom} (V^{\otimes n}, V)$ of multilinear maps on $V$. It carries a degree $-1$ graded Lie bracket (called the Gerstenhaber bracket) given by
\begin{align*}
[f, g]_\mathsf{G} := \sum_{i=1}^m (-1)^{(i-1)(n-1)} ~f \circ_i g - (-1)^{(m-1)(n-1)} \sum_{i=1}^n (-1)^{(i-1)(m-1)}~ g \circ_i f,
\end{align*}
for $f \in \mathrm{Hom} (V^{\otimes m}, V)$ and $g \in \mathrm{Hom} (V^{\otimes n}, V)$, where
\begin{align*}
(f \circ_i g) ( v_1, \ldots, v_{m+n-1}) = f ( v_1, \ldots, v_{i-1}, g (v_i, \ldots, v_{i+n-1}), v_{i+n}, \ldots, v_{m+n-1}).
\end{align*}
In other words, $\big( \oplus_{n \geq 0} \mathrm{Hom} (V^{\otimes n + 1}, V) , [~,~]_\mathsf{G} \big)$ is a graded Lie algebra.
Note that a multiplication $\mu \in \mathrm{Hom} (V^{\otimes 2}, V)$ defines an associative structure on $V$ if and only if $[\mu, \mu]_\mathsf{G} = 0$.
\medskip
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. We use the notations $\mu_A$ and $\mu_B$ for associative multiplications on $A$ and $B$, and $l, r$ for left and right $A$-actions on $B$, respectively. Take $V = A \oplus B$ and consider the graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom} (V^{\otimes n+1}, V), [~,~]_\mathsf{G} \big)$. Then it is easy to check that the graded subspace $\mathfrak{a} = \oplus_{n \geq 0} \mathrm{Hom} (B^{\otimes n+1}, A)$ is an abelian subalgebra. Moreover, we have the following observations.
\medskip
\medskip
\noindent {\bf Observation I.}
The element $\mu_A + l + r \in \mathrm{Hom}(V^{\otimes 2}, V)$ is a Maurer-Cartan element in the graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom} (V^{\otimes n+1}, V), [~,~]_\mathsf{G} \big)$. Hence it induces a differential $d_{\mu_A + l + r} := [\mu_A + l + r, -]_\mathsf{G}$ on $\oplus_{n \geq 0} \mathrm{Hom} (V^{\otimes n+1}, V)$. Therefore, by the derived bracket construction \cite{voro}, the shifted space $\mathfrak{a} [-1] = \oplus_{n \geq 1} \mathrm{Hom} (B^{\otimes n}, A)$
carries a graded Lie bracket (called the derived bracket) given by
\begin{align*}
\llbracket P, Q \rrbracket := (-1)^m ~ [ d_{\mu_A + l + r} (P), Q]_\mathsf{G} = (-1)^m~ [ ~[\mu_A + l + r, P]_\mathsf{G}, Q ]_\mathsf{G},
\end{align*}
for $P \in \mathrm{Hom}(B^{\otimes m}, A)$ and $Q \in \mathrm{Hom}(B^{\otimes n}, A)$. The explicit formula \cite{das-rota} is given by
\begin{align}\label{dgla-b}
&\llbracket P, Q \rrbracket (u_1, \ldots, u_{m+n}) \\
&= \sum_{i=1}^m (-1)^{(i-1)n}~ P ( u_1, \ldots, u_{i-1}, Q (u_i, \ldots, u_{i+n-1}) \cdot u_{i+n}, \ldots, u_{m+n}) \nonumber \\
&- \sum_{i=1}^m (-1)^{in} ~ P (u_1, \ldots, u_{i-1}, u_i \cdot Q (u_{i+1}, \ldots, u_{i+n}), u_{i+n+1}, \ldots, u_{m+n}) \nonumber \\
&- (-1)^{mn} \bigg\{ \sum_{i=1}^n (-1)^{(i-1)m}~ Q ( u_1, \ldots, u_{i-1}, P (u_i, \ldots, u_{i+m-1}) \cdot u_{i+m}, \ldots, u_{m+n}) \nonumber \\
&- \sum_{i=1}^n (-1)^{im} ~ Q (u_1, \ldots, u_{i-1}, u_i \cdot P (u_{i+1}, \ldots, u_{i+m}), u_{i+m+1}, \ldots, u_{m+n}) \bigg\} \nonumber \\
& + (-1)^{mn} \big[ P(u_1, \ldots, u_m) \cdot_A Q (u_{m+1}, \ldots, u_{m+n}) - (-1)^{mn} ~ Q (u_1, \ldots, u_n) \cdot_A P (u_{n+1}, \ldots, u_{m+n}) \big]. \nonumber
\end{align}
This graded Lie bracket can be extended to $\oplus_{n \geq 0} \mathrm{Hom} (B^{\otimes n}, A)$ by the following definitions:
\begin{align*}
&\llbracket P, a \rrbracket (u_1, \ldots, u_m) = \sum_{i=1}^m ~ P (u_1, \ldots, u_{i-1}, a \cdot u_i - u_i \cdot a, u_{i+1}, \ldots, u_m) \\
& \qquad \qquad \qquad \qquad \qquad + P (u_1, \ldots, u_m) \cdot_A a - a \cdot_A P (u_1, \ldots, u_m), \\
&\llbracket a, b \rrbracket = a \cdot_A b - b \cdot_A a, ~ \text{ for } P \in \mathrm{Hom} (B^{\otimes m}, A) \text{ and } a, b \in A.
\end{align*}
\medskip
\medskip
\noindent {\bf Observation II.}
For any $\lambda \in {\bf k}$, the associative multiplication $- \lambda \mu_B$ can be considered as an element in $\mathrm{Hom}( V^{\otimes 2}, V)$. This is infact a Maurer-Cartan element in the graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom} (V^{\otimes n+1}, V), [~,~]_\mathsf{G} \big)$. Therefore, $- \lambda \mu_B$ induces a differential $d_{ - \lambda \mu_B} := - [ \lambda \mu_B, -]_\mathsf{G}$ on $\oplus_{n \geq 1} \mathrm{Hom}(V^{\otimes n}, V)$. Moreover, the graded subspace $\oplus_{n \geq 1} \mathrm{Hom}(B^{\otimes n}, A)$ is closed under the differential $d_{- \lambda \mu_B}.$ We denote the restriction of the differential $d_{- \lambda \mu_B}$ to the subspace $\oplus_{n \geq 1} \mathrm{Hom}(B^{\otimes n}, A)$ by $d$, and it is given by
\begin{align}\label{dgla-d}
(df) (u_1, \ldots, u_{n+1}) = (-1)^{n-1} \sum_{i=1}^n (-1)^{i-1} ~f (u_1, \ldots, u_{i-1}, \lambda u_i \cdot_B u_{i+1}, u_{i+2}, \ldots, u_{n+1}).
\end{align}
The differential $d$ can be extended to $\oplus_{n \geq 0} \mathrm{Hom} (B^{\otimes n}, A)$ by $(da) (u) = T(u) \cdot_A a - a \cdot_A T(u)$, for $a \in A$ and $u \in B$.
\medskip
\medskip
\noindent {\bf Observation III.}
Finally, it is easy to see that the elements $\mu_A + l + r$ and $\lambda \mu_B$ satisfies the following compatibility
\begin{align*}
[\mu_A + l + r, \lambda \mu_B ]_\mathsf{G} = 0.
\end{align*}
Therefore, we have
\begin{align*}
&d \llbracket P, Q \rrbracket \\
&= (-1)^m ~ d~ [~[ \mu_A + l + r, P]_\mathsf{G}, Q ]_\mathsf{G} \\
&= (-1)^{m-1} ~[ \lambda \mu_B, [~[ \mu_A + l + r, P]_\mathsf{G}, Q ]_\mathsf{G} ]_\mathsf{G} \\
&= (-1)^{m-1} ~ [~[ \lambda \mu_B, [\mu_A + l + r, P]_\mathsf{G} ]_\mathsf{G}, Q ]_\mathsf{G} + (-1)^{m-1} (-1)^m ~ [~[\mu_A + l + r, P]_\mathsf{G}, [\lambda \mu_B, Q ]_\mathsf{G} ]_\mathsf{G} \\
&= (-1)^{m-1} (-1)^1 ~[~[\mu_A + l + r, [\lambda \mu_B, P ]_\mathsf{G} ]_\mathsf{G}, Q]_\mathsf{G} + [~[\mu_A + l + r, P]_\mathsf{G}, dQ ]_\mathsf{G} \\
&= - (-1)^{m} ~[~[ \mu_A + l + r, dP]_\mathsf{G}, Q ]_\mathsf{G} + [~[\mu_A + l + r, P]_\mathsf{G}, dQ ]_\mathsf{G} \\
&= \llbracket dP, Q \rrbracket + (-1)^m ~ \llbracket P, dQ \rrbracket.
\end{align*}
This shows that $d$ is a graded derivation for the derived bracket $\llbracket ~, ~ \rrbracket$ on the graded space $\oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n }, A)$.
\medskip
\medskip
As a summary of the above three observations, we get that $\big( \oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d \big)$ is a differential graded Lie algebra. The importance of this differential graded Lie algebra is given by the following.
\begin{thm}
A linear map $T : B \rightarrow A$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if $T$ is a Maurer-Cartan element in the differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d \big)$.
\end{thm}
\begin{proof}
For a linear map $T: B \rightarrow A$, we have from (\ref{dgla-b}) and (\ref{dgla-d}) that
\begin{align*}
(d T + \frac{1}{2} \llbracket T, T \rrbracket ) (u, v) = T ( \lambda ~ u \cdot_B v ) + T (T(u) \cdot v + u \cdot T(v)) - T(u) \cdot_A T(v).
\end{align*}
This shows that $T$ satisfies $d T + \frac{1}{2} \llbracket T, T \rrbracket = 0$ if and only if $T$ is a $\lambda$-weighted relative Rota-Baxter operator. Hence the proof.
\end{proof}
\medskip
Let $T : B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. Then $T$ induces a degree $1$ differential $d_T = d + \llbracket T, - \rrbracket$ on the graded space $\oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n}, A)$. The differential $d_T$ is explicitly given by
\begin{align}\label{t-diff}
&(d_T f) (u_1, \ldots, u_{n+1}) \\
&= T ( f (u_1, \ldots, u_n) \cdot u_{n+1} ) - (-1)^n ~ T (u_1 \cdot f (u_2, \ldots, u_{n+1})) \nonumber \\
&- (-1)^n \sum_{i=1}^n (-1)^{i-1} ~ f (u_1, \ldots, u_{i-1}, T(u_i) \cdot u_{i+1} + u_i \cdot T(u_{i+1}) + \lambda~ u_i \cdot_B u_{i+1}, \ldots, u_{n+1} ) \nonumber \\
&+ (-1)^n ~ T (u_1) \cdot_A f (u_2, \ldots, u_{n+1}) - f (u_1, \ldots, u_n) \cdot_A T (u_{n+1}), \nonumber
\end{align}
for $f \in \mathrm{Hom}(B^{\otimes n}, A)$ and $u_1, \ldots, u_{n+1} \in B$. We define
\begin{align*}
C^n_T (B,A) := \mathrm{Hom}(B^{\otimes n}, A), ~ \text{ for } n \geq 0.
\end{align*}
Then $\{ C^\ast_T (B,A), d_T \}$ is a cochain complex. If $Z^n_T (B,A) = \{ f \in C^n_T (B,A) |~ d_T f = 0 \}$ is the space of $n$-cocycles and $B^n_T (B,A) = \{ d_T f |~ f \in C^{n-1}_T (B,A) \}$ is the space of $n$-coboundaries, then we have $B^n_T (B, A) \subset Z^n_T (B,A)$, for $n \geq 0$. The corresponding quotients
\begin{align*}
H^n_T (B,A) := \frac{Z^n_T (B,A) }{ B^n_T (B,A) }, ~\text{ for } n \geq 0
\end{align*}
are called the cohomology groups of $T$.
\medskip
\medskip
Note that the differential $d_T$ makes the triple $\big( \oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d_T \big)$ into a new differential graded Lie algebra. This new structure governs Maurer-Cartan deformations of $T$ as described by the following.
\begin{thm}
Let $T: B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. For any linear map $T' : B \rightarrow A$, the sum $T + T'$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if $T'$ is a Maurer-Cartan element in the differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(B^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d_T \big)$.
\end{thm}
\begin{proof}
We observe that
\begin{align*}
d (T + T') + \frac{1}{2} \llbracket T + T', T + T' \rrbracket
&= dT + dT' + \frac{1}{2} \big( \llbracket T, T \rrbracket + 2 \llbracket T, T' \rrbracket + \llbracket T', T' \rrbracket \big) \\
&= dT' + \llbracket T, T' \rrbracket + \frac{1}{2} \llbracket T', T' \rrbracket \\
&= d_T (T') + \frac{1}{2} \llbracket T', T' \rrbracket.
\end{align*}
Hence the result follows.
\end{proof}
\subsection{A new interpretation of the cohomology}\label{subsec-hoch}
In this subsection, we show that the cohomology of a $\lambda$-weighted relative Rota-Baxter operator $T$ defined above can be expressed as the Hochschild cohomology of a suitable associative algebra with coefficients in an appropriate bimodule. We start with the following result.
\begin{prop}
Let $T: B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. Then $A$ is a bimodule over the induced associative algebra $(B, \cdot_T)$ with left and right actions given by
\begin{align*}
l^T_u ( a) := T(u) \cdot_A a - T( u \cdot a) ~~~~ \text{ and } ~~~~ r^T_u (a) := a \cdot_A T(u) - T(a \cdot u), ~ \text{ for } u \in B, a \in A.
\end{align*}
\end{prop}
\begin{proof}
For any $u, v \in B$ and $a \in A$, we have
\begin{align*}
&l^T_{u \cdot_T v} (a) - l^T_u l^T_v (a) \\
&= T (u \cdot_T v ) \cdot_A a - T ((u \cdot_T v) \cdot a) - l^T_u ( T(v) \cdot_A a - T (v \cdot a)) \\
&= (T(u) \cdot_A T(v) ) \cdot_A a - T \big( (T(u) \cdot v + u \cdot T(v) + \lambda~ u \cdot_B v) \cdot a \big) \\
& \qquad \qquad \qquad - T(u) \cdot_A (T(v) \cdot_A a - T (v \cdot a)) + T \big( u \cdot (T(v) \cdot_A a - T (v \cdot a)) \big) \\
&= - T \big( T(u) \cdot ( v \cdot a) + u \cdot (T(v) \cdot_A a) + \lambda ~ u \cdot_B (v \cdot a) \big) \\
& \qquad \qquad \qquad + T(u) \cdot_A T (v \cdot a) + T (u \cdot (T(v) \cdot_A a)) - T (u \cdot T (v \cdot a)) \\
&= - T \big( T(u) \cdot (v \cdot a) + \lambda ~u \cdot_B (v \cdot a) \big) + T \big( T(u) \cdot (v \cdot a) + u \cdot T(v \cdot a) + \lambda ~ u \cdot_B (v \cdot a) \big) - T (u \cdot T (v \cdot a) ) \\
&= 0.
\end{align*}
Similarly,
\begin{align*}
&r_v^T l_u^T (a) - l_u^T r_v^T (a) \\
&= r_v^T (T(u) \cdot_A a - T(u \cdot a)) - l_u^T (a \cdot_A T(v) - T(a \cdot v)) \\
&= (T(u) \cdot_A a - T(u \cdot a)) \cdot_A T(v) - T \big( (T(u) \cdot_A a - T(u \cdot a)) \cdot v \big) \\
& \qquad \qquad \qquad - T(u) \cdot_A (a \cdot_A T(v) - T (a \cdot v)) + T \big( u \cdot (a \cdot_A T(v) - T( a \cdot v) ) \big) \\
&= - T( u \cdot a) \cdot_A T(v) - T (T(u) \cdot (a \cdot v) - T(u \cdot a) \cdot v) + T (u) \cdot_A T (a \cdot v) + T \big( u \cdot (a \cdot_A T(v)) - u \cdot T (a \cdot v) \big) \\
&= - T \big( T(u \cdot a) \cdot v + (u \cdot a) \cdot T(v) + \lambda ~ (u \cdot a) \cdot_B v \big) - T (T(u) \cdot ( a \cdot v) - T (u \cdot a) \cdot v) \\
& \qquad \qquad \qquad + T \big( T(u) \cdot (a \cdot v) + u \cdot T ( a \cdot v) + \lambda~ u \cdot_B (a \cdot v) \big) + T((u \cdot a) \cdot T(v) - u \cdot T(a \cdot v)) \\
&= 0
\end{align*}
and
\begin{align*}
&r_v^T r_u^T (a) - r^T_{u \cdot_T v} (a) \\
&= r^T_v (a \cdot_A T(u) - T(a \cdot u)) - a \cdot_A T (u \cdot_T v) + T (a \cdot (u \cdot_T v)) \\
&= (a \cdot_A T(u) - T (a \cdot u)) \cdot_A T(v) - T \big( (a \cdot_A T(u) - T (a \cdot u)) \cdot v \big) \\
& \qquad \qquad \qquad - a \cdot_A (T(u) \cdot_A T(v)) + T (a \cdot (T(u) \cdot v + u \cdot T(v) + \lambda~ u \cdot_B v) ) \\
&= -T (a \cdot u) \cdot_A T(v) - T (a \cdot (T(u) \cdot v) - T(a \cdot u) \cdot v) \\
& \qquad \qquad \qquad + T (a \cdot (T(u) \cdot v)) + T (a \cdot (u \cdot T(v))) + \lambda~ T( a \cdot (u \cdot_B v)) \\
&= - T \big( T( a \cdot u) \cdot v + ( a \cdot u) \cdot T(v) + \lambda ~(a \cdot u) \cdot_B v \big) + T (T(a \cdot u) \cdot v + T ((a \cdot u) \cdot T(v)) + \lambda ~ T (( a \cdot u) \cdot_B v )\\
&= 0.
\end{align*}
This proves the result.
\end{proof}
\medskip
Let $T : B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. Then it follows from the previous proposition that one may consider the Hochschild complex $\{ C^\ast_\mathsf{H} (B,A) , d_\mathsf{H} \}$ of the associative algebra $(B, \cdot_T)$ with coefficients in the bimodule $(A, l^T, r^T).$ More precisely, the $n$-th cochain group
\begin{align*}
C^n_\mathsf{H} (B,A) := \mathrm{Hom}(B^{\otimes n}, A), ~ \text{ for } n \geq 0
\end{align*}
and the coboundary map $d_\mathsf{H} : C^n_\mathsf{H} (B,A) \rightarrow C^{n+1}_\mathsf{H} (B,A)$ given by
\begin{align}\label{hoch-diff}
(d_\mathsf{H} f ) (u_1, \ldots, u_{n+1}) =~& T(u_1) \cdot_A f (u_2, \ldots, u_{n+1}) - T (u_1 \cdot f (u_2, \ldots, u_{n+1})) \\
+~& \sum_{i=1}^n (-1)^i ~ f (u_1, \ldots, u_{i-1}, T(u_i) \cdot u_{i+1} + u_i \cdot T (u_{i+1}) + \lambda ~ u_i \cdot_B u_{i+1}, \ldots, u_{n+1} ) \nonumber \\
+~&(-1)^{n+1} ~ f(u_1, \ldots, u_n) \cdot_A T(u_{n+1}) - (-1)^{n+1} ~ T (f(u_1, \ldots, u_n) \cdot u_{n+1} ), \nonumber
\end{align}
for $f \in C^n_\mathsf{H}(B,A)$ and $u_1, \ldots, u_{n+1} \in B$. We denote the corresponding cohomology groups by $H^\ast_\mathsf{H}(B,A)$.
It follows from the expressions of (\ref{t-diff}) and (\ref{hoch-diff}) that $d_T f = (-1)^n ~d_\mathsf{H} f$, for $f \in \mathrm{Hom}(B^{\otimes n}, A)$. In other words, the differentials $d_T$ and $d_\mathsf{H}$ are same up to a sign, which implies that
\begin{align*}
H^\ast_T (B,A) \cong H^\ast_\mathsf{H} (B,A).
\end{align*}
\subsection{Particular case: $\lambda$-weighted Rota-Baxter operators}
Let $A$ be an associative algebra. Then $A$ is itself an associative $A$-bimodule, called the adjoint $A$-bimodule. Moreover, we have seen in Remark \ref{rel-not} that a $\lambda$-weighted Rota-Baxter operator on $A$ is a particular case of $\lambda$-weighted relative Rota-Baxter operator. Therefore, one may adopt the previous results in this particular case.
We summarize the results of Subsection \ref{subsection31} in the following theorem.
\begin{thm}
Let $A$ be an associative algebra and $\lambda \in {\bf k}$ be a fixed scalar. Then
\begin{itemize}
\item[(i)] there is a differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(A^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d \big)$ on the graded space of multilinear maps on $A$, where the bracket $\llbracket ~, ~ \rrbracket$ and the differential $d$ are given by the formulas (\ref{dgla-b}) and (\ref{dgla-d}), respectively.
\item[(ii)] A linear map $T: A \rightarrow A$ is a $\lambda$-weighted Rota-Baxter operator on $A$ if and only if $T \in \mathrm{Hom}(A, A)$ is a Maurer-Cartan element in the above differential graded Lie algebra.
\item[(iii)] Let $T$ be a $\lambda$-weighted Rota-Baxter operator on $A$. For any linear map $T' : A \rightarrow A$, the sum $T + T'$ is a $\lambda$-weighted Rota-Baxter operator on $A$ if and only if $T'$ is a Maurer-Cartan element in the differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(A^{\otimes n}, A), \llbracket ~, ~ \rrbracket, d_T = d + \llbracket T, - \rrbracket \big)$.
\end{itemize}
\end{thm}
It follows that a $\lambda$-weighted Rota-Baxter operator $T$ induces a cochain complex $\{ C^\ast_T (A,A), d_T \}$. The corresponding cohomology groups are called the cohomology of $T$.
\medskip
On the other hand, the $\lambda$-weighted Rota-Baxter operator $T$ induces a new associative structure on $A$ with the product given by
\begin{align*}
a \cdot_T b = T(a) \cdot_A b + a \cdot_A T(b) + \lambda ~ a \cdot_A b, ~ \text{ for } a, b \in A.
\end{align*}
The vector space $A$ can be given a bimodule structure over the associative algebra $(A, \cdot_T)$ with left and right actions
\begin{align*}
l_a^T (b) = T(a) \cdot_A b - T (a \cdot_A b) ~~~~ \text{ and } ~~~~ r_a^T (b) = b \cdot_A T(a) - T ( b \cdot_A a), ~ \text{ for } a, b \in A.
\end{align*}
Moreover, the cohomology of $T$ is isomorphic to the Hochschild cohomology of $(A, \cdot_T)$ with coefficients in the above bimodule $(A, l^T, r^T)$.
\section{Deformations of $\lambda$-weighted relative Rota-Baxter operators}\label{sec-4}
In this section, we study deformations of a $\lambda$-weighted relative Rota-Baxter operator $T$ from cohomological points of view. We introduce Nijenhuis elements associated to $T$ that generate trivial linear deformations of $T$. We also find a sufficient condition for the rigidity of $T$. Finally, given a finite order deformation of $T$, we construct a second cohomology class in the cohomology of $T$, called the obstruction class. The vanishing of the obstruction class ensures that the deformation is extensible.
\subsection{Linear deformations and Nijenhuis elements}
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Let $T : B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator.
\begin{defn}
A linear map $T_1 : B \rightarrow A$ is said to generate a linear deformation of $T$ if the linear sum $T_t := T + t T_1$ is a $\lambda$-weighted relative Rota-Baxter operator for all values of $t$. In this case, we say that $T_t$ is a linear deformation of $T$.
\end{defn}
Note that $T_1$ generates a linear deformation of $T$ if and only if the followings are hold
\begin{align}
T(u) \cdot_A T_1 (v) + T_1(u) \cdot_A T(v) =~& T (T_1(u) \cdot v + u \cdot T_1 (v)) + T_1 (T(u) \cdot v + u \cdot T(v) + \lambda u \cdot_B v), \label{lin1}\\
T_1 (u) \cdot_A T_1 (v) =~& T_1 (T_1 (u) \cdot v + u \cdot T_1(v)), \label{lin2}
\end{align}
for all $u, v \in B$. It follows from (\ref{lin1}) that $d_T (T_1) = 0$, i.e., $T_1$ is a $1$-cocycle in the cohomology complex of $T$. On the other hand, the identity (\ref{lin2}) implies that $T_1$ is a relative Rota-Baxter operator (of weight $0$).
\begin{defn}
Two linear deformations $T_t = T+ t T_1$ and $T_t' = T + t T_1'$ of a $\lambda$-weighted relative Rota-Baxter operator $T$ are said to be equivalent if there exists an element $a_0 \in A$ such that $(\phi = \mathrm{id}_A + t (l_{a_0}^{\mathrm{ad}} - r_{a_0}^{\mathrm{id}}) , ~ \psi = \mathrm{id}_B + t (l_{a_0} - r_{a_0}))$ is a morphism of $\lambda$-weighted relative Rota-Baxter operators from $T_t$ to $T_t'$.
\end{defn}
The condition in the above definition is equivalent to the followings (see \cite{das-rota} for similar observation)
\begin{align}
(a_0 \cdot_A a - a \cdot_A a_0) \cdot_A (a_0 \cdot_A b - b \cdot_A a_0) = 0 ~~~ \text{ and } ~~~ (a_0 \cdot u - u \cdot a_0) \cdot_B (a_0 \cdot v - v \cdot a_0) = 0, \label{equiv1}\\
\qquad \qquad \qquad \begin{cases}
T_1 (u ) - T_1' (u) = T (a_0 \cdot u - u \cdot a_0) - (a_0 \cdot_A T(u) - T(u) \cdot_A a_0 ), \\
a_0 \cdot_A T_1(u) - T_1 (u) \cdot_A a_0 = T_1' (a_0 \cdot u - u \cdot a_0),
\end{cases} \label{equiv2}\\
l_{(a_0 \cdot_A a - a \cdot_A a_0)} l_{a_0} = l_{(a_0 \cdot_A a - a \cdot_A a_0)} r_{a_0} ~~~~ \text{ and } ~~~~ r_{(a_0 \cdot_A a - a \cdot_A a_0)} l_{a_0} = r_{(a_0 \cdot_A a - a \cdot_A a_0)} r_{a_0}, \label{equiv3}
\end{align}
for all $a, b \in A$ and $u, v \in B$. It follows from (\ref{equiv2}) that $(T_1 - T_1') (u) = d_T (a_0)(u)$, for $u \in B$. Thus, we have the following.
\begin{prop}
Let $T_t = T+ t T_1$ and $T_t' = T + tT_1'$ be two equivalent linear deformations of $T$. Then $T_1$ and $T_1'$ corresponds to the same cohomology class in $H^1_T (B,A).$
\end{prop}
We now introduce Nijenhuis elements associated to $T$. This generalizes the similar notion defined for weight $0$ relative Rota-Baxter operators \cite{das-rota}.
\begin{defn}
An element $a_0 \in A$ is called a Nijenhuis element associated to a $\lambda$-weighted relative Rota-Baxter $T$ if $a_0$ satisfies (\ref{equiv1}), (\ref{equiv3}) and
\begin{align*}
a_0 \cdot_A (l_u^T (a_0) - r_u (a_0) ) - (l_u^T (a_0) - r_u^T (a_0) ) \cdot_A a_0 = 0, \text{ for all } u \in B.
\end{align*}
We denote the set of all Nijenhuis elements by $\mathrm{Nij}(T)$. A linear deformation $T_t$ is said to be trivial if $T_t$ is equivalent to $T_t' = T$. It follows from the above discussion that a trivial linear deformation gives rise to a Nijenhuis element. It was shown in \cite{das-rota} that Nijenhuis elements also generate trivial linear deformations of relative Rota-Baxter operator (of weight $0$). Similarly, we can prove the same result for a weighted relative Rota-Baxter operator. We only state the result and refer to \cite{das-rota} for the proof.
\begin{prop}
Let $T: B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. For any $a_0 \in \mathrm{Nij}(T)$, the linear map $T_1 = d_T(a) : B \rightarrow A$ generates a trivial linear deformation of $T$.
\end{prop}
\end{defn}
\subsection{Formal deformations}
In this subsection, we study formal deformations of a $\lambda$-weighted relative Rota-Baxter operator $T$ by keeping the underlying algebras $A$ and $B$ intact.
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Consider the space $A[[t]]$ of formal power series in $t$ with coefficients from $A$. The associative structure on $A$ induces an associative product on $A[[t]]$ by ${\bf k}[[t]]$-bilinearity. Similarly, $B[[t]]$ is an associative algebra over ${\bf k}[[t]]$, and $B[[t]]$ is an associative $A[[t]]$-bimodule.
\begin{defn}
Let $T: B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. A formal deformation of $T$ is given by a formal sum
\begin{align*}
T_t = T_0 + t T_1 + t^2 T_2 + \cdots ~\in \mathrm{Hom}(B,A)[[t]] ~~ \text{ with } T_0 = T
\end{align*}
such that the ${\bf k}[[t]]$-linear map $T_t : B[[t]] \rightarrow A[[t]]$ is a $\lambda$-weighted relative Rota-Baxter operator.
\end{defn}
Note that $T_t$ is a formal deformation of $T$ if and only if
\begin{align}\label{def-eqn-sys}
\sum_{i+j = n} T_i (u) \cdot_A T_j (v) = \sum_{i+j = n} T_i \big( T_j (u ) \cdot v + u \cdot T_j (v) \big) + \lambda ~ T_n (u \cdot_B v),~ \text{ for } n=0, 1, \ldots .
\end{align}
The identity (\ref{def-eqn-sys}) holds for $n=0$ as $T_0 = T$ is a $\lambda$-weighted relative Rota-Baxter operator. For $n =1$, we get
\begin{align*}
T(u) \cdot_A T_1 (v) + T_1 (u) \cdot_A T(v) = T ( T_1 (u) \cdot v + u \cdot T_1 (v)) ~+~ T_1 ( T (u) \cdot v + u \cdot T (v) + \lambda ~ u \cdot_B v),
\end{align*}
for $u, v \in B$. This implies that $T_1$ is a $1$-cocycle in the cohomology complex of $T$. This is called the infinitesimal of the deformation $T_t$. More generally, if $T_1 = \cdots = T_{k-1} = 0$ and $T_k$ is the first nonvanishing term after $T_0 = T$ in the expression of $T_t$, then $T_k$ is a $1$-cocycle in the cohomology complex of $T$.
\begin{defn}
Two formal deformations $T_t$ and $T_t'$ of a $\lambda$-weighted relative Rota-Baxter operator $T$ are said to be equivalent if there exists an element $a_0 \in A$, linear maps $\phi_i \in \mathrm{Hom}(A,A)$ and $\psi_i \in \mathrm{Hom}(B, B)$ for $i \geq 2$, such that
\begin{align*}
\big( \phi_t= \mathrm{id}_A + t (l_{a_0}^{\mathrm{ad}} - r_{a_0}^{\mathrm{ad}}) + \sum_{i \geq 2} t^i \phi_i, ~ \psi_{t} = \mathrm{id}_B + t (l_{a_0} - r_{a_0}) + \sum_{i \geq 2} t^i \psi_i \big)
\end{align*}
is a morphism of $\lambda$-weighted relative Rota-Baxter operators from $T_t$ to $T_t'$.
\end{defn}
The following result is similar to the case of linear deformations. Hence we omit the details.
\begin{prop}
Let $T_t$ and $T_t'$ be two equivalent formal deformations of a $\lambda$-weighted relative Rota-Baxter operator $T$. Then the corresponding infinitesimals are cohomologous, hence corresponds to the same cohomology class in $H^1_T (B,A).$
\end{prop}
\medskip
\begin{defn}
A $\lambda$-weighted relative Rota-Baxter operator $T$ is said to be rigid if any deformation $T_t$ is equivalent to the undeformed one $T_t' = T$.
\end{defn}
One may find the following sufficient condition for the rigidity of $T$ in terms of Nijenhuis elements. The proof is similar to \cite[Proposition 4.16]{das-rota}. Hence we will not repeat it here.
\begin{thm}
Let $T$ be a $\lambda$-weighted relative Rota-Baxter operator. If $Z^1_T (B,A) = d_T (\mathrm{Nij}(T))$ then $T$ is rigid.
\end{thm}
\subsection{Finite order deformations}
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. For any $N \geq 1$, consider the space $A[[t]]/(t^{N+1})$ which inherits an associative algebra structure over the base ring ${\bf k}[[t]]/(t^{N+1})$. Similarly, the space $B[[t]]/(t^{N+1})$ is an associative algebra over ${\bf k}[[t]]/(t^{N+1})$, and an associative $A[[t]]/(t^{N+1})$-bimodule.
\begin{defn}
Let $T: B \rightarrow A$ be a $\lambda$-weighted relative Rota-Baxter operator. An order $N$ deformation of $T$ consists of a polynomial of the form $T_t = \sum_{i=0}^N t^i T_i$ with $T_0 = T$, such that the ${\bf k}[[t]]/(t^{N+1})$-linear map $T_t : B[[t]]/(t^{N+1}) \rightarrow A[[t]]/(t^{N+1})$ is a $\lambda$-weighted relative Rota-Baxter operator.
\end{defn}
Note that $T_t = \sum_{i=0}^N t^i T_i$ is an order $N$ deformation of $T$ if and only if the identities (\ref{def-eqn-sys}) are hold for $n =0, 1, \ldots, N$. They can be equivalently expressed as
\begin{align}\label{fin-eq}
d_T (T_n ) = - \frac{1}{2} \sum_{i+j = n; i, j \geq 1}\llbracket T_i, T_j \rrbracket,~ \text{ for } n =0, 1, \ldots, N.
\end{align}
\begin{defn}
An order $N$ deformation $T_t = \sum_{i=0}^N t^i T_i$ of a $\lambda$-weighted relative Rota-Baxter operator $T$ is said to extensible if there exists a linear map $T_{N+1} : B \rightarrow A$ such that
\begin{align*}
\widetilde{T_t} = T_t + t^{N+1} T_{N+1} = \sum_{i=0}^{N+1} t^i T_i
\end{align*}
is a deformation of order $N+1$.
\end{defn}
Thus, in the case of extension, one more equation need to be satisfied, namely,
\begin{align}\label{exten-n}
d_T (T_{N+1}) = - \frac{1}{2} \sum_{i+j = N+1; i, j \geq 1} \llbracket T_i, T_j \rrbracket.
\end{align}
Observe that the right-hand side of the above equation does not contain the term $T_{N+1}$. Hence it depends only on the given order $N$ deformation $T_t$. Moreover, it is a $2$-cochain in the cohomology complex of $T$. We denote this $2$-cochain by $Ob_{T_t}.$
\begin{prop}
The cochain $Ob_{T_t}$ is a $2$-cocycle in the cohomology complex of $T$.
\end{prop}
\begin{proof}
We observe that
\begin{align*}
&d_T ( - \frac{1}{2} \sum_{i+j = N+1, i, j \geq 1} \llbracket T_i, T_j \rrbracket ) \\
&= - \frac{1}{2} \sum_{i+j = N+1, i, j \geq 1} ( d \llbracket T_i, T_j \rrbracket + \llbracket T, \llbracket T_i, T_j \rrbracket \rrbracket ) \\
&= - \frac{1}{2} \sum_{i+j = N+1, i, j \geq 1} \big( \llbracket d T_i, T_j \rrbracket - \llbracket T_i, dT_j \rrbracket + \llbracket \llbracket T, T_i \rrbracket, T_j \rrbracket - \llbracket T_i, \llbracket T, T_j \rrbracket \rrbracket \big) \\
&= - \frac{1}{2} \sum_{i+j = N+1, i, j \geq 1} \big( \llbracket d_T T_i, T_j \rrbracket - \llbracket T_i, d_T T_j \rrbracket \big) \\
&= \frac{1}{4} \sum_{i_1 + i_2 + j = N+1, i_1, i_2, j \geq 1} \llbracket \llbracket T_{i_1}, T_{i_2} \rrbracket, T_j \rrbracket - \frac{1}{4} \sum_{i + j_1 + j_2 = N+1, i, j_1, j_2 \geq 1} \llbracket T_i, \llbracket T_{j_1}, T_{j_2} \rrbracket \rrbracket ~~~ (\text{from } (\ref{fin-eq}))\\
&= \frac{1}{2} \sum_{i + j + k = N+1, i, j, k \geq 1} \llbracket \llbracket T_{i}, T_{j} \rrbracket, T_k \rrbracket = 0.
\end{align*}
This completes the proof.
\end{proof}
Thus, given an order $N$ deformation $T_t$, we obtain a second cohomology class $[Ob_{T_t} ] \in H^2_T (B,A)$ in the cohomology of $T$. This is called the obstruction class to extend the deformation $T_t$. Moreover, from the identity (\ref{exten-n}), we get the following.
\begin{thm}
An order $N$ deformation $T_t$ of a $\lambda$-weighted relative Rota-Baxter operator $T$ is extensible if and only if the corresponding obstruction class $[Ob_{T_t} ] \in H^2_T (B,A)$ is trivial.
\end{thm}
\begin{corollary}
\begin{itemize}
\item[(i)] If $H^2_T (B,A) = 0$ then any finite order deformation of $T$ is extensible.
\item[(ii)] If $H^2_T (B,A) = 0$ then every $1$-cocycle in the cohomology complex of $T$ is the infinitesimal of some formal deformation of $T$.
\end{itemize}
\end{corollary}
\begin{remark}
One may also study various aspects of deformations of $\lambda$-weighted Rota-Baxter operators on an algebra $A$. Since we will get similar results as above, we do not repeat them here.
\end{remark}
\section{The Lie case}\label{sec-5}
In this section, we define the cohomology of $\lambda$-weighted relative Rota-Baxter operators in the context of Lie algebras. We find its relation with the cohomology of $\lambda$-weighted relative Rota-Baxter operators on associative algebras.
\subsection{Weighted relative Rota-Baxter operators and their Cohomology}
Let $(\mathfrak{g}, [~,~]_\mathfrak{g})$ and $(\mathfrak{h}, [~,~]_\mathfrak{h})$ be two Lie algebras. Suppose the Lie algebra $\mathfrak{g}$ acts on $\mathfrak{h}$ by a Lie algebra homomorphism $\rho : \mathfrak{g} \rightarrow \mathrm{Der}(\mathfrak{h})$. In this case, we also say that $\mathfrak{h}$ is a Lie $\mathfrak{g}$-module.
\begin{defn}
\begin{itemize}
\item[(i)] Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras, and let $\mathfrak{g}$ acts on $\mathfrak{h}$ by a Lie algebra homomorphism $\rho : \mathfrak{g} \rightarrow \mathrm{Der}(\mathfrak{h})$. A linear map $T : \mathfrak{h} \rightarrow \mathfrak{g}$ is said to be a $\lambda$-weighted relative Rota-Baxter operator on $\mathfrak{h}$ over the Lie algebra $\mathfrak{g}$ if
\begin{align*}
[T(u), T(v)]_\mathfrak{g} = T \big( \rho (Tu) v - \rho (Tv) u + \lambda [u, v]_\mathfrak{h} \big), ~ \text{ for } u, v \in \mathfrak{h}.
\end{align*}
For simplicity, we say that $T$ is a $\lambda$-weighted relative Rota-Baxter operator.
\item[(ii)] Let $T, T' : \mathfrak{h} \rightarrow \mathfrak{g}$ be two $\lambda$-weighted relative Rota-Baxter operators. A morphism from $T$ to $T'$ is a pair $(\phi, \psi)$ consist of Lie algebra homomorphisms $\phi : \mathfrak{g} \rightarrow \mathfrak{g}$ and $\psi : \mathfrak{h} \rightarrow \mathfrak{h}$ satisfying additionally
\begin{align*}
\phi \circ T = T' \circ \psi ~~~~ \text{ and } \psi (\rho (x) u) = \rho (\phi (x)) \psi (u), ~ \text{ for } x \in \mathfrak{g}, u \in \mathfrak{h}.
\end{align*}
\end{itemize}
The set of all $\lambda$-weighted relative Rota-Baxter operators and morphisms between them forms a category. We denote this category by $\mathsf{rRB}_\lambda (\mathfrak{h}, \mathfrak{g}).$
\end{defn}
\medskip
In the following, we introduce $\lambda$-weighted modified Yang-Baxter equation in a Lie algebra and find a connection with $\lambda$-weighted Rota-Baxter operators.
\begin{defn}
Let $\mathfrak{g}$ be a Lie algebra. For a linear map $R : \mathfrak{g} \rightarrow \mathfrak{g}$, the equation
\begin{align*}
[R(x), R(y)]_\mathfrak{g} = R \big( [R(x), y]_\mathfrak{g} + [x, R(y)]_\mathfrak{g} \big) - \lambda^2 [x,y]_\mathfrak{g},~ \text{ for } x, y \in \mathfrak{g}
\end{align*}
is called the $\lambda$-weighted modified Yang-Baxter equation (modified YBE$_\lambda$).
\end{defn}
The following result is similar to Proposition \ref{rb-myb}. Hence we will omit the details.
\begin{prop}
Let $\mathfrak{g}$ be a Lie algebra. A linear map $T : \mathfrak{g} \rightarrow \mathfrak{g}$ is a $\lambda$-weighted Rota-Baxter operator if and only if $R = \lambda \mathrm{id}_\mathfrak{g} + 2T$ is a solution of modified YBE$_\lambda$.
\end{prop}
Let $\mathfrak{g}$ acts on $\mathfrak{h}$ by a Lie algebra homomorphism $\rho : \mathfrak{g} \rightarrow \mathrm{Der}(\mathfrak{h})$. Then for any $\lambda \in {\bf k}$, the direct sum $\mathfrak{g} \oplus \mathfrak{h}$ carries a Lie bracket given by
\begin{align*}
[(x,u), (u, v)]_{\ltimes_\lambda} := \big( [x,y]_\mathfrak{g},~ \rho (x) u - \rho(y) v + \lambda [u, v]_\mathfrak{h} \big), \text{ for } x, y \in \mathfrak{g} \text{ and } u, v \in \mathfrak{h}.
\end{align*}
This is called the $\lambda$-weighted semidirect product. With this notation, we have the following characterization of $\lambda$-weighted relative Rota-Baxter operators.
\begin{prop}
A linear map $T : \mathfrak{h} \rightarrow \mathfrak{g}$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if the graph $Gr (T) = \{ (T(u), u) |~ u \in \mathfrak{h} \}$ is a subalgebra of the $\lambda$-weighted semidirect product Lie algebra $(\mathfrak{g} \oplus \mathfrak{h}, [~,~]_{\ltimes_\lambda})$.
\end{prop}
As a consequence, we get the following.
\begin{prop}
Let $T: \mathfrak{h} \rightarrow \mathfrak{g}$ be a $\lambda$-weighted relative Rota-Baxter operator. Then $\mathfrak{h}$ carries a new Lie algebra structure with bracket
\begin{align*}
[u, v]_T := \rho (Tu) v - \rho (Tv) u + \lambda [u,v]_\mathfrak{h}, ~ \text{ for } u, v \in \mathfrak{h}.
\end{align*}
\end{prop}
\medskip
We will now define cohomology of a $\lambda$-weighted relative Rota-Baxter operator. Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras and let $\mathfrak{g}$ acts on $\mathfrak{h}$ by a Lie algebra homomorphism $\rho : \mathfrak{g} \rightarrow \mathrm{Der}(\mathfrak{h})$. Let $\pi_\mathfrak{g} \in \mathrm{Hom}(\wedge^2 \mathfrak{g}, \mathfrak{g})$ and $\pi_\mathfrak{h} \in \mathrm{Hom}(\wedge^2 \mathfrak{h}, \mathfrak{h})$ be the elements corresponding to the Lie brackets of $\mathfrak{g}$ and $\mathfrak{h}$, respectively. Let $W = \mathfrak{g} \oplus \mathfrak{h}$ be the direct sum vector space. Consider the Nijenhuis-Richardson bracket $[~,~]_\mathsf{NR}$ on the graded space $\oplus_{n \geq 1} \mathrm{Hom}(\wedge^n W, W)$ of skew-symmetric multilinear maps given by
\begin{align*}
[f, g ]_\mathsf{NR} = f \diamond g - (-1)^{(m-1)(n-1)}~ g \diamond f, ~~~ \text{ where }
\end{align*}
\begin{align*}
(f \diamond g ) (w_1, \ldots, w_{m+n-1}) = \sum_{\sigma \in \mathbb{S}_{(n, m-1)}} (-1)^\sigma ~ f \big( g (w_{\sigma (1)}, \ldots, w_{\sigma (n)}), w_{\sigma (n+1)}, \ldots, w_{\sigma (m+n-1)} \big).
\end{align*}
Then similar to the associative case, the graded space $\oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g})$ carries a graded Lie bracket
\begin{align}\label{derived-lie}
\{ \! [ P, Q ] \! \} := (-1)^m ~[[\pi_\mathfrak{g} + \rho, P]_\mathsf{NR}, Q ]_\mathsf{NR}, ~ \text{ for } P \in \mathrm{Hom}(\wedge^m \mathfrak{h}, \mathfrak{g}),~ Q \in \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g}).
\end{align}
The explicit formula of the bracket (\ref{derived-lie}) can be found in \cite[Equation (5)]{tang}. On the other hand, the differential $\delta_{ - \lambda \pi_\mathfrak{h}} := - [ \lambda \pi_\mathfrak{h}, - ]_\mathsf{NR}$ restricts to a differential $\delta$
on the graded space $\oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g})$. Moreover, we get that $\big( \oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g}) , \{ \! [ ~, ~ ] \! \}, \delta \big)$ is a differential graded Lie algebra.
\begin{thm}
Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras and $\mathfrak{g}$ acts on $\mathfrak{h}$ by a Lie algebra homomorphism $\rho : \mathfrak{g} \rightarrow \mathrm{Der}(\mathfrak{h})$. A linear map $T : \mathfrak{h} \rightarrow \mathfrak{g}$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if $T$ is a Maurer-Cartan element in the differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g}) , \{ \! [ ~, ~ ] \! \}, \delta \big)$.
\end{thm}
\begin{proof}
From the explicit formula \cite[Equation (5)]{tang} of the bracket $\{ \! [ ~, ~ ] \! \}$, we get that
\begin{align*}
\{ \! [ T, T ] \! \} (u, v) = 2 \big( T (\rho (Tu) v) - T (\rho (Tv)u) - [Tu, Tv]_\mathfrak{g} \big).
\end{align*}
On the other hand, We have $(\delta T)(u, v) = T (\lambda ~ [u, v]_\mathfrak{h}),$ for $u, v \in \mathfrak{h}$. Hence $T$ satisfies $\delta T + \frac{1}{2} \{ \! [ T, T ] \! \} = 0 $ if and only if $T$ is a $\lambda$-weighted relative Rota-Baxter operator.
\end{proof}
It follows from the above theorem that a $\lambda$-weighted relative Rota-Baxter operator $T$ induces a differential $\delta_T := \delta + \{ \! [ T, ~ ] \! \}$ on the graded space $\oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g})$. The corresponding cohomology groups are called the cohomology of $T$, denoted by $H^\ast_T (\mathfrak{h}, \mathfrak{g}).$
\medskip
The following result is similar to the associative case.
\begin{thm}
Let $T : \mathfrak{h} \rightarrow \mathfrak{g}$ be a $\lambda$-weighted relative Rota-Baxter operator. Then
\begin{itemize}
\item[(i)] the triple $\big( \oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g}) , \{ \! [ ~, ~ ] \! \}, \delta \big)$ is a differential graded Lie algebra.
\item[(ii)] For any linear map $T' : \mathfrak{h} \rightarrow \mathfrak{g}$, the sum $T + T'$ is a $\lambda$-weighted relative Rota-Baxter operator if and only if $T'$ is a Maurer-Cartan element in the differential graded Lie algebra $\big( \oplus_{n \geq 0} \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g}) , \{ \! [ ~, ~ ] \! \}, \delta_T \big)$.
\end{itemize}
\end{thm}
\medskip
\medskip
In the following, we will show that the cohomology of $T$ can be described in terms of Chevalley-Eilenberg cohomology. We define a map $\rho^T : \mathfrak{h} \rightarrow \mathrm{End}(\mathfrak{g})$ by
\begin{align*}
\rho^T (u) (x) := T (\rho (x) u) + [Tu, x]_\mathfrak{g}, ~ \text{ for } u \in \mathfrak{h}, x \in \mathfrak{g}.
\end{align*}
\begin{prop}
The map $\rho^T : \mathfrak{h} \rightarrow \mathrm{End}(\mathfrak{g})$ defines a representation of the Lie algebra $(\mathfrak{h}, [~,~]_T)$ on the vector space $\mathfrak{g}$.
\end{prop}
\begin{proof}
For any $u, v \in \mathfrak{h}$ and $x \in \mathfrak{g}$, we have
\begin{align*}
&[\rho^T (u), \rho^T (v) ] x \\
&= \rho^T (u) \rho^T (v) x - \rho^T (v) \rho^T (u) x \\
&= \rho^T (u) \big( T (\rho (x) v) + [ Tv, x]_\mathfrak{g} \big) -\rho^T (v) \big( T (\rho (x) u) + [ Tu, x]_\mathfrak{g} \big) \\
&= T \big( (\rho T (\rho (x) v)) u ) \big) + [Tu, T (\rho (x) v) ]_\mathfrak{g} + T \big( \rho ([Tv, x]_\mathfrak{g}) u \big) + [Tu, [Tv, x]_\mathfrak{g} ]_\mathfrak{g} \\
& \qquad \qquad - T \big( (\rho T (\rho (x) u)) v ) \big) - [Tv, T (\rho (x) u) ]_\mathfrak{g} - T \big( \rho ([Tu, x]_\mathfrak{g}) v \big) - [Tv, [Tu, x]_\mathfrak{g} ]_\mathfrak{g} \\
&= T \big( (\rho T (\rho (x) v)) u ) \big) +
T \big( \rho (Tu) \rho(x) v - \rho (T (\rho(x) v) )u + \lambda~[u, \rho(x) v]_\mathfrak{h} \big)
+ T \big( \rho ([Tv, x]_\mathfrak{g}) u \big) + [Tu, [Tv, x]_\mathfrak{g} ]_\mathfrak{g} \\
&- T \big( (\rho T (\rho (x) u)) v ) \big) -
T \big( \rho (Tv) \rho(x) u - \rho (T (\rho(x) u) )v + \lambda~[v, \rho(x) u]_\mathfrak{h} \big)
- T \big( \rho ([Tu, x]_\mathfrak{g}) v \big) - [Tv, [Tu, x]_\mathfrak{g} ]_\mathfrak{g} \\
&= T \big( \rho (Tu) \rho (x) v \big) + T \big( \rho ([Tv, x]_\mathfrak{g}) u \big) - T \big( \rho (Tv) \rho (x) u \big) - T \big( \rho ([Tu, x]_\mathfrak{g}) v \big) \\
& \qquad \qquad + \lambda T ([u, \rho (x) v]_\mathfrak{h}) - \lambda T ([v, \rho(x) u]_\mathfrak{h}) + [[Tu, Tv]_\mathfrak{g}, x ]_\mathfrak{g} \\
&= - T \big( \rho (x) \rho (Tv) u \big) + T \big( \rho (x) \rho (Tu) v \big) + \lambda T (\rho (x) [u, v]_\mathfrak{h}) + [T [u, v]_T, x]_\mathfrak{g} \\
&= T \big( \rho(x) [u, v]_T \big) + [T [u, v]_T, x]_\mathfrak{g} \\
&= \rho^T ([u, v]_T)x.
\end{align*}
This shows that $\rho^T$ is a representation. Hence the proof.
\end{proof}
Let $T : \mathfrak{h} \rightarrow \mathfrak{g}$ be a $\lambda$-weighted relative Rota-Baxter operator. Then it follows from the above proposition that one may consider the Chevalley-Eilenberg cohomology of the Lie algebra $(\mathfrak{h}, [~,~]_T)$ with coefficients in the representation $(\mathfrak{g}, \rho^T)$. More precisely, the $n$-th cochain group is $C^n_\mathsf{CE} (\mathfrak{h}, \mathfrak{g}) = \mathrm{Hom}(\wedge^n \mathfrak{h}, \mathfrak{g})$, for $n \geq 0$, and the coboundary map $\delta_\mathsf{CE} : C^n_\mathsf{CE} (\mathfrak{h}, \mathfrak{g}) \rightarrow C^{n+1}_\mathsf{CE} (\mathfrak{h}, \mathfrak{g})$ given by
\begin{align*}
&(\delta_\mathsf{CE} f)(u_1, \ldots, u_{n+1}) \\
&= \sum_{i=1}^{n+1} (-1)^{i+1} ~ T \big( \rho (f(u_1, \ldots, \widehat{u_i}, \ldots, u_{n+1})) u_i \big) + \sum_{i=1}^{n+1} (-1)^{i+1} ~ [ Tu_i, f(u_1, \ldots, \widehat{u_i}, \ldots, u_{n+1})]_\mathfrak{g} \\
&+ \sum_{i < i} (-1)^{i+j} ~ f \big( \rho (Tu_i) u_j - \rho (Tu_j) u_i + \lambda [u_i, u_j]_\mathfrak{h}, u_1, \ldots, \widehat{u_i}, \ldots, \widehat{u_j}, \ldots, u_{n+1} \big).
\end{align*}
The corresponding cohomology groups are denoted by $H^\ast_\mathsf{CE}(\mathfrak{h}, \mathfrak{g})$. Moreover, a direct computation says that
\begin{align*}
\delta_T f = (-1)^n ~\delta_\mathsf{CE} f, ~ \text{ for } f \in \mathrm{Hom} (\wedge^n \mathfrak{h}, \mathfrak{g}).
\end{align*}
Hence the cohomology $H^\ast_T (\mathfrak{h}, \mathfrak{g})$ is isomorphic to the Chevalley-Eilenberg cohomology $H^\ast_\mathsf{CE}(\mathfrak{h}, \mathfrak{g})$.
\begin{remark}
Similar to Section \ref{sec-4}, one may study deformations of $\lambda$-weighted relative Rota-Baxter operators on Lie algebras. The main results are similar to those given in Section \ref{sec-4}.
\end{remark}
\subsection{Relation with associative case}
Let $A$ be an associative algebra. Then there is a Lie algebra structure on $A$ given by the commutator bracket $[a, b]_c = a \cdot_A b - b \cdot_A a$, for $a, b \in A$. We denote this Lie algebra by $A_c$. If $A$ and $B$ are two associative algebras and $B$ is an associative $A$-bimodule, then it is easy to see that the Lie algebra $A_c$ acts on the Lie algebra $B_c$ by a Lie algebra homomorphism
\begin{align*}
\rho : A_c \rightarrow \mathrm{Der}(B_c), ~ \rho (a)(u) = a \cdot u - u \cdot a, ~ \text{ for } a \in A_c, u \in B_c.
\end{align*}
With the above notations, we have the following.
\begin{prop}\label{skew-prop}
\begin{itemize}
\item[(i)] Let $T \in \mathsf{rRB}_\lambda (B, A)$. Then $T \in \mathsf{rRB}_\lambda (B_c, A_c)$.
\item[(ii)] Let $T, T' \in \mathsf{rRB}_\lambda (B, A)$. If $(\phi, \psi)$ is a morphism from $T$ to $T'$ in the category $\mathsf{rRB}_\lambda (B, A)$, then $(\phi, \psi)$ is also a morphism from $T$ to $T'$ in the category $\mathsf{rRB}_\lambda (B_c, A_c)$.
\end{itemize}
\end{prop}
\begin{proof}
(i) For any $u, v \in B_c$, we have
\begin{align*}
[T(u), T(v)]_{A_c} =~& T(u) \cdot_A T(v) - T(v) \cdot_A T(u) \\
=~& T (T(u) \cdot v + u \cdot T(v) + \lambda ~ u \cdot_B v) - T (T(v) \cdot u + v \cdot T(u) + \lambda ~ v \cdot_B u) \\
=~& T ( \rho (Tu) v - \rho (Tv) u + \lambda ~ [u, v]_{B_c} ).
\end{align*}
This proves that $T \in \mathsf{rRB}_\lambda (B_c, A_c).$
(ii) Since $\phi : A \rightarrow A$ (resp. $\psi : B \rightarrow B$) is an algebra morphism, it follows that $\phi : A_c \rightarrow A_c$ (resp. $\psi: B_c \rightarrow B_c$) is a Lie algebra morphism. Moreover, we have $\phi \circ T = T' \circ \psi$. Finally,
\begin{align*}
\psi (\rho (a) u) = \psi (a \cdot u - u \cdot a) = \phi (a) \cdot \psi (u) - \psi (u) \cdot \phi (a) = \rho (\phi (a)) \psi (u).
\end{align*}
This shows that $(\phi, \psi)$ is a morphism from $T$ to $T'$ in the category $\mathsf{rRB}_\lambda (B_c, A_c)$.
\end{proof}
Let $T \in \mathsf{rRB}_\lambda (B, A)$.
In the following, we find the relation between the cohomologies $H^\ast_T (B,A)$ and $H^\ast_T (B_c, A_c)$. We start by recalling the following standard result. Let $A$ be an associative algebra and $M$ be an $A$-bimodule. Then $M$ can be considered as a representation of the Lie algebra $A_c$ with the action $\rho : A_c \rightarrow \mathrm{End}(M)$ given by $\rho (a)(m) = a \cdot m - m \cdot a$, for $a \in A_c$ and $m \in M$. We denote this representation by $M_c$.
\begin{prop}\label{sn-map}
The collection $\{ S_n \}_{n \geq 0}$ of maps $S_n : \mathrm{Hom}(A^{\otimes n}, M) \rightarrow \mathrm{Hom}(\wedge^n A_c, M_c)$ defined by
\begin{align*}
S_n (f) (a_1, \ldots, a_n ) = \sum_{\sigma \in \mathbb{S}_n} (-1)^\sigma ~ f (a_{\sigma(1)}, \ldots, a_{\sigma (n)})
\end{align*}
is a morphism from the Hochschild cochain complex of $A$ with coefficients in the $A$-bimodule $M$ to the Chevalley-Eilenberg cochain complex of $A_c$ with coefficients in the representation $M_c$.
\end{prop}
\medskip
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Let $T \in \mathsf{rRB}_\lambda (B, A)$ be a $\lambda$-weighted relative Rota-Baxter operator. Then we have seen in Subsection \ref{subsec-hoch} that the cohomology of $T$ is isomorphic to the Hochschild cohomology of $(B, \cdot_T)$ with coefficients in the bimodule $(A, l^T, r^T)$.
On the other hand, we have from Proposition \ref{skew-prop} that $T \in \mathsf{rRB}_\lambda (B_c, A_c)$ . The cohomology of $T$ (as a $\lambda$-weighted relative Rota-Baxter operator on Lie algebra) is isomorphic to the Chevalley-Eilenberg cohomology of $(B, [~,~]_T)$ with coefficients in the representation $(A, \rho^T)$, where
\begin{align*}
[u, v]_T := u \cdot_T v - v \cdot_T u ~~~~ \text{ and } ~~~~
\rho^T (u) (a) := l^T_u (a) - r^T_u (a).
\end{align*}
Thus, as a consequence of Proposition \ref{sn-map}, we get the following.
\begin{thm}
Let $A$ and $B$ be two associative algebras and $B$ be an associative $A$-bimodule. Let $T \in \mathsf{rRB}_\lambda (B, A)$ be a $\lambda$-weighted relative Rota-Baxter operator. Then the collection $\{ S_n \}_{n \geq 0}$ of maps
\begin{align*}
S_n : \mathrm{ Hom}(B^{\otimes n}, A) \rightarrow \mathrm{Hom}(\wedge^n B, A), ~~~ S_n (f) (u_1, \ldots, u_n) = \sum_{\sigma \in \mathbb{S_n}} (-1)^\sigma ~ f (u_{\sigma(1)}, \ldots, f_{\sigma (n)})
\end{align*}
induces a morphism of cohomologies from $H^\ast_T (B, A)$ to $H^\ast_T (B_c, A_c)$.
\end{thm}
\medskip
\noindent {\bf Acknowledgements.} The research is supported by the fellowship of Indian Institute of Technology (IIT) Kanpur. The author thanks the Institute for support.
\medskip
\medskip
| {
"timestamp": "2021-08-13T02:01:49",
"yymm": "2108",
"arxiv_id": "2108.05411",
"language": "en",
"url": "https://arxiv.org/abs/2108.05411",
"abstract": "Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\\lambda \\in {\\bf k}$, we construct a differential graded Lie algebra whose Maurer-Cartan elements are given by $\\lambda$-weighted relative Rota-Baxter operators. Using such characterization, we define the cohomology of a $\\lambda$-weighted relative Rota-Baxter operator $T$, and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal and finite order deformations of $T$ from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation which is the obstruction to extend the deformation. In the end, we also consider the cohomology of $\\lambda$-weighted relative Rota-Baxter operators in the Lie case and find a connection with the case of associative algebras.",
"subjects": "Representation Theory (math.RT); Rings and Algebras (math.RA)",
"title": "Cohomology and deformations of weighted Rota-Baxter operators",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465639,
"lm_q2_score": 0.7248702880639792,
"lm_q1q2_score": 0.7092019742071437
} |
https://arxiv.org/abs/quant-ph/0501177 | For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets | Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group D_n in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest. | \section{The Hidden Conjugate Problem}
Consider the following special case of the Hidden Subgroup Problem,
called the \emph{Hidden Conjugate Problem} in~\cite{MooreRRS04}. Let
$G$ be a group, and $H$ a non-normal subgroup of $G$; denote
conjugates of $H$ as $H^g = g^{-1} H g$. Then we are promised that
the hidden subgroup is $H^g$ for some $g$, and our goal is to find out
which one.
The usual approach is to prepare a uniform superposition over the
group, entangle the group element with a second register by
calculating or querying the oracle function, and then measure the
oracle function. This yields a uniform superposition over a random
left coset of the hidden subgroup,
\[ \ket{cH^g} = \frac{1}{\sqrt{|H|}} \sum_{h \in H^g} \ket{ch} \enspace . \]
Rather than viewing this as a pure state where $c$ is random, we may
treat this as a classical mixture over left cosets, giving the mixed
state with density matrix
\begin{equation}
\label{eq:rhog}
\rho_g = \frac{1}{|G|} \sum_{c \in G} \ket{cH^g} \bra{cH^g} \enspace .
\end{equation}
We then wish to find a positive operator-valued measurement (POVM) to
identify $g$.
A POVM consists of a set of positive \emph{measurement operators} $\{
E_i \}$ that obey the completeness condition
\begin{equation}
\label{eq:complete}
\sum_i E_i = \varmathbb 1 \enspace .
\end{equation}
If we are trying to distinguish $\ell$ density matrices $\{ \rho_i \mid 1 \leq
i \leq \ell \}$, and if $\rho_i$ is chosen with probability $p_i$, the
probability that the POVM gives the right answer is
\begin{equation}
\label{eq:success}
P_{\rm success} = \sum_i p_i \, \textbf{tr}\, E_i \rho_i \enspace .
\end{equation}
A theorem of Yuen, Kennedy and Max~\cite{YuenKM} and Holevo~\cite{Holevo} states that $P_{\rm success}$ is maximized if and only if the following conditions hold for every $i$:
\begin{gather}
\left( \sum_j p_j \rho_j E_j - p_i \rho_i \right) E_i
= 0\enspace, \label{eq:holevo}\\
\sum_j p_j \rho_j E_j = \sum_j p_j E_j \rho_j\enspace,\; \text{and} \label{eq:commute} \\
\sum_j p_j \rho_j E_j \geq p_i \rho_i\enspace, \label{eq:positive}
\end{gather}
where we write $A \geq B$ if $A-B$ is positive semidefinite.
These conditions come from recognizing that maximizing $P_{\rm success}$
subject to the completeness condition~\eqref{eq:complete} gives a semidefinite program.
Ip~\cite{Ip} used this fact to show that Shor's algorithm is optimal
for the hidden subgroup problem on the cyclic group $\mathbb{Z}_n$.
In a beautiful recent paper, Bacon, Childs and van Dam~\cite{BaconCvD}
consider the hidden conjugate problem for the dihedral group $D_n$
where $H$ is an order-2 subgroup; that is, when the hidden subgroup consists
of the identity and one of the $n$ involutions. They consider entangled
measurements over multiple registers each of which contains a coset
state, and show that the so-called ``pretty good
measurement''~\cite{PGM}, defined below, is in fact optimal for any
number of registers. They then show that this optimal measurement is
related to random cases of the Subset Sum problem (see also
Regev~\cite{Regev}). Other prior work includes Eldar, Megretski, and
Verghese~\cite{Eldar}, who showed that the PGM is optimal for certain
families of density matrices related by group symmetries; see references in that paper
for some other cases for which the PGM is optimal.
In this paper we point out that for any subgroup $H$ of a group $G$,
the pretty good measurement is the optimal one-register experiment for
finding a hidden conjugate of $H$ when all conjugates are equally
likely. We then write a general expression for the probability
$P_{\rm success}$ that this measurement identifies the hidden conjugate in
a single experiment on a coset state. We then go on to show that when $G$ and
$H$ form a Gel'fand pair, the pretty good measurement is in fact the
optimal measurement on \emph{any number of registers}. This recovers the optimality result
of~\cite{BaconCvD} as a special case, and establishes
optimal measurements for a variety of other interesting group-subgroup
pairs, including the subgroups of the affine groups studied
in~\cite{MooreRRS04}.
We use the machinery of representation theory; we refer the reader
to~\cite{FultonH91,Serre77} or to the review in our
paper~\cite{MooreRS} for an introduction and for notation.
\section{The one-register case}
\subsection{The pretty good measurement is optimal}
The \emph{pretty good measurement} (PGM), also known as
the \emph{least squares measurement}, is defined as
follows~\cite{PGM}. Given a set of density matrices $\rho_i$ with
associated probabilities $p_i$, let
\[ M = \sum_i p_i \rho_i . \]
Then the PGM associated with this family of mixed states is $\{ E_i \}$,
where the measurement operator $E_i$ is defined as
\begin{equation}
\label{eq:pgm}
E_i = p_i M^{-1/2} \rho_i M^{-1/2}
\end{equation}
where the inverse $M^{-1/2}$ is defined on the image of $M$; that is,
$M^{-1/2}$ is the unique positive operator such that $(M^{-1/2})^2 M$
is the projection operator $\Pi$ onto the image of $M$.
If $M$ has full rank, it is easy to see that this choice of
$\{ E_i \}$ satisfies the completeness condition~\eqref{eq:complete}; if the image
of $M$ is a proper subspace, we satisfy the completeness condition by adding
an additional measurement which projects onto its orthogonal complement.
We will show that the optimality
conditions~\eqref{eq:holevo},~\eqref{eq:commute},
and~\eqref{eq:positive} hold for the PGM for the family of density
matrices $\{ \rho_g \}$ defined above when endowed with the uniform
distribution $p_g = 1/|G|$. First we derive the structure of the PGM.
Observe that the mixed state $\rho_g = |G|^{-1} \sum_c \ket{cH^g}\bra{cH^g}$
has the property that
\begin{equation}
\label{eq:rhosquare}
\rho_g^2 = \frac{|H|}{|G|} \rho_g
\end{equation}
so that $\rho_g$ is a projection operator scaled by the constant
$|H|/|G|$ (this follows from the fact that the uniform distribution on
any subgroup is its own square under convolution).
As $\textbf{tr}\, \rho_g = 1$, we must have $\textbf{rk}\; \rho_g = |G|/|H|$,
the index of $H$ in $G$.
Note that $\rho_g$ commutes with the left action of $G$, since it is ``symmetrized''
over all left cosets. By Schur's lemma, $\rho_g$ is block diagonal, with blocks corresponding
to the decomposition of $\C[G]$ into bi-invariant spaces. Furthermore, the
block corresponding to the irreducible representation $\sigma$ has form
$A_\sigma \otimes \varmathbb 1_{d_\sigma}$, where $\varmathbb 1_{d_\sigma}$ acts within each
left-invariant space. Recalling~\eqref{eq:rhosquare}, $A_\sigma$ is a rescaled
projection operator, and we may write $\rho_g = \oplus_{\sigma \in \widehat{G}} \,\rho_g^\sigma$ where
\begin{equation}
\label{eq:rhogsigma}
\rho_g^\sigma = \frac{|H|}{|G|} \pi_g^\sigma \otimes \varmathbb 1_{d_\sigma}\enspace;
\end{equation}
here $\pi_g^\sigma$ is the projection operator
\begin{equation}
\label{eq:pi}
\pi_g^\sigma = \frac{1}{|H|} \sum_{h \in H^g} \sigma(h) \enspace .
\end{equation}
Since $\textbf{rk}\; \pi_g^\sigma$ does not depend on $g$, we denote it simply as
$\textbf{rk}\; \pi^\sigma$. Then the following lemma describes the structure of
the PGM for $\{ \rho_g \}$. Since $\rho_g = \rho_h$ whenever $g$ and $h$
are in the same coset of the normalizer ${\rm Norm}(H) = \{ g \mid H^g = H
\}$, without loss of generality we assume that the POVM gives a
uniformly random element of some coset of ${\rm Norm}(H)$, and we count it
as having succeeded if it gives an element of the correct coset. Thus
we will multiply $P_{\rm success}$ by the index of ${\rm Norm}(H)$ below.
\begin{lemma}
\label{lem:pgm}
For the family $\{ \rho_g \}$ of density matrices corresponding to random left cosets of conjugate subgroups $H^g$ with the uniform distribution on $g$, the pretty good measurement operators $E_g$ are given by
\[ E_g = \bigoplus_{\sigma \in \widehat{G}} E_g^\sigma \]
and, for each $\sigma \in \widehat{G}$,
\begin{equation}
\label{eq:eg}
E_g^\sigma = \frac{d_\sigma}{|G| \textbf{rk}\; \pi^\sigma} \pi_g^\sigma \otimes \varmathbb 1_{d_\sigma}
\end{equation}
where $\pi_g^\sigma$ is defined as in~\eqref{eq:rhogsigma}.
\end{lemma}
\begin{proof}
For each $x \in G$, let $L_x$ and $R_x$ denote the unitary operators
that carry out left and right group multiplication by $x$. Note that left cosets are mapped to each other by $L_x$ and, in particular, $\ket{cH} = L_c \ket{H}$. Note furthermore that $R_x$ maps left cosets of one conjugate onto left cosets of another conjugate, e.g., $\ket{H^g} = R_g \ket{g^{-1} H}$, and that $L_x$ commutes with $R_y$ for all $x,y$.
We have $p_g = 1/|G|$ for all $g$. Now if we write
\begin{align*}
M &= \frac{1}{|G|} \sum_g \rho_g = \frac{1}{|G|^2} \sum_{c,g} \ket{cH^g} \bra{cH^g}
= \frac{1}{|G|^2} \sum_{c,g} \ket{cHg} \bra{cHg} \\
&= \frac{1}{|G|^2} \sum_{c,g} L_c R_g \ket{H} \bra{H} R_g^\dagger L_c^\dagger
\end{align*}
(where in the third equality we replace $cg^{-1}$ with $c$),
we see that $M$ commutes with $L_x$ and $R_x$ for all $x \in G$. That is, summing over both the left coset and the choice of conjugate ``symmetrizes'' $M$ on both the left and the right. It follows by Schur's lemma that $M$ takes the form
\[ M = \bigoplus_{\sigma \in \widehat{G}} M^\sigma \]
where $M^\sigma$ is a scalar multiple of the identity operator for each $\sigma$. As
\[ M_\sigma = \frac{1}{|G|} \sum_g \rho_g^\sigma = \frac{|H|}{|G|^2} \sum_g \pi_g^\sigma \otimes \varmathbb 1_{d_\sigma} \enspace , \]
by taking traces we conclude that
$$
M^\sigma = \left( \frac{|H|}{|G|} \frac{\textbf{rk}\; \pi^\sigma}{d_\sigma}\right) \varmathbb 1_{d_\sigma^2}
\enspace .
$$
Similarly, $\rho_g$ is block-diagonal, as it commutes with $L_x$
(though not with $R_x$ unless $H$ is normal). Therefore, $M$ commutes with
$\rho_g$ for each $g$, and~\eqref{eq:pgm} becomes
\begin{equation}
\label{eq:infact}
E_g = \frac{1}{|G|} M^{-1} \rho_g
\end{equation}
giving, in each irreducible block,
\begin{equation}
\label{eq:pgm-operators-single}
E_g^\sigma = \frac{d_\sigma}{|H|\, \textbf{rk}\; \pi^\sigma} \rho_g^\sigma
= \frac{d_\sigma}{|G| \,\textbf{rk}\; \pi^\sigma} \pi_g^\sigma \otimes \varmathbb 1_{d_\sigma}
\end{equation}
which completes the proof.
\end{proof}
We now give our proof that the PGM is optimal for the hidden conjugate problem for any $G$ and $H$. This follows simply from the fact that $M$, and therefore $E_g$, commutes with $\rho_g$ for each $g$.
\begin{theorem}
\label{thm:pgm}
For the family $\{ \rho_g \}$ of density matrices corresponding to random left cosets of conjugate subgroups $H^g$ with the uniform distribution on $g$, the pretty good measurement $\{ E_g \}$ optimizes the probability of correctly measuring $g$.
\end{theorem}
\begin{proof}
Since the $E_g$ and $\rho_g$ are block-diagonal (according to the same
decomposition of $\C[G]$), it suffices to confirm the optimality
criteria~\eqref{eq:holevo}, \eqref{eq:commute},
and~\eqref{eq:positive} in each block, i.e., for each $\sigma \in \widehat{G}$.
First,~\eqref{eq:commute} holds trivially since $E_g^\sigma$ is proportional to $\rho_g^\sigma$
for each $g$, and so commutes with it.
As for condition~\eqref{eq:positive}, observe that from~\eqref{eq:rhosquare}
and~\eqref{eq:pgm-operators-single} we have
\begin{equation}
\label{eq:rhogeg}
\rho_g^\sigma E_g^\sigma = \frac{d_\sigma}{|G| \,\textbf{rk}\; \pi^\sigma} \rho_g^\sigma
\end{equation}
and so for any $h \in G$, recalling that $p_g = 1/|G|$ for all $g$, we have
\begin{align*}
\frac{1}{|G|} \sum_g \rho_g^\sigma E_g^\sigma
&= \frac{d_\sigma}{|G| \,\textbf{rk}\; \pi^\sigma} M^\sigma
= \frac{|H|}{|G|^2} \left( \varmathbb 1_{d_\sigma} \otimes \varmathbb 1_{d_\sigma} \right)
\geq \frac{|H|}{|G|^2} \left( \pi_h^\sigma \otimes \varmathbb 1_{d_\sigma} \right)
= \frac{1}{|G|} \rho_h
\end{align*}
since $\varmathbb 1_{d_\sigma} \geq \pi_h^\sigma$. Finally, for any $h \in G$
\begin{align*}
\left( \sum_g \rho_g^\sigma E_g^\sigma - \rho_h^\sigma \right) E_h^\sigma
&= \frac{|H|}{|G|} \left[ \left( \varmathbb 1_{d_\sigma} - \pi_h^\sigma \right) \otimes \varmathbb 1_{d_\sigma} \right] E_h^\sigma
= \frac{|H| \,d_\sigma}{|G|^2 \,\textbf{rk}\; \pi^\sigma} \left[ \left( \varmathbb 1_{d_\sigma} - \pi_h^\sigma\right) \pi_h^\sigma \right]
\otimes \varmathbb 1_{d_\sigma} = 0
\end{align*}
since $(\varmathbb 1-\pi)\pi = 0$ for any projection operator $\pi$. Again
recalling $p_g = 1/|G|$ establishes~\eqref{eq:holevo} and completes the proof.
\end{proof}
Note that Lemma~\ref{lem:pgm} and Theorem~\ref{thm:pgm} imply that, as pointed out before~\cite{Ip,Kuperberg03,MooreRS}, the optimal measurement consists of first measuring the representation name $\sigma$, and then performing an additional measurement $M_g^\sigma$ inside the bi-invariant space corresponding to $\sigma$.
\subsection{Partial measurements}
Suppose that rather than trying to identify the conjugate exactly, we wish to learn some partial information about it. To learn one bit, for instance, we would divide the set of conjugates into two equal subsets, and combine the $\rho_g$ into two mixed states $\rho_0$ and $\rho_1$ consisting of mixtures of those in the two subsets. The next theorem shows that the PGM is optimal for any such partial measurement as long as each subset of the set of conjugates has probability proportional to its size.
\begin{theorem}
\label{thm:partial}
Let $C(H)=\{ H^g \}$ be partitioned into disjoint sets $C_i$, $1 \leq i
\leq \ell$. Let
\[ \rho_i = \frac{1}{|C_i|} \sum_{g: H^g \in C_i} \rho_g \]
where $\rho_g$ is as in~\eqref{eq:rhog}, and let $p_i = |C_i|/|C(H)|$.
Then the pretty good measurement for the family $\{ \rho_i \}$ with
probabilities $p_i$ is the family of operators $\{ E_i \}$
\[ E_i = \sum_{g : H^g \in C^i} E_g \enspace , \]
where $E_g$ is given by Lemma~\ref{lem:pgm};
this measurement is optimal.
\end{theorem}
\begin{proof} The proof is exactly the same as that of Lemma~\ref{lem:pgm} and Theorem~\ref{thm:pgm}, except that for each $i$ we sum over the $g$ with $H^g \in C_i$.
\end{proof}
\subsection{The probability of success}
As a corollary to Theorem~\ref{thm:pgm}, we can determine the optimal
success probability. Let $C(H)$ denote the set of conjugates of $H$.
For a group $G$ the \emph{Plancherel measure} is the probability distribution on $\widehat{G}$
assigning $\sigma \in \widehat{G}$ the probability $d_\sigma^2/|G|$; this is the fraction, dimensionwise,
of $\C[G]$ consisting of the bi-invariant subspace corresponding to $\sigma$.
For a set $S \subset \widehat{G}$, we let $\textrm{Planch}(S)$ denote the probability of
observing an element of $S$ according to the Plancherel measure. We
remark that the Plancherel measure is precisely the probability
distribution obtained by performing weak Fourier sampling when the
hidden subgroup is trivial, since in that case the state $\rho$ is completely mixed.
Given a subgroup $H$ of $G$, let $S_H \subseteq \widehat{G}$ denote the
set of irreducible representations for which the projection operator
$\pi_H^\sigma = |H|^{-1} \sum_{h \in H} \sigma(h)$ is nonzero,
and let $\core{H}{G} = \cap_g H^g$ denote the largest normal subgroup
contained in $H$. Then the following theorem gives the success probability of the
optimal one-register experiment.
\begin{theorem}
\label{thm:pgm-success-single}
Given a group $G$ and a subgroup $H$, the probability that
the optimal single-register measurement correctly identifies a
uniformly random conjugate of $H$ is
\begin{equation}
\label{eq:psuccess-single}
P_{\rm success} = \frac{|H|}{|C(H)|} \textrm{Planch}(S_H) \leq \frac{|H|}{|C(H)| \cdot |\core{H}{G}|} \enspace .
\end{equation}
\end{theorem}
\begin{proof}
Let us say that $g \sim g'$ if $H^g = H^{g'}$ (and so $\rho_g = \rho_{g'}$).
Then $P_{\rm success}$ is the expectation
\[ P_{\rm success} = \Exp_g \sum_{g' \sim g} \textbf{tr}\, E_{g'} \rho_g \enspace , \]
where $g$ is selected uniformly in $G$. Since $g \sim g'$ if and only if
$g$ and $g'$ are in the same (right) coset of the normalizer
${\rm Norm}(H) = \{ g \mid H^g = H \}$, we have
\[
P_{\rm success} = |{\rm Norm}(H)| \cdot \Exp_g \textbf{tr}\, E_{g} \rho_g
= |{\rm Norm}(H)| \cdot \Exp_g \sum_\sigma \textbf{tr}\, E_{g}^\sigma \rho_g^\sigma \enspace .
\]
Considering~\eqref{eq:rhogsigma} and~\eqref{eq:rhogeg}, we have
\[ \textbf{tr}\, E_g^\sigma \rho_g^\sigma = \frac{|H| \,d_\sigma^2}{|G|^2}
= \frac{|H|}{|G|} \textrm{Planch}(\sigma) \enspace , \]
but only for those $\sigma$ where $\rho_g^\sigma$ and $E_g^\sigma$ are nonzero,
i.e., those for which $\textbf{rk}\; \pi^\sigma > 0$. Thus, we conclude that
\[
P_{\rm success} = \frac{|{\rm Norm}(H)| |H|}{|G|} \cdot \sum_{\sigma \in S_H} \textrm{Planch}(\sigma)
= \frac{|H|}{|C(H)|} \cdot\textrm{Planch}(S_H)
\]
where we recall that $|C(H)| = |G|/|{\rm Norm}(H)|$.
Now, note that for any subgroup $K \subseteq H$, we have $S_H \subseteq S_K$
since any $\sigma$ that annihilates $K$ also annihilates $H$.
In addition, if $K$ is normal, recall that for any $\sigma$
we have either $\pi_K^\sigma = 0$ or $\pi_K^\sigma = \varmathbb 1$. It follows that
$\sum_{\sigma \in S_K} d_\sigma^2 = \sum_{\sigma \in S_K} d_\sigma \textbf{rk}\; \pi_K^\sigma
= \textbf{rk}\; \pi_K^R$ where $R$ is the regular representation, and since
$\textbf{rk}\; \pi_K^R = |G|/|K|$ we have $\textrm{Planch}(S_K) = 1/|K|$.
Thus $\textrm{Planch}(S_H) \leq 1/|\core{H}{G}|$, completing the proof of~\eqref{eq:psuccess-single}.
\remove{
We remark, however, that $\textbf{rk}\; \pi^\sigma > 0$ exactly when $\langle \textrm{Res}_{H}
\chi_{\sigma}, 1\rangle_H > 0$ and hence, by Frobenius reciprocity
(see~\cite{Serre77}), that $\langle \chi_{\sigma}, \textrm{Ind}_{H} 1\rangle_G > 0$.
Specifically, $S$ consists of precisely those representations that
occur in the representation $\textrm{Ind}_{H} 1$. It is easy to argue that
$\ker (\textrm{Ind}_H^G 1) = \core{H}{G} =
\cap_{\ell \in L} H^\ell$; hence $S \subseteq \{ \sigma \mid \core{H}{L} < \ker \sigma\}$
and $\sum_{\sigma \in S} d_\sigma^2 \leq |G| / |\core{H}{G}|$. The
inequality of~\eqref{eq:psuccess-single} follows.
}
\end{proof}
Note that if we observe any $\sigma \notin S_H$, we know that the promise
that the hidden subgroup is a conjugate of $H$ has been violated. Thus,
as in~\cite{BaconCvD}, if we are promised that the hidden subgroup is
either trivial or a conjugate of $H$, we can complete the PGM with an
additional measurement operator $M_0$ that projects onto the orthogonal
complement of $S_H$, and conclude that the hidden subgroup is trivial if we observe
the outcome $M_0$.
It is interesting to compare Theorems~\ref{thm:pgm} and~\ref{thm:pgm-success-single}
with known results on the hidden subgroup
problem. For the dihedral group $D_n$ where $H$ is an order-2
subgroup, there are $n$ conjugates, and $S_H$ consists of all of $\widehat{G}$ except
for the sign representation. Thus we have
\[ P_{\rm success} = \frac{2}{n} \left( 1- \frac{1}{2n} \right) \enspace . \]
On the other hand, for the affine group $A_p$, the maximal subgroup $H=\mathbb{Z}_p^*$ has
$p$ conjugates, and $S_H$ includes all but the $p-2$ nontrivial one-dimensional
representations. This gives
\[ P_{\rm success} = \frac{p-1}{p} \left( 1-\frac{p-2}{p(p-1)} \right) = 1-\frac{2(p-1)}{p^2} \enspace . \]
Indeed, Moore, Rockmore, Russell and Schulman~\cite{MooreRRS04} gave an explicit algorithm using a von Neumann measurement that succeeds with constant probability. This algorithm can easily be modified to carry out the optimal POVM in polynomial time; see also Bacon, Childs and van Dam~\cite{BaconCvDHeisenberg}.
Now let us consider the case of the hidden subgroup problem relevant to Graph Isomorphism in the case of two rigid, connected graphs of size $n/2$. Here $G=S_n$ and $H$ is the order-2 subgroup consisting of $n/2$ disjoint transpositions,
\[ H = \{1, (1\,2)(3\,4)\cdots(n-1\,n) \} \]
of which there are $(n-1)!!$ conjugates, one for each perfect matching of $n$ items.
Using lemmas proved in~\cite{MooreRS}, it is easy to show that for almost all representations $\sigma$
(with respect to the Plancherel distribution) we have $\textbf{rk}\; \pi^\sigma = (1 \pm o(1)) d_\sigma / 2$,
so $\textrm{Planch}(S_H) = 1-o(1)$. Thus we have
\[ P_{\rm success} = \frac{2}{(n-1)!!} (1-o(1)) = n^{-n/2} e^{O(n)} \]
This can be generalized to other conjugacy classes using general character bounds due to Roichman~\cite{Roichman96}; see also Kempe and Shalev~\cite{KempeS}.
However, it should be emphasized that the fact that $P_{\rm success}$ is exponentially small does not mean that we need an exponential number of single-register experiments to solve the hidden conjugate problem. In particular, Ettinger and H{\o}yer~\cite{EttingerH98} showed that a polynomial number (i.e., $O(\log |G|) = O(\log n)$) of single-register experiments is enough to determine, information-theoretically, an involution in $D_n$. Thus our results here do not subsume the results of Moore, Russell and Schulman~\cite{MooreRS} and Moore and Russell~\cite{MooreR}, who showed that it takes an exponential number of single-register experiments, or a super-polynomial number of two-register experiments, to obtain \emph{even a single bit of information} about the conjugate of $H$ in $S_n$.
\section{Multiregister measurements and Gel'fand pairs}
For the multiregister experiment, we view states as elements of the
Hilbert space $\mathbb{C}[G^k] = \C[G]^{\otimes k}$. We now have a random left coset of the subgroup
$H^k \subset G^k$, and the corresponding mixed state is
$$
{\boldsymbol{\rho}}_g = \rho_g^{\otimes k} = \frac{1}{|G|^k} \sum_{\vec{c} \in G^k} \ket{ \vec{c} (H^g)^k}\bra{ \vec{c} (H^g)^k}\enspace.
$$
Since $\rho_g^{\otimes k}$ is symmetrized over left cosets, it commutes with
left multiplication in $G^k$. Thus by Schur's lemma it is block-diagonal,
where each block corresponds to
a representation ${\boldsymbol{\sigma}} = \sigma_1 \otimes \cdots \otimes \sigma_k$
of $G^k$, and each $\sigma_i$ is an irreducible representation of $G$.
Indeed, in a given such block we can write
\begin{equation}
\label{eq:vrho}
{\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}
= \rho_g^{\sigma_1} \otimes \cdots \otimes \rho_g^{\sigma_k}
\enspace .
\end{equation}
The situation in the multiregister case is complicated by the fact that, unlike the one-register case, $M$ does not generally commute with ${\boldsymbol{\rho}}_g$. Indeed, they do not commute even for the two-register case in the dihedral group. We note in passing that they do commute in a few special cases: for instance, when $H$ is generated by an involution that commutes with its conjugates. However, this is not a very interesting case, since then $H$ and its conjugates generate an Abelian subgroup $K \subset G$, and we can distinguish them by solving the hidden subgroup problem on $K$.
However, we can still prove that the PGM is optimal in the case that $G$ and $H$ form a Gel'fand pair; we review the definition here, and also refer the reader to~\cite{Terras99} for an introduction.
Given a group $G$ and a subgroup $H$, let $\mathcal{B} =
\mathcal{B}_H(G)$ denote the collection of functions $f: G \to \mathbb{C}$
that are invariant under both left and right multiplication by $H$,
i.e., such that $f(hg) = f(g) = f(gh)$ for all $g \in G$ and $h \in H$.
This collection of \emph{bi-invariant} functions forms a natural
algebra under convolution, and $\mathcal{B}$ can be identified
with the subalgebra of $\C[G]$ generated by elements corresponding
to double cosets,
$HgH = (\sum_{h \in H} h) \cdot g \cdot (\sum_{h \in H} h)$.
Then the following criteria are equivalent, and the pair $(G,H)$
is said to be \emph{Gel'fand} if any of them hold:
\begin{enumerate}
\item $\mathcal{B}$ is commutative.
\item The induced representation $\textrm{Ind}_H^G \varmathbb 1$ contains no more than
one copy of any particular $\sigma \in \widehat{G}$.
\item For any $\sigma \in \widehat{G}$ and any $f \in \mathcal{B}$, the Fourier transform $\hat{f}(\sigma) = \sum_g f(g) \sigma(g)$ has rank at most one.
\end{enumerate}
The third criterion is the one most relevant to our analysis.
Suppose $(G,H)$ is a Gel'fand pair; then since the uniform distribution on $H$ is an element of $\mathcal{B}$,
for any $\sigma \in \widehat{G}$ the projection operator $\pi^\sigma = |H|^{-1} \sum_{h \in H} \sigma(h)$
has rank at most one. Since this is also true of its conjugates $\pi_g^\sigma = \sigma(g)^{-1} \pi^\sigma \sigma(g)$,
we see that $(G,H^g)$ is Gel'fand for all $g \in G$.
As stated above, Bacon, Childs and van Dam~\cite{BaconCvD} showed that the pretty good measurement is optimal for the dihedral groups $D_n$ when the hidden subgroup $H$ is of order 2. Indeed, $(D_n,H)$ is Gel'fand for these subgroups, and we generalize their result as follows.
\begin{theorem}
\label{thm:gelfand}
For any number of registers $k > 0$, given the family $\{ {\boldsymbol{\rho}}_g \}$ of density matrices corresponding to random left cosets of conjugate subgroups $(H^g)^k \subset G^k$ with the uniform distribution on $g$, the pretty good measurement $\{ E_g \}$ optimizes the probability of correctly measuring $g$.
\end{theorem}
\begin{proof} As before, we will show that the optimality conditions
hold in each irreducible block, since the ${\boldsymbol{\rho}}_g$, and therefore
$M$ and the $E_g$, are block-diagonal. Since the tensor product of
rank-one operators has rank one, given ${\boldsymbol{\sigma}} = \sigma_1 \otimes \cdots \otimes \sigma_k$
with $\sigma_i \in \widehat{G}$ for all $i$, from~\eqref{eq:vrho} we have either
${\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} = 0$ or
\[ {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} = \ket{v_g} \bra{v_g} \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}} \]
for some vector $v_g \in {\boldsymbol{\sigma}}$. In the latter case we have
$M^{\boldsymbol{\sigma}} = m \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}}$ where
\[ m = \frac{1}{|G|} \left( \sum_g \ket{v_g} \bra{v_g} \right)\]
and
\[ {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} E_g^{\boldsymbol{\sigma}}
= \frac{1}{|G|} \left(\ket{v_g} \bra{v_g} m^{-1/2} \ket{v_g} \bra{v_g} m^{-1/2}
\right) \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}} = \frac{C}{|G|} {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}
(M^{\boldsymbol{\sigma}})^{-1/2}
\]
where $C$ is defined as the inner product
\[ C = \bra{v_g} m^{-1/2} \ket{v_g} \enspace . \]
Similarly, $E_g^{\boldsymbol{\sigma}} {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} = \frac{C}{|G|}
(M^{\boldsymbol{\sigma}})^{-1/2} {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}$.
It is easy to see that $C$ does not depend on $g$: if $R_g$ denotes
the unitary operator corresponding to right multiplication by the
diagonal element $(g,\ldots,g)$ in $\mathbb{C}[G^k]$, then $\ket{v_g} = R_g
\ket{v_1}$ and $m$ commutes with $R_g$. Thus $C = \bra{v_g}
m^{-1/2} \ket{v_g} = \bra{v_1} R_g m^{-1/2} R_g^\dagger \ket{v_1} =
\bra{v_1} m^{-1/2} \ket{v_1}$. Then we have
\[ \sum_g {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} E_g^{\boldsymbol{\sigma}}
= \frac{C}{|G|} \sum_g {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} (M^{\boldsymbol{\sigma}})^{-1/2}
= C (M^{\boldsymbol{\sigma}})^{1/2}
= \frac{C}{|G|} \sum_g (M^{\boldsymbol{\sigma}})^{-1/2} {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}
= \sum_g E_g^{\boldsymbol{\sigma}} {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}
\]
confirming~\eqref{eq:commute}.
As for condition~\eqref{eq:positive}, recalling the equality $\sum_g
{\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} E_g^{\boldsymbol{\sigma}} = C (M^{\boldsymbol{\sigma}})^{1/2}$ just above,
\eqref{eq:positive} is equivalent to the condition
$C (M^{{\boldsymbol{\sigma}}})^{1/2} \geq {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}}$, and hence to
$C m^{1/2} \geq \ket{v_g} \bra{v_g}$.
Furthermore, this holds if and only if
\[ \bra{v_g} C m^{1/2} \ket{v_g}
\geq \inner{v_g}{v_g} \inner{v_g}{v_g}
= \norm{v_g}^4 \]
or, expanding the definition of $C$,
\begin{equation}
\label{eq:positivity-gelfand}
\bra{v_g}m^{1/2}\ket{v_g}\bra{v_g}m^{-1/2}\ket{v_g}
\geq \norm{v_g}^4 \enspace .
\end{equation}
We remark that if a positive semidefinite operator $A$ is a linear combination
$A = \sum_\alpha a_\alpha B_\alpha$ of positive semidefinite operators $B_\alpha$
with $a_\alpha \in \mathbb{R}^+$, then the kernel of $A$ is $\bigcap_\alpha \ker B_\alpha$.
As $m$ is such a linear combination of operators $\ket{v_g}\bra{v_g}$,
it follows that $\ket{v_g}$ is orthogonal to $\ker m$, and therefore lies in the
image of $m$. Thus we can regard $m^{1/2}$ and $m^{-1/2}$
as inverses, and the inequality~\eqref{eq:positivity-gelfand} follows from the
following claim:
\begin{claim}
Let $A$ be a positive operator on a finite dimensional Hilbert space
$V$ and let $\ket{v} \in V$. Then
\[
\bra{v} A \ket{v} \bra{v} A^{-1} \ket{v} \geq
\norm{v}^4 \enspace .
\]
\end{claim}
\begin{proof}
For a vector $w = (w_1, \ldots, w_n)$ of non-negative weights and a real
number $r$, define
\[
\mathcal{M}_r^w(x_1, \ldots, x_n) = \left(\frac{\sum_i w_i x_i^r}{\sum_i
w_i}\right)^{1/r}
\]
to be the \emph{weighted $r$-mean} of the positive vector $x = (x_1,
\ldots, x_n)$. The \emph{power mean inequality}
(cf.~\cite[\S2.9]{HardyLP}) asserts that for $r < s$ we have
$\mathcal{M}_r^w(x) \leq \mathcal{M}_s^w(x)$.
The claim follows immediately from the from power mean inequality
with $r = -1$ and $s = 1$. Specifically, let $B = \{ \ket{b_i} \}$
be a spectral resolution of $A$, so that $B$ is an orthogonal basis
of eigenvectors for $V$, and let
$\lambda_i > 0$ be the associated eigenvalues, so that $\lambda_i \ket{b_i} =
A \ket{b_i}$. Writing $\ket{v} = \sum_i v_i \ket{b_i}$, we have
\[
\bra{v} A \ket{v} = \sum_i |v_i|^2 \lambda_i\qquad\text{and}\qquad\bra{v}
A^{-1} \ket{v} = \sum_i |v_i|^{2} \lambda_i^{-1}\enspace.
\]
If we adopt the weights $w_i = |v_i|^2$, then $\sum_i w_i = \norm{v}^2$
and the claim is equivalent to the the power mean
inequality $\mathcal{M}_{-1}^w(\lambda_1, \ldots, \lambda_n) \leq
\mathcal{M}_{1}^w(\lambda_1, \ldots, \lambda_n)$.
\end{proof}
Returning to the proof of Theorem~\ref{thm:gelfand}, it remains to
establish~\eqref{eq:holevo}. Consider
\begin{align}
\left( \sum_g {\boldsymbol{\rho}}_g^{{\boldsymbol{\sigma}}} E_g^{\boldsymbol{\sigma}} - {\boldsymbol{\rho}}_h^{{\boldsymbol{\sigma}}} \right) E_h^{\boldsymbol{\sigma}}
&= \frac{1}{|G|} \left[\left( C m^{1/2} - \ket{v_h} \bra{v_h} \right)
m^{-1/2} \ket{v_h} \bra{v_h} m^{-1/2}\right] \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}} \nonumber \\
&= \frac{C}{|G|} \left[ \left( \Pi - \varmathbb 1 \right) \ket{v_h} \bra{v_h}
m^{-1/2} \right] \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}}
\label{eq:why}
\end{align}
where $\Pi$ is the projection operator onto the image of $m$. Since $\ket{v_h}$
lies in this image as discussed above, we have $\Pi \ket{v_h} = \ket{v_h}$ and~\eqref{eq:why}
is identically zero.
\end{proof}
As in the one-register case, we can distinguish the trivial subgroup from the conjugates of $H$ by completing the PGM with a measurement $M_0$ that projects onto the complement of the image of $M$.
\subsection{The Probability of Success}
In this section we give an upper bound on the success probability of the
optimal multiregister experiment for Gel'fand pairs. To prepare for this,
we record a version of Holevo's theorem on
the capacity of a quantum channel~\cite{Holevo73b}.
\begin{lemma}
Let $R = \{ \rho_i \mid i \in I\}$ be a family of density matrices treated as
linear operators on the Hilbert space $V$. Let $E = \{ E_i \}$ be a
family of measurement operators on $V$ for which $\sum_i E_i = \varmathbb 1$
and, for each $i \in I$, $E_i$ is a scalar multiple of a projection
operator of rank $r$. Then
$$
\Exp_i \textbf{tr}\, E_i \rho_i \leq \frac{\dim H}{r |I|}\enspace,
$$
where the index $i$ is chosen uniformly at random in $I$.
\end{lemma}
\begin{proof}
By assumption, we may write $E_i = \alpha_i \Pi_i$ where $\Pi_i$ is a
projection operator of rank $r$ and $\alpha_i \geq 0$. As $\rho_i$ is a density
matrix, we may write $\rho_i = \sum_j p_j \ket{v_j}\bra{v_j}$ where each
$\ket{v_j}$ has unit length, each $p_j \in [0,1]$, and $\sum_j p_j = 1$.
Observe that
$$
\textbf{tr}\, E_i \rho_i = \sum_j p_j \bra{v_j} E_i \ket{v_j} \leq \sum_j p_j \|E_i\| = \|
E_i \| = \alpha_i\enspace,
$$
where $\| A \|$ is the operator norm of $A$, given by $\| A \| =
\max_{\vec{v} \neq 0} \| A\vec{v} \|/ \| \vec{v} \| $. Hence
$$
\Exp_i \textbf{tr}\, E_i \rho_i \leq \frac{\sum_i \alpha_i}{|I|}\enspace.
$$
Observe now that $\sum_i E_i = \varmathbb 1$ and hence that $\dim V = \textbf{tr}\, \varmathbb 1
= \textbf{tr}\, \sum_i E_i = \sum_i \alpha_i r$; evidently $\sum_i \alpha_i = \dim V / r$, which
completes the proof.
\end{proof}
\begin{theorem}
\label{thm:pgm-success-gelfand}
Let $H$ be a subgroup of $G$ for which $(G,H)$ is a Gel'fand pair
and let $\core{H}{G} = \cap_g H^g$ be the largest normal subgroup
contained in $H$. The probability that the optimal $k$-register
measurement correctly identifies a uniformly random conjugate of $H$
is
\begin{equation}
\label{eq:pgm-success-gelfand}
P_{\rm success} \leq \frac{1}{|C(H)|} \cdot \left(\frac{|H|}{|\core{H}{G}|} \right)^k
\end{equation}
\end{theorem}
\begin{proof}
Without sacrificing optimality, we may initially carry out weak
Fourier sampling, in which we observe a representation ${\boldsymbol{\sigma}}$
with probability
\[
P(\sigma) = \frac{|H|^k d_{\boldsymbol{\sigma}} \,\textbf{rk}\; \pi_g^{\boldsymbol{\sigma}}}{|G|^k} \enspace .
\]
Again accounting for the fact that ${\boldsymbol{\rho}}_g = {\boldsymbol{\rho}}_{g'}$ when $H^g =
H^{g'}$, the probability of success may then be written
\[
P_{\rm success} =
|{\rm Norm}(H)| \cdot \Exp_g \textbf{tr}\, E_g {\boldsymbol{\rho}}_g = |{\rm Norm}(H)| \cdot \Exp_g
\Exp_{{\boldsymbol{\sigma}}} a_{\boldsymbol{\sigma}} \textbf{tr}\, E_g^{\boldsymbol{\sigma}} {\boldsymbol{\rho}}_g^{\boldsymbol{\sigma}}
\]
where ${\boldsymbol{\sigma}}$ is distributed according to $P({\boldsymbol{\sigma}})$, $g$ is
uniform in $G$, and $a_{\boldsymbol{\sigma}} = 1/\textbf{tr}\, {\boldsymbol{\rho}}^\sigma_g$ normalizes
${\boldsymbol{\rho}}_g^{\boldsymbol{\sigma}}$ so that it is a density matrix.
Recall that ${\boldsymbol{\rho}}_g^{\boldsymbol{\sigma}}$ is proportional to $\pi_g^{\boldsymbol{\sigma}} \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}}$
where $\pi_g^{\boldsymbol{\sigma}}$ has rank one (or zero), that $M^{\boldsymbol{\sigma}} = m \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}}$,
and that the image of $\pi_g^{\boldsymbol{\sigma}}$ is contained in the image of $m$.
Therefore, $m^{-1/2} \pi_g^{\boldsymbol{\sigma}} m^{-1/2}$ has rank one or zero, and so
$E_g^{\boldsymbol{\sigma}} = m^{-1/2} \pi_g^{\boldsymbol{\sigma}} m^{-1/2} \otimes \varmathbb 1_{d_{\boldsymbol{\sigma}}}$
is either zero or a scalar multiple of a projection operator of rank $d_{\boldsymbol{\sigma}}$.
By the lemma above, for a representation ${\boldsymbol{\sigma}}$ we have $\Exp_g \textbf{tr}\,(a_{\boldsymbol{\sigma}}
E_g^{{\boldsymbol{\sigma}}} {\boldsymbol{\rho}}^{\boldsymbol{\sigma}}_g) \leq d_{\boldsymbol{\sigma}} / |G|$.
Let us again define $S_{H^k} \subset \widehat{G^k}$ to be the set
of representations for which $\pi^{\boldsymbol{\sigma}}_g$ is nonzero.
By commuting the two expectations above, we conclude that
\[
P_{\rm success} \leq |{\rm Norm}(H)| \Exp_{\boldsymbol{\sigma}} \frac{d_{\boldsymbol{\sigma}}}{|G|}
= \frac{|{\rm Norm}(H)|}{|G|} |H|^k \sum_{{\boldsymbol{\sigma}}} \textbf{rk}\; \pi^{\boldsymbol{\sigma}} \frac{d_{\boldsymbol{\sigma}}^2}{|G|^k}
= \frac{|H|^k}{|C(H)|} \textrm{Planch}(S_{H^k})
\]
where $\textrm{Planch}_{G^k}({\boldsymbol{\sigma}}) = d_{\boldsymbol{\sigma}}^2/|G|^k$ is the Plancherel distribution on
$\widehat{G^k}$. But this is just the product of the Plancherel distribution on $\widehat{G}$ over the
$\sigma_i$, so we have
\[
P_{\rm success} \leq \frac{|H|^k}{|C(H)|} \textrm{Planch}(S_H)^k
\]
and recalling from the proof of Theorem~\ref{thm:pgm-success-single} that
$\textrm{Planch}(S_H) \leq 1/|H_G|$ completes the proof.
\end{proof}
\subsection{Examples}
Theorem~\ref{thm:gelfand} applies to a number of group families that have appeared in the literature on the hidden subgroup problem. Here is a short list of examples of Gel'fand pairs:
\begin{itemize}
\item $(G,H)$ where $H$ is normal and $G/H$ is Abelian. Of course, whenever $H$ is normal the hidden conjugate problem becomes trivial.
\item $(D_n,H)$ where $H$ consists of the identity and an involution, as in Bacon, Childs and van Dam~\cite{BaconCvD}.
\item $(A_p,\mathbb{Z}_p^*)$ where $A_p$ is the affine group $\mathbb{Z}_p^* \ltimes
\mathbb{Z}_p$ and $\mathbb{Z}_p^*$ is a maximal non-normal subgroup. An efficient
quantum algorithm for the hidden conjugate problem in this case was given by~\cite{MooreRRS04}.
\item All the subgroups of the Heisenberg group, for which an information-theoretic reconstruction algorithm was given by Radhakrishnan, R\"{o}tteler and Sen~\cite{RadhakrishnanRS}.
\item $(SL_2(q),B)$ or $(GL_2(q),B)$ where $B$ is the Borel subgroup consisting of upper-triangular matrices.
\item $(S_n,H)$ where $H$ is the \emph{hyperoctahedral group}; this is the centralizer of $(1\,2)(3\,4) \cdots (n-1 \,n)$, or equivalently the wreath product $S_{n/2} \wr \mathbb{Z}_2$, or the symmetry group of the $(n/2)$-dimensional hyperoctahedron.
\item $(S_n,H)$ where $H=S_m \times S_{n-m}$ for some $0 \leq m \leq n$, i.e., the subgroup of permutations under which the set consisting of the first $m$ elements is invariant.
\end{itemize}
Note there is an efficient classical algorithm for the hidden conjugate problem for the last two examples in $S_n$: simply check for all ${n \choose 2}$ transpositions whether the oracle differs from its value on the identity. This allows us to determine the conjugate of $H$, which is associated with a matching (for the hyperoctahedral group) or a subset of size $m$ (for $S_m \times S_n$).
\section{Conclusion}
The hidden conjugate problem has important applications; in the dihedral group it is related to hidden shift problems~\cite{vanDamHI03} and lattice problems~\cite{Regev}. However, for problems such as Graph Isomorphism, we are typically interested in distinguishing one conjugacy class from another. While we can detect the trivial subgroup with the additional measurement $M_0$ defined here and in~\cite{BaconCvD}, the PGM is not generally optimal in this case [D. Bacon, personal communication]. Constructing the optimal measurement for the hidden subgroup problem, given a prior on the conjugacy classes, remains an important open question.
\section*{Acknowledgments.}
This work was supported by NSF grants CCR-0093065, PHY-0200909,
EIA-0218443, EIA-0218563, CCR-0220070, and CCR-0220264.
We are grateful to David Bacon, Andrew Childs, and Wim van Dam
for introducing us to the Pretty Good Measurement and alerting us to reference~\cite{YuenKM},
to the organizers of QIP 2005 at which much of this work was done, to Dan Rockmore and Martin R\"{o}tteler for thoughts on Gel'fand pairs,
and to Tracy Conrad and Sally Milius for their support and tolerance.
C.M.\ also thanks Rosemary Moore for her recent arrival, and for providing a
larger perspective.
| {
"timestamp": "2005-05-20T21:30:48",
"yymm": "0501",
"arxiv_id": "quant-ph/0501177",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0501177",
"abstract": "Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group D_n in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.",
"subjects": "Quantum Physics (quant-ph)",
"title": "For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860381,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7092019732934258
} |
https://arxiv.org/abs/1307.2198 | Zero forcing for sign patterns | We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an analo- gous bound for the maximum nullity of a matrix with a given sign pattern. This allows us to compute, for instance, the maximum nullity of a Z-matrix whose graph is L(K_{n}), the line graph of a clique. | \section{Introduction}
One of the most vibrant areas in algebraic graph theory in recent years has been the study of minimum rank/maximum nullity problems. If
$G$ is a simple graph on $n$ vertices, labelled $\{1,\ldots,n\}$, let $M^{\mathbb{F}}(G)$ be the maximum possible nullity of a symmetric $n \times n$ matrix over field $\mathbb{F}$ whose graph is $G$ (\emph{i.e.} for all $i \neq j$, the $ij$ entry of the matrix is non-zero if and only if $i,j$ are adjacent in $G$). The minimum possible rank of such a matrix is denoted $mr^{\mathbb{F}}(G)$ and it is not hard to see that $M^{\mathbb{F}}(G)+mr^{\mathbb{F}}(G)=n$. Therefore, lower bounds on $mr$ are equivalent to upper bounds on $M$. For many such results obtained by various authors to date we can direct the reader to the surveys \cite{FalHog07, FalHog11} by Fallat and Hogben.
A popular way of bounding the maximum nullity from above is by the so-called \emph{zero forcing} method. Zero forcing is a combinatorial game played on the vertices of a graph, during which the vertices are colored black and white. It was first formally defined and studied in \cite{AIM_zeroforce08}, although inchoate versions had been in use for some time before. Further variants of the basic idea were introduced and are surveyed in \cite{NewZF_with_TW}.
In this paper we address an issue that was raised by Hogben in \cite[p. 208]{Hog11}: how to adapt the zero forcing number argument to sign patterns?
A sign pattern $P$ is a $m \times n$ matrix whose entries are drawn from $\{+,-,0,?\}$ and a real $m \times n$ matrix $A$ is said to have sign pattern $P$ if the sign of $A_{ij}$ is equal to $P_{ij}$ whenever $P_{ij} \neq ?$. There is no restriction on the $ij$th entry of $A$ if $P_{ij}=?$.
\begin{rmrk}
All sign patterns in this paper are assumed to be square, \emph{i.e.} $m=n$.
\end{rmrk}
A square sign pattern naturally corresponds to a signed directed graph and the indices in $\{1,2,\ldots,n\}$ will be referred to as the vertices of the sign pattern.
\begin{defin}
Let $P$ be a square sign pattern. If $P_{ij} \neq ?$ whenever $i \neq j$ we will say that $P$ \emph{has fixed periphery}.
\end{defin}
\begin{rmrk}
All sign patterns in this paper are assumed to have fixed periphery.
\end{rmrk}
We shall denote by $M^{\mathbb{R}}(P)$ or simply $M(P)$ the maximum nullity of a real matrix whose sign pattern is $P$.
The problem of determining the ranks/nullities of sign patterns has been first posed by Johnson \cite{Joh82} in 1982, with the last decade or so witnessing a vigorous renewal of interest in it, motivated partly by applications in complexity theory. For further background information we can direct the reader to the introduction of \cite{Li_etal12} where a list of most of the relevant papers on the subject to date has been compiled.
We introduce a new variant, \emph{signed zero forcing}, which allows us in some cases to give non-trivial upper bounds on the maximum nullity of a sign pattern. Our main result is Theorem \ref{thm:main} which is the counterpart for sign patterns of the known upper bound on the maximum nullity given by classical zero forcing (we quote the latter bound here as Theorem \ref{thm:classical}).
The plan of the paper is: the classical game is described and analyzed in Section \ref{sec:classical} and our new game is defined in Section \ref{sec:newzf}. Section \ref{sec:proof} is devoted to the proof of Theorem \ref{thm:main} while in Section \ref{sec:branch} we very briefly sketch an extension of our new game that can yield even better bounds. Then, starting with Section \ref{sec:graphs} we turn our attention to graphs again and show how a corollary of Theorem \ref{thm:main} can be applied to bound the maximum possible nullity of a $Z$-matrix with a given graph.
\section{Classical zero forcing and nullity}\label{sec:classical}
We start the game by choosing a subset of vertices $S \subseteq V(G)$ and coloring them black. The other vertices remain white for now. Then the following color-change rule is applied until no more changes are possible. The resultant coloring is called the \emph{derived coloring of $S$}:
\begin{rul}[Classical zero forcing rule]\label{rul:zf_classical}
If $u$ is a black vertex of $G$ and exactly one neighbour $w$ of $u$ is white, then change the color of $w$ to black.
\end{rul}
A set $S \subseteq V(G)$ is said to be a \emph{zero forcing set for $G$} if all vertices of $G$ are black in the derived coloring of $S$. The \emph{zero forcing number of $G$}, $Z(G)$, is then the minimum size of a zero forcing set for $G$.
\begin{thm}\cite{AIM_zeroforce08}\label{thm:classical}
$M^{\mathbb{F}}(G) \leq Z(G)$, for any field $\mathbb{F}$.
\end{thm}
Let us briefly review here the proof of Theorem \ref{thm:classical} given in \cite[Propositions 2.2-2.4]{AIM_zeroforce08}. Some notations: we will denote the kernel of matrix $A$ by $\ker{A}$ and the subvector of vector $x$ formed on the entries indexed by a set $S$ by $x|_{S}$.
Consider a symmetric matrix $A$ whose graph is $G$ and a vector $x \in \mathbb{F}^{n}$. Denoting adjacency in $G$ by $\sim$, we have for any vertex $u$:
\begin{equation}\label{eq:basic}
(Ax)_{u}=a_{uu}x_{u}+\sum_{v \sim u}{a_{uv}x_{v}}.
\end{equation}
Now suppose that $S$ is a zero forcing set for $G$ and that $x \in \ker{A}$. We argue that if $x|_{S}=0$, then $x=0$. Indeed, there is some vertex $u \in S$ to which we can apply Rule \ref{rul:zf_classical} (or else, $S=V(G)$ and we are done). That is, $u$ has only one white neighbour, $w$. Looking at (\ref{eq:basic}), we obtain
\begin{equation}\label{eq:one}
0=(Ax)_{u}=a_{uw}x_{w}
\end{equation}
and thus $x_{w}=0$. An inductive argument shows that in fact all entries of $x$ are zero.
Finally, suppose, for the sake of contradiction, that $M^{\mathbb{F}}(G)>Z(G)$. Therefore there is a symmetric matrix $A$ (whose graph is $G$), such that the nullity of $A$ is greater than $Z(G)$. Let $S$ be a minimum zero forcing set, with $|S|=Z(G)$. A standard dimensional argument now implies the existence of a nonzero $x \in \ker{A}$ that vanishes on the vertices of $S$. By the foregoing argument if follows that $x=0$ - a contradiction.
\section{Signed zero forcing}\label{sec:newzf}
\subsection{Informal overview of the new method}\label{sec:newmot}
The main plank of the argument we discussed in the previous section was the deduction from Equation (\ref{eq:basic}) that some of the entries of the vector $x$ must be zero, given a set of known zero coordinates of $x$. To achieve that goal we had to reduce the expression on the right hand side of (\ref{eq:basic}) to just one summand in (\ref{eq:one}), since we had no information on the signs of the summands of \eqref{eq:basic}.
However, now that we are no longer considering zero-nonzero patterns but sign patterns (with fixed periphery), the signs of the $a_{uv}$s are fixed and known and therefore the signs of the summands of \eqref{eq:basic} depend only on the signs of the $x_{v}$s. We shall now see how to make good use of this fact.
Let $A$ be a matrix whose sign pattern is $P$ and write $j \rightarrow i$ whenever $P_{ij} \in \{+,-\}$. Let $u \in \{1,2,\ldots,n\}$ and assume for the moment that $x_{u}=0$ as before. We denote $$W=\{w | w \rightarrow u\}.$$ Then Equation \eqref{eq:one} takes on the form
\begin{equation}\label{eq:h}
0=\sum_{w \in W}{a_{uw}x_{w}}.
\end{equation}
Define now also the sets
$$W_{+}=\{w \in W| a_{uw}x_{w} \geq 0\},$$
$$W_{-}=\{w \in W| a_{uw}x_{w} \leq 0\}.$$
If, say, $W=W_{+}$ then the only way for Equation \eqref{eq:h} to hold is to have $x_{w}=0$ for all $w \in W$. This inference corresponds to the blackening of vertices in the zero forcing game - we deduce that some of the entries of the vector $x$ must be zero and blacken the corresponding vertices.
Another possible deduction from Equation \eqref{eq:h} is in the case that $W=W_{+} \cup \{w\}$. In this case we immediately deduce that $a_{uw}x_{w} \leq 0$ must be true for \eqref{eq:h} to hold and thus the weak sign of $x_{w}$ is the opposite of the sign of $a_{uw}$.
Thus we see that knowledge of the signs of some entries of $x$ can lead to the determination of the signs of other entries. (the case $W=W_{-} \cup \{w\}$
is, of course, symmetric).
What we need now is a way to incorporate this reasoning into the zero forcing game. To do this, we add to the game another element: white vertices may now be marked as $+$ or $-$ vertices. \emph{The markers at a given moment represent the information we have at this stage about the possible weak signs of the entries of $x$}. As the game progresses, marked white vertices can be colored black (when we learn that their entries are in fact zero) - in which case the sign marker disappears under the black paint.
Finally, we observe that the assumption $x_{u}=0$ is too stringent. In fact, if $P_{uu} \neq ?$ we can include the term $a_{uu}x_{u}$ in \eqref{eq:h} and analyze it on the same basis as the other terms. In combinatorial terms, this means that our new color-change rule can be applied either to black vertices or to white vertices whose "loop" has a known sign.
\subsection{Seeding markers}\label{sec:seeding}
We have seen in the previous subsection that given the signs of some entries in $x$ (which correspond to vertices of the sign pattern) we may be able to determine the signs of some other entries. But how does one, so to speak, gets the ball rolling? How does one obtain the first sign?
This is actually very simple: after the vertices of $S$ have been blackened, we can choose \emph{any} vertex and mark it with $+$!
Since the sign markers represent our belief about the signs of the entries of $x$, our seemingly arbitrary choice of $+$ amounts to choosing one vector from the pair $x,(-x)$. We can then continue the game, using the chosen vector, obtainining the same derived coloring for $x$ and $-x$.
One important observation remains to be made: we may seed the markers anew at some point in the game. This happens if all marked vertices had been blacked out, but some white vertices still remain.
\subsection{Formalization}
The \emph{signed zero forcing} game proceeds as the usual zero forcing game, but a new zero forcing rule is applied. Note that clause (a) of the new rule subsumes the classical rule and that clause (d) is a formalization of the seeding concept discussed above.
To state the rule formally a few extra notations are needed: the sign inversion function $\iota$ is defined by $\iota(+)=-$ and $\iota(-)=+$; the letters $s$ and $t$ will be used as indices taking values in $\{+,-\}$ and $s \cdot t$ will be defined as $+$ if $s=t$ and as $-$ if $s \neq t$. Finally, if $w$ is a white vertex, we will denote by $m(w)$ its marker if it is marked; otherwise we will write $m(w)=*$.
\begin{rul}[Signed zero forcing rule for sign pattern $P$]\label{rul:new}
Let $u$ be a vertex of $P$ such that either $u$ is black or $u$ is white and $P_{uu} \neq ?$.
Let $$W=\{w | w \ \textit{is white} \wedge w \rightarrow u \}\cup \{u\}.$$
Define $$W_{+}=\{w \in W| m(w)=P_{uw}\}, W_{-}=\{w \in W| m(w) \neq P_{uw}\}$$ and $$W_{*}=\{w \in W| m(w)=*\}.$$
\begin{enumerate}
\item
If $W=\{w\}$, color $w$ black.
\item
If either $W_{+}=W$ or $W_{-}=W$, color all vertices in $W$ black.
\item
If $W_{s} \neq \emptyset,W_{\iota(s)}=\emptyset$, and $W_{*}=\{w\}$, mark $w$ with $P_{uw} \cdot \iota(s)$.
\item
If no white vertices in the whole graph are marked and $u$ is white, then mark $u$ with $+$.
\end{enumerate}
\end{rul}
\begin{defin}
The \emph{signed zero forcing number} of $P$, $Z_{\pm}(P)$, is the size of the minimum forcing set when playing the game as outlined in this section.
\end{defin}
The arguments expounded in Sections \ref{sec:newmot} and \ref{sec:seeding} can be crystallized into the following statement, which is the main result of the paper.
\begin{thm}\label{thm:main}
Let $P$ be a sign pattern with fixed periphery. Then:
$$M^{\mathbb{R}}(P) \leq Z_{\pm}(P).$$
\end{thm}
Before proceeding to the formal proof of Theorem \ref{thm:main} (in effect, just a careful codification of the informal arguments given before) we wish to illustrate the operation of the game on a small example.
\begin{expl}\label{ex:hadamard}
Consider the $4 \times 4$ Hadamard sign pattern:
$$
P=\left(\begin{array}{cccc}
+&+&+&+\\
+&-&+&-\\
+&+&-&-\\
+&-&-&+
\end{array}\right).
$$
We are going to show that $Z_{\pm}(P)=2$, and therefore $M(P) \leq 2$. In fact, Hogben \cite[p.206]{Hog11} has shown that $mr(P)=3$, and therefore $M(P)=1$; we will see in Section \ref{sec:branch} how a natural extension of our technique enables us to prove that $M(P)=1$ just as well.
Let $S=\{1,2\}$ and colour the vertices of $S$ black. Now we apply Rule \ref{rul:new}(d) to mark vertex $3$ with $+$. Next we apply Rule \ref{rul:new}(c) to $u=3$. In this case we have $W=\{3,4\}$ and $W_{-}=\{3\},W_{*}=\{4\}$. We see that vertex $4$ can be marked with $P_{34} \cdot \iota(-)$, that is with $-$. Finally, we apply Rule \ref{rul:new}(b) to vertex $u=2$; now we have $W=W_{+}=\{3,4\}$ and so both $3$ and $4$ can be blackened, finishing the game.
Note that we could have replaced the last move with the application of Rule \ref{rul:new}(b) to $u=4$ instead, in which case we would have had $W=W_{-}=\{3,4\}$.
\end{expl}
\section{Proof of Theorem \ref{thm:main}}\label{sec:proof}
\begin{proof}
Suppose that $S$ is a minimum forcing set under Rule \ref{rul:new}, so that $|S|=Z_{\pm}(P)$. This means that the signed zero forcing game can be played in, say, $m$ moves: $\mathbb{M}_{1},\mathbb{M}_{2},\ldots,\mathbb{M}_{m}$. The first move $\mathbb{M}_{1}$ consists of coloring the vertices of $S$ in black. The remaining moves are repeated applications of Rule \ref{rul:new} - until all the vertices of $P$ are black.
Let $B^{k}$ be the sets of vertices that are black after the move $\mathbb{M}_{k}$ has been played. Observe that $B^{1}=S$ and $B^{m}=\{1,2,\ldots,n\}$. Let also $M_{k}$ be the set of white vertices with markers after move $\mathbb{M}_{k}$ and let $m_{k}(w)$ be the marker of $w$ at this stage, for a vertex $w \in M_{k}$.
Now let $A$ be a matrix whose sign pattern is $P$ and let $x \in \ker{A}$.
\emph{Claim} If $x|_{S}=0$, then for any $1 \leq k \leq m$ it holds that:
\begin{itemize}
\item
$x|_{B^{k}}=0$.
\item
If $w_{1},w_{2}\in M_{k}$ and $m(w_{1})=m(w_{2})$, then $x_{w_{1}}x_{w_{2}} \geq 0$.
\item
If $w_{1},w_{2}\in M_{k}$ and $m(w_{1}) \neq m(w_{2})$, then $x_{w_{1}}x_{w_{2}} \leq 0$.
\end{itemize}
The claim, once established, will show that for $x \in \ker{A}$, $x|_{S}=0$ entails $x=0$. The dimensional argument mentioned in Section \ref{sec:classical} will then finish the proof.
We proceed to prove the claim by induction on $k$. The claim is true for $k=1$ as $B^{1}=S$ and no vertices are marked at that stage. Suppose that the claim is true for some $k \geq 1$ and let us show it for $k+1$. The move $\mathbb{M}_{k+1}$ could have been the application of each of the four clauses of Rule \ref{rul:new} to a vertex $u \in B^{k}$.
Recall now the basic formula (\ref{eq:h}):
\begin{displaymath}
0=\sum_{w \in W}{a_{uw}x_{w}}.
\end{displaymath}
If $\mathbb{M}_{k+1}$ applied the first clause of Rule \ref{rul:new}, with $w$ being the sole white neighbour of $u$, then (\ref{eq:h}) simplifies to $0=a_{uw}x_{w}$ and therefore $x_{w}=0$. Since in this case
$B^{k+1}=B^{k} \cup \{w\}$ and $M_{k+1} \subseteq M_{k}$, the claim is upheld.
Now suppose that $\mathbb{M}_{k+1}$ applied the second clause of Rule \ref{rul:new} to $u$, with, say, $W_{+}=W$. Then \eqref{eq:h} reduces to $$0=\sum_{v \in W^{+}}{a_{uv}x_{v}}$$
which can be rewritten as:
\begin{equation}\label{eq:cl2}
0=\overbrace{\sum_{\substack{v \in W^{+} \\ m(v)=+}}{a_{uv}x_{v}}}^{A} +
\overbrace{\sum_{\substack{v \in W^{+} \\ m(v)=-}}{a_{uv}x_{v}}}^{B}.
\end{equation}
We claim that all the summands on the right hand side of \eqref{eq:cl2} have the same weak sign.
Indeed, all the $x_{v}$s that appear in $A$ have the same weak sign by the induction assumption and by the definition of $W^{+}$ the corresponding $a_{uv}$s are positive. On the other hand, by the induction assumption again, the weak sign of all the $x_{v}$s in $B$ is opposite to the weak sign of the $x_{v}$s in $A$, while the $a_{uv}$s in $B$ are negative (again by the definition of $W^{+}$). Summing up, we see that all summands, whether in $A$ or in $B$, have the same weak sign. However, this means that they must all be zero, as their sum is zero. Therefore $x_{v}=0$ for all $v \in W$ and the claim is upheld as $B^{k+1}=B^{k} \cup W$ and $M_{k+1} \subseteq M_{k}$.
Next we have to show that the claim remains valid when the the third clause of Rule \ref{rul:new} is applied. We shall only prove it for $s=+$ since the proof for $s=-$ is the same, \emph{mutatis mutandis}.
In this case Equation (\ref{eq:h}) takes the form
$$0=\overbrace{\sum_{v \in W^{+}}{a_{uv}x_{v}}}^{C} + a_{uw}x_{w},$$ where $w$ is the sole unmarked vertex in $W$. The same argument as before shows that all summands in $C$ have the same weak sign. Therefore, we deduce that the $a_{uw}x_{w}$ term has the opposite weak sign. If $v$ is some vertex in $W^{+}$ then $a_{uv}x_{v}$ and $a_{uw}x_{w}$ must have opposite weak signs. We now consider four possible cases and verify that in each of them the claim holds. Note that $m_{k+1}(w)=P_{uw} \cdot \iota(s)$; since $s=+$ it means in effect that $w$ is marked with $+$ if $a_{uw}>0$ and marked with $-$ if $a_{uw}<0$.
Case 1: $m_{k}(v)=+,m_{k+1}(w)=+$. Since $m_{k}(v)=+$, we see that $a_{uv}>0$. Also, $a_{uw}<0$, as discussed above. Therefore, for $a_{uv}x_{v}$ and $a_{uw}x_{w}$ to have opposite weak signs, $x_{v}$ and $x_{w}$ must have the same weak sign - which is just what the claim asserts.
Case 2: $m_{k}(v)=+,m_{k+1}(w)=-$. In this case, $a_{uv}>0$ and $a_{uw}>0$, so if $a_{uv}x_{v}$ and $a_{uw}x_{w}$ are to have opposite weak signs, then $x_{v}$ and $x_{w}$ must have opposite weak signs as well.
Case 3: $m_{k}(v)=-,m_{k+1}(w)=+$. Same kind of argument.
Case 4: $m_{k}(v)=-,m_{k+1}(w)=-$. Same kind of argument.
Finally, for the fourth clause of Rule \ref{rul:new} the claim is true in a trivial way.
\end{proof}
\section{Interlude - branching}\label{sec:branch}
Part of the charm of both classical and signed zero forcing games is that they proceed in a straightforward way (and so are easy to program). However, it might be possible to do better at the signed zero forcing game, at the price of introducing a branching element into the game.
To expound this idea, consider again Example \ref{ex:hadamard}. Colour the vertex $1$ in black and mark vertex $2$ with $+$, in accordance with Rule \ref{rul:new}(d). Now we create a branch in the game by considering three options for vertex $3$: marked with $+$, marked with $-$, or black. This corresponds to the three possible options for $x_{3}$: positive, negative, or zero. Let us consider all three options and see that they all lead to the blackening of all four vertices, \emph{viz.} to the conclusion that $x=0$.
Case 1: $m(3)=+$. Let $u=1$ - then we have $W_{+}=\{2,3\}$ and $W_{*}=\{4\}$. Therefore we can mark $4$ with $P_{14} \cdot \iota(+)= + \cdot -=-$. Now let $u=4$ and we obtain for it $W_{-}=\{2,3,4\}$. This means we can blacken all three white vertices!
Case 2: $m(3)=-$. Let $u=2$ and then $W_{-}=\{2,3\}$ and $W_{*}=\{4\}$. Therefore we get to mark vertex $4$ with $P_{24} \cdot \iota(-)=- \cdot +=-$. But now let $u=3$ and we see that $W_{+}=\{2,3,4\}$. Once again all three white vertices are blackened.
Case 3: $3$ is black. First consider $u=1$, leading to marking vertex $4$ with $-$. But now if we look at $u=3$ we see that $W_{+}=\{2,4\}$. Total blackout once again.
What have introduced here, in effect, a new variant of the game. A tentative name for it can be \emph{branched signed zero forcing}. If we denote it by $Z_{\pm}^{b}$ then we have shown that in our example $Z_{\pm}^{b}(P)=1$ and therefore $M(P) \leq Z_{\pm}^{b}(P) =1$.
It would be interesting to find more cases in which $Z_{\pm}^{b}<Z_{\pm}$.
\section{Signed zero forcing for graphs}\label{sec:graphs}
\begin{defin}
Let $\mathcal{Z}$ be a family of real symmetric matrices defined in the following way: $A \in \mathcal{Z}$ if and only if all non-zero off-diagonal entries of $A$ share the same weak sign.
\end{defin}
The class $\mathcal{Z}$ includes some important matrix classes, such as symmetric $Z$-matrices (in particular, Stieltjes matrices) and symmetric entrywise nonnegative matrices. These leads us to take interest in the following graph parameter:
\begin{defin}
The maximum possible nullity of a matrix in $\mathcal{Z}$ whose graph is $G$ will be denoted $M^{\mathbb{R}}_{\mathcal{Z}}(G)$.
\end{defin}
Clearly, we can bound $M^{\mathbb{R}}_{\mathcal{Z}}(G)$ from above by the following inequalities:
$$
M^{\mathbb{R}}_{\mathcal{Z}}(G) \leq M^{\mathbb{R}}(G) \leq Z(G).
$$
But now we are in a position to do better. Define a sign pattern $P_{\mathcal{Z}}(G)$ in the following way:
$$
P_{\mathcal{Z}}(G)_{ij}=\begin{cases}
? & \text{, if } i=j \\
- & \text{, if } i \sim j \\
0 & \text{, if } i \neq j, i \not\sim j. \\
\end{cases}
$$
The following result is then an immediate consequence of Theorem \ref{thm:main}:
\begin{thm}\label{thm:maing}
$M^{\mathbb{R}}_{\mathcal{Z}}(G) \leq Z_{\pm}(P_{\mathcal{Z}}(G))$.
\end{thm}
We will henceforth abuse notation and refer to $Z_{\pm}(P_{\mathcal{Z}}(G))$ as $Z_{\pm}(G)$.
\begin{expl}\label{ex:q3}
Let $Q_{3}$ be the hypercube of dimension $3$, having eight vertices, as shown in Figure \ref{fig:cube3}.
It is not difficult to verify directly that $Z(Q_{3})=4$. We are going to show that $Z_{\pm}(Q_{3})=3$. Consider the drawing in Figure \ref{fig:cube3} on the left. We take $S=\{1,3,7\}$. We seed vertex $5$ with $+$ and then apply Rule \ref{rul:new}(c) to $7$, allowing us to mark $8$ with $-$.
The state of the game is shown in Figure \ref{fig:cube3} on the right.
\begin{figure}[h]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=2.5in]{cube3} &
\includegraphics[width=2.5in]{cube3_b}
\end{array}$
\end{center}
\caption{Left - $Q_{3}$, Right - $Q_{3}$ after the first stage of the game}\label{fig:cube3}
\end{figure}
Our next step is to apply Rule \ref{rul:new}(c) to vertex $1$, allowing us to mark vertex $2$ with $-$. We also apply Rule \ref{rul:new}(a) to vertex $3$, allowing us to colour the vertex $4$ black. The state of the game now is pictured in Figure \ref{fig:state2} on the left.
Now comes the decisive blow - we apply Rule \ref{rul:new}(b) to vertex $4$ and blacken the vertices $2$ and $8$. The resultant state, depicted in Figure \ref{fig:state2} on the right, is such that six vertices out of eight are already black and the remaining two are dispatched easily using Rule \ref{rul:new}(a).
Finally, it is not hard to see that if we start with two vertices, it is impossible to blacken all vertices. Therefore $Z_{\pm}(Q_{3})=3$.
\begin{figure}[h]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=2.5in]{cube3_s2} &
\includegraphics[width=2.5in]{cube3_s3}
\end{array}$
\end{center}
\caption{}\label{fig:state2}
\end{figure}
\end{expl}
We can use Example \ref{ex:q3} to give an upper bound on the maximum $\mathcal{Z}$-multiplicity of a hypercube of higher dimensions.
For general hypercubes the following result is known:
\begin{thm}\cite{HuaChaYeh10} $M^{\mathbb{F}}(Q_{d})=Z(Q_{d})=2^{d-1}$, for any field $\mathbb{F}$.
\end{thm}
The argument for the upper bound $Z(Q_{d}) \leq 2^{d-1}$ had been given in \cite[Proposition 2.5]{AIM_zeroforce08} where it was shown that if $G \square H$ is a Cartesian graph product and $S$ is a classical zero forcing set for $G$, then by taking a copy of $S$ in each fiber of $G$ inside $G \square H$, we obtain a classical zero forcing set for $G \square H$.
\emph{Mutatis mutandis} the same argument works for the signed zero forcing game.
Therefore, we have:
\begin{thm}
For all $d \geq 3$, $M^{\mathbb{R}}_{\mathcal{Z}}(Q_{d}) \leq Z_{\pm}(Q_{d}) \leq 3 \cdot 2^{d-3}$.
\end{thm}
\section{Graphs with $Z_{\pm}(G)<Z(G)$}
Since Rule \ref{rul:new}(a) is equivalent to Rule \ref{rul:zf_classical}, it is always true that $Z_{\pm}(G) \leq Z(G)$. As we have seen for hypercubes, strict inequality may obtain, in which case $Z_{\pm}$ provides substantially new information over that already given by $Z$. In this section we will show another family of graphs for which $Z_{\pm} <Z$ and also report the result of a computer search over small graphs.
But our first result is a negative one - for trees the two parameters coincide:
\begin{thm}\label{thm:trees}
Let $T$ be a tree. Then $Z_{\pm}(T)=Z(T)$.
\end{thm}
\begin{proof}
It is a well-known fact (cf. \cite[Lemma 1.2]{Alba_etal06}) that any symmetric sign pattern is congruent via a positive diagonal matrix to a pattern all of whose nonzero off-diagonal entries are $+$. This implies that $M_{\mathcal{Z}}(T)=M(T)$. It is also known \cite[Proposition 4.2]{AIM_zeroforce08} that $M(T)=Z(T)$ for any tree $T$.
Therefore we can write:
$$M_{\mathcal{Z}}(T) \leq Z_{\pm}(T) \leq M(T)=M_{\mathcal{Z}}(T)$$
and equality must hold throughout.
\end{proof}
\subsection{Small graphs}
In order to find the smallest order of a graph with this property we ran a computer search over the catalogue of small graphs made publicly available by Brendan McKay \cite{McKayGraphsData}. We found out that the smallest order is $6$ and that there are exactly two graphs with $Z_{\pm}<Z$ on six vertices; for both of them $Z_{\pm}=3,Z=4$. They are shown in Figure \ref{fig:min}.
One of the referees has pointed out that both these graphs have also the properties $M=4$ and $M_{\mathcal{Z}}=3$. While $M=4$ follows for both from \cite[Proposition 4.3]{AIM_zeroforce08}, to see that $M_{\mathcal{Z}}=3$ for, say, the graph on the right in Figure \ref{fig:min}, the referee has suggested the following argument: a matrix from $\mathcal{Z}$ associated to it must be of the form
$$
\left(\begin{array}{ccccccc}
d_{1}&0&a&b&c&d\\
0&d_{2}&s&t&u&v\\
a&s&d_{3}&0&w&x\\
b&t&0&d_{4}&y&z\\
c&u&w&y&d_{5}&0\\
d&v&x&z&0&d_{6}
\end{array}\right),
$$
with all the off-diagonal variables being strictly positive. The minor formed on rows $\{1,3,5\}$ and on columns $\{2,4,6\}$ is:
$$
\left|\begin{array}{ccc}
0&b&d\\
s&0&x\\
u&y&0
\end{array}\right|=bx+dy>0.
$$
Thus we have found a $3 \times 3$ nonzero minor and so $M_{\mathcal{Z}}(G) \geq 3$. As $Z_{\pm}(G)=3$ we deduce that in fact $M_{\mathcal{Z}}(G)=3$.
\begin{figure}[h]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=2.5in]{g130} &
\includegraphics[width=2.5in]{g153}
\end{array}$
\end{center}
\caption{The smallest graphs for which $Z_{\pm}<Z$}\label{fig:min}
\end{figure}
The graph we have just considered is in fact a well-known one in a slight disguise, for it is isomorphic to the line graph $L(K_{4})$ of the clique on four vertices. Using a brute-force implementation we find that if $G=L(K_{5})$ then $Z(G)=7$ and $Z_{\pm}(G)=6$. In general the classical zero forcing number of $L(K_{n})$ is given by:
\begin{thm}\cite{EroKanYi12,AIM_zeroforce08}
For any $n \geq 4$, it holds that $$M(L(K_{n}))=Z(L(K_{n}))=\binom{n}{2}-(n-2).$$
\end{thm}
In the next subsection we are going to bound $M_{\mathcal{Z}}(L(K_{n})$ from above by computing $Z_{\pm}$. But before we do that we would like to find a lower bound that will coincide with the value of $Z_{\pm}$.
\subsection{Line graphs of cliques}
Recall that a \emph{clique cover} of a graph $G$ is a set $C_{1},C_{2},\ldots,C_{m}$ of cliques in $G$ such that every edge of $G$ belongs to at least one $C_{i}$. The minimum cardinality of a clique cover is called the \emph{clique-cover number} and denoted \emph{$cc(G)$}.
\begin{thm}\label{thm:cc}
For any graph $G$ on $n$ vertices it holds that $$M_{\mathcal{Z}}(G) \geq n-cc(G).$$
\end{thm}
\begin{proof}
Let $m=cc(G)$ and let $C_{1},C_{2},\ldots,C_{m}$ be a minimum clique cover of $G$. For every $1 \leq i \leq m$ let $v_{i}$ be the indicator vector of $C_{i}$, i.e. the entries of $v_{i}$ corresponding to vertices in $C_{i}$ are equal to $1$ and all the other entries are zero. Now define $$A=\sum_{i=1}^{r}{v_{i}v_{i}^{T}}.$$
Clearly, the graph of $A$ is $G$ and $r(A) \leq cc(G)$. Therefore $M(G) \geq n-r(A) \geq n-cc(G)$.
\end{proof}
\begin{thm}\label{thm:lkn}
For $n \geq 6$, let $G=L(K_{n})$. Then it holds that $$M_{\mathcal{Z}}(G)=Z_{\pm}(G)=\binom{n}{2}-n.$$
\end{thm}
\begin{proof}
The vertices of $G$ can be labelled as $(i,j)$ with $i,j \in \{1,2,\ldots,n\}$ and $i<j$. Define $$S=V(G) \setminus (A \cup B),$$
where
$$
A=\{(i,n)|1 \leq i \leq n-3\}, $$ $$B=\{(n-2,n-1),(n-2,n),(n-1,n)\}.$$
We claim that $S$ is a signed zero forcing set for $G$. To see this, let us start the game by colouring all vertices in $S$ black. We first seed vertex $(1,n)$ with a $+$ and then apply Rule \ref{rul:new}(c) to vertex $(1,2)$ which allows us to mark $(2,n)$ with $-$.
Now apply Rule \ref{rul:new}(c) to all vertices of the form $(1,j),j=3,4,\ldots,n-3$. Each one of them has two white neighbours: $(1,n)$ and $(j,n)$ and since $(1,n)$ is marked with $+$, we can mark $(j,n)$ with $-$.
Next we apply Rule \ref{rul:new}(b) to vertex $(2,3)$: its two white neighbours are $(2,n)$ and $(2,3)$, both marked $-$. Therefore we can blacken both of these two vertices. At this stage vertex $(1,2)$ has only one white neighbour, that is $(1,n)$ and therefore \ref{rul:new}(a), the classical clause, allows us to blacken $(1,n)$. Similarly, applying Rule \ref{rul:new}(a) to vertex $(2,j)$ for $4 \leq j \leq n-3$ allows us to blacken the remaining white vertices of $A$.
At the end of this process, we are left with only three white vertices - the vertices of $B$. The game can now be finished by seeding again from vertex $(n-3,n-2)$ which marks $(n-2,n-1)$ with $+$ and $(n-2,n)$ with $-$. Applying Rule \ref{rul:new}(c) to $(n-3,n-1)$ we can mark $(n-1,n)$ with $-$. Then Rule \ref{rul:new}(b) to $(n-2,n)$ allows us to blacken $(n-2,n-1)$ and $(n-2,n)$. Thus only $(n-2,n)$ is left white and there is any number of possible ways to deliver the \emph{coup de grace} to it. We have thus proved that
$$
Z_{\pm}(G) \leq |S|=\binom{n}{2}-n.
$$
On the other hand, if $H$ is any graph on $n$ vertices and $G=L(H)$ is its line graph, then $cc(G) \leq n$ since $G$ is clearly covered by the set of $n$ cliques of the form $C_{v}=\{e \in E(G)| v \in e\}$ for $v \in V(G)$. Therefore by Theorem \ref{thm:cc} we have:
$$
M_{\mathcal{Z}}(G) \geq \binom{n}{2}-n.
$$
An appeal to Theorem \ref{thm:main} finishes the proof.
\end{proof}
\section{Concluding remarks}\label{sec:remarks}
\begin{enumerate}
\item
Suppose now that the derived coloring of a set $S$ is not all black, but all white vertices have sign markers. If we somehow know in addition that $0$ is a low eigenvalue of $G$, we might be able to rule out the putative configuration of signs of the entries of $x \in \ker{A}$ resulting from our game by applying a suitable \emph{nodal domain theorem} (cf. \cite{DavGlaLeySta01}).
This idea has been used in \cite[Section 5.3]{Mythesis} for the Colin de Verdi\`{e}re number $\mu(G)$ (in which case $0$ is stipulated to be the second lowest eigenvalue of $G$), with the help of a nodal domain-type result for $\mu$ due to van der Holst, Lov\'{a}sz and Schrijver.
\begin{qstn}
Is it possible to find a non-trivial class $\Gamma$ of graphs for which $0$ is guaranteed to be a low eigenvalue of any matrix in $\mathcal{Z}$ whose graph is $G \in \Gamma$?
\end{qstn}
\item
Like the classical variety, our new game requires the sign pattern to have a decent helping of zeros to work well. The development of a method that works well for so-called \emph{full sign patterns} (i.e. without zeros) is still an open question. To try to reduce it to the sparse case, we pose a question:
\emph{Is it possible to find rank-preserving transformations of zero patterns that increase the number of zeros?}
We remark that class of transformations which preserve sign-nonsingular zero patterns (\emph{i.e} patterns all of whose corresponding matrices are nonsingular) has been studied in \cite{BeaYe95}.
\item
It seems that our method can be profitably applied to obtain bounds on the nullity of signed graphs, a problem which has been recently introduced and studied in \cite{AraHalLiHol13}.
\item
To handle patterns with unspecified off-diagonal entries, it is possible to define a further variant of the signed forcing game in which edges are also marked with $+$ and $-$ markers, in a similar way to the way we marked vertices. We leave the exploration of this variant to future efforts.
\end{enumerate}
\section{Acknowledgments}
We are grateful to Leslie Hogben for interesting discussions about zero forcing and to the two anonymous referees for comments which have greatly enchanced both the substance and the presentation of the paper. Graphviz was used to draw the figures.
\bibliographystyle{abbrv}
| {
"timestamp": "2013-07-09T02:10:51",
"yymm": "1307",
"arxiv_id": "1307.2198",
"language": "en",
"url": "https://arxiv.org/abs/1307.2198",
"abstract": "We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an analo- gous bound for the maximum nullity of a matrix with a given sign pattern. This allows us to compute, for instance, the maximum nullity of a Z-matrix whose graph is L(K_{n}), the line graph of a clique.",
"subjects": "Combinatorics (math.CO)",
"title": "Zero forcing for sign patterns",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846703886662,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.709201972046909
} |
https://arxiv.org/abs/0905.1660 | Möbius numbers of some modified generalized noncrossing partitions | In this paper we will give a Möbius number of $NC^{k}(W) \setminus \bf{mins} \cup \{\hat{0} \}$ for a Coxeter group $W$ which contains an affirmative answer for the conjecture 3.7.9 in Armstrong's paper [ Generalized noncrossing partitions and combinatorics of Coxeter groups.arXiv:math/0611106]. | \section{Introduction}
In this paper we will prove the following theorem which is conjectured in
\cite{armstrong}.
\begin{thm}\label{theorem}
For each finite Coxeter group $(W,S)$ with $|S| = n$ and
for all positive integers $k$, the M\"obius number of
$NC^{k}(W) \setminus \bf{mins} \cup \{ \widehat{0} \}$ equals to
$(-1)^{n} \Bigl( Cat_{+}^{(k)}(W) - Cat_{+}^{(k-1)}(W) \Bigr)$.
\end{thm}
Our method is using
the EL-labeling of $NC_{(k)}(W)$ introduced by Armstrong and Thomas
\cite{armstrong}.
If we give an EL-labeling for $NC(W)$ for any complex reflction group $W$,
then we can state our Theorem {\rmfamily \ref{theorem}} in the case of
any well-generated complex reflection group.
But it may be difficult to give a uniform proof
because Athanasiadis, Brady and Watt gave an EL-labeling for
$NC(W)$ using some properties of the root system derived from
a real reflection group $W$ \cite{abw}.
In \cite{armstrong-krattenthaler} they proved this result by counting
the multichains of $NC_{(k)}(W)$.
Moreover they proved in the case of well-generated complex reflection
groups.
Our approach is independent to theirs. It is surprising for us that
their paper \cite{armstrong-krattenthaler}
appeared in arXiv when we were typing this paper.
\section{Preliminaries}
\subsection{generalized noncrossing partition}
In this paper we put $(W,S)$ a Coxeter group $W$ with a set of generators
$S$ where $S=n$. Basic properties of Coxeter groups is
introduced in \cite{humphleys}.
We put $T:= \{ wsw^{-1} \ | \ s \in S, w \in W \}$ the
cojugate closure of the set of generators $S$.
Let $l_{T} \ : \ W \longrightarrow \mathbb{Z}$ denote the word length on $W$
with respect to the set $T$.
We call $l_{T}$ the absolute length on $W$.
Then the absolute length naturally
induces a partial order on $W$ as following:
$\pi \le \sigma$ if $l_{T}(\sigma) = l_{T}(\pi) + l_{T}(\pi^{-1} \sigma)$.
We call it the absolute order on $W$.
We fix a Coxeter element $\gamma \in W$ and call the poset $[e,\gamma]$
with the absolute order $NC(W)$.
Next we put $NC^{(k)}(W) := \{ (\pi_{1}, \ldots ,\pi_{k}) \ | \ \pi_{i} \in
NC(W) \ {\rm for \ } 1 \le i \le k \ {\rm with } \ \pi_{1}
\le \pi_{2} \le \cdots \le \pi_{k} \le \gamma \}$ and
$NC_{(k)}(W) := \{ (\delta_{1}, \ldots ,\delta_{k}) \ | \ \delta_{i} \in
NC(W) \ {\rm for } \ 1 \le i \le k \ {\rm with } \
l(\delta_{1} \cdots \delta_{i} ) = l(\delta_{1}) + \cdots l(\delta_{i})
\ {\rm for } \ 1 \le i \le k \}$. In \cite{armstrong} Armstrong introduced the
order structure of them is as follows:
For $(\pi)_{k}^{(1)} := (\pi_{1}^{(1)}, \ldots ,\pi_{k}^{(1)}) \ {\rm and} \
(\pi)_{k}^{(2)} := (\pi_{1}^{(2)}, \ldots ,\pi_{k}^{(2)}) \in
NC^{(k)}(W)$, $(\pi)_{k}^{(1)} \le (\pi)_{k}^{(2)}$ if
$ (\pi_{i}^{(2)})^{-1} (\pi_{i+1}^{(2)})
\le
(\pi_{i}^{(1)})^{-1} (\pi_{i+1}^{(1)}) $
for $1 \le i \le k$ where $\pi_{k+1}^{(1)} = \pi_{k+1}^{(2)} = \gamma$.
For $(\delta)_{k}^{(1)} := (\delta_{1}^{(1)}, \ldots ,\pi_{k}^{(1)}) \
{\rm and} \
(\delta)_{k}^{(2)} := (\delta_{1}^{(2)}, \ldots ,\delta_{k}^{(2)}) \in
NC_{(k)}(W)$,
$(\delta)_{k}^{(1)} \le (\delta)_{k}^{(2)} \ {\rm if } \
\delta_{i}^{(1)} \le \delta_{i}^{(2)}$ for $1 \le i \le k$.
It is easy to see that the poset $NC_{(k)}(W)$ is the dual poset of
$NC^{(k)}(W)$ (for more information, see \cite{armstrong}).
\subsection{EL-shellability}
Let $(P, \preceq )$ be a finite poset. Assume that $P$ is bounded, meaning
that $P$ has a minimum element and a maximum element, denoted
$\widehat{0}$ and $\widehat{1}$ respectively, and that it is graded,
meaning that all maximal chains in $P$ have the same length. This length is
called the $rank$ of $P$ and denoted rank$(P)$.
Let $\epsilon (P)$ be the set of covering relations of $P$, meaning pairs
$(x,y)$ of elements of $P$ such that $x \prec y$ in $P$.
Let $\Lambda$ be a totally ordered set. An $edge \ labeling $ of $P$ with
label set $\Lambda$ is a map $\lambda \ : \ \epsilon(P) \longrightarrow
\Lambda$.
Let $c$ be an unrefinable chain $x_{0} \prec x_{1} \prec \cdots \prec x_{r}$
of elements of $P$ so that $(x_{i-1},x_{i}) \in \epsilon(P) $
for all $1 \le i \le r$.
We let $\lambda(c) = (\lambda(x_{0},x_{1}), \lambda(x_{1},x_{2}), \cdots
\lambda(x_{r-1},x_{r}))$ be the label of $c$ with respect to $\lambda$ and
call $c$ $rising$ and $falling$ with respect to $\lambda$ if the entries
of $\lambda(c)$ strictly increase or weakly decrease, respectively, in the
total order of $\Lambda$.
We say that $c$ is $lexicographically \ smaller$ than an unrefinable chain
$\acute{c}$ in $P$ with respect to $\lambda$ if $\lambda(c)$ proceeds
$\lambda(\acute{c})$ in the lexicographic order induced by the total order
of $\Lambda$ \cite{abw}.
\begin{de}[\cite{bjo}]
An edge labeling $\lambda$ of $P$ is called an EL-labeling if for every
nonsingleton interval $[u,v]$ in $P$
(1) there is a unique rising maximal chain in $[u,v]$ and
(2) this chain is lexicographically smallest among all maximal chains in
$[u,v]$
with respect to $\lambda$.
\end{de}
The poset $P$ is called EL-shellable if it has EL-labeling for some
label set $\Lambda$.
For a graded and bounded poset $(P,\preceq)$, we denote by $\mu(P)$
the M\"obius number of $P$.
If $P$ is EL-shellable the M\"obius number of
$P$ is the number of falling maximal chains of $P$ up to
sign $(-1)^{rank(P)}$ \cite{stan}.
\section{Main result}
In this section we will prove the following Theorem.
\begin{thm}
For each finite Coxeter group $(W,S)$ with $|S| = n$ and
for all positive integers $k$, we have
$\mu(NC^{k}(W) \setminus \bf{mins} \cup \{ \widehat{0} \})= $
$(-1)^{n} \Bigl( Cat_{+}^{(k)}(W) - Cat_{+}^{(k-1)}(W) \Bigr)$.
\end{thm}
It is easy to see the following Lemma.
\begin{lem}
Let $P$ be a graded poset with a minimum element $\widehat{0}$.
We put $\bf{maxs(P)}$ the set of maximal elements of $P$.
Then the poset $P \setminus \bf{maxs(P)}$ is also graded.
We denote by $\mu(P \setminus \bf{maxs(P)} \cup \{\widehat{1}\})$
the M\"obius number of $P \setminus \bf{maxs(P)} \cup \{\widehat{1}\}$.
Then we have
$\mu(P \setminus \bf{maxs(P)} \cup \{\widehat{1}\}) =
\mu(P \cup \{\widehat{1}\})
+ \Sigma_{x \in \bf{maxs(P)}} \mu([\widehat{0}
, x])$
\end{lem}
For $k \in \mathbb{N}$ and an arbitrary finite Coxeter group $(W,S)$
we consider the poset $NC_{(k)}(W)$ which is the dual poset of
$NC^{(k)}(W)$.
We put ${\bf maxs}$ as a set of maximal elements of $NC_{(k)}(W)$.
To show our Theorem, it is sufficient to prove
$\mu (NC_{(k)}(W) \setminus {\bf maxs} \cup \{ \widehat{1} \}) =
(-1)^{n} \Bigl( Cat_{+}^{(k)}(W) - Cat_{+}^{(k-1)}(W) \Bigr) $
\vspace{5mm}
In \cite{armstrong} Armstrong and Thomas gave an EL-shelling of
$ NC_{k}(W) \cup \{ \widehat{1} \}$.
We will explain their method briefly.
We put $T$ the set of reflections of $W$.
Recall that the edges in the Hasse diagram of $NC(W)$ are naturally labelled
by reflections $T$. Athanasiadis, Brady and Watt defined a total order on
the set $T$ such that the natural edge-labelling by $T$ becomes an
EL-shelling of the poset $NC(W)$.
We put the EL-labeling $\lambda \ : \ \epsilon(NC(W)) \longrightarrow T$.
In \cite{abw} they called
the total order on $T$ the {\bf ABW} order.
They put $T:= \{ t_{1} , \cdots t_{N} \}$ with
the {\bf ABW} order $t_{1} < t_{2} < \cdots < t_{N}$.
Recall that $NC(W^{k})$ is edge-lebelled by the set of reflections
$T^{k} := \{ t_{i.j} = (1, 1, \cdots ,t_{i,j}, \cdots , 1 ) : 1 \le i,j \le
N \}$ where $t_{j}$ occurs in the $i$-th entry of $t_{i,j}$.
Then they defined the {\bf lex ABW order} on $T^{k}$ as
$t_{1,1} < t_{1,2} < \cdots < t_{1,N} < t_{2,1} < t_{2,2} < \cdots
t_{2,N} < \cdots t_{k,1} < t_{k,2} < \cdots t_{k,N}$.
This induces an EL-shelling of $NC(W^{k})$.
Now recall that $NC_{k}(W)$ is an order ideal in $NC(W^{k})$, so the
{\bf lex ABW order} on $T^{k}$ restricts to an EL-labelling of the Hasse
diagram of $NC_{k}(W)$.
They considered the set $T^{k} \cup \{ \theta \}$ with
$t_{1,1} < t_{1,2} < \cdots < t_{1,N} < \lambda < t_{2,1} < t_{2,2} < \cdots
t_{2,N} < \cdots t_{k,1} < t_{k,2} < \cdots t_{k,N}$.
For $x \in {\bf maxs } $ they put $\lambda(x, \widehat{1}) := \lambda$, where
$\lambda(x, \widehat{1})$ is the edge from $x$ to $\widehat{1}$. They showed
that the labeling as above induces an EL-shelling of
$NC_{k}(W) \cup \{\widehat{1}\}$.
Now we put their EL-labeling
$\widehat{\lambda} \ : \ \epsilon(NC_{k}(W) \cup \{\widehat{1}\})
\longrightarrow T^{k} \cup \{ \theta \}$.
We have
$\mu(NC^{(k)}(W) \setminus {\bf mins} \cup \{\widehat{0}\})$
$= \mu(NC_{(k)}(W) \setminus {\bf maxs} \cup \{\widehat{1}\})$
$\Sigma_{x \in {\rm maxis}} \mu(\widehat{0},x) +
\mu(NC_{(k)}(W) \cup \{\widehat{1}\})$
$= \Sigma_{x \in {\rm maxs}}
\mu(\widehat{0},x) + (-1)^{n-1} $Cat$_{+}^{(k-1)}(W)$.
It is sufficient to show
$\Sigma_{x \in {\rm maxs}} \mu(\widehat{0},x) = (-1)^{n} $Cat$_{+}^{(k)}(W)$ to
prove Theorem {\rmfamily \ref{theorem}}.
First we consider the EL-shelling introduced by Armstrong and Thomas.
Recall that $\mu(NC_{k}(W) \cup \{\widehat{1}\}) = (-1)^{n-1} \times$
the number of the falling maximal chains of
$NC_{k}(W) \cup \{\widehat{1}\}$ with respect to $\widehat{\lambda}$.
Now let $c$ be an unrefinable chain
$( e, \cdots ,e) \prec \cdots \prec (\delta_{1}, \cdots \delta_{k})
\prec \widehat{1} $ of elements of $NC_{k}(W) \cup \{\widehat{1}\}$.
If $c$ is a falling maximal chain with respect to $\widehat{\lambda}$,
we must have $\delta_{1} = e$ because
$\widehat{\lambda}((\delta_{1}, \cdots \delta_{k}),\widehat{1}) $ equals to
$ \lambda$
and $\lambda$ is bigger than $t_{1,i}$ for $1 \le i \le N$ in
the total order on $T^{k} \cup \{ \theta \}$.
Moreover we have
$c$ is a falling maximal chain
$\Longleftrightarrow$
$c \in \{ (e, \cdots e) \prec \cdots \prec (e, \cdots e, \delta_{k}) \prec
\cdots (e, \cdots \delta_{k-1},\delta_{k}) \prec \cdots
\prec (e,e,\delta_{3}, \cdots \delta_{k}) \prec
\cdots \prec (e,\delta_{2} \cdots \delta_{k}) \ | \
{\rm \ each \ part } \
( e, \cdots e) \prec \cdots \prec (e, \cdots e, \delta_{k}) ,
(e, \cdots e, \delta_{k}) \prec\cdots (e, \cdots \delta_{k-1},\delta_{k})
\cdots
(e,e,\delta_{3}, \cdots \delta_{k}) \prec \cdots \prec
(e,\delta_{2} \cdots \delta_{k}) \
{\rm are \ falling \ maximal \ chain}\}$
Hence we have
$\mu(NC_{k}(W) \cup \{\widehat{1}\})$
$= \Sigma_
{(e,\delta_{2}, \cdots ,\delta_{k}) \in {\rm maxs}}$
$(-1)^{{\rm rank}(\delta_{2})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{2}$ with respect to
$\lambda$ $\}$
$\times (-1)^{{\rm rank}(\delta_{3})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{3}$ with respect to
$\lambda$ $\}$
\begin{center}
$\vdots$
\end{center}
$ \times (-1)^{{\rm rank}(\delta_{k})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{k}$ with respect to
$\lambda$ $\}$
$= \Sigma_
{(e,\delta_{2}, \cdots ,\delta_{k}) \in {\rm maxs}}$
$\mu([e,\delta_{2}]) \times \mu([e,\delta_{3}]) \cdots \mu([e,\delta_{k}])$
$=\Sigma_
{(\delta_{1}, \cdots ,\delta_{k-1}) : \delta_{1} \cdots \delta_{k-1}=c \
{\rm and} l(\delta_{1}) + \cdots l(\delta_{k-1}) = n-1}$
$\mu([e,\delta_{1}]) \times \mu([e,\delta_{2}]) \cdots \mu([e,\delta_{k-1}])$.
Now we have the following proposition.
\begin{prop}
$\Sigma_{(\delta_{1}, \cdots ,\delta_{k-1}) , l( \delta_{1} \cdot \delta_{2}
\cdots \delta_{i}) = l(\delta_{1}) + \cdots l(\delta_{i}),
\delta_{1} \cdots \delta_{k-1} = c} \mu([e,\delta_{1}])
\cdots \mu(e,\delta_{k-1}) = (-1)^{n} {\rm Cat}_{+}^{k-1}(W)$.
\end{prop}
$Proof$
In \cite{armstrong}, Armstrong
showed that $\mu(NC^{(k)}(W) \cup \{ \widehat{0} \})=
\mu(NC_{(k)}(W) \cup \{\widehat{1}\})=
(-1)^{n} {\rm Cat}_{+}^{(k-1)}(W)$.
From the view of the
EL-labeling introduced by Armstrong and Thomas,
we have
$c$ is a falling maximal chain
$\Longleftrightarrow$
$c \in \{ (e, \cdots e) \prec \cdots \prec (e, \cdots e, \delta_{k-1}) \prec
\cdots (e, \cdots \delta_{k-2},\delta_{k-1}) \prec \cdots
\prec (e,\delta_{2}, \cdots \delta_{k-1}) \prec
\cdots \prec (\delta_{1} \cdots \delta_{k-1}) \ | \
{\rm \ each \ part } \
( e, \cdots e) \prec \cdots \prec (e, \cdots e, \delta_{k-1}) ,
(e, \cdots e, \delta_{k-1}) \prec\cdots (e, \cdots \delta_{k-2},\delta_{k-1}) \cdots
(e,\delta_{2}, \cdots \delta_{k-1}) \prec \cdots \prec
(\delta_{1} \cdots \delta_{k-1}) \
{\rm are \ falling \ maximal \ chain}\}$.
Hence we have
$(\mu(NC_{(k)}(W) \cup \{\widehat{1}\}))$
$= \Sigma_
{(\delta_{1}, \cdots ,\delta_{k-1}) ;
( \delta_{1} \cdot \delta_{2}
\cdots \delta_{i}) = l(\delta_{1}) + \cdots l(\delta_{i}),
\delta_{1} \cdots \delta_{k-1} = c}$
$(-1)^{{\rm rank}(\delta_{1})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{1}$ with respect to
$\lambda$ $\}$
$\times (-1)^{{\rm rank}(\delta_{2})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{2}$ with respect to
$\lambda$ $\}$
\begin{center}
$\vdots$
\end{center}
$ \times (-1)^{{\rm rank}(\delta_{k-1})} \times \{$
the number of falling maximal chains from $e$ to $\delta_{k-1}$ $\}$
$ = \Sigma_{(\delta_{1}, \cdots ,\delta_{k-1}) , l( \delta_{1} \cdot \delta_{2}
\cdots \delta_{i}) = l(\delta_{1}) + \cdots l(\delta_{i}),
\delta_{1} \cdots \delta_{k-1} = c} \mu([e,\delta_{1}])
\cdots \mu(e,\delta_{k-1})$.
Hence we obtain the derived result.
\ \ \ $\Box$
Now we have
$\Sigma_{x \in {\rm maxs}} \mu(\widehat{0},x) =
\Sigma_{(\delta_{1}, \cdots ,\delta_{k}) , l( \delta_{1} \cdot \delta_{2}
\cdots \delta_{i}) = l(\delta_{1}) + \cdots l(\delta_{i}) {\rm for } 1 \le i \le k,
\delta_{1} \cdots \delta_{k} = c} \mu([e,\delta_{1}])
\cdots \mu(e,\delta_{k-1}) = (-1)^{n} {\rm Cat}_{+}^{k-1}(W)
$
This complete the proof of our Theorem {\rmfamily \ref{theorem}}.
\vspace{5mm}
{\large \textbf{Acknowledgement}}
The author wishes to thank Professor Christian Krattenthaler,
Professor Jun Morita for their
valuable advices.
\renewcommand{\refname}{REFERENCE}
| {
"timestamp": "2009-05-11T22:42:49",
"yymm": "0905",
"arxiv_id": "0905.1660",
"language": "en",
"url": "https://arxiv.org/abs/0905.1660",
"abstract": "In this paper we will give a Möbius number of $NC^{k}(W) \\setminus \\bf{mins} \\cup \\{\\hat{0} \\}$ for a Coxeter group $W$ which contains an affirmative answer for the conjecture 3.7.9 in Armstrong's paper [ Generalized noncrossing partitions and combinatorics of Coxeter groups.arXiv:math/0611106].",
"subjects": "Combinatorics (math.CO)",
"title": "Möbius numbers of some modified generalized noncrossing partitions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846691281406,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019711331915
} |
https://arxiv.org/abs/2204.13936 | On the total versions of 1-2-3-conjecture for graphs and hypergraphs | In 2004, Karoński, Łuczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper.
After that, the total versions of this conjecture were suggested in the literature and recently, Kalkowski et al. have generalized this conjecture to hypergraphs. In this paper, some previously known results on the total versions are improved. Moreover, an affirmative answer is given to the conjecture for some well-known families of hypergraphs like complete $n$-partite hypergraphs, paths, cycles, theta hypergraphs and some geometric planes. Also, these hypergraphs are characterized based on the corresponding parameter. | \section{Introduction}
Through out the paper, we consider simple and undirected graphs. For a graph $G$, the notations $V(G)$ and $E(G)$ stand for the vertex set and the edge set of $G$, respectively. A graph is called \textit{nice}, if it contains no component isomorphic to $ K_2 $.
Let $ G $ be a nice graph and $ w:E(G)\rightarrow\{1,2,\ldots, k\} $ be an integer edge weighting function of $G$.
Karo\'nski, \L uczak and Thomason introduced a vertex coloring of $ G $ obtained from $w$ as follows \cite{KLT}.
For every $ v\in V(G) $, let $ \sigma^{e}(v) $ denotes the sum of the weights of edges incident on $ v $, i.e.
\begin{equation*}
\sigma^{e}(v) :=\sum_{e\ni v}w(e).
\end{equation*}
Whenever the induced vertex coloring is \textit{proper}, i.e. for every two adjacent vertices $ u $ and $ v $, $ \sigma^{e}(u)\neq\sigma^{e}(v), $ the corresponding edge weighting is called \textit{neighbor sum distinguishing}.
The minimum integer $ k $ for which $ G $ admits a neighbor sum distinguishing is denoted by $ \chi^{e}(G) $.
In $ 2008 $, Karo\'nski et al. proposed the following statement known as \textit{1-2-3-conjecture}.
\begin{conjecture}\label{conj:123}{\em\cite{KLT}}
For every nice graph $ G $, $ \chi^{e}(G)\leq 3 $.
\end{conjecture}
This conjecture has been explored in two ways. The first is to improve general upper bounds on $ \chi^{e}(G) $. The latter is to confirm it for some well-known families of graphs.
The chromatic number of a graph is the smallest number of colors required to give a proper vertex coloring for the graph.
A graph is called $ k $-colorable, if its chromatic number is at most $ k $.
In \cite{KLT}, it is shown that if $G$ is $ k $-colorable,
where $k$ is odd, then $ \chi^{e}(G) \leq k$. Consequently, the conjecture is true for
$3$-colorable graphs. In the literature, the first constant bound was obtained by
Addario-Berry et al. \cite{A2} who showed that $ \chi^{e}(G) \leq 30$ for every nice graph $ G $. This result was improved to $ \chi^{e}(G) \leq 16$ by Addario-Berry, Dalal, and Reed \cite{A1} and then to $ \chi^{e}(G) \leq 13$ by Wang
and Yu \cite{WY}. Currently, the best upper bound is $5$ due to Kalkowski, Karo\'nski and Pfender \cite{KKP}.
Parameter $ \chi^{e}$ has been studied for several classes of graphs
including paths, cycles, complete graphs, bipartite graphs, theta graphs and so on \cite{CH, DO, LYZ}. However, few number of graph classes are characterized by the invariant $ \chi^{e}$.
To see these results in details we refer the reader to the survey written by Seamone \cite{survey}.
The cycle of length six and the complete graph on three vertices are simple instances of nice graphs for which weights 1 and 2 do not
suffice to give a neighbor sum distinguishing.
Hence, the bound claimed in Conjecture \ref{conj:123} is best possible. On the other hand, the following theorem gives some intuitions for the correctness of Conjecture \ref{conj:123}.\\
The random graph $G_{n, p}$ is the graph on $n$ vertices such that there is an edge between any two vertices randomly and independently, with
probability $p$.
\begin{theorem}{\em\cite{A1}}\label{a.all}
Assume that $G=G_{n,p}$ is a random graph for any constant $p \in (0,1)$.
Asymptotically almost surely, $ \chi^{e}(G) \leq 2$.
\end{theorem}
Clearly, $ \chi^{e}(G) =1$, if and only if $G$ contains no adjacent vertices with the same degree. Therefore,
determining the exact value of $ \chi^{e}$ seems to be a more difficult problem for graphs with more regular structures. This observation motivates researchers to focus on some regular graphs and hypergraphs.
In the sequel, we present definitions of parameters $ \chi^{ve}$ and $ \chi^{ven}$ which are considered as the total versions of $ \chi^{e} $, introduced in \cite{PW} and \cite{B}, respectively.
Let $w:V(G)\cup E(G)\rightarrow\{1,2,\ldots, k\} $ be a total integer weighting function of $G$. For every vertex $v$, $ \sigma^{ve}(v) $ is defined by
\begin{equation*}
\sigma^{ve}(v) :=w(v)+ \sum_{e\ni v}w(e)= w(v)+\sigma^{e}(v).
\end{equation*}
If $ \sigma^{ve}$ is a proper vertex coloring, then $w$ is called a \textit{neighbor sum distinguishing total coloring} of $G$. The minimum integer $k$ for which there exists a neighbor
sum distinguishing total coloring is denoted by $ \chi^{ve}(G) $.
As a total version of 1-2-3-conjecture, the following statement was suggested by Przyby\l o and Wo\'{z}niak in \cite{PW}.
\begin{conjecture}\label{ve conj}{\em\cite{PW}}
For every graph $ G $, $ \chi^{ve}(G)\leq 2 $.
\end{conjecture}
Kalkowski in \cite{K} showed that $ \chi^{ve}(G)\leq 3 $ for every graph $G$. Note that, $ G $ is not necessarily a nice graph in Conjecture \ref{ve conj}.
Let $w:V(G)\cup E(G)\rightarrow\{1,2,\ldots, k\} $ be a total integer weighting function of $G$. For every vertex $v \in V(G)$,
\begin{equation*}
\sigma^{ven}(v) :=w(v)+ \sum_{e\ni v}w(e)+ \sum_{u\in N(v)}w(u)=\sigma^{ve}(v)+\sum_{u\in N(v)} w(u),
\end{equation*}
where $N(v)$ denotes the open neighborhood of $v$. The function $w$ is a \textit{neighbor full sum distinguishing
total k-coloring} of $ G, $ whenever $\sigma^{ven}$ is a proper vertex coloring. The minimum integer $k$ for which there exists
such a coloring is denoted by $ \chi^{ven}(G)$.
Recently, Baudon et al. have introduced this notation and have proposed the following conjecture \cite{B}.
\begin{conjecture}\label{ven conj}{\em\cite{B}}
For every nice graph $ G $, $ \chi^{ven}(G)\leq 3 $.
\end{conjecture}
They also applied a similar proof technique stated in \cite{KKP} and showed that $\chi^{ven}$ is bounded by $ 5 $ in general. Moreover, They gave an affirmative answer to Conjecture \ref{ven conj} for paths, cycles, bipartite graphs, complete graphs and split graphs.
In section \ref{graph}, we focus on graphs and present the main results of this section in theoremS \ref{g1} and \ref{g2}.
Let $ K_n(t) $ be the complete $ n$-partite graph each part of size $ t $. Davoodi and Omoomi in \cite{DO} showed that $ \chi^{e}(K_n(t)) =2$ for $n,t \geq 2$. Theorem \ref{g1} strengthens this result by determining of $ \chi^{e}$ for more complete $ n$-partite graphs.
The authors in \cite{KLT} asserted that for every $3$-partite graph $G$, $ \chi^{e}(G)\leq 3$ and consequently, it is proved that $ \chi^{ven}(G)\leq 3$ \cite{B}. In Theorem \ref{g2}, we determine the exact value of $\chi^{ven}$ for complete $3$-partite graphs.
\\
A {\it hypergraph} $\mathcal{H}$ is an ordered pair $\mathcal{H}=(V,E)$, where $V$ is a finite and non-empty set of vertices and $E$ is a collection of distinct non-empty subsets of $V$. The set of vertices and the set of edges of $\mathcal{H}$ are denoted by $V(\mathcal{H})$ and $E(\mathcal{H})$, respectively.
An {\it $r$-uniform
hypergraph} is a hypergraph such that all of its edges have size $r$. For a vertex $v \in V(\cal{H})$, $N(v):=\{u: \exists\ e \in E(\mathcal{H}), \{v,u\}\subseteq e\}$ stands for the neighborhood of $v$.
A hypergraph vertex coloring is called \textit{proper} when each edge of $\mathcal{H}$ contains at least two vertices with
distinct colors.
Similar to the graph case, $ \chi^{e}(\mathcal{H})$ is the minimum integer $ k $
for which there exists a weighting function $ w:E(\mathcal{H})\rightarrow\{1,2,\ldots, k\}$ inducing a proper vertex coloring. Also, parameters $ \chi^{ve}$ and $ \chi^{ven}$ are defined for hypergraphs, in a similar manner.
Conjecture \ref{conj:123} has been generalized to $3$-uniform hypergraphs by Kalkowski, Karon\'{s}ki, and Pfender in the following form \cite{KKP-H}.
\begin{conjecture}{\em\cite{KKP-H}}\label{conj:3-HYPER}
For every $3$-uniform hypergraph $ \mathcal{H} $ with no isolated edge, $ \chi^{e}(\mathcal{H})\leq 3 $.
\end{conjecture}
Let $r\geq 3$ be an integer. In \cite{KKP-H}, it is shown that for any $r$-uniform hypergraph $ \cal H $ with no
isolated edge, $\chi^{e}(\mathcal{H})\leq{\max}\{5, r + 1\}$. Also, the authors in \cite{KKP-H} have proved that $\chi^{ve}(\mathcal{H})\leq 3$ for every hypergraph $\mathcal{H}$ with no edges of size $0$ or $1$.
The random hypergraph $ \mathcal{H}^{r}_{n,p} $ is an $r$-uniform hypergraph on $n$ vertices such that each edge is appeared randomly and independently, with probability $p$.
Applying probabilistic arguments, Bennett et al. established the following theorem \cite{BDFL}.
\begin{theorem}\label{thm:almost}{\em\cite{BDFL}}
Assume that $ \mathcal{H}= \mathcal{H}^{r}_{n,p} $ is an $ r $-uniform random hypergraph. Asymptotically almost surely, $ \chi^{e}(\mathcal{H})=2 $, if $ r=3 $ and $ \chi^{e}(\mathcal{H})=1 $, whenever $ r\geq 4 $.
\end{theorem}
Moreover, they extended Conjecture \ref{conj:3-HYPER} to $r$-uniform hypergraphs as follows.
\begin{conjecture}\label{conj:r-HYPER}{\em\cite{BDFL}}
Let $r \geq 3$ and $\mathcal{H}$ be an $r$-uniform hypergraph with no isolated edge, then $ \chi^{e}(\mathcal{H})\leq 3. $
\end{conjecture}
It is inferred from Theorem \ref{thm:almost} that every edge in an $ r $-uniform random hypergraph with $ r\geq 4 $ contains two vertices of different degrees with high probability. This fact motivates researchers to explore regular $ r $-uniform hypergrphs in Conjecture \ref{conj:r-HYPER} as the most challenging structures.
In section \ref{hgraphs}, we determine the exact values of $\chi^{e}$, $\chi^{ve}$ and $\chi^{ven}$ for the well-known families of hypergraphs. Indeed, we will see that some $ r $-uniform hypergraphs can not be weighted by 1 and 2 and hence, three colors are required. Moreover, we show that three colors are sufficient for them.
Briefly, main results of section \ref{hgraphs} are as follows. Theorem \ref{complete} determines the parameter $ \chi^{e}$ for the complete $ n$-partite $ r$-uniform hypergraph with parts of equal sizes. This theorem is a generalization of the result in \cite{DO} to hypergraphs. Moreover, as a straightforward corollary, we get $ \chi^{e}(\mathcal{K}_n^r)=2 $ where $\mathcal{K}_n^r$ denotes the complete $ r$-uniform hypergraph of order $n$.
Also, correctness of Conjecture \ref{conj:r-HYPER} is confirmed for $t$-tight paths and $t$-tight cycles in Theorems \ref{path} and \ref{cycle}, respectively. Consequently, the exact value of $\chi^{e}$ is determined for loose paths, loose cycles, tight paths and tight cycles.
The theta hypergraph $\mathcal{H}_{\Theta}$ is considered as a generalization of $t$-tight cycle and the value of $\chi^{e}(\mathcal{H}_{\Theta})$ is determined by Theorem \ref{teta}.
Another regular structure that we explore is the family of geometric planes. In Theorem \ref{plane}, the parameter $\chi^{e}$ is determined for affine planes and projective planes which are known as the most applicable geometric planes.
\section{Main results for graphs\label{graph}}
In this section, we will prove two theorems in graph case.
Davoodi and Omoomi proved that $ \chi^{e}(K_n(t)) =2$, for $n,t \geq 2$ \cite{DO}. In the first theorem, we have eliminated the equality requirement on the parts. Precisely, we demonstrate that $ \chi^{e}(G) =2$, where $G$ is complete $n$-partite graph with parts of different sizes.
The second theorem specifies $\chi^{ven}$ for any complete $ 3$-partite graph $G$. This statement improves the result $ \chi^{ven}(G)\leq 3$ due to Baudon et al. in \cite{B}, as well.
\begin{theorem}\label{g1}
Let $G$ be a complete $ n$-partite graph with at least $\lfloor \frac{n-1}{2}\rfloor$ parts of size greater than one. Then $ \chi^{e}(G) =2$.
\end{theorem}
\begin{proof}
First, assume that $ n=2k $ for some $k\in \mathbb{N}$ and $X_1, X_2, \ldots ,X_k, Y_1,Y_2,\ldots ,Y_k$ are parts of $G$ such that $|X_i|=m_i$, $|Y_i|=n_i$, for $ 1\leq i\leq k $ and $m_1\geq m_2\geq \cdots \geq m_k\geq n_k\geq n_{k-1}\geq \cdots \geq n_1$.
Let $ B[X,Y] $ be the graph with parts $ X=\{x_1,x_2,\ldots, x_k\} $ and $ Y=\{y_1,y_2,\ldots, y_k\} $ in which $ B[Y]\cong K_k $ and $ N_B(x_i)=\{y_1,y_2,\ldots, y_i\} $ for $ 1\leq i \leq k $.
Suppose that $ B^* $ is the graph obtained from $ B $ by blowing up every vertex $x_i$ with a set of size $m_i$ and every vertex $y_i$ with a set of size $n_i$, for $1\leq i \leq k $. Note that $B^*$ is a subgraph of $G$.
We denote by $X_i $ ( resp. $Y_i$) the set of vertices corresponding to $x_i$ ( resp. $y_i$) in $V(B)$, for every $1\leq i \leq k $.\\
\textbf{Case 1.} $m_k > n_k$.\\
In this case we define $w(e)=2$ if $e\in E(B^*)$ and $w(e)=1$, otherwise.
Hence, for every vertex $ v\in V(G) $ we have,
\begin{center}
\begin{equation}\label{1}
\sigma^{e}(v)=
\left\{
\begin{array}{ll}
|V(G)|+\sum _{j=1}^i n_j-m_i & \ \ \ v\in X_i, \\
|V(G)|+\sum _{j=i}^k m_j + \sum_{j=1}^k n_j-2n_i & \ \ \ v\in Y_i.
\end{array}
\right.
\end{equation}
\end{center}
We claim that $ w $ induces a proper vertex coloring for $ G $. Let $1\leq \ell <\ell' \leq k $.
Suppose to the contrary that $\sigma^{e}(x_{\ell})=\sigma^{e}(x_{\ell'})$. Then, $n_{\ell +1}+\cdots +n_{\ell'}+m_{\ell}=m_{\ell'},$
which is impossible since $m_{\ell}\geq m_{\ell'}$ and $n_{\ell +1}+\cdots +n_{\ell'} >0$. Now, suppose that $\sigma^{e}(y_{\ell})=\sigma^{e}(y_{\ell'})$. Thus,
$m_{\ell}+\cdots +m_{\ell' -1}+2n_{\ell'}=2n_{\ell}.$
It is impossible since $n_{\ell'}\geq n_{\ell}$ and $m_{\ell}+\cdots +m_{\ell' -1} >0$.
If $\sigma^{e}(x_{\ell})=\sigma^{e}(y_{\ell'})$. Then, $\sum_{j=1}^k n_j+\sum _{j=\ell'}^k m_j+m_{\ell}=\sum _{j=1}^{\ell} n_j +2n_{\ell'}.$
It doesn't occur because $ \sum_{j=1}^k n_j >\sum _{j=1}^{\ell} n_j$ and $m_k+m_{\ell} \geq 2n_{\ell'}$.
Now, let $\sigma^{e}(x_{\ell'})=\sigma^{e}(y_{\ell})$. Hence, $\sum_{j=1}^k n_j+\sum _{j=\ell}^k m_j+m_{\ell'}=\sum _{j=1}^{\ell'} n_j +2n_{\ell}.$
It is impossible since $ \sum_{j=1}^k n_j \geq \sum _{j=1}^{\ell'} n_j$ and $m_{\ell}+m_k+m_{\ell'} >2n_{\ell}$.
If $\sigma^{e}(x_{\ell})=\sigma^{e}(y_{\ell})$, for some $1\leq \ell \leq k $. Then, $\sum _{j=1}^{\ell} n_j +2n_{\ell}=m_{\ell}+\sum _{j=\ell}^k m_j+\sum_{j=1}^k n_j$. It is impossible since $\sum_{j=1}^k n_j\geq \sum _{j=1}^{\ell} n_j $, $m_{\ell}+m_k \geq 2n_{\ell}$ and $m_k >n_k \geq n_{\ell}$.\\
\textbf{Case 2.} $n_k = m_k$ and $m_{k-1}>m_k.$\\
Consider a fixed vertex $z$ in $X_{k-1}$. Now, define weighting function $w$ as follows. Set $w(e)=2$ if $e\in E(B^*)$ or $e=zv$ for every $v\in X_k$ and $w(e)=1$, otherwise. Thus, for every vertex $ v\in V(G) $ we have,
\begin{center}
\begin{equation}\label{1}
\sigma^{e}(v)=
\left\{
\begin{array}{ll}
|V(G)|+\sum _{j=1}^i n_j-m_i & \ \ \ v\in X_i, i\neq k, v\neq z, \\
|V(G)|+\sum _{j=1}^{k-1} n_j +m_{k}-m_{k-1} & \ \ \ v=z,\\
|V(G)|+\sum _{j=1}^{k} n_j -m_{k}+1 & \ \ \ v\in X_k, \\
|V(G)|+\sum _{j=i}^k m_j + \sum_{j=1}^k n_j-2n_i & \ \ \ v\in Y_i.
\end{array}
\right.
\end{equation}
\end{center}
Similar to the previous case we show that $w$ induces a proper vertex coloring for $G$ in this case. Consider a vertex $u \in X_k$. Suppose that $\sigma^{e}(u)=\sigma^{e}(x)$, for some $x\in X_{\ell}$, $1\leq \ell \leq k-1 $ and $x \neq z$. Therefore, $\sum _{j=1}^{k} n_j + m_{\ell}+1=m_k +\sum _{j=1}^{\ell} n_j$. This is a contradiction since $\sum _{j=1}^{k} n_j >\sum _{j=1}^{\ell} n_j$ and $m_{\ell} > m_k$.
If $\sigma^{e}(u)=\sigma^{e}(y)$, for some $y \in Y_{\ell}$, $1\leq \ell \leq k $, then $2n_{\ell}+1=m_k + \sum _{j=\ell}^k m_j.$
If $\ell =k$, then we have $2n_{\ell}+1=2m_k$ that does not hold. Otherwise $1\leq \ell \leq k-1$. Since $m_{\ell}> m_k \geq n_{\ell}$ and $m_{\ell} >1$, it is clear that $2m_k +m_{\ell} >2n_{\ell}+1$ that is a contradiction.
Now let $\sigma^{e}(u)=\sigma^{e}(z)$. Thus, $n_k +m_{k-1}+1=2m_{k}.$
Equality $m_k=n_k$ implies $m_{k-1}+1=m_{k}$ that contradicts $m_{k-1}> m_{k}$.
If $\sigma^{e}(z)=\sigma^{e}(x)$ for some $x\in X_{\ell}$, $1\leq \ell \leq k-2 $, then
$\sum _{j=1}^{k-1} n_j +m_k+m_{\ell}=\sum _{j=1}^{\ell} n_j+m_{k-1}$. It does not occur since $\sum _{j=1}^{k-1} n_j >\sum _{j=1}^{\ell} n_j$ and $m_{\ell} \geq m_{k-1}$.
Finally, let $\sigma^{e}(z)=\sigma^{e}(y)$ for some $y\in Y_{\ell}$, $1\leq \ell \leq k $. Hence, $2n_{\ell}=m_{k-1}+\sum _{j=\ell}^k m_j $. It contradicts $m_{k-1} >m_k \geq n_{\ell}$.\\
\textbf{Case 3.} $n_k = m_k=m_{k-1}>1$\\
Let $X_{k}=\{u_1,u_2,\ldots ,u_p\}$, $X_{k-1}=\{u'_1,u'_2,\ldots ,u'_p\}$ and $A=\{u_1u'_1, u_2u'_2,\ldots ,u_pu'_{p}\}\subset E(G)$. Here, we present a weighting function as follows. We define $w(e)=2$ if $e\in E(B^*)\cup A$ and $w(e)=1$ otherwise. Therefore, for every vertex $ v\in V(G) $ we have,
\begin{center}
\begin{equation}\label{1}
\sigma^{e}(v)=
\left\{
\begin{array}{ll}
| V(G)|+\sum _{j=1}^i n_j-m_i & \ \ \ v\in X_i, 1\leq i\leq k-2, \\
| V(G)|+\sum _{j=1}^{k-1} n_j -m_{k-1}+1 & \ \ \ v\in X_{k-1},\\
|V(G)|+\sum _{j=1}^{k} n_j -m_{k}+1 & \ \ \ v\in X_k, \\
|V(G)|+\sum _{j=i}^k m_j + \sum_{j=1}^k n_j-2n_i & \ \ \ v\in Y_i.
\end{array}
\right.
\end{equation}
\end{center}
Consider two fixed vertices $u\in X_k$ and $u'\in X_{k-1}$. Suppose that $\sigma^{e}(u)=\sigma^{e}(x)$ for some $x\in X_{\ell}$, $1\leq \ell \leq k-2. $ Then, $ m_{k}=\sum _{j={\ell}+1}^k n_j+m_{\ell}+1$ that is a contradiction. If $\sigma^{e}(u')=\sigma^{e}(x)$ for some $x\in X_{\ell}$, $1\leq \ell \leq k-2 $, then $ m_{k-1}=m_{\ell}+\sum _{j={\ell}+1}^{k-1} n_j+1$. It is not possible since $m_{\ell}+1>m_{k-1}$. By the assumptions $m_{k}=m_{k-1}$ and $n_k >0$, we have $\sigma^{e}(u)>\sigma^{e}(u')$. Now, let $\sigma^{e}(u)=\sigma^{e}(y)$ for some $y\in Y_{\ell}$, $1\leq \ell \leq k. $ Hence, $2n_{\ell}+1=m_{k}+ \sum _{j={\ell}}^{k} m_j$. It does not occur since $m_{\ell}\geq m_k \geq n_{\ell}$ and $m_k >1$. If $\sigma^{e}(u')=\sigma^{e}(y)$ for some $y\in Y_{\ell}$, $1\leq \ell \leq k. $ Then, $2n_{\ell}+1=m_{k-1}+\sum _{j={\ell}}^{k} m_j +n_k$ which is not possible.
Since $u$ and $u'$ are chosen arbitrarily, we are done.
Now, we may assume that $n=2k+1$ for some integer $k$. Let $V_1,V_2,\ldots ,V_{n}$ be the parts of $G$ such that $|V_i|\geq |V_{i+1}|$ for every $1\leq i \leq n-1$. Therefore, $deg_G(v)\leq deg_G(u)$ for every $v\in V_1$ and $u \in V(G)\setminus V_1$ and graph $G-V_1$ is a complete $(n-1)$-partite graph, where $n-1=2k$. Now, construct graph $B^*$ on $G-V_1$, as mentioned above. Clearly, $deg_{B^*}(v)=0$ for every vertex $v\in V_1$ and $deg_{B^*}(v)>0$ for remaining vertices of $G$. Here, we consider weighting function $w$ similar to what applied in the previous cases. One can easily see that $\sigma^{e}(v)=deg_{G}(v)\leq deg_{G}(u)<\sigma^{e}(u)$ for every $v\in V_1$ and $u\in V(G)\setminus V_1$. Also, according to what mentioned above, we have $\sigma^{e}(u)\neq \sigma^{e}(u')$ for every $u,u'\in V(G)\setminus V_1$.
\end{proof}
\begin{theorem}\label{g2}
Let $ G \ncong K_3 $ be the complete $ 3$-partite graph with parts $ V_1, V_2, V_3 $. Then
\begin{equation*}
\chi^{ven}(G)=
\left\{
\begin{array}{cl}
1 & \ \ \ |V_i|\neq|V_j|,\ i\neq j,\ 1\leq i,j\leq 3, \\
2 & \ \ \ otherwise.
\end{array}
\right.
\end{equation*}
\end{theorem}
\begin{proof}
Obviously, if $ |V_i|\neq|V_j| $ for $ 1\leq i\neq j\leq 3 $, then $ w : V(G)\cup E(G) \rightarrow \{1\}$ turns out a proper vertex coloring and consequently, $ \chi^{ven}(G)=1 $. For the case $ |V_1|=|V_2|=|V_3|=t $, Theorem \ref{g1} implies that $\chi^{e}$ and consequently $ \chi^{ven} $ are equal to 2.
Thus, one may assume that $ G $ contains exactly two parts $ V_1 $ and $ V_2$ of a same size. We consider the following two cases and in each case present a total weighting function which induces a proper vertex coloring.\\
\textbf{Case 1.}
$ |V_1|= |V_2| =t $ and $ |V_3|\notin\{t,\frac{2t}{3}\} $.\\
Let $n=|V_1|+|V_2|+|V_3|$. Then $n>2t$. Define function $ w:V(G)\cup E(G)\rightarrow \{1,2\} $ as follows. For every vertex $ v\in V(G) $, $ w(v)=1 $ and
\begin{equation*}
w(e)=
\left\{
\begin{array}{ll}
1 & \ \ \ e\in [V_2,V_3], \\
2 & \ \ \ \rm{otherwise,}
\end{array}
\right.
\end{equation*}
in which notation $ e\in [V_2,V_3]$ means that edge $e$ is between parts $V_2$ and $V_3$.
Now, let $ v_i\in V_i $, $ 1\leq i\leq 3 $. By definition of $\sigma^{ven}$, we have
\begin{align*}
\sigma^{ven}(v_1) &=1+\big(2t+2(n-2t)\big)+\big(t+(n-2t)\big)=3n-3t+1,\\
\sigma^{ven}(v_2)&=1+\big(2t+(n-2t)\big)+\big(t+(n-2t)\big)=2n-t+1,\\
\sigma^{ven}(v_3)
&=1+3t+2t=5t+1.
\end{align*}
Note that the assumptions $ |V_1|= |V_2| =t $ and $ |V_3|\notin\{t,\frac{2t}{3}\} $ concludes that $ \big|\{3n-3t+1, 2n-t+1, 5t+1\}\big|=3 $. Hence, this coloring is proper.\\
\textbf{Case 2.}
$ |V_1|= |V_2| =t $ and $ |V_3|=\frac{2t}{3} $.\\
In this case, let
\begin{equation*}
w(e)=
\left\{
\begin{array}{cl}
2 & \ \ \ e\in [V_1,V_2], \\
1 & \ \ \ \text{otherwise,}
\end{array}
\right.
\qquad and \qquad
w(v)=
\left\{
\begin{array}{cl}
2 & \ \ \ v\in V_1, \\
1 & \ \ \ \text{otherwise.}
\end{array}
\right.
\end{equation*}
For $ v_i\in V_i $, $ 1\leq i\leq 3 $,
\begin{align*}
\sigma^{ven}(v_1)&=2+(2t+\frac{2t}{3})+(t+\frac{2t}{3})=4t+\frac{t}{3}+2,\\
\sigma^{ven}(v_2)&=1+(2t+\frac{2t}{3})+(2t+\frac{2t}{3})=5t+\frac{t}{3}+1,\\
\sigma^{ven}(v_3)&=1+2t+3t=5t+1.
\end{align*}
Since $ t, \frac{2t}{3}\in \mathbb{N} $, every pair of adjacent vertices receive different colors.
\end{proof}
\section{Main results for hypergraphs}\label{hgraphs}
In this section, first some preliminary definitions will be given and then a number of results will be presented for hypergraphs.
Let $\mathcal{H}$ be a hypergraph.
For a vertex $v \in V(\mathcal{H})$,\textit{ degree }of $v$ in $\mathcal{H}$, denoted by $deg_\mathcal{H}(v)$, is the number of edges in $\mathcal{H}$ containing $v$. We say that $\mathcal{H}'$ is a subhypergraph of $\mathcal{H}$, if $V(\mathcal{H}')\subseteq V(\mathcal{H})$ and $E(\mathcal{H}')\subseteq E(\mathcal{H})$. A subhypergraph $\mathcal{H}'$ of $\mathcal{H}$ is called spaning, whenever $V(\mathcal{H}')= V(\mathcal{H})$.
\subsection{Complete hypergraphs}
An $r$-uniform hypergraph is called \textit{$n$-partite} if there is a partition of the vertex set into $n$ parts such that each edge
has at most one vertex in each part.
A {\it complete $r$-uniform hypergraph} of
order $n$, denoted by { $\mathcal{K}_n^r$}, is a hypergraph consisting of
all the $r$-subsets of vertex set $V$ of cardinality $n$.
\begin{theorem}\label{complete}
Let $ r\geq 3 $ and $\mathcal{K}_n^r(t) $ be the complete $ r$-uniform $ n$-partite hypergraph on vertex sets $ V_1,\ldots, V_n $ such that $\vert V_i\vert = t$, for $1\leq i\leq n$. If $ n>2(r-1)^2 $, then $ \chi^{e}(\mathcal{K}_n^r(t))=2 $.
\end{theorem}
\begin{proof}
Let $\mathcal{H}=\mathcal{K}_n^r(t)$. We specify a subhypergraph $ \mathcal{H}^*\subset \mathcal{H} $ such that assigning weight $ 2 $ to the edges of $ \mathcal{H}^* $ and weight $ 1 $ to the remaining edges of $ \mathcal{H} $ yields a proper vertex coloring for $ \mathcal{H} $.
In the other words, every arbitrary edge in $\mathcal{H} $ has two vertices with different colors. Since $ \mathcal{H} $ is regular and for every vertex $ u \in V(\mathcal{H}) $ we have, $$ \sigma ^{e}(u)=\sum_{e\ni u}w(e)=2deg_{\mathcal{H}^*}(u)+deg_{\mathcal{H}\backslash \mathcal{H}^*}(u)=deg_{\mathcal{H}}(u)+deg_{\mathcal{H}^*}(u), $$
it suffices to demonstrate that every edge in $ \mathcal{H} $ contains two vertices $ u $ and $ u' $ such that
$ deg_{\mathcal{H}^*}(u)\neq deg_{\mathcal{H}^*}(u') $.
Suppose that $ n=(r-1)p+q $, where $ 0\leq q\leq r-2 $. Now, partition vertex sets $ V_1,\ldots, V_n $ into $ p $ disjoint classes $ \mathcal{A}_1, \mathcal{A}_2\ldots, \mathcal{A}_p $ each of size $ r-1 $ and a possible class $ \mathcal{A}_0 $ of size $ q $, i.e.
\begin{align*}
\mathcal{A}_0:&=\{V_1^0, V_2^0, \ldots, V_q^0\},\\
\mathcal{A}_j:&=\{V^j_1, V^j_2, \ldots, V^j_{r-1}\}, \qquad 1\leq j\leq p.
\end{align*}
Note that $ \mathcal{A}_0 $ is empty, whenever $ q=0 $. Hereafter, we handle the problem in the two cases based on parity of $ p $. \\
\textbf{Case 1.} $ p=2k $ for some integer $ k $.\\
Let $ E(\mathcal{H}^*)=\bigcup_{j=1}^{k}E_j $, where each
$ E_j $ is a collection of edges defined as follows. Note that by $ x\in \mathcal{A}_h $ we mean that $ x $ is in a $ t $-tuple of $ \mathcal{A}_h $.
\begin{align*}
E_j:=\bigg\{ (x, y_1, y_2,\ldots, y_{r-1}) \big|\ &x\in \mathcal{A}_{k+j},\ (y_1,y_2,\ldots, y_{r-1})\in V^s_{1}\times V^s_{2}\times \cdots\times V^s_{r-1},\\
& k-j+1\leq s\leq 2k,\ s\neq k+j
\bigg\}.
\end{align*}
Now, we show that $ \mathcal{A}_i $'s have different degrees in $ \mathcal{H}^* $. Since every edge in $ \mathcal{H} $ has size $ r $, it contains two vertices of different $ \mathcal{A}_i $'s. Therefore, every edge in $ \mathcal{H} $ contains two vertices with different degrees in $ \mathcal{H}^* $.
Let $ z\in \mathcal{A}_{j} $ and $ z'\in \mathcal{A}_{k+j'} $, for some $ 1\leq j,j'\leq k $. Then
\begin{align}
deg_{\mathcal{H}^*}(z)&=(r-1)jt^{r-1},\label{eq:dAj}\\
deg_{\mathcal{H}^*}(z')&=\big[
2k-(
k-j'+1
)
\big]t^{r-1}
+
(k-1)
(r-1)t^{r-1}\nonumber\\
&=\big(kr-r+j'\big)t^{r-1}.\label{eq:dAj'}
\end{align}
If $ z\in \mathcal{A}_k $ and $ z'\in \mathcal{A}_{k+1} $, then the assumptions
$ n>2(r-1)^2 $ and $ \lfloor\frac{n}{r-1}=2k\rfloor $ conclude that
\begin{align}\label{eq:dA nesf}
deg_{\mathcal{H}^*}(z)=(r-1)kt^{r-1}<(kr-r+1)t^{r-1}=deg_{\mathcal{H}^*}(z').
\end{align}
For every $ z''\in \mathcal{A}_0 $, $ deg_{\mathcal{H}^*}(z'')=0 $. On the other hand, by relations
\eqref{eq:dAj}, \eqref{eq:dAj'} and \eqref{eq:dA nesf}, it is clear that
$ deg_{\mathcal{H}^*}(u) $ is monotone increasing in $ j $, for every arbitrary vertex $ u\in \mathcal{A}_j $, $ 0\leq j \leq p $. It completes the proof when $ p $ is even.\\
\textbf{Case 2.} $ p=2k+1 $ for some integer $ k $.\\
The proof is similar to what mentioned above.
In this case, $ E(\mathcal{H}^*)=\bigcup_{j=1}^{k+1}E_j $, where
\begin{align*}
E_j:=\bigg\{ (x, y_1, y_2,\ldots, y_{r-1}) \big|\ &x\in \mathcal{A}_{k+j},\ (y_1,y_2,\ldots, y_{r-1})\in V^s_{1}\times V^s_{2}\times \cdots\times V^s_{r-1},\\
& k-j+2\leq s\leq 2k+1,\ s\neq k+j
\bigg\}.
\end{align*}
Let $ z\in \mathcal{A}_{j} $ and $ z'\in \mathcal{A}_{k+j'} $, for some $ 1\leq j\leq k $ and $ 1\leq j'\leq k+1 $. Then
\begin{align}
deg_{\mathcal{H}^*}(z)&=(r-1)jt^{r-1},\label{eq:dAj1}\\
deg_{\mathcal{H}^*}(z')&=
\big[2k+1-(
k-j'+2
)\big]t^{r-1}
+
k
(r-1)t^{r-1}\nonumber\\
&=\big(kr+j'-1\big)t^{r-1}.\label{eq:dAj'1}
\end{align}
For $ z\in \mathcal{A}_k $ and $ z'\in \mathcal{A}_{k+1} $, we have
\begin{align}\label{eq:dA nesf1}
deg_{\mathcal{H}^*}(z)=(r-1)kt^{r-1}<(kr+1-1)t^{r-1}=deg_{\mathcal{H}^*}(z').
\end{align}
By relations
\eqref{eq:dAj1}, \eqref{eq:dAj'1} and \eqref{eq:dA nesf1},
for every arbitrary vertex $ u\in \mathcal{A}_j $, $ deg_{\mathcal{H}^*}(u) $ is monotone increasing in $ j $, $ 0\leq j \leq 2k+1 $, that completes the proof for this case, as well.
\end{proof}
\begin{cor}
Let $ n>2(r-1)^2 $ and $r\geq 3$. Then, $ \chi^{e}(\mathcal{K}_n^r(t))=\chi^{ve}(\mathcal{K}_n^r(t))= \chi^{ven}(\mathcal{K}_n^r(t))=2. $ Also, for $n\geq r+1$ and $r\geq 3$,
$ \chi^{e}(\mathcal{K}_n^r)= \chi^{ve}(\mathcal{K}_n^r)=\chi^{ven}(\mathcal{K}_n^r)=2. $
\end{cor}
\begin{proof}
Regularity of hypergraph $ \mathcal{K}_n^r(t) $ implies that $ \chi^{e}(\mathcal{K}_n^r(t))$ and $ \chi^{ve}(\mathcal{K}_n^r(t))$ are strictly greater than one. Also, $ \chi^{ven}(\mathcal{K}_n^r(t))> 1$, since $ |N_{\mathcal{K}_n^r(t)}(u)|=|N_{\mathcal{K}_n^r(t)}(v)|$, for every $ u, v \in V({\mathcal{K}_n^r(t)}) $.
Let $ w: E(\mathcal{K}_n^r(t))\rightarrow \{1,2\} $ be the edge weighting which yields a proper vertex coloring. This function exists by Theorem \ref{complete}. It can be generalized to a total weighting function $ w'$ by setting $w'(v)=1$ for every vertex $v$, and $w'(e)=w(e)$, for all $ e\in E({\mathcal{K}_n^r(t)}) $.
Hence, $ \chi^{ven}({\mathcal{K}_n^r(t)}),\chi^{ve}({\mathcal{K}_n^r(t)})\leq\chi^{e}({\mathcal{K}_n^r(t)}) $. By Theorem \ref{complete}, $\chi^{ven}({\mathcal{K}_n^r(t)})=\chi^{ve}({\mathcal{K}_n^r(t)})=2 $.
However, Theorem \ref{complete} yields $ \chi^{e}(\mathcal{K}_n^r)=2 $ for $ n>2(r-1)^2 $, in what follows we bring an argument which shows by a combination of this theorem and the previous argument we have $ \chi^{ve}({\mathcal{K}_n^r})=\chi^{ven}({\mathcal{K}_n^r})=2$, for $ n\geq r+1 $ and $ r\geq 3 $.
To see this, we construct a hypergraph $ {\cal H}^*\subseteq{\mathcal{K}_n^r} $ such that for every selection of $ r $ vertices from $ V({\mathcal{K}_n^r}) $, always there exist $ i $ and $ j $, $ 1\leq i,j\leq r $, where $ \deg_{{\cal H}^*}(v_i)\neq\deg_{{\cal H}^*}(v_j) $. Now, we assign weight $ 2 $ to edge $ e $, if $ e\in E({\cal H}^*) $ and define weight $ e $ to be $ 1 $ for the remaining edges of $ \mathcal{K}_n^r $. This total weighting function yields the desired vertex coloring.
The configuration of $ H^* $ is as follows. Assume that if $ n $ divided by $ r-1 $ equals $ p $ with reminder $ q $, i.e. $ n=p(r-1)+q $. We define partition $ {\cal A}_0, {\cal A}_1, \ldots, {\cal A}_p $ of the vertex set $ V({\mathcal{K}_n^r}) $, where
$ {\cal A}_i=\{x_1^i,x_2^i,\ldots, x_{r-1}^i\} $, $ 1\leq i\leq p $ and $ {\cal A}_0=\{x_1^0,x_2^0,\ldots, x_{q}^0\} $.
First, suppose that $ p=1 $. By the assumption $ n\geq r+1 $, we have $ q\geq 2 $. In this case, let
$ E({\cal H}^*)=\{{\cal A}_1\cup{x_k^0} \ | \ 1\leq k\leq q\} $. Note that if $ v\in {\cal A}_1 $, then $ \deg_{{\cal H}^*}(v)=q\geq 2 $ and for $ v\in {\cal A}_0 $, $ \deg_{{\cal H}^*}(v)=1 $. Clearly, in this way there is no $ r $-tuple of vertices with a same degree in $ {\cal H}^* $.
Hence, we may assume that $ p\geq2 $. The edge set of $ {\cal H}^* $ is defined by $ E({\cal H}^*)=\bigcup_{i=1}^{p-1}\big\{{\cal A}_i\cup{x_k^j} \ | \ i+1\leq j\leq p, 1\leq k\leq r-1\big\} $. Therefore,
$$ \deg_{{\cal H}^*}(v)=
\left\{
\begin{array}{ll}
(i-1)+(r-1)(p-i) & \ \ \ v\in {\cal A}_i, 1\leq i\leq p \\
0 & \ \ \ v\in {\cal A}_0.
\end{array}
\right.
$$
It is readily to check that $ \deg_{{\cal H}^*}(v)\neq\deg_{{\cal H}^*}(u) $ for $ u\in {\cal A}_j $ and $ v\in {\cal A}_{j'} $, $ j\neq j' $ which completes the proof.
\end{proof}
\subsection{Paths and Cycles}
Let $t,r$ and $ n $ be integers, where $1\leq t <r$. An $r$-uniform $t$-tight path of order $n$, denoted by $ \mathcal{P}_{n,t}^{(r)}, $ is a hypergraph with vertex set $[n]=\{1,2,\ldots, n\}$ and edge set $\{e_1,e_2,\ldots ,e_{\ell}\}$, such that ${\ell}=\frac{n-t}{r-t}$ is an integer and for $1\leq i\leq \ell$, $e_i=\{ (i-1)(r-t)+
1, (i - 1)(r - t) + 2, \dots , (i - 1)(r - t) + r\}$. Note that the edges are intervals
of length $r$ in $[n]$ and consecutive edges intersect in exactly $t$ vertices. Whenever $t=1$, $ \mathcal{P}_{n,1}^{(r)} $
commonly referred as a loose path. Also, a $ \mathcal{P}_{n,r-1}^{(r)} $ is called a tight path.
An $r$-uniform $t$-tight cycle of order $n$, denoted by $ \mathcal{C}_{n,t}^{(r)} $, is a hypergraph with vertex set $[n]$ and edge set $\{e_1,e_2,\ldots ,e_{\ell}\}$, where ${\ell}=\frac{n}{r-t}$ is an integer and for $1\leq i\leq \ell$, $e_i =\{ (i-1)(r-t)+
1, (i - 1)(r - t) + 2, \dots , (i - 1)(r - t) + r\}$. Note that here the index of $e_i$'s is considered on module $n$ and every two consecutive edges have exactly $t$ vertices in common. Cycles $ \mathcal{C}_{n,1}^{(r)} $ and $ \mathcal{C}_{n,r-1}^{(r)} $ are called loose and tight cycles, respectively. The graphs path, $P_n$, and cycle, $C_n$, are special cases of $ \mathcal{P}_{n,t}^{(r)} $ and $ \mathcal{C}_{n,t}^{(r)} $, whenever $r=2$ and $t=1$. Hence, a restriction of every result for hypergraphs concludes an statement for graphs.
By notation $e=\{x_1,x_2,\ldots, x_r\}$ for an edge $e$, we mean that there is a total ordering on $x_i$'s, i.e. $x_i<x_{i+1}$ for every $1\leq i \leq r-1$.
In the two following theorems we characterize $\chi^{e}(\mathcal{P}_{n,t}^{(r)})$ and $\chi^{e}(\mathcal{C}_{n,t}^{(r)})$ for every $n$, $r$ and $t$.
\begin{theorem}\label{path}
Let $\mathcal{P}=\mathcal{P}_{n,t}^{(r)} $ be an $ r$-uniform $ t$-tight path of order $n$ and length $ \ell>2 $. Then,
\begin{equation*}
\chi^{e}(\mathcal{P})=
\left\{
\begin{array}{cl}
2 & \ \ \ t=\frac{r}{2}\ or\ t>\frac{r}{2},\ \text{r-t}\ divides \ \text{r}\ and\ \ell \geq 2\frac{r}{r-t}-1, \\
1 & \ \ \ \text{otherwise.}
\end{array}
\right.
\end{equation*}
\end{theorem}
\begin{proof}
Let $ \cal P $ be the path on edges $ e_1, e_2, \ldots, e_{\ell} $, where $ \ell>2 $.
Assume that $ t<\frac{r}{2} $ and $ e_i=\{x_1,\ldots, x_r\} $ is an arbitrary edge. If $i\neq \ell$, then $ 1=\deg_{\cal P}(x_{r-t})<\deg_{\cal P}(x_r)=2 $. Also, for edge $ e_\ell $, we have $ 1=\deg_{\cal P}(x_{t+1})<\deg_{\cal P}(x_1)=2 $. Therefore, every edge of $ \cal P $ contains two vertices of different degrees and $ \chi^{e}({\cal P})=1 $.
Let $ t=\frac{r}{2} $. Since degree of all the vertices in $ \cup^{\ell-1}_{i=2} e_i $ is two, $ \chi^e({\cal P})>1.$
Now, suppose that $ w:E(P)\rightarrow\{1,2\} $ is an edge weighting which induces a proper vertex coloring.
Clearly, $ w(e_i)\neq w(e_{i+2}) $, since otherwise, no matter what is $ w(e_{i+1}) $, all the vertices of $ e_{i+1} $ receive a same color.
Hence, every weight assignment to an edge forces weight of alternative edges on the path. Thus, it is enough to weight two consecutive edges on the path.
This fact implies that the only possible patterns to weight edges of a path are $ 1122\cdots $, $ 1221\cdots $, $ 2112\cdots $ and $ 2211\cdots $. Clearly, these weighting patterns yield a proper vertex coloring for $ \cal P $. Consequently, $ \chi^e({\cal P})=2.$
Now, let $ t>\frac{r}{2} $ and set $ k=\lfloor\frac{r}{r-t}\rfloor $. First assume that $ k=\frac{r}{r-t} $ is an integer and $ \ell\geq 2k-1 $. In this case, for every vertex $ x\in e_{i} $, $k\leq i\leq \ell-k+1 $, we have $ \deg_{\cal P}(x)=k $, thus $ \chi^e({\cal P})>1 $. On the other hand, the following edge weighting function shows $ \chi^e({\cal P})=2 $. For $ j\geq0 $, let
\begin{equation*}
w(e_{jk+1})=w(e_{jk+2})=\cdots=w(e_{jk+k})=
\left\{
\begin{array}{cl}
1 & \ \ \ j \ \text{is even,} \\
2 & \ \ \ j \ \text{is odd.}
\end{array}
\right.
\end{equation*}
Now, we suppose that $\frac{r}{r-t} $ is not an integer and $ \ell\geq 2k-1 $. Thus, $ r=k(r-t)+b_0 $, where $ b_0>0 $.
It is easy to check that if $ 1\leq i\leq k $ or $ \ell-k+1\leq i\leq\ell $, then $ e_i $ contains two vertices with different degrees.
On the other hand, for $ e_i=\{x_1, x_2, \ldots, x_r\} $, $ k+1\leq i\leq \ell-k $, we have $ k=\deg_{\cal P}(x_{r-b_0})<\deg_{\cal P}(x_r)=k+1 $.
It concludes that
$ \chi^e({\cal P})=1 $. Finally, for the case $ \ell<2k -1 $, every edge of $\mathcal{P}$ contains two vertices with distinct degrees and hence, $ \chi^e({\cal P})=1 $.
\end{proof}
The following statement is an straightforward conclusion of Theorem \ref{path}.
\begin{cor}
For every $ r$-uniform loose path $ \cal P $, we have $ \chi^e({\cal P})=1 $.
Also, let $ \cal P $ be the $ r$-uniform tight path of length $ \ell $. If $ \ell\geq 2r-1 $, then $ \chi^e({\cal P})=2 $. Otherwise $ \chi^e({\cal P})=1 $.
\end{cor}
\begin{theorem}\label{cycle}
Let $\mathcal{C}=\mathcal{C}_{n,t}^{(r)} $ be an $ r$-uniform $ t$-tight cycle of order $n$ and length $ \ell>2 $. Then,
\begin{equation*}
\chi^{e}(\mathcal{C})=
\left\{
\begin{array}{cl}
3 & \ \ \ t=\frac{r}{2}\ and\ \ell\overset{4}{\not\equiv}0,\\
2 & \ \ \ t=\frac{r}{2}\ and\ \ell\overset{4}{\equiv}0\ or\ t>\frac{r}{2}\ and\ \text{r-t}\ divides \ \text{r},\\
1 & \ \ \ \text{otherwise.}
\end{array}
\right.
\end{equation*}
\end{theorem}
\begin{proof}
Assume that $e_1,e_2,\ldots, e_{\ell}$ are the ordered edges of $\mathcal{C}$.
Let $ t<\frac{r}{2} $ and $ e=\{x_1,\ldots, x_r\} $ be an arbitrary edge. One can see that $ 2=\deg_{{\cal C}}(x_{1})>\deg_{{\cal C}}(x_{t+1})=1 $ and hence, $ \chi^e({{\cal C}})=~1 $.
If $ t=\frac{r}{2} $, then $ {\cal C}$ is a $ 2$-regular hypergraph. Therefore, $ \chi^e({\cal C})>1 $.
Let $ w:E(C)\rightarrow\{1,2\} $ be an edge weighting which induces a proper vertex coloring. Then, by the argument presented in the proof of Theorem \ref{path}, the only possible patterns to weight edges of $\mathcal{C}$ are $ 1122\cdots $, $ 1221\cdots $, $ 2112\cdots $ and $ 2211\cdots $.
If $ \ell\overset{4}{\equiv}0 $, then the first pattern yields a proper vertex coloring. One may readily check that no matter which pattern is applied, a monochromatic edge will be constructed when $ \ell\overset{4}{\not\equiv}0 $.
Now, we show that $ \chi^{e}({\cal C})=3 $.
We use the pattern $ 1122\cdots $ to weight the edges $ e_1,\ldots, e_{\ell-1} $ and put $ w(e_\ell)=3. $
If $ \ell\overset{4}{\equiv}1 $, then we are done. Otherwise,
it is enough to set $ w(e_{\ell-1}) = 3 $, as well.
Now, suppose that $ t>\frac{r}{2} $.
First, let $ k=\frac{r}{r-t} $, be an integer.
In this case,
$ {\cal C}$ is a $ k$-regular hypergraph.
Thus, $ \chi^e({\cal C})>1 $. Now, we prove that $ \chi^e({\cal C})=2 $. The assumption $ t>\frac{r}{2} $ implies that $ k\geq3 $.
If $ \ell \leq 2k$, we define $w(e_1)=w(e_2)=2$ and $w(e_i)=1$ for every $i$, $i \neq 1,2$. Clearly, this weighting function yields a proper vertex coloring in $\mathcal{C}$.
Otherwise, we may assume that $ \ell > 2k$. Let $ {\cal P}^* $ be the longest non-spaning $ (r-t)$-tight path in $ {\cal C}$ starting with edge $e_1$.
Since
$ r-t<\frac{r}{2} $, every $r$-consecutive vertices of $ {\cal C}$ contains two vertices with two different degrees $0$ and $1$ or $1$ and $2$ in $ {\cal P}^* $. Hence, the following weight function $ w $ causes a proper vertex coloring.
\begin{equation*}
w(e)=
\left\{
\begin{array}{cl}
2 &e \in E({\cal P}^*), \\
1 & \text{otherwise.}
\end{array}
\right.
\end{equation*}
It is easy to see that every edge in $\mathcal{C}$, contains two vertices with different weights $k$ and $k+1$ or $k+1$ and $k+2$.
Therefore,
$ \chi^e({\cal C})=2 $.
Finally, let $t> \frac{r}{2}$ and $ r=k(r-t)+b_0 $, where $ 0<b_0<r-t $. If $ e_i=\{x_1,\ldots, x_r\} $ is an arbitrary edge, then $ k=\deg_{\cal C}(x_{r-b_0})<\deg_{\cal C}(x_{r})=k+1 $. Hence, $ \chi^e({\cal C})=1 $.
\end{proof}
\subsection{Theta hypergraphs}
For integers $ r,s\geq 3 $ and $t$, let $p_1,p_2, \ldots, p_s$ be vertex disjoint $r$-uniform $ t$-tight paths of lengths $ \ell_1, \ell_2, \ldots, \ell_s$, respectively.
The $ r$-uniform \textit{theta hypergraph} $ \Theta_t^r(l_1,\ldots, l_s) $ is defined as follows. First, unify all the first $ t $ vertices of $ p_i $'s and call them $ x_1,\ldots, x_{t} $. Then, identify all the last $ t$ vertices of $ p_i $'s and nominate them $ y_1,\ldots, y_t$. Note that, if $s=2$, then $ \Theta_t^r(l_1, l_2) $ is the $r$-uniform $t$-tight cycle of length $ \ell_1+\ell_2$.
In the following, we classify theta hypergraphs based on the value of parameter $ \chi^{e}$.
\begin{theorem}\label{teta}
Let $r,s \geq 3$ and $t$ be integers and $ H_{\Theta}=\Theta_t^r(l_1,\ldots, l_s) $. Then,
\begin{itemize}
\item[i.] $ \chi^{e}(\mathcal{H}_{\Theta})=1$ when $t<\frac{r}{2}$.
\item[ii.] For $t>\frac{r}{2}$, if $r-t$ divides $r$ and $\ell _i > 2(k-1)$ for some $1\leq i \leq s$, then
$ \chi^{e}(\mathcal{H}_{\Theta})=2$. Otherwise, $ \chi^{e}(\mathcal{H}_{\Theta})=1$.
\item[iii.] If $t=\frac{r}{2}$, then
\begin{equation*}
\chi^{e}(\mathcal{H}_{\Theta})=
\left\{
\begin{array}{cl}
3 & \ \ \ \ell_1=1\ \text{and}\ \ell_i\overset{4}{\equiv}1,\ \ 2\leq i\leq s,\\
1 & \ \ \ \ell_i=2,\ \ 1\leq i\leq s,\\
2 & \ \ \ \text{otherwise.}
\end{array}
\right.
\end{equation*}
\end{itemize}
\end{theorem}
\begin{proof} Without loss of any generality, suppose that $\ell_1 \leq \ell_2 \leq \cdots \leq \ell_s$.\\
\begin{itemize}
\item[i.] Since every edge of $\mathcal{H}_{\Theta}$ contains two vertices with different degrees, we deduce that $ \chi^{e}(\mathcal{H}_{\Theta})=~1$.
\item[ii.] First, let $k=\frac{r}{r-t}$ be an integer. If $\ell _i \leq 2(k-1)$ for every $1\leq i \leq s$, then every edge $e$ of $\mathcal{H}_{\Theta}$ has nonempty intersection with set $\{x_1,\ldots, x_{t}, y_1,\ldots, y_{t}\}$. Also, $e$ contains two vertices with different degrees. Hence, $ \chi^{e}(\mathcal{H}_{\Theta})=1$.
Now, suppose that $\ell _j > 2(k-1)$ for some $1\leq j \leq s$. Thus, $k^{\text{th}}$ edge of $p_j$ is $k$-regular i.e. all of its vertices have degree $k$ in $\mathcal{H}_{\Theta}$. Therefore, $ \chi^{e}(\mathcal{H}_{\Theta})>1$. We show that $ \chi^{e}(\mathcal{H}_{\Theta})=2$.
Note that, for every $i,j$, $1\leq i\neq j\leq s$, the hypergraph $p_i \cup p_j$ is isomorphic to a $t$-tight cycle of length $\ell_i + \ell_j$ in $\mathcal{H}_{\Theta}$.
Let $\mathcal{H}^*=\cup_{j=2}^{s} E(\mathcal{P}^*_j)$, where $\mathcal{P}^*_j$ is the longest non-spaning $(r-t)$-tight path starting with the first edge of $p_1$, passing $p_1$ and then going through $p_j$ in cycle $p_1 \cup p_j$, whenever $\ell_j > 2(k-1)$.
Since
$ r-t<\frac{r}{2} $, every $r$-consecutive vertices of cycle $ p_1 \cup p_j$ contains two vertices with two different degrees $1$ and $2$, $1$ and $s$ or $1$ and $0$ in $\mathcal{H}^*$. Hence, the following weight function $ w $ yields a proper vertex coloring.
\begin{equation*}
w(e)=
\left\{
\begin{array}{cl}
2 &e \in E(\mathcal{H}^*), \\
1 & \text{otherwise.}
\end{array}
\right.
\end{equation*}
It is easy to see that every edge in $\mathcal{H}_{\Theta}$, contains two vertices with different weights.
Therefore,
$\chi^e(\mathcal{H}_{\Theta})=2 $.
Now, let $t>\frac{r}{2}$ and $r=k(r-t)+b_0$, where $0< b_0 <r-t$. One can readily see that every edge of $\mathcal{H}_{\Theta}$ contains two vertices of different degrees. Therefore, $ \chi^{e}(\mathcal{H}_{\Theta})=1$.
\item[iii.] First, let $ \ell_1=1 $. If $ \ell_i\overset{4}{\equiv}1 $, for $ 2\leq i\leq s $,
Clearly, $ \chi^{e}(\mathcal{H}_{\Theta})>1 $. To see $ \chi^{e}(\mathcal{H}_{\Theta})=3 $, suppose to the contrary that $ \chi^{e}(\mathcal{H}_{\Theta})=2 $ and $ w $ is the corresponding $ 2$-edge weighting. By the argument presented within the proof of Theorem \ref{path}, the only possible patterns to weight the edges of $ p_i $'s are $ 1122\cdots $, $ 1221\cdots $, $ 2112\cdots $ and $ 2211\cdots $.
Since $\ell_i\overset{4}{\equiv}1 $, for every path $ p_i $, $2\leq i \leq s$, always the first edge and the last edge receive a same weight. No matter weight of the edge $ \{x_1,\ldots,x_{\frac{r}{2}}, y_1,\ldots, y_{\frac{r}{2}}\} $ of $p_1$, this edge is monochromatic which is a contradiction. Hence, $ \chi^{e}(\mathcal{H}_{\Theta})>2 $.
Now, consider the following $ 3$-edge weighting for $ \mathcal{H}_{\Theta} $. Apply the pattern $ 1122\cdots $, for $p_i$'s, $ 2\leq i\leq s $, and weight $2$ to the edge $ \{x_1,\ldots,x_{\frac{r}{2}}, y_1,\ldots, y_{\frac{r}{2}}\} $ of $ p_1 $. Then, change weight of the first edge in $ p_2 $ to $ 3 $. It is easy to check that this weighting induces a proper vertex coloring for $ \mathcal{H}_{\Theta} $. Therefore, $ \chi^{e}(\mathcal{H}_{\Theta})=3$.
Now, suppose that $ \ell_1=1 $ and $ \ell_i\overset{4}{\not\equiv}1$, for some $ 2\leq i\leq s $. %
Assign $2$ to the edge $ \{x_1,\ldots,x_{\frac{r}{2}}, y_1,\ldots, y_{\frac{r}{2}}\} $ of $ p_1 $. Also, apply the pattern $ 1122\cdots $ for $p_i$'s, when $ \ell_i\overset{4}{\equiv}0$ or $1$ and the pattern $ 1221\cdots $ for $p_i$'s, whenever $ \ell_i\overset{4}{\equiv}2$ or $3$. This weight assignments implies a proper vertex coloring for $ \mathcal{H}_{\Theta} $.
Now, let $\ell_i \geq 2$ for all $1\leq i\leq s$. If $\ell_i = 2$ for all $i$, then every edge in $\mathcal{H}_{\Theta}$ contains two vertices with different degrees $2$ and $s$. The assumption $s \geq 3$ implies that $ \chi^{e}(\mathcal{H}_{\Theta})=1 $. Otherwise, $ \ell_i \geq 3$ for some $i$. We apply the pattern $ 2112\cdots $ for $p_i$, if $ \ell_i\overset{4}{\equiv}0$, the pattern $ 2211\cdots $, when $ \ell_i\overset{4}{\equiv}1$ or $2$ and the pattern $ 1122\cdots ,$ whenever $ \ell_i\overset{4}{\equiv}3$. This weighting yields a proper vertex coloring unless the case $s=3$, $ \ell_1,\ell_2\overset{4}{\equiv}3$ and $ \ell_3\overset{4}{\equiv}1$ or $2$. In this case, it is enough to change the pattern $p_3$ to $2112\cdots$.
\end{itemize}
\vspace*{-.5cm}
\end{proof}
\subsection{Geometric planes}
A projective plane of order $q$ is an ordered pair $(X,\mathcal{L})$ in which $X$ is a set of $q^2+q+1$ points and $\mathcal{L}$ is a subset of power set of $ X $, where $|\mathcal{L}|=q^2+q+1$ and satisfies the following conditions. Every member of $\mathcal{L}$ is a $(q+1)$-subset of $X$ called line, any point lies on $q + 1$ lines, every two points lie on a unique line and every two lines intersect in exactly one point.
An affine plane of order $q$ is an ordered pair $(X,\mathcal{L})$ in which $X$ is a set of $q^2$ points and $\mathcal{L}$ is a set of $q^2 + q$ lines, where
every line has $q$ points, any two points lie on a unique line and any point lies on $q + 1$ lines.
A parallel class in a geometric plane is a set of disjoint lines partitioning points set $X$.
It is a well known fact that the set of lines of an affine plane of order $ q $ is partitioned into $ q+1 $ parallel classes.
It is noteworthy that an affine plane of order $q$ exists if and only if there is a projective plane of order $q$.
To see this, let $(X,\mathcal{L})$ be a projective plane of order
$q$. Now, consider $(X',\mathcal{L}')$, where $X' =
X \setminus l_0$, $\mathcal{L}'= \{l \setminus {l}_0\ |\ l \in \mathcal{L}, l \neq l_0 \}$ and $l_0$ is a fixed line in $ \mathcal{L}$. It is easy to verify that structure $(X',\mathcal{L}')$ is an affine plane of order
$q$.
Conversely, assume that $(X,\mathcal{L})$ is an affine plane of order
$q$ with parallel classes $ \Pi_1,\ldots,\Pi_{q+1} $. We construct a projective plane $(X',\mathcal{L}')$ of order $q$ as follows. Add a new line $l_0=\{\infty_{1}, \infty_{2}, \ldots, \infty_{q+1}\}$ with $ q+1 $ new points and let $X'= X \cup l_0 $. Then, set $\mathcal{L}'=l_0\cup\big(\bigcup_{l\in \Pi_i} (l\cup \{ \infty_i \})\big) $. The structure $(X',\mathcal{L}')$ is a projective plane of order $q$.
In the following theorem, we determine the parameter $\chi^e(\mathcal{H})$ when $\mathcal{H}$ is an affine plane or a projective plane.
\begin{theorem}\label{plane}
Let $ \cal H $ be either an affine plane or a projective plane of order $ q $. Then $ \chi^e(\mathcal{H})=2 $.
\end{theorem}
\begin{proof}
First assume that $ \cal H $ is an affine plane of order $ q $ with parallel classes $ \Pi_1,\ldots,\Pi_{q+1} $.
By definition, for every line $ \ell\not\in \Pi_{q+1} $, $ |\ell\cap \ell_0|=1 $, where $ \ell_0=\{x_1,x_2,\ldots, x_q\} $ is a fixed line in $ \Pi_{q+1} $.
Let $ \ell_i^j $ be the unique line in class $ \Pi_j $ containing $ x_i $, $ 1\leq i,j\leq q $.
Define weighting function $ w:E({\cal H})\rightarrow\{1,2\} $ in which lines $ \ell_i^i $, $ 1\leq i\leq q-1 $, take weight $ 2 $ and all the remaining lines receive weight $ 1 $.
We claim that $ w $ induces a proper vertex coloring. For every $ \ell\in \Pi_{q}\cup\Pi_{q+1} $, there are $ q-1 $ lines of weight $ 2 $ each of them crossing $ \ell$ in exactly one point. Therefore, there are two elements in $ \ell$, say $ z,z' $, so that
$ \sigma^{e}(z)=q+1<q+2 \leq \sigma ^{e}(z') $.
Now, take an arbitrary line $ \ell'\in \Pi_{i} $, $ 1\leq i\leq q-1 $. There are $ q-2 $ lines of weight $ 2 $ crossing $ \ell' $ which implies existence of two points with different colors in $ \ell' $.
Let $ \mathcal {H}' $ be the projective plane of order $ q $ obtaining from $ \cal H $ by extending line set by the line $ \ell^*=\{\infty_{1}, \ldots, \infty_{q+1}\} $ and every $ \ell\in \Pi_i $, $ 1\leq i\leq q+1 $, by $ \ell\cup{\infty_{i}} $.
Now, we define weighting function $ w' $ for $ \cal H' $, using weighting function $ w $ as introduced above for affine plane $ \cal H $. Set $ w'(\ell^*)=1 $ and for every other line $ \ell'\in{\cal B(H')}\setminus \ell^* $, let the weight $ w'(\ell') $ be the same as its corresponding line in the affine plane. Then, $ \sigma ^{e}(\infty_{q+1})=q+1< q+2=\sigma ^{e}(\infty_{1}) $ that completes the proof.
\end{proof}
\section{Concluding Remark}
In Theorem \ref{g1}, some complete $n$-partite graphs with parts of arbitrary sizes are considered and the parameter $\chi^{e}$ is obtained for them. Theorem \ref{g2} determines $\chi^{ven}$ for complete $3$-partite graphs with parts of arbitrary sizes. Now, one may raise the following question.
\begin{problem}
Let $ G $ be the complete $n$-partite graph with parts of arbitrary sizes. What are the values of $ \chi^{e}(G) $ and $ \chi^{ven}(G) $?
\end{problem}
It was proved that if $ G $ is a $ 3$-colorable graph, then $ \chi^{e}(G)\leq 3$, \cite{KLT}. Consequently, the 1-2-3-conjecture is true for bipartite graphs. The only known bipartite graphs for which $ \chi^{e}$ reaches $3$ was an special subclass of theta graphs.
This observation motivated Khatirinejad et al. to ask the following question \cite{Khatir}.
\begin{problem} \em{\cite{Khatir}}\label{khatir}
Is it true that for every bipartite graph except an special subclass of theta graphs, we have $ \chi^{e}(G)\leq 2 $?
\end{problem}
Davoodi and Omoomi answered the problem in negative by extending this family to bipartite generalized polygon trees as a generalization of theta graphs (Theorem 4.4. in \cite{DO}).
Note that although there are many sufficient conditions for $ \chi^{e}(G)\leq 2 $, the problem of classifying graphs with $ \chi^{e}(G)\leq 2 $ is still an open problem, even for graphs where correctness of 1-2-3-conjecture was confirmed for them.
A similar observation can be considered for hypergraphs, as well.
Clearly, for a hypergraph $ \mathcal{H} $, $ \chi^e(\mathcal{H})=1 $ if and only if every edge of $\mathcal{H}$ contains two vertices with different degrees.
A pair of vertices $ u $ and $ v $ are called \textit{twins}, whenever the set
of edges containing $ u $ is the same as the set of edges containing $ v $.
A hypergraph is called \textit{twin-free} if it contains no twins. For a graph $ G $, let $ \mathcal{H}^r(G) $ be the family of $r$-uniform hypergraphs where every hypergraph $\mathcal{H}$ in $\mathcal{H}^r(G) $ is constructed as follow. Every vertex $v \in V( G) $ is replaced with a set $S_v$ of new vertices such that two vertices $u$ and $v$ are adjacent in $ G $ if and only if the set $S_u \cup S_v$ is an edge of size $r$ in $\mathcal{H}$.
For example, a cycle of length $4k+2$ can be transformed to an $r$-uniform hypergraph by setting $ |S_v|=r-1 $ for every other vertex on the cycle \cite{BDFL}.
This procedure can be generalized to every bipartite graph. It is sufficient to consider a breadth first search(BFS) tree of $G$ and then replace the vertices in odd levels with $ S_v $'s of size $p$ and the vertices in even levels with $ S_v $'s of cardinality $r-p$. Also, it is clear that a simple way to reach $ \cal H $, for an even $ r $, is to transform every vertex $ v $ to a set $ S_v $ of size $ \frac{r}{2} $.
Note that $|E(G)|=|E( \mathcal{H})|$ and $ \chi^{e}(G)= \chi^{e}( \mathcal{H})$ for any $\mathcal{H}\in \mathcal{H}^r(G) $. Also, every member of $ \mathcal{H}^r(G) $ has twins.
Define
$$ \mathfrak{A}^r=\{\mathcal{H} : \mathcal{H} \text{ is an } r\text{-uniform hypergraph and} \chi^{e}(\mathcal{H})\leq 2 \}, $$
$$ \mathfrak{B}^r=\{\mathcal{H} : \mathcal{H} \text{ is an } r\text{-uniform hypergraph and } \chi^{e}(\mathcal{H})> 2 \}. $$
Bennett and Dudek proved that $ \mathfrak{B}^r \neq\emptyset $ (Theorem 1 (ii) in \cite{BDFL}). Moreover, we presented examples of cycles and their generalizations to some theta hypergraphs $\mathcal{H}$ for which $ \chi^e(\mathcal{H})=3 $ in Theorems \ref{cycle} and \ref{teta}.
In \cite{KKP-H} Kalkowski et al. proposed the following conjecture.
\begin{conjecture}{\em\cite{KKP-H}}
There is no twin-free hypergraph in $ \mathfrak{B}^r$.
\end{conjecture}
It can be seen that all known $r$-uniform hypergraphs in $\mathfrak{B}^r$ (obtained in \cite{BDFL} and this paper), belong to $\mathcal{H}^r(G) $ for some graph $G$ with $ \chi^{e}(G)=3$.
By Theorem \ref{thm:almost}, for an $ r $-uniform random hypergraph $ \mathcal{H}= \mathcal{H}^{r}_{n,p} $, asymptotically almost surely, $ \chi^{e}(\mathcal{H})=1 $, if $ r\geq 4 $ and $ H\in \mathfrak{A}^r $, whenever $ r=3 $. Hence, motivated by Conjecture \ref{conj:3-HYPER} and Theorem \ref{thm:almost}, classifying hypergraphs in $ \mathfrak{B}^r$ is another interesting problem. We conclude this section by proposing the following question.
\begin{problem}
Is there any $ r$-uniform hypergraph $ {\cal H} \in {\mathfrak{B}}^r $, such that $ \mathcal{H}\notin \mathcal{H}^r(G)$ for some graph $G$?
\end{problem}
| {
"timestamp": "2022-05-02T02:10:55",
"yymm": "2204",
"arxiv_id": "2204.13936",
"language": "en",
"url": "https://arxiv.org/abs/2204.13936",
"abstract": "In 2004, Karoński, Łuczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\\rightarrow\\{1,2,3\\} $ such that the induced vertex coloring is proper. \nAfter that, the total versions of this conjecture were suggested in the literature and recently, Kalkowski et al. have generalized this conjecture to hypergraphs. In this paper, some previously known results on the total versions are improved. Moreover, an affirmative answer is given to the conjecture for some well-known families of hypergraphs like complete $n$-partite hypergraphs, paths, cycles, theta hypergraphs and some geometric planes. Also, these hypergraphs are characterized based on the corresponding parameter.",
"subjects": "Combinatorics (math.CO)",
"title": "On the total versions of 1-2-3-conjecture for graphs and hypergraphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846678676151,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019702194741
} |
https://arxiv.org/abs/1502.02611 | Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation | The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem. | \section{Introduction}
\setcounter{equation}{0}
Consider the quasilinear
second order wave equation
\begin{equation}\label{1.1}
u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0\,,
\qquad\qquad t\in [0,T], ~~x\inI\!\!R\,.\end{equation}
On the wave speed $c$ we assume
\begin{itemize}
\item[{\bf (A)}] The map
$c:I\!\!R\mapsto I\!\!R_+$ is
smooth and uniformly positive. The quotient $c'(u)/c(u)$ is uniformly
bounded. Moreover,
the following generic condition is satisfied:
\begin{equation}\label{morse} c'(u)~=~0\qquad\Longrightarrow\qquad c''(u)~\not=~ 0.\end{equation}
\end{itemize}
Notice that, by (\ref{morse}), the derivative $c'(u)$
vanishes only at isolated points.
The analysis in \cite{BZ, BCZ} shows that, for any initial data
\begin{equation}\label{1.2}
u(0,x)~=~u_0(x)\,,\qquad
u_t(0,x)~=~u_1(x)\,,
\end{equation}
with
$u_0\in H^1(I\!\!R)$, $u_1\in{\bf L}^2(I\!\!R)$, the Cauchy problem admits a unique
conservative solution $u=u(t,x)$, H\"older continuous in the $t$-$x$ plane.
We recall that conservative solutions satisfy an additional conservation
law for the energy, so that the total energy
$${\mathcal E}(t)~=~{1\over 2} \int [u_t^2 + c^2(u) u_x^2]\, dx$$
coincides with a constant for a.e.~time $t$.
A detailed construction of a global semigroup of these solutions,
including more singular
initial data, was carried out in \cite{HR}.
In the present paper we study the structure of these solutions.
Roughly speaking, we prove that, for generic smooth
initial data
$(u_0,u_1)$, the solution is piecewise smooth. Its gradient
$u_x$ blows up along finitely many smooth curves in the $t$-$x$ plane.
Our main result is
\vskip 1em
{\bf Theorem 1.}
{\it Let the function $u\mapsto c(u)$ satisfy the assumptions
{\bf (A)} and let $T>0$ be given.
Then there exists an open dense set of initial data
$${\mathcal D}~\subset~ \Big({\mathcal C}^3(I\!\!R)\cap H^1(I\!\!R)\Big) \times\Big({\mathcal C}^2(I\!\!R)\cap{\bf L}^2(I\!\!R)\Big)$$
such that, for $(u_0,u_1)\in{\mathcal D}$, the conservative solution
$u=u(t,x)$ of (\ref{1.1})-(\ref{1.2})
is twice continuously differentiable in the complement of finitely many characteristic curves
$\gamma_i$, within the domain $[0,T]\times I\!\!R$.
}
\vskip 1em
For the scalar conservation law in one space dimension, a well known
result by Schaeffer \cite{S} shows that
generic solutions are piecewise smooth, with finitely many shocks on any bounded domain
in the $t$-$x$ plane.
A similar result was proved by Dafermos and Geng \cite{DG}, for a special
$2\times 2$ Temple class
system of conservation laws. It remains an outstanding open problem
to understand whether generic solutions to
more general $2\times 2$ systems (such as the p-system of isentropic gas dynamics)
remain piecewise smooth, with finitely many shock curves.
The proof in \cite{S} relies on the Hopf-Lax representation formula,
while the proof in \cite{DG}
is based on the analysis of solutions along characteristics.
In the present paper we take a quite different approach, based on the representation
of solutions in terms of a semilinear system introduced in \cite{BZ}.
In essence, the analysis in \cite{BZ} shows that, after a suitable change of variables,
the quantities
$$w~\doteq~2\arctan\bigl(u_t +c(u) u_x\bigr),
\qquad\qquad z ~\doteq~2\arctan\bigl(u_t -c(u) u_x\bigr),$$
satisfy a semilinear system of equations, w.r.t.~new
independent variables $X$, $Y$. See (\ref{2.26})--(\ref{4.2})
in Section~2 for details.
Since this system has smooth coefficients, starting with smooth
initial data
one obtains a globally defined
smooth solution. To recover the singularities of the
solution $u$ of (\ref{1.1})
in the original $t$-$x$ plane, it now suffices to study the level sets
\begin{equation}\label{wzp} \{w(X,Y)=\pi\} \,,\qquad\qquad \{z(X,Y)=\pi\} \,. \end{equation}
Since $w$ and $z$ are smooth, the generic structure of these
level sets
can be analyzed by techniques of singularity theory \cite{Bloom, D, DD, G, T},
relying on Thom's transversality theorem.
One should be aware that, while the map
$(X,Y)\mapsto (t,x,u,w,z)$ is smooth,
the inverse map $(t,x)\mapsto (X,Y)$ can have singularities.
This variable transformation is indeed the source of
singularities in the solution $u=u(t,x)$ of (\ref{1.1}).
The present work was motivated by a research
program aimed at the construction of a distance which
renders Lipschitz continuous the semigroup of conservative solutions
of (\ref{1.1}).
Toward this goal, one needs
a dense set of piecewise smooth paths of solutions,
whose weighted length can be controlled in time.
In the final section of this paper we thus consider
a 1-parameter family of initial data
$\lambda\mapsto (u_0^\lambda, u_1^\lambda)$, with
$\lambda\in [0,1]$.
We show that it can be uniformly approximated by a second path
of initial data
$\lambda\mapsto (\tilde u_0^\lambda, \tilde u_1^\lambda)$,
such that the corresponding solutions $\tilde u^\lambda =\tilde
u^\lambda(t,x)$ of (\ref{1.1})
are piecewise smooth in the domain $[0,T]\timesI\!\!R$,
for all except at most finitely values of $\lambda\in [0,1]$.
An application of this result to
the construction of a Lipschitz metric will appear
in the forthcoming paper~\cite{BC}.
The remainder of the paper is organized as follows.
In Section~2 we review the variable change introduced in \cite{BZ} and
derive the semilinear system used in the construction
of conservative solutions
to (\ref{1.1}). In Section~3 we construct families of smooth solutions
to the semilinear system, depending on parameters.
By a transversality argument, in Section~4 we show that
for almost all of these solutions
the level sets (\ref{wzp}) satisfy a number of generic properties.
After these preliminaries, the proof of Theorem~1
is completed in Section~5.
Finally, in Section~6 we prove a theorem on generic regularity
for 1-parameter family of solutions.
For the nonlinear equation (\ref{1.1}), the formation of
singularities in finite time was first studied in \cite{GHZ}.
Based on the representations \cite{BZ, BH},
a detailed asymptotic description of structurally stable
singularities is given in \cite{BHY}, for
conservative as well as dissipative solutions.
We conjecture that the regularity property stated in Theorem~1
should also hold for generic dissipative solutions of (\ref{1.1}).
However, in the dissipative case
the corresponding semilinear system derived in \cite{BH}
contains discontinuous
terms,
and smooth initial data do not yield globally
smooth solutions. For this reason,
the techniques used in this paper can
no longer be applied. We remark that, at the present time,
the uniqueness and continuous dependence of
dissipative solutions to (\ref{1.1}) has not yet been proved,
for general initial data $(u_0, u_1)\in H^1(I\!\!R)\times {\bf L}^2(I\!\!R)$.
\vskip 1em
\section{Review of the main equations}
\setcounter{equation}{0}
Consider the variables
\begin{equation}
\left\{
\begin{array}{rcl}
R & \doteq &u_t+c(u)u_x\,, \\
S & \doteq &u_t-c(u)u_x\,,
\end{array} \right.\label{2.1}
\end{equation}
so that
\begin{equation}
u_t~=~{R+S\over 2}\,,\qquad\qquad u_x~=~{R-S\over 2c}\,.\label{2.2}
\end{equation}
For a smooth solution of (\ref{1.1}), these variables satisfy
\begin{equation}\label{2.3}
\left\{
\begin{array}{rcl}
R_t-cR_x &= & {c'\over 4c}(R^2-S^2), \\ [3mm]
S_t+cS_x &= & {c'\over 4c}(S^2-R^2).
\end{array} \right.
\end{equation}
In addition, $R^2$ and $S^2$ satisfy the balance laws
\begin{equation}\label{2.4}\left\{
\begin{array}{rcl}
(R^2)_t - (cR^2)_x & = & {c'\over 2c}(R^2S - RS^2)\, , \\ [3mm]
(S^2)_t + (cS^2)_x & = & {c'\over 2c}(S^2R-SR^2 )\,.
\end{array}
\right.
\end{equation}
As a consequence, for smooth solutions the following quantity is conserved:
\begin{equation}
E~\doteq ~{1\over 2}\big(u_t^2+c^2u_x^2\big)~=~{R^2+S^2\over 4}
\,.\label{2.5}
\end{equation}
One can think of $R^2$ and $S^2$ as the energy densities of backward and
forward moving waves, respectively. Notice that these are not separately conserved.
Indeed, by (\ref{2.4}) energy can be exchanged between forward and backward waves.
It is well known that, even for smooth
initial data, the quantities $u_t, u_x$ can blow up in finite time
\cite{GHZ}.
To deal with possibly unbounded values of $R,S$, following \cite{BZ}
it is convenient to introduce a new set of dependent variables:
\begin{equation}\label{wzdef}
w~\doteq ~2\arctan R\,,\qquad\qquad z~\doteq~ 2\arctan S\,.
\end{equation}
Using (\ref{2.3}), we obtain the equations
\begin{equation}
w_t-c\,w_x~=~{2\over 1+R^2}(R_t-c\,R_x)
~=~{c'\over 2c}{R^2-S^2\over 1+R^2}\,, \label{2.10}
\end{equation}
\begin{equation}
z_t+c\,z_x~=~{2\over 1+S^2}(S_t+c\,S_x)
~=~{c'\over 2c}{S^2-R^2\over 1+S^2}\,. \label{2.11}
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics[width=0.55\textwidth]{b77.eps}
\caption{\small Characteristic curves. As new coordinates of the point $(t,x)$
we choose the values $(X,Y) = \bigl(x^-(0, t,x)\,,~-x^+(0,t,x)\bigr)$.}
\label{f:b77}
\end{figure}
We now
perform a further change of independent variables (Fig.~\ref{f:b77}).
Consider
the equations for the backward and forward characteristics:
\begin{equation}
\dot x^-~=~-c(u)\,,\qquad\qquad \dot x^+~=~c(u)\,,\label{2.12}
\end{equation}
where the upper dot denotes a derivative w.r.t.~time.
The characteristics passing through the point $(t,x)$
will be denoted by
$$
s~\mapsto~ x^-(s,t,x)\,,\qquad\qquad s~\mapsto ~x^+(s,t,x)\,,
$$
respectively.
As coordinates $(X,Y)$ of a point $(t,x)$ we shall use the intersections
of these characteristics with the $x$-axis, namely
\begin{equation}
X~\doteq~ x^-(0,t,x)
\,,\qquad\qquad Y~\doteq ~- x^+(0,t,x)\,.\label{XY}
\end{equation}
Of course this implies
\begin{equation}
X_t- c(u)X_x~ =~0\,,\qquad\qquad Y_t+ c(u)Y_x ~=~0\,,\label{2.14}
\end{equation}
\begin{equation}
(X_x)_t- (c\,X_x)_x~=~0\,,\qquad\qquad (Y_x)_t+(c\,Y_x)_x~=~0\,.\label{2.15}
\end{equation}
For any smooth function $f$, using (\ref{2.14}) one finds
\begin{equation}\left\{
\begin{array}{ccccccr}
f_t+cf_x &=& f_XX_t+f_Y Y_t+cf_X X_x+cf_Y Y_x & = & (X_t+cX_x)f_X
& = & 2cX_x f_X\,,\cr\cr
f_t-cf_x &=& f_XX_t+f_Y Y_t-cf_X X_x-cf_Y Y_x & = & (Y_t-cY_x)f_Y
& = & -2cY_x f_Y\,.
\end{array}\right. \label{2.16}
\end{equation}
We now introduce the further variables
\begin{equation}\label{2.17}
p~\doteq~ {1+R^2\over X_x}\,,\qquad\qquad q~\doteq ~{1+S^2\over -Y_x}\,.
\end{equation}
Notice that the above definitions imply
\begin{equation}
{1\over X_x}~=~ {p\over 1+R^2}~=~p\,\cos^2{w\over 2}\,,\qquad\qquad
{-1\over Y_x}~=~{q\over 1+S^2}~=~q\,\cos^2{z\over 2}\,.\label{2.18}
\end{equation}
Starting with the nonlinear equation (\ref{1.1}),
using $X,Y$ as independent variables one obtains a semilinear
hyperbolic system with smooth coefficients for the variables $u,w,z,p,q$, namely
\begin{equation}
\left\{
\begin{array}{ccc}
u_X &= & {\sin w\over 4c} \, p\,,\\[4mm]
u_Y &= & {\sin z\over 4c} \, q\,,
\end{array}\right. \label{2.26}
\end{equation}
\begin{equation}
\left\{
\begin{array}{ccc}
w_Y&=&{c'\over 8c^2}\,( \cos z - \cos w)\,q\,,\\[4mm]
z_X&=&{c'\over 8c^2}\,( \cos w - \cos z)\,p\,,
\end{array}\right. \label{2.24}
\end{equation}
\begin{equation}
\left\{
\begin{array}{ccc}
p_Y &= & {c'\over 8c^2}\, \big(\sin z-\sin w\big)\,pq\,,\\[4mm]
q_X &= & {c'\over 8c^2}\, \big(\sin w-\sin z\big)\,pq\,.
\end{array}\right.\label{2.25}
\end{equation}
The map $(X,Y)\mapsto (t,x)$ can be constructed as follows.
Setting $f=x$, then $f=t$ in the two equations at
(\ref{2.16}), we find
$$
\left\{
\begin{array}{rcr}
c &=& 2cX_x\,x_X\,, \\[3mm]
-c&=& -2cY_x\,x_Y\,,
\end{array}\right.\qquad\qquad
\left\{
\begin{array}{rcr}
1&= &2cX_x\,t_X\,,\\[3mm]
1&=& -2cY_x\,t_Y\,,
\end{array}\right.
$$
respectively.
Therefore, using (\ref{2.18}) we obtain
\begin{equation}
\left\{
\begin{array}{ccccr}
x_X&~=~&{1\over 2X_x}&~=~&{(1+\cos w)\,p\over 4}\,,\\[4mm]
x_Y&~=~&{1\over 2Y_x}&~=~&-{(1+\cos z)\,q\over 4}\,,
\end{array}\right. \label{4.1}
\end{equation}
\begin{equation}
\left\{
\begin{array}{ccccr}
t_X&=&{1\over 2cX_x}&=&{(1+\cos w)\,p\over 4c}\,,\\[4mm]
t_Y&=&{1\over -2cY_x}&=&{(1+\cos z)\,q\over 4c}\,.
\end{array}\right. \label{4.2}\end{equation}
\vskip 1em
Given the initial data (\ref{1.2}), the corresponding boundary
data for (\ref{2.24})-(\ref{4.2}) can be determined as follows.
In the $X$-$Y$ plane, consider the line
$$\gamma_0~=~\{X+Y=0\}~\subset~I\!\!R^2$$
parameterized as
$x~\mapsto ~(\overline X(x), \,\overline Y(x))~\doteq~(x,\, -x)$.
Along $\gamma_0$ we can assign the boundary data
$(\overline u,\overline w, \overline z, \overline p, \overline q)$ by setting
\begin{equation}{\overline u }~ =~u_0(x)\,,\qquad\qquad
\left\{
\begin{array}{rcl}
{\overline w } &= & 2\arctan R(0,x)\,,\\ {\overline z} &= & 2\arctan S(0,x)\,,
\end{array}\right.\qquad \qquad
\left\{
\begin{array}{rcl}
{\overline p} &\equiv & 1+ R^2(0,x)\,,\\
{\overline q} &\equiv & 1+S^2(0,x)\,,
\end{array}\right. \label{2.28}
\end{equation}
at each point $(x,-x)\in\gamma_0$.
We recall that, at time $t=0$, by (\ref{1.2}) one has
$$\begin{array}{lr}R(0,x) &=~(u_t + c(u) u_x)(0,x) ~=~
u_1(x) + c(u_0(x)) u_{0,x}(x),\\[4mm]
S(0,x) &=~(u_t - c(u) u_x)(0,x) ~=~u_1(x) - c(u_0(x)) u_{0,x}(x).\end{array}$$
\vskip 1em
{\bf Remark 1.}
Since the semilinear system (\ref{2.24})--(\ref{4.2}) has smooth coefficients,
for smooth initial data all components of the solution remain smooth on the entire $X$-$Y$
plane. As proved in \cite{BZ},
the quadratic terms in (\ref{2.25}) (containing the product
$pq$) account
for transversal wave interactions and
do not produce finite time blow up of the
variables $p,q$. Moreover, if the values of $p,q$ are uniformly positive along a line $\{X+Y=\kappa\}$, then
they remain uniformly positive
on compact sets of the $X$-$Y$ plane. Throughout this paper, we always
consider solutions of (\ref{2.24})--(\ref{4.2}) where $p,q>0$.
\vskip 1em
By
expressing the solution $u(X,Y)$ in terms of the original variables
$(t,x)$, one obtains a solution of the Cauchy problem
(\ref{1.1})-(\ref{1.2}). Indeed, the following was proved in \cite{BZ}.
\vskip 1em
{\bf Lemma 1.} {\it Let $(u,w,z,p,q,x,t)$ be a smooth solution
to the system (\ref{2.26})--(\ref{4.2}), with $p,q>0$.
Then the set of points
\begin{equation}\label{graph}\Big\{ \bigl(t(X,Y), \, x(X,Y),
\, u(X,Y)\bigr)\,;~~(X,Y)\inI\!\!R^2\Big\}\end{equation}
is the graph of a conservative solution to
the variational wave equation (\ref{1.1}).}
\vskip 1em
We observe that, while the functions
\begin{equation}\label{Ldef}
(X,Y)~\mapsto~ u(X,Y),\qquad\qquad (X,Y)~\mapsto ~\Lambda(X,Y) ~\doteq
~\bigl(t(X,Y), \, x(X,Y)
\bigr)\end{equation}
are globally smooth,
the map $\Lambda:I\!\!R^2\mapstoI\!\!R^2$ may not have a smooth inverse.
Indeed, $\Lambda$ may not even
be one-to-one. Therefore, the solution $u(t,x) = u(\Lambda^{-1}(t,x))$
can fail to be smooth. This happens precisely at points where
the Jacobian matrix $D\Lambda$ is not invertible.
By (\ref{4.1})-(\ref{4.2}), singularities occur when
$\cos w= -1$ or $\cos z= -1$.
\vskip 1em
{\bf Remark 2.} The system (\ref{2.24})--(\ref{4.2}) is
invariant under translation by $2\pi$
in $w$ and $z$. We can thus think of
$w,z$ as points in the quotient manifold
${\mathbb T}=I\!\!R/2\pi {\mathbb Z}$.
Throughout the following we take advantage of this fact and
regard a solution of (\ref{2.24})--(\ref{4.2}) as a
map $(X,Y)\mapsto(u,w,z,p,q,x,t)$ from
$I\!\!R^2$ into $I\!\!R\times{\mathbb T}\times{\mathbb T} \timesI\!\!R\timesI\!\!R\timesI\!\!R$.
Observe that we have the implications
\begin{equation}\label{coswz}\begin{array}{l}
w~\not=~\pi\qquad\Longrightarrow\qquad \cos w ~>~ -1\,,\\[4mm]
z~\not=~\pi\qquad\Longrightarrow\qquad \cos z ~>~ -1\,.\end{array}\end{equation}
\vskip 1em
{\bf Remark 3.} In general, many distinct solutions to the system
(\ref{2.26})--(\ref{4.2}) can yield the same solution $u=u(t,x)$ of (\ref{1.1}).
Indeed, let $(u,w,z,p,q,x,t)(X,Y)$ be one particular solution.
Let $\phi,\psi:I\!\!R\mapstoI\!\!R$ be two ${\mathcal C}^2$ bijections,
with $\phi'>0$ and $\psi'>0$. Introduce the new independent and dependent variables
$(\widetilde X,\widetilde Y)$ and $(\tilde u,\tilde w,\tilde z,\tilde p,\tilde q,\tilde x,\tilde t)$
by setting
\begin{equation}\label{TXY}
X~=~\phi(\widetilde X)\,,\qquad\qquad Y~=~\psi(\widetilde Y),\end{equation}
\begin{equation}\label{TUWZ}
(\tilde u,\tilde w,\tilde z,\tilde x, \tilde t)(\widetilde X,\widetilde Y)~=~(u,w,z,p,q,x,t)(X,Y),\end{equation}
\begin{equation}\label{TPQ}\left\{\begin{array}{rl}
\tilde p(\widetilde X,\widetilde Y)&=~p(X,Y)\cdot \phi'(\widetilde X),\\[4mm]
\tilde q(\widetilde X,\widetilde Y)&=~q(X,Y)\cdot \psi'(\widetilde Y).\end{array}\right.\end{equation}
Then, as functions of $(\widetilde X, \widetilde Y)$, the
variables $(\tilde u,\tilde w,\tilde z,\tilde p,\tilde q,\tilde x,\tilde t)$
provide another solution of the same system (\ref{2.26})--(\ref{4.2}).
Moreover, by (\ref{TUWZ}) the set
\begin{equation}\label{graph2}\Big\{ \Bigl(\tilde t(\widetilde X,\widetilde Y), \, \tilde x(\widetilde X,\widetilde Y),
\,\tilde u(\widetilde X,\widetilde Y)\Bigr)\,;~~
(\widetilde X,\widetilde Y)\inI\!\!R^2\Big\}\end{equation}
coincides with the set (\ref{graph}). Hence it is the graph of the same solution
$u$ of (\ref{1.1}).
One can regard the variable transformation (\ref{TXY})
simply as a relabeling of forward and backward characteristics, in the solution $u$. A detailed analysis of relabeling symmetries, in
connection with the Camassa-Hom equation, can be found in \cite{GHR}.
For future reference we observe that
$$\tilde w_{\widetilde X}(\widetilde X,\widetilde Y)~=~w_X(X,Y)\cdot \phi'(\widetilde X)\,,$$
$$\tilde w_{\widetilde X\widetilde X}(\widetilde X,\widetilde Y)~=~w_{XX}(X,Y)\cdot [\phi'(\widetilde X)]^2
+w_X(X,Y) \cdot \phi''(\widetilde X)\,.$$
In particular, one has the equivalences
\begin{equation}\label{degen}
\begin{array}{rl}
\tilde w_{\widetilde X}(\widetilde X,\widetilde Y)~=~0\qquad &\Longleftrightarrow \qquad
w_X(X,Y)~=~0,\\[4mm]
\tilde z_{\widetilde Y}(\widetilde X,\widetilde Y)~=~0\qquad &\Longleftrightarrow \qquad
z_Y(X,Y)~=~0,\\[4mm]
(\tilde w_{\widetilde X},\,\tilde w_{\widetilde X\widetilde X})(\widetilde X,\widetilde Y)~=~(0,0)\qquad
&\Longleftrightarrow \qquad
(w_X, w_{XX})(X,Y)~=~(0,0),\\[4mm]
(\tilde z_{\widetilde Y},\,\tilde z_{\widetilde Y\widetilde Y})(\widetilde X,\widetilde Y)~=~(0,0)\qquad
&\Longleftrightarrow \qquad
(z_Y, z_{YY})(X,Y)~=~(0,0)\,.\end{array}\end{equation}
\vskip 1em
\subsection{Compatible boundary data}
More generally,
instead of (\ref{2.28}) we can assign boundary data
for the system (\ref{2.16})--(\ref{2.25}) on a line $\gamma=\{ X+Y=\kappa\}$.
Namely:
\begin{equation}\label{bdata} u(s,\kappa-s)~ =~\overline u(s)\,,\qquad\qquad
\left\{
\begin{array}{rl}
w(s,\kappa-s)&= ~\overline w(s)\,,\\ z(s,\kappa-s)&= ~ \overline z(s)\,,
\end{array}\right.\qquad \qquad
\left\{
\begin{array}{rcl}
p(s,\kappa-s)&=~ \overline p(s)\,,\\
q(s,\kappa-s)&=~\overline q(s)\,,
\end{array}\right.
\end{equation}
for suitable smooth functions $\overline u, \overline w,\overline z,\overline p,\overline q$.
If both identities in (\ref{2.26}) hold,
then
\begin{equation}\label{cc}
{d\over ds} \overline u(s) ~=~ {d\over ds } u(s,\kappa-s)~=~u_X - u_Y~=~
{\sin w\over 4c} \,p- {\sin z\over 4c}\,q\,.\end{equation}
The boundary data should thus satisfy
the compatibility condition
\begin{equation}\label{cc1} {d\over ds} \overline u(s)~=~{\sin \overline w(s)\over 4c(\overline u(s))} \,
\overline p(s)- {\sin \overline z(s)\over 4c(\overline u(s))}\,\overline q(s)\end{equation}
As remarked earlier, the system (\ref{2.26})--(\ref{2.25}) is overdetermined.
Indeed, the function $u=u(X,Y)$ could be recovered by either one of the
identities in (\ref{2.26}). We now prove that, if the compatibility condition
(\ref{cc1}) holds, then any smooth solution
satisfying one of the identities in (\ref{2.26}) satisfies the other as well.
\vskip 1em
{\bf Lemma 2.} {\it
Let $u,w,z,p,q$ be smooth functions on $I\!\!R^2$ which satisfy
(\ref{2.24})-(\ref{2.25})
together with the boundary conditions (\ref{bdata})
along the line $\gamma=\{ X+Y=\kappa\}$. Assume that the compatibility
condition (\ref{cc1}) holds.
Then one has
\begin{equation}\label{uY}u_Y~=~{\sin z\over 4c(u)}\,q \qquad\qquad
\hbox{for all }~ (X,Y)\in I\!\!R^2\end{equation}
if and only if
\begin{equation}\label{uX}u_X~=~{\sin w\over 4c(u)}\,p \qquad\qquad
\hbox{for all }~ (X,Y)\in I\!\!R^2.\end{equation}
}
\vskip 1em
{\bf Proof.} Consider the smooth, strictly increasing function
$$\Phi(u)~=~\int_0^u 4c(s)\, ds\,.$$
Observe that the identities (\ref{uY}), (\ref{uX}) are equivalent
respectively to
\begin{equation}\label{PXY}\Phi(u)_Y~=~\sin z\cdot q\,,\qquad\qquad
\Phi(u)_X~=~\sin w\cdot p\,. \end{equation}
Assume that (\ref{uY}) holds. Then
\begin{equation}\label{P5}\Phi(u( X, Y))~=~\Phi(u(X, \,\kappa- X))+
\int_{\kappa- X}^{ Y}
[\sin z\cdot q](X, s)\, ds\,.\end{equation}
Differentiating w.r.t.~$X$, and using the first equations
in (\ref{2.24})-(\ref{2.25}) together with the
compatibility condition (\ref{cc1}) we obtain
\begin{equation}\label{PX}
\begin{array}{rl}\Phi(u)_X(X,Y)
&\displaystyle=~\Phi'(u)\cdot[u_X-u_Y](X, \kappa-X)+
[\sin z\cdot q](X, \kappa-X)
\cr\cr
&\displaystyle\qquad\qquad+ \int_{\kappa-Y}^Y [ \cos z\cdot z_X q +
\sin z\cdot q_X](X,s)\, ds
\cr\cr
&\displaystyle=~[\sin w\cdot p](
X, \kappa-X) +\int_{\kappa-Y}^Y
\Big[{c'(u)\over 8 c^2(u)}\bigl(1+\sin(z+w)\bigr)pq\Big]
(X,s)\, ds\cr\cr
&\displaystyle=~[\sin w\cdot p](
X, \kappa-X) +\int_{\kappa-Y}^Y {\partial\over\partial Y}
[\sin w\cdot p]
(X,s)\, ds~=~[\sin w\cdot p](X,Y)\,.
\end{array}
\end{equation}
We have thus proved that the second identity in (\ref{PXY}) holds.
This is equivalent to (\ref{uX}).
The converse implication is proved in the same way.
\hphantom{MM}\hfill\llap{$\square$}\goodbreak
\vskip 1em
Next, consider initial data for $t,x$,
on the curve $\gamma=\{X+Y=\kappa\}$, say
\begin{equation}\label{bd2} x(s,\kappa-s)~ =~\overline x(s)\,,\qquad\qquad
t(s,\kappa-s)~=~\overline t(s)\,.
\end{equation}
Using (\ref{4.1})-(\ref{4.2}) we derive
the compatibility conditions
\begin{equation}\label{cc5}{d\over ds} \overline x(s) ~=~{(1+\cos \overline q(s)) \overline p(s)
+ (1+\cos \overline z(s) )\overline q(s)\over 4}\,,\end{equation}
\begin{equation}\label{cc6}
{d\over ds} \overline t(s) ~=~{(1+\cos \overline w(s)) \overline p(s)
- (1+\cos \overline z(s) )\overline q(s)
\over 4 c(\overline u(s))}\,.\end{equation}
\vskip 1em
{\bf Lemma 3.} {\it
Let $(u,w,z,p,q)(X,Y)$ be a solution of the system
(\ref{2.26})--(\ref{2.25}). Then there exists a solution
$(t,x)(X,Y)$ of (\ref{4.1})-(\ref{4.2}) with boundary
data (\ref{bd2}) if and only if the compatibility
conditions
(\ref{cc5})-(\ref{cc6}) are satisfied.}
\vskip 1em
{\bf Proof.} {\bf 1.} Assume that
the equations (\ref{4.1})-(\ref{4.2}) are satisfied
for all $(X,Y)\inI\!\!R^2$. In particular, they are satisfied
along the curve $\gamma=\{X+Y=\kappa\}$.
This implies
$${d\over ds}\, \overline x(s)~=~
{d\over ds}\, \overline x(s, \kappa-s)~=~[x_X-x_Y](s,\kappa-s)~
=~{(1+\cos w) p
+ (1+\cos z ) q\over 4}\,,$$
where the right hand side is evaluated at $(X,Y) = (x, \kappa-s)$.
Hence (\ref{cc5}) holds. The identity (\ref{cc6})
is derived in the same way.
\vskip 1em
{\bf 2.} Next, assume that the compatibility conditions
(\ref{cc5})-(\ref{cc6}) are satisfied. To prove that
(\ref{4.1}) admits a solution,
it then suffices to check
that the differential form
$${(1+\cos w)p\over 4}\, dX -{(1+\cos z)q\over 4}\, dY$$
is closed.
This is true because
\begin{equation}\label{xXY}
\begin{array}{rl}\displaystyle
\left[ {(1+\cos w)p\over 4}\right]_Y
&= ~\displaystyle-{\sin w\over 4} w_Y\,p+{1+\cos w\over 4}\,p_Y\\[4mm]
&\displaystyle=~ -{\sin w\over 4} \cdot {c'\over 8 c^2}(\cos z - \cos w)\, pq + {1+\cos w
\over 4} \cdot {c'\over 8c^2}(\sin z-\sin w) \, pq\\[4mm]
&\displaystyle=~ {c'\over 32 c^2}\Big[
(1+\cos w)\sin z-(1+\cos z)\sin w\Big]\, pq~=~
\left[ -{(1+\cos z)q\over 4}
\right]_X\,.
\end{array}
\end{equation}
Similarly, to prove that (\ref{4.2}) admits a solution,
it suffices to check that the
differential form
$${(1+\cos w)p\over 4c}\, dX +{(1+\cos z)q\over 4c}\, dY$$
is closed.
This is true because
\begin{equation}\label{tXY}
\begin{array}{l}\displaystyle
\left[ {(1+\cos w)p\over 4c}\right]_Y
\displaystyle= ~-{\sin w\over 4c} w_Y\,p - {1+\cos w\over 4c^2} c' u_Y\,p
+{1+\cos w\over 4c}\,p_Y\\[4mm]
=~\displaystyle -{\sin w\over 4c} \cdot {c'\over 8 c^2}(\cos z - \cos w)\, pq
- {1+\cos w\over 4c^2} c' {\sin z\over 4c}\,pq
+ {1+\cos w
\over 4c} \cdot {c'\over 8c^2}(\sin z-\sin w) \, pq\\[4mm]
=~\displaystyle - {c'\over 32 c^3}\Big[ (1+\cos w)\sin z+(1+\cos z)\sin w\Big]\, pq~=~
\left[ {(1+\cos z)q\over 4c}\right]_X \,.
\end{array}
\end{equation}
\hphantom{MM}\hfill\llap{$\square$}\goodbreak
\vskip 1em
\vskip 1em
{\bf Remark 4.} Let a solution $(u,w,z,p,q)$ of (\ref{2.26})--(\ref{2.25})
be given.
If we assign the values of $t,x$
at a single point $(X_0, Y_0)$, then by the compatibility conditions (\ref{cc5})-(\ref{cc6})
and the equations (\ref{4.1})-(\ref{4.2}) the functions $t(X,Y)$, $x(X,Y)$ are uniquely determined for all $(X,Y)\inI\!\!R^2$. Choosing different
values of $t,x$ at the point $(X_0, Y_0)$ we obtain the same solution
$u=u(t,x)$ of (\ref{1.1}), up to a shift of coordinates in the $t,x$ plane.
\vskip 1em
\section{Families of perturbed solutions}
\setcounter{equation}{0}
Let a point $(X_0, Y_0)$ be given and consider the
line
\begin{equation}\label{gd}\gamma~\doteq~\Big\{ (X,Y)\,;~~~X+Y=\kappa\Big\},
\qquad\qquad \kappa \doteq X_0+Y_0\,.\end{equation}
We can then arbitrarily assign the values of $w,z,p,q$
at every point $(X,Y)\in \gamma$. Moreover, we can arbitrarily
choose the values of $u,x,t$ at the single point $(X_0, Y_0)$.
In turn, these choices uniquely determine functions
$u,x,t$ on $\gamma$ which satisfy the compatibility
conditions (\ref{cc1}) and (\ref{cc5})-(\ref{cc6}).
Based on this observation, we can construct several families
of perturbations
of a given solution of (\ref{2.26})--(\ref{2.25}).
The main goal of this section is to prove
\vskip 1em
{\bf Lemma 4.} {\it Let the assumption {\bf (A)} hold.
Let $(u, w,z, p, q)$ be a smooth solution of the system
(\ref{2.26})-(\ref{2.25}) and
let a point $(X_0, Y_0)\inI\!\!R^2$ be given.
\begin{itemize}
\item[{\bf (1)}] If $(w,w_X,w_{XX})(X_0,Y_0)=(\pi,0,0)$,
then there exists a 3-parameter family of smooth
solutions $(u^\theta, w^\theta,z^\theta, p^\theta, q^\theta)$
of (\ref{2.26})-(\ref{2.25}),
depending smoothly on $\theta\inI\!\!R^3$,
such that the following holds.
\begin{itemize}
\item[(i)] When $\theta=0\inI\!\!R^3$ one recovers the original solution,
namely $(u^0, w^0,z^0, p^0, q^0)=(u, w,z, p, q)$.
\item[(ii)] At the point
$(X_0,Y_0)$, when $\theta=0$ one has
\begin{equation}\label{pert1}\hbox{\rm rank }D_\theta (w^\theta\,,~
w^\theta_X\,,~w^\theta_{XX})~=~3\,.\end{equation}
\end{itemize}
\item[{\bf (2)}] If $(w,z,w_X)(X_0,Y_0)=(\pi,\pi,0)$, then
there exists a 3-parameter family of smooth
solutions $(u^\theta, w^\theta,z^\theta, p^\theta, q^\theta)$
satisfying (i)-(ii) as above, with (\ref{pert1})
replaced by
\begin{equation}\label{pert2}\hbox{\rm rank }D_\theta (w^\theta\,,~
z^\theta\,,~w^\theta_X)~=~3\,.\end{equation}
\item[{\bf (3)}] If $(w,w_X, c'(u))(X_0,Y_0)=(\pi,0,0)$,
then there exists a 3-parameter family of smooth
solutions $(u^\theta, w^\theta,z^\theta, p^\theta, q^\theta)$ satisfying (i)-(ii) as above, with (\ref{pert1})
replaced by
\begin{equation}\label{pert3}\hbox{\rm rank }D_\theta \bigl(w^\theta\,,~
w^\theta_X\,,~c'(u^\theta)\bigr)~=~3\,.\end{equation}
\end{itemize}
}
\vskip 1em
For example (\ref{pert1}) means that we can construct perturbed solutions, depending on
parameters $\theta_1,\theta_2,\theta_3$, such that the Jacobian matrix
\begin{equation}\label{mat0}D_\theta (w^\theta\,,~
w^\theta_X\,,~w^\theta_{XX})~=~\left(\begin{array}{ccc} {\partial\over\partial\theta_1} w
& {\partial\over\partial\theta_2} w
& {\partial\over\partial\theta_3} w\cr\cr {\partial\over\partial\theta_1} w_X& {\partial\over\partial\theta_2} w_X
& {\partial\over\partial\theta_3} w_X\cr\cr
{\partial\over\partial\theta_1} w_{XX}& {\partial\over\partial\theta_2} w_{XX}
& {\partial\over\partial\theta_3} w_{XX}\end{array}\right),\end{equation}
computed at $\theta=0$, has full rank at the point $(X_0, Y_0)$.
\vskip 1em
\subsection{ Proof of Lemma 4}
Let $(u,w,z,p,q)$ be a ${\mathcal C}^\infty$ solution of the system
(\ref{2.26})-(\ref{2.25}).
Given the point $(X_0, Y_0)$,
consider the line $\gamma$ in (\ref{gd}) and
let $(\overline u,\overline w,\overline z,\overline p,\overline q)$
be the values of the solution along $\gamma$, as in
(\ref{bdata}).
For future use, we compute the values of $w_X, w_{XX}$
at the point $(X_0, Y_0)$. At any point $(s, \kappa-s)\in
\gamma$ we have
$$w_X- w_Y~=~\overline w'(s)\,,
\qquad\qquad z_X- z_Y~=~\overline z'(s)\,,\qquad\qquad
q_X- q_Y~=~\overline q'(s)\,.$$
Here and in the sequel, a prime denotes derivative w.r.t.~the
parameter $s$ along the curve $\gamma$.
Using (\ref{2.24})-(\ref{2.25})
we obtain
\begin{equation}\label{wX}
w_X(X_0, Y_0)~ =~\overline w' + {c'(\overline u)\over 8c^2(\overline u)}
(\cos \overline z-\cos \overline w)\overline q\,,\end{equation}
\begin{equation}\label{zY}z_Y(X_0, Y_0)~=~-\overline z'+ {c'(\overline u)\over 8c^2(\overline u)}
(\cos \overline w - \cos \overline z) \overline p\,,\end{equation}
\begin{equation}\label{qY}q_Y(X_0, Y_0)~=~-\overline q'+ {c'(\overline u)\over 8c^2(\overline u)}
(\sin \overline w - \sin \overline z) \overline p\overline q\,,\end{equation}
where all terms on the right hand sides are evaluated at $s=X_0$.
A further differentiation yields
$${d^2\over ds^2} \overline w(s)~=~{d\over ds}
\bigl[w_X(s, \kappa-s)-w_Y(s,\kappa-s)\bigr]~=~[w_{XX}+w_{YY}-2w_{XY}]
(s,\kappa-s).$$
Using (\ref{2.26})--(\ref{2.25}) together with (\ref{wX})--(\ref{qY})
we obtain
\begin{equation}\label{wXY}
\begin{array}{l}
w_{YX}(X_0, Y_0)\cr\cr
\quad =~
\left({c'(\overline u)\over 8c^2(\overline u)}\right)' {\overline u}_X\,( \cos\overline z - \cos\overline w)\,\overline q
+
{c'\over 8c^2}\,({\overline w}_X \sin\overline w- {\overline z}_X \sin \overline z)\,\overline q
+
{c'\over 8c^2}\,( \cos \overline z - \cos \overline w)\,{\overline q}_X\cr\cr
\quad =~\left({c'\over 8c^2}\right)' \frac{\sin \overline w}{4c}\,( \cos\overline z - \cos\overline w)\,\overline p \overline q
\cr\cr\qquad\qquad
+{c'\over 8c^2}\,\left\{\big(\overline w' + {c'\over 8c^2} (\cos \overline z-\cos \overline w)\overline q\big) \sin\overline w- {c'\over 8c^2}
(\cos \overline w - \cos \overline z) \overline p\sin \overline z\right\}\,\overline q
\cr\cr\qquad\qquad
+({c'\over 8c^2})^2\,( \cos \overline z - \cos \overline w)\,
(\sin \overline w - \sin \overline z) \overline p\overline q
\cr\cr \quad \doteq ~f_1
\end{array}
\end{equation}
and
\begin{equation}\label{wYY}
\begin{array}{l}
w_{YY}(X_0, Y_0)\cr\cr
\quad =~
\left({c'(\overline u)\over 8c^2(\overline u)}\right)'\overline u_Y\,( \cos\overline z - \cos\overline w)\,\overline q
+
{c'\over 8c^2}\,(\overline w_Y \sin \overline w- \overline z_Y \sin\overline z)\,\overline q
+
{c'\over 8c^2}\,( \cos\overline z - \cos\overline w)\,\overline q_Y\cr\cr
\quad =~\left({c'\over 8c^2}\right)' \frac{\sin \overline z}{4c}\,( \cos\overline z - \cos\overline w)\,\overline q^2
\cr\cr
\qquad\qquad
+{c'\over 8c^2}\,\left\{ {c'\over 8c^2} (\cos \overline z-\cos \overline w)\overline q\sin\overline w- \big(-\overline z'+{c'\over 8c^2}
(\cos \overline w - \cos \overline z) \overline p\big)\sin \overline z\right\}\,\overline q
\cr\cr
\qquad\qquad +
{c'\over 8c^2}\,( \cos \overline z - \cos \overline w)\,
\big(-\overline q'+ {c'\over 8c^2}
(\sin \overline w - \sin \overline z) \overline p\overline q\big)\cr\cr
\quad \doteq~ f_2\,.
\end{array}
\end{equation}
Hence
\begin{equation}\label{wXX}
w_{XX}(X_0, Y_0)~=~\overline w'' + 2f_1 - f_2\,.\end{equation}
\vskip 1em
We now construct families $(\overline u^\theta,
\overline q^\theta,\overline z^\theta, \overline p^\theta, \overline q^\theta)$
of perturbations of the data
(\ref{bdata}) along the curve $\gamma$, so that
at the point $(X_0, Y_0)$ the matrices in (\ref{pert1})--(\ref{pert3})
have full rank.
These perturbations will have the form
\begin{equation}\label{p00}\left\{\begin{array}{rl}
\overline w^\theta(s) &=~\overline w(s) + \sum_{i=1}^3\theta_i W_i(s)\,,\cr\cr
\overline z^\theta(s) &=~\overline z(s) + \sum_{i=1}^3\theta_i Z_i(s)\,,\end{array}
\right.
\qquad\qquad
\left\{\begin{array}{rl}
\overline p^\theta(s) &=~\overline p(s) + \sum_{i=1}^3\theta_i P_i(s)\,,\cr\cr
\overline q^\theta(s) &=~\overline q(s) + \sum_{i=1}^3\theta_i Q_i(s)\,,\end{array}
\right.\end{equation}
for suitable functions $W_i, Z_i, P_i, Q_i\in{\mathcal C}^\infty_c$.
Moreover, at the point $s=X_0$ we set
\begin{equation}\label{u00}
\overline u^\theta(X_0) ~=~\overline u(X_0) + \sum_{i=1,2,3}\theta_i U_i\,.\end{equation}
In turn, the above definitions together
with the compatibility conditions (\ref{cc1}) determine
the values of $\overline u^\theta(s)$ for all $s\inI\!\!R$.
In particular, for each $\theta\inI\!\!R^3$ we obtain
a unique solution of
the semilinear system (\ref{2.26})--(\ref{2.25}).
We observe that the functions $W_i, Z_i, P_i,Q_i$ can be chosen arbitrarily. Hence
at the point $s=X_0$, $\theta=0$,
we can arbitrarily assign all derivatives
$${d\over d\theta} {d^k\over ds^k}\overline w^\theta,
\qquad {d\over d\theta}{d^k\over ds^k} \overline z^\theta,
\qquad
{d\over d\theta}{d^k\over ds^k} \overline p^\theta\,,
\qquad{d\over d\theta} {d^k\over ds^k} \overline q^\theta\,,$$
with $k=0,1,2,\ldots$~ Moreover, we can
arbitrarily choose the quantity
${d\over d\theta} \overline u^\theta(X_0)$,
while all higher order derivatives ${d\over d\theta}{d^k\over ds^k}
\overline u^\theta$,
with $k\geq 1$, are then determined by the
compatibility condition (\ref{cc1}).
\vskip 1em
{\bf 1.}
To achieve (\ref{pert1}), we choose
perturbations $(\overline u^{\theta_i},
\overline w^{\theta_i},\overline z^{\theta_i}, \overline p^{\theta_i},
\overline q^{\theta_i})$, $i=1,2,3$, so that
the Jacobian matrix of first order derivatives
w.r.t.~$\theta_1,\theta_2,\theta_3$, computed
at $s= X_0$ and $\theta=0$, is given by
\[D_\theta \left(\begin{array}{c}
\overline u\cr
\overline w\cr
\overline z\cr
\overline z'\cr
\overline w'\cr
\overline w''\cr
\overline p\cr
\overline q\cr
\overline q'\end{array}\right)~=~
\left(\begin{array}{ccc}0&0&0\cr
1&0&0\cr
0&0&0\cr
0&0&0\cr
0&1&0\cr
0&0&1\cr
0&0&0\cr
0&0&0\cr
0&0&0
\end{array}\right)\,.\]
At the point $(X_0, Y_0)$, by (\ref{wX}) and (\ref{wXX}) this yields
\[D_\theta \left(\begin{array}{c}
w\cr w_X\cr w_{XX}\end{array}\right)~=~
\left(\begin{array}{ccc}
1&0&0\cr
*&1&0\cr
*&*&1\cr
\end{array}\right)\,.\]
Notice that, for the third family of perturbations (corresponding to the third column), the first order variations of
$f_1$ and $f_2$ in (\ref{wXY})-(\ref{wYY})
both vanish at $(X_0, Y_0)$.
This achieves (\ref{pert1}).
\vskip 1em
{\bf 2.} To achieve (\ref{pert2}), we choose
perturbations $(\overline u^{\theta_i},
\overline w^{\theta_i},\overline z^{\theta_i}, \overline p^{\theta_i},
\overline q^{\theta_i})$, $i=1,2,3$, so that at $s= X_0$ and $\theta=0$
one has
\begin{equation}\label{DT1}D_\theta \left(\begin{array}{c}
\overline u\cr
\overline w\cr
\overline z\cr\overline w'\cr
\overline p\cr\overline q\end{array}\right)~=~
\left(\begin{array}{ccc}0&0&0\cr
1&0&0\cr
0&1&0\cr
0&0&1\cr
0&0&0\cr
0&0&0
\end{array}\right)\,.\end{equation}
At the point $(X_0, Y_0)$, by (\ref{wX}) this yields
\begin{equation}\label{DT2}D_\theta \left(\begin{array}{c}
w\cr z\cr w_X\end{array}\right)~=~
\left(\begin{array}{ccc}
1&0&0\cr
0&1&0\cr
*&*&1\cr
\end{array}\right)\,.\end{equation}
Hence (\ref{pert2}) holds.
\vskip 1em
{\bf 3.} Finally, we construct three families of
perturbations satisfying (\ref{pert3}). If at $(X_0, Y_0)$
we have $c'(u(X_0,Y_0)) = 0 $, then the assumption
{\bf (A)} implies
\begin{equation}\label{c''}
c''\bigl(u(X_0, Y_0)\bigr)~\not= ~0\,.
\end{equation}
To achieve (\ref{pert3}), we choose three families of
perturbations such that, at $s= X_0$ and $\theta=0$,
\begin{equation}\label{DT5}D_\theta \left(\begin{array}{c}
\overline u\cr
\overline w\cr
\overline z\cr\overline w'\cr
\overline p\cr\overline q\end{array}\right)~=~
\left(\begin{array}{ccc}0&0&1\cr
1&0&0\cr
0&0&0\cr
0&1&0\cr
0&0&0\cr
0&0&0
\end{array}\right)\,.\end{equation}
At the point $(X_0, Y_0)$, by (\ref{wX}) and the first equation in (\ref{2.24}), this yields
\begin{equation}\label{DT6}D_\theta \left(\begin{array}{c}
w\cr w_X\cr c'(u)\end{array}\right)~=~
\left(\begin{array}{ccc}
1&0&0\cr
*&1&*\cr
*&0&c''(u)\cr
\end{array}\right)\,\end{equation}
This achieves (\ref{pert3}).
\hphantom{MM}\hfill\llap{$\square$}\goodbreak
\vskip 1em
\section{Generic solutions of the semilinear system}
\setcounter{equation}{0}
In this section we study smooth solutions to the semilinear
system (\ref{2.24})--(\ref{4.2}), determining the generic structure
of the level sets $\{(X,Y)\,;~~w(X,Y) =\pi\}$ and $\{(X,Y)\,;~~z(X,Y)=\pi\}$.
\vskip 1em
{\bf Lemma 5.} {\it Let the function $u\mapsto c(u)$ satisfy the assumptions
{\bf (A)} and consider a compact domain of the form
\begin{equation}\label{GTdef}\Gamma~\doteq~\Big\{(X,Y)\,;~~|X|+|Y|~\leq~M\Big\}.\end{equation}
Call ${\mathcal S}$ the family of all
${\mathcal C}^2$ solutions to
the system (\ref{2.26})--(\ref{2.25}), with $p,q>0$ for all
$(X,Y)\inI\!\!R^2$. Moreover, call ${\mathcal S}'\subset{\mathcal S}$ the subfamily
of all solutions $(u,w,z,p,q)$
such that, for $(X,Y)\in \Gamma$, none of the following
values is attained:
\begin{equation}\label{never1}\left\{\begin{array}{rl} (w,w_X,w_{XX}) &=~(\pi,0,0),\\[3mm]
(z,z_Y,z_{YY}) &=~(\pi,0,0),\end{array}\right.\end{equation}
\begin{equation}\label{never2}\left\{\begin{array}{rl} (w,z,w_X) &=~(\pi,\pi,0),\\[3mm]
(w,z,z_Y) &=~(\pi,\pi,0),\end{array}\right. \end{equation}
\begin{equation}\label{never3}
\left\{\begin{array}{rl} (w,w_X,c'(u)) &=~(\pi,0,0),\\[3mm]
(z,z_Y, c'(u)) &=~(\pi,0,0).\end{array}\right.\end{equation}
Then ${\mathcal S}'$ is a relatively open and dense subset of ${\mathcal S}$, in the topology induced by ${\mathcal C}^2(\Gamma)$.
}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{wa67.eps}
\caption{ \small Two level sets $\{w=\pi\}$ and $\{z=\pi\}$,
in a generic solution of (\ref{2.26})--(\ref{2.25}).
At $P_1$, $P_2$ one has $w=\pi$, $w_X=0$
while the generic conditions imply $w_Y\not=0$,
$w_{XX}\not= 0$. At the points
$Q_1$, $Q_2$ where the two singular curves cross, by (\ref{2.24}) one has $w_Y=z_X=0$, while the generic conditions imply
$w_X\not= 0$, $z_Y\not= 0$. Hence the two curves have a perpendicular intersection. }
\label{f:wa67}
\end{figure}
Some words of explanation are in order (Fig.~\ref{f:wa67}).
Asking that the values in (\ref{never1}) are never attained
is equivalent to the implications
\begin{equation}\label{impl1}\left\{\begin{array}{cl}
w~=~\pi\quad\hbox{and}\quad w_X~=~0\qquad &\Longrightarrow\qquad w_{XX}~\not= 0,\\[3mm]
z~=~\pi\quad\hbox{and}\quad z_Y~=~0\qquad &\Longrightarrow\qquad z_{YY}~\not= 0.\end{array}
\right.
\end{equation}
Writing the level curves in the form $\{w(X,Y)=\pi\}= \{ Y=\varphi(X)\}$ and
$\{z(X,Y)=\pi\}= \{ X=\psi(Y)\}$, this imposes some restrictions at the points
where $\varphi'=0$ or $\psi'=0$.
Asking that the values in (\ref{never2}) are never attained
is equivalent to the implication
\begin{equation}\label{impl2}
[w~=~\pi\quad\hbox{and}\quad z~=~\pi]\qquad \Longrightarrow\qquad
[w_X~\not= ~0\quad\hbox{and}\quad z_Y~\not=~0].
\end{equation}
This imposes restrictions at points where two level curves
$\{w=\pi\}$ and $\{z=\pi\}$ cross each other.
Finally, the lemma states the existence of a perturbed solution
such that values (\ref{never3})
are never attained. To understand the meaning of this condition,
consider a solution which never attains any of the values
in (\ref{never2})-(\ref{never3}). In this case, by
(\ref{coswz}) the
conditions
$w=\pi$ and $w_X=0$ together imply
$$w_Y~=~{c'(u)\over 8c^2(u)}(\cos z+1) q~\not= ~0\,.$$
This is equivalent to the implication
$$w~=~\pi~\qquad\Longrightarrow\qquad (w_X, w_Y)~\not=~(0,0).$$
By the implicit function theorem,
the level set $\{w=\pi\}$ is then the union of regular curves in the $X$-$Y$
plane (restricted to the domain $\Gamma$).
Similarly, the level set $\{z=\pi\}$ will be a
union of regular curves.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{wa68.eps}
\caption{ \small A generic solution $u=u(t,x)$
of (\ref{1.1}) with smooth initial data
remains smooth outside finitely many singular points and finitely many singular curves where
$u_x\to\pm\infty$. The curves where $u$ is singular are the images of the curves where
$w=\pi$ or $z=\pi$ in Fig.~\ref{f:wa67}, under the map $(t,x)=\Lambda(X,Y)$
at (\ref{Ldef}).
Here $p_i=\Lambda(P_i)$ are points where singular curves originate or terminate, while
$q_j=\Lambda(Q_j)$
are points where two singular curves cross.}
\label{f:wa68}
\end{figure}
We shall give a proof of Lemma 5,
using Lemma~4 together with Thom's transversality theorem.
For readers' convenience, we first review some basic definitions \cite{Bloom, GG, T}.
\vskip 1em
{\bf Definition (map transverse to a submanifold).}
Let $f : X \mapsto Y$ be a smooth map of manifolds and let
$W$ be a submanifold of $Y$. We say that
$f$ is {\it transverse} to $W$ at a point $p\in X$,
and write $f \cap\!\!\!\!|~ _p W$, if
\begin{itemize}
\item either $ f ( p ) \notin W$,
\item or else $f(p) \in W$
and $T_{f(p)}Y = (df)_p(T_pX) + T_{f(p)}W$.
\end{itemize}
Here $T_p X$ denotes the tangent space to $X$ at the point $p\in X$,
while $T_qY$ and $T_q W$ denote respectively the tangent spaces to $Y$ and to $W$ at the point
$q\in W\subset Y$. Finally, $(df)_p:T_pX\mapsto T_{f(p)}Y$ denotes the differential
of the map $f$ at the point $p$.
We say that $ f$ is {\it transverse} to $W$, and write $f \cap\!\!\!\!|~ W$, if $f \cap\!\!\!\!|~ _p W$ for every $p\in X$.
In the special case where $W=\{y\}$ consists of a single point, $f \cap\!\!\!\!|~ W$
if and only if $y$ is a regular value of $f$, in the following sense.
\vskip 1em
{\bf Definition (regular value).} Let $f : X \mapsto Y$ be a smooth map of manifolds.
A point $y\in Y$ is a {\it regular value} if,
for every $p\in X$ such that $f(p)=y$, one has
$$T_{f(p)}Y~ =~ (df)_p(T_pX).$$
\vskip 1em
{\bf Transversality Theorem.} {\it Let $X$, $\Theta$, and $Y$ be smooth manifolds,
$W$ a submanifold
of $Y$.
Let $\theta\mapsto \phi^\theta$ be a smooth map which to each
$\theta\in \Theta$ associates a
function $ \phi^\theta\in {\mathcal C}^\infty(X,Y)$, and define
$\Phi:X\times \Theta\mapsto Y$ by setting
$\Phi(x,\theta)=\phi^\theta(x)$.
If $\Phi\cap\!\!\!\!|~ W$ then the set
$\{\theta\in \Theta,;~~\phi^\theta\cap\!\!\!\!|~ W\}$
is dense in $\Theta$.}
For a proof, see \cite{Bloom, GG}.
\vskip 1em
\subsection{Proof of Lemma 5.}
{\bf 1.} We shall use the representation
\begin{equation}\label{Srep}
S'~=~S_1\cap S_2\cap S_3\cap S_4\cap S_5\cap S_6\,,\end{equation}
where $S_1,\ldots,S_6\subset{\mathcal S}$ are the families of solutions
for which one of the six values listed in
(\ref{never1})--(\ref{never3}) is never attained on $\Gamma$.
For example,
${\mathcal S}_1$ is the set of all solutions such that
\begin{equation}\label{S1}(w,w_X,w_{XX})(X,Y) ~\not=~(\pi,0,0)\qquad\qquad
\hbox{for all }~ (X,Y)\in\Gamma,\end{equation}
while
${\mathcal S}_6$ is the set of all solutions such that
\begin{equation}\label{S6}(z, z_Y, c'(u))(X,Y) ~\not=~(\pi,0,0)\qquad\qquad
\hbox{for all }~ (X,Y)\in\Gamma.\end{equation}
Since $\Gamma$ is a compact domain,
it is clear that each ${\mathcal S}_i$ is a relatively
open subset of ${\mathcal S}$, in the topology of ${\mathcal C}^2(\Gamma)$.
In the remainder of the proof we will show that each $S_i$ is dense
on ${\mathcal S}$.
\vskip 1em
{\bf 2.} Let $(u,w,z,p,q)$ be any ${\mathcal C}^2$ solution of
(\ref{2.26})--(\ref{2.25}), with $p,q>0$. By a smooth approximation
of the data along the line $\gamma_0=\{X+Y=0\}$, it is not
restrictive to assume that $u,w,z,p,q\in{\mathcal C}^\infty(I\!\!R^2)$.
We begin by looking at the first condition in (\ref{never1}).
Given any point $(X_0, Y_0)\in \Gamma$, two cases can occur.
CASE 1: $(w, w_X, w_{XX})(X_0, Y_0)~\not=~(\pi, 0,0)$.
In this case, by continuity, there exists a neighborhood ${\mathcal N}$ of $(X_0, Y_0)$
in the $X$-$Y$ plane where we still have $(w, w_X, w_{XX})~\not=~(\pi, 0,0)$.
CASE 2: $(w, w_X, w_{XX})(X_0, Y_0)~=~(\pi, 0,0)$. In this case,
by Lemma~4 we can find a 3-parameter family of
solutions $(u^\theta, w^\theta, z^\theta, p^\theta, q^\theta)$
such that the $3\times 3$ Jacobian matrix of the map
\begin{equation}\label{5map}
(\theta_1,\theta_2,\theta_3)~\mapsto~\Big(w^\theta(X,Y)\,, ~w_X^\theta(X,Y)\,,w_{XX}^\theta(X,Y)\Big)\end{equation}
has rank 3 at the point $(X_0, Y_0)$, when $\theta=0$.
By continuity, this matrix still has rank 3
on a neighborhood ${\mathcal N}$ of $(X_0,Y_0)$, for $\theta$ sufficiently
close to zero.
We now choose finitely many points $(X_i, Y_i)$, $i=1,\ldots,n$,
such that the corresponding open neighborhoods
${\mathcal N}_{(X_i,Y_i)}$ cover the compact set $\Gamma$.
Call $n_{\mathcal I}$ the cardinality of the set of indices
\begin{equation}\label{Idef}
{\mathcal I}~\doteq~\bigl\{ i\,;~~(w, w_X, w_{XX})(X_i, Y_i)~=~(\pi,0,0)\bigr\}
\end{equation}
so that CASE 2 applies, and set $N=3n_{\mathcal I}$.
\vskip 1em
{\bf 3.}
Let $\Omega\supset\Gamma$ be an open set contained in the union
of the neighborhoods ${\mathcal N}_{(X_i,Y_i)}$, and
call $B_\varepsilon\doteq\bigl\{ \theta\in I\!\!R^N\,;~~|\theta|<\varepsilon\bigr\}$
the open ball of radius $\varepsilon$ in $I\!\!R^N$.
We shall construct a family $(u^\theta, w^\theta,z^\theta,
p^\theta, q^\theta)$ of smooth solutions to (\ref{2.26})--(\ref{2.25}),
such that the map
\begin{equation}\label{Nmap}
(X,Y,\theta)~\mapsto~\Big(w^\theta(X,Y)\,, ~w_X^\theta(X,Y)\,,~w_{XX}^\theta(X,Y)\Big)\end{equation}
from $\Omega\times B_\varepsilon$
into $I\!\!R^3$ has $(\pi,0,0)$ as a regular value.
Toward this goal, we need
to combine perturbations based at possibly different points
$(X_i, Y_i)$ into a single $N$-parameter family
of perturbed solutions.
Let $(u,w,z,p,q)(X,Y)$ be a solution to the system
(\ref{2.26})--(\ref{2.25}).
For each $k=1,\ldots,N$,
let a point $(X_k,Y_k)$ be given,
together with a number $U_k\in I\!\!R$ and functions
$W_k, Z_k, P_k, Q_k\in {\mathcal C}^\infty_c(I\!\!R)$.
By the previous analysis, a 1-parameter family of perturbed solutions
to (\ref{2.26})--(\ref{2.25}) is then determined as follows.
For $|\varepsilon|<\varepsilon_k$ sufficiently small, let
\begin{equation}\label{solt}
(u^\varepsilon, w^\varepsilon, z^\varepsilon, p^\varepsilon, q^\varepsilon) ~\doteq~\Psi_k^\varepsilon(u,w,z,p,q)\end{equation}
be the unique solution of (\ref{2.26})--(\ref{2.25}) with
data assigned on the
line $\gamma_k\doteq \{X+Y=X_k+Y_k\}$ by setting
$$u^\varepsilon(X_k, Y_k) ~=~u(X_k, Y_k)+\varepsilon U_k\,,$$
while for $(X,Y)\in \gamma_k$
$$w^\varepsilon ~= ~w+\varepsilon W_k\,,\qquad
z^\varepsilon ~= ~z+\varepsilon Z_k\,,\qquad
p^\varepsilon ~= ~p+\varepsilon P_k\,,\qquad
q^\varepsilon ~= ~q+\varepsilon Q_k\,.$$
Given $(\theta_1,\ldots,\theta_{N})$, a perturbation of the
original solution $(u,w,z,p,q)$ is defined as the composition
of $N$ perturbations:
\begin{equation}\label{pert5}
(u^\theta,w^\theta,z^\theta,p^\theta,q^\theta)~\doteq~
\Psi_{N}^{\theta_{N}}\circ\cdots\circ\Psi_1^{\theta_1}(u,w,z,p,q).\end{equation}
\vskip 1em
{\bf 4.}
At each point $(X_i, Y_i)$ with $i\in{\mathcal I}$, we can apply Lemma~4
and obtain
three 1-parameter families of perturbed solutions
so that the Jacobian matrix (\ref{5map}) has rank 3 on
${\mathcal N}_{(X_i,Y_i)}$,
for all $\theta$ small enough.
Combining all these perturbations,
we obtain an $N$-parameter family
of solutions such that
the value
$(\pi,0,0)$ is a regular value
for the map (\ref{Nmap}), from $\Omega\times B_\varepsilon$ into $I\!\!R^3$.
By the transversality theorem, for a.e.~$\theta$ the value
$(\pi,0,0)$ is a regular value
for the map $(X,Y)~\mapsto~
\Big(w^\theta(X,Y)\,, ~w_X^\theta(X,Y)\,,~w_{XX}^\theta(X,Y)\Big)$
from $\Omega$ into $I\!\!R^3$.
Since $\Omega$ has dimension 2, for a.e.~$\theta$ the corresponding solution
$(u^\theta, w^\theta, z^\theta, p^\theta, q^\theta)$
has the property that
$$\bigl(w^\theta(X,Y)\,, ~w_X^\theta(X,Y)\,,~w_{XX}^\theta(X,Y)
\bigr)~\not=~(\pi,0,0)$$
for all $(X,Y)\in \Gamma$.
This proves that the set ${\mathcal S}_1$ of solutions for which (\ref{S1})
holds is dense on ${\mathcal S}$.
\vskip 1em
{\bf 5.} Repeating the above construction, we obtain that
each ${\mathcal S}_i$, $i=1,\ldots,6$, is a relatively open, dense
subset of ${\mathcal S}$. By (\ref{Srep}), the intersection $S'$ is
is a relatively open, dense
subset of ${\mathcal S}$.
\hphantom{MM}\hfill\llap{$\square$}\goodbreak
\section{Proof of Theorem 1.}
\setcounter{equation}{0}
Consider the product space
\begin{equation}\label{Udef}{\mathcal U}~\doteq~
\Big({\mathcal C}^3(I\!\!R)\cap H^1(I\!\!R)\Big)
\times\Big({\mathcal C}^2(I\!\!R)\cap{\bf L}^2(I\!\!R)\Big)\end{equation}
with norm
$$\bigl\| (u_0,u_1)\bigr\|_{\mathcal U}~\doteq~\|u_0\|_{{\mathcal C}^3} + \|u_0\|_{H^1} + \|u_1\|_{{\mathcal C}^2}
+ \|u_1\|_{{\bf L}^2}\,.$$
Given initial data $(\hat u_0, \hat u_1)~\in~ {\mathcal U}$,
consider the open ball
\begin{equation}\label{Bd}B_\delta~\doteq~\Big\{
(u_0, u_1)\in{\mathcal U}\,;~~
\bigl\|(u_0, u_1)-(\hat u_0,\hat u_1)\bigr\|_{\mathcal U}~<~\delta\Big\}.
\end{equation}
Theorem 1 will be proved by showing that, for any
$(\hat u_0, \hat u_1)~\in~ {\mathcal U}$ there exists a radius $\delta>0$
and an open dense subset
$\widehat{\mathcal D}\subseteq B_\delta$, with the following property:
For every initial data
$(u_0,u_1)\in\widehat {\mathcal D}$, the conservative solution
$u=u(t,x)$ of (\ref{1.1})-(\ref{1.2})
is twice continuously differentiable in the
complement of finitely many characteristic curves
$\gamma_i$, within the domain $[0,T]\times I\!\!R$.
\vskip 1em
{\bf 1.} Let $(\hat u_0, \hat u_1)~\in~ {\mathcal U}$ be given.
By the definition of the space ${\mathcal U}$ in (\ref{Udef}),
as $|x|\to\infty$ we have
\begin{equation}\label{00}
\hat u_0(x)\to 0,\qquad \hat u_{0,x}(x)\to 0,
\qquad \hat u_1(x)\to 0.\end{equation}
Hence the corresponding functions $R,S$ in (\ref{2.1})
satsfy
$$R(0,x)~\to ~0,\qquad\qquad S(0,x)~\to~0.$$
{}From (\ref{2.3}), it follows that the functions $R,S$ remain
uniformly bounded on a domain of the form
$\{ (t,x)\,;~~t\in [0,T],~~~|x|\geq r\}$, for $r$ sufficiently
large. More generally, we can choose
$\delta>0$ such that,
for every initial data $(u_0, u_1)\in B_\delta$,
the corresponding solution $u(t,x)$ remains
twice continuously differentiable
on the outer domain
\begin{equation}\label{outer}\Big\{ (t,x)\,;~~t\in [0,T],~~~|x|\geq \rho\Big\},\end{equation}
for some $\rho>0$ sufficiently large. Its singularities
can thus occur only on the compact domain
$[0,T]\times [-\rho, \,\rho]$.
The subset $\widehat D\subset B_\delta$ is now defined as follows.
~$(u_0, u_1)\in \widehat D$ if $(u_0, u_1)\in B_\delta$
and moreover, for the corresponding solution
$(u,w,z,p,q)$ of (\ref{2.26})--(\ref{4.2}) with boundary data
(\ref{2.28}), the values (\ref{never1})--(\ref{never3}) are never attained,
for any $(X,Y)$
such that
\begin{equation}\label{bad}(t(X,Y), \, x(X,Y))~\in~[0,T]\times [-\rho,\,\rho].\end{equation}
It is important to observe that, by (\ref{degen}),
the above condition is independent
of the relabeling (\ref{TXY}).
\vskip 1em
{\bf 2.} For any $(u_0, u_1)\in B_\delta$ we now consider the
corresponding solution $(t,x,u,w,z,p,q)$ of the system
(\ref{2.26})--(\ref{4.2}), with boundary data as in (\ref{2.28}).
Let $\Lambda$ be the map at (\ref{Ldef}) and let
$\Gamma$ be the square with side $2M$ in the $X$-$Y$ plane,
as in (\ref{GTdef}).
By choosing $M$ large enough, and by possibly shrinking the
radius $\delta$, we can achieve the inclusion
\begin{equation}\label{sur}[0,T]\times [-\rho,\rho]~\subset~
\Lambda(\Gamma)\,,\end{equation}
for every $(u_0, u_1)\in B_\delta$.
\vskip 1em
{\bf 3.} We begin by proving that $\widehat {\mathcal D}$ is open, in the
topology of ${\mathcal C}^3\times{\mathcal C}^2$.
Indeed, consider initial data $(u_0,u_1)\in\widehat {\mathcal D}$ and let
$(u_0^\nu, u_1^\nu)_{\nu\geq 1}$
be a sequence of initial data converging to $(u_0, u_1)$.
Assume, by contradiction, that $(u_0^\nu, u_1^\nu)\notin \widehat {\mathcal D}$ for all $\nu\geq 1$.
To fix the ideas, let $(X^\nu, Y^\nu)$ be points at which
the corresponding solutions $(u^\nu, w^\nu, z^\nu, p^\nu, q^\nu)$ satisfy
\begin{equation}\label{bad1}
(w^\nu, w^\nu_X, w^\nu_{XX}) (X^\nu, Y^\nu)~=~(\pi,\, 0,\,0),\qquad\quad (t^\nu,x^\nu)
(X^\nu, Y^\nu)~\in~[0,T]\times [-\rho,\,\rho],\end{equation}
for all $\nu\geq 1$.
By (\ref{sur}),
since the domain $\Gamma$ in (\ref{GTdef}) is compact,
by possibly taking a subsequence we can assume
$(X^{\nu}, Y^{\nu})\to (\overline X, \overline Y)$. By continuity, this
implies
$$(w, w_X, w_{XX})(\overline X, \overline Y) ~=~(\pi, \,0,\,0),
\qquad\qquad (t,x)
(\overline X,\, \overline Y)~\in~[0,T]\times [-\rho,\,\rho],$$
contradicting the assumption $(u_0, u_1)\in \widehat{\mathcal D}$.
The other cases in (\ref{never1})--(\ref{never3}) are handled in the same way.
This proves that $\widehat{\mathcal D}$ is open.
\vskip 1em
{\bf 4.} Next, we claim that $\widehat D$ is dense in $B_\delta$.
Indeed, let $(u_0,u_1)\in B_\delta$ be given.
By an arbitrarily small perturbation (measured in the norm of ${\mathcal U}$),
we can assume that $u_0, u_1\in{\mathcal C}^\infty$.
Using Lemma~5, we can construct a sequence of solutions
$(u^\nu, w^\nu, z^\nu, p^\nu, q^\nu, x^\nu, t^\nu)$ of (\ref{2.26})--(\ref{4.2})
such that:
\begin{itemize}
\item[(i)] For every bounded set $\Omega\subsetI\!\!R^2$ and any $k\geq 1$,
the ${\mathcal C}^k$ norm of the difference satisfies
\begin{equation}\label{lim0}
\lim_{\nu\to\infty}~\Big\|(u^\nu-u, ~w^\nu-w, ~z^\nu-z, ~p^\nu-p,
~q^\nu-q, ~x^\nu-x, ~t^\nu-t)
\Big\|_{{\mathcal C}^k(\Omega)}~=~0.\end{equation}
\item[(ii)] For every $\nu\geq 1$,
the values in (\ref{never1})--(\ref{never3}) are never attained,
for any $(X,Y)\in \Gamma$.
\end{itemize}
Consider the corresponding solutions
$u^\nu(t,x)$ of (\ref{1.1}), with graph
$$\Big\{ \bigl(u^\nu(X,Y),~ t^\nu(X,Y),~ x^\nu(X,Y)\bigr)\,;~~(X,Y)\inI\!\!R^2\Big\}
~\subset~I\!\!R^3.$$
For $t=0$, by (\ref{lim0}) the corresponding sequence of initial values satisfies
\begin{equation}\label{l5}\lim_{\nu\to\infty}~\bigl\| u^\nu(0,\cdot) - u_0\bigr\|_{{\mathcal C}^k([a,b])}~=~0,
\qquad\qquad
\lim_{\nu\to\infty}~\bigl\| u^\nu_t(0,\cdot) - u_1\bigr\|_{{\mathcal C}^k([a,b])}~=~0,\end{equation}
for every bounded interval $[a,b]$.
Next, consider a cutoff function $\eta\in {\mathcal C}^\infty_c$ such that
\begin{equation}\label{eta}
\begin{array}{l}\eta(x)~=~1\qquad\hbox{if}~~|x|\leq r,\\[4mm]
\eta(x)~=~0\qquad\hbox{if}~~|x|\geq r+1,\end{array}\end{equation}
with $r>\!>\rho$ sufficiently large.
For every $\nu\geq 1$, consider the initial data
$$\tilde u^\nu_0~\doteq~\eta u_0^\nu+(1-\eta)u_0\,,
\qquad \tilde u^\nu_1~\doteq~\eta u_1^\nu+(1-\eta)u_1\,.$$
By (\ref{l5}) we have
\begin{equation}\label{l6}\lim_{\nu\to\infty}~\bigl\| (\tilde u_0^\nu - u_0,
~\tilde u_1^\nu-u_1)\bigr\|_{\mathcal U}~=~0.\end{equation}
Moreover, if $r>0$ was chosen large enough, we have
$$\tilde u^\nu(t,x)~=~u^\nu(t,x)\qquad\qquad\hbox{for all }~
(t,x)\in [0,T]\times [-\rho,\rho]\,,$$
while $\tilde u^\nu$ remains ${\mathcal C}^2$ on the outer domain (\ref{outer}).
The above implies $(\tilde u^\nu_0, \tilde u^\nu_1)\in \widehat {\mathcal D}$
for all $\nu\geq 1$ sufficiently large,
proving that $\widehat {\mathcal D}$ is dense
on $B_\delta$.
\vskip 1em
{\bf 5.} To complete the proof we need to show that,
for every initial data $(u_0, u_1)\in \widehat{\mathcal D}$, the
solution $u(t,x)$ of (\ref{1.1}) is piecewise ${\mathcal C}^2$ on the
domain $[0,T]\timesI\!\!R$.
By the previous arguments, we already know that $u$
is ${\mathcal C}^2$ on the outer domain (\ref{outer}). It thus remains
to study the singularities of $u$ on the inner domain
$[0,T]\times [-\rho,\,\rho]$.
For this purpose,
call $(u,w,z,p,q,t,x)(X,Y)$ the corresponding solution of
(\ref{2.26})--(\ref{4.2}), with boundary data as in (\ref{2.28}).
By (\ref{sur}), every point of the inner domain is contained in the
image of the square $\Gamma$ in (\ref{GTdef}).
Consider a point $(X_0,Y_0)\in\Gamma$. Two cases can occur.
CASE 1: $w(X_0,Y_0)\not= \pi$ and $z(X_0,Y_0)\not= \pi$.
By (\ref{4.1})-(\ref{4.2}) it follows
$$\det\left(\begin{array}{cc}x_X&x_Y\cr t_X&t_Y\end{array}\right)~=~
{(1+\cos w)p\over 4}\cdot {(1+\cos z)p\over 4c}+{(1+\cos z)q\over 4}\cdot {(1+\cos w)p\over 4c}~>~0.$$
Hence the map $(X,Y)\mapsto (x,t)$ is locally invertible
in a neighborhood of $(X_0,Y_0)$. We can thus conclude that
the function $u$ is ${\mathcal C}^2$ in a neighborhood of the point
$\bigl(t(X_0, Y_0),
x(X_0,Y_0)\bigr)$.
CASE 2: $w(X_0, Y_0)=\pi$. In this case we have either $w_X(X_0,Y_0)\not= 0$, or else by (\ref{2.24})
\begin{equation}\label{wY0}w_Y(X_0,Y_0)~=~{c'(u)\over 8 c^2(u)}(\cos z +1)q~\not=~0.\end{equation}
Indeed, we always have $c(u)>0$ and $q>0$.
Moreover, by construction the values $(w,z,w_X)= (\pi,\pi,0)$
and $(w,w_X, c'(u))=(\pi,0,0)$ are never attained in $\Gamma$.
This implies (\ref{wY0}).
By the implicit function theorem, we thus conclude that the sets
\begin{equation}\label{SWZ}S^w~\doteq~\{(X,Y)\in \Gamma\,;~~w(X,Y)=\pi\},\qquad
\qquad S^z~\doteq~\{(X,Y)\in \Gamma\,;~~z(X,Y)=\pi\}\end{equation}
are the union of finitely many ${\mathcal C}^2$ curves.
The set of points $(t,x)$ where $u$ is singular
coincides with the image of
the two sets $S^w$, $S^z$ under the ${\mathcal C}^2$ map
$$(X,Y)~\mapsto~\Lambda(X,Y)~=~\bigl(t(X,Y),~x(X,Y)\bigr).$$
\vskip 1em
{\bf 6.} To complete the proof,
we study in more detail the images of the singular sets
$S^w,S^z$.
By (\ref{impl1}) there can be only finitely many
points inside
$\Gamma$
where $w=\pi$ and $w_X=0$, say $P_i=(X_i,Y_i)$, $i=1,\ldots,m$.
Moreover, by (\ref{impl2}), at a point $(X_0,Y_0)\in S^w\cap S^z$
we have
$$w_X\not=0, \qquad w_Y~=~0,\qquad z_X~=~0,\qquad z_Y~\not=~0.$$
Therefore, as shown in Fig.~\ref{f:wa67}, the two curves $\{w=\pi\}$
and $\{z=\pi\}$ intersect
perpendicularly. As a consequence, inside the compact set $\Gamma$, there can be only finitely many
such intersection points, say $Q_i = (X_i', Y_i')$, $i=1,\ldots,n$.
After removing these finitely many points $P_i, Q_i$,
we can thus write ${\mathcal S}^w$ as a finite union of curves $\gamma_j$
of the form
\begin{equation}\label{gj}\gamma_j~=~\bigl\{(X,Y)\,;~~X=\phi_j(Y),~~a_i<Y<b_j\bigr\}.\end{equation}
for suitable functions $\gamma_j$ of class ${\mathcal C}^2$.
We claim that the image of $\Lambda(\gamma_j)$ is a ${\mathcal C}^2$ curve
in the $t$-$x$ plane. To prove this, it suffices to show that,
on the open interval $]a_j, b_j[\,$ the differential of the map
$$Y~\mapsto~\bigl( x(\phi_j(Y),Y),~t(\phi_j(Y),Y)\bigr)$$
does not vanish.
This is true because, by (\ref{4.2})
$${d\over dY} \, t(\phi_j(Y),Y)~=~t_X\cdot \phi_j' + t_Y
~=~0\cdot \phi_j' + {(1+\cos z)q\over 4c(u)}~>~0.$$
Indeed, $z\not= \pi$ while $c(u), q >0$.
As shown in Fig.~\ref{f:wa68}, restricted to the
inner domain $[0,T]\times [-\rho,\,\rho]$ in the $t$-$x$ plane, the singular set $\Lambda(S^w)$
is thus the union of the finitely many points
$$\begin{array}{l} p_i~=~\Lambda(P_i), \qquad i=1,\ldots,m,\\[4mm]
q_i~=~\Lambda(Q_i), \qquad i=1,\ldots,n,\end{array}$$
together with finitely many ${\mathcal C}^2$ curves
$\Gamma_j~\doteq~\Lambda(\gamma_j)$.
The same representation is valid for the image $\Lambda(S^z)$.
This concludes the proof of Theorem~1. \hphantom{MM}\hfill\llap{$\square$}\goodbreak
\vskip 1em
\section{One-parameter families of solutions}
\setcounter{equation}{0}
In this section we study families of conservative solutions
$u=u(t,x,\lambda)$ of (\ref{1.1}) depending on an additional parameter
$\lambda\in [0,1]$.
We thus consider a 1-parameter family of initial data
\begin{equation}\label{6.1}
u(0,x,\lambda)~=~u_0(x,\lambda),\qquad\qquad u_t(0,x,\lambda)~=~u_1(x,\lambda),\end{equation}
smoothly depending on the additional parameter $\lambda\in [0,1]$.
More precisely, these paths of initial data will lie in the space
\begin{equation}\label{TD}{\mathcal X}~\doteq~
\Big({\mathcal C}^3([0,1]\times I\!\!R)\cap{\bf L}^\infty([0,1]; H^1(I\!\!R))\Big)
\times\Big({\mathcal C}^2([0,1]\timesI\!\!R)\cap{\bf L}^\infty([0,1];\,{\bf L}^2(I\!\!R))\Big).\end{equation}
In particular, the map $(x,\lambda)\mapsto u_0(x,\lambda)$ is
three times continuously differentiable and the $H^1$ norm
of $u_0(\cdot,\lambda)$ is uniformly bounded for all $\lambda$.
Moreover, the map $(x,\lambda)\mapsto u_1(x,\lambda)$ is
two times continuously differentiable and the ${\bf L}^2$ norm
of $u_1(\cdot,\lambda)$ is uniformly bounded for all $\lambda$.
By an adaptation of the previous arguments one obtains
\vskip 1em
{\bf Theorem 2.}
{\it Let the wave speed $c(u)$ satisfy the assumptions {\bf (A)}
and let $T>0$ be given.
Then, for any 1-parameter family of initial data
$(\hat u_0,\hat u_1)\in{\mathcal X}$
and any $\varepsilon>0$, there exists a perturbed family
$(x,\lambda)\mapsto( u_0, u_1)(x,\lambda)$
such that
\begin{equation}\label{ed}
\Big\|( u_0- \hat u_0\,,~ u_1 - \hat u_1)\Big\|_{\mathcal X}
~<~\varepsilon\,,\end{equation}
and moreover the following holds.
For all except at most finitely many $\lambda\in [0,1]$,
the conservative solution
$u=u(t,x;\lambda)$ of (\ref{1.1})
is smooth in the complement of finitely
many points $P_i$ and finitely many ${\mathcal C}^2$ curves
$\gamma_j$ in the domain $[0,T]\times I\!\!R$.}
\vskip 1em
Toward a proof, we shall need
\vskip 1em
{\bf Lemma 6.} {\it Let the function $u\mapsto c(u)$ satisfy the assumptions
{\bf (A)}, and let any $M>0$ be given.
Then there exists a dense set of paths of initial data
$ {\mathcal D}~\subset~{\mathcal X}$
such that, if $(x,\lambda)\mapsto( u_0, u_1)(x,\lambda)$
lies in ${\mathcal D}$, then the corresponding solutions
$(t,x,u,w,z,p,q)$ of (\ref{2.24})--(\ref{4.2})
with boundary data as in (\ref{2.28}) have the following properties.
On the domain $\Gamma$ in (\ref{GTdef})
one has
\begin{itemize}\item[(i)] The map $(X,Y,\lambda)\mapsto (w,w_X,w_{XX})$
is transversal to the point $(\pi,0,0)$.
\item[(ii)] The map $(X,Y,\lambda)\mapsto (z,z_Y,z_{YY}) $
is transversal to the point $(\pi,0,0)$.
\item[(iii)] The map $(X,Y,\lambda)\mapsto (w,z,w_X)$
is transversal to the point $(\pi,\pi,0)$.
\item[(iv)] The map $(X,Y,\lambda)\mapsto (w,z,z_Y)$
is transversal to the point $(\pi,\pi,0)$.
\item[(v)] The map $(X,Y,\lambda)\mapsto (w,w_X,c'(u))$
is transversal to the point $(\pi,0,0)$.
\item[(vi)] The map $(X,Y,\lambda)\mapsto (z,z_Y,c'(u))$
is transversal to the point $(\pi,0,0)$.
\end{itemize}}
\vskip 1em
{\bf Proof of Lemma~6.} Consider any point $(X_0, Y_0, \lambda_0)$.
Then, there exist
3-parameter families of perturbed initial data
$(u_0^\theta, u_1^\theta)$, $\theta\in I\!\!R^3$ such that
the properties {\bf (1)--(3)} in Lemma 4 hold.
Indeed, it suffices to repeat all the arguments in the proof of Lemma~4
regarding $\lambda_0$ as a constant. For a fixed
$\lambda=\lambda_0$, the perturbations
in (\ref{p00}) are thus functions of $s$ only, constant
w.r.t.~$\lambda$.
Combining these perturbations, as in the proof of
Lemma~5, we obtain a map $(X,Y,\lambda, \theta)\mapsto (u,w,z,p,q)$
for which all transversality conditions (i)--(vi) are satisfied.
By the transversality theorem, for a.e.~$\theta$ the corresponding
map $(X,Y,\lambda)\mapsto
(u^\theta,w^\theta,z^\theta,p^\theta,q^\theta)$ satisfies the
same transversality conditions. This achieves the proof. \hphantom{MM}\hfill\llap{$\square$}\goodbreak
{\bf Proof of Theorem 2.}
As in the proof of Theorem 1, we first
choose $\rho$ large enough so that all our solutions
will be ${\mathcal C}^2$ for $(t,x)$ in the outer domain
(\ref{outer}).
For each $\lambda\in [0,1]$, we denote by
$(u,w,z,p,q, x,t)(X,Y,\lambda)$ the corresponding solution
of the semilinear system (\ref{2.26})-(\ref{4.2}).
We choose $M$ sufficiently large such that,
for all $\lambda\in [0,1]$, the inner domain
$[0,T]\times [-\rho,\,\rho]$ is contained in the image
$$\Lambda^\lambda(\Gamma)~=~\Big\{\bigl( t(X,Y,\lambda)
,~x(X,Y,\lambda)\bigr)\,;~~|X|+|Y|\leq M\Big\}.$$
By performing an arbitrarily
small perturbation of the initial path of solutions
we obtain a second path $\lambda\mapsto u(\cdot,\lambda)$
such that, in the corresponding solution
$(u,w,z,p,q, x,t)(X,Y,\lambda)$,
the transversality relations (i)--(vi) in Lemma~6 hold.
Since the variables
$(X,Y,\lambda)\in \Gamma\times [0,1]$ range on a compact,
three dimensional set, this implies that the
values in (i)--(vi) are attained only at finitely many points,
say $(X_i, Y_i,\lambda_i)$, $i=1,\ldots,n$.
Hence, for $\lambda\notin \{\lambda_1,\ldots,\lambda_n\}$,
the solution $(t,x,u,w,z,p,q)(\cdot,\cdot,\lambda)$
does not attain any of the values in (i)--(vi),
for $(X,Y)\in \Gamma$.
As shown in steps
{\bf 5-6} in the proof of Theorem 1,
the corresponding solution
$u=u(t,x;\,\lambda)$
is then piecewise smooth on the inner domain
$[0,T]\times [-\rho,\,\rho]$. \hphantom{MM}\hfill\llap{$\square$}\goodbreak
\vskip 1em
{\bf Remark 5.} For a given solution $u=u(t,x)$,
define its {\it singular set} as
$$S^u~\doteq~\bigl\{ (t,x)\,;~~ u~\hbox{is not ${\mathcal C}^2$ on any
neighborhood of}~ (t,x)\bigr\}.$$
In the above construction, one
can regard $\lambda_1,\ldots,\lambda_n$
as {\it bifurcation values}, where the structure of the singular set
changes (Fig.~\ref{f:wa70}).
On the other hand, for $\lambda\notin\{\lambda_1,\ldots,\lambda_n\}$
the solution $u(\cdot,\cdot;\lambda)$ is
{\it structurally stable}. A small perturbation of the initial data
does not change the topology of the singular set.
Based on the present analysis, we speculate that a theory of
generic structural stability and a global classification
of solutions to (\ref{1.1}) can be developed, in analogy to
the classical theory for ODEs \cite{A, P}.
\begin{figure}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{wa70.eps}
\caption{ \small The singular set for a solution $u(t,x;\lambda)$.
When the parameter $\lambda$ crosses one of the critical
values $\lambda_i$, the topology of the singular set changes.}
\label{f:wa70}
\end{figure}
\vskip 1em
{\bf Acknowledgments.} This research was partially supported by NSF, with grant DMS-1411786: Hyperbolic Conservation Laws and Applications, and by the AMS Simons Travel Grant Program.
\vskip 1em
| {
"timestamp": "2015-02-10T02:24:31",
"yymm": "1502",
"arxiv_id": "1502.02611",
"language": "en",
"url": "https://arxiv.org/abs/1502.02611",
"abstract": "The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\\big(c(u) u_x\\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846672373524,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019697626153
} |
https://arxiv.org/abs/cs/0606056 | Fast and Simple Methods For Computing Control Points | The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement. |
\section{Introduction}
\label{sec1}
Polynomial curves and surfaces are used extensively in geometric
modeling and computer aided geometric design (CAGD) in
particular (see Ramshaw \cite{Ramshaw87}, Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek}, or Piegl and Tiller \cite{Piegl}).
One of the main reasons
why polynomial curves and surfaces are used so extensively in CAGD,
is that there is a very powerful and versatile algorithm to recursively
approximate a curve or a surface using repeated affine interpolation,
the {\it de Casteljau algorithm\/}. However, the de Casteljau algorithm
applies to curves and surfaces only if they are defined in terms
of {\it control points\/}. There are situations where
a curve or a surface is defined explicitly in terms of polynomials.
For example, the following polynomials define a surface
known as the Enneper surface:
$$\eqaligneno{
x(u, v) &= u - \frac{u^3}{3} + uv^2\cr
y(u, v) &= v - \frac{v^3}{3} + u^2v\cr
z(u, v) &= u^2 - v^2.\cr
}$$
Thus, the problem of computing
control points from polynomials (defined
as sums of monomials) arises.
If control points can be computed quickly from polynomials,
all the tools available
in CAGD for drawing curves and surfaces can be applied.
This could be very useful in problems
where a curve of a surface is obtained analytically
in terms of polynomials or rational functions, for example,
problems involving generalizations of Voronoi diagrams,
or motion planning problems. When dealing which such problems,
it is often necessary to decide whether curves segments or surface
patches intersect or not.
As is well known (for example, see \cite{Farin93,Farin95}),
there are effective methods based on subdivision
(exploiting the fact that a B\'ezier curve or surface patch
is contained within the convex hull of its control points)
for deciding whether B\'ezier curve segments or surface patches intersect.
Simple and fast methods for computing control points might also
be also useful to teach say, Math students, to learn computational tools
for drawing interesting curves and surfaces.
Now, it turns out that the problem of computing control points
can be viewed as a change of polynomial basis, more specifically
as a change of basis from the monomial basis to bases
of Bernstein polynomials.
Algorithms for performing such changes of basis have been
given by Piegl and Tiller \cite{Piegl}. More general algorithms
for performing changes of bases between progressive bases
and P\'olya bases are presented in
Goldman and Barry \cite{GoldBarry} and
Lodha and Goldman \cite{LodhaGold}.
These algorithms compute certain
triangles or tetrahedras whose nodes are labeled with
certain multisets, and are generalizations of the de Casteljau
and the de Boor algorithm.
In this paper, we present alternate and more direct methods
for computing control points from polynomial definitions
(in monomial form)
that run in the same low time complexity as the above algorithms
($O(m^2)$ for curves of degree $m$,
$O(p^2q^2)$ for rectangular surfaces of bidegree $\pairt{p}{q}$,
and $O(m^3)$ for triangular surfaces of total degree $m$).
Our algorithms are not as general as those of
Goldman and Barry \cite{GoldBarry} and
Lodha and Goldman \cite{LodhaGold}, but they are more direct
and very easy to implement.
\medskip
The paper is organized as follows.
In section \ref{sec2}, we review briefly the relationship between
polynomial definitions and control points. We begin with the polarization
of polynomials in one or two variables, and then we show how polynomial
curves and surfaces are completely determined by sets of control points.
In the case of surfaces, depending on the mode of
polarization, we get two kinds of surfaces, bipolynomial surfaces
(or rectangular patches) and total degree surfaces (or triangular patches).
Efficient methods for computing control points are given in the next
three section:
polynomial curves in section \ref{sec5},
bipolynomial surfaces in section \ref{sec3}, and
polynomial total degree surfaces in section \ref{sec4}.
Some examples are given in
section \ref{sec6}.
\section{Control Points}
\label{sec2}
\subsection{Polynomial Curves}
The deep reason why polynomial curves and surfaces can
be handled in terms of control points is that
polynomials in one or several variables can be {\it polarized\/}.
This means that every polynomial function arises from a unique
symmetric multiaffine map.
A detailed treatment of this approach can be found in
Ramshaw \cite{Ramshaw87}, Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek},
or Gallier \cite{Gallbook}. We simply review what is needed
to explain our algorithms.
\medskip
Recall that a map $\mapdef{f}{\mathbb{R}^d}{\mathbb{R}^n}$ is {\it affine\/} if
$$f((1 - \lambda) a + \lambda b) = (1 - \lambda) f(a) + \lambda f(b),$$
for all $a, b\in \mathbb{R}^d$, and all $\lambda\in\mathbb{R}$.
A map $\mapdef{f}{\underbrace{\mathbb{R}^d\times \cdots\times \mathbb{R}^d}_{m}}{\mathbb{R}^n}$
is {\it multiaffine\/} if it is affine in each of its arguments, and
a map $\mapdef{f}{\underbrace{\mathbb{R}^d\times \cdots\times \mathbb{R}^d}_{m}}{\mathbb{R}^n}$
is {\it symmetric\/} if it does
not depend on the order of its arguments, i.e.,
$f(a_{\pi(1)},\ldots, a_{\pi(m)}) = f(a_1,\ldots,a_m)$
for all $a_1, \ldots, a_m$, and all permutations $\pi$.
We also say that a map
$\mapdef{f}{\reals^{p}\times \reals^{q}}{\mathbb{R}^d}$
is {\it $\pairt{p}{q}$-symmetric\/} if
it is symmetric separately in its first $p$ arguments
and in its last $q$ arguments.
\medskip
Let us first treat the case of polynomials in one variable, which corresponds
to the case of curves.
Given a (plane) polynomial curve $\mapdef{F}{\reals}{\reals^2}$ of degree
$m$,
$$\eqaligneno{
x(t) &= F_1(t),\cr
y(t) &= F_2(t),\cr
}$$
where $F_1(t)$ and $F_2(t)$ are polynomials of degree $\leq m$,
it turns out that
$\mapdef{F}{\reals}{\reals^2}$ comes from a unique symmetric
multiaffine map $\mapdef{f}{\reals^m}{\reals^2}$,
the {\it polar form of $F$\/}, such that
$$F(t) = f(\underbrace{t, \ldots, t}_m),
\quad\hbox{for all $t\in\reals$.}$$
\medskip
Furthermore, given any interval $(r, s)$ (affine frame), the map
$\mapdef{f}{\reals^m}{\reals^2}$ is determined by
the sequence $(b_0,\ldots, b_m)$ of $m + 1$ {\it control points\/}
$$b_i = f(\underbrace{r, \ldots, r}_{m - i},
\underbrace{s, \ldots, s}_i),$$
where $0\leq i \leq m$.
Using linearity, in order to polarize a polynomial of one variable $t$,
it is enough to polarize a monomial $t^k$. Since there are
$\binomalal{m}{k}$ terms in the sum
$$ \sum_{I \subseteq \{1, \ldots, m\}\atop
\scriptstyle |I| = k}
\biggl(\prod_{i\in I} t_i\biggr),$$
the polar form $f_k^m(t_1,\ldots,t_{m})$
of the monomial $t^k$ with respect to the degree $m$
(where $k\leq m$) is given by
$$f_k^m(t_1,\ldots,t_{m}) =
\frac{1}{\binomalal{m}{k}}\, \sum_{I \subseteq \{1, \ldots, m\}\atop
\scriptstyle |I| = k}
\biggl(\prod_{i\in I} t_i\biggr).$$
\subsection{Polynomial Surfaces Polarization}
Given a polynomial surface $\mapdef{F}{\reals^2}{\mathbb{R}^3}$,
there are two natural ways to polarize the polynomials
defining $F$.
\medskip
The first way is to
polarize {\it separately\/} in $u$ and $v$.
If $p$ is the highest degree in $u$ and $q$
is the highest degree in $v$,
we get a unique $\pairt{p}{q}$-symmetric
degree $(p + q)$ multiaffine map
$$\mapdef{f}{\reals^{p}\times \reals^{q}}{\mathbb{R}^3},$$
such that
$$F(u,v) = f(\underbrace{u,\ldots,u}_{p};
\underbrace{v,\ldots,v}_{q}).$$
\medskip
We get what is traditionally called
a {\it tensor product surface\/}, or as we prefer to call it,
a {\it bipolynomial surface of bidegree $\pairt{p}{q}$\/}
(or a {\it rectangular surface patch\/}).
\medskip
The second way to polarize is to treat the variables
$u$ and $v$ {\it as a whole\/}.
This way, if $F$ is a polynomial surface such that the maximum total degree
of the monomials is $m$,
we get a unique symmetric degree $m$ multiaffine map
$$\mapdef{f}{(\reals^2)^{m}}{\mathbb{R}^3},$$
such that
$$F(u,v) = f(\underbrace{(u, v),\ldots,(u, v)}_{m}).$$
\medskip
We get what is called a {\it total degree surface\/}
(or a {\it triangular surface patch\/}).
\medskip
Using linearity, it is clear that
all we have to do is to polarize a monomial $u^{h}v^{k}$.
\medskip
It is easily verified that the unique $\pairt{p}{q}$-symmetric
multiaffine polar form of degree $p + q$
$$f^{p, q}_{h, k}(u_{1},\ldots, u_{p}; v_{1},\ldots,v_{q})$$
of the monomial $u^{h}v^{k}$ is given by
$$f^{p, q}_{h, k}(u_{1},\ldots, u_{p}; v_{1},\ldots,v_{q})
= \frac{1}{\binomalal{p}{h} \binomalal{q}{k}}\,
\sum_{{I\subseteq\{1,\ldots,p\}, |I|=h}\atop {J\subseteq\{1,\ldots, q\}, |J|= k}}
\left(\prod_{i \in I} u_{i}\right)\left(\prod_{j \in J}v_{j}\right).
$$
\medskip
The denominator $\binomalal{p}{h} \binomalal{q}{k}$ is the number of terms in the
above sum.
\medskip
It is also easily verified that the unique symmetric multiaffine
polar form of degree $m$
$$f^{m}_{h, k}((u_{1}, v_{1}), \ldots, (u_{m}, v_{m}))$$
of the monomial $u^{h}v^{k}$ is given by
$$f^{m}_{h, k}((u_{1}, v_{1}), \ldots, (u_{m}, v_{m}))
=\frac{1}{\binomalal{m}{h} \binomalal{m-h}{k}}
\sum_{{I \cup J\subseteq\{1,\ldots,m\}}\atop{|I|=h,|J|= k, I \cap J=\emptyset}}
\left(\prod_{i \in I} u_{i}\right)\left(\prod_{j \in J}v_{j}\right).
$$
\medskip
The denominator $\binomalal{m}{h} \binomalal{m-h}{k} = \binomalal{m}{h\> k\> (m - h - k)}$
is the number of terms in the above sum.
\subsection{Control Points For Polynomial Surfaces}
Let $\Delta_{m} = \{(i,\, j,\, k)\in\mathbb{N}^3
\ |\ i + j + k = m\}$.
Given an affine frame $\Delta rst$ in the plane
(where $r, s, t\in \mathbb{R}^2$ are affinely
independent points),
a polynomial surface $\mapdef{F}{\reals^2}{\mathbb{R}^3}$
of total degree $m$ specified by the symmetric
multiaffine map
$$\mapdef{f}{(\reals^2)^{m}}{\mathbb{R}^3}$$
is completely determined by the
family of $\frac{(m + 1)(m + 2)}{2}$ points in $\mathbb{R}^3$
$$b_{i,\,j,\,k} = f(\underbrace{r,\ldots,r}_{i},
\underbrace{s,\ldots,s}_{j},
\underbrace{t,\ldots,t}_{k}),$$
where $(i,j,k)\in\Delta_{m}$.
\medskip
These points are called {\it control points\/}, and the family
$\{b_{i,\,j,\,k}\ |\ (i,j,k)\in\Delta_{m}\}$
is called a {\it triangular control net\/}.
\medskip
Let $({r}_1, {s}_1)$ and $({r}_2, {s}_2)$ be
any two affine frames for the affine line $\reals$.
A bipolynomial surface $\mapdef{F}{\reals^2}{\mathbb{R}^3}$
of bidegree $\langle p, q\rangle$ specified by the
$\pairt{p}{q}$-symmetric multiaffine map
$$\mapdef{f}{\reals^p\times \reals^q}{\mathbb{R}^3},$$
is completely determined by the family of $(p + 1)(q + 1)$
points in $\mathbb{R}^3$
$$b_{i,\, j} = f(\underbrace{{r}_1,\ldots,{r}_1}_{p - i},
\underbrace{{s}_1,\ldots,{s}_1}_{i};
\underbrace{{r}_2,\ldots,{r}_2}_{q - j},
\underbrace{{s}_2,\ldots,{s}_2}_{j}),$$
where $0\leq i\leq p$ and $0\leq j\leq q$.
\medskip
These points are called {\it control points\/}, and the family
$\{b_{i,\, j}\ | \ 0\leq i\leq p,\, 0\leq j\leq q\}$
is called a {\it rectangular control net\/}.
\medskip
Thus, to compute control points, in principle, we need
to compute the polar forms of polynomials.
However, this method requires polarization, which is very expensive.
In the following sections, we give recurrence formulae for computing
control points efficiently.
As a corollary, in the case of any affine frame $(r, s)$ or of the
affine frame $((1, 0),\, (0, 1),\, (0, 0))$,
it is possible to give closed-form formulae
for calculating control points in terms of binomial coefficients.
\section{Computing Control Points For Curves}
\label{sec5}
We saw in section \ref{sec2} that the polar form of a monomial
$t^k$ with respect to the degree
$m$ is
$$f_k^m(t_1,\ldots,t_m) =
\frac{1}{\binomalal{m}{k}}\, \sum_{I \subseteq \{1, \ldots, m\}\atop
\scriptstyle |I| = k}
\biggl(\prod_{i\in I} t_i\biggr).$$
\medskip
Letting
$\sigma^{m}_{k} = \binomalal{m}{k}\, f^{m}_k$,
it is easily verified that we have the following recurrence
equations:
$$\sigma^{m}_{k} =
\cases{ \sigma^{m-1}_{k} + t_m\sigma^{m-1}_{k-1} & if $1\leq k\leq m$;\cr
1 & if $k = 0$ and $m \geq 0$;\cr
0 & otherwise.\cr
}$$
\medskip
The above formulae can be used to compute inductively the polar values
$$f^{m}_{k}(t_1,\ldots,t_m) =
\frac{1}{\binomalal{m}{k}}\sigma^{m}_{k}(t_1,\ldots,t_m).$$
The computation is reminiscent of the Pascal triangle.
Alternatively, we can compute $f^{m}_{k}$ directly using the recurrence formula
$$f^{m}_{k} = \frac{(m - k)}{m}\,f^{m-1}_{k} +
\frac{k}{m}t_m\, f^{m-1}_{k-1},$$
where $1\leq k\leq m$.
When writing computer programs implementing these recurrence equations,
we observed that computing $\sigma^{m}_{k}$ and dividing by $\binomalal{m}{k}$
is faster than computing $f^{m}_{k}$ directly using the above
formula. This is because the second method requires
more divisions.
\medskip
Given $(t_1,\ldots,t_m)$,
computing all the scaled polar values
$\sigma^{i}_{k}(t_1,\ldots,t_i)$, where
$1\leq k\leq i$ and $1\leq i \leq m$, requires
time $O(m^2)$. The naive method using polarization
requires computing $\sum_{i = 0}^{m}2^{i} = 2^{m + 1} - 1$ terms.
To compute the coordinates of control points, we simply combine the
$\sigma^{m}_{k}$. Specifically,
the coordinate value of the control point $b_j$ contributed by the polynomial
$\sum_{k = 0}^{m} a_kt^k$
(where $m \leq m$) is
$$\sum_{k = 0}^{m} a_k\binomalal{m}{k}^{-1} \sigma^{m}_{k}
(\underbrace{r,\ldots,r}_{m - j}, \underbrace{s,\ldots,s}_{j}),$$
where $(r, s)$ is an affine frame.
Given $(t_1,\ldots,t_m)$,
our algorithm computes the table of
values $\sigma^{i}_{k}(t_1,\ldots,t_i)$,
where
$1\leq k\leq i$ and $1\leq i \leq m$,
and thus, it is very cheap to compute these sums.
\medskip
\bigskip\noindent{\bf Remark:}\enspace Given a polynomial curve $F$ of degree $m$
specified by the sequence of control points $(b_0,\ldots,b_m)$
over $(0, 1)$,
it is well known (see Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek}, or Piegl and Tiller \cite{Piegl})
that $F(t)$ can be expressed in terms of the Bernstein
polynomials $B^m_k(t) = \binomalal{m}{k}(1 - t)^{m-k} t^k$ as
$$F(t) = B^m_0(t)\, b_0 + \cdots + B^m_m(t)\, b_m.$$
It is also well known that the Bernstein polynomials
$B^m_0(t),\ldots$, $B^m_m(t)$ form a basis of
the vector space of polynomials of degree $\leq m$.
Thus, it is also possible to compute the control points
of $F$ by expressing the polynomials involved in the explicit polynomial
definition of $F$ in term of the basis Bernstein polynomials.
Such algorithms were given by
Goldman and Barry \cite{GoldBarry}.
Our algorithm has the same complexity and is more direct.
\medskip
It is also easy to derive closed-form formulae
for any affine frame $(r, s)$.
\begin{thm}
If the total degree is $m$ and there are $r$ occurrences where $t = p$ and $s$ occurrences
where $t = q$, then
$$f^{m}_k
= \frac{\binomalal{p}{k}r^{k}+\binomalal{p}{k-1}\binomalal{q}{1}r^{k-1}s
+\binomalal{p}{k-2}\binomalal{q}{2}r^{k-2}s^{2}+,\ldots,+\binomalal{q}{k}s^{k}}{\binomalal{m}{k}}.$$
\end{thm}
\noindent{\it Proof\/}.\enspace It can be shown by induction using the above
recurrence equations. $\mathchoice\sqr76\sqr76\sqr{2.1}3\sqr{1.5}3$
\section{Computing Rectangular Control Nets}
\label{sec3}
As we saw in section \ref{sec2},
the polar form of the monomial $u^{h}v^{k}$ with respect to $(p,q)$ is
$$f^{p, q}_{h, k}(u_1,\ldots,u_{p};v_1,\ldots,v_{q})
= \frac{1}{\binomalal{p}{h} \binomalal{q}{k}}\,
\sum_{{I\subseteq\{1,\ldots,p\}, |I|=h}\atop {J\subseteq\{1,\ldots, q\}, |J|= k}}
\left(\prod_{i \in I} u_{i}\right)\left(\prod_{j \in J}v_{j}\right).
$$
\medskip
Letting
$\sigma^{p, q}_{h, k} =
\binomalal{p}{h} \binomalal{q}{k}\, f^{p, q}_{h, k}$,
it is easily verified that we have the following recurrence
equations:
$$\sigma^{p, q}_{h, k} =
\cases{ \sigma^{p-1, q-1}_{h, k} + u_p\sigma^{p-1, q-1}_{h-1, k}
+ v_q\sigma^{p-1, q-1}_{h, k-1} + u_pv_q\sigma^{p-1, q-1}_{h-1, k-1} & if $1\leq h \leq p$
and $1 \leq k \leq q$,\cr
\sigma^{p, q-1}_{0, k} + v_q\sigma^{p, q-1}_{0, k-1} &
if $h = 0\leq p$
and $1 \leq k \leq q$,\cr
\sigma^{p-1, q}_{h, 0} + u_p\sigma^{p-1, q}_{h-1, 0} &
if $1\leq h \leq p$
and $k = 0 \leq q$,\cr
1 & if $h = k = 0$, $p \geq 0$, and $q\geq 0$;\cr
0 & otherwise.\cr
}$$
\medskip
Observe that the recurrence formula is a sort of generalization of
the Pascal triangle.
Alternatively, prove that $f^{p, q}_{h, k}$ can be computed directly
using the recurrence formula
$$f^{p, q}_{h, k} =
\frac{(p - h)(q - k)}{pq}\, f^{p-1, q-1}_{h, k} +
\frac{h(q-k)}{pq}u_p\, f^{p-1, q-1}_{h-1, k}
+ \frac{(p-h)k}{pq}v_q\, f^{p-1, q-1}_{h, k-1} +
\frac{hk}{pq}u_pv_q\, f^{p-1, q-1}_{h-1, k-1},$$
where $1\leq h \leq p$ and $1 \leq k \leq q$,
$$f^{p, q}_{0, k} =
\frac{(q - k)}{q}\, f^{p, q-1}_{0, k} +
\frac{k}{q} v_q\, f^{p, q-1}_{0, k-1},$$
where $h = 0 \leq p$ and $1 \leq k \leq q$,
and
$$f^{p, q}_{h, 0} =
\frac{(p - h)}{p}\, f^{p-1, q}_{h, 0} +
\frac{h}{p}u_p\, f^{p-1, q}_{h-1, 0},$$
where $1 \leq h \leq p$ and $k = 0 \leq q$.
As in section \ref{sec5}, we found that computing $\sigma^{p, q}_{h, k}$
and dividing by $\binomalal{p}{h} \binomalal{q}{k}$
is faster than computing $f^{p, q}_{h, k}$ directly.
\medskip
Given $(u_1,\ldots,u_{p}; v_1,\ldots,v_{q})$, using the recurrence equations,
computing all the scaled polar values
$$\sigma^{i, j}_{h, k}(u_1,\ldots, u_{i}; v_1,\ldots,v_{j}),$$
where $1\leq h \leq i$, $1 \leq k\leq j$,
$1\leq i \leq p$, and $1 \leq j\leq q$,
can be done in time $O(p^2q^2)$. The naive method using polarization
requires computing
$\sum_{i = 0}^{p}\sum_{j = 0}^{q} 2^{i + j} = 2^{p + q + 1} - 1$
terms.
\medskip
To compute the coordinates of control points, we combine the
$\sigma^{p, q}_{h, k}$. Specifically,
the coordinate value of the control point $b_{i, j}$ contributed by the polynomial
$\sum_{h = 0}^{m}\sum_{k = 0}^{m} a_{h, k} u^hv^k$
(where $m \leq p$ and $m \leq q$) is
$$\sum_{h = 0}^{m}\sum_{k = 0}^{m} a_{h, k}
\binomalal{p}{h}^{-1} \binomalal{q}{k}^{-1}
\sigma^{p, q}_{h, k}
(\underbrace{r_1,\ldots,r_1}_{p - i}, \underbrace{s_1,\ldots,s_1}_{i};
\underbrace{r_2,\ldots,r_2}_{q - j}, \underbrace{s_2,\ldots,s_2}_{j}),$$
where $(r_1, s_1)$ and $(r_2, s_2)$ are affine frames.
Our algorithm computes the table of values
$\sigma^{m,m}_{h, k}(u_1,\ldots, u_{m}; v_1,\ldots,v_{m})$,
and thus, it is very cheap to compute these sums.
\medskip
When the affine frames $(0, 1)$ are used,
the following theorem gives closed-form formulae for the polar values
with respect to the bidegree $(p, q)$.
\begin{thm}
If there are $s$ occurrences where $u = 0$, $r$ occurrences where $v = 0$,
and all the other occurrences of $u$ and $v$ have the value $1$, then
$$ f^{p, q}_{h,k}(u_{1},u_{2},\ldots,u_{p}; v_{1},v_{2},\ldots, v_{q})
=\frac{\binomalal{p-s}{h} \binomalal{q-r}{k}}{\binomalal{p}{h} \binomalal{q}{k}}$$.
\end{thm}
\noindent{\it Proof\/}.\enspace It can be shown by induction using the above
recurrence equations. $\mathchoice\sqr76\sqr76\sqr{2.1}3\sqr{1.5}3$
\medskip
It is well known that $F$ can be expressed
in terms of control points
and (products of) Bernstein polynomials
(see Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek}, or Piegl and Tiller \cite{Piegl}).
As in the case of curves,
the methods of
Lodha and Goldman \cite{LodhaGold}
have the same complexity as ours, but our method is more direct.
\section{Computing Triangular Control Nets}
\label{sec4}
As we saw in section \ref{sec2},
the polar form of the monomial $u^{h}v^{k}$ with respect to the total degree $m$ is
$$f^{m}_{h, k}((u_1, v_1),\ldots, (u_m, v_m))
=\frac{1}{\binomalal{m}{h} \binomalal{m-h}{k}}
\sum_{{I \cup J\subseteq\{1,\ldots,m\}}\atop{|I|=h,|J|= k, I \cap J=\emptyset}}
\left(\prod_{i \in I} u_{i}\right)\left(\prod_{j \in J}v_{j}\right).
$$
\medskip
Letting
$\sigma^{m}_{h, k} = \binomalal{m}{h} \binomalal{m-h}{k}\, f^{m}_{h,k}$,
it is easily verified that we have the following recurrence
equations:
$$\sigma^{m}_{h, k} =
\cases{ \sigma^{m-1}_{h, k} + u_m\sigma^{m-1}_{h-1, k}
+ v_m\sigma^{m-1}_{h, k-1} & if
$h, k\geq 0$ and $1\leq h + k\leq m$,\cr
1 & if $h = k = 0$ and $m \geq 0$;\cr
0 & otherwise.\cr
}$$
\medskip
The above formulae can be used to compute inductively the polar values
$$\frac{1}{\binomalal{m}{h} \binomalal{m-h}{k}}
\sigma^{m}_{h,k}((u_1, v_1),\ldots, (u_m, v_m)).$$
The computation consists in building a tetrahedron of values
reminiscent of the Pascal triangle (but $3$-dimensional).
Alternatively, we can compute $f^{m}_{h, k}$ directly using the recurrence formula
$$f^{m}_{h, k} = \frac{(m - h - k)}{m}\,f^{m-1}_{h, k} +
\frac{h}{m}u_m\, f^{m-1}_{h-1, k}
+ \frac{k}{m}v_m\, f^{m-1}_{h, k-1},$$
where $h, k\geq 0$ and $1\leq h + k\leq m$.
Again, we found that computing $\sigma^{m}_{h, k}$
and dividing by $\binomalal{m}{h} \binomalal{m-h}{k}$
is faster than computing $f^{m}_{h, k}$ directly.
\medskip
Given $((u_1, v_1),\ldots,(u_m, v_{m}))$, using the recurrence equations,
computing all the scaled polar values
$$\sigma^{i}_{h, k}((u_1, v_1),\ldots, (u_i, v_i)),$$
where $h, k\geq 0$, $1\leq h + k\leq i$, and $1\leq i \leq m$,
can be done in time $O(m^3)$.
The naive method using polarization
requires computing
$\sum_{i=0}^{m} 3^{i} = (3^{m + 1} - 1)/2$
terms.
To compute the coordinates of control points, we simply combine the
$\sigma^{m}_{h, k}$. Specifically,
the coordinate value of the control point $b_{i, j, k}$
(where $i + j + k = m$)
contributed by the polynomial
$\sum_{h + l \leq m} a_{h, l} u^hv^l$
(where $m\leq m$) is
$$\sum_{h + l \leq m} a_{h, l} \binomalal{m}{h\> l\> d}^{-1}
\sigma^{m}_{h, l}
(\underbrace{r,\ldots,r}_{i}, \underbrace{s,\ldots,s}_{j};
\underbrace{t,\ldots,t}_{k}),$$
where $d = m - k - l$ and
$r = (r_1, r_2)$, $s = (s_1, s_2)$ and $t = (t_1, t_2)$ are the
vertices of the affine frame.
Our algorithm computes the table of values
$\sigma^{i}_{h, l}((u_1, v_1),\ldots, (u_i, v_i))$,
and thus, it is very cheap to compute these sums.
\medskip
When the affine frame $((1, 0),\, (0, 1),\, (0, 0))$ is used,
the following theorem gives closed-form formulae for the polar values
with respect to the total degree $m$.
\begin{thm}
Assume that $ m=r\ +\ s\ +\ t$, with $r$ occurrences of $(1,0)$,
$s$ occurrences of $ (0,1)$, and $t$
occurrences of $(0,0)$ (and no occurrences of $(1,1)$). Then
$$f^{m}_{h,k}((u_{1},v_{1}),\ldots, (u_{m}, v_{m}))
=\frac{\binomalal{r}{h}\binomalal{s}{k}}{\binomalal{m}{h}\binomalal{m-h}{k}}$$.
\end{thm}
\noindent{\it Proof\/}.\enspace It can be shown by induction using the above
recurrence equations. $\mathchoice\sqr76\sqr76\sqr{2.1}3\sqr{1.5}3$
\medskip
As in the previous case,
it is well known that $F$ can be expressed
in terms of control points
and (trivariate) Bernstein polynomials
(see Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek}, or Piegl and Tiller \cite{Piegl}).
The methods of
Lodha and Goldman \cite{LodhaGold}
have the same complexity as ours, but our method is more direct
and very easy to implement.
\section{Examples}
\label{sec6}
We wrote an implementation in {\it Mathematica\/} of a program
computing polar values for curves, using the recurrence equations
of section \ref{sec5}. It works for an arbitrary
affine frame $(r, s)$.
\medskip
As an example, consider the curve of degree $10$ given by
$$\eqaligneno{
x &= \frac{4 t (1 - t^2)^2 (1 - 14 t^2 + t^4)}{(1 + t^2)^5},\cr
y &= \frac{8 t^2 (1 - t^2) (3 - 10 t^2 + 3 t^4)}{(1 + t^2)^5}.\cr
}$$
Using the above program, the following
control polygon w.r.t. $[0, 1]$ is obtained:
\begin{verbatim}
rcpoly = {{0, 0, 1}, {2/5, 0, 1}, {18/25, 12/25, 10/9}, {1/2, 6/5, 4/3},
{-14/45, 71/45, 12/7}, {-45/37, 45/37, 148/63}, {-71/45, 14/45, 24/7},
{-6/5, -1/2, 16/3}, {-12/25, -18/25, 80/9}, {0, -2/5, 16}, {0, 0, 32}};
\end{verbatim}
Note that the control points also contain weights, since there are
denominators (see Ramshaw \cite{Ramshaw87}, Farin \cite{Farin93,Farin95},
Hoschek and Lasser \cite{Hoschek},
or Gallier \cite{Gallbook}).
Here is the rational curve.
\begin{figure}[H]
\centerline{
\psfig{figure=ctpointsps/rose23.ps,width=4truein}}
\caption{A rose}
\end{figure}
We also wrote an implementation in {\it Mathematica\/} of a program
computing polar values for triangular surface
patches, using the recurrence equations
of section \ref{sec4}. This algorithm works for any affine frame
$(r, s, t)$.
\medskip
In Hilbert and Cohn-Vossen \cite{Hilbert}
(and also do Carmo \cite{DoCarmo76}), an interesting
map $\s{H}$ from $\reals^3$ to $\reals^4$ is defined as
$$(x, y, z) \mapsto (xy, yz, xz, x^2 - y^2).$$
This map has the remarkable property that when restricted to the sphere
$S^2$, we have $\s{H}(x, y, z) = \s{H}(x', y', z')$ iff
$(x', y', z') = (x, y, z)$ or
$(x', y', z') = (-x, -y, -z)$. In other words,
the inverse image of every point in $\s{H}(S^2)$ consists
of two antipodal points.
Thus, the map $\s{H}$ induces an injective map
from the projective plane onto $\s{H}(S^2)$, which is
obviously continuous, and since the projective plane is
compact, it is a homeomorphism.
Thus, the map $\s{H}$ allows us to realize concretely
the projective plane in $\reals^4$, by choosing any
parameterization of the sphere $S^2$, and
applying the map $\s{H}$ to it.
For example, the following parametric definition
specifies the entire projective plane over $[-1, 1]\times [-1, 1]$:
$$\eqaligneno{
x &= \frac{16uv^2(1 - u^2)}{(u^2 + 1)^2(v^2 + 1)^2},\cr
y &= \frac{8uv(u^2 + 1)(v^2 - 1)}{(u^2 + 1)^2(v^2 + 1)^2},\cr
z &= \frac{4v(1 - u^4)(v^2 - 1)}{(u^2 + 1)^2(v^2 + 1)^2},\cr
t &= \frac{4v^2(u^4 - 6u^2 + 1)}{(u^2 + 1)^2(v^2 + 1)^2}.\cr
}$$
Using our algorithm,
the following net of degree $8$ over the affine frame
$((1, 0 ,0)$, $(0, 1, 0)$, $(0, 0, 1))$ is obtained:
\begin{verbatim}
proj8net =
{{0, 0, 0, 0, 1}, {0, 0, -1/2, 0, 1}, {0, 0, -14/15, 2/15, 15/14},
{0, 0, -20/17, 6/17, 17/14}, {0, 0, -120/101, 60/101, 101/70},
{0, 0, -1, 4/5, 25/14}, {0, 0, -11/16, 15/16, 16/7}, {0, 0, -1/3, 1, 3},
{0, 0, 0, 1, 4}, {0, 0, 0, 0, 1}, {0, -1/7, -1/2, 0, 1},
{4/45, -4/15, -14/15, 2/15, 15/14}, {4/17, -28/85, -20/17, 6/17, 17/14},
{40/101, -32/101, -120/101, 60/101, 101/70},
{8/15, -6/25, -1, 4/5, 25/14}, {5/8, -1/8, -11/16, 15/16, 16/7},
{2/3, 0, -1/3, 1, 3}, {0, 0, 0, 0, 15/14}, {0, -4/15, -7/15, 0, 15/14},
{20/121, -60/121, -105/121, 9/121, 121/105},
{10/23, -14/23, -25/23, 9/46, 46/35},
{240/331, -192/331, -360/331, 108/331, 331/210},
{80/83, -36/83, -75/83, 36/83, 83/42}, {10/9, -2/9, -11/18, 1/2, 18/7},
{0, 0, 0, 0, 17/14}, {0, -32/85, -7/17, 0, 17/14},
{9/46, -16/23, -35/46, -1/46, 46/35},
{27/53, -89/106, -50/53, -3/53, 53/35},
{36/43, -100/129, -40/43, -4/43, 129/70},
{12/11, -6/11, -25/33, -4/33, 33/14}, {0, 0, 0, 0, 101/70},
{0, -48/101, -34/101, 0, 101/70},
{56/331, -288/331, -204/331, -40/331, 331/210},
{56/129, -44/43, -98/129, -40/129, 129/70},
{16/23, -144/161, -120/161, -80/161, 23/10}, {0, 0, 0, 0, 25/14},
{0, -14/25, -6/25, 0, 25/14}, {8/83, -84/83, -36/83, -16/83, 83/42},
{8/33, -38/33, -6/11, -16/33, 33/14}, {0, 0, 0, 0, 16/7},
{0, -5/8, -1/8, 0, 16/7}, {0, -10/9, -2/9, -2/9, 18/7}, {0, 0, 0, 0, 3},
{0, -2/3, 0, 0, 3}, {0, 0, 0, 0, 4}};
\end{verbatim}
\medskip
Note that the control points also contain weights, since there are
denominators.
If we project the real projective plane onto
a hyperplane in $\reals^4$, either from a center or parallel to a direction,
we can see a ``3D shadow'' of the real projective plane in $\reals^3$.
For example, one of the projections is the cross-cap, whose
control net is
\begin{verbatim}
projnet4 =
{{0, 0, 0, 1}, {0, 0, 0, 1}, {0, 0, 2/15, 15/14}, {0, 0, 6/17, 17/14},
{0, 0, 60/101, 101/70}, {0, 0, 4/5, 25/14}, {0, 0, 15/16, 16/7},
{0, 0, 1, 3}, {0, 0, 1, 4}, {0, 0, 0, 1}, {0, -1/7, 0, 1},
{4/45, -4/15, 2/15, 15/14}, {4/17, -28/85, 6/17, 17/14},
{40/101, -32/101, 60/101, 101/70}, {8/15, -6/25, 4/5, 25/14},
{5/8, -1/8, 15/16, 16/7}, {2/3, 0, 1, 3}, {0, 0, 0, 15/14},
{0, -4/15, 0, 15/14}, {20/121, -60/121, 9/121, 121/105},
{10/23, -14/23, 9/46, 46/35}, {240/331, -192/331, 108/331, 331/210},
{80/83, -36/83, 36/83, 83/42}, {10/9, -2/9, 1/2, 18/7}, {0, 0, 0, 17/14},
{0, -32/85, 0, 17/14}, {9/46, -16/23, -1/46, 46/35},
{27/53, -89/106, -3/53, 53/35}, {36/43, -100/129, -4/43, 129/70},
{12/11, -6/11, -4/33, 33/14}, {0, 0, 0, 101/70}, {0, -48/101, 0, 101/70},
{56/331, -288/331, -40/331, 331/210}, {56/129, -44/43, -40/129, 129/70},
{16/23, -144/161, -80/161, 23/10}, {0, 0, 0, 25/14},
{0, -14/25, 0, 25/14}, {8/83, -84/83, -16/83, 83/42},
{8/33, -38/33, -16/33, 33/14}, {0, 0, 0, 16/7}, {0, -5/8, 0, 16/7},
{0, -10/9, -2/9, 18/7}, {0, 0, 0, 3}, {0, -2/3, 0, 3}, {0, 0, 0, 4}}
\end{verbatim}
\medskip
The cross-cap is shown below.
\medskip
\begin{figure}[H]
\centerline{
\epsfig{figure=ctpointsps/crosscap2.eps,width=4truein}}
\caption{The cross-cap surface}
\end{figure}
\bigskip
{\bf Acknowledgement\/}.
I wish to thank Deepak Tolani for very helpful comments, in particular
on the use of Bernstein polynomials.
| {
"timestamp": "2006-06-13T02:47:36",
"yymm": "0606",
"arxiv_id": "cs/0606056",
"language": "en",
"url": "https://arxiv.org/abs/cs/0606056",
"abstract": "The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae w.r.t. arbitrary affine frames. As a corollary, it is amusing that we can also give closed-form expressions in the case of the frame (r, s) for curves, and the frame ((1, 0, 0), (0, 1, 0), (0, 0, 1) for surfaces. Our methods have the same low polynomial (time and space) complexity as the other best known algorithms, and are very easy to implement.",
"subjects": "Computational Complexity (cs.CC); Graphics (cs.GR)",
"title": "Fast and Simple Methods For Computing Control Points",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846659768267,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.7092019688488976
} |
https://arxiv.org/abs/1609.00670 | An EM based Iterative Method for Solving Large Sparse Linear Systems | We propose a novel iterative algorithm for solving a large sparse linear system. The method is based on the EM algorithm. If the system has a unique solution, the algorithm guarantees convergence with a geometric rate. Otherwise, convergence to a minimal Kullback--Leibler divergence point is guaranteed. The algorithm is easy to code and competitive with other iterative algorithms. | \section{Introduction} \label{sec:intro}
An important problem is to find a solution to a system of linear equations
\begin{equation}\label{eq:ls}
Ax = b,
\end{equation}
where $A = (a_{ij})$ is an $m_1 \times m_2$ matrix and $b$ is an $m_1$-dimensional vector.
We mainly consider the square matrix where $m_1 = m_2 = m$, but the theory and computations presented in the paper works for general $A$.
If $A$ is nonsingular with inverse matrix $A^{-1}$, there exists a unique solution to \eqref{eq:ls}, denoted by $x^* = A^{-1} b$.
When the dimension $m$ is large, however, finding the inverse matrix $A^{-1}$ is computationally unfeasible.
As alternatives, a number of iterative methods have been proposed to find a sequence $(x_n)$ approximating $x^*$, and they are often implementable when $A$ is sparse, that is, most $a_{ij}$'s are zero.
For reviews of these iterative methods within a unified framework, we refer to the monograph \cite{saad2003iterative}.
Many sources of such large sparse linear systems come from the discretization of a partial differential equation; see Chapter 2 of \cite{saad2003iterative}.
Within statistical applications, sparse design matrices have been considered in \cite{kennedy1980gentle, koenker2003sparsem} and an algorithm sampling high-dimensional Gaussian random variables with sparse precision matrices has been developed in \cite{aune2013iterative}.
We start with a brief introduction of the most widely used iterative methods for solving \eqref{eq:ls}.
Current iterative methods are coordinate-wise updating algorithms.
Two of the most well-known methods are \emph{Jacobi} and \emph{Gauss--Seidel} which can be found in most standard textbooks.
Given $x_n = (x_{n,j})$, the Jacobi and Gauss--Seidel methods update $x_{n+1}$ via
\setlength\arraycolsep{2pt}\begin{eqnarray*}
x_{n+1,j} &=& \frac{1}{a_{jj}} \left(b_j - \sum_{i\neq j} a_{ji} \,x_{n,i} \right)
\end{eqnarray*}
and
\setlength\arraycolsep{2pt}\begin{eqnarray*}
x_{n+1,j} &=& \frac{1}{a_{jj}} \left(b_j - \sum_{i=1}^{j-1} a_{ji} \, x_{n+1,i} - \sum_{i=j+1}^{m} a_{ji} \,x_{n,i}\right),
\end{eqnarray*}
respectively.
Although they are simple and convenient, both of them are restrictive in practice because $(x_n)$ is not generally guaranteed to converge to $x^*$; see Chapter 4 of \cite{saad2003iterative}.
The \emph{Krylov subspace methods}, which are based on the \emph{Krylov subspace} of ${\mathbb R}^m$,
$$
\mathcal K_n = {\rm span} \Big\{r_0, Ar_0, A^2 r_0, \ldots, A^{n-1}r_0 \Big\},
$$
are the dominant approaches, where $x_0$ is an initial guess and $r_0 = b-Ax_0$.
Under the assumption that $A$ is sparse, matrix-vector multiplication is cheap to compute, so it is not difficult to handle $\mathcal K_n$ even when $m$ is very large.
If $A$ is symmetric and positive definite (SPD), the standard choice for solving \eqref{eq:ls} is the \emph{conjugate gradient method} (CG; \cite{hestenes1952methods}).
This is an \emph{orthogonal projection method}, see Chapter 5 of \cite{saad2003iterative}, onto $\mathcal K_n$, finding $x_n \in x_0 + \mathcal K_n$ such that $b - A x_n \perp \mathcal K_n$.
To be more specific, recall that two vectors $u,v\in{\mathbb R}^m$ are called \emph{$A$-conjugate} if $u^T A v = 0$.
If $A$ is symmetric and positive definite, then this quadratic form defines an inner product, and there is a basis for ${\mathbb R}^m$ consisting of mutually $A$-conjugate vectors.
The CG method sequentially generates mutually $A$-conjugate vectors $p_1, p_2, \ldots$, and approximates $x^* = \sum_{j=1}^m \alpha_j p_j$ as $x_n = \sum_{j=1}^n \alpha_j p_j$, where $\alpha_j = p_j^T b / p_j^T A p_j$.
Using the symmetry of $A$, the computation can be simplified as in Algorithm \ref{alg:cg}.
Here $\|\cdot\|_q$ denotes the $\ell_q$-norm on ${\mathbb R}^m$.
\begin{algorithm}
\caption{Conjugate gradient method for SPD $A$} \label{alg:cg}
\begin{algorithmic}[1]
\State {\bf Input}: $A, b, x_0$ and $\epsilon_{\rm tol} > 0$
\State $j\gets 0$
\State $r_0 \gets b - A x_0$
\State $p_0 \gets r_0$
\While{$\|r_j\|_2 > \epsilon_{\rm tol}$}
\State $\alpha_j \gets r_j^T r_j / p_j^T Ap_j$ \label{step:cg1}
\State $x_{j+1} \gets x_j + \alpha_j p_j$
\State $r_{j+1} \gets r_j - \alpha_j Ap_j$
\State $\beta_j \gets r_{j+1}^T r_{j+1} / r_j^T r_j$
\State $p_{j+1} \gets r_{j+1} + \beta_j p_j$
\State $j \gets j+1$\label{step:cg2}
\EndWhile
\State \Return $x_j$
\end{algorithmic}
\end{algorithm}
For a general matrix $A$, the \emph{generalized minimal residual method} (GMRES; \cite{saad1986gmres}) is the most popular.
It is an \emph{oblique projection method}, see Chapter 5 of \cite{saad2003iterative}, which finds $x_k \in \mathcal K_k$ satisfying $b - A x_k \perp A\mathcal K_k$, where $A\mathcal K_k = \{Av: v\in\mathcal K_k\}$.
When implementing GMRES, \emph{Arnoldi's method} \cite{arnoldi1951principle} is applied for computing an orthonormal basis of $\mathcal K_k$.
The method can be written as in Algorithm \ref{alg:gmres}.
For a given initial $x_0$, let us write the result of Algorithm \ref{alg:gmres} as $G_k(x_0)$.
Since the computational cost of Algorithm \ref{alg:gmres} is prohibitive for large $k$, a restart version of GMRES$(k)$, defined as $x_{n+1} = G_k(x_n)$, is applied with small $k$.
It should be noted that the generalized conjugate residual (GCR; \cite{elman1982iterative}), ORTHODIR \cite{young1980generalized} and Axelsson's method \cite{axelsson1980conjugate} are mathematically equivalent to GMRES; but it is known in \cite{saad1986gmres} that GMRES is computationally more efficient and reliable.
Further connections between these methods are discussed in \cite{saad1985conjugate}.
Convergence is guaranteed, but there are restrictions; see Section \ref{sec:comparison}.
\begin{algorithm}
\caption{GMRES$(k)$} \label{alg:gmres}
\begin{algorithmic}[1]
\State {\bf Input}: $A, b, x_0$ and $\epsilon_{\rm tol} > 0$
\State $\beta \gets \|b - A x_0\|_2$
\State $v_1 \gets (b-Ax_0)/\beta$
\For{$j=1, \ldots, k$} \label{step:gmres0}
\State $w_j \gets Av_j$ \label{step:gmres1}
\For {$i=1, \ldots, j$}
\State $h_{ij} \gets w_j^T v_i$ \label{step:gmres2}
\State $w_j \gets w_j - h_{ij} v_i$ \label{step:gmres3}
\EndFor
\State $h_{j+1,j} \gets \|w_j\|_2$ \label{step:gmres4}
\If {$h_{j+1,j} < \epsilon_{\rm tol}$ } set $k\gets j$ and {\bf break}\EndIf
\State $v_{j+1} \gets w_j / h_{j+1,j}$ \label{step:gmres5}
\EndFor
\State $y_k \gets \argmin_y \|\beta e_1 - H_k y\|_2$, where $H_k = (h_{ij})_{i \leq k+1, j \leq k}$ and $e_1 = (1, 0, \ldots, 0)^T$ \label{step:gmres6}
\State $x_k = x_0 + V_k y_k$, where $V_k = (v_1, \ldots, v_k) \in {\mathbb R}^{m\times k}$ \label{step:gmres7}
\State \Return $x_k$
\end{algorithmic}
\end{algorithm}
The \emph{minimum residual method} (MINRES; \cite{paige1975solution}) can be understood as a special case of GMRES when $A$ is a symmetric matrix.
In this case, Arnoldi's method (steps \ref{step:gmres0}-\ref{step:gmres5}) in Algorithm \ref{alg:gmres} can be replaced by the simpler \emph{Lanczos algorithm} \cite{lanczos1950iteration}, described in Algorithm \ref{alg:lanczos}, where $\alpha_j = h_{jj}$ and $\beta_j = h_{j-1, j}$.
\begin{algorithm}
\caption{Lanczos algorithm} \label{alg:lanczos}
\begin{algorithmic}[1]
\State $\beta_1 \gets 0$
\State $v_0 \gets 0$
\For{$j=1, \ldots, k$}
\State $w_j \gets A v_j - \beta_j v_{j-1}$
\State $\alpha_j \gets w_j^T v_j$
\State $w_j \gets w_j - \alpha_j v_j$
\State $\beta_{j+1} \gets \|w_j\|_2$
\If {$\beta_{j+1} < \epsilon_{\rm tol}$ } set $k\gets j$ and {\bf break}\EndIf
\State $v_{j+1} \gets w_j / \beta_{j+1}$
\EndFor
\end{algorithmic}
\end{algorithm}
In summary, standard approaches for solving \eqref{eq:ls} are (i) CG for SPD $A$; (ii) MINRES for symmetric $A$; and (iii) GMRES for general $A$.
However, convergence is not guaranteed for GMRES.
As an alternative, one can solve the normal equation
\begin{equation} \label{eq:normal}
A^T Ax = A^T b,
\end{equation}
where iterative algorithms guarantee convergence.
However, this approach is often avoided in practice because the matrix $A^T A$ is less well conditioned than the original $A$; see Chapter 8 of \cite{saad2003iterative}.
There are a large number of other general approaches, and many of them are variations and extensions of Krylov subspace methods.
Each method has some appealing properties, but it is difficult in general to analyze them theoretically.
See Chapter 7 of \cite{saad2003iterative}.
Also, there are some algorithms which are devised to solve a structured linear system \cite{ho2012fast, golub2003solving, bostan2008solving}.
To the best of our knowledge, however, there is no efficient iterative algorithm that can solve an arbitrary sparse linear system.
In particular, the most popular, GMRES, often has quite strange convergence properties, see \cite{embree2003tortoise} and \cite{greenbaum1996any}, making the algorithm difficult to use in practice.
In this paper, we propose an iterative method which guarantees convergence for an arbitrary linear system.
Under the assumption that $A, b$ and $x^*$ are nonnegative, the basic algorithm is known in \cite{vardi1993image} as an EM algorithm with an infinite number of observations.
Although the EM algorithm satisfies certain monotonicity criteria, see \cite{dempster1977maximum}, a detailed convergence analysis is omitted in \cite{vardi1993image}.
Independently from \cite{vardi1993image}, \citet{walker2017iterative} studied the same algorithm viewing it as a Bayesian updating algorithm and provided the proof for convergence.
The innovation of this paper is to extend the algorithm to general linear systems where $A, b$ and $x^*$ are not necessarily nonnegative, and to provide more detailed convergence analysis.
In particular, our convergence results include inconsistent systems, {\it i.e.}\ the linear system \eqref{eq:ls} has no solution.
In this case, it is shown that $(x_n)$ converges to a certain minimal Kullback--Leibler divergence point.
The algorithm is easy to implement and requires small storage.
The proposed algorithm can serve as a suitable alternative to the Krylov subspace methods.
The new algorithm and its theoretical properties are studied in Section \ref{sec:main}.
A comparison to existing methods is provided in Section \ref{sec:comparison} and concluding remarks are given in Section \ref{sec:discussion}.
\subsection*{Notation}
Every vector such as $b$ and $x$ are column vectors, and components are denoted with a subscript, {\it e.g.}\ $b=(b_i)$.
Dots in subscripts present the summation in those indices, {\it i.e.}\ $a_{\cdot j} = \sum_{i=1}^m a_{ij}$.
The $j$th column of $A$ is denoted by $a^{(j)}$.
For $X$, which may be a vector or a matrix, is said to be nonnegative (positive, resp.) and denoted $X \geq 0$ ($x > 0$, resp.) if each component of $X$ is nonnegative (positive, resp.).
The number of nonzero elements of $X$ is denoted $\mathcal N_X$.
\section{An iterative algorithm with guaranteed convergence}
\label{sec:main}
\subsection{Algorithm for solving nonnegative systems}
Assume that $A$ is a nonsingular square matrix and $A, b$ and $x^*$ are nonnegative.
In this case, \citet{vardi1993image} and \citet{walker2017iterative} proposed the iterative algorithm
\begin{equation}\label{eq:bayes-update}
x_{n+1,j} = \frac{x_{n,j}}{a_{\cdot j}} \sum_{i} a_{ij} \frac{b_i}{b_{n,i}}, \quad n\geq 0,
\end{equation}
where $b_n = (b_{n,i}) = Ax_n$ and $x_0 \geq 0$ is an initial guess.
To briefly introduce the main idea, assume that $b,x$ and $a^{(j)}$ are probability vectors, {\it i.e.}\ a vector with non-negative entries that sum to one.
Now consider discrete random variables $I$ and $J$ whose joint distribution is given by
$$
\mathbb{P}(J=j) = x_j^* \quad {\rm and} \quad \mathbb{P}(I=i | J=j) = a_{ij}.
$$
Then, the marginal probability of $I$ is
$$
\mathbb{P}(I=i) = \sum_j \mathbb{P}(I=i|J=j) \mathbb{P}(J=j) = \sum_j x_j^* a_{ij} = b_i.
$$
Note that
\begin{equation}\label{eq:bayes-prob}
\mathbb{P}(J=j|I=i) = \frac{\mathbb{P}(J=j) \mathbb{P}(I=i|J=j)}{\sum_{j^\prime}\mathbb{P}(J=j^\prime) \mathbb{P}(I=i|J=j^\prime)} = \frac{x_j^* a_{ij}}{\sum_{j^\prime} a_{ij^\prime} x_{j^\prime}^*}
\end{equation}
by Bayes theorem.
\citet{vardi1993image} constructed the iteration \eqref{eq:bayes-update} through an EM algorithm.
With known $A$ and $b$, consider the problem of estimating $x^*$ based on the observation $I_1, \ldots, I_N$, where $(I_k, J_k)_{1 \leq k \leq N}$ are i.i.d. copies of $(I,J)$.
Since we do not directly observe $J_1, \ldots, J_N$, a standard method to find a maximum likelihood estimator is the EM algorithm.
Let $N_{ij}$ be the number of $k$'s such that $(I_k, J_k) = (i,j)$.
Then, the complete log-likelihood is
$$
L^c(x) = \sum_{i,j} N_{ij} \log a_{ij} + \sum_j N_{\cdot j} \log x_j,
$$
so we have
$$
Q(x|x_n) \stackrel{\rm def}{=} {\mathbb E}_{x_n}[L^c(x) | I_1, \ldots, I_k]
= C + \sum_j {\mathbb E}_{x_n}[N_{\cdot j}|I_1, \ldots, I_n] \log x_j,
$$
where $C$ does not depend on $x$.
Thus, the EM iteration $x_{n+1} = \argmax_x Q(x|x_n)$ is given as
$$
x_{n+1,j} = \frac{{\mathbb E}_{x_n}[N_{\cdot j}|I_1, \ldots, I_n]}{\sum_{j'}{\mathbb E}_{x_n}[N_{\cdot j'}|I_1, \ldots, I_n]}.
$$
Since
$$
{\mathbb E}_{x_n}[N_{ij}|I_1, \ldots, I_n] =
\frac{x_{n,j} a_{ij}}{\sum_{j^\prime} a_{ij^\prime} x_{n, j^\prime}} N_{i\cdot}
$$
by \eqref{eq:bayes-prob}, we have
$$
x_{n+1,j} = x_{n,j} \sum_i \frac{a_{ij}}{\sum_{j^\prime} a_{ij^\prime} x_{j^\prime}} \frac{N_{i\cdot}}{N_{\cdot\cdot}}.
$$
Note that $N_{i\cdot}/N_{\cdot\cdot} \rightarrow b_i$ almost surely as $N \rightarrow \infty$, reducing the iteration \eqref{eq:bayes-update}.
Therefore, \eqref{eq:bayes-update} can be interpreted as an EM algorithm with infinite number of observations.
\citet{walker2017iterative} viewed the iteration \eqref{eq:bayes-update} as Bayesian updating.
Given a prior $x_n$ and an observation $I$, the posterior update of $x_{n,j}$ is given by
$$
\frac{x_{n,j} a_{Ij}}{\sum_{j'} a_{Ij'} x_{n,j'}}
$$
by \eqref{eq:bayes-prob}.
Since we do not have data, a natural choice is to use average update
$$
x_{n+1,j} = \sum_i \frac{x_{n,j} a_{ij}}{\sum_{j'} a_{ij'} x_{n,j'}} b_i
$$
which is exactly \eqref{eq:bayes-update}.
The update \eqref{eq:bayes-update} can also be understood as a fixed-point iteration.
From the identity
\setlength\arraycolsep{2pt}\begin{eqnarray*}
x_j^* = \mathbb{P}(J=j) = \sum_i \mathbb{P}(J=j | I=i) \mathbb{P}(I=i) = x_j^* \sum_i \frac{a_{ij} b_i}{\sum_{j^\prime} a_{ij^\prime} x_{j^\prime}^*},
\end{eqnarray*}
we consider an equation $\phi(x) = x$, where
$$\phi_j(x)=\sum_i \frac{a_{ij}b_i }{\sum_{j^\prime} a_{ij^\prime} x_{j^\prime}}$$
and $\phi(x)$ is the corresponding vector.
Then, it is not difficult to see that $x=x^*$ if and only if $\phi_j(x)=1$ for every $j$.
Thus, if the recursive update
$$x_{n+1}=x_n\circ \phi(x_n),$$
where $\circ$ denotes elementwise product, converges, it does so to $x^*$.
If $A,b,x \geq 0$ but some of $b,x$ and $a^{(j)}$'s are not probability vectors, we can easily rescale the problem as
\begin{equation}\label{eq:reformula}
\widetilde A \widetilde x = \widetilde b
\end{equation}
with the update \eqref{eq:bayes-update}, where $\widetilde A = (a_{ij}/a_{\cdot j})_{i,j\leq m}$, $\widetilde x = (x_j a_{\cdot j} / b_\cdot)_{j=1}^m$ and $\widetilde b = (b_i / b_\cdot)_{i=1}^m$.
Theorem \ref{thm:bayes-rate} assures the convergence of the update \eqref{eq:bayes-update} with geometric rate.
We need well-known bounds for probability metrics for the proof.
For $m$-dimensional vectors $u, v \geq 0$, define the \emph{Kullback--Leibler (KL) divergence} $D(u,v) = \sum_{i=1}^m u_i \log(u_i / v_i)$ and \emph{total variation} $V(u,v) = \sum_{i=1}^m |u_i - v_i|$.
In the definition of the KL divergence, we let $u_i \log(u_i/v_i)=0$ if $u_i=0$ and $D(u,v) = \infty$ if $u_i>0$ and $v_i=0$ for some $i$.
It is well-known that $D(u,v) \geq 0$ for every pair of probability vectors $(u,v)$, and equality holds if and only if $u=v$.
Let $\|\cdot\|_1$ denotes the $\ell_1$-operator norm ({\it i.e.}\ maximum absolute column sum) of a matrix.
\begin{theorem}\label{thm:bayes-rate}
Assume that $A \geq 0$, $x^*, b > 0, x_0 > 0$ and $A$ is nonsingular.
Then, for $(x_n)$ defined by \eqref{eq:bayes-update}, there exists $N$ such that
$D(\widetilde x^*, \widetilde x_{n+1}) \leq (1-\delta) D(\widetilde x^*, \widetilde x_n)$ for all $n \geq N$, where $\widetilde x^* = (x^*_j a_{\cdot j}/b_\cdot)_{j=1}^m$ and $\widetilde x_n = (x_{n,j} a_{\cdot j}/b_\cdot)_{j=1}^m$ and
$$
\delta = \frac{1}{3\|A^{-1}\|_1^2} \min_{1\leq j\leq m} \widetilde x^*_j.
$$
\end{theorem}
\begin{proof}
If some of $b, x^*$ and $a^{(j)}$'s are not probability vectors, we can reformulate the problem using \eqref{eq:reformula}.
Therefore, we may assume without loss of generality that $b, x^*$ and $a^{(j)}$'s are probability vectors.
For any $x_0 > 0$, it is easy to see that $x_n > 0$ and $\sum_{j=1}^m x_{n,j} = 1$ for every $n\geq 1$.
Thus, $b_n$ and $x_n$ are also probability vectors for every $n\geq 1$.
From \eqref{eq:bayes-update} we have
\setlength\arraycolsep{2pt}\begin{eqnarray*}
\log x_{n+1,j} = \log x_{n,j} + \log \sum_{i=1}^m \left( \frac{b_i}{b_{n,i}} a_{ij} \right)
\geq \log x_{n,j} + \sum_{i=1}^m a_{ij} \log \left( \frac{b_i}{b_{n,i}} \right),
\end{eqnarray*}
where the inequality holds by Jensen.
Therefore,
$$
\sum_{j=1}^m x_j^* \log x_{n+1,j} \geq \sum_{j=1}^m x_j^* \log x_{n,j} + D(b, b_n).
$$
This implies that
\begin{equation} \label{eq:KL-inequality}
D(x^*, x_{n+1}) \leq D(x^*, x_n) - D(b, b_n),
\end{equation}
and $D(x^* x_n)$ converges, by the monotone convergence theorem.
Thus, $D(b, b_n) \rightarrow 0$, which in turn implies that $x_n \rightarrow x^*$.
Note that
$$
V(x^*, x_n) = V(A^{-1}b, A^{-1}b_n) \leq \|A^{-1}\|_1 V(b, b_n),
$$
where $\|\cdot\|_1$ denotes the $\ell_1$-operator norm ({\it i.e.} maximum absolute column sum) of the matrix.
Therefore,
$$
D(b, b_n) \geq \hbox{$1\over2$} V^2(b, b_n) \geq \frac{1}{2\|A^{-1}\|_1^2} V^2(x^*, x_n),
$$
where the first inequality holds by Pinsker's inequality (\cite{pinsker1964information, csiszar2011information}).
Since
\setlength\arraycolsep{2pt}\begin{eqnarray*}
D(x^*, x_n) &=& \sum_{j=1}^m x^*_j \log \frac{x^*_j}{x_{n,j}} \leq \sum_{j=1}^m x^*_j \left( \frac{x^*_j}{x_{n,j}} - 1\right)
= \sum_{j=1}^m \left(1 + \frac{x^*_j - x_{n,j}}{x_{n,j}} \right) (x^*_j - x_{n,j})
\\
&=& \sum_{j=1}^m \frac{(x^*_j - x_{n,j})^2}{x_{n,j}}
\leq \left(\sum_{j=1}^m \frac{|x^*_j - x_{n,j}|}{\sqrt{x_{n,j}}} \right)^2
\leq V^2(x^*, x_n) \max_{1 \leq j \leq m} x_{n,j}^{-1}
\end{eqnarray*}
and $x_n \rightarrow x^*$, we have $D(b, b_n) \geq \delta D(x^*, x_n)$ for all large enough $n$, where
$$
\delta = \frac{1}{3\|A^{-1}\|_1^2} \min_{1\leq j\leq m} x^*_j.
$$
Therefore, by \eqref{eq:KL-inequality},
$$
\delta D(x^*, x_n) \leq D(b, b_n) \leq D(x^*, x_n) - D(x^*, x_{n+1})
$$
for all large enough $n$.
It follows that $D(x^*, x_{n+1}) \leq (1-\delta) D(x^*, x_n)$ for all large $n$.
\end{proof}
\bigskip
Note that for any nonnegative vectors $p$ and $q$ with the same $\ell_1$-norm, the Kullback--Leibler divergence and the Euclidean norm are related as
$$
\frac{1}{\|p\|_1}\sum_{j} p_j\log (p_j/q_j) \geq \frac{\|p-q \|_1^2}{2 \|p\|_1^2} \geq \frac{\|p-q \|_2^2}{2 \|p\|_1^2}.
$$
Thus, $\|x_n - x^*\|_2^2 \leq 2 \|x^*\|_1^2 D(\widetilde x^*, \widetilde x_n)$.
The key to the proof of Theorem \ref{thm:bayes-rate} is inequality \eqref{eq:KL-inequality}.
This inequality implies that the larger $D(b, b_n)$ is the larger we gain at the $n$th iteration.
It should be noted that $x^*, b > 0$ is essential for the convergence of the algorithm.
When $b_i \leq 0$ for some $i$, we can easily reformulate the problem as
\begin{equation}\label{eq:reformula2}
A x_t = b_t,
\end{equation}
where $x_t = x + t{\bf 1}_m, b_t = b + tA{\bf 1}_m, {\bf 1}_m = (1, \ldots, 1)^T$ and $t>0$ is a constant such that $b_t > 0$.
Note that $A {\bf 1}_m > 0$ because $A$ is nonsingular and nonnegative.
Note also that $A\geq 0$ and $b > 0$ does not imply that $x \geq 0$.
If $t$ is large enough, however, we have $x^* + t{\bf 1}_m > 0$, leading to Algorithm \ref{alg:bayes} which guarantees the convergence for any $A \geq 0$ and $b > 0$.
We call this algorithm as the \emph{nonnegative algorithm (NNA)}.
As seen in Section \ref{ssec:illustration}, $t$ can be chosen as a very large constant without being detrimental to the algorithm.
\begin{algorithm}
\caption{Nonnegative algorithm for $A \geq 0$ and $b > 0$} \label{alg:bayes}
\begin{algorithmic}[1]
\State {\bf Input}: $A, b, x_0, \epsilon_{\rm tol}$ and $t >0$
\State $b \gets b + t A{\bf 1}_m$
\State $x_0 \gets x_0 + t{\bf 1}_m$
\State $n \gets 0$
\While {$\|Ax_n - b\|_2 > \epsilon_{\rm tol}$ }
\State $b_n \gets Ax_n$
\State $c_n \gets b/b_n$ (componentwise division)
\State $d_n = \widetilde{A}^T c_n$
\State $x_{n+1} = d_n \circ x_n$ (componentwise multiplication)
\State $n \gets n+1$
\EndWhile
\State $x_n \gets x_n - t{\bf 1}_m$
\State \Return $x_n$
\end{algorithmic}
\end{algorithm}
Here we consider the computational complexity of Algorithm \ref{alg:bayes}.
In \eqref{eq:bayes-update}, we first need to compute $b_n = Ax_n$, and then compute $c_n = b/b_n$, where $/$ represents componentwise division.
Finally, we compute $x_{n+1} = (\widetilde A^T c_n) \circ x_n$, where $\widetilde A = (a_{ij} / a_{\cdot j})_{i,j\leq m}$.
In summary, we need two matrix-vector multiplications and two vector-vector componentwise operations.
Assume that the sparsity structure of $A$ is known and $\mathcal N_A \geq m$.
Then, the number of flops (floating-point operations; addition, subtraction, multiplication, or division) for matrix multiplication is less than $2 \mathcal N_A$.
Also, for a vector-vector multiplication (or division), $2m$ flops are required.
Therefore, the total number of flops for one iteration of \eqref{eq:bayes-update} is less than $4(\mathcal N_A + m)$.
We compare the number of flops with other algorithms in Section \ref{sec:comparison}.
We can apply Algorithm \ref{alg:bayes} for any linear system even when $A$ is not invertible or no solution exists.
For the remainder of this subsection, we assume that $A \in {\mathbb R}^{m_1\times m_2}$, $b \in {\mathbb R}^{m_2}$, $x \in {\mathbb R}^{m_1}$.
We first consider the case that a solution $x^*$ exists.
Since a solution may not be unique, it is not guaranteed that $x_n \rightarrow x^*$.
Theorem \ref{thm:consistent-system} assures the convergence of $b_n$ to $b$ with an upper bound of order $O(1/\epsilon)$ for the number of iterations to achieve $D(b_n, b) \leq \epsilon$.
\begin{theorem}\label{thm:consistent-system}
Assume that $A \geq 0$, $x^*, b > 0$ and $Ax^* = b$.
For any $x_0 > 0$, the sequence $(x_n)$ defined as \eqref{eq:bayes-update} satisfies $D(b, b_n) \rightarrow 0$.
In particular, for every $\epsilon > 0$ there exists $N \leq D(x^*, x_1) / \epsilon + 1$ such that $D(b_N, b) \leq \epsilon$.
\end{theorem}
\begin{proof}
As in the proof of Theorem \ref{thm:bayes-rate}, we may assume that $x^*, b$ and $a^{(j)}$, $1 \leq j \leq m_2$ are probability vectors without loss of generality.
Then, $b_n$ and $x_n$ are probability vectors for every $n \geq 1$, so the inequality \eqref{eq:KL-inequality} holds in the same way.
Thus, $D(x^*, x_n)$ converges by the monotone convergence theorem and
it follows that $D(b, b_n) \rightarrow 0$.
For a given $\epsilon > 0$, let $N$ be the largest integer less than or equal to $D(x^*, x_1) / \epsilon + 1$ and assume that $D(b, b_n) > \epsilon$ for every $n \leq N$.
Then, since
$$
0 \leq D(x^*, x_{N+1}) \leq D(x^*, x_1) - \sum_{n=1}^N D(b, b_n),
$$
using \eqref{eq:KL-inequality}, we have $N < D(x^*, x_1)/\epsilon$.
This makes a contradiction and completes the proof.
\end{proof}
Assume that the linear system \eqref{eq:ls} do not have a solution.
In this case, the iteration \eqref{eq:bayes-update} converges to a minimal KL divergence points as Theorem \ref{thm:am}.
For the proof, we view the iteration \eqref{eq:bayes-update} as an alternating minimization for which powerful tools have been developed in \cite{csiszar1984information} to study its convergence.
\begin{theorem} \label{thm:am}
Assume that $A \geq 0$ and $b > 0$.
For any $x_0 > 0$, the sequence $(x_n)$ defined as \eqref{eq:bayes-update} satisfies $\lim_n D(b, Ax_n) \downarrow \inf_x D(b, Ax)$, where $x$ ranges over every positive vector with $\sum_j x_j a_{\cdot j} = b_\cdot$.
\end{theorem}
\begin{proof}
Without loss of generality, we may assume that $b$ and $a^{(j)}$, $1 \leq j \leq m_2$ are probability vectors.
Let $\mathcal P$ and $\mathcal Q$ be the set of every bivariate probability mass functions $(i,j) \mapsto p(i,j)$ and $(i,j) \mapsto q(i,j)$ such that $\sum_j p(i,j) = b_i$ and $q(i,j) = a_{ij} x_j$ for some probability vector $x$, respectively.
Then, it is obvious that $\mathcal P$ and $\mathcal Q$ are convex.
Let
\setlength\arraycolsep{2pt}\begin{eqnarray*}
q_n(i,j) = x_{n,j} a_{ij}
\quad {\rm and} \quad
p_n(i,j) = \frac{b_i q_n(i,j)}{\sum_{j'} q_n(i,j')}.
\end{eqnarray*}
Then, $D(p_n, q_n) = D(b, b_n) \leq D(p, q_n)$ for every $p \in \mathcal P$, where the inequality holds because $b$ and $b_n$ are marginal probabilities of $p$ and $q_n$.
Thus, $p_n = \argmin_{p \in \mathcal P} D(p, q_n)$.
For a probability vector $x$ let $q(i,j) = a_{ij} x_j$.
Then,
\setlength\arraycolsep{2pt}\begin{eqnarray*}
D(p_n, q) = \sum_{i,j} p_n(i,j) \log \frac{p_n(i,j)}{q(i,j)} = C_1 - \sum_{i,j} p_n(i,j) \log x_j
\\
= C_2 - \sum_j p_n(\cdot, j) \log \frac{x_j}{p_n(\cdot,j)},
\end{eqnarray*}
where $p_n(\cdot, j) = \sum_i p_n(i,j)$ and $C_j$'s are terms independent of $x$.
Since
\setlength\arraycolsep{2pt}\begin{eqnarray*}
p_n(\cdot, j) = \sum_i \frac{b_i x_{n,j} a_{ij}}{\sum_{j'} x_{n,j'} a_{ij'}}
= x_{n,j} \sum_i a_{ij} \frac{b_i}{b_{n,i}} = x_{n+1,j},
\end{eqnarray*}
we have $D(p_n, q) = C_2 + D(x_{n+1}, x)$.
It follows that $q_{n+1} = \argmin_{q\in\mathcal Q} D(p_n, q)$.
In summary, the sequences $(p_n)$ and $(q_n)$ are obtained by alternating minimization.
By Theorem 3 of \cite{csiszar1984information}, $D(p_n, q_n) \downarrow \inf_{p\in\mathcal P, q\in\mathcal Q} D(p, q)$.
Note that when
\setlength\arraycolsep{2pt}\begin{eqnarray*}
q(i,j) = a_{ij} x_j
\quad {\rm and} \quad
p(i,j) = \frac{b_i q(i,j)}{\sum_{j'} q(i,j')},
\end{eqnarray*}
we have $D(p,q) = D(b, Ax)$.
Therefore, $D(p_n, q_n) = D(b, b_n) \downarrow \inf_x D(b, Ax)$.
\end{proof}
\subsection{General linear systems}
For convenience, we only consider a square matrix $A$, but the approach introduced in this subsection can also be applied to any linear system.
The main idea is to embed the original system \eqref{eq:ls} into a larger nonnegative system, and then apply Algorithm \ref{alg:bayes}.
This kind of slack variable techniques are well-known in linear algebra and optimization.
The enlarged system should be minimal to reduce any additional computational burden.
As an illustrative example, consider the system of linear equations
\setlength\arraycolsep{2pt}\begin{eqnarray*}
\left.\begin{array}{ccccccc}
a_{11} x_1 &-& a_{12} x_2 &+& a_{13} x_3 &=& b_1,
\\
a_{21} x_1 &+& a_{22} x_2 &-& a_{23} x_3 &=& b_2,
\\
a_{31} x_1 &+& a_{32} x_2 &+& a_{33} x_3 &=& b_3,
\end{array}\right.
\end{eqnarray*}
where $a_{ij} \geq 0$ for every $i$ and $j$, so $A$ has negative elements.
We consider two more equations
\setlength\arraycolsep{2pt}\begin{eqnarray*}
x_2 + x_4 = 0 \quad {\rm and} \quad x_3 + x_5 = 0,
\end{eqnarray*}
where each equation contains only two nonzero elements.
Then, it is easy to see that solving the linear system consisting of the above five equations is equivalent to solving the following five equations:
\begin{equation}\label{eq:ls3}
\left.\begin{array}{rrrrrrrrrrr}
a_{11} x_1 & & &+& a_{13} x_3 &+& a_{12} x_4 & & &=& b_1,
\\
a_{21} x_1 &+& a_{22} x_2 & & & & &+& a_{23} x_5 &=& b_2,
\\
a_{31} x_1 &+& a_{32} x_2 &+& a_{33} x_3 & & & & &=& b_3,
\\
& & x_2 & & &+& x_4 & & &=& 0,
\\
& & & & x_3 & & &+& x_5 &=& 0.
\end{array}\right.
\end{equation}
Let $Py = c$ be the matrix form of \eqref{eq:ls3}, then we have $P \geq 0$, so NNA can be applied.
This can be generalized as in the following theorem.
\begin{theorem} \label{thm:consistent}
For $A \in {\mathbb R}^{m\times m}$ and $b \in {\mathbb R}^m$, assume that $Ax^* = b$.
Then, there exists a linear system $Py = c$ with solution $y^*$, such that $P$ is a $(m+J)\times (m+J)$ matrix with $J \leq m$, $\mathcal N_P = \mathcal N_A + 2J$ and the first $m$ components of $y^*$ are equal to $x^*$.
\end{theorem}
\begin{proof}
Let $\mathcal J = \{j \leq m: a_{ij} < 0 \;\textrm{for some $i\leq m$}\}$ and $J$ be the cardinality of $\mathcal J$.
If $J > 0$, we can write $\mathcal J = \{j_1, \ldots, j_J\}$ with $j_1 < \cdots < j_J$.
Let $A^+ = (\max\{a_{ij}, 0\})_{i,j\leq m}$, $A^- = -(\min\{a_{ij}, 0\})_{i,j\leq m}$ and $\widetilde A^-$ be the $m\times J$ sub-matrix of $A^-$ consisting of all nonzero columns.
Let $D = (d_{ij})$ be the $J \times m$ matrix defined as
$$
d_{ij} = \left\{ \begin{array}{cc} 1 & \textrm{if $j=j_i$} \\ 0 & \textrm{otherwise.} \end{array}\right.
$$
Define a $(m+J) \times (m+J)$ matrix $P$ as
$$
P =
\left(\begin{matrix}
A^+ & \widetilde A^-
\\
D & I_J
\end{matrix}\right),
$$
where $I_J$ denotes the $J\times J$ identity matrices.
It is obvious that $\mathcal N_P = \mathcal N_A + 2J$.
Consider the linear system
\begin{equation}\label{eq:ls2}
Py = c,
\end{equation}
where $c = (b^T, {\bf 0}_J^T)^T$ and ${\bf 0}_J \in{\mathbb R}^J$ is the zero vector.
Then it is easy to see that $y^* = ((x^*)^T, -(x^*_\mathcal J)^T)^T$ is a solution of \eqref{eq:ls2}, where $x^*_\mathcal J = (x^*_j)_{j\in\mathcal J}$.
\end{proof}
\bigskip
Hence, from the proof, we see that both $P$ and $c$ are easy to find.
The corresponding algorithm is summarized in Algorithm \ref{alg:general}, where $J=m$ is assumed for simplification.
\begin{algorithm}
\caption{General algorithm} \label{alg:general}
\begin{algorithmic}[1]
\State {\bf Input}: $A, b, x_0, \epsilon_{\rm tol}$ and $t >0$
\State $P \gets {\bf 0}_{2m\times 2m}$
\For {$i=1, \ldots, m$} {}
\For {$j=1, \ldots, m$} {}
\If {$a_{ij} > 0$} {$p_{ij} = a_{ij}$}
\ElsIf {$a_{ij} < 0$} $p_{i,m+j} = -a_{ij}$
\EndIf
\EndFor
\State $p_{m+i,i} \gets 1$
\State $p_{m+i, m+i} \gets 1$
\EndFor
\State $c \gets (b^T, {\bf 0}_m^T)^T$
\State $y_0 \gets (x_0^T, -x_0^T)^T$
\State $y \gets$ NNA($P, c, y_0, \epsilon_{\rm tol}, t$) (Algorithm \ref{alg:bayes})
\State \Return $(y_1, \ldots, y_m)^T$
\end{algorithmic}
\end{algorithm}
\subsection{Illustrations} \label{ssec:illustration}
Firstly, we illustrate the effect of $t$ with a small dimensional example.
We set $m=10$ and generate a matrix $A$ by sampling $a_{ij}$ independently from the uniform distribution on the unit interval $[0,1]$.
Hence, with probability one, $A$ will be invertible.
Each component $b_i$ is also generated from the uniform distribution.
We then ran 100 iterations of Algorithm \ref{alg:bayes} with $t=10, 100$ and $1000$.
At each step, we obtain $\|Ax_n - b\|_2$, which are drawn in Figure \ref{fig:t-effect} with natural logarithmic scale.
The results are robust to the value of $t$, which is a common phenomenon with all our experiments.
Therefore, we can choose $t$ sufficiently large in practice.
\begin{figure} \begin{center}
\includegraphics[width=100 mm, height=100 mm]%
{./t-effect.jpeg}
\caption{The effect of the value of $t$. Residual norms $\|Ax_n - b\|_2$ are plotted on log scale for $t=10$ (black solid), $100$ (red dashed) and $1000$ (green dotted). The three lines are almost overlapped.}
\label{fig:t-effect}
\end{center} \end{figure}
We next consider a large sparse random matrix.
We set $m=1000$ and randomly generated $5m$ nonzero nondiagonal elements from the uniform distribution on $[0,1]$.
Each diagonal element of $A$ is generated from the uniform distribution on the interval $[0,100]$.
We compare the NNA algorithm ($t=0$) with GMRES, applied to the original system, and normal equation given in \eqref{eq:normal}.
The result is given in Figure \ref{fig:comparison}, showing the better convergence for the NNA compared to GMRES.
\begin{figure} \begin{center}
\includegraphics[width=100 mm, height=100 mm]%
{./comparison.jpeg}
\caption{Residual norms $\|A_s x_n - b\|_2$ are plotted in log scale for NNA (black solid), GMRES applied to the original system (red dashed) and GMRES to the normal equation (green dotted).}
\label{fig:comparison}
\end{center} \end{figure}
Finally, we consider a real example, known as GRE-1107, which can be found in \cite{matrixmarket}.
It is a nonsymmetric indefinite matrix, and the number of non-zero components is 5664 with $m=1107$.
NNA converges quickly without preconditioning while preconditioned (by the incomplete LU decomposition) GMRES and BI-CGSTAB \cite{van1992bi} fail to converge.
Residual norms are plotted in Figure \ref{fig:real}.
\begin{figure} \begin{center}
\includegraphics[width=100 mm, height=100 mm]%
{./real.jpeg}
\caption{Residual plot of NNA for the matrix GRE-1107.}
\label{fig:real}
\end{center} \end{figure}
In the next section we compare our algorithm with those mentioned in Section \ref{sec:intro}.
We do this under the conditions of guaranteed convergence, which impose a restriction on all algorithms, save our own.
In particular, we will compare flops per iteration and convergence rate.
\section{Comparison with other iterative methods} \label{sec:comparison}
As mentioned in the introduction, there is a vast amount of literature for solving sparse linear systems, but difficult to study theoretically.
As a consequence, only a few algorithms possess convergence properties, but even then under restrictive conditions.
In this section, we compare widely used iterative methods and their convergence properties.
Under the assumption that the arithmetic is exact, the result of this section is summarized in Table \ref{tab:comparison}.
Note that the computational complexities of MINRES$(k)$ and GMRES$(k)$ are not directly comparable to those of other methods because they depend on the number of step size $k$.
\begin{table}
\caption{Comparison of iterative methods with known convergence properties. The second column represents sufficient conditions guaranteeing the convergence: DD (diagonally dominant), PD (positive definite) and SPD (symmetric and PD).}
\label{tab:comparison}
\begin{center}
\begin{tabular}{c|ccc}
\hline
& Conditions for & \multirow{2}{*}{FLOPs} & \multirow{2}{*}{Storage} \\
& convergence & & \\
\hline
\multirow{2}{*}{NNA} & \multirow{2}{*}{-} & \multirow{2}{*}{$O(\mathcal N_A + m)$} & \multirow{2}{*}{$O(\mathcal N_A + m)$} \\
& \\
\hline
\multirow{2}{*}{Jacobi} & \multirow{2}{*}{DD} & \multirow{2}{*}{$O(\mathcal N_A + m)$} & \multirow{2}{*}{$O(\mathcal N_A + m)$} \\
& &&\\
\hline
Gauss- & \multirow{2}{*}{DD or SPD} & \multirow{2}{*}{$O(\mathcal N_A + m)$} & \multirow{2}{*}{$O(\mathcal N_A + m)$} \\
Seidel &&&\\
\hline
Conjugate & \multirow{2}{*}{SPD} & \multirow{2}{*}{$O(\mathcal N_A + m)$} & \multirow{2}{*}{$O(\mathcal N_A + m)$} \\
gradient &&& \\
\hline
\multirow{2}{*}{MINRES$(k)$} & \multirow{2}{*}{symmetric} & \multirow{2}{*}{$O(k \mathcal N_A + k m)$} & \multirow{2}{*}{$O(\mathcal N_A + k m)$} \\
&&& \\
\hline
\multirow{2}{*}{GMRES$(k)$} & \multirow{2}{*}{PD} & \multirow{2}{*}{$O(k \mathcal N_A + k^2 m)$} & \multirow{2}{*}{$O(\mathcal N_A + km)$} \\
&&& \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Basic methods: Jacobi and Gauss--Seidel}
It is easy to see that the numbers of flops for each step of the Jacobi and Gauss--Seidel methods are $2 (\mathcal N_A + m)$.
Also, required storages is $\mathcal N_A + 3m$ for Jacobi and $\mathcal N_A + 2m$ for Gauss--Seidel.
Let $L, U$ and $D$ be the lower, upper triangular and diagonal parts of $A = L + U + D$, respectively.
Then, the Jacobi and Gauss--Seidel methods can be expressed in matrix forms as
$$
x_{n+1} = D^{-1} \{ b - (L+U) x_n\} \quad \textrm{and} \quad
x_{n+1} = (L+D)^{-1} (b-U x_n),
$$
respectively.
It is well-known (see Chapter 4 of \cite{saad2003iterative}) that updates of the form $x_{n+1} = G x_n + f$ for some $G\in{\mathbb R}^{m\times m}$ and $f\in{\mathbb R}^m$ assures convergence if $\rho(G) < 1$, where $\rho(G)$ is the spectral radius of $G$.
More specifically, $x_n$ obtained by the Jacobi and Gauss--Seidel methods satisfy
$$
\|x_n - x^*\|_2 \leq \{\rho(D^{-1} (L+U))\}^n \|x_0 - x^*\|_2
$$
and
$$
\|x_n - x^*\|_2 \leq \{\rho((L+D)^{-1} U)\}^n \|x_0 - x^*\|_2,
$$
respectively.
It follows that $x_n \rightarrow x^*$ if the corresponding spectral radius is strictly smaller than 1.
For both methods, $x_n$ sometimes converges to $x^*$ even when the spectral radius is larger than 1.
It can be expensive to compute the spectral radius of a given large matrix.
Fortunately, there are well-known sufficient conditions which are easy to check.
A matrix $A\in{\mathbb R}^{m\times m}$ is called \emph{diagonally dominant} if $|a_{jj}| \geq \sum_{i\neq j} a_{ji}$ for every $i \geq 1$, and \emph{strictly diagonally dominant} if every inequality is strict.
A matrix $A$ is called \emph{irreducible} if the graph representation of $A$ is irreducible,
and \emph{irreducibly diagonally dominant} if it is irreducible, diagonally dominant and $|a_{jj}| > \sum_{i\neq j} |a_{ji}|$ for some $j \geq 1$.
If $A$ is strictly or irreducibly diagonally dominant, then $\rho(D^{-1} (L+U)) < 1$ and
$\rho((L+D)^{-1} U) < 1$; see Chapter 4 of \cite{saad2003iterative}.
Another sufficient condition for $\rho((L+D)^{-1} U) < 1$ is that $A$ is symmetric and positive definite; see \cite{golub2012matrix}.
\subsection{Conjugate gradient method}
It is easy to see that the number of flops in steps \ref{step:cg1}--\ref{step:cg2} of Algorithm \ref{alg:cg} is $2 \mathcal N_A + 12m$, and the required storage is $\mathcal N_A + 4m$.
Let $x_n$ be the approximate solution obtained at the $n$th step of the conjugate gradient method.
If the arithmetic is exact, we have $x_m = x^*$, so the exact solution can be found in $m$ steps.
If $m$ is prohibitively large, let $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ be the maximum and minimum eigenvalues of $A$, respectively.
Then, an upper bound on the conjugate norm between $x_n$ and $x^*$ is given as
$$
(x_n - x^*)^T A (x_n - x^*) \leq 4 \left( \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{2n} (x_0 - x^*)^T A (x_0 - x^*),
$$
where $\kappa = \lambda_{\max}(A) / \lambda_{\min}(A)$ (Chapter 6 of \cite{saad2003iterative}).
In practice, the improvement is typically linear in the step size; see \cite{liesen2004convergence}.
\subsection{MINRES and GMRES}
Ignoring the computational complexity of step \ref{step:gmres6}, that is relatively small for $k \ll m$, the numbers of flops required for steps \ref{step:gmres1}, \ref{step:gmres2}, \ref{step:gmres3}, \ref{step:gmres4}, \ref{step:gmres5} and \ref{step:gmres7} of Algorithm \ref{alg:gmres} are $2N_A, 2m, 2m, 2m, m$ and $(2k+1)m$, respectively.
Thus, the number of flops is $2k \mathcal N_A + (2k^2 + 7k +1)m$.
Since we only need to save $A$, the orthonormal matrix $V_k \in {\mathbb R}^{m\times k}$, the approximate solution and vector for $A v_i$, the required storage is $\mathcal N_A + (k+2)m$.
Here, storage for the Hassenberg matrix $H_k$ is ignored because $k$ is relatively small.
For the Lanczos algorithm (Algorithm \ref{alg:lanczos}), it is not difficult to see that the number of flops is $k(2\mathcal N_A + 9m)$.
In general, Algorithm \ref{alg:gmres} does not guarantee convergence unless $k=m$.
In particular, it is shown in \cite{greenbaum1996any} that for any decreasing sequence $\epsilon_0 > \epsilon_1 > \cdots > \epsilon_m = 0$, there exists a matrix $A \in {\mathbb R}^{m\times m}$ and vectors $b, x_0 \in {\mathbb R}^m$ such that $\|G_k(x_0)\|_2 = \epsilon_k$.
Define $(x_n)$ as $x_{n+1} = G_k(x_n)$, a sequence generated by the restarted GMRES.
Then, if $A$ is positive definite, $x_n$ converges for any $k \geq 1$; see \cite{eisenstat1983variational}.
In particular, the rate is given by
$$
\|Ax_n - b\|_2^2 \leq \left\{ 1 - \frac{\lambda_{\min}^2( (A+A^T)/2)}{\lambda_{\max} (A^T A)} \right\}^{nk} \|A x_0 - b\|_2^2.
$$
Some other convergence criteria of GMRES can be found in \cite{chronopoulos1991s}.
Also, more general upper bounds for residual norms, but not guaranteeing convergence, can be found in \cite{liesen2004convergence} and \cite{saad2003iterative}.
If $A$ is symmetric (not necessarily positive definite),
$$
\|Ax_n - b\|_2^2 \leq \left\{ 1 - \frac{\lambda_{\min}^2(A^2)}{\lambda_{\max}(A^4)} \right\}^n \|Ax_0 - b\|_2^2
$$
for every $k \geq 2$; see \cite{chronopoulos1991s}, assuring the convergence of restarted MINRES.
Under a certain condition on the spectrum of $A$, a different type of upper bound can be found in \cite{liesen2004convergence}.
\subsection{$s$-step methods}
A number of $s$-step methods and their convergence properties are studied in \cite{chronopoulos1991s}.
In particular, it is shown that $s$-step generalized conjugate residual, Orthomin$(k)$ and minimal residual methods converge for all positive definite and some indefinite matrices.
Here, $s$-step minimal residual method is mathematically equivalent to GMRES$(s)$.
However, it is not easy in practice to check conditions for convergence of indefinite matrices.
Furthermore, computational costs for $s$-step methods can be expensive because they require more matrix-vector multiplications in each step.
\section{Discussion}
\label{sec:discussion}
The main contribution of the paper is to describe an algorithm which guarantees convergence for indefinite linear systems of equations.
The key idea is that arbitrary systems can be embedded within a nonnegative system.
Other algorithms, such as CG and GMRES$(k)$, guarantee convergence under certain conditions, but it is difficult in general to transform an arbitrary system into a guaranteed convergent one for them.
Finally, we could do the updates using parallel computing which would provide faster convergence times.
\section*{Acknowledgement}
The second author is partially supported by NSF grant DMS No.\ 1612891.
\bibliographystyle{apalike}
| {
"timestamp": "2018-08-03T02:01:49",
"yymm": "1609",
"arxiv_id": "1609.00670",
"language": "en",
"url": "https://arxiv.org/abs/1609.00670",
"abstract": "We propose a novel iterative algorithm for solving a large sparse linear system. The method is based on the EM algorithm. If the system has a unique solution, the algorithm guarantees convergence with a geometric rate. Otherwise, convergence to a minimal Kullback--Leibler divergence point is guaranteed. The algorithm is easy to code and competitive with other iterative algorithms.",
"subjects": "Numerical Analysis (math.NA)",
"title": "An EM based Iterative Method for Solving Large Sparse Linear Systems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465639,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019683920389
} |
https://arxiv.org/abs/1610.02532 | On uniform closeness of local times of Markov chains and i.i.d. sequences | In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11]. | \section{Introduction}
\label{intro}
The purpose of this paper is to compare the field of
local times of a discrete-time
Markov process with the corresponding field of i.i.d.\
random variables distributed
according to the stationary measure of this process,
in total variation distance.
We mention that local times (also called occupation times)
of Markov processes is a very well studied subject.
It is frequently possible to obtain a complete
characterization of the law of this field
in terms of some Gaussian random field or process,
especially in continuous time (and space) setup.
The reader is probably familiar with
Ray-Knight theorems as well as Dynkin’s and Eisenbaum’s
isomorphism theorems; cf.\ e.g.\ \cite{R,Szn12}.
One should observe, however, that these theorems
usually work in the case when the underlying Markov
process is reversible and/or symmetric in some sense.
To explain what we are doing in this paper,
let us start by considering the following example:
let $(X_j)_{j\geq 1}$ be a Markov chain
on the state space $\Sigma=\{0,1\}$,
with the following transition probabilities:
$\mathbb{P}[X_{n+1}=k\mid X_n=k]=1-\mathbb{P}[X_{n+1}=1-k\mid X_n=k]
=\frac{1}{2}+\varepsilon$ for $k=0,1$,
where~$\varepsilon\in(0,\frac{1}{2})$ is small.
Clearly, by symmetry, $(\frac{1}{2},\frac{1}{2})$ is
the stationary distribution of this Markov chain.
Next, let $(Y_j)_{j\geq 1}$
be a sequence of i.i.d.\ Bernoulli random variables
with success probability~$\frac{1}{2}$.
What can we say about the distance in total variation between
the laws
of $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$? Note that the
``na\"\i{}ve'' way of trying to force
the trajectories to be equal (given $X_1=Y_1$, use the maximal
coupling of $X_2$ and $Y_2$;
if it happened that $X_2=Y_2$, then try to couple $X_3$
and $Y_3$, and so on) works
only up to $n=O(\varepsilon^{-1})$. Even though this method is probably
not optimal, in this case it is easy to obtain that the total
variation distance converges to~$1$
as $n\to\infty$. This is because of the
following: consider the event
\[
\Xi^Z = \Big\{\frac{1}{n}\sum_{j=1}^{n-1}
\mathds{1}_{\{Z_j = Z_{j+1}\}} > \frac{1}{2} + \frac{\varepsilon}{2} \Big\},
\]
where $Z$ is~$X$ or~$Y$.
Clearly, the random variables $\mathds{1}_{\{Z_j = Z_{j+1}\}}$,
$j\in\{1,\ldots,n-1\}$ are i.i.d.\ Bernoulli,
with success probabilities $\frac{1}{2}+\varepsilon$ and $\frac{1}{2}$
for $Z=X$ and~$Z=Y$
correspondingly. Therefore, if $n\gg \varepsilon^{-2}$,
it is elementary to obtain that that
$\mathbb{P}[\Xi^X]\approx 1$ and $\mathbb{P}[\Xi^Y]\approx 0$,
and so the total variation distance
between the \emph{trajectories} of~$X$ and~$Y$
is almost~$1$ in this case.
So, even in the case when the Markov chain gets quite close
to the stationary distribution
just in one step, usually it is not possible to couple its trajectory
with an i.i.d.\ sequence,
unless the length of the trajectory is relatively short.
Assume, however, that we are not interested in the exact
trajectory of~$X$ or~$Y$,
but rather, say, in the number of visits to~$0$ up to time~$n$.
That is, denote
\[
L_n^Z(0) = \sum_{j=1}^n \mathds{1}_{\{Z_j=0\}}
\]
for $Z=X$ or $Y$. Are $L_n^X(0)$ and $L_n^Y(0)$
close in total variation distance
for \emph{all}~$n$?
Well, the random variable $L_n^Y(0)$ has the binomial
distribution with parameters~$n$
and~$\frac{1}{2}$, so it is approximately Normal with
mean~$\frac{n}{2}$
and standard deviation~$\frac{\sqrt{n}}{2}$. As for $L_n^X(0)$,
it is elementary to obtain that it is
approximately Normal with mean~$\frac{n}{2}$ and
standard deviation~$\sqrt{n}\big(\frac{1}{2}+O(\varepsilon)\big)$.
Then, it is also elementary to obtain that the total variation distance between
these two Normals is~$O(\varepsilon)$, \emph{uniformly} in~$n$
(indeed, that total variation distance equals the total variation distance
between the Standard Normal and the centered Normal with variance
$(1+O(\varepsilon))^2$; that distance is easily verified to be of order~$\varepsilon$).
This \emph{suggests} that the total variation distance between~$L_n^X(0)$
and~$L_n^Y(0)$ should be also of order~$\varepsilon$ uniformly in~$n$.
Observe, by the way, that the distribution of the local times
of a two-state Markov chain can be explicitly written
(cf.~\cite{BG}), so one can obtain a rigorous proof
of the last statement in a direct way, after some work.
Let us define the \emph{local time} of a stochastic process~$Z$
at site~$x$ at time~$n$ as the number of visits
to~$x$ up to time~$n$:
\begin{equation*}
L^Z_n(x) = \sum_{j=1}^n \mathds{1}_{\{Z_j=x\}}
\end{equation*}
(sometimes we omit the upper index when it is clear which process
we are considering).
The above example shows that, if one is only interested in the local
times of the
Markov chain (and not the complete trajectory),
then there is hope to obtain a coupling
with the local times of an i.i.d.\ random sequence
(which is much easier to handle).
Observe that there are many quantities of interest that
can be expressed in terms
of local times only
(and do not depend on the order), such as, for instance,
\begin{itemize}
\item hitting time of a site~$x$: $\tau(x) = \min\{n: L_n(x)>0\}$;
\item cover time: $\min\{n: L_n(x)>0 \text{ for all }x\in\Sigma\}$,
where $\Sigma$ is the space where the process lives;
\item blanket time~\cite{DLP}: $\min\{n\geq 1: L_n(x)\geq \delta n \pi(x)\}$,
where $\pi$ is the stationary measure of the process
and $\delta\in (0,1)$ is a parameter;
\item disconnection time~\cite{DSzn,Szn10}:
loosely speaking, it is the time~$n$ when the set $\{x: L_n(x)>0\}$
becomes ``big enough'' to ``disconnect''
the space~$\Sigma$ in some precise sense;
\item the set of favorite (most visited) sites (e.g.~\cite{HS,T}):
$\{x: L_n(x)\geq L_n(y)\text{ for all }y\in\Sigma\}$;
\item and so on.
\end{itemize}
Therefore, if it is possible to obtain a coupling as above that
works with high probability, then that coupling may be useful.
Note also that, although
not every Markov chain comes close to
the stationary distribution in just one step, that
can be sometimes circumvented by considering
the process at times $k, 2k, 3k, \ldots$ with a large~$k$.
In particular, we expect that our results may be useful when
dealing with
\emph{excursion processes} (i.e., when~$\Sigma$ is a set
of excursions of a random walk).
One may e.g.~refer to \cite{MS}, cf.\ Lemma~2.2 there
(note that the order of excursions does not matter,
so one would be able to get rid of the factor~$m$
in the right-hand side);
also, we are working now on applications of our results to
the decoupling for random
interlacements~\cite{BGP}.
\section{Notations and results}
\label{notations}
We start describing the assumptions under which we will prove our main result.
Let~$(\Sigma,d)$ be a compact metric space, with~${\mathcal{B}}(\Sigma)$
representing its Borel~$\sigma$-algebra.
\begin{assump}\label{assump_1}
Assume that $(\Sigma,d)$ is of \emph{polynomial class}: there
exist some $\beta\geq0$ and~$\phi\geq 1$ such that
for all $r\in(0,1]$, the number of open balls of radius at most~$r$
needed to cover $\Sigma$ is smaller than or equal to $\phi r^{-\beta}$.
\end{assump}
As an example of metric space of polynomial class, consider first a
finite space~$\Sigma$, endowed with the discrete metric
\[
d(x,y)=\mathds{1}_{\{x\neq y\}}, \text{ for } x,y\in\Sigma.
\]
In this case, we can choose~$\beta=0$ and~$\phi=|\Sigma|$ (where~$|\Sigma|$
represents the cardinality of~$\Sigma$). As a second example, let us
consider~$\Sigma$ to be a compact $k$-dimensional Lipschitz
submanifold of~$\mathbb{R}^m$ with metric induced by the Euclidean norm of~$\mathbb{R}^m$.
In this case we can take~$\beta=k$, but $\phi$ will in general depend
on the precise structure of~$\Sigma$.
It is important to observe that, for a finite~$\Sigma$, it may not
be the best idea to use the above discrete metric; one may be better off
with another one, e.g., the metric inherited from the Euclidean
space where~$\Sigma$ is immersed (see e.g.\ the proof
of Lemma~2.9 of~\cite{CP}).
We consider a Markov chain~$X=(X_i)_{i\geq 1}$ with transition
kernel~$\mathop{\mathfrak{P}}(x,dy)$ on~$(\Sigma,{\mathcal{B}}(\Sigma))$, and we suppose that
the chain has a unique invariant probability measure~$\pi$.
Moreover, we assume the transition kernel to be absolutely continuous
with respect to~$\pi$, $\mathop{\mathfrak{P}}(x,\cdot)\ll\pi(\cdot)$ for all~$x\in\Sigma$.
Let us denote by~$p(x,\cdot)$ the Radon-Nikodym derivative
(i.e., \emph{density}) of~$\mathop{\mathfrak{P}}(x,\cdot)$
with respect to~$\pi$: for~$x\in\Sigma$,
\begin{align*}
\mathop{\mathfrak{P}}(x,A) = \int_{A} p(x,y) \pi(dy), \text{ for all } A\in{\mathcal{B}}(\Sigma).
\end{align*}
We also consider
\begin{assump}\label{assump_2}
Assume that the density $p(x,\cdot)$ is
\emph{uniformly H\"older continuous}, that is, there exist constants $\kappa>0$
and $\gamma\in(0,1]$ such that for all $x,z,z'\in \Sigma$,
\begin{equation*}
|p(x,z)-p(x,z')|\leq \kappa d^{\gamma}(z,z').
\end{equation*}
\end{assump}
In the rest of this paper, we assume that the chain~$X$ starts
with some probability law absolutely continuous with respect to~$\pi$
and we denote by~$\nu$ its density. We also work under
\begin{assump}\label{assump_3}
Let $\varepsilon_0\in (0,1)$. Suppose that there exists $\varepsilon\in (0,\varepsilon_0]$ such that
\begin{equation}
\label{max_eps}
\sup_{x,y\in\Sigma}|p(x,y)-1| \leq \varepsilon,
\end{equation}
and
\begin{equation}
\label{cond_nu}
\sup_{x\in \Sigma}|\nu(x)-1| \leq \varepsilon.
\end{equation}
\end{assump}
\noindent
Observe that~\eqref{cond_nu} is not very restrictive because,
due to \eqref{max_eps}, the chain will anyway
come quite close to stationarity already on step~$2$.
Additionally, let us denote by $Y=(Y_i)_{i\geq 1}$ a sequence
of i.i.d.\ random variables with law~$\pi$.
Before stating our main result, we recall the definition of the total
variation distance between
two probability measures~$\bar{\mu}$ and~$\hat{\mu}$ on some measurable
space $(\Omega, \mathcal{T})$,
\begin{equation*}
\|\bar{\mu}-\hat{\mu}\|_{\text{TV}}
=\sup_{A\in\mathcal{T}}|\bar{\mu}(A)-\hat{\mu}(A)|.
\end{equation*}
When dealing with random elements $U$ and $V$, we will write (with a slight abuse of notation) $\|U-V\|_{\text{TV}}$ to denote the total variation distance between the laws of $U$ and $V$.
Denoting by $L_n^Z:=(L_n^Z(x))_{x\in \Sigma}$ the local time field of the process $Z=X$ or $Y$ at time $n$, we are now ready to state
{\thm \label{Main_Thm} Under Assumptions~\ref{assump_1}--\ref{assump_3},
there exists a positive cons\-tant~$K=K(\varepsilon_0)$
such that, for all $n\geq 1$, it holds that
\begin{align*}
\|L_n^X - L_n^Y\|_{\emph{TV}} \leq K\varepsilon
\displaystyle\sqrt{1 + \ln(\phi 2^{\beta})
+ \frac{\beta}{\gamma}\ln\Big(\frac{\kappa\vee (2\varepsilon)}{\varepsilon}\Big)}.
\end{align*}
}
As an application of our main theorem, consider a
finite state space~$\Sigma$, endowed with the discrete metric.
As we have already mentioned, in this case we can choose~$\beta=0$
and~$\phi=|\Sigma|$. Additionally, observe that, for any
Markov chain~$X$ on $\Sigma$, under Assumption \ref{assump_3}, we can always take~$\kappa=2$ and~$\gamma=1$,
so the H\"older continuity of~$p$
is automatically verified here. Thus, Theorem~\ref{Main_Thm} leads to
\begin{equation*}
\|L_n^X - L_n^Y \|_{\text{TV}}
\leq K\varepsilon \displaystyle\sqrt{1 + \ln|\Sigma|},
\end{equation*}
for all~$n\geq 1$.
Observe that, since~$K$ is unknown, Theorem~\ref{Main_Thm} becomes interesting only when $\varepsilon$ is small enough. Therefore, it is also relevant to check if we can obtain a uniform control (in $n$) of
$\|L_n^X - L_n^Y\|_{\text{TV}}$ away from~$1$, for all~$\varepsilon\in(0,1)$.
In this direction, we obtain the following
{\thm \label{Thm2} Under Assumptions~\ref{assump_1}--\ref{assump_3}, there exists a positive cons\-tant~$K'=K'(\beta,\varphi,\kappa,\gamma,\varepsilon_0)$
such that, for all $n\geq 1$, it holds that
\begin{equation*}
\|L_n^X - L_n^Y\|_{\emph{TV}} \leq 1-K'.
\end{equation*}
}
Such a result may be useful e.g.\ in the following context:
if we are able to prove that, for the i.i.d.\ sequence,
something happens with probability close to~$1$, then
the same happens for the field of local time of the Markov chain
with at least uniformly positive probability.
Observe that it is not unusual that the fact
that the probability of something is uniformly positive
implies that it should be close to~$1$ then.
The rest of the paper is organized in the following way.
In Section~\ref{Sim}, among other things, we show how the soft local time method
can be applied to the Markov chain~$X$ for constructing its
local time field.
In Section~\ref{coupling} we present the construction of a
coupling between the local time fields of the two processes~$X$ and~$Y$ at time~$n$.
In Section~\ref{TV_binomial} we estimate the total variation
distance between two binomial point processes. This auxiliary result
will be useful to bound from above the probability of the complement
of the coupling event introduced in Section~\ref{coupling}.
In Section~\ref{Premres} we use a concentration inequality due to~\cite{Adam08}
together with the machinery of empirical processes to obtain some
intermediate results. In Section~\ref{Main_Thm_proof}
we give the proof of Theorem~\ref{Main_Thm}. Finally, in Section~\ref{Second_Thm}, we give the proof of Theorem~\ref{Thm2}.
We end this section with considerations on the notation for constants
used in this paper. Throughout the text, in general, we
use capital~$C_1,C_2,\dots$ to denote global constants that
appear in the results.
When these constants depend on some parameter(s),
we will explicitly put (or mention) the dependence, otherwise the constants are
considered universal. Moreover, we will use small~$c_1,c_2,\dots$ to
denote local constants that appear locally in the proofs,
restarting the enumeration at the beginning of each proof.
\section{Constructions using soft local times}
\label{Sim}
We assume that the reader is familiar with the
general idea of using Poisson point processes for constructing
general adapted stochastic processes, also known as
the \emph{soft local time} method. We refer to Section~4
of~\cite{PT15} for the general theory, and also
to Section~2 of~\cite{CGPV13} for a simplified introduction.
In this paper we use a modified version of
this technique to couple the local time fields of
both processes, and we do this precisely in Section~\ref{coupling}.
In this section
we first present two different constructions of the Markov chain~$X$,
and then we present a construction of the local time field of~$X$ only.
Then, in Section~\ref{Sim_IID}, we present a construction
of the i.i.d.\ sequence~$Y$. All these constructions
consist of applying the method of soft local times
in a (relatively) straightforward way.
Let~$\alpha$ be the \emph{regeneration coefficient} of the
chain~$X$ with respect to~$\pi$ (see Definition~4.28 of~\cite{FG}) defined by
\begin{align*}
\alpha := \inf_{x,y\in\Sigma} p(x,y).
\end{align*}
Note that $\alpha\leq 1$ since $p(x,\cdot)$ is a probability density
for all $x\in \Sigma$.
Moreover, \eqref{max_eps} implies
that~$\alpha\geq 1-\varepsilon =:q$.
Hence, we consider the following decomposition
\begin{align*}
p(x,\cdot) = q + (1-q)\mu(x,\cdot), \text{ for all } x\in\Sigma,
\end{align*}
where $\mu(x,\cdot)=1+\frac{p(x,\cdot) -1}{1-q}\geq 0$
is a probability density with respect to~$\pi$.
On some probability
spa\-ce~$(\tilde{\Omega},\tilde{\mathcal{T}},\mathbb{P})$, suppose that we are
given the following independent random elements:
\begin{itemize}
\item A sequence~$(I_j)_{j\geq 1}$ with~$I_1=1$ and~$(I_j)_{j\geq 2}$
i.i.d.~Bernoulli($q$) random variables;
\item A Poisson point process~$\eta$ on~$\Sigma\times\mathbb{R}_+$
with intensity measure~$\pi\otimes\lambda_+$, where~$\lambda_+$
is the Lebesgue measure on~$\mathbb{R}_+$ and $\pi$ is the invariant
probability measure of~$X$ (cf.\ Section~\ref{notations}).
\end{itemize}
Then, we define the sequence~$(\rho_j)_{j\geq 0}$ such that
\begin{align*}
\rho_0 &= 1,\\
\rho_{k+1} &= \inf\{ j>\rho_k : I_j=1 \} \text{ for } k\geq 1.
\end{align*}
We interpret the elements of the sequence~$(\rho_j)_{j\geq 1}$ as being
the random regeneration times of the Markov chain~$X$: at each random
time~$\rho_j$, the chain~$X$ starts afresh with law~$\pi$.
In this way, the chain will be viewed as a sequence of
(independent) blocks (called {\it regeneration blocks}) with starting law~$\pi$
and transitions
according to~$\mu(\cdot,\cdot)$. Such blocks thus have lengths given by
the differences of the subsequent elements of~$(\rho_j)_{j\geq 1}$.
\subsection{Construction of the local time field
of the Markov chain~$X$}
\label{Sim_MC}
We first give two ways to construct the Markov chain~$X$ up to time~$n$
using soft local times.
For that,
we consider the Poisson point process described above,
\begin{align*}
\eta = \sum_{\lambda\in\Lambda} \boldsymbol{\delta}_{(z_{\lambda}, t_{\lambda})},
\end{align*}
(where $\Lambda$ is a countable index set),
and we proceed with the soft local time scheme in the classical way first.
Denote by~$(x_i)_{i\geq 1}$ the elements of~$\Sigma$
which we will consecutively construct.
We begin with the construction of~$x_1$ by defining
\begin{align*}
\xi_1 &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\text{ such that } \ell \nu(z_{\lambda}) \geq t_{\lambda}\big\}, \\
G^X_1(x) &= \xi_1 \nu(x), \;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(x_1,t_1)$ to be the unique pair~$(z_{\lambda},t_{\lambda})$
satisfying~$G^X_1(z_{\lambda}) = t_{\lambda}$.
Then, once we have obtained the first state~$x_1$ visited by the chain~$X$,
we proceed to the construction of the other ones. For~$i=2,3,\dots$, let
\begin{align*}
\xi_i &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\notin\{(x_k,t_k)\}_{k=1}^{i-1} \text{ such that } \\
& \hspace{5cm} G^X_{i-1}(z_{\lambda}) + \ell p(x_{i-1},z_{\lambda})
\geq t_{\lambda}\big\}, \\
G^X_i(x) &= G^X_{i-1}(x) + \xi_i p(x_{i-1},x), \;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(x_i,t_i)$ to be the unique pair~$(z_{\lambda},t_{\lambda})$ out
of the set~$\{(x_k,t_k)\}_{k=1}^{i-1}$ satisfying~$G^X_i(z_{\lambda})
= t_{\lambda}$.
Thus, after performing this iterative scheme for~$n$ iterations,
we obtain the accumulated soft local time of the Markov chain~$X$
at time~$n$, which is given by
\begin{align*}
G^X_n(x) = \xi_1 \nu(x) + \sum_{k=2}^{n} \xi_k p(x_{k-1},x) ,
\end{align*}
for~$x\in\Sigma$.
Next, we present an alternative construction of the same Markov chain,
taking into account the regeneration times of~$X$, using the Poisson point
process $\eta$ and the sequence~$(I_j)_{j\geq 1}$ of Bernoulli random variables introduced at
the beginning of this section. Denote now by~$(\hat{x}_i)_{i\geq 1}$
the elements of~$\Sigma$
which we will consecutively construct in this alternative way.
By construction, the sequence~$(\hat{x}_i)_{i\geq 1}$ will also
have the law of the Markov chain~$X$.
We begin with the construction of~$\hat{x}_1$ by defining
\begin{align*}
\hat{\xi}_1 &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\text{ such that } \ell \nu(z_{\lambda}) \geq t_{\lambda}\big\}, \\
\hat{G}^X_1(x) &= \hat{\xi}_1 \nu(x), \;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(\hat{x}_1,\hat{t}_1)$ to be the unique pair~$(z_{\lambda},t_{\lambda})$
satisfying~$\hat{G}^X_1(z_{\lambda}) = t_{\lambda}$.
Then, once we have obtained the first state~$\hat{x}_1$
visited by the chain~$X$, we proceed to the construction
of the other ones. For~$i=2,3,\dots$, define
\begin{align*}
g_i(\hat{x}_{i-1},\cdot) &=
\begin{cases}
\mu(\hat{x}_{i-1},\cdot) , & \text{ if } I_i=0, \\
1 , & \text{ if } I_i=1,
\end{cases}
\end{align*}
and then
\begin{align*}
\hat{\xi}_i &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\notin\{(\hat{x}_k,\hat{t}_k)\}_{k=1}^{i-1} \text{ such that } \\
& \hspace{5cm} \hat{G}^X_{i-1}(z_{\lambda})
+ \ell g_i(\hat{x}_{i-1},z_{\lambda}) \geq t_{\lambda}\big\}, \\
\hat{G}^X_i(x) &= \hat{G}^X_{i-1}(x)
+ \hat{\xi}_i g_i(\hat{x}_{i-1},x),\;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(\hat{x}_i,\hat{t}_i)$ to be the unique
pair~$(z_{\lambda},t_{\lambda})$ out of the
set~$\{(\hat{x}_k,\hat{t}_k)\}_{k=1}^{i-1}$
satisfying~$\hat{G}^X_i(z_{\lambda}) = t_{\lambda}$.
Thus, after performing this iterative scheme for~$n$ iterations,
we obtain the accumulated soft local time at
time~$n$
\begin{align*}
\hat{G}^X_n(x) = \hat{\xi}_1 \nu(x) + \sum_{k=2}^{n} \hat{\xi}_k
\big(I_k +(1-I_k)\mu(\hat{x}_{k-1},x) \big),
\end{align*}
for~$x\in\Sigma$. Observe that, under~$\mathbb{P}$, $\hat{G}^X_n$
and $G^X_n$ have the same law.
Since in this paper we are interested in the random
field of local times of the chain until time~$n$,
the order of appearance of the states of~$X$ is not relevant
for us and we will use the soft local time scheme in a slightly
different way from that described above.
Specifically, we use the random variables $I_1,\dots, I_n$ as in
the previous construction but now we first construct all the regeneration
blocks of size strictly greater than one and then the regeneration blocks
of size one.
We proceed by considering the Poisson point process~$\eta$ and
the random variables~$I_1, I_2, \dots, I_n$.
Then, we define the random set~${\mathfrak{H}}\subset\{1,2,\dots,n\}$ as
\begin{align}
{\mathfrak{H}}= \big\{j\in\{2,3,\dots,n-1\} : I_j I_{j+1}=1 \}
\cup \{j\in\{n\} : I_j=1\big\},
\label{set_H}
\end{align}
and the random permutation~${\mathfrak{S}}:\{1,2,\dots,n\}
\rightarrow\{1,2,\dots,n\}$ in the following way:
\begin{itemize}
\item for~$j\in {\mathfrak{H}}^c$, ${\mathfrak{S}}(j) = j-\sum_{i=2}^{j-1}
I_i I_{i+1}$,
\item for~$j\in {\mathfrak{H}}$, ${\mathfrak{S}}(j)
= |{\mathfrak{H}}^c| + |\{i\in{\mathfrak{H}} : i\leq j\}|$,
\end{itemize}
with the convention that~$\sum_{i=2}^{k} =0$ if~$k<2$.
Now, define
\begin{align*}
\tilde{\xi}_1 &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\text{ such that } \ell \nu(z_{\lambda}) \geq t_{\lambda}\big\}, \\
\tilde{G}^X_1(x) &= \tilde{\xi}_1 \nu(x),\;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(\tilde{x}_1,\tilde{t}_1)$ to be the unique
pair~$(z_{\lambda},t_{\lambda})$
satisfying~$\tilde{G}^X_1(z_{\lambda}) = t_{\lambda}$.
Next, for~$i=2,3,\dots,n$, define
\begin{align*}
\tilde{\xi}_i &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\notin\{(\tilde{x}_k,\tilde{t}_k)\}_{k=1}^{i-1} \text{ such that } \\
& \hspace{2cm} \tilde{G}^X_{i-1}(z_{\lambda})
+ \ell (I_{{\mathfrak{S}}^{-1}(i)} + (1-I_{{\mathfrak{S}}^{-1}(i)})
\mu(\tilde{x}_{i-1},z_{\lambda})) \geq t_{\lambda}\big\}, \\
\tilde{G}^X_i(x) &= \tilde{G}^X_{i-1}(x) + \tilde{\xi}_i
(I_{{\mathfrak{S}}^{-1}(i)} + (1-I_{{\mathfrak{S}}^{-1}(i)})
\mu(\tilde{x}_{i-1},x)),\;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(\tilde{x}_i,\tilde{t}_i)$ to be the unique
pair~$(z_{\lambda},t_{\lambda})$ out of the
set~$\{(\tilde{x}_k,\tilde{t}_k)\}_{k=1}^{i-1}$
satisfying~$\tilde{G}^X_i(z_{\lambda}) = t_{\lambda}$.
At the end of this procedure, we obtain the accumulated soft
local time until time~$n$,
\begin{align*}
\tilde{G}^X_n(x) = \tilde{\xi}_1\nu(x) +\sum_{i=2}^{n} \tilde{\xi}_i
(I_{{\mathfrak{S}}^{-1}(i)} + (1-I_{{\mathfrak{S}}^{-1}(i)})
\mu(\tilde{x}_{i-1},x)),
\end{align*}
for~$x\in\Sigma$, observing that~$\tilde{G}^X_n$ has the same
law as~$G^X_n$ and~$\hat{G}^X_n$, under~$\mathbb{P}$. Also, observe
that when proceeding in this way, we obtain the decomposition
\begin{align*}
\tilde{G}^X_n(x) = \tilde{G}^X_{|{\mathfrak{H}}^c|}(x)
+ (\tilde{G}^X_n(x)-\tilde{G}^X_{|{\mathfrak{H}}^c|}(x)),
\end{align*}
where~$\tilde{G}^X_{|{\mathfrak{H}}^c|}(x)$ is the accumulated soft local time
corresponding to the construction of the first block plus the
regeneration blocks of size strictly greater than one, until time~$n$.
By implementing this last scheme, one produces a sequence~$(\tilde{x}_i)_{i}$
with~$n$ elements, the local time field of which has the law of the local time
field of~$X$ at time~$n$, just as we wanted. Also, we recall the property
(of the soft local times) that the elements in the family~$(\tilde{\xi}_i)_i$
are all i.i.d.~Exponential($1$) random variables, independent
of all other random elements (cf.~\cite{PT15}).
\subsection{Construction of the i.i.d.~sequence~$Y$}
\label{Sim_IID}
Now, we describe how we can construct the i.i.d.~sequence~$Y_1,\ldots,Y_n$
using the soft local time technique.
We denote by~$(y_i)_{i\geq 1}$ the elements of~$\Sigma$ which we
will consecutively construct.
Considering the same Poisson point process~$\eta$ described
above, we begin with the construction of~$y_1$ by defining
\begin{align*}
\xi'_1 &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\text{ such that } \ell \geq t_{\lambda}\big\}, \\
G^Y_1(x) &= \xi'_1,\;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(y_1,t_1)$ to be the unique pair~$(z_{\lambda},t_{\lambda})$
satisfying~$G^Y_1(z_{\lambda}) = t_{\lambda}$.
Then, we proceed to the construction of~$y_2,y_3,\dots,y_n$:
for~$i=2,3,\dots,n$, define
\begin{align*}
\xi'_i &= \inf\big\{\ell\geq 0: \exists (z_{\lambda},t_{\lambda})
\notin\{(y_k,t_k)\}_{k=1}^{i-1} \text{ such that }G^Y_{i-1}(z_{\lambda})
+ \ell \geq t_{\lambda}\big\}, \\
G^Y_i(x) &= G^Y_{i-1}(x) + \xi'_i, \;\text{for all}\; x\in \Sigma,
\end{align*}
and~$(y_i,t_i)$ to be the unique pair~$(z_{\lambda},t_{\lambda})$
out of the set~$\{(y_k,t_k)\}_{k=1}^{i-1}$ satisfying~$G^Y_i(z_{\lambda}) = t_{\lambda}$.
At the end of this iterative scheme,
we obtain the first~$n$ elements of the sequence~$Y$.
As before, the elements in the family~$(\xi'_i)_i$ are all
i.i.d.\ Exponential($1$) random variables, and independent of
all the other quantities.
The use of the soft local times to construct the sequence~$Y$,
as described above, produces the accumulated soft local time until time~$n$,
\begin{align*}
G_n^Y(x) = \sum_{k=1}^{n} \xi'_k, \text{ for } x\in\Sigma.
\end{align*}
\section{The coupling}
\label{coupling}
Before starting the construction of our coupling we need to introduce some notation. For this, consider the following independent
random elements: a sequence $(e_n)_{n\geq 1}$ of Exponential(1)
independent random variables, a stationary version of the Markov chain~$(X_n)_{n\geq 1}$, which we call~$(M_n)_{n\geq 1}$
(that is, $(M_n)_{n\geq 1}$ has transition density~$p$ and initial law~$\pi$), and a Geometric($q$) random variable~$T$. Then,
we define the random function
\begin{equation}
\label{widehatW}
\widehat{W}(x)=\sum_{k=1}^{T} e_k(1-p(M_{k-1},x)),
\;\text{for all}\; x\in \Sigma,
\end{equation}
and consider a sequence of i.i.d.\ random functions
$(\widehat{W}_n)_{n\geq 1}$ with the same law as $\widehat{W}$.
We will show in Section~\ref{Est_F} that
\begin{equation}
\label{EPE}
\sup_{n\in\mathbb{N}}\frac{1}{\sqrt{n}}E\Big[\sup_{x\in \Sigma}\Big|\sum_{k=1}^{n}\widehat{W}_k(x)\Big|
\Big]\leq F
\end{equation}
where
\begin{align}
F := C_4\varepsilon \displaystyle\sqrt{1 + \ln(\phi 2^{\beta}) + \frac{\beta}{\gamma}\ln\Big(\frac{\kappa\vee (2\varepsilon)}{\varepsilon}\Big)}
\label{F_exp}
\end{align}
and $C_4=C_4(\varepsilon_0)$ is a positive constant depending on~$\varepsilon_0$.
Now, we present the construction of a coupling between
the local time fields of the Markov
chain~$X$ and the i.i.d.\ sequence~$Y$
at time~$n$. We will use the random element
${\mathcal{W}}:=((I_1,\dots,I_n),\eta)$ (introduced in Section~\ref{Sim})
and, the auxiliary random elements~$V$, $V'$, $V''$ and~$\eta'$
(that we define later in this section),
to construct a coupling
between two copies
of~$\eta$ which we call~$\eta_X$ and~$\eta_Y$.
These copies will be such that
the third construction of Section~\ref{Sim_MC}
applied to~$\eta_X$ and the
construction of Section~\ref{Sim_IID} applied to~$\eta_Y$ will
give high
probability of successful coupling of
the local time fields of~$X$ and~$Y$, for~$\varepsilon$ sufficiently small.
It is important to stress that the ``na\"\i{}ve''
coupling (that is, using the same realization of the
Poisson marks for constructing both the Markov chain
and the i.i.d.\ sequence)
does not work, because it is not probable that both
constructions will pick \emph{exactly} the same marks
(look at the two marks at the upper
right part of the top pictures on Figure~\ref{f_resampling}).
To circumvent this, we proceed as shown on Figure~\ref{f_resampling}:
we first remove all the
marks which are above the ``dependent'' part (that is, the marks strictly above the curve~$\tilde{G}^{X}_{|{\mathfrak{H}}^c|}$), and then resample
them using the maximal coupling of the ``projections''.
In the following, we describe this construction
in a rigorous way.
In order to construct the coupling we are looking for, we assume that,
in addition to the Bernoulli sequence
$(I_j)_{j\geq 1}$ and the Poisson point
process~$\eta$, the triple $(\tilde{\Omega}, \tilde{\mathcal{T}}, \mathbb{P})$
(from Section~\ref{Sim}) also support an independent copy of~$\eta$,
which we call~$\eta'$.
We will also need other random elements on $(\tilde{\Omega}, \tilde{\mathcal{T}}, \mathbb{P})$
to be defined later. We assume that $(\tilde{\Omega}, \tilde{\mathcal{T}}, \mathbb{P})$
is large enough to support all these random elements.
\begin{figure}
\begin{center}
\includegraphics{resampling_2}
\caption{Resampling of the ``independent parts''}
\label{f_resampling}
\end{center}
\end{figure}
We start with the construction of~$\eta_X$.
As explained in Section~\ref{Sim_MC},
we first use~${\mathcal{W}}$ to construct the local time field of
the Markov chain~$X$
up to time~$n$. In this way,
we obtain the soft local time curves
$\tilde{G}^X_i$, for $1\leq i\leq n$,
together with the sequences $\tilde{\xi}_1,\dots, \tilde{\xi}_n$
and $\tilde{x}_1,\dots, \tilde{x}_n$.
Then, we define the random function
\begin{align}
\Psi(\cdot) = \frac{\displaystyle\sum_{i=1}^n
\tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(\cdot)}
{\displaystyle\sum_{i=|{\mathfrak{H}}^c|+1}^{n}\tilde{\xi}_i}
\label{fct_psi}
\end{align}
and for all~$i\in\mathbb{N}$ the events
\begin{equation}
{\mathsf{A}}_i=\Big\{\sup_{x\in \Sigma}|\Psi(x)-1|\leq \frac{(1+i)F}{\sqrt{n}}\Big\},
\label{event_Av}
\end{equation}
where~$F$ is defined in~\eqref{F_exp}.
Now, we partition $\tilde{\Omega}$ using the events ${\mathsf{B}}_1:={\mathsf{A}}_1$ and ${\mathsf{B}}_{i+1}={\mathsf{A}}_{i+1}\setminus{\mathsf{A}}_i$, for $i\geq 1$ and define
\[
\mathsf{G}=\bigcup_{i\in \mathbb{N}:(1+i)F\leq 1}{\mathsf{B}}_i.
\]
Observe that, on $\mathsf{G}$, $\Psi$ is actually a (random) probability density with respect to~$\pi$.
Note that,
under~$\mathbb{P}$, $G'_n:=\sum_{i=1}^n\tilde{\xi}_i$ has the same law
as~$G^Y_n$, the soft local time of $Y$ at time~$n$
(cf.\ Section~\ref{Sim_IID} and the middle right picture on Figure~\ref{f_resampling}). Anticipating on what is coming, on~$\mathsf{G}$, the law
$\Psi \text{d}\pi$ will serve as the ``compensating'' law to reconstruct
the~$|{\mathfrak{H}}|$ marks of~$\eta_Y$ between $\tilde{G}^{X}_{|{\mathfrak{H}}^c|}$
and~$G'_n$.
Now, going back to the construction of~$\eta_X$,
on~$\mathsf{G}$, we adopt a resampling scheme:
we first ``erase''
all the marks of the point process~$\eta$ in the space~$\Sigma\times\mathbb{R}_+$ that are on the curves
$\tilde{G}^X_{|{\mathfrak{H}}^c|+1},\dots,\tilde{G}^X_n$,
then we reconstruct the marks as follows.
We introduce the random vector $V:=(V_1,\dots,V_{_{|{\mathfrak{H}}|}})$
such that under $\mathbb{P}[\;\cdot \mid {\mathcal{W}}]$, its coordinates are
independent and distributed according to the invariant measure~$\pi$.
We use the random vector~$V$ to place the (new) marks
\[
\Big(V_1,\tilde{G}^X_{|{\mathfrak{H}}^c|+1}(V_1)\Big),
\Big(V_2,\tilde{G}^X_{|{\mathfrak{H}}^c|+2}(V_2)\Big),\dots,
\Big(V_{|{\mathfrak{H}}|},\tilde{G}^X_{n}(V_{|{\mathfrak{H}}|})\Big),
\]
on the curves $\tilde{G}^X_{|{\mathfrak{H}}^c|+1},\dots,\tilde{G}^X_n$ (see Figure~\ref{f_resampling}, bottom left picture).
On~$\mathsf{G}^c$, we keep the original marks. Hence, $\eta_X$
is the point process obtained using this resampling
procedure.
We continue with the construction of $\eta_Y$. We will
construct the marks of $\eta_Y$ below $G'_n$ and then
``glue''~$\eta'$ above~$G'_n$ to complete the marks of~$\eta_Y$.
We now need the following two random vectors.
First, consider the random vector
$V':=(V'_1,\dots,V'_{|{\mathfrak{H}}|})$ such that
under $\mathbb{P}[\;\cdot \mid {\mathcal{W}}=w]$:\\
For $w\in \mathsf{G}$,
\begin{itemize}
\item $V'$ has independent coordinates distributed
according to $\Psi \text{d}\pi$;
\item the elements $\big(\sum_{i=1}^{|{\mathfrak{H}}|}
\mathds{1}_{\{x\}}(V'_i)\big)_{x\in \Sigma }$
and
$\big(\sum_{i=1}^{|{\mathfrak{H}}|}\mathds{1}_{\{x\}}(V_i)\big)_{x\in \Sigma }$
are maximally coupled;
\end{itemize}
and for $w\in \mathsf{G}^c$,
\begin{itemize}
\item $V'$ has independent coordinates distributed
according to~$\pi$;
\item the vectors $(V'_1,\dots,V'_{|{\mathfrak{H}}|})$
and $(V_1,\dots,V_{|{\mathfrak{H}}|})$ are independent.
\end{itemize}
We also introduce the random vector
$V'':=(V''_1,\dots,V''_{|{\mathfrak{H}}^c|})$,
such that under $\mathbb{P}[\;\cdot \mid {\mathcal{W}}]$, $V''$ has
independent coordinates distributed according to~$\pi$
and is independent of the pair $(V,V')$.
On $\mathsf{G}$, we construct the marks of the point
process~$\eta_Y$ below $G'_n$ in the following way:
we keep the marks obtained below $\tilde{G}^{X}_{|{\mathfrak{H}}^c|}$
and we use the law $\Psi\text{d}\pi$ to complete the process
until~$G'_n$.
For this, we adopt a resampling scheme just as before.
We first erase all the marks of the point process~$\eta$
that are (strictly) above $\tilde{G}^{X}_{|{\mathfrak{H}}^c|}$,
then we resample the part of the process $\eta$ up to~$G'_n$,
using the marks:
\begin{align*}
\Big(V'_1,\tilde{G}^X_{|{\mathfrak{H}}^c|}(V'_1)+\Psi(V'_1)\tilde{\xi}_{|{\mathfrak{H}}^c|+1}\Big),&\dots, \Big(V'_j,\tilde{G}^X_{|{\mathfrak{H}}^c|}(V'_j)+\Psi(V'_j)\sum_{i=1}^j\tilde{\xi}_{|{\mathfrak{H}}^c|+i}\Big),\dots\nonumber\\
&\dots,\Big(V'_{|{\mathfrak{H}}|},\tilde{G}^X_{|{\mathfrak{H}}^c|}(V'_{|{\mathfrak{H}}|})+\Psi(V'_{|{\mathfrak{H}}|})\sum_{i=1}^{|{\mathfrak{H}}|}\tilde{\xi}_{|{\mathfrak{H}}^c|+i}\Big)
\end{align*}
(see Figure~\ref{f_resampling}, bottom right picture). On $\mathsf{G}^c$, we construct the points below~$G'_n$
as follows. We consider the decomposition
$G'_n=\sum_{i=1}^{|{\mathfrak{H}}^c|}\tilde{\xi}_i
+\sum_{i=|{\mathfrak{H}}^c|+1}^n\tilde{\xi}_i$.
First, we use the random vector~$V''$ to sample the marks
$$\Big(V''_1,\tilde{\xi}_1\Big),\Big(V''_2,\sum_{i=1}^{2}\tilde{\xi}_i\Big),\dots, \Big(V''_{|{\mathfrak{H}}^c|},\sum_{i=1}^{|{\mathfrak{H}}^c|}\tilde{\xi}_i\Big),$$
on the first $|{\mathfrak{H}}^c|$ curves.
Then, on the second part, we use the random vector~$V'$
to sample the marks
$$\Big(V'_1,\sum_{i=1}^{|{\mathfrak{H}}^c|+1}\tilde{\xi}_i\Big),\Big(V'_2,\sum_{i=1}^{|{\mathfrak{H}}^c|+2}\tilde{\xi}_i\Big),\dots, \Big(V'_{|{\mathfrak{H}}|},\sum_{i=1}^{n}\tilde{\xi}_i\Big),$$
on the $n-|{\mathfrak{H}}^c|$ last curves.
For the sake of brevity, let us denote by $m^X_1,\dots, m^X_n$ and $m^Y_1,\dots, m^Y_n$, the first coordinates ($\in \Sigma$) of the marks of $\eta_X$ and $\eta_Y$ below the curves
$\tilde{G}^X_n$ and~$G'_n$ respectively. Let~$\tilde{L}_n$ and~$L'_n$ be the fields of
local times associated to these first coordinates, that is, for all $x\in \Sigma$,
\begin{equation*}
\tilde{L}_n(x)=\sum_{i=1}^n\mathds{1}_{\{x\}}(m^X_i)\phantom{**} \text{and}\phantom{**} L'_n(x)=\sum_{i=1}^n\mathds{1}_{\{x\}}(m^Y_i).
\end{equation*}
By construction, we have the following
{\prop
We have that~$\eta_X\stackrel{\text{\tiny law}}{=}\eta_Y\stackrel{\text{\tiny law}}{=}\eta$.
Furthermore, it holds that
$\tilde{L}_n\stackrel{\text{\tiny law}}{=} L^X_n$ and
$L'_n\stackrel{\text{\tiny law}}{=} L^Y_n$
(where $\stackrel{\text{\tiny law}}{=} $ stands for equality in law).
}
\medskip
\noindent
Consequently, we obtain a coupling between~$L^X_n$ and~$L^Y_n$.
We will denote by~$\Upsilon$ the coupling event associated to
this coupling (that is, $\Upsilon = \{\tilde{L}_n=L'_n\}$).
In Section~\ref{Main_Thm_proof},
we will obtain an upper bound for $\mathbb{P}[\Upsilon^c]$.
\section{Total variation distance between binomial point processes} \label{TV_binomial}
In this section, we estimate the total variation distance
between two binomial point processes on some measurable
space $(\Omega, \mathcal{T})$ with laws~${{\bf P}}_n$ and~${{\bf Q}}_n$
of respective parameters $({\bf p}_n,n)$ and $({\bf q}_n,n)$,
where $n\in \mathbb{N}$ and ${\bf p}_n$, ${\bf q}_n$ are two probability
laws on $(\Omega, \mathcal{T})$. We also assume
that ${\bf q}_n\ll {\bf p}_n$ and that~${\bf p}_n$ and~${\bf q}_n$
are close in a certain sense to be defined below.
For two probability measures~$\bar{\mu}$ and~$\hat{\mu}$
on $(\Omega, \mathcal{T})$, we recall that
if $\bar{\mu}\ll \hat{\mu}$,
\begin{equation}
\label{TVdiscrete}
\|\bar{\mu}-\hat{\mu}\|_{\text{TV}}
=\frac{1}{2}\int_{\Omega}
\Big|\frac{\text{d}\bar{\mu}}{\text{d}\hat{\mu}}-1\Big|d\hat{\mu}.
\end{equation}
We will prove the following result, which is actually
a little bit more than we need in this paper.
{\prop
\label{Propmulti}
Let $\delta_0\in (0,1]$ and $\delta \in [0,\delta_0)$
such that for all $n\in \mathbb{N}$,
$|\frac{\emph{d}{\bf q}_n}{\emph{d}{\bf p}_n}(x)-1|\leq\delta n^{-1/2}$
for all~$x\in \Omega$. Then, for
$C_1(\delta_0)=\exp(\delta_0^2)
\frac{\sinh(\delta_0^2)}{\delta_0}+\sqrt{2\pi}\exp(\frac{5}{2}\delta_0^2)$
we have, for all $n\in \mathbb{N}$,
\begin{equation*}
\|{\bf P}_n-{\bf Q}_n\|_{\emph{TV}}\leq C_1(\delta_0)\delta.
\end{equation*}
}
\begin{proof}
In this proof, when we want to emphasize the probability
law under which we take the expectation we will indicate the
law as a subscript. For example, the expectation under some
probability law~$\bar{\mu}$ will be denoted by~$E_{\bar{\mu}}$.
To begin, let us suppose that $n\geq 2$. We first observe
that~${{\bf P}}_n$ and~${{\bf Q}}_n$ can be seen as probability measures
on the space of $n$-point measures ${\mathcal{M}}_n=\{m:m=\sum_{i=1}^{n}\boldsymbol{\delta}_{x_i},
x_i\in \Omega, 1\leq i\leq n\}$ endowed with
the $\sigma$-algebra generated by the
mappings $\Phi_B:{\mathcal{M}}_n\to \mathbb{Z}_+$ defined by $\Phi_B(m)=m(B)=\sum_{i=1}^n\boldsymbol{\delta}_{x_i}(B)$,
for all $B\in \mathcal{T}$. Observe that the law of~${\bf P}_n$
(respectively,~${\bf Q}_n$) is completely characterized by its values
on the sets of the form $\{m\in{\mathcal{M}}_n:m(B_1)=n_1,\dots, m(B_J)=n_J\}$,
where $J\in \mathbb{Z}_+$, $B_1,\dots, B_J$ are disjoint sets
in~$\mathcal{T}$ and $n_1,\dots,n_J$ are non-negative integers
such that $n_1+\dots+n_J=n$.
With this observation it is easy to deduce that ${\bf Q}_n\ll {\bf P}_n$
and check that its Radon-Nikodym derivative with respect
to~${\bf P}_n$ is given by
\[
\frac{\text{d}{\bf Q}_n}{\text{d}{\bf P}_n}(m)
=\prod_{i=1}^n\frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x_i)
\]
where $m=\sum_{i=1}^{n}\boldsymbol{\delta}_{x_i}$.\\
By (\ref{TVdiscrete}) we obtain that
\begin{align*}
\|{\bf P}_n-{\bf Q}_n\|_{\text{TV}}&=\frac{1}{2}\int_{{\mathcal{M}}_n}\Big|
\frac{\text{d}{\bf Q}_n}{\text{d}{\bf P}_n}(m)-1\Big|\text{d}{\bf P}_n(m)\nonumber\\
&=\frac{1}{2}\int_{{\mathcal{M}}_n}\Big|\prod_{i=1}^n
\frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x_i)-1\Big|\text{d}{\bf P}_n(m).
\end{align*}
Now, for all $n\in \mathbb{N}$, we define the function
$f_n:\Omega\to \mathbb{R}$ such that, for~$x\in \Omega$,
we have $f_n(x)=\frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x)-1$.
Observe that $E_{{\bf p}_n}[f_n]=0$ and that
$\|f_n\|_{\infty}\leq\delta n^{-1/2}$ for all $n\geq 2$. We have that
\begin{align}
\label{TV1}
\|{\bf P}_n-{\bf Q}_n\|_{\text{TV}}&=\frac{1}{2}\int_{{\mathcal{M}}_n}\Big|
\prod_{i=1}^n(1+f_n(x_i))-1\Big|\text{d}{\bf P}_n(m)\nonumber\\
&=\frac{1}{2}\int_{{\mathcal{M}}_n}\Big|
\exp\Big(\sum_{i=1}^n\ln(1+f_n(x_i))\Big)-1\Big|
\text{d}{\bf P}_n(m)\nonumber\\
&=\frac{1}{2}E_{{\bf P}_n}|\exp\{m(g_n)\}-1|
\end{align}
where, for all $n\geq 2$, $g_n$ is the function $\Omega\to \mathbb{R}$
defined by $g_n=\ln(1+f_n)$
(by using $|\frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x)-1|\leq
\delta n^{-1/2}$ for all~$x\in \Omega$,
we can observe that $g_n$ is well defined)
and $m(g_n):=\int g_n dm$. Using the fact that $|\ln(1+x)|\leq 2|x|$
for $x\in (-1/\sqrt{2},1/\sqrt{2})$, we deduce that
$\|g_n\|_{\infty}\leq 2\|f_n\|_{\infty}\leq 2\delta n^{-1/2}$,
for all $n\geq 2$.
Now observe that under ${\bf P}_n$, $m(g_n)$ has the same law as
the sum $g_n(X_1)+\dots+ g_n(X_n)$, where the random
variables $X_1,\dots,X_n$ are i.i.d.\ with law~${\bf p}_n$.
We deduce that
\begin{align*}
E_{{\bf P}_n}|\exp\{m(g_n)\}-1|
=E|\exp\{g_n(X_1)+\dots+g_n(X_n)\}-1|.
\end{align*}
Observe that $|E[g_n(X_1)]|=|E[(g_n-f_n)(X_1)]|$
since $E[f_n(X_1)]=E_{{\bf p}_n}[f_n]=0$.
Now we use the fact that, for all $x\in \mathbb{R}$ such
that $|x|\leq 1/\sqrt{2}$, we have that
\[
|\ln(1+x)-x|\leq 2x^2.
\]
Since $\|f_n\|_{\infty} \leq \delta n^{-1/2}$ we obtain that
$\|g_n-f_n\|_{\infty}\leq 2\delta^2 n^{-1}$. We deduce that
\begin{align}
\label{EST1}
|E[g_n(X_1)]|\leq \|g_n-f_n\|_{\infty}\leq 2\delta^2 n^{-1}.
\end{align}
Let $Z_n:=\sum_{k=1}^{n}f_n(X_k)$. Using the fact that
$|\exp(x) -1|\leq \exp(|x|)-1$ for all $x\in \mathbb{R}$ and~(\ref{EST1})
we obtain that
\begin{align}
\label{TV2}
E_{{\bf P}_n}|\exp\{(m(g_n)\}-1| &\leq E\Big[\exp\Big(|Z_n + \sum_{k=1}^{n}(g_n-f_n)(X_k)|\Big)-1\Big]\nonumber\\
&\leq \exp(2\delta^2) E\Big[\exp(|Z_n|)\Big]-1.
\end{align}
Now let us obtain an upper bound for the expectation of the
right-hand side of~(\ref{TV2}). Using integration by
parts, we have
\begin{align}
\label{Parts}
E\Big[\exp(|Z_n|)\Big]=1+\int_{0}^{\infty}e^tP[|Z_n|
\geq t]dt.
\end{align}
By Hoeffding's inequality, we have, for all $n\geq 2$ and
for all $t\geq 0$,
\begin{equation*}
P[|Z_n|\geq t]\leq 2\exp\Big(-\frac{t^2}{2\delta^2}\Big).
\end{equation*}
Using this last inequality in~(\ref{Parts}), we obtain that
\begin{align*}
E\Big[\exp(|Z_n|)\Big]&=1+2\int_{0}^{\infty}
\exp\Big(t-\frac{t^2}{2\delta^2}\Big)dt\nonumber\\
&\leq 1+2\delta \exp\Big(\frac{\delta^2}{2}\Big)\sqrt{2\pi}.
\end{align*}
Therefore, going back to~(\ref{TV2}) and using the fact that
$\delta< \delta_0$, we deduce that for all $n\geq 2$,
\begin{align}
\label{EST3}
E_{{\bf P}_n}|\exp\{m(g_n)\}-1| &\leq \exp(2\delta^2)
\Big[1+2\delta \exp\Big(\frac{\delta^2}{2}\Big)\sqrt{2\pi}\Big]-1\nonumber\\
&\leq \delta\Big[2\exp(\delta^2)
\frac{\sinh(\delta^2)}{\delta}+2\sqrt{2\pi}\exp\Big(\frac{5}{2}\delta^2\Big)\Big]\nonumber\\
&\leq c_1(\delta_0) \delta,
\end{align}
where $c_1(\delta_0):=2\exp(\delta_0^2)
\frac{\sinh(\delta_0^2)}{\delta_0}+2\sqrt{2\pi}\exp(\frac{5}{2}\delta_0^2)$.
Gathering~(\ref{TV1}), (\ref{EST3}) and considering the fact that
$\|{\bf P}_1-{\bf Q}_1\|_{\text{TV}}\leq \frac{\delta}{2}$,
we conclude the proof of Proposition~\ref{Propmulti}
by taking $C_1= c_1/2$.
\end{proof}
\section{Controlling the ``dependent part'' of the soft
local time}
\label{Premres}
We recall that, for $i\in\mathbb{N}$, we defined in Section~\ref{coupling}
the events
\begin{equation*}
{\mathsf{A}}_i=\Big\{\sup_{x\in \Sigma}\big|\Psi(x)-1\big|
\leq \frac{(1+i)F}{\sqrt{n}} \Big\}.
\end{equation*}
The goal of this section is to prove the following
{\prop \label{goodenv} There exist a positive constant~$C_2=C_2(\varepsilon_0)$ and $n_0=n_0(\varepsilon_0)\in\mathbb{N}$
such that, for all integer~$n\geq n_0$ and
$i\in\mathbb{N}$, it holds that
\begin{align*}
\mathbb{P}\Big[{\mathsf{A}}_i^c\;\Big|\; |{\mathfrak{H}}|> \frac{q_0^2}{6}n \Big] &\leq \frac{C_2}{(1+i)^3}
\end{align*}
where $q_0:=1-\varepsilon_0$.
}
We postpone the proof of this proposition to
Section~\ref{Proof_Av}. Before that, in Section~\ref{Est_F} we show~\eqref{EPE} and in Section~\ref{concentr}
we use a concentration inequality to obtain a tail estimate on the numerator of $\Psi-1$ (see~\eqref{Psidecomp}).
\subsection{Proof of inequality~\eqref{EPE}}
\label{Est_F}
In this section, we present a standard method based on bracketing numbers to prove~(\ref{EPE}).
Without loss of generality, we assume
in this section that~$\kappa$ from
Assumption~\ref{assump_2} is greater than or equal to~$2\varepsilon$.
We start introducing
the space~${\mathcal{S}}=\mathbb{R}^{\Sigma}$ and the class~${\mathfrak{F}}=(f_x)_{x\in\Sigma}$ of
functions~$f_x:{\mathcal{S}}\rightarrow \mathbb{R}$ such that~$f_x(\omega)=\omega(x)$,
for~$\omega\in{\mathcal{S}}$.
In this setting, for~$s>0$ and~$\mathfrak{E}:{\mathcal{S}}\rightarrow\mathbb{R}$
an envelope function of the class~${\mathfrak{F}}$ (that is, a function such that~$\mathfrak{E}\geq |f_x|$ for all~$f_x\in{\mathfrak{F}}$), we need to estimate the bracketing number
\begin{align*}
N_{[\,]}\Big(s\|\mathfrak{E}\|_{2},{\mathfrak{F}},L_2\Big),
\end{align*}
which is defined to be the minimum number of brackets
\begin{align*}
[f_1,f_2] := \big\{f:{\mathcal{S}}\rightarrow \mathbb{R}; f_1\leq f\leq f_2\big\},
\end{align*}
satisfying~$\|f_2-f_1\|_{2} < s\|\mathfrak{E}\|_{2}$, that are
needed to cover the class~${\mathfrak{F}}$, where the given
functions~$f_1$ and~$f_2$ have finite $L_2$-norms
(see Definition~2.1.6 of~\cite{VW96}).
For that, we consider an (initially arbitrary) exhaustive and
finite collection of subsets of the space~$\Sigma$, that is,
a finite collection~$\{{\mathsf{D}}_i\}_i$ such that~${\mathsf{D}}_i\subset\Sigma$
for each~$i$ and~$\bigcup_i{\mathsf{D}}_i=\Sigma$, and for each such set~${\mathsf{D}}_i$
we define two functions~$f_{{\mathsf{D}}_i},\hat{f}_{{\mathsf{D}}_i}:{\mathcal{S}}\rightarrow\mathbb{R}$,
\begin{align*}
f_{{\mathsf{D}}_i}(\omega) = \inf_{z_0\in{\mathsf{D}}_i} \omega(z_0) ~\mbox{ and }~ \hat{f}_{{\mathsf{D}}_i}(\omega) = \sup_{z_0\in{\mathsf{D}}_i} \omega(z_0),
\end{align*}
so that, if~$x\in{\mathsf{D}}_i$ then~$f_x\in[f_{{\mathsf{D}}_i},\hat{f}_{{\mathsf{D}}_i}]$.
Thus, to each set in the family~$\{{\mathsf{D}}_i\}_i$ we associate
a bracket, and so any particular finite exhaustive family of subsets
of~$\Sigma$ induces a finite collection of
brackets~$\{[f_{{\mathsf{D}}_i},\hat{f}_{{\mathsf{D}}_i}]\}_i$ which cover the class~${\mathfrak{F}}$.
So, in order to properly estimate the bracketing number,
the task is to determine a suitable collection~$\{{\mathsf{D}}_i\}_i$
of subsets of~$\Sigma$ in such a way that the induced brackets have their
sizes all smaller than~$s\|\mathfrak{E}\|_{2}$. The number of sets in that
collection will serve as an upper bound for~$N_{[\,]}$.
Through the following lemma we better characterize the size
(in~$L_2$) of the induced brackets we are considering.
{\lem Under Assumption~\ref{assump_2}, it holds that, for any
set~${\mathsf{D}}\subset\Sigma$,
\begin{align*}
\|\hat{f}_{{\mathsf{D}}}-f_{{\mathsf{D}}}\|_{2} \leq \frac{\sqrt{2}\kappa}
{q}\displaystyle\max_{z,z'\in{\mathsf{D}}}d^{\gamma}(z,z').
\end{align*}
\label{bracket_size}
}
\begin{proof}
Recalling the notation introduced at the beginning of Section~\ref{coupling},
we want to bound the~$L_2$-norm
\begin{align*}
\Big\|\hat{f}_{{\mathsf{D}}}\Big(\widehat{W}_1(\cdot)\Big)-f_{{\mathsf{D}}}\Big(\widehat{W}_1(\cdot)
\Big)\Big\|_{2}
= \Big\|\sup_{z\in{\mathsf{D}}}\widehat{W}_1(z)-\inf_{z\in{\mathsf{D}}}\widehat{W}_1(z)\Big\|_{2}
\end{align*}
which is
{\allowdisplaybreaks
\begin{align*}
\Big\| \sup_{z\in{\mathsf{D}}} \sum_{k=1}^{T} e_k (1- &p(M_{k-1},z))
- \inf_{z\in{\mathsf{D}}} \sum_{k=1}^{T} e_k (1 - p(M_{k-1},z))\Big\|_{2}\\
&= \Big\| \sup_{z\in{\mathsf{D}}} \sum_{k=1}^{T} e_k p(M_{k-1},z) -
\inf_{z\in{\mathsf{D}}} \sum_{k=1}^{T} e_k p(M_{k-1},z)\Big\|_{2} \\
&\leq \Big\| \sum_{k=1}^{T} e_k \Big(\sup_{z\in{\mathsf{D}}} p(M_{k-1},z)
- \inf_{z\in{\mathsf{D}}} p(M_{k-1},z)\Big) \Big\|_{2} \\
&\leq \kappa\Big\| \sum_{k=1}^{T_1} e_k \Big\|_{2} \displaystyle\max_{z,z'\in{\mathsf{D}}}d^{\gamma}(z,z'),
\end{align*}
where we used Assumption~\ref{assump_2} to establish
the second inequality.
}
Thus, we conclude the proof by using the fact
that~$\sum_{k=1}^{T} e_k$ is exponentially distributed
with parameter~$q$, so that
\begin{align*}
\Big\| \sum_{k=1}^{T} e_k \Big\|_{2} = \frac{\sqrt{2}}{q}.
\end{align*}
\end{proof}
In view of the above result,
we must impose the sets we are constructing,
$\{{\mathsf{D}}_i\}_i$, to be such that
\begin{align}
\max_{z,z'\in{\mathsf{D}}_i}d(z,z') < \Big(s\|\mathfrak{E}\|_{2}\frac{q}{\sqrt{2}\kappa}\Big)^{1/\gamma}, \mbox{ for each } i,
\label{condition_max}
\end{align}
in order to obtain, from Lemma~\ref{bracket_size}, that
\begin{align}
\|\hat{f}_{{\mathsf{D}}_i}-f_{{\mathsf{D}}_i}\|_{2} < s\|\mathfrak{E}\|_{2}, \mbox{ for each } i.
\label{condition_max_2}
\end{align}
Then, we prove
{\prop Under Assumptions~\ref{assump_1} and~\ref{assump_2},
there exists a universal positive constant~$C_3$ such that, if
\begin{align}
\frac{\gamma}{\beta} \Big[1 + \ln(\phi 2^{\beta})
+ \frac{\beta}{\gamma}\ln\Big(\frac{\sqrt{2}\kappa}
{q\|\mathfrak{E}\|_{2}}\Big)\Big] \geq \frac{1}{2},
\label{cond_geq1}
\end{align}
then it holds that, for all~$n\in\mathbb{N}$,
\begin{align*}
E\Big[\sup_{x\in \Sigma}\Big|\sum_{k=1}^{n}\widehat{W}_k(x)\Big|
\Big] \leq C_3 \displaystyle\sqrt{1
+ \ln(\phi 2^{\beta}) + \frac{\beta}{\gamma}
\ln\Big(\frac{\sqrt{2}\kappa}{q\|\mathfrak{E}\|_{2}}\Big)} \|\mathfrak{E}\|_{2}\sqrt{n}.
\end{align*}
\label{expect_R_tilde}
}
\begin{proof}
For~$s>0$, we just define the sets~$\{{\mathsf{D}}_i\}_i$ to
be open balls of radius at most
\begin{align*}
\frac{1}{2}\Big(s\|\mathfrak{E}\|_{2}\frac{q}{\sqrt{2}\kappa}\Big)^{1/\gamma},
\end{align*}
so that~\eqref{condition_max} is verified (and
consequently~\eqref{condition_max_2} too,
by Lemma~\ref{bracket_size}).
Then, under Assumption~\ref{assump_1}, we can say that
the number of such balls that are needed to cover~$\Sigma$
is at most
\begin{align*}
\phi 2^{\beta}\Big[\frac{\sqrt{2}\kappa}
{qs\|\mathfrak{E}\|_{2}}\Big]^{\beta/\gamma},
\end{align*}
and therefore, for any~$s>0$,
\begin{align*}
N_{[\,]}\Big(s\|\mathfrak{E}\|_{2},{\mathfrak{F}},L_2\Big)
\leq \phi 2^{\beta}\Big[\frac{\sqrt{2}\kappa}
{qs\|\mathfrak{E}\|_{2}}\Big]^{\beta/\gamma}.
\end{align*}
Taking this bound on the bracketing number into account,
we can estimate the bracketing entropy integral (of the class~${\mathfrak{F}}$)
\begin{align*}
J_{[\,]}\Big(1,{\mathfrak{F}},L_2\Big) := \int_{0}^{1}
\sqrt{1+\ln N_{[\,]}\Big(s\|\mathfrak{E}\|_{2},{\mathfrak{F}},L_2\Big)} ds,
\end{align*}
(see its definition e.g.\ in Section~2.14.1
of~\cite{VW96}, page 240). We do that by just bounding it above by
\begin{align*}
\int_{0}^{1} \sqrt{1+\ln \Big(\phi 2^{\beta}
\Big[\frac{\sqrt{2}\kappa}{qs\|\mathfrak{E}\|_{2}}\Big]^{\beta/\gamma}\Big)} ds,
\end{align*}
which, after some changes of variables, can be shown to be equal to
\begin{align*}
\Big(\frac{\beta}{\gamma}\Big)^{1/2}
(\phi 2^{\beta}e)^{\gamma/\beta}
\Big(\frac{\sqrt{2}\kappa}{q\|\mathfrak{E}\|_{2}}\Big)
\int_{\frac{\gamma}{\beta} [1
+ \ln (\phi 2^{\beta}[\frac{\sqrt{2}\kappa}
{q\|\mathfrak{E}\|_{2}}]^{\beta/\gamma})]}^{\infty}
\sqrt{x}e^{-x} dx.
\end{align*}
Now, using the asymptotic behaviour of the (upper) incomplete Gamma function
\begin{align*}
\Gamma(y) = \int_{y}^{\infty} \sqrt{x}e^{-x} dx
\end{align*}
as~$y\rightarrow\infty$, it is elementary to see that there exists
a universal positive constant~$c_1$ such
that~$\Gamma(y)\leq c_1\sqrt{y}e^{-y}$ for all~$y\geq 1/2$.
Thus, since we are assuming~\eqref{cond_geq1}, we have
\begin{align*}
J_{[\,]}\Big(1,{\mathfrak{F}},L_2\Big) \leq c_1 \sqrt{1 + \ln(\phi 2^{\beta})
+ \frac{\beta}{\gamma}\ln\Big(\frac{\sqrt{2}\kappa}{q\|\mathfrak{E}\|_{2}}\Big)}.
\end{align*}
Finally, we use Theorem 2.14.2 of~\cite{VW96} to obtain that,
if~\eqref{cond_geq1} holds, then (for a universal positive
constant~$c_2$)
\begin{align*}
E\Big[\sup_{x\in \Sigma}\Big|\sum_{k=1}^{n}\widehat{W}_k(x)\Big|
\Big] &\leq c_2 J_{[\,]}\Big(1,{\mathfrak{F}},L_2\Big)
\|\mathfrak{E}\|_{2} \sqrt{n} \\
&\leq c_2 c_1 \sqrt{1 + \ln(\phi 2^{\beta})
+ \frac{\beta}{\gamma}\ln\Big(\frac{\sqrt{2}\kappa}
{q\|\mathfrak{E}\|_{2}}\Big)} \|\mathfrak{E}\|_{2} \sqrt{n},
\end{align*}
which completes the proof with~$C_3=c_2 c_1$.
\end{proof}
We now investigate the~$L_2$-norm of the envelope~$\mathfrak{E}$. If we take, for example, an envelope of~${\mathfrak{F}}$ given by
\begin{align}
\mathfrak{E}(\omega) = \varepsilon\sum_{k=1}^{T} e_k ,
\mbox{ for any } \omega \in {\mathcal{S}}
\label{envelope}
\end{align}
(so that~$\mathfrak{E}\geq |f_x|$ for all~$f_x\in{\mathfrak{F}}$), then we have
{\prop If~$\mathfrak{E}$ is given by~\eqref{envelope}, then it holds that
\begin{align*}
\|\mathfrak{E}\|_{2} = \frac{\sqrt{2}}{q}\varepsilon.
\end{align*}
\label{envelope_norm}
}
\begin{proof}
Use the fact that~$\sum_{i=1}^{T}e_i$ is
exponentially distributed with parameter~$q$.
\end{proof}
Thus, using Propositions~\ref{expect_R_tilde}, \ref{envelope_norm} and the fact that~$q\geq 1-\varepsilon_0$, we have under Assumptions~\ref{assump_1} and~\ref{assump_2}, for all~$n\in\mathbb{N}$,
\begin{align*}
E\Big[\sup_{x\in \Sigma}\Big|\sum_{k=1}^{n}\widehat{W}_k(x)\Big|
\Big] \leq F\sqrt{n}
\end{align*}
where
\begin{align*}
F := C_4\varepsilon \displaystyle\sqrt{1 + \ln(\phi 2^{\beta}) + \frac{\beta}{\gamma}\ln\Big(\frac{\kappa\vee (2\varepsilon)}{\varepsilon}\Big)},
\end{align*}
and~$C_4=C_4(\varepsilon_0)$ is a positive constant depending on~$\varepsilon_0$.
\subsection{A tail bound involving~$\Psi$}
\label{concentr}
Recall the definition of~$\Psi$ from~\eqref{fct_psi}:
for~$x\in\Sigma$,
\begin{equation}
\label{Psidecomp}
\Psi(x) = \frac{\displaystyle\sum_{i=1}^n
\tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(x)}
{\displaystyle\sum_{i=|{\mathfrak{H}}^c|+1}^{n}\tilde{\xi}_i}
=1 + \frac{\displaystyle\sum_{i=1}^{|{\mathfrak{H}}^c|}
\tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(x)}
{\displaystyle\sum_{i=|{\mathfrak{H}}^c|+1}^{n}\tilde{\xi}_i}.
\end{equation}
In this section, we will use a concentration inequality from~\cite{Adam08},
to estimate the tail of the numerator of the last expression above.
Recalling the notation of Section \ref{Sim_MC}, if we define for~$x\in\Sigma$
\begin{align*}
Z_n(x)=\sum_{i=1}^{n}\xi_i-G^X_n(x),
\end{align*}
we observe that $Z_n(\cdot)\stackrel{\text{\tiny law}}{=} \sum_{i=1}^{|{\mathfrak{H}}^c|} \tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(\cdot)$.
Now we use the fact that the Markov chain~$X$ has
its regenerations at times~$(\rho_j)_{j\geq 1}$
and we approximate~$Z_n$ by a sum of independent random elements.
Specifically, we define
\begin{align*}
W_j(\cdot) = Z_{\rho_{j+1}-1}(\cdot) - Z_{\rho_j-1}(\cdot), \text{ for } j\geq 0,
\end{align*}
with the convention that~$Z_0(\cdot)$=0, and we intend
to approximate~$Z_n(\cdot)$ by a suitably chosen sum
of $W_j(\cdot)$'s.
Write $T_0=\rho_1-\rho_0$ for the length of the first block,
and
\begin{align*}
T_j = \rho_{j+1} - \rho_j, \qquad j\geq 1,
\end{align*}
for the lengths of the subsequent regeneration blocks.
The random variables~$(T_j)_{j\geq 0}$
are i.i.d.\ Geometric($q$). The random
elements~$(W_j(\cdot))_{j\geq 0}$ are independent and,
additionally, the elements~$(W_j(\cdot))_{j\geq 1}$
are identically distributed. Also, for any~$x\in\Sigma$, we have
\begin{align*}
W_0(x) = \xi_1(1 -\nu(x)) + \sum_{k=2}^{T_0}
\xi_k (1 - p(x_{k-1},x) ),
\end{align*}
and, recalling~\eqref{widehatW},
\begin{align}
W_1(x) \stackrel{\text{\tiny law}}{=} \widehat{W}(x) = \sum_{k=1}^{T} e_k (1 - p(M_{k-1},x) ).
\label{W1_law}
\end{align}
Moreover, let us observe the fact that~$\mathbb{E}[W_1(x)]=0$ for any~$x\in\Sigma$.
Let us denote, for~$n,m\in\mathbb{N}$,
\begin{align}
\label{Rn}
R_n &= \sup_{x\in\Sigma}\Big|Z_n(x)\Big|, \\
\tilde{R}_m &= \sup_{x\in\Sigma}\Big|\sum_{j=1}^m W_j(x)\Big|, \nonumber
\end{align}
with the convention that~$\tilde{R}_0=0$.
Observe that in Section~\ref{Est_F} we actually proved that
\begin{align}
\mathbb{E}[\tilde{R}_m] \leq F\sqrt{m},
\label{EPE_F_exp}
\end{align}
where~$F$ is defined in~\eqref{F_exp}.
We now obtain
{\prop For all~$\theta>0$, it holds that
\begin{align*}
\mathbb{P}\Big[R_n \geq 8 F \sqrt{n} + 7\theta n\Big] & \leq 2\exp\Big\{-\frac{3}{8}\frac{q^2\theta^2 n}{\varepsilon^2}\Big\} + 6\exp\Big\{-3C_5\frac{q\theta n}{\varepsilon\ln(3n+1)}\Big\}\\
&\phantom{***} + 2\exp\Big\{-\frac{q\theta n}{2\varepsilon}\Big\},
\end{align*}
where $C_5$ is a universal positive constant.
\label{Prop_Rn}
}
\begin{proof}
We will use an argument analogous to the one used in the proof of Lemma 2.9 in \cite{CP}. To begin, let us assume that there exists a positive constant~$C_5$ such that, for all~$\theta>0$,
\begin{align}
\mathbb{P}\Big[\tilde{R}_m \geq \frac{3}{2} F \sqrt{m} + \theta m\Big] \leq \exp\Big\{-\frac{q^2\theta^2 m}{8\varepsilon^2}\Big\}
+ 3\exp\Big\{-C_5\frac{q\theta m}{\varepsilon\ln(m+1)}\Big\}.
\label{assump}
\end{align}
(This statement will be proved later, in Proposition~\ref{Prop_Rn_tilde}).
Using Markov's inequality and~\eqref{EPE_F_exp}, it is elementary to show that, for~$c_1= 4/3$,
\begin{align*}
\mathbb{P}\Big[\tilde{R}_m \geq \frac{3}{2} c_1 F \sqrt{m} + \theta m\Big] \leq \frac{1}{2},
\end{align*}
so that
\begin{align}
\mathbb{P}\Big[\tilde{R}_m \geq& 2 F \sqrt{m} + \theta m\Big] \nonumber \\
&\leq \frac{1}{2} \wedge \Big(\exp\Big\{-\frac{q^2\theta^2 m}{8\varepsilon^2}\Big\}
+ 3\exp\Big\{-C_5\frac{q\theta m}{\varepsilon\ln(m+1)}\Big\}\Big).
\label{EP2}
\end{align}
Now, let us define
\begin{align*}
\tilde{M} = \min\Big\{ i\in[0,3m] : \tilde{R}_i \geq 8 F \sqrt{m} + 6\theta m\Big\},
\end{align*}
with the convention that $\min\emptyset=\infty$, so that
\begin{align*}
\mathbb{P}\Big[\tilde{M}\in[0,3m]\Big] = \mathbb{P}\Big[\max_{i\in[0,3m]}\tilde{R}_i \geq& 8 F \sqrt{m} + 6\theta m\Big].
\end{align*}
Then, using~\eqref{EP2} one gets
\begin{align*}
\exp\Big\{&-\frac{3}{8}\frac{q^2\theta^2 m}{\varepsilon^2}\Big\} + 3\exp\Big\{-3C_5\frac{q\theta m}{\varepsilon\ln(3m+1)}\Big\} \\
&\geq \mathbb{P}\big[\tilde{R}_{3m} \geq 2 F \sqrt{3m} + 3\theta m\big] \\
&\geq \sum_{j=0}^{3m} \mathbb{P}[\tilde{M}=j] \mathbb{P}\Big[\tilde{R}_{3m-j} < 2 F \sqrt{3m} + 3\theta m\Big] \\
&\geq \mathbb{P}\big[\tilde{M}\in[0,3m]\big] \min_{j\in[0,3m]} \mathbb{P}\Big[\tilde{R}_{3m-j} < 2 F \sqrt{3m-j} + \theta (3m-j)\Big] \\
&\geq \frac{1}{2} \mathbb{P}\big[\tilde{M}\in[0,3m]\big],
\end{align*}
which in turn proves that
\begin{align}
\mathbb{P}\Big[\max_{i\in[0,3m]} & \tilde{R}_i \geq 8 F \sqrt{m} + 6\theta m\Big] \nonumber \\
&\leq 2\exp\Big\{-\frac{3}{8}\frac{q^2\theta^2 m}{\varepsilon^2}\Big\} + 6\exp\Big\{-3C_5\frac{q\theta m}{\varepsilon\ln(3m+1)}\Big\}.
\label{Max_inqty}
\end{align}
Now observe that, if we define
\begin{align*}
\sigma_n = \min\{j\geq 1 : \rho_j > n\},
\end{align*}
then~$\sigma_n - 1$ is a Binomial($n-1,q$) random variable, and, under Assumption~\ref{assump_3},
\begin{align}
R_n \leq \tilde{R}_{\sigma_n-1} + \varepsilon\sum_{i=n+1}^{\rho_{\sigma_n}} \xi_i + \varepsilon\sum_{i=1}^{\rho_1-1}\xi_i.
\label{Rn_inqty}
\end{align}
Note that~$\rho_{\sigma_n}-n$ and~$\rho_1-1$ are both Geometric($q$) distributed random variables. Consequently, $\sum_{i=n+1}^{\rho_{\sigma_n}} \xi_i$ and~$\sum_{i=1}^{\rho_1-1} \xi_i$ are both Exponential($q$) distributed and we deduce that
\begin{align}
\mathbb{P}\Big[ \varepsilon\sum_{i=n+1}^{\rho_{\sigma_n}} \xi_i + \varepsilon\sum_{i=1}^{\rho_1-1} \xi_i \geq \theta n\Big]
&\leq \mathbb{P}\Big[ \varepsilon\sum_{i=n+1}^{\rho_{\sigma_n}}\xi_i \geq \frac{\theta n}{2}\Big] + \mathbb{P}\Big[\varepsilon\sum_{i=1}^{\rho_1-1} \xi_i \geq \frac{\theta n}{2}\Big] \nonumber\\
&= 2\exp\Big\{-\frac{q\theta n}{2\varepsilon}\Big\}.
\label{Exp_inqty}
\end{align}
Finally, using~\eqref{Rn_inqty} together
with~\eqref{Max_inqty} and~\eqref{Exp_inqty}, we obtain that
\begin{align*}
\mathbb{P}\big[& R_n \geq 8 F \sqrt{n} + 7\theta n\big] \\
& \leq \mathbb{P}\Big[ \max_{i\in[0,3n]}\tilde{R}_i \geq 8 F \sqrt{n}
+ 6\theta n\Big] + \mathbb{P}\Big[ \sum_{i=n+1}^{\rho_{\sigma_n}} \xi_i
+ \sum_{i=1}^{\rho_1-1} \xi_i \geq \frac{\theta n}{\varepsilon}\Big] \\
&\leq 2\exp\Big\{-\frac{3}{8}\frac{q^2\theta^2 n}{\varepsilon^2}\Big\}
+ 6\exp\Big\{-3C_5\frac{q\theta n}{\varepsilon\ln(3n+1)}\Big\}
+ 2\exp\Big\{-\frac{q\theta n}{2\varepsilon}\Big\} .
\end{align*}
\end{proof}
Then, we deduce the following
{\cor For all $i,n\in\mathbb{N}$, it holds that
\begin{align*}
\mathbb{P}\big[R_n \geq 8 F \sqrt{n} (1+i)\big] &
\leq 2\exp\Big\{-C_6\frac{F^2 i^2 }{\varepsilon^2}\Big\}
+ 8\exp\Big\{-C_7\frac{Fi}{\varepsilon}\frac{\sqrt{n}}{\ln(3n+1)}\Big\},
\end{align*}
where $C_6$ and~$C_7$ are positive constants that depend on $\varepsilon_0$.
\label{Cor_Rn}
}
\begin{proof}
Take~$\theta=\frac{8}{7}\frac{Fi}{\sqrt{n}}$ in
Proposition~\ref{Prop_Rn}, and use the fact that~$q\geq 1-\varepsilon_0$.
\end{proof}
Before proving assertion~\eqref{assump},
which was assumed to be true in the beginning of the proof
of Proposition~\ref{Prop_Rn}, we must prove some
preliminary results.
{\lem It holds that
\begin{align*}
\sigma^2 := \sup_{x\in\Sigma} \sum_{j=1}^m \mathbb{E}\big[W_j(x)^2\big]
\leq \frac{2}{q^2}\varepsilon^2m.
\end{align*}
\label{lem_sigma2}
}
\begin{proof}
Since, for any~$x\in\Sigma$, the elements of~$(W_j(x))_{j\geq 1}$
are i.i.d., we only have to prove that, for any~$x\in\Sigma$,
\begin{align*}
\mathbb{E}\big[W_1(x)^2\big] \leq \frac{2}{q^2}\varepsilon^2.
\end{align*}
We use~\eqref{W1_law} and~\eqref{max_eps} to write, for any~$x\in\Sigma$,
\begin{align*}
\mathbb{E}\big[W_1(x)^2\big] \leq \varepsilon^2
\mathbb{E}\Big[\Big(\sum_{k=1}^{T} e_k \Big)^2 \Big].
\end{align*}
Since~$T$ is Geometric($q$), we use the fact that~$\sum_{i=1}^{T}e_i$ is exponentially
distributed with parameter~$q$ to see that
\begin{align*}
\mathbb{E}\big[W_1(x)^2\big] \leq \varepsilon^2 \frac{2}{q^2},
\end{align*}
and this completes the proof.
\end{proof}
In order to formulate the next lemma,
we define the so-called $\psi_1$-Orlicz norm of a
random variable~$X$, in the following way:
\begin{align*}
\|X\|_{\psi_{1}} = \inf\big\{t>0 : \mathbb{E} e^{|X|/t} \leq 2\big\},
\end{align*}
see Definition 1 of~\cite{Adam08}.
{\lem It holds that
\begin{align*}
\Big\| \max_{1\leq j\leq m} \sup_{x\in\Sigma}|W_j(x)|
\Big\|_{\psi_1} \leq C_8 \frac{\varepsilon}{q} \ln(m+1),
\end{align*}
where~$C_8$ is a universal positive constant.
\label{lem_orlicz}
}
\begin{proof}
First, Lemma 2.2.2 of~\cite{VW96} provides the inequality
\begin{align*}
\Big\|\max_{1\leq j \leq m} \sup_{x\in\Sigma} |W_j(x)|
\Big\|_{\psi_{1}} \leq c_1 \max_{1\leq j \leq m} \Big\|\sup_{x\in\Sigma}
|W_j(x)| \Big\|_{\psi_{1}} \ln (m+1),
\end{align*}
for a universal positive constant~$c_1$.
But, due to~\eqref{max_eps},
\begin{align*}
\sup_{x\in\Sigma}\Big|\sum_{k=1}^{T} e_k
(1 - p(M_{k-1},x) ) \Big| \leq \varepsilon \sum_{k=1}^{T} e_k,
\end{align*}
so that (recall~\eqref{W1_law})
\begin{align*}
\Big\|\sup_{x\in\Sigma} |W_1(x)| \Big\|_{\psi_1} \leq \varepsilon
\Big\|\sum_{k=1}^{T} e_k \Big\|_{\psi_1}.
\end{align*}
Then, since~$\sum_{k=1}^{T} e_k$ is exponentially
distributed with mean~$1/q$ and the~$\psi_1$-Orlicz norm of an
exponential random variable equals twice its mean, we
obtain the result (recalling that the~$W_j$'s, for $j\geq 1$,
are independent and identically distributed).
\end{proof}
Now, in order to address the problem of estimating the probability
involving~$\tilde{R}_m$ (whose bound was postulated in~\eqref{assump})
from the viewpoint of the theory of empirical processes, we recall
the space~${\mathcal{S}}=\mathbb{R}^{\Sigma}$ and the class~${\mathfrak{F}}=(f_x)_{x\in\Sigma}$ of
functions~$f_x:{\mathcal{S}}\rightarrow \mathbb{R}$ such that~$f_x(\omega)=\omega(x)$,
for~$\omega\in{\mathcal{S}}$, so that the above mentioned probability can be
rewritten as
\begin{align}
\mathbb{P}\Big[ \sup_{f_x\in{\mathfrak{F}}} \Big|\sum_{j=1}^{m} f_x(W_j(\cdot))\Big|
\geq \frac{3}{2} F \sqrt{m} + \theta m \Big],
\label{prob_int}
\end{align}
with~$W_j(\cdot)$ being interpreted as a vector in~${\mathcal{S}}$ whose components
are~$W_j(x)$ for each~$x\in\Sigma$. In this setting, we are able to apply
Theorem~4 of~\cite{Adam08} to prove~\eqref{assump},
and this is done in the next proposition.
{\prop There exists a universal positive constant~$C_5$ such that,
for all~$\theta>0$,
\begin{align*}
\mathbb{P}\Big[\tilde{R}_m \geq \frac{3}{2} F \sqrt{m} + \theta m\Big]
\leq \exp\Big\{-\frac{q^2\theta^2 m}{8\varepsilon^2}\Big\}
+ 3\exp\Big\{-C_5\frac{q\theta m}{\varepsilon\ln(m+1)}\Big\}.
\end{align*}
\label{Prop_Rn_tilde}
}
\begin{proof}
We use~\eqref{EPE_F_exp} and just apply Theorem 4 of~\cite{Adam08}
(with~$\delta=1$, $\eta=1/2$ and~$\alpha=1$ there)
together with Lemmas~\ref{lem_sigma2} and~\ref{lem_orlicz},
to see that there exist universal positive constants~$C$ and~$C_8$
($C$ is from Theorem 4 of~\cite{Adam08} and~$C_8$ is
from Lemma~\ref{lem_orlicz}) such that, for all~$t>0$,
\begin{align*}
\mathbb{P}\Big[\tilde{R}_m \geq \frac{3}{2} F \sqrt{m} + t \Big]
\leq \exp\Big\{-\frac{q^2t^2}{8\varepsilon^2 m}\Big\}
+ 3\exp\Big\{-\frac{qt}{C_8C\varepsilon\ln(m+1)}\Big\}.
\end{align*}
We conclude the proof by setting~$t=\theta m$,
for~$\theta>0$, and~$C_5=(C_8C)^{-1}$.
\end{proof}
\subsection{Proof o Proposition \ref{goodenv}} \label{Proof_Av}
We begin this section obtaining a tail estimate for the cardinality of the random set~${\mathfrak{H}}$ introduced in~\eqref{set_H}, which verifies
\begin{align}
\label{Xidef}
|{\mathfrak{H}}| =
\sum_{j=2}^{n-1} (I_j I_{j+1}) + I_n.
\end{align}
{\prop \label{Prop_H_comp} There exist a positive
constant~$C_9=C_9(\varepsilon_0)$ and $n_2=n_2(\varepsilon_0)\in \mathbb{N}$, such that, for all $\varepsilon\in (0,\varepsilon_0]$ and~$n\geq n_2$,
it holds that
\begin{align*}
\mathbb{P}\Big[|{\mathfrak{H}}|\leq \frac{q_0^2}{6}n\Big] \leq \min\Big\{C_9 \varepsilon,
\frac{1}{2}\Big\}
\end{align*}
where $q_0:=1-\varepsilon_0$.
}
\begin{proof}
First, observe that for $n\geq 4 $ we have that
\begin{align*}
\mathbb{P}\Big[|{\mathfrak{H}}| \leq \frac{q_0^2}{6}n \Big]
&\leq \mathbb{P}\Big[\sum_{j=2}^{n-1} (I_j I_{j+1})
\leq \frac{q_0^2}{6}n \Big]
\leq \mathbb{P}\Big[\sum_{k=1}^{\lfloor\frac{n-2}{2}\rfloor}
(I_{2k} I_{2k+1}) \leq \frac{q_0^2}{6}n \Big].
\end{align*}
Moreover, observe that the random variables~$(I_{2k} I_{2k+1})$,
$k=1,2,\dots,\lfloor\frac{n-2}{2}\rfloor$, are independent
Bernoulli($q^2$). Now, we recall the standard lower tail bound
for the binomial law: for~$\mathcal{X}\sim$ Binomial($m,p$)
and~$\delta\geq 0$, we have that
\begin{equation*}
\mathbb{P}\big[\mathcal{X}\leq (1-\delta) m p\big] \leq \exp\{-m I(p,\delta)\},
\end{equation*}
where
\begin{equation*}
I(p,\delta):=p(1-\delta)\ln(1-\delta)
+p\Big(\frac{1-p}{p}+\delta\Big)
\ln\Big(1+\frac{\delta p}{1-p}\Big).
\end{equation*}
Applying the above formula to the random variable
\begin{align*}
\sum_{k=1}^{\lfloor\frac{n-2}{2}\rfloor} (I_{2k} I_{2k+1})
\end{align*}
with~$1-\delta=q_0^2/(2q^2)$, and using the fact that~$q\geq q_0$,
we obtain that
\begin{align*}
\mathbb{P}\Big[\sum_{k=1}^{\lfloor\frac{n-2}{2}\rfloor} (I_{2k} I_{2k+1})
\leq \frac{q_0^2}{6}n \Big] \leq \exp\Big\{-\frac{n'_1}{3}I\Big(q^2,1-\frac{q_0^2}{2q^2}\Big)\Big\}\leq \frac{1}{2},
\end{align*}
for~$n\geq 12\vee n'_1$, where $n'_1:=\Big\lceil \frac{3\ln 2}{I(q_0^2, 1/2)}\Big\rceil$. Finally, recall that $q=1-\varepsilon$ to conclude the proof.
\end{proof}
Next, we obtain an upper bound for
\[
\mathbb{E}\Big[\sup_{x\in \Sigma}|\Psi(x)-1|^3 \; \Big|\; |{\mathfrak{H}}|> \frac{q_0^2}{6}n\Big].
\]
Then, the upper bound for $\mathbb{P}[{\mathsf{A}}_i^c\mid |{\mathfrak{H}}|> \frac{q_0^2}{6}n]$
will be a direct application of Markov's inequality.
Applying Proposition~\ref{Prop_H_comp} and recalling~\eqref{Rn}, we have
\begin{align}
\label{EXPPSI}
\mathbb{E}\Big[\sup_{x\in \Sigma}|\Psi(x)-1|^3\; \Big|\; |{\mathfrak{H}}|> \frac{q_0^2}{6}n\Big]
&\leq 2\mathbb{E}\Big[\sup_{x\in\Sigma}\Big|
\frac{\sum_{i=1}^{|{\mathfrak{H}}^c|}
\tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(x)}
{\sum_{i=|{\mathfrak{H}}^c|+1}^{n}\tilde{\xi}_i}\Big|^3
\mathds{1}_{\{|{\mathfrak{H}}|> \frac{q_0^2}{6}n\}}\Big]\nonumber\\
&\leq 2\mathbb{E}\Big[\sup_{x\in\Sigma}\Big|
\sum_{i=1}^{|{\mathfrak{H}}^c|}
\tilde{\xi}_i-\tilde{G}^{X}_{|{\mathfrak{H}}^c|}(x)\Big|^3\Big]
\mathbb{E}\Big[\Big(\sum_{i=1}^{\lfloor
\frac{q_0^2}{6}n\rfloor}\tilde{\xi}_i\Big)^{-3}\Big]\nonumber\\
&=2\mathbb{E}[R_n^3]\mathbb{E}\Big[\Big(
\sum_{i=1}^{\lfloor \frac{q_0^2}{6}n\rfloor}\tilde{\xi}_i\Big)^{-3}\Big],
\end{align}
where at the second step, we used the fact that, conditionally
on $\sigma(I_j,j\geq 1)$ the numerator and the denominator of
the ratio in the first line are independent, and the event
$\{|{\mathfrak{H}}|> \frac{q_0^2}{6}n\}$ is measurable with respect to
$\sigma(I_j,j\geq 1)$. Then, using an integration by parts,
Corollary~\ref{Cor_Rn}, and the fact that the square root
in~(\ref{F_exp}) is greater than one, we obtain that
\begin{align*}
\mathbb{E}\Big[\Big(\frac{R_n}{8F\sqrt{n}}\Big)^3\Big]&=3\int_0^{\infty}t^2
\mathbb{P}\Big[\frac{R_n}{8F\sqrt{n}}>t\Big]dt\nonumber\\
&\leq 1+3\int_1^{\infty}t^2
\mathbb{P}\Big[\frac{R_n}{8F\sqrt{n}}>t\Big]dt\nonumber\\
&\leq c_1,
\end{align*}
where $c_1$ is a universal positive constant. Therefore, we have
\begin{equation}
\label{RICO}
\mathbb{E}[R_n^3]\leq c_2F^3n^{3/2},
\end{equation}
where $c_2$ is a universal positive constant.
On the other hand, since
$\big(\sum_{i=1}^{\lfloor \frac{q_0^2}{6}n\rfloor}\tilde{\xi}_i\big)^{-1}$
is an Inverse Gamma random variable with parameters
$(\lfloor \frac{q_0^2}{6}n\rfloor,1)$, we obtain that
\begin{equation}
\label{InvGam}
\mathbb{E}\Big[\Big(\sum_{i=1}^{\lfloor \frac{q_0^2}{6}n\rfloor}\tilde{\xi}_i\Big)^{-3}\Big]= \frac{1}{(\lfloor \frac{q_0^2}{6}n\rfloor-1)(\lfloor \frac{q_0^2}{6}n\rfloor-2)(\lfloor \frac{q_0^2}{6}n\rfloor-3)}\leq c_3 n^{-3}
\end{equation}
for $n\geq 24/q_0^2$ and $c_3$ a positive constant depending on $\varepsilon_0$.
Gathering (\ref{EXPPSI}), (\ref{RICO}), and~(\ref{InvGam})
and applying Markov's inequality we finally obtain
\begin{equation*}
\mathbb{P}\Big[{\mathsf{A}}_i^c \; \Big|\; |{\mathfrak{H}}|> \frac{q_0^2}{6}n\Big]\leq \frac{C_2}{(1+i)^3}
\end{equation*}
for some positive constant $C_2=C_2(\varepsilon_0)$.
\qed
\section{Proof of Theorem~\ref{Main_Thm}}
\label{Main_Thm_proof}
For $n\geq n_0$ (where $n_0$ is from Proposition~\ref{goodenv}), we estimate $\mathbb{P}[\Upsilon^c]$ from above
(recall that $\Upsilon$ is the coupling event from
Section~\ref{coupling}) to obtain an upper bound on the total
variation distance between $L_n^X$ and $L_n^Y$.
At this point, we mention that we will use the notation from Section~\ref{coupling}.
By definition of the total variation distance, we have that
\begin{equation}
\label{Theoproof1}
\|L_n^X-L_n^Y\|_{\text{TV}}\leq \mathbb{P}[\Upsilon^c].
\end{equation}
First, let us decompose $\Upsilon$ according to
$\mathsf{C}:=\{|{\mathfrak{H}}|> \frac{q_0^2}{6}n\}$ (recall~\eqref{set_H})
and its complement:
\begin{equation}
\label{Theoproof2}
\mathbb{P}[\Upsilon^c]=\mathbb{P}[\Upsilon^c, \mathsf{C}]+\mathbb{P}[\Upsilon^c, \mathsf{C}^c]\leq \mathbb{P}[\Upsilon^c,\mathsf{C}]+\mathbb{P}[\mathsf{C}^c].
\end{equation}
Now we use the partition $\mathsf{B}_i$, $i\geq 1$, defined in Section \ref{coupling}, to write
\begin{equation*}
\mathbb{P}[\Upsilon^c,\mathsf{C}]=\sum_{i=1}^{\infty}\mathbb{P}[\Upsilon^c,\mathsf{B}_i,\mathsf{C}].
\end{equation*}
Since $\mathsf{B}_i\cap\mathsf{C}$ is $\sigma({\mathcal{W}})$-measurable (we recall that ${\mathcal{W}}$ was introduced in
Section~\ref{coupling}), we have that
\begin{equation*}
\mathbb{P}[\Upsilon^c, \mathsf{B}_i,\mathsf{C}]=\mathbb{E}[{\bf 1}_{\mathsf{B}_i\cap\mathsf{C}}\mathbb{P}[\Upsilon^c\mid {\mathcal{W}}]].
\end{equation*}
Then, observe that from the coupling construction
of Section~\ref{coupling}, we have on the set $\mathsf{G}\cap\mathsf{C}$,
\begin{equation*}
\mathbb{P}[\Upsilon^c\mid {\mathcal{W}}]\leq \big\|\mathbb{P}[{V}\in \cdot\mid {\mathcal{W}}]-\mathbb{P}[{V'}\in \cdot\mid {\mathcal{W}}]\big\|_{\text{TV}}.
\end{equation*}
Hence, applying Proposition \ref{Propmulti} to the term in the
right-hand side with $\delta_0=1$ we obtain, on the sets
$\mathsf{B}_i\cap\mathsf{C}$ such that
$\mathsf{B}_i\subset\mathsf{G}$,
\begin{equation*}
\mathbb{P}[\Upsilon^c\mid {\mathcal{W}}]\leq C_1(1)(1+i)F.
\end{equation*}
Since $C_1(1)>1$ this last upper bound is still valid on the sets $\mathsf{B}_i\cap\mathsf{C}$ such that $\mathsf{B}_i\subset\mathsf{G}^c$.
We deduce that
\begin{equation*}
\mathbb{P}[\Upsilon^c, \mathsf{C}]\leq C_1(1)F\sum_{i=1}^{\infty}(i+1)\mathbb{P}[\mathsf{B}_i,\mathsf{C}]\leq C_1(1)F\Big(2+\sum_{i=2}^{\infty}(i+1)\mathbb{P}[\mathsf{A}^c_{i-1}\mid \mathsf{C}]\Big),
\end{equation*}
which by Proposition \ref{goodenv} implies
\begin{equation}
\label{Theoproof3}
\mathbb{P}[\Upsilon^c, \mathsf{C}]\leq c_1 F
\end{equation}
for some positive constant~$c_1=c_1(\varepsilon_0)$.
Finally, combining~(\ref{Theoproof1}),
(\ref{Theoproof2}),
(\ref{Theoproof3}), Proposition~\ref{Prop_H_comp}
and~(\ref{F_exp}),
we obtain Theorem~\ref{Main_Thm} for $n\geq n_0$.
For $n<n_0$, we simply perform a step-by-step coupling between
the Markov chain~$X$ and the sequence~$Y$
(as described in the introduction)
to obtain $\|L_n^X-L_n^Y\|_{\text{TV}}\leq n_0 \varepsilon$
and thus prove Theorem~\ref{Main_Thm}.
\qed
\section{Proof of Theorem~\ref{Thm2}}
\label{Second_Thm}
First, we prove a preliminary lemma. As in Section~\ref{TV_binomial},
consider again two binomial point processes on some measurable space
$(\Omega, \mathcal{T})$ with laws~${{\bf P}}_n$ and~${{\bf Q}}_n$
of respective parameters $({{\bf p}}_n,n)$ and $({\bf q}_n,n)$,
where $n\in \mathbb{N}$ and~${\bf p}_n$, ${\bf q}_n$ are
two probability laws on $(\Omega, \mathcal{T})$ such that
${\bf q}_n\ll {\bf p}_n$. Then, we have
{\lem
\label{Propmulti_2}
Let~$\delta>0$ such that, for all $n\in \mathbb{N}$,
$|\frac{\mathrm{d}{\bf q}_n}{\mathrm{d}{\bf p}_n}(x)-1|\leq\delta n^{-1/2}$
for all~$x\in \Omega$. Then
\begin{equation*}
\sup_{n\geq 1}\|{\bf P}_n-{\bf Q}_n\|_{\emph{TV}}\leq 1 - C_{10}(\delta),
\end{equation*}
where~$C_{10}(\delta)$ is a positive constant depending on $\delta$.
}
\begin{proof}
As pointed out in the proof of Proposition~\ref{Propmulti},
${\bf P}_n$ and~${\bf Q}_n$ can be seen as probability measures
on the space of $n$-point measures
\[
{\mathcal{M}}_n=\big\{m:m=\sum_{i=1}^{n}\boldsymbol{\delta}_{x_i},
x_i\in \Omega, 1\leq i\leq n\big\}
\]
endowed with the $\sigma$-algebra generated by the mappings
$\Phi_B:{\mathcal{M}}_n\to \mathbb{Z}_+$ defined by $\Phi_B(m)=m(B)
=\sum_{i=1}^n\boldsymbol{\delta}_{x_i}(B)$,
for all $B\in \mathcal{T}$. Also, recall that ${\bf Q}_n\ll {\bf P}_n$
and its Radon-Nikodym derivative with respect to ${\bf P}_n$ is given by
\[
\frac{\text{d}{\bf Q}_n}{\text{d}{\bf P}_n}(m)
=\prod_{i=1}^n\frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x_i)
\]
where $m=\sum_{i=1}^{n}\boldsymbol{\delta}_{x_i}$.
Moreover, for~$n\in \mathbb{N}$, recall
the functions~$f_n,g_n:\Omega\to \mathbb{R}$ given by
\begin{align*}
f_n(x) = \frac{\text{d}{\bf q}_n}{\text{d}{\bf p}_n}(x)-1 \ \ \mbox{ and } \ \ g_n(x) = \ln(f_n(x)+1), \ \ \mbox{ for } x\in \Omega.
\end{align*}
We start by proving the lemma for all large enough~$n$.
It is convenient to introduce now two new distinct elements~$\textbf{0}_1$ and~$\textbf{0}_2$ in order to define a new space~$\hat{\Omega}=\Omega\cup\{\textbf{0}_1,\textbf{0}_2\}$ (we assume that
$\textbf{0}_1, \textbf{0}_2\notin \Omega$), endowed with the
$\sigma$-algebra~$\hat{\mathcal{T}}:=\sigma(\mathcal{T},\{\textbf{0}_1\})$.
Then, on~$(\hat{\Omega},\hat{\mathcal{T}})$ we consider a
new binomial point process with law~${\hat{{\bf P}}}_{n,k}$ of parameters
$(\hat{{\bf p}}_n,k)$, where $k\in \mathbb{N}$ and~$\hat{{\bf p}}_n$
is the probability law on $(\hat{\Omega}, \hat{\mathcal{T}})$ given by
\begin{align*}
\hat{{\bf p}}_n(A) =
\begin{cases}
\frac{{\bf p}_n(A)}{2}, &\mbox{ for } A\in\mathcal{T}, \\
\frac{1}{4}, &\mbox{ for } A\in \{\{\textbf{0}_1\},\{\textbf{0}_2\}\},
\end{cases}
\end{align*}
Additionally, for $n>\delta^2 $, consider another binomial point
process on~$(\hat{\Omega},\hat{\mathcal{T}})$ with
law~${\hat{{\bf Q}}}_{n,k}$ and parameters~$(\hat{{\bf q}}_n,k)$,
where~$\hat{{\bf q}}_n$ is the probability law on $(\hat{\Omega}, \hat{\mathcal{T}})$ such that~$\hat{{\bf q}}_n(A)=\frac{{\bf q}_n(A)}{2}$
for all~$A\in\mathcal{T}$ and
\begin{align*}
\hat{{\bf q}}_n(\{\textbf{0}_1\})=\frac{1}{4}\Big(1+\frac{\delta}{\sqrt{n}}\Big),\; \hat{{\bf q}}_n(\{\textbf{0}_2\}) = \frac{1}{4}\Big(1-\frac{\delta}{\sqrt{n}}\Big)
\end{align*}
so that $\hat{{\bf q}}_n(\{\textbf{0}_1\})+\hat{{\bf q}}_n(\{\textbf{0}_2\}) = 1/2$.
Thus, $\hat{{\bf P}}_{n,k}$ and~$\hat{{\bf Q}}_{n,k}$ can be seen as probability
measures on the space
\[
\hat{{\mathcal{M}}}_k=\big\{\hat{m}:\hat{m}=\sum_{i=1}^{k}\boldsymbol{\delta}_{x_i},
x_i\in \hat{\Omega}, 1\leq i\leq k\big\}.
\]
We need to introduce the corresponding functions~$\hat{f}_n,\hat{g}_n,:\hat{\Omega}\to \mathbb{R}$ given by
\begin{align*}
\hat{f}_n(x) = \frac{\text{d}\hat{{\bf q}}_n}{\text{d}\hat{{\bf p}}_n}(x)-1 \ \
\mbox{ and } \ \ \hat{g}_n(x) = \ln(\hat{f}_n(x)+1).
\end{align*}
Also, define~$\hat{h}_n:\hat{\Omega}\to \mathbb{R}$ as~$\hat{h}_n = \delta^{-1}n^{1/2}\hat{f}_n$ and the set~$\hat{{\mathcal{M}}}^{\hat{{\bf Q}}}_k
=\big\{\hat{m}\in\hat{{\mathcal{M}}}_k:\frac{\text{d}\hat{{\bf Q}}_{n,k}}
{\text{d}\hat{{\bf P}}_{n,k}}(\hat{m}) \geq 1\big\}$.
Since
\begin{align*}
\frac{\text{d}\hat{{\bf Q}}_{n,k}}{\text{d}\hat{{\bf P}}_{n,k}}(\hat{m})
\geq 1 \Leftrightarrow \sum_{i=1}^k
\ln\Big(\frac{\text{d}\hat{{\bf q}}_n}{\text{d}\hat{{\bf p}}_n}(x_i)\Big)
\geq 0,
\end{align*}
for~$\hat{m}\in\hat{{\mathcal{M}}}_k$, we have for any~$n,k\in\mathbb{N}$
\begin{align*}
\|\hat{{\bf P}}_{n,k}-\hat{{\bf Q}}_{n,k}\|_{\text{TV}}
&=\int_{\hat{{\mathcal{M}}}_k}\Big(\frac{\text{d}\hat{{\bf Q}}_{n,k}}
{\text{d}\hat{{\bf P}}_{n,k}}(\hat{m})-1\Big)^+\text{d}\hat{{\bf P}}_{n,k}(\hat{m})\nonumber\\
&=\int_{\hat{{\mathcal{M}}}^{\hat{{\bf Q}}}_k}\Big(\frac{\text{d}\hat{{\bf Q}}_{n,k}}
{\text{d}\hat{{\bf P}}_{n,k}}(\hat{m})-1\Big)\text{d}\hat{{\bf P}}_{n,k}(\hat{m})\nonumber\\
&\leq 1 - \hat{{\bf P}}_{n,k}\Big[\hat{m}(\hat{g}_n) \geq 0\Big].
\end{align*}
Next, we will bound~$\hat{{\bf P}}_{n,2n}[\hat{m}(\hat{g}_n) \geq 0]$ from above.
If we define~$n_2=n_2(\delta)=\lceil 3\delta^2\rceil$ then using the
fact that~$\ln(1+x)\geq x-x^2$ for~$x\in(-1/\sqrt{3},1/\sqrt{3})$, we
have that, for~$n\geq n_2$,
\begin{align*}
\hat{{\bf P}}_{n,2n}\Big[\hat{m}(\hat{g}_n) \geq 0\Big] \geq \hat{{\bf P}}_{n,2n}\Big[\hat{m}(\hat{f}_n) \geq \hat{m}(\hat{f}^2_n)\Big] = \hat{{\bf P}}_{n,2n}\Bigg[\frac{\hat{m}(\hat{h}_n)}{\sqrt{n}} \geq \delta\frac{\hat{m}(\hat{h}^2_n)}{n}\Bigg].
\end{align*}
On the other hand,
observe that, under~$\hat{{\bf P}}_{n,2n}$,
the random variables $\hat{m}(\hat{h}_n)$
and~$\hat{m}(\hat{h}^2_n)$ have the same law
as~$\hat{h}_n(\hat{X}_1)+\dots+ \hat{h}_n(\hat{X}_{2n})$ and~$\hat{h}^2_n(\hat{X}_1)+\dots+ \hat{h}^2_n(\hat{X}_{2n})$, respectively, where the random variables $\hat{X}_1,\dots,\hat{X}_{2n}$ are i.i.d.~with law~$\hat{{\bf p}}_n$. Moreover, we have
that~$|\hat{h}_n(\hat{X}_1)|\leq 1$, $\hat{{\bf p}}_n$-a.s., $E_{\hat{{\bf p}}_n}[\hat{h}_n(\hat{X}_1)]=0$
and~$\sigma^2:=E_{\hat{{\bf p}}_n}[\hat{h}^2_n(\hat{X}_1)]\geq 1/2$. If we denote the standard Normal distribution function by~$\Phi$,
and take~$n_3=n_3(\delta)=4\lceil\frac{1}{(1-\Phi(\delta))^2}\rceil\vee n_2$, then,
by using the Berry-Esseen theorem (with~$1/2$ as an upper bound
for the Berry-Esseen constant, see for example~\cite{Tyurin}),
we obtain that, for~$n\geq n_3$,
\begin{align*}
\hat{{\bf P}}_{n,2n}\Bigg[\frac{\hat{m}(\hat{h}_n)}{\sqrt{2n}} \geq \delta\frac{\hat{m}(\hat{h}^2_n)}{n\sqrt{2}}\Bigg] \geq \hat{{\bf P}}_{n,2n}\Bigg[\frac{\hat{m}(\hat{h}_n)}{\sigma\sqrt{2n}} \geq \frac{\delta}{\sigma\sqrt{2}}\Bigg] \geq c_1,
\end{align*}
where
\begin{align*}
c_1=c_1(\delta) = \frac{1}{2}\Big(1-\Phi(\delta)\Big).
\end{align*}
Then observe that the above implies that
\begin{align}
\|\hat{{\bf P}}_{n,2n}-\hat{{\bf Q}}_{n,2n}\|_{\text{TV}} \leq 1-c_1,
\label{TVD_2n}
\end{align}
for all~$n\geq n_3$.
Now, denote by~$\hat{\mu}_{n,k}$ the maximal coupling of~$\hat{{\bf P}}_{n,k}$ and~$\hat{{\bf Q}}_{n,k}$,
and by $(\hat{m}_1, \hat{m}_2)$ the elements of
$\hat{{\mathcal{M}}}_{k}\times \hat{{\mathcal{M}}}_{k}$.
Let~$\mathsf{K}$ be the coupling event (that is, $\mathsf{K}=\{(\hat{m}_1, \hat{m}_2): \hat{m}_1 = \hat{m}_2\}$), $\mathsf{K}_1$
the coupling event of~$(\hat{m}_i(\{\textbf{0}_1\}), \hat{m}_i(\{\textbf{0}_2\}))$,
for $i=1,2$, and $\mathsf{K}_2$ the coupling event of~$(\hat{m}_i(A))_{A\in\mathcal{T}}$, for $i=1,2$,
and also observe that~$\mathsf{K}=\mathsf{K}_1\cap\mathsf{K}_2$.
Thus, we deduce that, for all~$n,k\in\mathbb{N}$,
\begin{align*}
\|\hat{{\bf P}}_{n,k}-\hat{{\bf Q}}_{n,k}\|_{\text{TV}} &= 1- \hat{\mu}_{n,k}[\mathsf{K}] \\
&= 1- \sum_{\ell=0}^{k}\hat{\mu}_{n,k}[\mathsf{K}_1 \cap \mathsf{K}_2, \hat{m}_1(\{\textbf{0}_1,\textbf{0}_2\})= \hat{m}_2(\{\textbf{0}_1,\textbf{0}_2\})=\ell ] \\
&\geq \sum_{\ell=0}^{k} \Big(1-\hat{\mu}_{n,k}[\mathsf{K}_2 \mid \hat{m}_1(\{\textbf{0}_1,\textbf{0}_2\})= \hat{m}_2(\{\textbf{0}_1, \textbf{0}_2\})=\ell] \Big) \mathfrak{p}_{\ell}^{k} \\
&= \sum_{\ell=0}^{k} \hat{\mu}_{n,k}[\mathsf{K}_2^c \mid \hat{m}_1(\{\textbf{0}_1,\textbf{0}_2\})= \hat{m}_2(\{\textbf{0}_1,\textbf{0}_2\})=\ell] \mathfrak{p}_{\ell}^{k} \\
&\geq \sum_{\ell=0}^{k} \mathfrak{p}_{\ell}^{k} \|{\bf P}_{n,\ell}-{\bf Q}_{n,\ell}\|_{\text{TV}},
\end{align*}
where~$\mathfrak{p}_{\ell}^{k}$ is the probability mass function of a Binomial($k,1/2$) random variable at~$\ell$ and ${\bf P}_{n,\ell}$ (respectively,~${\bf Q}_{n,\ell}$) is a binomial process with paramenters $({\bf p}_n,\ell)$ (respectively,~$({\bf q}_n,\ell)$).
Using~\eqref{TVD_2n} and the fact
that~$\|\hat{{\bf P}}_{n,k}-\hat{{\bf Q}}_{n,k}\|_{\text{TV}}$ is non decreasing
in~$k$ (this follows from the fact
that~$\|\hat{{\bf P}}_{n,k}-\hat{{\bf Q}}_{n,k}\|_{\text{TV}}
= \frac{1}{2}E[|1-\mathcal{L}_k(X_1, \dots, X_k)|]$,
where~$(X_i)_{i\geq 1}$ are i.i.d random variables with
law~${\bf p}_n$ and~$\mathcal{L}_k(X_1, \dots, X_k)
:=\Pi_{i=1}^k\frac{\mathrm{d}{\bf q}_n}{\mathrm{d}{\bf p}_n}(X_i)$
is a martingale under the canonical filtration), we obtain that,
for all~$n\geq n_3$ and~$i \leq n$,
\begin{align}
\sum_{k=0}^{2i} \mathfrak{p}_{k}^{2i} \|{\bf P}_{n,k}-{\bf Q}_{n,k}\|_{\text{TV}}
\leq 1 - c_1.
\label{bound_sum}
\end{align}
Using again the Berry-Esseen theorem
(once again with~$1/2$ as an upper bound for the Berry-Esseen constant),
we can deduce that there exist $n_4=n_4(\delta)
=\lceil(\frac{12}{1-\Phi(\delta)})^2\rceil$ and $c_2=c_2(\delta)=-\frac{1}{2}\Phi^{-1}(\frac{1-\Phi(\delta)}{24})\geq1$,
such that for all $i\geq n_4$, we have
\begin{align*}
\sum_{k\in \mathcal{I}_i(\delta)} \mathfrak{p}_{k}^{2i}
\geq 1 - \frac{c_1}{3},
\end{align*}
where~$\mathcal{I}_i(\delta) := [i-c_2\sqrt{i}, i+c_2\sqrt{i}]$.
On the other hand, if $i\geq n_3$, by~\eqref{bound_sum} it follows that
\begin{align*}
\sum_{k\in \mathcal{I}_i(\delta)}
\mathfrak{p}_{k}^{2i} \|{\bf P}_{n,k}-{\bf Q}_{n,k}\|_{\text{TV}}
\leq 1 - c_1(\delta),
\end{align*}
so that, if $i\geq n_3\vee n_4$, there exists~$i_0\in\mathcal{I}_i(\delta)$ such that
\begin{align*}
\|{\bf P}_{n,i_0}-{\bf Q}_{n,i_0}\|_{\text{TV}}
\leq \frac{1 - c_1}{1 - \frac{c_1}{3}} \leq 1 - \frac{2}{3}c_1.
\end{align*}
To conclude the proof of the lemma, observe that for any~$n$ large enough, there exists~$i\geq n_3\vee n_4\vee c_2^2$ such that~$n-(i+\lfloor c_2\sqrt{i} \rfloor)\in[0,3]$,
the above argument allows to obtain~$i_0\in\mathbb{N}$ such that~$n-5c_2\sqrt{n} \leq i_0 \leq n$ and
\begin{align}
\|{\bf P}_{n,i_0}-{\bf Q}_{n,i_0}\|_{\text{TV}} \leq 1 - \frac{2}{3}c_1.
\label{i0}
\end{align}
Now, if $n-i_0\leq (n_3\vee n_4\vee c_2^2)+5\sqrt{5} c_2^{3/2}n^{1/4}$,
using (\ref{i0}) and making a point-by-point coupling between
the $n-i_0$ remaining points of the binomial processes~${\bf P}_{n}$
and~${\bf Q}_{n}$, we obtain that
\begin{align}
\|{\bf P}_{n}-{\bf Q}_{n}\|_{\text{TV}} \leq 1-\frac{2}{3}c_1(\delta)\Bigg(1-\frac{[(n_3\vee n_4\vee c_2^2)+5\sqrt{5} c_2^{3/2}n^{1/4}]\delta}{2\sqrt{n}}\Bigg)^+.
\label{Case1}
\end{align}
On the other hand, if
$n-i_0> (n_3\vee n_4\vee c_2^2)+5\sqrt{5} c_2^{3/2}n^{1/4}$,
we first consider $j\in \mathbb{N}$ such that
\begin{equation*}
n-i_0 - ( j-i_0+ \lfloor c_2\sqrt{j-i_0}\rfloor) \in [0,3].
\end{equation*}
Observe that in this case, $j-i_0> n_3\vee n_4\vee c_2^2$
and thus by the former analysis we obtain that there exists $j_0>i_0$
such that~$n-5\sqrt{5}c_2^{3/2}n^{1/4} \leq j_0 \leq n$ and
\begin{align}
\|{\bf P}_{n,j_0-i_0}-{\bf Q}_{n,j_0-i_0}\|_{\text{TV}} \leq 1 - \frac{2}{3}c_1.
\label{j0}
\end{align}
Using (\ref{i0}), (\ref{j0}) and performing a point-by-point
coupling between the $n-j_0$ remaining points of the binomial
processes~${\bf P}_{n}$ and~${\bf Q}_{n}$, we obtain that
\begin{align}
\|{\bf P}_{n}-{\bf Q}_{n}\|_{\text{TV}} \leq 1-\Big(\frac{2}{3}c_1\Big)^2\Bigg(1-\frac{5\sqrt{5} c_2^{3/2}n^{1/4}\delta}{2\sqrt{n}}\Bigg)^+.
\label{Case2}
\end{align}
Finally, using~(\ref{Case1}) and~(\ref{Case2}),
we obtain that there exists $n_5=n_5(\delta)$ such that,
for $n\geq n_5$ we have that
\begin{align}
\label{OPIO}
\|{\bf P}_{n}-{\bf Q}_{n}\|_{\text{TV}} \leq 1-c_3,
\end{align}
where $c_3=c_3(\delta)$ is a positive constant depending on~$\delta$.
To finish the proof of Lemma~\ref{Propmulti_2},
we consider the case $n<n_5$.
Since ${\bf q}_n\ll{\bf p}_n$ and
$|\frac{\mathrm{d}{\bf q}_n}{\mathrm{d}{\bf p}_n}(x)-1|\leq\delta$
for all~$x\in \Omega$, we first observe that
\begin{equation*}
\max_{1\leq n< n_5}\|{\bf p}_n-{\bf q}_n\|_{\text{TV}}\leq 1-\frac{1}{1+\delta}.
\end{equation*}
Then, from this last fact we obtain that
\begin{equation*}
\max_{1\leq n< n_5} \|{\bf P}_{n}-{\bf Q}_{n}\|_{\text{TV}}
\leq 1-\Big(\frac{1}{1+\delta}\Big)^{n_5}.
\end{equation*}
Together with~(\ref{OPIO}), this concludes the proof of
Lemma~\ref{Propmulti_2}.
\end{proof}
We now prove Theorem~\ref{Thm2}.
For this proof, we consider~$\varepsilon_0=\varepsilon$ in Assumption~\ref{assump_3}.
We start taking $i=i_1:=\lceil (2C_2)^{1/3}\rceil-1$
in Proposition~\ref{goodenv}, so that
\begin{equation*}
\mathbb{P}\Big[{\mathsf{A}}_{i_1} \;\Big|\; |{\mathfrak{H}}|> \frac{q_0^2}{6}n\Big]
\geq \frac{1}{2},
\end{equation*}
for all~$n\geq n_0$ (where~$n_0$ is from
Proposition~\ref{goodenv}). Then, using Proposition~\ref{Prop_H_comp},
we obtain that $\mathbb{P}[{\mathsf{A}}_{i_1}]\geq 1/4$, for $n\geq n_0$.
Now, observe that, by Lemma~\ref{Propmulti_2}, for all
$n\geq n'_0 = n'_0(\beta,\varphi, \kappa, \gamma, \varepsilon_0)
:=n_0 \vee [F(1+i_1)]^2$ (where $F$ is given in~(\ref{F_exp})
with~$\varepsilon=\varepsilon_0$) we obtain that
\begin{equation}
\label{TIRIO}
\|L_n^X - L_n^Y\|_{\text{TV}}\leq 1-\mathbb{P}[\Upsilon]
\leq 1-\mathbb{P}[\Upsilon,{\mathsf{A}}_{i_1}]\leq 1-\frac{1}{4}C_{10}((1+i_1)F).
\end{equation}
Then, to complete the proof we just observe that
\begin{equation}
\label{TYREL}
\max_{1\leq n< n'_0}\|L_n^X - L_n^Y\|_{\text{TV}}
\leq 1-\Big(\frac{1}{1+\varepsilon_0}\Big)^{n'_0}.
\end{equation}
Finally, using~(\ref{TIRIO}) and~(\ref{TYREL}), we conclude
the proof of Theorem~\ref{Thm2}.
\qed
\section*{Acknowledgements}
Diego F.~de Bernardini thanks São Paulo Research Foundation (FAPESP) (grant \#2016/13646-4) and Fundo de Apoio ao Ensino,
\`a Pesquisa e \`a Extensão (FAEPEX) (grant \#2866/16). Christophe Gallesco was partially supported by CNPq (grant 313496/2014-5).
Serguei Popov was partially supported by CNPq (grant 300886/2008--0).
| {
"timestamp": "2016-10-11T02:03:00",
"yymm": "1610",
"arxiv_id": "1610.02532",
"language": "en",
"url": "https://arxiv.org/abs/1610.02532",
"abstract": "In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of [11].",
"subjects": "Probability (math.PR)",
"title": "On uniform closeness of local times of Markov chains and i.i.d. sequences",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465639,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.7092019683920388
} |
https://arxiv.org/abs/1903.08646 | Bachet's game with lottery moves | Bachet's game is a variant of the game of Nim. There are $n$ objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number $m$. The player who takes the last object loses. We consider a variant of Bachet's game in which each move is a lottery over set $\{1,2,\ldots, m\}$. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to $1/2$ as $n$ tends to infinity. | \section{Introduction and main result}
Bachet's game was formulated in~\cite{Bachet} as follows. Starting from 1, two players add one after another some integer number not exceeding 10 to the sum. The player who is the first to reach 100, wins. This game can be considered as a variant of the game of Nim \cite{Bouton} (other variants can be found, for example, in \cite{Boros2018, Boros2019, Gray, Li, Moore}). One can easily find subgame perfect Nash equilibrium (SPNE) in Bachet's game with backward induction~\cite{Bachet}.
Now assume that at every move instead of choosing the exact number not exceeding some $m$, the player chooses some lottery (i.e. probability distribution)
over numbers $\{1, 2, \ldots, m\}$ from some set of available lotteries, observes realization of the lottery and then makes the corresponding move. Below we provide formal rules of the game that is considered in this paper.
\textbf{Bachet's game with lottery moves} (BGLM). The game is defined by the natural number $n$ of objects in the pile, the natural number $m$ and a set of available lotteries $K\subset S_m$, where $S_m$ is a simplex of all lotteries over numbers $\{1, 2, \ldots, m\}$. Two players take turns to choose a lottery from the set $K$. After making the choice, the player observes realization of the lottery and then takes the corresponding number of objects from the pile. The player who takes the last object loses, including the case when they have to take more objects than remains in the pile. Both players want to maximize the probability of their own victory.
Our main result is the following theorem.
\begin{theorem}\label{thm:main}
Fix arbitrary integer $m>1$ and some compact
set $K\subset S_m$ with the following properties:
\begin{equation}
\eta := \max_{(\pi_1,\ldots ,\pi_m) \in K } \max_{i \in \{1, \ldots, m\}} \pi_i < 1;
\label{eta}
\end{equation}
\begin{equation}
\nu := \min_{i\in \{1, \ldots, m\}} \max_{(\pi_1,\ldots ,\pi_m) \in K} \pi_i>0.
\label{nu}
\end{equation}
For any initial number of objects $n$, consider BGLM
with parameters $n$, $m$, $K$. This game has a non-empty set of SPNE.
Denote by $p_n$ the probability that the first player wins in arbitrary
SPNE.
Then $p_n$ does not depend on the choice of
SPNE and
\begin{equation}\label{eq:main}
\lim_{n\to\infty}p_n=\frac{1}{2}.
\end{equation}
\end{theorem}
\begin{remark}
It can be easily proved that if limit~\eqref{eq:main} exists, it has
to be equal to~$\frac12$. Assume by contradiction that
limit~\eqref{eq:main} exists and equals $a\neq\frac{1}{2}$. Take some
$\varepsilon<|a-\frac{1}{2}|$. Then, for some $N$ and all $i\geqslant
1$, it is true that $|p_{N+i}-a|<\varepsilon$. Consider two cases. If
$a>\frac12$, then $p_{N+i}>\frac{1}{2}$ for all $i=1,\ldots,m$ and it
follows that $p_{N+m+1}<\frac{1}{2}$. Indeed, if any move from the
initial position leads to a state with winning probability greater than
$\frac12$, then the winning probability for the initial position is less
than $\frac12$; formally it follows from~\eqref{eq:pk} below. Similarly,
if $a<\frac12$, then $p_{N+i}<\frac{1}{2}$ for all $i=1,\ldots,m$ and it
follows that $p_{N+m+1}>\frac{1}{2}$. This leads us to a contradiction
with the definition of $N$. Hence, the interesting part is the existence
of this limit.
\end{remark}
\begin{remark}
\autoref{thm:main} allows the following interpretation. Assume that the players play classical Bachet's game, but after choosing their move, they make mistakes and play other moves (including suboptimal ones) with some positive probability. Condition \eqref{eta} says that mistakes are unavoidable: there are no pure (i.e., not mixed) moves in the set of all possible moves $K$. This condition is an essential characteristic of BGLM; \eqref{eta} does not hold for classical Bachet's game ($\eta=1$ for the latter). It follows from~\autoref{thm:main} that the presence of unavoidable mistakes drastically changes the outcome of the game for large $n$. Condition~\eqref{nu} says that it is possible to take $i$ objects from the pile, $i=1,\ldots, m$, with positive probability. Condition \eqref{nu} also holds for classical Bachet's game since $\nu=1$ (one can take any number of objects with probability 1).
\end{remark}
\begin{conjecture*}
Though condition \eqref{nu} plays an important technical role in our proof, we believe~\autoref{thm:main} holds true even if this condition is violated.
\end{conjecture*}
\begin{remark}
In order to refine the set of all Nash equilibria in games in extensive form,
Selten introduced the concept of the trembling hand \cite{Selten}. This concept
takes into account the lack of perfect rationality and possibility of random
mistakes. If $\Gamma$ is a game in extensive form, construct a perturbed game by
assuming that in each information set of $\Gamma$ a player must mix all
available moves (including suboptimal ones) with some positive weight not less
than the predetermined value (which is a parameter of a particular information
set in a particular perturbed game). Thus, the set of all admissible mixed moves
in a particular information set is a compact subset of the simplex of all
lotteries over pure moves in this information set. This is similar to set $K$ in
the definition of BGLM. The difference is that in BGLM the set of admissible
mixed moves is the same in all information sets. Another major difference is
that in the trembling hand equilibrium concept, the key object is the limit of
the sequence of perturbed games as the severity of random mistakes tends to 0.
We keep this severity parameter ($1-\eta$ in our notation) fixed and vary the
number of objects in the pile, considering infinite horizon limit. Therefore, we
get different perspective on the role of slight mistakes.
\end{remark}
\section{Proof of the main result}
\subsection{Existence of SPNE}
We find SPNE by backward induction. Fix $m$ and $K$. Obviously, for
$n=1$, any move leads to losing, as the player has to
take at least one object in any case. Therefore, any move
of the first player is in the set of all SPNE and $p_1=0$.
For convenience reasons, let $p_s=1$ for any $s\le 0$.
Now assume we proved the existence of SPNE for all BGLM with no more than
$n=k-1$ objects. Consider BGLM with $n=k$ objects. Assume that after the
move of the first player, $i$ objects are taken from the pile. The second
player now plays BGLM with $n=k-i$ objects (becoming `first player' in this
subgame) and wins it with probability $p_{k-i}$ by induction hypothesis. If
the second player wins, the first player loses. Therefore, the probability
that the first player wins in this case is $1-p_{k-i}$.
By the law of total probility, for move $\boldsymbol{\pi}=(\pi_1, \ldots, \pi_m) \in
K$, the probability
that the first player wins is given by:
\begin{equation}\label{eq:wtpk}
\widetilde{p}_k(\boldsymbol{\pi})=1-\sum_{i=1}^m \pi_i p_{k-i}.
\end{equation}
The player wants to maximize this probability by choosing optimal $\boldsymbol{\pi}$. Function
$\widetilde{p}_k$ is continuous with respect to
$\boldsymbol{\pi}$ and therefore attains its maximum value on
compact set $K$. Then
\begin{equation}\label{eq:pk}
p_k=\max_{\boldsymbol{\pi} \in K}\widetilde{p}_k(\boldsymbol{\pi})
\end{equation}
and $\argmax_{\boldsymbol{\pi}} \widetilde{p}_k(\boldsymbol{\pi})$ is non-empty. Obviously, $p_k$
does not depend on the choice of the move. After the move, the number of objects in the pile will be reduced, hence, the
existence of SPNE now follows from the induction hypothesis.
\subsection{Limit behaviour}
In this section we prove~\eqref{eq:main}.
\subsubsection{The notation and the idea of the proof}
First, we introduce some notation. Let
\begin{gather*}
\mathcal{D}_n:=p_n-\frac{1}{2},\quad \Delta_n:=|\mathcal{D}_n|,\\
W_k=\{k, k-1, \ldots, k-m + 1\},\quad \overline{\Delta}_k=\max_{j \in
W_k} \Delta_j.
\end{gather*}
It is easy to show that sequence $\{\overline{\Delta}_k\}$ is non-increasing
(see~\autoref{lem:monot} and~\autoref{cor:monot}). Our goal is to show that
it is strictly decreasing and has zero limit.
Consider the state of a game with $k+1$ objects in the pile. Due to \eqref{eq:wtpk}-\eqref{eq:pk}, $\mathcal
D_{k+1}$ is a convex combination of values~$\mathcal D_j$, $j\in W_k$, taken
with a negative sign. If some of these values taken with nontrivial weights are less
by absolute value than their maximum possible value $\overline{\Delta}_k$,
their convex combination is also less than $\overline{\Delta}_k$ by absolute
value and $\Delta_{k+1} < \overline{\Delta}_k$. Moreover, the gap can be
estimated from below. This suggests a way to prove that
sequence $\{\overline{\Delta}_k\}$ is strictly decreasing and tends to zero.
However, it is also possible that the convex combination for $\mathcal
D_{k+1}$ includes (with nontrivial weights) only those $\mathcal D_j$
whose absolute values are (almost) equal to $\overline{\Delta}_k$. In this
case, $\Delta_{k+1}\approx \overline{\Delta}_{k}$ and no significant drop
occurs. Such cases should be considered separately.
Due to condition~\eqref{nu}, the player is allowed to put nontrivial
weight on any move $j$. Due to rationality, the player tends to put larger
weights on moves with smaller $\mathcal D_j$. The `worst case'
scenario
is when all $\mathcal D_j$'s, $j\in W_k$, are positive and (almost) equal to
$\overline{\Delta}_k$. We show that in this case $\mathcal D_{k-m}$ should
be negative and significantly larger by absolute value than
$\overline{\Delta}_k$, see details in \autoref{lem:k-m}.
This gives us a drop between $\overline{\Delta}_{k-m}$ and $\Delta_{k+1}$.
Another case that needs special attention is when there are several negative
values of $\mathcal D_j \approx -\overline{\Delta}_j$, $j\in W_k$. This case
is covered by \autoref{lem:drop}. There we prove that significant drops in
$\Delta_k$ occur at least for every additional $3m$ objects in the pile, and
the sequence $\{\Delta_k\}$ can be estimated from above by a decreasing
geometric progression and obtain the main result.
\subsubsection{Preliminary considerations}
\begin{lemma}[Monotonicity lemma]
For every integer $k>1$, $\Delta_{k} \le \overline{\Delta}_{k-1}$.
\label{lem:monot}
\end{lemma}
\begin{proof}
It follows from~\eqref{eq:wtpk}-\eqref{eq:pk} that
\[
p_{k}=1-\sum_{i=1}^{m}\pi_i p_{k-i}.
\]
for some $\boldsymbol{\pi} \in S$. We have:
\begin{multline}
\Delta_k=|\mathcal D_k|=\left|p_k-\frac{1}{2}\right|=
\left|\frac{1}{2}-\sum_{i=1}^{m}\pi_i p_{k-i}\right|=
\left|\sum_{i=1}^{m}\pi_i\left(\frac{1}{2}-p_{k-i}\right)\right|\le\\
\sum_{i=1}^m
\pi_i\left|\frac{1}{2}-p_{k-i}\right|=\sum_{i=1}^m \pi_i
\Delta_{k-i} \le \sum_{i=1}^m \pi_i
\overline{\Delta}_{k-1}=\overline{\Delta}_{k-1}.
\end{multline}
\end{proof}
\begin{corollary}\label{cor:monot}
For every integer $k>1$, $\overline{\Delta}_k \le
\overline{\Delta}_{k-1}$.
\end{corollary}
\begin{proof}
Indeed,
\begin{multline}
\overline{\Delta}_k=\max\{\Delta_k, \Delta_{k-1}, \ldots,
\Delta_{k-m+1}\} \le
\max\{\overline{\Delta}_{k-1}, \Delta_{k-1}, \ldots, \Delta_{k-m+1}\}= \\
\max\{\max\{\Delta_{k-1},\ldots, \Delta_{k-m}\},\Delta_{k-1},\ldots,
\Delta_{k-m+1}\}=\\
\max\{\Delta_{k-1},\ldots, \Delta_{k-m}\}=\overline{\Delta}_{k-1}.
\end{multline}
\end{proof}
\begin{lemma}[No long winning series]\label{lem:before-and-after}
Assume that for some integer $k>m$ and for all $j\in W_k$, $p_j>
\frac{1}{2}$. Then
\begin{equation}
\label{eq:after}
p_{k+1}<\frac{1}{2}
\end{equation}
and
\begin{equation}
\label{eq:before}
p_{k-m}\le \frac{1}{2}.
\end{equation}
\end{lemma}
\begin{proof}
First, let us prove~\eqref{eq:after}. For some $\boldsymbol{\pi} \in K$,
$$
p_{k+1}=1-\sum_{i=1}^m \pi_i p_{k-i+1} <1-\sum_{i=1}^m \pi_i
\frac{1}{2}=1-\frac{1}{2}=\frac{1}{2}.
$$
Now prove~\eqref{eq:before} by contradiction. Assume
$p_{k-m}>\frac{1}{2}$. Then one can apply~\eqref{eq:after} with $k$
decreased by 1
and prove that $p_{k}$ has to be less than $\frac{1}{2}$.
Contradiction.
\end{proof}
\subsubsection{Worst case analysis}
\begin{lemma}\label{lem:k-m}
Assume that for some $\varkappa\in (0,1)$, for some integer $k>1$ and for all $j\in W_k$ the following inequality holds:
\begin{equation}
p_{j}\geqslant \frac{1}{2} + (1-\varkappa)\Delta_{k+1}.
\label{assumption}
\end{equation}
Then the following inequality holds:
\begin{equation}
\Delta_{k+1} \leqslant \frac{\eta}{(2-\eta)(1-\varkappa)}\Delta_{k-m}.
\label{theta}
\end{equation}
\end{lemma}
\begin{proof}
Consider strategy $\boldsymbol{\pi}=(\pi_1, \ldots, \pi_m) \in K$ that allows the
player facing $k$ objects to reach the winning probability of $p_k$. It
follows from the definition (see~\eqref{eq:wtpk}) that
\begin{equation}\label{eq:pk1}
p_k = 1 - \sum_{i=1}^m p_{k-i} \pi_i.
\end{equation}
Note that due to~\autoref{lem:before-and-after}, $p_{k-m}\le
\frac{1}{2}$ and therefore $p_{k-m}=\frac{1}{2}-\Delta_{k-m}$. Put it
into~\eqref{eq:pk1}:
\begin{multline}
\label{pik}
p_k = 1-\left(\pi_m\left(\frac{1}{2}-\Delta_{k-m}\right) +
\sum\limits_{i=1}^{m-1}p_{k-i}\pi_i\right)=\\
1-\frac{\pi_m}{2}-\sum_{i=1}^{m-1}p_{k-i}\pi_i+\pi_m \Delta_{k-m}.
\end{multline}
Therefore,
\begin{equation}\label{eq:pmdkm}
\pi_m\Delta_{k-m} = p_k - 1 +\frac{\pi_m}{2} +
\sum\limits_{i=1}^{m-1}p_{k-i}\pi_i.
\end{equation}
Estimate $p_k$ and $p_{k-i}$ in~\eqref{eq:pmdkm} from below with
$\frac12 + (1-\varkappa)\Delta_{k+1}$ using
lemma assumption~\eqref{assumption}:
\begin{equation}\label{eq:pmdkm-le}
\pi_m\Delta_{k-m} \geqslant \frac{1}{2}+(1-\varkappa)\Delta_{k+1}- 1 +\frac{\pi_m}{2} + (1-\pi_m)\left(\frac{1}{2} + (1-\varkappa)\Delta_{k+1}\right).
\end{equation}
Here we also used the relation $\sum_{i=1}^{m-1} =1-\pi_m$. Simplifying the right-hand side of inequality~\eqref{eq:pmdkm-le}, we
get:
$$
\pi_m\Delta_{k-m} \geqslant \Delta_{k+1} (1-\varkappa)(2-\pi_m),
$$
or
\begin{equation}
\Delta_{k-m}\geqslant (1-\varkappa)\frac{2-\pi_m}{\pi_m}\Delta_{k+1}\geqslant (1-\varkappa)\frac{2-\eta}{\eta}\Delta_{k+1}
\label{seventeen}
\end{equation}
(from definition of $\eta$ and Theorem assumption (see \eqref{eta}), it follows that $\pi_m\leqslant\eta<1$). Then \eqref{theta} follows from \eqref{seventeen}.
\end{proof}
\subsubsection{Drop down for losing positions}
In this part we show that for every \emph{losing position} (i.e. position
with winning probability less than $1/2$), there is a `drop down' in the
value of $\Delta_k$.
\begin{lemma}\label{lem:drop-down}
There exists $\delta<1$ such that the following holds: if $p_{k+1}< 1/2$ for
some $k$, then
\begin{equation}\label{eq:drop-down}
\Delta_{k+1} \leqslant \delta \overline{\Delta}_{k-m}.
\end{equation}
\end{lemma}
We need the following lemma for the proof.
\begin{lemma}[Corridor lemma]\label{lem:corridor}
Assume that $p_{k+1}<1/2$.
Then
\begin{equation}
\label{corridor}
\max_{\substack{i\in W_k}}
\left(p_{i}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)
\ge \frac{\nu}{1-\nu}\max_{\substack{i\in
W_k}}\left(\frac{1}{2}+\Delta_{k+1}-p_{i}\right).
\end{equation}
\end{lemma}
The proof of \autoref{lem:corridor} is rather technical and is relegated to Appendix.
\begin{proof}[Proof of \autoref{lem:drop-down}]
Fix arbitrary $\tau$ such that
\begin{equation}\label{eq:tau-def}
0 < \tau < \frac{\nu}{1-\nu}\frac{2-2\eta}{2-\eta}.
\end{equation}
Such $\tau$ exists since $\nu \in (0,1)$ and $\eta \in (0,1)$. We show that
$$
\delta:=\max\left\{\frac{\eta}{2-\eta}\frac{\nu}{\nu-\tau+\nu\tau},
\frac{1}{1+\tau}\right\}
$$
satisfies~\eqref{eq:drop-down}. Due to~\eqref{eq:tau-def}, $0<\delta<1$.
Consider separately two cases.
\paragraph{Case 1.} For all $j\in W_k$
\begin{equation}
p_j-\frac{1}{2}\leqslant (1+\tau)\Delta_{k+1}.
\end{equation}
This inequality can be rewritten as
\begin{equation}
p_j-\left(\frac{1}{2}+\Delta_{k+1}\right)\leqslant \tau\Delta_{k+1}.
\end{equation}
Since the latter inequality is true for any $j\in W_k$, we obtain:
\begin{equation}
\max_{\substack{j\in W_k}} \left(p_{j}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\leqslant \tau\Delta_{k+1}.
\label{upper}
\end{equation}
According to Corridor lemma \ref{lem:corridor},
\begin{equation}
\max_{\substack{j\in W_k}} \left(p_{j}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\geqslant \frac{\nu}{1-\nu} \max_{\substack{j \in W_k}} \left(\frac{1}{2}+\Delta_{k+1}-p_{j}\right).
\label{lower}
\end{equation}
From \eqref{upper} and \eqref{lower} it follows that
\begin{equation}
\max_{\substack{j \in W_k}} \left(\frac{1}{2}+\Delta_{k+1}-p_{j}\right)\leqslant \frac{1-\nu}{\nu}\tau\Delta_{k+1}.
\end{equation}
Hence, for any $j\in W_k$ it is true that
\begin{equation}
\frac{1}{2}+\Delta_{k+1}-p_j\leqslant \frac{1-\nu}{\nu}\tau\Delta_{k+1},
\end{equation}
or
\begin{equation}
p_j\geqslant \frac{1}{2} + \left(1-\frac{1-\nu}{\nu}\tau\right)\Delta_{k+1}.
\end{equation}
Applying \autoref{lem:k-m} with $\varkappa=\frac{1-\nu}{\nu}\tau$, we obtain that
\begin{equation}
\Delta_{k+1}\leqslant \frac{\eta}{(2-\eta)\left(1-\frac{1-\nu}{\nu}\tau\right)}\Delta_{k-m},
\end{equation}
or
\begin{equation}
\Delta_{k+1}\leqslant
\frac{\eta}{2-\eta}\frac{\nu}{\nu-\tau+\nu\tau}\Delta_{k-m}\leqslant \delta
\Delta_{k-m}\le \delta \overline{\Delta}_{k-m}.
\end{equation}
\paragraph{Case 2.} There exists $i\in W_k$ such that
\begin{equation}
p_i-\frac{1}{2}> (1+\tau)\Delta_{k+1}.
\end{equation}
Then,
\begin{equation}
\Delta_{k+1}<\frac{1}{1+\tau}\left(p_i-\frac{1}{2}\right) \leqslant \delta
\Delta_i\leqslant \delta \overline{\Delta}_i \leqslant \delta
\overline{\Delta}_{k-m}.
\end{equation}
The last inequality is due to \autoref{cor:monot} and the fact that $i>k-m$.
\end{proof}
\subsubsection{Drop down for any positions}
\begin{lemma}\label{lem:drop}
For $\delta$ from~\autoref{lem:drop-down} and for all integer
$k>2m$,
\begin{equation}
\Delta_{k+1}\leqslant \delta \overline{\Delta}_{k-{2m}}.
\end{equation}
\end{lemma}
To prove~\autoref{lem:drop} we have to introduce new notation and prove an
auxiliary proposition. Let
\newcommand{\Delta^-}{\Delta^-}
\newcommand{\Delta^+}{\Delta^+}
\newcommand{\overline{\Delta}^-}{\overline{\Delta}^-}
\newcommand{\overline{\Delta}^+}{\overline{\Delta}^+}
\begin{align*}
\Delta^-_k =\max\left\{0, \frac{1}{2}-p_k\right\},&\quad
\Delta^+_k =\max\left\{0, p_k-\frac{1}{2}\right\},\\
\overline{\Delta}^-_k =\max_{i\in W_k} \Delta^-_i,&\quad
\overline{\Delta}^+_k =\max_{i\in W_k} \Delta^+_i.
\end{align*}
Obviously, $\overline{\Delta}_k=\max\{\overline{\Delta}^-_k, \overline{\Delta}^+_k\}$.
\begin{proposition}\label{prop:dkpdkm}
For any natural $k$ the following holds:
$$\Delta_{k+1}^+ \le \overline{\Delta}^-_k.$$
\end{proposition}
\begin{proof}
If $p_{k+1}\le 1/2$, then $\Delta^+_{k+1}=0\le \overline{\Delta}^-_{k}$ by definition of $\overline{\Delta}^-_k$. Consider case $p_{k+1}\ge 1/2$. Then for some $\boldsymbol{\pi} \in K$,
\begin{multline}
p_{k+1}-\frac{1}{2}=\frac{1}{2}-\sum_{i=1}^m \pi_i p_{k-i+1}=
\sum_{i=1}^m\pi_i \left(\frac{1}{2}- p_{k-i+1}\right) \\
\le \sum_{\substack{i=1,\\ p_{k-i+1} \le 1/2}}^{m}
\pi_i\left(\frac{1}{2}- p_{k-i+1}\right)
\le \sum_{\substack{i=1,\\ p_{k-i+1} \le 1/2}}^{m}
\pi_i \bar{\Delta}_k^- \\
\le \sum_{i=1}^m \pi_i \bar{\Delta}_k^-=\bar{\Delta}_k^-.
\end{multline}
\end{proof}
Now we can prove \autoref{lem:drop}.
\begin{proof}[Proof of \autoref{lem:drop}]
If $p_{k+1}<1/2$, \autoref{lem:drop-down} implies:
$$
\Delta_{k+1} \leqslant \delta \overline{\Delta}_{k-m} \leqslant
\delta \overline{\Delta}_{k-2m}
$$
and the lemma is proved. (The last inequality is due
to~\autoref{cor:monot}.)
Now assume $p_{k+1}\geqslant 1/2$. In this case
$\Delta_{k+1}=\Delta_{k+1}^+ \leqslant \overline{\Delta}^-_k$ due to
\autoref{prop:dkpdkm}. For all $j \in W_k$ such that $p_j<1/2$,
\autoref{lem:drop-down} implies:
$$
\Delta_j^-=\Delta_j\leqslant \delta
\overline{\Delta}_{j-1-m}\leqslant
\delta\overline{\Delta}_{k-2m}.
$$
Again, the last inequality is due to \autoref{cor:monot} since $j\geqslant
k-m+1$. Therefore, $\overline{\Delta}^-_k\leqslant \delta \overline{\Delta}_{k-2m}$.
This finishes the proof of \autoref{lem:drop}.
\end{proof}
\begin{corollary}\label{cor:final-drop}
For all integer $k>3m$, $\overline{\Delta}_{k}\leqslant
\delta \overline{\Delta}_{k-3m}$.
\end{corollary}
\begin{proof}
From definition of $\overline{\Delta}_{k}$, \autoref{lem:drop} and \autoref{cor:monot} it follows that
$$
\overline{\Delta}_{k} = \max(\Delta_k,\ldots , {\Delta}_{k-m+1})\leqslant \delta\max(\overline{\Delta}_{k-2m-1},\ldots , \overline{\Delta}_{k-3m})= \delta \overline{\Delta}_{k-3m}.
$$
\end{proof}
Now we are ready to finish the proof of the main result. Let $k_N=1+3mN$ for
arbitrary integer $N$. Inductive application of \autoref{cor:final-drop}
implies:
$$
\overline{\Delta}_{k_N}\leqslant
\delta^{N}\overline{\Delta}_1=\frac{1}{2}\delta^N\to 0\text{ as }N\to \infty.
$$
Due to monotonicity of $\overline{\Delta}_k$, this implies:
$$
\lim_{k\to \infty} \overline{\Delta}_k \to 0.
$$
By definition of $\overline{\Delta}_k$, $\Delta_k \leqslant
\overline{\Delta}_k$ and therefore:
$$
\lim_{k\to \infty} {\Delta}_k \to 0
$$
which is equivalent to~\eqref{eq:main}. \autoref{thm:main} is proved modulo~\autoref{lem:corridor}.
\bigskip
\textit{This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Declarations of interest: none.}
\section*{Appendix}
In this Appendix, we prove \autoref{lem:corridor}.
\begin{proof}
Take any $\boldsymbol{\pi}=(\pi_1, \ldots, \pi_m) \in K$. Since the players are
rational~\eqref{eq:pk},
$$
p_{k+1}\geqslant 1 - \sum_{i=1}^{m}\pi_ip_{k-i+1},
$$
or equivalently,
$$
\sum_{i=1}^{m}\pi_ip_{k-i+1} \geqslant 1 - p_{k+1}.
$$
Due to Lemma assumption, $p_{k+1}<\frac12$ and therefore
$p_{k+1}=\frac12-\Delta_{k+1}$. We have:
$$
\sum_{i=1}^{m}\pi_i p_{k-i+1} \geqslant
1 - \left(\frac{1}{2} - \Delta_{k+1}\right) =
\frac{1}{2} + \Delta_{k+1}.
$$
Then, the following inequality holds:
\begin{multline}
\label{zero}
\sum_{i=1}^{m}\pi_i\left(p_{k-i+1}-
\left(\frac{1}{2}+\Delta_{k+1}\right)\right)=\\
\sum_{i=1}^{m}\pi_i p_{k-i+1}-
\sum_{i=1}^{m}\pi_i\left(\frac{1}{2}+\Delta_{k+1}\right)\geqslant\\
\left(\frac{1}{2}+\Delta_{k+1}\right)-\left(\frac{1}{2}+\Delta_{k+1}\right)=0.
\end{multline}
Now take arbitrary
\begin{equation}
j\in \argmax_{1\leqslant i\leqslant m}\left(\frac{1}{2}+\Delta_{k+1}-p_{k-i+1}\right).
\label{jargmax}
\end{equation}
By definition of $\nu$ and Theorem assumption $\nu>0$ (see \eqref{nu}), there exists a strategy $\widehat\boldsymbol{\pi}=(\widehat\pi_1, \ldots, \widehat\pi_m) \in K$ such that
\begin{equation}
\widehat\pi_j\geqslant \nu > 0.
\label{pij}
\end{equation}
Inequality \eqref{zero} holds for arbitrary $\boldsymbol{\pi}$ and therefore it
holds for
$\widehat\boldsymbol{\pi}$. Rewrite it in the following way, separating the term with
$i=j$ from the rest of the sum:
$$
\sum_{\substack{1\leqslant i\leqslant m \\ i\neq j}}\widehat\pi_i\left(p_{k-i+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)+ \widehat\pi_j\left(p_{k-j+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\geqslant 0.
$$
Then we have the following sequence of estimates:
\begin{multline}
-\widehat\pi_j\left(p_{k-j+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\leqslant\sum_{\substack{1\leqslant i\leqslant m \\ i\neq j}}\widehat\pi_i\left(p_{k-i+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right) \leqslant
\\
\sum_{\substack{1\leqslant i\leqslant m \\ i\neq j}}\widehat\pi_i\max_{\substack{1\leqslant t\leqslant m}} \left(p_{k-t+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right) =
\\
\left(\sum_{\substack{1\leqslant i\leqslant m \\ i\neq j}}\widehat\pi_i\right)\cdot \max_{\substack{1\leqslant t\leqslant m }} \left(p_{k-t+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)=
\\
(1-\widehat\pi_j) \max_{\substack{1\leqslant t\leqslant m}} \left(p_{k-t+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right),
\label{pimax}
\end{multline}
where the last equality follows from the fact that
$$\sum_{i=1}^m\widehat\pi_i=1.$$
From \eqref{pimax} we derive the lower estimate for the left-hand side of the Corridor lemma inequality \eqref{corridor}:
\begin{multline}
\max_{\substack{1\leqslant t\leqslant m}} \left(p_{k-t+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\geqslant -\frac{\widehat\pi_j}{1-\widehat\pi_j} \left(p_{k-j+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)=
\\
\frac{\widehat\pi_j}{1-\widehat\pi_j} \left(\frac{1}{2}+\Delta_{k+1}-p_{k-j+1}\right).
\label{almost}
\end{multline}
Note that
$$
\frac{1}{2}+\Delta_{k+1}-p_{k-j+1}\geqslant 0.
$$
Indeed, otherwise, from the definition of $j$ (see \eqref{jargmax}) it
would follow that for all $i=1,\ldots, m$,
$$
\frac{1}{2}+\Delta_{k+1}-p_{k-i+1}< 0
$$
or
$$
p_{k-i+1}> \frac{1}{2}+\Delta_{k+1}.
$$
However, this is impossible because for optimal strategy
$(\pi_1,\ldots,\pi_m)$ we have:
$$p_{k+1}=1-\sum_{i=1}^m p_{k-i+1} \pi_i <1 -\sum_{i=1}^m \pi_i \left(\frac{1}{2}+\Delta_{k+1}\right) = \frac{1}{2}-\Delta_{k+1}$$
whereas $p_{k+1}=\frac{1}{2}-\Delta_{k+1}$ by definition.
Note that function $x\mapsto \frac{x}{1-x}$ is increasing for $x \in (0,1)$. Thus
we can estimate $\frac{\widehat \pi_j}{1-\widehat \pi_j}$ by
$\frac{\nu}{1-\nu}$ from below in~\eqref{almost} and obtain
\begin{multline}
\max_{\substack{1\leqslant t\leqslant m}} \left(p_{k-t+1}-\left(\frac{1}{2}+\Delta_{k+1}\right)\right)\geqslant \frac{\nu}{1-\nu} \left(\left(\frac{1}{2}+\Delta_{k+1}\right)-p_{k-j+1} \right)=
\\
\frac{\nu}{1-\nu}\max_{\substack{1\leqslant t\leqslant m}}\left(\frac{1}{2}+\Delta_{k+1}-p_{k-t+1}\right).
\end{multline}
The last equality follows from the definition of $j$ (see \eqref{jargmax}). This finishes the proof
of \autoref{lem:corridor} and the main result (\autoref{thm:main}).
\end{proof}
| {
"timestamp": "2019-10-16T02:04:53",
"yymm": "1903",
"arxiv_id": "1903.08646",
"language": "en",
"url": "https://arxiv.org/abs/1903.08646",
"abstract": "Bachet's game is a variant of the game of Nim. There are $n$ objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number $m$. The player who takes the last object loses. We consider a variant of Bachet's game in which each move is a lottery over set $\\{1,2,\\ldots, m\\}$. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to $1/2$ as $n$ tends to infinity.",
"subjects": "Optimization and Control (math.OC)",
"title": "Bachet's game with lottery moves",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465639,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.7092019683920388
} |
https://arxiv.org/abs/1001.1323 | Spectral clustering based on local linear approximations | In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higher-order spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairwise distances) of Ng, Jordan and Weiss (NIPS '01). The optimal choice for some of the tuning parameters depends on the dimension and thickness of the clusters. We provide estimators that come close enough for our theoretical purposes. We also discuss the cases of clusters of mixed dimensions and of clusters that are generated from smoother surfaces. In our experiments, this algorithm is shown to outperform pairwise spectral clustering on both simulated and real data. | \section{Introduction}
In a number of modern applications, the data appear to cluster near some low-dimensional structures. In the particular setting of manifold learning~\cite{Tenenbaum00ISOmap,Roweis00LLE,Belkin03,survey-kernel-spectral,DG05}, the data are assumed to lie near manifolds embedded in Euclidean space. When multiple manifolds are present, the foremost task is separating them, meaning the recovery of the different components of the data associated with the different manifolds.
Manifold clustering naturally occurs in the human visual cortex, which excels at grouping points into clusters of various shapes~\cite{Neumann2007189,Field1993173}. It is also relevant for a number of modern applications. For example, in cosmology, galaxies seem to cluster forming various geometric structures such as one-dimensional filaments and two-dimensional walls~\cite{galaxy-nonrandom,MarSaa}. In motion segmentation, feature vectors extracted from moving objects and tracked along different views cluster along affine or algebraic surfaces~\cite{Ma07,1530127,vidal2006unified,AtevKSCC}. In face recognition, images of faces in fixed pose under varying illumination conditions cluster near low-dimensional affine subspaces~\cite{Ho03,Basri03,Epstein95}, or along low-dimensional manifolds when introducing additional poses and camera views.
In the last few years several algorithms for multi-manifold clustering were introduced; we discuss them individually in Section~\ref{subsec:other_methods}. We focus here on spectral clustering methods, and in particular, study a prototypical multiway method relying on local linear approximations, with precursors appearing in~\cite{spectral_applied,Agarwal06,Shashua06,Agarwal05,Govindu05}. We refer to this method as Higher-Order Spectral Clustering (HOSC). We establish theoretical guarantees for this method within a standard mathematical framework for multi-manifold clustering. Compared with all other algorithms we are aware of, HOSC is able to separate clusters that are much closer together; equivalently, HOSC is accurate under much lower sampling rate than any other algorithm we know of. Roughly speaking, a typical algorithm for multi-manifold clustering relies on local characteristics of the point cloud in a way that presupposes that all points, or at least the vast majority of the points, in a (small enough) neighborhood are from a single cluster, except in places like intersections of clusters. In contrast, though HOSC is also a local method, it can work with neighborhoods where two or more clusters coexist.
\subsection{Higher-Order Spectral Clustering (HOSC)}
\label{sec:spectral}
We introduce our higher-order spectral clustering algorithm in this section, tracing its origins to the spectral clustering algorithm of Ng et al.~\cite{Ng02} and the spectral curvature clustering of Chen and Lerman~\cite{spectral_applied,spectral_theory}.
Spectral methods are based on building a neighborhood graph on the data points and partitioning the graph using its Laplacian~\cite{survey-kernel-spectral,1288832}, which is closely related to the extraction of connected components. The version introduced by Ng et al.~\cite{Ng02} is an emblematic example---we refer to this approach as SC. It uses an affinity based on pairwise distances. Given a scale parameter $\epsilon > 0$ and a kernel $\phi$, define
\begin{equation}
\label{eq:pair-affinity}
\alpha(\mathbf{x}_1,\mathbf{x}_2) = \left\{\begin{array}{ll}
\phi(\|\mathbf{x}_1 - \mathbf{x}_2\|/\epsilon), & \mathbf{x}_1 \neq \mathbf{x}_2; \\
0, & \mathbf{x}_1 = \mathbf{x}_2.
\end{array} \right.
\end{equation}
($\| \cdot \|$ denotes the Euclidean norm.)
Standard choices include the heat kernel $\phi(s) = \exp(-s^2)$ and the simple kernel $\phi(s) = {\bf 1}_{\{|s| < 1\}}$.
Let $\mathbf{x}_1, \dots, \mathbf{x}_N \in \mathbb R^D$ denote the data points.
SC starts by computing all pairwise affinities $\mathbf{W} = (W_{ij})$, with $W_{ij} = \alpha(\mathbf{x}_i, \mathbf{x}_j)$, for $i,j = 1, \dots, N$. It then computes the matrix $\mathbf{Z} = (Z_{ij}): Z_{ij} = W_{ij}/(D_i D_j)^{1/2}$, where $D_i = \sum_{1\leq j\leq N} W_{ij}$ is the degree of the $i$th point in the graph with similarity matrix $\mathbf{W}$. Note that $\mathbf{I} - \mathbf{Z}$ is the corresponding normalized Laplacian. Providing the algorithm with the number of clusters $K$, SC continues by extracting the top $K$ eigenvectors of $\mathbf{Z}$, obtaining a matrix $\mathbf{U} \in \mathbb R^{N \times K}$, and after normalizing its rows, uses them to embed the data into $\mathbb R^K$. The algorithm concludes by applying $K$-means to the embedded points. See Algorithm~\ref{algo:NJW} for a summary.
\begin{algo}[ht]
\caption{Spectral Clustering (SC)~\cite{Ng02} }
\centering
\begin{tabular}{p{4in}}
\textbf{Input:}\\
\hspace*{.3in} $\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_{\scriptscriptstyle N}$: the data points\\
\hspace*{.3in} $\epsilon$: the affinity scale\\
\hspace*{.3in} $K$: the number of clusters\\
\textbf{Output:}\\
\hspace*{.3in} A partition of the data into $K$ disjoint clusters\\[.1in]
\textbf{Steps:}\\
{\bf 1:} Compute the affinity matrix $\mathbf{W} = (W_{ij})$, with $W_{ij} = \alpha(\mathbf{x}_i, \mathbf{x}_j)$. \\
{\bf 2:} Compute the $\mathbf{Z} = (Z_{ij}): Z_{ij} = W_{ij}/(D_i D_j)^{1/2}$, where $D_i = \sum_j W_{ij}$. \\
{\bf 3:} Extract $\mathbf{U} = [\mathbf{u}_1, \dots, \mathbf{u}_{K}]$, the top $K$ eigenvectors of $\mathbf{Z}$. \\
{\bf 4:} Renormalize each {\it row} of $\mathbf{U}$ to have unit norm, obtaining a matrix $\mathbf{V}$. \\
{\bf 5:} Apply $K$-means to the row vectors of $\mathbf{V}$ in $\mathbb{R}^{K}$ to find $K$ clusters. \\
{\bf 6:} Accordingly group the original points into $K$ disjoint clusters. \\
\end{tabular}
\label{algo:NJW}
\end{algo}
Spectral methods utilizing multiway affinities were proposed to better exploit additional structure present in the data.
The spectral curvature clustering (SCC) algorithm of Chen and Lerman~\cite{spectral_applied, spectral_theory} was designed for the case of hybrid linear modeling where the manifolds are assumed to be affine, a setting that arises in motion segmentation~\cite{Ma07}. Assuming that the subspaces are all of dimension $d$---a parameter of the algorithm, SCC starts by computing the (polar) curvature of all $(d+2)$-tuples, creating an $N^{\otimes (d+2)}$-tensor. The tensor is then flattened into a matrix $\mathbf{A}$ whose product with its transpose, $\mathbf{W}=\mathbf{A}\bA'$, is used as an affinity matrix for the spectral algorithm SC.
(In practice, the algorithm is randomized for computational tractability.) Kernel spectral curvature clustering (KSCC)~\cite{AtevKSCC} is a kernel version of SCC designed for the case of algebraic surfaces.
The SCC algorithm (and therefore KSCC) is not localized in space as it fits a parametric model that is global in nature. The method we study here may be seen as a localization of SCC, which is appropriate in our nonparametric setting since the manifolds resemble affine surfaces locally. This type of approach is mentioned in publications on affinity tensors~\cite{Agarwal06,Shashua06,Agarwal05,Govindu05} and is studied here for the first time, to our knowledge. As discussed in Section~\ref{sec:discussion}, all reasonable variants have similar theoretical properties, so that we choose one of the simplest versions to ease the exposition. Concretely, we consider a multiway affinity that combines pairwise distances between nearest neighbors and the residual from the best $d$-dimensional local linear approximation. Formally, given a set of $m \geq d+2$ points, $\mathbf{x}_1, \dots, \mathbf{x}_{m}$, define
\begin{equation} \label{eq:Lambda}
\Lambda_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m}) = \min_{L \in \mathcal{A}_{d}} \, \max_{j=1,\dots,m} \, {\mathrm{dist}}(\mathbf{x}_j, L),
\end{equation}
where ${\mathrm{dist}}(\mathbf{x}, S) := \inf_{\mathbf{s} \in S} \|\mathbf{x} -\mathbf{s}\|$ for a subset $S \subset \mathbb R^D$ and $\mathcal{A}_{d}$ denotes the set of $d$-dimensional affine subspaces in $\mathbb R^D$.
In other words, $\Lambda_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m})$ is the width of the thinnest tube (or band) around a $d$-dimensional affine subspace that contains $\mathbf{x}_1, \dots, \mathbf{x}_{m}$.
(In our implementation, we use the mean-square error; see Section~\ref{sec:numerics}.)
Given scale parameters $\epsilon>\eta>0$ and a kernel function $\phi$, define the following affinity: $\alpha_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m}) = 0$ if $\mathbf{x}_1, \dots, \mathbf{x}_{m}$ are not distinct; otherwise:
\begin{equation}
\label{eq:linear-affinity}
\alpha_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m}) =
\phi\left(\frac{{\mbox{\rm diam}}(\mathbf{x}_1, \dots, \mathbf{x}_{m})}{\epsilon}\right) \cdot \phi\left(\frac{\Lambda_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m})}{\eta}\right),
\end{equation}
where ${\mbox{\rm diam}}(\mathbf{x}_1, \dots, \mathbf{x}_{m})$ is the diameter of $\{\mathbf{x}_1, \dots, \mathbf{x}_{m}\}$.
See Figure~\ref{fig:alpha} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.30\linewidth]{figures/affinity1.pdf} \
\includegraphics[width=.30\linewidth]{figures/affinity2.pdf} \
\includegraphics[width=.30\linewidth]{figures/affinity3.pdf}
\caption{The circle is of radius $\epsilon/2$ and the band is of half-width $\eta$. Assuming we use the simple kernel, the $m$-tuple on the left has affinity $\alpha_d$ equal to one, while the other two $m$-tuples have affinity equal to zero, the first one for having a diameter exceeding $\epsilon$ and the second one for being `thicker' than $\eta$.}
\label{fig:alpha}
\end{figure}
Given data points $\mathbf{x}_1, \dots, \mathbf{x}_{\scriptscriptstyle N}$ and approximation dimension $d$, we compute all $m$-way affinities, and then obtain pairwise similarities by clique expansion~\cite{Agarwal05} (note that several other options are possible~\cite{spectral_applied,Shashua06,Govindu05}):
\begin{equation}
\label{eq:W_def}
W_{ij} = \sum_{i_1, \dots, i_{m-2}} \alpha_{d}(\mathbf{x}_{i}, \mathbf{x}_{j}, \mathbf{x}_{i_1},\ldots,\mathbf{x}_{i_{m-2}}).
\end{equation}
Though it is tempting to choose $m$ equal to $d+2$, a larger $m$ allows for more tolerance to weak separation and small sampling rate. The down side is what appears to be an impractical computational burden, since the mere computation of $\mathbf{W}$ in \eqref{eq:W_def} requires order $O(N^m)$ flops. In Section~\ref{sec:complexity}, we discuss how to reduce the computational complexity to $O(N^{1 + o(1)})$ flops, essentially without compromising performance.
Once the affinity matrix $\mathbf{W}$ is computed, the SC algorithm is applied. We call the resulting procedure higher-order spectral clustering (HOSC), summarized in Algorithm~\ref{algo:linear}.
Note that HOSC is (essentially) equivalent to SC when $\eta \geq \epsilon$, and equivalent to SCC when $\epsilon = \infty$.
\begin{algo}[ht]
\caption{Higher Order Spectral Clustering (HOSC)}
\centering
\begin{tabular}{p{4in}}
\textbf{Input:}\\
\hspace*{.3in} $\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_{\scriptscriptstyle N}$: the data points\\
\hspace*{.3in} $d, m$: the approximation dimension and affinity order \\
\hspace*{.3in} $\epsilon, \eta$: the affinity scales\\
\hspace*{.3in} $K$: the number of clusters\\
\textbf{Output:}\\
\hspace*{.3in} A partition of the data into $K$ disjoint clusters\\[.1in]
\textbf{Steps:}\\
{\bf 1:} Compute the affinity matrix $\mathbf{W} = (W_{ij})$ according to (\ref{eq:W_def}). \\
{\bf 2:} Apply SC (Algorithm~\ref{algo:NJW}).
\end{tabular}
\label{algo:linear}
\end{algo}
\subsection{Generative Model}
\label{sec:setting}
It is time to introduce our framework. We assume a generative model where the clusters are the result of sampling points near surfaces embedded in an ambient Euclidean space, specifically, the $D$-dimensional unit hypercube $(0,1)^D$.
For a surface $S \subset (0,1)^D$ and $\tau > 0$, define its $\tau$-neighborhood as
\[
B(S, \tau) = \{\mathbf{x} \in (0,1)^D: {\mathrm{dist}}(\mathbf{x}, S) < \tau\}.
\]
The reach of $S$ is the supremum over $\tau > 0$ such that, for each $\mathbf{x} \in B(S, \tau)$, there is a unique point realizing $\inf\{\|\mathbf{x} - \mathbf{s}\|: \mathbf{s} \in S\}$~\cite{MR0110078}. It is well-known that, for $C^2$ submanifolds, the reach bounds the radius of curvature from below~\cite[Lem.~4.17]{MR0110078}.
For a connection to computational geometry, the reach coincides with the condition number introduced in~\cite{1349695} for submanifolds without boundary.
Let ${\rm vol}_d(S)$ denote the $d$-dimensional Hausdorff measure,
and $\partial S$ the boundary of $S$ within $(0,1)^D$.
For an integer $1 \leq d \leq D-1$ and a constant $\kappa \geq 1$, let $\mathcal{S}_d^2(\kappa)$ be the class of $d$-dimensional, connected, $C^2$ submanifolds $S \subset (0,1)^D$ of $1/\kappa \leq {\mbox{\rm diam}}(S) \leq \kappa$ and ${\rm reach}(S) \geq 1/\kappa$, and if $S$ has a boundary, $\partial S$ is a $(d-1)$-dimensional $C^2$ submanifold with ${\rm reach}(\partial S) \geq 1/\kappa$.
Given surfaces $S_1, \dots, S_{\scriptscriptstyle K} \in \mathcal{S}_d^2(\kappa)$ and $\tau < 1/\kappa$, we generate clusters $\mathcal{X}_1, \dots, \mathcal{X}_{\scriptscriptstyle K}$ by sampling $N_k$ points uniformly at random in $B(S_k, \tau)$, the $\tau$-neighborhood of $S_k$ in $(0,1)^D$, for all $k = 1, \dots, K$. We call $\tau$ the jitter level.
Except for Section~\ref{sec:intersect}, where we allow for intersections, we assume that the surfaces are separated by a distance of at least $\delta \geq 0$, i.e.
\begin{equation} \label{eq:delta}
{\mathrm{dist}}(S_k, S_\ell) := \inf_{\mathbf{x} \in S_k} \inf_{\mathbf{y} \in S_\ell} \|\mathbf{x} - \mathbf{y}\| \geq \delta, \quad \forall k \neq \ell.
\end{equation}
In that case, by the triangle inequality, the actual clusters are separated by at least $\delta - 2 \tau$, i.e.
\[
{\mathrm{dist}}(\mathcal{X}_k, \mathcal{X}_\ell) \geq \delta -2\tau.
\]
We assume that the clusters are comparable in size by requiring that $N_k \leq \zeta N_\ell$ for all $k \neq \ell$, for some finite constant $\zeta$. Let $\mathbf{x}_1, \dots, \mathbf{x}_{\scriptscriptstyle N}$ denote the data points thus generated. See Figure~\ref{fig:setting} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.45\linewidth]{figures/higher-cluster-sim-1} \
\includegraphics[width=.45\linewidth]{figures/higher-cluster-sim-2} \
\vspace{-.3in}
\caption{This figure illustrates the generative model. Left: Three surfaces (here curves) with their $\tau$-neighborhood. The curves are separated by at least $\delta$. Right: Points sampled within the tubular neighborhoods of the surfaces.}
\label{fig:setting}
\end{figure}
Given data $\mathcal{X} := \{\mathbf{x}_1, \dots, \mathbf{x}_{\scriptscriptstyle N}\}$, we aim at recovering the clusters $\mathcal{X}_1, \dots, \mathcal{X}_{\scriptscriptstyle K}$.
Formally, a clustering algorithm is a function taking data $\mathcal{X}$, and possibly other tuning parameters, and outputs a partition of $\mathcal{X}$. We say that it is `perfectly accurate' if the output partition coincides with the original partition of $\mathcal{X}$ into $\mathcal{X}_1, \dots, \mathcal{X}_{\scriptscriptstyle K}$.
Our main focus is on relating the sample size $N$ and the separation requirement in \eqref{eq:delta} (in order for HOSC to cluster correctly), and in particular we let $\tau$ and $\delta$ vary with $N$. This dependency is left implicit. In contrast, we assume that $d, K, \zeta$ are fixed. Also, we assume that $d, \tau, K$ are known throughout the paper (except for Section~\ref{sec:param} where we consider their estimation).
Though our setting is already quite general, we discuss some important extensions in Section~\ref{sec:discussion}.
We will also consider the situation where outliers may be present in the data. By outliers we mean points that were not sampled near any of the underlying surfaces. We consider a simple model where outliers are points sampled uniformly in $(0,1)^D \setminus \bigcup_k B(S_k, \delta_0)$ for some $\delta_0 > 0$, in general different from $\delta$. That is, outliers are at least a distance $\delta_0$ away from the surfaces. We let $N_0$ denote the number of outliers, while $N$ still denotes the total number of data points, including outliers. See Figure~\ref{fig:outliers} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\linewidth]{figures/higher-cluster-sim-10} \
\vspace{-.3in}
\caption{This figure illustrates the generative model with outliers included in the data.}
\label{fig:outliers}
\end{figure}
\subsection{Performance in terms of Separation and Robustness}
\subsubsection{Performance of SC}
A number of papers analyze SC under generative models similar to ours~\cite{pairwise,vonLuxburg08,pelletier11,LDS_NIPS_06}, and the closely related method of extracting connected components of the neighborhood graph~\cite{pairwise,1519716,brito,maier2007cluster}. The latter necessitates a compactly supported kernel $\phi$ and may be implemented via a union-of-balls estimator for the support of the density~\cite{MR579432}.
Under the weaker (essentially Lipschitz) regularity assumption
\begin{equation} \label{eq:S-vol}
C^{-1}\, \epsilon^d \leq {\rm vol}_d(B(\mathbf{s}, \epsilon) \cap S) \leq C\, \epsilon^d, \quad \forall \epsilon \in (0, 1/C), \, \forall \mathbf{s} \in S,
\end{equation}
Arias-Castro~\cite{pairwise} shows that SC with a compactly supported kernel is accurate if
\begin{equation} \label{eq:pair-sep}
\delta - 2 \tau \gg {\rm sep}_{\scriptscriptstyle N} := \left(\frac{\log N}{N}\right)^{1/d} \vee \, \tau^{1 - d/D} \left(\frac{\log N}{N}\right)^{1/D}.
\end{equation}
($a \vee b$ denotes the maximum of $a$ and $b$ and $a_{\scriptscriptstyle N} \gg b_{\scriptscriptstyle N}$ if $a_{\scriptscriptstyle N}/b_{\scriptscriptstyle N} \to \infty$ as $N \to \infty$).
With the heat kernel, the same result holds up to a $\sqrt{\log N}$ multiplicative factor.
See also~\cite{1519716,maier2007cluster}, which prove a similar result for the method of extracting connected components under stronger regularity assumptions. At the very least, \eqref{eq:pair-sep} is necessary for the union-of-balls approach and for SC with a compactly supported kernel, because ${\rm sep}_{\scriptscriptstyle N}$ is the order of magnitude of the largest distance between a point and its closest neighbor from the same cluster~\cite{penrose}.
Note that \eqref{eq:S-vol} is very natural in the context of clustering as it prevents $S$ from being too narrow in some places and possibly confused with two or more disconnected surfaces. And, when $C$ in \eqref{eq:S-vol} is large enough and $\kappa$ is small enough, it is satisfied by any surface $S$ belonging to $\mathcal{S}_d^2(\kappa)$. Indeed, such a surface resembles an affine subspace locally and \eqref{eq:S-vol} is obviously satisfied for an affine surface.
When outliers may be present in the data, as a preprocessing step, we identify as outliers data points with low connectivity in the graph with affinity matrix $\mathbf{W}$, and remove these points from the data before proceeding with clustering. (This is done between Steps 1 and 2 in Algorithm~\ref{algo:NJW}.) In the context of spectral clustering, this is very natural; see, e.g.,~\cite{spectral_applied,1519716,pairwise}.
Using the pairwise affinity \eqref{eq:pair-affinity}, outliers are properly identified if $\delta_0 -\tau$ satisfies the lower bound in \eqref{eq:pair-sep} and if the sampling is dense enough, specifically~\cite{pairwise},
\begin{equation} \label{eq:N-cond-lb}
N_k \geq (N^{d/D} \vee N \tau^{D-d}) \log(N), \quad \forall k=1,\dots,K.
\end{equation}
When the surfaces are only required to be of Lipschitz regularity as in \eqref{eq:S-vol}, we are not aware of any method that can even detect the presence of clusters among outliers if the sampling is substantially sparser.
\subsubsection{Performance of HOSC}
Methods using higher-order affinities are obviously more complex than methods based solely on pairwise affinities. Indeed, HOSC depends on more parameters and is computationally more demanding than SC. One, therefore, wonders whether this higher level of complexity is justified. We show that HOSC does improve on SC in terms of clustering performance, both in terms of required separation between clusters and in terms of robustness to outliers.
Our main contribution in this paper is to establish a separation requirement for HOSC which is substantially weaker than~\eqref{eq:pair-sep} when the jitter $\tau$ is small enough. Specifically, HOSC operates under the separation
\begin{equation} \label{eq:sep}
\delta - 2 \tau \gg (\tau \wedge {\rm sep}_{\scriptscriptstyle N}) \vee {\rm sep}_{\scriptscriptstyle N}^2,
\end{equation}
where $a \wedge b$ denotes the minimum of $a$ and $b$, and ${\rm sep}_{\scriptscriptstyle N}$ is the separation required for SC with a compactly supported kernel, defined in \eqref{eq:pair-sep}. This is proved in Theorem~\ref{th:linear} of Section~\ref{sec:same}.
In particular, in the jitterless case (i.e.~$\tau = 0$), the magnitude of the separation required for HOSC is (roughly) the square of that for SC at the same sample size; equivalently, at a given separation, HOSC requires (roughly) the square root of the sample size needed by SC to correctly identify the clusters.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/threecircles_data.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_njw.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_lscc.pdf}
\caption{Left: data. Middle: output from SC. Right: output from HOSC. The sampling is much sparser than in the original paper of Ng et al.~\cite{Ng02}, which is why SC fails. This figure is part of Figure~\ref{fig:artificial_data} in Section~\ref{sec:numerics}, which displays more numerical experiments. }
\label{fig:sep}
\end{figure}
That HOSC requires less separation than SC is also observed numerically. In Figure~\ref{fig:sep} we compare the outputs of SC and HOSC on the emblematic example of concentric circles given in~\cite{Ng02} (here with three circles).
While the former fails completely, the latter is perfectly accurate.
Indeed, SC requires that the majority of points in an $\epsilon$-ball around a given data point come from the cluster containing that point. In contrast, HOSC is able to properly operate in situations where the separation between clusters is so small, or the sampling rate is so low, that any such neighborhood is empty of data points except for the one point at the center.
To further illustrate this point, consider the simplest possible setting consisting of two parallel line segments in dimension $D = 2$, separated by a distance $\delta > 0$, specifically, $S_1 := \{(t,0): t \in [0,1]\}$ and $S_2 := \{(t,\delta): t \in [0,1]\}$. Suppose $N/2$ points are sampled uniformly on each of these line segments. It is well-known that the typical distance between a point on $S_k$ and its nearest neighbor on $S_k$ is of order $O(1/N)$; see~\cite{penrose}. Hence, a method computing local statistics requires neighborhoods of radius at least of order $1/N$, for otherwise some neighborhoods are empty. From \eqref{eq:sep}, HOSC is perfectly accurate when $\delta = (\log N)^3/N^2$, say. When the separation $\delta$ is that small, typical ball of radius of order $1/N$ around a data point contains about as many points from $S_1$ as from $S_2$ (thus SC cannot work). See Figure~\ref{fig:sep_small} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.48\linewidth]{figures/twolines_smallsep_njw.pdf}
\includegraphics[width=.48\linewidth]{figures/twolines_smallsep_hosc.pdf}
\caption{Clustering results obtained by SC (left) and HOSC (right) on a data set of two lines with small separation ($\delta=0.005$). 100 points are sampled from each line, equally spaced (at a distance $0.01$). Note that the inter-point separation on the same cluster is twice as large as the separation between clusters. In this case, SC cannot separate the two lines correctly, as we have argued. In contrast, HOSC performs perfectly when clustering the data, which again agrees with the theory and our expectation. We have also tried increasing the separation $\delta$ from $0.005$ to $0.025$, in which case both SC and HOSC perform correctly.}
\label{fig:sep_small}
\end{figure}
As a bonus, we also show that HOSC is able to resolve intersections in some (very) special cases, while SC is incapable of that. See Proposition~\ref{prop:intersect} and also Figure~\ref{fig:inter-sim}.
To make HOSC robust to outliers, we do exactly as described above, identifying outliers as data points with low connectivity in the graph with affinity matrix $\mathbf{W}$, this time computed using the multiway affinity \eqref{eq:linear-affinity}. The separation and sampling requirements are substantially weaker than \eqref{eq:N-cond-lb}, specifically, $\delta_0 -\tau$ is required to satisfy the lower bound in \eqref{eq:sep} and the sampling
\begin{equation} \label{eq:N-cond-lb-linear}
N_k \gg (N^{d/(2D -d)} \vee N \tau^{D-d}) \log(N), \quad \forall k=1,\dots,K.
\end{equation}
This is established in Proposition~\ref{prop:linear-outliers-2}, and again, we are not aware of any method for detection that is reliable when the sampling is substantially sparser.
For example, when $\tau = 0$ and we are clustering curves ($d=1$) in the plane ($D=2$) (with background outliers), the sampling requirement in \eqref{eq:N-cond-lb} is roughly $N_k \gg N^{1/2} \log(N)$, compared to $N_k \gg N^{1/3} \log(N)$ in \eqref{eq:N-cond-lb-linear}.
In Figure~\ref{fig:sep_out} below we compare both SC and HOSC on outliers detection, using the data in Figure~\ref{fig:sep} but further corrupted with $33.3\%$ outliers.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/threecircles_data_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_njw_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_lscc_outliers.pdf}
\caption{Left: data with outliers. Middle: outliers (black dots) detected by SC. Right: outliers (black dots) detected by HOSC. This figure is part of Figure~\ref{fig:artificial_data_outliers} in Section~\ref{sec:numerics}, where more outliers-removal experiments are conducted.
}
\label{fig:sep_out}
\end{figure}
\subsubsection{Other Methods} \label{subsec:other_methods}
We focus on comparing HOSC and SC to make a strong point that higher-order methods may be preferred to simple pairwise methods when the underlying clusters are smooth and the jitter level is small. In fact, we believe that no method suggested in the literature is able to compete with HOSC in terms of separation requirements. We quickly argue why.
The algorithm of Kushnir et al.~\cite{kushnir} is multiscale in nature and is rather complex, incorporating local information (density, dimension and principal directions) within a soft spectral clustering approach. In the context of semi-supervised learning, Goldberg et al.~\cite{goldberg2009multi} introduce a spectral clustering method based on a local principal components analysis (PCA) to utilize the unlabeled points. Both methods rely on local PCA to estimate the local geometry of the data and they both operate by coarsening the data, eventually applying spectral clustering to a small subset of points acting as representative hubs for other points in their neighborhoods. They both implicitly require that, for the most part, the vast majority of data points in each neighborhood where the statistics are computed come from a single cluster.
Souvenir and Pless~\cite{souvenir} suggest an algorithm that starts with ISOMAP and then alternates in EM-fashion between the cluster assignment and the computation of the distances between points and clusters---this is done in a lower dimensional Euclidean space using an MDS embedding. Though this iterative method appears very challenging to be analyzed, it relies on pairwise distances computed as a preprocessing step to derive the geodesic distances, which implicitly assumes that the points in small enough neighborhoods are from the same manifold.
Thus, like the SC algorithm, all these methods effectively rely on neighborhoods where only one cluster dominates. This is strong evidence that their separation requirements are at best similar to that of SC.
The methods of Haro et al.~\cite{Haro06} and Gionis et al.~\cite{gionis} are solely based on the local dimension and density, and are powerless when the underlying manifolds are of same dimension and sampled more or less uniformly, which is the focus of this paper.
The method of Guo et al.~\cite{energy} relies on minimizing an energy that, just as HOSC, incorporates the diameter and local curvature of $m$-tuples, with $m = 3$ for curves and $m = 4$ for surfaces in 3D, and the minimization is combinatorial over the cluster assignment. In principle, this method could be analyzed with the arguments we deploy here. That said, it seems computationally intractable.
\subsection{Computational Considerations}
\label{sec:complexity}
Thus it appears that HOSC is superior to SC and other methods in terms of separation between clusters and robustness to outliers, when the clusters are smooth and the jitter is small. But is HOSC even computationally tractable?
Assume $K$ and $D$ are fixed. The algorithm starts with building the neighborhood graph (i.e., computing the matrix $\mathbf{W}$). This may be done by brute force in $O(m N^m)$ flops. Clearly, this first step is prohibitive, in particular since we recommend using a (moderately) large $m$. However, we may restrict computations to points within distance $\epsilon$, which essentially corresponds to using a compactly supported kernel $\phi$. Hence, we could apply a range search algorithm to reduce computations. Alternatively, at each point we may restrict computations to its $\ell = \omega_{\scriptscriptstyle N} \log(N)$ nearest neighbors, with $\omega_{\scriptscriptstyle N} \to \infty$, or in a slightly different fashion, adapt the local scaling method proposed in~\cite{Zelnik-Manor04} by replacing $\epsilon$ in $\alpha_{d}(\mathbf{x}_{i_1}, \dots, \mathbf{x}_{i_m})$ by $(\epsilon_{i_1} \cdots \epsilon_{i_m})^{1/m}$, where $\epsilon_i$ denotes the distance between $\mathbf{x}_i$ and its $\ell$th nearest neighbor. The reason is that the central condition (\ref{eq:eps}) effectively requires that the degree at each point be of order $\log(N)^{m-1}$ (roughly), which is guaranteed if the $\ell$-nearest neighbors are included in the computations; see~\cite{pairwise,maier2007cluster} for rigorous arguments leading to that conclusion.
In low dimensions, $D = O(\log \log N)$, a range search and $\ell$-nearest-neighbor search may be computed effectively with kd-trees in $O(N {\rm poly}(\log N))$ flops. In higher dimensions, it is essential to use methods that adapt to the intrinsic dimensionality of the data. Assuming that $d$ is small, the method suggested in~\cite{1143857} has a similar computational complexity. Hence, the (approximate) affinity matrix $\mathbf{W}$ can be computed in order $O(N {\rm poly}(\log N)) + O(N \cdot \ell^m)$; assuming $m \leq \log(N)/(\omega_{\scriptscriptstyle N}\log\log(N))$, this is of order $O(N^{1 + 1/\omega_{\scriptscriptstyle N}})$.
This is within the possible choices for $m$ in Theorem~\ref{th:linear}.
Assume we use the $\ell$-nearest-neighbor approximation to the neighborhood graph, with $\ell = \omega_{\scriptscriptstyle N} \log(N)$. Then computing $\mathbf{Z}$ may be done in $O(N^{1 + 1/\omega_{\scriptscriptstyle N}})$ flops, since the affinity matrix $\mathbf{W}$ has at most $\ell^m = O(N^{1/\omega_{\scriptscriptstyle N}})$ non-zero coefficients per row. Then extracting the leading $K$ eigenvectors of $\mathbf{Z}$ may be done in $O(K N^{1 + 1/\omega_{\scriptscriptstyle N}})$ flops, using Lanczos-type algorithms~\cite{MR1948689}.
Thus we may run the $\ell$-nearest neighbor version of HOSC in $O(N^{1 + 1/\omega_{\scriptscriptstyle N}})$ flops, and it may be shown to perform comparably.
We actually implemented the $\ell$-nearest-neighbor variant of HOSC and tried it on a number of simulated datasets and a real dataset from motion segmentation. The results are presented in Section~\ref{sec:numerics}. The code is publicly available online~\cite{hosc}.
\subsection{Content}
The rest of the paper is organized as follows.
The main theoretical results are in Section~\ref{sec:same} where we provide theoretical guarantees for HOSC, including in contexts where outliers are present or the underlying clusters intersect. We emphasize that HOSC is only able to separate intersecting clusters under very stringent assumptions.
In the same section we also address the issue of estimating the parameters that need to be provided to HOSC. In theory at least, they may be chosen automatically.
In Section~\ref{sec:numerics} we implemented our own version of HOSC and report on some numerical experiments involving both simulated and real data.
Section~\ref{sec:discussion} discusses a number of important extensions, such as when the surfaces self-intersect or have boundaries, which are excluded from the main discussion for simplicity. We also discuss the case of manifolds of different intrinsic dimensions, suggesting an approach that runs HOSC multiple times with different $d$. And we describe a kernel version of HOSC that could take advantage of higher degrees of smoothness. Other extensions are also mentioned, including the use of different kernels.
The proofs are postponed to the Appendix.
\section{Theoretical Guarantees}
\label{sec:same}
Our main result provides conditions under which HOSC is perfectly accurate with probability tending to one in the framework introduced in Section~\ref{sec:setting}.
Throughout the paper, we state and prove our results when the surfaces have no boundary and for the simple kernel $\phi(s) = {\bf 1}_{\{|s| < 1\}}$, for convenience and ease of exposition. We discuss the case of surfaces with boundaries in Section~\ref{sec:boundary} and the use of other kernels in Section~\ref{sec:extensions}.
\begin{theorem}
\label{th:linear}
Consider the generative model of Section~\ref{sec:setting}. For $\rho_{\scriptscriptstyle N} \to \infty$ slowly (e.g., $\rho_{\scriptscriptstyle N} = \log \log N$), assume the parameters of HOSC satisfy
\begin{equation} \label{eq:m}
\log N \geq m \geq \frac{\log N}{\sqrt{\log \rho_{\scriptscriptstyle N}}},
\end{equation}
\begin{equation} \label{eq:eps}
\epsilon \geq \left(\rho_{\scriptscriptstyle N}^2 \frac{\log N}{N}\right)^{1/d} \vee \tau^{1 - d/D} \left(\rho_{\scriptscriptstyle N}^2 \frac{\log N}{N}\right)^{1/D}.
\end{equation}
and
\begin{equation} \label{eq:eta}
\eta \geq \epsilon \wedge (\tau + \rho_{\scriptscriptstyle N} \epsilon^2)
\end{equation}
Assume that (\ref{eq:delta}) holds with
\begin{equation} \label{eq:delta-lb}
\delta - 2 \tau > \epsilon \wedge \rho_{\scriptscriptstyle N} \eta.
\end{equation}
Under these conditions, when $N$ is large enough, HOSC is perfectly accurate with probability at least $1 - N^{-\rho_{\scriptscriptstyle N}}$.
\end{theorem}
To relate this to the separation requirement stated in the Introduction, the condition \eqref{eq:sep} is obtained from \eqref{eq:delta-lb} by choosing $\epsilon$ and $\eta$ equal to their respective lower bounds in \eqref{eq:eps} and \eqref{eq:eta}.
We further comment on the theorem. First, the result holds if $\rho_{\scriptscriptstyle N} = \rho$ and $\rho$ is sufficiently large. We state and prove the result when $\rho_{\scriptscriptstyle N} \to \infty$ as a matter of convenience. Also, by \eqref{eq:m} and \eqref{eq:delta-lb}, the weakest separation requirement is achieved when $m$ is at least of order slightly less than $O(\log N)$ so that $\rho_N$ is of order $O(1)$. However, as discussed in Section~\ref{sec:complexity}, the algorithm is not computationally tractable unless $m = o(\log N)$. This is another reason why we focus on the case where $\rho_{\scriptscriptstyle N} \to \infty$.
Regarding the constraints \eqref{eq:eps}-\eqref{eq:eta} on $\epsilon$ and $\eta$, they are there to guarantee that, with probability tending to one, each cluster is `strongly' connected in the neighborhood graph. Note that the bound on $\epsilon$ is essentially the same as that required by the pairwise spectral method SC~\cite{pairwise,maier2007cluster}.
In turn, once each cluster is `strongly' connected in the graph, clusters are assumed to be separated enough that they are `weakly' connected in the graph. The lower bound \eqref{eq:delta-lb} quantifies the required separation for that to happen. Note that it is specific to the simple kernel. For example, the heat kernel would require a multiplicative factor proportional to $\sqrt{\log N}$.
So how does HOSC compare with SC? When the jitter is large enough that $\tau \gg (\log(N)/N)^{1/d}$, we have $\eta \geq \epsilon$ and the local linear approximation contribution to (\ref{eq:linear-affinity}) does not come into play. In that case, the two algorithms will output the same clustering (see Figure~\ref{fig:tau_large} for an example).
\begin{figure}[htbp]
\centering
\includegraphics[width=.48\linewidth]{figures/twolines_largetau_njw.pdf}
\includegraphics[width=.48\linewidth]{figures/twolines_largetau_hosc.pdf}
\caption{Clustering results obtained by SC (left) and HOSC (right) on the data set of Figure~\ref{fig:sep_small}, but with separation $\delta=0.025$ and jitter $\tau=0.01$. In this example, neither SC nor HOSC can successfully separate the two lines. This example supports our claim that when the jitter is large enough (relative to separation), HOSC does not improve over SC and the two algorithms will output the same clustering.}
\label{fig:tau_large}
\end{figure}
When the jitter is small enough that $\tau \ll (\log(N)/N)^{1/d}$, HOSC requires less separation, as demonstrated in Figure~\ref{fig:sep_small}.
Intuitively, in this regime the clusters are sampled densely enough relative to the thickness $\tau$ that the smoothness of the underlying surfaces comes into focus and each cluster, as a point cloud, becomes locally well-approximated by a thin band.
We provide some numerical experiments in Section~\ref{sec:numerics} showing HOSC outperforming SC in various settings.
Thus, HOSC improves on SC only when the jitter is small. This condition is quite severe, though again, we do not know of any other method that can accurately cluster under the weak separation requirement displayed here, even in the jitterless case. It is possible that some form of scan statistic (i.e., matched filters) may be able to operate under the same separation requirement without needing the jitter to be small, however, we do not know how to compute it in our nonparametric setting---even in the case of hybrid linear modeling where the surfaces are affine, computing the scan statistic appears to be computationally intractable.
At any rate, the separation required by HOSC is essentially optimal when $\tau$ is of order $O(N^{-1/d})$ or smaller. A quick argument for the case $d=1$ and $D=2$ goes as follows. Consider a line segment of length one and sample $N$ points uniformly at random in its $\tau$-neighborhood, with $\tau = O(1/N)$. The claim is that this neighborhood contains an empty band of thickness of order slightly less than $O(1/N^2)$, and therefore cannot be distinguished from two parallel line segments. Indeed, such band of half-width $\lambda$ inside that neighborhood is empty of sample points with probability $(1 - \lambda/\tau)^N$, which converges to 1 if $N \lambda/\tau \to 0$, and when $\tau = O(1/N)$, this is the case if $\lambda = o(1/N^2)$.
In regards to the choice of parameters, the recommended choices depend solely on $(d, \tau, K)$. These model characteristics are sometimes unavailable and we discuss their estimation in Section~\ref{sec:param}. Afterwards, we discuss issues such as outliers (Section~\ref{sec:outliers}) and intersection (Section~\ref{sec:intersect}).
\subsection{Parameter Estimation}
\label{sec:param}
In this section, we propose some methods to estimate the intrinsic dimension $d$ of the data, the jitter $\tau$ and the number of clusters $K$. Though we show that these methods are consistent in our setting, further numerical experiments are needed to determine their potential in practice.
Compared to SC, HOSC requires the specification of three additional parameters. This is no small issue in practice. In theory, however, we recommend choosing $d$ and $K$ consistent with their true values, $\epsilon$ and $\eta$ as functions of $\tau$, and $m$ of order slightly less than $\log(N)$. The true unknowns are therefore $(d, \tau, K)$. We provide estimators for $d$ and $K$ that are consistent, and an estimator for $\tau$ that is accurate enough for our purposes. Specifically, we estimate $d$ and $\tau$ using the correlation dimension~\cite{cor-dim} and an adaptation of our own design.
The number of clusters $K$ is estimated via the eigengap of the matrix $\mathbf{Z}$.
\subsubsection{The Intrinsic Dimension and the Jitter Level}
A number of methods have been proposed to estimate the intrinsic dimensionality; we refer the reader to~\cite{levina-bickel} and references therein. The correlation dimension, first introduced in~\cite{cor-dim}, is perhaps the most relevant in our context, since surfaces may be close together. Define the pairwise correlation function
$$
{\rm Cor}(\epsilon) = \sum_{i} \sum_{j \neq i} {\bf 1}_{\{\| \mathbf{x}_i - \mathbf{x}_j \| \leq \epsilon\}}.
$$
The authors of~\cite{cor-dim} recommend plotting $\log {\rm Cor}(\epsilon)$ versus $\log \epsilon$ and estimating the slope of the linear part. We use a slightly different estimator that allows us to estimate $\tau$ too, if it is not too small. The idea is to regress $\log {\rm Cor}(\epsilon)$ on $\log \epsilon$ and identify a kink in the curve. See Figure~\ref{fig:cor-eps} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.50\linewidth]{figures/logcorr_eps.pdf}
\caption{A correlation curve for a simulated data set of 240 points sampled from the $\tau$-neighborhood of three disjoint one-dimensional curves ($d=1$) in dimension ten ($D=10)$ crossing all dimensions. The jitter is $\tau=0.01$. We see that the linear part of the curve has slope (near) 1, which coincides with the intrinsic dimension of the curves. The kink appears near $\hat{\tau} := \exp(-4.5) = 0.0111$, a close approximation to $\tau$.}
\label{fig:cor-eps}
\end{figure}
Though several (mostly ad hoc) methods have been proposed for finding kinks, we describe a simple method for which we can prove consistency.
Fix $\rho_{\scriptscriptstyle N} \to \infty$, with $\rho_{\scriptscriptstyle N} \ll \log N$. Define
$$
r_{\scriptscriptstyle N} = -\left[\frac{\log \log(N) -\log N}{d \log \rho_{\scriptscriptstyle N}}\right] -2.
$$
Let $A_r = \log {\rm Cor}(\rho_{\scriptscriptstyle N}^{-r})$.
If there is $r \in \{3, \dots, r_{\scriptscriptstyle N} - 2D-1\}$ such that
$$(A_r - A_{r+1})/\log \rho_{\scriptscriptstyle N} > D - 1/2,$$
then let $\hat{r} \geq 0$ be the smallest such $r$; otherwise, let $\hat{r} = r_{\scriptscriptstyle N} - 2D$.
Define $\hat{\tau} = \rho_{\scriptscriptstyle N}^{-\hat{r}}$; and also $\hat{d} = D$, if $\hat{r} = 3$, and $\hat{d}$ the closest integer to $(A_{3} - A_{\hat{r}})/(\hat{r} \log \rho_{\scriptscriptstyle N})$, otherwise.
\begin{proposition} \label{prop:tau-1}
Consider the generative model described in Section~\ref{sec:setting} with $S_1, \dots, S_{\scriptscriptstyle K} \in \mathcal{S}_{d}^2(\kappa)$. Assume that $\tau \leq \rho_{\scriptscriptstyle N}^{-3}$ and, if there are $N_0$ outliers, assume that $N -N_0 \geq N/\rho_{\scriptscriptstyle N}$. Then the following holds with probability at least $1 - N^{-\sqrt{\rho_{\scriptscriptstyle N}}}$: if $\hat{r} < r_{\scriptscriptstyle N} - 2D$, then $\tau \in [\hat{\tau}/\rho_{\scriptscriptstyle N}, \rho_{\scriptscriptstyle N} \hat{\tau}]$; if $\hat{r} = r_{\scriptscriptstyle N} - 2D$, then $\tau \leq \hat{\tau}$; moreover, if $\hat{r} > 3$, $\hat{d} = d$.
\end{proposition}
In the context of Proposition~\ref{prop:tau-1}, the only time that $\hat{d}$ is inconsistent is when $\tau$ is of order $\rho_{\scriptscriptstyle N}^{-3}$ or larger, in which case $\hat{d} = D$; this makes sense, since the region $\bigcup_k B(S_k, \tau)$ is in fact $D$-dimensional if $\tau$ is of order 1. Also, $\hat{\tau}$ is within a $\rho_{\scriptscriptstyle N}$ factor of $\tau$ if $\tau$ is not much smaller than $(\log(N)/N)^{1/d}$.
We now extend this method to deal with a smaller $\tau$. Consider what we just did. The quantity ${\rm Cor}(\epsilon)$ is the total degree of the $\epsilon$-neighborhood graph built in SC. Fixing $(d, m)$, we now consider the total degree of the $\eta$-neighborhood graph built in HOSC. Define the multiway correlation function
$${\rm Cor}_{d,m}(\epsilon, \eta) = \sum_{i} D_i^{1/(m-1)}.$$
Similarly, we shall regress $\log{\rm Cor}_{d,m}(\epsilon, \eta)$ on $\log \eta$ and identify a kink in the curve (Figure~\ref{fig:cor-eps-smalltau} displays such a curve).
\begin{figure}[htbp]
\centering
\includegraphics[width=.48\linewidth]{figures/logcorr_eps_njw_bad.pdf}
\includegraphics[width=.48\linewidth]{figures/logcorr_eps_hosc_good.pdf}
\caption{Correlation curves corresponding to SC (left) and HOSC (right) for the data set of Figure~\ref{fig:cor-eps}, but with a much smaller $\tau=1e-4$. We see that the pairwise correlation function works poorly in this case, while the multiway correlation curve has a kink near $\hat{\tau} := \exp(-10.5) = 2.754e-5$, within a factor of $\frac{1}{4}$ of the true $\tau$.}
\label{fig:cor-eps-smalltau}
\end{figure}
Using the multiway correlation function, we then propose an estimator $\hat{\tau}$ as follows.
We assume that the method of Proposition~\ref{prop:tau-1} returned $\hat{r} = r_{\scriptscriptstyle N} - 2D $, for otherwise we know that $\hat{\tau}$ is accurate. Choose $d = \hat{d}$ and $m \geq \log(N) (\log \rho_{\scriptscriptstyle N})^2$. Note that this is the only time we require $m$ to be larger than $\log N$. Let $B_{s} = \log {\rm Cor}_{d, m}(\rho_{\scriptscriptstyle N}^{-\hat{r}}, \rho_{\scriptscriptstyle N}^{-\hat{r}-s})$.
If there is $s \in \{0, \dots, \hat{r}-1\}$ such that
$$(B_s - B_{s+1})/\log \rho_{\scriptscriptstyle N} > D - d - 1/2,$$
then let $\hat{s}$ be the smallest one; otherwise, let $\hat{s} = \hat{r}$.
We then redefine $\hat{\tau}$ as $\hat{\tau} = \rho_{\scriptscriptstyle N}^{-\hat{r}-\hat{s}+1}$.
\begin{proposition} \label{prop:tau-2}
In the context of Proposition~\ref{prop:tau-1}, assume that $\hat{r} = r_{\scriptscriptstyle N} - 2D$. Then redefining $\hat{\tau}$ as done above, the following holds with probability at least $1 - N^{-\sqrt{\rho_{\scriptscriptstyle N}}}$: if $\hat{s} < \hat{r}$, then $\tau \in [\hat{\tau}/\rho_{\scriptscriptstyle N}, \rho_{\scriptscriptstyle N} \hat{\tau}]$; if $\hat{s} = \hat{r}$, then $\tau \leq \hat{\tau}$.
\end{proposition}
Now, $\hat{\tau}$ comes close to $\tau$ if $\tau$ is not much smaller than $(\log(N)/N)^{2/d}$. Whether this is the case, or not, the statement of Theorem~\ref{th:linear} applies with $\hat{\tau}$ in place of $\tau$ in \eqref{eq:eta}.
Though our method works in theory, it is definitely asymptotic. In practice, we recommend using other approaches for determining the location of the kink and the slope of the linear part of the pairwise correlation function (in log-log scale). Robust regression methods with high break-down points, like least median of squares and least trimmed squares, worked well in several examples. We do not provide details here, as this is fairly standard, but the figures are quite evocative.
\subsubsection{The Number of Clusters}
\label{sec:K}
HOSC depends on choosing the number of clusters $K$ appropriately. A common approach consists in choosing $K$ by inspecting the eigenvalues of $\mathbf{Z}$. We show that, properly tuned, this method is consistent within our model.
\begin{proposition}
\label{prop:K-choice}
Compute the matrix $\mathbf{Z}$ in HOSC with the same choice of parameters as in Theorem~\ref{th:linear}, except that knowledge of $K$ is not needed.
Set the number of clusters equal to the number of eigenvalues of $\mathbf{Z}$ (counting multiplicity) exceeding $1 - N^{-2}/\rho_{\scriptscriptstyle N}$. Then with probability at least $1 - N^{-\rho_{\scriptscriptstyle N}}$, this method chooses the correct number of clusters.
\end{proposition}
We implicitly assumed that $d$ and $\tau$ are known, or have been estimated as described in the previous section.
The proof of Proposition~\ref{prop:K-choice} is parallel to that of~\cite[Prop.~4]{pairwise}, this time using the estimate provided in part \ref{a1} of the proof of Theorem~\ref{th:linear}. Details are omitted.
Figure~\ref{fig:eigen-graph} illustrates a situation where the number of clusters is correctly chosen by inspection of the eigenvalues, more specifically, by counting the number of eigenvalue $1$ in the spectrum of $\mathbf{Z}$ (up to numerical error). This success is due to the fact that the clusters are well-separated, and even then, the eigengap is quite small.
\begin{figure}[htbp]
\centering
\includegraphics[width=.50\linewidth]{figures/eigs_Z_hosc}
\caption{The top six eigenvalues of the weight matrix $\mathbf{Z}$ obtained by HOSC in Step 2 for the same data used in Figure~\ref{fig:cor-eps}. Though in this example the clusters are well-separated, the eigengap is still very small (about 0.005).
}
\label{fig:eigen-graph}
\end{figure}
We apply this strategy to more data later in Section~\ref{sec:numerics}, and show that it can correctly identify the parameter $K$ in some cases (see Figure \ref{fig:eigs_hosc_syndata}).
In general we do not expect this method to work well when the data has large noise or intersecting clusters,
though we do not know of any other method that works in theory under our very weak separation requirements.
\subsection{When Outliers are Present}
\label{sec:outliers}
So far we have only considered the case where the data is devoid of outliers. We now assume that some outliers may be included in the data as described at the end of Section~\ref{sec:setting}. As stated there, we label as outlier any data point with low degree in the neighborhood graph, as suggested in~\cite{spectral_applied,1519716,pairwise}. Specifically, we compute $\mathbf{D}$ as in Step 2 of HOSC, and then label as outliers points $\mathbf{x}_i$ with degree $D_i$ below some threshold.
Let $\rho_{\scriptscriptstyle N} \to \infty$ slower than any power of $N$, e.g., $\rho_{\scriptscriptstyle N} = \log N$.
We propose two thresholds:
\renewcommand{\theenumi}{(O\arabic{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\begin{enumerate}
\item \label{o1} Identify as outliers points with degree:
$$D_i^{1/(m-1)} \leq \rho_{\scriptscriptstyle N}^{-1} \max_j D_j^{1/(m-1)}.$$
\item \label{o2} Identify as outliers points with degree:
$$D_i^{1/(m-1)} \leq \rho_{\scriptscriptstyle N} N \epsilon^{d} \eta^{D-d}.$$
\end{enumerate}
Taking up the task of identifying outliers, only the separation between outliers and non-outliers is relevant, so that we do not require any separation between the actual clusters.
We first analyze the performance of \ref{o1}, which requires about the same separation between outliers and non-outliers as HOSC requires between points from different clusters in \eqref{eq:delta-lb}.
\begin{proposition}
\label{prop:linear-outliers-1}
Consider the generative model described in Section~\ref{sec:setting}.
Assume that $N -N_0 \geq N/\rho_{\scriptscriptstyle N}$ and that \eqref{eq:m}-\eqref{eq:eta} hold.
In terms of separation, assume that $\delta_0 - \tau > \epsilon \wedge \rho_{\scriptscriptstyle N} \eta.$
Then with probability at least $1 - N^{-\rho_{\scriptscriptstyle N}}$, the procedure \ref{o1} identifies outliers without error.
\end{proposition}
We now analyze the performance of \ref{o2}, which requires a stronger separation between outliers and non-outliers, but operates under very weak sampling requirements.
\begin{proposition}
\label{prop:linear-outliers-2}
Assume that $m$ is as in \eqref{eq:m}, and
\begin{equation} \label{eq:eps-eta-O2}
\epsilon = (\rho_{\scriptscriptstyle N} \log(N)/N)^{1/(2D-d)}, \quad \eta = (\rho_{\scriptscriptstyle N} \log(N)/N)^{2/(2D-d)}.
\end{equation}
In terms of separation, assume that $\delta_0 - \tau > \epsilon$. In addition, suppose that
\begin{equation} \label{eq:eps-lb}
N_k \geq \rho_{\scriptscriptstyle N} \log(N) N^{d/(2D-d)} \vee N \tau^{D-d}, \ \forall k = 1, \dots, K.
\end{equation}
Then with probability at least $1 - N^{-\rho_{\scriptscriptstyle N}}$, the procedure \ref{o2} identifies outliers without error.
\end{proposition}
If $\delta_0 = \tau$, so that outliers are sampled everywhere but within the $\tau$-tubular regions of the underlying surfaces, then both \ref{o1} and \ref{o2} may miss some outliers within a short distance from some $B(S_k, \tau)$. Specifically, \ref{o1} (resp.~\ref{o2}) may miss outliers within $\epsilon \wedge \rho_{\scriptscriptstyle N} \eta$ (resp.~within $\epsilon$) from some $B(S_k, \tau)$. Using Weyl's tube formula~\cite{MR1507388}, we see that there are order $N_0 (\epsilon \wedge \rho_{\scriptscriptstyle N} \eta)^{D-d}$ (resp.~$N_0 \epsilon^{D-d}$) such outliers, a small fraction of all outliers.
The sampling requirement (\ref{eq:eps-lb}) is weaker than the corresponding requirement for pairwise methods displayed in \eqref{eq:N-cond-lb}. In fact, (\ref{eq:eps-lb}) is only slightly stronger than what is required to just detect the presence of a cluster hidden in noise. We briefly explain this point. Instead of clustering, consider the task of detecting the presence of a cluster hidden among a large number of outliers. Formally, we observe the data $\mathbf{x}_1, \dots, \mathbf{x}_{\scriptscriptstyle N}$, and want to decide between the following two hypotheses: under the null, the points are independent, uniformly distributed in the unit hypercube $(0,1)^D$; under the alternative, there is a surface $S_1 \in \mathcal{S}_d^2(\kappa)$ such that $N_1$ points are sampled from $B(S_1, \tau)$ as described in Section~\ref{sec:setting}, while the rest of the points, $N-N_1$ of them, are sampled from the unit hypercube $(0,1)^D$, again uniformly.
Assuming that the parameters $d$ and $\tau$ are known, it is shown in~\cite{AriasCastro2009,CTD} that the scan statistic is able to separate the null from the alternative if
\begin{equation} \label{eq:detect}
N_1 \gg N^{d/(2D-d)} \vee N \tau^{D-d}.
\end{equation}
We are not aware of a method that is able to solve this detection task at a substantially lower sampling rate, and (\ref{eq:eps-lb}) comes within a logarithmic factor from (\ref{eq:detect}).
We thus obtain the remarkable result that accurate clustering is possible within a log factor of the best (known) sampling rate that allows for accurate detection in the same setting.
\subsection{When Clusters Intersect}
\label{sec:intersect}
We now consider the setting where the underlying surfaces may intersect. The additional conditions we introduce are implicit constraints on the dimension of, and the incidence angle at, the intersections. We suppose there is an integer $0 \leq d_{\rm int} \leq d-1$ and a finite constant $C > 0$ such that
\begin{equation} \label{eq:d-cap}
{\rm vol}_{d}(B(S_k \cap S_\ell, \epsilon) \cap S_k) \leq C \epsilon^{d-d_{\rm int}}, \ \forall \epsilon \in (0,1/\kappa), \ \forall k \neq \ell.
\end{equation}
(The subscript $_{\rm int}$ stands for `intersection'.)
In addition, we assume that for some $\theta_{\rm int} \in (0, \pi/2]$,
\begin{equation} \label{eq:delta-cap}
{\mathrm{dist}}(\mathbf{x}, S_\ell) \geq \delta \wedge \sin(\theta_{\rm int}) {\mathrm{dist}}(\mathbf{x}, S_k \cap S_\ell), \ \forall \mathbf{x} \in S_k, \ \forall k \neq \ell \text{ with } S_k \cap S_\ell \neq \emptyset.
\end{equation}
(\ref{eq:d-cap}) is slightly stronger than requiring that $S_k \cap S_\ell$ has finite $d_{\rm int}$-dimensional volume. If the surfaces are affine, it is equivalent to the condition $\dim(S_k \cap S_\ell) \leq d_{\rm int}, \ \forall k \neq \ell.$
(\ref{eq:delta-cap}), on the other hand, is a statement about the minimum angle at which any two surfaces intersect. For example, if the surfaces are affine within distance $\delta$ of their intersection, then (\ref{eq:delta-cap}) is equivalent to their maximum (principal) angle being bounded from below by $\theta_{\rm int}$. See Figure~\ref{fig:intersect} for an illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=.450\linewidth]{figures/higher-cluster-sim-8} \
\includegraphics[width=.450\linewidth]{figures/higher-cluster-sim-9} \
\vspace{-.3in}
\caption{Illustration of intersecting surfaces. Though the human eye easily distinguishes the two clusters, the clustering task is a lot harder for machine learning algorithms. The main issue is that there are too many data points at the intersection of the two tubular regions. However, in very special cases HOSC {\em is} able to separate intersecting clusters (see Figure~\ref{fig:inter-sim} for such an example).}
\label{fig:intersect}
\end{figure}
\begin{proposition}
\label{prop:intersect}
Consider the setting of Theorem~\ref{th:linear}, with (\ref{eq:delta}) replaced by (\ref{eq:delta-cap}). In addition, assume that (\ref{eq:d-cap}) holds. Define
$$\gamma_{\scriptscriptstyle N} := N^2 \epsilon^{d} (\epsilon \wedge \rho_{\scriptscriptstyle N} \eta)^{d-d_{\rm int}} (\sin \theta_{\rm int})^{d_{\rm int}-d}.$$
Then there is a constant $C > 0$ such that, with probability at least $1 - C\, \gamma_{\scriptscriptstyle N}$, HOSC is perfectly accurate.
\end{proposition}
The most favorable case is when $\tau = 0$ and $\theta_{\rm int} = \pi/2$. Then with our choice of $\epsilon$ and $\eta$ in Theorem~\ref{th:linear}, assuming $\rho_{\scriptscriptstyle N}$ increases slowly, e.g., $\rho_{\scriptscriptstyle N} \prec \log N$, we have $\gamma_{\scriptscriptstyle N} \to 0$ if $2 d_{\rm int} < d$, and partial results suggest this cannot be improved substantially. This constraint on the intersection of two surfaces is rather severe. Indeed, a typical intersection between two (smooth) surfaces of same dimension $d$ is of dimension $d-1$, and if so, only curves satisfy this condition. Figure~\ref{fig:inter-sim} provides a numerical example showing the algorithm successfully separating two intersecting one-dimensional clusters.
Thus, even with no jitter and the surfaces intersecting at right angle, HOSC is only able to separate intersecting clusters under exceptional circumstances. Moreover, even when the conditions of Proposition~\ref{prop:intersect} are fulfilled, the probability of success is no longer exponentially small, but is at best of order $(1/N)^{1-2d_{\rm int}/d}$.
That said, SC does not seem able to properly deal with intersections at all (see also Figure~\ref{fig:inter-sim}). It essentially corresponds to taking $\eta = \epsilon$ in HOSC, in which case $\gamma_{\scriptscriptstyle N}$ never tends to zero.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/fourcircles_data.pdf}
\includegraphics[width=.32\linewidth]{figures/fourcircles_lscc.pdf}
\includegraphics[width=.32\linewidth]{figures/fourcircles_njw.pdf}
%
\caption{Left: data. Middle: output from HOSC. Right: Output from SC. This example shows that HOSC is able to separate intersecting curvilinear clusters when the incidence angle is perpendicular and there is no jitter ($\tau=0$). In particular, the conditions of Proposition~\ref{prop:intersect} are satisfied. On the contrary, SC fails in this case.}
\label{fig:inter-sim}
\end{figure}
Though the implications of Proposition~\ref{prop:intersect} are rather limited, we do not know of any other clustering method which provably separates intersecting clusters under a similar generative model. This is a first small step towards finding such a method.
\section{Software and Numerical Experiments}
\label{sec:numerics}
We include in this section a few experiments where a preliminary implementation of HOSC outperforms SC, to demonstrate that higher-order affinities can bring a significant improvement over pairwise affinities in the context of manifold clustering.
In our implementation of HOSC, we used the heat kernel $\phi(s)=\exp(-s^2)$. Following the discussion in Section~\ref{sec:complexity},
at each point we restrict the computations to its $\ell$ nearest neighbors so that we practically remove the locality parameter $\epsilon$ from the affinity function of \eqref{eq:linear-affinity} and obtain
\begin{equation}
\label{eq:linear-affinity_noeps}
\alpha_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m}) = \begin{cases}
\phi\left({\Lambda_{d}(\mathbf{x}_1, \dots, \mathbf{x}_{m})}/{\eta}\right), & \mathrm{if}\, \mathbf{x}_2, \ldots, \mathbf{x}_{m} \in \textrm{$\ell$-NN}(\mathbf{x}_1) \textrm{ distinct}; \\ 0, & \textrm{otherwise},
\end{cases}
\end{equation}
where $\textrm{$\ell$-NN}(\mathbf{x}_1)$ is the set of the $\ell$ nearest neighbors of $\mathbf{x}_1$ .
For computational ease, we used
\begin{equation}
\Lambda_{d}^{(2)}(\mathbf{x}_1, \dots, \mathbf{x}_{m}) = \min_{L \in \mathcal{A}_{d}} \ \sqrt{\frac{1}{m} \ \sum_{j=1}^m {\mathrm{dist}}(\mathbf{x}_j, L)^2},
\end{equation}
which can be easily computed using the bottom $m-d$ singular values of the $m$ points. Note that, since
$\Lambda_{d}/\sqrt{m} \leq \Lambda_{d}^{(2)} \leq
\Lambda_{d},$ the results we obtained apply, with $\eta$ changed
by a $\sqrt{m}$ factor, at most. (In the paper, the standard choice for $\eta$ is a power of $N$, while $m$ is of order at most $\log N$, so this factor is indeed negligible.) In practice, we always search a subinterval of $[0,1]$ for the best working $\eta$ (e.g., $[.001, .1]$), based on the smallest variance of the corresponding clusters
in the eigenspace (the row space of the matrix $\mathbf{V}$), as suggested in~\cite{Ng02}. When the given data contains outliers, the optimal choice of $\eta$ is based on the largest gap between the means of the two sets of degrees (associated to the inliers and outliers), normalized by the maximum degree. The code is available online~\cite{hosc}.
\subsection{Synthetic Data}
We first generate five synthetic data sets in the unit cube $(0,1)^D$ ($D=2$ or $3$), shown in Figure~\ref{fig:artificial_data}. In this experiment, the actual number of clusters (i.e.~$K$) and dimension of the
underlying manifolds (i.e.~$d$) are assumed known to all algorithms. For HOSC, we fix $\ell = 10, m=d+2$, and use the subinterval $[0.001, 0.1]$ as the search interval of $\eta$.
For SC, we considered
two ways of tuning the scale parameter $\epsilon$: directly, by
choosing a value in the interval $[0.001, 0.25]$ (SC-NJW); and by the local scaling method of~\cite{Zelnik-Manor04} (SC-LS), with the number of nearest neighbors $\ell
= 5, \dots, 15$. The final choices of these parameters were also based on the same criterion as used by HOSC.
Figure~\ref{fig:artificial_data} exhibits the clusters found by each algorithm when applied to the five data sets, respectively. Observe that HOSC succeeded in
a number of difficult situations for SC, e.g.,
when the sampling is sparse, or when the separation is small at
some locations.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/threecircles_data.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_njw.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_lscc.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_data.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_njw.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_lscc.pdf}
%
\includegraphics[width=.32\linewidth]{figures/squiggles_data.pdf}
\includegraphics[width=.32\textwidth]{figures/squiggles_njw.pdf}
\includegraphics[width=.32\textwidth]{figures/squiggles_lscc.pdf}
%
\includegraphics[width=.32\linewidth]{figures/sphere_ellipsoid_data.pdf}
\includegraphics[width=.32\textwidth]{figures/sphere_ellipsoid_njw.pdf}
\includegraphics[width=.32\textwidth]{figures/sphere_ellipsoid_lscc.pdf}
\includegraphics[width=.32\linewidth]{figures/twomoons_data.pdf}
\includegraphics[width=.32\linewidth]{figures/twomoons_njw.pdf}
\includegraphics[width=.32\linewidth]{figures/twomoons_lscc.pdf}
%
\caption{Left column: data. (The third example shows a sphere containing an ellipsoid inside.) Middle column: best output from SC with the scale parameter chosen by both searching the interval $[0.001, 0.25]$ and applying local scaling~\cite{Zelnik-Manor04} with at most 15 nearest neighbors. Right column: output from HOSC. The optimal value of $\eta$ is selected from the interval $[0.001, 0.1]$. We also tried the simple kernel instead of the heat kernel, and obtained same results except in data set 3.}
\label{fig:artificial_data}
\end{figure}
We also plot the leading eigenvalues of the matrix $\mathbf{Z}$ obtained by HOSC on each data set; see Figure~\ref{fig:eigs_hosc_syndata}. We see that in data sets 1, 2, 5, the number of eigenvalue 1 coincides with the true number of clusters, while in 3 and 4 there is some discrepancy between the $K$th eigenvalue and the number 1. Though we do not expect the eigengap method to work well in general, Figure~\ref{fig:eigs_hosc_syndata} shows that it can be useful in some cases.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/eigs_Z_hosc_threecircles.pdf}
\includegraphics[width=.32\linewidth]{figures/eigs_Z_hosc_curvesline.pdf}
\includegraphics[width=.32\linewidth]{figures/eigs_Z_hosc_squiggles.pdf}\\
\includegraphics[width=.32\linewidth]{figures/eigs_Z_hosc_sphereellipsoid.pdf}
\includegraphics[width=.32\linewidth]{figures/eigs_Z_hosc_twomoons.pdf}\\
\caption{Top eigenvalues of the matrix $\mathbf{Z}$ obtained by HOSC on each of the five data sets in Figure~\ref{fig:artificial_data} (in same order).}
\label{fig:eigs_hosc_syndata}
\end{figure}
Figure~\ref{fig:artificial_data_outliers} displays some experiments including outliers. We simply sampled points from the unit square $(0,1)^2$ uniformly at random and added them as outliers to the first three data sets in Figure
\ref{fig:artificial_data}, with percentages 33.3\%, 60\% and 60\%, respectively.
We applied SC and HOSC assuming knowledge of the proportion of outliers, and labeled points with smallest degrees as outliers. Choosing the threshold automatically remains a challenge; in particular, we
did not test the theory.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\linewidth]{figures/threecircles_data_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_njw_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/threecircles_lscc_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_data_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_njw_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/curvesline_lscc_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/squiggles_data_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/squiggles_njw_outliers.pdf}
\includegraphics[width=.32\linewidth]{figures/squiggles_lscc_outliers.pdf}
\caption{Outlier-removal experiments. Left column: data with outliers.
The percentages of outliers are 33.3\%, 60\% and 60\%, respectively.
Middle: outliers (black dots) detected by pairwise spectral clustering (both SC-NJW and SC-LS, but only the better result is shown).
Right: outliers (black dots) detected by HOSC.
The use of the simple kernel (instead of the heat kernel) in HOSC gives very similar results.}
\label{fig:artificial_data_outliers}
\end{figure}
We observe that HOSC could successfully
remove most of the true outliers, leaving out smooth structures in
the data; in contrast, SC tended to keep
isolated high-density regions, being insensitive to sparse smooth
structures. A hundred replications of this experiment (i.e., fixing the
clusters and adding randomly generated outliers) show that the True
Positive Rates (i.e., percentages of correctly identified outliers)
for (SC, HOSC) are (58.1\% vs 67.7\%), (75.4\% vs 86.8\%) and (76.8\% vs 88.0\%), respectively.
\subsection{Real Data}
We next compare SC and HOSC using the two-view motion data studied in \cite{AtevKSCC,RAS}. This data set contains 13 motion sequences:
(1) \emph{boxes}, (2) \emph{carsnbus3}, (3) \emph{deliveryvan}, (4) \emph{desk}, (5) \emph{lightbulb}, (6) \emph{manycars}, (7) \emph{man-in-office}, (8) \emph{nrbooks3}, (9) \emph{office}, (10) \emph{parking-lot}, (11) \emph{posters-checkerboard}, (12) \emph{posters-keyboard}, and (13) \emph{toys-on-table}; and each sequence consists of two image frames of a 3-D dynamic scene taken
by a perspective camera (see Figure \ref{fig:twoview_examples} for a few such sequences). Suppose that several feature points have been extracted from the moving objects in the two camera views of the scene. The task is to separate
the trajectories of the feature points according to different motions. This application, which lies in the field of \emph{structure from motion}, is one of the fundamental problems in computer vision.
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\textwidth]{figures/desk_view1.pdf}
\includegraphics[width=.32\textwidth]{figures/manycars_view1.pdf}
\includegraphics[width=.32\textwidth]{figures/maninoffice_view1.pdf}\\
\includegraphics[width=.32\textwidth]{figures/desk_view2.pdf}
\includegraphics[width=.32\textwidth]{figures/manycars_view2.pdf}
\includegraphics[width=.32\textwidth]{figures/maninoffice_view2.pdf} \\
\caption{Three exemplary two-view motion sequences (arranged in columns): (4) \emph{desk}, (6) \emph{manycars} and (7) \emph{man-in-office}. The true clusters are displayed in different colors and markers (the black dots are outliers).}
\label{fig:twoview_examples}
\end{figure}
Given a physical point $\mathbf{x} \in\mathbb{R}^3$ and its image correspondences in the two views $(x_1,y_1)', (x_2,y_2)' \in \mathbb{R}^2$, one can always form a joint image sample $\mathbf{y}=(x_1,y_1,x_2,y_2,1)'\in \mathbb{R}^5$. It is shown in \cite{RAS} that, under perspective camera projection,
all the joint image samples $\mathbf{y}$ corresponding to different motions live on different manifolds in $\mathbb{R}^5$, some having dimension 2 and others having dimension 4. Exploratory analysis applied to these data suggests that the manifolds in this dataset mostly have dimension 2 (see Figure \ref{fig:twoview_trueclusters}). Therefore, we will apply our algorithm (HOSC) with $d=2$ to these data sets in order to compare with pairwise spectral clustering (SC-NJW, SC-LS).
\begin{figure}[htbp]
\centering
\includegraphics[width=.32\textwidth]{figures/desk_truth.pdf}
\includegraphics[width=.32\textwidth]{figures/manycars_truth.pdf}
\includegraphics[width=.32\textwidth]{figures/maninoffice_truth.pdf}\\
\caption{The true clusters of the three sequences in Figure \ref{fig:twoview_examples} (in same order), shown in top three principal dimensions. (The outliers have been removed from the data and thus are not displayed). These plots clearly indicate that the underlying manifolds are two dimensional.}
\label{fig:twoview_trueclusters}
\end{figure}
We use the following parameter values for the two algorithms. In HOSC, we choose $\ell=20, m=d+2, \eta\in [.0001,.1]$, while in SC we try both searching the interval $[.001, .5]$ (SC-NJW) and local scaling with at most 24 nearest neighbors (SC-LS).
The original data contains some outliers. In fact, 10 sequences out of the 13 are corrupted with outliers, with the largest percentage being about 32\%. We first manually remove the outliers from those sequences and solely focus on the clustering aspects of the two algorithms. Next, we add outliers back and compare them regarding outliers removal. (Note that we need to provide both algorithms with the true percentage of outliers in each sequence.) By doing so we hope to evaluate the clustering and outliers removal aspects of an algorithm separately and thus in the most accurate way.
\begin{table}[htbp]
\caption{\small The misclassification rates and the numbers of true outliers detected by HOSC, SC-NJW and SC-LS. In the clustering experiment, the outliers-free data is used; then the outliers are added back so that each of these algorithms can be applied to detect them. For SC-NJW, the tuning parameter is selected from the interval $[.001, .5]$; for SC-LS, a maximum of $24$ nearest neighbors are used; for HOSC, $20$ nearest neighbors are used and the flatness parameter $\eta$ is selected from the interval $[.0001, .1]$.}
\vspace{.1in}
\begin{tabular}{|l|l|l||r|r|r||l|l|l|}
\hline
\multicolumn{3}{|c||}{Data}
&\multicolumn{3}{c||}{Clustering Errors}
&\multicolumn{3}{c|}{ \# True Outliers Detected}\\
\hline
seq. & \#samples & \#out. & SC-NJW & SC-LS & HOSC & SC-NJW & SC-LS & HOSC\\
\hline\hline
1 & 115,121 & 2 & 0.85\% & 0.85\% & 0.85\% & 1 & 1 & 1\\
2 & 85,45,89 & 28 & 0\% & 0\% & 0\% & 24 & 24 &24\\
3 & 62,192 & 0 & 30.3\% & 23.6\% & 30.3\% & N/A & N/A & N/A\\
4 & 50,50,55 & 45 & 0.65\% & 2.58\% & 1.29\% & 35 & 30 &37\\
5 & 51,121,33 & 0 & 0\% & 0\% & 0\% &N/A & N/A & N/A\\
6 & 54,24,23,43 & 0 & 18.8\% & 0\% & 0\% &N/A &N/A & N/A \\
7 & 16,57 & 34 & 19.2\% & 19.2\% & 0\% & 17 & 12 &26\\
8 & 129,168,91 & 32 & 22.9\% & 17.8\% & 22.9\% & 12 & 17 &23\\
9 & 76,109,74 & 48 & 0\% & 0\% & 0\% & 36 & 28 &36\\
10 & 19,117 & 4 & 0\% & 47.8\% & 0\% & 0 & 0 & 1\\
11 & 100,99,81 & 99 & 0\% & 1.79\% & 0\% & 42 & 39 &73 \\
12 & 99,99,99 & 99 & 0.34\% & 0.34\% & 0\% & 80 & 43 & 91 \\
13 & 49,42 & 35 &33.0\% & 15.4\% & 2.20\% & 7 & 6 & 21\\
\hline
\end{tabular}
\label{tab:comparison_2view}
\end{table}
Table~\ref{tab:comparison_2view} presents the results from the experiments above. Observe that HOSC achieved excellent clustering results in all but two sequences, with zero error on eight sequences, one mistake on sequence (13), and two mistakes on each of (1) and (4). We remark that HOSC also outperformed the algorithms in \cite[Table 1]{AtevKSCC}, in terms of clustering accuracy, but due to the main aim of this paper, we do not include those results in Table~\ref{tab:comparison_2view}. In contrast, each of SC-NJW and SC-LS failed on at least five sequences (with over 15\% misclassification rates), both containing the two bad sequences for HOSC. As a specific example, we display in Figure \ref{fig:maninoffice_hosc_sc} the clusters obtained by both HOSC and SC on sequence (7), demonstrating again that higher order affinities can significantly improve over pairwise affinities in the case of manifold data.
Regarding outliers removal, HOSC is also consistently better than SC-NJW and SC-LS (if not equally good).
\begin{figure}[htbp]
\centering
\includegraphics[width=.45\textwidth]{figures/maninoffice_hosc.pdf}
\includegraphics[width=.45\textwidth]{figures/maninoffice_njw.pdf}
\caption{Clustering results of both HOSC and SC (left to right) on sequence (7). (The truth is shown in Figure~\ref{fig:twoview_trueclusters}, rightmost plot). In this example, HOSC correctly found the two clusters, using geometric information; in contrast, SC failed because it solely relies on pairwise distances.}
\label{fig:maninoffice_hosc_sc}
\end{figure}
\section{Extensions}
\label{sec:discussion}
\subsection{When the Underlying Surfaces Self-Intersect}
\label{sec:self}
In our generative model described in Section~\ref{sec:setting} we assume that the surfaces are submanifolds, implying that they do not self-intersect. This is really for convenience as there is essentially no additional difficulty arising from self-intersections. If we allow the surfaces to self-intersect, then we bound the maximum curvature (from above) and not the reach.
We could, for example, consider surfaces of the form $S = f(B_d(0,1))$, where $f:B_d(0,1) \to (0,1)^D$ is locally bi-Lipschitz and has bounded second derivative. A similar model is considered in~\cite{MR1332579} in the context of set estimation. Clearly, proving that each cluster is connected in the neighborhood graph in this case is the same. The only issue is in situations where a surface comes within distance $\epsilon$ from another surface at a location where the latter intersects itself. The geometry involved in such a situation is indeed complex. If we postulate that no such situation arises, then our results generalize immediately to this setting.
\subsection{When the Underlying Surfaces Have Boundaries}
\label{sec:boundary}
When the surfaces have boundaries, points near the boundary of a surface may be substantially connected with points on a nearby surface. See Figure~\ref{fig:right-angle} for an illustration. This is symptomatic of the fact that the algorithm is not able to resolve intersections in general, as discussed in Section~\ref{sec:intersect}, with the notable exception of clusters of dimension $d=1$, as illustrated in the `two moons' example of Figure~\ref{fig:artificial_data}.
\begin{figure}[htbp]
\centering
\includegraphics[width=.45\linewidth]{figures/higher-cluster-sim-6} \
\includegraphics[width=.45\linewidth]{figures/higher-cluster-sim-7} \
\vspace{-.3in}
\caption{An example of a surface with a boundary coming close to another surface. This is a potentially problematic situation for HOSC as the points near the boundary of one surface and close to the other surface may be strongly connected to points from both clusters. Numerically, we show in Figure~\ref{fig:artificial_data} such an example where HOSC is successful.}
\label{fig:right-angle}
\end{figure}
If we require a stronger separation between the boundary of a surface and the other surfaces, specifically,
\begin{equation} \label{eq:delta-partial}
{\mathrm{dist}}(\partial S_k, S_\ell) \geq \delta_\ddag, \quad \forall k \neq \ell,
\end{equation}
with $\delta_\ddag - 2 \tau > \epsilon$, no point near the boundary of a cluster is close to a point from a different cluster. (A corresponding requirement in the context of outliers would be that outliers be separated from the boundary of a cluster by at least $\delta_{0, \ddag}$, with $\delta_{0, \ddag} - \tau > \epsilon$.)
\subsection{When the Data is of Mixed Dimensions}
\label{sec:mixed}
In a number of situations, the surfaces may be of different intrinsic dimensions. An important instance of that is the study of the distribution of galaxies in space, where the galaxies are seen to cluster along filaments ($d=1$) and walls ($d=2$)~\cite{MarSaa}. We propose a top-down approach, implementing HOSC for each dimension $d$ starting at $D-1$ and ending at $1$ (or between any known upper and lower bounds for $d$).
At each step, the algorithm is run on each cluster obtained from the previous step, including the set of points identified as outliers. Indeed, when the dimension parameter of the algorithm is set larger than the dimension of the underlying surfaces, HOSC may not be able to properly separate clusters. For example, two parallel segments satisfying the separation requirement of Theorem~\ref{th:linear} still belong to a same plane and HOSC with dimension parameter $d=2$ would not be able to separate the two line segments. Another reason for processing the outlier bin is the greater disparity in the degrees of the data points in the neighborhood graph often observed with clusters of different dimensions. At each step, the number of clusters is determined automatically according to the procedure described in Section~\ref{sec:param}, for such information is usually not available. The parameters $\epsilon $ and $\eta$ are chosen according to \eqref{eq:eps-eta-O2}.
Partial results suggest that, under some additional sampling conditions, this top-down procedure is accurate under weaker separation requirements than required by pairwise methods, which handle the case of mixed dimensions seamlessly~\cite{pairwise}.
The key is that an actual cluster $\mathcal{X}_k$, as defined in Section~\ref{sec:setting}, is never cut into pieces.
Indeed, properties \ref{a1} and \ref{a4} in the proof of Theorem~\ref{th:linear}, which guarantee the connectivity and regularity (in terms of comparable degrees) of the subgraph represented by $\mathcal{X}_k$, are easily seen to also be valid when the dimension parameter of the algorithm is set larger than $d$.
(This observation might explain the success of the SCC algorithm of~\cite{spectral_applied} in some mixed settings when using an upper bound on the intrinsic dimensions.)
\subsection{Clustering Based on Local Polynomial Approximations}
\label{sec:poly}
For $1 \leq d \leq D-1$ and an integer $r \geq 3$, let $\mathcal{S}_d^{r}(\kappa)$ be the subclass of $\mathcal{S}_d^2(\kappa)$ of $d$-dimensional submanifolds $S$ such that, for every $\mathbf{x} \in S$ with tangent $T_\mathbf{x}$, the orthogonal projection $S \cap B(\mathbf{x}, 1/\kappa) \to T_\mathbf{x}$ is a $C^{r}$-diffeomorphism with all partial derivatives of order up to $r$ bounded in supnorm by $\kappa$.
For example, $\mathcal{S}_d^r(\kappa)$ includes a subclass of surfaces of the form $S = f(B_d(0,1))$, where $f:B_d(0,1) \to (0,1)^D$ is locally bi-Lipschitz and has its first $r$ derivatives bounded. (We could also consider surfaces of intermediate, i.e., H\"older smoothness, a popular model in function and set estimation~\cite{MR0358168,MR1332579}.)
Given that surfaces in $\mathcal{S}_d^{r}$ are well-approximated locally by polynomial surfaces, it is natural to choose an affinity based on the residual of the best $d$-dimensional polynomial approximation of degree at most $r-1$ to a set of points $\mathbf{x}_1, \dots, \mathbf{x}_{m}$. This may be implemented via the ``kernel trick" with a polynomial kernel, as done in~\cite{AtevKSCC} for the special case of algebraic surfaces.
The main difference with the case of $C^2$ surfaces that we consider in the rest of the paper is the degree of approximation to a surface $S \in \mathcal{S}_d^r$ by its osculating algebraic surface of order $r-1$; within a ball of radius $\epsilon$, it is of order $O(\epsilon^r)$.
Partial results suggest that, under similar conditions, the kernel version of HOSC with $r$ known may be able to operate under a separation of the form (\ref{eq:sep}), with the exponent $2/d$ replaced by $r/d$ and, in the presence of outliers, within a logarithmic factor of the best known sampling rate ratio achieved by any detection method~\cite{AriasCastro2009,CTD}:
\begin{equation}
\min_k N_k \geq N^{d/(r D - (r-1) d)} \vee N \tau^{D-d}.
\end{equation}
Regarding the estimation of $\tau$, defining the correlation dimension using the underlying affinity defined here allows to estimate $\tau$ accurately down to (essentially) $(\log(N)/N)^{r/d}$, if the surfaces are all in $\mathcal{S}_d^{r}(\kappa)$. The arguments are parallel and we omit the details.
Thus, using the underlying affinity defined here may allow for higher accuracy, if the surfaces are smooth enough. However, this comes with a larger computational burden and at the expense of introducing a new parameter $r$, which would need to be estimated if unknown, and we do not know a good way to do that.
\subsection{Other Extensions}
\label{sec:extensions}
The setting we considered in this paper, introduced in Section~\ref{sec:setting}, was deliberately more constrained than needed for clarity of exposition. We list a few generalizations below, all straightforward extensions of our work.
\begin{itemize}
\item {\it Sampling.} Instead of the uniform distribution, we could use any other distribution with a density bounded away from $0$ and $\infty$, or with fast decaying tails such as the normal distribution.
\item {\it Kernel.} The rate of decay of the kernel $\phi$ dictates the range of the affinity (\ref{eq:linear-affinity}). Let $\omega_{\scriptscriptstyle N}$ be a non-decreasing sequence such that $N^{3m} \phi(\omega_{\scriptscriptstyle N}) \to 0.$
For a compactly supported kernel, $\omega_{\scriptscriptstyle N} = \sup\{s: \phi(s) > 0\}$, while for the heat kernel, we can take $\omega_{\scriptscriptstyle N} = 2 \sqrt{m \log N}$. As we will take $m \to \infty$, $\phi$ is essentially supported in $[0, \omega_{\scriptscriptstyle N}]$ so that points that are further than $\omega_{\scriptscriptstyle N} \epsilon$ apart have basically zero affinity. Specifically, we use the following bounds:
$$
\phi(1) {\bf 1}_{\{|s| < 1\}} \leq \phi(s) \leq {\bf 1}_{\{|s| < \omega_{\scriptscriptstyle N}\}} + \phi(\omega_{\scriptscriptstyle N}).
$$
The results are identical, except that statements of the form $\delta - 2 \tau > Z$ are replaced with $\delta - 2 \tau > \omega_{\scriptscriptstyle N} Z$.
\item {\it Measure of flatness.} As pointed out in the introduction, any reasonable measure of linear approximation could be used instead. Our choice was driven by convenience and simplicity.
\end{itemize}
| {
"timestamp": "2011-11-30T02:02:30",
"yymm": "1001",
"arxiv_id": "1001.1323",
"language": "en",
"url": "https://arxiv.org/abs/1001.1323",
"abstract": "In the context of clustering, we assume a generative model where each cluster is the result of sampling points in the neighborhood of an embedded smooth surface; the sample may be contaminated with outliers, which are modeled as points sampled in space away from the clusters. We consider a prototype for a higher-order spectral clustering method based on the residual from a local linear approximation. We obtain theoretical guarantees for this algorithm and show that, in terms of both separation and robustness to outliers, it outperforms the standard spectral clustering algorithm (based on pairwise distances) of Ng, Jordan and Weiss (NIPS '01). The optimal choice for some of the tuning parameters depends on the dimension and thickness of the clusters. We provide estimators that come close enough for our theoretical purposes. We also discuss the cases of clusters of mixed dimensions and of clusters that are generated from smoother surfaces. In our experiments, this algorithm is shown to outperform pairwise spectral clustering on both simulated and real data.",
"subjects": "Machine Learning (stat.ML); Statistics Theory (math.ST)",
"title": "Spectral clustering based on local linear approximations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.978384664716301,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.70920196793518
} |
https://arxiv.org/abs/1610.05355 | A simple finite element method for the Stokes equations | The goal of this paper is to introduce a simple finite element method to solve the Stokes and the Navier-Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate and robust. | \section{Introduction}
The Stokes problem is to
seek a pair of unknown functions $({\bf u}; p)$ satisfying
\begin{eqnarray}
-\nu\Delta{\bf u}+\nabla p &=&{\bf f}\quad \mbox{in}\;\Omega,\label{moment}\\
\nabla\cdot{\bf u}&=&0\quad\mbox{in}\;\Omega,\label{cont}\\
{\bf u} &=& {\bf 0}\quad\mbox{on}\;\partial\Omega,\label{bc}
\end{eqnarray}
where $\nu$ denotes the fluid viscosity; $\Delta$, $\nabla$, and
$\nabla\cdot$ denote the Laplacian, gradient, and divergence
operators, respectively; $\Omega \subset \mathbb{R}^d$ is the
region occupied by the fluid; ${\bf f}={\bf f}({\bf x})\in ([L^2(\Omega))]^d$
is the unit external volumetric force acting on the fluid at
${\bf x}\in\Omega$. For simplicity, we let $\nu=1$.
The weak formulation of the Stokes equations seeks ${\bf u}\in [H_0^1(\Omega)]^d$ and $p\in L_0^2(\Omega)$ satisfying
\begin{eqnarray}
(\nabla{\bf u},\;\nabla{\bf v})-(\nabla\cdot{\bf v},\;p)&=&({\bf f},\;{\bf v}),\quad {\bf v}\in [H_0^1(\Omega)]^d\label{wf-m} \\
(\nabla\cdot{\bf u},\;q)&=&0,\quad\quad\quad q\in L_0^2(\Omega).\label{wf-c}
\end{eqnarray}
The linear Stokes equations are the limiting case of zero Reynolds number for the Navier-Stokes equations. The Stokes equations have attracted a substantial attention from researchers because of its close relation with the Navier-Stokes equations. Numerical solutions of the Stokes equations have been investigated intensively and many different numerical schemes have been developed such as conforming/noconforming finite element methods, MAC method and finite volume methods. It is impossible to cite all the references. Therefore we just cite some classic ones \cite{cr,gr,gun,ht}.
In this paper, we present a finite element scheme for the Stokes equations and its equivalent divergence free formulation. In this method, velocity is approximated by weak Galerkin element of degree $k=0$ and pressure is approximated by piecewise polynomials of degree $k=0$. Weak Galerkin refers to a general finite element technique for
partial differential equations in which differential operators are approximated by weak forms as distributions for generalized functions. The weak Galerkin finite element method first introduced in \cite{wy, wy-mixed} is a natural extension of the standard Galerkin finite element method for functions with discontinuity.
One of the main difficulties in solving the the Stokes and the Navier-Stokes equations
is that the velocity and the pressure variables are coupled in a saddle point system. Many methods are developed to overcome
this difficulty. Divergence free finite element methods are such methods by approximating velocity from weakly or exactly divergence free subspaces. As a consequence, pressure is eliminated from a saddle point system, along with the incompressibility
constraint resulting in a symmetric and positive definite system with a significantly
smaller number of unknowns. For this simple finite element formulation, a divergence free basis is constructed explicitly.
The rest of the paper is organized as follows.
The finite element formulation of this weak Galerkin method is introduced in Section \ref{section-wg}. Implementation issues of the method are discussed in Section \ref{section-implementation}. In Section \ref{section-error}, we prove optimal order convergence rate of the method. Divergence free basis functions are constructed in Section \ref{section-divfree}. Using these basis functions, we can derive a divergence free weak Galerkin finite element formulation that will reduce a saddle point problem to a symmetric and positive definite system. Numerical examples are presented in Section \ref{section-ne} to demonstrate the robustness and accuracy of the method.
\section{Finite Element Scheme}\label{section-wg}
Let ${\cal T}_h$ be a shape-regular triangulation of the domain
$\Omega$ with mesh size $h$. Denote by ${\cal E}_h$
the set of all edges or faces in ${\cal T}_h$, and let ${\cal
E}_h^0={\cal E}_h\backslash\partial\Omega$ be the set of all
interior edges or faces. let ${\cal V}_h$ be the set of all interior vertices in ${\mathcal T}_h$. Define $N_E=card ({\mathcal E}_h^0)$, $N_V=card ({\cal V}_h)$ and $N_T=card ({\mathcal T}_h)$. For every element $T\in {\mathcal T}_h$, we
denote by $h_T$ its diameter and mesh size $h=\max_{T\in{\mathcal T}_h} h_T$
for ${\cal T}_h$.
The weak Galerkin methods create a new way to define a function $v$ that allows $v$ taking different forms in the interior and on the boundary of the element:
$$
v=
\left\{
\begin{array}{l}
\displaystyle
v_0,\quad {\rm in}\; T^0
\\ [0.08in]
\displaystyle
v_b,\quad {\rm on}\;\partial T
\end{array}
\right.
$$
where $T_0$ denotes the interior of $T$.
Since weak function $v$ is formed by two parts $v_0$ and $v_b$, we write $v$ as $v=\{v_0,v_b\}$ in short without confusion. Let $P_k(T)$ denote the set consisting all the polynomials of degree less or equal to $k$.
Associated with ${\mathcal T}_h$, we define finite element spaces $V_h$ for velocity
\begin{equation}\label{vhspace}
V_h=\{{\bf v}=\{{\bf v}_0,{\bf v}_b\}:\; {\bf v}_0|_T\in [P_0(T)]^d,\ {\bf v}_b|_e\in [P_0(e)]^d,\ e\in{\partial T}, T\in {\mathcal T}_h\}
\end{equation}
and $W_h$ for pressure
\begin{equation}\label{phspace}
W_h=\{q\in L^2_0(\Omega):\; q|_T\in P_0(T),\; T\in{\mathcal T}_h\},
\end{equation}
where $L^2_0(\Omega)$ is the subspace of $L_2(\Omega)$ consisting of functions with mean value zero.
We define $V_h^0$ a subspace of $V_h$ as
\begin{equation}\label{vh0space}
V^0_h=\{{\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h:\ {\bf v}_b=0 \mbox{ on } \partial\Omega\}.
\end{equation}
We would like to emphasize that any function $v\in V_h$ has a single
value $v_b$ on each edge $e\in{\mathcal E}_h$.
Since the functions in $V_h$ are discontinuous polynomials, gradient operator $\nabla$ and divergence operator $\nabla\cdot$ in (\ref{wf-m})-(\ref{wf-c}) cannot be applied to them. Therefore we defined weak gradient and weak divergence for the functions in $V_h$. Let $RT_0(T)=[P_0(T)]^d+{\bf x}P_0(T)$ introduced in \cite{rt}. Let ${\bf n}$ denote the unit outward normal.
For ${\bf v}\in V_h$ and $T\in{\mathcal T}_h$, we define weak gradient
$\nabla_{w}{\bf v} \in [RT_0(T)]^d$ as the unique polynomial
satisfying the following
equation
\begin{equation}\label{dwg}
(\nabla_{w}{\bf v}, \tau)_T = -({\bf v}_0,\nabla\cdot \tau)_T+ \langle {\bf v}_b,
\tau\cdot{\bf n}\rangle_{\partial T},\qquad \forall \tau\in [RT_0(T)]^d,
\end{equation}
and define weak divergence $\nabla_{w}\cdot{\bf v} \in P_0(T)$ as the unique polynomial
satisfying
\begin{equation}\label{dwd}
(\nabla_{w}\cdot{\bf v}, q)_T = -({\bf v}_0,\nabla q)_T+ \langle {\bf v}_b,
q{\bf n}\rangle_{\partial T},\qquad \forall q\in P_0(T).
\end{equation}
Define two bilinear forms as
\begin{eqnarray*}
a({\bf v},{\bf w})=\sum_{T\in{\mathcal T}_h}(\nabla_w{\bf v},\nabla_w{\bf w})_T, \quad b({\bf v},q)=\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\bf v},q)_T.
\end{eqnarray*}
For each element $T\in {\mathcal T}_h$, denote by $Q_0$ and ${\bf Q}_0$ the $L^2$ projections from $L^2(T)$ to $P_0(T)$ and
from $[L^2(T)]^d$ to $[P_0(T)]^d$ respectively. Denote by ${\bf Q}_b$ the $L^2$ projection from
$[L^2(e)]^d$ to $[P_{0}(e)]^d$.
\medskip
\begin{algorithm}
A weak Galerkin method for the Stokes equations seeks
${\bf u}_h=({\bf u}_0,{\bf u}_b)\in V_h^0$ and $p_h\in W_h$ satisfying the following equation:
\begin{eqnarray}
a({\bf u}_h,{\bf v})-b({\bf v},p_h)&=&({\bf f},\;{\bf v}_0), \quad\forall {\bf v}=\{{\bf v}_0, {\bf v}_b\}\in V_h^0\label{wg-m}\\
b({\bf u}_h,q)&=&0,\quad\quad\quad\forall q\in W_h.\label{wg-c}
\end{eqnarray}
\end{algorithm}
\section{Implementation of the method}\label{section-implementation}
The linear system associated with the algorithm (\ref{wg-m})-(\ref{wg-c}) is a saddle point problem with the form,
\begin{equation}\label{matrix}
\left(
\begin{array}{cc}
A & -B \\
B^T&0
\end{array}
\right)
\left(\begin{array}{c}U \\P \end{array}\right)=\left(\begin{array}{c}
F_1 \\ F_2
\end{array}
\right).
\end{equation}
The methodology of implementing this weak Galerkin method is the same as that for continuous
Galerkin finite element method except that computing standard gradient $\nabla$ and divergence $\nabla\cdot$ are replaced by computing weak gradient $\nabla_w$ and weak divergence $\nabla_w\cdot$. For basis function $\Theta_l$, we will show that $\nabla_w\cdot\Theta_l$ and $\nabla_w\Theta_l$ can be calculated explicitly.
The procedures of implementing the method (\ref{wg-m})-(\ref{wg-c}) can be described as following steps. Here we let $d=2$ for simplicity.
\medskip
1. Find basis functions for $V_h$ and $W_h$. First we define two types of scalar piecewise constant basis functions $\phi_i$ associated with the interior of the triangle $T_i\in{\mathcal T}_h$ and $\psi_j$ associated with an edge $e_j\in {\mathcal E}_h$ respectively,
\begin{equation}\label{phi}
\phi_i=\left\{
\begin{array}{l}
1\quad\mbox{on} \;\; T_i^0,\\ [0.08in]0\quad\mbox{therwise},
\end{array}
\right.
\psi_j=\left\{
\begin{array}{l}
1\quad\mbox{on} \;\; e_j.\\ [0.08in]0\quad\mbox{therwise},
\end{array}
\right.
\end{equation}
Please note that $\phi_i$ and $\psi_j$ are functions defined over the whole domain $\Omega$. Then we can define the vector basis functions for velocity as
\begin{equation}\label{b1}
\Phi_{i,1}=\left(\begin{array}{c}\phi_i \\0 \end{array}\right),\;\;\Phi_{i,2}=\left(\begin{array}{c}0 \\\phi_i \end{array}\right),\;\; i=1,\cdots,N_T,
\end{equation}
and
\begin{equation}\label{b2}
\Psi_{i,1}=\left(\begin{array}{c}\psi_i \\0 \end{array}\right),\;\;\Psi_{i,2}=\left(\begin{array}{c}0 \\\psi_i \end{array}\right), \;\; i=1,\cdots,N_E.
\end{equation}
Let $n=N_T+N_E$. These $2n$ vector functions will form a basis for $V_h$,
\begin{eqnarray}
V_h&=&{\rm span} \{\Phi_{i,j}, i=1,\cdots, N_T,\;\Psi_{k,j}, k=1,\cdots,N_E, \;j=1,2\}\nonumber\\
&=&{\rm span} \{\Theta_1,\cdots,\Theta_{2n}\}.\label{vh}
\end{eqnarray}
Define $\overline{W}_h$ as
\[
\overline{W}_h={\rm span} \{\phi_1,\cdots,\phi_{N_T}\}.
\]
The pressure space is a subspace of $\overline{W}_h$,
\[
W_h=\{q\in \overline{W}_h, \int_\Omega qdx=0\}.
\]
\medskip
2. Compute weak gradient $\nabla_w$ and weak divergence $\nabla_w\cdot$ for the basis function $\Theta_l$ defined in (\ref{vh}). By the definition of $\Theta_l$, to compute $\nabla_w\cdot\Theta_l$, we compute $\nabla_w\cdot\Phi_{i,j}$ and $\nabla_w\cdot\Psi_{i,j}$ instead. To find $\nabla_w\Theta_l$, we just need to compute $\nabla_w\phi_i$ and $\nabla_w\psi_i$.
\medskip
\begin{itemize}
\item Computing $\nabla_w\cdot\Phi_{i,j}$.\\
Using (\ref{dwd}), we have $\nabla_w\cdot\Phi_{i,j}|_T=0$ for all $T\in{\mathcal T}_h$.\\
\item Computing $\nabla_w\cdot\Psi_{i,j}$.\\
Assume that $i^{th}$ edge $e_i$ is on ${\partial T}$ and $\Psi_{i,j}$ is defined in (\ref{b2}). Then
\begin{equation}\label{cwd}
\nabla_w\cdot\Psi_{i,j}|_T=\frac1{|T|}\int_{e_i}\Psi_{i,j,b}\cdot{\bf n} ds,
\end{equation}
where $|T|$ is the area of $T$ and $\Psi_{i,j}=\{\Psi_{i,j,0},\Psi_{i,j,b}\}$.
Note that $\nabla_w\cdot\Psi_{i,j}$ is only nonzero on two triangles that share $e_i$.\\
\item Computing $\nabla_w\phi_i$.\\
Let $T$ be the $i^{th}$ triangle in ${\mathcal T}_h$ and $\phi_i$ be defined as in (\ref{phi}). Then $\nabla_w\phi_i$ is only nonzero on $T$ and can be calculated by
\begin{eqnarray*}
\nabla_w\phi_i|_T&=&-C_T({\bf x}-{\bf x}_T),
\end{eqnarray*}
where ${\bf x}_T$ is the centroid of $T$ and $C_T= \frac{2|T|}{\|{\bf x}-{\bf x}_T\|_T^2}$.\\
\item Computing $\nabla_w\psi_i$.\\
Assume that $i^{th}$ edge $e_i$ is on ${\partial T}$ and $\psi_{i}$ is defined in (\ref{phi}). Then
\begin{eqnarray*}
\nabla_w\psi_i|_T&=&\frac{C_T}{3}({\bf x}-{\bf x}_T)+\frac{|e_i|}{|T|}{\bf n},
\end{eqnarray*}
Note that $\nabla_w\psi_i$ is only nonzero on two triangles that share $e_i$.
\end{itemize}
\medskip
3. Form the stiffness matrix (\ref{matrix}) with
\[
A=(a_{ij})=(a(\Theta_i,\Theta_j)),\quad B=(b_{ij})=(b(\Theta_i,\phi_j)).
\]
Note that
$$
a(\Theta_i,\Theta_j)=\sum_{T\in{\mathcal T}_h}(\nabla_w\Theta_i,\nabla_w\Theta_j)_T,\;\; b(\Theta_i,\phi_j)=\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot\Theta_i,\phi_j)_T.
$$
\section{Error estimate}\label{section-error}
Denote by $\pi_h$ a $L^2$ projection from $[L^2(T)]^{d\times d} $ to $[RT_0(T)]^d$.
We also define a projection $\Pi_h$ such that
$\Pi_h{\bf q}\in [H({\rm div},\Omega)]^d$, and on each $T\in {\cal T}_h$,
one has $\Pi_h{\bf q} \in [RT_0(T)]^d$ and the following equation satisfied:
$$
(\nabla\cdot{\bf q},\;{\bf v}_0)_T=(\nabla\cdot\Pi_h{\bf q},\;{\bf v}_0)_T, \qquad
\forall {\bf v}_0\in [P_0(T)]^d.
$$
For any $\tau\in [H({\rm div},\Omega)]^d$, we have (see \cite{bf})
\begin{equation}\label{4.200}
\sum_{T\in {\cal T}_h}(-\nabla\cdot\tau, \;{\bf v}_0)_T=\sum_{T\in {\cal T}_h}(\Pi_h\tau, \;\nabla_w{\bf v})_T,\quad\forall {\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h.
\end{equation}
The following two identities can be verified easily and also can be found in \cite{wy, wy-mixed}.
\begin{eqnarray}
\nabla_w {\bf Q}_h {\bf u} &=& \pi_h (\nabla {\bf u}),\label{4.88}\\
\nabla_w\cdot {\bf Q}_h {\bf u} &=& Q_0 (\nabla \cdot{\bf u}).\label{4.99}
\end{eqnarray}
We introduce two semi-norms ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}$ and $\|\cdot\|_{1,h}$ as follows:
\begin{eqnarray}
{|\hspace{-.02in}|\hspace{-.02in}|} {\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}^2 &=& a({\bf v},{\bf v}), \label{norm2}\\
\|{\bf v}\|_{1,h}^2&=&\sum_{T\in {\mathcal T}_h}\left(\|\nabla{\bf v}_0\|_T^2+h_T^{-1}\|{\bf v}_0-{\bf v}_b\|_{{\partial T}}^2\right).\label{norm3}
\end{eqnarray}
The following norm equivalences is proved in \cite{mwwy} that there exist two constants $C_1$ and $C_2$ independent of $h$ satisfying
\begin{equation}\label{norm-e}
C_1\|{\bf v}\|_{1,h}\le {|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}\le C_2 \|{\bf v}\|_{1,h}.
\end{equation}
\begin{lemma}
The semi-nome ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot{|\hspace{-.02in}|\hspace{-.02in}|}$ defined in (\ref{norm2}) is a norm in $V_h^0$.
\end{lemma}
\medskip
\begin{proof}
We only need to prove $v=0$ if ${|\hspace{-.02in}|\hspace{-.02in}|} v{|\hspace{-.02in}|\hspace{-.02in}|}=0$ for all $v\in V_h^0$. Let $v\in V_h^0$ and $\|v\|_{1,h}=0$. Then we have $\nabla v_0=0$ on each $T\in{\mathcal T}_h$ , $v_0=v_b$ on $e\in{\mathcal E}_h^0$ and $v_b=0$ for $e$ on $\partial\Omega$. $\nabla v_0=0$ on $T$ implies that $v_0$ is a constant on each $T$. $v_0=v_b$ on $e$ means that $v_0$ is continuous. With $v_0=v_b=0$ on $\partial\Omega$, we conclude $v=0$ and prove that $\|\cdot\|_{1,h}$ is a norm in $V_h$. Combining it with (\ref{norm-e}), we have proved that ${|\hspace{-.02in}|\hspace{-.02in}|}\cdot{|\hspace{-.02in}|\hspace{-.02in}|}$ is a norm in $V_h^0$.
\end{proof}
\medskip
Define two linear functionals on $V_h^0$ by
\begin{eqnarray*}
\ell_{{\bf u}}({\bf v})=\sum_{T\in{\mathcal T}_h}(\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf v})_T,\;\;\;
\ell_p({\bf v})=\sum_{T\in{\mathcal T}_h}{\langle} {\bf v}_0-{\bf v}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}.
\end{eqnarray*}
\begin{lemma}\label{e-e}
Let ${\bf e}_h={\bf Q}_h{\bf u}-{\bf u}_h=\{{\bf e}_0,\;{\bf e}_b\}=\{{\bf Q}_0{\bf u}-{\bf u}_0,\;{\bf Q}_b{\bf u}-{\bf u}_b\}$ and $\varepsilon_h=Q_0p-p_h$.
Then, the following equations hold
true
\begin{eqnarray}
a({\bf e}_h,{\bf v})-b({\bf v},\varepsilon_h)&=&-\ell_{{\bf u}}({\bf v})-\ell_p({\bf v}),\;\forall{\bf v}\in V_h^0,\label{ee-m}\\
b({\bf e}_h,q)&=&0,\quad\qquad\qquad\forall q\in W_h.\label{ee-c}
\end{eqnarray}
\end{lemma}
\begin{proof}
Testing (\ref{moment}) by ${\bf v}=\{{\bf v}_0,{\bf v}_b\}\in V_h^0$ gives
\begin{equation}\label{mmm1}
(-\nabla\cdot(\nabla{\bf u}), \;{\bf v}_0)+(\nabla p,\;{\bf v}_0)=({\bf f}, {\bf v}_0).
\end{equation}
Equations (\ref{4.200}) and (\ref{4.88}) imply
\begin{eqnarray*}
\sum_{T\in{\mathcal T}_h}(-\nabla\cdot(\nabla{\bf u}), \;{\bf v}_0)_T&=&\sum_{T\in{\mathcal T}_h}((\Pi_h\nabla{\bf u}, \nabla_w{\bf v})_T=\sum_{T\in{\mathcal T}_h}((\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf v})_T+(\nabla_w{\bf Q}_h{\bf u}, \nabla_w{\bf v})_T).
\end{eqnarray*}
It follows from (\ref{dwd}), the integration by parts and the fact $\sum_{K\in{\mathcal T}_h}\langle{\bf v}_b,\; p\;{\bf n}\rangle_{\partial K}=0$ that for $v\in V_h^0$
\begin{eqnarray*}
(\nabla p,\;{\bf v}_0)&&=\sum_{T\in{\mathcal T}_h}(-(p,\; \nabla\cdot{\bf v}_0)_T +{\langle} p{\bf n},\; {\bf v}_0{\rangle}_T)\\
&&=\sum_{T\in{\mathcal T}_h}(-(Q_0p,\; \nabla\cdot{\bf v}_0)_T +{\langle} p{\bf n},\; {\bf v}_0-{\bf v}_b{\rangle}_T)\\
&&=\sum_{T\in{\mathcal T}_h}(({\bf v}_0,\; \nabla Q_0p)_T -{\langle} Q_0p{\bf n},\; {\bf v}_0{\rangle}_T+{\langle} p{\bf n},\; {\bf v}_0-{\bf v}_b{\rangle}_T)\\
&&=-b({\bf v}, Q_0p)+\sum_{T\in{\mathcal T}_h}{\langle} {\bf v}_0-{\bf v}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}.
\end{eqnarray*}
Combining two equations above with (\ref{mmm1}), we have
\begin{eqnarray}
a({\bf Q}_h{\bf u},{\bf v})-b({\bf v},Q_0p)=({\bf f}, {\bf v}_0)-\ell_{{\bf u}}({\bf v})-\ell_p({\bf v}).\label{t-m}
\end{eqnarray}
Using (\ref{4.99}) and (\ref{cont}), we have that for any $q\in W_h$
\begin{eqnarray}
b({\bf Q}_h{\bf u},q)&=&\sum_{T\in{\mathcal T}_h}(\nabla_w\cdot{\bf Q}_h{\bf u}, q)_T=\sum_{T\in{\mathcal T}_h}(Q_0(\nabla\cdot{\bf u}), q)_T\nonumber\\
&=&\sum_{T\in{\mathcal T}_h}(\nabla\cdot{\bf u}, q)_T=b({\bf u},q)=0,\label{t-c}
\end{eqnarray}
The differences of (\ref{wg-m})-(\ref{wg-c}) and (\ref{t-m})-(\ref{t-c}) yield (\ref{ee-m})-(\ref{ee-c}). The proof is completed.
\end{proof}
\medskip
\begin{lemma}
For any $\rho\in W_h$, then there exists a constant $C$ independent of $h$ such that
\begin{equation}\label{inf-sup}
\sup_{{\bf v}\in V_h^0}\frac{(\nabla_w\cdot{\bf v},\;\rho)}{{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}}\ge C\|\rho\|.
\end{equation}
\end{lemma}
\begin{proof}
For a given $\rho\in W_h\subset L_0^2(\Omega)$, it is well known that there exists $\tilde{\bf v}\in [H_0^1(\Omega)]^d$ such that
\begin{equation}\label{c-inf-sup}
\frac{(\nabla\cdot\tilde{\bf v},\rho)}{\|\nabla\tilde{\bf v}\|}\ge C\|\rho\|.
\end{equation}
Let ${\bf v}={\bf Q}_h\tilde{{\bf v}}$.
(\ref{4.88}) implies
\begin{equation}\label{m2}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}=\|\nabla_w{\bf v}\|=\|\nabla_w({\bf Q}_h\tilde{{\bf v}})|=\|\pi_h\nabla \tilde{{\bf v}}\|\le \|\nabla\tilde{\bf v}\|.
\end{equation}
It follows from (\ref{4.99}) and the definition of $\pi_h$
\begin{eqnarray*}
(\nabla_w\cdot{\bf v},\;\rho)&=&(\nabla_w\cdot {\bf Q}_h\tilde{\bf v},\;\rho)=(Q_0(\nabla\cdot\tilde{\bf v}),\;\rho)=(\nabla\cdot\tilde{\bf v},\;\rho).
\end{eqnarray*}
Using the equation above, (\ref{c-inf-sup}) and (\ref{m2}), we have
\begin{eqnarray*}
\frac{(\nabla_w{\bf v},\rho)} {{|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}} &\ge & \frac{(\nabla\cdot\tilde{\bf v},\rho)}{\|\nabla\tilde{\bf v}\|}\ge C\|\rho\|.
\end{eqnarray*}
We proved the lemma.
\end{proof}
For any function $\varphi\in H^1(T)$, the following trace
inequality holds true
\begin{equation}\label{trace}
\|\varphi\|_{e}^2 \leq C \left( h_T^{-1} \|\varphi\|_T^2 + h_T
\|\nabla \varphi\|_{T}^2\right).
\end{equation}
\begin{theorem}\label{h1-bd}
Let $({\bf u}, p)\in [H_0^1(\Omega)\cap H^{2}(\Omega)]^d\times L_0^2(\Omega)\cap H^1(\Omega)$ and $({\bf u}_h,p_h)\in V_h\times W_h$ be the solutions of (\ref{moment})-(\ref{bc}) and (\ref{wg-m})-(\ref{wg-c}) respectively. Then
\begin{eqnarray}
\|\nabla_w ({\bf Q}_h{\bf u}- {\bf u}_h)\|+\|Q_0p-p_h\|&\le& Ch(\|{\bf u}\|_{2}+\|p\|_1),\label{error1}\\
\|{\bf Q}_0{\bf u}-{\bf u}_0\|&\le& Ch^2(\|{\bf u}\|_{2}+\|p\|_1).\label{error2}
\end{eqnarray}
\end{theorem}
\begin{proof}
Letting ${\bf v}={\bf e}_h$ and $q=\varepsilon_h$ in (\ref{ee-m})-(\ref{ee-c}) and adding the two equations yield
\begin{equation}\label{eee1}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}^2=a({\bf e}_h,{\bf e}_h)=|\ell_{{\bf u}}({\bf e}_h)+\ell_{p}({\bf e}_h)|.
\end{equation}
It follows from the definitions of $\Pi_h$ and $\pi_h$ that
\begin{eqnarray}
|\ell_{{\bf u}}({\bf e}_h)|&=&|\sum_{T\in{\mathcal T}_h}(\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}, \nabla_w{\bf e}_h)_T|\nonumber\\
&\le& \sum_{T\in{\mathcal T}_h}\|\Pi_h\nabla {\bf u}-\pi_h\nabla{\bf u}\|_T\|\nabla_w{\bf e}_h\|_T\|\le Ch\|{\bf u}\|_2{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}.\label{eee2}
\end{eqnarray}
Using the trace inequality (\ref{trace}), the definition of $Q_0$ and norm equivalence (\ref{norm-e}), we have
\begin{eqnarray}
|\ell_{p}({\bf e}_h)|&=&|\sum_{T\in{\mathcal T}_h}{\langle} {\bf e}_0-{\bf e}_b,\;(p-Q_0p){\bf n}{\rangle}_{\partial T}|\nonumber\\
&\le&\sum_{T\in{\mathcal T}_h}\|p-Q_0p\|_{{\partial T}}\|{\bf e}_0-{\bf e}_b\|\nonumber\\
&\le&(\sum_{T\in{\mathcal T}_h}h\|p-Q_0p\|_{{\partial T}}^2)^{1/2}(\sum_{T\in{\mathcal T}_h}h^{-1}\|{\bf e}_0-{\bf e}_b\|_{\partial T}^2)^{1/2}\nonumber\\
&\le& Ch\|p\|_1{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}.\label{eee3}
\end{eqnarray}
Combining the estimates (\ref{eee2})-(\ref{eee3}) and (\ref{eee1}) gives
\begin{equation}\label{eee4}
{|\hspace{-.02in}|\hspace{-.02in}|}{\bf e}_h{|\hspace{-.02in}|\hspace{-.02in}|}\le Ch(\|{\bf u}\|_2+\|p\|_1).
\end{equation}
Using (\ref{ee-m}) and (\ref{eee2})-(\ref{eee4}), we have
\begin{equation}\label{eee5}
|b({\bf v},\varepsilon_h)|=|a({\bf e}_h,{\bf v})+\ell_{{\bf u}}({\bf v})+\ell_p({\bf v})|\le Ch(\|{\bf u}\|_2+\|p\|_1){|\hspace{-.02in}|\hspace{-.02in}|}{\bf v}{|\hspace{-.02in}|\hspace{-.02in}|}.
\end{equation}
It follows from (\ref{inf-sup}) and (\ref{eee5})
\[
\|Q_0p-p_h\|\le Ch(\|{\bf u}\|_{2}+\|p\|_1).
\]
The estimate (\ref{error2}) can be derived by using the standard duality argument. The main goal of this paper is about introducing a new method and how to implement it. We skip the proof of (\ref{error2}).
\end{proof}
\section{Discrete Divergence Free Basis}\label{section-divfree}
The finite element formulations (\ref{wg-m})-(\ref{wg-c}) lead to a large saddle point system (\ref{matrix})
for which most existing numerical solvers are less effective and robust than for definite
systems. Such a saddle-point system can be reduced to a definite problem
for the velocity unknown defined in a divergence-free subspace $D_h$ of $V_h^0$,
\begin{equation}\label{D_h}
D_h=\{{\bf v}\in V_h;\;b({\bf v},q) =0,\quad\forall q\in W_h\}.
\end{equation}
By taking the test function from $D_h$,
the discrete formulation (\ref{wg-m})-(\ref{wg-c}) is equivalent to the following divergence-free finite element scheme:
\medskip
\begin{algorithm}\label{algorithm2}
A discrete divergence free approximation for (\ref{moment})-(\ref{bc}) with homogeneous boundary condition is to find
${\bf u}_h=\{{\bf u}_0,{\bf u}_b\}\in D_h$
\begin{eqnarray}
a({\bf u}_h,\ {\bf v})&=&(f,\;{\bf v}_0),\quad \forall {\bf v}=\{{\bf v}_0,{\bf v}_b\}\in D_h.\label{df-wg}
\end{eqnarray}
\end{algorithm}
The system (\ref{df-wg}) is symmetric and positive definite with much fewer unknowns. To implement Algorithm \ref{algorithm2}, we need to construct basis functions for the divergence free subspace $D_h$. There are three types of functions in $V_h$ that are divergence free.
\medskip
{\bf Type 1}. All $\Phi_{i,j}$ defined in (\ref{b1}) are divergence free. This can be verified easily. Since all the functions $\Phi_{i,j}$ defined in (\ref{b1}) take zero value on ${\partial T}$ for all $T\in{\mathcal T}_h$, it follows from (\ref{cwd}) that $\nabla_w\cdot\Phi_{i,j}|_T=\frac1{|T|}\int_{\partial T}\Phi_{i,j,b}\cdot{\bf n} ds=0$.
{\bf Type 2}. For any $e_i\in{\mathcal E}_h^0$, let ${\bf n}_{e_i}$ and ${\bf t}_{e_i}$ be a normal vector and a tangential vector to $e_i$ respectively. Define $\Upsilon_i=C_1\Psi_{i,1}+C_2\Psi_{i,2}$ such that $\Upsilon_i|_{e_i}={\bf t}_{e_i}$. Thus $\Upsilon_i$ is only nonzero on $e_i$. It is easy to see that $\nabla_w\cdot\Upsilon_i|_T=\frac1{|T|}\int_{\partial T}\Upsilon_{i,b}\cdot{\bf n} ds=0$.
{\bf Type 3}. For a given interior vertex $P_i\in {\cal V}_h$, there are $r$ elements having $P_i$ as a vertex which form a hull ${\cal H}_{P_i}$ as shown in Figure \ref{fig1}. Then there are $r$ interior edges $e_j$ ($j=1,\cdots, r$) associated with ${\cal H}_{P_i}$. Let ${\bf n}_{{e_j}}$ be a normal vector on $e_j$ such that normal vectors ${\bf n}_{e_j}$ $j=1,\cdots,r$ are counterclockwise around vertex $P_i$ as shown in Figure \ref{fig1}. For each $e_j$, let $\Psi_{j,1}$ and $\Psi_{j,2}$ be the two basis functions of $V_h$ which is only nonzero on $e_j$. Define $\Theta_j=C_1\Psi_{j,1}+C_2\Psi_{j,2}\in V_h$ such that $\Theta_j|_{e_j}={\bf n}_{e_j}$. Define $\Lambda_i=\sum_{j=1}^r \frac{1}{|e_j|}\Theta_j$. It can be shown that $\nabla_w\cdot\Lambda_i=0$ (see \cite{mwy-divfree} for the details)
\begin{figure}
\begin{center}
\includegraphics[width=4cm]{div-free_stokes}
\caption{Hull ${\cal H}_{P_i}$ .} \label{fig1}
\end{center}
\end{figure}
\medskip
\begin{lemma}
These three types of divergence free functions form a basis for $D_h$, i.e.
\begin{equation}\label{b5}
D_h={\rm span}\{\Phi_{i,j},i=1,\cdots,N_T,j=1,2;\Upsilon_i, i=1,\cdots, N_E; \Lambda_i, i=1,\cdots, N_V\}.
\end{equation}
\end{lemma}
\begin{proof}
It is easy to check that all the weakly divergence free functions in (\ref{b5}) are linearly independent. It is left to check that the number of the basis functions in (\ref{b5}) is equal to the dimension of $D_h$. Obviously, the number of the functions in (\ref{b5}) is $2N_T+N_E+N_V$. On the other hand we have
$$\rm{dim} (D_h)=\rm{dim} (V_h)-\rm{dim}(W_h)=2N_T+2N_E-N_T+1.$$
For ${\mathcal T}_h$, it is well known as Euler formula that
\begin{equation}\label{key}
N_E+1=N_V+N_K.
\end{equation}
Using (\ref{key}), we have
$$\rm{dim} (D_h)=\rm{dim} (V_h)-\rm{dim}(W_h)=2N_T+2N_E-N_T+1=2N_T+N_E+N_V.$$
We have proved the lemma.
\end{proof}
\section{Numerical Examples}\label{section-ne}
In this section, six numerical examples are tested for
the two dimensional Stokes equations (\ref{moment})-(\ref{bc}). The numerical experiments indicate that the weak Galerkin methods are robust, accurate and easy to implement.
\subsection{Example 1}\label{Num_ex1}
We first consider the Stokes equations with homogeneous boundary condition defined on a square $(0, 1)\times (0, 1)$.
The exact solutions are given by
$$
{\bf u} = \begin{pmatrix}2\pi\sin(\pi x)\sin(\pi x)\cos(\pi y)\sin(\pi y) \\
-2\pi\sin(\pi x)\sin(\pi y)\cos(\pi x)\sin(\pi y)
\end{pmatrix},
$$
and
\[
p=\cos(\pi x)\cos(\pi y).
\]
The uniform triangular mesh is used for testing. Denote mesh size by $h.$ The numerical results of Algorithm 1 are presented in Table \ref{ex1_1}. These results show the $O(h)$ error of the velocity in the $H^1-$norm and pressure in the $L^2-$norm as predicted by Theorem \ref{h1-bd}. Convergence rate of $O(h^2)$ for velocity in the $L^2-$norm is observed.
Furthermore, the divergence free weak Galerkin Algorithm 2 is tested for this example. The weakly divergence-free subspace $D_h$ is constructed as described in previous section. By using the basis functions in $D_h$, the saddle-point system (\ref{wg-m})-(\ref{wg-c}) is reduced to a definite system (\ref{df-wg}) only depending on velocity unknowns. The numerical performance of velocity measured in $H^1-$norm and $L^2-$norm is shown in Table \ref{ex1_2}.
\begin{table}
\caption{Example \ref{Num_ex1}. Numerical results of Algorithm 1.}
\label{ex1_1}
\center
\begin{tabular}{||c||ccc||}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & $||Q_0p-p_h||$ \\
\hline\hline
1/4 &4.0478 &3.7181e-1 &1.7906 \\ \hline
1/8 &1.8723 &9.8624e-2 &8.7513e-1 \\ \hline
1/16 &9.1907e-1 &2.5276e-2 &4.1211e-1 \\ \hline
1/32 &4.5785e-1 &6.3793e-3 &2.0019e-1 \\ \hline
1/64 &2.2874e-1 &1.5992e-3 &9.9207e-2 \\ \hline
1/128 &1.1435e-1 &4.0009e-4 &4.9486e-2 \\ \hline\hline
$O(h^r),r=$ & 1.0238 & 1.9750 & 1.0386 \\ \hline\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Example \ref{Num_ex1}. Numerical results of Algorithm 2.}
\label{ex1_2}
\center
\begin{tabular}{||c||cc||}
\hline\hline
$h$ &${|\hspace{-.02in}|\hspace{-.02in}|}{\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ \\
\hline\hline
1/4 &6.3120 &2.6300e-1 \\ \hline
1/8 &3.3499 &6.9789e-2 \\ \hline
1/16 &1.7174 &1.7890e-2 \\ \hline
1/32 &8.6696e-1 &4.5154e-3 \\ \hline
1/64 &4.3468e-1 &1.1320e-3 \\ \hline
1/128 &2.1750e-1 &2.8320e-4 \\ \hline\hline
$O(h^r),r=$ &9.7484e-01 &1.9748 \\ \hline\hline
\end{tabular}
\end{table}
\subsection{Example 2}\label{Num_ex2}
The purpose of this example is to test the robustness and accuracy of this WG method for handling non-homogeneous boundary condition and complicated geometry.
Consider the Stokes equations with non-homogeneous boundary condition that have the exact solutions
$$
{\bf u} = \begin{pmatrix}x+x^2-2xy+x^3-3xy^2+x^2y \\
-y-2xy+y^2-3x^2y+y^3-xy^2
\end{pmatrix},
$$
and
\[
p=xy+x+y+x^3y^2-4/3.
\]
In this example, domain $\Omega$ is derived from a square $(0,1)\times(0,1)$ by taking out three circles centered at $(0.5,0.5),\ (0.2,0.8),\ (0.8,0.8) $ with radius $0.1$.
We start the weak Galerkin simulation on the initial mesh as shown in Figure \ref{IniMesh_ex2}. Then each refinement is obtained by dividing each triangle into four congruent triangles.
Table \ref{tab:inhomogeneous0} displays the errors and convergence rate of the numerical solution of Algorithm 1. Optimal order convergence rates for velocity and pressure are observed in corresponding norms.
\begin{table}
\caption{Example \ref{Num_ex2}. Numerical results of Algorithm 1.}
\label{tab:inhomogeneous0}
\center
\begin{tabular}{||c||ccc||}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|} {\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & $||Q_0p-p_h||$ \\
\hline\hline
Level 1 &1.5123e-1 &6.5055e-3 &2.2512e-1\\ \hline
Level 2 &7.6271e-2 &1.7736e-3 &9.6785e-2\\ \hline
Level 3 &3.8276e-2 &4.6167e-4 &4.0673e-2\\ \hline
Level 4 &1.9168e-2 &1.1707e-4 &2.3700e-2\\ \hline
Level 5 &9.5900e-3 &2.9394e-5 &1.9385e-2\\ \hline\hline
$O(h^r),r=$ & 9.9506e-1 & 1.9501 & 9.1053e-1\\ \hline\hline
\end{tabular}
\end{table}
\begin{figure}[!tb]
\centering
\begin{tabular}{c}
\resizebox{2.45in}{2.1in}{\includegraphics{holesquare_1_mesh.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex2}: Initial mesh.}
\label{IniMesh_ex2}
\end{figure}
\subsection{Example 3}\label{Num_ex3}
Let $({\bf u},p)$ as follows
\begin{eqnarray*}
{\bf u} = \begin{pmatrix}1-e^{\lambda x}\cos(2\pi y) \\
\frac{\lambda}{2\pi}e^{\lambda x}\sin(2\pi y)
\end{pmatrix},
p=\frac{1}{2}e^{2\lambda x}+C,
\end{eqnarray*}
be the exact solution of the Stokes equations,
\begin{eqnarray}
-\frac{1}{\mathrm {Re}}\Delta{\bf u}+\nabla p={\bf f},\quad \nabla\cdot{\bf u}=0\mbox{ in }\Omega,
\end{eqnarray}
where $\lambda=\mathrm {Re}/2-\sqrt{\mathrm {Re}^2/4+4\pi^2}$ and $\mathrm {Re}$ is the Reynolds number.
Let $\Omega=(-1/2,3/2)\times(0,2).$ The Dirichlet boundary condition for velocity is considered in this example.
In Table \ref{ex3}, we demonstrate the error profiles and convergence rates of the numerical solution of Algorithm 1 with $\mathrm {Re}=1,\ 10,\ 100,\ 1000.$ The streamlines of velocity and color contour of pressure for $\mathrm {Re}=1,\ 10,\ 100,\ 1000$ are plotted in Figure \ref{fig:ex3}.
\begin{table}
\caption{Example \ref{Num_ex3}: Numerical results of Algorithm 1.}\label{ex3}
\center
\begin{tabular}{|c|cc|cc|cc|}
\hline\hline
$h$ & ${|\hspace{-.02in}|\hspace{-.02in}|}{\bf u}_h-{\bf Q}_{h}{\bf u}{|\hspace{-.02in}|\hspace{-.02in}|}$ & order& $\|{\bf u}_0-{\bf Q}_{0}{\bf u}\|$ & order &$||Q_0p-p_h||$ & order\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=1$}\\ \hline
1/8 &4.2375e+1 & &4.2372 & &2.9223e+1& \\
1/16 &2.4722e+1 &7.7742e-1 &1.3963 &1.6015 &1.2713e+1&1.2008\\
1/32 &1.3100e+1 &9.1623e-1 &3.9686e-1 &1.8149 &5.2018 &1.2892\\
1/64 &6.6667e &9.7452e-1 &1.0374e-1 &1.9357 &2.3142 &1.1685\\
1/128 &3.3504e &9.9264e-1 &2.6294e-2 &1.9802 &1.1550 &1.0026\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=10$}\\ \hline
1/8 &6.0606 & &2.0457 & &7.6173e-1 & \\
1/16 &3.2851 &8.8352e-1 &5.9724e-1 &1.7762 &2.9472e-1 &1.3699\\
1/32 &1.6896 &9.5926e-1 &1.5926e-1 &1.9069 &1.1379e-1 &1.3730\\
1/64 &8.5296e-1 &9.8613e-1 &4.0832e-2 &1.9636 &4.5851e-2 &1.3113\\
1/128 &4.2787e-1 &9.9531e-1 &1.0306e-2 &1.9862 &1.9770e-2 &1.2136\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=100$}\\ \hline
1/8 &5.5209e-1 & &6.7127e-1 & &1.5818e-2 & \\
1/16 &2.7981e-1 &9.8046e-1 &1.7946e-1 &1.9032 &6.7914e-3 &1.2198\\
1/32 &1.4063e-1 &9.9254e-1 &4.5955e-2 &1.9654 &2.9102e-3 &1.2226\\
1/64 &7.0434e-2 &9.9756e-1 &1.1575e-2 &1.9892 &1.3386e-3 &1.1204\\
1/128 &3.5235e-2 &9.9926e-1 &2.9001e-3 &1.9968 &6.4795e-4 &1.0468\\
\hline
\multicolumn{7}{|c|}{$\mathrm {Re}=1000$}\\ \hline
1/8 &2.0636e-1 & &8.1461e-1 & &1.8694e-3 & \\
1/16 &1.0436e-1 &9.8359e-1 &2.1625e-1 &1.9134 &7.6097e-4 &1.2967\\
1/32 &5.2395e-2 &9.9407e-1 &5.5149e-2 &1.9713 &3.2176e-4 &1.2419\\
1/64 &2.6230e-2 &9.9821e-1 &1.3868e-2 &1.9916 &1.4850e-4 &1.1155\\
1/128 &1.3120e-2 &9.9945e-1 &3.4726e-3 &1.9977 &7.2192e-5 &1.0406\\ \hline\hline
\end{tabular}
\end{table}
\begin{figure}[!tb]
\centering
\begin{tabular}{cc}
\resizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-1.pdf}} \quad
\resizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-10.pdf}} \\
\resizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-100.pdf}}\quad
\resizebox{2.45in}{2.1in}{\includegraphics{Kovasznay_80-1000.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex3} Streamlines of velocity and color contour of pressure for $\mathrm {Re}=1,\ 10,\ 100,\ 1000$ (top left, top right, bottom left, bottom right.)}\label{fig:ex3}
\end{figure}
\subsection{Example 4}\label{Num_ex4}
Two dimensional channel flow around a circular obstacle is simulated in this problem.
We consider the Stokes equations with non-homogeneous boundary condition:
\begin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}=\begin{cases}
(1,0)^t,&\mbox{ if } x=0;\\
(1,0)^t,&\mbox{ if } x=1;\\
(0,0)^t,&\mbox{ else. }
\end{cases}
\end{eqnarray*}
Let $\Omega=(0,1)\times(0,1)$ with one circular hole centered at $(0.5,0.5),$ with radius 0.1.
We start with initial mesh and then refined the mesh uniformly. Level 1 mesh and level 2 mesh are shown in Figure \ref{IniMesh_Ex4}. The velocity fields of Algorithm 1 on level 1 and level 2 meshes are shown in Figures \ref{Num_Ex4_1}. The streamlines of velocity and color contour of pressure are plotted in Figure \ref{Num_Ex4_2}.
\begin{figure}[!tb]
\centering
\begin{tabular}{cc}
\resizebox{2.45in}{2.1in}{\includegraphics{holes1quare_1_mesh.pdf}} \quad
\resizebox{2.45in}{2.1in}{\includegraphics{holes1quare_2_mesh.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex4} Meshes of Level 1 (Left) and Level 2 (Right)}
\label{IniMesh_Ex4}
\end{figure}
\begin{figure}[!tb]
\centering
\begin{tabular}{cc}
\resizebox{2.45in}{2.1in}{\includegraphics{holes1quare_1_quiver.pdf}} \quad
\resizebox{2.45in}{2.1in}{\includegraphics{holes1quare_2_quiver.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex4}: Vector fields of Velocity on Mesh Level 1 (Left) and Mesh Level 2 (Right)}
\label{Num_Ex4_1}
\end{figure}
\begin{figure}[!tb]
\centering
\begin{tabular}{c}
\resizebox{2.7in}{2.45in}{\includegraphics{squarehole.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex4}: Streamlines of velocity and color contour of pressure.}
\label{Num_Ex4_2}
\end{figure}
\subsection{Example 5}\label{Num_ex5}
This example is used to test the backward facing step problem. This example is a benchmark problem. Let $\Omega=(-2,8)\times(-1,1)\backslash [-2,0]\times[-1,0]$, consider the Stokes problem with ${\bf f}=0,$ and Dirichlet boundary condition as:
\begin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}=\begin{cases}
(-y(y-1)/10,0)^t,&\mbox{ if } x=-2;\\
(-(y+1)(y-1)/80,0)^t,&\mbox{ if } x=8;\\
(0,0)^t, &\mbox{else}.
\end{cases}
\end{eqnarray*}
\begin{figure}[!tb]
\centering
\begin{tabular}{c}
\resizebox{4.45in}{1.1in}{\includegraphics{backstepe.pdf}} \\
\resizebox{4.45in}{1.1in}{\includegraphics{backstepe_zoom2.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex5}: Streamlines of velocity and color contour of pressure (top); Zoom in plot.}
\label{Num_Ex5_1}
\end{figure}
Figure \ref{Num_Ex5_1} plots the streamlines of velocity and color contour of pressure. The plot shows that the pressure is high on the left and low on the right. A zoom figure of lower left corner $[0,0.5]\times[-1,-0.65]$ is plotted in Figure \ref{Num_Ex5_1}, which shows one eddy.
\subsection{Example 6}\label{Num_ex6}
The lid-driven cavity flow is considered in this example with $\Omega=(0,1)\times(0,1)$, ${\bf f}=0$, and the Dirichlet boundary condition as:
\begin{eqnarray*}
{\bf u}|_{\partial\Omega}={\bf g}=\begin{cases}
(1,0)^t,&\mbox{ if } y=1;\\
(0,0)^t, &\mbox{else}.
\end{cases}
\end{eqnarray*}
\begin{figure}[!tb]
\centering
\begin{tabular}{ccc}
\resizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_p.pdf}} \quad
\resizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80.pdf}} \\
\resizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_left.pdf}}\quad
\resizebox{2.45in}{2.1in}{\includegraphics{lidcavity_80_right.pdf}}
\end{tabular}
\caption{Example \ref{Num_ex6}: Color contour of pressure (top left); Streamlines of velocity (top right); Zoom in plot of two bottom corners.}\label{Num_ex6_1}
\end{figure}
Figure \ref{Num_ex6_1} displays the color contour of pressure $p_h$ and streamlines of velocity ${\bf u}_h$ on a uniform mesh. Two Moffat eddies at the bottom corners are detected in Figure \ref{Num_ex6_1}.
| {
"timestamp": "2016-10-19T02:01:01",
"yymm": "1610",
"arxiv_id": "1610.05355",
"language": "en",
"url": "https://arxiv.org/abs/1610.05355",
"abstract": "The goal of this paper is to introduce a simple finite element method to solve the Stokes and the Navier-Stokes equations. This method is in primal velocity-pressure formulation and is so simple such that both velocity and pressure are approximated by piecewise constant functions. Implementation issues as well as error analysis are investigated. A basis for a divergence free subspace of the velocity field is constructed so that the original saddle point problem can be reduced to a symmetric and positive definite system with much fewer unknowns. The numerical experiments indicate that the method is accurate and robust.",
"subjects": "Numerical Analysis (math.NA)",
"title": "A simple finite element method for the Stokes equations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860381,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019674783212
} |
https://arxiv.org/abs/2012.15689 | Airy eigenstates and their relation to coordinate eigenstates | We study the eigenvalue problem for a linear potential Hamiltonian and, by writing Airy equation in terms of momentum and position operators define Airy states. We give a solution of the Schrödinger equation for the symmetrical linear potential in terms of the squeeze and displacement operators. Finally, we write the unit operator in terms of Airy states and find a relation between them and position and momentum eigenstates. | \section{Introduction}
The study of the Airy functions has attracted a lot attention in many branches of physics and applied mathematics. This because Airy functions are eigenfunctions of a Hamiltonian with a linear potential \cite{sakurai,gea_1999}, as well as for its non-spreading and bending properties \cite{ZHUKOVSKY20117966} in free space as well as in inhomogeneous, {\it i.e.,} time-varying linear potential \cite{berry_1979}. Airy beams have been introduced in optics due to their intriguing properties: self-healing \cite{Broky:08} and self-accelerating along a parabolic trajectory \cite{PhysRevLett.99.213901,Siviloglou:07,ChavezCerda2011}. Taking into account these unique characteristics, the Airy beams and Airy functions have various applications in many areas, such as light-sheet microscopy \cite{Vettenburg2014}, electron Airy waves \cite{Voloch-Bloch2013} and Stark effect \cite{Avron1977,Matin2014}.\\
The coordinate representation of the Schr\"odinger equation is the more usual way to board and to set problems in quantum mechanics. For instance the eigenfunctions of the harmonic oscillator, are represented by the braket $\langle x\vert n\rangle$, where the kets $\vert n\rangle$ with $n=0,1,2,\cdots$, are called number or Fock states (see for instance \cite{Leonhardt}). And any ket $\vert g\rangle$ can be expanded in terms of these number states. The basis set of kets $\vert n\rangle$ is discrete, however, there are also continuous bases. The questions we want to answer here are, Is it possible to have an Airy basis similar to number states? and Is it possible to relate Airy eigenstates to position eigenstates?
In general, those are complex questions. The main aim of this paper is to try to answer both. We present an explicit eigenstate of the linear symmetrical linear potential Hamiltonian. These states are orthogonal and their eigenvalues are not equally spaced. We present two numerical examples in order to illustrate our findings and the relation between the Airy basis with the usual coordinate eigenstates. Finally, we present a relation that allows to obtain the Airy states from the application of an exponential operator to the vacuum state.
\section{Airy states}
Airy functions $\hbox{Ai}(x)$ may be defined by the following Fourier transform,
\begin{equation} \label{eq001}
\hbox{Ai}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dt\;\hbox{e}^{i\left(\frac{t^{3}}{3}+xt\right)}\;.
\end{equation}
It is well known that Airy functions cannot be normalized, however they have the interesting property
\begin{equation} \label{eq003}
\int_{-\infty}^{+\infty}dx\;\hbox{Ai}(y-x)\hbox{Ai}(y'-x)=\delta(y-y')\;,
\end{equation}
where $\delta(y-y')$ is the Dirac delta function.\\
In order to obtain the Airy state, we consider the Schr\"odigner equation for a single particle under a linear potential $V(x)=\vert k\vert x$, and $-\infty<x<\infty$, (for the sake of simplicity we set $m=\hbar=1$), that is
\begin{equation} \label{eq004}
-\frac{1}{2}\frac{d^{2}}{dx^{2}}\psi(x)+ \vert k\vert x\psi(x)=E\psi(x)\;.
\end{equation}
By the substitution, $z=\sqrt[3]{2\vert k\vert}\left(x-E/\vert k\vert\right)$, the above equation is transformed into the Airy equation
\begin{equation} \label{eq005}
\frac{d^{2} \phi(z)}{dz^{2}}-z\phi(z)=0\;,
\end{equation}
whose solution is determined by an Airy function, $\hbox{Ai}(z)$ or $\hbox{Bi}(z)$. In this situation the solution $\hbox{Bi}(z)$ is not acceptable, since $\hbox{Bi}(z)$ goes to infinity as $z$ grows. Then the solution of equation (\ref{eq004}) is given by
\begin{equation} \label{eq006}
\psi_{{}_E}(x)=\sqrt[6]{2\vert k\vert}\hbox{Ai} \left(\sqrt[3]{2\vert k\vert}\left(x-\frac{E}{\vert k\vert}\right)\right)\;,
\end{equation}
where we have added the constant $\sqrt[6]{2\vert k\vert}$ in order to the equation (\ref{eq006}) in order to satisfy relation (\ref{eq003}).
Although the spectral properties of operators related to the linear potential (Stark effect), was proposed in reference \cite{Avron1977}, we prefer to use the squeeze $\hat{S}(r)$ and displacement $\hat{D}(\alpha)$ operators, with parameters $r= \ln\sqrt[3]{2\vert k\vert}$ and $\alpha =E/({\sqrt{2}\vert k\vert})$, respectively. Where the action of the above operators on arbitrary function is well known \cite{Leonhardt}. We may rewrite the solution of the Schr\"odinger equation (\ref{eq006}) as
\begin{equation} \label{eq007}
\psi_{_{E}}(x)=\hat{D}\left(\frac{E}{\sqrt{2}\vert k\vert}\right)\hat{S} \left(\ln\sqrt[3]{2\vert k\vert}\right) \hbox{Ai}(x)\;.
\end{equation}
If we set $\psi_{_{E}}(x)=\langle x\vert \psi_{_{E}}\rangle$, we have
\begin{equation} \label{eq008}
\vert\psi_{_{E}} \rangle=\hat{D}\left(\alpha\right)\hat{S} \left(r\right)\vert\hbox{Ai} \rangle\;,
\end{equation}
where $\vert\hbox{Ai}\rangle$ is the Airy state. We wish to express the Schr\"odinger equation (\ref{eq004}) in terms of the ket $\vert\hbox{Ai}\rangle$, in order to establish an eigenvalue equation. Inserting equation (\ref{eq008}) into equation (\ref{eq004}) we obtain
\begin{equation} \label{eq011}
\hat{S}^{\dagger}\left(r\right)\hat{D}^{\dagger}\left(\alpha\right)\left(\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}\right)\hat{D}\left(\alpha\right)\hat{S} \left(r\right)\vert\hbox{Ai}\rangle=E\vert\hbox{Ai}\rangle\;,
\end{equation}
where we rewrite the Hamiltonian (\ref{eq004}) in terms of the coordinate and momentum operators $\hat x$ and $\hat{p}$, respectively. It may be shown that \cite{Leonhardt}
\begin{eqnarray} \label{eq0012}
\hat{D}^{\dagger}\left(\alpha\right)\left(\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}\right)\hat{D}\left(\alpha\right)&=&\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}+E\;,\nonumber\\
\hat{S}^{\dagger}\left(r\right)\left(\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}\right)\hat{S}\left(r\right)&=&\frac{(2\vert k\vert)^{\frac{2}{3}}}{2}\left(\hat{p}^{2}+\hat{x}\right)\;,
\end{eqnarray}
where we have
\begin{equation} \label{eq0013}
(\hat{p}^{2}+\hat{x})\vert\hbox{Ai}\rangle=0\;,
\end{equation}
that confirms that the ket $\vert\hbox{Ai}\rangle$ satisfies the Airy equation. But does the Airy equation represents an eigenvalue problem? This question may be answered by using the displaced ket $\vert\gamma,\hbox{Ai}\rangle=\hbox{e}^{-i\gamma \hat{p}}\vert\hbox{Ai}\rangle$, and it is easy to see that
\begin{equation} \label{eq0014}
\left(\hat{p}^{2}+\hat{x}\right)\vert\gamma,\hbox{Ai}\rangle=\gamma\vert\gamma,\hbox{Ai}\rangle\;,
\end{equation}
we have thus shown that any displaced Airy ket $\vert\gamma,\hbox{Ai}\rangle$ are eigenvectors of the Airy equation with eigenvalue $\gamma$, in particular when $\gamma=0$ as in equation (\ref{eq0013}). If we assume that
\begin{equation} \label{eq0015}
\left(\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}\right)\vert\psi_{_{E'}} \rangle=E'\vert\psi_{_{E'}} \rangle\;,
\end{equation}
where $\vert\psi_{_{E'}} \rangle=\hat{D}\left(\alpha\right)\hat{S} \left(r\right)\vert\gamma,\hbox{Ai}\rangle$, we have
\begin{equation} \label{eq0016}
E'=\frac{(2\vert k\vert)^{\frac{2}{3}}}{2}\gamma+E\;,
\end{equation}
and, by making $\gamma=0$, we recover equation $\ref{eq011}$.
On the other hand, we have a completeness relation
\begin{equation}
\label{eq0019}
\int_{-\infty}^{+\infty} d\gamma\vert\gamma,\hbox{Ai}\rangle\langle \gamma,\hbox{Ai}\vert=\hat{I}\;,
\end{equation}
that may be verified by using $\langle x\vert\hat{I}\vert x'\rangle=\delta(x-x')$, and equation (\ref{eq003}).\\
To finish this section, let us consider again the Schr\"odigner equation for a single particle under a linear potential $V(x)=-\vert k\vert x$, for $-\infty<x<\infty$, whose Hamiltonian may be reduced to the last one by considering
\begin{equation} \label{eq0020}
(-1)^{\hat{n}} \left( \frac{\hat{p}^{2}}{2} -\vert k\vert\hat{x} \right)(-1)^{\hat{n}}=\frac{\hat{p}^{2}}{2}+\vert k\vert\hat{x}\;,
\end{equation}
where $(-1)^{\hat{n}}$ is the so-called parity operator. Then, we have
\begin{equation} \label{eq0021}
\vert\psi_{_{E}}\rangle=(-1)^{\hat{n}}\hat{D}\left(\frac{E}{\sqrt{2}\vert k\vert}\right)\hat{S}\left(\ln\sqrt[3]{2\vert k\vert}\right)\vert\hbox{Ai}\rangle\;.
\end{equation}
Therefore, we may conclude that the algebraic Airy problem does not depend on the sign of $k$.
\section{Airy basis state}
The fact that the optical paraxial Helmholtz equation is mathematically equivalent to a Schr\"odinger equation, may be responsible for the many analogies that are found between quantum physics and classical optics. We will now consider the collapses and revivals occurring for Airy-beam propagation
in a symmetrical linear potential and waveguide respectively, using an Airy bases states.\\
Let us consider a symmetrical linear potential $V=\lambda\vert x\vert$, where $\lambda>0$, the Schr\"odigner equation for a single particle under this potential reads,
\begin{equation}
\label{eq3.1}
-\frac{1}{2}\frac{d^{2}}{dx^{2}}\psi(x)+\lambda\vert x\vert\psi(x)=E\psi(x)\;,
\end{equation}
again, we have used appropriately scaled units $m=\hbar=1$ to simplify the equation. Now we are going to solve for all values of $x$.\\
\noindent Let us assume $x\geq 0$, the above Schr\"odinger equation reads:
\begin{equation}
\label{eq3.1b}
-\frac{1}{2}\frac{d^{2}}{dx^{2}}\psi_{{}_+}(x)+ \lambda x\psi_{{}_+}(x)=E\psi_{{}_+}(x)\;.
\end{equation}
whose solution is given by
\begin{equation}
\label{eq3.5}
\psi_{{}_+}(x)=N_{{}_+}\hbox{Ai} \left(\sqrt[3]{2k}\left(x-\frac{E}{\lambda}\right)\right)\;,
\end{equation}
where $N_{{}_+}$ is an appropiate normalization constant. On the other hand, if $x<0$ the Schr\"odinger equation reads:
\begin{equation}
\label{eq3.1c}
-\frac{1}{2}\frac{d^{2}}{dx^{2}}\psi_{{}_-}(x)- \lambda x\psi_{{}_-}(x)=E\psi_{{}_-}(x)\;.
\end{equation}
and its solution can be written as,
\begin{equation}
\label{eq3.5b}
\psi_{{}_-}(x)=N_{{}_-}\hbox{Ai} \left(\sqrt[3]{2k}\left(-x-\frac{E}{\lambda}\right)\right)\;,
\end{equation}
where $N_{{}_-}$ is another appropiate normalization constant.\\
From wave functions (\ref{eq3.5}) and (\ref{eq3.5b}), it is easy to see that if $\psi_{{}_+}(x)=\psi_{{}_-}(-x)$, then $N_{{}_+}=N_{{}_-}$, and we have the even wave solutions. Similarly, if $\psi_{{}_+}(x)=-\psi_{{}_-}(-x)$, then $N_{{}_+}=-N_{{}_-}$ and we have odd wave solutions. In order to solve equation (\ref{eq3.1}), we must connect the two wave functions (\ref{eq3.5}) and (\ref{eq3.5b}) along with their derivates at $x=0$. From $\psi_{{}_-}(0)=\psi_{{}_+}(0)$, and $\psi'_{{}_-}(0)=\psi'_{{}_+}(0)$, we write
\begin{eqnarray}
(N_{{}_-}-N_{{}_+})\hbox{Ai}\left(-\sqrt[3]{\frac{2}{\lambda^2}}E\right)&=&0\;,\label{eq3.5d}\\
(N_{{}_-}+N_{{}_+})\hbox{Ai}'\left(-\sqrt[3]{\frac{2}{\lambda^2}}E\right)&=&0\;,\label{eq3.5e}
\end{eqnarray}
respectively. If $N_{{}_-}=N_{{}_+}$, from equation (\ref{eq3.5e}) we define the energy of the even states as,
\begin{equation} \label{eq3.5f}
E_{2n}=-\sqrt[3]{\frac{\lambda^2}{2}}a'_{n+1}\;,
\end{equation}
where $a'_n$ are the $n$-th zeros of the Airy function derivate \cite{abramowitz}. Then the even states will be,
\begin{equation} \label{eq3.5g}
\psi_{{}_{2n}}(x)=N_{{}_{2n}}\hbox{Ai}\left(\sqrt[3]{2\lambda}\left(\vert x\vert-\frac{E_{2n}}{\lambda}\right)\right)\;.
\end{equation}
Similarly, if $N_{{}_-}\neq N_{{}_+}$, from equation (\ref{eq3.5d}) we define the energy of the odd states as,
\begin{equation} \label{eq3.5h}
E_{2n+1}=-\sqrt[3]{\frac{\lambda^2}{2}}a_{n+1}\;,
\end{equation}
where $a_n$ are the $n$-th zeros of the of Airy function \cite{abramowitz} ($n\ge 1$), and the odd states reads,
\begin{equation} \label{eq3.5i}
\psi_{{}_{2n+1}}(x)=N_{{}_{2n+1}} \hbox{sgn}(x)\hbox{Ai} \left(\sqrt[3]{2\lambda}\left(\vert x\vert-\frac{E_{2n+1}}{\lambda}\right)\right)\;.
\end{equation}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\textwidth]{figure01.png}
\caption{The Airy eigenfunctions for six lowest energy levels.}
\label{fig1}
\end{figure}
In figure \ref{fig1}, we show the Airy eigenfunctions for $\lambda=1$ and six lowest energy levels, (a) $E_0=$0.808616, (b) $E_1=$1.855757, (c) $E_2=$2.578096, (d) $E_3=$3.244607, (e) $E_4=$3.825715 and (f) $E_5=$4.381671, obtained from the equations (\ref{eq3.5f}) and (\ref{eq3.5h}). We have calculated numerically the normalization factors for each eigenfunction.\\
It is worth noting that Airy eigenfuctions were obtained from a sequence of the squeeze $\hat{S}(r)$ and displacement $\hat{D}(\alpha)$ operators, with appropriate choice for displaced parameter in terms of the energy equations (\ref{eq3.5f}) and (\ref{eq3.5h}). In case of odd functions, for $x<0$ the parity operator was used by means of sign function $\hbox{sgn}(x)$.\\
It may be numerically shown that these functions form and orthonormal basis with a weight function equal to one, i.e.,
\begin{equation} \label{eq3.5j}
\int^{+\infty}_{-\infty}dx'\psi_m(x')\psi_n(x')=\delta_{mn}\;.
\end{equation}
\section{Applications}
The time-dependent Sch\"odinger equation with a symmetrical linear potential
is,
\begin{equation} \label{eq3.100}
-\frac{1}{2}\frac{d^{2}}{dx^{2}}\phi(x,t)+ \lambda\vert x\vert\phi(x,t)=i\frac{\partial}{\partial t}\phi(x,t)\;.
\end{equation}
As the Hamiltonian is time independent, the equation (\ref{eq3.100}) can be integrated with respect to time, and the formal solution for any arbitrary initial condition $\phi(x,0)$ at time $t=0$, is
\begin{equation} \label{eq3.90}
\phi(x,t)=\exp\left[-it\left(\frac{1}{2}\hat{p}^2+\lambda\vert x\vert\right)\right]\phi(x,0)\;.
\end{equation}
Therefore, $\phi(x,0)$ can be expanded as
\begin{equation} \label{eq.392}
\phi(x,0)=\sum^{\infty}_{n=0}c_n\psi_n(x)\;,
\end{equation}
where $\psi_n(x)$ are the Airy eigenfunctions and $c_n$ are the expansion coefficients. Considering the expansion (\ref{eq.392}), we have the propagated wave function
\begin{equation} \label{eq3.101}
\phi(x,t)=\sum^{\infty}_{n=0}c_n\exp(-itE_n)\psi_n(x)\;,
\end{equation}
where $E_n$ are the corresponding eigenvalues of the orthogonal Airy eigenfunctions. The expansion coefficients are determined from the initial condition as
\begin{equation} \label{eq3.102}
c_n=\int^{+\infty}_{-\infty}dx'\phi(x',0)\psi_n(x')\;,
\end{equation}
which we may obtain numerically.
\subsection[revival]{Collapses and revivals}
Because of their charge neutrality and long lifetime, neutrons were promising candidates with which to observe experimentaly gravitational quantum bound states \cite{Nesvizhevsky2002}. The dynamics of a bouncing wave packet under the influence of a gravity potential has been studied in Ref. \cite{gea_1999}, and the falling packet or neutrons do not move continuously along the vertical direction, but rather jump from one height to another, as predicted by quantum theory with the colapses and revivals \cite{PhysRevLett.44.1323}.\\
By employing a similar methodology as in reference \cite{gea_1999}, we consider an initial state as a Gaussian wave packet with a width $\sigma$ and localized at $x=x_0$ with zero initial momentum,
\begin{equation} \label{eq3.103}
\phi(x,0)=\left(\frac{2}{\pi\sigma^2}\right)^{1/4}\exp\left[-\frac{(x-x_0)^2}{\sigma^2}\right]\;,
\end{equation}
in a symmetrical linear potencial for $\lambda=1$. In order to have the trayectory of this wave packet, we obtain te mean value of coordinate $x$,
\begin{equation} \label{eq3.104}
\langle\hat x\rangle=\int^{+\infty}_{-\infty}dx'x'\vert\phi(x',t)\vert^2\;,
\end{equation}
as is shown in figure \ref{fig2}, where $\phi(x,t)$ was obtained by using equation (\ref{eq3.101}) with $x_0=10$, $\sigma=2.0$, gravity $g=2$ and $t_g=1 /\sqrt[3]{2}$, as in reference (\cite{gea_1999}).\\
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\textwidth]{figure02.png}
\caption{Mean value of the position as a funtion of time for a wave packet defined in equation (\ref{eq3.103}).}
\label{fig2}
\end{figure}
The particle has a few well defined bounces, after a while the bounces cease around the mean value of position zero. But at later times the oscillations revive an the particle begins to bounce again. This revival oscillations are purely quantum and a consequence of the discrete energy levels of this problem, obtained from the equations (\ref{eq3.5f}) and (\ref{eq3.5h}), see figure \ref{fig1}.
\subsection[light]{Light propagation}
Laser beam propagation has been a subject of active research for several years. Quantum mechanics describes similar physics as the physical optics, providing new methods to describe and characterize the beam propagation. Paraxial wave propagation of Airy beams has been the object of several investigations. In particular, the Huygens-Fresnel integral yields the propagated wave function Airy-type wavelets \cite{Torre_2015}, defined as product of two Airy functions, i.e.,
\begin{equation} \label{eq3.105}
E(x)=\sqrt{C}\;\hbox{Ai}\left(x+q\right)\hbox{Ai}\left(-x+q\right)\;,
\end{equation}
where $q$ is called a shift parameter and $\sqrt{C}$ is the normalization constant. It is worth mentioning that Airy patterns are very sensitive to such shift parameter. Airy wavelets spread faster than the Gaussian beam, for this reason we will assume a symmetrical linear GRIN medium as $n^2(x)=n^2_0(1-\alpha\vert x\vert)$, in order to overrule the spreading. The one-dimensional Helmholtz equation for a GRIN medium is,
\begin{equation} \label{eq3.106}
-\frac{\partial^2E}{\partial z^2}=\left[\frac{\partial^2}{\partial x^2}+\tilde{k}^2n^2(x)\right]E\;,
\end{equation}
where $\tilde{k}$ is the wave number, and $n(x)$ the variable refraction index. Introducing the momentun operator $\hat p$, the the Helmholtz equation is
expressed
\begin{equation} \label{eq3.107}
\frac{\partial^2E}{\partial z^2}=-\left[\kappa^2-\left(\hat{p}^2+2\lambda\vert x\vert\right)\right]E\;,
\end{equation}
where we have defined $\kappa=\tilde{k}n_0$ and $2\lambda=\tilde{k}^2n^2_0\alpha$, whose formal solution may written as
\begin{eqnarray}
E(x,z)&=&\exp\left[-iz\sqrt{\kappa^2-2\left(\frac{1}{2}\hat{p}^2+\lambda\vert x\vert\right)}\;\right]E(x,0)\;,\label{eq3.108}\\
&\approx&\exp\left(-i\kappa z\right)\exp\left[\frac{iz}{\kappa}\left(\frac{1}{2}\hat{p}^2+\lambda\vert x\vert\right)\right]E(x,0)\;,\label{eq3.109}
\end{eqnarray}
here we have developed the square root as a first order Taylor series. Comparing the propagation operator in equation (\ref{eq3.90}) with the corresponding operator in equation (\ref{eq3.109}), it is noted that $z=-\kappa t$.\\
\begin{figure}[ht!]
\centering
\includegraphics[width=0.6\textwidth]{figure03.png}
\caption{(a) Initial intensity. (b) Intensity distribution of the propagation of Airy-type wavelet beam $\hbox{Ai}\left(x+q\right)\hbox{Ai}\left(-x+q\right)$ for $q=-1.472910$.}
\label{fig3}
\end{figure}
Figure \ref{fig3}(b) depicts intensity cross-sections dynamics under symmetrical linear GRIN medium propagation of Airy-type wavelet as a function of distance $z$, which can be calculated from equation (\ref{eq3.101}) with parameters $\lambda=0.1$ and $\kappa=1.0$. The intensity of this initial field (\ref{eq3.105}) is shown in figure \ref{fig3}(a), where $q=2^{-2/3}a_1$ where $a_1$ is the first zero of the Airy function. As the propagation goes on, the initial field splits into two parts, and the intensity distribution in figure \ref{fig3}(b) is the result of interference of these two beams symmetrically reflected with respect to the plane $x=0$, and may resemble a Pearcey beam \cite{Ring:12}.
\section{Airy states in terms of position eigenstates}
Finally, in this section we will show how the Airy states may be obtained from position and momentum eigenstates. Given an arbitrary state $\langle\psi\vert$, with the completeness relation in mind, we have
\begin{eqnarray} \label{eq4.1}
\langle\psi\vert\hbox{e}^{-ix\hat{p}}\vert\hbox{Ai}\rangle &=& \int_{-\infty}^{+\infty}dp\;\hbox{e}^{-ixp} \langle \psi\vert p\rangle \langle p\vert\hbox{Ai}\rangle\;,\nonumber\\
&=&\int_{-\infty}^{+\infty}dp\; \langle \psi\vert\hbox{e}^{i \frac{\hat{p}^{3}}{3}}\vert p\rangle\langle p\vert x\rangle=\langle \psi\vert\hbox{e}^{i \frac{\hat{p}^{3}}{3}}\vert x\rangle\;,
\end{eqnarray}
where we have used $\exp(-ixp)=\sqrt{2\pi}\langle x\vert p\rangle$, and $\langle p\vert\hbox{Ai}\rangle=\exp(ip^3/3)/\sqrt{2\pi}$. Then
\begin{equation} \label{eq4.2}
\vert\hbox{Ai}\rangle=\exp\left(\frac{i\hat{p}^{3}}{3}\right)\exp\left(ix\hat{p}\right)\vert x\rangle\;.
\end{equation}
It is possible to check the above relation, let us consider
\begin{eqnarray} \label{eq4.3}
\langle x'\vert\hbox{Ai}\rangle &=&\langle x'\vert\exp\left(\frac{i\hat{p}^{3}}{3}\right)\exp\left(ix\hat{p}\right)\vert x\rangle\;,\nonumber\\
&=&\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\; \exp\left[i\left( \frac{p^3}{3}+x'p\right)\right]\;,
\end{eqnarray}
which recover the Airy's integral (\ref{eq001}). On the other hand, we know already that $\hat{x}\vert x\rangle=x\vert x\rangle$, from equation (\ref{eq4.2}), yield
\begin{eqnarray} \label{eq4.3a}
\hat{x}\vert x\rangle&=&\hat{x}\;\exp\left(-\frac{i\hat{p}^{3}}{3}\right)\exp\left(-ix\hat{p}\right)\vert\hbox{Ai}\rangle\;,\nonumber\\
&=&x\;\exp\left(-\frac{i\hat{p}^{3}}{3}\right)\exp\left(-ix\hat{p}\right)\vert\hbox{Ai}\rangle\;,
\end{eqnarray}
where the equation (\ref{eq0013}). The above equation allows to write position eigenstates as the application of an exponential operator to the Airy states
\begin{eqnarray} \label{eq4.3a}
\vert x\rangle=\;\exp\left(-\frac{i\hat{p}^{3}}{3}\right)\exp\left(-ix\hat{p}\right)\vert\hbox{Ai}\rangle\;,
\end{eqnarray}
Therefore Airy states may also be related to the vacuum state via an exponential operator as position eigenstates also may be written as \cite{Soto-Eguibar2013}
\begin{equation}
|x\rangle = \frac{e^{-x^2/2}}{\pi^{1/4}}e^{-\frac{a^{\dagger 2}}{2}+\sqrt{2}xa^{\dagger}}|0\rangle .
\end{equation}
\section{Conclusions}
In conclusion, to our best knowledge, we have obtained a new solution of the Schr\"odinger equation for a symmetrical linear potential. This solution were obtained using the squeeze $\hat{S}(r)$ and displacement $\hat{D}(\alpha)$ operators, with appropriate parameters related with the energy. This Airy ket is related to the usual coordinate eigenstate.
| {
"timestamp": "2021-06-01T02:05:51",
"yymm": "2012",
"arxiv_id": "2012.15689",
"language": "en",
"url": "https://arxiv.org/abs/2012.15689",
"abstract": "We study the eigenvalue problem for a linear potential Hamiltonian and, by writing Airy equation in terms of momentum and position operators define Airy states. We give a solution of the Schrödinger equation for the symmetrical linear potential in terms of the squeeze and displacement operators. Finally, we write the unit operator in terms of Airy states and find a relation between them and position and momentum eigenstates.",
"subjects": "Quantum Physics (quant-ph)",
"title": "Airy eigenstates and their relation to coordinate eigenstates",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860381,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019674783212
} |
https://arxiv.org/abs/1710.03108 | The structure of multiplicative tilings of the real line | Suppose $\Omega, A \subseteq \RR\setminus\Set{0}$ are two sets, both of mixed sign, that $\Omega$ is Lebesgue measurable and $A$ is a discrete set. We study the problem of when $A \cdot \Omega$ is a (multiplicative) tiling of the real line, that is when almost every real number can be uniquely written as a product $a\cdot \omega$, with $a \in A$, $\omega \in \Omega$. We study both the structure of the set of multiples $A$ and the structure of the tile $\Omega$. We prove strong results in both cases. These results are somewhat analogous to the known results about the structure of translational tiling of the real line. There is, however, an extra layer of complexity due to the presence of sign in the sets $A$ and $\Omega$, which makes multiplicative tiling roughly equivalent to translational tiling on the larger group $\ZZ_2 \times \RR$. | \section{Introduction}
\label{sec:intro}
Tilings have long fascinated mathematicians \cite{grunbaum1986tilings}.
The case where one moves a single object by translation in an abelian group (translational tiling)
has proved both challenging and full of connections to Functional Analysis \cite{kolountzakis2004milano},
such as connections to the so-called Fuglede Conjecture or Spectral Set Conjecture \cite{fuglede1974operators,tao2004fuglede,kolountzakis2006tiles}.
Simultaneous tiling of a set by both translation and multiplication (with linear operators on the space where tiling takes place)
has also been studied mainly because of its connections to wavelets
\cite{wang2002wavelets,speegle2003dilation,olafsson2004wavelets,dobrescu2006wavelet,ionascu2006simultaneous}.
\begin{definition}[Translational tiling, multiplicative tiling]\label{def:tiling}\ \\
Suppose $f:{\mathbb R}^d\to{\mathbb C}$ is measurable and $A \subseteq {\mathbb R}^d$. We say that $f+A$ is a tiling
of ${\mathbb R}^d$ at level $\ell$ if
$$
\sum_{a \in A} f(x-a) = \ell
$$
for almost all $x \in {\mathbb R}^d$ with the sum converging absolutely.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $\Omega+A$ is a tiling.
If $A \subseteq {\mathbb R}\setminus\Set{0}$ and
$$
\sum_{a \in A} f(a^{-1} x) = \ell
$$
for almost all $x \in {\mathbb R}^d$, with absolute convergence, then we say that $A\cdot f$ is a tiling.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $A\cdot \Omega$ is a tiling.
\end{definition}
Our purpose here is to study the structure of multiplicative tilings.
We are guided in this by previous work on the structure of translational tilings of the real line or of the integer line.
In \cites{newman1977tesselations,leptin1991uniform,lagarias1996tiling,kolountzakis1996structure}
it is proved, under very broad conditions, that any translational tiling of the real line (or of the integer line) must be periodic.
The main tool in the study of translational tilings by a single tile has long been (see e.g.\ \cite{kolountzakis2004milano}) Fourier Analysis,
where the problem is expressed roughly as a support condition of the Fourier Transform of the set of translates
on the zero set of the Fourier Transform of the tile, an approach that will also be used extensively in this paper.
Suppose then that $A \subseteq {\mathbb R}\setminus\Set{0}$ is a discrete set and $\Omega \subseteq {\mathbb R}$ is a measurable set.
We want to derive properties of $\Omega$ and $A$ from the assumption of multiplicative tiling $A\cdot \Omega = {\mathbb R}$ (multiplicative tiling at level 1).
\noindent{\bf The importance of sign.}
It is important to emphasize that if $A$ or $\Omega$ are nonnegative (or of one sign, more generally) then the problem
quickly reduces to translational tiling.
Indeed, suppose that $\Omega \subseteq (0,+\infty)$.
Then, writing $A = A^+ \cup (-A^-)$, with $A^\pm \subseteq (0,+\infty)$, we see that
the tiling condition $A\cdot\Omega = {\mathbb R}$ is equivalent to the two tiling conditions
$$
{\mathbb R}^+ = A^+\cdot \Omega\ \ \ \mbox{and}\ \ \ {\mathbb R}^+ = A^-\cdot\Omega.
$$
Taking logarithms of both we reduce $A\cdot\Omega={\mathbb R}$ to the two independent translational tilings
$$
{\mathbb R} = \log\Omega + \log A^+\ \ \ \mbox{and}\ \ \ {\mathbb R} = \log\Omega + \log A^-.
$$
So if one can understand translational tiling by the set $\log\Omega$ then all results transfer back to our multiplicative
tiling $A\cdot\Omega = {\mathbb R}$ if $\Omega$ is of one sign.
Similarly, if $A \subseteq (0, +\infty)$ then, writing again $\Omega = \Omega^+ \cup (-\Omega^-)$, with $\Omega^\pm \subseteq (0, +\infty)$,
we obtain that $A\cdot \Omega = {\mathbb R}$ is equivalent to the two translational tilings
$$
{\mathbb R} = \log\Omega^+ + \log A \ \ \ \mbox{and}\ \ \ {\mathbb R} = \log\Omega^- + \log A.
$$
If however the two sets $A$ and $\Omega$ have both a positive and a negative part the multiplicative tiling $A\cdot \Omega = {\mathbb R}$
becomes a lot more complicated. It still reduces to tiling by translation but not of the ordinary kind with one set being
translated around to fill space.
Indeed, when $A = A^+ \cup (-A^-)$ and $\Omega = \Omega^+ \cup (-\Omega^-)$ then the multiplicative tiling $A \cdot \Omega = {\mathbb R}$
reduces to the two simultaneous tilings
$$
{\mathbb R}^+ = A^+\Omega^+ \cup A^-\Omega^-\ \ \ \mbox{and}\ \ \ {\mathbb R}^+ = A^-\Omega^+ \cup A^+\Omega^-,
$$
and, after taking logarithms, to the sumultaneous translational tiling
\begin{align}
{\mathbb R} &= (\log\Omega^++\log A^+) \cup (\log\Omega^-+\log A^-)\ \ \ \mbox{and}\ \ \ \label{cti}\\
{\mathbb R} &= (\log\Omega^++\log A^-) \cup (\log\Omega^-+\log A^+).\nonumber
\end{align}
The meaning of the notation here should be obvious. For instance, the meaning of the first equation in \eqref{cti} is that
almost every point in ${\mathbb R}$ can be written, uniquely, either in the form $\log\omega+\log a$, with $\omega\in\Omega^+, a \in A^+$,
or in the form $\log\omega+\log a$, with $\omega \in \Omega^-, a \in A^-$.
Put differently, the translates of the set $\log\Omega^+$ by the numbers in $\log A^+$ together with the translates of the set $\log\Omega^-$
by the numbers in $\log A^-$ cover almost all of ${\mathbb R}$ exactly once and any two of these sets intersect at a set of measure zero.
The purpose of this paper is first, to exploit \eqref{cti} in order to derive structural properties of the
set $\Omega$ (the tile) and the set $A$ (the set of multiples) and, second, to study \eqref{cti} (which we call {\em cross-tiling})
in itself, and in the case of a finite cyclic group, where things are simpler. In particular, we want to make some
connections and distinctions to ordinary translational tiling where only one set is translated.
The structure of the paper is as follows.
In \S\ref{sec:rationality} we restrict ourselves to translational tilings and generalize some periodicity and rationality results
(from \cite{kolountzakis1996structure,lagarias1996tiling}) to the extent that they become useful to us in the analysis of \S\ref{sec:structure}
and \S\ref{sec:tile-structure}
where structure results are proved for the logarithms of the sets $A$ and $\Omega$ respectively.
In \S\ref{sec:ct} the problem of cross tiling is studied in cyclic groups (we show in \S\ref{sec:structure} that multiplicative tiling of ${\mathbb R}$
reduces to cross tiling in cyclic groups), and we provide examples of cross tilings which differ significantly from ordinary translational
tilings as well as a Fourier condition for cross tiling, analogous to the one for ordinary translational tilings.
\section{The structure of multiple translational tiling by a set}
\label{sec:rationality}
\begin{lemma}\label{lm:poly-zeros}
Suppose $\Lambda$ is a finite subset of the torus ${\mathbb T}={\mathbb R}/{\mathbb Z}$ and
\beql{expoly}
f(n) = \sum_{\lambda\in\Lambda} c_\lambda e^{2\pi i \lambda n},\ \ \ (n\in{\mathbb Z})
\end{equation}
is an exponential polynomial on the integers ($c_\lambda \in {\mathbb C}$).
Suppose that
$$
\Lambda = \Lambda_1 \cup \cdots \cup \Lambda_r,\ \ \ (r\ge 1)
$$
is the decomposition of $\Lambda$ into rational equivalence classes (two points in $\Lambda$ have rational difference
if and only if they belong to the same $\Lambda_j$).
Write also $f_j(n) = \sum_{\lambda\in\Lambda_j} c_\lambda e^{2\pi i \lambda n}$ so that $f=f_1+\cdots+f_r$.
Then the zeros of $f$ are the common zeros of the $f_j$ plus a finite set (possibly empty).
\end{lemma}
\begin{proof}
Write $Z(\phi)$ for the zero set of a function $\phi$ on its domain.
Define the set of integers
$$
X = Z(f) \setminus \bigcap_{j=1}^r Z(f_j).
$$
We have to show that $X$ is finite.
By the Skolem-Mahler-Lech Theorem \cite{lech1953note} the integer zero set of every exponential polynomial, such as \eqref{expoly},
is a periodic set plus a finite set (possibly empty). Since $\Abs{f_1}^2+\cdots+\Abs{f_r}^2$ is also an exponential
polynomial it follows that both $Z(f)$ and $\bigcap_{j=1}^r Z(f_j)$ are periodic sets plus a finite set.
Therefore $X$ is also a periodic set, give or take a finite set.
It suffices therefore to prove that $X$ does not contain arithmetic progressions of arbitrary length, as then
it follows that $X$ has no periodic part and is just a finite set.
For $j=1,2,\ldots,r$
write $\Lambda_j = a_j + Q_j$, where $Q_j \subseteq {\mathbb Q}$ is a finite set and $a_i-a_j \notin {\mathbb Q}$ for $i \neq j$.
Let $N \in {\mathbb N}$ be the least common multiple of all denominators in all the $Q_j$, so that $Nq \in {\mathbb Z}$ for all $q \in \bigcup_{j=1}^r Q_j$.
If $X$ contains arbitrarily long arithmetic progressions then it contains a progression of the form
\beql{assumption1}
a+bNk,\ \ \ (k=0, 1, 2,\ldots, r)
\end{equation}
for some $a, b \in {\mathbb Z}$, $b>0$.
For each $k=1,2,\ldots,r$ we then have
\begin{align}
f(a+bNk) &= \sum_{j=1}^r \sum_{q\in Q_j} c_{a_j+q} e^{2\pi i (a_j+q) (a+bNk)} \nonumber \\
&= \sum_{j=1}^r z_j^k x_j \label{vandermonde}
\end{align}
with
\begin{align*}
z_j &= e^{2\pi i a_j b N} \\
x_j &= e^{2\pi i a_j a} \sum_{q \in Q_j} c_{a_j+q} e^{2\pi i qa} = f_j(a).
\end{align*}
All numbers $z_j$ are different since the differences of the $a_j$'s are irrational, so the Vandermonde linear system
$$
\sum_{j=1}^r z_j^k x_j = 0,\ \ \ (j=1,2,\ldots,r)
$$
which we obtain if we assume that $a+bNk \in X$, for $k=1,2,\ldots,r$, has only the all-zero solution $x_1=\cdots=x_r=0$, which implies
that $f_1(a)=\cdots=f_r(a)=0$, hence $a$ is a common zero of all $f_j$, hence not in $X$, a contradiction with \eqref{assumption1} for $k=0$.
So $X$ does not contain arbitrarily long arithmetic progressions and is, therefore, a finite set.
\end{proof}
\begin{lemma}\label{lm:torus-tiling}
Suppose $\Lambda$ is a finite subset of the torus ${\mathbb T}={\mathbb R}/{\mathbb Z}$ and
$$
\Lambda = \Lambda_1 \cup \cdots \cup \Lambda_r,
$$
is the decomposition of $\Lambda$ into rational equivalence classes.
Suppose also that $F \in L^1({\mathbb T})$ and $c_\lambda \in {\mathbb C}$ are such that
\beql{torus-tiling}
\sum_{\lambda\in\Lambda} c_\lambda F(x-\lambda) = \text{const.\ \ for almost all $x \in {\mathbb T}$}.
\end{equation}
If the function $F$ takes only countably many values then for each $j=1,2,\ldots,r$ we also have
$$
\sum_{\lambda \in \Lambda_j} c_\lambda F(x-\lambda) = \text{const.${}_j$\ \ for almost all $x \in {\mathbb T}$}.
$$
\end{lemma}
\begin{proof}
Our assumption \eqref{torus-tiling} is equivalent to
$$
\ft{F}(n) = 0\ \ \text{or}\ \ \sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n} = 0,\ \ \ (n \neq 0).
$$
In other words we must have
$$
Z(\ft{F}) \cup Z\left(\sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n}\right) \supseteq {\mathbb Z}\setminus\Set{0}.
$$
But, from Lemma \ref{lm:poly-zeros},
$$
Z\left(\sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n}\right) \setminus \bigcap_{j=1}^r\ Z\left(\sum_{\lambda\in\Lambda_j}c_\lambda e^{2\pi i \lambda n}\right)
$$
is a finite set.
This implies that
$Z(\ft{F}) \cup Z(\sum_{\lambda\in\Lambda_j}c_\lambda e^{2\pi i \lambda n})$ contains all but finitely many integers, for each $j=1,2,\ldots,r$.
Consequently the function
\beql{torus-tiling-j}
\sum_{\lambda \in \Lambda_j} c_\lambda F(x-\lambda)
\end{equation}
is a trigonometric polynomial of $x$. But, as $F$ takes only countably many values and this is a finite sum, the function
in \eqref{torus-tiling-j} has a countable range too, and this can only happen
if the function is a constant, which is exactly what we wanted to prove.
\end{proof}
\begin{lemma}\label{lm:integer-tiling}
Suppose that $V \subseteq {\mathbb Z}\setminus\Set{0}$ is a finite set of non-zero integers, $A \subseteq {\mathbb R}$ is a
discrete set of bounded density and $v_t \in V$, for $t \in A$, are such that
\beql{int-tiling}
\sum_{t \in A} v_t f(x-t) = k,\ \ \ \mbox{for almost all $x \in {\mathbb R}$},
\end{equation}
where $f \in L^1({\mathbb R})$ is an integer-valued function of compact support and $k$ is an integer.
Then the measure
$$
\mu = \sum_{t \in A} v_t \delta_t
$$
is a periodic measure (and, therefore, $A$ is a periodic set).
\end{lemma}
\begin{proof}
It follows from the proof of \cites[Theorem 3.1]{kolountzakis1996structure} that
\beql{mu-ft}
{\rm supp\,}\ft{\mu} \subseteq \Set{0} \cup \Set{\xi: \ft{f}(\xi)=0},
\end{equation}
with the right-hand side of the above equation being a discrete set (since $f$ has compact support $\ft{f}$ is analytic).
For $v \in V$ write $A^v = \Set{t \in A: v_t = v}$.
It follows from the proof of \cites[Theorem 5.1]{kolountzakis1996structure}
that each $A^v$ has the form
\beql{av-qp-1}
A^v = F^v \triangle \bigcup_{j=1}^{J^v} \left( \alpha_j^v {\mathbb Z} + \beta_j^v \right),
\end{equation}
for some $J^v \in {\mathbb N}$, $\alpha_j^v > 0, \beta_j^v \in {\mathbb R}$ and finite set $F^v \subseteq {\mathbb R}$.
(Let us only indicate that, as in \cites{kolountzakis1996structure}, the main ingredient
in the proof of \eqref{av-qp-1} is a theorem of Meyer \cites[Theorem 4.2]{kolountzakis1996structure}, \cites{meyer1970nombres},
itself a consequence of the Idempotent Theorem of P.J. Cohen \cites[Theorem 4.1]{kolountzakis1996structure}, \cites{cohen1959homomorphisms}.)
By merging together the $a_j^v$ which are commensurable we can rewrite \eqref{av-qp-1} as
\beql{av-qp}
\delta_{A^v} = \sum_{i=1}^{I^v} \delta_{\gamma_i^v{\mathbb Z}}*\nu_i^v + \nu^v,
\end{equation}
where all the $\gamma_i^v$ have irrational ratios and the measures $\nu_i^v$ and $\nu^v$
are each a finite sum of integer point masses on ${\mathbb R}$.
Using \eqref{av-qp} we can now write
$$
\mu = \sum_{v\in V} \delta_{A^v} = \sum_{v \in V} v \sum_{i=1}^{I^v} \delta_{\gamma_i^v{\mathbb Z}}*\nu_i^v + \sum_{v \in V} v \nu^v.
$$
Taking Fourier Transforms above we observe that the first summand on the right contributes a discrete measure to $\ft{\mu}$
(by the Poisson Summation Formula) and the second summand contributes a trigonometric polynomial. But since, by \eqref{mu-ft},
the Fourier Transform of $\mu$ must have a discrete support it follows that the second summand is 0 and we have
\beql{mu-decomp}
\mu = \sum_{k=1}^K \delta_{\zeta_k {\mathbb Z}}*\tau_k,
\end{equation}
where (having, again, merged the arithmetic progressions with commensurable periods) the ratio of any two $\zeta_k$ is irrational
and the $\tau_k$ are finite sums of integer point masses on ${\mathbb R}$.
Taking Fourier Transforms we get by the Poisson Summation Formula that
$$
\ft{\mu}(\xi) = \sum_{k=1}^K \ft{\tau_k}(\xi) \zeta_k^{-1} \delta_{\zeta_k^{-1}{\mathbb Z}},
$$
and observe that the measures $\ft{\tau_k}(\xi) \zeta_k^{-1} \delta_{\zeta_k^{-1}{\mathbb Z}}$ have disjoint supports except at $0$.
Using our assumption \eqref{int-tiling} that $f*\mu = k$ we obtain the tilings
$$
f*(\delta_{\zeta_k {\mathbb Z}}*\tau_k) = C_k,
$$
where $C_k$ is also an integer constant.
Integrating this over one period $[0,\zeta_k)$ we obtain
$$
C_k \zeta_k = \int f \cdot \tau_k([0,\zeta_k)).
$$
This shows that all $\zeta_k$ are rational multiples of $\int f$
so all summands in \eqref{mu-decomp} can be merged to one
$$
\mu = \delta_{\zeta{\mathbb Z}}*\tau,
$$
where $\tau$ is, again, a finite sum of integer point masses, hence $\mu$ is a periodic measure with period $\zeta$, as
we had to prove.
\end{proof}
\begin{theorem}\label{th:integer-tiling}
Suppose that $V \subseteq {\mathbb Z}$ is a finite set of non-zero integers, $A \subseteq {\mathbb R}$ is a
discrete set of bounded density and $v_t \in V$, for $t \in A$, are such that
\beql{integer-tiling}
\sum_{t \in A} v_t f(x-t) = k,\ \ \ \mbox{for almost all $x \in {\mathbb R}$},
\end{equation}
where $f \in L^1({\mathbb R})$ is an integer-valued function of compact support and $k$ is an integer.
Then
\begin{enumerate}[(i)]
\item\label{en:periodic}
The measure $\mu = \sum_{t\in A} v_t \delta_t$ is periodic and can be written in the form
$$
\mu = \delta_{\zeta{\mathbb Z}}*\tau,
$$
where $\zeta>0$ and $\tau$ is a finite sum of integer point masses
$$
\tau = \sum_{s=1}^S c_s \delta_{x_s},
$$
where $c_s \in {\mathbb Z}$, $x_s \in [0,\zeta)$.
\item\label{en:split}
Write $X = \Set{x_1, x_2, \ldots, x_S}$ and
$$
X = X_1 \cup \cdots \cup X_r
$$
for the partition of $X$ into equivalence classes mod $\zeta{\mathbb Q}$.
Then for $j=1,2,\ldots,r$ and with $\tau_j = \sum_{x \in X_j} c_x \delta_x$ we have
the tilings
$$
f*\delta_{\zeta{\mathbb Z}}*\tau_j = k_j,
$$
for some integers $k_j$, $j=1,2,\ldots,r$.
\end{enumerate}
\end{theorem}
Theorem \ref{th:integer-tiling} was proved in \cites{lagarias1996tiling} for
$f$ being the indicator function of a bounded, measurable subset of ${\mathbb R}$,
and with $v_t = 1$ for all $t \in A$.
In this case the number $r$ of classes in \eqref{en:split} is 1, and the tiling set $A$ is therefore rational, i.e.\ the differences of its
elements are rational multiples of the period.
The proof does not readily extend to the more general case of Theorem \ref{th:integer-tiling} and this is what we show here.
\begin{proof}
Part \eqref{en:periodic} of the Theorem is merely a restatement of Lemma \ref{lm:integer-tiling}.
Notice that we can assume from now on that $\zeta$ (the period of the tiling) is 1, as we can dilate the axis.
Define the ${\mathbb Z}$-periodization of $f$
$$
F(x) = \sum_{n \in {\mathbb Z}} f(x-n) = f*\delta_{\mathbb Z}(x),
$$
which is in $L^1({\mathbb T})$,
and observe that the tiling $f*(\delta_{\mathbb Z}*\tau)(x)=k$ is equivalent to the tiling of ${\mathbb T}$
$$
F*\tau(x)=k,\ \ \ \mbox{for almost all $x \in {\mathbb T}$}.
$$
Since $F$ is also integer-valued Lemma \ref{lm:torus-tiling} applies and we conclude that
$$
F*\tau_j(x)=k_j,\ \ \ \mbox{for almost all $x \in {\mathbb T}$ and some integer $k_j$},
$$
which is equivalent to $f*(\delta_{\mathbb Z}*\tau_j)=k_j$ as we had to prove.
This concludes the proof of \eqref{en:split}.
\end{proof}
\section{The structure of the set of multiples of a multiplicative tile}
\label{sec:structure}
\begin{theorem}\label{th:mult-structure}
(Structure of the set of multiples)\\
Suppose $\Omega \subseteq {\mathbb R}$ is a bounded measurable set such that $\Omega \cap (-\epsilon, \epsilon) = \emptyset$ for some $\epsilon>0$.
Suppose also $A \subseteq {\mathbb R}\setminus\Set{0}$ is a discrete set such that
$$
A \cdot \Omega
$$
is a (multiplicative) tiling of ${\mathbb R}$ at level 1.
Let $\Omega^+, \Omega^-, A^+, A^- \subseteq {\mathbb R}^+$ be the positive and negative parts of $\Omega$ and $A$
$$
\Omega^+ = \Omega \cap (0, +\infty),\ \Omega^- = -(\Omega \cap (-\infty, 0)),\
A^+ = A \cap (0, +\infty),\ A^- = -(A \cap (-\infty, 0)).
$$
Then
\begin{enumerate}[(i)]
\item\label{symmetric-case} If $\Omega$ is \underline{essentially symmetric} (i.e.\ if $\Abs{\Omega^+ \, \triangle\, \Omega^-} = 0$) then
$A^+ \cap A^- = \emptyset$ and the union $\log A^+ \cup \log A^-$ is periodic of the form
\beql{periodic-rational}
\alpha{\mathbb Z}+\Set{r_1, r_2, \ldots, r_s}\ \ \ \text{with $r_i-r_j$ rational multiples of $\alpha>0$}.
\end{equation}
The partition of the set $\log A^+ \cup \log A^-$ into its component sets $\log A^+$ and $\log A^-$
can be completely arbitrary.
\item\label{nonsymmetric-case} If $\Omega$ is \underline{not essentially symmetric} with respect to the origin then
the sets $\log A^+$ and $\log A^-$ are both periodic and of the form \eqref{periodic-rational} with the same period $\alpha$.
\end{enumerate}
\end{theorem}
\begin{proof}
In order to transfer the problem to the translational case, which is much better understood, it is natural to take logarithms.
Allowing ourselves a slight abuse of terminology,
we then have that $A\cdot \Omega={\mathbb R}$ is a tiling if and only if
$$
{\mathbb R}^+ = A^+ \Omega^+ \cup A^- \Omega^-,\ \ \ {\mathbb R}^+ = A^- \Omega^+ \cup A^+ \Omega^-
$$
are both tilings (where ${\mathbb R}^+$ is the right half-line).
Taking logarithms of both we obtain the additive (translational) tilings
\beql{log-tilings}
{\mathbb R} = (a^+ + \omega^+) \cup (a^- + \omega^-) = (a^-+\omega^+) \cup (a^+ + \omega^-),
\end{equation}
where we write the lower case letter for the set of logarithms of a set written with the corresponding capital letter,
e.g.\ $a^+ = \log A^+$.
Identifying, further, the sets $\omega^\pm$ with their indicator functions
and the discrete sets $a^{\pm}$ with a collection of unit point masses at their points (for instance, we write $a^+$ instead of $\delta_{a^+}$),
we may rewrite the above tilings using convolution as
\beql{tilings}
1 = a^+*\omega^+(x) + a^-*\omega^-(x) = a^-*\omega^+(x) + a^+*\omega^-(x),\ \ \ \mbox{for almost all $x \in {\mathbb R}$}.
\end{equation}
Adding and subtracting the two identities in \eqref{tilings} we get the equivalent set of identities (valid for almost all $x \in {\mathbb R}$)
\beql{sum}
2 = (\omega^++\omega^-)*(a^++a^-)(x)
\end{equation}
and
\beql{diff}
0 = (\omega^+ - \omega^-)*(a^+-a^-)(x).
\end{equation}
Notice that $\omega^+(x)+\omega^-(x)$ is a function that only takes the values 0, 1 and, possibly, 2 and that $a^++a^-$ is a measure,
which is a collection of point masses of weight 1 or 2.
Similarly $a^+-a^-$ is a measure which is a collection of point masses of weight $\pm 1$.
There is obviously no problem with the definition of the convolutions in \eqref{tilings} and \eqref{sum}, as there are
only nonnegative quantities involved. A moment's thought should convince us that there is no problem in \eqref{diff} either,
as all sums involved have finitely many terms,
the functions $\omega^\pm(x)$ being of compact support and the sets $a^\pm$ being discrete.
{\bf Periodicity.}
From Theorem \ref{th:integer-tiling}\eqref{en:periodic} applied to the tiling \eqref{sum} we obtain that
$a$ (viewed as a multiset when the point mass at a point has weight 2)
is a periodic set
\beql{periodic-set}
a = a^+ \cup a^- = \gamma{\mathbb Z} + \Set{\beta_1, \beta_2, \ldots, \beta_J},
\end{equation}
for some $\gamma>0$, $\beta_j \in {\mathbb R}$.
As a consequence the set
$$
a^+ \cap a^-
$$
is also periodic, as this is where the multiplicity of $a$ is equal to 2.
Assume now that $\omega^+$ is not identical to $\omega^-$ (that is $\Omega$ is not symmetric with respect to 0)
so that the function $\omega^+-\omega^-$ that appears in \eqref{diff} is not equal to zero almost everywhere.
The measure $a^+-a^-$ is a collection of Dirac point masses of weight $\pm 1$. The weight 1 appears exactly on
the points of the set $a^+ \setminus a^-$ and the weight -1 exactly on the set $a^- \setminus a^+$.
Given that we have already established the periodicty of $a^+ \cap a^-$
the periodicity of the set $a^+$ and the periodicity of the set $a^-$ will follow if we manage to show
the periodicity of $a^+ \setminus a^-$ and of $a^- \setminus a^+$ with a period commensurable to a period of $a^+ \cap a^-$.
It follows again from Theorem \ref{th:integer-tiling}\eqref{en:periodic}, applied to tiling \eqref{diff},
with $f = \omega^+-\omega^-$ (this is a compactly supported function, because of our assumption
that the bounded set $\Omega$ avoids an open neighborhood of $0$),
that the sets $a^+\setminus a^-$ and $a^-\setminus a^+$ are periodic with the same period.
Since we have already shown that $a^+ \cap a^-$ is also periodic it follows that each of the sets $a^+$ and $a^-$
is a union of two periodic sets, and these must be of commensurable periods, as, otherwise, the set
$a^+ \cup a^-$, which is already known to be periodic, would contain two arithmetic progressions with inocommensurable step,
an impossibility.
It follows that $a^+$ and $a^-$ are periodic too, and with commensurable periods.
{\bf Rationality.}
Dilating space we may assume now that the sets $a^\pm$ have period 1.
It remains to prove that the set $a^+ \cup a^-$ has rational differences.
Write
\beql{one-period-plus}
a^+ + a^- = \delta_{\mathbb Z}*\sigma,\ \ \ \mbox{with}\ \ \ \sigma = \sum_{j=1}^J n_j \delta_{x_j},
\end{equation}
where $n_j \in \Set{1,2}$ and $x_j \in [0,1)$, for $j=1, 2, \ldots, J$.
It follows that we can also write
\beql{one-period-minus}
a^+-a^- = \delta_{\mathbb Z}*\tau,\ \ \ \mbox{with}\ \ \ \tau = \sum_{j=1}^J m_j \delta_{x_j},
\end{equation}
where now $m_j \in \Set{-1, 0, 1}$, for $j=1, 2, \ldots, J$.
Contrary to what we want to prove, assume that not all the $x_j$s are rationally equivalent, and
let $x_{i_1}, x_{i_2}, \ldots, x_{i_r}$ be a rationally equivalent class of the points $x_1, x_2, \ldots, x_J$ in \eqref{one-period-plus}.
We now apply Theorem \ref{th:integer-tiling}\eqref{en:split} to the two tilings \eqref{sum} and \eqref{diff}.
It follows that we have the two tilings
\beql{sum-1}
(\omega^++\omega^-)*(\delta_{\mathbb Z}+\sum_{j=1}^r n_{i_j} \delta_{x_{i_j}}) = k_1
\end{equation}
and
\beql{diff-1}
(\omega^+-\omega^-)*(\delta_{\mathbb Z}+\sum_{j=1}^r m_{i_j} \delta_{x_{i_j}}) = k_2,
\end{equation}
for some two integers $k_1, k_2$.
Write $a^{'\pm}$ for the restriction of $a^\pm$ to the set ${\mathbb Z}+\Set{x_{i_1}, \ldots, x_{i_r}}$
and observe that we now have
\beql{sum-2}
a^{'+}+a^{'-} = \delta_{\mathbb Z} + \sum_{j=1}^r n_{i_j} \delta_{x_{i_j}}
\end{equation}
and
\beql{diff-2}
a^{'+}-a^{'-} = \delta_{\mathbb Z} + \sum_{j=1}^r m_{i_j} \delta_{x_{i_j}},
\end{equation}
so that the tilings \eqref{sum-1} and \eqref{diff-1} can now be written as
\beql{sum-3}
(\omega^++\omega^-)*(a^{'+}+a^{'-}) = k_1
\end{equation}
and
\beql{diff-3}
(\omega^+-\omega^-)*(a^{'+}-a^{'-}) = k_2.
\end{equation}
By our assumption that not all points in $\Set{x_1, x_2 \ldots, x_J}$ are rationally equivalent it
follows that \eqref{sum-3} is a proper subtiling of \eqref{sum}, which forces $k_1=1$.
Adding and subtracting \eqref{sum-3} and \eqref{diff-3} and dividing by 2 we obtain the tilings
\beql{tilings-3}
\frac{1+k_2}{2} = a^{'+}*\omega^+(x) + a^{'-}*\omega^-(x),\ \ \ \frac{1-k_2}{2} = a^{'-}*\omega^+(x) + a^{'+}*\omega^-(x),
\end{equation}
analogous to \eqref{tilings} and proper subtilings of those in \eqref{tilings}.
But the tilings in \eqref{tilings} are minimal, as they are tilings by translates of sets, at level 1, a contradiction.
In case \eqref{symmetric-case} of the Theorem,
if $\omega^+ \equiv \omega^-$ and $\omega^+$ tiles ${\mathbb R}$ with some set $a$, then \eqref{diff} is trivial and \eqref{sum} is clearly valid.
From \eqref{sum} it follows using the same tiling structure theorems that $a$ is a periodic multiset,
and \eqref{tilings} can be satisfied by arbitrarily breaking up the multiset $a$ into two sets (not multisets)
$a^+$ and $a^-$. Nothing more can be said about the sets $a^+$ and $a^-$ in this case.
\end{proof}
\section{The structure of a multiplicative tile}
\label{sec:tile-structure}
\subsection{Symmetric tile}\label{sec:symmetric-tile}
Suppose that the set $\Omega$ is symmetric with respect to $0$. In the notation of Theorem \ref{th:mult-structure}
this means that $\Omega^+ = \Omega^-$ (up to measure 0).
In this case the tilings of \eqref{log-tilings} become just one tiling:
\beql{single-tiling}
{\mathbb R} = (a^+ \cup a^-) + \omega^+.
\end{equation}
(Remember that the lower case letters denote the logarithms of the sets written in the corresponding upper case.)
So in this case the problem of multiplicative tiling becomes exactly the problem of translational tiling of the real line
by the tile $\omega^+$ and with set of translates the multiset $a^+ \cup a^-$.
The structure of $\omega^+$ in this case has been completely characterized in \cites[Theorem 3]{lagarias1996tiling}.
\subsection{Non-symmetric tile}\label{sec:non-symmetric-tile}
Suppose now that we are in the case where $\omega^+ \nequiv \omega^-$ and, according to Theorem \ref{th:mult-structure},
$a^+, a^-$ are both periodic with the same period obeying \eqref{periodic-rational}.
We can thus write (after scaling)
\beql{a-shape}
a^+ = {\mathbb Z}+\frac{1}{L}\Set{0=\alpha_1^+, \alpha_2^+, \ldots , \alpha_m^+},\ \ \
a^- = {\mathbb Z}+\frac{1}{L}\Set{\alpha_1^-, \alpha_2^-, \ldots , \alpha_n^+},
\end{equation}
for some positive integer $L$, where $\alpha_j^\pm \in {\mathbb Z}_L = {\mathbb Z} / (L{\mathbb Z})$.
In order to express our tiling problem on the torus ${\mathbb T} = {\mathbb R} / {\mathbb Z}$ we identify
$\omega^\pm$ with the indicator function of the set that arises when taking their
projection mod 1.
(Because of the tiling assumption the points of each $\omega^\pm$ are different mod 1.)
Write
$$
\alpha^+ = \Set{\alpha_1^+, \alpha_2^+, \ldots, \alpha_m^+},\ \ \ \alpha^- = \Set{\alpha_1^-, \alpha_2^-, \ldots, \alpha_n^-},
$$
and also $\alpha^\pm$ for the collection of unit Dirac masses at the points of $\alpha^\pm$.
The tiling conditions \eqref{tilings} now become equivalently the tilings of the torus
\beql{mod-tilings}
1 = \frac{\alpha^+}{L}*\omega^+(t) + \frac{\alpha^-}{L}*\omega^-(t) = \frac{\alpha^-}{L}*\omega^+(t) + \frac{\alpha^+}{L}*\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
\end{equation}
Write for $x \in [0,1)$
$$
C_x = \Set{x+\frac{k}{L} \bmod 1:\ k=0,1,\ldots,L-1} \subseteq {\mathbb T}.
$$
Multiplying \eqref{mod-tilings} by ${\bf 1}_{C_x}(t)$ we get
$$
{\bf 1}_{C_x}(t) =
\frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^-(t) =
\frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
$$
Restricting to $t \in C_x$ we can rewrite this as the two tiling conditions on ${\mathbb Z}_L$, valid for all $x \in [0, \frac{1}{L})$,
\begin{align}
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^+ \ \ \cup\ \ (\omega^-\cap C_x) + \alpha^-,\nonumber\\
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^- \ \ \cup\ \ (\omega^-\cap C_x) + \alpha^+\label{cycle-tilings},
\end{align}
where we are now identifying the sets $\omega^\pm\cap C_x$ with the obvious subset of ${\mathbb Z}_L$.
The sets $\omega^\pm \cap C_x$ can be chosen independently for all $x \in [0, 1/L)$ as tiling is not affected by
what happens on different cosets of $\frac{1}{L}{\mathbb Z}$.
We conclude that the sets $\omega^\pm$ are of the form
\begin{align}
\omega^+ &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^+_x \right)\nonumber\\
\omega^- &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^-_x \right) \ \ \ \mbox{(disjoint unions)} \label{decomposition}
\end{align}
where $b^\pm_x \subseteq {\mathbb Z}_L$ are such that for each $x$ we have the two tilings
\begin{align}
{\mathbb Z}_L &= b^+_x + \alpha^+ \cup b^-_x + \alpha^-\nonumber\\
{\mathbb Z}_L &= b^+_x + \alpha^- \cup b^-_x + \alpha^+\label{cycle-tilings-final}
\end{align}
In the next section we are trying to understand better the kind of tiling described in \eqref{cycle-tilings-final}.
\section{Cross tiling}
\label{sec:ct}
\begin{definition}[Cross tiling]\label{def:cross-tiling}\ \\
Suppose $N>1$ is a positive integer and $A, B, X, Y \subseteq {\mathbb Z}_N$. We say that the pair $A, B$ admits
{\em cross tiling} with complements $X, Y$ if the following tilings hold:
\begin{align}
{\mathbb Z}_N = (A + X) \cup (B + Y)\nonumber\\
{\mathbb Z}_N = (A + Y) \cup (B + X) \label{cross-tiling}
\end{align}
A cross tiling \eqref{cross-tiling} is called {\em trivial} if
$(A \cup B) + X = {\mathbb Z}_N$ is a tiling and $X=Y$, or the same with the roles of
$A, B$ exchanged with those of $X, Y$.
(It is obvious that in this case conditions \eqref{cross-tiling} are indeed satisfied.)
\end{definition}
\noindent{\bf Remark.}
It is interesting to observe that cross tiling is really an ordinary tiling by translation, although of a larger group.
With $A, B, X, Y \subseteq {\mathbb Z}_N$ as in Definition \ref{def:cross-tiling} above write
\begin{align*}
\Gamma &= {\mathbb Z}_N \times {\mathbb Z}_2,\\
C &= A\times\Set{0} \cup B\times\Set{1},\\
Z &= X\times\Set{0} \cup Y\times\Set{1}.
\end{align*}
It is easy to see that the cross tiling condition \eqref{cross-tiling} is equivalent to the tiling by translation
$$
\Gamma = C + Z.
$$
This alternative characterization of cross tiling may be of interest but is not exploited in this paper
The following two examples are non-trivial cross tilings.
\begin{example}\label{ex:first-non-trivial}
Let $N=2ab$, with odd $a, b \in {\mathbb N}$ and view $G={\mathbb Z}_N$ as the cross-product ${\mathbb Z}_N = {\mathbb Z}_{ab}\times{\mathbb Z}_2$.
Define
\begin{align*}
A &= \Set{0, 1, 2, \ldots, a-1} \times \Set{0},\\
B &= \Set{0, 1} \cup \Set{a+2, a+3, \ldots, 2a-1} \times \Set{0},\\
X &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{0},\\
Y &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{1}.
\end{align*}
It follows that
$$
A + X = B + X = {\mathbb Z}_{ab}\times\Set{0}\ \ \ \text{and}\ \ \ B + Y = A + Y = {\mathbb Z}_{ab}\times\Set{1}
$$
so \eqref{cross-tiling} follows. See Figure \ref{fig:easy-example}.
\end{example}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=5, b=3, N=2*a*b;
pair A[]; for(i=0; i<a; ++i) { A[i] = (i,0); }
pair B[]; B[0] = (0,0); B[1] = (1,0); for(j=2, i=a+2; i<2*a; ++i, ++j) { B[j] = (i,0); }
pair X[]; for(i=0; i<b; ++i) { X[i] = (i*a, 0); }
pair Y[]; for(i=0; i<b; ++i) { Y[i] = (i*a, 1); }
pair u=(0,4);
// Draw group
picture pg;
for(i=0; i<a*b; ++i) {
for(j=0; j<2; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
add(pg);
label("Group $G$ ", (0,.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A$ ", u+(0,.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B$ ", 2*u+(0,.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X$ ", 3*u+(0,.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets cross tiling in Example \ref{ex:first-non-trivial} with $a=5, b=3$} \label{fig:easy-example}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=15, b=4;
real F1[]={0,1,2}, F2[]={0,4,5}, C[]={0, 3, 6, 9, 12}, CC[]={3, 6, 9, 12};
pair A[]; for(i=0; i<F1.length; ++i) {A.push((F1[i], 0)); A.push((F1[i], 1)); }
pair B[]; for(i=0; i<F2.length; ++i) {B.push((F2[i], 0)); B.push((F2[i], 1)); }
pair X[]; for(i=0; i<C.length; ++i) {X.push((C[i], 0)); X.push((C[i], 2)); }
pair Y[]; Y.push((0,1)); Y.push((0,3)); for(i=0; i<CC.length; ++i) {Y.push((CC[i], 0)); Y.push((CC[i], 2)); }
pair u=(0,5);
// Draw group
picture pg;
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
//draw(pg, (i+0.5,j), 1bp+black);
//draw(pg, (i+0.5,j+0.5), 1bp+black);
draw(pg, (i,j+0.5), 1bp+black);
}
}
add(pg);
label("Group $G$ ", (0,1.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A'$ ", u+(0,1.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B'$ ", 2*u+(0,1.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X'$ ", 3*u+(0,1.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y'$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets $A', B', X', Y' \subseteq H$ in Example \ref{ex:second-non-trivial} giving rise to the sets $A, B, X, Y \subseteq G$} \label{fig:better-example}
\end{figure}
\begin{example}\label{ex:second-non-trivial}
In Example \ref{ex:first-non-trivial} the sets $X$ and $Y$ are translates of each other, which is perhaps not very
satisfactory in terms of deviating from triviality.
Now we give an example where this does not happen.
We work in the group $G = {\mathbb Z}_{15}\times{\mathbb Z}_8 = {\mathbb Z}_{120}$
which contains the subgroup
$$
H = {\mathbb Z}_{15} \times \Set{0, 2, 4, 6}.
$$
Next define the subsets of ${\mathbb Z}_{15}$
$$
F_1 = \Set{0, 1, 2},\ \ \ F_2 = \Set{0, 4, 5}
$$
and notice that they both tile ${\mathbb Z}_{15}$ with the complement $\Set{0, 3, 6, 9, 12}$.
Define the subsets of $H$
\begin{align*}
A' &= F_1 \times \Set{0, 2},\\
B' &= F_2 \times \Set{0, 2},\\
X' &= \Set{0, 3, 6, 9, 12} \times \Set{0, 4},\\
Y' &= (\Set{0}\times\Set{1, 3}) \cup (\Set{3, 6, 9, 12}) \times \Set{0, 4}.
\end{align*}
See Figure \ref{fig:better-example} where the group $H$ is shown as thick dots
while its single coset in $G$ is shown as thin dots.
The difference between $X'$ and $Y'$ is that the first ``column'' of $Y'$ is ``raised'' by 1.
It is easy to verify that each of $A', B'$ tiles $H$ with each of $X', Y'$ as a tiling complement.
Finally define the subsets of $G$
\begin{align*}
A &= A', \\
B &= B', \\
X &= X', \\
Y &= Y'+(0,1),
\end{align*}
and observe that they do satisfy the cross-tiling conditions
\begin{align}
G &= (A+X) \cup (B+Y) \nonumber \\
G &= (A+Y) \cup (B+X). \label{ct}
\end{align}
The sets in parentheses in \eqref{ct} are tilings of each of the two $H$-cosets in $G$.
The sets with $+X$ tile $H$ and those with $+Y$ tile $H+(0,1)$.
None of the sets $A, B, X, Y$ is a translate of another.
\end{example}
\subsection{Fourier condition for cross tiling}
Translational tiling $A + X = {\mathbb Z}_N$ has a simple equivalent Fourier condition (we use the same letter for a set and its indicator function):
\beql{tiling-ft}
\ft{A}(0) \ft{X}(0) = N,\ \ \ \mbox{and}\ \ \ \forall k \in {\mathbb Z}_N\setminus\Set{0}: \ft{A}(k) \neq 0 \Longrightarrow \ft{X}(k) = 0.
\end{equation}
Adding and subtracting the cross-tiling defining conditions
$$
A*X+B*Y \equiv 1,\ \ \ A*Y+B*X \equiv 1,
$$
we obtain the equivalent conditions
\beql{cross-tiling-equiv}
(A+B)*(X+Y) \equiv 2,\ \ \ (A-B)*(X-Y) \equiv 0.
\end{equation}
Taking Fourier Transforms these conditions lead to the following equivalent Fourier condition for cross-tiling:
\begin{align}
&(\ft{A}(0) + \ft{B}(0)) (\ft{X}(0)+\ft{Y}(0)) = 2N,\nonumber\\
\forall k \in {\mathbb Z}_N\setminus\Set{0}:\ \ \ &\ft{A}(k) \neq -\ft{B}(k) \Longrightarrow \ft{X}(k) = -\ft{Y}(k). \label{cross-tiling-ft}\\
\forall k \in {\mathbb Z}_N:\ \ \ &\ft{A}(k) \neq \ft{B}(k) \Longrightarrow \ft{X}(k) = \ft{Y}(k).\nonumber
\end{align}
Using $k=0$ in the last set of equations we obtain that necessarily
$$
\Abs{A} = \Abs{B}\ \ \ \text{or}\ \ \ \Abs{X} = \Abs{Y}.
$$
\printbibliography
\end{document}
\section{Introduction}
\label{sec:intro}
Tilings have long fascinated mathematicians \cite{grunbaum1986tilings}.
The case where one moves a single object by translation in an abelian group (translational tiling)
has proved both challenging and full of connections to Functional Analysis \cite{kolountzakis2004milano},
such as connections to the so-called Fuglede Conjecture or Spectral Set Conjecture \cite{fuglede1974operators,tao2004fuglede,kolountzakis2006tiles}.
Simultaneous tiling of a set by both translation and multiplication (with linear operators on the space where tiling takes place)
has also been studied mainly because of its connections to wavelets
\cite{wang2002wavelets,speegle2003dilation,olafsson2004wavelets,dobrescu2006wavelet,ionascu2006simultaneous}.
\begin{definition}[Translational tiling, multiplicative tiling]\label{def:tiling}\ \\
Suppose $f:{\mathbb R}^d\to{\mathbb C}$ is measurable and $A \subseteq {\mathbb R}^d$. We say that $f+A$ is a tiling
of ${\mathbb R}^d$ at level $\ell$ if
$$
\sum_{a \in A} f(x-a) = \ell
$$
for almost all $x \in {\mathbb R}^d$ with the sum converging absolutely.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $\Omega+A$ is a tiling.
If $A \subseteq {\mathbb R}\setminus\Set{0}$ and
$$
\sum_{a \in A} f(a^{-1} x) = \ell
$$
for almost all $x \in {\mathbb R}^d$, with absolute convergence, then we say that $A\cdot f$ is a tiling.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $A\cdot \Omega$ is a tiling.
\end{definition}
While translational tiling or more generally tiling using congruent tiles has been studied extensively, one particular tiling, namely {\em multiplicative tiling}, has not. Such tiling arise rather ubiquitously in the study wavelets and wavelet sets. In the standard setting, a function $f(x) \in L^2({\mathbb R})$ is a {\em wavelet} if $\{2^{j/2}f(2^jx-k): ~j,k\in{\mathbb Z}\}$ form an orthonormal basis for $L^2({\mathbb R})$. A set $\Omega\subset {\mathbb R}$ is a {\em wavelet set} if $\ft\chi_\Omega$ is a wavelet. It was first shown by Dai and Larson \cite{dai1998wandering} that $\Omega$ is a wavelet set if and only if it tiles ${\mathbb R}$ translationally by ${\mathbb Z}$ and multiplicatively by the set $\{2^j:~j\in{\mathbb Z}\}$, see also \cite{dai1997wavelet,speegle2003dilation}. The more general multiplicative tiling, which we aim to study here, was first introduced in Wang \cite{wang2002wavelets} to study a more general form of wavelet sets.
Our purpose here is to study the structure of multiplicative tilings.
We are guided in this by previous work on the structure of translational tilings of the real line or of the integer line.
In \cites{newman1977tesselations,leptin1991uniform,lagarias1996tiling,kolountzakis1996structure}
it is proved, under very broad conditions, that any translational tiling of the real line (or of the integer line) must be periodic.
The main tool in the study of translational tilings by a single tile has long been (see e.g.\ \cite{kolountzakis2004milano}) Fourier Analysis,
where the problem is expressed roughly as a support condition of the Fourier Transform of the set of translates
on the zero set of the Fourier Transform of the tile, an approach that will also be used extensively in this paper.
Suppose then that $A \subseteq {\mathbb R}\setminus\Set{0}$ is a discrete set and $\Omega \subseteq {\mathbb R}$ is a measurable set.
We want to derive properties of $\Omega$ and $A$ from the assumption of multiplicative tiling $A\cdot \Omega = {\mathbb R}$ (multiplicative tiling at level 1).
\noindent{\bf The importance of sign.}
It is important to emphasize that if $A$ or $\Omega$ are nonnegative (or of one sign, more generally) then the problem
quickly reduces to translational tiling.
Indeed, suppose that $\Omega \subseteq (0,+\infty)$.
Then, writing $A = A^+ \cup (-A^-)$, with $A^\pm \subseteq (0,+\infty)$, we see that
the tiling condition $A\cdot\Omega = {\mathbb R}$ is equivalent to the two tiling conditions
$$
{\mathbb R}^+ = A^+\cdot \Omega\ \ \ \mbox{and}\ \ \ {\mathbb R}^+ = A^-\cdot\Omega.
$$
Taking logarithms of both we reduce $A\cdot\Omega={\mathbb R}$ to the two independent translational tilings
$$
{\mathbb R} = \log\Omega + \log A^+\ \ \ \mbox{and}\ \ \ {\mathbb R} = \log\Omega + \log A^-.
$$
So if one can understand translational tiling by the set $\log\Omega$ then all results transfer back to our multiplicative
tiling $A\cdot\Omega = {\mathbb R}$ if $\Omega$ is of one sign.
Similarly, if $A \subseteq (0, +\infty)$ then, writing again $\Omega = \Omega^+ \cup (-\Omega^-)$, with $\Omega^\pm \subseteq (0, +\infty)$,
we obtain that $A\cdot \Omega = {\mathbb R}$ is equivalent to the two translational tilings
$$
{\mathbb R} = \log\Omega^+ + \log A \ \ \ \mbox{and}\ \ \ {\mathbb R} = \log\Omega^- + \log A.
$$
If however the two sets $A$ and $\Omega$ have both a positive and a negative part the multiplicative tiling $A\cdot \Omega = {\mathbb R}$
becomes a lot more complicated. It still reduces to tiling by translation but not of the ordinary kind with one set being
translated around to fill space.
Indeed, when $A = A^+ \cup (-A^-)$ and $\Omega = \Omega^+ \cup (-\Omega^-)$ then the multiplicative tiling $A \cdot \Omega = {\mathbb R}$
reduces to the two simultaneous tilings
$$
{\mathbb R}^+ = A^+\Omega^+ \cup A^-\Omega^-\ \ \ \mbox{and}\ \ \ {\mathbb R}^+ = A^-\Omega^+ \cup A^+\Omega^-,
$$
and, after taking logarithms, to the sumultaneous translational tiling
\begin{align}
{\mathbb R} &= (\log\Omega^++\log A^+) \cup (\log\Omega^-+\log A^-)\ \ \ \mbox{and}\ \ \ \label{cti}\\
{\mathbb R} &= (\log\Omega^++\log A^-) \cup (\log\Omega^-+\log A^+).\nonumber
\end{align}
The meaning of the notation here should be obvious. For instance, the meaning of the first equation in \eqref{cti} is that
almost every point in ${\mathbb R}$ can be written, uniquely, either in the form $\log\omega+\log a$, with $\omega\in\Omega^+, a \in A^+$,
or in the form $\log\omega+\log a$, with $\omega \in \Omega^-, a \in A^-$.
Put differently, the translates of the set $\log\Omega^+$ by the numbers in $\log A^+$ together with the translates of the set $\log\Omega^-$
by the numbers in $\log A^-$ cover almost all of ${\mathbb R}$ exactly once and any two of these sets intersect at a set of measure zero.
The purpose of this paper is first, to exploit \eqref{cti} in order to derive structural properties of the
set $\Omega$ (the tile) and the set $A$ (the set of multiples) and, second, to study \eqref{cti} (which we call {\em cross-tiling})
in itself, and in the case of a finite cyclic group, where things are simpler. In particular, we want to make some
connections and distinctions to ordinary translational tiling where only one set is translated.
The structure of the paper is as follows.
In \S\ref{sec:rationality} we restrict ourselves to translational tilings and generalize some periodicity and rationality results
(from \cite{kolountzakis1996structure,lagarias1996tiling}) to the extent that they become useful to us in the analysis of \S\ref{sec:structure}
and \S\ref{sec:tile-structure}
where structure results are proved for the logarithms of the sets $A$ and $\Omega$ respectively.
In \S\ref{sec:ct} the problem of cross tiling is studied in cyclic groups (we show in \S\ref{sec:structure} that multiplicative tiling of ${\mathbb R}$
reduces to cross tiling in cyclic groups), and we provide examples of cross tilings which differ significantly from ordinary translational
tilings as well as a Fourier condition for cross tiling, analogous to the one for ordinary translational tilings.
\section{The structure of multiple translational tiling by a set}
\label{sec:rationality}
\begin{lemma}\label{lm:poly-zeros}
Suppose $\Lambda$ is a finite subset of the torus ${\mathbb T}={\mathbb R}/{\mathbb Z}$ and
\beql{expoly}
f(n) = \sum_{\lambda\in\Lambda} c_\lambda e^{2\pi i \lambda n},\ \ \ (n\in{\mathbb Z})
\end{equation}
is an exponential polynomial on the integers ($c_\lambda \in {\mathbb C}$).
Suppose that
$$
\Lambda = \Lambda_1 \cup \cdots \cup \Lambda_r,\ \ \ (r\ge 1)
$$
is the decomposition of $\Lambda$ into rational equivalence classes (two points in $\Lambda$ have rational difference
if and only if they belong to the same $\Lambda_j$).
Write also $f_j(n) = \sum_{\lambda\in\Lambda_j} c_\lambda e^{2\pi i \lambda n}$ so that $f=f_1+\cdots+f_r$.
Then the zeros of $f$ are the common zeros of the $f_j$ plus a finite set (possibly empty).
\end{lemma}
\begin{proof}
Write $Z(\phi)$ for the zero set of a function $\phi$ on its domain.
Define the set of integers
$$
X = Z(f) \setminus \bigcap_{j=1}^r Z(f_j).
$$
We have to show that $X$ is finite.
By the Skolem-Mahler-Lech Theorem \cite{lech1953note} the integer zero set of every exponential polynomial, such as \eqref{expoly},
is a periodic set plus a finite set (possibly empty). Since $\Abs{f_1}^2+\cdots+\Abs{f_r}^2$ is also an exponential
polynomial it follows that both $Z(f)$ and $\bigcap_{j=1}^r Z(f_j)$ are periodic sets plus a finite set.
Therefore $X$ is also a periodic set, give or take a finite set.
It suffices therefore to prove that $X$ does not contain arithmetic progressions of arbitrary length, as then
it follows that $X$ has no periodic part and is just a finite set.
For $j=1,2,\ldots,r$
write $\Lambda_j = a_j + Q_j$, where $Q_j \subseteq {\mathbb Q}$ is a finite set and $a_i-a_j \notin {\mathbb Q}$ for $i \neq j$.
Let $N \in {\mathbb N}$ be the least common multiple of all denominators in all the $Q_j$, so that $Nq \in {\mathbb Z}$ for all $q \in \bigcup_{j=1}^r Q_j$.
If $X$ contains arbitrarily long arithmetic progressions then it contains a progression of the form
\beql{assumption1}
a+bNk,\ \ \ (k=0, 1, 2,\ldots, r)
\end{equation}
for some $a, b \in {\mathbb Z}$, $b>0$.
For each $k=1,2,\ldots,r$ we then have
\begin{align}
f(a+bNk) &= \sum_{j=1}^r \sum_{q\in Q_j} c_{a_j+q} e^{2\pi i (a_j+q) (a+bNk)} \nonumber \\
&= \sum_{j=1}^r z_j^k x_j \label{vandermonde}
\end{align}
with
\begin{align*}
z_j &= e^{2\pi i a_j b N} \\
x_j &= e^{2\pi i a_j a} \sum_{q \in Q_j} c_{a_j+q} e^{2\pi i qa} = f_j(a).
\end{align*}
All numbers $z_j$ are different since the differences of the $a_j$'s are irrational, so the Vandermonde linear system
$$
\sum_{j=1}^r z_j^k x_j = 0,\ \ \ (j=1,2,\ldots,r)
$$
which we obtain if we assume that $a+bNk \in X$, for $k=1,2,\ldots,r$, has only the all-zero solution $x_1=\cdots=x_r=0$, which implies
that $f_1(a)=\cdots=f_r(a)=0$, hence $a$ is a common zero of all $f_j$, hence not in $X$, a contradiction with \eqref{assumption1} for $k=0$.
So $X$ does not contain arbitrarily long arithmetic progressions and is, therefore, a finite set.
\end{proof}
\begin{lemma}\label{lm:torus-tiling}
Suppose $\Lambda$ is a finite subset of the torus ${\mathbb T}={\mathbb R}/{\mathbb Z}$ and
$$
\Lambda = \Lambda_1 \cup \cdots \cup \Lambda_r,
$$
is the decomposition of $\Lambda$ into rational equivalence classes.
Suppose also that $F \in L^1({\mathbb T})$ and $c_\lambda \in {\mathbb C}$ are such that
\beql{torus-tiling}
\sum_{\lambda\in\Lambda} c_\lambda F(x-\lambda) = \text{const.\ \ for almost all $x \in {\mathbb T}$}.
\end{equation}
If the function $F$ takes only countably many values then for each $j=1,2,\ldots,r$ we also have
$$
\sum_{\lambda \in \Lambda_j} c_\lambda F(x-\lambda) = \text{const.${}_j$\ \ for almost all $x \in {\mathbb T}$}.
$$
\end{lemma}
\begin{proof}
Our assumption \eqref{torus-tiling} is equivalent to
$$
\ft{F}(n) = 0\ \ \text{or}\ \ \sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n} = 0,\ \ \ (n \neq 0).
$$
In other words we must have
$$
Z(\ft{F}) \cup Z\left(\sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n}\right) \supseteq {\mathbb Z}\setminus\Set{0}.
$$
But, from Lemma \ref{lm:poly-zeros},
$$
Z\left(\sum_{\lambda\in\Lambda}c_\lambda e^{2\pi i \lambda n}\right) \setminus \bigcap_{j=1}^r\ Z\left(\sum_{\lambda\in\Lambda_j}c_\lambda e^{2\pi i \lambda n}\right)
$$
is a finite set.
This implies that
$Z(\ft{F}) \cup Z(\sum_{\lambda\in\Lambda_j}c_\lambda e^{2\pi i \lambda n})$ contains all but finitely many integers, for each $j=1,2,\ldots,r$.
Consequently the function
\beql{torus-tiling-j}
\sum_{\lambda \in \Lambda_j} c_\lambda F(x-\lambda)
\end{equation}
is a trigonometric polynomial of $x$. But, as $F$ takes only countably many values and this is a finite sum, the function
in \eqref{torus-tiling-j} has a countable range too, and this can only happen
if the function is a constant, which is exactly what we wanted to prove.
\end{proof}
\begin{lemma}\label{lm:integer-tiling}
Suppose that $V \subseteq {\mathbb Z}\setminus\Set{0}$ is a finite set of non-zero integers, $A \subseteq {\mathbb R}$ is a
discrete set of bounded density and $v_t \in V$, for $t \in A$, are such that
\beql{int-tiling}
\sum_{t \in A} v_t f(x-t) = k,\ \ \ \mbox{for almost all $x \in {\mathbb R}$},
\end{equation}
where $f \in L^1({\mathbb R})$ is an integer-valued function of compact support and $k$ is an integer.
Then the measure
$$
\mu = \sum_{t \in A} v_t \delta_t
$$
is a periodic measure (and, therefore, $A$ is a periodic set).
\end{lemma}
\begin{proof}
It follows from the proof of \cites[Theorem 3.1]{kolountzakis1996structure} that
\beql{mu-ft}
{\rm supp\,}\ft{\mu} \subseteq \Set{0} \cup \Set{\xi: \ft{f}(\xi)=0},
\end{equation}
with the right-hand side of the above equation being a discrete set (since $f$ has compact support $\ft{f}$ is analytic).
For $v \in V$ write $A^v = \Set{t \in A: v_t = v}$.
It follows from the proof of \cites[Theorem 5.1]{kolountzakis1996structure}
that each $A^v$ has the form
\beql{av-qp-1}
A^v = F^v \triangle \bigcup_{j=1}^{J^v} \left( \alpha_j^v {\mathbb Z} + \beta_j^v \right),
\end{equation}
for some $J^v \in {\mathbb N}$, $\alpha_j^v > 0, \beta_j^v \in {\mathbb R}$ and finite set $F^v \subseteq {\mathbb R}$.
(Let us only indicate that, as in \cites{kolountzakis1996structure}, the main ingredient
in the proof of \eqref{av-qp-1} is a theorem of Meyer \cites[Theorem 4.2]{kolountzakis1996structure}, \cites{meyer1970nombres},
itself a consequence of the Idempotent Theorem of P.J. Cohen \cites[Theorem 4.1]{kolountzakis1996structure}, \cites{cohen1959homomorphisms}.)
By merging together the $a_j^v$ which are commensurable we can rewrite \eqref{av-qp-1} as
\beql{av-qp}
\delta_{A^v} = \sum_{i=1}^{I^v} \delta_{\gamma_i^v{\mathbb Z}}*\nu_i^v + \nu^v,
\end{equation}
where all the $\gamma_i^v$ have irrational ratios and the measures $\nu_i^v$ and $\nu^v$
are each a finite sum of integer point masses on ${\mathbb R}$.
Using \eqref{av-qp} we can now write
$$
\mu = \sum_{v\in V} \delta_{A^v} = \sum_{v \in V} v \sum_{i=1}^{I^v} \delta_{\gamma_i^v{\mathbb Z}}*\nu_i^v + \sum_{v \in V} v \nu^v.
$$
Taking Fourier Transforms above we observe that the first summand on the right contributes a discrete measure to $\ft{\mu}$
(by the Poisson Summation Formula) and the second summand contributes a trigonometric polynomial. But since, by \eqref{mu-ft},
the Fourier Transform of $\mu$ must have a discrete support it follows that the second summand is 0 and we have
\beql{mu-decomp}
\mu = \sum_{k=1}^K \delta_{\zeta_k {\mathbb Z}}*\tau_k,
\end{equation}
where (having, again, merged the arithmetic progressions with commensurable periods) the ratio of any two $\zeta_k$ is irrational
and the $\tau_k$ are finite sums of integer point masses on ${\mathbb R}$.
Taking Fourier Transforms we get by the Poisson Summation Formula that
$$
\ft{\mu}(\xi) = \sum_{k=1}^K \ft{\tau_k}(\xi) \zeta_k^{-1} \delta_{\zeta_k^{-1}{\mathbb Z}},
$$
and observe that the measures $\ft{\tau_k}(\xi) \zeta_k^{-1} \delta_{\zeta_k^{-1}{\mathbb Z}}$ have disjoint supports except at $0$.
Using our assumption \eqref{int-tiling} that $f*\mu = k$ we obtain the tilings
$$
f*(\delta_{\zeta_k {\mathbb Z}}*\tau_k) = C_k,
$$
where $C_k$ is also an integer constant.
Integrating this over one period $[0,\zeta_k)$ we obtain
$$
C_k \zeta_k = \int f \cdot \tau_k([0,\zeta_k)).
$$
This shows that all $\zeta_k$ are rational multiples of $\int f$
so all summands in \eqref{mu-decomp} can be merged to one
$$
\mu = \delta_{\zeta{\mathbb Z}}*\tau,
$$
where $\tau$ is, again, a finite sum of integer point masses, hence $\mu$ is a periodic measure with period $\zeta$, as
we had to prove.
\end{proof}
\begin{theorem}\label{th:integer-tiling}
Suppose that $V \subseteq {\mathbb Z}$ is a finite set of non-zero integers, $A \subseteq {\mathbb R}$ is a
discrete set of bounded density and $v_t \in V$, for $t \in A$, are such that
\beql{integer-tiling}
\sum_{t \in A} v_t f(x-t) = k,\ \ \ \mbox{for almost all $x \in {\mathbb R}$},
\end{equation}
where $f \in L^1({\mathbb R})$ is an integer-valued function of compact support and $k$ is an integer.
Then
\begin{enumerate}
\item[(i)]\label{en:periodic}
The measure $\mu = \sum_{t\in A} v_t \delta_t$ is periodic and can be written in the form
$$
\mu = \delta_{\zeta{\mathbb Z}}*\tau,
$$
where $\zeta>0$ and $\tau$ is a finite sum of integer point masses
$$
\tau = \sum_{s=1}^S c_s \delta_{x_s},
$$
where $c_s \in {\mathbb Z}$, $x_s \in [0,\zeta)$.
\item[(ii)]\label{en:split}
Write $X = \Set{x_1, x_2, \ldots, x_S}$ and
$$
X = X_1 \cup \cdots \cup X_r
$$
for the partition of $X$ into equivalence classes mod $\zeta{\mathbb Q}$.
Then for $j=1,2,\ldots,r$ and with $\tau_j = \sum_{x \in X_j} c_x \delta_x$ we have
the tilings
$$
f*\delta_{\zeta{\mathbb Z}}*\tau_j = k_j,
$$
for some integers $k_j$, $j=1,2,\ldots,r$.
\end{enumerate}
\end{theorem}
Theorem \ref{th:integer-tiling} was proved in \cites{lagarias1996tiling} for
$f$ being the indicator function of a bounded, measurable subset of ${\mathbb R}$,
and with $v_t = 1$ for all $t \in A$.
In this case the number $r$ of classes in \eqref{en:split} is 1, and the tiling set $A$ is therefore rational, i.e.\ the differences of its
elements are rational multiples of the period.
The proof does not readily extend to the more general case of Theorem \ref{th:integer-tiling} and this is what we show here.
\begin{proof}
Part \eqref{en:periodic} of the Theorem is merely a restatement of Lemma \ref{lm:integer-tiling}.
Notice that we can assume from now on that $\zeta$ (the period of the tiling) is 1, as we can dilate the axis.
Define the ${\mathbb Z}$-periodization of $f$
$$
F(x) = \sum_{n \in {\mathbb Z}} f(x-n) = f*\delta_{\mathbb Z}(x),
$$
which is in $L^1({\mathbb T})$,
and observe that the tiling $f*(\delta_{\mathbb Z}*\tau)(x)=k$ is equivalent to the tiling of ${\mathbb T}$
$$
F*\tau(x)=k,\ \ \ \mbox{for almost all $x \in {\mathbb T}$}.
$$
Since $F$ is also integer-valued Lemma \ref{lm:torus-tiling} applies and we conclude that
$$
F*\tau_j(x)=k_j,\ \ \ \mbox{for almost all $x \in {\mathbb T}$ and some integer $k_j$},
$$
which is equivalent to $f*(\delta_{\mathbb Z}*\tau_j)=k_j$ as we had to prove.
This concludes the proof of \eqref{en:split}.
\end{proof}
\section{The structure of the set of multiples of a multiplicative tile}
\label{sec:structure}
\begin{theorem}\label{th:mult-structure}
(Structure of the set of multiples)\\
Suppose $\Omega \subseteq {\mathbb R}$ is a bounded measurable set such that $\Omega \cap (-\epsilon, \epsilon) = \emptyset$ for some $\epsilon>0$.
Suppose also $A \subseteq {\mathbb R}\setminus\Set{0}$ is a discrete set such that
$$
A \cdot \Omega
$$
is a (multiplicative) tiling of ${\mathbb R}$ at level 1.
Let $\Omega^+, \Omega^-, A^+, A^- \subseteq {\mathbb R}^+$ be the positive and negative parts of $\Omega$ and $A$
$$
\Omega^+ = \Omega \cap (0, +\infty),\ \Omega^- = -(\Omega \cap (-\infty, 0)),\
A^+ = A \cap (0, +\infty),\ A^- = -(A \cap (-\infty, 0)).
$$
Then
\begin{enumerate}
\item[(i)]\label{symmetric-case} If $\Omega$ is \underline{essentially symmetric} (i.e.\ if $\Abs{\Omega^+ \, \triangle\, \Omega^-} = 0$) then
$A^+ \cap A^- = \emptyset$ and the union $\log A^+ \cup \log A^-$ is periodic of the form
\beql{periodic-rational}
\alpha{\mathbb Z}+\Set{r_1, r_2, \ldots, r_s}\ \ \ \text{with $r_i-r_j$ rational multiples of $\alpha>0$}.
\end{equation}
The partition of the set $\log A^+ \cup \log A^-$ into its component sets $\log A^+$ and $\log A^-$
can be completely arbitrary.
\item[(ii)]\label{nonsymmetric-case} If $\Omega$ is \underline{not essentially symmetric} with respect to the origin then
the sets $\log A^+$ and $\log A^-$ are both periodic and of the form \eqref{periodic-rational} with the same period $\alpha$.
\end{enumerate}
\end{theorem}
\begin{proof}
In order to transfer the problem to the translational case, which is much better understood, it is natural to take logarithms.
Allowing ourselves a slight abuse of terminology,
we then have that $A\cdot \Omega={\mathbb R}$ is a tiling if and only if
$$
{\mathbb R}^+ = A^+ \Omega^+ \cup A^- \Omega^-,\ \ \ {\mathbb R}^+ = A^- \Omega^+ \cup A^+ \Omega^-
$$
are both tilings (where ${\mathbb R}^+$ is the right half-line).
Taking logarithms of both we obtain the additive (translational) tilings
\beql{log-tilings}
{\mathbb R} = (a^+ + \omega^+) \cup (a^- + \omega^-) = (a^-+\omega^+) \cup (a^+ + \omega^-),
\end{equation}
where we write the lower case letter for the set of logarithms of a set written with the corresponding capital letter,
e.g.\ $a^+ = \log A^+$.
Identifying, further, the sets $\omega^\pm$ with their indicator functions
and the discrete sets $a^{\pm}$ with a collection of unit point masses at their points (for instance, we write $a^+$ instead of $\delta_{a^+}$),
we may rewrite the above tilings using convolution as
\beql{tilings}
1 = a^+*\omega^+(x) + a^-*\omega^-(x) = a^-*\omega^+(x) + a^+*\omega^-(x),\ \ \ \mbox{for almost all $x \in {\mathbb R}$}.
\end{equation}
Adding and subtracting the two identities in \eqref{tilings} we get the equivalent set of identities (valid for almost all $x \in {\mathbb R}$)
\beql{sum}
2 = (\omega^++\omega^-)*(a^++a^-)(x)
\end{equation}
and
\beql{diff}
0 = (\omega^+ - \omega^-)*(a^+-a^-)(x).
\end{equation}
Notice that $\omega^+(x)+\omega^-(x)$ is a function that only takes the values 0, 1 and, possibly, 2 and that $a^++a^-$ is a measure,
which is a collection of point masses of weight 1 or 2.
Similarly $a^+-a^-$ is a measure which is a collection of point masses of weight $\pm 1$.
There is obviously no problem with the definition of the convolutions in \eqref{tilings} and \eqref{sum}, as there are
only nonnegative quantities involved. A moment's thought should convince us that there is no problem in \eqref{diff} either,
as all sums involved have finitely many terms,
the functions $\omega^\pm(x)$ being of compact support and the sets $a^\pm$ being discrete.
{\bf Periodicity.}
From Theorem \ref{th:integer-tiling}\eqref{en:periodic} applied to the tiling \eqref{sum} we obtain that
$a$ (viewed as a multiset when the point mass at a point has weight 2)
is a periodic set
\beql{periodic-set}
a = a^+ \cup a^- = \gamma{\mathbb Z} + \Set{\beta_1, \beta_2, \ldots, \beta_J},
\end{equation}
for some $\gamma>0$, $\beta_j \in {\mathbb R}$.
As a consequence the set
$$
a^+ \cap a^-
$$
is also periodic, as this is where the multiplicity of $a$ is equal to 2.
Assume now that $\omega^+$ is not identical to $\omega^-$ (that is $\Omega$ is not symmetric with respect to 0)
so that the function $\omega^+-\omega^-$ that appears in \eqref{diff} is not equal to zero almost everywhere.
The measure $a^+-a^-$ is a collection of Dirac point masses of weight $\pm 1$. The weight 1 appears exactly on
the points of the set $a^+ \setminus a^-$ and the weight -1 exactly on the set $a^- \setminus a^+$.
Given that we have already established the periodicty of $a^+ \cap a^-$
the periodicity of the set $a^+$ and the periodicity of the set $a^-$ will follow if we manage to show
the periodicity of $a^+ \setminus a^-$ and of $a^- \setminus a^+$ with a period commensurable to a period of $a^+ \cap a^-$.
It follows again from Theorem \ref{th:integer-tiling}\eqref{en:periodic}, applied to tiling \eqref{diff},
with $f = \omega^+-\omega^-$ (this is a compactly supported function, because of our assumption
that the bounded set $\Omega$ avoids an open neighborhood of $0$),
that the sets $a^+\setminus a^-$ and $a^-\setminus a^+$ are periodic with the same period.
Since we have already shown that $a^+ \cap a^-$ is also periodic it follows that each of the sets $a^+$ and $a^-$
is a union of two periodic sets, and these must be of commensurable periods, as, otherwise, the set
$a^+ \cup a^-$, which is already known to be periodic, would contain two arithmetic progressions with inocommensurable step,
an impossibility.
It follows that $a^+$ and $a^-$ are periodic too, and with commensurable periods.
{\bf Rationality.}
Dilating space we may assume now that the sets $a^\pm$ have period 1.
It remains to prove that the set $a^+ \cup a^-$ has rational differences.
Write
\beql{one-period-plus}
a^+ + a^- = \delta_{\mathbb Z}*\sigma,\ \ \ \mbox{with}\ \ \ \sigma = \sum_{j=1}^J n_j \delta_{x_j},
\end{equation}
where $n_j \in \Set{1,2}$ and $x_j \in [0,1)$, for $j=1, 2, \ldots, J$.
It follows that we can also write
\beql{one-period-minus}
a^+-a^- = \delta_{\mathbb Z}*\tau,\ \ \ \mbox{with}\ \ \ \tau = \sum_{j=1}^J m_j \delta_{x_j},
\end{equation}
where now $m_j \in \Set{-1, 0, 1}$, for $j=1, 2, \ldots, J$.
Contrary to what we want to prove, assume that not all the $x_j$s are rationally equivalent, and
let $x_{i_1}, x_{i_2}, \ldots, x_{i_r}$ be a rationally equivalent class of the points $x_1, x_2, \ldots, x_J$ in \eqref{one-period-plus}.
We now apply Theorem \ref{th:integer-tiling}\eqref{en:split} to the two tilings \eqref{sum} and \eqref{diff}.
It follows that we have the two tilings
\beql{sum-1}
(\omega^++\omega^-)*(\delta_{\mathbb Z}+\sum_{j=1}^r n_{i_j} \delta_{x_{i_j}}) = k_1
\end{equation}
and
\beql{diff-1}
(\omega^+-\omega^-)*(\delta_{\mathbb Z}+\sum_{j=1}^r m_{i_j} \delta_{x_{i_j}}) = k_2,
\end{equation}
for some two integers $k_1, k_2$.
Write $a^{'\pm}$ for the restriction of $a^\pm$ to the set ${\mathbb Z}+\Set{x_{i_1}, \ldots, x_{i_r}}$
and observe that we now have
\beql{sum-2}
a^{'+}+a^{'-} = \delta_{\mathbb Z} + \sum_{j=1}^r n_{i_j} \delta_{x_{i_j}}
\end{equation}
and
\beql{diff-2}
a^{'+}-a^{'-} = \delta_{\mathbb Z} + \sum_{j=1}^r m_{i_j} \delta_{x_{i_j}},
\end{equation}
so that the tilings \eqref{sum-1} and \eqref{diff-1} can now be written as
\beql{sum-3}
(\omega^++\omega^-)*(a^{'+}+a^{'-}) = k_1
\end{equation}
and
\beql{diff-3}
(\omega^+-\omega^-)*(a^{'+}-a^{'-}) = k_2.
\end{equation}
By our assumption that not all points in $\Set{x_1, x_2 \ldots, x_J}$ are rationally equivalent it
follows that \eqref{sum-3} is a proper subtiling of \eqref{sum}, which forces $k_1=1$.
Adding and subtracting \eqref{sum-3} and \eqref{diff-3} and dividing by 2 we obtain the tilings
\beql{tilings-3}
\frac{1+k_2}{2} = a^{'+}*\omega^+(x) + a^{'-}*\omega^-(x),\ \ \ \frac{1-k_2}{2} = a^{'-}*\omega^+(x) + a^{'+}*\omega^-(x),
\end{equation}
analogous to \eqref{tilings} and proper subtilings of those in \eqref{tilings}.
But the tilings in \eqref{tilings} are minimal, as they are tilings by translates of sets, at level 1, a contradiction.
In case \eqref{symmetric-case} of the Theorem,
if $\omega^+ \equiv \omega^-$ and $\omega^+$ tiles ${\mathbb R}$ with some set $a$, then \eqref{diff} is trivial and \eqref{sum} is clearly valid.
From \eqref{sum} it follows using the same tiling structure theorems that $a$ is a periodic multiset,
and \eqref{tilings} can be satisfied by arbitrarily breaking up the multiset $a$ into two sets (not multisets)
$a^+$ and $a^-$. Nothing more can be said about the sets $a^+$ and $a^-$ in this case.
\end{proof}
\section{The structure of a multiplicative tile}
\label{sec:tile-structure}
\subsection{Symmetric tile}\label{sec:symmetric-tile}
Suppose that the set $\Omega$ is symmetric with respect to $0$. In the notation of Theorem \ref{th:mult-structure}
this means that $\Omega^+ = \Omega^-$ (up to measure 0).
In this case the tilings of \eqref{log-tilings} become just one tiling:
\beql{single-tiling}
{\mathbb R} = (a^+ \cup a^-) + \omega^+.
\end{equation}
(Remember that the lower case letters denote the logarithms of the sets written in the corresponding upper case.)
So in this case the problem of multiplicative tiling becomes exactly the problem of translational tiling of the real line
by the tile $\omega^+$ and with set of translates the multiset $a^+ \cup a^-$.
The structure of $\omega^+$ in this case has been completely characterized in \cites[Theorem 3]{lagarias1996tiling}.
\subsection{Non-symmetric tile}\label{sec:non-symmetric-tile}
Suppose now that we are in the case where $\omega^+ \nequiv \omega^-$ and, according to Theorem \ref{th:mult-structure},
$a^+, a^-$ are both periodic with the same period obeying \eqref{periodic-rational}.
We can thus write (after scaling)
\beql{a-shape}
a^+ = {\mathbb Z}+\frac{1}{L}\Set{0=\alpha_1^+, \alpha_2^+, \ldots , \alpha_m^+},\ \ \
a^- = {\mathbb Z}+\frac{1}{L}\Set{\alpha_1^-, \alpha_2^-, \ldots , \alpha_n^+},
\end{equation}
for some positive integer $L$, where $\alpha_j^\pm \in {\mathbb Z}_L = {\mathbb Z} / (L{\mathbb Z})$.
In order to express our tiling problem on the torus ${\mathbb T} = {\mathbb R} / {\mathbb Z}$ we identify
$\omega^\pm$ with the indicator function of the set that arises when taking their
projection mod 1.
(Because of the tiling assumption the points of each $\omega^\pm$ are different mod 1.)
Write
$$
\alpha^+ = \Set{\alpha_1^+, \alpha_2^+, \ldots, \alpha_m^+},\ \ \ \alpha^- = \Set{\alpha_1^-, \alpha_2^-, \ldots, \alpha_n^-},
$$
and also $\alpha^\pm$ for the collection of unit Dirac masses at the points of $\alpha^\pm$.
The tiling conditions \eqref{tilings} now become equivalently the tilings of the torus
\beql{mod-tilings}
1 = \frac{\alpha^+}{L}*\omega^+(t) + \frac{\alpha^-}{L}*\omega^-(t) = \frac{\alpha^-}{L}*\omega^+(t) + \frac{\alpha^+}{L}*\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
\end{equation}
Write for $x \in [0,1)$
$$
C_x = \Set{x+\frac{k}{L} \bmod 1:\ k=0,1,\ldots,L-1} \subseteq {\mathbb T}.
$$
Multiplying \eqref{mod-tilings} by ${\bf 1}_{C_x}(t)$ we get
$$
{\bf 1}_{C_x}(t) =
\frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^-(t) =
\frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
$$
Restricting to $t \in C_x$ we can rewrite this as the two tiling conditions on ${\mathbb Z}_L$, valid for all $x \in [0, \frac{1}{L})$,
\begin{align}
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^+ \ \ \cup\ \ (\omega^-\cap C_x) + \alpha^-,\nonumber\\
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^- \ \ \cup\ \ (\omega^-\cap C_x) + \alpha^+\label{cycle-tilings},
\end{align}
where we are now identifying the sets $\omega^\pm\cap C_x$ with the obvious subset of ${\mathbb Z}_L$.
The sets $\omega^\pm \cap C_x$ can be chosen independently for all $x \in [0, 1/L)$ as tiling is not affected by
what happens on different cosets of $\frac{1}{L}{\mathbb Z}$.
We conclude that the sets $\omega^\pm$ are of the form
\begin{align}
\omega^+ &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^+_x \right)\nonumber\\
\omega^- &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^-_x \right) \ \ \ \mbox{(disjoint unions)} \label{decomposition}
\end{align}
where $b^\pm_x \subseteq {\mathbb Z}_L$ are such that for each $x$ we have the two tilings
\begin{align}
{\mathbb Z}_L &= b^+_x + \alpha^+ \cup b^-_x + \alpha^-\nonumber\\
{\mathbb Z}_L &= b^+_x + \alpha^- \cup b^-_x + \alpha^+\label{cycle-tilings-final}
\end{align}
In the next section we are trying to understand better the kind of tiling described in \eqref{cycle-tilings-final}.
\section{Cross tiling}
\label{sec:ct}
\begin{definition}[Cross tiling]\label{def:cross-tiling}\ \\
Suppose $N>1$ is a positive integer and $A, B, X, Y \subseteq {\mathbb Z}_N$. We say that the pair $A, B$ admits
{\em cross tiling} with complements $X, Y$ if the following tilings hold:
\begin{align}
{\mathbb Z}_N = (A + X) \cup (B + Y)\nonumber\\
{\mathbb Z}_N = (A + Y) \cup (B + X) \label{cross-tiling}
\end{align}
A cross tiling \eqref{cross-tiling} is called {\em trivial} if
$(A \cup B) + X = {\mathbb Z}_N$ is a tiling and $X=Y$, or the same with the roles of
$A, B$ exchanged with those of $X, Y$.
(It is obvious that in this case conditions \eqref{cross-tiling} are indeed satisfied.)
\end{definition}
\noindent{\bf Remark.}
It is interesting to observe that cross tiling is really an ordinary tiling by translation, although of a larger group.
With $A, B, X, Y \subseteq {\mathbb Z}_N$ as in Definition \ref{def:cross-tiling} above write
\begin{align*}
\Gamma &= {\mathbb Z}_N \times {\mathbb Z}_2,\\
C &= A\times\Set{0} \cup B\times\Set{1},\\
Z &= X\times\Set{0} \cup Y\times\Set{1}.
\end{align*}
It is easy to see that the cross tiling condition \eqref{cross-tiling} is equivalent to the tiling by translation
$$
\Gamma = C + Z.
$$
This alternative characterization of cross tiling may be of interest but is not exploited in this paper
The following two examples are non-trivial cross tilings.
\begin{example}\label{ex:first-non-trivial}
Let $N=2ab$, with odd $a, b \in {\mathbb N}$ and view $G={\mathbb Z}_N$ as the cross-product ${\mathbb Z}_N = {\mathbb Z}_{ab}\times{\mathbb Z}_2$.
Define
\begin{align*}
A &= \Set{0, 1, 2, \ldots, a-1} \times \Set{0},\\
B &= \Set{0, 1} \cup \Set{a+2, a+3, \ldots, 2a-1} \times \Set{0},\\
X &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{0},\\
Y &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{1}.
\end{align*}
It follows that
$$
A + X = B + X = {\mathbb Z}_{ab}\times\Set{0}\ \ \ \text{and}\ \ \ B + Y = A + Y = {\mathbb Z}_{ab}\times\Set{1}
$$
so \eqref{cross-tiling} follows. See Figure \ref{fig:easy-example}.
\end{example}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=5, b=3, N=2*a*b;
pair A[]; for(i=0; i<a; ++i) { A[i] = (i,0); }
pair B[]; B[0] = (0,0); B[1] = (1,0); for(j=2, i=a+2; i<2*a; ++i, ++j) { B[j] = (i,0); }
pair X[]; for(i=0; i<b; ++i) { X[i] = (i*a, 0); }
pair Y[]; for(i=0; i<b; ++i) { Y[i] = (i*a, 1); }
pair u=(0,4);
// Draw group
picture pg;
for(i=0; i<a*b; ++i) {
for(j=0; j<2; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
add(pg);
label("Group $G$ ", (0,.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A$ ", u+(0,.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B$ ", 2*u+(0,.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X$ ", 3*u+(0,.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets cross tiling in Example \ref{ex:first-non-trivial} with $a=5, b=3$} \label{fig:easy-example}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=15, b=4;
real F1[]={0,1,2}, F2[]={0,4,5}, C[]={0, 3, 6, 9, 12}, CC[]={3, 6, 9, 12};
pair A[]; for(i=0; i<F1.length; ++i) {A.push((F1[i], 0)); A.push((F1[i], 1)); }
pair B[]; for(i=0; i<F2.length; ++i) {B.push((F2[i], 0)); B.push((F2[i], 1)); }
pair X[]; for(i=0; i<C.length; ++i) {X.push((C[i], 0)); X.push((C[i], 2)); }
pair Y[]; Y.push((0,1)); Y.push((0,3)); for(i=0; i<CC.length; ++i) {Y.push((CC[i], 0)); Y.push((CC[i], 2)); }
pair u=(0,5);
// Draw group
picture pg;
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
//draw(pg, (i+0.5,j), 1bp+black);
//draw(pg, (i+0.5,j+0.5), 1bp+black);
draw(pg, (i,j+0.5), 1bp+black);
}
}
add(pg);
label("Group $G$ ", (0,1.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A'$ ", u+(0,1.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B'$ ", 2*u+(0,1.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X'$ ", 3*u+(0,1.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y'$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets $A', B', X', Y' \subseteq H$ in Example \ref{ex:second-non-trivial} giving rise to the sets $A, B, X, Y \subseteq G$} \label{fig:better-example}
\end{figure}
\begin{example}\label{ex:second-non-trivial}
In Example \ref{ex:first-non-trivial} the sets $X$ and $Y$ are translates of each other, which is perhaps not very
satisfactory in terms of deviating from triviality.
Now we give an example where this does not happen.
We work in the group $G = {\mathbb Z}_{15}\times{\mathbb Z}_8 = {\mathbb Z}_{120}$
which contains the subgroup
$$
H = {\mathbb Z}_{15} \times \Set{0, 2, 4, 6}.
$$
Next define the subsets of ${\mathbb Z}_{15}$
$$
F_1 = \Set{0, 1, 2},\ \ \ F_2 = \Set{0, 4, 5}
$$
and notice that they both tile ${\mathbb Z}_{15}$ with the complement $\Set{0, 3, 6, 9, 12}$.
Define the subsets of $H$
\begin{align*}
A' &= F_1 \times \Set{0, 2},\\
B' &= F_2 \times \Set{0, 2},\\
X' &= \Set{0, 3, 6, 9, 12} \times \Set{0, 4},\\
Y' &= (\Set{0}\times\Set{1, 3}) \cup (\Set{3, 6, 9, 12}) \times \Set{0, 4}.
\end{align*}
See Figure \ref{fig:better-example} where the group $H$ is shown as thick dots
while its single coset in $G$ is shown as thin dots.
The difference between $X'$ and $Y'$ is that the first ``column'' of $Y'$ is ``raised'' by 1.
It is easy to verify that each of $A', B'$ tiles $H$ with each of $X', Y'$ as a tiling complement.
Finally define the subsets of $G$
\begin{align*}
A &= A', \\
B &= B', \\
X &= X', \\
Y &= Y'+(0,1),
\end{align*}
and observe that they do satisfy the cross-tiling conditions
\begin{align}
G &= (A+X) \cup (B+Y) \nonumber \\
G &= (A+Y) \cup (B+X). \label{ct}
\end{align}
The sets in parentheses in \eqref{ct} are tilings of each of the two $H$-cosets in $G$.
The sets with $+X$ tile $H$ and those with $+Y$ tile $H+(0,1)$.
None of the sets $A, B, X, Y$ is a translate of another.
\end{example}
\subsection{Fourier condition for cross tiling}
Translational tiling $A + X = {\mathbb Z}_N$ has a simple equivalent Fourier condition (we use the same letter for a set and its indicator function):
\beql{tiling-ft}
\ft{A}(0) \ft{X}(0) = N,\ \ \ \mbox{and}\ \ \ \forall k \in {\mathbb Z}_N\setminus\Set{0}: \ft{A}(k) \neq 0 \Longrightarrow \ft{X}(k) = 0.
\end{equation}
Adding and subtracting the cross-tiling defining conditions
$$
A*X+B*Y \equiv 1,\ \ \ A*Y+B*X \equiv 1,
$$
we obtain the equivalent conditions
\beql{cross-tiling-equiv}
(A+B)*(X+Y) \equiv 2,\ \ \ (A-B)*(X-Y) \equiv 0.
\end{equation}
Taking Fourier Transforms these conditions lead to the following equivalent Fourier condition for cross-tiling:
\begin{align}
&(\ft{A}(0) + \ft{B}(0)) (\ft{X}(0)+\ft{Y}(0)) = 2N,\nonumber\\
\forall k \in {\mathbb Z}_N\setminus\Set{0}:\ \ \ &\ft{A}(k) \neq -\ft{B}(k) \Longrightarrow \ft{X}(k) = -\ft{Y}(k). \label{cross-tiling-ft}\\
\forall k \in {\mathbb Z}_N:\ \ \ &\ft{A}(k) \neq \ft{B}(k) \Longrightarrow \ft{X}(k) = \ft{Y}(k).\nonumber
\end{align}
Using $k=0$ in the last set of equations we obtain that necessarily
$$
\Abs{A} = \Abs{B}\ \ \ \text{or}\ \ \ \Abs{X} = \Abs{Y}.
$$
\printbibliography
\end{document}
\section{Introduction}
\label{sec:intro}
\begin{definition}[Translational and multiplicative tiling]\label{def:tiling}\ \\
Suppose $f:{\mathbb R}^d\to{\mathbb C}$ is measurable and $A \subseteq {\mathbb R}^d$. We say that $f+A$ is a tiling
of ${\mathbb R}^d$ at level $\ell$ if
$$
\sum_{a \in A} f(x-a) = \ell
$$
for almost all $x \in {\mathbb R}^d$ with the sum converging absolutely.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $\Omega+A$ is a tiling.
If $A \subseteq {\mathbb R}\setminus\Set{0}$ and
$$
\sum_{a \in A} f(a^{-1} x) = \ell
$$
for almost all $x \in {\mathbb R}^d$, with absolute convergence, then we say that $A\cdot f$ is a tiling.
If $\Omega \subseteq {\mathbb R}^d$ is a measurable set and $f = {\bf 1}_\Omega$ we also say that $A\cdot \Omega$ is a tiling.
\end{definition}
Our purpose here is to study the structure of multiplicative tilings.
We are guided in this by previous work on the structure of translational tilings of the real line or of the integer line.
In \cites{newman1977tesselations,leptin1991uniform,lagarias1996tiling,kolountzakis1996structure}
it is proved, under very broad conditions, that any translational tiling of the real line (or of the integer line) must be periodic.
\section{The structure of multiple translational tiling by a set}
\label{sec:rationality}
In this section we prove a structure theorem for translational tilings of the line by any function that
is integer valued.
\begin{theorem}[Multiple translational tiling by an integer-valued function]\label{th:set-tiling}
Suppose $f\ge 0$ is a measurable integrable function on the real line, of compact support,
which takes only integer values and that $A \subseteq {\mathbb R}$ is such that $0 \in A$ and
\beql{set-tiling}
\sum_{a\in A} f(x-a) = k,\ \ \ \mbox{for almost all $x\in{\mathbb R}$},
\end{equation}
where $k>0$ is an integer constant.
Then $A$ is of the form
\beql{a-structure}
A = a{\mathbb Z} + \Set{0=r_1, r_2, \ldots, r_l}
\end{equation}
where $k a = l\int f$.
If the tiling \eqref{set-tiling} is indecomposable then we also have $r_j/a \in {\mathbb Q}$ for $j=1,2,\ldots,l$.
\end{theorem}
Theorem \ref{th:set-tiling} was proved in \cites{lagarias1996tiling} for
for $f$ being the indicator function of a bounded, measurable subset of ${\mathbb R}$,
and $k=1$ (in this case indecomposability is automatic).
The proof does not readily extend to the more general case of Theorem \ref{th:set-tiling} and this is what we show here.
\begin{proof}
The periodicity of the set $A$ follows
from the results in \cites{lagarias1996tiling,kolountzakis1996structure}.
For instance, it is proved in \cite[Theorem 1.1]{kolountzakis1996structure}, that every
tiling of the real line by a compactly supported function $f$ splits into a finite number of
indecomposable tilings, each of them with a periodic tile set.
When $f$ is an integer-valued function, as in our case, the level $\ell$ of every indecomposable tiling is an integer
and clearly, if $a$ is a period of the tiling, we have
$$
a\cdot \ell = \int f \cdot l,
$$
where $l$ is the number of copies of the tile per period. This shows that the period of any indecomposable tiling
is always a rational multiple of $\int f$, hence the decomposition of a general tiling by such a function $f$ into finitely many
indecomposable tilings can always be rewritten in such a way that all periods are the same integral multiple of $\int f$ (in the process
we may of course have to enlarge the periods, so we do not claim that the minimal period of the tiling is an integral
multiple of $\int{f}$, though it is a rational multiple).
So in the case of tiling by integer-valued functions all tilings, not just indecomposable ones, are periodic.
By scaling both the values and the domain of $f$ we may
assume that $\int{f}=1$ and that $A$ is already written in the form \eqref{a-structure} with $a=1$.
What remains to be shown is that all $r_j$ are rational.
Contrary to what we want to show suppose then that
\beql{r-decomposition}
R = \Set{0=r_1, r_2, \ldots, r_l} = R_1 \cup \cdots \cup R_\nu,\ \ \ (\nu > 1, \mbox{disjoint union})
\end{equation}
where the $R_j$ are the equivalence classes of $R$ modulo differing by a rational number.
Next we observe that the tiling condition \eqref{set-tiling} may be rewritten, as a tiling of the torus,
\beql{set-tiling-torus}
\sum_{j=1}^l F(x-r_j) = k,\ \ \ \mbox{(for almost all $x \in {\mathbb R}/{\mathbb Z}$)}.
\end{equation}
The function $F$ that appears in \eqref{set-tiling-torus} is
the periodization of the function $f$ by ${\mathbb Z}$,
$$
F(x) = \sum_{n \in {\mathbb Z}} f(x-n),
$$
which is also integer-valued.
Computing the Fourier coefficients of both sides in the above identity \eqref{set-tiling-torus} we obtain
\beql{ft-tiling}
\ft{F}(n) \phi(n) = k \,\delta_0(n),\ \ \ \ (n \in {\mathbb Z})
\end{equation}
where $\phi(n) = \sum_{j=1}^l e^{-2\pi i r_j n}$.
We also write $\phi_s(n) = \sum_{r \in R_s} e^{-2\pi i r n}$, for $s=1,2,\ldots,\nu$,
satisfying of course
$$
\phi(n) = \sum_{s=1}^{\nu} \phi_s(n).
$$
Writing $Z(h)$ for the integer zero set of function $h$ and
$$
X = \bigcap_{s=1}^{\nu} Z(\phi_s),
$$
for the common set of zeros of the $\phi_s$ (over the integers)
we finally define the set $Y$ to be disjoint from $X$ and be such that
$$
X \subseteq Z(\phi) = X \cup Y.
$$
It is proved in \cites[\S 4]{lagarias1996tiling} that the set $Y$ is finite.
We do not reprove this important part of the proof (which is based on the Skolem-Mahler-Lech theorem) but
refer to \cites{lagarias1996tiling} instead.
From \eqref{ft-tiling} it follows that
$$
{\rm supp\,}{\ft{F}} \subseteq \Set{0} \cup Z(\phi) = \Set{0} \cup Y \cup X,
$$
so, for each $s = 1, 2, \ldots, \nu$, we have
$$
{\rm supp\,}{\ft{F}} \subseteq \Set{0} \cup Y \cup Z(\phi_s),
$$
so that
$$
\ft{F}(n) \cdot \phi_s(n)
$$
has finite support $\subseteq \Set{0} \cup Y$ and its Fourier Transform
\beql{tmp-1}
\sum_{r \in R_s} F(x-r),\ \ \ \ (x \in {\mathbb R}/{\mathbb Z})
\end{equation}
is, therefore, a trigonometric polynomial.
But the function in \eqref{tmp-1} takes only integer values at each point $x$,
so the only way this can be a trigonometric polynomial is if it is a constant.
This would imply that $f$ tiles with ${\mathbb Z}+R_s$, at some level,
and that the initial tiling of $f$ with ${\mathbb Z}+R$ is not indecomposable as assumed, a contradiction.
It follows that $\nu = 1$ for the decomposition \eqref{r-decomposition} and that $R \subseteq {\mathbb Q}$
as required.
\end{proof}
\section{The structure of the set of multiples of a multiplicative tile}
\label{sec:structure}
\begin{theorem}\label{th:mult-structure}
(Structure of the set of multiples)\\
Suppose $\Omega \subseteq {\mathbb R}$ is a bounded measurable set such that $\Omega \cap (-\epsilon, \epsilon) = \emptyset$ for some $\epsilon>0$.
Suppose also $A \subseteq {\mathbb R}\setminus\Set{0}$ is a discrete set such that
$$
A \cdot \Omega
$$
is a (multiplicative) tiling of ${\mathbb R}$ at level 1.
Let $\Omega^+, \Omega^-, A^+, A^- \subseteq {\mathbb R}^+$ be the positive and negative parts of $\Omega$ and $A$
$$
\Omega^+ = \Omega \cap (0, +\infty),\ \Omega^- = -(\Omega \cap (-\infty, 0)),\
A^+ = A \cap (0, +\infty),\ A^- = -(A \cap (-\infty, 0)).
$$
Then
\begin{enumerate}[(i)]
\item\label{symmetric-case} If $\Omega$ is essentially symmetric (i.e.\ if $\Abs{\Omega^+ \, \triangle\, \Omega^-} = 0$) then
the union $\log A^+ \cup \log A^-$ is a periodic multiset (of multiplicity up to 2)
of the form
\beql{periodic-rational}
\alpha{\mathbb Z}+\Set{r_1, r_2, \ldots, r_s}\ \ \ \text{with $r_i-r_j$ rational multiples of $\alpha>0$}.
\end{equation}
The partition of the multiset $\log A^+ \cup \log A^-$ into its component sets $\log A^+$ and $\log A^-$
can be completely arbitrary.
\item\label{nonsymmetric-case} If $\Omega$ is not essentially symmetric with respect to the origin then
the sets $\log A^+$ and $\log A^-$ are both periodic and of the form \eqref{periodic-rational} with the same $\alpha$.
\end{enumerate}
\end{theorem}
\begin{proof}
In order to transfer the problem to the translational case, which is much better understood, it is natural to take logarithms.
Allowing ourselves a slight abuse of terminology,
we then have that
$$
{\mathbb R}^+ = A^+ \Omega^+ \cup A^- \Omega^-,\ \ \ {\mathbb R}^+ = A^- \Omega^+ \cup A^+ \Omega^-
$$
are both tilings (where ${\mathbb R}^+$ is the right half-line).
Taking logarithms of both we obtain the additive (translational) tilings
\beql{log-tilings}
{\mathbb R} = (a^+ + \omega^+) \cup (a^- + \omega^-) = (a^-+\omega^+) \cup (a^+ + \omega^-),
\end{equation}
where we write the lower case letter for the set of logarithms of a set written with the corresponding capital letter.
Identifying, further, the sets $\omega^\pm$ with their indicator functions
and the discrete sets $a^{\pm}$ with a collection of unit point masses at their points,
we may rewrite the above tilings using convolution as
\beql{tilings}
1 = a^+*\omega^+(x) + a^-*\omega^-(x) = a^-*\omega^+(x) + a^+*\omega^-(x),\ \ \ \mbox{for almost all $x \in {\mathbb R}$}.
\end{equation}
Adding and subtracting the two identities in \eqref{tilings} we get the identities (valid for almost all $x \in {\mathbb R}$)
\beql{sum}
2 = \omega*a(x)
\end{equation}
and
\beql{diff}
0 = (\omega^+ - \omega^-)*(a^+-a^-)(x),
\end{equation}
where $\omega(x) = \omega^+(x) + \omega^-$(x) and $a(x) = a^+(x) + a^-(x)$.
Notice that $\omega(x)$ is a function that only takes the values 0, 1 and, possibly, 2 and that $a$ is a measure,
which is a collection of point masses of weight 1 or 2.
There is obviously no problem with the definition of the convolutions in \eqref{tilings} and \eqref{sum}, as there are
only nonnegative quantities involved. A moment's thought should convince us that there is no problem in \eqref{diff} either,
as all sums involved have finitely many terms.
From Theorem \ref{th:set-tiling} applied to the tiling \eqref{sum} we obtain that
that $a$ (viewed as a multiset when the point mass at a point has weight 2)
is a periodic set
\beql{qp}
a = a^+ \cup a^- = \gamma{\mathbb Z} + \Set{\beta_1, \beta_2, \ldots, \beta_J},
\end{equation}
for some $\gamma>0$, $\beta_j \in {\mathbb R}$.
As a consequence the set
$$
a^+ \cap a^-
$$
is also periodic, as this is where the multiplicity of $a$ is equal to 2.
Assume now that $\omega^+$ is not identical to $\omega^-$ (that is $\Omega$ is not symmetric with respect to 0)
so that the function $\omega^+-\omega^-$ that appears in \eqref{diff} is not equal to zero almost everywhere.
The measure $a^+-a^-$ is a collection of Dirac point masses of weight $\pm 1$. The weight 1 appears exactly on
the points of the set $a^+ \setminus a^-$ and the weight -1 exactly on the set $a^- \setminus a^+$.
Given that we have already established the periodicty of $a^+ \cap a^-$
the periodicity of the set $a^+$ and the periodicity of the set $a^-$ will follow if we manage to show
the periodicity of $a^+ \setminus a^-$ and of $a^- \setminus a^+$.
This is a consequence of the following result, which is a variation of \cites[Theorem 5.1]{kolountzakis1996structure}.
\begin{theorem}\label{th:structure}
Let $f \in L^1({\mathbb R})$ have a Fourier Trasnform $\ft{f} \in C^\infty({\mathbb R})$ which has a discrete zero set satisfying
the bound
\beql{linear-bound}
\#\Set{\xi: \ft{f}(\xi) = 0 \ \mbox{and}\ \Abs{\xi} \le R} \le cR,
\end{equation}
for some positive constant $c$ and for $R>1$.
(For instance, this is true if $f$ is integrable and has compact support.)
Suppose that $M, L$ are disjoint, discrete subsets of ${\mathbb R}$ of bounded density
and assume the tiling condition
$$
\sum_{m\in M} f(x-m) - \sum_{l \in L} f(x-l) = C,\ \ \ \text{for almost all $x \in {\mathbb R}$},
$$
where $C \in {\mathbb R}$ is a constant.
Then each of the sets $M$ and $L$ is a finite union of complete arithmetic progressions.
\end{theorem}
We do not supply the full proof of Theorem \ref{th:structure} here, as it is not essentially
different from the proof of \cites[Theorem 5.1]{kolountzakis1996structure}.
Let us only indicate that, as in \cites{kolountzakis1996structure}, the main ingredient
in the proof of Theorem \ref{th:structure} is a theorem of Meyer \cites[Theorem 4.2]{kolountzakis1996structure}, \cites{meyer1970nombres},
itself a consequence of the Idempotent Theorem of P.J. Cohen \cites[Theorem 4.1]{kolountzakis1996structure}, \cites{cohen1959homomorphisms}.
It follows from Theorem \ref{th:structure}, applied to tiling \eqref{diff},
with $f = \omega^+-\omega^-$ (this is a compactly supported function, because of our assumption
that the bounded set $\Omega$ avoids an open neighborhood of $0$), $M = a^+\setminus a^-$, $L = a^-\setminus a^+$,
that the sets $a^+\setminus a^-$ and $a^-\setminus a^+$ are individually quasi-periodic (i.e.\ of the form
\eqref{qp}).
Since we have already shown that $a^+ \cap a^-$ is also quasi-periodic it follows that each of the sets $a^+$ and $a^-$
are quasi-periodic.
And since $a = a^+ \cup a^-$ is already known to be periodic \eqref{qp} it follows that $a^+$ and $a^-$ are periodic too.
In case \eqref{symmetric-case} of the Theorem,
if $\omega^+ \equiv \omega^-$ and $\omega^+$ tiles ${\mathbb R}$ with some set $a$, then \eqref{diff} is trivial and \eqref{sum} is clearly valid.
From \eqref{sum} it follows using the same tiling structure theorems that $a$ is a periodic multiset,
and \eqref{tilings} can be satisfied by arbitrarily breaking up the multiset $a$ into two sets (not multisets)
$a^+$ and $a^-$. Nothing more can be said about the sets $a^+$ and $a^-$ in this case.
\end{proof}
\section{The structure of a multiplicative tile}
\subsection{Symmetric tile}\label{sec:symmetric-tile}
Suppose that the set $\Omega$ is symmetric with respect to $0$. In the notation of Theorem \ref{th:mult-structure}
this means that $\Omega^+ = \Omega^-$ (up to measure 0).
In this case the tilings of \eqref{log-tilings} become just one tiling:
\beql{single-tiling}
{\mathbb R} = (a^+ \cup a^-) + \omega^+.
\end{equation}
(Remember that the lower case letters denote the logarithms of the sets written in the corresponding upper case.)
So in this case the problem of multiplicative tiling becomes exactly the problem of translational tiling of the real line
by the tile $\omega^+$ and with set of translates the multiset $a^+ \cup a^-$.
The structure of $\omega^+$ in this case has been completely characterized in \cites[Theorem 3]{lagarias1996tiling}.
\subsection{Non-symmetric tile}\label{sec:non-symmetric-tile}
Suppose now that we are in the case where $\omega^+ \nequiv \omega^-$ and, according to Theorem \ref{th:mult-structure},
$a^+, a^-$ are both periodic with the same period obeying \eqref{periodic-rational}.
We can thus write (after scaling)
\beql{a-shape}
a^+ = {\mathbb Z}+\frac{1}{L}\Set{0=\alpha_1^+, \alpha_2^+, \ldots , \alpha_m^+},\ \ \
a^- = {\mathbb Z}+\frac{1}{L}\Set{0=\alpha_1^-, \alpha_2^-, \ldots , \alpha_n^+},
\end{equation}
for some positive integer $L$, where $\alpha_j^\pm \in {\mathbb Z}_L = {\mathbb Z} / (L{\mathbb Z})$.
In order to express our tiling problem on the torus ${\mathbb T} = {\mathbb R} / {\mathbb Z}$ we identify
$\omega^\pm$ with the indicator function of the set that arises when taking their
projection mod 1.
(Because of the tiling assumption the points of each $\omega^\pm$ are different mod 1.)
Write
$$
\alpha^+ = \Set{\alpha_1^+, \alpha_2^+, \ldots, \alpha_m^+},\ \ \ \alpha^- = \Set{\alpha_1^-, \alpha_2^-, \ldots, \alpha_n^-},
$$
and also $\alpha^\pm$ for the collection of unit Dirac masses at the points of $\alpha^\pm$.
The tiling conditions \eqref{tilings} now become equivalently the tilings of the torus
\beql{mod-tilings}
1 = \frac{\alpha^+}{L}*\omega^+(t) + \frac{\alpha^-}{L}*\omega^-(t) = \frac{\alpha^-}{L}*\omega^+(t) + \frac{\alpha^+}{L}*\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
\end{equation}
Write for $x \in [0,1)$
$$
C_x = \Set{x+\frac{k}{L} \bmod 1:\ k=0,1,\ldots,L-1} \subseteq {\mathbb T}.
$$
Multiplying \eqref{mod-tilings} by ${\bf 1}_{C_x}(t)$ we get
$$
{\bf 1}_{C_x}(t) =
\frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^-(t) =
\frac{\alpha^-}{L}*{\bf 1}_{C_x}\omega^+(t) + \frac{\alpha^+}{L}*{\bf 1}_{C_x}\omega^-(t),\ \ \ \text{for $t \in {\mathbb T}$.}
$$
Restricting to $t \in C_x$ we can rewrite this as the two tiling conditions on ${\mathbb Z}_L$, valid for all $x \in [0, \frac{1}{L})$,
\begin{align}
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^+ \cup (\omega^-\cap C_x) + \alpha^-,\nonumber\\
{\mathbb Z}_L &= (\omega^+\cap C_x) + \alpha^- \cup (\omega^-\cap C_x) + \alpha^+\label{cycle-tilings},
\end{align}
where we are now identifying the sets $\omega^\pm\cap C_x$ with the obvious subset of ${\mathbb Z}_L$.
The sets $\omega^\pm \cap C_x$ can be chosen independently for all $x \in [0, 1/L)$ as tiling is not affected by
what happens on different cosets of $\frac{1}{L}{\mathbb Z}$.
We conclude that the sets $\omega^\pm$ are of the form
\begin{align}
\omega^+ &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^+_x \right)\nonumber\\
\omega^- &= \bigcup_{x \in [0,1/L)} \left(x + \frac{1}{L} b^-_x \right) \ \ \ \mbox{(disjoint unions)} \label{decomposition}
\end{align}
where $b^\pm_x \subseteq {\mathbb Z}_L$ are such that for each $x$ we have the two tilings
\begin{align}
{\mathbb Z}_L &= b^+_x + \alpha^+ \cup b^-_x + \alpha^-\nonumber\\
{\mathbb Z}_L &= b^+_x + \alpha^- \cup b^-_x + \alpha^+\label{cycle-tilings-final}
\end{align}
In the next section we are trying to understand better the kind of tiling described in \eqref{cycle-tilings-final}.
\section{Cross tiling}
\begin{definition}[Cross tiling]\ \\
Suppose $N>1$ is a positive integer and $A, B, X, Y \subseteq {\mathbb Z}_N$. We say that the pair $A, B$ admits
{\em cross tiling} with complements $X, Y$ if the following tilings hold:
\begin{align}
{\mathbb Z}_N = (A + X) \cup (B + Y)\nonumber\\
{\mathbb Z}_N = (A + Y) \cup (B + X) \label{cross-tiling}
\end{align}
A cross tiling \eqref{cross-tiling} is called {\em trivial} if
$(A \cup B) + X = {\mathbb Z}_N$ is a tiling and $X=Y$, or the same with the roles of
$A, B$ exchanged with those of $X, Y$.
(It is obvious that in this case conditions \eqref{cross-tiling} are indeed satisfied.)
\end{definition}
The following two examples are non-trivial cross tilings.
\begin{example}\label{ex:first-non-trivial}
Let $N=2ab$, with odd $a, b \in {\mathbb N}$ and view $G={\mathbb Z}_N$ as the cross-product ${\mathbb Z}_N = {\mathbb Z}_{ab}\times{\mathbb Z}_2$.
Define
\begin{align*}
A &= \Set{0, 1, 2, \ldots, a-1} \times \Set{0},\\
B &= \Set{0, 1} \cup \Set{a+2, a+3, \ldots, 2a-1} \times \Set{0},\\
X &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{0},\\
Y &= \Set{0, a, 2a, \ldots, (b-1)a} \times \Set{1}.
\end{align*}
It follows that
$$
A + X = B + X = {\mathbb Z}_{ab}\times\Set{0}\ \ \ \text{and}\ \ \ B + Y = A + Y = {\mathbb Z}_{ab}\times\Set{1}
$$
so \eqref{cross-tiling} follows. See Figure \ref{fig:easy-example}.
\end{example}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=5, b=3, N=2*a*b;
pair A[]; for(i=0; i<a; ++i) { A[i] = (i,0); }
pair B[]; B[0] = (0,0); B[1] = (1,0); for(j=2, i=a+2; i<2*a; ++i, ++j) { B[j] = (i,0); }
pair X[]; for(i=0; i<b; ++i) { X[i] = (i*a, 0); }
pair Y[]; for(i=0; i<b; ++i) { Y[i] = (i*a, 1); }
pair u=(0,4);
// Draw group
picture pg;
for(i=0; i<a*b; ++i) {
for(j=0; j<2; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
add(pg);
label("Group $G$ ", (0,.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A$ ", u+(0,.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B$ ", 2*u+(0,.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X$ ", 3*u+(0,.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets cross tiling in Example \ref{ex:first-non-trivial} with $a=5, b=3$} \label{fig:easy-example}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{asy}
size(10cm,0);
int i, j, a=15, b=4;
real F1[]={0,1,2}, F2[]={0,4,5}, C[]={0, 3, 6, 9, 12}, CC[]={3, 6, 9, 12};
pair A[]; for(i=0; i<F1.length; ++i) {A.push((F1[i], 0)); A.push((F1[i], 1)); }
pair B[]; for(i=0; i<F2.length; ++i) {B.push((F2[i], 0)); B.push((F2[i], 1)); }
pair X[]; for(i=0; i<C.length; ++i) {X.push((C[i], 0)); X.push((C[i], 2)); }
pair Y[]; Y.push((0,1)); Y.push((0,3)); for(i=0; i<CC.length; ++i) {Y.push((CC[i], 0)); Y.push((CC[i], 2)); }
pair u=(0,5);
// Draw group
picture pg;
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
draw(pg, (i,j), 3bp+black);
}
}
for(i=0; i<a; ++i) {
for(j=0; j<b; ++j) {
draw(pg, (i+0.5,j), 1bp+black);
draw(pg, (i+0.5,j+0.5), 1bp+black);
draw(pg, (i,j+0.5), 1bp+black);
}
}
add(pg);
label("Group $G$ ", (0,1.5), W);
// Sets
picture pa;
for(i=0; i<A.length; ++i) {
draw(pa, A[i], 5bp+red);
}
add(shift(u)*pg);
add(shift(u)*pa);
label("Set $A'$ ", u+(0,1.5), W);
picture pb;
for(i=0; i<B.length; ++i) {
draw(pb, B[i], 5bp+blue);
}
add(shift(2*u)*pg);
add(shift(2*u)*pb);
label("Set $B'$ ", 2*u+(0,1.5), W);
picture px;
for(i=0; i<X.length; ++i) {
draw(px, X[i], 5bp+green);
}
add(shift(3*u)*pg);
add(shift(3*u)*px);
label("Set $X'$ ", 3*u+(0,1.5), W);
picture py;
for(i=0; i<Y.length; ++i) {
draw(py, Y[i], 5bp+cyan);
}
add(shift(4*u)*pg);
add(shift(4*u)*py);
label("Set $Y'$ ", 4*u+(0,.5), W);
\end{asy}
\end{center}
\caption{The sets $A', B', X', Y' \subseteq H$ in Example \ref{ex:second-non-trivial} giving rise to the sets $A, B, X, Y \subseteq G$} \label{fig:better-example}
\end{figure}
\begin{example}\label{ex:second-non-trivial}
In Example \ref{ex:first-non-trivial} the sets $X$ and $Y$ are translates of each other, which is perhaps not very
satisfactory in terms of deviating from triviality.
Now we give an example where this does not happen.
We work in the group $G = {\mathbb Z}_{15}\times{\mathbb Z}_8 = {\mathbb Z}_{120}$
which contains the subgroup
$$
H = {\mathbb Z}_{15} \times \Set{0, 2, 4, 6}.
$$
Next define the subsets of ${\mathbb Z}_{15}$
$$
F_1 = \Set{0, 1, 2},\ \ \ F_2 = \Set{0, 4, 5}
$$
and notice that they both tile ${\mathbb Z}_{15}$ with the complement $\Set{0, 3, 6, 9, 12}$.
Define the subsets of $H$
\begin{align*}
A' &= F_1 \times \Set{0, 2},\\
B' &= F_2 \times \Set{0, 2},\\
X' &= \Set{0, 3, 6, 9, 12} \times \Set{0, 4},\\
Y' &= (\Set{0}\times\Set{1, 3}) \cup (\Set{3, 6, 9, 12}) \times \Set{0, 4}.
\end{align*}
See Figure \ref{fig:better-example} where the group $H$ is shown with thick dots
while its remaining three cosets in $G$ are shown in thin dots.
The difference between $X'$ and $Y'$ is that the first ``column'' of $Y'$ is ``raised'' by 1.
It is easy to verify that each of $A', B'$ tiles $H$ with each of $X', Y'$ as a tiling complement.
Finally define the subsets of $G$
\begin{align*}
A &= A', \\
B &= B', \\
X &= X', \\
Y &= Y'+(0,1),
\end{align*}
and observe that they do satisfy the cross-tiling conditions
\begin{align}
G &= (A+X) \cup (B+Y) \nonumber \\
G &= (A+Y) \cup (B+X). \label{ct}
\end{align}
The sets in parentheses in \eqref{ct} are tilings of each of the two $H$-cosets in $G$.
The sets with $+X$ tile $H$ and those with $+Y$ tile $H+(0,1)$.
None of the sets $A, B, X, Y$ are a translate of another.
\end{example}
\subsection{Fourier condition for cross tiling}
Translational tiling $A + X = {\mathbb Z}_N$ has a simple equivalent Fourier condition (we use the same letter for a set and its indicator function):
\beql{tiling-ft}
\ft{A}(0) \ft{X}(0) = N,\ \ \ \mbox{and}\ \ \ \forall k \in {\mathbb Z}_N\setminus\Set{0}: \ft{A}(k) \neq 0 \Longrightarrow \ft{X}(k) = 0.
\end{equation}
Adding and subtracting the cross-tiling defining conditions
$$
A*X+B*Y \equiv 1,\ \ \ A*Y+B*X \equiv 1,
$$
we obtain the equivalent conditions
\beql{cross-tiling-equiv}
(A+B)*(X+Y) \equiv 2,\ \ \ (A-B)*(X-Y) \equiv 0.
\end{equation}
Taking Fourier Transforms these conditions lead to the following equivalent Fourier condition for cross-tiling:
\begin{align}
&(\ft{A}(0) + \ft{B}(0)) (\ft{X}(0)+\ft{Y}(0)) = 2N,\nonumber\\
\forall k \in {\mathbb Z}_N\setminus\Set{0}:\ \ \ &\ft{A}(k) \neq -\ft{B}(k) \Longrightarrow \ft{X}(k) = -\ft{Y}(k). \label{cross-tiling-ft}\\
\forall k \in {\mathbb Z}_N:\ \ \ &\ft{A}(k) \neq \ft{B}(k) \Longrightarrow \ft{X}(k) = \ft{Y}(k).\nonumber
\end{align}
Using $k=0$ in the last set of equations we obtain that necessarily
$$
\Abs{A} = \Abs{B}\ \ \ \text{or}\ \ \ \Abs{X} = \Abs{Y}.
$$
\printbibliography
\end{document}
| {
"timestamp": "2017-10-10T02:15:15",
"yymm": "1710",
"arxiv_id": "1710.03108",
"language": "en",
"url": "https://arxiv.org/abs/1710.03108",
"abstract": "Suppose $\\Omega, A \\subseteq \\RR\\setminus\\Set{0}$ are two sets, both of mixed sign, that $\\Omega$ is Lebesgue measurable and $A$ is a discrete set. We study the problem of when $A \\cdot \\Omega$ is a (multiplicative) tiling of the real line, that is when almost every real number can be uniquely written as a product $a\\cdot \\omega$, with $a \\in A$, $\\omega \\in \\Omega$. We study both the structure of the set of multiples $A$ and the structure of the tile $\\Omega$. We prove strong results in both cases. These results are somewhat analogous to the known results about the structure of translational tiling of the real line. There is, however, an extra layer of complexity due to the presence of sign in the sets $A$ and $\\Omega$, which makes multiplicative tiling roughly equivalent to translational tiling on the larger group $\\ZZ_2 \\times \\RR$.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "The structure of multiplicative tilings of the real line",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846703886661,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.709201966231804
} |
https://arxiv.org/abs/1507.02268 | Optimal approximate matrix product in terms of stable rank | We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows. Here $\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future.Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15].We also separately point out that the proof of the "BSS" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\tilde{r}/\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10]. | \section{Introduction}\SectionName{intro}
Much recent work has successfully utilized randomized dimensionality reduction techniques to speed up solutions to linear algebra problems, with applications in machine learning, statistics, optimization, and several other domains; see the recent monographs \cite{HMT11,Mahoney11,Woodruff14} for more details. In our work here, we give new spectral norm guarantees for approximate matrix multiplication (AMM). Aside from AMM being interesting in its own right, it has become a useful primitive in the literature for analyzing algorithms for other large-scale linear algebra problems as well. We show applications of our new guarantees to speeding up standard algorithms for generalized regression and low-rank approximation problems. We also describe applications of our results to $k$-means clustering (discovered in \cite{CohenEMMP14}) and nonparametric regression \cite{YangPW15}.
In AMM we are given $A, B$ each with a large number of rows $n$, and the goal is to compute some matrix $C$ such that $\|C - A^T B\|_X$ is ``small'', for some norm $\|\cdot\|_X$. Furthermore, we would like to compute $C$ much faster than the usual time required to exactly compute $A^T B$.
Work on randomized methods for AMM began with \cite{DrineasKM06}, which focused on $\|\cdot\|_X = \|\cdot\|_F$, i.e.,\ Frobenius norm error. They showed by picking an appropriate sampling matrix $\Pi\in\mathbb{R}^{m\times n}$,
\begin{equation}
\|(\Pi A)^T(\Pi B) - A^T B\|_F \le \varepsilon\|A\|_F\|B\|_F \EquationName{frob-error}
\end{equation}
with good probability, if $m = \Omega(1/\varepsilon^2)$. By a {\em sampling matrix}, we mean the rows of $\Pi$ are independent, and each row is all zero except for a $1$ in a random location according to some appropriate distribution. If $A\in\mathbb{R}^{n\times d}$ and $B\in\mathbb{R}^{n\times p}$, note $(\Pi A)^T (\Pi B)$ can be computed in $O(mdp)$ time once $\Pi A$ and $\Pi B$ are formed, as opposed to the straightforward $O(ndp)$ time to compute $A^T B$.
The Frobenius norm error guarantee of \Equation{frob-error} was also later achieved in \cite[Lemma 6]{Sarlos06} via a different approach, with some later optimizations to the parameters in \cite[Theorem 6.2]{KaneN14}. The approach of Sarl\'{o}s was not via sampling, but rather to use a matrix $\Pi$ drawn from a distribution satisfying an ``oblivious Johnson-Lindenstrauss (JL)'' guarantee, i.e.\ a distribution $\mathcal{D}$ over $\mathbb{R}^{m\times n}$ satisfying the following condition for some $\varepsilon, \delta\in(0, 1/2)$:
\begin{equation}
\forall x\in\mathbb{R}^n,\ \Pr_{\Pi\sim\mathcal{D}}\left(|\|\Pi x\|_2^2 - \|x\|_2^2| > \varepsilon \|x\|_2^2\right) < \delta .
\end{equation}
Such a matrix $\Pi$ can be taken with $m = O(\varepsilon^{-2}\log(1/\delta))$ \cite{JL84}. Furthermore, one can take $\Pi$ to be a Fast JL transform \cite{AilonC09} (or any of the follow-up improvements \cite{AilonL13,KrahmerW11,NelsonPW14,Bourgain14,HavivR16}) or a sparse JL transform \cite{dks10,KaneN14} to speed up the computation of $\Pi A$ and $\Pi B$. One could also use the Thorup-Zhang sketch \cite{ThorupZ12} combined with a certain technique of \cite{LiangBKW14} (see \cite[Theorem 2.10]{Woodruff14} for details) to efficiently boost success probability.
Other than Frobenius norm error, the main other error guarantee investigated in previous work is spectral error. That is, we would like $\|C - A^T B\|$ to be small, where $\|M\|$ denotes the largest singular value of $M$. If one is interested in applying $A^T B$ to some set of input vectors then this type of error is the most meaningful, since $\|C - A^T B\|$ being small is equivalent to $\|Cx\| \approx \|A^T B x\|$ for any $x$. The first work along these lines was again by \cite{DrineasKM06}, who gave a procedure based on entry-wise sampling of the entries of $A$ and $B$. The works \cite{DrineasMM06,SpielmanS11} showed that row-sampling according to leverage scores also provides the desired guarantee with few samples.
Then \cite{Sarlos06}, combined with a quantitative improvement in \cite{ClarksonW13}, showed that one can take a $\Pi$ drawn from an oblivious JL distribution with $\delta = 2^{-\Theta(r)}$ where $r(\cdot)$ denotes rank and $r = r(A) + r(B)$. Then for $\Pi$ with $m = O((r+\log(1/\delta))/\varepsilon^2)$, with probability at least $1-\delta$ over $\Pi$,
\begin{equation}
\|(\Pi A)^T (\Pi B) - A^T B\| \le \varepsilon \|A\|\|B\| . \EquationName{op-error}
\end{equation}
As we shall see shortly via a very simple lemma (\Lemma{simple-mmult}), a sufficient deterministic condition implying \Equation{op-error} is that $\Pi$ is an $O(\varepsilon)$-{\em subspace embedding} for the $r$-dimensional subspace spanned by the columns of $A, B$. The notion of a subspace embedding was introduced by \cite{Sarlos06}.
\begin{definition}
$\Pi$ is an {\em $\varepsilon$-subspace embedding} for $U\in\mathbb{R}^{n\times r}$, $U^T U = I$, if $\Pi$ satisfies \Equation{op-error} with $A = B = U$, i.e. $\|(\Pi U)^T (\Pi U) - I\| \le \varepsilon$. This is equivalent to $\forall x\in \mathbb{R}^{r},\ (1-\varepsilon)\|x\|_2^2 \le \|\Pi Ux\|_2^2 \le (1+\varepsilon)\|x\|_2^2$, i.e.\ $\Pi$ preserves norms of all vectors in the subspace spanned by the columns $U$.
An {\em $(\varepsilon, \delta, r)$-oblivious subspace embedding (OSE)} is a distribution $\mathcal{D}$ over $\mathbb{R}^{m\times n}$ such that
$$
\forall U\in\mathbb{R}^{n\timesr},\ U^TU = I,\ \Pr_{\Pi\sim \mathcal{D}}(\|(\Pi U)^T (\Pi U) - I\| > \varepsilon) < \delta .
$$
\end{definition}
Fast subspace embeddings $\Pi$, i.e.\ such that the products $\Pi A$ and $\Pi B$ can be computed quickly, are known using variants on the Fast JL transform such as the Subsampled Randomized Hadamard Transform (SRHT) \cite{Sarlos06,LibertyWMRT07,Tropp11,LuDFU13} (also see a slightly improved analysis of the SRHT in \Section{srht}) or via sparse subspace embeddings \cite{ClarksonW13,MengM13,NelsonN13,LiMP13,CohenLMMPS15,Cohen16}. In most applications it is important to have a fast subspace embedding to shrink the time it takes to transform the input data to a lower-dimensional form. The SRHT is a construction of a $\Pi$ such that $\Pi A$ can be computed in time $O(nd\log n)$ (see \Section{srht} for details of the construction). The sparse subspace embedding constructions have some parameter $m$ rows and exactly $s$ non-zero entries per column, so that $\Pi A$ can be computed in time $O(s\cdot \mathop{nnz}(A))$, where $\mathop{nnz}(\cdot)$ is the number of non-zero entries, and there is a tradeoff in the upper bounds between $m$ and $s$.
An issue addressed by the work of \cite{MagenZ11} is that of robustness. As stated above, achieving \Equation{op-error} requires $\Pi$ be a subspace embedding for an $r$-dimensional subspace. However, consider the case when $A$ (and similarly for $B$) is of high rank but can be expressed as the sum of a low-rank matrix plus high-rank noise of small magnitude, i.e.,\ $A = \tilde{A} + E_A$ for $\tilde{A}$ of rank $r(\tilde{A}) \ll r$, and where $\|E_A\|$ is very small but $E_A$ has high (even full) rank. One would hope the noise could be ignored, but standard results require $\Pi$ to have a number of rows at least as large as $r$, regardless of how small the magnitude of the noise is. Another case of interest (as we will see in \Section{applications}) is when $A$ and $B$ are each of high rank, but their singular values decay at some appropriate rate. As discussed in \Section{applications}, in several applications where AMM is not the final goal but rather is used as a primitive in analyzing an algorithm for some other problem (such as $k$-means clustering or nonparametric regression), the matrices that arise do indeed have such decaying singular values.
The work \cite{MagenZ11} remedied this by considering the {\em stable ranks} $\tilde{r}(A), \tilde{r}(B)$ of $A$ and $B$. Define $\tilde{r}(A) = \|A\|_F^2 / \|A\|^2$. Note $\tilde{r}(A) \le r(A)$ always, but can be much less if $A$ has a small tail of singular values. Let $\tilde{r}$ denote $\tilde{r}(A) + \tilde{r}(B)$. Among other results, \cite{MagenZ11} showed that to achieve \Equation{op-error} with good probability, one can take $\Pi$ to be a random (scaled) sign matrix with either $m = \Omega(\tilde{r} / \varepsilon^4)$ or $m = \Omega(\tilde{r}\log(d+p)/\varepsilon^2)$ rows. As noted in follow-up work \cite{KyrillidisVZ14}, both the $1/\varepsilon^4$ dependence and the $\log(d+p)$ factor are undesirable. In their data-driven low dimensional embedding application, they wanted a dimension $m$ independent of the original dimensions, which are assumed much larger than the stable rank, and also wanted lower dependence on $1/\varepsilon$. To this end, \cite{KyrillidisVZ14} defined the {\em nuclear rank} as $\tilde{nr}(A) = \|A\|_* / \|A\|$ and showed $m = \Omega(\tilde{nr}/\varepsilon^2)$ rows suffice for $\tilde{nr} = \tilde{nr}(A) + \tilde{nr}(B)$. Here $\|A\|_*$ is the nuclear norm, i.e.,\ sum of singular values of $A$. Since $\|A\|_F^2$ is the sum of squared singular values, it is straightforward to see that $\tilde{nr}(A) \ge \tilde{r}(A)$ always. Thus there is a tradeoff: the stable rank guarantee is worsened to nuclear rank, but dependence on $1/\varepsilon$ is improved to quadratic.
We show switching to the weaker $\tilde{nr}$ guarantee is unnecessary by showing quadratic dependence on $1/\varepsilon$ holds even with stable rank. This answers the main open question of \cite{MagenZ11,KyrillidisVZ14}.
To state our results in a more natural way, we rephrase our main result to say that we achieve
\begin{equation}
\|(\Pi A)^T (\Pi B) - A^T B\| \le \varepsilon \sqrt{\left(\|A\|^2 + \frac{\|A\|_F^2}k\right)\left(\|B\|^2 + \frac{\|B\|_F^2}k\right)}. \EquationName{mmult-general}
\end{equation}
for an arbitrary $k\ge 1$, and we do so by using subspace embeddings for $O(k)$-dimensional subspaces in a certain black box way (which will be made precise soon) regardless of the ranks of $A, B$.
\begin{remark}\RemarkName{equiv}
\textup{
Note that our previously stated main contribution is equivalent, since one could set $k = \tilde{r}(A) + \tilde{r}(B)$ to arrive at the conclusion that subspace embeddings for $O(\tilde{r})$-dimensional subspaces yield the guarantee in \Equation{op-error}. Alternatively one could obtain the \Equation{mmult-general} guarantee via \Equation{op-error} with error parameter $\varepsilon' = \Theta(\varepsilon\cdot \min\{1, \sqrt{(\tilde{r}(A)\cdot \tilde{r}(B))/k}\})$.
}
\end{remark}
Henceforth, we use the following definition.
\begin{definition}
For conforming matrices $A^T, B$, we say $\Pi$ satisfies the {\em $(k, \varepsilon)$-approximate spectral norm matrix multiplication property ($(k,\varepsilon)$-AMM) for $A, B$} if \Equation{mmult-general} holds. If $\Pi$ is random and satisfies $(k,\varepsilon)$-AMM with probability $1-\delta$ for any fixed $A, B$, then we say $\Pi$ satisfies $(k,\varepsilon,\delta)$-AMM.
\end{definition}
\paragraph{Our main contribution:} We give two different characterizations for $\Pi$ supporting $(k,\varepsilon)$-AMM, both of which imply $(k,\varepsilon,\delta)$-AMM $\Pi$ having $m = O((k+\log(1/\delta))/\varepsilon^2)$ rows. The first characterization applies to any OSE distribution for which a moment bound has been proven for $\|(\Pi U)^T(\Pi U) - I\|$ (which is true for the best analyses of all known OSE's). In this case, we show a black box theorem: any $(\varepsilon, \delta, 2k$)-OSE provides $(k,\varepsilon,\delta)$-AMM. Since matrices with subgaussian entries and $m = \Omega((k+\log(1/\delta))/\varepsilon^2)$ are $(\varepsilon, \delta, 2k)$-OSE's, our originally stated main result follows. This result is optimal, since \cite{nn14} shows any randomized distribution over $\Pi$ with $m$ rows having the $(k,\varepsilon, \delta)$-AMM property must have $m = \Omega((k + \log(1/\delta))/\varepsilon^2)$ (the hard instance there is when $A = B = U$ has orthonormal columns, and thus rank and stable rank are equal).
Our second characterization identifies certain deterministic conditions which, if satisfied by $\Pi$, imply the desired $(k,\varepsilon)$-AMM property. These conditions are of the form: (1) $\Pi$ should preserve a certain set of $O(\log(1/\varepsilon))$ different subspaces of varying dimensions (all depending on $k, \varepsilon$ and not on the ranks of $A, B$) with varying distortions, and (2) for a certain two matrices in our analysis, left-multiplication by $\Pi$ should not increase their operator norms by more than an $O(1)$ factor. These conditions are chosen carefully so that matrices with subgaussian entries and $m = \Omega(k/\varepsilon^2)$ satisfy all conditions simultaneously with high probability, again thus proving our main result while also suggesting that the conditions we have identified are the ``right'' ones.
Due to the black box reliance on the subspace embedding primitive in our proofs, $\Pi$ need not only be a subgaussian map. Thus not only do we improve on $m$ compared with previous work, but also in terms of the general class of $\Pi$ our result applies to. For example given our first characterization, not only does it suffice to use a random sign matrix with $\Omega(k/\varepsilon^2)$ rows, but in fact one can apply our theorem to more efficient subspace embeddings such as the SRHT or sparse subspace embeddings, or even constructions discovered in the future. That is, one can automatically transfer bounds proven for the subspace embedding property to the $(k,\varepsilon)$-AMM property. Thus, for example, the best known SRHT analysis (in our appendix, see \Theorem{best-srht}) implies $(k,\varepsilon, \delta)$-AMM for $m = \Omega((k + \log(1/(\varepsilon\delta))\log(k/\delta))/\varepsilon^2)$ rows. For sparse subspace embeddings, the analysis in \cite{Cohen16} implies $m = \Omega(k\log(k/\delta)/\varepsilon^2)$ suffices with $s = O(\log(k/\delta)/\varepsilon)$ non-zeroes per column of $\Pi$. The only reason for the $\log k$ loss in $m$ for these particular distributions is not due to our theorems, but rather due to the best analyses for the simpler {\em subspace embedding} property in previous work already incurring the extra $\log k$ factor (note being a subspace embedding for a $k$-dimensional subspace is simply a special case of $(k,\varepsilon)$-AMM where $A = B = U$ has $k$ orthonormal columns). In the case of the SRHT, this extra $\log k$ factor is actually necessary \cite{Tropp11}; for sparse subspace embeddings, it is conjectured that the $\log k$ factor can be removed and that $m = \Omega((k+\log(1/\delta))/\varepsilon^2)$ actually suffices to obtain an OSE \cite[Conjecture 14]{NelsonN13}. We also discuss in \Remark{efficiency} that one can set $\Pi$ to be $\Pi_1\cdot \Pi_2$ where $\Pi_1$ has subgaussian entries with $O(k/\varepsilon^2)$ rows, and $\Pi_2$ is some other fast OSE (such as the SRHT or sparse subspace embedding), and thus one could obtain the best of both worlds: (1) $\Pi$ has $O(k/\varepsilon^2)$ rows, and (2) can be applied to any $A\in\mathbb{R}^{n\times d}$ in time $T + O(km'd/\varepsilon^2)$, where $T$ is the (fast) time to apply $\Pi_2$ to $A$, and $m'$ is the number of rows of $\Pi_2$. For example, by appropriate composition as discussed in \Remark{efficiency}, $\Pi$ can have $O(k/\varepsilon^2)$ rows and support multiplying $\Pi A$ for $A\in\mathbb{R}^{n\times d}$ in time $O(\mathop{nnz(A)}) + \tilde{O}(\varepsilon^{-O(1)}(k^3 + k^2d))$.
We also observe the proof of the main result of \cite{BatsonSS12} can be modified to show that given any $A, B$ each with $n$ rows, and given any $\varepsilon\in(0,1/2)$, there exists a diagonal matrix $\Pi\in\mathbb{R}^{n\times n}$ with $O(k/\varepsilon^2)$ non-zero entries, and that can be computed by a deterministic polynomial time algorithm, achieving $(k,\varepsilon)$-AMM. The original work of \cite{BatsonSS12} achieved \Equation{op-error} with $m = O(r/\varepsilon^2)$ for $r$ being the sum of ranks of $A, B$. The work \cite{BatsonSS12} stated their result for the case $A = B$, but the general case of potentially unequal matrices reduces to this case; see \Section{bss}. Our observation also turns out to yield a stronger form of \cite[Theorem 3.3]{KollaMST10}; also see \Section{bss}.
\bigskip
As mentioned, aside from AMM being interesting on its own, it is a useful primitive widely used in analyses of algorithms for several other problems, including $k$-means clustering \cite{BoutsidisZMD15,CohenEMMP14}, nonparametric regression \cite{YangPW15}, linear least squares regression and low-rank approximation \cite{Sarlos06}, approximating leverage scores \cite{DrineasMMW12}, and several other problems (see \cite{Woodruff14} for a recent summary). For all these, analyses of correctness for algorithms based on dimensionality reduction via some $\Pi$ rely on $\Pi$ satisfying AMM for certain matrices in the analysis.
After making certain quantitative improvements to connections between AMM and applications, and combining them with our main result, in \Section{applications} we obtain the following new results.
\begin{enumerate}
\item \textbf{Generalized regression:} Given $A\in\mathbb{R}^{n\times d}$ and $B\in\mathbb{R}^{n\times p}$, consider the problem of computing $X^* = \mathop{argmin}_{X\in\mathbb{R}^{d\times p}} \|AX - B\|$. It is standard that $X^* = (A^T A)^+ A^T B$ where $(\cdot)^+$ is the Moore-Penrose pseudoinverse. The bottleneck here is computing $A^T A$, taking $O(nd^2)$ time. A popular approach is to instead compute $\tilde{X} = ((\Pi A)^T(\Pi A))^+ (\Pi A)^T \Pi B$, i.e.,\ the minimizer of $\|\Pi A X - \Pi B\|$. Note that computing $(\Pi A)^T (\Pi A)$ (given $\Pi A$) only takes a smaller $O(md^2)$ amount of time. We show that if $\Pi$ satisfies $(k, O(\sqrt{\varepsilon}))$-AMM for $U_A, P_{\bar{A}} B$, and is also an $O(1)$-subspace embedding for a certain $r(A)$-dimensional subspace (see \Theorem{gen-regression}), then
$$
\| A \tilde X - B \|^2 \leq (1 + \varepsilon) \| P_A B - B \|^2 + (\varepsilon / k) \| P_A B - B \|_F^2
$$
where $P_A$ is the orthogonal projection onto the column space of $A$, $P_{\bar{A}} = I - P_A$, and $U_A$ has orthonormal columns forming a basis for the column space of $A$. The punchline is that if the regression error $P_{\bar{A}} B$ has high actual rank but stable rank only on the order of $r(A)$, then we obtain multiplicative spectral norm error with $\Pi$ having fewer rows. Generalized regression is a natural extension
of the case when $B$ is a vector,
and arises for example
in Regularized Least Squares Classification, where one has multiple (non-binary) labels, and for
each label one creates a column of $B$; see e.g.\ \cite{CLLLH10} for this and variations.
\item \textbf{Low-rank approximation:} We are given $A\in\mathbb{R}^{n\times d}$ and integer $k\ge 1$, and we want to compute $A_k = \mathop{argmin}_{r(X) \le k} \|A - X\|$. The Eckart-Young theorem implies $A_k$ is obtained by truncating the SVD of $A$ to the top $k$ singular vectors. The standard way to use dimensionality reduction for speedup, introduced in \cite{Sarlos06}, is to let $S = \Pi A$ then compute $\tilde{A} = A P_S$. Then return $\tilde{A}_k$, the best rank-$k$ approximation of $\tilde{A}$, instead of $A_k$ (it is known $\tilde{A}_k$ can be computed more efficiently than $A_k$; see \cite[Lemma 4.3]{ClarksonW09}). We show if $\Pi$ satisfies $(k, O(\sqrt{\varepsilon}))$-AMM for $U_k$ and $A-A_k$, and is a $(1/2)$-subspace embedding for the column space of $A_k$, then
$$
\|\tilde{A}_k - A\|^2 \le (1+\varepsilon)\|A - A_k\|^2 + (\varepsilon/k)\|A - A_k\|_F^2 .
$$
The punchline is that if the stable rank of the tail $A - A_k$ is on the same order as the rank parameter $k$, then standard algorithms from previous work for Frobenius multiplicative error actually in fact also provide {\em spectral} multiplicative error. This property indeed holds for any $k$ for popular kernel matrices in machine learning such as the gaussian and Sobolev kernels (see \cite{ReyhaniHV11} and Examples 2 and 3 of \cite{YangPW15}), and low-rank approximation of kernel matrices has been applied to several machine learning problems; see \cite{GittensM13} for a discussion.
\end{enumerate}
We also explain in \Section{applications} how our result has already been applied in recent work on dimensionality reduction for $k$-means clustering \cite{CohenLMMPS15}, and how it generalizes results in \cite{YangPW15} on dimensionality reduction for nonparametric regression to use a larger class of embeddings $\Pi$.
\subsection{Preliminaries and notation}
We frequently use the singular value decomposition (SVD). For a matrix $A\in\mathbb{R}^{n\times d}$ of rank $r$, consider the compact SVD $A = U_A \Sigma_A V_A^T$ where $U_A \in \mathbb{R}^{n\times r}$ and $V_A\in\mathbb{R}^{d\times r}$ each have orthonormal columns, and $\Sigma_A$ is diagonal with strictly positive diagonal entries (the singular values of $A$). We assume $(\Sigma_A)_{i,i} \ge (\Sigma_A)_{j,j}$ for $i< j$. We let $P_A = U_AU_A^T$ denote the orthogonal projection operator onto the column space of $A$. We use $\mathrm{span}(A)$ to refer to the subspace spanned by $A$'s columns.
Often for a matrix $A$ we write $A_k$ as the best rank-$k$ approximation to $A$ under Frobenius or spectral error (obtained by writing the SVD of $A$ then setting all $(\Sigma_A)_{i,i}$ to $0$ for $i>k$). We often denote $A - A_k$ as $A_{\bar{k}}$. For matrices with orthonormal columns, such as $U_A$, $(U_A)_k$ denotes the $n\times k$ matrix formed by removing all but the first $k$ columns of $U$. When $A$ is understood from context, we often write $U \Sigma V^T$ instead of $U_A \Sigma_A V_A^T$, and $U_k$ to denote $(U_A)_k$ (and $\Sigma_k$ for $(\Sigma_A)_k$, etc.).
\section{Analysis of matrix multiplication for stable rank}\SectionName{analysis}
First we record a simple lemma relating subspace embeddings and AMM.
\begin{lemma}\LemmaName{simple-mmult}
Let $E = \mathrm{span}\{A, B\}$, and let $\Pi$ be an $\varepsilon$-subspace embedding for $E$. Then \Equation{op-error} holds.
\end{lemma}
\begin{proof}
First, without loss of generality we may assume $\|A\| = \|B\| = 1$ since we can divide both sides of
\Equation{op-error} by $\|A\| \cdot \|B\|$. Let $U$ be a matrix whose columns form an orthonormal basis for $E$. Then note for any $x, y$ we can write $Ax = Uw, By = Uz$ where $\|w\| \le \|x\|, \|z\| \le \|y\|$. Then
\allowdisplaybreaks
\begin{align*}
\|(\Pi A)^T (\Pi B) - A^T B\| &= \sup_{\|x\|=\|y\|=1} |\inprod{\Pi A x, \Pi By} - \inprod{Ax, By}|\\
{}&= \sup_{\|w\|, \|z\| \le 1} |\inprod{\Pi U z, \Pi U w} - \inprod{Uz, Uw}|\\
{}&= \|(\Pi U)^T (\Pi U) - I\|\\
{}& < \varepsilon
\end{align*}
\end{proof}
\Lemma{simple-mmult} implies that if $A, B$ each have rank at most $r$, it suffices for $\Pi$ to have $\Omega(r/\varepsilon^2)$ rows.
In the following two subsections, we give two different characterizations for $\Pi$ to provide $(k,\varepsilon)$-AMM, both only requiring $\Pi$ to have $\Omega(k/\varepsilon^2)$ rows, independent of $r$.
\subsection{Characterization for $(k,\varepsilon,\delta)$-AMM via a moment property}\SectionName{moments}
Here we provide a way to obtain $(k,\varepsilon)$-AMM for any $\Pi$ whose subspace embedding property has been established using the moment method, e.g.\ sparse subspace embeddings \cite{MengM13,NelsonN13,Cohen16}, dense subgaussian matrices as analyzed in \Section{subgaussian-ose-moments}, or even the SRHT as analyzed in \Section{srht}. Our approach in this subsection is inspired by the introduction of the ``JL-moment property'' in \cite{KaneN14} to analyze approximate matrix multiplication with Frobenius error. The following is a generalization of \cite[Definition 6.1]{KaneN14}, which was only concerned with $d=1$.
\begin{definition}
A distribution $\mathcal{D}$ over $\mathbb{R}^{m\times n}$ has {\em $(\varepsilon,\delta,d,\ell)$-OSE moments} if for all matrices $U\in\mathbb{R}^{n\times d}$ with orthonormal columns,
$$
\E_{\Pi\sim\mathcal{D}} \left\|(\Pi U)^T(\Pi U) - I\right\|^\ell < \varepsilon^\ell \cdot \delta
$$
\end{definition}
Note that this is just a special case of bounding the expectation of an arbitrary function of $\|(\Pi U)^T(\Pi U) - I\|$. The arguments below will actually apply to any nonnegative, convex, increasing function of $\|(\Pi U)^T(\Pi U) - I\|^2$, but we restrict to moments for simplicity of presentation. The acronym ``OSE'' refers to {\em oblivious subspace embedding}, a term coined in \cite{NelsonN13} to refer to distributions over $\Pi$ yielding a subspace embedding for any fixed subspace of a particular bounded dimension with high probability. We start with a simple lemma.
\begin{lemma}\LemmaName{matmult}
Suppose $\mathcal{D}$ satisfies the $(\varepsilon,\delta,2d,\ell)$-OSE moment property and $A, B$ are matrices with (1) the same number of rows, and (2) sum of ranks at most $2d$. Then
$$
\E_{\Pi\sim\mathcal{D}} \left\|(\Pi A)^T(\Pi B) - A^T B\right\|^\ell < \varepsilon^\ell \|A\|^\ell \|B\|^\ell \cdot \delta
$$
\end{lemma}
\begin{proof}
First, we apply \Lemma{simple-mmult} to $A$ and $B$, where $U$ forms an orthonormal basis for the subspace $\mathrm{span}\{\mathrm{columns}(A),\mathrm{columns}(B)\}$, showing that
$$
\left\|(\Pi A)^T(\Pi B) - A^T B\right\| \le \left\|(\Pi U)^T(\Pi U) - I\right\| \|A\| \|B\| .
$$
Therefore
$$
\E_{\Pi\sim\mathcal{D}} \left\|(\Pi A)^T(\Pi B) - A^T B\right\|^\ell \le \E_{\Pi\sim\mathcal{D}} \left\|(\Pi U)^T(\Pi U) - I\right\|^\ell \|A\|^\ell \|B\|^\ell < \varepsilon^\ell \|A\|^\ell \|B\|^\ell \cdot \delta
$$
\end{proof}
Then, just as \cite[Theorem 6.2]{KaneN14} showed that having OSE moments with $d=1$ implies approximate matrix multiplication with Frobenius norm error, here we show that having OSE moments for larger $d$ implies approximate matrix multiplication with operator norm error.
\begin{theorem}\TheoremName{main2}
Given $k,\varepsilon,\delta \in (0, 1/2)$, let $\mathcal{D}$ be any distribution over matrices with $n$ columns with the $(\varepsilon,\delta,2k,\ell)$-OSE moment property for some $\ell\ge 2$. Then, for any $A,B$,
\begin{equation}
\Pr_{\Pi\sim\mathcal{D}} \left ( \|(\Pi A)^T(\Pi B) - A^T B\| > \varepsilon \sqrt{(\|A\|^2 + \|A\|_F^2 / k) (\|B\|^2 + \|B\|^2_F / k)} \right ) < \delta \EquationName{moment-thm}
\end{equation}
\end{theorem}
\begin{proof}
We can assume $A, B$ each have orthogonal columns. This is since, via the full SVD, there exist orthogonal matrices $R_A, R_B$ such that $A R_A$ and $B R_B$ each have orthogonal columns. Since neither left nor right multiplication by an orthogonal matrix changes operator norm,
$$
\|(\Pi A)^T(\Pi B) - A^T B\| = \|(\Pi A R_A)^T(\Pi B R_B) - (A R_A)^T B R_B\| .
$$
Thus, we replace $A$ by $A R_A$ and similarly for $B$. We may also assume the columns $a_1, a_2, \ldots$ of $A$ are sorted so that $\|a_i\|_2 \ge \|a_{i+1}\|_2$ for all $i$. Henceforth we assume $A$ has orthogonal columns in this sorted order (and similarly for $B$, with columns $b_i$). Now, treat $A$ as a block matrix in which the columns are blocked into groups of size $k$, and similarly for $B$ (if the number of columns of either $A$ or $B$ is not divisible by $k$, then pad them with all-zero columns until they are, which does not affect the claim). Let the spectral norm of the $i$th block of $A$ be $s_i = \|a_{(i-1)\cdot k + 1}\|_2$, and for $B$ denote the spectral norm of the $i$th block as $t_i = \|b_{(i-1)\cdot k + 1}\|_2$. These equalities for $A, B$ hold since their columns are orthogonal and sorted by norm. We claim $\sum_i s_i^2 \le \|A\|^2 + \|A\|_F^2 / k$ (and similarly for $\sum_i t_i^2$). To see this, let the blocks of $A$ be $A'_1, \ldots, A'_q$ where $s_i = \|A'_i\|$. Note $s_1^2 = \|A'_1\| \le \|A\|$. Also, for $i>1$ we have
$$
s_i^2 = \|a_{(i-1)\cdot k + 1}\|_2^2\le \frac 1{k} \sum_{(i-2)\cdot k + 1 \le j \le (i-1)\cdot k} \|a_j\|_2^2 = \frac 1{k} \|A'_{i-1}\|_F^2 .
$$
Thus
$$
\sum_{i>1} s_i^2 \le \|A\|_F^2 / k.
$$
Define $C = (\Pi A)^T(\Pi B) - A^T B$. Let $v_{\{ i \}}$ denote the $i$th block of a vector $v$ (the $k$-dimensional vector whose entries consist of entries $(i-1)\cdot k + 1$ to $i\cdot k$ of $v$), and $C_{\{i\},\{j\}}$ the $(i,j)$th block of $C$, a $k\times k$ matrix (the entries in $C$ contained in the $i$th block of rows and $j$th block of columns).
Now, $\|C\| = \sup_{\|x\|=\|y\|=1} x^T C y$. For any such vectors $x$ and $y$, we define new vectors $x'$ and $y'$ whose coordinates correspond to entire blocks: we let $x'_i = \| x_{\{ i \}} \|$, with $y'$ defined analogously. We similarly define $C'$ with entries corresponding to blocks of $C$, where $C'_{i,j} = \| C_{\{i\},\{j\}} \|$. Then $x^T C y \le x'^T C' y'$, simply by bounding the contribution of each block. Thus it suffices to upper bound $\|C'\|$, which we bound by its Frobenius norm $\|C'\|_F$. Now, recalling for a random variable $X$ that $\|X\|_\ell$ denotes $(\E|X|^\ell)^{1/\ell}$ and using Minkowski's inequality (that $\|\cdot\|_\ell$ is a norm for $\ell \ge 1$),
\allowdisplaybreaks
\begin{align*}
\|\|C'\|_F^2\|_{\ell/2} &= \left \| \sum_{i,j} \|(\Pi A'_i)^T(\Pi B'_j) - A_i'^T B'_j\|^2 \right \|_{\ell/2} \\
{}&\le \sum_{i,j} \|\|(\Pi A'_i)^T(\Pi B'_j) - A_i'^T B'_j\|^2 \|_{\ell/2}\\
{}&\le \sum_{i,j} \varepsilon^2 s_i^2 t_j^2\cdot \delta^{2/\ell} \text{ (\Lemma{matmult})}\\
{}&= \varepsilon^2 \left ( \sum_i s_i^2 \right ) \cdot \left ( \sum_j t_j^2 \right ) \delta^{2/\ell} \\
{}&\le \left ( \varepsilon \sqrt{(\|A\|^2 + \|A\|_F^2 / k) (\|B\|^2 + \|B\|^2_F / k)} \delta^{1/\ell} \right )^2
\end{align*}
Now, $\E \|C'\|_F^\ell = \| \|C'\|_F^2 \|_{\ell/2}^{\ell/2}$, implying
\begin{align*}
\Pr \left ( \|C'\| > \varepsilon \sqrt{(\|A\|^2 + \frac{\|A\|_F^2}k) (\|B\|^2 + \frac{\|B\|^2_F}k)} \right ) &\le \Pr \left ( \|C'\|_F > \varepsilon \sqrt{(\|A\|^2 + \frac{\|A\|_F^2}k) (\|B\|^2 + \frac{\|B\|^2_F} k)} \right ) \\
{}&< \frac {\E\|C'\|_F^\ell}{\left ( \varepsilon \sqrt{(\|A\|^2 + \frac{\|A\|_F^2}k) (\|B\|^2 + \frac{\|B\|^2_F}k)} \right)^\ell}\\
{}&\le \delta .
\end{align*}
\end{proof}
We now discuss the implications of applying \Theorem{main2} to specific OSE's.
\paragraph{Subgaussian maps:} In \Section{subgaussian-ose-moments} we show that if $\Pi$ has independent subgaussian entries and $m = \Omega((k+\log(1/\delta))/\varepsilon^2)$ rows, then it satisfies the $(\varepsilon,\delta,2k,\Theta(k + \log(1/\delta)))$ OSE moment property. Thus \Theorem{main2} applies to show that such $\Pi$ will satisfy $(k,\varepsilon,\delta)$-AMM.
\paragraph{SRHT:} The SRHT is the matrix product $\Pi = SHD$ where $D\in\mathbb{R}^{n\times n}$ is $n\times n$ diagonal with independent $\pm 1$ entries on the diagonal, $H$ is a ``bounded orthonormal system'' (i.e.\ an orthogonal matrix in $\mathbb{R}^{n\times n}$ with $\max_{i,j} |H_{i,j}| = O(1/\sqrt{n})$), and the $m$ rows of $S$ are independent and each samples a uniformly random element of $[n]$. Bounded orthonormal systems include the discrete Fourier matrix and the Hadamard matrix; thus such $\Pi$ exist supporting matrix-vector multiplication in $O(n\log n)$ time. Thus when computing $\Pi A$ for some $n\times d$ matrix $A$, this takes time $O(nd\log n)$ (by applying $\Pi$ to $A$ column by column). In \Theorem{best-srht} we show that the SRHT with $m = \Omega((k + \log(1/(\varepsilon\delta))\log(k/\delta))/\varepsilon^2)$ satisfies the $(\varepsilon, \delta, 2k, \log(k/\delta))$-OSE moment property, and thus provides $(k,\varepsilon,\delta)$-AMM. Interestingly our analysis of the SRHT in \Section{srht} seems to be asymptotically tighter than any other analyses in previous work even for the basic subspace embedding property, and even slightly improves the by now standard analysis of the Fast JL transform given in \cite{AilonC09}.
\paragraph{Sparse subspace embeddings:} The sparse embedding distribution with parameters $m, s$ is as follows \cite{ClarksonW13,NelsonN13,KaneN14}. The matrix $\Pi$ has $m$ rows and $n$ columns. The columns are independent, and for each column exactly $s$ uniformly random entries are chosen without replacement and set to $\pm 1/\sqrt{s}$ independently; other entries in that column are set to zero. Alternatively, one could use the CountSketch \cite{CharikarCF04}: the $m$ rows are equipartitioned into $s$ sets of size $m/s$ each. The columns are independent, and in each column we pick exactly one row from each of the $s$ partitions and set the corresponding entry in that column to $\pm 1/\sqrt{s}$ uniformly; the rest of the entries in the column are set to $0$. Note $\Pi A$ can be multiplied in time $O(s\cdot \mathop{nnz}(A))$, and thus small $s$ is desirable.
It was shown in \cite{MengM13,NelsonN13}, slightly improving \cite{ClarksonW13}, that either of the above distributions satisfies the $(\varepsilon,\delta,k, 2)$-OSE moment property for $m = \Omega(k^2/(\varepsilon^2 \delta))$, $s = 1$, and hence $(k,\varepsilon,\delta)$-AMM (though this particular conclusion follows easily from \cite[Theorem 6.2]{KaneN14}). It was also shown in \cite{Cohen16}, improving upon \cite{NelsonN13}, that they satisfy the $(\varepsilon, \delta, k, \log(k/\delta))$-OSE moment property, and hence also $(k,\varepsilon,\delta)$-AMM, for $m = \Omega(B k\log(k/\delta)/\varepsilon^2), s = \Omega(\log_B(k/\delta)/\varepsilon)$ for any $B > 2$. It is conjectured that for $B = O(1)$, $m = \Omega((k + \log(1/\delta))/\varepsilon^2)$ should suffice \cite[Conjecture 14]{NelsonN13}.
\begin{remark}\RemarkName{cohen-osnap}
\textup{
The work \cite{Cohen16} does not explicitly discuss the OSE moment property for sparse subspace embeddings. Rather, \cite{Cohen16} bounds $\E e^{E 2\ell/\varepsilon} = O(k)$ for $E = \|(\Pi U)^T(\Pi U) - I\|$ and $\ell = \log(k/\delta)$. Note though for $x\ge 0$ and integer $\ell \ge 1$, $x^\ell \le \ell!\cdot e^x\le \ell^\ell\cdot e^x$ by Taylor expansion of the exponential. Setting $x = 2\ell E/\varepsilon$, \cite{Cohen16} thus implies $\E (2\ell E/\varepsilon)^\ell \le \ell^\ell \cdot \E e^{E 2\ell/\varepsilon} = O(k)$. Thus $\E E^\ell = O(k)\cdot (\varepsilon/2)^\ell < \delta$ by choice of $\ell$, which is the $(\varepsilon, \delta, k, \log(k/\delta))$-OSE moment property.
}
\end{remark}
\begin{remark}\RemarkName{efficiency}
\textup{
Currently there appears to be a tradeoff: one can either use $\Pi$ such that $\Pi A$ can be computed quickly, such as sparse subspace embeddings or the SRHT, but then the number of rows $m$ is at least $k\log k$. Alternatively one could achieve the optimal $m = O(k/\varepsilon^2)$ using subgaussian $\Pi$, but then multiplying by $\Pi$ is slower: $O(mnd)$ time for $A\in\mathbb{R}^{n\times d}$. However, settling for a tradeoff is unnecessary. One can actually obtain the ``best of both worlds'' by composition, i.e.\ the multiplication $\Pi = \Pi_1\cdot \Pi_2$ of two matrices both supporting AMM. Thus $\Pi_2$ could be a fast matrix providing AMM to low (but suboptimal) dimension, and $\Pi_1$ a ``slow'' (e.g.\ subgaussian) matrix with the optimal $O(k/\varepsilon^2)$ number of rows. In fact one can even set $\Pi = \Pi_1\Pi_2\Pi_3$ where $\Pi_3$ is the sparse subspace embedding with $O(k^2/\varepsilon^2)$ rows and $s = 1$, $\Pi_2$ is the SRHT, and $\Pi_1$ is a subgaussian matrix. Then $\Pi A$ will have the desired $O(k/\varepsilon^2)$ rows and can be computed in time $O(\mathop{nnz(A)}) + \tilde{O}(\varepsilon^{-O(1)}(k^3 + k^2d))$; see \Section{composition} for justification.
}
\end{remark}
\subsection{Characterization for $(k,\varepsilon,)$-AMM via deterministic events}\SectionName{conditioning}
Here we provide a different characterization for achieving $(k,\varepsilon)$-AMM. Without loss of generality we assume $\max\{\|A\|^2, \|A\|_F^2 / k\} = \max\{\|B\|^2, \|B\|_F^2 / k\} = 1$ (so $\|A\|^2, \|B\|^2 \le 1$ and $\|A\|_F^2, \|B\|_F^2 \le k$).
Let $w,w'$ each be minimal such that $\|A_{\bar{w}}\|, \|B_{\bar{w'}}\| \le \varepsilon / C'$ for some sufficiently large constant $C'$ (which will be set in the proof of \Theorem{main}). It was shown that $w,w' = O(k/\varepsilon^2)$ in the proof of Theorem 3.2 (i.b) in \cite{MagenZ11}. Write the SVDs $A_w = U_{A_w} \Sigma_{A_w} V_{A_w}^T$, $B_{w'} = U_{B_{w'}} \Sigma_{B_{w'}} V_{B_{w'}}^T$.
For $0\le i\le \log_2(1/\varepsilon^2)$ define $D_i'$ as set of all columns of $U_{A_w},U_{B_{w'}}$ whose corresponding squared singular values (from $\Sigma_{A_w},\Sigma_{B_{w'}}$) are at least $1/2^i$. Let $D_{A_w}$ be the set of $\min\{k,w\}$ largest singular vectors from $U_{A_w}$, and define $D_{B_{w'}}$ similarly. Define $D_i = D_i'\cup D_{A_w}\cup D_{B_{w'}}$. Let $s_i$ denote the dimension of $\mathrm{span}(D_i)$, and note the $s_i$ are non-decreasing.
Let $\tilde{s}_i$ be $s_i$ after rounding up to the nearest power of $2$. Group all $i$ with the same value of $\tilde{s}_i$ into groups $G_1,G_2,\ldots,G_{\log_2(1/\varepsilon^2)}$. For example if for $i=0,1,2,3$ the $s_i$ are $3,4,15,16$ then the $\tilde{s}_i$ are $4,4,16,16$ and $G_1 = \{0,1\}$, $G_2 = \{2,3\}$. Let $v_j$ be the common value of $\tilde{s}_i$ for $i$ in $G_j$.
\begin{lemma}\LemmaName{si-sumbound}
$\sum_i s_i/2^i \le 8k$.
\end{lemma}
\begin{proof}
Define $s = |D_{A_w}\cup D_{B_{w'}}|\le 2k$ and let $s_i'$ denote the dimension of $\mathrm{span}(D_i')$. Then the above summation is at most $\sum_i (s/2^i + s_i'/2^i) \le 4k + \sum_i s_i'/2^i$. It thus suffices to bound the second summand by $4k$.
Note that we can find a basis for $D_i'$ among the columns of $U_{A_w}, U_{B_{w'}}$ with corresponding squared singular value at least $1/2^i$, so let $a_i + b_i = s_i'$, where $a_i$ is the number of columns of $U_{A_w}$ in the basis and $b_i$ the number of columns of $U_{B_{w'}}$ in the basis. Then by averaging, if the inequality of the lemma statement does not hold then either $\sum_i a_i/2^i > 2k$ or $\sum_i b_i/2^i > 2k$. Without loss of generality assume the former.
Consider an arbitrary column of $U_{A_w}$, and suppose it has squared singular value in the range $[1/2^i, 1/2^{i-1})$. Then it is in $\mathrm{span}(D_j')$ for all $j \ge i$. Its contribution to $\sum_i a_i/2^i$ is therefore $1/2^i + 1/2^{i+1} + \ldots$ which is at most $2/2^i = 1/2^{i-1}$. It follows that $\sum_i a_i/2^i \le 2k$, since the squared Frobenius norm of $A_w$ is at most $k$. This is a contradiction to $\sum_i a_i/2^i > 2k$.
\end{proof}
Now we prove the main theorem of this subsection.
\begin{theorem}\TheoremName{main}
Suppose that the following conditions hold:
\begin{enumerate}
\item[(1)] If $w+w' \le k$, then $\Pi$ is an $\varepsilon/C$-subspace embedding for the subspace spanned by the columns of $A_w,B_{w'}$. Otherwise if $w+w' > k$, then for each $0\le i\le \log_2(1/\varepsilon^2)$, $\Pi$ is an $\varepsilon_i/C$-subspace embedding for $\mathrm{span}(D_{i'})$ with
$$
\varepsilon_i = \min\left\{\frac 12, \varepsilon \sqrt{\frac{v_j}{k}}\right\}
$$
where $i'$ is the largest $i$ with $s_i$ in $G_j$.
\item[(2)] $\|\Pi A_{\bar{w}}\|, \|\Pi B_{\bar{w'}}\| \le \varepsilon/C$.
\end{enumerate}
Then \Equation{mmult-general} holds as long as $C$ is smaller than some fixed universal constant.
\end{theorem}
\begin{proof}
We would like to bound
\begin{align}
\nonumber \|(\Pi A)^T(\Pi B) - A^T B\| & \le \underbrace{\|(\Pi A_w)^T\Pi B_{w'} - A_w^T B_{w'}\|}_\alpha + \underbrace{\|(\Pi A_{\bar{w}})^T\Pi B_{w'}\|}_\beta + \underbrace{\|(\Pi A_w)^T\Pi B_{\bar{w'}}\|}_\gamma\\
&\hspace{.2in}{}+ \underbrace{\|(\Pi A_{\bar{w}})^T\Pi B_{\bar{w'}}\|}_\Delta + \underbrace{\|A_{\bar{w}}^T B_{w'}\|}_\zeta + \underbrace{\|A_w^T B_{\bar{w'}}\|}_\eta + \underbrace{\|A_{\bar{w}}^T B_{\bar{w'}}\|}_\Theta \EquationName{final-bound}
\end{align}
Using $\|XY\| \le \|X\|\cdot \|Y\|$ for any conforming matrices $X,Y$, we see $\Delta \le \varepsilon^2/C^2$ by condition (2). Furthermore by the definition of $w,w'$ we know $\|A_{\bar{w}}\|, \|B_{\bar{w'}}\| \le \varepsilon/C'$, and thus $\zeta + \eta + \Theta \le 2\varepsilon/C' + (\varepsilon/C')^2$. Note condition (1) implies that $\Pi$ is a $(1/2)$-subspace embedding for the subspace spanned by columns of $A_w,B_{w'}$ (by taking $i$ maximal). Thus by both conditions we have $\beta, \gamma \le (\varepsilon/C)(1+1/2)$.
It only remains to bound $\alpha$. If $w+w' \le k$, then we are done by condition (1) and \Lemma{simple-mmult}. Thus assume $w+w' > k$. Then we have
$$
\|(\Pi A_w)^T\Pi B_{w'} - A_w^T B_{w'}\| = \sup_{\|x\|=\|y\|=1} \left|\inprod{\Pi U_{A_w} \Sigma_{A_w} x, \Pi U_{B_{w'}} \Sigma_{B_{w'}} y} - \inprod{U_{A_w}\Sigma_{A_w} x, U_{B_{w'}} \Sigma_{B_{w'}} y}\right|
$$
Let $x,y$ be any unit norm vectors. Write $x = x^1 + x^2 + \ldots + x^b$ for $b = \log_2(1/\varepsilon^2)$, where $x^i$ is the restriction of $x$ to coordinates for which the corresponding squared singular values in $\Sigma_{A_w}$ are in $(1/2^i, 1/2^{i-1}]$. Similarly define $y^1,\ldots,y^b$. Then $|\inprod{\Pi U_{A_w} \Sigma_{A_w} x, \Pi U_{B_{w'}} \Sigma_{B_{w'}} y} - \inprod{U_{A_w}\Sigma_{A_w} x, U_{B_{w'}} \Sigma_{B_{w'}} y}|$ equals
\allowdisplaybreaks
\begin{align}
\nonumber &\left|\sum_{i=1}^b\sum_{j=1}^b \inprod{\Pi U_{A_w}\Sigma_{A_w} x^i, \Pi U_{B_{w'}}\Sigma_{B_{w'}} y^j} - \inprod{U_{A_w}\Sigma_{A_w} x^i, U_{B_{w'}}\Sigma_{B_{w'}}y^j}\right|\\
\nonumber &\hspace{.4in}{}\le \sum_{i=1}^b \left|\inprod{\Pi U_{A_w}\Sigma_{A_w}x^i, \Pi U_{B_{w'}}\Sigma_{B_{w'}}\sum_{j\le i}y^j} - \inprod{U_{A_w}\Sigma_{A_w}x^i, \sum_{j\le i}U_{B_{w'}}\Sigma_{B_{w'}}y^j}\right|\\
&\hspace{.6in}{} + \sum_{j=1}^b \left|\inprod{\Pi U_{A_w}\Sigma_{A_w}\sum_{i\le j}x^i, \Pi U_{B_{w'}}\Sigma_{B_{w'}} y^j} - \inprod{\sum_{i\le j} x^i, y^j}\right| \EquationName{will-cs}
\end{align}
We bound the first sum, as bounding the second is similar. Note $U_{A_w}\Sigma_{A_w} x^i, U_{B_{w'}}\Sigma_{B_{w'}}\sum_{j\le i} y^j\in D_i$. Therefore by property (1) and \Lemma{simple-mmult},
\begin{align}
\nonumber \Bigg|\inprod{\Pi U_{A_w}\Sigma_{A_w}x^i, \Pi U_{B_{w'}}\Sigma_{B_{w'}}\sum_{j\le i}y^j}& - \inprod{U_{A_w}\Sigma_{A_w}x^i, U_{B_{w'}}\Sigma_{B_{w'}}\sum_{j\le i}y^j}\Bigg| \le \frac{\varepsilon_i}{C 2^{(i-1)/2}} \cdot \|x^i\|\cdot \|y\|\\
{}&\le \frac{\varepsilon}{C 2^{(i-1)/2}} \cdot \sqrt\frac{2s_i}{k} \cdot \|x^i\| \EquationName{vs}
\end{align}
where \Equation{vs} used that the corresponding $v$ value in property (1) is at most $2s_i$. Returning to \Equation{will-cs} and applying Cauchy-Schwarz and \Lemma{si-sumbound},
\begin{align*}
\sum_{i=1}^b \Bigg|\inprod{\Pi U_{A_w}\Sigma_{A_w}x^i, \Pi U_{B_{w'}}\Sigma_{B_{w'}}\sum_{j\le i}y^j}& - \inprod{U_{A_w}\Sigma_{A_w}x^i, \sum_{j\le i}U_{B_{w'}}\Sigma_{B_{w'}}y^j}\Bigg| \le \sum_{i=1}^b \frac{\varepsilon}{C 2^{(i-1)/2}} \cdot \sqrt\frac{2s_i}{k} \cdot \|x^i\|\\
{}&\le \frac{2\varepsilon}{C\sqrt{k}}\cdot \left(\sum_{i=1}^b \frac{s_i}{2^i}\right)^{1/2} \cdot \left(\sum_{i=1}^b \|x^i\|^2\right)^{1/2} \\
{}&\le \frac{2\sqrt{8}\varepsilon}C
\end{align*}
We thus finally have that \Equation{final-bound} is at most $(2\sqrt{8} + 3)\varepsilon/C + + (\varepsilon/C)^2 + 2\varepsilon/C' + (\varepsilon/C')^2$, which is at most $\varepsilon$ for $C, C'$ sufficiently large constants.
\end{proof}
Now we discuss some implications of \Theorem{main} for specific $\Pi$.
\paragraph{Example 1:} Let $\Pi$ have $O(k/\varepsilon^2)$ rows forming an orthonormal basis for the span of the columns of $A_w,B_{w'}$. Property (1) is satisfied for every $i$ in fact with $\varepsilon_i = 0$. Property (2) is also satisfied since $\|\Pi A_{\bar{w}}\| \le \|\Pi\| \cdot \|A_{\bar{w}}\| \le \varepsilon$, and similarly for bounding $\|\Pi B_{\bar{w'}}\|$.
\paragraph{Example 2:} Let $\Pi$ be a random $m\times n$ matrix with independent entries that are subgaussian with variance $1/m$. For example, the entries of $\Pi$ may be $\mathcal{N}(0,1/m)$, or uniform in $\{-1/\sqrt{m}, 1/\sqrt{m}\}$. Let $m$ be $\Theta((k+\log(1/\delta))/\varepsilon^2)$. As mentioned in \Section{subgaussian-ose-moments}, such $\Pi$ is an $\varepsilon$-subspace embedding for a $k$-dimensional subspace with failure probability $\delta$. For property (1) of \Theorem{main}, if $w+w'\le k$ then we would like $\Pi$ to be an $\varepsilon$-subspace embedding for a subspace of dimension at most $k$, which holds with failure probability $\delta$. If $w+w' > k$ then we would like $\Pi$ to be an $\varepsilon_i$-subspace embedding for $\mathrm{span}(D_{i'})$ for all $1\le i\le \log_2(1/\varepsilon^2)$ simultaneously. Note $\max_j v_j \le 2(w+w') = O(k/\varepsilon^2)$, and thus $\max_j v_j \le m$. Thus for a subspace under consideration $\mathrm{D_{i'}}$ for $i' \in G_j$, we have failure probability $\delta^{v_j/k}$ for our choice of $m$. By construction every $v_j$ is at least $k$, and the $v_j$ increase at least geometrically. Thus our total failure probability is, by a union bound, $\sum_j \delta^{v_j/k} \le \sum_j \delta^{2^{j-1}} = O(\delta)$. Property (2) of \Theorem{main} is satisfied with failure probability $\delta$ by \cite[Theorem 3.2]{RudelsonV13}.
\section{Applications}\SectionName{applications}
Spectral norm approximate matrix multiplication with dimension bounds depending on stable rank has immediate applications for the analysis of generalized regression and low-rank approximation problems. We also point out to the reader recent applications of this result to kernelized ridge regression \cite{YangPW15} and $k$-means clustering \cite{CohenEMMP14}.
\subsection{Generalized regression}\SectionName{gen-reg}
Here we consider generalized regression: attempting to approximate a matrix $B$ as $AX$, with $A$ of rank at most $k$. Let $P_A$ be the orthogonal projection operator to the column space of $A$, with $P_{\bar A} = I-P$; then the natural best approximation
will satisfy
$$
AX = P_A B.
$$
This minimizes both the Frobenius and spectral norms of $AX - B$. A standard approximation algorithm for this is to replace $A$ and $B$ with sketches $\Pi A$ and $\Pi B$, then
solve the reduced problem exactly (see e.g. \cite{ClarksonW09}, Theorem 3.1). This will produce
\begin{align*}
\tilde X &= ((\Pi A)^T \Pi A)^{-1} (\Pi A)^T \Pi B \\
A \tilde X &= A ((\Pi A)^T \Pi A)^{-1} (\Pi A)^T \Pi B \\
{}&= U_A ((\Pi U_A)^T\Pi U_A)^{-1} (\Pi U_A)^T\Pi B.
\end{align*}
Below we give a lemma on the guarantees of the sketched solution in terms of properties of $\Pi$.
\begin{theorem}\TheoremName{gen-regression}
If $\Pi$
\begin{enumerate}
\item
satisfies the $(k, \sqrt{\varepsilon / 8})$-approximate spectral norm matrix multiplication property for $U_A, P_{\bar A} B$
\item
is a $(1/2)$-subspace embedding for the column space of $A$ (which is implied by $\Pi$ satisfying the spectral norm approximate matrix multiplication property for $U_A$ with itself)
\end{enumerate}
then
\begin{equation}
\| A \tilde X - B \|^2 \leq (1 + \varepsilon) \| P_A B - B \|^2 + (\varepsilon/k)\cdot \| P_A B - B \|_F^2.\EquationName{regress}
\end{equation}
\end{theorem}
\begin{proof}
We may write:
\begin{align*}
\| A \tilde X - B \|_2^2 &= \| U_A ((\Pi U_A)^T\Pi U_A)^{-1} (\Pi U_A)^T \Pi B - B \|^2 \\
{}&= \| U_A ((\Pi U_A)^T\Pi U_A)^{-1} (\Pi U_A)^T \Pi ( P_A B + P_{\bar A} B ) - P_A B - P_{\bar A} B \|^2 \\
{}&= \| P_A B + U_A ((\Pi U_A)^T \Pi U_A)^{-1} (\Pi U_A)^T\Pi P_{\bar A} B - P_A B - P_{\bar A} B \|^2 \\
{}&= \| U_A ((\Pi U_A)^T \Pi U_A)^{-1} (\Pi U_A)^T\Pi P_{\bar A} B - P_{\bar A} B \|^2.
\end{align*}
So far, we have shown that the error depends only on $P_{\bar A} B$ and not $P_A B$ (with the third line following from the fact that the sketched regression is exact on $P_A B$). Now, in the last line, we can see that the two terms lie in orthogonal column spaces (the first in the span of $A$, the second orthogonal to it). For matrices $X$ and $Y$ with orthogonal column spans, $\| X+Y \|^2 \leq \| X \|^2 + \| Y \|^2$, so this is at most
$$
\| U_A ((\Pi U_A)^T \Pi U_A)^{-1} (\Pi U_A)^T\Pi P_{\bar A} B \|^2 + \| P_{\bar A} B \|^2.
$$
Spectral submultiplicativity then implies the first term is at most
$$
( \| U_A \| \cdot \| ((\Pi U_A)^T \Pi U_A)^{-1} \| \cdot \| (\Pi U_A)^T \Pi P_{\bar A} B \| )^2.
$$
$\| U_A \|$ is 1, since $U_A$ is orthonormal. $((\Pi U_A)^T\Pi U_A)^{-1}$ is at most 2, since $\Pi$ is a subspace embedding for $U_A$. Finally, $\| (\Pi U_A)^T \Pi P_{\bar A} B \|$ is at most
$$
\sqrt{\varepsilon / 8} \cdot \sqrt{(\|U_A\|^2 + \|U_A\|_F^2 / k) (\|P_{\bar A} B\|^2 + \|P_{\bar A} B\|^2_F / k)} = \sqrt{(\varepsilon / 8) \cdot 2 \cdot ( \| P_A B - B \|^2 + \| P_A B - B \|^2 / k )}.
$$
Multiplying these together, squaring, and adding the remaining $\| P_{\bar A} B \|^2$ term gives a bound of
$$
(1 + \varepsilon) \| P_A B - B \|^2 + (\varepsilon / k)\cdot \| P_A B - B \|_F^2
$$
as desired.
\end{proof}
\subsection{Low-rank approximation}
Now we apply the generalized regression result from \Section{gen-reg} to obtain a result on low-rank approximation: approximating a matrix $A$ in the form $\tilde U_k \tilde \Sigma_k \tilde V_k^T$, where $\tilde U_k$ has only $k$ columns and both $\tilde U_k$ and $\tilde V_k$ have orthonormal columns. Here, we consider a previous approach (see e.g.\ \cite{Sarlos06}):
\begin{enumerate}
\item
Let $S = \Pi A$.
\item
Let $P_S$ be the orthogonal projection operator to the row space of $S$. Let $\tilde A = A P_S$.
\item
Compute a singular value decomposition of $\tilde A$, and keep only the top $k$ singular vectors. Return the resulting low rank approximation $\tilde{A}_k$ of $\tilde A$.
\end{enumerate}
It turns out computing $\tilde{A}_k$ can be done much more quickly than computing $A_k$; see details in \cite[Lemma 4.3]{ClarksonW09}.
Let $A_k$ be the exact $k$-truncated SVD approximation of $A$ (and thus the best rank-$k$ approximation, in the spectral and Frobenius norms), and let $U_k$ be the top $k$ column singular vectors, and $A_{\bar k} = A - A_k$ be the tail.
\begin{theorem}
If $\Pi$
\begin{enumerate}
\item
satisfies the $(k, \sqrt{\varepsilon / 8})$-approximate spectral norm matrix multiplication property for $U_k, A_{\bar k}$
\item
is a $(1/2)$-subspace embedding for the column space of $U_k$
\end{enumerate}
then
\begin{equation}
\| A - \tilde{A}_k \|^2 \leq (1 + \varepsilon) \| A - A_k \|^2 + (\varepsilon / k) \| A - A_k \|_F^2
\end{equation}
\end{theorem}
\begin{proof}
Note that this procedure chooses the best possible (in the spectral norm) rank-$k$ approximation to $A$ subject to the constraint of lying in the row space of $S$. Thus, the spectral norm error can be no worse than the error of a specific such matrix we exhibit.
We simply choose the matrix obtained by running our generalized regression algorithm from $A$ onto $U_k$, with $\Pi$:
$$
U_k ((\Pi U_k)^T \Pi U_k)^{-1} (\Pi U_k)^T \Pi A
$$
This is rank-$k$ by construction, since it is multiplied by $U_k$, and it lies in the row space of $S = \Pi A$ since that is the rightmost factor. On the other hand, it is an application of the regression algorithm to $A$ where the optimum output is $A_k$ (since that is the projection of $A$ onto the space of $U_k$). Plugging this into \Equation{regress} gives the desired result.
\end{proof}
\subsection{Kernelized ridge regression}
In nonparametric regression one is given data $y_i = f^*(x_i) + w_i$ for $i=1,\ldots,n$, and the goal is to recover a good estimate for the function $f^*$. Here the $y_i$ are scalars, the $x_i$ are vectors, and the $w_i$ are independent noise, often assumed to be distributed as mean-zero gaussian with some variance $\sigma^2$. Unlike linear regression where $f^*(x_i)$ is assumed to take the form $\inprod{\beta, x}$ for some vector $\beta$, in nonparametric regression we allow $f^*$ to be an arbitrary function from some function space. Naturally the goal then is to recover some $\tilde{f}$ from the data so that, as $n$ grows, the probability that $\tilde{f}$ is ``close'' to $f^*$ increases at some good rate.
The recent work \cite{YangPW15} considers the well studied problem of obtaining $\tilde{f}$ so that $\|\tilde{f} - f^*\|_n^2$ is small with high probability over the noise $w$, where one uses the definition
$$
\|f - g\|_n^2 = \frac 1n\sum_{i=1}^n (f(x_i) - g(x_i))^2 .
$$
The work \cite{YangPW15} considers the case where $f^*$ comes from a Hilbert space $\mathcal{H}$ of functions $f$ such that $f$ is guaranteed to be square integrable, and the map $x\mapsto f(x)$ is a bounded linear functional. The function $\tilde{f}$ is then defined to be the optimal solution to the {\em Kernel Ridge Regression (KRR)} problem of computing
\begin{equation}
f^{LS} = \mathop{argmin}_{f\in\mathcal{H}}\left\{ \frac 1{2n}\sum_{i=1}^n (y_i - f(x_i))^2 + \lambda_n \cdot \|f\|_{\mathcal{H}}^2\right\} \EquationName{krr}
\end{equation}
for some parameter $\lambda_n$. It is known that any $\mathcal{H}$ as above can be written as the closure of the set of all functions
\begin{equation}
g(\cdot) = \sum_{i=1}^N \alpha_i k(\cdot, z_i) ,\EquationName{kernelized}
\end{equation}
over all $\alpha\in\mathbb{R}^N$ and vectors $z_1,\ldots,z_N$ for some positive semidefinite {\em kernel function} $k$. Furthermore, the optimal solution to \Equation{krr} can be expressed as $f^{LS} = \sum_{i=1}^n \alpha^{LS}_i\cdot k(\cdot, x_i)$ for some choice of weight vector $\alpha^{LS}$, and it is known that $\|f^{LS} - f^*\|_n$ will be small with high probability, over the randomness in $w$, if $\lambda_n$ is chosen appropriately (see \cite{YangPW15} for background references and precise statements).
After rewriting \Equation{krr} using \Equation{kernelized} and defining a matrix $K$ with $K_{i,j} = k(x_i, x_j)$, one arrives at a reformulation for KRR of computing
$$
\alpha^{LS} = \mathop{argmin}_{\alpha\in\mathbb{R}^n}\left\{ \frac 1{2n} \alpha^T K^2 \alpha - \frac 1n\alpha^T K y + \lambda_n \alpha^T K \alpha\right\} = \left(\frac 1n K^2 + 2\lambda_n K\right)^{-1}\cdot \frac 1n Ky ,
$$
which can be computed in $O(n^3)$ time. The work \cite{YangPW15} then focuses on speeding this up, by instead computing a solution to the lower-dimensional problem
$$
\tilde{\alpha}^{LS} = \mathop{argmin}_{\alpha\in\mathbb{R}^m}\left\{ \frac 1{2n} \alpha^T \Pi K^2\Pi^T \alpha - \frac 1n\alpha^T \Pi K y + \lambda_n \alpha^T \Pi K \Pi^T \alpha\right\} = \left(\frac 1n \Pi K^2\Pi^T + 2\lambda_n \Pi K\Pi^T\right)^{-1}\cdot \frac 1n \Pi Ky
$$
and then returning as $\tilde{f}$ the function specified by the weight vector $\tilde{\alpha} = \Pi^T \tilde{\alpha}^{LS}$. Note that once various matrix products are formed (where the running time complexity depends on the $\Pi$ being used), one only needs to invert an $m\times m$ matrix thus taking $O(m^3)$ time. They then prove that $\|\tilde{f} - f^*\|_n$ is small with high probability as long as $\Pi$ satisfies two deterministic conditions (see the proof of Lemma 2 \cite[Section 4.1.2]{YangPW15}, specifically equation (26) in that work):
\begin{itemize}
\item $\Pi$ is a $(1/2)$-subspace embedding for a particular low-dimensional subspace
\item $\|\Pi B\| = O(\|B\|)$ for a particular matrix $B$ of low stable rank ($B$ is $U D_2$ in \cite{YangPW15}). Note
$$
\|\Pi B\| = \|(\Pi B)^T\Pi B\|^{1/2} \le \left(\|(\Pi B)^T\Pi B - B^T B\| + \|B^T B\|\right)^{1/2} \le \|(\Pi B)^T\Pi B - B^T B\|^{1/2} + \|B\| ,
$$
and thus it suffices for $\Pi$ to provide the approximate matrix multiplication property for the product $B^T B$, where $B$ has low stable rank.
\end{itemize}
The first bullet simply requires a subspace embedding in the standard sense, and for the second bullet \cite{YangPW15} avoided AMM by obtaining a bound on $\|\Pi B\|$ directly by their own analyses for gaussian $\Pi$ and the SRHT (in the gaussian case, it also follows from \cite[Theorem 3.2]{RudelsonV13}). Our result thus provides a unifying analysis which works for a larger and general class of $\Pi$, including for example sparse subspace embeddings.
\subsection{$k$-means clustering}
In the works \cite{BoutsidisZMD15,CohenEMMP14}, the authors considered dimensionality reduction methods for $k$-means clustering. Recall in $k$-means clustering one is given $n$ points $x_1,\ldots,x_n\in\mathbb{R}^d$, as well as an integer $k\ge 1$, and the goal is to find $k$ points $y_1,\ldots,y_k\in\mathbb{R}^d$ minimizing
$$
\sum_{i=1}^n \min_{j=1}^k \|x_i - y_j\|_2^2 .
$$
That is, the $n$ points can be partitioned arbitrarily into $k$ clusters, then a ``cluster center'' should be assigned to each cluster so as to minimize sums of squared Euclidean distances of each of the $n$ points to their cluster centers. It is a standard fact that once a partition $\mathcal{P} = \{P_1,\ldots,P_k\}$ of the $n$ points into clusters is fixed, the optimal cluster centers to choose are the centroids of the points in each of the $k$ partitions, i.e.\ $y_j = (1/|P_j|)\cdot \sum_{i \in P_j} x_i$.
One key observation common to both of the works \cite{BoutsidisZMD15,CohenEMMP14} is that $k$-means clustering is closely related to the problem of low-rank approximation. More specifically, given a partition $\mathcal{P} = \{P_1, \ldots, P_k\}$, define the $n\times k$ matrix $X_{\mathcal{P}}$ by
$$
(X_{\mathcal{P}})_{i, j} =
\begin{cases}
\frac 1{\sqrt{|P_j|}}, &\ \text{if } i \in P_j\\
0, &\ \text{otherwise}
\end{cases}
$$
Let $A\in\mathbb{R}^{n\times d}$ have rows $x_1,\ldots,x_n$. Then the $k$-means problem can be rewritten as computing
$$
\mathrm{argmin}_{\mathcal{P}} \|A - X_{\mathcal{P}} X_{\mathcal{P}}^T A\|_F^2
$$
where $\mathcal{P}$ ranges over all partitions of $\{1,\ldots,n\}$ into $k$ sets. It is easy to verify that the non-zero columns of $X_{\mathcal{P}}$ are orthonormal, so $X_{\mathcal{P}}X_{\mathcal{P}}^T$ is the orthogonal projection onto the column space of $X_{\mathcal{P}}$. Thus if one defines $\mathcal{S}$ as the set of all rank at most $k$ orthogonal projections obtained as $X_{\mathcal{P}}X_{\mathcal{P}}^T$ for some $k$-partition $\mathcal{P}$, then the above can be rewritten as the {\em constrained rank-$k$ projection problem} of computing
\begin{equation}
\mathrm{argmin}_{P\in\mathcal{S}} \|(I-P)A\|_F^2 .
\end{equation}
One can verify this by hand, since the rows of $A$ are the points $x_i$, and the $i$th row of $PA$ for $P = X_{\mathcal{P}}X_{\mathcal{P}}^T$ is the centroid of the points in $i$'s partition in $\mathcal{P}$.
The work \cite{CohenEMMP14} showed that if $\mathcal{S}$ is any subset of projections of rank at most $k$ (henceforth {\em rank-$k$ projections}) and $\Pi\in\mathbb{R}^{m\times d}$ satisfies certain technical conditions to be divulged soon, then if $\tilde{P}\in\mathcal{S}$ satisfies
\begin{equation}
\|(I-\tilde{P})A\Pi^T\|_F^2 \le \gamma \cdot \mathrm{min}_{P\in\mathcal{S}} \|(I-P)A\Pi^T\|_F^2 , \EquationName{pre-condition}
\end{equation}
then
\begin{equation}
\|(I-\tilde{P})A\|_F^2 \le \frac{(1+\varepsilon)}{(1-\varepsilon)}\cdot \gamma \cdot \mathrm{min}_{P\in\mathcal{S}} \|(I-P)A\|_F^2 . \EquationName{post-condition}
\end{equation}
One set of sufficient conditions for $\Pi$ is as follows (see \cite[Lemma 10]{CohenEMMP14}). Let $A_k$ denote the best rank-$k$ approximation to $A$ and let $A_{\bar{k}} = A - A_k$. Define $Z \in \mathbb{R}^{d\times r}$ for $r = 2k$ by $Z = V_r$, i.e.\ the top $r$ right singular vectors of $A$ are the columns of $Z$. Define $B_1 = Z^T$ and $B_2 = \frac{\sqrt{k}}{\|A_{\bar{k}}\|_F}\cdot (A - AZZ^T)$. Define $B\in\mathbb{R}^{(n+r)\times d}$ as having $B_1$ as its first $r$ rows and $B_2$ as its lower $n$ rows. Then \cite[Lemma 10]{CohenEMMP14} states that \Equation{pre-condition} implies \Equation{post-condition} as long as
\begin{align}
\|(\Pi B^T)^T (\Pi B^T) - BB^T\| &< \varepsilon, \EquationName{condition1-sr}\\
\text{and }\left| \|\Pi B_2\|_F^2 - \|B_2\|_F^2\right| & \le \varepsilon k \EquationName{easier-condition}
\end{align}
One can easily check $\|B\|^2 = 1$ and $\|B\|_F^2 \le 3k$, so the stable rank $\tilde{r}(B)$ is at most $3k$. Thus \Equation{condition1-sr} is implied by the $(3k, \varepsilon/2)$-AMM property for $B^T, B^T$, and our results apply to show that $\Pi$ can be taken to have $m = O((k + \log(1/\delta))/\varepsilon^2)$ rows to have success probability $1-\delta$ for \Equation{condition1-sr}. Obtaining \Equation{easier-condition} is much simpler and can be derived from the JL moment property (see the proof of \cite[Theorem 6.2]{KaneN14}).
Without our results on stable-rank AMM provided in this current work, \cite{CohenEMMP14} gave a different analysis, avoiding \cite[Lemma 10]{CohenEMMP14}, which required $\Pi$ to have $m = \Theta(k\cdot \log(1/\delta)/\varepsilon^2)$ rows (note the product between $k$ and $\log(1/\delta)$ instead of the sum).
\section{Stable rank and row selection}\SectionName{bss}
As well as random projections, approximate matrix multiplication (and subspace embeddings) by row selection are also common in algorithms. This corresponds to setting $\Pi$ to a diagonal matrix $S$ with relatively few nonzero entries. Unlike random projections, there are no \emph{oblivious} distributions of such matrices $S$ with universal guarantees. Instead, $S$ must be determined (either randomly or deterministically) from the matrices being embedded.
There are two particularly algorithmically useful methods for obtaining such $S$. The first is importance sampling: independent random sampling of the rows, but with nonuniform sampling probabilities. This is analyzed using matrix Chernoff bounds \cite{AhlswedeW02}, and for the case of $k$-dimensional subspace embedding or approximate matrix multiplication of rank-k matrices, it can produce $O(k (\log k) / \varepsilon^2)$ samples \cite{SpielmanS11}. The second method is the deterministic selection method given in \cite{BatsonSS12}, often called ``BSS'', choosing only $O(k / \varepsilon^2)$ rows. This still runs in polynomial time, but originally required many relatively expensive linear algebra steps and thus was slower in general; see \cite{LeeS15} for runtime improvements.
The matrix Chernoff methods can be extended to the stable-rank case, making even the log factor depend only on the stable rank, using ``intrinsic dimension'' variants of the bounds as presented in Chapter 7 of \cite{Tropp15}. Specifically, Theorem 6.3.1 of that work can be applied with each $n$ summands each equal to $\frac{1}{n} \left ( \frac{1}{p_i} a_i^T b_i - A^T B \right )$, where $a_i$ is the $i$th row of $A$, and $i$ is random with the probability of choosing a particular row $i$ equal to
\begin{equation*}
p_i = \frac{\| a_i \|^2 + \| b_i \|^2}{\sum_j \| a_j \|^2 + \| b_j \|^2}
\end{equation*}
We here give an extension of BSS that covers low stable rank matrices as well.
\begin{theorem}\TheoremName{bss-thm}
Given an $n$ by $d$ matrix $A$ such that $\| A \|^2 \le 1$ and $\| A \|_F^2 \le k$, and an $\varepsilon \in (0, 1)$, there exists a diagonal matrix $S$ with $O(k / \varepsilon^2)$ nonzero entries such that
$$
\| (SA)^T (SA) - A^T A \| \le \varepsilon
$$
Such an $S$ can be computed by a polynomial-time algorithm.
\end{theorem}
When $A^TA$ is the identity, this is just the original BSS result. It is also stronger than Theorem 3.3 of \cite{KollaMST10}, implying it when $A$ is the combination of the rows $\sqrt{N/T}\cdot v_i$ from that theorem statement with an extra column containing the costs, and a constant $\epsilon$. The techniques in that paper, on the other hand, can prove a result comparable to \Theorem{bss-thm}, but with the row count scaling as $k/\varepsilon^3$ rather than $k/\varepsilon^2$.
\medskip
\begin{proof}
The proof closely follows the original proof of BSS. However, for simplicity, and because the tight constants are not needed for most applications, we do not include \cite[Claim 3.6]{BatsonSS12} and careful parameter-setting.
At each step, the algorithm will maintain a partial approximation $Z = (SA)^T (SA)$ (the matrix ``$A$'' in \cite{BatsonSS12}), with $S$ beginning as 0. Additionally, we keep track of upper and lower ``walls'' $X_u$ and $X_l$; in the original BSS these are just multiples of the identity. The final $S$ will be returned by the algorithm (rescaled by a constant so that the average of the upper and lower walls is $A^T A$).
We will maintain the invariants
\begin{align}
\mathop{tr}(A (X_u - Z)^{-1} A^T) &\le 1 \\
\mathop{tr}(A (Z - X_l)^{-1} A^T) &\le 1.
\end{align}
These are the so-called upper and lower potentials from BSS. We also require $X_u \prec Z \prec X_l$; recall $M\prec M'$ means that $M'-M$ is positive definite. Note that unlike \cite{BatsonSS12}, here we do not apply a change of variables making $A^T A$ the identity (to avoid confusion, since that would change the Frobenius norm). This is the reason for the slightly more complicated form of the potentials.
In the original BSS, $X_u$ and $X_l$ were always scalar multiples of the identity (here, without the change of variables, that would correspond to always being multiples of $A^T A$). \cite{BatsonSS12} thus simply represented them with scalars. Like BSS, we will increase $X_u$ and $X_l$ by multiples of $A^T A$--however, the key difference from BSS is that they are \emph{initialized} to multiples of the identity, rather than $A^T A$. In particular, we may initialize $X_u$ to $k I$ and $X_l$ to $-k I$. This is still good enough to get the spectral norm bounds we require here (as opposed to the stronger multiplicative approximation guaranteed by BSS).
We will have two scalar values, $\delta_u$ and $\delta_l$, depending only on $\varepsilon$; they will be set later. One step consists of
\begin{enumerate}
\item
Choose a row $a_i$ from $A$ and a positive scalar $t$, and add $t a_i a_i^T$ to $Z$ (via increasing the $i$ component of $S$).
\item
Add $\delta_u A^T A$ to $X_u$ and $\delta_l A^T A$ to $X_l$.
\end{enumerate}
We will show that with suitable values of $\delta_u$ and $\delta_l$, for any $Z$ obeying the invariants there always exists a choice of $i$ and $t$ such that the invariants will still be true after the step is complete. This corresponds to Lemmas 3.3 through 3.5 of BSS.
For convenience, we define, at a given step, the matrix functions of $y$
\begin{align*}
M_u(y) &= ((X_u + y A^T A) - Z)^{-1} \\
M_l(y) &= (Z - (X_l + y A^T A))^{-1}.
\end{align*}
The upper barrier value, after making a step of $t a_i a_i^T$ and increasing $X_u$, is
\begin{equation*}
\mathop{tr}(A ((X_u + \delta_u A^T A) - (Z + t a_i a_i^T))^{-1} A^T).
\end{equation*}
Applying the Sherman-Morrison formula, and cyclicity of trace, to the rank-1 update $t a_i a_i^T$, this can be rewritten as
\begin{equation*}
\mathop{tr}(A M_u(\delta_u) A^T) + \frac{t a_i^T M_u(\delta_u) A^T A M_u(\delta_u) a_i}{1 - t a_i^T M_u(\delta_u) a_i}.
\end{equation*}
Since the function $f(y) = \mathop{tr}(A M_u(y) A^T)$ is a convex function of $y$ with derivative
$$f'(y) = -\mathop{tr}(A M_u(y) A^T A M_u(y) A^T) ,$$
we have $f(\delta_u) - f(0) \le -\delta_u \mathop{tr}(A M_u(\delta_u) A^T A M_u(\delta_u) A^T)$. Then the difference between the barrier before and after the step is at most
\begin{align*}
&\frac{t a_i^T M_u(\delta_u) A^T A M_u(\delta_u) a_i}{1 - t a_i^T M_u(\delta_u) a_i} - \delta_u \mathop{tr}(A M_u(\delta_u) A^T A M_u(\delta_u) A^T).
\end{align*}
Constraining this to be no greater than zero, rewriting in terms of $\frac{1}{t}$ and pulling it out gives
\begin{equation*}
\frac{1}{t} \ge \frac{a_i^T M_u(\delta_u) A^T A M_u(\delta_u) a_i}{\delta_u \mathop{tr}(A M_u(\delta_u) A^T A M_u(\delta_u) A^T)} + a_i^T M_u(\delta_u) a_i.
\end{equation*}
Furthermore, as long as $\frac{1}{t}$ is at least this, $Z$ will remain below $X_u$, since the barrier must approach infinity as $t$ approaches the smallest value passing $X_u$.
For the lower barrier value after the step, we get
\begin{equation*}
\mathop{tr}(A ((Z + t a_i a_i^T) - (X_l + \delta_l A^T A))^{-1} A^T).
\end{equation*}
Again, applying Sherman-Morrison rewrites it as
\begin{equation*}
\mathop{tr}(A M_l(\delta_l) A^T) - \frac{t a_i^T M_l(\delta_l) A^T A M_l(\delta_l) a_i}{1 + t a_i^T M_l(\delta_l) a_i}.
\end{equation*}
Again, due to convexity the increase in the barrier from raising $X_l$ is at most $\delta_l$ times the local derivative. The difference in the barrier after the step is then at most
\begin{align*}
&-\frac{t a_i^T M_l(\delta_l) A^T A M_l(\delta_l) a_i}{1 + t a_i^T M_l(\delta_l) a_i} + \delta_l \mathop{tr}(A M_l(\delta_l) A^T A M_l(\delta_l) A^T).
\end{align*}
This is not greater than zero as long as
\begin{equation*}
\frac{1}{t} \le \frac{a_i^T M_l(\delta_l) A^T A M_l(\delta_l) a_i}{\delta_l \mathop{tr}(A M_l(\delta_l) A^T A M_l(\delta_l) A^T)} - a_i^T M_l(\delta_l) a_i.
\end{equation*}
There is some value of $t$ that works for $a_i$ as long as the lower bound for $\frac{1}{t}$ is no larger than the upper bound. To show that there is at least one choice of $i$ for which this holds, we look at the sum of all the lower bounds and compare to the sum of all the upper bounds. Summing the former over all $i$ gets
\begin{equation*}
\frac{\mathop{tr}(A M_u(\delta_u) A^T A M_u(\delta_u) A^T)}{\delta_u \mathop{tr}(A M_u(\delta_u) A^T A M_u(\delta_u) A^T)} + \mathop{tr}(A M_u(\delta_u) A^T)
\end{equation*}
and the latter gets
\begin{equation*}
\frac{\mathop{tr}(A M_l(\delta_l) A^T A M_l(\delta_l) A^T)}{\delta_l \mathop{tr}(A M_l(\delta_l) A^T A M_l(\delta_l) A^T)} - \mathop{tr}(A M_l(\delta_l) A^T).
\end{equation*}
Finally, note that
\begin{equation*}
\mathop{tr}(A M_u(\delta_u) A^T) = \mathop{tr}(A ((X_u + \delta_u A^T A) - Z)^{-1} A^T) \le \mathop{tr}(A (X_u - Z)^{-1} A^T) \le 1
\end{equation*}
and the lower barrier implies $Z - X_l \succ A^T A$, implying that as long as $\delta_l \leq \frac{1}{2}$,
\begin{equation*}
\mathop{tr}(A M_l(\delta_l) A^T) = \mathop{tr}(A (Z - (X_l + \delta_l A^T A))^{-1} A^T) \le 2 \mathop{tr}(A (Z - X_l)^{-1} A^T) \le 2.
\end{equation*}
Thus, we can always make a step as long as $\delta_u$ and $\delta_l$ are set so that
\begin{equation*}
\frac{1}{\delta_u} + 1 \leq \frac{1}{\delta_l} - 2
\end{equation*}
and $\delta_l \leq \frac{1}{2}$.
This is satisfied by
\begin{align*}
\delta_u &= \varepsilon + 2 \varepsilon^2 \\
\delta_l &= \varepsilon - 2 \varepsilon^2.
\end{align*}
Before the first step, $X_u$ and $X_l$ can be initialized as $kI$ and $-kI$, respectively. If the algorithm is then run for $\frac{k}{\varepsilon^2}$ steps, we have:
\begin{align*}
X_u &= \frac{k}{\varepsilon} A^T A + 2k A^T A + k I \\
&\preceq \frac{k}{\varepsilon} A^T A + 3k I \\
X_l &= \frac{k}{\varepsilon} A^T A - 2k A^T A - k I \\
&\succeq \frac{k}{\varepsilon} A^T A - 3k I.
\end{align*}
$\frac{\varepsilon}{k} X_u$ and $\frac{\varepsilon}{k} X_l$ both end up within $3 \varepsilon I$ of $A^T A$, so $\frac{\varepsilon}{k} Z$ (from $\sqrt{\frac{\varepsilon}{k}} S$) satisfies the requirements of the output for $3 \varepsilon$ (one can simply apply this argument for $\varepsilon / 3$). Furthermore, all the computations required to verify the preservation of invariants and compute explicit $t$s can be performed in polynomial time.
\end{proof}
This obtains more general AMM as a corollary:
\begin{corollary}\CorollaryName{bss-amm}
Given two matrices $A$ and $B$, each with $n$ rows, and an $\varepsilon \in (0, 1)$, there exists a diagonal matrix $S$ with $O(k / \varepsilon^2)$ nonzero entries satisfying the $(k, \varepsilon)$-AMM property for $A$, $B$.
Such an $S$ can be computed by a polynomial-time algorithm.
\end{corollary}
\begin{proof}
Apply \Theorem{bss-thm} to a matrix $X$ consisting of the columns of $\frac{A}{\sqrt{2} \max(\| A \|_2, \| A \|_F / \sqrt{k})}$ appended to the columns of $\frac{B}{\sqrt{2} \max(\| B \|_2, \| B \|_F / \sqrt{k})}$, and use the resulting $S$.
Note that $X$ satisfies the conditions of that theorem, since concatenating the sets of columns at most adds the squares of their spectral and Frobenius norms. $(SA)^T (SB) - A^T B$ is a submatrix of $2 \max(\| A \|_2, \| A \|_F / \sqrt{k}) \max(\| B \|_2, \| B \|_F / \sqrt{k}) ((SX)^T (SX) - X^T X)$, so its spectral norm is upper bounded by the spectral norm of that matrix, which in turn is bounded by the guarantee of \Theorem{bss-thm}.
\end{proof}
\section*{Acknowledgments}
We thank Jaros{\l}aw B{\l}asiok for pointing out the connection between low stable rank approximate matrix multiplication and the analyses in \cite{YangPW15}.
\bibliographystyle{alpha}
| {
"timestamp": "2016-03-03T02:09:35",
"yymm": "1507",
"arxiv_id": "1507.02268",
"language": "en",
"url": "https://arxiv.org/abs/1507.02268",
"abstract": "We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\\tilde{r}/\\varepsilon^2)$ rows. Here $\\tilde{r}$ is the maximum stable rank, i.e. squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [MZ11, KVZ14], and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future.Our main theorem, via connections with spectral error matrix multiplication shown in prior work, implies quantitative improvements for approximate least squares regression and low rank approximation. Our main result has also already been applied to improve dimensionality reduction guarantees for $k$-means clustering [CEMMP14], and implies new results for nonparametric regression [YPW15].We also separately point out that the proof of the \"BSS\" deterministic row-sampling result of [BSS12] can be modified to show that for any matrices $A, B$ of stable rank at most $\\tilde{r}$, one can achieve the spectral norm guarantee for approximate matrix multiplication of $A^T B$ by deterministically sampling $O(\\tilde{r}/\\varepsilon^2)$ rows that can be found in polynomial time. The original result of [BSS12] was for rank instead of stable rank. Our observation leads to a stronger version of a main theorem of [KMST10].",
"subjects": "Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Machine Learning (stat.ML)",
"title": "Optimal approximate matrix product in terms of stable rank",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846621952494,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7092019661077448
} |
https://arxiv.org/abs/1511.06404 | Bounded tiles in $\mathbb{Q}_p$ are compact open sets | Any bounded tile of the field $\mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact. | \section{Introduction}
Let $G$ be a locally compact abelian group and let $\Omega\subset G$ be a Borel set of positive and finite Haar measure.
We say that the set $\Omega$ is a {\em tile} of $G$ if there exists a set $T \subset G$ of translates
such that $\sum_{t\in T} 1_\Omega(x-t) =1$ for almost all $x\in G$, where $1_A$ denotes the indicator function of a set $A$. Such a set $T$ is called a
{\em tiling complement} of $\Omega$ and $(\Omega, T)$ is called a {\em tiling pair}.
In the case of $G= \mathbb{R}$ considered as an additive group, compact sets of positive measure that tile $\mathbb{R}$ by translation were extensively studied. The simplest case concerns compact sets consisting of finite number of unit intervals all of whose endpoints are integers. This tiling problem can be reformulated in terms of finite subsets of $\mathbb{Z}$ which tile the group $\mathbb{Z}$. See \cite{Tijdeman} for references on the study of tiling problem in the group $\mathbb{Z}$.
There are also investigations on the existence of tiles having infinitely many connected components. For example, self-affine tiles are studied by Bandt \cite{Ban}, Gr\"ochenig and Haas \cite{GH94}, Kenyon \cite{kenyonthesis}, Lagarias and Wang \cite{Lagarias-Wang1996}.
A structure theorem for bounded tiles in $\mathbb{R}$ is obtained by Lagarias and Wang in \cite{Lagarias-WangInvent} where it is proved that all tilings of $\mathbb{R}$ by a bounded region (compact set with zero boundary measure) must be periodic, and that the corresponding tiling complements are rational up to affine transformations.
As we shall see, the situation the field of $p$-adic numbers is relatively simple.
Let $\mathbb{Q}_p$ be the field of $p$-adic numbers and let $\mathfrak{m}$
be the Haar measure of $\mathbb{Q}_p$ such that $\mathfrak{m}(\mathbb{Z}_p) = 1$ where $\mathbb{Z}_p$ is the ring of $p$-adic integers.
A Borel set $\Omega \subset \mathbb{Q}_p$ is said to be {\em almost compact open} if there exists a compact open set $\Omega^{\prime}$ such that $\mathfrak{m}(\Omega\setminus \Omega^{\prime})=\mathfrak{m}(\Omega^{\prime}\setminus\Omega)=0$.
In this note, we prove the following theorem.
\begin{theorem} \label{mainthm}
Assume that $\Omega\subset \mathbb{Q}_p$ is a bounded Borel set of positive and finite Haar measure. If $\Omega$ tiles $\mathbb{Q}_{p}$ by translation, then it is an almost compact open set.
\end{theorem}
Compact open tiles in $\mathbb{Q}_p$ are characterized in \cite{FFS} by the $p$-homogeneity which is easy to check.
Actually, it is proved in \cite{FFLS2015} that tiles in $\mathbb{Q}_p$ are always compact open without the boundedness assumption. But the proof in \cite{FFLS2015} was based on the theory of Bruhat-Schwartz distributions and on the theory of the Colombeau algebra of generalized functions. The proof of Theorem \ref{mainthm} given in this note is more direct and easier to understand.
\section{Proof of Theorem \ref{mainthm}}
The proof is based on Fourier analysis. We first recall some notation and some useful facts on Fourier analysis and on the $p^n$-th roots of unity.
\subsection{Recall of notation}\label{definition}
We denote by $|\cdot|_p$ the {\em absolute value} on $\mathbb{Q}_p$.
A typical element of $\mathbb{Q}_p$ is expanded as a convergent series
$$
x= \sum_{n= v}^\infty a_n p^{n} \qquad (v\in \mathbb{Z}, a_n \in \{0,1,\cdots, p-1\}, a_{v}\neq 0),
$$
where $v$ is the valution of $x$, which satisfies $|x|_p=p^{-v}$.
We wirte $\{x\}:= \sum_{n=v}^{-1} a_n p^n$, which is called the {\em fractional part} of $x$.
A non-trivial additive {\em character} of $\mathbb{Q}_p$ is the following function
$$
\chi(x) = e^{2\pi i \{x\}},
$$
which produces all characters $\chi_y$ of $\mathbb{Q}_p$ $(y \in \mathbb{Q}_p)$, where
$\chi_y(x) =\chi(yx)$.
Remark that each $\chi_y(\cdot)$ is uniformly locally constant, i.e.
\begin{equation}\label{chi}
\chi_y(x)=\chi_y(x^{\prime}), \hbox{ if } |x-x^{\prime}|_p\leq \frac{1}{|y|_p} .
\end{equation}
\noindent Notation:
\\ \indent
$\mathbb{Z}_{p}^\times := \mathbb{Z}_{p}\setminus p\mathbb{Z}_{p}=\{x\in \mathbb{Q}_{p}: |x|_p=1\}$.
It is the group of units of $\mathbb{Z}_{p}$.
$B(0, p^{n}): = p^{-n} \mathbb{Z}_{p}$. It is the (closed) ball centered at $0$ of radius $p^n$.
$B(x, p^{n}): = x + B(0, p^{n})$.
$1_A:$ the indicator function of a set $A$.
$\mathbb{L}:=\left\{\{x\}, x\in \mathbb{Q}_p\right\}.$ It is a complete set of representatives of the cosets of the additive subgroup $\mathbb{Z}_p$.
\subsection{Fourier transformation}
Let $\mu$ be a finite Borel measure on $\mathbb{Q}_{p}$. The {\em Fourier transform} of
$\mu$ is classically defined to be
$$
\widehat{\mu} (y)
= \int_{\mathbb{Q}_{p}} \overline{\chi}_y(x) d\mu(x) \qquad (y \in \mathbb{Q}_{p}).
$$
The Fourier transform $\widehat{f}$ of $f \in L^1(\mathbb{Q}_{p})$ is that of $\mu_f$ where $\mu_f$ is the measure
defined by $d\mu_f = f d\mathfrak{m}$.
The following lemma shows that the Fourier transform of the indicator function of a ball centered at $0$
is a function of the same type.
\begin{lemma}[\cite{Fan,FFS}]\label{FourierIntegral} Let $\gamma\in \mathbb{Z}$ be an integer.
We have $$
\forall \xi \in \mathbb{Q}_{p}, \ \ \
\widehat{1_{B(0, p^\gamma)}}(\xi)= p^\gamma 1_{B(0, p^{-\gamma})} (\xi).$$
\end{lemma}
The following lemma shows that the Fourier transform of a compactly supported integrable function is uniformly locally constant.
\begin{lemma}\label{localconstant}
Let $f\in L^1(\mathbb{Q}_p)$ be a complex-value integrable function. If $f$ is supported by $B(0,p^\gamma)$, then $$ \widehat{f}(x+u)=\widehat{f}(x),\quad \forall x\in \mathbb{Q}_p \text{ and } \forall u\in B(0, p^{-\gamma}).$$
\end{lemma}
\begin{proof} It suffices to observe that
\begin{align*}
\widehat{f}(x+u)-\widehat{f}(x)
&=\int_{B(0,p^\gamma)}f(y)\overline{\chi(xy)}(\overline{\chi(uy)-1})dy
\end{align*}
and that $u \in B(0, p^{-\gamma})$ implies $|uy|_p\le 1$
for all $y\in B(0,p^\gamma)$, so that $\chi(uy)-1=0$, by (\ref{chi}).
\end{proof}
\subsection{$\mathbb{Z}$-module generated by $p^n$-th roots of unity}
Let $\omega_\gamma = e^{2\pi i/p^\gamma}$, which is a primitive $p^\gamma$-th root of unity.
\begin{lemma}
\cite{Lagarias-Wang1996,Schoenberg1964}]\label{root}
Let $(a_0,a_1,\cdots, a_{p^\gamma-1})\in \mathbb{Z}^{p^\gamma}$.
Suppose $$\sum_{i=0}^{p^\gamma-1 }a_i\omega_{\gamma}^i=0.$$ Then for any integer $0\leq i\leq p^{\gamma-1}-1$, we have $a_i=a_{i+j p^{\gamma-1}}$ for all $j=0,1,\cdots, p-1$.
\end{lemma}
Lemma \ref{root} immediately leads to the following
consequence.
\begin{lemma}\label{Cor-Sch}
Let $S\subset \mathbb{Z}_{p}$ be a set of $p$ points and let $\mu_S = \sum_{s \in S} \delta_s$, , where $\delta_{s} $ is the dirac measure concentrated at the point $s$. Then $\widehat{\mu_{S}}(\xi)=0$
if and only if $|(s-s^{\prime})\xi|_p=p$ for all distinct $s,s^{\prime}\in S $.
\end{lemma}
\begin{proof}
It suffices to notice that
$
\widehat{\mu_{S}}(\xi)=\sum_{s\in S} e^{-2\pi i \{s \xi\}}.
$
\end{proof}
For a finite set $T\subset \mathbb{Z}_{p}$, let $$\gamma_{T}:=\max_{\substack {t,t^{\prime}\in T\\t\neq t^{\prime}}}\{-\log_{p}(|t-t^{\prime}|_p)\}.$$
\begin{lemma}\label{nonzero}Let $T \subset Z_p$ be a finite set.
We have
$\widehat{\mu_{T}}(\xi)\neq 0$ for all $ \xi \not\in B(0, p^{\gamma_T+1})$.
\end{lemma}
\begin{proof}
We give a proof by contradiction. Suppose that $\widehat{\mu_{T}}(\xi)= 0$ for some $\xi \in \mathbb{Q}_p$ with $|\xi|_p>p^{\gamma_T+1} $. Then from $\widehat{\mu_{T}}(\xi)=0$ and Lemma \ref{root}, we deduce that the cardinality of $T$ is a multiple of $p$ and $T$ is partitioned into ${\rm Card}(T) /p$ subsets $T_0, T_1,\cdots ,T_{{\rm Card}(T)/p}$ such that each $T_i$ contains $p$ elements and $\widehat{\mu_{T_i}}(\xi)= 0$.
By Lemma \ref{Cor-Sch} applied to $S=T_0$, we get $|(t-t^{\prime})\xi|_p=p$ for distinct
$t, t^{\prime}\in T_0$.
However, by the definition of $\gamma_T$ and the fact $|\xi|_p>p^{\gamma_T +1}$, we have
\[
|t-t^{\prime}|_p\geq p^{-\gamma_T}>\frac{p}{|\xi|_p},
\]
which contradicts $|(t-t^{\prime})\xi|_p=p$.
\end{proof}
\subsection{Proof of Theorem \ref{mainthm}}
It is clear that the translation and the
dilation don't change the tiling property of a tile in $\mathbb{Q}_p$. Then, without loss of generality, we assume that $\Omega\subset \mathbb{Z}_p$.
Observe that for any $a\in \mathbb{Q}_p$, either $\Omega+a \subset \mathbb{Z}_p$ or $(\Omega+a)\cap \mathbb{Z}_p= \emptyset$. Also observe that $\mathbb{Z}_p$ is a tile of $\mathbb{Q}_p$ with tiling complement $\mathbb{L}$ (see Section \ref{definition} for the definition of $\mathbb{L}$). These observations imply that $\Omega$ is a tile of $\mathbb{Z}_p$ if and only if it is a tile of $\mathbb{Q}_p$.
Now assume that $\Omega$ is a tile of $\mathbb{Z}_p$ with tiling complement $T$. Since $\mathfrak{m}(\Omega)>0$ and $\mathfrak{m}(\mathbb{Z}_p)=1$, the tiling complement $T$ is a finite subset of $\mathbb{Z}_p$. Let $\mu_{T} := \sum_{t\in T } \delta_{t}$. By definition, $(\Omega, T)$ is a tiling pair in $\mathbb{Z}_p$ iff the following convolution equality holds
\begin{align}\label{convo}
1_{\Omega}*\mu_{T}(x) =1_{\mathbb{Z}_{p}}(x), \quad a.e. \ x\in \mathbb{Q}_{p}.
\end{align}
Taking Fourier transform of both sides of the equality,
by Lemma \ref{FourierIntegral}, we have
\begin{align}\label{zero}
\forall\ \xi\in \mathbb{Q}_p\setminus\mathbb{Z}_p, \quad \quad \widehat{{1}_{\Omega}}(\xi)\cdot\widehat{{\mu}_{T}}(\xi)=0.
\end{align}
By Lemma \ref{nonzero}, we deduce from (\ref{zero}) that
\begin{align}\label{zero1}
\widehat{{1}_{\Omega}}(\xi)=0 \quad \hbox{for all $\xi\in \mathbb{Q}_p$ with $|\xi|_p>p^{\gamma_T+1} $.}
\end{align}
Then, by using Lemma \ref{localconstant}, we get
$$ \widehat{\widehat{1_{\Omega}}}(x+u)= \widehat{\widehat{1_{\Omega}}}(x) \quad (\forall x\in \mathbb{Q}_p, \forall u\in B(0, p^{-(\gamma_T+1)})),$$
that is to say, the continuous function $\widehat{\widehat{1_{\Omega}}}$ is constant on each ball of radius $p^{-(\gamma_T+1)}$. Therefore,
the support of $\widehat{\widehat{1_{\Omega}}}$ is a union of disjoint balls of radius $p^{-(\gamma_T+1)}$.
That is the same for $\Omega$ but up to a set of zero measure because
$ \widehat{\widehat{1_{\Omega}}}(x)=1_{\Omega}(-x), \ a.e. \ x\in \mathbb{Q}_{p}. $ Then, using the fact that $\mathfrak{m}(\Omega)<\infty $, we conclude that the support of $\Omega$ is equal almost everywhere to a finite number of disjoint balls of radius $p^{-(\gamma_T+1)}$.
\qed
\setcounter{equation}{0}
| {
"timestamp": "2015-11-23T02:01:40",
"yymm": "1511",
"arxiv_id": "1511.06404",
"language": "en",
"url": "https://arxiv.org/abs/1511.06404",
"abstract": "Any bounded tile of the field $\\mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact.",
"subjects": "Number Theory (math.NT); Classical Analysis and ODEs (math.CA)",
"title": "Bounded tiles in $\\mathbb{Q}_p$ are compact open sets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846697584033,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7092019657749454
} |
https://arxiv.org/abs/1601.07736 | An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices | A square matrix is called stochastic (or row-stochastic) if it is non-negative and has each row sum equal to unity. Here, we constitute an eigenvalue localization theorem for a stochastic matrix, by using its principal submatrices. As an application, we provide a suitable bound for the eigenvalues, other than unity, of the Randić matrix of a connected graph. | \section{Introduction}
Stochastic matrices occur in many fields of research, such as, computer-aided-geometric designs \cite{Pen}, computational biology \cite{New}, Markov chains \cite{Sene}, etc. A stochastic matrix $\s$ is irreducible if its underlying directed graph is strongly connected.
In this paper, we consider $\s$ to be irreducible.
Let \textbf{\textit{e}} be the column vector whose all entries are equal to 1. Clearly, 1 is an eigenvalue of $\s$ with the corresponding eigenvector \textbf{\textit{e}}. By Perron-Frobenius theorem (see Theorem 8.4.4 in \cite{Horn}), the multiplicity of the eigenvalue 1 is one and all other eigenvalues of $\s$ lie in the closed unit disc $\{z\in\mathbb{C}:|z|\leq1\}$. The eigenvalue 1 is called the Perron eigenvalue (or Perron root) of the matrix $\s$, whereas, the eigenvalues other than 1 are known as non-Perron eigenvalues of $\s$.\\
Here, we describe a method for localization of the non-Perron eigenvalues of $\s$. The eigenvalue localization problem for stochastic matrices is not new. Many researchers gave significant contribution to this context \cite{Cve1,Kir1,Kir2,LiLi1,LiLi2}. In this paper, we use Ger{\v s}gorin disc theorem \cite{Ger} to localize the non-Perron eigenvalues of $\s$. Cvetkovi\'c et al.~\cite{Cve1} and Li et al.~\cite{LiLi1,LiLi2} derived some usefull results, using the fact that any non-Perron eigenvalue of $\s$ is also an eigenvalue of the matrix $\s$-$(\textbf{\textit{e}}\e^T)diag(c_1,c_2,\cdots,c_n)$, where $ c_1,c_2,\cdots,c_n\in\mathbb{R} $.
In \cite{Cve1}, Cvetkovi\'c et al. found a disc which contains all the non-Perron eigenvalues of $\s.$
\begin{theorem}
\label{st:th3a}
\cite{Cve1}
Let $\s=[s_{ij}]$ be a stochastic matrix, and let
$s_i$ be the minimal element among the off-diagonal entries of the i-th column of $\s$. Taking $\gamma=\max_{i\in \mathbb{N}}(s_{ii}-s_i)$, for any $\lambda\in\sigma(\s)\setminus\{1\}$, we have
$$|\lambda-\gamma|\leq 1-trace(\s)+(n-1)\gamma.$$
\end{theorem}
Theorem \ref{st:th3a} was further modified by Li and Li \cite{LiLi1}. They found another disc with different center and different radius.
\begin{theorem}
\label{st:th3b}
\cite{LiLi1}
Let $\s=[s_{ij}]$ be a stochastic matrix, and let
$S_i=\max_{j\neq i}s_{ji}$. Taking $\gamma'=\max_{i\in \mathbb{N}}(S_i-s_{ii})$, for any $\lambda\in\sigma(\s)\setminus\{1\}$, we have
$$|\lambda+\gamma'|\leq trace(\s)+(n-1)\gamma'-1.$$
\end{theorem}
In this paper, we show that there exist square matrices of order $n-1$, whose eigenvalues are the non-Perron eigenvalues of $\s$. We apply Ger{\v s}gorin disc theorem to those matrices in order to obtain our results. We provide an example where our result works better than Theorem \ref{st:th3a} and Theorem \ref{st:th3b}. \\
Let $\Gamma=(V,E)$ be a simple, connected, undirected graph on $n$ vertices. Two vertices $i,j\in V$ are called neighbours, written as $i\sim j$, if they are connected by an edge in $E$. For a vertex
$i\in V$, let $d_i$
be its degree and $N_i$ be the set neighbours of the vertex $i$. For two vertices $i,j\in V$, let $N(i,j)$ be the number of common neighbours of $i$ and $j$, that is, $N(i,j)=|N_{i}\cap N_{j}|$. Let $\textbf{\textit{A}}$ denote the adjacency matrix \cite{Cve} of $\Gamma$ and let $\textbf{\textit{D}}$ be the diagonal matrix of vertex degrees of $\Gamma$. The \textit{Randi\'c} matrix \textbf{\textit{R}} of $\Gamma$ is defined by $\textit{\textbf{R}}=\textbf{\textit{D}}^{-\frac{1}{2}}\textbf{\textit{A}}\textbf{\textit{D}}^{-\frac{1}{2}}$ which is similar to the matrix $\mathcal{R}=\textbf{\textit{D}}^{-1}\textbf{\textit{A}}$. Thus, the matrices \textbf{\textit{R}} and $\mathcal{R}$ have the same eigenvalues. The matrix $\mathcal{R}$ is an irreducible stochastic matrix and its (i,j)-th entry is
\begin{eqnarray*}
\r_{ij}=\begin{cases}
\frac{1}{{d_i}}, & \text{ if } i\sim j,\\
0, & \text{ otherwise. }
\end{cases}
\end{eqnarray*}
The name \textit{Randi\'c matrix} was introduced by Bozkurt et al.~\cite{Boz2} because \textbf{\textit{R}} has a connection with Randi\'c index \cite{Li,Ran}. In recent days, Randi\'c matrix becomes more popular to researchers. The Randi\'c matrix has a direct connection with \textit{normalized Laplacian matrix} $\mathcal{L}=\textbf{\textit{I}}_n-\textbf{\textit{R}}$ studied in \cite{Chung} and with $\Delta=\textbf{\textit{I}}_n-\r$ studied is \cite{Ban1,Ban2}. Thus, for any graph $\Gamma$, if $\lambda$ is an eigenvalue of the normalized Laplacian matrix, then $1-\lambda$ is an eigenvalue of the Randi\'c matrix.\\
In Section 3, we localize non-Perron eigenvalues of $\mathcal{R}$. We provide an upper bound for the largest non-Perron eigenvalue and a lower bound for the smallest non-Perron eigenvalue of $\r$ in terms of common neighbours of two vertices and their degrees. The eigenvalue bound problem was studied previously in many articles \cite{Bau,Chung,LiGu,Rojo}, but the lower bound of the smallest eigenvalue of $\r$ given by Rojo and Soto \cite{Rojo} is the only one which involves the same parameters as in our bound. We recall the Rojo-Soto bound for Randi\'c matrix.
\begin{theorem}\cite{Rojo}
\label{st:th10}
Let $\Gamma$ be a simple undirected connected graph. If $\rho_n$ is the eigenvalue
with the largest modulus among the negative Randi\'c eigenvalues of $\Gamma$, then
\begin{equation}
\label{st:eq1}
|\rho_n|\leq1-\min_{i\sim j}\Big{\{}\frac{N(i,j)}{\max\{d_i,d_j\}}\Big{\}},
\end{equation}
where the minimum is taken over all pairs $(i, j)$ , $1\leq i<j\leq n$, such that the vertices i
and j are adjacent.
\end{theorem}
One of the drawbacks of Theorem \ref{st:th10} is that it always produces the trivial lower bound of $\rho_n$, if the graph contains an edge which does not participate in a triangle. Though the bound in Theorem \ref{st:th10} and our bound (Theorem \ref{st:th5}) are incomparable but, in many occasions, our bound works better than Rojo-Soto bound. We illustrate this by a suitable example.
\section{Localization of the eigenvalues of an irreducible stochastic matrix}
Let $\textbf{\textit{e}}_1, \textbf{\textit{e}}_2,\ldots,\textbf{\textit{e}}_n$ be the standard orthonormal basis for $\mathbb{R}^n$ and let $\textbf{\textit{e}}'=\left[\begin{array}{ccccc}
1&-1&-1&\cdots&-1
\end{array}\right]^T$. For $k\geq 1$, let $\textbf{\textit{j}}_k$ be the $k\times 1$ matrix with each entry equal to 1 and $\textbf{\textit{0}}_k$ be the $k\times 1$ zero matrix. We define the matrix $\p$ as
$$\p=\left[\begin{array}{ccccc}
\textbf{\textit{e}}&\textbf{\textit{e}}_2&\textbf{\textit{e}}_3&\ldots&\textbf{\textit{e}}_n
\end{array}\right].$$
It is easy to verify that the matrix $\p$ is nonsingular and its inverse is $$\p^{-1}=\left[\begin{array}{ccccc}
\textbf{\textit{e}}'&\textbf{\textit{e}}_2&\textbf{\textit{e}}_3&\ldots&\textbf{\textit{e}}_n
\end{array}\right].$$
We use $\s(i|i)$ to denote the principal submatrix of $\s$ obtained by deleting $i$-th row and the $i$-th column. Now we have the following theorem.
\begin{theorem}
\label{st:th1}
Let $\s$ be a stochastic matrix of order n. Then $\s$ is similar to the matrix
$$\left[\begin{array}{cc}
1&\x^T\\
\textbf{\textit{0}}_{n-1}&\textbf{\textit{B}}\end{array}\right]$$
where $\x^T=\left[\begin{array}{cccc}
s_{12}&s_{13}&\cdots&s_{1n}
\end{array}\right]$, and
$\textbf{\textit{B}}=\s(1|1)-\textbf{\textit{j}}_{n-1}\x^T.$
\end{theorem}
\begin{proof}
Let $\y=\left[\begin{array}{cccc}
s_{21}&s_{31}&\cdots&s_{n1}
\end{array}\right]^T$.
Then the matrices $\s$, $\p$, $\p^{-1}$ can be partitionoid as,
$$\s=\left[\begin{array}{cc}
s_{11}&\x^T\\
\y&\s(1|1)\end{array}\right],$$
$$\p=\left[\begin{array}{cc}
1&\textbf{\textit{0}}^T_{n-1}\\
\textbf{\textit{j}}_{n-1}&\textbf{\textit{I}}_{n-1} \end{array}\right],$$
$$\p^{-1}=\left[\begin{array}{cc}
1&\textbf{\textit{0}}^T_{n-1}\\
-\textbf{\textit{j}}_{n-1}&\textbf{\textit{I}}_{n-1} \end{array}\right].$$
Now
\begin{eqnarray*}
\p^{-1}\s\p&=&\left[\begin{array}{cc}
1&\textbf{\textit{0}}^T_{n-1}\\
-\textbf{\textit{j}}_{n-1}&\textbf{\textit{I}}_{n-1}\end{array}\right]
\left[\begin{array}{cc}
s_{11}&\x^T\\
\y&\s(1|1)\end{array}\right]
\left[\begin{array}{cc}
1&\textbf{\textit{0}}^T_{n-1}\\
\textbf{\textit{j}}_{n-1}&\textbf{\textit{I}}_{n-1}\end{array}\right]\\
&=&\left[\begin{array}{cc}
s_{11}&\x^T\\
\y-s_{11}\textbf{\textit{j}}_{n-1}&\s(1|1)-\textbf{\textit{j}}_{n-1}\x^T\end{array}\right]
\left[\begin{array}{cc}
1&\textbf{\textit{0}}_{n-1}\\
\textbf{\textit{j}}_{n-1}&\textbf{\textit{I}}_{n-1}\end{array}\right]\\
&=&\left[\begin{array}{cc}
\sum_{j=1}^n s_{1j}&\x^T\\
\y-s_{11}\textbf{\textit{j}}_{n-1}+\s(1|1)\textbf{\textit{j}}_{n-1}-\textbf{\textit{j}}_{n-1}\x^T\textbf{j}_{n-1}&\s(1|1)-\textbf{\textit{j}}_{n-1}\x^T\end{array}\right].
\end{eqnarray*}
For $i=2,3,\ldots,n$, we have
$(\p^{-1}\s\p)_{i1}=s_{i1}-s_{11}+\sum_{j=2}^ns_{ij} -\sum_{j=2}^ns_{1j}=0$ and hence the result follows.
\end{proof}
\begin{theorem}
\label{st:th2}
Let $\s=[s_{ij}]$ be a stochastic matrix of order n. Then any eigenvalue other than 1 is also an eigenvalue of the matrix
$$\s(k)=\s(k|k)-\textbf{\textit{j}}_{n-1}\textbf{s}(k)^T,\textbf{ }k=1,2,\ldots, n$$
where $\textbf{s}(k)^T=\left[\begin{array}{cccccc}
s_{k1}&\cdots&s_{k,k-1}&s_{k,k+1}&\cdots&s_{kn}
\end{array}\right]$ is the k-deleted row of $\s$.
\end{theorem}
\begin{proof}
If $k=1$ then the proof is straightforward from Theorem \ref{st:th1}.\\
For $k>1$, consider the permutation matrix $\p_k=\left[\begin{array}{cccccccc}
\textbf{\textit{e}}_2&\textbf{\textit{e}}_3&\cdots&\textbf{\textit{e}}_{k}&\textbf{\textit{e}}_1&\textbf{\textit{e}}_{k+1}&\cdots&\textbf{\textit{e}}_n
\end{array}\right]$. \\
Therefore, the matrix $\s$ is similar to the matrix
\begin{eqnarray*}
\p_k^{-1}\s\p_k=\left[\begin{array}{cc}
s_{kk}&\x^T\\
\y&\s(k|k)\end{array}\right],
\end{eqnarray*}
where $\x=\textbf{s}(k)=\left[\begin{array}{cccccc}
s_{k1}&\cdots&s_{k,k-1}&s_{k,k+1}&\cdots&s_{kn}
\end{array}\right]^T$\\ and $\y=\left[\begin{array}{cccccc}
s_{1k}&\cdots&s_{k-1,k}&s_{k+1,k}&\cdots&s_{nk}
\end{array}\right]^T$.\\
Now, applying Theorem \ref{st:th1} to $\p_k^{-1}\s\p_k$, we get that $\s$ is similar to the matrix
$$\left[\begin{array}{cc}
1&\textbf{s}(k)\\
\textbf{\textit{0}}_{n-1}&\s(k|k)-\textbf{\textit{j}}_{n-1}\textbf{s}(k)^T\end{array}\right].$$
Thus, any eigenvalue of $\s$, other than $1$, is also an eigenvalue of the matrix $\s(k)$, $k=1,2,\ldots,n$.
\end{proof}
\begin{theorem}
\label{st:th3}
(\textbf{Ger{\v s}gorin}\cite{Ger})
Let $\a=[a_{ij}]$ be an $n\times n$ complex matrix. Then the eigenvalues of $\a$ lie in the region
$$G_\a=\bigcup_{i=1}^n\Big{\{}z\in\mathbb{C}:|z-a_{ii}|\leq \sum_{j\neq i}|a_{ij}|\Big{\}}.$$
\end{theorem}
\begin{theorem}
\label{st:th4}
Let $\s$ be a stochastic matrix of order n. Then the eigenvalues of $\s$ lie in the region
$$\bigcap_{i=1}^n \Big{[}G_{\s(i)}\cup\{1\}\Big{]},$$
where
$G_{\s(i)}=\bigcup_{k\neq i}\{z\in\mathbb{C}:|z-s_{kk}+s_{ik}|\leq\sum_{j\neq k}|s_{kj}-s_{ij}|\}.$
\end{theorem}
\begin{proof}
By Theorem \ref{st:th2}, we have, for all $i$,
$$\sigma(\s)=\sigma(\s(i))\cup\{1\}.$$
By Ger{\v s}gorin disc theorem, $\sigma(\s(i))\subseteq G_{\s(i)}$, for $i=1,2,\ldots,n$.
Therefore, $$\sigma(\s)\subseteq\bigcap_{i=1}^n \Big{[}G_{\s(i)}\cup\{1\}\Big{]}.$$
Again, applying Theorem \ref{st:th3} to $G_{\s(i)}$, we get
\begin{eqnarray*}
G_{\s(i)}&=&\bigcup_{\substack{k=1,\\k\neq i}}^{n} \Big{\{}z\in\mathbb{C}:|z-\s(i)_{kk}|\leq \sum_{j\neq k}|\s(i)_{kj}|\Big{\}}\\
&=&\bigcup_{\substack{k=1,\\k\neq i}}^n\Big{\{}z\in\mathbb{C}:|z-s_{kk}+s_{ik}|\leq\sum_{j\neq k}|s_{kj}-s_{ij}|\Big{\}}.
\end{eqnarray*}
Hence, the proof is completed.
\end{proof}
\begin{rem}
Theorem \ref{st:th4} works nicely in some occasions even if Ger{\v s}gorin disc theorem fails to provide a non-trivial result. For example, let $\s$ be an irreducible stochastic matrix with at least one diagonal element zero. Then, by Ger{\v s}gorin disc theorem, $G_\s\supseteq \{z\in\mathbb{C}:|z|\leq1\}$. But, in this case, Theorem \ref{st:th4} may provide a non-trivial eigenvalue inclusion set (see Example~\ref{st:ex1} and Example~\ref{st:ex2}). Again, Theorem~\ref{st:th3a} and Theorem~\ref{st:th3b} always provide larger single discs, whereas, the eigenvalue inclusion set in Theorem~\ref{st:th4} is a union of smaller regions. Example~\ref{st:ex1} gives a numerical explanation to this interesting fact.
\end{rem}
\begin{example}
\label{st:ex1}Consider the $4\times 4$ stochastic matrix$$\s=\left[\begin{array}{cccc}
0.25&0.25&0.3&0.2\\
0&0.5&0.33&0.17\\
0.6&0.4&0&0\\
0.1&0.2&0.3&0.4
\end{array}\right].$$
Then we have $$\s(1)=\left[\begin{array}{ccc}
0.25&0.03&-0.03\\
0.15&-0.3&-0.2\\
-0.05&0&0.2
\end{array}\right],$$
$$\s(2)=\left[\begin{array}{ccc}
0.25&-0.03&0.03\\
0.6&-0.33&-0.17\\
0.1&-0.03&0.23
\end{array}\right],$$
$$\s(3)=\left[\begin{array}{ccc}
-0.35&-0.15&0.2\\
-0.6&0.1&0.17\\
-0.5&-0.2&0.4
\end{array}\right],$$ and
$$\s(4)=\left[\begin{array}{ccc}
0.15&0.05&0\\
-0.1&0.3&0.03\\
0.5&0.2&-0.3
\end{array}\right].$$
\begin{figure}[h]
\label{st:fig1}
\centering
\includegraphics[width=\textwidth]{stfig1.pdf}
\caption{The regions $G_\s(k)$, $k=1,2,3,4.$}
\end{figure}
The eigenvalues of $\s$ are $-0.307$, 0.174, 0.282, 1. Figure 1 shows that any eigenvalue other than 1 lies in each $G_{\s(k)}$. Also, from Figure 1, it is clear that $\sigma(\s)\subseteq \bigcap_{k=1}^4 [G_{\s(k)}\cup\{1\}]=G_{\s(1)}\cup \{1\}$. \\
Now, we estimate the eigenvalue inclusion sets in Theorem \ref{st:th3a} and Theorem \ref{st:th3b}. We have $s_1=0$, $s_2=0.2$, $s_3=0.3$, $s_4=0$ and $S_1=0.6$, $S_2=0.4$, $S_3=0.33$, $S_4=0.2$. Therefore,
$$\gamma=\max\{0.25,0.3,-0.3,0.4\}=0.4$$
and
$$\gamma'=max\{0.35,-0.1,0.33,-0.2\}=0.35.$$
By Theorem \ref{st:th3a}, any eigenvalue $\lambda\neq 1$ of $\s$ satisfies
$$|\lambda-0.4|\leq1.05.$$
Again, by Theorem \ref{st:th3b}, for any $\lambda\in\sigma(\s)\setminus\{1\}$, we have
$$|\lambda+0.35|\leq1.2.$$
It is easy to verify that $G_{\s(1)}$ is contained in both the discs.
Therefore, in this example, Theorem \ref{st:th4} works better than the other two.
\end{example}
\section{Bound for Randi\'c eigenvalues}
In this section, we give a nice bound for non-Perron eigenvalues of the Randi\'c matrix of a connected graph $\Gamma$. Since \textbf{\textit{R}} is symmetric, the eigenvalues of \textbf{\textit{R}}(or $\r$) are all real and lie in the closed interval $[-1,1]$. We arrange the eigenvalues of $\r$ as $$-1\leq\lambda_n\leq\lambda_{n-1}\leq\cdots\leq \lambda_2<\lambda_1=1.$$ Now we have the following theorem.
\begin{theorem}
\label{st:th5}
Let $\Gamma$ be a simple connected graph of order n. Then
$$-2+\max_{i\in\Gamma}\{\min_{k\neq i}\{\alpha_{ik}\},1\}\leq \lambda_n(\r)\leq\lambda_2(\r)\leq 2-\max_{i\in\Gamma}\{\min_{k\neq i}\{\beta_{ik}\},1\},$$
where, for $k\neq i$, $\alpha_{ik}$ and $\beta_{ik}$ are given by
$$\alpha_{ik}=\begin{cases}
\frac{1}{d_k}+\frac{2N(i,k)}{\max\{d_i,d_k\}},&\textit{ if }k\sim i\\\
\frac{2N(i,k)}{\max\{d_i,d_k\}},&\textit{ if }k\nsim i
\end{cases}$$
and
$$\beta_{ik}=\begin{cases}
\frac{1}{d_k}+\frac{2}{d_i}+\frac{2N(i,k)}{\max\{d_i,d_k\}},&\textit{ if }k\sim i\\
\frac{2N(i,k)}{\max\{d_i,d_k\}},&\textit{ if }k\nsim i.
\end{cases}$$
\end{theorem}
\begin{proof}
Let $\lambda$ be a non-Perron eigenvalue of $\r$.
By Theorem \ref{st:th2}, $\lambda$ is also an eigenvalue of $\r(i)=\r(i|i)-\textbf{\textit{j}}_{n-1}\textbf{r}(i)^T$, where $\textbf{r}(i)^T$ is the $i$-deleted row of $\r$, for $i=1,2,\ldots,n$. So $\lambda$ lies in the regions $G_{\r(i)}$ with $$G_{\r(i)}=\bigcup_{k\neq i}\Big{\{}z\in\mathbb{C}:|z+r_{ik}|\leq\sum_{j\neq k}|r_{kj}-r_{ij}|\Big{\}}=\bigcup_{\substack{k=1\\k\neq i}}^nG_{\r(i)}(k),$$
where $G_{\r(i)}(k)$ are the Ger{\v s}gorin discs for $\r(i)$. Now, we consider each individual disc of $G_{\r(i)}$. For the vertex $k\in \Gamma$, $k\neq i$, we calculate the centre and the radius of $G_{\r(i)}(k)$. Here two cases may arise.\\
\textbf{Case I: }Let $k\sim i$. Then $r_{ik}=\frac{1}{d_i}$ and $r_{ki}=\frac{1}{d_k}$. Thus, the disc $G_{\r(i)}(k)$ is given by
\begin{eqnarray*}
|z+\frac{1}{d_i}|&\leq&\sum_{j\neq i,k}|r_{kj}-r_{ij}|\\
&=&\sum_{\substack{j\sim i,\\j\sim k}}|r_{kj}-r_{ij}|+\sum_{\substack{j\nsim i,\\j\sim k}}|r_{kj}-r_{ij}|+ \sum_{\substack{j\sim i,\\j\nsim k}}|r_{kj}-r_{ij}|+\sum_{\substack{j\nsim i,\\j\nsim k}}|r_{kj}-r_{ij}|\\
&=&N(i,k)|\frac{1}{d_k}-\frac{1}{d_i}|+\frac{d_k-N(i,k)-1}{d_k}+\frac{d_i-N(i,k)-1}{d_i}+0\\
&=&2-\frac{1}{d_k}-\frac{1}{d_i}-\frac{2N(i,k)}{\max\{d_i,d_k\}}.
\end{eqnarray*}
\textbf{Case II: }If $k\nsim i$. Then $r_{ik}=0$ and $r_{ki}=0$. Thus, we have the disc
\begin{eqnarray*}
|z|&\leq&\sum_{j\neq i,k}|r_{kj}-r_{ij}|\\
&=&\sum_{\substack{j\sim i,\\j\sim k}}|r_{kj}-r_{ij}|+\sum_{\substack{j\nsim i,\\j\sim k}}|r_{kj}-r_{ij}|+ \sum_{\substack{j\sim i,\\j\nsim k}}|r_{kj}-r_{ij}|+\sum_{\substack{j\nsim i,\\j\nsim k}}|r_{kj}-r_{ij}|\\
&=&N(i,k)|\frac{1}{d_k}-\frac{1}{d_i}|+\frac{d_k-N(i,k)}{d_k}+\frac{d_i-N(i,k)}{d_i}+0\\
&=&2-\frac{2N(i,k)}{\max\{d_i,d_k\}}.
\end{eqnarray*}
Now, we consider the whole region $G_{\r(i)}$. Since the eigenvalues of $\r$ are real, by combining Case I and Case II, we obtain that any non-Perron eigenvalue $\lambda$ of $\r$ must satisfy
$$-2+\min_{k\neq i}\{\alpha_{ik}\}\leq \lambda\leq 2-\min_{k\neq i}\{\beta_{ik}\},$$
for all $i=1,2,\ldots,n.$\\
Therefore, by Theorem \ref{st:th4}, we obtain our required result.
\end{proof}
\begin{corollary}
Let $\Gamma$ be a simple connected graph. If $\rho_2$ and $\rho_n$ are the smallest and the largest nonzero normalized Laplacian eigenvalue of $\Gamma$, then $$-1+\max_{i\in\Gamma}\{\min_{k\neq i}\{\beta_{ik}\},1\}\leq\rho_2\leq\rho_n\leq 3-\max_{i\in\Gamma}\{\min_{k\neq i}\{\alpha_{ik}\},1\}, $$
where $\alpha_{ik}$, $\beta_{ik}$ are the constants defined as in Theorem \ref{st:th5}.
\end{corollary}
\begin{corollary}
Let $\gamma$ be a connected $r$-regular graph on n vertices. If $\lambda\neq 1$ be any eigenvalue of $\r$, then
$$-2+\frac{1}{r}\max_i\{\min_{k\neq i}\{\gamma_{ik}\},1\}\leq \lambda\leq2-\frac{1}{r}\max_i\{\min_{k\neq i}\{\delta_{ik}\},1\},$$
where
\begin{equation*}
\gamma_{ik}=\begin{cases}
1+2N(i,k),&\textit{ if }k\sim i\\
2N(i,k),&\textit{ if }k\nsim i
\end{cases}
\end{equation*}
and
\begin{equation*}
\delta_{ik}=\begin{cases}
3+2N(i,k),&\textit{ if }k\sim i\\
2N(i,k),&\textit{ if }k\nsim i.
\end{cases}
\end{equation*}
\end{corollary}
\begin{figure}[h]
\label{st:fig2}
\centering
\includegraphics[width=7cm]{stfig2.pdf}
\caption{A graph containing an edge which is not a part of a triangle. }
\end{figure}
Below we give an example where Theorem~\ref{st:th10} is improved by Theorem~\ref{st:th5}.
\begin{example}
\label{st:ex2}
Let $\Gamma$ be the graph as in Figure 2. The vertex degrees of $\Gamma$ are $d_1=4,$ $d_2=5,$ $d_3=d_4=d_5=d_6=4$, $d_7=3.$ The sets of neighbours of each vertex are given by
$$N_1=\{2,3,6,7\},$$
$$N_2=\{1,3,4,5,6\},$$
$$N_3=\{1,2,4,7\},$$
$$N_4=\{2,3,5,6\},$$
$$N_5=\{2,4,6,7\},$$
$$N_6=\{1,2,4,5\},$$
$$N_7=\{1,3,5\}.$$
Let $\alpha_i=\displaystyle \min_{k\neq i}\{\alpha_{ik}\}$ and $\beta_i=\displaystyle \min_{k\neq i}\{\beta_{ik}\}.$\\
The numbers of common neighbours of the vertex $2\in \Gamma$ with all other vertices are
$N(2,1)=2$, $N(2,3)=2$, $N(2,4)=3$, $N(2,5)=2$, $N(2,6)=3$ and $N(2,7)=3$. Also note that the vertex $2$ is adjacent to all other vertices other than the vertex $7$. Thus we obtain
\begin{eqnarray*}
\alpha_2&=&\min\Big{\{}\frac{1}{4}+\frac{4}{5},\frac{1}{4}+\frac{6}{5},\frac{6}{5}\Big{\}}\\
&=&1.05
\end{eqnarray*} and
\begin{eqnarray*}
\beta_2&=&\min\Big{\{}\frac{1}{4}+\frac{2}{5}+\frac{4}{5},\frac{1}{4}+\frac{2}{5}+\frac{6}{5},\frac{6}{5}\Big{\}}\\
&=&1.2
\end{eqnarray*}
Similarly, for all other vertices of $\Gamma$ we get,
$\alpha_1=0.75$, $\beta_1=1.25$, $\alpha_3=0.75$, $\beta_3=1.25$, $\alpha_4=0.75$, $\beta_4=1$, $\alpha_5=0.333$, $\beta_5=0.833$, $\alpha_6=0.75$, $\beta_6=1$, $\alpha_7=1$, $\beta_7=1$.\\
Therefore, using Theorem~\ref{st:th5}, we get
$$\lambda_2\leq 0.75\textit{ and }\lambda_7\geq -0.95.$$
Note that, since $N(5,7)=0$, the lower bound for $\lambda_7$ in (\ref{st:eq1}) becomes $-1$.
\end{example}
\section{Acknowledgement}
We are very grateful to the referees for detailed comments and suggestions, which helped to improve the manuscript. We also thankful to Ashok K.~Nanda for his kind suggestions during writing the manuscript. Ranjit Mehatari is supported by CSIR, India, Grant No.~09/921(0080)/2013-EMR-I.
| {
"timestamp": "2016-05-02T02:03:51",
"yymm": "1601",
"arxiv_id": "1601.07736",
"language": "en",
"url": "https://arxiv.org/abs/1601.07736",
"abstract": "A square matrix is called stochastic (or row-stochastic) if it is non-negative and has each row sum equal to unity. Here, we constitute an eigenvalue localization theorem for a stochastic matrix, by using its principal submatrices. As an application, we provide a suitable bound for the eigenvalues, other than unity, of the Randić matrix of a connected graph.",
"subjects": "Combinatorics (math.CO)",
"title": "An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846678676152,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7092019644043693
} |
https://arxiv.org/abs/1010.0232 | Eigenvectors for a random walk on a hyperplane arrangement | We find explicit eigenvectors for the transition matrix of a random walk due to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and Diaconis' analysis of its stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelfand and Varchenko. | \section{Introduction}
In 1999, Bidegare, Hanlon and Rockmore~\cite{BHR99} introduced a
simultaneous generalization of several well-studied discrete Markov chains.
Let ${\mathcal A}$ be an arrangement of $n$ linear
hyperplanes in $W={\mathbb R}^\ell$, and let ${\mathcal C}$ denote the set of chambers:
i.e., the connected components of the complement of ${\mathcal A}$ in $W$.
They construct a random walk on ${\mathcal C}$, which we will follow \cite{BD98}
in calling the BHR random walk, by means of the face product,
defined below. By choosing the hyperplane arrangement suitably,
one obtains as special cases the Tsetlin library (``move-to-front rule''),
Ehrenfests' urn, and various card-shuffling models~\cite{BHR99,BD98}.
A key insight in this construction and main result of \cite{BHR99}
is that the eigenvalues of the transition matrix
can be expressed simply in terms of the combinatorics
of the arrangement ${\mathcal A}$. This is useful for bounding the rate of
convergence to the stationary distribution: Brown and Diaconis
make this analysis in \cite{BD98}, show that the transition matrix is
diagonalizable, and give an explicit
description of the random walk's stationary distribution.
Subsequently Brown~\cite{Br00} generalized the BHR random walk
to the setting of certain semigroup algebras, showing in particular
that the diagonalization result held there as well. He
describes projection operators onto each eigenspace, which in principle
provides a description of the eigenvectors of the random walk's
transition matrix. However, the general expression is necessarily somewhat
complicated.
The purpose of this paper, then, is to provide a relatively straightforward
description of the original BHR random walk's eigenvectors. It turns out
that the combinatorial Heaviside functions of Gel$'$ fand and
Varchenko~\cite{VG87} behave well with respect to the face product and
the BHR random walk, so we highlight their role in this problem.
Using the description of the stationary distribution from \cite{BD98},
the main result here, Theorem~\ref{th:main}, describes a spanning set for
each eigenspace in terms of flags in the intersection lattice.
\section{Background and notation}
\subsection{The face algebra}\label{ss:face}
Let ${\mathcal A}$ be an arrangement of $n$ hyperplanes in ${\mathbb R}^\ell$.
We will assume throughout that ${\mathcal A}$ is central and essential: that
is, the intersection of the hyperplanes ${\mathcal A}$ equals the origin in ${\mathbb R}^\ell$.
We will follow the notational conventions of the standard reference
for hyperplane arrangements, \cite{OTbook}. In particular, let $L({\mathcal A})$
denote the lattice of intersections, ordered by reverse inclusion, and
${\mathcal F}={\mathcal F}({\mathcal A})$ the face semilattice, also ordered by reverse inclusion.
For $0\leq p\leq \ell$, let $L_p({\mathcal A})$ and ${\mathcal F}_p({\mathcal A})$ denote the subspaces
and faces, respectively, of codimension $p$. In particular, the set of
chambers ${\mathcal C}={\mathcal C}({\mathcal A})={\mathcal F}_0({\mathcal A})$.
For a face $F\in{\mathcal F}$, let $\abs{F}$ denote the smallest subspace in
$L({\mathcal A})$ containing $F$. Recall that ${\mathcal A}^X$ denotes the arrangement
in $X$ of hyperplanes $\set{H\cap X\colon H\in{\mathcal A},X\not\subseteq H}$,
whenever $X\in L({\mathcal A})$, and ${\mathcal A}_X$ denotes the subarrangement
$\set{H\in{\mathcal A}\colon X\subseteq H}$. We will identify faces (and chambers) of
${\mathcal A}^X$ with faces $F\in{\mathcal F}({\mathcal A})$ for which $\abs{F}\geq X$ (and
$\abs{F}=X$, respectively):
\begin{equation}\label{eq:loc}
{\mathcal F}({\mathcal A}^X)\cong \set{F\in{\mathcal F}({\mathcal A})\colon\abs{F}\geq X}.
\end{equation}
For hyperplanes $H\in {\mathcal A}$, let $F_H$ be $0,\pm 1$
depending on
whether $F$ is contained in $H$, on the positive side, or the
negative side, respectively. We will abbreviate the values of $F_H$ with
$0,\px,\mx$. A face $F$ is uniquely determined
by the sign sequence $(F_H)_{H\in{\mathcal A}}$.
We recall the face product of two faces $F$ and $G$
can be described by its sign sequence:
\begin{equation}\label{eq:vprod}
(FG)_H=\begin{cases}
F_H & \text{if $F_H\neq0$;}\\
G_H & \text{otherwise.}
\end{cases}
\end{equation}
From the definition, $FF=F$ for any $F$, and $FGF=FG$ for any faces $F,G$
(the ``deletion property'': see \cite{Br00}.)
For a fixed arrangement ${\mathcal A}$, let $A=R[{\mathcal F}]$ denote its face algebra,
introduced in \cite{BD98}. Additively, $A$ is the free $R$-module on
${\mathcal F}$, and multiplication is given by linearly extending the face product.
Following \cite{BD98}, let $V({\mathcal A})$ denote the free $R$-module on ${\mathcal C}({\mathcal A})$:
then the face product makes $V({\mathcal A})$ a left $A$-module. More generally,
following \cite[Section~5C]{BD98}, we can make $V({\mathcal A}^X)$ a left $A$-module
for any $X\in L({\mathcal A})$: for $F\in{\mathcal F}$ and $G\in{\mathcal C}({\mathcal A}^X)$, set
\begin{equation}\label{eq:localaction}
FG=\begin{cases}
FG & \text{if $\abs{F}\geq X$;}\\
0 & \text{otherwise.}
\end{cases}
\end{equation}
\subsection{The zonotope of a real arrangement}
\begin{figure}
\includegraphics[height=1.5in]{zono1.pstex}
\caption{Zonotope for $f_1=x$, $f_2=y$, $f_3=x-y$.}\label{fig:one}
\end{figure}
As Brown and Diaconis~\cite{BD98} observe,
it is useful in this setting
to consider the zonotope of a real arrangement. Let
$f_H$ denote a linear equation defining the hyperplane $H$, for each
$H\in{\mathcal A}$. The zonotope $Z_{\mathcal A}$ is, by definition, the Minkowski sum
of intervals,
\[
Z_{\mathcal A}=\sum_{H\in{\mathcal A}}[-f_H,f_H].
\]
This is a polytope in $W^*$: we refer to \cite{Ziebook} for details.
Its most relevant feature here
is that the face poset of $Z_{\mathcal A}$ (ordered by inclusion)
is isomorphic to ${\mathcal F}({\mathcal A})$: the isomorphism arises by identifying the
outer normal fan of the zonotope with the arrangement. If $F$ is a
face of ${\mathcal A}$, then, let $\hat{F}$ denote the corresponding face of the
zonotope. Each face $\hat{F}$ is an affine translate of a
zonotope of a closed subarrangement: for $F\in{\mathcal F}$, let $X=\abs{F}$ and
$v_F=\sum_{H\in{\mathcal A}}(F_H)f_H$. Then $\hat{F}=v_F+Z_{{\mathcal A}_X}$:
see, for example, discussion in \cite{McM71}.
For the sake of intuition, we indicate one way to visualize the face
product in the zonotope world.
\begin{definition}\label{def:retract}
Suppose $P$ is a polytope in $W^*$ and $v\in W^*$ is nonzero. Define a
map $r_{P,v}\colon P+[-v,v]\to P+v$ as follows. If $p\in P+[-v,v]$,
let $\lambda$ be maximal for which $p=x+\lambda v$, where $x\in P$
and $\lambda\leq 1$. Set $r_{P,v}(p)=x+v$.
Clearly $r_{P,v}$ is piecewise-linear and fixes $P+v$ pointwise.
\end{definition}
\begin{proposition}\label{prop:Zprod}
For each face $F$ of ${\mathcal A}$, there is a piecewise-linear retract $p_F$
of $Z_{\mathcal A}$ onto $\hat{F}$, with $p_F(\hat{G})=\widehat{FG}$, for all
$G\in {\mathcal F}({\mathcal A})$.
\end{proposition}
\begin{proof}
We will write $p_F$ as the composition of the retracts from
Definition~\ref{def:retract},
one for each hyperplane of ${\mathcal A}$ not containing $F$.
Again, let $X=\abs{F}$. Let $S_1=\set{f_H\colon F_H=0}$ and
$S_2=\set{(F_H)f_H\colon F_H\neq 0}$. Then
\begin{eqnarray*}
Z_{\mathcal A}&=&\sum_{v\in S_1}[-v,v]+\sum_{v\in S_2}[-v,v]\\
&=&Z_{{\mathcal A}_X}+\sum_{v\in S_2}[-v,v],
\end{eqnarray*}
and $v_F=\sum_{v\in S_2}v$. Put the elements of $S_2$ in any order,
writing $S_2=\set{v_1,v_2,\ldots,v_k}$. Let $P_0=Z_{{\mathcal A}_X}+v_F$, and let
$P_i=P_{i-1}+[-2v_i,0]$ for $1\leq i\leq k$.
Then
\[
\hat{F}=P_0\subseteq P_1\subseteq\cdots\subseteq P_k=Z_{\mathcal A},
\]
and the following composite collapses along
the vectors in $S_2$, one at a time:
\[
p_F=r_{P_0-v_1,v_1}\circ r_{P_1-v_2,v_2}\circ\cdots\circ r_{P_{k-1}-v_k,v_k}
\colon Z_{\mathcal A}\to \hat{F}.
\]
By construction, this is a piecewise-linear retract onto $\hat{F}$.
To see that $p_F(\hat{G})=\widehat{FG}$ for any face $G$, let $S\in(\pm1)^{\mathcal A}$
be an arbitrary sequence of nonzero signs,
and define $v_S=\sum_{H\in{\mathcal A}}S_H f_H$. Then
$
r_{P_{i-1}-v_i,v_i}(v_S)=v_{S'},
$
where
\[
(S')_H=\begin{cases} S_H&\text{if $f_H\neq \pm v_i$;}\\
c & \text{if $f_H=c v_i$.}\end{cases}
\]
It follows that $p_F(\hat{C})=\widehat{FC}$ for chambers $C\in{\mathcal C}({\mathcal A})$.
By writing an arbitrary face $\hat{G}$ as the convex hull of the
vertices $\hat{C}$
for which $C\leq G$, one then obtains $p_F(\hat{G})=\widehat{FG}$ for all $G$.
\end{proof}
Note that the map $p_F$ is not uniquely defined, and depends on a choice of
order. See Figure~\ref{fig:retracts} for examples.
\begin{figure}
\centering
\subfigure[$p_F$ with $F=\mx0\px\px$]{
\includegraphics[height=1.5in]{zono6.pstex}
}
\subfigure[$p_F$ with $F=00\px\px0$]{
\includegraphics[height=1.6in]{zono3d.pstex}
}
\caption{Retracts onto faces of zonotopes of dimension $2$ and $3$}
\label{fig:retracts}
\end{figure}
\subsection{The BHR random walk}
Let ${\mathcal A}$ be an arrangement, and $w\colon{\mathcal F}\to{\mathbb R}$ a discrete probability
distribution on the faces of ${\mathcal A}$. The random walk introduced in
\cite{BHR99} is a random walk on chambers, taking $C\in{\mathcal C}$ to $FC$ with
probability $w_F$. Its transition matrix $K=K_{\mathcal A}$ can be regarded as a
linear endomorphism of the vector space with basis ${\mathcal C}$, given by
\begin{equation}\label{eq:BHR}
K(C)=\sum_{F\in{\mathcal F}}w_FFC.
\end{equation}
An eigenvector of $K$ with eigenvalue $1$ gives a stationary distribution
for the random walk. Brown and Diaconis~\cite[Theorem~2]{BD98} find that,
with an assumption of nondegeneracy, the eigenspace
is $1$-dimensional, which is to say
the stationary distribution is unique. In this
case, it is given by sampling faces without replacement: explicitly, one
sums over all permutations of the faces to obtain
\begin{equation}\label{eq:stationary}
\pi_C=\sum_{\substack{\sigma\in S_N\colon \\
C=F_{\sigma(1)}\cdots F_{\sigma(N)}}} \prod_{1\leq p\leq N}
\frac{w_{F_{\sigma(p)}}}{1-\sum_{i<p}w_{F_{\sigma(i)}}},
\end{equation}
where $N=\abs{{\mathcal F}}$, the number of faces. The complete list of eigenvalues
is given as follows, where $\mu$ denotes the M\"obius function of $L({\mathcal A})$.
For each $X\in L({\mathcal A})$, let
\begin{equation}\label{eq:lambda}
\lambda_X=\sum_{F\colon\abs{F}\geq X} w_F.
\end{equation}
(Equivalently, $\lambda_X=\sum_{F\in{\mathcal F}({\mathcal A}^X)} w_F$, by \eqref{eq:loc}.)
\begin{theorem}[\cite{BHR99,BD98}]\label{th:BHR}
For each $X\in L_p({\mathcal A})$, for $0\leq p\leq \ell$,
the matrix $K$ has an eigenspace of multiplicity
$(-1)^p\mu(W,X)$ with eigenvalue $\lambda_X$.
\end{theorem}
The main result here is a corresponding basis of eigenvectors for each
eigenvalue, Theorem~\ref{th:main}. For this, it is convenient to
regard the distribution weights $w_F$ as indeterminates as in \cite{Br00},
diagonalize, and then specialize afterwards: let
$R={\mathbb R}(w_F\colon F\in {\mathcal F})$, the fraction field
of polynomials in the variables $w_F$. In doing so, we relax the
condition that the weights sum to $1$, so \eqref{eq:stationary}
needs to be adjusted accordingly. Let $q\in V({\mathcal A})$ be the vector
whose coordinate on chamber $C$ is given by
\begin{equation}\label{eq:1eivec}
q_C=\sum_{\substack{\sigma\in S_N\colon \\
C=F_{\sigma(1)}\cdots F_{\sigma(N)}}} \prod_{p=1}^N
\big(\sum_{i=p}^N w_{F_{\sigma(i)}}\big)^{-1}.
\end{equation}
\begin{lemma}\label{lem:eivec}
For any arrangement ${\mathcal A}$, the
vector $q$ is a $\lambda_W$-eigenvector of $K$.
\end{lemma}
\begin{proof}
We provide a direct calculation in lieu of adapting the corresponding
result from \cite{BD98}. For succinctness, let
\[
f(x_1,\ldots,x_N)=\prod_{i=1}^N\big(\sum_{i=p}^N x_i\big)^{-1}
\]
for any choice of $x_i$'s, and abbreviate $f(\sigma):=f(w_{F_{\sigma(1)}},
\ldots,w_{F_{\sigma(N)}})$ for any permutation of the faces $\sigma\in S_N$.
By clearing denominators, one may verify the identity
\begin{align}\label{eq:identity}
f(x_1,\ldots,x_N)+&f(x_2,x_1,x_3,\ldots,x_N)+\cdots\nonumber \\
&+f(x_2,\ldots,x_N,x_1)=f(x_1,\ldots,x_N)\cdot\big(\sum_{i=1}^N x_i\big)/x_1.
\end{align}
For $1\leq i\leq N$, let $\sigma_i$ denote the $i$-cycle
$(1,2,\ldots,i)\in S_N$. Then \eqref{eq:identity} states that
\begin{equation}\label{eq:identity2}
\sum_{i=1}^N f(\sigma\sigma_i^{-1})=(\lambda_W/x_1)f(\sigma),
\end{equation}
since $\lambda_W=\sum_{i=1}^N w_{F_i}$.
Now $Kq= \sum_{F\in{\mathcal F}} w_Fq_C(FC)$. For each $C\in {\mathcal C}$, we compute:
\begin{eqnarray*}
(Kq)_C&=& \sum_{F,C'\colon C=FC'} w_Fq_{C'}\\
&=&
\sum_{\substack{F\in{\mathcal F},\;\sigma\in S_N\colon
\\ C=FF_{\sigma(1)}\cdots F_{\sigma(N)}}}
w_F f(\sigma)\\
&=& \sum_{\substack{\sigma\in S_N\colon \\
C=F_{\sigma(1)}\cdots F_{\sigma(N)}}}
\sum_{i=1}^N w_{F_{\sigma(1)}}f(\sigma\sigma_i^{-1}),\text{~by the deletion property,
\S\ref{ss:face},}\\
&=&\lambda_W q_C,\text{~by \eqref{eq:identity2}.}
\end{eqnarray*}
\end{proof}
\begin{remark}
As written, the weights $\set{w_F}$ in \eqref{eq:1eivec} only admit specializations
to nonzero real numbers. One may clear denominators to obtain a general
polynomial expression for an eigenvector, however this is clumsy to write
in general.
\end{remark}
\subsection{Combinatorial Heaviside functions}
We will recover Theorem~\ref{th:BHR}, together with eigenvectors, by
exploiting some fundamental structural results of Varchenko and
Gel$'$ fand~\cite{VG87}, which we briefly describe here. For
more details, see \cite{De02}.
The Varchenko-Gel$'$ fand ring of ${\mathcal A}$ is defined additively to
be simply the space of linear functionals on chambers, $V^*$. The
ring structure is given by coordinatewise multiplication.
The interest lies in the choice of generators: for each hyperplane
$H\in{\mathcal A}$, let $x_H\in V^*$ be the function defined by
\[
x_H(C)=\begin{cases}
1& \text{if $C$ is on the positive side of $H$;}\\
0& \text{otherwise.}
\end{cases}
\]
For a set of hyperplanes $I\subseteq{\mathcal A}$, let $x_I$ be the monomial
$x_I=\prod_{H\in I} x_H$. (Since each $x_H$ is idempotent,
we only need to consider square-free monomials.) Let $1\in V^*$ be the
function given by $1(C)=1$ for all chambers $C\in{\mathcal C}$.
In \cite{VG87}, it is shown that the Varchenko-Gel$'$ fand ring
admits a presentation much like the Orlik-Solomon algebra, with
generators $x_H$ and certain combinatorial relations: see
\cite{Pr06} for an interpretation that compares the two.
Unlike the Orlik-Solomon algebra, however,
the relations amongst the generators are inhomogeneous, so it is
useful to define a degree filtration by letting
\[
P_pV^*=\set{f\in V^*\colon \text{$f$ can be written as a polynomial in
$x_H$'s of degree at most $p$.}}
\]
By \cite[Theorem~1]{VG87},
\[
V^*=P_\ell V^*\supseteq P_{\ell-1} V^*\supseteq\cdots P_0 V^*\supsetneq 0,
\]
with the function $1$ spanning $P_0$. The filtration is ``natural'' in
the sense that, if ${\mathcal B}$ is a subarrangement of ${\mathcal A}$, containment
gives a map of chambers ${\mathcal C}({\mathcal A})\to {\mathcal C}({\mathcal B})$. This induces a map
$V({\mathcal B})^*\to V({\mathcal A})^*$, which is easily seen to preserve the degree filtration.
Now let $\gr_p V^*=P_p V^*/P_{p-1} V^*$, for $0\leq p\leq \ell$.
Let $b_p=(-1)^p\sum_{X\in L_p({\mathcal A})}\mu(W,X)$, the $p$th Betti number of
the arrangement. Then the degree filtration satisfies an analogue of
Brieskorn's Lemma for arrangements:
\begin{theorem}[Theorem~3, Corollaries 2,3 in \cite{VG87}]\label{th:vg}
For $0\leq p\leq \ell$, $\gr_p V^*$ is a free module, of rank equal
to $b_p({\mathcal A})$. More precisely, there are isomorphisms
\begin{equation}\label{eq:brieskorn}
\gr_p V({\mathcal A})^*\cong\bigoplus_{X\in L_p({\mathcal A})}\gr_p V({\mathcal A}_X)^*.
\end{equation}
induced by the inclusion of the arrangement ${\mathcal A}_X$ into ${\mathcal A}$.
\end{theorem}
\begin{example}\label{ex:threelines}
Consider the arrangement of lines $\set{x,y,x-y}$ through the origin in ${\mathbb R}^2$,
as in Figure~\ref{fig:one}.
Call the lines $H_1,H_2,H_3$ in this order. Bases for $\gr_p V^*$ are:
\[
\begin{array}{r|l}
p=2 & x_{H_1}x_{H_2},\;x_{H_1}x_{H_3},\\
p=1 & x_{H_i}\colon 1\leq i\leq 3,\\
p=0 & 1.
\end{array}
\]
For example,
the function $x_{H_1}x_{H_3}$ takes the value $1$ on the chamber $x>0$, $x>y$,
and zero elsewhere.
\end{example}
\subsection{The dual filtration}\label{ss:dualfiltr}
The degree filtration defines an orthogonal,
decreasing filtration on the dual space,
$V^{**}\cong V$: following \cite{VG87}, let
\[
W^pV=\set{v\in V\colon f(v)=0\text{~for all $f\in P_{p-1}V^*$.}}
\]
Then
\[
V=W^0V\supseteq W^1V\supseteq\cdots\supseteq W^{\ell+1}V=0.
\]
If ${\mathcal B}$ is a subarrangement of ${\mathcal A}$ and $i\colon {\mathcal C}({\mathcal A})\to
{\mathcal C}({\mathcal B})$ is the induced map of chambers, the ``natural'' map
$V({\mathcal A})\to V({\mathcal B})$ extends $i$ linearly. Dually, this map
preserves the $W$-filtration.
Let $\gr^pV=W^pV/W^{p+1}V$, for $0\leq p\leq\ell$. The dual version
of Theorem~\ref{th:vg} reads as follows:
\begin{proposition}
For $0\leq p\leq \ell$, we have $\gr_p V^*\cong(\gr^p V)^*$.
So $\gr^p V$ is also free of rank $b_p$, and admits a decomposition
\begin{equation}\label{eq:brieskorn'}
\gr^p V({\mathcal A})\cong\bigoplus_{X\in L_p({\mathcal A})}\gr^p V({\mathcal A}_X).
\end{equation}
\end{proposition}
\begin{proof}
We prove the first assertion, from which the rest follows directly.
Let $e\colon V\to V^{**}$ be the natural isomorphism.
If $x\in W^pV$, then $e(x)$ restricts to a map $P_pV^*\to R$ by $x(f)=f(x)$.
If $e(x)=0$, this means $f(x)=0$ for all $f\in P_pV^*$, so $x\in W^{p+1}V$.
On the other hand, restriction of functions gives a surjective map
${\rm res}\colon (P_pV^*)^*\to (P_{p-1}V^*)^*$. Putting this together, we have
shown that the sequence
\begin{equation}\label{eq:dualityseq}
\xymatrix{
0\ar[r] & W^{p+1}V\ar[r] & W^pV\ar[r]^{e} &
(P_pV^*)^*\ar[r]^{{\rm res}} & (P_{p-1}V^*)^*\ar[r] & 0
}
\end{equation}
is exact except possibly at $(P_pV^*)^*$.
For this, suppose $\phi$ is in the kernel of ${\rm res}$. Let $x=e^{-1}(\phi)$.
That $\phi$ restricts to zero means
$f(x)=0$ for all $f\in P_{p-1}V^*$, which is to say that $x\in W^pV$,
and \eqref{eq:dualityseq} is exact.
The cokernel of $W^{p+1}V\hookrightarrow W^pV$ is $\gr^pV$, by definition.
Since $\gr_pV^*$ is
free (Theorem~\ref{th:vg}), the kernel of the map ${\rm res}$ is $(\gr_pV^*)^*$.
It follows that the restriction of $e$ induces an isomorphism
$\gr^pV\cong(\gr_pV^*)^*$.
\end{proof}
\subsection{Flag cochains}
The dual counterparts of the combinatorial Heaviside functions are
the {\em flag cochains} of \cite{VG87}, which we reformulate slightly
following \cite{De02}. First, for a ranked poset $P$, for $p\geq0$
let $\flag_p(P)$ be the set of $p$-flags: that is,
chains $x_0\lx x_1\lx \cdots\lx x_p$ in $P$ where $x_i$ has rank $i$ for
$0\leq i\leq p$.
For a fixed arrangement ${\mathcal A}$, let $\Fl_p=\Fl_p({\mathcal A})$
be the free abelian group on $\flag_p(L({\mathcal A}))$, modulo the relations
\[
\sum_{Y\colon X_{i-1}< Y< X_{i+1}}(X_0\lx \cdots\lx X_{i-1}\lx Y\lx X_{i+1}
\cdots\lx X_p),
\]
for each flag $(X_0\lx\cdots\lx X_p)$ and index $i$, $0<i<p$.
The groups $\Fl_p$ are isomorphic to the homology groups of
the complexified complement of ${\mathcal A}$: see \cite{SV91}. In particular,
\begin{equation}\label{eq:dualb}
\Fl_p({\mathcal A})\cong\bigoplus_{X\in L_p({\mathcal A})}\Fl_p({\mathcal A}_X),
\end{equation}
a dual formulation of Brieskorn's Lemma. The rank of the summand
indexed by $X$ is $\abs{\mu(W,X)}$.
Let $\FF$ be a flag in ${\mathcal F}({\mathcal A})$: then
$\FF=(F_0\lx F_1\lx\cdots\lx F_p)$ for some $p$, where each
$F_i$ is a face of codimension $i$. Since we continue to assume that
${\mathcal A}$ is a central arrangement, each face $F$ has an antipodally opposite
face, which we denote by $\overline{F}$.
Define an element of $V$ using the following expression in the face algebra:
\begin{equation}\label{eq:b}
b(\FF)=F_p(F_{p-1}-\overline{F_{p-1}})(F_{p-2}-\overline{F_{p-2}})
\cdots(F_0-\overline{F_0}).
\end{equation}
Varchenko and Gel$'$ fand~\cite{VG87} find that the flag cochains
$b(\FF)$ span $W^pV$, for each $p$. In fact, they do so in a way
compatible with the Brieskorn decompositions \eqref{eq:brieskorn'},
\eqref{eq:dualb}.
In order to indicate how this goes, we need some additional notation.
\begin{example}\label{ex:b}
For the arrangement of Example~\ref{ex:threelines},
let $\FF=(\px\px\mx\lx0\px\mx\lx000)$.
Then $b(\FF)=C_{\px\px\mx}-C_{\mx\px\mx}-C_{\px\mx\px}+C_{\mx\mx\px}$.
\end{example}
For any $\FF\in\flag_p({\mathcal F})$, let $\abs{\FF}$ denote the flag $\XX$
in the intersection lattice with $X_i=\abs{F_i}$. Regard $F_p$ as a
chamber of ${\mathcal A}^{X_p}$, in order to define a map
\[
f\colon \flag_p({\mathcal F}({\mathcal A}))\to
\bigsqcup_{X\in L_p({\mathcal A})}\flag_p(L({\mathcal A}_X))\times {\mathcal C}({\mathcal A}^X)
\]
by $f(\FF)=(\abs{\FF},F_p)$. Consider the fibres of $f$.
If $f(\FF)=(\XX,F)$, then $F_p=F$. There are two possibilities for
$F_{p-1}$, however, obtained by moving away from $F$ inside $X_{p-1}$
in either of two directions. Once $F_{p-1}$ is chosen, there are
two possiblities for $F_{p-2}$, and so on. By inspecting
\eqref{eq:b}, one finds that flag cochains $b(\FF)$ differ at most by
a sign on flags in the same fibre of the map $f$.
In order to reconcile the choice of signs, choose an orientation of each
face of the zonotope $Z_{\mathcal A}$ so that parallel faces have the same orientation,
but arbitrarily otherwise. (This is equivalent to choosing a coorientation
of each element of $L({\mathcal A})$, as in \cite{VG87}, but easier to draw.)
Then for each covering relation $F<G$ in ${\mathcal F}({\mathcal A})$, let
${\varepsilon}(F,G)=\pm1$ according to whether or not the orientations on $F$ and
$G$ agree. Let ${\varepsilon}(\FF)=\prod_{i=0}^{p-1}{\varepsilon}(F_i,F_{i+1})$.
The last result we need to recall is the dual formulation of
\cite[Theorem~8]{VG87}.
\begin{theorem}
For $0\leq p\leq \ell$, there is a well-defined map depending on the
choices of orientations,
\begin{equation}\label{eq:noncanonical}
\pi_p\colon W^p V({\mathcal A})\to \Fl_p({\mathcal A}),
\end{equation}
for which $\pi_p(b(\FF))={\varepsilon}(\FF)\abs{\FF}$,
for all flags $\FF\in\flag_p({\mathcal F})$.
The kernel of $\pi_p$ is $W^{p+1}V({\mathcal A})$.
\end{theorem}
There is also a map in the other direction, given by \cite[Theorem 18.3.3]{Va93}:
\begin{proposition}[\cite{Va93}]
For each $p$, the map
\begin{equation}\label{eq:defphi}
\phi_p\colon\bigoplus_{X\in L_p({\mathcal A})}
\Fl_p(L({\mathcal A}_X))\otimes_R V({\mathcal A}^X)\to W^pV({\mathcal A})
\end{equation}
given by sending $\XX\otimes F$ to ${\varepsilon}(\FF)b(\FF)$ is well-defined, where
$\FF$ is any flag with $f(\FF)=(\XX,F)$.
\end{proposition}
\begin{example}[Example~\ref{ex:b}, continued]\label{ex:c}
Orient the faces of the zonotope as shown in Figure~\ref{fig:b}. For
the flag $\XX=({\mathbb R}^2,H_1,0)$, the only choice for $F$ is the face $000$.
Picking $\FF=(\px\px\mx\lx0\px\mx\lx000)$, we see ${\varepsilon}(\FF)=-1$, and
coordinates of $\phi(\XX\otimes C_{000})$ are as indicated. The
simplex corresponding to $\FF$ in the barycentric subdivision of $Z_{\mathcal A}$ is
shaded in Figure~\ref{fig:b1}.
For $\XX=({\mathbb R}^2,H_1)$, there are two choices for $F$. We find
$\phi(\XX\otimes C_{0\mx\px})=C_{\px\mx\px}-C_{\mx\mx\px}$ and
$\phi(\XX\otimes C_{0\px\mx})=C_{\px\px\mx}-C_{\mx\px\mx}$.
\end{example}
\begin{figure}
\subfigure[A flag cochain]{\includegraphics[height=1.9in]{zono4.pstex}
\label{fig:b1}}\qquad
\subfigure[A relation in $\Fl_2({\mathcal A})$]{\includegraphics[height=1.9in]{zono5.pstex}}
\caption{Flag cochains}\label{fig:b}
\end{figure}
\section{Eigenvectors for the random walk}
\subsection{The face algebra action on flag cochains}
We recall that $A=R[{\mathcal F}]$ denoted the face algebra. The domain of
the map $\phi_p$ in \eqref{eq:defphi} has the structure of a left $A$-module
by the action of $A$ on each $V({\mathcal A}^X)$. We see in this section that the
codomain of $\phi_p$ is also an $A$-module (Corollary~\ref{cor:submodule}), and
$\phi_p$ is an $A$-module homomorphism (Lemma~\ref{lem:hom}).
First, consider
applying the face product to each element of a flag $\FF\in\flag_p({\mathcal F})$.
Clearly for each $F\in{\mathcal F}$, we have $FF_0\leq \cdots\leq FF_p$.
By \cite[Lemma~3.2]{De02}, however,
this is a flag if and only if $\abs{F}\geq\abs{F_p}$; otherwise,
$FF_k=FF_{k+1}$ for some $k$ with $0\leq k\leq p-1$.
As usual, we regard the faces $F$ with $\abs{F}\geq\abs{F_p}$
as chambers of ${\mathcal A}^\abs{F_p}$.
In this case, let $F\FF$ denote the flag $FF_0\lx\cdots\lx FF_p$.
\begin{proposition}\label{prop:dotwithb}
For each $F\in{\mathcal F}$ and $\FF\in\flag_p({\mathcal F})$, we have
\[
Fb(\FF) = \begin{cases}
b(F\FF) & \text{if $\abs{F}\geq \abs{F_p}$;}\\
0 & \text{otherwise.}
\end{cases}
\]
\end{proposition}
\begin{proof}
By the deletion property,
\begin{eqnarray*}
Fb(\FF) &=& FF_p(F_{p-1}-\overline{F_{p-1}})(F_{p-2}-\overline{F_{p-2}})
\cdots(F_0-\overline{F_0})\\
&=&FF_p(FF_{p-1}-F\overline{F_{p-1}})(FF_{p-2}-F\overline{F_{p-2}})
\cdots(FF_0-F\overline{F_0}).\\
\end{eqnarray*}
If $\abs{F}\geq \abs{F_p}$, then $\abs{F}\geq\abs{F_i}$ for each $i$,
so $F\overline{F_i}=\overline{FF_i}$ for each $i$, and we obtain $b(F\FF)$.
Otherwise, $FF_k=FF_{k+1}$ for
some $k$. Since $F\overline{F}=F$ for any $F$, it is easy to check
that $(FF_k-F\overline{F_k})(FF_k-F\overline{F_k})=0$, from which it follows
that $Fb(\FF)=0$ as well.
\end{proof}
Since $W^pV$ is spanned by vectors $b(\FF)$ for $\FF\in\flag_p({\mathcal F})$, we
see the filtration is compatible with the face product. That is,
\begin{corollary}\label{cor:submodule}
For $0\leq p\leq \ell$, $W^pV$ is an $A$-submodule of $V$.
\end{corollary}
\begin{lemma}\label{lem:hom}
The map $\phi$ of \eqref{eq:defphi} is an $A$-module homomorphism.
\end{lemma}
\begin{proof}
For $\XX\otimes F\in \Fl_p({\mathcal A}_X)\otimes V({\mathcal A}^X)$, choose a flag $\FF$ with
$\abs{\FF}=\XX$ and $F_p=F$. Recall that we chose the orientations in $Z_{\mathcal A}$
to agree on parallel faces. Note the face $\widehat{GF_i}$ is a translate
of $\hat{F_i}$ (Proposition~\ref{prop:Zprod}), for all $0\leq i\leq p$ and
$\abs{G}\geq\abs{F_p}$, so their orientations agree. It follows that
${\varepsilon}(\FF)={\varepsilon}(G\FF)$ for all faces $G$ with $\abs{G}\geq X$. So
for $G\in{\mathcal C}({\mathcal A}^X)$,
\begin{eqnarray*}
\phi(G(\XX\otimes F)) &=&{\varepsilon}(G\FF) b(G\FF)\\
&=&{\varepsilon}(\FF)G b(\FF),\\
&=&G\phi(\XX\otimes F)
\end{eqnarray*}
using Proposition~\ref{prop:dotwithb} at the second step. On the other
hand, if $\abs{G}\not \geq X$, both sides are zero, by \eqref{eq:localaction} and
Proposition~\ref{prop:dotwithb}.
\end{proof}
\subsection{The main result}
Now we are able to state and prove a description of each eigenspace of
the BHR random walk's transition matrix $K$. The main idea is to
use the stationary distribution \eqref{eq:stationary} on each
the arrangement ${\mathcal A}^X$, for each $X$, together with Lemma~\ref{lem:hom}.
Recall that eigenvalues of $K$ were indexed by subspaces $X\in L({\mathcal A})$,
from \eqref{eq:lambda}.
For each $X\in L({\mathcal A})$, then let $q^X\in V({\mathcal A}^X)$ be the eigenvector
\eqref{eq:1eivec} for the arrangement ${\mathcal A}^X$, a vector over the ring $R$:
by Lemma~\ref{lem:eivec},
\begin{equation}\label{eq:local}
K_{{\mathcal A}^X}\cdot q^X=\lambda_X q^X.
\end{equation}
For each
$C\in{\mathcal C}({\mathcal A}^X)$, let $q^X_C$ denote the $C$th coordinate of $q^X$.
For each $X\in L_p({\mathcal A})$, define a map $\psi_X\colon \Fl_p({\mathcal A}_X)\to
W^pV({\mathcal A})\subseteq V({\mathcal A})$ by letting
\begin{equation}\label{eq:eigenvector}
\psi_X(\XX)=\sum_{C\in{\mathcal C}({\mathcal A}^X)}q^X_C\phi(\XX\otimes C).
\end{equation}
Let $\psi\colon\Fl({\mathcal A})\to V$ be given on $\Fl_p({\mathcal A})$
by composing the isomorphism
\eqref{eq:dualb} with the maps $\psi_X$.
\begin{theorem}\label{th:main}
The map $\psi_X$ is one-to-one, and its image is the eigenspace of
$K_{\mathcal A}$ with eigenvalue $\lambda_X$.
\end{theorem}
In other words, for each flag $\XX$ ending at $X$,
and each face $F$ with $\abs{F}=X$, choose any flag $\FF\in\flag({\mathcal F})$
with $\abs{\FF}=\XX$, ending at $F$.
Then the vector
\[
\sum_{F\colon\abs{F}=X}{\varepsilon}(\FF)q^X_F b(\FF)
\]
is an eigenvector for $K$ with eigenvalue $\lambda_X$. Moreover, we see now that the
Varchenko-Gel$'$ fand dual filtration (\S\ref{ss:dualfiltr}) can be used
to keep track of the eigenspace multiplicities (Theorem~\ref{th:BHR}):
\begin{corollary}
The map $\psi$ is an isomorphism, taking the Brieskorn decomposition \eqref{eq:dualb}
of $\Fl({\mathcal A})$ to the eigenspace decomposition of $V$.
\end{corollary}
\begin{proof}[Proof of Theorem~\ref{th:main}]
First check $\psi_X(\XX)$ is an eigenvector, for any flag $\XX$. We have
\begin{eqnarray*}
K\cdot\psi(\XX) &=&\sum_{F\in{\mathcal F}({\mathcal A})}w_F F\sum_{C\in{\mathcal C}({\mathcal A}^X)}q^X_C
\phi(\XX\otimes C)\\
&=&\sum_{F\in{\mathcal F}({\mathcal A}^X)}w_F \sum_{C\in{\mathcal C}({\mathcal A}^X)} q^X_C\phi(\XX\otimes FC)
\\
&=&\lambda_X\psi(\XX),
\end{eqnarray*}
using first Lemma~\ref{lem:hom} then \eqref{eq:local}.
To see that the kernel of $\psi_X$ is zero, consider the two short
exact sequences
\[
\xymatrix{
0\ar[r]& W^{p+1}V({\mathcal A})\ar[r]\ar[d]_{i^*} & W^pV({\mathcal A})\ar[r]^{\pi_p}\ar[d]_{i^*} &
\Fl_p({\mathcal A})\ar[r]\ar[d] & 0\\
0\ar[r]& W^{p+1}V({\mathcal A}_X)\ar[r] & W^pV({\mathcal A}_X)\ar[r]^{\pi_p} &
\Fl_p({\mathcal A}_X)\ar[r]\ar@{.>}[ul]_{\psi_X} & 0,\\
}
\]
where the rows are given by \eqref{eq:noncanonical}, and the vertical
maps are the natural ones (\S\ref{ss:dualfiltr}). Note that
$i^*b(\FF)=i^*b(\FF')$ for any two flags $\FF$ and $\FF'\in \flag_p({\mathcal F}({\mathcal A}_X))$.
It follows that
$i^*\circ\phi(\XX\otimes C)=i^*\circ\phi(\XX\otimes C')$ for any $C,C'\in{\mathcal C}({\mathcal A}^X)$, so
the composite
\begin{eqnarray*}
\pi_p\circ i^*\circ \psi_X(\XX) &=& \pi_p\Big(\big(\sum_{C\in {\mathcal C}({\mathcal A}^X)}
q^X_C\big)i^*\phi(\XX\otimes C)\Big)\\
&=& (\sum_{C\in {\mathcal C}({\mathcal A}^X)}p_C^X)\XX.
\end{eqnarray*}
Since the vector $q^X\in V({\mathcal A}^X)$ is a rescaling of a probability
distribution,
the sum of its coordinates must be nonzero. Since our coefficient ring
$R$ is a domain, this means that the composite has zero kernel, so
$\psi_X$ is one-to-one.
\end{proof}
\begin{example}[Example~\ref{ex:c}, continued]\label{ex:d}
Here is a basis of eigenvectors in the case of three lines in the plane
given by Theorem~\ref{th:main}.
The $\lambda_{{\mathbb R}^2}$-eigenvector is provided by $q$ of \eqref{eq:1eivec}:
if we specialize the weights to a probability distribution, recall
$1=\lambda_{{\mathbb R}^2}$, and the eigenvector is the stationary distribution
\eqref{eq:stationary}.
For an arrangement of one point in a line, the vector \eqref{eq:1eivec}
equals
\begin{equation}\label{eq:onepoint}
\frac{w_\px+w_0+w_\mx}{w_0w_\px w_\mx(w_\px+w_\mx)}\cdot(w_\px,w_\mx),
\end{equation}
ordering the chambers with $\px$ first.
Let $\XX=({\mathbb R}^2,H_1)$. Then $\Phi_{H_1}(\XX)$ is a unit multiple of
$w_{0\px\mx}\phi(\XX\otimes C_{0\px\mx})
+w_{0\mx\px}\phi(\XX\otimes C_{0\mx\px})$,
as in \eqref{eq:onepoint}.
Using the calculation in Example~\ref{ex:c} gives the vector shown
in Figure~\ref{fig:c}.
We order the basis of $V$ counterclockwise, starting with the chamber
$\px\px\px$. For clarity, the weights of the codimension-$0$ and
-$1$ faces are relabelled $w_0,\ldots,w_6$ as shown in Figure~\ref{fig:c}.
Then the eigenvectors given by Theorem~\ref{th:main} are
\[
\begin{array}{c|c|c}
\XX & \Psi(\XX)\text{~proportional to} & \lambda\\ \hline
({\mathbb R}^2) & p,\text{~above} & \sum_{i=0}^6 w_i\\ \hline
({\mathbb R}^2,H_1) & (0,w_2,-w_2,0,-w_5,w_5) &
w_0+w_2+w_5\\
({\mathbb R}^2,H_2) & (w_6,0,w_3,-w_3,0,-w_6) & w_0+w_3+w_6\\
({\mathbb R}^2,H_3) & (-w_1,w_1,0,w_4,-w_4,0) & w_0+w_1+w_4\\ \hline
({\mathbb R}^2,H_1,0) & (0,-1,1,0,-1,1) & w_0\\
({\mathbb R}^2,H_2,0) & (1,0,-1,1,0,-1) & w_0
\end{array}
\]
\end{example}
\begin{figure}
\includegraphics[height=1.9in]{zono7.pstex}
\caption{$\Psi_{H_1}({\mathbb R}^2,H_1)$ in Example~\ref{ex:d}}\label{fig:c}
\end{figure}
\begin{remark}
The results of \cite{VG87} on which Theorem~\ref{th:main} is based
can likely be generalized to arbitrary oriented matroids without
change, in which case the results here would generalize in the same
way: see the discussion in \cite{BD98}.
In \cite{Br00}, Brown shows that many of the BHR random walk's properties
(such as diagonalizability) can also be generalized by replacing the face
algebra with any semigroup with the left-regular band property. The same
paper describes idempotents for irreducible modules, which implicitly
give eigenvectors for the generalized random walk. However, that description
is somewhat more complicated than the one here.
Recent work of
Saliola~\cite{sa10} gives a quite different description of the eigenvectors of
the BHR random walk, using methods that hold for any left-regular
band. His construction is complementary to the one given here. The
cost of working with the general setting of \cite{Br00} is apparently
no longer to have an eigenbasis with such convenient labelling as
non-broken circuits. It would be interesting to know, then, if the
main results of \cite{VG87} would admit left-regular band generalizations.
\end{remark}
\begin{ack}
The author would like to thank Phil Hanlon for helpful discussions at the
start of this project,
and the Institut de G\'eom\'etrie, Alg\`ebre et Topologie at the EPFL
for its hospitality during its completion.
\end{ack}
\bibliographystyle{amsplain}
\def$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
| {
"timestamp": "2011-10-14T02:04:57",
"yymm": "1010",
"arxiv_id": "1010.0232",
"language": "en",
"url": "https://arxiv.org/abs/1010.0232",
"abstract": "We find explicit eigenvectors for the transition matrix of a random walk due to Bidegare, Hanlon and Rockmore. This is accomplished by using Brown and Diaconis' analysis of its stationary distribution, together with some combinatorics of functions on the face lattice of a hyperplane arrangement, due to Gelfand and Varchenko.",
"subjects": "Combinatorics (math.CO)",
"title": "Eigenvectors for a random walk on a hyperplane arrangement",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846659768267,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7092019630337929
} |
https://arxiv.org/abs/2007.01218 | On Koopman Operator for Burgers' Equation | We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD). | \section{\label{sec:level1}Introduction}
The Koopman operator is a linear operator defined by Koopman in \cite{Koopman1931} to `linearize' nonlinear flows. This tool linearly evolves a set of observables (functionals) of the system state and allows for defining a spectrum for nonlinear flows. Such a spectrum is often considered as the generalization of `normal modes' for linear systems to `Koopman modes' for nonlinear systems \citep{Mezic2005,Mezic2013}.
The decomposition of data on the Koopman modes is commonly referred to as Koopman Decomposition and offers promising applications in reduced-order modeling, feature extraction, and control \citep{Budisic2012,KutzDMD}. The Koopman framework is also invoked as a background for the Dynamic Mode Decomposition (DMD \cite{SCHMID2010}), a decomposition which finds the best linear dynamical system approximating snapshots of a flow (best in the least square sense). The link between DMD and Koopman decomposition was first revealed by \cite{ROWLEY2009} and is described by various authors (e.g., \textcolor{black}{\cite{Brunton2016,Rowley2017}}) on simple nonlinear ODEs. A systematic analysis of more complex problems is limited by the difficulties in deriving Koopman eigenfunctions for PDEs. The DMD is thus often considered as a data-driven method to compute Koopman modes under the assumption that the chosen observables and the available dataset are sufficiently `rich', in a sense defined by \cite{Tu2014}. Kernel methods to heuristically augment the richness of available data and observables have been proposed by \cite{Williams2015,Williams2015a} and reviewed by \cite{Kutz2018} in the framework of support vector machines (SVM).
The recent contribution by \cite{Page2018} has opened new avenues for Koopman analysis.
These authors \textcolor{black}{proposed a procedure to derive} the Koopman decomposition for a nonlinear PDE, namely the Burgers' equation. \textcolor{black}{This procedure encounters a geometric multiplicity problem which is not solved. However, it} enabled the analysis of the (non-trivial) requirements for their data-driven identification via DMD. The main difficulties in identifying Koopman modes via DMD for problems with multiple invariant solutions are further discussed by \cite{Page2019}. More examples of Koopman analysis for PDEs are given by \cite{Nakao2020}.
In this work, we compute explicitly the set of eigen-observables of the Koopman operator for the \textcolor{black}{Burgers'} equation. We construct all modes and eigenfunctionals, including the eigenspaces involved
in geometric multiplicity. This goes beyond the summation formulas provided by
\cite{Page2018}, where only one term per eigenvalue was given. We prove convergence of the Koopman decomposition, and the completeness of the Koopman modes.
The problem set and the relevant definitions are given in Section \ref{SecII}. \textcolor{black}{Section \ref{SecIII} recalls the Cole-Hopf transform, the methodology firstly proposed by Page and Kerswell \cite{Page2018}, and the geometric multiplicity problem they encounter. In section \ref{SecIV}, we propose a procedure that computes as many independent Koopman modes as needed by the encountered geometric multiplicity. We thus give an explicit Koopman decomposition. We link the coefficients of the Koopman decomposition to the eigenfunctionals of the Koopman operator. This leads to a `completeness' property of the Koopman eigenfunctionals. We give a numerical example of Koopman approximations in section \ref{SecVI} and compare its eigenvalues to those of a DMD. Estimates needed in the proofs are given in section \ref{SecV}, where the smallness of the Cauchy data required for the decomposition's convergence is quantified. Concluding remarks are given in section \ref{SecVII}.}
\section{Definitions and Scope}\label{SecII}
We consider the Burgers' equation on $[0,1]$:
\begin{equation}
\label{F1}
\partial_t u =F_{\mathcal{B}}(u):= -u\partial_xu+\partial_{xx}u\quad u(t,0)=u(t,1)=0\quad u(0,x)=u_0(x)\,.
\end{equation}
Any Burgers' equation, with $x \in [0,L]$ and viscosity $\nu$, can be reduced to this form with a proper re-scaling of $x$, $u$, and $t$. Equation (\ref{F1}) translates to:
\begin{equation}
\label{flow}
u(t,x)=\Phi^t_{\mathcal{B}}(u_0)=u_0(x)+\int^{t}_{0} F_{\mathcal{B}} (u)(s)ds\,,
\end{equation}
\textcolor{black}{We define $\Omega_B$ as a subset of $L^2(]0,1[)$ invariant under $\Phi_{\mathcal{B}}^t$ and where a Koopman decomposition converges for $t>0$. The precise definition of $\Omega_B$ is given in Section \ref{SecIV} in formula (\ref{Omega_0})}. \noindent We consider the set $\mathcal {O}_{\mathcal{B}}$ of continuous observables (functionals) on $\Omega_{\mathcal{B}}$, i.e. the set of continuous maps $\phi$ from ${\Omega_{\mathcal{B}}}$ to $\mathbb{R}$. For any $t>0$, the Koopman operator $\mathcal{K}^t_{\mathcal{B}}$ of the Burgers' flow is the map from $\mathcal {O}_{\mathcal{B}}$ to itself defined by
\begin{equation}
\forall \phi \in \mathcal {O}_{\mathcal{B}}, \, \forall u_0 \in {\Omega_{\mathcal{B}}},\, (\mathcal{K}^t_{\mathcal{B}}(\phi))(u_0)=\phi(\Phi^t_{\mathcal{B}}(u_0))\label{F2}\,.
\end{equation}
\noindent $(\mathcal{K}_{B}^t)_{t\geq 0}$ is a flow on $\mathcal{O}_{B}$. $\mathcal{K}^t_{\mathcal{B}}$ is a linear map, and fulfills the multiplicative property
\begin{equation} \label{Fmult}\textcolor{black}{\forall \phi, \phi' \in \mathcal{O}_{\mathcal{B}},} \quad \forall u_0 \in {\Omega_{\mathcal{B}}}, \quad (\mathcal{K}^t_{\mathcal{B}}(\phi\phi'))(u_0)= (\mathcal{K}^t_{\mathcal{B}}(\phi))(u_0)(\mathcal{K}^t_{\mathcal{B}}(\phi'))(u_0)\,.\end{equation}
The eigen-observables (eigen-functionals) $\varphi_\nu(u_0)$ \textcolor{black}{of the Koopman operator} are observables whose evolution by the Koopman flow is:
\begin{equation}
\label{eig}
(\mathcal{K}^t_{\mathcal{B}}(\varphi_\nu))(u_0)=\varphi_\nu(u_0) e^{\lambda_{\nu} t}\,,
\end{equation}
\textcolor{black}{
\noindent with $\lambda_\nu$ the associated eigenvalue and $e^{\lambda_{\nu} t}$ the corresponding temporal evolution. Combining \eqref{Fmult} and \eqref{eig} shows that the product of two eigen-observables $\varphi_\nu$ and $\varphi_{\nu'}$, with eigenvalues $\lambda_\nu$ and $\lambda_{\nu'}$, is also an eigen-observable $\varphi_{\nu''}=\varphi_\nu\varphi_{\nu'}$ with eigenvalue $\lambda_{\nu''}=\lambda_\nu+\lambda_{\nu'}$.}
\medskip
The usual assumption in the Koopman decomposition is that the set of eigen-observables is sufficiently large to represent \emph{any} observable. Therefore, the value of \textit{any} observable $\phi$ at any function $f$ can be written as
\begin{equation}
\label{phi}
\phi(f)=\sum_{\nu } a_{\nu}\textcolor{black}{(\phi)}\varphi_{\nu}(f) \,.
\end{equation}
\textcolor{black}{
Combining \eqref{phi} with \eqref{F2} and \eqref{eig}, the observable $\phi$ evolves along a Burgers \textcolor{black}{orbit} through the Koopman flow: }
\begin{equation}
\label{Koopman_DE2}
\phi(\Phi^t_{\mathcal{B}}(u_0))=(K^t_{\mathcal{B}} \phi) (u_0)=\sum_{\nu } a_{\nu}(\phi) \varphi_{\nu}(\Phi_{\mathcal{B}}^t (u_0))=\sum_{\nu } a_{\nu}(\phi) (K^t_{\mathcal{B}}\,\varphi_{\nu})(u_0) =\sum_{\nu } a_{\nu}(\phi)\varphi_{\nu}(u_0) e^{\lambda_v t} \,.
\end{equation}
Equation \eqref{Koopman_DE2} is known as Koopman decomposition with respect to the observables $\phi$ and $a_\nu(\phi)$ are the associated Koopman modes. We consider the Dirac observables $\phi:=\delta_x$, where $\delta_x(u_0)=u_0(x)$: this observable maps any state variable $u$ to its value at location $x$. Equation \eqref{Koopman_DE2} becomes:
\begin{equation}
\label{Koopman_DE}
u(t,x)=\Phi^t_{\mathcal{B}}(u_0) =\sum_{\nu } e^{\lambda_v t} \varphi_{\nu}(u_0) a_{\nu}(x)\,.
\end{equation}
While this decomposition looks linear, the nonlinearity of Burgers equation shows in the nonlinearity of the coefficients $\varphi_\nu(u_0)$.
We derive the Koopman decomposition \eqref{Koopman_DE} for Burgers equation (\ref{F1}) for $u_0 \in \Omega_{\mathcal{B}}$. We prove its convergence in section \ref{SecIV}. We give explicit formulas for the Koopman modes $a_{\nu}(x)$, the eigenvalues $\lambda_{\nu}$ and the coefficients $\varphi_{\nu}(u_0)$ in (\ref{Koopman_DE}). We prove that $(e^{\lambda_\nu t},\varphi_\nu)$ are the eigen-elements of the Koopman operator associated with the Burgers equation. It is important to stress that the convergence of \eqref{Koopman_DE} requires smallness of the Cauchy data as shown in section \ref{SecIV}, hence the restriction of the Burgers flow to $\Omega_{\mathcal{B}}$.
\section{Problem Statement via Cole-Hopf Transform}\label{SecIII}
\textcolor{black}{The Burgers equation \eqref{F1} is one of the few examples of nonlinear PDEs amenable to Koopman Analysis, thanks to the linearizing Cole-Hopf transformations. This was noticed by Kutz \emph{et al} \cite{Kutz2018} and exploited by Page and Kerswell \cite{Page2018}. }
\textcolor{black}{The Cole-Hopf transforms are defined as:}
\textcolor{black}{
\begin{equation}
u:=C(v)=-2{\partial_xv\over v}\quad {\rm and} \quad v:=H(u)={e^{-{1\over 2}\int_0^xu(s)ds}\over \int_0^1e^{-{1\over 2}\int_0^xu(s)ds}dx}\,.
\label{colehopf}
\end{equation}}
$H$ is defined for all functions in $L^2([0,1])$, fulfills $\int_0^1H(u)(s)ds=1$, and $H(u) > 0$. We restrict $C$ to the set of functions $v$ having first weak derivative in $L^2([0,1])$, strictly positive, and with $\int_0^1v(s)ds=1$. This makes $H$ and $C$ inverse transforms.
The Cole-Hopf transforms are central tools because if $u(t,.)$ solves Burgers equation (\ref{F1}), then $v(t,.)=H(u(t,.))$ solves the linear heat equation:
\textcolor{black}{\begin{equation}
\partial_tv=\partial_{xx}v\quad \partial_xv(t,0)=\partial_xv(t,1)=0\quad v(0,x)=H(u_0):=v_0 \label{F5}\,,
\end{equation}
and the converse is true. }
Writing the Cauchy data in the Fourier basis with $e_m(x):=\sqrt{2}\cos(m\pi x)$ for $m\not= 0$ as: $$ v_0(x) = 1 + \sum_{m=1}^\infty c_m(v_0)e_m(x) \quad { \rm with} \quad c_m(v_0)= \int_0^1 v_0(s) e_m(s) ds\,,$$
the solution of the heat equation is:
\begin{equation}\label{vfourier}
v(t,x)=\Phi^t_{\mathcal{C}}(v_0)= 1 + \sum_{m=1}^\infty e^{-m^2\pi^2t} c_m(v_0)e_m(x)\,.
\end{equation}
If $\Phi^t_{\mathcal{C}}$ denotes the flow associated with the heat equation, then $\Omega_{\mathcal{C}}:=H(\Omega_{\mathcal{B}})$ is invariant under this flow and we have:
\begin{equation}
\textcolor{black}{
H(\Phi^t_{\mathcal{B}}(u_0))=\Phi^t_{\mathcal{C}}( H(u_0))\quad {\rm and}\quad C(\Phi^t_{\mathcal{C}}(v_0))=\Phi^t_{\mathcal{B}}(C(v_0))}\,.
\label{F6_M}
\end{equation}
\textcolor{black}{Using the composition law $(A\circ B)(z):= A(B(z))$, }equation \eqref{F6_M} can be written as
\begin{equation}
H\circ \Phi^t_{\mathcal{B}}=\Phi^t_{\mathcal{C}}\circ H\quad {\rm and}\quad C\circ \Phi^t_{\mathcal{C}}=\Phi^t_{\mathcal{B}}\circ C\,.
\label{F6}
\end{equation}
These equations give the conjugacy properties linking the flows $\Phi^t_{\mathcal{B}}$ and $\Phi^t_{\mathcal{C}}$ via the Cole-Hopf transform \eqref{colehopf}. This intertwining of the Burgers flow with the linear heat flow gives the Koopman decomposition for Burgers' equation.
Page and Kersell's procedure in \cite{Page2018} consists in writing \eqref{colehopf} as $u v = -2\partial_x v$, plugging the Fourier decomposition \eqref{vfourier} of $v$ and a formal Koopman decomposition \eqref{Koopman_DE} of $u$, and then identifying the Koopman modes, amplitudes and eigenvalues by inspection. Except for modes with multiplicity one, this procedure encounters a problem.
In the next sections, we propose an approach which solves the issue of geometric multiplicity. We explicitly give the Koopman decomposition and prove its convergence.
\section{An Explicit Koopman Decomposition}\label{SecIV}
\subsection{An Analytical Decomposition into Exponentials}\label{SubSecIV_I}
\noindent For any $u_0\in \Omega_{\mathcal{B}}$, let $v_0=H(u_0)$. \textcolor{black}{We note that the steady state solution $v=1$ is a sink for the heat equation, i.e. $v(t,x)\rightarrow1$ for $t\rightarrow +\infty$ for all $v_0 \in \Omega_C$.} \textcolor{black}{We restrict our decomposition to the vicinity of this steady state and write $v_0=1+w_0$, with $w_0$ a function such that $\int_0^1w_0(x)dx=0$ and $\Vert \partial_x w_0 \Vert_{L^2}\leq {1\over 4}$. From Section \ref{lemmi} and the choice of $\Omega_B$, we get that $\Vert \partial_x w_0 \Vert_{L^2}\leq {1\over 4}$, hence $\sup_x\vert w_0(x)\vert \leq 1/4$ (by property 1 in section \ref{lemmi})}.
\medskip
\noindent For $t>0$, \textcolor{black}{we} write $\Phi^t_{\mathcal{C}}(v_0)=v(t,.)=1+ w(t,.)$. $\partial_xw$ fulfills the heat equation \textcolor{black}{ with Dirichlet boundary conditions. Energy decay due to diffusion gives} $\Vert \partial_x w\Vert_{L^2}\leq \Vert \partial_x w_0\Vert_{L^2}$ so $\sup_x\vert w(t,x)\vert \leq {1\over 4}$. One can therefore apply the Cole-Hopf transforms, \textcolor{black}{and have the following asymptotic expression for the solution of the Burgers's equation:}
\begin{equation} \label{Fdev}\Phi^t_{\mathcal{B}}(u_0)=u(t,.)=C(v(t,.))=-2{ \partial_x w\over {1+ w}}=-2 \partial_x w\sum_{q=0}^\infty (-1)^q w^q\,.
\end{equation}
\textcolor{black}{Using the expression for $w$ given by formula (\ref{vfourier}) we compute} $w^q$, the product of $q$ identical sums, by taking term \textcolor{black}{of rank} $n_1$ in the first sum, term \textcolor{black}{of rank} $n_2$ in the second sum, \textcolor{black}{and so on up to $m$-th} sum. We get:
$$w^q=\sum_{(n_1,..,n_q)\in {(\textcolor{black}{N\backslash \{0\}})}^q }e^{-\sum_{k=1}^qn_k^2 \pi^2t}\prod_{k=1}^{q}c_{n_k}(v_0)\prod_{k=1}^{q}e_{n_k}(x)\,.
$$ \noindent \textcolor{black}{On the other hand:}
$$\partial_xw(t,x)=\sum_{n_0=1}^\infty e^{-n_0^2\pi^2t} c_{n_0}(v_0)\partial_xe_{n_0}(x)\,.
$$
\noindent
Replacing the above expressions in formula (\ref{Fdev}) gives
\begin{equation}
\begin{split}
\Phi^t_{\mathcal{B}}(u_0)(x)=-2
\sum_{n_0=1}^\infty e^{-n_0^2\pi^2t} c_{n_0}(v_0)\partial_xe_{n_0}(x)
\\-2\sum_{q=1}^\infty \sum_{n_0=1}^\infty \sum_{\stackrel{n_1,..,n_q}{\in {(\textcolor{black}{N \backslash \{0\}} )}^q}}
(-1)^{q}e^{-\pi^2 t\sum_{k=0}^qn_k^2} \prod_{k=0}^{q}c_{n_k}(v_0)\,\,\partial_xe_{n_0}(x)\prod_{k=1}^{q}e_{n_k}(x)\,.
\end{split}
\end{equation}
\textcolor{black}{This decomposition is more easily handled by taking the following set of indices:}
\begin{equation}
\mathcal{A} = \{\nu = (n_0,n_1,...n_{\alpha(\nu)}); \alpha(\nu) \in \mathbb{N}, n_0 \in \mathbb{N}, n_i \in \mathbb{N}\backslash \{0\} \,\,{\rm for}\,\,i=1,..,\alpha(\nu)\,.
\end{equation}\label{indices}
Then, the above formula for $\Phi^t_{\mathcal{B}}(u_0)$ looks familiar if one introduces:
\begin{equation}
\label{lambdas}
\lambda_\nu=-\pi^2\sum_{k=0}^{\alpha(\nu)}n_k^2\,,
\end{equation}
\textcolor{black}{
\begin{equation}
a_{\nu}(x)=(-1)^{\alpha(\nu)}2^{\alpha(\nu)+3\over 2}
n_0\pi \sin{(n_0\pi x)}\prod_{k=1}^{\alpha(\nu)}{\cos{(n_k\pi x)} }
\label{F_a_nu}\,,
\end{equation} }
\textcolor{black}{
\begin{equation}
\varphi_{\nu}(u_0)=\prod_{k=0}^{\alpha(\nu)}l_{n_k}(u_0)\quad {\rm with}\quad l_n(u_0)=c_n(H(u_0))\,.
\label{Phi_nu}
\end{equation} }
\noindent \textcolor{black}{With these notations, we have derived the following Koopman decomposition:}
\textcolor{black}{
\begin{equation}
\forall u_0\in \Omega_{\mathcal{B}},\quad \forall t>0 \quad \forall x\in [0,1]\quad \Phi^t_{\mathcal{B}}(u_0)(x)=u(t,x)=\sum_{\nu\in \mathcal{A}} e^{\lambda_\nu t}\,\varphi_{\nu}(u_0)a_{\nu}(x)\,.
\label{F3}\end{equation} }
\noindent \textcolor{black}{
Note that the length of $\nu$ is $\alpha(\nu)+1$ and in formula (\ref{F_a_nu}) the last product should be taken as $1$ for $\alpha(\nu)=0$.}
\textcolor{black}{The proper set of initial conditions $u_0$ granting convergence of this series is $\Omega_\mathcal{B}$, that we now define:}
\textcolor{black}{
\begin{equation}
\Omega_\mathcal{B}=\{u_0\in L^2([0,1]); 2e^{\Vert u_0\Vert_{L^2}}\Vert u_0\Vert_{L^2}<1\}\,.
\label{Omega_0}
\end{equation} }
\textcolor{black}{Section \ref{proof2} shows that the convergence of \eqref{F3} is uniform and in absolute values.}
\subsection{The Koopman Eigenfunctionals for Burgers' Equation}\label{SubSecIV_II}
\textcolor{black}{We define $\mathcal{O}_{\mathcal{C}}$ as the set of observables on $\Omega_\mathcal{C}$. Let $\mathcal{K}_{\mathcal{C}}^t$ be the Koopman operator of the heat equation \eqref{vfourier}}.
The conjugacy formula (\ref{F6}) leads to the following adjoint identities linking $\mathcal{K}^t_{\mathcal{B}}$ and $\mathcal{K}^t_{\mathcal{C}}$:
\textcolor{black}{\begin{subequations}
\label{F7}
\begin{equation}
\label{F7a}
\forall \phi \in \mathcal{O}_{\mathcal{B}}\,,\,\, (\mathcal{K}^t_{\mathcal{B}}\phi)\circ C=\mathcal{K}^t_{\mathcal{C}}(\phi\circ C)\,,
\end{equation}
\begin{equation}
\label{F7b}
\forall \psi\in \mathcal{O_{\mathcal{C}}}\,,\,\, (\mathcal{K}^t_{\mathcal{C}}\psi)\circ H=\mathcal{K}^t_{\mathcal{B}}(\psi\circ H)\,.
\end{equation}
\end{subequations} }
Using previous notations, formula \eqref{F7a} is true because of the following:
$$(\mathcal{K}^t_{\mathcal{B}} \phi)(C(v_0)) = (\mathcal{K}^t_{\mathcal{B}} \phi)(u_0) = \phi(\Phi^t_{\mathcal{B}}u_0) = \phi(u(t,.)) = \phi(C (v(t,.))) = (\phi \circ C)(v(t,.)) = (\phi \circ C) ( \Phi^t_{\mathcal{C}}(v_0)) = \mathcal{K}^t_{\mathcal{C}}( \phi \circ C) (v_0)\,.$$
This also applies to \eqref{F7b}.
We now prove that the functions $\varphi_\nu(u_0)$ and the exponents $\lambda_\nu$ of decomposition (\ref{F3}) are eigenfunctionals and the associated eigenvalues of the Koopman operator in formula \eqref{F2}, i.e. they satisfy property (\ref{eig}).
The proof goes as follows:
\textcolor{black}{the linearity of $c_n$ and $\Phi^t_{\mathcal{C}}$ in \eqref{vfourier} gives }
\begin{equation}
(\mathcal{K}^t_{\mathcal{C}}(c_n))(v_0) = c_n(\Phi^t_{\mathcal{C} }(v_0)) = c_n \biggl(1 + \sum_{m=1}^\infty e^{-m^2\pi^2t} c_m(v_0)e_m(x)\biggr)=c_n(v_0)e^{-n^2\pi^2 t} \,,
\end{equation}
\textcolor{black}{so, $\mathcal{K}^t_{\mathcal{C}}(c_n) = e^{-n^2 \pi^2 t} c_n$, and by \eqref{F7b} $\mathcal{K}^t_{\mathcal{B}}(c_n \circ H )= e^{-n^2 \pi^2 t }(c_n \circ H)$. Then:}
\begin{equation} \label{F9}
\forall n\in N \quad \mathcal{K}_{\mathcal{B}}^t(l_n)=e^{-n^2\pi^2t}l_n\,.
\end{equation}
The multiplicative property (\ref{Fmult}) gives, for all $\nu=(n_0,..,,n_{\alpha(\nu)}) \in \mathcal{A}$,
\begin{equation} \label{Ffi}
\mathcal{K}^t_{\mathcal{B}}(\prod_{k=0}^{\alpha(\nu)} l_{n_k}) = e^{\lambda_{\nu} t} \prod_{k=0}^{\alpha(\nu)} l_{n_k}. \end{equation}
\noindent \textcolor{black}{This proves that $\lambda_{\nu}$ is an eigenvalue of ${\mathcal{K}}^t_{\mathcal{B}}$} with \textcolor{black}{eigen-functional} $\varphi_\nu$.
\textcolor{black}{This identifies coefficients and exponentials in formula (\ref{F3}) as spectral elements of $\mathcal{K}^t_{\mathcal{B}}$: the coefficients are the value taken by the eigenfunctionals at the Cauchy data.}
\textcolor{black}{The $a_{\nu}$ will be identified in the next section}.
Notice that the definition of these eigenfunctionals needs no assumption on the smallness of the state variables.
\subsection{On completeness}\label{SubSecIV_III}
\textcolor{black}{We call weak-completeness of these eigenfunctionals the property that any Dirac functional $\delta_x$ can be decomposed on these eigenfunctionals.} \textcolor{black}{To prove this weak-completeness property}, it is worth examining under what assumption
the convergence of formula (\ref{F3}) is valid at $t=0$. \textcolor{black}{A suitable assumption is a regularity assumption on initial conditions: square integrability of the first derivative of the Cauchy data. That leads to define the following set of initial conditions: }
$$\omega_{\mathcal{B}}=\biggl \{u_0\in \Omega_{\mathcal{B}};\quad u_0(0)=u_0(1)=0;\quad \int^1_0\vert \partial_xu_0\vert^2<\infty\biggr\}.$$
We prove in section \ref{proof3} that for $u_0\in\omega_{\mathcal{B}}$ the convergence of formula (\ref{F3}) is uniform, for $t\geq 0$ and $x\in [0,1]$. This implies:
\begin{equation} \forall u_0\in \omega_{\mathcal{B}},\quad \forall x\in [0,1], \quad u_0(x)=\sum_{\nu \in \mathcal{A}} \varphi_{\nu }(u_0) a_{\nu}(x).\label{F4}\end{equation}
If one takes for observables the values at specific locations $x$, denoted by the Dirac notation $\delta_x$, formula (\ref{F4}) can be written as:
\begin{equation}
\forall x\in [0,1]\quad \delta_{x}=\sum_{\nu \in \mathcal{A}} a_{\nu}(x)\varphi_{\nu }.
\label{F4bis}\end{equation}
\textcolor{black}{Comparing this with \eqref{phi}, we identify the Koopman modes $a_{\nu}(x)$ as the coefficients of the decomposition of $\delta_x$ on eigenfunctionals of the Koopman operator.}
\medskip
To illustrate this weak-completeness, one can show that formula \eqref{F4bis} implies the decomposition of the kinetic energy observable:
$$E(u)= \int_0^1u^2(s)ds=\int_0^1(\delta_s(u))^2ds.$$
For $\nu=(n_0,\cdots n_m)\in \mathcal{A}$ and $\nu'=(n'_{0},\cdots n'_{m})\in \mathcal{A}$, we define the concatenation of these indices as
$$
c (\nu,\nu ')=\begin{dcases}
(n_0, \dots, n_m, n'_{0},\cdots n'_{m}) \quad \mbox {if} \quad n'_0\neq 0\\
(n_0, \dots, n_m, n'_{1},\cdots n'_{m}) \quad \mbox {if} \quad n'_0= 0
\end{dcases}
$$
\textcolor{black}{The multiplicative property of the Koopman eigen-observables,
$\varphi_{\nu} \varphi_{\nu'} =\varphi_{c(\nu,\nu')}$ and the uniform convergence in the $x$ variable at $t=0$ of formula (\ref{F4}) implies: }
$$E(u_0)= \textcolor{black}{ \sum_{\mathcal{A} \times \mathcal{A}} \varphi_{\nu}(u_0) \varphi_{\nu'}(u_0) \int_0^1 a_{\nu}(s) a_{\nu'}(s) ds }= \sum_\mathcal{A\times A}b_{c(\nu,\nu')}\varphi_{c(\nu,\nu')}(u_0) \quad {\rm with}\quad b_{c(\nu,\nu')}=\int_0^1a_\nu(s)a_{\nu'}(s)ds\,.
$$
\noindent The Koopman decomposition \eqref{Koopman_DE} \textcolor{black}{of the kinetic energy is then, by \eqref{Koopman_DE2}: }
\textcolor{black}{
$$E(u(t,.))=\sum_\mathcal{A\times A}e^{\lambda_{c(\nu,\nu')}t}b_{c(\nu,\nu')}\varphi_{c(\nu,\nu')}(u_0).
$$}
\section{Numerical Illustrations of Koopman Approximations}\label{SecVI}
\textcolor{black}{
We present a numerical illustration of the Koopman decomposition \eqref{F3} for the Burger's flow \eqref{F1} on the interval $[0,1]$. We consider initial conditions with $c_m=0$ for $m>2$ in \eqref{vfourier}:}
\begin{equation}
\quad u_0(x)=C(v_0)\quad \mbox{with} \quad v_0(x)=1+{1\over2}\cos{\pi x}+{1\over4}\cos{2\pi x}.
\label{C1}
\end{equation}
\textcolor{black}{
The Burger's flow is computed from \eqref{colehopf} as $u(t,x)=C(v(t,x))$ with $v(t,x)$ obtained from \eqref{vfourier}.
The computations are performed on a uniform mesh $x_i=(i-1)\Delta x$ with $i\in[1,1024]$, computing all integrals via the trapezoidal method. }
\textcolor{black}{
We consider the set of Koopman modes originating from the indices of length one, up to length six ($0\leq\alpha(\nu)\leq 5$). This includes all the sets $\nu=(n_0)$, $\nu=(n_0,n_1)$, up to $\nu=(n_0,n_1,n_2,n_3,n_4,n_5)$, with $n_k\in[1,2]$ for all $k\leq5$. We thus consider a total of $126$ Koopman modes with repetitions, out of which $30$ are independent. All indices with $n_k>2$ leads to zero amplitude according to \eqref{Phi_nu} for this particular initial conditions given in \eqref{C1}, where $c_m=0$ for $m>2$.}
Figure \ref{Mig1}(a) shows the dynamics $u(x,t)$ (blue continuous lines) for $t=0,0.02, 0.04, 0.06, 0.14, 0.24$ along with the Koopman approximation (black dashed lines). While the completeness of the (infinite) Koopman basis has been proven, these results highlight the challenges in the convergence up to $t\rightarrow 0$: because $u_0$ does not fulfil the smallness property $u_0 \in \Omega_B$, convergence does not extend to $t=0$, as shown by the part of the Cauchy data close to $x=0$ (zoomed axis).
\textcolor{black}{
Figure \ref{Mig1}(b) shows the Koopman coefficients $\varphi_{\nu}(u_0)$ as a function of the associated Koopman eigenvalue $\lambda_\nu$. The markers in the figures, labeled in the legend, recall the length of the set $\nu$ to which each mode corresponds. Notice that only 30 markers are visible. The multiplicity of the eigenvalues (hence the temporal evolution) depends on the set of indices (e.g, $\nu=(2)$ and $\nu=(1,1,1,1)$ lead to the same $\lambda_\nu$) and increases with $-\lambda_v$. The problem encountered by Page \& Kerswell \cite{Page2018} is that they identify the Koopman modes by combining their formulas (8) and (9) to get (10) and construct only one Koopman mode irrespective of whether its multiplicity is greater than one.}
\begin{figure}[h!]
\centering
\subfigure[\ ]{
\includegraphics[trim=5 30 5 30,clip,width=8.5cm]{Koopman_Cos_n0_2_n1_2_n2_2_n3_2.pdf}}
\subfigure[\ ]{
\includegraphics[trim=5 30 5 30,clip,width=8.5cm]{Koopman_Spectra_Cos_n0_2_n1_2_n2_2_n3_2.pdf}}
\vspace{-2em}
\caption{Koopman approximations of the Burgers' flow with initial condition in \eqref{C1}. Figure (a) shows the dynamics $u(x,t)$ (continuous blue line) along with the Koopman approximations (dashed lines). This initial condition does not satisfy the smallness condition needed in the proofs and this is why it does not converge up to $t=0$; Figure (b) plots the amplitudes of the Koopman modes $\varphi_\nu(u_0)$ as a function of $-\lambda_\nu$. }
\label{Mig1}
\end{figure}
\textcolor{black}{
Because of the exponential evolution of the Koopman modes, the amplitudes $\varphi_\nu(u_0)$ weigh the contribution of each mode in the decomposition of the initial data. To analyze the relative contribution to the Burgers' flow $u(t,x)$, and to consider the role of the spatial basis $a_\nu (x)$, we introduce a measure of the mode relevance in the form of relative $L^2$ norm in space and time:}
\begin{equation}
\label{Eq_Importance}
\sigma_{\nu}=\frac{||e^{-\lambda_{\nu}t} \varphi_{\nu}(u_0)a_{\nu}(x) ||}{||u(t,x)||} \quad \mbox{with} \quad ||f||^2=\int^{t_2}_{t_1}\int_0^1 \bigl(f(x,t)\bigr)^2 dx \,dt\,.
\end{equation}
\textcolor{black}{
It is worth highlighting that this parameter only serves illustrative purposes, as the lack of orthogonality of the Koopman basis does not allow to recover the energy of the flow by summing the contribution of each mode.}
\textcolor{black}{
The relative importance of the modes is analyzed for two time intervals, namely $[t_1,t_2]=[0,0.12]$ and $[t_1,t_2]=[0.12,0.24]$. The contribution of each mode according to \eqref{Eq_Importance} is shown in Figure \ref{Mig2}. Only modes with $\sigma_\nu>10^{-3}$ are considered.}
\begin{figure}[h!]
\centering
\subfigure[\ ]{
\includegraphics[trim=5 30 5 30,clip,width=8.5cm]{Importance_First_Half_Cos_n0_2_n1_2_n2_2_n3_2.pdf}}
\subfigure[\ ]{
\includegraphics[trim=5 30 5 30,clip,width=8.5cm]{Importance_Last_Half_Cos_n0_2_n1_2_n2_2_n3_2.pdf}}
\vspace{-1em}
\caption{Contribution of each Koopman mode according to \eqref{Eq_Importance}, taking $[t_1,t_2]=[0,0.12]$ (a) and $[t_1,t_2]=[0.12,0.24]$ (b).}
\label{Mig2}
\end{figure}
\textcolor{black}{
A comparison of these figures shows how the relative importance of the Koopman modes changes in time, depending on the relative weight of advection and diffusion. Seven modes have $\sigma_\nu>0.05$ in the first time interval while only two have the same relevance in the second interval. The same plot considering $[t_1,t_2]=[0,24]$ is indistinguishable from Fig \ref{Mig2}(a) and it is thus not shown. This practically illustrates how the initial conditions lead to the definition of the decomposition. Moreover, this figure shows how adding the contribution of the spatial structure $a_\nu(x)$ results in the spreading of the plot in Figure \ref{Mig1}(a), as modes having the same amplitude ($\varphi(u_0)$) and temporal evolution ($\lambda_\nu$) might have different $a_\nu(x)$.}
\textcolor{black}{
The seven leading modes (with $\sigma_\nu>0.05$) taking the full time interval are shown in Figure \ref{a_nus}, with the legend recalling the associated set of indices $\nu$. Except for the first two, all the other modes display nonlinear interactions through the multiplicative property: for instance, the interaction of a mode with itself, that is self-interaction, gives rise to a squared mode.}
\begin{figure}[htbp]
\centering
\includegraphics[width=13cm]{Koopman_MODES_n0_5_n1_5_n2_5_n3_5.pdf}
\caption{Five Koopman modes $a_\nu(x)$ of the Burgers' flow. Indices are indicated in the legend.}
\label{a_nus}
\end{figure}
Finally, we conclude this illustrative section by analyzing the accuracy of a data-driven approximation of these Koopman modes using a Dynamic Mode Decomposition (DMD) (see \cite{SCHMID2010,ROWLEY2009,Tu2014}). We consider the SVD-based approach. A set of $n_t=101$ snapshots uniformly sampled in time discretization $t_k=(k-1)\Delta t$ with $\Delta t=0.002$ is used to construct the dataset matrix $\mathbf{X}\in\mathbb{R}^{1023\times 101}$.
\begin{figure}[!htbp]
\centering
\includegraphics[trim=5 30 5 30,clip,width=8.5cm]{DMD_Approximation.pdf}
\includegraphics[width=8cm]{DMD_Spectra_Cosine.pdf}\\
\hspace{1.7cm} a) \hspace{8.5cm} b)
\caption{Results from a DMD analysis of the Burgers' flow $u(t,x)$. Figure (a) shows the accuracy of the DMD. Figure (b) shows the computed DMD eigenvalues. Figure (c) shows the spatial structure of the leading DMD modes.}
\label{DMD_Migu}
\end{figure}
\textcolor{black}{
The DMD solves the regression problem of identifying the eigendecomposition of the best linear propagator $\mathbf{P}$ such that $\mathbf{X}_2=\mathbf{P}\mathbf{X_1}$, with $\mathbf{X}_1,\mathbf{X}_2\in \mathbb{R}^{1023\times 100}$ containing the columns of $\mathbf{X}$ from $1$ to $n_t-1$ and from $2$ to $n_t$ respectively.
The hoped for approximation of the Koopman eigenvalues are the eigenvalues of a reduced propagator $\mathbf{S}=\mathbf{U}^T \mathbf{P}\mathbf{U}$, with $\mathbf{U}$ the matrix of left singular vectors from the singular value decomposition $\mathbf{X}_1=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^T$. Given $\mathbf{S}=\mathbf{W}\mathbf{\Lambda}\mathbf{W}^{-1}$ the eigenvalue decomposition of $\mathbf{S}$, the approximation of the Koopman modes are computed as the columns of the matrix $\mathbf{\Phi}=\mathbf{X}_2 \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{W}$ as proposed in \cite{Tu2014}. }
\textcolor{black}{
The main results of the DMD analysis are shown in Figure \ref{DMD_Migu}. Figure \ref{DMD_Migu}a) compares the dynamics $u(t,x)$ with the DMD reconstruction using $\mbox{rank}(\mathbf{X})=15$ modes, showing the excellent convergence of the DMD. The associated DMD eigenvalues are shown in \ref{DMD_Migu}(b) (full markers) and compared to the actual Koopman eigenvalues (empty markers). In this figure, the size of the markers is proportional to the mode contribution, computed from formula \eqref{Eq_Importance}. }
\textcolor{black}{
With the only exception of the eigenvalue for $\nu=(1)$, $\lambda_\nu=-\pi^2$, the DMD eigenvalues differ significantly from the actual Koopman ones, with the leading DMD modes (marked as 1 and 2 in the figure) having non-negligible imaginary parts. Consequently, the coefficients and the spatial structures of the DMD differ from those of the Koopman decomposition.
A detailed analysis of the reasons for such discrepancy is out of the scope of this work and will be presented in a later contribution. In general, an accurate approximation of an operator leads to an accurate approximation of its eigenvalues of multiplicity one, but this does not extend to eigenvalues with higher multiplicity. This example gives a practical illustration of the difficulties one faces in the data-driven identification of the Koopman modes.
}
\section{Concluding Remarks}\label{SecVII}
\textcolor{black}{
This paper gives an explicit Koopman decomposition (formula \eqref{F3}) for the Burgers' flow. It identifies the coefficients ($\varphi_\nu(u_0)$ in formula \eqref{Phi_nu}) of the decomposition as values of the eigenfunctionals of the Koopman operator (formula \eqref{Ffi}) and the temporal evolution as its eigenvalues. It also identifies the Koopman modes as the coefficients of the decomposition of the Dirac functionals on the Koopman eigenfunctionals (formula \eqref{F4bis}).
\vskip 0.1cm}
\textcolor{black}{
This work builds on that of Page and Kerswell \cite{Page2018} and overcomes the intrinsic multiplicity issue they encounter: we construct as many Koopman modes as the geometric multiplicity}.
\textcolor{black}{The convergence of the Koopman decomposition is here proven. To authors' knowledge, this is the first proof and explicit computation of Koopman modes for the flow given by a PDE.}
\vskip 0.1cm
This paper highlights the need for localization in the function space of the state variable and shows that the convergence of the decomposition is a local property. One must thus restrict the decomposition to its convergence set, which should contain the considered orbits for all $t>t_0$.
In the case of the Burgers equation, the only invariant set, where the decomposition converges, is the vicinity of the sink (the zero solution). This is where the necessary smallness condition shows up, quantified by the definition $\Omega_{\mathcal{B}}$ in \eqref{Omega_0}. For an attempt to localize in a neighborhood of an unstable singular location, an interesting numerical experiment is made by Page and Kerswell in \cite{Page2019}.
\vskip 0.1cm
\textcolor{black}{The need for regularity of the initial condition to have convergence of the Koopman decomposition at $t =0$ is a classic PDE ingredient. It is used in this paper to prove that the Dirac functionals can be decomposed on the eigenfunctionals of the Koopman operator, hence to prove \eqref{phi} for the case $\phi:=\delta_x$.}
\vskip 0.1cm
\textcolor{black}{An illustration of the Koopman decomposition is shown. This illustration gives a practical demonstration of the key features of this decomposition, including the multiplicity of the eigenvalues and the difficulties in convergence for $t\rightarrow 0$. These features make the data-driven identification of the actual Koopman modes challenging. To further illustrate this, a classic DMD of the Burgers flow is presented and the DMD eigenvalues compared to the actual Koopman eigenvalues. While the DMD outperforms the Koopman decomposition in terms of convergence and shows no difficulties as $t\rightarrow 0$, only one of the DMD eigenvalues correspond to a Koopman eigenvalue: the leading eigenvalue of multiplicity one.}
\vskip 0.1cm
\textcolor{black}{The explicit Koopman decomposition presented in this work enables a systematic analysis of the feasibility of a data-driven identification of Koopman modes via DMD.}
\section{Appendix: Estimates}\label{SecV}
\subsection {Three Preliminary Properties}\label{lemmi}
Here is the estimate used \textcolor{black}{to derive (\ref{Fdev})}:
{\bf Property 1}: If $v_0$ fulfills $\int_0^1v_0(s)ds=1$ and $\,\Vert \partial_xv_0\Vert_{L^2}<{1\over 4}$ then
$$\sup_x\vert 1-v_0(x)\vert< {1\over 4}\quad {\rm and} \quad \Vert u_0 \Vert_{L^2}<{2\over 3}\quad {\rm for} \quad u_0=C(v_0).$$
\noindent {\it Proof:} $v_0$ is given by:
$$v_0(x)=1+\int_0^xs\partial_xv_0(s)ds-\int_x^1(1-s)\partial_xv_0(s)ds
$$
By Cauchy-Schwartz, $\sup_x\vert1-v_0(x)\vert \leq \Vert \partial_xv_0\Vert_{L^2}<{1\over 4}$ so $\vert u_0(x)\vert\leq {8\over 3}\vert \partial_xv_0(x)\vert$.
\noindent
\textcolor{black}{We now give the assumption needed on $u_0 = C(v_0)$ in order that $v_0$ fulfills assumptions of property 1. This assumption on $u_0$ leads to the definition of $\Omega_{\mathcal{B}}$ given by formula (\ref{Omega_0}) that is needed in the proof of formula (\ref{F3}) below (section \ref{proof2}).}
{\bf Property 2:}
Let $u_0 \in L^2$ and $v_0=H(u_0)$ then
$$\sup_x\vert v_0(x)\vert\leq e^{\Vert u_0\Vert_{L^2}} \quad {\rm and}\quad \Vert \partial_xv_0\Vert_{L^2}\leq {1\over 2}e^{ \Vert u_0\Vert _{L^2}}\Vert u_0\Vert_{L^2}$$
\noindent {\it proof:} because $\int_0^xu_0(s)ds\leq \Vert u_0\Vert_{L^2}$ one has $v_0(x)\leq e^{{1\over 2} \Vert u_0\Vert _{L^2}}e^{-{1\over 2} \int_0^xu_0(s)ds}\leq e^{ \Vert u_0\Vert _{L^2}}$. From $u_0v_0=-2\partial_xv_0$ follows ${ \Vert \partial_xv_0\Vert _{L^2}}\leq {1\over 2}\sup_x\vert v_0(x)\vert \Vert u_0\Vert _{L^2}\leq {1\over 2}e^{ \Vert u_0\Vert _{L^2}}\Vert u_0\Vert _{L^2}$.
\\
\noindent \textcolor{black}{Here is the estimate needed to prove uniform and absolute convergence in formula (\ref{F3}) for all $t\ge 0$ for regular initial conditions. It is needed below in section \ref{proof3} to prove formula \eqref{F4}.}
{\bf Property 3:}
Let $u_0\in L^2$, $\partial_xu_0\in L^2$ and $u_0(0)=u_0(1)=0$. If $v_0=H(u_0)$ then
$$\Vert \partial^2_{xx}v_0\Vert_{L^2}\leq {1\over 2}e^{ \Vert u_0\Vert _{L^2}}\Vert \partial_{x}u_0\Vert_{L^2}(1+{\Vert u_0\Vert_{L^2}\over 2})$$
{\it Proof}: boundary conditions on $u_0$ give $\sup\vert u_0(x)\vert\leq \Vert \partial_{x}u_0\Vert_{L^2}$.
\noindent Because
$\partial^2_{xx}v_0=-{1\over 2}(v_0\partial_xu_0+u_0\partial_xv_0)$ we get $\Vert \partial^2_{xx}v_0\Vert_{L^2}\leq {1\over 2}(\sup_x\vert v_0(x)\vert \Vert \partial_xu_0\Vert_{L^2} $ $ +\sup_x\vert u_0(x)\vert \Vert \partial_xv_0\Vert_{L^2}) $ and the result follows from the estimates given in property 2.
\subsection {Convergence of formula (\ref{F3})}\label{proof2}
\medskip
\noindent We have, through integration by parts, for all $n\geq 1$:
$$ l_n(u_0) = \sqrt{2}\int_0^1H(u_0)(s)\cos{(n\pi s)}ds =
{\sqrt{2}\over n\pi}\int_0^1\partial_xH(u_0)(s)\sin{(n\pi s)}ds$$
so, because $\Vert u_0 \Vert_{L^2}\leq 1$ for $u_0\in \Omega_{\mathcal{B}}$, we get through Parseval formula and \textcolor{black}{property 2}:
$$\sum_1^\infty n^2\pi^2\vert l_n(u_0)\vert^2=\Vert \partial_xH(u_0)\Vert_{L^2}^2=\Vert \partial_xv_0\Vert_{L^2}^2
\leq {1\over 4}e^{2\Vert u_0\Vert_{L^2}}\Vert u_0\Vert_{L^2}^2\leq {e^2\over 4}\Vert u_0\Vert_{L^2}^2$$
We now prove the absolute convergence of formula (\ref{F3}): because
$$\forall \nu \in \mathcal{A} , \quad\vert a_\nu(x)\vert \leq 2^{\alpha(\nu)+3\over 2}\pi n_0$$
$$I:=\sum_{\nu\in \mathcal{A}}e^{\lambda_{\nu}t}\vert a_\nu(x)\vert \varphi_\nu(u_0)\vert
\leq
2^{3\over 2}\pi\sum_{m=0}^\infty 2^{m\over 2}\sum_{\nu\in \mathcal{A};\alpha(\nu)=m}\prod_{k=0}^me^{-n_k^2\pi^2 t}n_0\prod_{k=0}^m\vert l_{n_k}(u_0)\vert
$$
$$\leq 2^{3\over 2}\pi \,\sum_{n=1}^\infty e^{-n^2\pi^2t}n\vert l_n(u_0)\vert\,\,\sum_{m=0}^\infty 2^{m\over 2}(\sum_{k=1}^\infty \vert l_k(u_0)\vert)^m
$$
We apply the discrete Cauchy-Schwartz inequality and get:
$$I \leq 2^{3\over 2}\pi \sqrt{\sum_{n=1}^\infty e^{-2n^2\pi^2t} }\sqrt{\sum_{n=1}^\infty n^2\vert l_n(u_0)\vert^2 }
\,\, \sum_{m=0}^\infty 2^{m\over 2} \beta_2^m \,\,(\sum_{n=1}^\infty n^2 \vert l_n(u_0)\vert^2 ) ^{m\over 2}
$$
\noindent we use the estimate $\sum n^2\vert l_n(u_0)\vert^2\leq {e^2\over 4\pi^2}\Vert u_0\Vert^2$ proved above and get:
$$I \leq {t^{-{1\over 4}}} e\Vert u_0\Vert_{L^2}\sum_{m=0}^\infty (2^{-{1\over 2}} \pi^{-1}e\beta_2\Vert u_0\Vert_{L^2})^m
$$
\noindent with $\beta_2=\sqrt{\sum_1^\infty {1/ n^2}}={\pi /\sqrt{6}}$. This series converge because $u_0\in \Omega_{\mathcal{B}}$ so $\Vert u_0\Vert_{L^2}<{ 2\sqrt{3}/ e}$.
We proved convergence of absolute values, hence commutative convergence, as well as uniform convergence in the $x$ variable, \textcolor{black}{for $t >0$.}
\subsection{Uniform convergence in formula (\ref{F4})} \label{proof3}
One only needs to prove $(t,x)$-uniform convergence of absolute values for all $t\geq 0, x\in [0,1]$. Let $v_0=H(u_0)$. $v_0$ fulfills Neumann boundary conditions because $u_0 \in \omega_{\mathcal{B}}$ fulfills Dirichlet boundary conditions. Moreover $u_0$ has a square integrable weak derivative, so first and second weak derivatives of $v_0$ are square integrable as shown by property 3. Two integration by parts give:
$$\sum_1^\infty n^4\pi^4\vert l_n(u_0)\vert^2 = \Vert \partial_{xx}^2v_0\Vert_{L^2}^2$$
\textcolor{black}{The estimate goes as follows:}
$$J:=\sum_{\nu\in \mathcal{A}}e^{\lambda_{\nu}t}\vert a_\nu(x)\vert \varphi_\nu(u_0)\vert \leq
2^{3\over 2}\pi\sum_{m=0}^\infty 2^{m\over 2}\sum_{\nu\in \mathcal{A};\alpha(\nu)=m}n_0\prod_{k=0}^m\vert l_{n_k}(u_0\vert)\leq
$$
$$ 2^{3\over 2}\pi\sum_{m=0}^\infty 2^{m\over 2}(\sum_{n=1}^\infty n\vert l_n(u_0)\vert)\,\,(\sum_{k=1}^\infty \vert l_k(u_0)\vert)^m
$$
We use the discrete Cauchy-Schwartz inequality to get:
$$J\leq {2\sqrt{2}\over \pi}\beta_2\sqrt{\sum_{n=1}^\infty n^4\vert l_n\vert^2}\sum_{m=0}^\infty (\sqrt{2}\beta_2\sqrt{\sum_{n=1}^\infty n^2\vert l_n\vert^2}\,\,)^m \leq {2^{3\over 2}\beta_2\over \pi}\Vert \partial^2_{xx}v_0\Vert_{L^2}\sum_{m=0}^\infty ({\beta_2e\over\sqrt{2}\pi}\Vert u_0\Vert_{L^2})^m
$$
\noindent this is because $\Vert u_0\Vert_{L^2}\leq 1$ for $u_0\in \Omega_{\mathcal{B}}$. We use property 3 to get
$$J\leq {3e\beta_2\over \sqrt{2}\pi}\Vert \partial_{x}u_0\Vert_{L^2}\sum_{m=0}^\infty ({\beta_2e\over\sqrt{2}\pi}\Vert u_0\Vert_{L^2})^m
$$
This series converges for $\Vert u_0\Vert_{L^2}<{2\sqrt{3}/ e}$. This condition is fulfilled by any $u_0\in\Omega_{\mathcal{B}}$.
| {
"timestamp": "2021-04-27T02:07:52",
"yymm": "2007",
"arxiv_id": "2007.01218",
"language": "en",
"url": "https://arxiv.org/abs/2007.01218",
"abstract": "We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).",
"subjects": "Dynamical Systems (math.DS); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)",
"title": "On Koopman Operator for Burgers' Equation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846722794542,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7092019617872752
} |
https://arxiv.org/abs/1712.02485 | The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods | We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the algorithm converges. We show that in continuous time enforcement of an invariant that this approximate duality gap decreases at a certain rate exactly recovers a wide range of first-order continuous-time methods. We characterize the discretization errors incurred by different discretization methods, and show how iteration-complexity-optimal methods for various classes of problems cancel out the discretization error. The techniques are illustrated on various classes of problems -- including convex minimization for Lipschitz-continuous objectives, smooth convex minimization, composite minimization, smooth and strongly convex minimization, solving variational inequalities with monotone operators, and convex-concave saddle-point optimization -- and naturally extend to other settings. |
\section{Introduction}\label{sec:intro}
First-order optimization methods have recently gained high popularity due to their applicability to large-scale problem instances arising from modern datasets, their relatively low computational complexity, and their potential for parallelizing computation \cite{sra2012optimization}. Moreover, such methods have also been successfully applied in discrete optimization leading to faster numerical methods~\cite{ST04,KOSZ13}, graph algorithms~\cite{KLOS2014,Sherman2013,LRS2013}, and submodular optimization methods~\cite{Ene}.
Most first-order optimization methods can be obtained from the discretization of continuous-time dynamical systems that converge to optimal solutions. In the case of mirror descent, the continuous-time view was the original motivation for the algorithm~\cite{nemirovskii1983problem}, while more recent work has focused on deducing continuous-time interpretations of accelerated methods~\cite{wibisono2016variational,wilson2016lyapunov,krichene2015accelerated,SuBC16,Scieur2017}.
Motivated by these works, we focus on providing a unifying theory of first-order methods as discretizations of continuous-time dynamical systems. We term this general framework
the \emph{Approximate Duality Gap Technique (\textsc{adgt}\xspace)}. In addition to providing an intuitive and unified convergence analysis of various first-order methods that is often only a few lines long, \textsc{adgt}\xspace is also valuable in developing new first-order methods with tight convergence bounds%
~\cite{AXGD,cohen2018acceleration}, {in clarifying interactions between the acceleration and noise~\cite{cohen2018acceleration}, and in obtaining width-independent\footnote{{Width-independent algorithms enjoy poly-logarithmic dependence of their convergence times on the constraints matrix width -- namely, the ratio between the constraint matrix maximum and minimum non-zero elements. By contrast, standard first-order methods incur (at best) linear dependence on the matrix width, which is not even considered to be polynomial-time convergence~\cite{nesterov2005smooth}.}} algorithms for problems with positive linear constraints~\cite{LP-jelena-lorenzo,diakonikolas2018fairpc}. Further, we have extended \textsc{adgt}\xspace to the setting of block coordinate descent methods~\cite{diakonikolas2018alternating}.}
Unlike traditional approaches that start from an algorithm description and then use arguments such as
Lyapunov stability criteria to prove convergence bounds~\cite{nemirovskii1983problem,wibisono2016variational,wilson2016lyapunov,krichene2015accelerated,SuBC16},
our approach takes the opposite direction: \emph{continuous-time methods are obtained from the analysis}, using purely optimization-motivated arguments.
{In particular, \textsc{adgt}\xspace can be summarized as follows. Given a convex optimization problem $\min_{\mathbf{x} \in X}f(\mathbf{x})$, to show that a method converges to a minimizer $\mathbf{x}^* \in \argmin_{\mathbf{x} \in X} f(\mathbf{x})$ at rate $1/\a^{(t)}$ (e.g., $\a^{(t)} = t$ or $\a^{(t)}=t^2$), we need to show that $f(\mathbf{x}^{(t)}) - f(\mathbf{x}^*) \leq \nicefrac{Q}{\a^{(t)}}$, where $\mathbf{x}^{(t)}\in X$ is the solution produced by the method at time $t$ and $Q \in \mathbb{R_+}$ is some bounded quantity that is independent of time $t$. In general, keeping track of the true optimality gap $f(\mathbf{x}^{(t)}) - f(\mathbf{x}^*)$ is challenging, as the minimum function value $f(\mathbf{x}^*)$ is typically not known to the method. Instead, the main idea of \textsc{adgt}\xspace is to create an estimate of the optimality gap $G^{(t)}$ that can be easily tracked and controlled and ensure that $\a^{(t)}G^{(t)}$ is a non-increasing function of time. The estimate corresponds to the difference between an upper bound on $f(\mathbf{x}^{(t)})$, $U^{(t)}\geq f(\mathbf{x}^{(t)}),$ and a lower bound on $f(\mathbf{x}^*),$ $L^{(t)}\leq f(\mathbf{x}^*),$ so that $f(\mathbf{x}^{(t)}) - f(\mathbf{x}^*)\leq U^{(t)}-L^{(t)} = G^{(t)}$. Since \textsc{adgt}\xspace ensures that $\a^{(t)}G^{(t)}$ is a non-increasing function of time, it follows that $f(\mathbf{x}^{(t)}) - f(\mathbf{x}^*)\leq G^{(t)} \leq \nicefrac{\alpha^{(0)}G^{(0)}}{\alpha^{(t)}}$, which is precisely what we want to show, as long as we ensure that $\alpha^{(0)}G^{(0)}$ is bounded.}
To illustrate the power and generality of the technique, we show how to obtain and analyze several well-known first-order methods, such as gradient descent, dual averaging~\cite{nesterov2009primal}, mirror-prox/extra-gradient method~\cite{extragradient-descent,Nesterov2007dual-extrapolation,Mirror-Prox-Nemirovski}, accelerated methods~\cite{Nesterov1983,nesterov2005smooth}, composite minimization methods \cite{duchi2010composite,nesterov2015universal}, and Frank-Wolfe methods~\cite{nesterov2015cgm}. The same ideas naturally extend to other classes of convex optimization problems and their corresponding optimal first-order methods. Here, ``optimal'' is in the sense that the methods yield worst-case iteration complexity bounds for which there is a matching lower bound (i.e., ``optimal'' is in terms of iteration complexity).
\subsection{Related Work}\label{sec:related-work}
There exists a large body of research in optimization and first-order methods in particular, and, while we cannot provide a thorough literature review, we refer the reader to recent books \cite{sra2012optimization,Bubeck2015,ben2001lectures,nesterov2013introductory}.
Multiple approaches to unifying analyses of first-order methods have been developed, with a particular focus on explaining the acceleration phenomenon. Tseng gives a formal framework that unifies all the different instantiations of accelerated gradient methods~\cite{tseng2008}. More recently, Allen-Zhu and Orecchia~\cite{AO-survey-nesterov} interpret acceleration as coupling of mirror descent and gradient descent steps. Bubeck~\textit{et al}.~provide an elegant geometric interpretation of the Euclidean instantiation of Nesterov's method~\cite{Bubeck2015}. Drusvyatskiy~\textit{et al}.~\cite{drusvyatskiy2016optimal} interpret the geometric descent of Bubeck~\textit{et al}.~\cite{Bubeck2015} as a sequence minimizing quadratic lower-models of the objective function and obtain limited-memory extensions with improved performance. Lin~\textit{et al}.~\cite{lin2015universal} provide a universal scheme for accelerating non-accelerated first-order methods.
Su~\textit{et al}.~\cite{SuBC16} and Krichene~\textit{et al}.~\cite{krichene2015accelerated} interpret Nesterov's accelerated method as a discretization of a certain continuous-time dynamics and analyze it using Lyapunov stability criteria. Scieur~\textit{et al}.~\cite{Scieur2017} interpret acceleration as a multi-step integration method from numerical analysis applied to the gradient flow differential equation.
Wibisono~\textit{et al}.~\cite{wibisono2016variational} and Wilson~\textit{et al}.~\cite{wilson2016lyapunov} interpret accelerated methods using Lyapunov stability analysis and drawing ideas from Lagrangian mechanics.
A recent note of Bansal and Gupta~\cite{bansal2017potential} provides an intuitive, potential-based interpretation of many of the commonly-known convergence proofs for first-order methods.
The two references~\cite{wibisono2016variational,wilson2016lyapunov} are most closely related to our work, as they are motivated by the Lagrangian view from classical mechanics that leads to the description of system dynamics through the principle of stationary action. In a similar spirit, most dynamics described in our work maintain the invariant that $\frac{\d}{\d t}(\a^{(t)}G^{(t)})=0.$ However, unlike our work, which relies on enforcing an invariant that both leads to the algorithms and their convergence analysis, the results from~\cite{wibisono2016variational,wilson2016lyapunov} rely on the use of separate Lyapunov functions to obtain convergence results. It is unclear how these Lyapunov functions relate to the problems' optimality gaps, which makes them harder to generalize, especially in non-standard settings such as, e.g.,~\cite{LP-jelena-lorenzo,diakonikolas2018fairpc,diakonikolas2018alternating}.
The approximate duality gap presented here is closely related to Nesterov's estimate sequence (see e.g.,~\cite{nesterov2013introductory}). In particular, up to the regularization term $\phi(\mathbf{x}^*)/\a^{(t)}$, our lower bound $L^{(t)}$ is equivalent to Nesterov's estimate sequence, providing a natural interpretation of this powerful and commonly used technique.
\subsection{Notation}\label{sec:notation}
We use non-boldface letters to denote scalars and boldface letters to denote vectors. Superscript index $(\cdot)^{(t)}$ denotes the value of $(\cdot)$ at time $t$. The ``dot'' notation is used to denote the time derivative, i.e., $\dot{x} = \frac{d x}{d t}$. Given a measure $\alpha^{(\tau)}$ defined on $\tau \in [0, t]$, we use the Lebesgue-Stieltjes notation for the integral. In particular, given $\phi^{(\tau)}$ defined on $\tau\in [0, t]$:
\begin{equation*}
\int_{0}^{t} \phi^{(\tau)} \dot{\alpha}^{(\tau)}d\tau = \int_{0}^{t} \phi^{(\tau)} \mathrm{d}\alpha^{(\tau)}.
\end{equation*}
{
In the discrete-time setting, we will assume that $\a^{(\tau)}$ is an increasing piecewise constant function, with discontinuities occurring only at discrete time points $i \in Z_+$, and such that $\a^{(t)}=0$ for $t <0$. Hence, $\dot{\a}^{(\tau)}$ can be expressed as a train of Dirac Delta functions: $\dot{\a}^{(\tau)} = \sum_{i=0}^\infty a_i\delta(\tau-i),$ where $a_i = \a^{(i+\Delta)} - \a^{(i-\Delta)}$ for $\Delta \in (0, 1)$. This means that $\dot{\a}^{(\tau)}$ samples the function under the integral, so that $\int_{0}^t \phi^{(\tau)}\d\a^{(\tau)} = \sum_{i=0}^{\lfloor t \rfloor}a_i\phi^{(i)}$.}
We denote $A^{(t)} = \int_{0}^t \mathrm{d}\alpha^{(\tau)}$, so that $\frac{1}{A^{(t)}}\int_{0}^t \mathrm{d}\alpha^{(\tau)} = 1$. {In continuous time, $A^{(t)} = \a^{(t)}-\a^{(0)},$ while in the discrete time $A^{(t)} = \sum_{i=0}^{\lfloor t \rfloor}a_i=\a^{(t)}.$}
We assume throughout the paper that {$\a^{(0)}>0$ and} $\dot{\alpha}^{(t)}>0, \forall t\geq 0$, and use the following notation for the aggregated negative gradients:
\begin{equation}\label{eq:def-of-z}
\mathbf{z}^{(t)}\stackrel{\mathrm{\scriptscriptstyle def}}{=} -\int_{0}^t \nabla f(\mathbf{x}^{(\tau)})\mathrm{d}\alpha^{(\tau)}.
\end{equation}
For all considered problems, we assume that the feasible region is a closed convex set $X\subseteq \mathbb{R}^n$, for a finite $n$. We assume that there is a (fixed) norm $\|\cdot\|$ associated with $X$ and define its dual norm in a standard way: $\|\mathbf{z}\|_* = \max_{\mathbf{x} \in X}\{\innp{\mathbf{z}, \mathbf{x}}: \|\mathbf{x}\|\leq 1\}$, where $\innp{\cdot, \cdot}$ denotes the inner product.
\subsection{Preliminaries}\label{sec:prelims}
We focus on minimizing a continuous and differentiable\footnote{The differentiability assumption is not always necessary and can be relaxed to subdifferentiability in the case of dual averaging/mirror descent methods. Nevertheless, we will assume differentiability throughout the paper for simplicity of exposition.} convex function $f(\cdot)$ defined on a convex set $X \subseteq \mathbb{R}^n$, and we let $\mathbf{x}^* = \arg\min_{\mathbf{x} \in X}f(\mathbf{x})$ denote the minimizer of $f(\cdot)$ on $X$. The following definitions will be useful in our analysis, and thus we state them here for completeness.
\begin{definition}\label{def:convexity}
A function $f:X\rightarrow \mathbb{R}$ is convex on $X$, if for all $\mathbf{x}, \mathbf{\hat{x}} \in X$: $f(\mathbf{\hat{x}}) \geq f(\mathbf{x}) + \innp{\nabla f(\mathbf{x}), \mathbf{\hat{x}} - \mathbf{x}}$.
\end{definition}
\begin{definition}\label{def:smoothness}
A function $f:X\rightarrow \mathbb{R}$ is smooth on $X$ with smoothness parameter $L$ and with respect to a norm $\|\cdot\|$, if for all $\mathbf{x}, \mathbf{\hat{x}} \in X$: $f(\mathbf{\hat{x}}) \leq f(\mathbf{x}) + \innp{\nabla f(\mathbf{x}), \mathbf{\hat{x}} - \mathbf{x}} + \frac{L}{2}\|\mathbf{\hat{x}} - \mathbf{x}\|^2$. Equivalently: $\|\nabla f(\mathbf{x})-\nabla f(\mathbf{\hat{x}})\|_*\leq L\|\mathbf{x} - \mathbf{\hat{x}}\|$.
\end{definition}
\begin{definition}\label{def:strong-convexity}
A function $f:X\rightarrow \mathbb{R}$ is strongly convex on $X$ with strong convexity parameter $\sigma$ and with respect to a norm $\|\cdot\|$, if for all $\mathbf{x}, \mathbf{\hat{x}} \in X$: $f(\mathbf{\hat{x}}) \geq f(\mathbf{x}) + \innp{\nabla f(\mathbf{x}), \mathbf{\hat{x}} - \mathbf{x}} + \frac{\sigma}{2}\|\mathbf{\hat{x}} - \mathbf{x}\|^2$. Equivalently: $\|\nabla f(\mathbf{x})-\nabla f(\mathbf{\hat{x}})\|_*\geq \sigma\|\mathbf{x} - \mathbf{\hat{x}}\|$.
\end{definition}
\begin{definition}\label{def:bregman-divergence}(Bregman Divergence) Bregman divergence of a function $\psi$ is defined as:
$D_{\psi}(\mathbf{x}, \mathbf{\hat{x}}) \stackrel{\mathrm{\scriptscriptstyle def}}{=} \psi(\mathbf{x}) - \psi(\mathbf{\hat{x}})-\innp{\nabla \psi(\mathbf{\hat{x}}), \mathbf{x} - \mathbf{\hat{x}}}$.
\end{definition}
\begin{definition}\label{def:cxv-conj}(Convex Conjugate) Function $\psi^*(\cdot)$ is the convex conjugate of $\psi: X \rightarrow \mathbb{R}$, if $\psi^* (\mathbf{z}) = \sup_{\mathbf{x} \in X}\{\innp{\mathbf{z}, \mathbf{x}} - \psi(\mathbf{x})\}$, $\forall \mathbf{z} \in \mathbb{R}$.
\end{definition}
As $X$ is assumed to be closed, $\sup$ in Definition~\ref{def:cxv-conj} can be replaced by $\max$.
We will assume that there is a differentiable strictly convex function $\phi:X\rightarrow \mathbb{R}$, possibly dependent on $t$ (in which case we denote it as $\phi_t$), such that $\max_{\mathbf{x} \in X} \{\innp{\mathbf{z}, \mathbf{x}} - \phi (\mathbf{x})\}$ is easily solvable, possibly in a closed form. Notice that this problem defines the convex conjugate of $\phi(\cdot)$, i.e., $\phi^* (\mathbf{z}) = \max_{\mathbf{x} \in X} \{\innp{\mathbf{z}, \mathbf{x}} - \phi (\mathbf{x})\}$. {We will further assume without loss of generality that $\min_{\mathbf{x} \in X} \phi(\mathbf{x}) \geq 0$.\footnote{{This assumption can be easily satisfied by taking $\phi(\cdot)$ to be, for example, a Bregman divergence: $\phi(\mathbf{x}) = D_{\psi}(\mathbf{x}, \mathbf{x}^{(0)})$ for some strictly convex and differentiable $\psi$ and fixed $\mathbf{x}^{(0)} \in X$.}} The role of function $\phi$ will be to regularize the lower bound in the construction of the approximate duality gap.}
The following standard fact, based on Danskin's Theorem (see, e.g.~\cite{bertsekas1971control,Bertsekas2003}), will be extremely useful in carrying out the analysis of the algorithms in this paper.
\begin{fact}\label{fact:danskin}
Let $\phi: X \to \mathbb{R}$ be a differentiable strongly convex function. Then:
$$
\nabla \phi^*(\mathbf{z}) = \arg\max_{\mathbf{x} \in X} \left\{ \innp{\mathbf{z}, \mathbf{x}} - \phi(\mathbf{x})\right\} = \arg\min_{\mathbf{x} \in X} \left\{ -\innp{\mathbf{z}, \mathbf{x}} + \phi(\mathbf{x})\right\}.
$$
\end{fact}
In particular, Fact \ref{fact:danskin} implies:
\begin{equation}\label{eq:mirror-map-z-t}
\nabla\phi^*(\mathbf{z}^{(t)}) = \arg\min_{\mathbf{x} \in X}\left\{\int_{0}^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x} - \mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)} +\phi(\mathbf{x})\right\}
\end{equation}
Some other useful properties of Bregman divergence can be found in Appendix~\ref{sec:breg-div-prop}.
\paragraph{Overview of Continuous-Time Operations}
In continuous time, changes in the variables are described by differential equations. Of particular interest are (weighted) aggregation and averaging. Aggregation of a function $g(x)$ is $\dot{y}^{(t)} = \dot{\alpha}^{(t)}g(x^{(t)})$.
Observe that, by integrating both sides from $0$ to $t$, this is equivalent to: $y^{(t)} = y^{(0)} + \int_{0}^t g(x^{(\tau)})\mathrm{d}\alpha^{(\tau)}$. Averaging of a function $g(x)$ is $\dot{y}^{(t)} = \dot{\alpha}^{(t)}\frac{g(x^{(t)})-y^{(t)}}{\alpha^{(t)}}$. This can be equivalently written as $\frac{\d}{\d t}(\alpha^{(t)}y^{(t)}) = \dot{\alpha}^{(t)}g(x^{(t)})$, implying $y^{(t)} = \frac{\alpha^{(0)}}{\alpha^{(t)}}y^{(0)} + \frac{1}{\alpha^{(t)}}\int_{0}^t g(\mathbf{x}^{(\tau)})\mathrm{d}\alpha^{(\tau)}$.
The following simple proposition will be useful in our analysis.
\begin{proposition}\label{prop:ct-differentiation-inside-min}
$\frac{\d}{\d t} \min_{\mathbf{x} \in X}\left\{ -\innp{\mathbf{z}^{(t)}, \mathbf{x}} + \phi(\mathbf{x}) \right\} = -\innp{\dot{\mathbf{z}}^{(t)}, \nabla\phi^*(\mathbf{z}^{(t)})}$.
\end{proposition}
\begin{proof}
Follows by observing that $\phi^*(\mathbf{z}^{(t)}) = - \min_{\mathbf{x} \in X}\left\{ -\innp{\mathbf{z}^{(t)}, \mathbf{x}} + \phi(\mathbf{x}) \right\}$ and applying the chain rule.
\end{proof}
\section{The Approximate Duality Gap Technique}\label{sec:agt}
{As already mentioned in the introduction, to unify the analysis of a large class of first-order methods, we will show how to construct an upper estimate $G^{(t)}$ of the optimality gap $f(\mathbf{\hat{x}}^{(t)})-f(\mathbf{x}^*)$, where $\mathbf{\hat{x}}^{(t)}$ is the output of a first-order method at time $t$. This upper estimate is defined as $G^{(t)} = U^{(t)}-L^{(t)},$ where $U^{(t)}\geq f(\mathbf{\hat{x}}^{(t)})$ is an upper bound on $f(\mathbf{\hat{x}}^{(t)})$ and $L^{(t)}\leq f(\mathbf{x}^*)$ is a lower bound on $f(\mathbf{x}^*)$. We refer to $G^{(t)}$ as the approximate duality gap, due to the connections between the lower bound $L^{(t)}$ and the Fenchel dual of a certain approximation of the objective function $f(\mathbf{x}^{(t)}),$ further discussed in Section~\ref{sec:lb}. To show that the method converges at some rate $\a^{(t)}$ (e.g., $\a^{(t)} = t$), we will show that $\a^{(t)}G^{(t)}$ is a non-increasing function of time, so that $\a^{(t)}G^{(t)}\leq \a^{(0)}G^{(0)},$ and, consequently, $f(\mathbf{\hat{x}}^{(t)})-f(\mathbf{x}^*)\leq G^{(t)}\leq \nicefrac{\a^{(0)}G^{(0)}}{\a^{(t)}}$.
}
\subsection{Upper Bound}
{The simplest upper bound on $f(\mathbf{\hat{x}}^{(t)})$ is $f(\mathbf{\hat{x}}^{(t)})$ itself: i.e., $U^{(t)} = f(\mathbf{\hat{x}}^{(t)})$. In this case, $\mathbf{\hat{x}}^{(t)}$ will be the last point constructed by the algorithm, i.e., $\mathbf{\hat{x}}^{(t)} = \mathbf{x}^{(t)}$. We will make this choice of the upper bound whenever we can assume that $f(\cdot)$ is differentiable (e.g., in the setting of accelerated and Frank-Wolfe methods), so that in the continuous time setting we can differentiate $\a^{(t)}U^{(t)}$ with respect to $t$ and write $\frac{\d}{\d t}(\a^{(t)}U^{(t)}) = \dot{\a}^{(t)}f(\mathbf{\hat{x}}^{(t)}) + \a^{(t)}\innp{\nabla f(\mathbf{\hat{x}}^{(t)}), \frac{\d}{\d t}{\mathbf{\hat{x}}}^{(t)}}$. In the settings where $f(\cdot)$ is typically not assumed to be differentiable but only subdifferentiable (e.g., in the setting of dual averaging/mirror descent methods), $\mathbf{\hat{x}}^{(t)}$ will be a weighted average of the points $\mathbf{x}^{(\tau)}$ constructed by the method up to time $t$: $\mathbf{\hat{x}}^{(t)} = \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}\mathbf{x}^{(0)} + \frac{1}{\a^{(t)}}\int_{0}^t \mathbf{x}^{(\tau)}\d\a^{(\tau)}$ and we will choose $U^{(t)} = \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}f(\mathbf{x}^{(0)})+\frac{1}{\a^{(t)}}\int_0^t f(\mathbf{x}^{(\tau)})\d\a^{(\tau)}$. Due to Jensen's inequality, $f(\mathbf{\hat{x}}^{(t)})\leq U^{(t)},$ i.e., $U^{(t)}$ is a valid upper bound on $f(\mathbf{\hat{x}}^{(t)})$. Observe that this choice of $U^{(t)}$ allows us to differentiate $\a^{(t)}U^{(t)}$ with respect to $t$ in the continuous time setting, and, thus, we can write $\frac{\d}{\d t}(\a^{(t)}U^{(t)}) = \dot{\a}^{(t)}f(\mathbf{x}^{(t)}),$ as $\a^{(t)}-A^{(t)} = \a^{(0)}$ is a constant. These choices of upper bounds easily extend to the setting of composite
objectives $\bar{f}(\cdot) = f(\cdot) + \psi(\cdot)$ (see Section~\ref{sec:ct-algos} for more details).}
\subsection{Lower Bound}\label{sec:lb}
{The simplest lower bound on $f(\mathbf{x}^*)$ is $f(\mathbf{x}^*)$ itself. However, it is not clear how to use $L^{(t)} = f(\mathbf{x}^*)$ and guarantee $\frac{\d}{\d t}(\a^{(t)}G^{(t)})\leq 0,$ as in that case in the continuous-time domain $\frac{\d}{\d t}(\alpha^{(t)}L^{(t)}) = \dot{\a}^{(t)}f(\mathbf{x}^*)$ is not possible to evaluate, as we do not know $f(\mathbf{x}^*)$ (recall that, by assumption, $\dot{\a}^{(t)}>0$). Observe that if instead $f(\mathbf{x}^*)$ appeared in the lower bound as $\frac{c}{\a^{(t)}}f(\mathbf{x}^*)$ for some constant $c,$ we would not have this problem anymore, as $f(\mathbf{x}^*)$ would not appear in $\frac{\d}{\d t}(\alpha^{(t)}L^{(t)})$. This is true because $\frac{\d}{\d t}\left(\a^{(t)}\frac{c}{\a^{(t)}}f(\mathbf{x}^*)\right) = \frac{\d}{\d t}\left({c}f(\mathbf{x}^*)\right)=0.$ On the other hand, convexity of $f(\cdot)$ leads to the following lower-bounding hyperplanes for all $\mathbf{x} \in X$: $f(\mathbf{x}^*) \geq f(\mathbf{x}) + \innp{\nabla f(\mathbf{x}), \mathbf{x}^* - \mathbf{x}}$.\footnote{{Observe here that if $f(\cdot)$ is not differentiable but only subdifferentiable, we can still obtain lower-bounding hyperplanes by using subgradients in place of the gradients.}} In particular, taking a convex combination of the trivial lower bound $f(\mathbf{x}^*)$ and the lower-bounding hyperplanes defined by points $\mathbf{x}^{(\tau)}$ constructed by the method up to time $t$, we have:}
\begin{equation}\label{eq:lb-simple}
{f(\mathbf{x}^*) \geq \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}f(\mathbf{x}^*) + \frac{1}{\a^{(t)}}\int_0^t \left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^*-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}}.
\end{equation}
{
As in the continuous-time domain $\a^{(t)}-A^{(t)} = \a^{(0)}$ is a positive constant, the lower bound equal to the right-hand side of~\eqref{eq:lb-simple} is well-defined at $t=0$ and $f(\mathbf{x}^*)$ does not appear in $\frac{\d}{\d t}(\alpha^{(t)}L^{(t)})$. However, it would still not be possible to evaluate $\frac{\d}{\d t}(\alpha^{(t)}L^{(t)})$, as $\mathbf{x}^*$ is not known. One way of addressing this issue is to use that:
\begin{equation}\label{eq:ineq-for-lb-fw}
\begin{aligned}
&\int_0^t \left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^*-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}\\
&\hspace{2cm}\geq \int_0^t \min_{\mathbf{u} \in X}\left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u}-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}, \text{ or }
\end{aligned}
\end{equation}
\begin{equation}\label{eq:ineq-for-lb-gen}
\begin{aligned}
&\int_0^t \left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^*-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}\\
&\hspace{2cm}\geq \min_{\mathbf{u} \in X}\int_0^t \left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u}-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}.
\end{aligned}
\end{equation}
While the inequality~\eqref{eq:ineq-for-lb-gen} is tighter than~\eqref{eq:ineq-for-lb-fw}, as a minimum of affine functions it is not differentiable w.r.t.~$\mathbf{u}$ (and consequently not differentiable w.r.t.~$t$). The use of~\eqref{eq:ineq-for-lb-fw} leads to the continuous-time version of the standard Frank-Wolfe method~\cite{frank1956algorithm}.
\begin{example}\label{ex:frank-wolfe}
Standard continuous-time Frank-Wolfe method. Using~\eqref{eq:ineq-for-lb-fw}, we have the following lower bound:
\begin{equation}\label{eq:fw-lb}
\begin{aligned}
f(\mathbf{x}^*) \geq L^{(t)} \stackrel{\mathrm{\scriptscriptstyle def}}{=} &\frac{\int_0^t \min_{\mathbf{u} \in X}\left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u}-\mathbf{x}^{(\tau)}}\right)\d\a^{(\tau)}}{\a^{(t)}}\\
&+ \frac{+ (\a^{(t)}-A^{(t)})f(\mathbf{x}^*)}{\a^{(t)}}.
\end{aligned}
\end{equation}
Since the standard assumption in this setting is that $f(\cdot)$ is smooth (or, at the very least, continuously differentiable), we take $U^{(t)} = f(\mathbf{x}^{(t)})$ and $\mathbf{\hat{x}}^{(t)} = \mathbf{x}^{(t)}$. Denoting $\mathbf{v}^{(t)} \in \argmin_{\mathbf{u} \in X}\left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u}-\mathbf{x}^{(\tau)}}\right)$ and computing $\a^{(t)}G^{(t)},$ we get:
\begin{align*}
\frac{\d}{\d t}(\a^{(t)}G^{(t)}) =& \innp{\nabla f(\mathbf{x}^{(t)}), \a^{(t)}\dot{\mathbf{x}}^{(t)} - \dot{\a}^{(t)}(\mathbf{v}^{(t)}-\mathbf{x}^{(t)})}.
\end{align*}
Setting $\a^{(t)}\dot{\mathbf{x}}^{(t)} - \dot{\a}^{(t)}(\mathbf{v}^{(t)}-\mathbf{x}^{(t)})=0$ gives $\frac{\d}{\d t}(\a^{(t)}G^{(t)}) = 0$ and precisely recovers the continuous-time version of the Frank-Wolfe algorithm, as in that case $\dot{\mathbf{x}}^{(t)} = \frac{\dot{\a}^{(t)}(\mathbf{v}^{(t)}-\mathbf{x}^{(t)})}{\a^{(t)}},$ i.e., (as explained in Section~\ref{sec:prelims}) $\mathbf{x}^{(t)}$ is a weighted average of $\mathbf{v}^{(t)}$'s.
\end{example}
Notice that the use of~\eqref{eq:ineq-for-lb-fw} in the construction of the lower bound makes sense only when linear minimization over $X$ is possible. However, there are insights we can take from the construction of the lower bound~\eqref{eq:fw-lb}. In particular, we can alternatively view~\eqref{eq:fw-lb} as being constructed as a lower bound on $f(\mathbf{x}^*) + \psi(\mathbf{x}^*),$ where $\psi(\mathbf{x}^*)$ is the indicator of $X$. Hence we can view $L^{(t)}$ from~\eqref{eq:fw-lb} as being constructed as follows:
\begin{align*}
f(\mathbf{x}^*) + \psi(\mathbf{x}^*) \geq &\frac{\int_0^t \left(f(\mathbf{x}^{(\tau)}) + \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^*-\mathbf{x}^{(\tau)}} + \psi(\mathbf{x}^*)\right)\d\a^{(\tau)}}{\a^{(t)}}\\
&+ \frac{(\a^{(t)}-A^{(t)})f(\mathbf{x}^*)}{\a^{(t)}}\geq L^{(t)}.
\end{align*}
We will see later in Section~\ref{sec:ct-algos} how this leads to a more general version of Frank-Wolfe method for composite functions, along the lines of the method from~\cite{nesterov2015cgm}.}
{Constructing a lower bound on $f(\mathbf{x}^*) + \psi(\mathbf{x}^*)$ when $\psi(\mathbf{x}^*)$ was an indicator function had no effect, as in that case $f(\mathbf{x}^*) + \psi(\mathbf{x}^*) = f(\mathbf{x}^*)$. To generalize this idea, we can construct a lower bound on a function that closely approximates $f(\cdot)$ around $\mathbf{x}^*$. Since we want to obtain convergent methods, any error we introduce into this approximation should vanish at rate $1/\a^{(t)}$. Hence, a natural choice is to create a lower bound on $f(\mathbf{x}^*) + \frac{1}{\a^{(t)}}\phi(\mathbf{x}^*)$, where $\phi(\mathbf{x}^*)$ is bounded\footnote{{A common choice of $\phi(\cdot)$ that ensures boundedness of $\phi(\mathbf{x}^*)$ and non-negativity of $\phi(\cdot)$ is Bregman divergence of some function $\psi;$ namely, $\phi(\cdot) = D_{\psi}(\cdot, \mathbf{x}^{(0)})$. Hence, in this case we can interpret $\phi(\mathbf{x}^*)$ as a generalized notion of the initial distance to the optimal solution.}}:
\begin{align*}
f(\mathbf{x}^*) + \frac{1}{\a^{(t)}}\phi(\mathbf{x}^*) \geq &\frac{\int_0^t f(\mathbf{x}^{(\tau)})\d\a^{(\tau)} + \int_0^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^* - \mathbf{x}^{(\tau)}}\d\a^{(\tau)} + \phi(\mathbf{x}^*)}{\a^{(t)}}\\
&+ \frac{(\a^{(t)}-A^{(t)})f(\mathbf{x}^*)}{\a^{(t)}}.
\end{align*}
Now, if $\phi(\cdot)$ is strictly convex, $\min_{\mathbf{u} \in X}\{\int_0^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u} - \mathbf{x}^{(\tau)}}\d\a^{(\tau)} + \phi(\mathbf{u})\}$ is differentiable (i.e., we can generalize the stronger inequality from~\eqref{eq:ineq-for-lb-gen}). We can view this as regularization of the minimum from~\eqref{eq:ineq-for-lb-gen}, leading to the following lower bound:
\begin{equation}\label{eq:lb-general}
\begin{aligned}
f(\mathbf{x}^*) +& \frac{1}{\a^{(t)}}\phi(\mathbf{x}^*) \geq L^{(t)} + \frac{\phi(\mathbf{x}^*)}{\a^{(t)}}\\
=& \frac{\int_0^t f(\mathbf{x}^{(\tau)})\d\a^{(\tau)} + \min_{\mathbf{u} \in X}\{\int_0^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{u} - \mathbf{x}^{(\tau)}}\d\a^{(\tau)} + \phi(\mathbf{u})\}}{\a^{(t)}}\\
&+ \frac{(\a^{(t)}-A^{(t)})f(\mathbf{x}^*)}{\a^{(t)}}.
\end{aligned}
\end{equation}
Another advantage of the lower bound from~\eqref{eq:lb-general} over the previous one is that for many feasible sets $X$ there are natural choices of $\phi$ for which the minimization inside the lower bound from~\eqref{eq:lb-general} is easily solvable, often in a closed form (see, e.g.,~\cite{ben2001lectures}).}
\paragraph{Dual View of the Lower Bound}
{An alternative view of the lower bound from~\eqref{eq:lb-general} is through the concept of Fenchel Duality, which is defined for the sum of two convex functions (or the difference of a convex and a concave function). In particular, the Fenchel dual of $f(\mathbf{x})+\phi_t(\mathbf{x})$ is defined as $-f^*(-\mathbf{u})-\phi_t^*(\mathbf{u})$ (see, e.g., Chapter~15.2 in~\cite{bauschke2011convex}). Let $\phi_t(\cdot) = \frac{1}{A^{(t)}}\phi(\cdot)$ and $\mathbf{u}^{(t)} = - \frac{\int_0^t \nabla f(\mathbf{x}^{(\tau)})\d\a^{(\tau)}}{A^{(t)}}$. Observe that the minimization problem from~\eqref{eq:lb-general} defines a convex conjugate of $\phi_t.$ Thus, we can equivalently write~\eqref{eq:lb-general} as:
\begin{align*}
& f(\mathbf{x}^*) + \frac{A^{(t)}}{\a^{(t)}}\phi_t(\mathbf{x}^*) \geq L^{(t)} + \frac{A^{(t)}}{\a^{(t)}}\phi_t(\mathbf{x}^*)\\
&\hspace{1cm} = \frac{\int_0^t (f(\mathbf{x}^{(\tau)})-\innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x}^{(\tau)}})\d\a^{(\tau)}}{\a^{(t)}} - \frac{A^{(t)}}{\a^{(t)}}\phi_t^*(\mathbf{u}^{(t)}) + \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}f(\mathbf{x}^*)\\
&\hspace{1cm} = \frac{-\int_0^t f^*(\nabla f(\mathbf{x}^{(\tau)}))\d\a^{(\tau)}}{\a^{(t)}} - \frac{A^{(t)}}{\a^{(t)}}\phi_t^*(\mathbf{u}^{(t)}) + \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}f(\mathbf{x}^*).
\end{align*}
Rearranging the terms in the last equality, we can equivalently write:
\begin{align*}
& L^{(t)} + \phi_t(\mathbf{x}^*)\\
&\hspace{.5cm}= -\frac{A^{(t)}}{\a^{(t)}}\Big(\frac{\int_0^t f^*(\nabla f(\mathbf{x}^{(\tau)}))\d\a^{(\tau)}}{A^{(t)}} + \phi_t^*(\mathbf{u}^{(t)})\Big) + \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}(f(\mathbf{x}^*) + \phi_t(\mathbf{x}^*))\\
&\hspace{.5cm}\geq \frac{A^{(t)}}{\a^{(t)}}\Big(-f^*(-\mathbf{u}^{(t)}) - \phi_t^*(\mathbf{u}^{(t)})\Big) + \frac{\a^{(t)}-A^{(t)}}{\a^{(t)}}(f(\mathbf{x}^*) + \phi_t(\mathbf{x}^*)),
\end{align*}
where the last line is by Jensen's inequality. Hence, we can view the general lower bound from~\eqref{eq:lb-general} as being slightly stronger than the weighted average of $f(\mathbf{x}^*) + \phi_t(\mathbf{x}^*)$ and its Fenchel dual $-f^*(-\mathbf{u}^{(t)})-\phi_t^*(\mathbf{u}^{(t)})$ evaluated at the average negative gradient $\mathbf{u}^{(t)},$ and corrected by the introduced approximation error $\phi_t(\mathbf{x}^*)$. This means that we can think about the lower bound $L^{(t)}$ as encoding the Fenchel dual of $f(\mathbf{x}^*) + \phi_t(\mathbf{x}^*)$ -- an approximation to $f(\mathbf{x}^*)$ that converges to $f(\mathbf{x}^*)$ at rate $1/\a^{(t)}$ -- and constructing dual solutions from the history of the gradients of $f$.}
\paragraph{Extension to Strongly Convex Objectives} When the objective is $\sigma$-strongly convex for some $\sigma > 0$, we can use $\sigma$-strong convexity (instead of just regular convexity) in the construction of the lower bound. This will generally give us a better lower bound which will lead to the better convergence guarantees in the discrete-time domain. It is not hard to verify (by repeating the same arguments as above) that in this case we have the following lower bound:
\begin{equation}\label{eq:lb-strongly-cvx}
\begin{aligned}
L^{(t)} =& \frac{\int_{0}^t f(\mathbf{x}^{(\tau)}) \mathrm{d}\alpha^{(\tau)}}{\alpha^{(t)}}+ \frac{( \alpha^{(t)} - A^{(t)})f(\mathbf{x}^*) - \phi(\mathbf{x}^*)}{\alpha^{(t)}}\\
&+ \frac{\min_{\mathbf{x} \in X} \left\{\int_{0}^t \left(\innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x} - \mathbf{x}^{(\tau)}} + \frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(\tau)}\|^2\right)\mathrm{d}\alpha { + \phi(\mathbf{x})}\right\}}{\alpha^{(t)}}.
\end{aligned}
\end{equation}
\begin{remark}
{Note that, due to the strong convexity of $f$, we do not need additional regularization in the lower bound to ensure that the minimum inside it is differentiable, i.e., we could use $\phi(\cdot) = 0$. This choice of $\phi$ will have no effect in the continuous-time convergence. In the discrete time, however, if we chose $\phi(\cdot) = 0$ the initial gap (and, consequently, the convergence bound) would scale with $\|\mathbf{x}^{(1)} - \mathbf{x}^{(0)}\|^2$. Adding a little bit of regularization (i.e., choosing a non-zero $\phi$) will allow us to replace $\|\mathbf{x}^{(1)} - \mathbf{x}^{(0)}\|^2$ with $\|\mathbf{x}^{*} - \mathbf{x}^{(0)}\|^2$.}
\end{remark}
\paragraph{Extension to Composite Objectives} Suppose now that we have a composite objective $\bar{f}(\mathbf{x}) = f(\mathbf{x}) + \psi(\mathbf{x})$. Then, we can choose to apply the convexity argument only to $f(\cdot)$ and use $\psi(\cdot)$ as a regularizer (this will be particularly useful in the discrete-time domain in the settings where $f(\cdot)$ has some smoothness properties while $\psi(\cdot)$ is generally non-smooth). Therefore, we could start with $\bar{f}(\mathbf{x}) \geq f(\mathbf{\hat{x}}) + \innp{\nabla f(\mathbf{\hat{x}}), \mathbf{x} - \mathbf{\hat{x}}} + \psi(\mathbf{x})$. Repeating the same arguments as in the general construction of the lower bound presented earlier in this subsection:
\begin{equation}\label{eq:lb-composite}
\begin{aligned}
L^{(t)} \stackrel{\mathrm{\scriptscriptstyle def}}{=} &\frac{\int_{0}^t f(\mathbf{x}^{(\tau)}) \mathrm{d}\alpha^{(\tau)}}{\alpha^{(t)}}+ \frac{(\alpha^{(t)}-A^{(t)})\bar{f}(\mathbf{x}^*) - \phi(\mathbf{x}^*)}{\alpha^{(t)}}\\
&+ \frac{\min_{\mathbf{x} \in X} \left\{\int_{0}^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x} - \mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)} + A^{(t)}\psi(\mathbf{x}) + \phi(\mathbf{x})\right\}}{\alpha^{(t)}}.
\end{aligned}
\end{equation}
\subsection{Extension to Monotone Operators and Saddle-Point Formulations} The notion of the approximate gap can be defined for problem classes beyond convex minimization. Examples are monotone operators and convex-concave saddle-point problems. {More details are provided in Appendix~\ref{app:mon-ops}}.
\section{First-Order Methods in Continuous Time}\label{sec:ct-algos}
We now show how different {assumptions about the problem leading to the different} choices of the upper and lower bounds (and, consequently, the gap) yield different first-order methods.
\subsection{Mirror Descent/Dual Averaging Methods}\label{sec:ct-md}
{Let us start by making minimal assumptions about the objective function $f$: we will assume that $f$ is convex and subdifferentiable (with the abuse of notation, in this case $\nabla f(\mathbf{x}^{(t)})$ denotes an arbitrary but fixed subgradient of $f$ at $\mathbf{x}^{(t)}$). As discussed in the previous section, since we are not assuming that $f$ is differentiable, we will take $U^{(t)} = \frac{\a^{(0)}}{\a^{(t)}}f(\mathbf{x}^{(0)}) + \frac{\int_0^t f(\mathbf{x}^{(\tau)})\d\a^{(\tau)}}{\a^{(t)}},$ so that $\a^{(t)}U^{(t)}$ is differentiable w.r.t.~the time and well-defined at the initial time point. As there are no additional assumptions about the objective (such as composite structure and strong convexity), we will use the generic lower bound from~\eqref{eq:lb-general}. Hence, we have the following expression for the gap:}
\begin{equation}
\begin{aligned}
G^{(t)} = &\frac{-\min_{\mathbf{x} \in X}\left\{ \int_{0}^t\innp{\nabla f(\mathbf{x}^{(\tau)}), \mathbf{x} - \mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)} + \phi(\mathbf{x}) \right\}}{\alpha^{(t)}}\\
&+ \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \phi(\mathbf{x}^*) }{\alpha^{(t)}}.
\end{aligned}
\end{equation}
Observe that $G^{(0)} \leq \frac{\phi(\mathbf{x}^*)}{\alpha^{(0)}} + f(\mathbf{x}^{(0)}) - f(\mathbf{x}^*)$. Thus, if we show that $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)})\leq 0$, this would immediately imply:
$$
f(\mathbf{\hat{x}}^{(t)}) - f(\mathbf{x}^*) \leq U^{(t)} - L^{(t)} \leq \frac{\alpha^{(0)}}{\alpha^{(t)}}G^{(0)} \leq \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \phi(\mathbf{x}^*)}{\alpha^{(t)}}.
$$
Now we show that ensuring the invariance $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) = 0$ exactly produces the continuous-time mirror descent dynamics. Using (\ref{eq:mirror-map-z-t}) and Proposition (\ref{prop:ct-differentiation-inside-min}) with $\mathbf{z}^{(t)} = -\int_{0}^t\nabla f(\mathbf{x}^{(\tau)})\mathrm{d}\alpha^{(t)}$:
\begin{equation*}
\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) = -\innp{\nabla f(\mathbf{x}^{(t)}), \nabla\phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}\dot{\alpha}^{(t)}.
\end{equation*}
Thus, to have $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) = 0$, we can set $\mathbf{x}^{(t)} = \nabla \phi^*(\mathbf{z}^{(t)})$, which is precisely the mirror descent dynamics from \cite{nemirovskii1983problem}
\begin{equation}\label{eq:ct-md}\tag{CT-MD}
\begin{gathered}
\dot{\mathbf{z}}^{(t)} = - \dot{\alpha}^{(t)}\nabla f(\mathbf{x}^{(t)}), \\
\mathbf{x}^{(t)} = \nabla\phi^*(\mathbf{z}^{(t)}), \\
\dot{\hat{\mathbf{x}}}^{(t)} = \dot{\alpha}^{(t)}\frac{\mathbf{x}^{(t)}-\mathbf{\hat{x}}^{(t)}}{\alpha^{(t)}},\\
\mathbf{z}^{(0)} = 0,\; \mathbf{\hat{x}}^{(0)} = \mathbf{x}^{(0)}, \text{ for an arbitrary initial point } \mathbf{x}^{(0)}\in X.
\end{gathered}
\end{equation}
{We immediately obtain the following lemma:}
\begin{lemma}\label{lemma:ct-md-conv}
Let $\mathbf{x}^{(t)}, \mathbf{\hat{x}}^{(t)}$ evolve according to (\ref{eq:ct-md}) for some convex function $f:X\rightarrow\mathbb{R}$. Then, $\forall t > 0$:
$$
f(\mathbf{\hat{x}}^{(t)}) - f(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \phi(\mathbf{x}^*)}{\alpha^{(t)}}.
$$
\end{lemma}
\subsection{Accelerated Convex Minimization}\label{sec:ct-acc}
{Let us now assume more about the objective function: we will assume that $f$ is continuously differentiable, which means that our choice of the upper bound will be $U^{(t)} = f(\mathbf{\hat{x}}^{(t)}) = f(\mathbf{x}^{(t)}).$ Since there are no additional assumptions about $f,$ the lower bound we use is the generic one from Equation~\eqref{eq:lb-general}. Differentiating $\a^{(t)}G^{(t)},$ we have:}
\begin{align*}
\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) =& \frac{\d}{\d t}(\alpha^{(t)}f(\mathbf{x}^{(t)})) - \dot{\alpha}^{(t)}\left(f(\mathbf{x}^{(t)}) + \innp{\nabla f(\mathbf{x}^{(t)}), \nabla \phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}\right)\\
=& \innp{\nabla f(\mathbf{x}^{(t)}), \alpha^{(t)}\dot{\mathbf{x}}^{(t)}- \dot{\alpha}^{(t)}(\nabla \phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)})},
\end{align*}
where we have used $\frac{\d}{\d t}(f(\mathbf{x}^{(t)})) = \innp{\nabla f(\mathbf{x}^{(t)}), \dot{\mathbf{x}}^{(t)}}$. Choosing $\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla \phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}}$, we get $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) = 0$. This is precisely the accelerated mirror descent dynamics \cite{krichene2015accelerated,wibisono2016variational}, and we immediately get the convergence result stated as Lemma \ref{lemma:ct-amd-conv} below.
\begin{equation}\label{eq:ct-amd}\tag{CT-AMD}
\begin{gathered}
\dot{\mathbf{z}}^{(t)} = -\dot{\alpha}^{(t)}\nabla f(\mathbf{x}^{(t)}),\\
\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla \phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}},\\
\mathbf{z}^{(0)} = 0,\; \mathbf{x}^{(0)}\in X \text{ is an arbitrary initial point}.
\end{gathered}
\end{equation}
\begin{lemma}\label{lemma:ct-amd-conv}
Let $\mathbf{x}^{(t)}, \mathbf{z}^{(t)}$ evolve according to (\ref{eq:ct-amd}), for some {continuously differentiable} convex function $f:X\rightarrow\mathbb{R}$. Then, $\forall t > 0$:
$$
f(\mathbf{x}^{(t)})-f(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \phi(\mathbf{x}^*)}{\alpha^{(t)}}.
$$
\end{lemma}
\subsection{Gradient Descent}
Using the same approximate gap as in the previous subsection, now consider the special case when $\phi(\mathbf{x}) = \frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(0)}\|_2^2$ for an arbitrary initial point $\mathbf{x}^{(0)} = \mathbf{x}^{(0)}\in X$, $X = \mathbb{R}^n$, and some $\sigma > 0$. Then, $\nabla \phi^*(\mathbf{z}^{(t)}) = \mathbf{x}^{(0)} + \nicefrac{\mathbf{z}^{(t)}}{\sigma}$. Using the result for $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)})$ from the previous subsection and setting $\mathbf{x}^{(t)} = \nabla\phi^*(\mathbf{z}^{(t)})$, we have:
\begin{align*}
\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) &= \alpha^{(t)}\innp{\nabla f(\mathbf{x}^{(t)}), \dot{\mathbf{x}}^{(t)}} = -\frac{\alpha^{(t)}\dot{\alpha}^{(t)}}{\sigma}\|\nabla f(\mathbf{x}^{(t)})\|_2^2 \leq 0.
\end{align*}
The choice $\mathbf{x}^{(t)} = \nabla\phi^*(\mathbf{z}^{(t)}) = \mathbf{x}^{(0)} + \nicefrac{\mathbf{z}^{(t)}}{\sigma}$ precisely defines the gradient descent algorithm, and the convergence result stated as Lemma~\ref{lemma:ct-grad-desc} immediately follows.
\begin{equation}\label{eq:ct-gd}\tag{CT-GD}
\begin{gathered}
\dot{\mathbf{z}}^{(t)} = - \dot{\alpha}^{(t)}\nabla f(\mathbf{x}^{(t)})\\
\mathbf{x}^{(t)}= \nabla \phi^*(\mathbf{z}^{(t)}) = \mathbf{x}^{(0)} + \nicefrac{\mathbf{z}^{(t)}}{\sigma},\\
\mathbf{z}^{(0)} = 0, \; \mathbf{x}^{(0)}\in \mathbb{R}^n \text{ is an arbitrary initial point}.
\end{gathered}
\end{equation}
\begin{lemma}\label{lemma:ct-grad-desc}
Let $\mathbf{x}^{(t)}, \mathbf{z}^{(t)}$ evolve according to (\ref{eq:ct-gd}), for some {continuously differentiable} convex function $f:\mathbb{R}^n\rightarrow\mathbb{R}$. Then, $\forall t > 0$:
$$
f(\mathbf{x}^{(t)})-f(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \frac{\sigma}{2}\|\mathbf{x}^* - \mathbf{x}^{(0)}\|_2^2}{\alpha^{(t)}}.
$$
\end{lemma}
\subsection{Accelerated Strongly Convex Minimization}
{Let us now assume that, in addition to being continuously differentiable, $f$ is strongly convex. }
In that case, we use $U^{(t)} = f(\mathbf{x}^{(t)})$ and the lower bound from (\ref{eq:lb-strongly-cvx}). Let $\phi_t(\mathbf{x}) = \int_{0}^t \frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(\tau)}\|^2\mathrm{d}\a^{(\tau)}{ + \phi(\mathbf{x})}$. Observe that $\frac{\d}{\d t}\phi_t(\mathbf{x}) \geq 0$, $\forall \mathbf{x} \in X$. Then, we have the following result for the change in the gap:
\begin{align*}
\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) =& \frac{\d}{\d t}(\alpha^{(t)}f(\mathbf{x}^{(t)})) - \dot{\alpha}^{(t)}\left(f(\mathbf{x}^{(t)}) + \innp{\nabla f(\mathbf{x}^{(t)}), \nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}\right)\\
&- \frac{\d}{\d\tau}\left.(\phi_{\tau}(\nabla \phi_t^*(\mathbf{z}^{(t)})))\right|_{\tau = t}\\
\leq& \innp{\nabla f(\mathbf{x}^{(t)}), \alpha^{(t)}\dot{\mathbf{x}}^{(t)}- \dot{\alpha}^{(t)}(\nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)})}.
\end{align*}
Therefore, choosing $\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}}$ gives $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)})\leq 0$, and the convergence result stated as Lemma \ref{lemma:ct-amd-sc-conv} below follows.
\begin{equation}\label{eq:ct-amd-sc}\tag{CT-ASC}
\begin{gathered}
\dot{\mathbf{z}}^{(t)} = -\dot{\alpha}^{(t)}\nabla f(\mathbf{x}^{(t)}),\\
\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}},\\
\mathbf{z}^{(0)} = 0,\; \mathbf{x}^{(0)}\in \mathbb{R}^n \text{ is an arbitrary initial point}.
\end{gathered}
\end{equation}
\begin{lemma}\label{lemma:ct-amd-sc-conv}
Let $\mathbf{x}^{(t)}$ evolve according to (\ref{eq:ct-amd-sc}) {for some continuously differentiable and $\sigma-$strongly convex function $F$}, where $\phi_t(\mathbf{x}) = \int_{0}^t \frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(\tau)}\|^2{ + \phi(\mathbf{x})}$. Then, $\forall t > 0$:
$$
f(\mathbf{x}^{(t)})-f(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(f(\mathbf{x}^{(0)})-f(\mathbf{x}^*)) + \phi(\mathbf{x}^*)}{\alpha^{(t)}}.
$$
\end{lemma}
Note that, while there is no difference in the convergence bound for (\ref{eq:ct-amd}) and (\ref{eq:ct-amd-sc}), in the discrete time domain these two algorithms lead to very different convergence bounds, due to the different discretization errors they incur.
\subsection{Composite Dual Averaging}
Now assume that the objective is composite: $\bar{f}(\mathbf{x}) = f(\mathbf{x}) + \psi(\mathbf{x})$, where $f(\mathbf{x})$ is convex and $\nabla\phi_t^*(\cdot)$ is easily computable, for $\phi_t(\mathbf{x}) = A^{(t)}\psi(\mathbf{x}) + \phi(\mathbf{x})$. Then, we can use the lower bound for composite functions~(\ref{eq:lb-composite}). {Since we are not assuming that either $f$ or $\psi$ is continuously differentiable, the upper bound of choice is:}
\begin{align*}
U^{(t)} &= \frac{1}{\alpha^{(t)}}\int_{0}^t \bar{f}(\mathbf{x}^{(\tau)})\mathrm{d}\alpha^{(\tau)} + \frac{\alpha^{(0)}}{\alpha^{(t)}}\bar{f}(\mathbf{x}^{(0)})\\
&= \frac{1}{\alpha^{(t)}}\int_{0}^t (f(\mathbf{x}^{(\tau)})+\psi(\mathbf{x}^{(\tau)}))\mathrm{d}\alpha^{(\tau)} + \frac{\alpha^{(0)}}{\alpha^{(t)}}(f(\mathbf{x}^{(0)})+\psi(\mathbf{x}^{(0)})).
\end{align*}
Then, the change in the gap is:
\begin{equation}\label{eq:ct-gap-change-comp}
\begin{aligned}
&\frac{\d}{\d t}(\alpha^{(t)}G^{(t)})\\
&\hspace{1cm}= \dot{\alpha}^{(t)}\psi(\mathbf{x}^{(t)}) - \dot{\alpha}^{(t)}\innp{\nabla f(\mathbf{x}^{(t)}), \nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}} - \dot{\alpha}^{(t)}\psi(\nabla\phi_t^*(\mathbf{z}^{(t)})).
\end{aligned}
\end{equation}
Thus, when $\dot{\mathbf{x}}^{(t)} = \nabla\phi_t^*(\mathbf{z}^{(t)})$, where $\phi_t(\cdot) = \alpha^{(t)}\psi(\cdot)$, we have $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)}) = 0$, and Lemma \ref{lemma:ct-composite-conv} follows immediately. The algorithm can be thought of as mirror descent (or dual averaging) for composite minimization.
\begin{equation}\label{eq:ct-comp-md}\tag{CT-CMD}
\begin{gathered}
\dot{\mathbf{z}}^{(t)} = -\dot{\alpha}^{(t)}\nabla f(\mathbf{x}^{(t)}),\\
\mathbf{x}^{(t)} = \nabla\phi_t^*(\mathbf{z}^{(t)}),\\
\mathbf{\hat{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\mathbf{x}^{(t)}-\mathbf{\hat{x}}^{(t)}}{\alpha^{(t)}},\\
\mathbf{z}^{(0)} = 0,\; \mathbf{\hat{x}}^{(0)} = \mathbf{x}^{(0)}, \text{ for arbitrary initial point } \mathbf{x}^{(0)}\in X.
\end{gathered}
\end{equation}
\begin{lemma}\label{lemma:ct-composite-conv}
Let $\mathbf{x}^{(t)}, \mathbf{\hat{x}}^{(t)}$ evolve according to (\ref{eq:ct-comp-md}), for some convex composite function $\bar{f} = f+ \psi: X\rightarrow \mathbb{R}$. Then, $\forall t > 0$:
$$
\bar{f}(\mathbf{\hat{x}}^{(t)})-\bar{f}(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(\bar{f}(\mathbf{x}^{(0)})-\bar{f}(\mathbf{x}^*)) + \phi(\mathbf{x}^*)}{\alpha^{(t)}}.
$$
\end{lemma}
\subsection{{Generalized} Frank-Wolfe Method}
{We have already discussed the standard version of Frank-Wolfe method in Example~\ref{ex:frank-wolfe}. As discussed there, we can view standard Frank-Wolfe method as minimizing a composite objective $\bar{f} = f + \psi,$ where $\psi$ is the indicator function of set $X$, and using the assumption that problems of the form $\min_{\mathbf{u} \in X} \{\innp{\mathbf{z}, \mathbf{u}} + \psi(\mathbf{u})\}$ are easily solvable for any fixed $\mathbf{z}$. We will now show how the method generalizes for any (possibly non-differentiable) $\psi$. The lower bound from~\eqref{eq:fw-lb} derived for standard Frank-Wolfe method immediately generalizes to:}
\begin{equation}\label{eq:gen-lb-fw}
\begin{aligned}
{
L^{(t)}} = &{\frac{\int_0^t f(\mathbf{x}^{(\tau)})\d\a^{(\tau)} + \int_0^t \min_{\mathbf{u} \in X}\{\innp{\nabla f(\mathbf{x}^{(\tau)})\d\a^{(\tau)}, \mathbf{u} - \mathbf{x}^{(\tau)}} + \psi(\mathbf{u})\}}{\a^{(t)}}}\\
&{+ \frac{(\a^{(t)}-A^{(t)})\bar{f}(\mathbf{x}^*)}{\a^{(t)}}.
}
\end{aligned}
\end{equation}
{By (generalized) Danskin's Theorem~\cite{bertsekas1971control}, we have that $$\nabla\psi^*(-\nabla f(\mathbf{x}^{(\tau)}))\in \argmin_{\mathbf{u} \in X}\Big\{\innp{\nabla f(\mathbf{x}^{(\tau)})\d\a^{(\tau)}, \mathbf{u} - \mathbf{x}^{(\tau)}} + \psi(\mathbf{u})\Big\}$$
(note that the minimizer may not be unique as $\psi$ is not necessarily strictly convex; with the abuse of notation, $\nabla\psi^*(-\nabla f(\mathbf{x}^{(\tau)}))$ may be a subgradient of $\psi^*$). Since $f$ is assumed to be continuously differentiable, we would like to use $f(\mathbf{x}^{(t)})$ in the upper bound. On the other hand, $\psi$ is not necessarily differentiable, and hence we cannot use $\psi(\mathbf{x}^{(t)})$ in the upper bound. Instead, we need to have $\frac{\a^{(0)}}{\a^{(t)}}\psi(\mathbf{x}^{(0)}) + \frac{\int_0^t \psi(\mathbf{v}^{(\tau)})\d\a^{(\tau)}}{\a^{(t)}}$ for some points $\mathbf{v}^{(\tau)} \in X$ to ensure that $\a^{(t)}U^{(t)}$ is differentiable. Hence, based on the rules for choosing the upper bound discussed in Section~\ref{sec:agt}, we would like the upper bound to be of the form:}
\begin{equation}\label{eq:ub-fw}
{U^{(t)} = f(\mathbf{x}^{(t)}) + \frac{\a^{(0)}}{\a^{(t)}}\psi(\mathbf{x}^{(0)}) + \frac{\int_0^t \psi(\mathbf{v}^{(\tau)})\d\a^{(\tau)}}{\a^{(t)}}.}
\end{equation}
{Using Jensen's Inequality, this is a valid upper bound on $\bar{f}(\mathbf{x}^{(t)})$ if $\mathbf{x}^{(t)} = \frac{\a^{(0)}}{\a^{(t)}}\mathbf{x}^{(0)} + \frac{\int_0^t \mathbf{v}^{(\tau)}\d\a^{(\tau)}}{\a^{(t)}}.$ Taking a leap of faith, let us consider the gap constructed based on the lower and upper bounds from~\eqref{eq:gen-lb-fw}, \eqref{eq:ub-fw}. Differentiating $\a^{(t)}G^{(t)}:$}
\begin{align*}
{\frac{\d}{\d t}(\a^{(t)}G^{(t)}) =}&{\innp{\nabla f(\mathbf{x}^{(t)}), \a^{(t)}\dot{\mathbf{x}}^{(t)} - \dot{\a}^{(t)}(\nabla \psi^*(-\nabla f(\mathbf{x}^{(t)}))-\mathbf{x}^{(t)})}} \\
&{+ \dot{\a}^{(t)}\psi(\mathbf{v}^{(t)}) - \dot{\a}^{(t)}\psi(\nabla \psi^*(-\nabla f(\mathbf{x}^{(t)}))).}
\end{align*}
{For the terms from the second line to cancel out, we need $\mathbf{v}^{(t)} = \nabla \psi^*(-\nabla f(\mathbf{x}^{(t)})).$ Then, if we set $\a^{(t)}\dot{\mathbf{x}}^{(t)} - \dot{\a}^{(t)}(\mathbf{v}^{(t)}-\mathbf{x}^{(t)}) = 0$, we get $\frac{\d}{\d t}(\a^{(t)}G^{(t)}) = 0$. But this precisely defines $\mathbf{x}^{(t)}$ as $\mathbf{x}^{(t)} = \frac{\a^{(0)}}{\a^{(t)}}\mathbf{x}^{(0)} + \frac{\int_0^t \mathbf{v}^{(\tau)}\d\a^{(\tau)}}{\a^{(t)}}$, which is what we needed for the upper bound to be valid. Hence, we exactly recover the continuous-time counterpart of the generalized Frank-Wolfe method from~\cite{nesterov2015cgm} and Lemma~\ref{lemma:ct-fw-conv} follows.}
\begin{equation}\label{eq:ct-fw}\tag{CT-FW}
\begin{gathered}
\mathbf{\hat{z}}^{(t)} = -\nabla f(\mathbf{x}^{(t)}),\\
\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla\psi^*(\mathbf{\hat{z}}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}},\\
\mathbf{x}^{(0)}\in X \text{ is an arbitrary initial point}.
\end{gathered}
\end{equation}
\begin{lemma}\label{lemma:ct-fw-conv}
Let $\mathbf{x}^{(t)}, \mathbf{\hat{z}}^{(t)}$ evolve according to (\ref{eq:ct-fw}), for some convex composite function $\bar{f} = f+ \psi: X\rightarrow \mathbb{R}${, where $f$ is continuously differentiable}. Then:
$$
\bar{f}(\mathbf{x}^{(t)})-\bar{f}(\mathbf{x}^*) \leq \frac{\alpha^{(0)}(\bar{f}(\mathbf{x}^{(0)})-\bar{f}(\mathbf{x}^*))}{\alpha^{(t)}}, \; \forall t \geq 0.
$$
\end{lemma}
\section{Discretization and Incurred Errors}\label{sec:discretization}
Suppose now that $\alpha^{(t)}$ is a discrete measure. In particular, let $\alpha^{(t)}$ be an increasing piecewise constant function, with $\alpha^{(t)}= 0$ for $t < 0$, $\alpha^{(t)}$ constant in intervals $(0 + i,\; 0 + i + 1)$ for $i \in \mathbb{Z}_+$, and $\alpha^{((0 + i)+)} - \alpha^{((0 + i)-)} = a_i$ for some $a_i > 0$ and $i \in \mathbb{Z}_+${, as discussed in Section~\ref{sec:notation}}.
For the continuous-time algorithms (and their analyses) presented in Section \ref{sec:ct-algos}, there are generally two causes of the discretization error: (i) different integration rules applying to continuous and discrete measures, and (ii) discontinuities in the algorithm updates. We discuss these two causes in more details below.
\paragraph{Integration errors}
To understand where the integration errors occur, we first note that such errors cannot occur in integrals whose sole purpose is weighted averaging, since for these integrals there is no functional difference in the continuous- and discrete-time domains.
Thus, the only place where the integration errors can occur is in the integral appearing under the minimum in the lower bound. In $\alpha^{(t)}G^{(t)} = A^{(t)}G^{(t)}$, the integral appears as:
\begin{equation}
I^{(0, t)} = -\int_{0}^t \innp{\nabla f(\mathbf{x}^{(\tau)}), \nabla \phi_t^*(\mathbf{z}^{(t)})-\mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)},\notag
\end{equation}
where $\phi_t(\cdot) = \phi(\cdot)$ in the case of mirror descent and accelerated convex minimization. Let $I_c^{(0, t)}$ denote the value $I^{(0, t)}$ {would take if} $\alpha$ was a continuous measure, {i.e., if rules of continuous integration applied}.
Observe that, as between times $i-1$ and $i$ $\dot{\alpha}^{(\tau)}$ samples the function under the integral at time $i$, we have:
\begin{equation}\label{eq:I-i-1-i}
I^{(i-1, i)} = -{a_i}\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\phi_i^*(\mathbf{z}^{(i)}) - \mathbf{x}^{(i)}}.
\end{equation}
\paragraph{Discontinuities in the algorithm updates}
In all of the described algorithms, the updates for $\mathbf{x}^{(t)}$ (and possibly $\mathbf{\hat{x}}^{(t)}$) depend on $\nabla \phi_t^*(\mathbf{z}^{(t)})$. Recall that $\mathbf{z}^{(t)}$ aggregates negative gradients up to time $t$ and thus also depends on $\nabla f(\mathbf{x}^{(t)})$. In the continuous time domain, this is not a problem, since the updates in $\mathbf{x}^{(t)}$ can follow updates in $\mathbf{z}^{(t)}$ with an arbitrarily small delay, meaning that in the limit we can take that $\mathbf{x}^{(t)}$ changes simultaneously with $\mathbf{z}^{(t)}$. In the discrete time, however, the delay between the two updates cannot be neglected, and {using implicit updates of the form} $\mathbf{x}^{(t)} = g(\nabla \phi_t^*(\mathbf{z}^{(t)}))$ for some function $g(\cdot)$ is in general either not possible or requires many fixed-point iterations.
Apart from affecting the value of $I^{(i-1, i)}$ described above, the discontinuities will also contribute additional discretization error in the case of composite minimization. The reason for the additional discretization error is that the analysis of the gap reduction relies on bounding the change in $\psi(\mathbf{x}^{(t)})$ (or the $\dot{\alpha}^{(\tau)}$-weighted average of $\psi(\mathbf{x}^{(\tau)})$'s for $\tau \in [0, t]$) from the upper bound by $\dot{\alpha}^{(t)}\psi(\nabla\phi_t^*(\mathbf{z}^{(t)}))$ from the lower bound. For composite dual averaging, this discretization error at time $i$ will amount to $a_i\left(\psi(\mathbf{x}^{(i)})-\psi(\nabla\phi_i^*(\mathbf{z}^{(i)}))\right)$.
Similar to composite dual averaging, Frank-Wolfe will accrue discretization error $a_i\left(\psi(\mathbf{x}^{(i)})-\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)}))\right)$.
\paragraph{Effect of discretization errors on the gap}
Since in continuous time we had that $\frac{\d}{\d t}(\alpha^{(t)}G^{(t)})\leq 0$, if the discretization error between discrete time points $i-1$ and $i$ is $E_d^{(i)}$, then $A^{(i)}G^{(i)}-A^{(i-1)}G^{(i-1)}\leq E_d^{(i)}$, and we can conclude that:
\begin{equation}\label{eq:gap-w-discr-err}
G^{(k)} \leq \frac{a_0 G^{(0)}}{A^{(k)}} + \frac{\sum_{i=1}^k E_d^{(i)}}{A^{(k)}}.
\end{equation}
{Note that $E_d^{(i)}$'s contain both the integration error and the error due to the discontinuities in the algorithm updates discussed above. These discretization errors will ultimately determine the algorithms' convergence rate: while in the continuous time domain we could choose $\a^{(t)}$ (and $A^{(t)}$) to grow arbitrarily fast as a function of time, in the discrete time domain, the discretization errors $E_d^{(k)}$ will depend on the choice of $A^{(k)}$. In particular, this co-dependence between $A^{(k)}$ and $E_d^{(k)}$ will determine the choice of $A^{(k)}$ leading to the highest decrease in the bound on $G^{(k)}$ from~\eqref{eq:gap-w-discr-err} as a function of $k$.}
We are now ready to bound the discretization errors of the algorithms from Section~\ref{sec:ct-algos}. Before doing so, we make the following two remarks:
\begin{remark}The versions of mirror descent and mirror prox presented here are in fact the ``lazy'' versions of these methods that are known as Nesterov's dual averaging \cite{nesterov2009primal}.{The ``lazy'' and standard versions of the methods are equivalent whenever $X = \mathbb{R}^n$. The reason we chose to work with the ``lazy'' versions of the methods is that they follow more directly from the discretization of continuous-time dynamics presented in the previous section.}
\end{remark}
\begin{remark}
It is possible to obtain the discretization error $E_d^{(i)}$ by directly computing $A^{(k)}G^{(k)}-A^{(k-1)}G^{(k-1)}$ for the discrete version of the gap. We have chosen the approach presented here to illustrate the effects of discretization.
\end{remark}
\subsection{Dual Averaging (Lazy Mirror Descent)}
Recall that in mirror descent, $\phi_i(\cdot) = \phi(\cdot)$. The discretization error can be bounded as follows.
\begin{proposition}\label{prop:md-discr-err}
The discretization error for (\ref{eq:ct-md}) is:
\begin{equation
E_d^{(i)} = -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\phi^*(\mathbf{z}^{(i)})-\mathbf{x}^{(i)}} - D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).\notag
\end{equation}
\end{proposition}
\begin{proof}
In
(\ref{eq:ct-md}), $\mathbf{x}^{(\tau)}=\nabla\phi^*(\mathbf{z}^{(\tau)})$, and thus:
\begin{align}
I_c^{(i-1, i)} &= \int_{i-1}^{i}\innp{\dot{\mathbf{z}}^{(\tau)}, \nabla \phi^*(\mathbf{z}^{(i)})-\nabla\phi^*(\mathbf{z}^{(\tau)})}d\tau\notag\\
&= - \int_{i-1}^{i} \frac{d D_{\phi^*}(\mathbf{z}^{(\tau)}, \mathbf{z}^{(i)})}{d\tau}d\tau = D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}). \label{eq:md-I_c}
\end{align}
Since for non-composite functions $E_d^{(i)} = I^{(i-1, i)}-I_c^{(i-1, i)}$, combining (\ref{eq:I-i-1-i}) and (\ref{eq:md-I_c}) completes the proof.
\end{proof}
We now consider two different discretization methods that lead to discrete-time algorithms known as (lazy) mirror descent and mirror prox.
\paragraph{Forward Euler Discretization: Lazy Mirror Descent}
Forward Euler discretization leads to the following algorithm updates:
\begin{equation}\label{eq:classical-md}\tag{MD}
\begin{gathered}
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)} - a_{i}\nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \nabla\phi^*(\mathbf{z}^{(i-1)}),\\
\mathbf{\hat{x}}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{\hat{x}}^{(i-1)} + \frac{a_i}{A^{(i)}}\mathbf{x}^{(i)},\\
\mathbf{z}^{(0)} = -a_0 f(\mathbf{x}^{(0)}), \; \mathbf{\hat{x}}^{(0)} = \mathbf{x}^{(0)}, \text{ for arbitrary } \mathbf{x}^{(0)}\in X.
\end{gathered}
\end{equation}
It follows (from Proposition \ref{prop:md-discr-err}) that in this case the discretization error is given as:
\begin{equation}
E_d^{(i)} = -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}} - D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).
\end{equation}
When $f(\cdot)$ is Lipschitz-continuous, we recover the classical mirror descent/dual averaging convergence result \cite{nemirovskii1983problem}:
\begin{theorem} \label{thm:md}
Let $f:X\rightarrow \mathbb{R}$ be an $L$-Lipschitz-continuous convex function, and let $\psi:X\rightarrow \mathbb{R}$ be $\sigma$-strongly convex for some $\sigma >0$. Let $\mathbf{x}^{(i)}, \mathbf{\hat{x}}^{(i)}$ evolve according to (\ref{eq:classical-md}) for $i\leq k$ and $k\geq 1$. Then, if $a_i = \frac{1}{L}\sqrt{\frac{2\sigma D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})}{k+1}}$ and
$\phi(\cdot) = D_{\psi}(\cdot, \mathbf{x}^{(0)})$:
$$
f(\mathbf{\hat{x}}^{(k)}) - f(\mathbf{x}^*) \leq \sqrt{\frac{2D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})}{\sigma}}\cdot \frac{L}{\sqrt{k+1}}.
$$
\end{theorem}
\begin{proof}
By Proposition \ref{prop:cvx-conj-bd-is-strongly-cvx-too}, $D_{\psi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})\geq \frac{\sigma}{2}\|\nabla\psi^*(\mathbf{z}^{(i-1)})-\nabla\psi^*(\mathbf{z}^{(i)})\|^2 = \frac{\sigma}{2}\|\mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}\|^2$. As $D_{\phi}(\cdot, \cdot) = D_{\psi}(\cdot, \cdot)$ and $f(\cdot)$ is Lipschitz continuous with parameter $L$, using Cauchy-Schwartz Inequality:
\begin{align*}
E_d^{(i)} \leq a_i L\|\mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}\| - \frac{\sigma}{2}\|\mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}\|^2\leq \frac{{a_i}^2 L^2}{2\sigma},
\end{align*}
where the second inequality follows from $2ab - b^2 \leq a^2$, $\forall a, b$. Therefore, from (\ref{eq:gap-w-discr-err}):
\begin{equation}\label{eq:md-gap-w-error}
G^{(k)} \leq \frac{a_0 G^{(0)}}{A^{(k)}} + \frac{L^2}{2\sigma}\cdot \frac{\sum_{i=1}^k {a_i}^2}{A^{(k)}}.
\end{equation}
Similarly, we can bound the initial gap as:
\begin{align}
a_0 G^{(0)} &= -a_0\innp{\nabla f(\mathbf{x}^{(0)}), \nabla \phi^*(\mathbf{z}^{(0)})-\mathbf{x}^{(0)}} - \phi(\nabla \phi^*(\mathbf{z}^{(0)})) + \phi(\mathbf{x}^*)\notag\\
&= -a_0\innp{\nabla f(\mathbf{x}^{(0)}), \mathbf{x}^{(1)}-\mathbf{x}^{(0)}} -D_{\psi}(\mathbf{x}^{(1)}, \mathbf{x}^{(0)}) + D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})\notag\\
&\leq \frac{{a_0}^2 L^2}{2\sigma} + D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)}). \label{eq:md-initial-gap}
\end{align}
Finally, combining (\ref{eq:md-gap-w-error}), (\ref{eq:md-initial-gap}), the choice of $a_i$'s, and $f(\mathbf{\hat{x}}^{(k)})-f(\mathbf{x}^*)\leq G^{(k)}$, the result follows.
\end{proof}
\paragraph{Approximate Backward Euler Discretization: Mirror Prox/Extra-gradient}
Observe that if we could set $\mathbf{x}^{(i)}=\nabla\phi^*(\mathbf{z}^{(i)})$ (i.e., if we were using backward Euler discretization for $\mathbf{x}$), then the discretization error would be negative: $E_d^{(i)} = - D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})$. However, backward Euler is only an implicit discretization method, as it involves solving $\mathbf{x}^{(i)} = \nabla\phi^*(\mathbf{z}^{(i-1)}-a_i \nabla f(\mathbf{x}^{(i)}))$. Fortunately, the fact that the discretization error is negative enables an approximate implementation of the method, where only two fixed-point iteration steps are performed.\footnote{{This discretization can also be viewed as the predictor-corrector method.}} The resulting discrete-time method is known as mirror prox \cite{Mirror-Prox-Nemirovski} or extra-gradient descent \cite{extragradient-descent}.
\begin{equation}\label{eq:mp}\tag{MP}
\begin{gathered}
\mathbf{\tilde{x}}^{(i-1)} = \nabla\phi^*(\mathbf{z}^{(i-1)}),\\
\mathbf{\tilde{z}}^{(i-1)} = \mathbf{z}^{(i-1)} - a_i \nabla f(\mathbf{\tilde{x}}^{(i-1)}),\\
\mathbf{x}^{(i)} = \nabla\phi^*(\mathbf{\tilde{z}}^{(i-1)}),\\
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)} - a_i \nabla f(\mathbf{x}^{(i)}), \\
\mathbf{\hat{x}}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{\hat{x}}^{(i)} + \frac{a_i}{A^{(i)}}\mathbf{x}^{(i)},\\
\mathbf{z}^{(0)} = - a_0 \nabla f(\mathbf{x}^{(0)}), \text{ and } \mathbf{x}^{(0)}\in X \text{ is an arbitrary initial point.}
\end{gathered}
\end{equation}
{This method is typically used for solving variational inequalities with monotone operators~\cite{Mirror-Prox-Nemirovski}. Its convergence bound is provided in Theorem~\ref{thm:mp-conv} in Appendix~\ref{app:mon-ops}.}
\subsection{Accelerated Smooth Minimization}\label{sec:amd}
In this and in the following subsection, we will only consider forward Euler discretization of the accelerated dynamics, which corresponds to the Nesterov's accelerated algorithm. Approximate backward Euler discretization using similar ideas as in the proof of convergence of mirror prox from the previous subsection is also possible and leads to the recent accelerated extra-gradient descent (\textsc{axgd}\xspace) algorithm that we presented in \cite{AXGD}.
As before, we can bound the discretization error by computing $I^{(i-1, i)}$ to obtain the following result.
\begin{proposition}\label{prop:amd-discr-err}
The discretization for (\ref{eq:ct-amd}) is:
\begin{equation}\label{eq:amd-fe-discr-error}
\begin{aligned}
E_d^{(i)} =& -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi^*(\mathbf{z}^{(i)})-\mathbf{x}^{(i)}} + A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)}))\\
&- D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})\\
\leq & \innp{\nabla f(\mathbf{x}^{(i)}), A^{(i)} \mathbf{x}^{(i)} - A^{(i-1)}\mathbf{x}^{(i-1)}-a_i \nabla \phi^*(\mathbf{z}^{(i)})} - D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).
\end{aligned}
\end{equation}
\end{proposition}
\begin{proof}
Recall continuous-time accelerated dynamics (\ref{eq:ct-amd}), where $\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla\phi^*(\mathbf{z}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}}$ and $\phi_i(\cdot) = \phi(\cdot)$. We have:
\begin{align*}
I_c^{(i-1, i)} &= -\int_{i-1}^{i}\innp{\nabla f(\mathbf{x}^{(\tau)}), \nabla \phi^*(\mathbf{z}^{(i)})-\mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)}\\
&= -\int_{i-1}^{i}\innp{\nabla f(\mathbf{x}^{(\tau)}), \alpha^{(\tau)}\dot{\mathbf{x}}^{(\tau)}}d\tau + \int_{i-1}^{i}\innp{\dot{\mathbf{z}}^{(\tau)}, \nabla \phi^*(\mathbf{z}^{(i)}) - \nabla \phi^*(\mathbf{z}^{(\tau)})}d\tau.
\end{align*}
Integrating by parts, the first integral is $-A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)}))$, while the second integral is (as we have seen in the previous subsection) $D_{\psi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})$. Thus, using (\ref{eq:I-i-1-i}), the discretization error is:
\begin{equation*
\begin{aligned}
E_d^{(i)} =& -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi^*(\mathbf{z}^{(i)})-\mathbf{x}^{(i)}} + A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)}))\\
&- D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})\\
\leq &\innp{\nabla f(\mathbf{x}^{(i)}), A^{(i)} \mathbf{x}^{(i)} - A^{(i-1)}\mathbf{x}^{(i-1)}-a_i \nabla \phi^*(\mathbf{z}^{(i)})} - D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}),
\end{aligned}
\end{equation*}
where we used $f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})\leq \innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x}^{(i)}-\mathbf{x}^{(i-1)}}$, by $f(\cdot)$'s convexity.
\end{proof}
Standard forward Euler discretization sets $\mathbf{x}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{x}^{(i-1)} + \frac{a_i}{A^{(i)}}\nabla \phi^*(\mathbf{z}^{(i-1)})$, which results in the discretization error equal to $D_{\phi^*}(\mathbf{z}^{(i)}, \mathbf{z}^{(i-1)})$. We cannot bound such a discretization error, since we are not assuming that $f(\cdot)$ is Lipschitz-continuous. However, since $f(\cdot)$ is $L$-smooth, we can introduce an additional gradient step whose role is to cancel out the discretization error by reducing the upper bound. The algorithm then becomes the familiar Nesterov's accelerated method~\cite{Nesterov1983}:
\begin{equation}\label{eq:amd}\tag{AMD}
\begin{gathered}
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)} - a_i \nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{\hat{x}}^{(i-1)} + \frac{a_i}{A^{(i)}}\nabla\phi^*(\mathbf{z}^{(i-1)}),\\
\mathbf{\hat{x}}^{(i)} = \mathrm{Grad}(\mathbf{x}^{(i)}), \\
\mathbf{z}^{(0)} = -a_0 f(\mathbf{x}^{(0)}), \; \mathbf{\hat{x}}^{(0)} = \mathrm{Grad}(\mathbf{x}^{(0)}), \text{ for arbitrary } \mathbf{x}^{(0)}\in X,
\end{gathered}
\end{equation}
where
\begin{equation}\label{eq:grad-step-smooth}
\mathrm{Grad}(\mathbf{x}^{(i)}) = \arg\min_{\mathbf{x} \in X}\left\{\innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x} - \mathbf{x}^{(i)}} + \frac{L}{2}\|\mathbf{x} - \mathbf{x}^{(i)}\|^2\right\}.
\end{equation}
The introduced gradient steps only affect the upper bound, changing it from $U^{(i)} = f(\mathbf{x}^{(i)})$ to $U^{(i)} = f(\mathbf{\hat{x}}^{(i)})$. Thus, correcting (\ref{eq:amd-fe-discr-error}) for the change in the upper bound:
\begin{align}
E_d^{(i)} =& -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi^*(\mathbf{z}^{(i)})-\mathbf{x}^{(i)}} + A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)}))\notag\\
&- D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})
+ A^{(i)}(f(\mathbf{\hat{x}}^{(i)}) - f(\mathbf{x}^{(i)})) - A^{(i-1)}(f(\mathbf{\hat{x}}^{(i-1)})-f(\mathbf{x}^{(i-1)}))\notag\\
\leq & A^{(i)}(f(\mathbf{\hat{x}}^{(i)}) - f(\mathbf{x}^{(i)})) + \innp{\nabla f(\mathbf{x}^{(i)}), A^{(i)} \mathbf{x}^{(i)} - A^{(i-1)}\mathbf{\hat{x}}^{(i-1)}-a_i \nabla \phi^*(\mathbf{z}^{(i)})}\notag\\
&- D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})\notag\\
=& A^{(i)}(f(\mathbf{\hat{x}}^{(i)}) - f(\mathbf{x}^{(i)})) + a_i \innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi^*(\mathbf{z}^{(i-1)}) - \nabla \phi^*(\mathbf{z}^{(i)})}\notag\\
&- D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}). \label{eq:amd-discr-err}
\end{align}
We are now ready to prove the convergence of Nesterov's algorithm for smooth functions \cite{Nesterov1983}:
\begin{theorem}\label{thm:amd-conv}
Let $f:X\rightarrow \mathbb{R}$ be an $L$-smooth function, $\psi:X\rightarrow \mathbb{R}$ be a $\sigma$-strongly convex function, and let $\phi(\cdot) = D_{\psi}(\cdot, \mathbf{x}^{(0)})$. If $\mathbf{x}^{(t)}, \mathbf{\hat{x}}^{(t)}$ evolve according to (\ref{eq:amd}) for $a_i = {\frac{\sigma}{L}}\frac{i+1}{2}$, then $\forall k \geq 1$:
$$
f(\mathbf{\hat{x}}^{(k)}) - f(\mathbf{x}^*) \leq \frac{4L}{\sigma}\cdot\frac{D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})}{(k+1)(k+2)}.
$$
\end{theorem}
\begin{proof}
As $f(\cdot)$ is $L$-smooth, by the definition of $\mathbf{\hat{x}}^{(i)}$:
\begin{equation}\label{eq:amd-f-smoothness}
f(\mathbf{\hat{x}}^{(i)}) \leq f(\mathbf{x}^{(i)}) + \min_{\mathbf{x} \in X}\left\{\innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x} - \mathbf{x}^{(i)}} + \frac{L}{2}\|\mathbf{x} - \mathbf{x}^{(i)}\|^2\right\}.
\end{equation}
Since the gradient step was introduced to cancel out the discretization error, intuitively, it is natural to try to cancel out the second two terms from (\ref{eq:amd-discr-err}) (that correspond to the original discretization error (\ref{eq:amd-fe-discr-error})) by $A^{(i)}(f(\mathbf{\hat{x}}^{(i)})-f(\mathbf{x}^{(i)}))$, the decrease due to the gradient step. A point $\mathbf{x}\in X$ that would charge the gradient term from (\ref{eq:amd-discr-err}) to the gradient term in (\ref{eq:amd-f-smoothness}) is $\mathbf{x} = \mathbf{x}^{(i)} - \frac{a_i}{A^{(i)}}(\nabla \phi^*(\mathbf{z}^{(i-1)}) - \nabla \phi^*(\mathbf{z}^{(i)})) = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{\hat{x}}^{(i-1)} + \frac{a_i}{A^{(i)}}\nabla \phi^*(\mathbf{z}^{(i)}) \in X$. It follows from (\ref{eq:amd-f-smoothness}) that:
\begin{equation}\label{eq:amd-f-particular-x}
\begin{aligned}
A^{(i)}f(\mathbf{\hat{x}}^{(i)}) \leq & A^{(i)}f(\mathbf{x}^{(i)}) - {a_i}\innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi^*(\mathbf{z}^{(i-1)}) - \nabla \phi^*(\mathbf{z}^{(i)})}\\
&+ \frac{L {a_i}^2}{2A^{(i)}}\|\nabla \phi^*(\mathbf{z}^{(i-1)}) - \nabla \phi^*(\mathbf{z}^{(i)})\|^2.
\end{aligned}
\end{equation}
By Proposition \ref{prop:cvx-conj-bd-is-strongly-cvx-too}, $D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}) \geq \frac{\sigma}{2}\|\nabla \phi^*(\mathbf{z}^{(i-1)}) - \nabla\phi^*(\mathbf{z}^{(i)})\|^2$. Therefore, for the quadratic term in (\ref{eq:amd-f-particular-x}) to cancel the remaining term in (\ref{eq:amd-discr-err}), $D_{\phi^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})$, it suffices to have $\frac{{a_i}^2}{A^{(i)}}\leq \frac{\sigma}{L}$. It is easy to verify that $a_i = \frac{\sigma}{L}\frac{i+1}{2}$ from the theorem statement satisfies $\frac{{a_i}^2}{A^{(i)}}\leq \frac{\sigma}{L}$, and thus it follows that $G^{(k)}\leq \frac{a_0G^{(0)}}{A^{(k)}}$.
It remains to bound the initial gap, while the final bound will follow by simple computation of $A^{(k)}$. We have:
\begin{align*}
a_0G^{(0)} =& a_0(f(\mathbf{\hat{x}}^{(0)})-f(\mathbf{x}^{(0)})) - a_0\innp{\nabla f(\mathbf{x}^{(0)}), \nabla \phi^*(\mathbf{z}^{(0)})-\mathbf{x}^{(0)}}\\
&- D_{\psi}(\nabla\phi^*(\mathbf{z}^{(0)}), \mathbf{x}^{(0)}) + D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})\\
\leq & D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)}),
\end{align*}
by the same arguments as in bounding the discretization error above.
\end{proof}
\subsection{Gradient Descent} The discretization error of gradient descent is the same as the discretization error of the accelerated method from the previous subsection (Eq.~(\ref{eq:amd-fe-discr-error})), since the two methods use the same approximate duality gap. Classical gradient descent uses Forward Euler discretization, which sets $\mathbf{x}^{(i)} = \mathbf{x}^{(0)} - \nicefrac{\mathbf{z}^{(i-1)}}{\sigma}$. Thus, the algorithm can be stated as:
\begin{equation}\label{eq:grad-desc}\tag{GD}
\begin{gathered}
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)} - a_i \nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \nabla \phi^*(\mathbf{z}^{(i-1)}) = \mathbf{x}^{(0)} + \frac{\mathbf{z}^{(i-1)}}{\sigma},\\
\mathbf{z}^{(0)} = 0, \; \mathbf{x}^{(0)} \in \mathbb{R}^n \text{ is an arbitrary initial point.}
\end{gathered}
\end{equation}
To cancel out the discretization error, we only need to use the fact that gradient steps (corresponding to the steps of the algorithm) reduce the function value, assuming that the function is $L$-smooth for some $L \in \mathbb{R}_{++}$. This is achieved by setting $U^{(i)} = f(\mathbf{x}^{(i+1)})$. Correcting the discretization error by the change in the upper bound:
\begin{proposition}\label{prop:gd-discr-err}
The discretization error of (\ref{eq:grad-desc}) is:
\begin{equation*}
\begin{aligned}
E_d^{(i)} =& A^{(i)}\left(f(\mathbf{x}^{(i+1)}) - f(\mathbf{x}^{(i)})\right) - a_i \innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x}^{(i+1)} - \mathbf{x}^{(i)}} - \frac{{a_i}^2}{2\sigma}\|\nabla f(\mathbf{x}^{(i)})\|_2^2.
\end{aligned}
\end{equation*}
\end{proposition}
\begin{proof}
Follows directly by combining (\ref{eq:amd-fe-discr-error}), $U^{(i)} = f(\mathbf{x}^{(i+1)})$, and (\ref{eq:grad-desc}).
\end{proof}
We can now recover the classical bound for gradient descent (see, e.g., \cite{Bube2014}).
\begin{theorem}\label{thm:grad-desc}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be an $L$-smooth function for $L \in \mathbb{R}_{++}$, $\phi(\mathbf{x}) = \frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(0)}\|^2$, and let $\mathbf{x}^{(i)}$ evolve according to (\ref{eq:grad-desc}). If $a_i = \frac{\sigma}{L}$, $\forall i \geq 0$, then, $\forall k \geq 0$:
$$
f(\mathbf{x}^{(k+1)}) - f(\mathbf{x}^*) \leq \frac{L\|\mathbf{x}^* - \mathbf{x}^{(0)}\|^2}{2(k+1)}.
$$
\end{theorem}
\begin{proof}
By the smoothness of $f(\cdot)$ and (\ref{eq:grad-desc}):
\begin{align*}
E_d^{(i)} &\leq A^{(i-1)}\innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}} + A^{(i)}\frac{L}{2}\|\mathbf{x}^{(i+1)} - \mathbf{x}^{(i)}\|_2^2 - \frac{{a_i}^2}{2\sigma}\|\nabla f(\mathbf{x}^{(i)})\|_2^2\\
&= \left(-\frac{a_i A^{(i-1)}}{\sigma} + \frac{A^{(i)}{a_i}^2 L}{2\sigma^2} - \frac{{a_i}^2}{2\sigma}\right)\|\nabla f(\mathbf{x}^{(i)})\|_2^2.
\end{align*}
By type-checking the last expression, it follows that $a_i$ needs to be proportional to $\frac{\sigma}{L}$ to obtain $E_d^{(i)}\leq 0$. Choose $a_i = \frac{\sigma}{L}$. It remains to bound the initial gap. Observing that for $a_i = \frac{\sigma}{L}$, $U^{(0)} = f(\mathbf{x}^{(0)})- \frac{1}{2L}\|\nabla f(\mathbf{x}^{(0)})\|_2^2$ and $L^{(0)} = f(\mathbf{x}^{(0)}) - \frac{1}{2L}\|\nabla f(\mathbf{x}^{(0)})\|_2^2 + \frac{L}{2a_0}\|\mathbf{x}^* - \mathbf{x}^{(0)}\|_2^2$, the claimed bound on the gap follows.
\end{proof}
\subsection{Accelerated Smooth and Strongly Convex Minimization}
Recall the accelerated dynamics for $\sigma$-strongly convex objectives (\ref{eq:ct-amd-sc}). The dynamics is almost the same as (\ref{eq:ct-amd}), except that instead of a fixed $\phi_i(\cdot)$, we now have: $\phi_i(\mathbf{x}) = \int_0^i \frac{\sigma}{2}\|\mathbf{x}-\mathbf{x}^{(\tau)}\|^2\mathrm{d}\alpha^{(\tau)} + \phi(\mathbf{x})$. Observe that for $i \geq j$, $\phi_i(\mathbf{x})\geq \phi_j(\mathbf{x})$, $\forall \mathbf{x} \in X$. We can compute the discretization error for (\ref{eq:ct-amd-sc}) as follows.
\begin{proposition}\label{prop:amd-sc-fe-discr-error}
The discretization error for (\ref{eq:ct-amd-sc}) is:
\begin{align*}
E_d^{(i)} \leq & A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})) - \innp{\mathbf{z}^{(i)} - \mathbf{z}^{(i-1)}, \mathbf{x}^{(i)} - \nabla\phi_{i-1}^*(\mathbf{z}^{(i)})}\notag\\
&- D_{\phi_{i-1}^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).
\end{align*}
\end{proposition}
\begin{proof}
To compute the discretization error, we first need to compute $I_c^{(i-1, i)}$:
\begin{align}
I_c^{(i-1, i)} =& -\int_{i-1}^i \innp{\nabla f(\mathbf{x}^{(\tau)}), \nabla \phi_i^*(\mathbf{z}^{(i)})-\mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)}\notag\\
=& -\int_{i-1}^i \innp{\nabla f(\mathbf{x}^{(\tau)}), \alpha^{(\tau)}\dot{\mathbf{x}}^{(\tau)}}d\tau + \int_{i-1}^i \innp{\dot{\mathbf{z}}^{(\tau)}, \nabla \phi_i^*(\mathbf{z}^{(i)})}d\tau\notag\\
&- \int_{i-1}^i \innp{\dot{\mathbf{z}}^{(\tau)}, \nabla \phi_{\tau}^*(\mathbf{z}^{(\tau)})}d\tau\notag\\
=& -A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})) + \innp{\mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}, \nabla\phi_i^*(\mathbf{z}^{(i)})}\label{eq:amd-sc-I-c}\\
&- \phi_i^*(\mathbf{z}^{(i)})+\phi_{i-1}^*(\mathbf{z}^{(i-1)}).\notag
\end{align}
Combining (\ref{eq:I-i-1-i}), (\ref{eq:amd-sc-I-c}), and the fact that $-a_i\nabla f(\mathbf{x}^{(i)}) = \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}$:
\begin{equation}
E_d^{(i)} = A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})) - \innp{\mathbf{z}^{(i)} - \mathbf{z}^{(i-1)}, \mathbf{x}^{(i)}} + \phi_i^*(\mathbf{z}^{(i)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)}). \label{eq:amd-sc-discr-err-gen}
\end{equation}
As $\phi_i(\mathbf{x})\geq \phi_{i-1}(\mathbf{x})$, $\forall \mathbf{x} \in X$, it follows that also $\phi_i^*(\mathbf{z}) \leq \phi_{i-1}^*(\mathbf{z})$, $\forall \mathbf{z}$. Using the definition of Bregman divergence and convexity of $f(\cdot)$:
\begin{align}
E_d^{(i)} \leq & A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})) - \innp{\mathbf{z}^{(i)} - \mathbf{z}^{(i-1)}, \mathbf{x}^{(i)} - \nabla\phi_{i-1}^*(\mathbf{z}^{(i)})}\notag\\
&- D_{\phi_{i-1}^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).
\end{align}
\end{proof}
Comparing the discretization error from Proposition \ref{prop:amd-sc-fe-discr-error} with the discretization error (\ref{eq:amd-fe-discr-error}) from previous subsection, we can observe that they take the same form, with the only difference of $\phi^*$ being replaced by $\phi_{i-1}^*$. Thus, introducing a gradient descent step into the discrete algorithm leads to the same changes in the discretization error, and we can use the same arguments to analyze the convergence. The algorithm is given as:
\begin{equation}\label{eq:amd-sc}\tag{ASC}
\begin{gathered}
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)} - a_i \nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{\hat{x}}^{(i-1)} + \frac{a_i}{A^{(i)}}\nabla\phi_{i-1}^*(\mathbf{z}^{(i-1)}),\\
\mathbf{\hat{x}}^{(i)} = \mathrm{Grad}(\mathbf{x}^{(i)}),\\
\mathbf{z}^{(0)} = -a_0 f(\mathbf{x}^{(0)}),\; \mathbf{\hat{x}}^{(0)} = \mathrm{Grad}(\mathbf{x}^{(0)}), \text{ for arbitrary }\mathbf{x}^{(0)} \in X,
\end{gathered}
\end{equation}
while the discretization error for (\ref{eq:amd-sc}) becomes:
\begin{equation}\label{eq:amd-sc-discr-err}
\begin{aligned}
E_d^{(i)} &\leq A^{(i)}(f(\mathbf{\hat{x}}^{(i)}) - f(\mathbf{x}^{(i)})) + a_i \innp{\nabla f(\mathbf{x}^{(i)}), \nabla \phi_{i-1}^*(\mathbf{z}^{(i-1)}) - \nabla \phi_{i-1}^*(\mathbf{z}^{(i)})}\\
&- D_{\phi_{i-1}^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)}).
\end{aligned}
\end{equation}
We have the following convergence result:
\begin{theorem}\label{thm:amd-sc-conv}
Let $f:X\rightarrow \mathbb{R}$ be an $L$-smooth and $\sigma$-strongly convex function, $\psi:X\rightarrow\mathbb{R}$ be a $\sigma_0$-strongly convex function, for $\sigma_0 = L - \sigma$, $\phi(\cdot) = D_{\psi}(\cdot, \mathbf{x}^{(0)})$, and let $\mathbf{x}^{(i)}, \mathbf{\hat{x}}^{(i)}, \mathbf{z}^{(i)}$ evolve according to (\ref{eq:amd-sc}), where $\phi_i(\mathbf{x}) = \sum_{j=0}^i a_j\frac{\sigma}{2}\|\mathbf{x} - \mathbf{x}^{(j)}\|^2 + \phi(\mathbf{x})$, $\forall \mathbf{x} \in X$. If $a_0 = 1$ and $\frac{a_i}{A^{(i)}} = \frac{\sqrt{4\kappa + 1} - 1}{2\kappa}$, where $\kappa = L/\sigma$ is $f(\cdot)$'s condition number, then:
$$
f(\mathbf{\hat{x}}^{(k)}) - f(\mathbf{x}^*) \leq \left(1-\frac{\sqrt{4\kappa + 1} - 1}{2\kappa}\right)^k D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)}).
$$
\end{theorem}
\begin{proof}
The proof follows by applying the same arguments as in the proof of Theorem \ref{thm:amd-conv}. To obtain the convergence bound, we observe that $\phi_i(\cdot)$ is $\sigma_i$-strongly convex for $\sigma_i = \sigma\sum_{j=0}^ia_j + \sigma_0 = A^{(i)}\sigma + \sigma_0$. Thus, we only need to show that $\frac{{a_i}^2}{A^{(i)}}\leq \frac{\sigma_{i-1}}{L}$. A sufficient condition is that $\frac{{a_i}^2}{A^{(i)}A^{(i-1)}}\leq \frac{\sigma}{L} = \frac{1}{\kappa}$, which is equivalent to $\frac{{a_i}^2}{(A^{(i)})^2}\leq \frac{1}{\kappa}(1-\frac{a_i}{A^{(i)}})$. Solving $\frac{{a_i}^2}{(A^{(i)})^2} = \frac{1}{\kappa}(1-\frac{a_i}{A^{(i)}})$ gives the $a_i$'s from the theorem statement for $i\geq 1$. The choice of $a_0 = 1$, $\sigma_0 = L-\sigma$ ensures $a_0G^{(0)} \leq \phi(\mathbf{x}^*) = D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})$.
\end{proof}
\begin{remark}
When $X = \mathbb{R}^n$, we obtain a tighter convergence bound. Namely, assuming that $\|\cdot\| = \|\cdot\|_2$, we can recover the standard guarantee $f(\mathbf{\hat{x}}^{(k)}) - f(\mathbf{x}^*) \leq \left(1-\frac{1}{\sqrt{\kappa}}\right)^k D_{\psi}(\mathbf{x}^*, \mathbf{x}^{(0)})$ \cite{nesterov2013introductory}. More details can be found in Appendix~\ref{sec:amd-sc-unconstrained}.
\end{remark}
\subsection{Composite Dual Averaging}
Consider the forward Euler discretization of (\ref{eq:ct-comp-md}), recovering updates similar to \cite{duchi2010composite}\footnote{(\ref{eq:comp-md}) is the ``lazy'' (dual averaging) version of the COMID algorithm from \cite{duchi2010composite}.}:
\begin{equation}\label{eq:comp-md}\tag{CMD}
\begin{gathered}
\mathbf{z}^{(i)} = \mathbf{z}^{(i-1)}- a_i \nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \nabla\phi_{i}^*(\mathbf{z}^{(i-1)}),\\
\mathbf{\hat{x}}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}} + \frac{a_i}{A^{(i)}}\mathbf{x}^{(i)},\\
\mathbf{z}^{(0)} = -a_0 f(\mathbf{x}^{(0)}),\; \mathbf{\hat{x}}^{(0)} = \mathbf{x}^{(0)}, \text{ for arbitrary } \mathbf{x}^{(0)} \in X.
\end{gathered}
\end{equation}
Unlike in the standard (non-composite) convex minimization, as discussed at the beginning of the section, in the composite case the discretization error needs to take into account an extra term. The additional term appears due to the discontinuous solution updates and $\psi(\cdot)$ in the objective; in the continuous-time case $\mathbf{x}^{(t)}=\nabla \phi_t^*(\mathbf{z}^{(t)})$ and the change in the upper bound term $\int_{0}^t\psi(\mathbf{x}^{(\tau)})\mathrm{d}\alpha^{(\tau)}$ matches the change in $\psi(\nabla\phi_t^*(\mathbf{z}^{(t)}))$. In the discrete time, however, $\mathbf{x}^{(i)} = \nabla\phi_{i}^*(\mathbf{z}^{(i-1)})$, and thus the error also includes $a_i(\psi(\nabla\phi_{i}^*(\mathbf{z}^{(i-1)}))-\psi(\nabla\phi_{i}^*(\mathbf{z}^{(i)})))$, leading to the following bound on the discretization error.
\begin{proposition}\label{prop:comp-md-discr-err}
The discretization error for forward Euler discretization of (\ref{eq:ct-comp-md}) is:
$$
E_d^{(i)} \leq D_{\phi_i^*}(\mathbf{z}^{(i)}, \mathbf{z}^{(i-1)}).
$$
\end{proposition}
\begin{proof}
In continuous-time regime, $\mathbf{x}^{(\tau)} = \nabla\phi_{\tau}^*(\mathbf{z}^{(\tau)})$, and thus:
\begin{align*}
I_c^{(i-1, i)} =& \int_{i-1}^i \innp{\dot{\mathbf{z}}^{(\tau)}, \nabla\phi_i^*(\mathbf{z}^{(i)})-\nabla\phi_{\tau}^*(\mathbf{z}^{(\tau)})}d\tau\\
=& \innp{\nabla\phi_i^*(\mathbf{z}^{(i)}), \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}} - \int_{i-1}^i \innp{\nabla\phi^*_{\tau}(\mathbf{z}^{(\tau)}), \dot{\mathbf{z}}^{(\tau)}}d\tau\\
=& \innp{\nabla\phi_i^*(\mathbf{z}^{(i)}), \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}} - \int_{i-1}^i \left(\frac{\d}{\d\tau}\phi_{\tau}^*(\mathbf{z}^{(\tau)}) - \frac{d}{d s}\left.\phi_{s}^*(\mathbf{z}^{(\tau)})\right|_{s=\tau}\right)d\tau\\
=& \innp{\nabla\phi_i^*(\mathbf{z}^{(i)}), \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}} - \phi_i^*(\mathbf{z}^{(i)}) + \phi_{i-1}^*(\mathbf{z}^{(i-1)})\\
&+ \int_{i-1}^{i}\frac{d}{d s}\left.\phi_{s}^*(\mathbf{z}^{(\tau)})\right|_{s=\tau}d\tau.
\end{align*}
Recalling that $\phi_t(\mathbf{x}) = \alpha^{(t)}\psi(\mathbf{x}) + \phi(\mathbf{x})$ and using Danskin's theorem, we have:
\begin{align*}
\int_{i-1}^{i}\frac{d}{d s}\left.\phi_{s}^*(\mathbf{z}^{(\tau)})\right|_{s=\tau}d\tau &= \int_{i-1}^i \frac{d}{d s}\left.\max_{\mathbf{x} \in X}\left\{\innp{\mathbf{z}^{(\tau)}, \mathbf{x}}-\alpha^{(s)}\psi(\mathbf{x})-\phi(\mathbf{x})\right\}\right|_{s=\tau}d\tau\\
&= -\int_{i-1}^i \dot{\alpha}^{(\tau)}\psi(\nabla\phi^*_{\tau}(\mathbf{z}^{(\tau)}))d\tau = -a_i\psi(\nabla\phi^*_i(\mathbf{z}^{(i)})).
\end{align*}
Therefore:
$$
I_c^{(i-1, i)} = \innp{\nabla\phi_i^*(\mathbf{z}^{(i)}), \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}} - \phi_i^*(\mathbf{z}^{(i)}) + \phi_{i-1}^*(\mathbf{z}^{(i-1)})-a_i\psi(\nabla\phi^*_i(\mathbf{z}^{(i)})).
$$
On the other hand, as:
\begin{align*}
I^{(i-1, i)} &= -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\phi_i^*(\mathbf{z}^{(i)})-\mathbf{x}^{(i)}}\\
&= \innp{\mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}, \nabla\phi_i^*(\mathbf{z}^{(i)}) - \nabla\phi_{i}^*(\mathbf{z}^{(i-1)})},
\end{align*}
the discretization error is:
\begin{align*}
E_d^{(i)} &= \phi_i^*(\mathbf{z}^{(i)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)}) - \innp{\nabla\phi_{i}^*(\mathbf{z}^{(i-1)}), \mathbf{z}^{(i)}-\mathbf{z}^{(i-1)}} + a_i\psi(\nabla\phi^*_{i}(\mathbf{z}^{(i-1)}))\\
&= D_{\phi_i^*}(\mathbf{z}^{(i)}, \mathbf{z}^{(i-1)}) + \phi_{i}^*(\mathbf{z}^{(i-1)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)})+ a_i\psi(\nabla\phi^*_{i}(\mathbf{z}^{(i-1)})).
\end{align*}
It remains to show that $\phi_{i}^*(\mathbf{z}^{(i-1)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)})+ a_i\psi(\nabla\phi^*_{i}(\mathbf{z}^{(i-1)}))\leq 0$. Observing that $a_i\psi(\nabla\phi^*_{i}(\mathbf{z}^{(i-1)})) = \phi_i(\nabla \phi_i^*(\mathbf{z}^{(i-1)}))-\phi_{i-1}(\nabla \phi_i^*(\mathbf{z}^{(i-1)}))$ and using the definition of a convex conjugate together with Fact \ref{fact:danskin}:
\begin{align*}
&\phi_{i}^*(\mathbf{z}^{(i-1)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)})+ a_i\psi(\nabla\phi^*_{i}(\mathbf{z}^{(i-1)}))\\
&\hspace{1cm}= \phi_{i}^*(\mathbf{z}^{(i-1)}) - \phi_{i-1}^*(\mathbf{z}^{(i-1)}) + \phi_i(\nabla \phi_i^*(\mathbf{z}^{(i-1)}))-\phi_{i-1}(\nabla \phi_i^*(\mathbf{z}^{(i-1)}))\\
&\hspace{1cm}= \innp{\mathbf{z}^{(i-1)}, \nabla\phi_i^*(\mathbf{z}^{(i-1)})-\nabla\phi_{i-1}^*(\mathbf{z}^{(i-1)})}\\
&\hspace{1.3cm}+\phi_{i-1}(\nabla \phi_{i-1}^*(\mathbf{z}^{(i-1)}))-\phi_{i-1}(\nabla \phi_{i}^*(\mathbf{z}^{(i-1)}))\\
&\hspace{1cm}\leq 0,
\end{align*}
where the inequality is by Fact \ref{fact:danskin}, as $\nabla\phi_{i-1}(\mathbf{z}^{(i-1)})=\arg\min_{\mathbf{x}\in X}\{-\innp{\mathbf{z}^{(i-1)}, \mathbf{x}} + \phi_{i-1}(\mathbf{x})\}.$
\end{proof}
Finally, we can obtain the following convergence result for the composite functions, similar to the classical case of mirror descent.
\begin{theorem}\label{thm:composite-md}
Let $\bar{f} = f + \psi: X\rightarrow \mathbb{R}$ be a composite function, such that $f(\cdot)$ is $L$-Lipschitz-continuous and convex, and $\psi(\cdot)$ is ``simple'' and convex. Here, ``simple'' means that $\nabla\phi_i^*(\mathbf{z})$ is easily computable for $\phi_i(\cdot) = A^{(i)}\psi(\cdot) + D_\phi(\cdot, \mathbf{x}^{(0)})$ and some $\sigma$-strongly convex $\phi(\cdot)$ where $\sigma > 0$. Fix any $k\geq 1$ and let $\mathbf{x}^{(i)}, \mathbf{\hat{x}}^{(i)}$ evolve according to (\ref{eq:comp-md}) for $a_i = \frac{1}{L}\sqrt{\frac{2\sigma \phi(\mathbf{x}^*)}{k+1}}$. Then:
$$
\bar{f}(\mathbf{\hat{x}}^{(k)}) - \bar{f}(\mathbf{x}^*) \leq \sqrt{\frac{2D_{\phi}(\mathbf{x}^*, \mathbf{x}^{(0)})}{\sigma}}\frac{L}{\sqrt{k+1}}.
$$
\end{theorem}
\begin{proof}
Observe that since $\phi(\cdot)$ is $\sigma$-strongly convex, $\phi_i(\cdot)$ is also $\sigma$-strongly convex. The rest of the proof follows the same argument as the proof of Theorem \ref{thm:md} (dual averaging convergence), as $D_{\phi_i^*}(\mathbf{z}^{(i)}, \mathbf{z}^{(i-1)}) = -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \mathbf{x}^{(i+1)}-\mathbf{x}^{(i)}}-D_{\phi_i^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})$, and is omitted.
\end{proof}
\begin{remark}
Observe that if $\psi(\cdot)$ was $\sigma$-strongly convex for some $\sigma > 0$, we could have obtained a stronger convergence result, as in that case we would have $D_{\phi_i^*}(\mathbf{z}^{(i-1)}, \mathbf{z}^{(i)})\geq A^{(i)}\frac{\sigma}{2}\|\mathbf{x}^{(i)}-\mathbf{x}^{(i+1)}\|^2$, which would allow choosing larger steps $a_i$.
\end{remark}
\subsection{Frank-Wolfe Method}
For the discretization of continuous-time Frank-Wolfe method (\ref{eq:ct-fw}), we need to take into account the different lower bound we obtained in Equation~\eqref{eq:gen-lb-fw}. In particular, the integral that accrues a discretization error is $-\int_{i-1}^i \innp{\nabla f(\mathbf{x}^{(\tau)}), \nabla\psi^*(\mathbf{\hat{z}}^{(\tau)})-\mathbf{x}^{(\tau)}}\mathrm{d}\alpha^{(\tau)}$, where $\mathbf{\hat{z}}^{(\tau)} = -\nabla f(\mathbf{x}^{(\tau)})$. The forward Euler discretization gives the following algorithm:
\begin{equation}\label{eq:fw}\tag{FW}
\begin{gathered}
\mathbf{\hat{z}}^{(i)} = - \nabla f(\mathbf{x}^{(i)}),\\
\mathbf{x}^{(i)} = \frac{A^{(i-1)}}{A^{(i)}}\mathbf{x}^{(i-1)} + \frac{a_i}{A^{(i)}}\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)}),\\
\mathbf{x}^{(0)} \in X \text{ is an arbitrary initial point}.
\end{gathered}
\end{equation}
As discussed before, the discretization error needs to include $a_i(\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)}))-\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)})))$ in addition to $I^{(i-1, i)}-I_c^{(i-1, i)}$, and is bounded as follows.
\begin{proposition}\label{prop:fw-discr-err}
The discretization error for forward Euler discretization of (\ref{eq:ct-fw}) is:
$$
E_d^{(i)} \leq a_i \innp{\nabla f(\mathbf{x}^{(i)})-\nabla f(\mathbf{x}^{(i-1)}), \nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})-\nabla\psi^*(\mathbf{\hat{z}}^{(i)})}.
$$
\end{proposition}
\begin{proof}
In the discrete-time case, $I^{(i-1, i)} = -a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\psi^*(\mathbf{\hat{z}}^{(i)})-\mathbf{x}^{(i)}}$, while in continuous-time case, as $\dot{\mathbf{x}}^{(t)} = \dot{\alpha}^{(t)}\frac{\nabla\psi^*(\mathbf{\hat{z}}^{(t)})-\mathbf{x}^{(t)}}{\alpha^{(t)}}$ and integrating by parts:
$$
I_c^{(i-1, i)} = -\int_{i-1}^{i}\innp{\nabla f(\mathbf{x}^{(\tau)}), \alpha^{(\tau)}\dot{\mathbf{x}}^{(\tau)}}d\tau = -A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})).
$$
Therefore:
\begin{equation}\label{eq:fw-discr-err-1}
\begin{aligned}
E_d^{(i)} =& A^{(i-1)}(f(\mathbf{x}^{(i)})-f(\mathbf{x}^{(i-1)})) - a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\psi^*(\mathbf{\hat{z}}^{(i)})-\mathbf{x}^{(i)}}\\
&+ a_i(\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)}))-\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)})))\\
\leq & a_i\innp{\nabla f(\mathbf{x}^{(i)}), \nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})-\nabla\psi^*(\mathbf{\hat{z}}^{(i)})}\\
&+ a_i(\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)}))-\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)}))),
\end{aligned}
\end{equation}
where we have used the convexity of $f(\cdot)$ and $\mathbf{x}^{(i)}=\frac{A^{(i-1)}}{A^{(i)}}\mathbf{x}^{(i-1)}+\frac{a_i}{A^{(i)}}\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})$.
Further, by Fact \ref{fact:danskin},
\begin{align}
-\innp{\mathbf{\hat{z}}^{(i-1)}, \nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})} + \psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)}))\leq -\innp{\mathbf{\hat{z}}^{(i-1)}, \nabla\psi^*(\mathbf{\hat{z}}^{(i)})} + \psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)})),\notag
\end{align}
and, therefore, as $\mathbf{\hat{z}}^{(i-1)} = -\nabla f(\mathbf{x}^{(i-1)})$:
\begin{align}
\psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})) - \psi(\nabla\psi^*(\mathbf{\hat{z}}^{(i)})) \leq \innp{\nabla f(\mathbf{x}^{(i-1)}), \nabla\psi^*(\mathbf{\hat{z}}^{(i)}) - \nabla\psi^*(\mathbf{\hat{z}}^{(i-1)})}. \label{eq:fw-discr-err-psi-terms}
\end{align}
Combining (\ref{eq:fw-discr-err-1}) and (\ref{eq:fw-discr-err-psi-terms}), the claimed bound on discretization error follows.
\end{proof}
We can now recover the convergence result from \cite{nesterov2015cgm}.
\begin{theorem}\label{thm:fw}
Let $\bar{f} = f+\psi: X\rightarrow \mathbb{R}$ be a composite function, where $\psi(\cdot)$ is convex and $f(\cdot)$ is convex with H\"{o}lder-continuous gradients, i.e., for some fixed $L_{\nu}<\infty$, $\nu \in (0, 1]$\footnote{Observe that when $\nu = 1$, $f(\cdot)$ is $L_{\nu}$-smooth.}:
\begin{equation}
\|\nabla f(\mathbf{x}) - \nabla f(\mathbf{\hat{x}})\|_*\leq L_{\nu}\|\mathbf{x} - \mathbf{\hat{x}}\|^{\nu}, \; \forall \mathbf{x}, \mathbf{\hat{x}} \in X.\label{eq:f-holder-cont}
\end{equation}
Let $D \stackrel{\mathrm{\scriptscriptstyle def}}{=} \max_{\mathbf{x}, \mathbf{\hat{x}} \in X}\|\mathbf{x} - \mathbf{\hat{x}}\|$ denote the diameter of $X$.
If $\mathbf{x}^{(i)}$ evolves according to (\ref{eq:fw}), then, $\forall k \geq 1$:
$$
\bar{f}(\mathbf{x}^{(k)}) - \bar{f}(\mathbf{x}^*) \leq L_{\nu}D^{1+\nu}\frac{1}{A^{(k)}}{\sum_{i=0}^k \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}}}.
$$
In particular, if $a_i = i+1$, then
$$
\bar{f}(\mathbf{x}^{(k)}) - \bar{f}(\mathbf{x}^*) \leq 2^{1+ \nu}\frac{L_{\nu}D^{1+\nu}}{(k+1)^{\nu}}.
$$
\end{theorem}
\begin{proof}
Applying Cauchy-Schwartz Inequality to the discretization error given by Proposition \ref{prop:fw-discr-err}, we have:
\begin{align*}
E_d^{(i)} &\leq a_i \|\nabla f(\mathbf{x}^{(i)}) - \nabla f(\mathbf{x}^{(i-1)})\|_*\cdot \|\nabla \psi^*(\mathbf{\hat{z}}^{(i)}) - \nabla \psi^*(\mathbf{\hat{z}}^{(i-1)})\|\\
&\leq \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}}L_{\nu}\|\mathbf{x}^{(i-1)} - \nabla \psi^*(\mathbf{\hat{z}}^{(i-1)})\|^{\nu}\cdot \|\nabla \psi^*(\mathbf{\hat{z}}^{(i)}) - \nabla \psi^*(\mathbf{\hat{z}}^{(i-1)})\|\\
&\leq \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}}L_{\nu} D^{1+\nu},
\end{align*}
where the second inequality follows from (\ref{eq:f-holder-cont}) and $\mathbf{x}^{(i)}-\mathbf{x}^{(i-1)} = \frac{a_i}{A^{(i)}}(\mathbf{x}^{(i-1)} - \nabla \psi^*(\mathbf{\hat{z}}^{(i-1)}))$ (by (\ref{eq:fw})). Therefore: $G^{(k)}\leq \frac{a_0 G^{(0)}}{A^{(k)}} + L_{\nu}D^{1+\nu}\frac{1}{A^{(k)}}{\sum_{i=1}^k \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}}}$.
We now use the same arguments to bound $G^{(0)}$. As $\mathbf{x}^{(0)}$ can be mapped to $\nabla\psi^*(\mathbf{\hat{z}}^{(-1)})$, for some $\mathbf{\hat{z}}^{(-1)}$, we have:
\begin{align*}
G^{(0)} &= - \innp{\nabla f(\mathbf{x}^{(0)}), \nabla \psi^*(\mathbf{\hat{z}}^{(0)})-\mathbf{x}^{(0)}} - \psi(\nabla\psi^*(\mathbf{\hat{z}}^{(0)})) + \psi(\mathbf{x}^{(0)})\leq {L_{\nu}}D^{1+\nu}.
\end{align*}
Therefore, $G^{(k)}\leq L_{\nu}D^{1+\nu}\frac{1}{A^{(k)}}{\sum_{i=0}^k \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}}}$. In particular, when $a_i = i+1$, then $A^{(i)} = \frac{(i+1)(i+2)}{2}$. Finally, $\sum_{i=0}^k \frac{{a_i}^{1+\nu}}{(A^{(i)})^{\nu}} < 2^{\nu}\sum_{i=0}^k (i+1)^{1-\nu}< 2^{\nu}(k+1)^{2-\nu}$, and the convergence bound follows.
\end{proof}
\section{Conclusion}\label{sec:conclusion}
We presented a general technique for the analysis of first-order methods. The technique is intuitive and follows the argument of reducing approximate duality gap at the rate equal to the convergence rate. Besides the unified interpretation of many first-order methods, the technique is generally useful for obtaining new optimization methods~\cite{AXGD,LP-jelena-lorenzo,diakonikolas2018fairpc,cohen2018acceleration,diakonikolas2018alternating}. An interesting direction for future is extending this technique to other settings, such as, e.g., geodesically convex optimization.
\section*{Acknowledgements}
We thank the anonymous reviewers for their thoughtful comments and suggestions, which have greatly improved the presentation of this paper. We also thank Ziye Tang for pointing out several typos in the earlier version of the paper and providing useful suggestions for improving its presentation.
\bibliographystyle{siamplain}
{\small
| {
"timestamp": "2018-12-07T02:00:37",
"yymm": "1712",
"arxiv_id": "1712.02485",
"language": "en",
"url": "https://arxiv.org/abs/1712.02485",
"abstract": "We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the algorithm converges. We show that in continuous time enforcement of an invariant that this approximate duality gap decreases at a certain rate exactly recovers a wide range of first-order continuous-time methods. We characterize the discretization errors incurred by different discretization methods, and show how iteration-complexity-optimal methods for various classes of problems cancel out the discretization error. The techniques are illustrated on various classes of problems -- including convex minimization for Lipschitz-continuous objectives, smooth convex minimization, composite minimization, smooth and strongly convex minimization, solving variational inequalities with monotone operators, and convex-concave saddle-point optimization -- and naturally extend to other settings.",
"subjects": "Optimization and Control (math.OC); Data Structures and Algorithms (cs.DS)",
"title": "The Approximate Duality Gap Technique: A Unified Theory of First-Order Methods",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860382,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7092019616632165
} |
https://arxiv.org/abs/1909.06792 | Fundamental domains in ${\rm PSL}(2,{\mathbb R})$ for Fuchsian groups | In this paper, we provide a necessary and sufficient condition for a set in ${\rm PSL}(2,{\mathbb R})$ or in $T^1{\mathbb H}^2$ to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane ${\mathbb H}^2$. | \section{Introduction}
Fundamental domains arise naturally in the study of group actions on topological spaces.
The concept {\em fundamental domain} is used to describe a set in a topological space under a group action
of which the images tessellate the whose space.
The term {\em fundamental domain} is well-known in the model of the hyperbolic plane $\mathbb H^2$ for
the action of Fuchsian groups via M\"obius transformations. If there exists a point in $\mathbb H^2$ that is not a fixed point for all elements different from the unity in a Fuchsian group $\Gamma$ then there always exists a convex and connected fundamental domain for $\Gamma$ named Dirichlet domain (see \cite{einsward,katok}). Other examples of fundamental domains are Ford domains (see \cite{series}). {\em Poincar\'e's polygon theorem} \cite{hubbard,katok,series} provides a fundamental domain, which is a polygon for the Fuchsian group generated
by the side-pairing transformations. In this case, if the polygon has finite edges (and hence it is relatively compact), the Fuchsian group
is finitely generated and the space of $\Gamma$-orbits denoted by $\Gamma\backslash\mathbb H^2$ is compact. Fundamental
domains have several applications for the study of $\Gamma\backslash\mathbb H^2$. If the action of $\Gamma$ has no fixed points,
the quotient space $\Gamma\backslash\mathbb H^2$ has a Riemann surface structure that is a closed Riemann surface of genus at least $2$
and has the hyperbolic plane $\mathbb H^2$ as the universal covering. Furthermore,
it is well-known that any compact orientable surface
with constant negative curvature is isometric to a factor $\Gamma\backslash \mathbb H^2$.
If the Fuchsian group has a finite-area fundamental domain then all the fundamental domains have finite area and
have the same area. This area is defined for the measure of the quotient space $\Gamma\backslash\mathbb H^2$.
In addition, the space $\Gamma\backslash\mathbb H^2$ is compact if and only if every fundamental domain in $\mathbb H^2$ for $\Gamma$ is relatively compact (see \cite{katok}).
There exists a bijection $\Theta: T^1\mathbb H^2\to{\rm PSL}(2,\mathbb R)$. The natural Riemannian metric on ${\rm PSL}(2,\mathbb R)$ induces
a left-invariant metric function (a metric in the usual sense). The topology induced from this metric is the same as the quotient topology induced from the one in ${\rm SL}(2,\mathbb R)$. The Sasaki metric on the unit tangent bundle $T^1\mathbb H^2$ with respect to
the hyperbolic metric on $\mathbb H^2$ makes $\Theta$ an isometry.
This induces an isometry from $T^1(\Gamma\backslash\mathbb H^2)$ to
$\Gamma\backslash{\rm PSL}(2,\mathbb R)$, where $\Gamma\backslash{\rm PSL}(2,\mathbb R)$ denotes the collection of right co-sets $\Gamma g$ of $\Gamma$ in ${\rm PSL}(2,\mathbb R)$, which is also obtained from a left action of Fuchsian group $\Gamma$
on ${\rm PSL}(2,\mathbb R)$. Furthermore, there is an action of $\Gamma$ on the unit tangent bundle $T^1\mathbb H^2$ by derivative operators and this arises fundamental domains for $\Gamma$ in $T^1\mathbb H^2$ also. However, up to now there have not been any results about fundamental domains for $\Gamma$ in ${\rm PSL}(2,\mathbb R)$ or in $T^1\mathbb H^2$.
The aim of this paper is to study fundamental domains in ${\rm PSL}(2,\mathbb R)$ and in $T^1\mathbb H^2$ for Fuchsian groups. A necessary and sufficient condition
for a set in ${\rm PSL}(2,\mathbb R)$ or in $T^1\mathbb H^2$ to be a fundamental domain via its respective fundamental domain in $\mathbb H^2$ is provided.
The paper is organized as follows. In the next section we present the actions of Fuchsian groups
on the hyperbolic plane $\mathbb H^2$, the unit tangent bundle $T^1\mathbb H^2$ and the group ${\rm PSL}(2,\mathbb R)$. The main results are stated and proved in Section 3.
\setcounter{equation}{0}
\section{Preliminaries}
In this section we introduce the necessary background material which can be found in \cite{hubbard,katok,series}.
The unity of an arbitrary group is always denoted by $e$.
\subsection{Fundamental domains}
Let $X$ be a non-empty set and let $G$ be a group. Let $\rho: G\times X \to X$ be a (left) group action,
that is, $\rho(e,x)=x$ and $\rho(g_1,\rho(g_2,x))=\rho(g_1g_2,x)$ for all $x\in X$ and $g_1,g_2\in G$.
For a subset $A\subset X$, define $\rho(g,A)=\{\rho(g,x), x\in A \}$.
\begin{definition}\label{funddo-def}\rm
Let $G$ be a group and let $X$ be a topological space. Suppose that $\rho:G\times X\to X$ is group action.
A non-empty open set $F\subset X$ is said to be a {\em fundamental domain} for $G$, if
\begin{itemize}
\item[(a)] $\bigcup_{g\in G} \rho(g,\overline{F})=X$ and
\item[(b)] $\rho(g, F)\cap F=\varnothing$ for all $g\in G\setminus\{e\}$.
\end{itemize}
Here $e$ is the unity of $G$ and $\overline{F}$ denotes the closure of $F$ in $X$.
\end{definition}
Due to the fact that $G$ is a group,
condition (b) is equivalent to
\[\rho(g_1, F)\cap\rho(g_2,F)=\varnothing
\quad\mbox{for all}\quad g_1, g_2\in G,
\,\,g_1\neq g_2. \]
We will introduce some examples in the next subsection.
\subsection{$\mathbb H^2$ and ${\rm PSL}(2,\mathbb R)$}
The hyperbolic plane is the upper half plane
$\mathbb H^2=\{(x, y)\in\mathbb R^2: y>0\}$, endowed with the Riemannian metric $(g_z)_{z\in\mathbb H^2}$,
where $g_z(\xi,\eta)=\frac{\xi_1\eta_1+\xi_2\eta_2}{({\rm Im\,} z)^2}$
for $\xi=(\xi_1,\xi_2), \eta=(\eta_1,\eta_2)\in T_z\mathbb H^2\cong \mathbb C$.
The group of M\"obius transformations ${\rm M\ddot ob}(\mathbb H^2)=\{z\mapsto \frac{az+b}{cz+d}\,:\, a,b,c,d\in\mathbb R\,, ad-bc =1\}$
can be identified with the projective group ${\rm PSL}(2,\mathbb R)={\rm SL}(2,\mathbb R)/\{\pm E_2\}$ by means of the isomorphism
\begin{equation}\label{Phi}
\Phi\Bigg(\pm\bigg(\begin{array}{cc}a&b\\c&d\end{array}\bigg)\Bigg)=z\mapsto\frac{az+b}{cz+d},
\end{equation}
where ${\rm SL}(2,\mathbb R)$ is the group of all real $2\times 2$ matrices with unity determinant,
and $E_2$ denotes the unit matrix.
Let $\Gamma$ be a Fuchsian group, which is a discrete subgroup in ${\rm PSL}(2,\mathbb R)$. We consider the action
$\rho: \Gamma\times \mathbb H^2\to \mathbb H^2,\,\rho(\gamma,z)=\Phi(\gamma)(z)\ \mbox{for}\ (\gamma,z)\in \Gamma\times\mathbb H^2$.
The action is called free if
$\Phi(\gamma)(z)=z$ for some $z\in\mathbb H^2$ then $\gamma=e$. In this case, there always exist fundamental domains for $\Gamma$ as follows.
\begin{proposition}
Let $\Gamma\subset {\rm PSL}(2, \mathbb R)$ be a Fuchsian group
and take $z_0\in\mathbb H^2$ such that $z_0\neq\Phi(\gamma)(z_0)$ holds
for all $\gamma\in\Gamma\setminus\{e\}$. Then the Dirichlet region
\[ D_{z_0}(\Gamma)=\Big\{z\in\mathbb H^2: d_{\mathbb H^2}(z, z_0)<d_{\mathbb H^2}(z, T(z_0))
\,\,\mbox{for all}\,\,T=\Phi(\gamma), \gamma\in\Gamma\setminus\{e\}\Big\} \]
is a fundamental domain for $\Gamma$ which contains $z_0$.
\end{proposition}
See \cite[Lemma 11.5]{einsward} for a proof. Note that such a $z_0$ does exist if the action of $\Gamma$ on $\mathbb H^2$ is free.
For $g=\Big\{\pm\big(\scriptsize\begin{array}{cc}a&b\\c&d\end{array} \big)\Big\}\in{\rm PSL}(2,\mathbb R)$,
the trace of $g$ is defined by \[{\rm tr}(g)=|a+d|.\]
Element $g$ is called hyperbolic if $|{\rm tr}(g)|>2$, elliptic if $|{\rm tr}(g)|<2$ and parabolic if $|{\rm tr}(g)|=2$.
It is well-known that the action of $\Gamma$ on $\mathbb H^2$ is free if and only if $\Gamma$ does not contain any elliptic elements (see \cite[Remark 4.26]{series}).
For any $g\in{\rm PSL}(2,\mathbb R)$, the cyclic group $\langle g\rangle =\{g^n:n\in\mathbb Z \}\subset {\rm PSL}(2,\mathbb R)$
is a Fuchsian group. We will consider fundamental domains for $\langle g\rangle$ with some special classes of $g$.
For $t\in\mathbb R$, let
\begin{eqnarray*}
A_t=\bigg(\begin{array}{cc} e^{t/2} & 0 \\
0 & e^{-t/2}\end{array}\bigg),
B_t=\bigg(\begin{array}{cc} 1 & t \\
0 & 1\end{array}\bigg), D_t=\bigg(\begin{array}{cc} \cos\frac{t}2 & \sin\frac{t}2 \\
-\sin\frac{t}2 & \cos\frac{t}2\end{array}\bigg)\in{\rm SL}(2,\mathbb R)
\end{eqnarray*}
and respectively
\[ a_t=[A_t], b_t=[B_t],
d_t=[D_t]\in {\rm PSL}(2,\mathbb R) .\]
\begin{proposition}\label{examfun}
(a) For any $t>0$, the set
\[F_t=\{z=x+iy\in\mathbb H^2: 1<y<e^t\}\]
is a fundamental domain in $\mathbb H^2$ for
the Fuchsian group
$\langle a_t\rangle $.
(b) For any $t>0$, the set
\[E_t=\{z=x+iy\in\mathbb H^2: 0< x< t \} \]
is a fundamental domain in $\mathbb H^2$ for the Fuchsian group $\langle b_t\rangle$.
\end{proposition}
\noindent{\bf Proof\,:} (a) Obviously $F_t$ is open and
\[\overline {F_t}
=\{z=x+iy\in\mathbb H^2: 1\leq y \leq e^t \}.\]
We have
$\Phi(a_{jt})
=e^{jt}\,{\rm id}$, with ${\rm id}: \mathbb H^2\to\mathbb H^2$ denoting the identity map. Here
\[ \Phi(a_{jt})(\overline{F_t})=e^{jt}\,{\rm id}(\{z=x+iy\in\mathbb H^2: 1\le y\le e^t\})
=\{z\in\mathbb H^2: e^{jt}\le y\le e^{(j+1)t}\}, \]
so that $\bigcup_{j\in\mathbb Z}\Phi(a_{jt})(\overline{F_t})=\mathbb H^2$
and $\Phi(a_{jt})({F_t})\cap\Phi(a_{kt})({F_t})=\varnothing$
for $j\neq k$.
(b) It is proved analogously to (a).
{\hfill$\Box$}\bigskip
The collection of right co-sets $\Gamma g$ of $\Gamma$ in ${\rm PSL}(2,\mathbb R)$ denoted by $\Gamma\backslash{\rm PSL}(2,\mathbb R)$ can be also obtained by $\Gamma$-orbits of the left action
\begin{equation}\label{varrho}\varrho: \Gamma \times {\rm PSL}(2,\mathbb R)\rightarrow {\rm PSL}(2,\mathbb R),\,
\varrho(\gamma,g)=\gamma g\ \mbox{for}\ \gamma\in\Gamma,g\in{\rm PSL}(2,\mathbb R).\end{equation}
This leads to the concept {\em fundamental domain} in ${\rm PSL}(2,\mathbb R)$.
\begin{remark}\rm If $\mathcal F\subset {\rm PSL}(2,\mathbb R)$ is a fundamental domain for $\Gamma$ and $\gamma\in\Gamma\setminus\{e\}$,
then $\gamma \mathcal F$ is a fundamental domain disjoint from $\mathcal F$. For, it is obvious that $\gamma \mathcal F$ is open since $\mathcal F$ open
and $\overline{\gamma\mathcal F}=\gamma\overline{\mathcal F}$
in ${\rm PSL}(2,\mathbb R)$. Therefore
\[ \bigcup_{\gamma'\in\Gamma} \gamma'\overline{\gamma\mathcal F}
=\bigcup_{\gamma'\in\Gamma} \gamma'(\gamma\overline{\mathcal F})
=\bigcup_{\gamma'\in\Gamma}(\gamma'\gamma)(\overline{\mathcal F})
=\bigcup_{\eta\in\Gamma}\eta \overline{\mathcal F}={\rm PSL}(2,\mathbb R), \]
and for $\gamma'\in\Gamma\setminus\{e\}$,
\[ (\gamma\mathcal F)\cap \gamma'(\gamma\mathcal F)=\gamma \mathcal F\cap( \gamma'\gamma)\mathcal F=\varnothing \]
by Definition \ref{funddo-def} (a), due to $\gamma\neq\gamma'\gamma$. {\hfill$\diamondsuit$}
\end{remark}
\subsection{$T^1\mathbb H^2$}
The unit tangent bundle of $\mathbb H^2$ is defined by
\begin{equation}\label{T1H2-def}
T^1\mathbb H^2=\{(z, \xi): z\in\mathbb H^2, \xi\in T_z\mathbb H^2,
{\|\xi\|}_z=g_z(\xi, \xi)^{1/2}=1\}.
\end{equation}
For $g\in {\rm PSL}(2, \mathbb R)$ we consider the derivative operator
\begin{equation*}\label{cald-def}
{\mathfrak D}g: T^1\mathbb H^2\to T^1\mathbb H^2
\end{equation*}
defined as
\begin{equation*}\label{bursetal}
{\mathfrak D}g(z, \xi)=(T(z), T'(z)\xi),
\end{equation*}
where $T=\Phi(g)$; recall $\Phi$ in \eqref{Phi}. Then ${\mathfrak D}$ is well-defined. Explicitly,
if $g=\scriptsize\Big\{\pm\big(\begin{array}{cc} a & b \\ c & d\end{array}\big)\Big\}$,
then $T(z)=\frac{az+b}{cz+d}$ and $ad-bc=1$, whence
\begin{equation}\label{calDexpl}
{\mathfrak D}g(z, \xi)=\Big(\frac{az+b}{cz+d},\,\frac{\xi}{(cz+d)^2}\Big).
\end{equation}
Let $\Gamma\subset{\rm PSL}(2,\mathbb R)$ be a subgroup. Consider the group action
$$\kappa: \Gamma\times T^1\mathbb H^2\to T^1\mathbb H^2,\ \kappa(\gamma, (z,\xi))={\mathfrak D }\gamma (z,\xi)
\
\mbox{for}\ \gamma\in\Gamma, (z,\xi)\in T^1\mathbb H^2.$$
If $\Gamma={\rm PSL}(2,\mathbb R)$ then the action is simply transitive (see \cite[Lemma 9.2]{einsward}), that is,
for given $(z,\xi), (w,\eta)\in T^1\mathbb H^2$, there exists a unique $g\in{\rm PSL}(2,\mathbb R)$ such that
$\kappa(g,(z,\xi)) =\mathfrak D g(z,\xi)=(w,\eta)$.
In particular, we have the following property.
\begin{lemma}\label{unig}
For each $(z,\xi)\in T^1\mathbb H^2$, there is a unique $g\in{\rm PSL}(2,\mathbb R)$ such that $\mathfrak D g (i,i)=(z,\xi).$
\end{lemma}
Explicitly, if $(z,\xi)\in T^1\mathbb H^2$ then $g=\Big\{\pm\scriptsize\Big(\begin{array}{cc}a&b\\c&d \end{array}\Big)\Big\}$
is defined by
\begin{equation}\label{dg}\frac{ac+bd}{c^2+d^2}={\rm Re\,} z, \frac{1}{c^2+d^2}={\rm Im\,} z, \frac{2cd}{(c^2+d^2)^2}={\rm Re\,}\xi,
\quad \frac{d^2-c^2}{(c^2+d^2)^2}={\rm Im\,}\xi.
\end{equation}
We will use these relations afterwards.
\section{Main results}
This section deals with the relation of fundamental domains for a Fuchsian group in $\mathbb H^2$, ${\rm PSL}(2,\mathbb R)$ and in $T^1\mathbb H^2$.
The main result of this paper is the following:
\begin{theorem}\label{fdPSL}
Let $\Gamma\subset {\rm PSL}(2,\mathbb R)$ be a Fuchsian group. For $F\subset \mathbb H^2$, denote
\[\mathcal F=\{g=b_x a_{\ln y}d_\theta\, :\, x+iy\in F, \theta\in [0,2\pi) \} \subset {\rm PSL}(2,\mathbb R).\]
Then $F$ is a fundamental domain for $\Gamma$ in $\mathbb H^2$ if and only if $\mathcal F$
is a fundamental domain for $\Gamma$ in ${\rm PSL}(2,\mathbb R)$.
\end{theorem}
\begin{remark}\rm
Recall that if $\Gamma$ contains no elliptic elements then there always exist fundamental domains in $\mathbb H^2$ for
$\Gamma$ and hence fundamental domains in ${\rm PSL}(2,\mathbb R)$ do always exist. The collection of $\Gamma$-orbits of
the action $\varrho$ (see \eqref{varrho}) denoted by $\Gamma\backslash{\rm PSL}(2,\mathbb R)=\{\Gamma g, g\in{\rm PSL}(2,\mathbb R) \}$
is compact if and only if the quotient space $\Gamma\backslash\mathbb H^2$ is compact
if and only if there is a relatively compact fundamental domain (in $\mathbb H^2$ or in ${\rm PSL}(2,\mathbb R)$) for $\Gamma$.
In this case all fundamental domains of $\Gamma$ are relatively compacts.
For proofs of the case in $\mathbb H^2$, see \cite[Chapter 3]{katok}. {\hfill$\diamondsuit$}
\end{remark}
In order to prove Theorem \ref{fdPSL}, we need the following factorization, which is called NAK decomposition (so-called Iwasawa decomposition).
\begin{lemma}[\cite{sieber}]\label{siber1}
If $G= \scriptsize\Big(\begin{array}{cc} a&b\\c&d\end{array}\Big)\in{\rm SL}(2,\mathbb R)$ then
$G=B_xA_{\ln y}d_\theta$ with
\begin{equation}
x=\frac{ac+bd}{c^2+d^2},\ \ y=\frac{1}{c^2+d^2},\ \ \theta=-2\arg(d+ic).
\end{equation}
\end{lemma}
\begin{lemma}\label{lemnak}
(a) If $g=[G]\in{\rm PSL}(2,\mathbb R)$ for $G= \scriptsize\Big(\begin{array}{cc} a&b\\c&d\end{array}\Big)\in{\rm SL}(2,\mathbb R)$ then
$g= b_x a_{\ln y}d_\theta$ with
\begin{equation}\label{nak1}
x=\frac{ac+bd}{c^2+d^2},\ \ y=\frac{1}{c^2+d^2},\ \ \theta=-2\arg(d+ic).
\end{equation}
(b) \begin{eqnarray*}
{\rm PSL}(2,\mathbb R)&=&\{b_xa_{\ln y}d_\theta : x+iy\in\mathbb H^2, \theta\in [0,2\pi) \}\\
&=&
\{b_xa_{\ln y}d_\theta : x+iy\in\mathbb H^2, \theta\in\mathbb R \}.
\end{eqnarray*}
\end{lemma}
\noindent
{\bf Proof\,:}
(a) This follows directly from Lemma \ref{siber1}. (b) According to (a), every element $g=\Big\{\pm \Big({\scriptsize\begin{array}{cc} a&b\\c&d\end{array}}\Big)\Big\}\in{\rm PSL}(2,\mathbb R)$
has the decomposition $g=b_xa_{\ln y}d_\theta$ for $x+iy\in\mathbb H^2$ and $\theta=-2\arg(d+ic)\in (-2\pi,0]$. It remains to
verify that we can find some $\theta'\in [0,2\pi)$ such that $d_{\theta'}=d_\theta$
and as a consequence, $g=b_xa_{\ln y}d_{\theta'}$.
Indeed, the matrix $D_\theta$ changes by an overall sign
if $\theta$ changes by $2\pi$ and so does the matrix $G=B_{x}A_{\ln y}D_\theta$.
Therefore we can find a unique $k\in\mathbb Z$ such that $\theta':= 2k\pi+\theta\in [0,2\pi)$
to have $d_{\theta'}=d_\theta$. This implies the first equality in (b). The latter follows from
$d_{\theta+2k\pi}=d_\theta$ for all $\theta\in [0,2\pi)$ and $k\in\mathbb Z$.
{\hfill$\Box$}\bigskip
\noindent
{\bf Proof of Theorem \ref{fdPSL}.} First,
denote
\[\hat F=\{G=B_x A_{\ln y} D_\theta \in{\rm SL}(2,\mathbb R) : x+iy\in F, \theta\in [0, 2\pi)\}.\]
It is easy to see that $\hat F$ is open in ${\rm SL}(2,\mathbb R)$ and since the projection
$\pi: {\rm SL}(2,\mathbb R)\rightarrow {\rm PSL}(2,\mathbb R)$ is an open map and $\pi(\hat F)=\mathcal F$, it
follows that ${\cal F}$ is open in ${\rm PSL}(2,\mathbb R)$ (note that $d_{\theta+2k\pi}=d_\theta$ for $\theta\in [0,2\pi)$ and $k\in\mathbb Z$).
To establish part (a) in the definition, we first claim that the closure of $\mathcal F$ in ${\rm PSL}(2,\mathbb R)$ is
\[\overline \mathcal F=
\{g=b_x a_{\ln y}d_\theta\,\in{\rm PSL}(2,\mathbb R) :\, x+iy\in \overline F, \theta\in [0, 2\pi) \}.\]
Indeed, it suffices to check that
\begin{equation}\label{hatF}
\overline {\hat F}=\{G=B_x A_{\ln y}D_\theta\, :\, x+iy\in \overline F, \theta\in [0, 2\pi)\},
\end{equation}
where $\overline F$ denotes the closure of $F$ in $\mathbb H^2$
and $\overline{\hat F}$ denotes the closure of $\hat F$ in ${\rm SL}(2,\mathbb R)$.
The set in the right-hand side of \eqref{hatF} is denoted by $cl(\hat F)$.
For every $G\in cl\hat F$, we show that $G$ is the limit for some sequence $(G_n)\subset \hat F$.
Writing $G=B_x A_{\ln y} D_\theta$, we have $x+iy\in \overline F$ by the definition of $cl(\hat F)$.
Let $(x_n+iy_n)_n\subset F$ be such that
$x_n+iy_n\rightarrow x+iy $ in $\mathbb H^2$ as $n\rightarrow \infty$. Then $x_n\rightarrow x$
as well as $y_n\rightarrow y$ in $\mathbb R$. Taking
$G_n = B_{x_n}A_{\ln y_n}D_\theta\in cl(\hat F)$, we obtain
$G_n\rightarrow G$ in ${\rm SL}(2,\mathbb R)$ after a short check.
Next, for any $g=b_xa_{\ln y}d_\theta \in {\rm PSL}(2,\mathbb R)$ we have
$z:=x+iy\in \Phi(\gamma)(\overline F)$ for some $\gamma\in\Gamma$
as $F\subset \mathbb H^2$ is a fundamental domain for $\Gamma$.
Take $\tilde z=\tilde x+i\tilde y=\Phi(\gamma^{-1})(z)\in\overline F$ and write $\gamma=[T]$ with
$T=\scriptsize\Big(\begin{array}{cc}t_{11}&t_{12}\\t_{21}&t_{22}\end{array} \Big)
\in{\rm SL}(2,\mathbb R)$. Let
\[\tilde \theta=\theta+2\arg(t_{21}\tilde z+t_{22})+2k\pi\in[0,2\pi)\]
for a unique $k\in\mathbb Z$ to obtain $h:=b_{\tilde x}a_{\ln \tilde y}d_{\tilde\theta}\in\overline{\mathcal F}$.
Thus
\[ x+iy=\frac{t_{11}\tilde z +t_{12}}{t_{21}\tilde z+t_{22}}
\quad \mbox{and}\quad \theta =\tilde \theta-2\arg(t_{21}\tilde z+t_{22})-2k\pi\]
imply $g=\gamma h\in\gamma\overline{\mathcal F}$ after a short
calculation. This completes the proof for (a) in the definition.
For part (b), suppose on the contrary that there exists
$g\in \mathcal F \cap \gamma\mathcal F$ for some $\gamma\in\Gamma\setminus\{e\}$.
Then
$g=b_xa_{\ln y}d_\theta$ and $g=\gamma b_{x'}a_{\ln y'} d_{\theta'}$
for $x+iy\in F$ and $x'+iy'\in F$.
A short calculation shows that
$x+iy=\Phi(\gamma)(x'+iy')\in F \cap \Phi(\gamma)(F)$, which however contradicts the fact that $F$ is a fundamental domain. Thus $\mathcal F\cap \gamma \mathcal F=\varnothing$ for all $\gamma\in\Gamma\setminus\{e\}$.
Conversely, assume that $\mathcal F$ is a fundamental domain for $\Gamma$. Then $F\subset \mathbb H^2$ is open
since $\mathcal F$ is open.
For any $z=x+iy\in\mathbb H^2$, then $g=b_xa_{\ln y}\in{\rm PSL}(2,\mathbb R)=\cup_{\gamma\in\Gamma}\gamma \overline{\mathcal F}$
implies that
$g=\gamma h$ for some $\gamma\in\Gamma$ and $h\in\overline{\mathcal F}=\{b_xa_{\ln y}d_\theta, x+iy\in\overline F, 0\leq \theta\leq 2\pi\}$.
Write $h=b_{\tilde x}a_{\ln\tilde y}d_{\tilde\theta}$. Then $\tilde z=\tilde x+i\tilde y\in \overline F$ and $z=\Phi(\gamma)(\tilde z)$ yield $z\in \Phi(\gamma)(\overline F)$. This proves (a) in Definition \ref{funddo-def}.
Finally, assume that $z=x+iy\in F$ and $z=\Phi(\gamma)(z')$ for some $\gamma\in\Gamma\setminus\{e\} $ and $z'=x'+iy'\in F$.
Then take $g=b_xa_{\ln y}d_{\pi}$ and $h=b_{x'}a_{\ln y'}d_\theta$ with $\theta=2\arg(h_{21}z+h_{22})+2k\pi$
for a unique $k\in\mathbb Z$ such that $\theta \in [0,2\pi)$; here $h=\pi(H)$, $H=\scriptsize\Big(\begin{array}{cc}h_{11}&h_{12}\\h_{21}&h_{22}\end{array} \Big)$. Then $g=\gamma h$ after a short computation.
This means that $\mathcal F\cap\gamma \mathcal F\ne \varnothing$, which is impossible since $\mathcal F$ is a fundamental domain.
{\hfill$\Box$}\bigskip
The next result follows directly from Proposition \ref{examfun} and Lemma \ref{fdPSL}.
\begin{corollary}\label{fundaF0}
(a) For $t>0$, the set
\begin{equation}\label{et}
\mathcal F_t=\{g=b_x a_{\ln y}d_\theta\in{\rm PSL}(2,\mathbb R) \,:\, x\in\mathbb R,
1<y<e^t, \theta\in [0,2\pi)\}
\end{equation}
is a fundamental domain in ${\rm PSL}(2,\mathbb R)$ for the Fuchsian group $\langle a_t\rangle.$
(b) For $t>0$, the set
\begin{equation}\label{et2}
{\cal E}_t=\{g=b_x a_{\ln y}d_\theta\in{\rm PSL}(2,\mathbb R) \,:\,0< x<t,
y>0, \theta\in [0,2\pi)\}
\end{equation} is a fundamental domain in ${\rm PSL}(2,\mathbb R)$ for the Fuchsian group
$\langle b_t\rangle$.
\end{corollary}
It is well-known that ${\rm PSL}(2,\mathbb Z)$ is a Fuchsian group and the set
\[F=\Big\{z\in\mathbb H^2: |z|>1, |{\rm Re\,} z|<\frac{1}{2} \Big\} \]
is a fundamental domain of ${\rm PSL}(2,\mathbb Z)$ in $\mathbb H^2$ (see \cite[Proposition 9.18]{einsward}).
The following result follows from Lemma \ref{lemnak} and Theorem \ref{fdPSL}.
\begin{corollary}
The set
\begin{eqnarray*}
{\cal F}&=&\Big\{g=[G]\in{\rm PSL}(2,\mathbb R), G=\scriptsize\Big(\begin{array}{cc} a&b\\c&d \end{array}\Big)\in{\rm SL}(2,\mathbb R):\\
&& \ \ \ 2|ac+bd|<{c^2+d^2}, (ac+bd)^2+1> (c^2+d^2)^2 \Big\}
\end{eqnarray*} is a fundamental domain in ${\rm PSL}(2,\mathbb R)$ for ${\rm PSL}(2,\mathbb Z)$.
\end{corollary}
The next result shows us how to find a fundamental domain for a cyclic group
as we know a fundamental domain for the cyclic group generated by
a conjugate element of its generator.
\begin{lemma}\label{fundaconj}
Let $g_1$ and $g_2$ be conjugate in ${\rm PSL}(2,\mathbb R)$ and $g_2=h g_1 h^{-1}$ for $h\in{\rm PSL}(2,\mathbb R)$.
Then if $\mathcal F_1\subset {\rm PSL}(2,\mathbb R)$ is a fundamental domain for
$\langle g_1 \rangle $
then $\mathcal F_2=h \mathcal F_1$ is a fundamental domain for
$\langle g_2 \rangle$.
\end{lemma}
\noindent
{\bf Proof.} Obviously $\overline{ \mathcal F_2}=h\overline{\mathcal F_1}$.
Since $\mathcal F_1$ is a fundamental domain for $\langle g_1\rangle$, we have
\begin{eqnarray*}
\bigcup_{j\in\mathbb Z} g_2^j \overline{\mathcal F_2}
=\bigcup_{j\in\mathbb Z} hg_1^jh^{-1}h\overline{\mathcal F_1}
=\bigcup_{j\in\mathbb Z} hg_1^j\overline{\mathcal F_1}
= h(\bigcup_{j\in\mathbb Z} g_1^j\overline{\mathcal F_1})
=h {\rm PSL}(2,\mathbb R)={\rm PSL}(2,\mathbb R),
\end{eqnarray*}
and if $j\in\mathbb Z, g_2^j\ne e$, then $g_1^j\ne e$ yields
\begin{eqnarray*}
\mathcal F_2\cap g_2^j \mathcal F_2
=h \mathcal F_1\cap hg_1^jh^{-1}h \mathcal F_1
=h\mathcal F_1\cap hg_1^j \mathcal F_1
=h(\mathcal F_1\cap g_1^j \mathcal F_1)=\varnothing.
\end{eqnarray*}
Also both $\mathcal F_1\subset {\rm PSL}(2,\mathbb R)$
and $\mathcal F_2\subset {\rm PSL}(2,\mathbb R)$ are open.
{\hfill$\square$}\bigskip
Recall that every hyperbolic (resp. parabolic) element is conjugate with $a_t$ (resp. $b_t$) for some $t\in\mathbb R$.
Note that $\langle a_t\rangle=\langle a_{-t}\rangle$ and $\langle b_t\rangle=\langle b_{-t}\rangle$.
The next result follows from the preceding lemma.
\begin{proposition} Let $g\in {\rm PSL}(2,\mathbb R)$ be a hyperbolic element
(resp. parabolic element). If $h\in{\rm PSL}(2,\mathbb R)$ and $t\in\mathbb R$ are such that $g=h^{-1}a_th$
(resp. $g=h^{-1}b_t h$) then ${\cal F}=h{\cal F}_{|t|}$
(resp. ${\cal E}=h{\cal E}_{|t|}$) is a fundamental domain for
$\Gamma=\langle g\rangle $, where ${\cal F}_{|t|}$ (resp. ${\cal E}_{|t|}$) is a fundamental domain
for $\langle a_t\rangle$ (resp. $\langle b_t\rangle$) given by \eqref{et} (resp. \eqref{et2}).
\end{proposition}
Next we define $\Theta:T^1\mathbb H^2\to {\rm PSL}(2,\mathbb R)$ by $\Theta(z,\xi)=g$ for $(z,\xi)\in T^1\mathbb H^2$,
where $g\in{\rm PSL}(2,\mathbb R)$ satisfies $\mathfrak D g(i,i)=(z,\xi)$.
Then $\Theta$ is well-defined and bijective owing to Lemma \ref{unig}.
Note that there exist metrics on ${\rm PSL}(2,\mathbb R)$ and $T^1\mathbb H^2$ such that $\Theta$ is an isometry.
\begin{lemma}\label{cc}
Let $F\subset\mathbb H^2$ and denote $T^1F=\{(z,\xi)\in T^1\mathbb H^2: z\in F \}$.
Then \[\Theta(T^1F)=\{g\in{\rm PSL}(2,\mathbb R):g=b_x a_{\ln y}d_\theta: x+iy\in F,\theta\in [0,2\pi) \}.\]
\end{lemma}
\noindent
{\bf Proof\,:} For any $g=b_x a_{\ln y}d_\theta\in{\rm PSL}(2,\mathbb R)$
with $x+iy\in F$, if $g=\big\{\pm \big({\scriptsize\begin{array}{cc}a&b\\c&d \end{array}}\big) \big\}$ then
we take $z=x+iy\in F$ and $\xi ={\rm Re\,} \xi +i{\rm Im\,} \xi$ satisfying
\[ x=\frac{ac+bd}{c^2+d^2},\ y=\frac{1}{c^2+d^2},\ {\rm Re\,}\xi=\frac{2cd}{(c^2+d^2)^2},
\ {\rm Im\,}\xi=\frac{d^2-c^2}{(c^2+d^2)^2}. \]
Then $\|\xi\|_z=\frac{|\xi|}{y}=1$ means that $(z,\xi)\in T^1F$
and \eqref{dg} shows $\Theta(z,\xi)=g.$
On the other hand, for $(z,\xi)\in T^1F$ and $\Theta(z,\xi)=g\in{\rm PSL}(2,\mathbb R)$. If $g=\Big\{\pm \scriptsize\Big(\begin{array}{cc}a&b\\c&d \end{array}\Big)\Big\}=b_xa_{\ln y}d_\theta$
then $ x=\frac{ac+bd}{c^2+d^2}, y=\frac{1}{c^2+d^2}$ by Lemma \ref{lemnak} (a).
Once again \eqref{dg} implies that $z=x+iy\in F$. This completes the proof.
{\hfill $\Box$}
The relation of fundamental domains in $\mathbb H^2$ and in $T^1\mathbb H^2$ is the following:
\begin{theorem} Let $\Gamma$ be a Fuchsian group.
A set $F\subset\mathbb H^2$ is a fundamental domain for $\Gamma$ if and only if $T^1F\subset T^1\mathbb H^2$ is a
fundamental domain for $\Gamma$.
\end{theorem}
\noindent{\bf Proof\,:}
Let $F\subset \mathbb H^2$ and ${\cal F}=\{ g=b_xa_{\ln y}d_\theta, x+iy\in F, \theta\in [0,2\pi)\}\subset {\rm PSL}(2,\mathbb R)$.
Then $\Theta^{-1}({\cal F})= T^1F$ by Lemma \ref{cc} and this follows from Theorem \ref{fdPSL}
and the fact that $\Theta$ is an isometry.
{\hfill$\Box$}
\bibliographystyle{mystyle}
\noindent {\bf Acknowledgments:} This work is supported by Vietnam National Foundation
for Science and Technology Development (Grant No. 101.02-2020.21).
| {
"timestamp": "2020-10-21T02:07:01",
"yymm": "1909",
"arxiv_id": "1909.06792",
"language": "en",
"url": "https://arxiv.org/abs/1909.06792",
"abstract": "In this paper, we provide a necessary and sufficient condition for a set in ${\\rm PSL}(2,{\\mathbb R})$ or in $T^1{\\mathbb H}^2$ to be a fundamental domain for a given Fuchsian group via its respective fundamental domain in the hyperbolic plane ${\\mathbb H}^2$.",
"subjects": "Differential Geometry (math.DG); Geometric Topology (math.GT)",
"title": "Fundamental domains in ${\\rm PSL}(2,{\\mathbb R})$ for Fuchsian groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860381,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7092019616632164
} |
https://arxiv.org/abs/1708.07756 | An undetermined time-dependent coefficient in a fractional diffusion equation | In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\alpha u(x,t)-a(t)\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\Omega$ and right-hand side $F(x,t)$. For the inverse problem--recovering $a(t),$ we introduce an operator $K$ one of whose fixed points is $a(t)$ and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for $a(t)$ is created and some numerical results are provided to illustrate the theories. | \section{Introduction}
\par This paper considers the fractional diffusion equation (FDE) with
a continuous and positive coefficient function $a(t):$
\begin{equation}\label{fde}
\begin{aligned}
^C\!D_t^{\alpha} u(x,t)-a(t)\mathcal{L} u(x,t)&=F(x,t),\ &&x\in \Omega,\ t\in (0,T];\\
u(x,t)&=0,\ &&(x,t)\in \partial\Omega \times (0,T];\\
u(x,0)&=u_0(x),\ &&x\in \Omega,
\end{aligned}
\end{equation}
where $\Omega$ is a bounded and smooth subset of $R^n, n=1,2,3,$ $-\mathcal{L}$ is a symmetric uniformly elliptic operator defined as
$$-\mathcal{L} u=-\sum_{i,j=1}^n (a^{ij}(x)u_{x_i})_{x_j}+c(x)u$$ with conditions
\begin{equation}\label{sobolev assumption}
a^{ij},c\in C^2(\overline{\Omega}) \ (i,j=1,\dots,n),\ \partial \Omega\ \text{is}\ C^3,
\end{equation}
and $^C\!D_t^{\alpha}$ is the left-sided Djrbashian-–Caputo $\alpha$-th order derivative with respect to time $t.$ The definition for $^C\!D_t^{\alpha}$ is $$^C\!D_t^{\alpha} u(x,t)=\frac{1}{\Gamma(n-\alpha)}
\int_{0}^{t} (t-\tau)^{n-\alpha-1}
\frac{d^n}{d\tau^n}u(x,\tau) {\rm d}\tau$$
with Gamma function $\Gamma(\cdot)$ and the nearest integer $n$ with
$\alpha\le n.$ In this paper, we are assuming a subdiffusion process,
i.e. $\alpha\in(0,1).$ This simplifies the definition of $^C\!D_t^{\alpha}$ as
$$ ^C\!D_t^{\alpha} u(x,t)=\frac{1}{\Gamma(1-\alpha)}
\int_{0}^{t} (t-\tau)^{-\alpha}\frac{d}{d\tau}u(x,\tau) {\rm d}\tau.$$
This work is an extension of \cite{Zhang2016undetermined}
from a simple space domain $\Omega$ to $\mathbb{R}^n$, considers the
more general analysis for the direct problem and contains an existence
argument for the inverse problem of recovering $a(t).$
\par This paper consists of two parts; the direct problem and
the inverse problem. For the direct problem, we build the
spectral representation of the weak solution $u(x,t;a).$ The notation $u(x,t;a)$ is used for displaying the dependence of the solution $u$ on the diffusivity $a(t).$
Then the existence, uniqueness and regularity results are proved
with several assumptions on the coefficient function $a(t).$
Unlike \cite{Zhang2016undetermined},
the right hand side function $F(x,t)$ is not of the form $f(x)g(t)$, so
that the proof of regularity is more delicate. For the inverse problem, we use the single point flux data
$$a(t)\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)=g(t),\ x_0\in \partial \Omega$$
to recover the coefficient $a(t)$
(We choose the data $a(t)\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)=g(t)$ instead of
the classical flux $\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)$ because
in practice, $a(t)\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)$ is usually
measured as the flux).
For the reconstruction, we only consider to recover a
continuous and positive $a(t)$ to match the assumptions set in the direct
problem. Acting a flux data, we introduce an operator $K$
one of whose fixed points is the coefficient $a(t).$ Using
the weak maximum principle \cite{luchko2009maximum}, we establish the monotonicity and uniqueness of the fixed points of operator $K$, and the proof of uniqueness leads to a numerical reconstruction algorithm. Since we consider a multidimensional domain $\Omega$ here, the Sobolev Embedding Theorem yields that we need to add the condition \eqref{sobolev assumption} on the operator $-\mathcal{L}$ to ensure the $C^1$-regularity of the series representation of $u$. Then the operator $K$ is well-defined, where the proofs can be seen in section 4. This is a significant difference from \cite{Zhang2016undetermined}.
Furthermore, an existence argument of the fixed points of $K$ is included by this paper, which \cite{Zhang2016undetermined} does not contain.
\par The rest of this paper follows the following structure. In section 2, we collect
some preliminary results about fractional calculus and the eigensystem
of $-\mathcal{L}$. The direct problem
is discussed in section 3, i.e. we establish the existence, uniqueness
and some regularity results of the weak solution for FDE \eqref{fde}.
Then section 4 deals with the inverse problem of recovering $a(t)$. Specifically, an operator
$K$ is introduced at the beginning of this section, then its monotonicity
and uniqueness of its fixed points give an algorithm to recover the
coefficient $a(t)$. In particular, the existence argument of the fixed
points of $K$ is included by this section. In section 5, some
numerical results are presented to illustrate the theoretical basis.
\section{Preliminary material}
\subsection{Mittag-Leffler function}
\par In this part, we describe the Mittag-Leffler function which plays
an important role in fractional diffusion equations. This
is a two-parameter function defined as
\begin{equation*}
E_{\alpha,\beta}(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(k\alpha+\beta)},
\ z\in \mathbb{C}.
\end{equation*}
It generalizes the natural exponential function in the sense that
$E_{1,1}(z)=e^z$. We list some important properties of the
Mittag-Leffler function for future use.
\begin{lemma}\label{mittag_bound}
Let $0<\alpha<2$ and $\beta\in\mathbb{R}$ be arbitrary, and $\frac{\alpha\pi}{2}
<\mu<\min(\pi,\alpha\pi)$. Then there exists a constant $C=C(\alpha,\beta,\mu)>0$ such
that
\begin{equation*}
|E_{\alpha,\beta}(z)|\leq \frac{C}{1+|z|},\quad \mu\leq|\mathrm{arg}(z)|\leq \pi.
\end{equation*}
\end{lemma}
\begin{proof}
This proof can be found in \cite{kilbas2006theory}.
\end{proof}
\begin{lemma}\label{mittag_derivative}
\par For $\lambda>0,\ \alpha>0$ and $\N+,$ we have
$$\frac{d^n}{dt^n}E_{\alpha,1}(-\lambda t^\alpha)
=-\lambda t^{\alpha-n}E_{\alpha,\alpha-n+1}(-\lambda t^\alpha),
\ t>0.$$
In particular, if we set $n=1,$
then there holds
$$\frac{d}{dt}E_{\alpha,1}(-\lambda t^\alpha)
=-\lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^\alpha),
\ t>0.$$
\end{lemma}
\begin{proof}
\par This is \cite[Lemma $3.2$]{sakamoto2011initial}.
\end{proof}
\begin{lemma}\label{mittag_positive}
If $0<\alpha<1$ and $z>0,$ then
$E_{\alpha,\alpha}(-z)\ge 0.$
\end{lemma}
\begin{proof}
This proof can be found in \cite{miller2001completely,pollard1948completely,schneider1996completely}.
\end{proof}
\begin{lemma}\label{mittag_positive_1}
\par For $0<\alpha<1,$ $E_{\alpha,1}(-t^\alpha)$ is completely monotonic,
that is,
$$(-1)^n \frac{d^n}{dt^n}E_{\alpha,1}(-t^\alpha)\ge 0,\ for\ t>0
\ and\ n=0,1,2,\cdots.$$
\end{lemma}
\begin{proof}
See \cite{gorenflofractional}.
\end{proof}
\subsection{Fractional calculus}
\par In this part, we collect some results of fractional calculus.
The next lemma states the extremal principle of ${^C\!D_t^{\alpha}}.$
\begin{lemma}\label{extreme}
Fix $0<\alpha<1$ and given $f(t)\in C[0,T]$ with $^C\!D_t^{\alpha} f \in C[0,T].$
If $f$ attains its maximum (minimum) over the interval $[0,T]$ at the point
$t=t_0,\ t_0\in (0,T],$ then
$^C\!D_{t_0}^{\alpha} f \ge(\le) 0.$
\end{lemma}
\begin{proof}
\par Even though the conditions are different from the ones of \cite[Theorem 1]{luchko2009maximum}, the maximum case can be proved following the proof of \cite[Theorem 1]{luchko2009maximum}.
For the minimum case, we only need to set $\overline{f}=-f.$
\end{proof}
\par The following lemma about the composition between
$^C\!D_t^{\alpha}$ and the fractional integral $I_t^\alpha$ is presented in
\cite{samko1993fractional}.
\begin{lemma}\label{I_alpha}
Define the Riemann-–Liouville $\alpha$-th order integral $I_t^\alpha$ as
$$
I_t^\alpha u=\frac{1}{\Gamma(\alpha)}\int_{0}^{t} (t-\tau)^{\alpha-1}
u(\tau) {\rm d}\tau.
$$
For $0<\alpha<1,$ $u(t),{^C\!D_t^{\alpha}} u\in C[0,T],$ we have
$$(^C\!D_t^{\alpha} \circ I_t^\alpha u)(t)=u(t),\quad (I_t^\alpha\circ {^C\!D_t^{\alpha}} u)(t)=u(t)-u(0),\ t\in[0,T].$$
\end{lemma}
\subsection{Eigensystem of $-\mathcal{L}$}
\par Since $-\mathcal{L}$ is a symmetric uniformly elliptic operator,
we denote the eigensystem of $-\mathcal{L}$ by $\{(\lambda_n,\phi_n):\N+\}.$
Then we have $0<\lambda_1\le \lambda_2\le \cdots$ where finite multiplicity is possible, $\lambda_n\to \infty$ and $\{\phi_n:\N+\}\subset H^2(\Omega)\cap
H_0^1(\Omega)$ forms an orthonormal basis of $L^2(\Omega).$
\par Moreover, with the condition \eqref{sobolev assumption}, for each $\N+$, it holds that $\phi_n \in H^3(\Omega)$ \cite{Evans2010}.
Then by the Sobolev Embedding Theorem, we have $\phi_n\in C^1(\overline{\Omega})$ and
$\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)$ is well-defined for each $\N+$. Hence, without loss of generality, we can suppose
\begin{equation}\label{sign_eigenfunction_derivative}
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)\ge 0,\ \text{for each}\ \N+.
\end{equation}
Otherwise, if $\frac{\partial \phi_k}{\partial {\overrightarrow{\bf n}}}(x_0)< 0$ for some
$k\in \mathbb{N}^{+},$ we can replace $\phi_k$ by $-\phi_k.$
$-\phi_k$ satisfies all the properties we need, such as it is
an eigenfunction of $-\mathcal{L}$ corresponding to the eigenvalue $\lambda_k,$
composes an orthonormal basis of $L^2(\Omega)$ together with
$\{\phi_n:\N+,n\ne k\}$ and $\frac{\partial (-\phi_k)}{\partial {\overrightarrow{\bf n}}}(x_0)\ge 0.$
The assumption \eqref{sign_eigenfunction_derivative} will be used in Section 4.
\section{Direct Problem--Existence, Uniqueness and Regularity}
\par Throughout this section, we suppose $a(t),$ $u_0(x)$ and $F(x,t)$
satisfy the following assumptions:
\begin{assumption}\label{assumption_direct}
~
\begin{itemize}
\item[(a)] $a(t)\in C^{+}[0,T]:=\{\psi\in C[0,T]:\psi(t)>0,\ t\in[0,T]\};$
\item[(b)] $F(x,t) \in C([0,T];L^2(\Omega));$
\item[(c)] $u_0(x) \in H_0^1(\Omega).$
\end{itemize}
\end{assumption}
\subsection{Spectral Representation}
\begin{definition}\label{weak solution}
We call $u(x,t;a)$ a weak solution of
FDE \eqref{fde} in $L^2(\Omega)$ corresponding to the
coefficient $a(t)$ if
$u(\cdot,t;a)\in H_0^1(\Omega)$ for $t\in(0,T]$ and
for any $\psi(x)\in H^2(\Omega)\cap H_0^1(\Omega),$ it holds
\begin{equation*}
\begin{split}
&(^C\!D_t^{\alpha} u(x,t;a),\psi(x))-(a(t)\mathcal{L} u(x,t;a),\psi(x))=(F(x,t),\psi(x)),
\ t\in(0,T];\\
&(u(x,0;a),\psi(x))=(u_0(x),\psi(x)),
\end{split}
\end{equation*}
where $(\cdot,\cdot)$ is the inner product in $L^2(\Omega).$
\end{definition}
\par With the above definition, we give a spectral representation
for the weak solution in the following lemma.
\begin{lemma}\label{spectral representation}
Define $b_n:=(u_0(x),\phi_n(x)), F_n(t)=(F(x,t),\phi_n(x)),
\ \N+.$
The spectral representation of the weak solution of FDE
\eqref{fde} is
\begin{equation}\label{solution}
u(x,t;a)=\sum_{n=1}^\infty u_n(t;a)\phi_n(x),\ (x,t)\in \Omega\times [0,T],
\end{equation}
where
$u_n(t;a)$ satisfies the fractional ODE
\begin{equation}\label{ODE}
^C\!D_t^{\alpha} u_n(t;a)+\lambda_n a(t) u_n(t;a)=F_n(t),
\ u_n(0;a)=b_n,\ \N+.
\end{equation}
\end{lemma}
\begin{proof}
\par For each $\N+,$ multiplying $\phi_n(x)$ on both sides of
FDE \eqref{fde} and integrating it on $x$ over $\Omega$
allow us to deduce that
\begin{equation}\label{equality_1}
^C\!D_t^{\alpha}( u(x,t;a), \phi_n(x))+\lambda_n a(t)(u(x,t;a),\phi_n(x))=F_n(t),
\end{equation}
where
$(-\mathcal{L} u(x,t;a),\phi_n(x))=(u(x,t;a),-\mathcal{L}\phi_n(x))
=\lambda_n (u(x,t;a),\phi_n(x))$ follows from the symmetricity of $-\mathcal{L}.$
Set $u_n(t;a)=(u(x,t;a), \phi_n(x))$ and define
$u(x,t;a)=\sum\limits_{n=1}^\infty u_n(t;a)\phi_n(x).$
Then \eqref{equality_1} and the completeness of
$\{\phi_n(x):\N+\}$ lead to the desired result.
\end{proof}
\subsection{Existence and Uniqueness}
\par In order to show the existence and uniqueness
of the weak solution \eqref{solution}, we state the following lemma
\cite[Theorem 3.25]{kilbas2006theory}.
\begin{lemma}\label{kilbassrivastavatrujillo}
For the Cauchy-type problem
$$^C\!D_t^{\alpha} y=f(y,t),\ y(0)=c_0,$$
if for any continuous $y(t),$ $f(y,t)\in C[0,T]$, $\exists A>0$
which is independent of $y\in C[0,T]$ and $t\in[0,T]$ s.t.
$\lvert f(t,y_1)-f(t,y_2)\rvert \le A\lvert y_1-y_2\rvert,$
then there exists a unique solution $y(t)$ for the Cauchy-type
problem, which satisfies $^C\!D_t^{\alpha} y\in C[0,T]$.
\end{lemma}
\par The theorem of existence and uniqueness for $u(x,t;a)$ follows from
Lemma \ref{kilbassrivastavatrujillo}.
\begin{theorem}[Existence and Uniqueness]\label{existenceuniqueness}
Suppose Assumption \ref{assumption_direct} holds. Under
Definition \ref{weak solution}, there exists a unique weak solution
$u(x,t;a)$ of FDE \eqref{fde} with the spectral
representation \eqref{solution} and
for each $\N+,$ $u_n(t;a)\in C[0,T]$ is the unique solution of
the fractional ODE \eqref{ODE} with $^C\!D_t^{\alpha} u_n(t;a) \in C[0,T].$
\end{theorem}
\begin{proof}
\par From the spectral representation \eqref{solution}, it
suffices to show the existence and uniqueness of $u_n(t;a),\N+.$
Fix $\N+,$ Assumption \ref{assumption_direct} $(a)$ and $(b)$ yield
that the fractional ODE \eqref{ODE} satisfies the conditions of
Lemma \ref{kilbassrivastavatrujillo}. Hence the existence and uniqueness
for $u_n(t;a)$ hold.
\end{proof}
\subsection{Sign of $u_n(t;a)$}
\par In this part, we state two properties of $u_n(t;a)$ which play important
roles in building the regularity of $u(x,t;a).$
\begin{lemma}\label{sign}
Given $h\in C^{+}[0,T],$ $f\in C[0,T]$ with $^C\!D_t^{\alpha} f \in C[0,T],$
if $f(0)\le (\ge)0$ and $^C\!D_t^{\alpha} f+h(t)f(t)\le (\ge)0,$ then $f\le
(\ge)0$ on $[0,T]$.
\end{lemma}
\begin{proof}
\par Since $f(t)\in C[0,T]$, $f(t)$ attains its maximum over $[0,T]$
at some point $t_0\in [0,T]$.
If $t_0=0,$ then $f(t)\le f(0)\le 0.$
If $t_0\in (0,T],$ with Lemma \ref{extreme}, we have
${}^C\!D_{t}^{\alpha} f(t_0)\ge 0$, which yields $h(t_0)f(t_0)\le 0,$ i.e.
$f(t_0)\le 0$ due to $h>0$ on $[0,T].$
The definition of $t_0$ assures $f\le 0$.
\par For the case of ``$\ge 0$", let $\overline{f}(t)=-f(t),$ then the
above proof gives $\overline{f}\le 0,$ i.e. $f\ge 0$.
\end{proof}
\par The following corollary, which concerns the sign of $u_n(t;a)$, follows
from Lemma \ref{sign} directly.
\begin{corollary}\label{sign_eigenfunction}
Set $u_n(t;a)$ be the unique solution of
the fractional ODE \eqref{ODE}. Then
$^C\!D_t^{\alpha} u_n(t;a)+\lambda_n a(t) u_n(t;a)\le (\ge) 0$ on $[0,T]$ and
$u_n(0;a) \le (\ge) 0$ imply $u_n(t;a)\le (\ge)0$ on $[0,T],\ \N+$.
\end{corollary}
\begin{proof}
Assumption \ref{assumption_direct} gives that $\lambda_na(t)\in
C^{+}[0,T].$ Then the proof is completed by applying Lemma \ref{sign}
to the fractional ODE \eqref{ODE}.
\end{proof}
\subsection{Regularity}
\par In this part, we establish the regularity of $u(x,t;a)$.
To this end, we split FDE \eqref{fde} into
\begin{equation}\label{model_right}
\begin{aligned}
^C\!D_t^{\alpha} u(x,t)-a(t)\mathcal{L} u(x,t)&=F(x,t),\ &&x\in \Omega,\ t\in (0,T];\\
u(x,t)&=0,\ &&(x,t)\in \partial\Omega \times (0,T];\\
u(x,0)&=0,\ &&x\in \Omega,
\end{aligned}
\end{equation}
and
\begin{equation}\label{model_initial}
\begin{aligned}
^C\!D_t^{\alpha} u(x,t)-a(t)\mathcal{L} u(x,t)&=0,\ &&x\in \Omega,\ t\in (0,T];\\
u(x,t)&=0,\ &&(x,t)\in \partial\Omega \times (0,T];\\
u(x,0)&=u_0(x),\ &&x\in \Omega.
\end{aligned}
\end{equation}
\par Denote the weak solutions of FDEs \eqref{model_right} and
\eqref{model_initial} by $u^r(x,t;a)$ and $u^i(x,t;a)$, respectively
(``r" and ``i" denote the initials of ``right-hand side" and ``initial
condition").
The following lemma about $u^r(x,t;a)$ and $u^i(x,t;a)$
follows from Lemma \ref{spectral representation} and
Theorem \ref{existenceuniqueness}.
\begin{lemma}
Suppose Assumption \ref{assumption_direct} holds. Then
$u^r(x,t;a)$ and $u^i(x,t;a)$ are the unique solutions for FDEs
\eqref{model_right} and \eqref{model_initial}, respectively, with the
spectral representations as
\begin{equation}\label{u^r u^i}
u^r(x,t;a)=\sum_{n=1}^\infty u^r_n(t;a)\phi_n(x),\
u^i(x,t;a)=\sum_{n=1}^\infty u^i_n(t;a)\phi_n(x),
\end{equation}
where $u^r_n(t;a),\ u^i_n(t;a)$ satisfy the following fractional ODEs
\begin{equation}\label{ODE_right}
^C\!D_t^{\alpha} u_n^r(t;a)+\lambda_na(t)u_n^r(t;a)=F_n(t),\ u_n^r(0;a)=0,\ \N+;
\end{equation}
\begin{equation}\label{ODE_initial}
^C\!D_t^{\alpha} u_n^i(t;a)+\lambda_na(t)u_n^i(t;a)=0,\ u_n^i(0;a)=b_n,\ \N+.
\end{equation}
\par Moreover, Theorem \ref{existenceuniqueness} ensures the weak
solution $u(x,t;a)$ of FDE \eqref{fde}
can be written as $u(x,t;a)=u^r(x,t;a)+u^i(x,t;a),$ i.e.
$u_n(t;a)=u^r_n(t;a)+u^i_n(t;a),\ \N+.$
\end{lemma}
\subsubsection{Regularity of $u^r$}
\par For each $\N+,$ define
\begin{equation}\label{+-}
F^{+}_n(t)=\begin{cases}
F_n(t), &\text{if}\ F_n(t)\ge0;\\
0, &\text{if}\ F_n(t)<0,
\end{cases}
\quad F^{-}_n(t)=\begin{cases}
F_n(t),&\text{if}\ F_n(t)<0;\\
0,&\text{if}\ F_n(t)\ge0.
\end{cases}
\end{equation}
It is obvious that $F_n=F^{+}_n+F^{-}_n,$ the supports
of $F^{+}_n$ and $F^{-}_n$ are disjoint and
$F^{+}_n,F^{-}_n \in C[0,T]$ which follows from $F_n\in C[0,T]$.
Split $u_n^r(t;a)$ as $u_n^r(t;a)=u_n^{r,+}(t;a)+u_n^{r,-}(t;a),$
where $u_n^{r,+}(t;a),\ u_n^{r,-}(t;a)$ satisfy
\begin{equation}\label{ODE_right_+}
^C\!D_t^{\alpha} u_n^{r,+}(t;a)+\lambda_na(t)u_n^{r,+}(t;a)=F^{+}_n(t),\ u_n^{r,+}(0;a)=0,\ \N+;
\end{equation}
\begin{equation}\label{ODE_right_-}
^C\!D_t^{\alpha} u_n^{r,-}(t;a)+\lambda_na(t)u_n^{r,-}(t;a)=F^{-}_n(t),\ u_n^{r,-}(0;a)=0,\ \N+,
\end{equation}
respectively. The existence and uniqueness of
$u_n^{r,+}(t;a)$ and $u_n^{r,-}(t;a)$ hold due to Lemma
\ref{kilbassrivastavatrujillo} and we can write
\begin{equation}\label{u^r u^{r,+} u^{r,-}}
u^r(x,t;a)=u^{r,+}(x,t;a)+u^{r,-}(x,t;a),
\end{equation}
where
\begin{equation}\label{u^{r,+} u^{r,-}}
u^{r,+}(x,t;a)=\sum\limits_{n=1}^{\infty}u_n^{r,+}(t;a)\phi_n(x),
\ u^{r,-}(x,t;a)=\sum\limits_{n=1}^{\infty}u_n^{r,-}(t;a)\phi_n(x).
\end{equation}
\par Then we state some properties of $u_n^{r,+}(t;a)$
and $u_n^{r,-}(t;a).$
\begin{lemma}\label{sign_right_+-}
For any $\N+,$
$u_n^{r,+}(t;a)\ge 0$ and $u_n^{r,-}(t;a)\le 0$ on $[0,T].$
\end{lemma}
\begin{proof}
This proof follows from Corollary \ref{sign_eigenfunction} directly.
\end{proof}
\begin{lemma}\label{monotone_right}
Given $a_1(t), a_2(t)\in C^{+}[0,T]$ with $a_1(t)\le a_2(t)$ on $[0,T]$,
we have
$$0\le u_n^{r,+}(t;a_2)\le u_n^{r,+}(t;a_1),
\ u_n^{r,-}(t;a_1)\le u_n^{r,-}(t;a_2)\le 0,
\ t\in [0,T],\ \N+.$$
\end{lemma}
\begin{proof}
Pick $\N+,$ $u_n^{r,+}(t;a_1)$ and $u_n^{r,+}(t;a_2)$ satisfy
the following system:
\begin{equation*}
\begin{cases}
^C\!D_t^{\alpha} u_n^{r,+}(t;a_1)+\lambda_na_1(t)u_n^{r,+}(t;a_1)=F^{+}_n(t);\\
^C\!D_t^{\alpha} u_n^{r,+}(t;a_2)+\lambda_na_2(t)u_n^{r,+}(t;a_2)=F^{+}_n(t);\\
u_n^{r,+}(0;a_1)=u_n^{r,+}(0;a_2)=0,
\end{cases}
\end{equation*}
which leads to
$$^C\!D_t^{\alpha} w+\lambda_na_1(t)w(t)=\lambda_n u_n^{r,+}(t;a_2)(a_2(t)-a_1(t))\ge0,
\ w(0)=0,$$
where $w(t)=u_n^{r,+}(t;a_1)-u_n^{r,+}(t;a_2)$ and the
last inequality follows from Lemma \ref{sign_right_+-} and $a_1\le a_2.$
Hence, Corollary \ref{sign_eigenfunction} shows that
$w(t)\ge 0,$ i.e. $u_n^{r,+}(t;a_2)\le u_n^{r,+}(t;a_1)$ and
Lemma \ref{sign_right_+-} gives
$0\le u_n^{r,+}(t;a_2)\le u_n^{r,+}(t;a_1),\ t\in [0,T].$
\par Similarly, we have
$u_n^{r,-}(t;a_1)\le u_n^{r,-}(t;a_2)\le 0,\ t\in [0,T],$
completing the proof.
\end{proof}
\par Assumption \ref{assumption_direct} $(a)$ implies
there exists constants $q_a, Q_a$ s.t.
\begin{equation}\label{q_a Q_a}
0<q_a<a(t)<Q_a\ \text{on}\ [0,T].
\end{equation}
From Lemma \ref{monotone_right}, we obtain
\begin{equation}\label{inequality_1}
|u_n^{r,+}(t;a)|\le |u_n^{r,+}(t;q_a)|,\
|u_n^{r,-}(t;a)|\le |u_n^{r,-}(t;q_a)|\ \text{on}\ t\in [0,T],\ \N+,
\end{equation}
where $u_n^{r,+}(t;q_a),u_n^{r,-}(t;q_a)$ are the unique solutions of
fractional ODEs \eqref{ODE_right_+} and \eqref{ODE_right_-} respectively
with $a(t)\equiv q_a$ on $[0,T]$.
The next two lemmas concern the regularity of
$u^{r,+}(x,t;a)$ and $^C\!D_t^{\alpha} u^{r,+}(x,t;a),$ respectively.
\begin{lemma}\label{regularity_u^{r,+}}
$$\|u^{r,+}\|_{L^2(0,T;H^2(\Omega))}
\le C\|F\|_{L^2([0,T]\times\Omega)}.$$
\end{lemma}
\begin{proof}
\par Calculating $\|u^{r,+}(x,t;a)\|_{L^2(0,T;H^2(\Omega))}^2$ directly
yields
\begin{equation*}
\begin{split}
\|u^{r,+}(x,t;a)\|_{L^2(0,T;H^2(\Omega))}^2
&=\int_0^T\|u^{r,+}(x,t;a)\|_{H^2(\Omega))}^2 {\rm d}t
\le \int_0^T C\|(-\mathcal{L} u^{r,+})(x,t;a)\|_{L^2(\Omega)}^2 {\rm d}t\\
&=C\int_0^T\|\sum\limits_{n=1}^{\infty}\lambda_nu_n^{r,+}(t;a)\phi_n(x)\|
_{L^2(\Omega)}^2 {\rm d}t\\
&=C\int_0^T\sum\limits_{n=1}^{\infty}\lambda_n^2 |u_n^{r,+}(t;a)|^2
{\rm d}t
\le C\int_0^T\sum\limits_{n=1}^{\infty}\lambda_n^2
|u_n^{r,+}(t;q_a)|^2 {\rm d}t,
\end{split}
\end{equation*}
where the last inequality is obtained from \eqref{inequality_1}.
By the Monotone Convergence Theorem, we have
\begin{equation}\label{inequality_5}
\quad\|u^{r,+}(x,t;a)\|_{L^2(0,T;H^2(\Omega))}^2
\le C\int_0^T\sum\limits_{n=1}^{\infty}\lambda_n^2
|u_n^{r,+}(t;q_a)|^2 {\rm d}t
=C \sum\limits_{n=1}^{\infty} \int_0^T
|\lambda_nu_n^{r,+}(t;q_a)|^2 {\rm d}t.
\end{equation}
\par For each $\N+$, \cite{sakamoto2011initial} gives the explicit
representation of $u_n^{r,+}(t;q_a)$
\begin{equation*}\label{u_n^{r,+}(t;q_a)}
u_n^{r,+}(t;q_a)=\int_0^t F^+_n(\tau) (t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau,
\end{equation*}
which together with Young's inequality leads to
\begin{equation*}\label{inequality_2}
\begin{split}
\int_0^T|\lambda_nu_n^{r,+}(t;q_a)|^2 {\rm d}t
&=\|F_n^{+}(t)*(\lambda_nt^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_nq_at^\alpha))\|_{L^2[0,T]}^2\\
&\le \|F_n^{+}\|_{L^2[0,T]}^2 \|\lambda_nt^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_nq_at^\alpha)\|_{L^1[0,T]}^2.
\end{split}
\end{equation*}
Lemmas \ref{mittag_derivative}, \ref{mittag_positive} and
\ref{mittag_positive_1} give the bound of $\|\lambda_nt^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_nq_at^\alpha)\|_{L^1[0,T]}$
\begin{equation*}
\begin{split}
\|\lambda_nt^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_nq_at^\alpha)\|_{L^1[0,T]}
&=\int_0^T \big|\lambda_n\tau^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a\tau^\alpha)\big|{\rm d}\tau\\
&=\int_0^T \lambda_n\tau^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a\tau^\alpha){\rm d}\tau\\
&=-q_a^{-1}\int_0^T \frac{d}{d\tau}
E_{\alpha,1}(-\lambda_nq_a\tau^\alpha){\rm d}\tau\\
&=q_a^{-1}(1-E_{\alpha,1}(-\lambda_nq_aT^\alpha))
\le q_a^{-1};
\end{split}
\end{equation*}
while the definition \eqref{+-} provides the bound of
$\|F_n^{+}\|_{L^2[0,T]}$ as
$\|F_n^{+}\|_{L^2[0,T]}\le \|F_n\|_{L^2[0,T]}.$
Consequently, it holds $\int_0^T|\lambda_nu_n^{r,+}(t;q_a)|^2 {\rm d}t\le q_a^{-2}
\|F_n\|_{L^2[0,T]}^2,\ \N+,$
i.e.
$$\sum_{n=1}^\infty\int_0^T|\lambda_nu_n^{r,+}(t;q_a)|^2 {\rm d}t
\le q_a^{-2} \sum_{n=1}^\infty\|F_n\|_{L^2[0,T]}^2,$$
which together with \eqref{inequality_5} and the completeness
of $\{\phi_n(x):\N+\}$ in $L^2(\Omega)$ gives
\begin{equation*}\label{inequality_6}
\begin{split}
\|u^{r,+}(x,t;a)\|_{L^2(0,T;H^2(\Omega))}^2
&\le C\sum\limits_{n=1}^{\infty} \int_0^T
|\lambda_nu_n^{r,+}(t;q_a)|^2 {\rm d}t\\
&\le C\sum_{n=1}^\infty\|F_n\|_{L^2[0,T]}^2
=C\|F\|_{L^2([0,T]\times\Omega)}^2,
\end{split}
\end{equation*}
where the constant $C$ only depends on $a(t).$ This completes the proof.
\end{proof}
\begin{lemma}\label{regularity_D_u^{r,+}}
$$
\|^C\!D_t^{\alpha} u^{r,+}\|_{L^2([0,T]\times\Omega)}
\le C\|F\|_{L^2([0,T]\times\Omega)}.
$$
\end{lemma}
\begin{proof}
\par \eqref{ODE_right_+}, \eqref{u^{r,+} u^{r,-}},
definition \eqref{+-} and the Monotone Convergence Theorem give
\begin{equation}\label{inequality_17}
\begin{split}
\|^C\!D_t^{\alpha} u^{r,+}\|_{L^2([0,T]\times\Omega)}^2
&=\int_0^T \|\sum_{n=1}^{\infty} {^C\!D_t^{\alpha}} u_n^{r,+}(\cdot;a) \phi_n(x)\|
_{L^2(\Omega)}^2{\rm d}t
=\sum_{n=1}^{\infty} \int_0^T |^C\!D_t^{\alpha} u_n^{r,+}(\cdot;a)|^2 {\rm d}t\\
& \le \sum_{n=1}^{\infty} \int_0^T \left(2|\lambda_n a(t) u_n^{r,+}(t;a)|^2
+2|F_n^{+}(t)|^2\right){\rm d}t\\
&\le2\sum_{n=1}^{\infty} \int_0^T |\lambda_n a(t) u_n^{r,+}(t;a)|^2
{\rm d}t +2\sum_{n=1}^{\infty} \int_0^T |F_n(t)|^2{\rm d}t.
\end{split}
\end{equation}
The estimate of $\sum\limits_{n=1}^{\infty} \int_0^T
|\lambda_n a(t) u_n^{r,+}(t;a)|^2{\rm d}t$ follows from
\eqref{q_a Q_a}, \eqref{inequality_1} and the proof of Lemma
\ref{regularity_u^{r,+}}
\begin{equation*}\label{inequality_8}
\sum_{n=1}^{\infty} \int_0^T|\lambda_n a(t) u_n^{r,+}(t;a)|^2{\rm d}t
\le Q_a \sum_{n=1}^{\infty} \int_0^T
|\lambda_n u_n^{r,+}(t;q_a)|^2{\rm d}t
\le C\|F\|_{L^2([0,T]\times\Omega)}^2;
\end{equation*}
while the completeness of $\{\phi_n(x):\N+\}$ gives
$ \sum\limits_{n=1}^{\infty} \int_0^T |F_n(t)|^2{\rm d}t
=\|F\|_{L^2([0,T]\times\Omega)}^2.$
Hence, \eqref{inequality_17} develops
$\|^C\!D_t^{\alpha} u^{r,+}\|_{L^2([0,T]\times\Omega)}^2
\le C\|F\|_{L^2([0,T]\times\Omega)}^2,$
which implies the indicated conclusion.
\end{proof}
\par The following corollary follows immediately from the proofs of
Lemmas \ref{regularity_u^{r,+}} and \ref{regularity_D_u^{r,+}}.
\begin{corollary}\label{regularity_u^{r,-}}
\begin{equation*}
\|u^{r,-}\|_{L^2(0,T;H^2(\Omega))}
\le C\|F\|_{L^2([0,T]\times\Omega)},
\ \|^C\!D_t^{\alpha} u^{r,-}\|_{L^2([0,T]\times\Omega)}
\le C\|F\|_{L^2([0,T]\times\Omega)}.
\end{equation*}
\end{corollary}
\par From Lemmas \ref{regularity_u^{r,+}}, \ref{regularity_D_u^{r,+}},
Corollary \ref{regularity_u^{r,-}} and \eqref{u^r u^{r,+} u^{r,-}},
we are able to deduce the regularity for $u^r(x,t;a)$ and
$^C\!D_t^{\alpha} u^r(x,t;a)$.
\begin{lemma}[Regularity of $u^r$]\label{regularity_u^r}
$$\|u^r\|_{L^2(0,T;H^2(\Omega))}
+\|^C\!D_t^{\alpha} u^r\|_{L^2([0,T]\times\Omega)}
\le C\|F\|_{L^2([0,T]\times\Omega)}.$$
\end{lemma}
\begin{proof}
\eqref{u^r u^{r,+} u^{r,-}} gives
$u^r(x,t;a)=u^{r,+}(x,t;a)+u^{r,-}(x,t;a),$ which leads to
\begin{equation*}
\begin{split}
&\|u^r\|_{L^2(0,T;H^2(\Omega))}
+\|^C\!D_t^{\alpha} u^r\|_{L^2([0,T]\times\Omega)}\\
\le&\ \|u^{r,+}\|_{L^2(0,T;H^2(\Omega))}
+\|u^{r,-}\|_{L^2(0,T;H^2(\Omega))}\\
&+\|^C\!D_t^{\alpha} u^{r,+}\|_{L^2([0,T]\times\Omega)}
+\|^C\!D_t^{\alpha} u^{r,-}\|_{L^2([0,T]\times\Omega)}\\
\le &\ C\|F\|_{L^2([0,T]\times\Omega)}.
\end{split}
\end{equation*}
\end{proof}
\par If we impose a higher regularity on $F,$ we can obtain the regularity estimate of $\|u^{r}\|_{C([0,T];H^2(\Omega))}$.
\begin{corollary}\label{Cregularity_u^r}
Under Assumption \ref{assumption_direct}, if $F\in C^\theta([0,T];L^2(\Omega)),\ 0<\theta<1,$ then $$\|u^{r}\|_{C([0,T];H^2(\Omega))}
+\|^C\!D_t^{\alpha} u^r\|_{C([0,T];L^2(\Omega))}\le C \|F\|_{C^\theta([0,T];L^2(\Omega))},$$ where $C$ depends on $\Omega$, $-\mathcal{L}$ and $a(t).$
\end{corollary}
\begin{proof}
For each $t\in[0,T],$ we have
\begin{equation*}
\begin{split}
\|u^{r,+}(x,t;a)\|^2_{H^2(\Omega)}
&\le C \|-\mathcal{L} u^{r,+}\|^2_{L^2(\Omega)}
\le C \sum_{n=1}^\infty
|\lambda_nu^{r,+}_n(t;a)|^2\\
&\le C \sum_{n=1}^\infty
\left|\lambda_n \int_0^t F^+_n(\tau) (t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau \right|^2\\
&\le C \sum_{n=1}^\infty
\left|\lambda_n \int_0^t |F^+_n(\tau) -F^+_n(t)|(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau\right|^2\\
&\quad +C \sum_{n=1}^\infty
\left|F_n^+(t)\int_0^t \lambda_n(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau\right|^2.
\end{split}
\end{equation*}
The definition of $F_n^+(t)$ yields that
$|F^+_n(\tau) -F^+_n(t)|\le |F_n(\tau) -F_n(t)|$;
Lemma \ref{mittag_derivative} gives
$$0<\int_0^t \lambda_n(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau=
q_a^{-1}(1-E_{\alpha,1}(-\lambda_n q_a t^\alpha))<q_a^{-1}.$$
Hence,
\begin{equation*}
\begin{split}
\|u^{r,+}(x,t;a)\|^2_{H^2(\Omega)}
&\le C \sum_{n=1}^\infty
\left|\lambda_n \int_0^t |F_n(\tau) -F_n(t)|(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_a(t-\tau)^\alpha){\rm d}\tau\right|^2\\
&\quad +C \sum_{n=1}^\infty
\left|F_n(t)\right|^2.
\end{split}
\end{equation*}
By \cite[Lemma 3.4]{sakamoto2011initial}, we have
$$\|u^{r,+}(x,t;a)\|^2_{H^2(\Omega)}
\le C \|F\|_{C^\theta([0,T];L^2(\Omega))}^2+C\| F(\cdot,t)\|_{L^2(\Omega)}^2,
\ t\in[0,T],
$$
which gives
$$\|u^{r,+}\|_{C([0,T];H^2(\Omega))}\le C \|F\|_{C^\theta([0,T];L^2(\Omega))},$$
and the constant $C$ depends on $\Omega$, $-\mathcal{L}$ and $a(t).$ Similarly, we can show\\ $\|u^{r,-}\|_{C([0,T];H^2(\Omega))}\le C \|F\|_{C^\theta([0,T];L^2(\Omega))}.$
\par For $^C\!D_t^{\alpha} u^r,$ by \eqref{ODE_right}, we have
$^C\!D_t^{\alpha} u^{r,+}=\sum_{n=1}^\infty [-\lambda_n a(t) u_n^{r,+}(t;a)+F_n^+(t)]\phi_n(x).$
Then for each $t\in[0,T],$
\begin{equation*}
\begin{split}
\|^C\!D_t^{\alpha} u^{r,+} \|^2_{L^2(\Omega)}&\le
C\sum_{n=1}^\infty Q_a^2|\lambda_n u_n^{r,+}(t;a)|^2+C\sum_{n=1}^\infty |F_n(t)|^2 \\
&\le C\sum_{n=1}^\infty |\lambda_n u_n^{r,+}(t;a)|^2+C\|F(\cdot,t)\|_{L^2(\Omega)}^2.
\end{split}
\end{equation*}
From the above proof for $\|u^{r,+}\|^2_{H^2(\Omega)},$ it holds
$$\|^C\!D_t^{\alpha} u^{r,+} \|^2_{L^2(\Omega)}
\le C \|F\|^2_{C^\theta([0,T];L^2(\Omega))}+C\|F(\cdot,t)\|_{L^2(\Omega)}^2,\ t\in[0,T],$$
which gives
$$\|^C\!D_t^{\alpha} u^{r,+}\|_{C([0,T];L^2(\Omega))}
\le C \|F\|_{C^\theta([0,T];L^2(\Omega))}.$$
Analogously, we can show
$\|^C\!D_t^{\alpha} u^{r,-}\|_{C([0,T];L^2(\Omega))}
\le C \|F\|_{C^\theta([0,T];L^2(\Omega))}.$
\par The estimates of $u^{r,+},\ u^{r,-},\ ^C\!D_t^{\alpha} u^{r,+}$ and $^C\!D_t^{\alpha} u^{r,-}$
yield the desired result and complete this proof.
\end{proof}
\subsubsection{Regularity of $u^i$}
\par In this part we consider the regularity of $u^i.$ Just as in
the regularity results for $u^r,$ we first state two lemmas which
concern the positivity and monotonicity of $u^i,$ respectively.
\begin{lemma}\label{sign_u^i}
With the representation \eqref{u^r u^i} and the fractional ODE
\eqref{ODE_initial}, for each $\N+,$ $b_n \le (\ge)0$ implies that
$u^i_n(t;a)\le (\ge)0$ on $[0,T]$.
\end{lemma}
\begin{proof}
This is a directly result of Corollary \ref{sign_eigenfunction}.
\end{proof}
\begin{lemma}\label{monotone_initial}
Given $a_1, a_2\in C^{+}[0,T]$ with $a_1\le a_2$ on $[0,T]$,
for each $\N+,$ we have
\begin{equation*}
\begin{cases}
0\le u_n^i(t;a_2)\le u_n^i(t;a_1),\ \text{if}\ b_n\ge 0;\\
u_n^i(t;a_1)\le u_n^i(t;a_2) \le 0,\ \text{if}\ b_n\le 0.
\end{cases}
\end{equation*}
\end{lemma}
\begin{proof}
\par Fix $\N+,$ from the fractional ODE \eqref{ODE_initial},
the functions $u^i_n(t;a_1)$ and $u^i_n(t;a_2)$ satisfy the following system
\begin{equation*}
\begin{cases}
^C\!D_t^{\alpha} u_n^i(t;a_1)+\lambda_na_1(t)u_n^i(t;a_1)=0;\\
^C\!D_t^{\alpha} u_n^i(t;a_2)+\lambda_na_2(t)u_n^i(t;a_2)=0;\\
u_n^i(0;a_1)=u_n^i(0;a_2)=b_n.
\end{cases}
\end{equation*}
This gives
\begin{equation}\label{equality_2}
^C\!D_t^{\alpha} w+\lambda_na_1(t)w(t)=\lambda_nu_n^i(t;a_2)(a_2(t)-a_1(t)),
\ w(0)=0,
\end{equation}
where $w(t)=u_n^i(t;a_1)-u_n^i(t;a_2)$.
\par If $b_n\ge 0,$ Corollary \ref{sign_eigenfunction}
shows that $u_n^i(t;a_1),u_n^i(t;a_2)\ge 0.$
Also, Lemma \ref{sign_u^i} and
$a_1\le a_2$ ensures the right side of \eqref{equality_2} is nonnegative,
which together with Corollary \ref{sign_eigenfunction} implies $w\ge 0$,
i.e. $0\le u_n^i(t;a_2)\le u_n^i(t;a_1).$
The similar argument yields $u_n^i(t;a_1)\le u_n^i(t;a_2) \le 0$
for the case $b_n\le 0.$
\end{proof}
\begin{lemma}[Regularity for $u^i$]\label{regularity_u^i}
$$\|u^i\|_{L^2(0,T;H^2(\Omega))}+\|^C\!D_t^{\alpha} u^i\|_{L^2([0,T]\times\Omega)}
\le C T^{\frac{1-\alpha}{2}}\|u_0\|_{H^1(\Omega)}.$$
\end{lemma}
\begin{proof}
\par Given $t\in [0,T],$ the direct calculation and Lemma \ref{monotone_initial}
yield that
\begin{equation*}
\begin{split}
\|u^i(x,t;a)\|_{H^2(\Omega)}^2
&\le C\|-\mathcal{L} u^i(x,t;a)\|_{L^2(\Omega)}^2
=C\|\sum_{n=1}^\infty \lambda_nu^i_n(t;a)\phi_n(x) \|_{L^2(\Omega)}^2\\
&=C\sum_{n=1}^\infty|\lambda_nu^i_n(t;a)|^2
\le C\sum_{n=1}^\infty|\lambda_nu^i_n(t;q_a)|^2.
\end{split}
\end{equation*}
Recall that \cite{sakamoto2011initial} established the representation as
$u^i_n(t;q_a)=b_n E_{\alpha,1}(-\lambda_nq_at^\alpha),\ \N+.$
Hence, by Lemma \ref{mittag_bound},
\begin{equation}\label{inequality_11}
\begin{split}
\|u^i(x,t;a)\|_{H^2(\Omega)}^2
&\le C\|-\mathcal{L} u^i(x,t;a)\|_{L^2(\Omega)}^2
\le C\sum_{n=1}^\infty|\lambda_nb_n
E_{\alpha,1}(-\lambda_nq_at^\alpha)|^2\\
&\le C\sum_{n=1}^\infty |\frac{1}{1+\lambda_nq_at^\alpha}|^2\lambda_n^2b_n^2
= C\sum_{n=1}^\infty|\frac{(\lambda_nq_at^\alpha)^{\frac{1}{2}}}
{1+\lambda_nq_at^\alpha}|^2t^{-\alpha}q_a^{-1}\lambda_nb_n^2\\
&\le Ct^{-\alpha}\sum_{n=1}^\infty((-\mathcal{L})^{\frac{1}{2}}u_0,\phi_n)^2
\le Ct^{-\alpha}\|u_0\|_{H^1(\Omega)}^2,
\end{split}
\end{equation}
which leads to
$ \|u^i\|_{L^2(0,T;H^2(\Omega))}^2
\le C\int_0^T t^{-\alpha}\|u_0\|_{H^1(\Omega)}^2 {\rm d}t
=CT^{1-\alpha}\|u_0\|_{H^1(\Omega)}^2,$
i.e.
\begin{equation}\label{inequality_12}
\|u^i\|_{L^2(0,T;H^2(\Omega))}
\le C T^{\frac{1-\alpha}{2}}\|u_0\|_{H^1(\Omega)}.
\end{equation}
\par For the estimate of $^C\!D_t^{\alpha} u^i(x,t;a),$
\eqref{u^r u^i} and \eqref{ODE_initial} yield
$$^C\!D_t^{\alpha} u^i(x,t;a)=\sum_{n=1}^{\infty} {^C\!D_t^{\alpha}} u^i_n(t;a)\phi_n(x)
=-\sum_{n=1}^\infty \lambda_n a(t) u^i_n(t;a)\phi_n(x),$$
which together with \eqref{q_a Q_a} gives
\begin{equation*}
\begin{split}
\|^C\!D_t^{\alpha} u^i(x,t;a)\|_{L^2(\Omega)}^2
&\le Q_a^2 \sum_{n=1}^\infty |\lambda_n u^i_n(t;a)|^2\\
&=Q_a^2 \|-\mathcal{L} u^i(x,t;a)\|_{L^2(\Omega)}^2
\le Ct^{-\alpha}\|u_0\|_{H^1(\Omega)}^2,\ t\in[0,T],
\end{split}
\end{equation*}
where the last inequality follows from \eqref{inequality_11}.
This result implies that
\begin{equation*}
\|^C\!D_t^{\alpha} u^i(x,t;a)\|_{L^2([0,T]\times\Omega)}^2
=\int_0^T \|^C\!D_t^{\alpha} u^i(x,t;a)\|_{L^2(\Omega)}^2 {\rm d}t
\le CT^{1-\alpha}\|u_0\|_{H^1(\Omega)}^2,
\end{equation*}
i.e. $\|^C\!D_t^{\alpha} u^i\|_{L^2([0,T]\times\Omega)}
\le C T^{\frac{1-\alpha}{2}}\|u_0\|_{H^1(\Omega)},$ which together
with \eqref{inequality_12} completes the proof.
\end{proof}
\par Moreover, with a stronger condition on $u_0,$ such as assuming
$u_0\in H^2(\Omega)\cap H^1_0(\Omega),$
we can deduce the $C$-regularity estimate of $u^i.$
\begin{corollary}\label{regularity_u^i_H^2}
With Assumption \ref{assumption_direct} and
$u_0\in H^2(\Omega)\cap H^1_0(\Omega),$ then
\begin{equation*}
\|u^i\|_{C([0,T];H^2(\Omega))}+\|^C\!D_t^{\alpha} u^i\|_{C([0,T];L^2(\Omega))}
\le C\|u_0\|_{H^2(\Omega)}.
\end{equation*}
\end{corollary}
\begin{proof}
\par Lemma \ref{mittag_bound} yields that
\begin{equation*}\label{inequality_15}
\sum_{n=1}^\infty|\lambda_nb_n
E_{\alpha,1}(-\lambda_nq_at^\alpha)|^2
\le C\sum_{n=1}^\infty|\lambda_nb_n|^2
=C\|-\mathcal{L} u_0\|_{L^2(\Omega)}^2 \le C \|u_0\|_{H^2(\Omega)}^2,\ t\in[0,T];
\end{equation*}
meanwhile, the following estimates have been shown in
the proof of Theorem \ref{regularity_u^i}
\begin{equation*}\label{inequality_14}
\begin{cases}
\|u^i(x,t;a)\|_{H^2(\Omega)}^2
\le C\|-\mathcal{L} u^i(x,t;a)\|_{L^2(\Omega)}^2
\le C\sum_{n=1}^\infty|\lambda_nb_n
E_{\alpha,1}(-\lambda_nq_at^\alpha)|^2,\\
\|^C\!D_t^{\alpha} u^i(x,t;a)\|_{L^2(\Omega)}^2
\le Q_a^2 \sum_{n=1}^\infty |\lambda_n u^i_n(t;a)|^2
=C \|-\mathcal{L} u^i(x,t;a)\|_{L^2(\Omega)}^2.
\end{cases}
\end{equation*}
Hence, it holds that
\begin{equation*}
\|u^i(x,t;a)\|_{H^2(\Omega)}+\|^C\!D_t^{\alpha} u^i(x,t;a)\|_{L^2(\Omega)}
\le C\|u_0\|_{H^2(\Omega)},\ t\in[0,T],
\end{equation*}
which leads to the claimed result.
\end{proof}
\subsection{Main theorem for the direct problem}
\par The main theorem for the direct problem follows from Theorem
\ref{existenceuniqueness}, Lemmas \ref{regularity_u^r} and \ref{regularity_u^i},
Corollaries \ref{Cregularity_u^r} and \ref{regularity_u^i_H^2}, and the relation
$u(x,t;a)=u^r(x,t;a)+u^i(x,t;a).$
\begin{theorem}[Main theorem for the direct problem]\label{main_direct}
\par Let Assumption \ref{assumption_direct} be valid, then under Definition
\ref{weak solution}, there exists a unique weak solution $u(x,t;a)$
of FDE \eqref{fde} with the spectral representation
\eqref{solution} and the following regularity estimates:
$$\|u\|_{L^2(0,T;H^2(\Omega))}+\|^C\!D_t^{\alpha} u\|_{L^2([0,T]\times\Omega)}
\le C(\|F\|_{L^2([0,T]\times\Omega)}
+T^{\frac{1-\alpha}{2}}\|u_0\|_{H^1(\Omega)}).$$
Moreover, if the conditions $u_0\in H^2(\Omega)\cap H^1_0(\Omega)$ and $F\in C^\theta([0,T];L^2(\Omega)),\ 0<\theta<1$ are added, we have:
\begin{equation*}
\|u\|_{C([0,T];H^2(\Omega))}+\|^C\!D_t^{\alpha} u\|_{C([0,T];L^2(\Omega))}
\le C(\|F\|_{C^\theta([0,T];L^2(\Omega))}+\|u_0\|_{H^2(\Omega)}).
\end{equation*}
\end{theorem}
\section{Inverse Problem--Reconstruction of the diffusion coefficient $a(t)$}
\par In this section, we discuss how to recover the coefficient $a(t)$
through the output flux data
$$a(t)\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)=g(t),\ x_0\in \partial\Omega.$$
All cross the inverse problem work, the operator $-\mathcal{L}$ is assumed to satisfy the condition \eqref{sobolev assumption}, then the expression $\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)$ makes sense.
We only consider this reconstruction in the space $C^+[0,T],$ which
can be regarded as the admissible set for $a(t).$
To this end, we introduce an operator $K,$ which will be shown to have
a fixed point consisting of the desired coefficient $a(t).$
\subsection{Operator $K$}
\par The operator $K$ is defined as
$$K \psi(t):=\frac{g(t)}{\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;\psi)}
=\frac{g(t)}{\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)},
\ t\in [0,T]$$
with domain
$$\mathcal{D}(K):=\{\psi\in C^{+}[0,T]:\psi(t)\ge g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1},\ t\in[0,T] \}.$$
\par To analyze $K$, we make the following assumptions.
\begin{assumption}\label{assumption_inverse}
$u_0,$ $F$ and $g$ should satisfy the following restrictions:
\begin{itemize}
\item [(a)] $u_0\in H^3(\Omega)\cap H_0^1(\Omega)$ with
$b_n:=(u_0,\phi_n)\ge 0,\ \N+;$
\item [(b)] $\exists \theta\in (0,1)$ s.t. $F(x,t)\in C^\theta([0,T];H^3(\Omega)\cap H_0^1(\Omega))$ with
$F_n(t):=(F(\cdot,t),\phi_n)\ge 0$ on $[0,T]$ for each $\N+;$
\item [(c)] $\exists N\in \mathbb{N}^{+}$ s.t.
$\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0,$ $b_N>0$ and $F_N(t)>0$ on $[0,T];$
\item [(d)] $g\in C^+[0,T].$
\end{itemize}
\end{assumption}
\par The next remark shows that the equality in the definition of $K$ is valid.
\begin{remark}\label{partial derivative of u}
Given $\psi\in C^+[0,T]$ and for each $t\in[0,T],$ by the proofs of Corollaries \ref{Cregularity_u^r} and \ref{regularity_u^i_H^2}, we have
\begin{equation*}
\begin{split}
\|u^{r,+}(x,t;\psi)\|^2_{H^3(\Omega)}
&\le C \|(-\mathcal{L})^{3/2}u^{r,+}\|^2_{L^2(\Omega)}
\le C \sum_{n=1}^\infty
|\lambda_n^{3/2}u^{r,+}_n(t;\psi)|^2\\
&\le C \sum_{n=1}^\infty
\left|\lambda_n^{3/2} \int_0^t F^+_n(\tau) (t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_\psi(t-\tau)^\alpha){\rm d}\tau \right|^2\\
&\le C \sum_{n=1}^\infty
\left|\lambda_n \int_0^t \lambda_n^{1/2}|F^+_n(\tau) -F^+_n(t)|(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_\psi(t-\tau)^\alpha){\rm d}\tau\right|^2\\
&\quad +C \sum_{n=1}^\infty
\left|\lambda_n^{1/2}F_n^+(t)(1-E_{\alpha,1}(-\lambda_n q_\psi t^\alpha))\right|^2\\
&\le C \sum_{n=1}^\infty
\left|\lambda_n \int_0^t \lambda_n^{1/2}|F_n(\tau) -F_n(t)|(t-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n q_\psi(t-\tau)^\alpha){\rm d}\tau\right|^2\\
&\quad +C \sum_{n=1}^\infty
\left|\lambda_n^{1/2}F_n(t)(1-E_{\alpha,1}(-\lambda_n q_\psi t^\alpha))\right|^2\\
&\le C \|(-\mathcal{L})^{1/2} F\|_{C^\theta([0,T];L^2(\Omega))}^2+C\|(-\mathcal{L})^{1/2} F(\cdot,t)\|_{L^2(\Omega)}^2\\
&\le C \| F\|_{C^\theta([0,T];H^1(\Omega))}^2+C\|F(\cdot,t)\|_{H^1(\Omega)}^2
\end{split}
\end{equation*}
and
$$\|u^{r,-}(x,t;\psi)\|^2_{H^3(\Omega)}
\le C \| F\|_{C^\theta([0,T];H^1(\Omega))}^2+C\|F(\cdot,t)\|_{H^1(\Omega)}^2,$$
which give
$\|u^r\|_{C([0,T];H^3(\Omega))}
\le C\| F\|_{C^\theta([0,T];H^1(\Omega))}$;
\begin{equation*}
\begin{split}
\|u^i(x,t;\psi)\|^2_{H^3(\Omega)}
&\le C \|(-\mathcal{L})^{3/2}u^i\|^2_{L^2(\Omega)}
\le C \|\sum_{n=1}^\infty\lambda_n^{3/2}u_n^i(t;\psi)\phi_n(x)\|^2_{L^2(\Omega)}\\
&\le C\sum_{n=1}^\infty|\lambda_n^{3/2}b_n
E_{\alpha,1}(-\lambda_nq_\psi t^\alpha)|^2
\le C\sum_{n=1}^\infty|\lambda_n^{3/2}b_n|^2\\
&=C\|(-\mathcal{L})^{3/2} u_0\|_{L^2(\Omega)}^2 \le C \|u_0\|_{H^3(\Omega)}^2,
\end{split}
\end{equation*}
which gives $\|u^i\|_{C([0,T];H^3(\Omega))}\le C \|u_0\|_{H^3(\Omega)}.$
Combining the above two results yields that
$$\|u\|_{C([0,T];H^3(\Omega))}\le C (\| F\|_{C^\theta([0,T];H^1(\Omega))}+\|u_0\|_{H^3(\Omega)})<\infty,$$
which means for each $t\in [0,T],$
$\|u\|_{H^3(\Omega)}<\infty.$ Recall that $\Omega\subset R^n, n=1,2,3$, then the Sobolev Embedding Theorem gives
$$u(x,t;\psi)=\sum_{n=1}^\infty u_n(t;\psi)\phi_n(x)\in C^1(\overline{\Omega})\ \text{for each}\ t\in [0,T].$$
Hence,
$\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)$ is well-defined and
$$\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;\psi)=\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0),\quad t\in[0,T].$$
\end{remark}
\par The following two remarks will explain the reasonableness and reason
for Assumption \ref{assumption_inverse}.
\begin{remark}\label{resonable_assumption_inverse}
\par For the inverse problem, the right-hand side function $F(x,t)$
and the initial condition $u_0(x)$ are input data, which,
at least in some circumstance, can be assumed to be controlled.
Even though Assumption \ref{assumption_inverse}
$(a),$ $(b)$ and $(c)$ appear restrictive, it is not hard to construct
functions that satisfy them. For example, in $(a)$ if $u_0=c\phi_k$ for
some $c>0,$ then Assumption \ref{assumption_inverse} $(a)$ will be
satisfied. This will also be true if $u_0=\sum_{k=1}^{M} c_k\phi_k$
with all $c_k>0.$ Similarly, $(b)$ is satisfied if $F(x,t)$ is also a
linear combination of $\{\phi_n:\N+\}$ with positive coefficients.
For $(c),$ by the completeness of $\{\phi_n:\N+\}$ in $L^2(\Omega),$ there should exist
$N\in \mathbb{N}^{+}$ s.t. $\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0.$ Otherwise, for each
$\psi\in H^3(\Omega)\subset L^2(\Omega),$ $\frac{\partial \psi}{\partial {\overrightarrow{\bf n}}}(x_0)=0$ and obviously
it is incorrect. Then for this $N,$ we only need to set the
coefficients of $u_0$ and $F$ upon $\phi_N$ be strictly positive.
\par The output flux data $g(t),$ it is not under our control.
However, if there exists $a\in C^+[0,T]$ s.t. $a(t)\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)=g(t),$
Assumption \ref{assumption_inverse} $(a),$ $(b)$ and Corollary
\ref{sign_eigenfunction} yield that $u_n(t;a)\ge 0;$
\eqref{sign_eigenfunction_derivative} gives
$\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)\ge 0, \N+;$
Assumption \ref{assumption_inverse} $(c)$ ensures
$\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0$ and
$u_N(t;a)>0$ on $[0,T],$ where the proof can be seen in
Lemma \ref{well_definedness}.
Consequently, $$\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)
=\sum\limits_{n=1}^\infty u_n(t;a)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\ge u_N(t;a)\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0,\ t\in[0,T].$$
This together with $a\in C^+[0,T]$ gives that $g>0.$
The continuity of $g$ follows from the ones of $a$ and $u_n(t;a),\ \N+,$
which are derived from the admissible set $C^+[0,T]$ and
Theorem \ref{existenceuniqueness}, respectively.
Therefore, Assumption \ref{assumption_inverse} $(d)$ is
reasonable and can be attained.
\end{remark}
\begin{remark}\label{remark_assumption_inverse}
The well-definedness of the domain $\mathcal{D}(K)$ is guaranteed by
Assumption \ref{assumption_inverse} $(a)$, $(b)$, $(c)$ and $(d)$ in the
sense that the $H^3$-regularity of $u_0,$ $F$ and the Sobolev Embedding Theorem support that $\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)$ and $\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)$ are well defined, and
the dominator of the lower bound of $\mathcal{D}(K)$
$$
\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)+I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]
=\sum_{n=1}^{\infty} (b_n+I_t^\alpha F_n) \frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\ge (b_N+I_t^\alpha F_N) \frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0
$$
on $[0,T].$ Recall that the numerator $g>0,$ so that the lower bound
$g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1}>0,$ which
gives that $\mathcal{D}(K)$ is a subspace of $C^+[0,T].$
Also, $F(x,t)\in C^\theta([0,T];H^3(\Omega)\cap H_0^1(\Omega))$ yields that
$F_N(t)$ is continuous on $[0,T],$ so is
$(b_N+I_t^\alpha F_N) \frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0).$ Then $\exists C>0$
s.t. $(b_N+I_t^\alpha F_N) \frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>C>0,$ which leads to
the dominator $$\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]>C>0\ \text{on}\ [0,T].$$
The strict positivity of the dominator avoids $\mathcal{D}(K)$ degenerating
to an empty set.
\par In order to show the well-definedness of $K,$
Assumption \ref{assumption_inverse} $(a),$ $(b)$ and $(c)$ will
be used. Furthermore, Assumption \ref{assumption_inverse}
$(a)$ and $(b)$ are crucial to build the monotonicity of operator $K$;
meanwhile, Assumption \ref{assumption_inverse} $(c)$ is stated for
the uniqueness of fixed points of $K$.
\end{remark}
\par For the operator $K$, we have the following lemmas.
\begin{lemma}\label{well_definedness}
The operator $K$ is well-defined.
\end{lemma}
\begin{proof}
\par For each $\psi\in \mathcal{D}(K),$ Theorem \ref{existenceuniqueness} ensures that
there exists a unique $u_n(t;\psi)$ for $\N+,$
which implies the existence and uniqueness of $K\psi.$
\par Then it is suffice to show the dominator $\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)>0$ on $[0,T].$
With \eqref{ODE}, Lemma \ref{sign_eigenfunction} and Assumption
\ref{assumption_inverse} $(a)$ and $(b),$ we have
$u_n(t;\psi)\ge 0$ on $[0,T],$ which together with
$\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)\ge0$ gives
$\sum\limits_{n=1}^\infty u_n(t;\psi)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\ge u_N(t;\psi)\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0).$
Due to the assumption $\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0,$
we claim that $u_N(t;\psi)>0.$
Assume not, i.e. $\exists t_0\in [0,T]$ s.t. $u_N(t_0;\psi)\le 0.$
The result $u_N(t;\psi)\ge 0$ yields that $u_N(t_0;\psi)=0$ so that
$u_N(t;\psi)$ attains its minimum at $t=t_0.$
$u_N(0;\psi)=b_N>0$ implies $t_0\ne 0,$ i.e. $t_0\in(0,T].$
Then Lemma \ref{extreme}, $u_N(t_0;\psi)=0$ and the ODE \eqref{ODE}
show that $^C\!D_{t}^{\alpha}u_N(t_0;\psi)=F_N(t_0)\le 0,$
which contradicts with Assumption \ref{assumption_inverse} $(c)$
and confirms the claim. Hence,
$$\sum\limits_{n=1}^\infty u_n(t;\psi)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\ge u_N(t;\psi)\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0,$$
which completes the proof.
\end{proof}
\begin{lemma}\label{itself}
$K$ maps $\mathcal{D}(K)$ into $\mathcal{D}(K).$
\end{lemma}
\begin{proof}
\par Given $\psi\in \mathcal{D}(K).$ The continuity of $K\psi$ follows from the continuity
of $u_n(t;\psi)$ for each $\N+$ and the continuity of $g$, which are
established by Theorem \ref{existenceuniqueness} and Assumption
\ref{assumption_inverse} $(d)$ respectively.
\par For each $\N+,$ \eqref{ODE} ensures $u_n(t;\psi)$ satisfies
\begin{equation*}
^C\!D_t^{\alpha} u_n(t;\psi)+\lambda_n \psi(t) u_n(t;\psi)=F_n(t),
\ u_n(0;\psi)=b_n.
\end{equation*}
Taking $I_t^\alpha$ on both sides of the above ODE and using
Lemma \ref{I_alpha} yield that
\begin{equation*}
u_n(t;\psi)+\lambda_n I_t^\alpha[\psi(t) u_n(t;\psi)]=I_t^\alpha F_n+b_n.
\end{equation*}
From the proof of Lemma \ref{well_definedness}, we have
$u_n(t;\psi)\ge 0$ on $[0,T],$ which together with
$\lambda_n>0,$ the positivity of $\psi$ and the definition of
$I_t^\alpha$ yields that $\lambda_n I_t^\alpha[\psi(t) u_n(t;\psi)]\ge 0.$
Since $u_n(t;\psi)\ge 0$ and
$\lambda_n I_t^\alpha[\psi(t) u_n(t;\psi)]\ge 0,$
we deduce that $0\le u_n(t;\psi)\le I_t^\alpha F_n+b_n$ on $[0,T].$
Hence, with $\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)\ge 0$ and
the smoothness assumptions $u_0\in H^3(\Omega)\cap H_0^1(\Omega),\
F\in C^\theta([0,T];H^3(\Omega)\cap H_0^1(\Omega))$
stated in Assumption \ref{assumption_inverse} $(a)$ and $(b)$ respectively,
the following inequality holds
\begin{equation*}\label{inequality_16}
\sum_{n=1}^\infty u_n(t;\psi)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\le \sum_{n=1}^\infty (I_t^\alpha F_n+b_n) \frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
=\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)+I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)],
\end{equation*}
which together with $g>0$ yields that
$$K\psi(t) = \frac{g(t)}{\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)}\ge
g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1}>0,\ t\in[0,T],
$$
where the last inequality follows from Remark \ref{remark_assumption_inverse}.
The above result and the continuity of $K\psi$ lead to
$K\psi \in \mathcal{D}(K),$ which is the expected result.
\end{proof}
\subsection{Monotonicity}
\par In this part, we show the monotonicity of the operator $K$.
\begin{theorem}[Monotonicity]\label{monotonicity}
Given $a_1,a_2\in \mathcal{D}(K)$ with $a_1\le a_2,$ then
$Ka_1\le Ka_2$ on $[0,T].$
\end{theorem}
\begin{proof}
\par Pick $\N+,$ due to \eqref{ODE}, $u_n(t;a_1)$ and $u_n(t;a_2)$
satisfy
\begin{equation*}
\begin{cases}
^C\!D_t^{\alpha} u_n(t;a_1)+\lambda_n a_1(t) u_n(t;a_1)=F_n(t),
\ u_n(0;a_1)=b_n;\\
^C\!D_t^{\alpha} u_n(t;a_2)+\lambda_n a_2(t) u_n(t;a_2)=F_n(t),
\ u_n(0;a_2)=b_n,
\end{cases}
\end{equation*}
which together with $a_1\le a_2$ and Lemma \ref{sign} yields
\begin{equation}\label{w}
^C\!D_t^{\alpha} w+\lambda_na_1(t)w(t)=\lambda_nu_n(t;a_2)(a_2(t)-a_1(t))\ge 0,
\ w(0)=0,
\end{equation}
where $w(t)=u_n(t;a_1)-u_n(t;a_2).$
Applying Lemma \ref{sign} to the above ODE yields that
$w\ge 0,$ i.e. $u_n(t;a_1)\ge u_n(t;a_2)\ge0,$ which together with
assumption \eqref{sign_eigenfunction_derivative} leads to
$$\sum\limits_{n=1}^\infty u_n(t;a_1)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
\ge\sum\limits_{n=1}^\infty u_n(t;a_2)\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
> 0,\ t\in[0,T].$$
Therefore, with the condition $g>0$ stated in
Assumption \ref{assumption_inverse} $(d)$,
$$Ka_1(t)=\frac{g(t)}{\sum\limits_{n=1}^\infty u_n(t;a_1)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)}
\le \frac{g(t)}{\sum\limits_{n=1}^\infty u_n(t;a_2)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)}=Ka_2(t),\ t\in[0,T],$$
which completes this proof.
\end{proof}
\subsection{Uniqueness}
\par In order to show the uniqueness, we state two lemmas.
\begin{lemma}\label{uniqueness_monotone}
If $a_1,a_2\in \mathcal{D}(K)$ are both fixed points of $K$ with $a_1\le a_2,$
then $a_1\equiv a_2$.
\end{lemma}
\begin{proof}
\par Pick a fixed point $a(t),$ then
$$a(t)\sum\limits_{n=1}^\infty u_n(t;a)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
=\sum\limits_{n=1}^\infty a(t)u_n(t;a)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)=g(t),$$
which gives
\begin{equation}\label{equality_3}
\sum\limits_{n=1}^\infty I_t^\alpha[a(t)u_n(t;a)]
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)=I_t^\alpha g
\end{equation}
by taking $I_t^\alpha$ on both sides.
Similarly, taking $I_t^\alpha$ on the both sides of \eqref{ODE}
and applying Lemma \ref{I_alpha} yield that
\begin{equation*}\label{equality_4}
I_t^\alpha[a(t)u_n(t;a)]=\lambda_n^{-1}I_t^\alpha F_n+\lambda_n^{-1}b_n
-\lambda_n^{-1}u_n(t;a),\ \N+,
\end{equation*}
which together with \eqref{equality_3} generates
\begin{equation}\label{equality_5}
\sum\limits_{n=1}^\infty \lambda_n^{-1}u_n(t;a)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
=\sum\limits_{n=1}^\infty \lambda_n^{-1}(I_t^\alpha F_n+b_n)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)-I_t^\alpha g.
\end{equation}
In \eqref{equality_5}, the convergence of the two series in $C[0,T]$ is supported by Assumption \ref{assumption_inverse}, Remark \ref{partial derivative of u} and the fact that $0<\lambda_1\le\lambda_2\le\cdots.$
\par Given two fixed points $a_1,a_2$ with $a_1\le a_2,$ then $a_1$ and $a_2$
should satisfy \eqref{equality_5} simultaneously, which gives
\begin{equation}\label{equality_6}
\sum\limits_{n=1}^\infty \lambda_n^{-1}
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)(u_n(t;a_1)-u_n(t;a_2))=0.
\end{equation}
In the proof of Theorem \ref{monotonicity}, we have shown that
$u_n(t;a_1)\ge u_n(t;a_2)\ge0$. Also recall that
$\lambda_n^{-1} \frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)\ge 0,\ \N+,$
then $\lambda_n^{-1}\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
(u_n(t;a_1)-u_n(t;a_2))\ge 0$ on $[0,T]$ for $\N+.$
Hence, \eqref{equality_6} implies that
$$\lambda_n^{-1}\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)
(u_n(t;a_1)-u_n(t;a_2))= 0,\ t\in[0,T],\ \N+.$$
Let $n=N,$ $\lambda_N^{-1}\frac{\partial \phi_N}{\partial {\overrightarrow{\bf n}}}(x_0)>0$
gives $u_N(t;a_1)\equiv u_N(t;a_2)$ on $[0,T].$
Set $w(t)=u_N(t;a_1)-u_N(t;a_2)=0.$ Then \eqref{w} yields that
$$
0={^C\!D_t^{\alpha}} w+\lambda_Na_1(t)w(t)=\lambda_Nu_N(t;a_2)(a_2(t)-a_1(t)),
$$
i.e. $u_N(t;a_2)(a_2(t)-a_1(t))\equiv 0$ on $[0,T]$; while the proof of Lemma \ref{well_definedness} yields that $u_N(t;a_2)>0.$ Hence, we have
$a_1= a_2$ on $[0,T],$ which completes the proof.
\end{proof}
\par Before showing uniqueness, we introduce a successive iteration procedure
which will generate a sequence converging to a fixed point if it exists. Set
$$\overline{a}_0(t)=g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1},\
\overline{a}_{n+1}=K\overline{a}_n,\ n\in\mathbb{N}.$$ Then this iteration reproduces
a sequence $\{\overline{a}_n:n\in\mathbb{N}\}$ which is contained by
$\mathcal{D}(K)$ due to Lemma \ref{itself}.
\begin{lemma}\label{uniqueness_series}
If there exists a fixed point $a(t)\in \mathcal{D}(K)$ of operator $K,$ then the
sequence $\{\overline{a}_n:n\in\mathbb{N}\}$ will converge to $a(t).$
\end{lemma}
\begin{proof}
\par $\overline{a}_0$ is the lower bound of $\mathcal{D}(K)$ and
$\{\overline{a}_n:n\in\mathbb{N}\}\subset\mathcal{D}(K)$ yield that
$\overline{a}_0\le \overline{a}_1.$ Using Theorem \ref{monotonicity},
we have $\overline{a}_1=K\overline{a}_0\le K\overline{a}_1=\overline{a}_2,$
i.e. $\overline{a}_1\le \overline{a}_2.$ The same argument
gives $\overline{a}_2=K\overline{a}_1\le K\overline{a}_2=
\overline{a}_3.$ Continue this process, we can deduce
$\overline{a}_0\le\overline{a}_1\le\overline{a}_2\le \dots,$
which means $\{\overline{a}_n:n\in\mathbb{N}\}$ is increasing.
Since the results that $\overline{a}_0$ is the lower bound of $\mathcal{D}(K)$
and $a(t)\in \mathcal{D}(K),$ it holds $\overline{a}_0\le a.$
Applying Theorem \ref{monotonicity}
to this inequality, we obtain $\overline{a}_1=K\overline{a}_0\le Ka=a,$
i.e. $\overline{a}_1\le a.$
This argument generates
$\overline{a}_n\le a,\ n\in\mathbb{N},$ which means $a(t)$ is an upper bound
of $\{\overline{a}_n:n\in\mathbb{N}\}.$
\par We have proved $\{\overline{a}_n:n\in\mathbb{N}\}$ is an increasing sequence
in $\mathcal{D}(K)$ with an upper bound $a(t),$ which leads to $\{\overline{a}_n:n\in\mathbb{N}\}$
is convergent in $\mathcal{D}(K)$ and the limit is smaller than $a(t).$
Denote the limit of $\{\overline{a}_n:n\in\mathbb{N}\}$ by $\overline{a}$.
We have $\overline{a}\in \mathcal{D}(K),$ $\overline{a}\le a$ and
$\overline{a}$ is a fixed point of $K$ in $\mathcal{D}(K).$
Hence, Lemma \ref{uniqueness_monotone} yields
$\overline{a}=a,$ which is the desired result.
\end{proof}
\par Now, we are able to prove the uniqueness of fixed points of $K$.
\begin{theorem}[Uniqueness]\label{uniqueness}
There is at most one fixed point of $K$ in $\mathcal{D}(K).$
\end{theorem}
\begin{proof}
\par Let $a_1, a_2\in \mathcal{D}(K)$ be both fixed points of $K.$
Lemma \ref{uniqueness_series} implies that
$\overline{a}_n\to a_1$ and $\overline{a}_n\to a_2,$ which
leads to $a_1=a_2$ and completes this proof.
\end{proof}
\subsection{Existence}
\par Assumption \ref{assumption_inverse} is not sufficient to deduce the
existence of the fixed points of $K$ since $\mathcal{D}(K)$ has
no upper bound so that an increasing sequence in $\mathcal{D}(K)$ may not be convergent.
In this part, we discuss the existence of fixed points, by providing
some extra conditions.
\begin{assumption}\label{assumption_inverse_existence}
Additional assumptions on $u_0$, $F$ and $g$:
\begin{itemize}
\item [(a)] $-\mathcal{L} u_0\in H^3(\Omega)\cap H_0^1(\Omega);$
\item [(b)] $F(x,t)=-\mathcal{L} u_0(x)\cdot f(t)$ s.t. $f\in C^\theta[0,T], 0<\theta<1$ and $f(t)\ge g(t)
\big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\big]^{-1}$ on $[0,T].$
\end{itemize}
\end{assumption}
\begin{remark}
Assumption \ref{assumption_inverse_existence} is set up to make sure that
$F(x,t)=-\mathcal{L} u_0(x)\cdot f(t)\in C^\theta([0,T];H^3(\Omega)\cap H_0^1(\Omega))$, so that
$F(x,t)$ also satisfies Assumption \ref{assumption_inverse}.
Fix $u_0$ and $f,$ if the measured data $g$ does not satisfy
Assumption \ref{assumption_inverse_existence} $(b),$ then we can modify
$u_0$ by increasing the value of $u_0$ in a very small neighborhood
of the point $x_0$ so that the value of $\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)$
becomes larger. Meanwhile, since $u_0$ is changed in a small domain,
the coefficients $\{b_n:\N+\}$ only vary slightly, so do $u_n(t;a)$ and $u(x,t;a).$
Hence, $\frac{\partial u}{\partial {\overrightarrow{\bf n}}}(x_0,t;a)$ and $g(t)$ will not appear
a significant change that can violate Assumption
\ref{assumption_inverse_existence} $(b).$
\end{remark}
\par Define the subspace $\mathcal{D}(K)'$ of $\mathcal{D}(K)$ as
\begin{equation*}
\begin{split}
\mathcal{D}(K)':=\Big\{\psi\in C^{+}[0,T]:\ & g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1}\\
&\le \psi(t)\le g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\Big]^{-1},
\ t\in[0,T]\Big\}.
\end{split}
\end{equation*}
We have proved the lower bound of $\mathcal{D}(K)'$ is positive in Remark
\ref{remark_assumption_inverse} and clearly the upper bound
of $\mathcal{D}(K)'$ is larger than the lower bound. Consequently,
$\mathcal{D}(K)'$ is well-defined.
\par The next lemma concerns the range of $K$ with domain $\mathcal{D}(K)'$.
\begin{lemma}\label{itself_existence}
With Assumptions \ref{assumption_inverse} and
\ref{assumption_inverse_existence}, K maps $\mathcal{D}(K)'$ into $\mathcal{D}(K)'$.
\end{lemma}
\begin{proof}
\par Given $\psi \in \mathcal{D}(K)'$, we have proved
$K \psi \in C^{+}[0,T]$ and
$$K\psi(t) \ge g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1},\ t\in[0,T]$$
in the proof of Lemma \ref{itself}, so that
it is sufficient to show $K\psi \le
g(t)\big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\big]^{-1}$ on $[0,T].$
\par For each $\N+,$ let $w_n(t;\psi)=u_n(t;\psi)-b_n$,
\eqref{ODE} yields the following ODE by direct calculation
$$
^C\!D_t^{\alpha} w_n(t;\psi)+\lambda_n \psi(t)w_n(t;\psi)=\lambda_nb_n(f(t)-\psi(t))\ge0,
\ w_n(0,\psi)=0,
$$
where $\lambda_nb_n(f(t)-\psi(t))\ge 0$ follows from
the fact $\psi(t) \le g(t)\big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\big]^{-1}$
and Assumption \ref{assumption_inverse_existence} $(b)$.
Applying Corollary \ref{sign_eigenfunction} to the above ODE gives
$w_n(t;\psi)\ge 0,$ i.e. $u_n(t;\psi)\ge b_n\ge 0$ on $[0,T].$
Hence,
$$
K\psi(t)=\frac{g(t)}{\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)} \le
\frac{g(t)}{\sum\limits_{n=1}^\infty b_n\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)}
=g(t)\big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\big]^{-1}
$$
and this proof is complete.
\end{proof}
\par The existence conclusion is derived from Lemmas \ref{uniqueness_series}
and \ref{itself_existence}.
\begin{theorem}[Existence]\label{existence}
\par Suppose Assumptions \ref{assumption_inverse} and
\ref{assumption_inverse_existence} be valid, then there exists a
fixed point of $K$ in $\mathcal{D}(K)'.$
\end{theorem}
\begin{proof}
\par Lemma \ref{uniqueness_series} yields the sequence
$\{\overline{a}_n:n\in\mathbb{N}\}$ is increasing, while Lemma \ref{itself_existence}
gives $\{\overline{a}_n:n\in\mathbb{N}\}\subset \mathcal{D}(K)'.$
Then $\{\overline{a}_n:n\in\mathbb{N}\}$ is an increasing sequence with an
upper bound $g(t)\big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)\big]^{-1},$
which implies the convergence of $\{\overline{a}_n:n\in\mathbb{N}\}.$
Denote the limit by $\overline{a},$ clearly $\overline{a}$
is a fixed point of $K.$ Also, the closedness of $\mathcal{D}(K)'$
yields that $\overline{a}\in \mathcal{D}(K)'.$
Therefore, $\overline{a}$ is a fixed point of $K$ in $\mathcal{D}(K)',$ which
confirms the existence.
\end{proof}
\subsection{Main theorem for the inverse problem and reconstruction algorithm}
\par Lemma \ref{uniqueness_series}, Theorems \ref{uniqueness} and
\ref{existence} allow us to deduce the main theorem for this
inverse problem.
\begin{theorem}[Main theorem for the inverse problem]\label{main_inverse}
Suppose Assumption \ref{assumption_inverse} holds.
\begin{itemize}
\item [(a)] If there exists a fixed point of K in $\mathcal{D}(K),$ then it is
unique and coincides with the limit of $\{\overline{a}_n:n\in\mathbb{N}\};$
\item [(b)] If Assumption \ref{assumption_inverse_existence} is also valid,
then there exists a unique fixed point of K in $\mathcal{D}(K)',$ which is the
limit of $\{\overline{a}_n:n\in\mathbb{N}\}.$
\end{itemize}
\end{theorem}
\par The following reconstruction algorithm for $a(t)$ is based on
Theorem \ref{main_inverse}.
\begin{table}[h!]
\caption{Numerical Algorithm}\label{algorithm}
\begin{tabular*}{16cm}{l}
\hline
Iteration algorithm to recover
the coefficient $a(t)$ \\
\hline
1: Set up the right-hand side function $F(x,t)$ and the initial
condition $u_0(x)$,\\
then measure the output flux data $g(t).$
$F$, $u_0$ and $g$ should satisfy Assumption \ref{assumption_inverse};\\
2: Set the initial guess as
$\overline{a}_0(t)=g(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1};$\\
3: {\bf for k = 1,...,N do}\\
4: Using the L1 time-stepping \cite{JinLazarovZhou:L1}
to compute $u(x,t;\overline{a}_{k-1})$,\\
which is the weak solution of FDE \eqref{fde}
with coefficient function $\overline{a}_{k-1}$;\\
5: Update the coefficient $\overline{a}_{k-1}$ by
$ \overline{a}_{k}=K\overline{a}_{k-1};$\\
6: Check stopping criterion $\|\overline{a}_{k}-
\overline{a}_{k-1}\|_{L^2[0,T]}\le \epsilon_0$
for some $\epsilon_0>0$;\\
7: {\bf end for}\\
8: {\bf output} the approximate coefficient function $\overline{a}_{N}$.\\
\hline
\end{tabular*}
\end{table}
\section{Numerical Results for inverse problem}
\subsection{L1 time-stepping of $^C\!D_t^{\alpha}$}
\par The fourth step of Algorithm \ref{algorithm} includes
solving the direct problem of FDE \eqref{fde} numerically.
To this end, we choose L1 time
stepping \cite{JinLazarovZhou:L1,LinXu:2007} to discretize the term
$^C\!D_t^{\alpha} u(x,t):$
\begin{equation*}
\begin{aligned}
^C\!D_t^{\alpha} u(x,t_N) &= \frac{1}{\Gamma(1-\alpha)}\sum^{N-1}_{j=0}
\int^{t_{j+1}}_{t_j} \frac{\partial u(x,s)}{\partial s}
(t_N-s)^{-\alpha}\, ds \\
&\approx \frac{1}{\Gamma(1-\alpha)}\sum^{N-1}_{j=0}
\frac{u(x,t_{j+1})-u(x,t_j)}{\tau}\int_{t_j}^{t_{j+1}}
(t_N-s)^{-\alpha}ds\\
&=\sum_{j=0}^{N-1}b_j\frac{u(x,t_{N-j})-u(x,t_{N-j-1})}
{\tau^\alpha}\\
&=\tau^{-\alpha} [b_0u(x,t_N)-b_{N-1}u(x,t_0)
+\sum_{j=1}^{N-1}(b_j-b_{j-1})u(x,t_{N-j})] ,
\end{aligned}
\end{equation*}
where
\begin{equation*}
b_j=((j+1)^{1-\alpha}-j^{1-\alpha})/\Gamma(2-\alpha),\ j=0,1,\ldots,N-1.
\end{equation*}
\subsection{Numerical results for noise free data}
\par In this part, we set $\Omega=(0,1),\ x_0=0,\ T=1,\ \mathcal{L} u=u_{xx},$
pick $u_0(x)=-\sin{\pi x}, \ F(x,t)=-(t+1)\sin{\pi x}$
and consider the following two coefficients:
\begin{itemize}
\item [(a1)] smooth coefficient: $a(t)=\sin{5\pi t}+1.3;$
\item [(a2)] nonsmooth coefficient (``smile'' function):
\begin{equation*}
\begin{split}
a(t)&=[0.8\sin{3\pi t}+1.5]\chi_{[0,1/3]}
+[-0.5\sin{(3\pi t-\pi)}+0.6]\chi_{(1/3,2/3)}\\
&\quad+[0.8\sin{(3\pi t-2\pi)}+1.5]\chi_{[2/3,1]}.
\end{split}
\end{equation*}
\end{itemize}
\par In experiment (a1), the exact coefficient we pick is a smooth
function. Figure \ref{smooth_monotone} shows the initial guess and the
first three iterations, while Figure \ref{smooth_unique_9} presents
the exact and approximate coefficients. From these two figures, we
observe that $\{\overline{a}_n:n\in\mathbb{N}\}$ converges to $a(t)$ monotonically,
which illustrates Theorems \ref{monotonicity} and \ref{main_inverse}.
Moreover, the $L^2$ error of the approximation in Figure
\ref{smooth_unique_9} is
$\|a-\overline{a}_N\|_{L^2[0,T]}=1.04\times 10^{-6},$
which implies us the $L^2$ error of this approximation may be bounded
by the stopping criterion number $\epsilon_0.$ This guess is confirmed
by Figure \ref{error_epsilon} and can be expressed as
$$\|a-\overline{a}_N\|_{L^2[0,T]}=O(\epsilon_0).$$
Several attempts of experiment (a1) for different $\alpha\in(0,1)$
are taken to find the dependence of the convergence rate of Algorithm
\ref{algorithm} on the fractional order $\alpha,$ which is shown in Figure
\ref{alpha_N}. This figure shows the amounts of iterations
required, i.e. $N,$ corresponding to different $\alpha,$ which imply
that restricted $\alpha\in(0,1),$ the larger $\alpha$ is, the faster the
convergence rate of Algorithm \ref{algorithm} is. This phenomenon
is explained in \cite{jin2012inverse} by a property of the Mittag-Leffler
function; for $\alpha\in(0,1),$ the larger $\alpha$ is,
the faster the decay rate of $E_{\alpha,1}(-z)$ is as $z\to \infty.$
\begin{figure}[th!]
\center
\subfigure[$\alpha=0.3$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_monotone_3.eps}
}
\subfigure[$\alpha=0.5$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_monotone_5.eps}
}\\
\subfigure[$\alpha=0.7$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_monotone_7.eps}
}
\subfigure[$\alpha=0.9$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_monotone_9.eps}
}
\caption{Experiment (a1): the initial guess and first three iterations }
\label{smooth_monotone}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{smooth_unique_9.eps}
\end{tabular}
\vspace{-.2cm}
\caption{Experiment (a1): the exact and approximate coefficients for $\alpha=0.9$ and
$\epsilon_0=10^{-6}$}
\label{smooth_unique_9}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{alpha_N.eps}
\end{tabular}
\vspace{-.2cm}
\caption{the amounts of iterations $N$ for different $\alpha$}
\label{alpha_N}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{error_epsilon.eps}
\end{tabular}
\vspace{-.2cm}
\caption{$\|a-\overline{a}_N\|_{L^2[0,T]}$ for different $\epsilon_0$ under $\alpha=0.9$}
\label{error_epsilon}
\end{figure}
\par The definition of $\mathcal{D}(K)$ restricts the coefficient $a(t)$ in the space
$C^+[0,T],$ however, the results of experiment (a2) indicate that Algorithm
\ref{algorithm} still works for nonsmooth $a(t),$ which means the numerical
restriction on $a(t)$ can possibly be extended from $a(t)\in C^{+}[0,T]$ to
$a(t)\in L^\infty [0,T].$
For discontinuous $a(t),$ Figures \ref{nonsmooth_monotone}
and \ref{nonsmooth_unique_9} explain that
Theorems \ref{monotonicity} and \ref{main_inverse} still hold,
while Figures \ref{alpha_N} and \ref{error_epsilon}
illustrate the similar conclusions as
the larger $\alpha$ is, the faster the convergence rate of Algorithm
\ref{algorithm} is, and
$$\|a-\overline{a}_N\|_{L^2[0,T]}=O(\epsilon_0).$$
\begin{figure}[th!]
\center
\subfigure[$\alpha=0.3$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_monotone_3.eps}
}
\subfigure[$\alpha=0.5$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_monotone_5.eps}
}\\
\subfigure[$\alpha=0.7$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_monotone_7.eps}
}
\subfigure[$\alpha=0.9$]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_monotone_9.eps}
}
\caption{Experiment (a2): the initial guess and first three iterations }
\label{nonsmooth_monotone}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{nonsmooth_unique_9.eps}
\end{tabular}
\vspace{-.2cm}
\caption{Experiment (a2): the exact and approximate coefficients for $\alpha=0.9$ and
$\epsilon_0=10^{-6}$}
\label{nonsmooth_unique_9}
\end{figure}
\subsection{Numerical results for noisy data}
\par In this subsection, we will consider data polluted by noise.
Set $g$ be the exact data and denote the noisy data by $g_\delta$
with relative noise level $\delta,$ i.e.
$\|(g-g_\delta)/g\|_{L^\infty[0,T]}\le \delta.$
Then the perturbed operator $K_\delta$ is
$$K_\delta \psi(t)
=\frac{g_\delta(t)}{\sum\limits_{n=1}^\infty u_n(t;\psi)
\frac{\partial \phi_n}{\partial {\overrightarrow{\bf n}}}(x_0)}$$
with domain
$$\mathcal{D}(K_\delta):=\{\psi\in C^{+}[0,T]:g_\delta(t)\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1}\le \psi(t),\ t\in[0,T] \}.$$
Also, the sequence $\{\overline{a}_{\delta,n}:n\in \mathbb{N}\}$ can be
obtained from the iteration
$$\overline{a}_{\delta,0}=g_\delta\Big[\frac{\partial u_0}{\partial {\overrightarrow{\bf n}}}(x_0)
+ I_t^\alpha[\frac{\partial F}{\partial {\overrightarrow{\bf n}}}(x_0,t)]\Big]^{-1},\
\overline{a}_{\delta,n+1}=K_\delta\overline{a}_{\delta,n},\ n\in \mathbb{N}.$$
Since $\delta$ is a small positive number and $g$ is a strictly
positive function, we can assume $g_\delta$ is still positive, which
means Theorem \ref{main_inverse} still holds for $K_\delta.$
Hence, if there exists a fixed point $a_\delta\in \mathcal{D}(K_\delta)$,
the sequence $\{\overline{a}_{\delta,n}:n\in \mathbb{N}\}$ will converge to
$a_\delta$ monotonically and we denote the limit by $\overline{a}_\delta.$
Algorithm \ref{algorithm} is still able to be used to recover
$\overline{a}_\delta$ after a slightly modification$-$replacing
$g$ and $K$ by $g_\delta$ and $K_\delta,$ respectively.
\par We take the experiments (a1) and (a2) with
noise level $\delta>0.$ Figures \ref{smooth_unique_9_noise_3} and
\ref{nonsmooth_unique_9_noise_3} present
the exact and approximate coefficients under $\delta=3\%$ for
experiments (a1) and (a2) respectively. From figures
\ref{smooth_unique_9_noise_3} and \ref{nonsmooth_unique_9_noise_3},
we observe that the smaller $|a(t)|$ is, the better the approximation is.
This can be explained by $\delta$ means the relatively noise level, i.e.
we pick $g_\delta=(1+\zeta\delta)g$ in the codes, where
$\zeta$ follows a uniform distribution on $[-1,1].$
Figure \ref{error_delta} illustrates that
$$\|a-\overline{a}_{\delta,N}\|_{L^2[0,T]}/\|a\|_{L^2[0,T]}=O(\delta),$$
showing the domination of the noise level $\delta$ in
relatively $L^2$ error with the reason that $\epsilon_0 \ll \delta.$
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{smooth_unique_9_noise_3.eps}
\end{tabular}
\vspace{-.2cm}
\caption{Experiment (a1): the exact and approximate coefficients with $\alpha=0.9,$
$\epsilon_0=10^{-6}$ and $\delta=3\%$}
\label{smooth_unique_9_noise_3}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{nonsmooth_unique_9_noise_3.eps}
\end{tabular}
\vspace{-.2cm}
\caption{Experiment (a2): the exact and approximate coefficients with $\alpha=0.9,$
$\epsilon_0=10^{-6}$ and $\delta=3\%$}
\label{nonsmooth_unique_9_noise_3}
\end{figure}
\begin{figure}[h!]
\center
\begin{tabular}{cc}
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=5cm,width=8cm]{error_delta.eps}
\end{tabular}
\vspace{-.2cm}
\caption{$\|a-\overline{a}_{\delta,N}\|_{L^2[0,T]}/\|a\|_{L^2[0,T]}$
for different $\delta$ under $\alpha=0.9$ and $\epsilon_0=10^{-6}$}
\label{error_delta}
\end{figure}
\subsection{Numerical results in two dimensional case}
In this part, the numerical experiments on a two dimensional domain will be considered. We set $\alpha=0.9,\ \epsilon_0=10^{-6},\ \Omega=(0,1)^2,\ x_0=(0,1/2),\ T=1,\ \mathcal{L} u=\triangle u,$ choose $u_0(x,y)=-\sin{[\pi xy(1-x)(1-y)]},$ $F(x,y)=-(t+1)\cdot\sin{[\pi xy(1-x)(1-y)]},$ and consider experiments (a1) and (a2).
Figures \ref{smooth_monotone_2d} and \ref{nonsmooth_monotone_2d} confirm the theoretical conclusions in section 4.
\begin{figure}[th!]
\center
\subfigure[Initial guess and first three iterations]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_monotone_2d.eps}
}
\subfigure[Exact and approximate coefficients]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{smooth_unique_2d.eps}
}
\caption{Experiment (a1) in two dimensional case}
\label{smooth_monotone_2d}
\end{figure}
\begin{figure}[th!]
\center
\subfigure[Initial guess and first three iterations]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_monotone_2d.eps}
}
\subfigure[Exact and approximate coefficients]{
\includegraphics[trim = .5cm .15cm .5cm .3cm, clip=true,height=4.5cm,width=6cm]
{nonsmooth_unique_2d.eps}
}
\caption{Experiment (a2) in two dimensional case}
\label{nonsmooth_monotone_2d}
\end{figure}
\section*{Acknowledgment}
The author is indebted to William Rundell for assistance in this work
and acknowledges partial support from NSF-DMS 1620138.
\bibliographystyle{abbrv}
| {
"timestamp": "2017-08-28T02:05:47",
"yymm": "1708",
"arxiv_id": "1708.07756",
"language": "en",
"url": "https://arxiv.org/abs/1708.07756",
"abstract": "In this work, we consider a FDE (fractional diffusion equation) $${}^C D_t^\\alpha u(x,t)-a(t)\\mathcal{L} u(x,t)=F(x,t)$$ with a time-dependent diffusion coefficient $a(t)$. For the direct problem, given an $a(t),$ we establish the existence, uniqueness and some regularity properties with a more general domain $\\Omega$ and right-hand side $F(x,t)$. For the inverse problem--recovering $a(t),$ we introduce an operator $K$ one of whose fixed points is $a(t)$ and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for $a(t)$ is created and some numerical results are provided to illustrate the theories.",
"subjects": "Analysis of PDEs (math.AP); Mathematical Physics (math-ph)",
"title": "An undetermined time-dependent coefficient in a fractional diffusion equation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846634557752,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7092019612063576
} |
https://arxiv.org/abs/1710.02287 | Explicit Methods for Hilbert Modular Forms of Weight 1 | In this article we present an algorithm that uses the graded algebra structure of Hilbert modular forms to compute the adelic $q$-expansion of Hilbert modular forms of weight one as the quotient of Hilbert modular forms of higher weight. The main improvement to existing methods is that our algorithm can be applied in weight $1$, which fills a gap left by standard computational methods. Additionally, the algorithm can be used to compute Hilbert modular forms over finite fields in all characteristic simultaneously. We use this algorithm to compute a first candidate of a Hilbert modular form of parallel weight $1$ that is non-liftable and specify the exact conditions under which our candidate $q$-expansion corresponds to a non-liftable Hilbert Modular form. | \section{Introduction}
Established methods to compute Hilbert modular forms (HMFs) are restricted to weight at least $2$. As with classical modular forms, the weight $1$ case is more intricate. However, one can use the graded algebra structure of HMFs to compute the adelic $q$-expansion of a HMF of (partial) weight $1$ as the quotient of HMFs of higher weights.
If $f$ and $E$ are HMFs of level $\mathfrak N$, weights $k$ and $k'$ and characters $\mathcal E$ and $\mathcal E'$ respectively, then their product $f\cdot E$ is a HMF of weight $k+k'$ and character $\mathcal E\cdot \mathcal E'$. So, if the adelic $q$-expansion of $E$ is invertible, then any HMF of weight $k$ is the quotient of a HMF of weight $k+k'$ by the form $E$, i.e.
$$\mathcal{M}_k(\mathfrak N,\mathcal E) \subset \frac 1 E \mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\cdot\mathcal E').$$
Moreover, the left hand side is stable under the action of the Hecke algebra of weight $k$, so we can shrink the right hand side by taking the largest subspace of $\frac 1 E \mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\cdot \mathcal E')$ that is stable under the action of the Hecke algebra. The underlying philosophy is that we can reduce this candidate space until it only contains HMFs of weight $k$. This approach was used for the first time for classical modular forms in 1978 by Joe Buhler in \cite{Buh78} and later by Kevin Buzzard in \cite{Buz14} and George Schaeffer in \cite{Sch12}, and for classical HMFs over quadratic fields with narrow class number $1$ by Richard Moy and Joel Specter in \cite{MoSp15}.
The key observation that enables these results to extend to HMFs is that we can verify whether or not a candidate fraction of HMFs is indeed a HMF by using its truncated adelic $q$-expansion. That is, we prove that the square of the truncated adelic $q$-expansion of such a fraction $g/E$ coincides with the adelic $q$-expansion of a HMF (of higher weight) if and only if the quotient of the truncated adelic $q$-expansions of $g$ and $E$ coincides with the adelic $q$-expansion of a HMF (of lower weight).
We apply this approach to develop an algorithm that computes HMFs with coefficients in $\mathds{C}$ and any weight as well as HMFs over finite fields with parallel weight. In particular, we prove that our algorithm computes in almost all characteristics simultaneously, in the sense that given weight $k$, level $\mathfrak N$ and character $\mathcal E$, the output of our algorithm to compute HMFs with coefficients in a number field with trivial class group includes a finite set of primes $\mathcal{L}$ such that for all primes $p$ not contained in $\mathcal{L}$ satisfying certain conditions in higher weight, all HMFs of weight $k$, level $\mathfrak N$ and character $\mathcal E$ over $\overline \mathds{F}_p$ lift to characteristic zero. Finally, we use our algorithm to find explicit examples of non-liftable HMFs of parallel weight $1$ by running the algorithms for primes contained in $\mathcal{L}$.
\subsection*{Notation}
Throughout this article, $K$ will denote a totally real number field of degree $n>1$ and $\mathcal{O}_K$ its ring of integers. If $a$ is an element of $K$ we will denote the image of $a$ under the $n$ distinct embeddings of $K$ into $\mathds{R}$ by $a^{(1)},...,a^{(n)}$. An element $a$ of $K$ is said to be \emph{totally positive}, denoted $ a \gg 0$, if $a^{(i)} > 0$ for all embeddings of $K$ into $\mathds{R}$. If $\mathfrak a$ is a subset of $K$, we will denote $\mathfrak a^+$ the subset of totally positive elements of $\mathfrak a$. For example $\mathcal{O}_K^{\times,+}$ is the set of totally positive units in $\mathcal{O}_K$. We denote the set of all integral ideals of $K$ by $I_K$, the narrow class group of $K$ by $\mathrm{Cl}^+$ and the narrow class number of $K$ by $h ^+$.
For $n$-tuples $z=(z_1,...,z_n) \in \mathds{C}^n$ and $k=(k_1,...,k_n) \in \mathds{Z}^n$ we write
\begin{align*}
z^k &= \prod_{i=1}^n z_i^{k_i}& \text{ and }& & \trace z &= \sum_{i=1}^n z_i.
\intertext{We extend this notation to $ K$ by identifying $\xi \in K$ with the $n$-tuple \mbox{$\big(\xi^{(1)},...,\xi^{(n)}\big)$} in $\mathds{R}^n$, i.e.}
\xi^k &= \prod_{i=1}^n \big(\xi^{(i)}\big)^{k_i}&\text{ and }&& \trace \xi &= \sum_{i=1}^n \xi^{(i)}.
\end{align*}
Moreover, we write $k_0 = \text{max}_i\{k_i\}$. If $\ell$ is an integer we write $\underline \ell= (\ell,...,\ell)$ to distinguish the integer $\ell$ from the parallel vector $\underline \ell$. If $\mathfrak N$ is an ideal of $\mathcal{O}_K$, $\mathcal E$ a (Dirichlet) character mod $\mathfrak N$ and $R$ a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra}, then we denote the $R$-module of HMFs and cuspidal HMFs of weight $k$, level $\mathfrak N$ and character $\mathcal E$ over $R$ by $\mathcal{M}_k(\mathfrak N,\mathcal E;R)$ and $\mathcal{S}_k(\mathfrak N,\mathcal E;R)$ respectively. If $W$ is a subset of $\mathds{Z}^n$ that contains $0=(0,0,...,0)$ and that is closed under addition, then $\mathcal{M}_W(\Gamma_ 1 (\mathfrak N);R)$ denotes the graded $R$-algebra of HMFs of weights $k$ in $W$ and congruence subgroup $\Gamma_1(\mathfrak N)$. We will write $\mathds{T}_k(\mathfrak N,\mathcal E;R)$ for the $R$-algebra of Hecke operators acting on $\mathcal{M}_k(\mathfrak N,\mathcal E;R)$. For a detailed approach of HMFs and Hecke operators see for example \cite{AnGo05}, \cite{Hida04}, \cite{Shi78} or \cite{vdG88}.
\section{Adelic $q$-expansion}
In this section we construct the graded $R$-algebra of adelic power series (of weights in $W$) and formulate conditions on the ring $R$ such that this construction is well defined. Finally, we will show that a suitable $q$-expansion principle from $\mathcal{M}_W(\Gamma_ 1 (\mathfrak N);R)$ to this graded $R$-algebra of adelic power series exists.
Let $\mathfrak N$ be an integral ideal of $K$ and $W$ a subset of $\mathds{Z}^n$ that contains $0=(0,0,...,0)$ and is closed under addition. Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying:
\begin{align}\label{eq:condGe}
&\text{The element $\varepsilon^{k/2}$ is a unit in $R$ for all $\varepsilon$ in $\mathcal{O}_K^{\times, +}$ and all $k$ in $W$;} \\\label{eq:condAd}
&\text{The element $\xi^{(\underline k_0-k)/2}$ is a unit in $R$ for all $\xi$ in $\mathcal{O}_K^{+}$ and all $k$ in $W$.}
\end{align}
\begin{remark}
\begin{enumerate}
\item The fields $\overline \mathds{Q}$ and $\mathds{C}$ both satisfy conditions \eqref{eq:condGe} and \eqref{eq:condAd} for any $W$.
\item If $W$ is the subset of parallel weights, then any $\mathds{Z}[\tfrac 1 {\Norm \mathfrak N}]$-algebra satisfies both \eqref{eq:condGe} and \eqref{eq:condAd}.
\item If $W$ is not contained in the set of parallel weights and Char$(R) = p>0$ then $R$ does not satisfy \eqref{eq:condAd} since $p^{(\underline k_0-k)/2}$ is not contained in $R^\times$ for any non-parallel weight vector $k$. However, this does not necessarily imply that the adelic $q$-expansion principle is not valid in this setting, only that our specific construction is not applicable.
\end{enumerate}
\end{remark}
Let $\{\mathfrak t_\lambda\}_{\mathrm{Cl}^+}$ be a full set of representatives of the narrow class group of $K$.
We define the $R$-module of \emph{adelic power series over $R$}, denoted $\Adqm R$, as the $R$-module whose elements consist of an $h^+$-tuple $a_{(0)}$ in $R^{\mathrm{Cl}^+}$ together with a rule associating to every non trivial ideal $\mathfrak m$ of $\mathcal{O}_K$ an element $a_\mathfrak m$ of $R$. To emphasise that these adelic power series will be the adelic $q$-expansions of HMFs, we write
\begin{align*}
\Adqm R & := R ^{\mathrm{Cl}^+} \oplus \prod_{0\ne\mathfrak b \lhd \mathcal{O}_K} R\cdot q^\mathfrak b
\\&= \Big\{ \big(a_{(0),[\mathfrak t_\lambda]}\big)_{[\mathfrak t_\lambda] \in \mathrm{Cl}^+}+ \sum_{0\ne\mathfrak b \lhd \mathcal{O}_K} a_\mathfrak b q^\mathfrak b\ \big |\ \text{ with all } a_\star \in R \Big\}.
\intertext{For any weight vector $k$ in $W$ we define the $R$-module of \emph{geometric power series of weight $k$ over $R$} as the $R$-module of $h^+$-tuples of formal power series where the coefficients of the power series at $\lambda$ are indexed by the totally positive elements of $\mathfrak t_\lambda$ and satisfy $ a_{\lambda,\varepsilon\xi} =\varepsilon^{k/2} a_{\lambda,\xi} $ for all $\varepsilon$ in $ \mathcal{O}_K^{\times,+} $, i.e. }
\Geq R &:= \Big\{\Big(a_{\lambda,0} + \sum_{\xi \in \mathfrak t_\lambda^+} a_{\lambda,\xi} q^\xi \Big)_{\mathrm{Cl}^+}\ \Big \vert \ a_{\lambda,\varepsilon\xi} = \varepsilon^{k/2} a_{\lambda,\xi} \text{ for all } \varepsilon \in \mathcal{O}_K^{\times,+} \Big\},
\end{align*}
where all $a_{\xi,\mathfrak t_\lambda}$ lie in $R$. Note that the module of adelic power series does not depend on the choice of representatives $\{\mathfrak t_\lambda\}_{\mathrm{Cl}^+}$. The module of geometric power series over $R$ does depend on the choice of representatives $\{\mathfrak t_\lambda\}_{\mathrm{Cl}^+}$. However, we will show that the modules obtained by different choices of representatives are isomorphic.
Moreover, the $R$-module $\bigoplus_{k\in W}\Geq R$ has a natural structure of a graded $R$-algebra by componentwise multiplication. The following proposition will allow us to view the adelic power series as a graded $R$-algebra.
\begin{proposition}\label{prop:GeomIsAdel}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying conditions \eqref{eq:condGe} and \eqref{eq:condAd}.
The choice of representatives $\{\mathfrak t_\lambda\}_ {\mathrm{Cl}^+}$ induces an isomorphism of $R$-modules $\Psi_{k,\{\mathfrak t_\lambda\}}$ between geometric power series and adelic power series.
\end{proposition}
\begin{proof}
We construct the map $\Psi_{k,\{\mathfrak t_\lambda\}}$ by
\begin{align*}
\Psi_{k,\{\mathfrak t_\lambda\}}&: \Geq R \rightarrow \Adqm R:\\& \Big(a_{\lambda,0} + \sum_{\xi \in \mathfrak t_\lambda^+} a_{\lambda,\xi} q^\xi \Big)_{\mathrm{Cl}^+}\mapsto (a_{\lambda,0})_{\mathrm{Cl}^+} + \sum_{0\ne\mathfrak b \lhd \mathcal{O}_K} a_\mathfrak b q^\mathfrak b,
\intertext{where $a_\mathfrak b = a_{\lambda,\xi}\, \xi^{\left(\underline k_0- k\right)/2}$ with $\xi$ and $\lambda$ such that $ \mathfrak b = \xi \mathfrak t_\lambda^{-1}$. One checks that $\Psi_{k,\{\mathfrak t_\lambda\}}$ is a well defined morphism of $R$-modules and that its inverse is given by}
\Phi_{k,\{\mathfrak t_\lambda\}}&:\Adqm R \rightarrow \Geq R: \\& (a_{(0),[\mathfrak t_\lambda]})_{\mathrm{Cl}^+} + \sum_{0\ne\mathfrak b \lhd \mathcal{O}_K} a_\mathfrak b q^\mathfrak b \mapsto \Big(a_{(0),[\mathfrak t_\lambda]} + \sum_{\xi \in \mathfrak t_\lambda^+} a_{\lambda,\xi} q^\xi \Big)_{\mathrm{Cl}^+},
\end{align*}
with $a_{\lambda,\xi} = a_{\xi \, \mathfrak t_\lambda^{-1}} \cdot \xi^{\left(k-\underline k_0 \right)/2}$. Note that $\xi \mathfrak t_\lambda^{-1}$ is an integral ideal of $K$ since $\xi$ is an element of $\mathfrak t_\lambda$, hence $a_{\xi\mathfrak t_{\lambda}^{-1}}$ is well defined.
\end{proof}
\begin{definition}\label{def:AdPoSe}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying conditions \eqref{eq:condGe} and \eqref{eq:condAd}.
The graded $R$-algebra of \emph{adelic power series} (of weights $W$) is defined as the $R$-module
$$\Adq R :=\bigoplus_{k \in W} \Adqm R$$ equipped with the graded $R$-algebra structure induced by the isomorphisms $\Psi_{k,\{\mathfrak t_\lambda\}}$ and $\Phi_{k,\{\mathfrak t_\lambda\}}$, i.e. if $f$ and $g$ are adelic power series of weight $k$ and $k'$ respectively then
$$f\cdot g = \Psi_{k+k', \{\mathfrak t_\lambda\}} \left( \Phi_{k, \{\mathfrak t_\lambda\}}(f) \cdot \Phi_{k', \{\mathfrak t_\lambda\}}(g)\right) $$ by definition.
\end{definition}
\begin{theorem}\label{thm:adisind}
The graded $R$-algebra $\Adq R$ is independent of the choice of representatives $\{\mathfrak t_\lambda\}_{\mathrm{Cl}^+}$.
\end{theorem}
\begin{proof}
Let $\{\mathfrak t_\lambda\}_{\mathrm{Cl}^+}$ and $\{\mathfrak t'_\lambda\}_{\mathrm{Cl}^+}$ be two choices of representatives of the narrow class group of $K$ and let $\{\xi_\lambda\}_{\mathrm{Cl}^+}$ be totally positive elements of $K$ such that
$$\xi_\lambda\mathfrak t_\lambda = \mathfrak t'_\lambda\text{ for all }[\mathfrak t_\lambda] \text{ in } \mathrm{Cl}^+.$$
We define an isomorphism of $R$-modules as follows
\begin{align*}
\phi_{k,\{\xi_\lambda\}}&: \Geq R
\rightarrow \Geqp R: \\&
\Big( \sum_{\xi' \in \mathfrak t_\lambda^+} a_{\lambda,\xi'} q^{\xi'} \Big)_{\mathrm{Cl}^+}\mapsto\Big( \sum_{\xi' \in \mathfrak t_\lambda^+} a_{\lambda,\xi'}\xi_\lambda^{(k- \underline k_0)/2} q^{\xi_\lambda\xi'} \Big)_{\mathrm{Cl}^+}.
\end{align*}
By construction of $\phi_{k,\{\xi_\lambda\}}$ the following diagram commutes.
$$ \xymatrix{
\Geq R \ar@/_/[rd]_{\Psi_{k,\{\mathfrak t_\lambda\}} } \ar[rr]^{\phi_{k,\{\xi_\lambda\}}}& &
\Geqp R\ar@/^/[ld]^{\Psi_{k,\{\mathfrak t'_\lambda\}} } \\
& \Adq R
} $$
Finally, one checks that the isomorphism $\phi_{k,\{\xi_\lambda\}}$ induces an isomorphism of graded $R$-algebras.
\end{proof}
\begin{theorem}[The geometric $q$-expansion principle]\label{thm:geomq}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying condition \eqref{eq:condGe}. Then there exists a natural injective morphism of graded $R$-algebras $$\mathcal{M}_W(\Gamma_1(\mathfrak N); R) \rightarrow \bigoplus_{k\in W} \Geq R$$ called the \emph{geometric $q$-expansion principle}.
If $R'$ is a $\mathds{Z}[\tfrac 1 {\Norm \mathfrak N}]$-algebra containing $R$, then
$$\mathcal{M}_{ W}(\Gamma_1(\mathfrak N);R) = \left\{ f \in \mathcal{M}_{ W}(\Gamma_1(\mathfrak N);R')\ \Big\vert \ a_{\lambda,\xi}(f) \in R\, \text{ for all } \xi\in \mathfrak t_\lambda^+\right \}.$$
\end{theorem}
\begin{proof}
See \cite[Theorem 6.7]{Rap78}.
\end{proof}
\begin{corollary}[The adelic $q$-expansion principle] \label{thm:adelisuniqueR}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying conditions \eqref{eq:condGe} and \eqref{eq:condAd}.
\begin{enumerate}
\item The geometric $q$-expansion principle composed with the isomorphims $\{\Phi_{k,\{\mathfrak t_\lambda\}}\}_k$ induces an injective morphism of graded $R$-algebras from $\mathcal{M}_{{W}}(\Gamma_1(\mathfrak N);R)$ to $\Adq R$ called the \emph{adelic $q$-expansion map}.
\item The image of $\mathcal{M}_W(\Gamma_1(\mathfrak N);R)$ in $\Adq R$ is independent of the choice of representatives $\{\mathfrak t_\lambda\}$.
\item Let $R'$ be a $\mathds{Z}[\tfrac 1 {\Norm \mathfrak N}]$-algebra containing $R$. Then
$$\mathcal{M}_{ W}(\Gamma_1(\mathfrak N);R) = \left\{ f \in \mathcal{M}_{ W}(\Gamma_1(\mathfrak N);R')\ \Big\vert \ \begin{matrix} a_{(0)}(f) \in R^{\mathrm{Cl}^+} \text{ and } a_\mathfrak b \in R\\ \text{ for all } 0\ne\mathfrak b \lhd \mathcal{O}_K \end{matrix}\right \}.$$
\end{enumerate}
\end{corollary}
\begin{proof}
This follows from Proposition \ref{prop:GeomIsAdel}, Theorem \ref{thm:adisind} and Theorem \ref{thm:geomq}.
\end{proof}
\begin{proposition}\label{prop:HeckeGeo}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying conditions \eqref{eq:condGe} and \eqref{eq:condAd} and $\mathcal E$ an $R$-valued character mod $\mathfrak N$. The action of the Hecke operators on the adelic $q$-expansion of $\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R)$ is given by
\begin{align*}
a_{(0),[\mathfrak t_\lambda]}\big(T_\mathfrak a(f)\big) &= \sum_{\mathfrak a \subset \mathfrak b} \mathcal E(\mathfrak b) \Norm \mathfrak b ^{k_0-1} a_{(0),[\mathfrak t_\lambda \mathfrak a/\mathfrak b^2]}(f), \\
a_\mathfrak m\big(T_\mathfrak a(f)\big) &= \sum_{\mathfrak m+\mathfrak a \subset \mathfrak b} \mathcal E(\mathfrak b) \Norm \mathfrak b ^{k_0-1} a_{\mathfrak m\mathfrak a/\mathfrak b^2}(f).
\end{align*}
\end{proposition}
\begin{proof}
See \cite[Section 2]{Shi78} .
\end{proof}
\begin{corollary}\label{cor:AdelInSubR}
Let $R'$ be a ring that contains $R$ and let $B$ be a positive integer such that the Hecke algebra $\mathds{T}_{ k}(\mathfrak N,\mathcal E;R')$ is generated by the Hecke operators $T_\mathfrak b$ with $\Norm \mathfrak b \le B$ as an $R$-module. Then
$$\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R) = \left\{ f \in \mathcal{M}_{ k}(\mathfrak N,\mathcal E;R')\ \Big\vert \ \begin{matrix} a_{(0)}(f) \in R^{\mathrm{Cl}^+} \text{ and } a_\mathfrak b \in R\\ \text{ for all } 0\ne\mathfrak b \lhd \mathcal{O}_K \text{ with }\Norm \mathfrak m \le B \end{matrix}\right \}.$$
\end{corollary}
\begin{proof}
Let $f$ be a HMF in $\mathcal{M}_k(\mathfrak N,\mathcal E;R')$ such that $a_{(0)}(f) \in R^{\mathrm{Cl}^+}$ and $a_\mathfrak b (f) \in R$ for all non trivial ideals $\mathfrak b$ with $\Norm \mathfrak b \le B$. By Corollary \ref{thm:adelisuniqueR} it suffices to show that $a_\mathfrak m (f) \in R$ for all non trivial ideals $\mathfrak m$. Let $\mathfrak m$ be a non trivial ideal of $K$. Then $T_\mathfrak m$ is an $R$-linear combination of Hecke operators $T_{\mathfrak b_i}$ with $\Norm {\mathfrak b_i}\le B$, hence
\begin{align*}
a_\mathfrak m(f) & = a_{\mathcal{O}_K}(T_\mathfrak m f)
= a_{\mathcal{O}_K}\left(\sum_{i} r_{i}T_{\mathfrak b_i} f \right)
= \sum_{i} r_{i} a_{\mathfrak b_i}(f).
\end{align*}
Since all $a_{\mathfrak b_i}(f)$ and $r_i$ are elements of $R$, so is $a_\mathfrak m(f)$.
\end{proof}
\subsection*{Truncated power series}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying condition \eqref{eq:condGe} and \eqref{eq:condAd} and let $B$ be a positive number, then by multiplicativity of the norm and by the definition of multiplication of adelic power series, Definition \ref{def:AdPoSe}, the following subgroup is an ideal of $\Adq R$
$$\big(q^{B}\big) = \left\{ \sum_{\mathfrak m \lhd \mathcal{O}_{K} } a_{\mathfrak m } q^{\mathfrak m} \ \Big\vert \ a _{\mathfrak m}=0 \text{ for all } \Norm \mathfrak m< B \right\}.$$
We call the quotient, $\Adqmod R B$, \emph{the ring of adelic power series mod $q^{B}$}.
We denote $\pi_B$ for the projection map onto $\Adqmod R B$.
\begin{definition} \label{def:Sturmad}
The \emph{Sturm bound of weight $k$, level $\mathfrak N$ and character $\mathcal E$ over $R$} is the smallest positive integer $B$ such that the adelic $q$-expansion map followed by the natural projection of $q$-expansions
$$ \mathcal{M}_{ k}(\mathfrak N,\mathcal E;R) \rightarrow \Adqmod R B$$
is an injective morphism of $R$-modules.
\end{definition}
We define the $R$-module of \emph{fractional Hilbert modular forms of weight $ k$, level $\mathfrak N$, character $\mathcal E$ over $R$} as
$$\mathcal{M}^f_{ k}(\mathfrak N,\mathcal E;R):=\left\{ \frac{f}{g}\ \Big \vert\ \begin{matrix} k_1- k_2 = k,\ \mathcal E_1/\mathcal E_2=\mathcal E,\ f\in \mathcal{M}_{ k_1}(\mathfrak N,\mathcal E_1;R), \\ 0\ne g \in \mathcal{M}_{ k_2}(\mathfrak N,\mathcal E_2;R) \text{ and } g \vert f \text{ in } \Adq R \end{matrix}\right\}.$$
Since the HMFs $f$ and $g$ are elements of respectively $H^0\big(X_1(\mathfrak N)\times \text{Spec}(R), \omega^{\otimes k_1}_\mathcal E\big)$ and $H^0\big(X_1(\mathfrak N)\times \text{Spec}(R), \omega^{\otimes k_2}_\mathcal E)$ with $X_1(\mathfrak N)$ the Hilbert modular variety of level $\Gamma_1(\mathfrak N)$, $\omega_\mathcal E^{\otimes k}$ the modular line bundle of weight $k$ and character $\mathcal E$, their quotient $f/g$ is an element of $H^0\big(X_1(\mathfrak N)\times \text{Spec}(R), \omega^{\otimes k}_\mathcal E\otimes \mathcal{K}\big)$ with $\mathcal{K}$ the sheaf of meromorphic functions on $X_1(\mathfrak N)\times \text{Spec}(R)$.
The condition $g \vert f \text{ in } \Adq R$ means precisely that $ f/ g $ has a well defined adelic $q$-expansion in $\Adq R$. In particular, the $q$-expansion is over integral ideals of $K$ rather than fractional ideals. The following lemma and theorem will give necessary and sufficient conditions on a fractional HMF to be a HMF.
\begin{lemma}\label{lemma:FracIsHolR}
Let $f$ be a fractional HMF over $R$. Then $f^2$ is a HMF if and only if $f$ is a HMF.
\end{lemma}
\begin{proof}
This follows from the fact that the $R$-module of HMF is precisely the $R$-submodule of the fractional HMFs that do not admit any poles.
\end{proof}
The following theorem is the key observation that enables us to develop our algorithm. It shows that we only need a limited amount of precision in order to conclude that a fractional HMF is a HMF.
\begin{theorem}\label{cor:FracIsHolQexpR}
Let $k$ and $k'$ be weight vectors in $W$, let $\mathcal E$ and $\mathcal E'$ be $R$-valued characters mod $\mathfrak N$, let $E$ be a HMF in $\mathcal{M}_{ k'}(\mathfrak N,\mathcal E';R)$, let $B$ be the Sturm bound of weight $2 k+2 k'$, level $\mathfrak N$ and character $\mathcal E^2\mathcal E'^2$ and let $\tilde f$ be the truncated adelic $q$-expansion mod $q^B$ of a fractional HMF in $E^{-1} \mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps';R)$.
Then
$\tilde f^2$ agrees with the adelic $q$-expansion mod $q^B$ of a HMF in $\mathcal{M}_{2 k}(\mathfrak N,\mathcal E^2;R)$ if and only if $\tilde f$ agrees with the adelic $q$-expansion mod $q^B$ of a HMF in $\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R)$.
\end{theorem}
\begin{proof}
The `only if' part is immediate.
Conversely, let $g\in \mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ and $h \in \mathcal{M}_{2k}(\mathfrak N,\mathcal E^2;R)$ such that
$$ \tilde f \equiv \frac g E \mod q^B \phantom{MG} \text{ and }\phantom{MG}
\tilde f ^2 \equiv h \mod q^B.$$
Then $\frac {g^2} {E^2} E^2$ and $hE^2$ are HMFs in $\mathcal{M}_{2k+2k'}(\mathfrak N,\mathcal E^2\mathcal E'^2;R)$. Moreover, the adelic $q$-expansions of $g^2$ and $hE^2$ agree up to $q^B$, indeed
$$ g^2 \equiv \tilde f ^2 E^2 \equiv h E^2 \mod q^B.$$
By Definition \ref{def:Sturmad} we obtain
$$g^2 = h E^2.$$
In particular, $\frac{g^2}{E^2} = h$ is a HMF in $\mathcal{M}_{2k}(\mathfrak N,\mathcal E^2;R)$, hence $\frac g E$ is a fractional HMF such that $(\frac g E) ^2$ is a HMF and Lemma \ref{lemma:FracIsHolR} implies that $\frac g E$ is a HMF. By construction, the adelic $q$-expansion mod $q^B$ of the HMF $\frac g E$ agrees with $\tilde f$, completing the proof.
\end{proof}
\section{The Algorithm}\label{sect:AlgR}
Our algorithm is based on the philosophy used by Joe Buhler in \cite{Buh78}, by Kevin Buzzard in \cite{Buz14} and George Schaeffer in \cite{Sch12} for classical modular forms and by Richard Moy and Joel Specter in \cite{MoSp15} for classical HMFs. The underlying idea is that a space of meromorphic functions that satisfy the modularity condition and which is stable under the action of Hecke operators, should be contained in the space of modular forms. We will give conditions under which such a candidate space equals the space of HMFs. Although the algorithm can be used in arbitrary weight, the interesting applications are to compute spaces of HMFs of (partial) weight $1$ since there are other algorithms known in higher weights, see for example \cite{DeVo04}. In fact, we will make use of such algorithms to compute in weight $1$.
Suppose that $E$ is a HMF in $\mathcal{M}_{k'}(\mathfrak N,\mathcal E';R)$ with invertible adelic $q$-expansion then by the graded $R$-algebra structure of HMFs we have the following inclusion of $R$-modules
$$ \mathcal{M}_{k}(\mathfrak N,\mathcal E;R) \subset E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R).$$
So $ E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ is a candidate module of finite rank. Moreover, $\mathcal{M}_{k}(\mathfrak N,\mathcal E;R)$ is stable under the action of the Hecke operators so we can shrink our candidate module $E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ by computing the largest Hecke stable submodule. Finally, we verify that all adelic power series in the candidate module do coincide with the adelic $q$-expansion of a HMF in $\mathcal{M}_{k}(\mathfrak N,\mathcal E;R)$ by squaring the $q$-expansion and applying Theorem \ref{cor:FracIsHolQexpR}.
\begin{algorithm}[H]
\caption{Given an invertible adelic $q$-expansion mod $q^B$ of a HMF $E$ in $\mathcal{M}_{ k'}(\mathfrak N,\mathcal E';R)$ and the adelic $q$-expansion mod $q^B$ of a finite set of generators of $\mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps';R)$, computes the largest Hecke stable submodule of the image of $ E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ in $\Adqmod R B$.}\label{alg:alg}
\begin{algorithmic}
\State $V_0 \gets \text{the image of the adelic $q$-expanion map of } E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R) \mod q^B$
\State $i \gets 1$
\While{ $V_{i-1}$ is not $T_{\mathfrak m_{i}}$-stable for some $T_{\mathfrak m_{i}}$}
\State $V_{i} \gets\big({T_{\mathfrak m_{i}}}\vert_ {V_{i-1}}\big)^{-1} \big(\pi_{ B/\Norm {\mathfrak m_i}}(V_{i-1})\big) $
\State $i \gets i+1$
\EndWhile \\
\Return{$V_i$}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{Given an invertible adelic $q$-expansion mod $q^B$ of a HMF $E$ in $\mathcal{M}_{ k'}(\mathfrak N,\mathcal E';R)$ and the adelic $q$-expansion mod $q^B$ of a basis of $\mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps';R)$, computes the adelic $q$-expansion mod $q^B$ of all normalised eigenforms in $\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R)$.} \label{alg:eig}
\begin{algorithmic}
\State $V_0 \gets \text{the image of the adelic $q$-expanion map of } E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R) \mod q^B$
\State $V \gets \text{the largest Hecke stable submodule of } V_0\text{ using Algorithm \ref{alg:alg}}$
\State $<\beta_1>,...,<\beta_\ell>\gets$ simultaneous eigenspaces of $V$ \\
\Return {$\left\{\frac{1}{a_1(\beta_i)}\beta_i\ \big\vert\ \beta_i^2 \in \mathcal{M}_{2 k+2 k'}(\mathfrak N,\mathcal E^2\mathcal E'^2;R)\right\} $}
\end{algorithmic}
\end{algorithm}
\begin{theorem}\label{thm:MAIN}
Let $R$ be a \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} satisfying conditions \eqref{eq:condGe} and \eqref{eq:condAd}.
Let $ k$ and $ k'$ be weight vectors in $W$, let $\mathcal E $ and $\mathcal E'$ be $R$-valued characters and let $E$ be a HMF in $\mathcal{M}_{k'}(\mathfrak N,\mathcal E;R)$ such that $a_{(0)}(E)$ is invertible in $R^{\mathrm{Cl}^+}$. Let $V$ be the largest Hecke stable submodule of the image of the adelic $q$-expansion map of $E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ in $\Adqmod R B$.
\begin{enumerate}
\item The submodule $V$ contains the image of the adelic $q$-expansion map of $\mathcal{M}_k(\mathfrak N,\mathcal E;R)$.
\item If the bound $B$ is larger than the Sturm bound for $\mathcal{M}_{2 k+2 k'}(\mathfrak N,\mathcal E^2\mathcal E'^2;R)$, then the $R$-module of adelic $q$-expansions mod $q^B$ of $\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R)$ is precisely the $R$-module of mod $q^B$-adelic power series $v$ in $V$ such that $v^2$ agrees with the adelic $q$-expansion of a HMF in $M_{2 k}(\mathfrak N,\mathcal E^2;R)$ mod $q^B$.
\item If either $R$ is a field or an $S$-algebra for some \textsc{pid} $S$ such that $\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ is free of finite rank as an $S$-module and the bound $B$ is such that the Hecke algebra $\mathds{T}_k(\mathfrak N,\mathcal E;\text{Frac}(S)\otimes_S R)$ is generated as an $R$-module by the Hecke operators $T_\mathfrak m$ with $\Norm \mathfrak m ^2 \le B$,
then Algorithm \ref{alg:alg} computes the largest Hecke stable submodule $V$ of the image of the adelic $q$-expansion map of $E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ in $\Adqmod R B$ in finite time. \label{thm:MAIN3}
\item If $R$ is an algebraically closed field, the bound $B$ is larger than the Sturm bound for $\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ and the Hecke algebra $\mathds{T}_k(\mathfrak N,\mathcal E;R)$ is generated by all Hecke operators $T_\mathfrak m$ with $\Norm \mathfrak m \le B$, then Algorithm \ref{alg:eig} computes the adelic $q$-expansion mod $q^B$ of all normalised eigenforms of $\mathcal{M}_{k}(\mathfrak N,\mathcal E;R)$.
\end{enumerate}
Finally, the analogous statements hold for $R$-modules of cuspidal HMFs.
\end{theorem}
\begin{proof}
\begin{enumerate}
\item The $R$-module of adelic $q$-expansions mod $q^B$ of $\mathcal{M}_{ k}(\mathfrak N,\mathcal E;R)$ is a Hecke stable submodule of the image of the adelic $q$-expansion map of $E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ in $\Adqmod R B$, hence it is contained in the largest Hecke stable submodule.
\item This follows from Theorem \ref{cor:FracIsHolQexpR}.
\item As in Algorithm \ref{alg:alg}, we take $V_0$ to be the $R$-module of adelic $q$-expansions mod $q^B$ of $E^{-1}\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ and $\{\mathfrak m_i\}_{i}$ the ideals of $\mathcal{O}_K$ such that
$$ V_{i} := \big({T_{\mathfrak m_i}}\vert_ {V_{i-1}}\big)^{-1} \big(\pi_{ B/\Norm {\mathfrak m_i}}(V_{i-1})\big) \subset V_{i-1}. $$
Clearly, the largest Hecke-stable submodule $V$ is contained in each of the spaces $V_i$. Hence, if the algorithm terminates, the result is the largest Hecke-stable subspace of $V$. It remains to show that the algorithm does terminate after a finite number of iterations.
Note that by construction, each of the inclusion $V_{i} \subset V_{i-1}$ is strict, so we obtain a descending chain of $R$-modules
$$V_0 \supset V_1 \supset ... \supset V_{i-1} \supset V_{i} \supset ...\supset V.$$
If $R$ is a field we obtain a strictly decreasing sequence of positive integers
$$\text{dim}_R( V_0) >\text{dim}_R( V_1)>\cdots > \text{dim}_R(V_{i-1}) > \text{dim}_R (V_{i})>....> \text{dim}_R(V).$$
Since, $V_0$ is finite dimensional, the descending chain $V_i$ terminates after at most $\text{dim}_R( V_0)$ iterations.
Let $S$ be the \textsc{pid} such that $\mathcal{M}_{k+k}(\mathfrak N,\mathcal E\eps';R)$ is free of finite rank as an $S$-module, by the above argument, it suffices to show that
$$ \text{rank}_S (V_{i-1}) = \text{rank}_S (V_{i})\text{ if and only if } V_{i-1} = V_{i}.$$
We will show that the following two statements hold for any bound $B$ such that the Hecke algebra $\mathds{T}_k(\mathfrak N,\mathcal E;\text{Frac}(S)\otimes R)$ is generated as an $R$-module by the Hecke operators $T_\mathfrak m$ with $\Norm \mathfrak m ^2 \le B$.
\begin{enumerate}
\item[(i)]\label{claim:1} For all $v_0$ in $\text{Frac}(S)\otimes_S \Adqmod R B$ and all $s \in S\setminus \{0\}$,
$$s\cdot v_0 \in V_0 \text{ and } \pi_{\sqrt B}(v_0) \in\Adqmod R {\sqrt B} \text{ if and only if } v_0\in V_0.$$
\item[(ii)]
For all $v$ in $\Adqmod R B$, all $s \in S \setminus \{0\}$ and all integers $i\ge 0$,
$$s \cdot v \in V_i \text{ if and only if }v \in V_i.$$
\end{enumerate}
The `only if' part of (i) is immediate. Conversely, let $v_0$ be an element of $\text{Frac}(S) \otimes_S \Adqmod R B $ and $s$ a non-zero element of $S$ such that $s\cdot v_0 \in V_0$ and $\pi_{\sqrt B}(v_0) \in \Adqmod R {\sqrt B}$. By definition of $V_0$, there exists a HMF $f$ in $\mathcal{M}_{ k + k'}(\mathfrak N,\mathcal E\eps';R)$ such that
$$ s\cdot v_0 \cdot E \equiv f \mod q^{ B}.$$
Now $f/s$ is a HMF in $\mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps',\text{Frac}(S)\otimes_S R)$ whose coefficients mod $q^{ B}$ agree with those of $v_0\cdot E$. The coefficients of $v_0$ and $E$ mod $q^{\sqrt B}$ are elements of $R$, hence the coefficients of $f/s$ mod $q^{\sqrt B}$ are contained in $R$. Since the $R$-module $\mathds{T}_{ k+ k'}(\mathfrak N,\mathcal E\eps';\text{Frac}(S)\otimes R)$ is generated by the Hecke operators $T_\mathfrak b$ with $\Norm \mathfrak b \le \sqrt B$, Corollary \ref{cor:AdelInSubR} implies that $f/s$ is a HMF in $\mathcal{M}_{ k + k'}(\mathfrak N,\mathcal E\eps';R)$. So $f/(s\cdot E)$ is a fractional HMF in $E^{-1}\mathcal{M}_{ k + k'}(\mathfrak N,\mathcal E\eps';R)$ whose adelic $q$-expansion mod $q^B$ agrees with $v_0$, i.e. $v_0$ is an element of $V_0$.
Next, we prove statement (ii). Again, the `only if' part is immediate. We prove the converse by induction on $i \ge 0$. Conversely, the case $i=0$ is a special case of statement (i).
Let $i>0$ be an integer, $v \in\Adqmod R B$ and $s \in S\setminus \{0\}$ such that $s\cdot v \in V_{i}$. By construction of $V_i$ this means that $s\cdot v \in V_{i-1}$ and $T_{\mathfrak m_i}(s\cdot v) \in\pi_{B/\Norm {\mathfrak m_i}}( V_{i-1})$, i.e. $s \cdot v \in V_{i-1}$ and there exists an element $v'$ in $V_{i-1}$ such that
$$T_{\mathfrak m_i}(s\cdot v) = \pi_{B/\Norm {\mathfrak m_i}}(v').$$
The element $v'/s$ is an element of $\text{Frac}(S) \otimes_S V_{i-1} \subset \text{Frac}(S) \otimes _S V_0 $ such that $$\pi_{\sqrt B}(v'/s) = \pi_{\sqrt B} \circ T_{\mathfrak m_i}(v) \in \Adqmod R {\sqrt B}$$. Note that the projection mod $q^{\sqrt B}$ of $T_{\mathfrak m_i} (v)$ is well defined since $ B > \Norm {\mathfrak m_i}^2$. So by statement (i), $v'/s$ is an element of $V_0$. In particular, $v'/s$ is an element of $\Adqmod R B$ and $s \cdot v'/s = v' \in V_{i-1}$, the induction hypothesis implies that $v'/s$ is an element of $V_{i-1}$. So $$T_{\mathfrak m_i} (v) = \pi_{B/\Norm {\mathfrak m_i}}(v'/s) \in \pi_{B/\Norm {\mathfrak m_i}}(V_{i-1})$$ and $v \in V_{i-1}$, hence $v$ is an element of $V_i$. This completes the proof by induction of part (ii).
Finally, we show that (ii) implies that $$ \text{rank}_S (V_{i-1}) = \text{rank}_S (V_{i})\text{ if and only if } V_{i-1} = V_{i}$$ for all $i \ge 0$.
Suppose that $\text{rank}_S (V_{i-1}) = \text{rank}_S (V_{i})$, then $\text{Frac}(S) \otimes_S V_{i-1} = \text{Frac}(S) \otimes_S V_{i}$. Let $v$ be an element of $V_{i-1}$, then there exists an element $s$ in $S$ such that $s\cdot v \in V_{i}$. By statement (ii), this implies that $v \in V_{i}$.
\item To show the algorithm is well defined, it suffices to prove that all simultaneous eigenspaces of $V$ are $1$-dimensional. By Proposition \ref{prop:HeckeGeo} any normalised eigenvector $f$ in $\Adqmod R B$ satisfies
$$T_\mathfrak m(f) = a_\mathfrak m(f) \cdot f.$$
So if $\beta$ and $\beta'$ are normalised simultaneous eigenvectors in the same simultaneous eigenspace, then $a_\mathfrak m(\beta) = a_\mathfrak m(\beta')$ for all ideals $\mathfrak m$ with $\Norm \mathfrak m \le B$, so $\beta = \beta'$ mod $q^B$. Since $\beta^2 \in \mathcal{M}_{2 k+2 k'}(\mathfrak N,\mathcal E^2\mathcal E'^2;R)$ for all $\beta$ in the output of the algorithm and since $B$ is larger than the Sturm bound for $\mathcal{M}_{2 k+2 k'}(\mathfrak N,\mathcal E^2\mathcal E'^2;R)$, we can conclude by Theorem \ref{cor:FracIsHolQexpR} that each of the adelic power series mod $q^B$ in the output of the algorithm agrees with the adelic $q$-expansion of some normalised HMF $f$ in $\mathcal{M}_{k}(\mathfrak N,\mathcal E;R)$, with $f$ a normalised eigenvector for all Hecke operators $T_\mathfrak m$ with $\Norm B \le \mathfrak m$. Since these Hecke operators generate the full Hecke algebra, the form $f$ is a normalised eigenform.
\end{enumerate}
\end{proof}
One application of our algorithm is to compute examples of non-liftable HMFs of parallel weight $1$. The following corollary to Theorem \ref{thm:MAIN} will allow us to compute the $q$-expansion of HMFs with coefficients in $\mathds{F}_p$ for almost all primes $p$ simultaneously, under certain conditions. That is, it will compute a space of HMFs and a finite list of primes such that the projection morphism is surjective for all primes $p$ not in the list.
This enables us to find explicit non-liftable HMFs by rerunning the algorithm in characteristic $p$ for primes in the finite list.
\begin{corollary}\label{cor:nonlift}
Let $\mathcal{O}$ be a ring of integers with trivial class group, $ k$ and $ k'$ be parallel weight vectors, $\mathfrak N$ an integral ideal, $\mathcal E$ and $\mathcal E'$ characters mod $\mathfrak N$ with values in $\mathcal{O}$ and $E$ a HMF in $\mathcal{M}_{k'}(\mathfrak N,\mathcal E;\mathcal{O}[\tfrac 1{\Norm \mathfrak N}])$ whose constant coefficient $a_{(0)}(E)$ has no component equal to $0$. Let $\widetilde R$ be the smallest $\mathds{Z}[\tfrac 1 {\Norm \mathfrak N}]$-algebra containing $\mathcal{O}[\tfrac 1{\Norm \mathfrak N}]$ and such that all components of $a_{(0)}(E)$ are invertible in $\widetilde R$, i.e $$\widetilde R := \mathcal{O}\left[\tfrac{1}{\Norm \mathfrak N},\tfrac 1 {a_{(0)}(E)^{\underline 1}}\right].$$
If Algorithm \ref{alg:alg} over the ring $\widetilde R $ yields a candidate submodule $V$ of $\Adq {\widetilde R}$ such that the adelic $q$-expansion map followed by the natural projection mod $q^B$ is an isomorphism of $\widetilde R$-modules from $\mathcal{M}_{ k}(\mathfrak N,\mathcal E; \widetilde R)$ to $V$, then it also yields a finite set of prime ideals $\mathcal{L}$ such that for all primes $\mathfrak p$ not contained in $\mathcal{L}$ the conditions
\begin{enumerate}
\item The projection morphism $\mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps'; \widetilde R) \rightarrow \mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps'; \widetilde R/\mathfrak p)$ is surjective and
\item The Sturm bound for $\mathcal{M}_{k}(\mathfrak N,\mathcal E; \widetilde R/\mathfrak p)$ is smaller than the precision $B$,
\end{enumerate}
imply that the projection $ \mathcal{M}_{ k}(\mathfrak N,\mathcal E; \widetilde R) \rightarrow \mathcal{M}_{ k}(\mathfrak N,\mathcal E; \widetilde R/\mathfrak p)$ is surjective.
The analogous statement holds for $R$-modules of cuspidal HMFs.
\end{corollary}
\begin{proof}
Note that the ring $\widetilde R$ is a \textsc{pid} since it is the localisation of a \textsc{pid}. Moreover, $a_{(0)}(E)$ is a unit in $\widetilde R^{\mathrm{Cl}^+}$ and both $\mathcal E$ and $\mathcal E'$ are $\widetilde R$-valued characters. In particular we can apply Theorem \ref{thm:MAIN} and Algorithm \ref{alg:alg} both over $\widetilde R$ and any quotient $\widetilde R / \mathfrak p$ with $\mathfrak p$ a prime ideal of $\widetilde R$.
Let $V(\widetilde R)$ be the output of Algorithm \ref{alg:alg}. Note that by construction $V(\widetilde R)$ is the solution of a system of linear equations defined over $\widetilde R$. Let us denote $V(\widetilde R / \mathfrak p)$ for the $\widetilde R / \mathfrak p$-vectorspace of solutions of the mod $\mathfrak p$-reduced linear system. Then for almost all primes $\mathfrak p$ we have
$$V(\widetilde R / \mathfrak p) = V(\widetilde R)/\mathfrak p,$$
i.e. for almost all primes $\mathfrak p$ the solution of the reduced system is the reduction of the solution of the system. We define $\mathcal{L}$ to be the set of primes for which this equality does not hold. Note that since $\widetilde R$ is a \textsc{pid}, we can compute the Smith normal form of the matrix representation of the linear system of equations defining $V$. Moreover the set $\mathcal{L}$ is contained in the set of primes dividing the pivot elements of the Smith normal form of the matrix representation of the linear system of equations defining $V$, hence we can explicitly determine all primes $\mathfrak p$ such that $V(\widetilde R / \mathfrak p) \ne V(\widetilde R)/\mathfrak p$.
Let $\mathfrak p$ be a prime not contained in $\mathcal{L}$ and such that the projection morphism $$\mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps'; \widetilde R) \rightarrow \mathcal{M}_{ k+ k'}(\mathfrak N,\mathcal E\eps'; \widetilde R / \mathfrak p)$$ is surjective, then the first part of Theorem \ref{thm:MAIN} implies that the candidate space $V(\widetilde R / \mathfrak p)$ contains the adelic $q$-expansions mod $q^B$ of $\mathcal{M}_{ k}(\mathfrak N,\mathcal E,\widetilde R / \mathfrak p)$.
Let us write $q^B(-)$ for the image of the adelic $q$-expansion map followed by the natural projection mod $q^B$.
Then, the first assumption of the corollary says that
$$ q^B(-):\mathcal{M}_{ k }(\mathfrak N,\mathcal E,\widetilde R) \rightarrow V(\widetilde R)$$
is an isomorphims of $\widetilde R$-modules. Hence, we obtain the following commutative diagram
$$\xymatrix{
q^B\big(\mathcal{M}_{ k }(\mathfrak N,\mathcal E;\widetilde R) \big)/\mathfrak p\ \ar@{=}[rd] \ar@{^{(}->}[r] & q^B\big(\mathcal{M}_{ k }(\mathfrak N,\mathcal E;\widetilde R / \mathfrak p )\big)\,\ar@{^{(}->}[r] & V(\widetilde R / \mathfrak p)\\
& V\big(\widetilde R)/\mathfrak p \ar@{=}[ru] &
}$$
where all injections are inclusions. In particular, we obtain
$$q^B\big(\mathcal{M}_{ k }(\mathfrak N,\mathcal E;\widetilde R)\big)/\mathfrak p = q^B\big(\mathcal{M}_{ k }(\mathfrak N,\mathcal E;\widetilde R /\mathfrak p)\big).$$
Finally, if $\mathfrak p$ is a prime such that the Sturm bound for weight $ k$, level $\mathfrak N$ and character $\mathcal E$ over $\widetilde R/\mathfrak p$ is less than the precision $B$, then this implies that the projection $$ \mathcal{M}_{ k}(\mathfrak N,\mathcal E;\widetilde R) \rightarrow \mathcal{M}_{ k}(\mathfrak N,\mathcal E; \widetilde R/\mathfrak p)$$ is surjective.
\end{proof}
\begin{remark}
\begin{enumerate}
\item If the \mbox{$\mathds{Z}[ \frac 1 {\Norm \mathfrak N}]$-algebra} $R$ in Theorem \ref{thm:MAIN}.\ref{thm:MAIN3} is itself free of finite rank as an $S$-module for some \textsc{pid} $S$, then $\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E\eps';R)$ is free of finite rank as an $S$-module. In particular we can apply Algorithm \ref{alg:alg} over the ring of integers of any number field since any such ring is free of finite rank as a $\mathds{Z}$-module.
\item There are $46$ cyclotomic fields with class number one, see for example \cite[Theorem 11.1]{Law83}. This classification gives an indication of the orders of the characters $\mathcal E$ and $\mathcal E'$ in Corollary \ref{cor:nonlift}. In particular, $\mathds{Q}(\zeta_n)$ has class number one for $n$ up to $22$.
\item If $F$ is a number field of class number greater than one, we cannot apply Corrolary \ref{cor:nonlift} to compute in all characteristics simulataniously. However, we can compute HMFs over any quotient $\mathcal{O}_F/\mathfrak p$ using Algorithm \ref{alg:alg}.
\end{enumerate}
\end{remark}
\section{Numerical Examples}
In this section, we discuss explicit results obtained by our implementation of the above algorithms in Magma.
\subsection*{Limitations}
In order to apply Theorem \ref{thm:MAIN} and Corollary \ref{cor:nonlift} to prove that an adelic power series obtained from Algorithms \ref{alg:alg} or \ref{alg:eig} corresponds to the adelic $q$-expansion of a HMF in $\mathcal{M}_{ k}(\mathfrak N,\mathcal E,\overline\mathds{F}_ p)$ the precision $B$ must be larger than the Sturm bound for $\mathcal{M}_{ 2k+2 k'}(\mathfrak N,\mathcal E^4,\overline\mathds{F}_ p)$. However, the best known bound on the Sturm bound, $2(k+k') \Norm \mathfrak N ^3 $, is beyond what we can compute in a reasonable time with the available computing power.
This remark holds in all weights, levels and characters. More precisely, our computations in characteristic $0$ yield a candidate space that is a upper bound, i.e. the output of the algorithm contains the space of adelic $q$-expansions of HMFs, but this inclusion could be strict if the precision is lacking.
A second obstruction to proving that the computed space of adelic power series agrees with the adelic $q$-expansions of HMFs arises from the first condition in Corollary \ref{cor:nonlift} which requires the projection morphism
$$\mathcal{M}_{k+k'}(\mathfrak N,\mathcal E^2,\widetilde R) \rightarrow \mathcal{M}_{k+k'}(\mathfrak N,\mathcal E^2,\widetilde R/\mathfrak p)$$
to be surjective. For parallel weight at least $3$ this is precisely the main result of \cite{LaSu14}. However, no such result is proven for parallel weight $2$.
If the projection morphism is not surjective for a given prime $\mathfrak p$, the computed candidate space is only a lower bound, i.e. there could, a priori, exist more HMFs in $\mathcal{M}_{k}(\mathfrak N,\mathcal E,\widetilde R/\mathfrak p)$.
Given enough time and computational power, one could use our algorithms to compute a bound on the Sturm bound in individual cases. One could also use Corollary \ref{cor:nonlift} to verify that the projection morphism in parallel weight $2$ is indeed surjective for all primes $\mathfrak p$ since all HMFs in parallel weight larger than $2$ are liftable. These computations would then yield proven adelic $q$-expansions. However, the time required is beyond reasonable.
Instead, we run our algorithm with increasing precision in steps of $500$. If the number of linearly independent eigenforms remains the same after increasing the bound we are led to believe that we have computed with sufficient precision. In the example that follows, using coefficients with ideals up to norm at most $2000$ sufficed.
\subsection*{A Non-liftable Example}
The real quadratic field $\mathds{Q}(\sqrt 6)$ has class number $1$ and narrow class number $2$. Let $\omega$ denote $\sqrt 6$, let $\mathfrak N_{331} = ( 25 + 7\omega)$ be a prime ideal above $331$ and let $\mathcal E$ be the unique quadratic character mod $\mathfrak N_{331}$.
The Eisenstein series $E_{\underline 1}(\mathcal E,1)$, see \cite[Proposition 2.1]{DDP11} is cuspidal. However, the Eisenstein series $E_{\underline 1}(\mathcal E',1)$ with $\mathcal E'$ the primitive character inducing $\mathcal E$ has constant term $a_{\mathcal{O}} = [\frac 1 {12},\frac 1 {12}]$. In particular, $12\cdot E_{\underline 1}(\mathcal E',1)$ is an (old) HMF in $\mathcal{M}_{\underline 1}(\mathfrak N_{331},\mathcal E,\mathds{Z}[\frac 1 {\Norm { \mathfrak N_{331} } } ])$ whose adelic $q$-expansion is invertible in $\Adq {(\mathds{F}_p)}$ for all primes $p$ and $W$ the set of parallel weight vectors.
So we can apply algorithms \ref{alg:alg} and \ref{alg:eig} to obtain
\begin{align*} \text{dim}\big(\mathcal{S}_{\underline 1}(\mathfrak N_{331},\mathcal E,\mathds{C})\big) &= 0 \text{ and } \\
\text{dim}\big( \mathcal{S}_{\underline 1}(\mathfrak N_{331},\mathcal E,\overline \mathds{F}_p) \big) &= \begin{cases} 2 & \text{ if } p = 3 \\ 0 & \text{ else.} \end{cases}
\end{align*}
Moreover, we can compute normalised eigenforms $f$ and $f^\sigma$ with coefficients in $\mathds{F}_9 = \mathds{F}_3(\zeta)$ where $\zeta^4 = 1$. The eigenform $f$ and its Galois conjugate $f^\sigma$ span the space $\mathcal{S}_{\underline 1}(\mathfrak N_{331},\mathcal E, \overline \mathds{F}_3)$. For primes $\mathfrak p$ with norm up to $25$ we list the norm of the prime, a generator $\alpha_\mathfrak p$ of the prime and the coefficient, hence Hecke eigenvalue, of the normalised eigenforms $a_\mathfrak p(f)$ and $a_\mathfrak p(f^\sigma)$ in Table \ref{table:S1_331}.
The absolute and relative frequencies of elements of $\mathds{F}_9$ occurring as a Hecke eigenvalue for primes up to norm $2000$ are given in Table \ref{table:S1_331_Freq}. These frequencies suggest that the image of the associated Galois representation in $\GL {\mathds{F}_9}$ is one of the following subgroups $H_1\cong \mathds{Z}/8\mathds{Z}\times \mathds{Z}/2\mathds{Z} $, $H_2=\text{SmallGroup}(48,33)$ or $H_3=\text{SmallGroup}(144,130)$. The images of these subgroups in $\text{PGL}_2(\mathds{F}_9)$ are respectively isomorphic to $ \mathds{Z}_4$, $A_4$ and $\text{SmallGroup}(36,9)$.
\begin{table}[H]
\captionsetup{font=small}
\caption{ The Hecke eigenvalues for primes $\mathfrak p =(\alpha_\mathfrak p)$ with norm less than $25$ for the normalised eigenforms $f$ and $f^\sigma$ spanning the space $\mathcal{S}_{\underline 1}(\mathfrak N_{331},\mathcal E, \overline \mathds{F}_3)$.}
$$\begin{tabu}{c|cccccccccccccccc}
\Norm \mathfrak p & 2 & 3 & 5 & 5 & 19 &19 &23& 23
\\ \alpha_\mathfrak p & 2 - \omega & 3-\omega & 1 +\omega & 1-\omega & 5- \omega & 5+\omega & 1+2\omega & 1-2\omega
\\ \hline a_\mathfrak p(f) & -\zeta & 1 & \zeta & 0 & 2 & 0 & \zeta & 0
\\ a_\mathfrak p(f^\sigma) & \zeta & 1 & -\zeta &0 & 2 & 0 &-\zeta & 0
\end{tabu}$$
\label{table:S1_331}
\end{table}
\begin{table}[H]\captionsetup{font=small}
\caption{The absolute and relative frequencies of elements $ x \in \mathds{F}_9=\mathds{F}_3(\zeta)$ occurring as the Hecke eigenvalues for primes $\mathfrak p$ with norm less than $2000$ for the normalised eigenforms $f$ and $f^\sigma$ spanning the space $\mathcal{S}_{\underline 1}(\mathfrak N_{331},\mathcal E, \overline \mathds{F}_3)$.}
\label{table:S1_331_Freq}
$$\begin{tabu}{c|ccccc}
x & a_\mathfrak p(f) &a_\mathfrak p(f) & a_\mathfrak p(f^\sigma) & a_\mathfrak p(f^\sigma)
\\ & \text{abs. freq. } & \text{rel. freq. } & \text{abs. freq. } &\text{abs. freq. }
\\\hline 0 & 71 & 0.24 & 71 & 0.24
\\ 1 & 56 & 0.19& 56 & 0.19
\\ -1 & 52 & 0.18 & 52 & 0.18
\\\zeta & 63& 0.21 & 53 & 0.18
\\ -\zeta & 53 & 0.18 & 63 & 0.21
\end{tabu}$$
\end{table}
| {
"timestamp": "2017-10-09T02:04:56",
"yymm": "1710",
"arxiv_id": "1710.02287",
"language": "en",
"url": "https://arxiv.org/abs/1710.02287",
"abstract": "In this article we present an algorithm that uses the graded algebra structure of Hilbert modular forms to compute the adelic $q$-expansion of Hilbert modular forms of weight one as the quotient of Hilbert modular forms of higher weight. The main improvement to existing methods is that our algorithm can be applied in weight $1$, which fills a gap left by standard computational methods. Additionally, the algorithm can be used to compute Hilbert modular forms over finite fields in all characteristic simultaneously. We use this algorithm to compute a first candidate of a Hilbert modular form of parallel weight $1$ that is non-liftable and specify the exact conditions under which our candidate $q$-expansion corresponds to a non-liftable Hilbert Modular form.",
"subjects": "Number Theory (math.NT)",
"title": "Explicit Methods for Hilbert Modular Forms of Weight 1",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846634557751,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7092019612063575
} |
https://arxiv.org/abs/1907.08646 | Fair quantile regression | Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to "unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the target quantiles are brought into balance, in a statistical sense that we call $\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a dataset of 200,000 live births, where the objective is to characterize the dependence of the birth weights of the babies on demographic attributes of the birth mother; the protected attribute is the mother's race. | \section*{Acknowledgment}
Research supported in part by ONR grant N00014-12-1-0762 and NSF grant DMS-1513594.
\section{Background}
\label{sec:background}
In this section we review the essentials of quantile
regression that will be relevant to our analysis. We also
briefly discuss definitions of fairness.
\subsection{Linear programming formulation}
The formulation of quantile estimates as solutions to linear programs starts with the ``check'' or ``hockey stick'' function $\rho_\tau(u)$ defined by $\rho_\tau(u)= (\tau-1) u\mathds{1}\{u\leq 0\}+ \tau u\mathds{1}\{u>0\}$.
For the median, $\rho_{1/2}(u) = \frac{1}{2} |u|$. If $Y\sim F$ is a random variable,
define $\hat\alpha(\tau)$ as the solution to the optimization $\hat\alpha(\tau) = \mathop{\text{arg\,min}}_a \mathbb{E} \rho_\tau(Y-a)$. Then the stationary condition is seen to be
\begin{equation*}
0 = (\tau-1) \int_{-\infty}^\alpha dF(u) + \tau \int_\alpha^\infty dF(u) = (\tau-1) F(\alpha) + \tau (1-F(\alpha)),
\end{equation*}
from which we conclude $\hat\alpha(\tau) = F^{-1}(\tau)$ is the $\tau$-quantile of $F$. Similarly the conditional quantile of $Y$ given random variable $X\in\mathbb{R}^p$ can be written as the solution to the optimization
$
q_\tau(x) = \mathop{\text{arg\,min}}_q \mathbb{E}\left( \rho_\tau(Y - q) \,|\, X=x\right).
$
For a linear estimator $q_\tau(X) = X^T \hat{\beta}_\tau$, minimizing the
empirical check function loss leads to a convex optimization $\hat{\beta}_\tau=\mathop{\text{arg\,min}}_{\beta} \sum_{i\leq n}\rho_\tau (Y_i-X_i^T\beta)$. Dividing the residual $Y_i-X_i^T\beta$ into positive part $u_i$ and negative part $v_i$ yields the linear program
\[
\min_{u,v\in\mathbb{R}^n, \beta\in\mathbb{R}^p} \; \tau \mathds{1}^T u + (1-\tau)\mathds{1}^T v,\;\;\;
\text{such that} \; Y = X\beta + u - v ,\;
u \geq 0, \; v\geq 0.
\]
The dual linear program is then formulated as
\begin{equation}
\label{eq:dual}
\max_{b} \; Y^T b \;\;\;
\text{such that} \; X^T b = (1-\tau) X^T \mathds{1},\;
b \in [0,1]^n.
\end{equation}
When $n>p$, the primal
solution is obtained from a set of $p$ observations $X_h\in \mathbb{R}^{p\times p}$ for
which the residuals are exactly zero, through
the correspondence
$
\hat\beta_\tau = X_h^{-1} Y_h.
$
The dual variables $\hat b_\tau\in [0,1]^n$, also known as regression rank scores, play the role of ranks. In particular, the quantity $\int_0^1 \hat b_{\tau,i} d\tau$ can be interpreted as the quantile
at which $Y_i$ lies for the conditional distribution of $Y$ given
$X_i$ \citep{gutenbrunner1992regression}. As seen below,
the stochastic process $\hat b_\tau$ plays an important role in fairness and inference for quantile regression.
\subsection{Notions of fairness}
\label{sec:fair.def}
\cite{hardt2016equality} introduce the notion of
{\it equalized odds} to assess fairness of
classifiers. Suppose a classifier $\widehat{Y}$ serves to estimate
some unobserved binary outcome variable $Y$. Then the estimator is
said to satisfy the equalized odds property with respect to a
protected attribute $A$ if
\begin{equation}\label{eo.def}
\widehat{Y}\Perp A \,|\, Y.
\end{equation}
This fairness property requires that the true positive
rates~$\mathbb{P}\{\widehat{Y}=1 \,|\, Y=1, A\}$ and the false
positive rates $\mathbb{P}\{\widehat{Y}=1 \,|\, Y=0, A\}$ are
constant functions of~$A$. In other words,
$\widehat{Y}$ has the same proportion of type-I and type-II errors
across the subpopulations determined by the different values of $A$.
This could be extended to a related notion of fairness for quantile
regression estimators. Denote the
true conditional quantiles for outcome $Y$ given attributes $X$ as
$q_\tau(X)$. Analogous to the definition of equalized odds in~\eqref{eo.def}, we
would call a quantile estimator $\widehat{q}_\tau(X)$ fair if
\begin{equation}
\mathds{1} \left\{Y>\widehat{q}_\tau(X)\right\}\Perp A \,|\, \mathds{1}\left\{Y>q_\tau(X)\right\}.
\end{equation}
Conditioned on the event $\left\{Y
\leq q_\tau(X)\right\}$, we say that
$\left\{Y > \widehat{q}_\tau(X)\right\}$ is a false positive.
Conditioned on the complementary event $\left\{Y>q_\tau(X)\right\}$, we
say that $\left\{Y \leq \widehat{q}_\tau(X)\right\}$ is a false
negative. Thus, an estimator is fair if the false positive and false
negative rates do not depend on the protected attribute $A$.
The notion of fairness that we focus on in this paper is a natural
one. Considering binary $A$, we ask if the average quantiles conditional on the protected
attribute agree for $A=0$ and $A=1$. More precisely, define the effective quantiles as
\begin{equation}
\label{eq:effective.quantile}
\hat{\tau}_a=\mathbb{P}\left\{Y\leq \hat{q}_\tau(X)\,|\, A=a\right\},\;\;\; a = 0,1.
\end{equation}
We say that the estimator $\hat q_\tau$ is fair if $\hat{\tau}_0=\hat{\tau}_1$.
Typically when $\hat{q}_\tau$ is trained on a sample of size $n$, exact equality is too strong to ask for.
If the estimators are accurate, each of the effective quantiles should be approximately
$\tau$, up to stochastic error that decays at rate $1/\sqrt{n}$.
We say $\hat{q}_\tau$ is $\sqrt{n}$-fair if $\hat{\tau}_0=\hat{\tau}_1+O_p(1/\sqrt{n})$.
As shall be seen,
this fairness property follows from the linear programming
formulation when $A$ is included in the regression. As seen from the
birth weight example in Section~\ref{sec:birth},
if $A$ is not available at training time, the quantiles can be
severely under- or over-estimated for a subpopulation.
This formulation of fairness is closely related
to calibration by group, and demographic parity
\citep{kleinberg2016inherent,hardt2016equality,chouldechova2017fair}.
An advantage of this fairness definition is that it can be
evaluated empirically, and does not require a correctly specified model.
\subsection{Birthweight data analysis}
\label{sec:birth}
\afterpage{
\begin{figure}[t!]
\begin{center}
\def.16{.16}
\begin{tabular}{ccccc}
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-1} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-2} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-3} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-14} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-16} \\
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-4} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-5} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-6} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-7} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-12} \\
\end{tabular}
\end{center}
\caption{\small\it Quantile regression coefficients for birth data. The quantile $\tau$ runs along horizontal axis; curves are the coefficients $\hat \beta_\tau$; unit is grams. Solid/salmon: race is included in the model; dashed/blue: race excluded. When race is excluded, the strongly correlated variable ``married'' can be seen as serving as a kind of proxy.
}
\vskip-10pt
\label{fig:all}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[t]{0.35\textwidth}
\centering
\begin{footnotesize}
\begin{tabular}{| c || c | c | c | c | c |}
\hline\hline
target & $5$ & $25$ & $50$ & $75$ \\ \hline\hline
$\hat{\tau}_0$(before) & $4.42$ & $23.14$ & $47.87$ & $73.12$ \\
$\hat{\tau}_1$(before) & $7.91$ & $33.80$ & $60.02$ & $82.46$ \\
$\hat{\tau}_0$(after) & $5.03$ & $25.01$ & $49.77$ & $74.61$ \\
$\hat{\tau}_1$(after) & $5.02$ & $24.02$ & $49.95$ & $74.62$ \\
\hline
\end{tabular}
\end{footnotesize}
\caption{Quantiles before and after adjustment.}
\end{subfigure}%
~
\begin{subfigure}[t]{0.7\textwidth}
\centering
\begin{tabular}{cc}
\quad\quad\includegraphics[width=.35\textwidth]{fig/pred_wob} &
\includegraphics[width=.35\textwidth]{fig/pred_corrected}
\end{tabular}
\caption{Birth weights before and after adjustment.}
\end{subfigure}
\caption{\small \it Left: the effect of the adjustment procedure. Right: scatter plots of observed birth weights against estimated $20\%$ conditional quantiles over the test set, before and after adjustment.}
\label{fig:effective}
\end{figure}
}
The birth weight dataset from
\cite{abrevaya:01}, which is analyzed by
\cite{koenker2001quantile}, includes the weights
of 198,377 newborn babies, and other attributes of the babies
and their mothers, such as the baby's gender,
whether or not the mother is married, and the mother's age. One of the
attributes includes information about the race of the mother, which we treat as the
protected attribute $A$. The variable $A$ is binary---black ($A=1$) or
not black ($A=0$). The birth weight is reported in grams. The other attributes include education of the mother,
prenatal medical care, an indicator of whether the mother smoked during pregnancy,
and the mother's reported weight gain during pregnancy.
Figure~\ref{fig:all} shows the coefficients $\widehat{\beta}_\tau$ obtained
by fitting a linear quantile regression model, regressing birth weight
on all other attributes. The model is fit two ways,
either including the protected
race variable $A$ (solid, salmon confidence bands), or excluding $A$ (long dashed,
light blue confidence bands).
The top-right figure shows that babies of black mothers weigh less on
average, especially near the lower quantiles where they weigh nearly
300 grams less compared to babies of nonblack mothers. A description
of other aspects of this linear model is given by \cite{koenker2001quantile}.
A striking aspect of the plots is the disparity between birth weights of infants born to black and nonblack
mothers, especially at the left tail of the
distribution. In particular, at the 5th percentile of the conditional distribution,
the difference is more than 300 grams. Just as striking is the
observation that when the race attribute $A$ is excluded from the model, the variable
``married,'' with which it has a strong negative correlation, effectively serves
as a proxy, as seen by the upward shift in its regression
coefficients. However, this and the other variables do not completely
account for race, and as a result the model overestimates the weights
of infants born to black mothers, particularly at the lower
quantiles.
To correct for the unfairness of $\widehat{q}_\tau$, we apply the correction procedure described in Section~\ref{sec:results}. For the target quantile $\tau=20\%$, the corrected estimator $\tilde{q}_\tau$ achieves effective quantiles $20.4\%$ for the black population and $20.1\%$ for the nonblack population. Table~\ref{fig:effective} (left) shows the effective quantiles at a variety of quantile levels. We see that the correction procedure consistently pulls the effective quantiles for both subpopulations closer to the target quantiles.
For 1000 randomly selected individuals from the test set, Figure~\ref{fig:effective} (right) shows their observed birth weights plotted against the conditional quantile estimation at $\tau=20\%$ before (left) and after (right) the correction. The dashed line is the identity. When $A$ is not included in the quantile regression, the conditional quantiles for the black subpopulation are overestimated. Our procedure achieves fairness correction by shifting the estimates for the $A_i=1$ data points smaller (to the left) and shifting the $A_i=0$ data points larger (to the right). After the correction, the proportion of data points that satisfy $Y\leq\tilde{q}_\tau$ are close to the target $20\%$ for both subpopulations.
\section{Discussion}
In this paper we have studied the effects of excluding a distinguished
attribute from quantile regression estimates, together with procedures
to adjust for the bias in these estimates through post-processing.
The linear programming basis for quantile regression
leads to properties and analyses that complement what has
appeared previously in the fairness literature.
Several extensions of the work presented here
should be addressed in future work. For example, the generality of
the concentration result of Lemma~\ref{lmm:emp} could allow the extension
of our results to multiple attributes of different types. In the fairness
analysis in Section~\ref{sec:proof} we used a linear quantile regression in the adjustment step, which
allows us to more easily leverage previous statistical analyses
\cite{gutenbrunner1992regression} on quantile rank scores. Nonparametric
methods would be another interesting direction to explore.
The birth data studied here has been instrumental in developing our
thinking on fairness for quantile regression. It will be interesting
to investigate the ideas introduced here for other data sets. If the tail
behaviors, including outliers, of the conditional distributions for a
set of subpopulations are very different, and the identification of
those subpopulations is subject to privacy restrictions or other
constraints that do not reveal them in the data, the issue of bias in
estimation and decision making will come into play.
\section{Experiments}
\label{sec:experiments}
\subsection{Experiments on synthetic data}
In this section we show experiments on synthetic data that verify our theoretical claims.
\footnote{Code and data for all experiments are available online at \url{https://drive.google.com/file/d/1Ibaq5VWaAE4539hec4-UdIOgPsNv0x_t/view?usp=sharing}}
The experiment is carried out in $N=10{,}000$ independent repeated trials. In each trial, $n=1{,}000$ data points $(X,A,Y)\in \mathbb{R}^p\times \{0,1\}\times \mathbb{R}$ are generated independently as follows:
\begin{itemize}
\item Let $p=20$. Generate $X$ from the multivariate distribution with correlated attributes: $X\sim \mathcal{N}(0,\Sigma)$, where the the covariance matrix $\Sigma\in \mathbb{R}^{p\times p}$ takes value $1$ for diagonal entries and $0.3$ for off-diagonal entries.
\item The protected attribute $A$ depends on $X$ through a logistic
model: $A\,|\, X\sim \text{Bernoulli}(b)$ with
\begin{equation*}
b=\exp\left(X^T\gamma\right)/\left(1+\exp\left(X^T\gamma\right)\right).
\end{equation*}
\item Given $A,X$, generate $Y$ from a heteroscedastic model:
$Y\,|\, A,X\sim \mathcal{N}\left(X^T\beta+\mu A, (X^T\eta)^2\right)$.
\end{itemize}
The parameters $\beta$, $\gamma$, $\eta$ are all generated independently from $\mathcal{N}(0,I_p)$ and stay fixed throughout all trials. The coefficient $\mu$ is set to be 3.
In each of the $N$ trials, conditional quantile estimators are trained on a training set of size $n/2$ and evaluated on the remaining size $n/2$ held out set. We train three sets of quantile estimators at $\tau=0.5$:
\begin{enumerate}
\item Full quantile regression of $Y$ on $A$ and $X$.
\item Quantile regression of $Y$ on $X$ only.
\item Take the estimator from procedure 2 and correct it with the method described in Section~\ref{sec:results}.
\end{enumerate}
The average residuals $Y-\hat{q}_\tau(X,A)$ are then evaluated on the test set for the $A=0$ and $A=1$ subpopulations. In Figure~\ref{fig:synthetic} we display the histograms of these average residuals across all $N$ trials for the quantile regression estimator on $X$ (\ref{fig:residual.before}) and the corrected estimator (\ref{fig:residual.after}). In the simulation we are running, $A$ is positively correlated with the response $Y$. Therefore when $A$ is excluded from the regression, the quantile estimator underestimates when $A=1$ and overestimates when $A=0$. That is why we observe different residual distributions for the two subpopulations. This effect is removed once we apply the correction procedure, as shown in Figure~\ref{fig:fair.measure}.
We also test whether our correction procedure corrects the unbalanced effective quantiles of an unfair initializer. In each trial we measure the fairness level of an estimator $\widehat{q}_\tau$ by the absolute difference $|\hat\tau_1 - \hat\tau_0|$ between the effective quantiles of the two subpopulations on a heldout set $S$, where $\hat{\tau}_a$ is defined as in~\eqref{eq:effective.quantile}.
We established in previous sections that quantile regression excluding attribute $A$ is in general not fair with respect to $A$. A histogram of the fairness measure obtained from this procedure is shown in Figure~\ref{fig:fair.measure} (salmon). Plotted together are the fairness measures after the correction procedure (light blue). For comparison we also include the histogram obtained from the full regression (black). Note that the full regression has the ``unfair" advantage of having access to all observations of $A$. Figure~\ref{fig:fair.measure} shows that the correction procedure pulls the fairness measure to a level comparable to that of a full regression, which as we argued in Section~\ref{sec:training}, produces $\sqrt{n}$-fair estimators.
\begin{figure}
\centering
\begin{subfigure}[b]{0.27\textwidth}
\begin{centering}
\includegraphics[width=\textwidth]{fig/hist_residual_wob}
\caption[justification = centering]{}
\label{fig:residual.before}
\end{centering}
\end{subfigure}
\begin{subfigure}[b]{0.27\textwidth}
\begin{centering}
\includegraphics[width=\textwidth]{fig/hist_residual_corrected}
\caption[justification = centering]{}
\label{fig:residual.after}
\end{centering}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.30\textwidth}
\begin{centering}
\includegraphics[width=\textwidth]{fig/hist_fairness}
\caption[justification = centering]{}
\label{fig:fair.measure}
\end{centering}
\end{subfigure}
\caption{\small \it From left to right: (a): histograms of average residuals for quantile regression of $Y$ on $X$ only; (b): histograms of average residuals for the corrected quantile estimators; (c): histograms and density estimates of the fairness measures obtained by running quantile regression on $X$, before (salmon) and after (light blue) the adjustment procedure. The histogram from the full regression (black) serves as a benchmark for comparison.}
\label{fig:synthetic}
\end{figure}
\subsection{Birthweight Analysis}
\label{sec:birth}
\begin{figure*}[t]
\begin{center}
\def.16{.16}
\begin{tabular}{ccccc}
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-1} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-2} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-3} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-14} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-16} \\
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-4} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-5} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-6} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-7} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-12} \\
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-8} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-9} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-10} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-11} &
\includegraphics[width=.16\textwidth]{fig/qrcoeff-w-wo-black-13} \\
\end{tabular}
\end{center}
\caption{\small\it Quantile regression coefficients for the birth weight data. In each plot, the quantile $\tau$ runs along the horizontal axis, and the curves are the coefficients $\hat \beta_\tau$ for the variable noted on the vertical axis; the unit is grams. Solid/salmon coefficients are obtained when the race variable $A$ is included in the model, and dashed/blue coefficients are obtained when it is not included. When race is excluded, the strongly correlated variable ``married'' can be seen as serving as a kind of proxy.
}
\label{fig:all}
\end{figure*}
The birth weight dataset from
\cite{abrevaya:01}, which is analyzed by
\cite{koenker2001quantile}, includes the weights
of 198,377 newborn babies, and other attributes of the babies
and their mothers, such as the baby's gender,
whether or not the mother is married, and the mother's age. One of the
attributes includes information about the race of the mother, which we treat as the
protected attribute $A$. The variable $A$ is binary---black ($A=1$) or
not black ($A=0$).
The birth weight is reported in grams. Education of the mother is divided
into four categories: less than high school, high school, some college
and college graduate. The default or omitted category is less than high school so
coefficients may be interpreted relative to this value. The
prenatal medical care of the mother is also divided into four
categories: no prenatal visit, and first prenatal
visit in the first, second or trimester of the pregnancy.
The omitted category is the group with a first visit in the
first trimester, making up almost 85 percent of the sample. An
indicator of whether the mother smoked during pregnancy is included in the model, as well
as mother's reported average number of cigarettes smoked per day. The
mother's reported weight gain during pregnancy (in pounds) is included
as a quadratic effect.
Figure~\ref{fig:all} shows the coefficients $\widehat{\beta}_\tau$ obtained
by fitting a linear quantile regression model, regressing birth weight
on all other attributes. The model is fit two ways,
either including the protected
race variable $A$ (solid, salmon confidence bands), or excluding $A$ (long dashed,
light blue confidence bands).
The top-right figure shows that babies of black mothers weigh less on
average, especially near the lower quantiles where they weigh nearly
300 grams less compared to babies of nonblack mothers. A description
of other aspects of this linear model is given by \cite{koenker2001quantile}.
For instance, the intercept of the model (upper left plot) may be
interpreted as ``the estimated conditional quantile function of the
birth weight distribution of a girl born to an unmarried, white mother
with less than a high school education, who is 27 years old and had a
weight gain of 30 pounds, didn't smoke, and had her first prenatal
visit in the first trimester of the pregnancy.'' A striking aspect
of the plots is the disparity between birth weights of infants born to black and nonblack
mothers, especially at the left tail of the
distribution. In particular, at the 5th percentile of the conditional distribution,
the difference is more than 300 grams. Just as striking is the
observation that when the race attribute $A$ is excluded from the model, the variable
``married,'' with which it has a strong negative correlation, effectively serves
as a proxy, as seen by the upward shift in its regression
coefficients. However, this and the other variables do not completely
account for race, and as a result the model overestimates the weights
of infants born to black mothers, particularly at the lower
quantiles. For instance, at the target level $\tau=0.2$, the effective quantile for black mothers on a heldout set is $\hat\tau_1 \approx .29$. In other words, nearly 30\% of the births to black mothers have weights below the estimated level $\hat q_\tau(X)$, when the target is $20\%$. The target level of $20\%$ is more closely attained for the births to nonblack mothers, with $\hat\tau_0 \approx 0.184$.
We return to these data and analyses in Section~\ref{sec:experiments}, when we consider adjusting for race.
\section{Introduction}
Recent research on fairness has formulated interesting new perspectives on
machine learning methodologies and their deployment, through work
on definitions, axiomatic characterizations, case studies, and
algorithms
\citep{hardt2016equality,dwork12,kleinberg2016inherent,chouldechova2017fair,woodworth2017learning}.
Much of the work on fairness in machine learning has
been focused on classification, although the influential paper
of \cite{hardt2016equality} considers general frameworks that include regression.
Just as the mean gives a coarse summary of
a distribution, the regression curve gives a rough summary
of a family of conditional distributions \citep{MostellerTukey77}.
Quantile regression targets a more complete
understanding of the dependence between a response variable and a
collection of explanatory variables.
Given a conditional distribution $F_X(y) = \mathbb{P}(Y\leq y \,|\, X)$,
the quantile function $q_\tau(X)$ is characterized by $F_X(q_\tau(X))
= \tau$, or $q_\tau(X) = F_X^{-1}(\tau) = \inf \{y: F_X(y) \geq \tau\}$.
We consider the setting where an estimate $\what q_\tau(X)$ is formed
using a training set $\{(X_i, Y_i)\}$ for which a protected
attribute $A$ is unavailable. The estimate $\what q_\tau(X)$ will often give
quantiles that are far from $\tau$, when conditioned on
the protected variable. We study methods that adjust the estimator using a heldout
sample for which the protected attribute $A$ is observed.
As example, to be developed at length below, consider forecasting
the birth weight of a baby as a function of
the demographics and personal history of the birth mother, including
her prenatal care, smoking history, and educational background. As
will be seen, when the race of mother is excluded, the quantile
function may be very inaccurate, particularly at the lower quantiles
$\tau < 0.2$ corresponding to low birth weights. If used
as a basis for medical advice, such inaccurate forecasts could conceivably have health consequences for the
mother and infant. It would be important to adjust the estimates if the race of the mother became available.
In this paper we study the simple procedure that adjusts an initial estimate
$\what q_\tau(X)$ by adding $\what \mu_\tau A + \hat\nu_\tau$, by carrying out a quantile regression of $Y-\what q_\tau(X)$ onto $A$.
We show that this leads to an estimate
$\tilde{q}_\tau(X,A) = \what q_\tau(X) + \what \mu_\tau A + \hat\nu_\tau$ for which the conditional quantiles are
close to the target level $\tau$ for both subpopulations $A=1$ and $A=0$. This result follows
from an empirical process analysis that exploits the special dual structure of quantile regression as a linear program.
The main technical result of our paper is that our adjustment procedure is
$\sqrt{n}$-fair at the population level. Roughly speaking, this means
that the effective quantiles for the two subpopulations agree, up to a
stochastic error that decays at a parametric $1/\sqrt{n}$ rate. We establish this result using empirical process techniques that generalize to more general types of attributes, not just binary.
In the following section we provide
technical background on quantile regression, including its formulation
in terms of linear programming, the dual program, and methods for
inference. We also provide background on notions of fairness that are related to this work and give our definition of fairness. In Section~\ref{sec:results} we formally state the methods and results.
The key steps in the proof are given in Section~\ref{sec:proof}. We illustrate these results on synthetic data and birth weight data in Section \ref{sec:experiments}.
We finish with a discussion of the results and possible directions for future work.
Full proofs of the technical results are provided in Section~\ref{sec:pop}
\section{Proof Techniques}\label{sec:proof}
In this section we outline the main steps in the proofs of Theorems~\ref{thm:fair} and~\ref{thm:risk} on the fairness and risk properties of the adjustment procedure. We defer the details of the proofs to Section~\ref{sec:pop}.
We first establish some necessary notation.
From the construction of $\tilde{q}_\tau$, the event $\{y>\tilde{q}_\tau\}$ is equivalent to $\{r>\hat{\mu}_\tau a+\hat{\nu}_\tau\}$ for $r=y-\hat{q}_\tau(a,x)$, which calls for analysis of stochastic processes of the following form. For $d\in\mathbb{R}^n$, let
\[
W_d(\mu,\nu)=\frac{1}{n}\sum_{i\leq n}d_i\left\{R_i>\mu A_i+\nu\right\}.
\]
Let $\overline{W}_d(\mu,\nu)=\mathbb{E}W_d(\mu,\nu)$. It is easy to check that
\[
E_F(\{y>\tilde{q}_\tau\})=\overline{W}_{\mathbbm{1}}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right),\;\;\;\;\;\text{Cov}_F(a,\{y>\tilde{q}_\tau\})=\overline{W}_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right).
\]
The following lemma is essential for establishing concentration results of the $W$ processes around $\overline{W}$, on which the proofs of Theorem~\ref{thm:fair} and Theorem~\ref{thm:risk} heavily rely.
\begin{lemma}\label{lmm:emp}
Suppose $\mathcal{F}$ is a countable family of real functions on $\mathcal{X}$ and $P$ is some probability measure on $\mathcal{X}$. Let $X_1, ..., X_n \stackrel{i.i.d.}{\sim} P$. If
\begin{enumerate}
\item there exists $F:\mathcal{X}\rightarrow \mathbb{R}$ such that $|f(x)|\leq F(x)$ for all $x$ and $C^2:=\int F^2 dP<\infty$;
\item the collection $\text{Subgraph}(\mathcal{F})=\bigl\{\{(x,t)\in\mathcal{X}\times \mathbb{R}: f(x)\leq t\}: f\in \mathcal{F}\bigr\}$ is a Vapnik-Chervonenkis(VC) class of sets,
\end{enumerate}
then there exist positive constant $C_1,C_2$ for which
\begin{equation}
\label{eq:emp.lemma}
\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}f(X_i)-\int f dP\right|>t\right\}\leq 4\mathbb{E}\exp\left(-\left(\frac{t}{C_2\|F\|_n}-1\right)^2\right)+4\mathbb{P}\Bigl\{2\|F\|_n> t/C_2\Bigr\},\;\;\; \forall t\geq C_1,
\end{equation}
where $\|F\|_n =n^{-1/2}\sqrt{\sum F(X_i)^2}$.
In particular,
$\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}f(X_i)-\int f dP\right|=O_p(1)$.
\end{lemma}
We note that more standard results could be used for concentration of measure over VC classes of Boolean functions, or over bounded classes of real functions. We use the lemma above because of its generality and to make our analysis self-contained. The proof of this result is included in Section~\ref{sec:pop}.
Recall that $\text{Cov}_F(a,\{y>\tilde{q}_\tau\})=\overline{W}_{A-\mathbb{E}A}(\hat{\mu}_\tau,\hat{\nu}_\tau)$.
We have
\begin{align*}
\sup_\tau\left|\text{Cov}_F\left(a,\{y>\tilde{q}_\tau(a,x)\}\right)\right|
\leq
\sup_{\mu,\nu\in\mathbb{R}}\left|\left(W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right)\right|+ \sup_\tau\left|W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right)\right|.
\end{align*}
We use Lemma~\ref{lmm:emp} to control the tail of the first term
using the VC class $\text{Subgraph}(\mathcal{F})$ where $f_{\mu,\nu}(a,r)=(a-\mathbb{E}a)\mathds{1}\{r>\mu a+\nu\}$ and $\mathcal{F}=\{f_{\mu,\nu}:\mu,\nu\in\mathbb{Q}\}$.
For the second term we have
\[
W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right) = \frac{1}{n}\sum_{i\leq n}\left(A_i-\mathbb{E}A_i\right)\left\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\right\}.
\]
and we exploit the dual form of quantile regression in terms of rank scores
together with large deviation bounds for sub-Gaussian random variables.
The proof of Theorem~\ref{thm:risk} similarly exploits Lemma~\ref{lmm:emp}, but using
the VC class $\text{Subgraph}(\mathcal{F})$ where $f_{\mu,\nu}(a,r)=\rho_\tau(r-\mu a-\nu)$ and $\mathcal{F}=\{f_{\mu,\nu}:\mu,\nu\in\mathbb{Q}\}$.
\section{Proofs}\label{sec:pop}
\begin{proof}[Proof of Lemma~\ref{lmm:emp}]
To prove the lemma we first transform the problem into bounding the tail of a Rademacher process via a symmetrization technique. Let $\epsilon_i$ be distributed {\it i.i.d.} Rademacher ($\mathbb{P}\{\epsilon_i=1\}=\mathbb{P}\{\epsilon_i=-1\}=1/2$). Write $P_n$ for the empirical (probability) measure that puts mass $n^{-1}$ at each $X_i$. We claim that for all $t>2\sqrt{2}C=:C_1$,
\begin{equation}
\label{eq:symm}
\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\int f dP_n-\int f dP\right|>t\right\}\leq
4\mathbb{P}\left\{\sqrt{n}\sup_{f\in \mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_i f(X_i)\right|>\frac{t}{4}\right\}.
\end{equation}
{\bf Proof of~\eqref{eq:symm}}: Let $\tilde{X}_1,...,\tilde{X}_n$ be independent copies of $X_1,..., X_n$ and let $\tilde{P}_n$ be the corresponding empirical measure. Define events
\[
\mathcal{A}_f = \left\{\sqrt{n}\left|\int f dP_n- \int f dP\right|>t\right\};\;\;\;\text{and }\mathcal{B}_f=\left\{\sqrt{n}\left|\int f d\tilde{P}_n-\int f dP\right|\leq \frac{t}{2}\right\}.
\]
For all $t>C_1$,
\[
\mathbb{P}\mathcal{B}_f= 1-\mathbb{P}\left\{\sqrt{n}\left|\int f d\tilde{P}_n-\int f dP\right|>\frac{t}{2}\right\}
\geq 1-\frac{\text{Var} f(X_1)}{(t/2)^2}
\geq 1-\frac{\int F^2 dP}{(t/2)^2}
\geq \frac{1}{2}.
\]
On the other hand, because $\mathcal{F}$ is countable, we can always find mutually exclusive events $\mathcal{D}_f$ for which
\[
\mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{A}_f=\mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{D}_f = \sum_{f\in\mathcal{F}}\mathbb{P}\mathcal{D}_f.
\]
Since $2\mathbb{P}\mathcal{B}_f\geq 1$ for all $f$, the above is upper bounded by $2\sum_{f\in\mathcal{F}}\mathbb{P}\mathcal{D}_f\mathbb{P} \mathcal{B}_f$. From independence of $X$ and $\tilde{X}$, it can be rewritten as
\[
2\sum_{f\in\mathcal{F}}\mathbb{P}(\mathcal{D}_f\cap \mathcal{B}_f)=2\mathbb{P} \cup_{f\in\mathcal{F}}(\mathcal{D}_f\cap \mathcal{B}_f)\leq 2\mathbb{P}\cup_{f\in\mathcal{F}}(\mathcal{A}_f \cap \mathcal{B}_f),
\]
which is no greater than 2$\mathbb{P}\{\sqrt{n}\sup_f |\int f dP_n-\int f d\tilde{P}_n|>t/2\}$ since
\[
\mathcal{A}_f\cap \mathcal{B}_f\subset \left\{\sqrt{n}\left|\int f dP_n-\int f d\tilde{P}_n\right|>t/2\right\}.
\]
Because $\tilde{X}_i$ is an independent copy of $X_i$, by symmetry $f(X_i)-f(\tilde{X}_i)$ and $\epsilon_i (f(X_i)-f(\tilde{X}_i))$ are equal in distribution. Therefore
\begin{align*}
& \mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\int f dP_n-\int f dP\right|>t\right\}\\
= & \mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{A}_f \leq 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_i (f(X_i)-f(\tilde{X}_i))\right|>\frac{t}{2}\right\}\\
\leq & 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(X_i)\right|>\frac{t}{4}\right\} + 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(\tilde{X}_i)\right|>\frac{t}{4}\right\}\\
= & 4\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(X_i)\right|>\frac{t}{4}\right\}.
\end{align*}
That concludes the proof of~\eqref{eq:symm}.
Denote as $Z_n(f)$ the Rademacher process $n^{-1/2}\sum \epsilon_i f(X_i)$. Let $\mathbb{P}_X$ be the probability measure of $\epsilon$ conditioning on $X$. By independence of $\epsilon$ and $X$, $\epsilon_i$ is still Rademacher under $\mathbb{P}_X$, and it is sub-Gaussian with parameter 1. This implies that for all $f,g\in\mathcal{F}$, $Z_n(f)-Z_n(g)$ is sub-Gaussian with parameter$\sqrt{\int (f-g)^2 dP_n}$ under $\mathbb{P}_X$. In other words,
\[
\mathbb{P}_X\left\{\left| Z_n(f)-Z_n(g)\right|>2\sqrt{\int (f-g)^2 dP_n}\sqrt{u}\right\}\leq 2e^{-u},\;\;\;\forall u>0.
\]
We have shown that conditioning on $X$, $Z_n(f)$ is a process with sub-Gaussian increments controlled by the $\mathcal{L}^2$ norm with respect to $P_n$. For brevity write $\|f\|$ for $\sqrt{\int f^2 dP_n}$. Apply Theorem 3.5 in~\cite{dirksen2015tail} to deduce that there exists positive constant $C_3$, such that for all $f_0\in\mathcal{F}$,
\begin{equation}
\label{eq:chaining}
\mathbb{P}_X\left\{\sup_{f\in\mathcal{F}}\left|Z_n(f)-Z_n(f_0)\right|\geq C_3\left(\Delta(\mathcal{F},\|\cdot\|)\sqrt{u}+\gamma_2(\mathcal{F},\|\cdot\|)\right)\right\}\leq e^{-u}\;\;\;\forall u\geq 1,
\end{equation}
where $\Delta(\mathcal{F},\|\cdot\|)$ is the diameter of $\mathcal{F}$ under the metric $\|\cdot\|$, and $\gamma_2$ is the generic chaining functional that satisfies
\[
\gamma_2(\mathcal{F},\|\cdot\|)\leq C_4 \int_0^{\Delta(\mathcal{F},\|\cdot\|)} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta
\]
for some constant $C_4$. Here $N(\mathcal{F},\|\cdot\|,\delta)$ stands for the $\delta$-covering number of $\mathcal{F}$ under the metric $\|\cdot\|$. We should comment that the generic chaining technique by~\cite{dirksen2015tail} is a vast overkill for our purpose. With some effort the large deviation bounds we need can be derived using the classical chaining technique.
Because $|f|\leq F$ for all $f\in\mathcal{F}$, we have $\Delta(\mathcal{F},\|\cdot\|)\leq 2\|F\|$, so that
\begin{align}
\nonumber & \int_0^{\Delta(\mathcal{F},\|\cdot\|)} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta\\
\label{eq:c.o.v.} \leq & \int_0^{2\|F\|} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta
= 2\|F\|\int_0^1 \sqrt{\log N(\mathcal{F},\|\cdot\|,2\delta\|F\|)}d\delta
\end{align}
via change of variables. To bound the covering number, invoke the assumption that $\text{Subgraph}(\mathcal{F})$ is a VC class of sets. Suppose the VC dimension of $\text{Subgraph}(\mathcal{F})$ is $V$. By Lemma 19 in~\cite{nolan1987u}, there exists positive constant $C_5$ for which the $\mathcal{L}^1(Q)$ covering numbers satisfy
\[
N\left(\mathcal{F},\mathcal{L}^1(Q),\delta \int F dQ\right)\leq (C_5/\delta)^V
\]
for all $0<\delta\leq 1$ and any $Q$ that is a finite measure with finite support on $\mathcal{X}$. Choose $Q$ by $dQ/dP_n=F$. Choose $f_1,..., f_N\in\mathcal{F}$ with $N=N(\mathcal{F},\mathcal{L}^1(Q),\delta \int F dQ)$ and $\min_i \int |f-f_i| dQ \leq \delta \int F dQ$ for each $f\in\mathcal{F}$. Suppose $f_i$ achieves the minimum. Since $F$ is an envelope function for both $f$ and $f_i$,
\[
\int \left|f-f_i\right|^2 dP_n \leq \int 2F \left|f-f_i\right| dP_n,
\]
which by definition of $Q$, is equal to
\[
2\int \left|f-f_i\right| dQ\leq 2\delta \int F dQ = 2\delta\int F^2 dP_n.
\]
Take square roots on both sides to deduce that
\[
N\left(\mathcal{F},\|\cdot\|, 2\delta \|F\|\right)\leq (C_5/\delta^2)^V.
\]
Plug into~\eqref{eq:c.o.v.} this upper bound on the covering number to deduce that the integral in~\eqref{eq:c.o.v.} converges, and $\gamma_2(\mathcal{F},\|\cdot\|)$ is no greater than a constant multiple of $\|F\|$. Recall that we also have $\Delta(\mathcal{F},\|\cdot\|)\leq 2\|F\|$. From~\eqref{eq:chaining}, there exists positive constant $C_6$ for which
\[
\mathbb{P}_X\left\{\sup_{f\in\mathcal{F}}\left|Z_n(f)-Z_n(f_0)\right|\geq C_6 \|F\|\left(\sqrt{u}+1\right)\right\}\leq e^{-u}\;\;\;\forall u\geq 1.
\]
Take $f_0=0$ so we have $Z_n(f_0)=0$. If the zero function does not belong in $\mathcal{F}$, including it in $\mathcal{F}$ does not disrupt the VC set property, and all previous analysis remains valid for $\mathcal{F}\cup \{0\}$. Letting $u=(t/4C_6 \|F\|-1)^2$ yields
\[
\mathbb{P}_X \left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\}\leq \exp\left(-\left(\frac{t}{4C_6\|F\|}-1\right)^2\right),\;\;\; \forall t\geq 8C_6 \|F\|.
\]
Under $\mathbb{P}$, $\|F\|$ is no longer deterministic. Divide the probability space according to the event $\{t\geq 8C_6 \|F\|\}$:
\begin{align*}
\mathbb{P}\left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\}
\leq &\mathbb{E}\mathds{1}\{t\geq 8C_6 \|F\|\}\mathbb{P}_X \left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\} + \mathbb{P}\{t< 8C_6 \|F\|\}\\
\leq &\mathbb{E}\mathds{1}\{t\geq 8C_6 \|F\|\}\exp\left(-\left(\frac{t}{4C_6\|F\|}-1\right)^2\right)+
\mathbb{P}\{t< 8C_6 \|F\|\}.
\end{align*}
Choose $C_2=4C_6$ and~\eqref{eq:emp.lemma} follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:fair}]
Recall that $\text{Cov}_F(a,\{y>\tilde{q}_\tau\})=\overline{W}_{A-\mathbb{E}A}(\hat{\mu}_\tau,\hat{\nu}_\tau)$. Therefore
\begin{align}
\nonumber& \sup_\tau\left|\text{Cov}_F\left(a,\{y>\tilde{q}_\tau(a,x)\}\right)\right|\\
\nonumber= & \sup_\tau\left|\overline{W}_{A-\mathbb{E}A}(\hat{\mu}_\tau,\hat{\nu}_\tau)\right|\\
\label{eq:triangle}\leq & \sup_{\mu,\nu\in\mathbb{R}}\left|\left(W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right)\right|+ \sup_\tau\left|W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right)\right|.
\end{align}
Use Lemma~\ref{lmm:emp} to control the tail of the first term. Apply Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\left\{f: (a,r)\mapsto (a-\mathbb{E}a)\mathds{1}\{r>\mu a+\nu\}: \mu,\nu\in\mathbb{Q}\right\}.
\]
Note that we are only allowing $\mu,\nu$ to take rational values because Lemma~\ref{lmm:emp} only applies to countable sets of functions. This restriction will not hurt us because the supremum of the $W$ processes over all $\mu,\nu\in\mathbb{R}$ equals the supremum over all $\mu,\nu\in\mathbb{Q}$. Let $F(a,r)=|a-\mathbb{E}a|$ be the envelope function. We need to check that $\text{Subgraph}(\mathcal{F})$ is a VC class of sets.
\begin{align}
\nonumber\text{Subgraph}(\mathcal{F}) =& \left\{\{(a,r,t): (a-\mathbb{E}a)\mathds{1}\{r>\mu a+\nu\}\leq t\} : \mu,\nu\in\mathbb{R}\right\}\\
\label{eq:vc.check}= & \left\{\{(a,r,t): \left(\{r>\mu a+\nu\}\cap \{a-\mathbb{E}a\leq t\}\right)\cup \left(\{r\leq \mu a+\nu\}\cap \{t\geq 0\}\right)\} : \mu,\nu\in\mathbb{Q}\right\}.
\end{align}
Since half spaces in $\mathbb{R}^2$ are of VC dimension 3~\cite[p~221]{alon2004probabilistic}, the set $\{\{r\leq\mu a+\nu\}:\mu,\nu\in\mathbb{Q}\}$ forms a VC class. By the same arguments all four events in~\eqref{eq:vc.check} form VC classes. Deduce that $\text{Subgraph}(\mathcal{F})$ is also a VC class because the VC property is stable under any finitely many union/intersection operations. The assumptions of Lemma~\ref{lmm:emp} are satisfied, which gives that for all $t\geq 2C_1/\sqrt{n}$,
\begin{align*}
&\mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|>\frac{t}{2}\right\}\\
\leq & 4\mathbb{E}\exp\left(-\left(\frac{nt}{2C_2|A-\mathbb{E}A|}-1\right)^2\right)+4\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\}\\
\leq & 4\exp\left(-\left(\frac{\sqrt{n}t}{2C_2 u}-1\right)^2\right)+4\mathbb{P}\left\{|A-\mathbb{E}A|>\sqrt{n}u\right\}+4\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\},\;\;\;\forall u>0.
\end{align*}
Here $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^n$. Under the assumption that $A_i$ has finite second moment, we could pick $u$ to be a large enough constant, and pick $t$ to be a large enough constant multiple of $1/\sqrt{n}$ to make the above arbitrarily small. In other words,
\[
\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|=O_p\left(\frac{1}{\sqrt{n}}\right).
\]
Under the stronger assumption that $A_i-\mathbb{E}A_i$ is sub-Gaussian, we have that $(A_i-\mathbb{E}A_i)^2-\text{Var}(A_i)$ is sub-exponential. Choose $u$ to be a large enough constant and we have
\[
\mathbb{P}\left\{|A-\mathbb{E}A|>\sqrt{n}u\right\}= \mathbb{P}\left\{\frac{1}{\sqrt{n}}\sum_{i\leq n}(A_i-\mathbb{E}A_i)^2>\sqrt{n}u^2\right\}\leq \exp(-C_4(\sqrt{n}u^2-1)).
\]
Similarly if $t>C_1/\sqrt{n}$ for some large enough constant $C_1$, there exists $C_5>0$ such that
\[
\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\}\leq \exp\left(-C_5(nt^2-1)\right).
\]
Organizing all the terms yields for some positive constants $C,C_1,C_2,C_3$ whose values may have changed from previous lines,
\[
\mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|>\frac{t}{2}\right\}\leq C\left(\exp\left(-C_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)\right),\;\;\;\forall t>C_1/\sqrt{n}.
\]
For the second term of~\eqref{eq:triangle}, write
\[
W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right) = \frac{1}{n}\sum_{i\leq n}\left(A_i-\mathbb{E}A_i\right)\left\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\right\}.
\]
By the dual form of quantile regression~\cite[p~308]{gutenbrunner1992regression}, there exists regression rank scores $b_\tau\in [0,1]^n$ such that
\[
A^T b=(1-\tau)A^T\mathbbm{1},\;\;\; \mathbbm{1}^T b=(1-\tau)n,\;\;\;\text{and}
\]
\[
b_{\tau,i} =\mathbbm{1}\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\},\;\;\;\forall i\notin M_\tau,
\]
for some $M_\tau\subset[n]$ of size at most $p$. As a result,
\begin{align*}
& \sup_\tau\left|W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right)\right|\\
\leq & \frac{1}{n}\sup_\tau \left|A^T b_\tau-\mathbb{E}A_1 \frac{1}{n}\mathbbm{1}^T b_\tau\right|+\frac{1}{n}\sup_\tau\left|\sum_{i\in M_\tau}(A_i-\mathbb{E}A_i)\left(b_{\tau,i}-\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
= & \frac{1}{n}\sup_\tau\left|(1-\tau)A^T\mathbbm{1}-\mathbb{E}A_1 (1-\tau)n\right|+\frac{1}{n}\sup_\tau\left|\sum_{i\in M_\tau}(A_i-\mathbb{E}A_i)\left(b_{\tau,i}-\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
\leq & \left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|+\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|.
\end{align*}
If $A_i$ has finite second moment, the above is clearly of order $O_p(1/\sqrt{n})$. If we have in addition that $A_i-\mathbb{E}A_i\sim \mbox{SubG}(\sigma)$, then $|\sum_i (A_i-\mathbb{E}A_i)/n|\sim \mbox{SubG}(\sigma/\sqrt{n})$. For all $t>0$,
\[
\mathbb{P}\left\{\left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|>\frac{t}{4}\right\}\leq 2\exp\left(-\frac{nt^2}{32\sigma^2}\right).
\]
We also have
\[
\mathbb{P}\left\{\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|>\frac{t}{4}\right\}\leq n\mathbb{P}\left\{|A_1-\mathbb{E}A_1|>\frac{tn}{4p}\right\}\leq 2\exp\left(-\frac{n^2t^2}{32\sigma^2p^2}+\log n\right).
\]
Hence
\begin{align*}
&\mathbb{P}\left\{\sup_\tau\left|\text{Cov}_F\left(a,\{y>\tilde{q}_\tau(a,x)\}\right)\right|>t\right\}\\
\leq & \mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|\left(W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right)\right|>\frac{t}{2}\right\}\\
& + \mathbb{P}\left\{\left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|>\frac{t}{4}\right\} + \mathbb{P}\left\{\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|>\frac{t}{4}\right\}\\
\leq &C\left(\exp\left(-C_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)\right)+ 2\exp\left(-\frac{nt^2}{32\sigma^2}\right)+2\exp\left(-\frac{n^2t^2}{32\sigma^2p^2}+\log n\right)\\
\leq & C\left(\exp\left(-C'_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)+n\exp\left(-C_4 n^2t^2\right)\right).
\end{align*}
That concludes the proof of~\eqref{eq:fair}. The proof of~\eqref{eq:faithful} is similar. Simply note that
\begin{align*}
& \sup_\tau\left|E_F\{y>\tilde{q}_\tau(a,x)\}-(1-\tau)\right|\\
= & \sup_\tau \left|\overline{W}_{\mathds{1}}(\hat{\mu}_\tau,\hat{\nu}_\tau)-(1-\tau)\right|\\
\leq & \sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|+ \sup_\tau \left|\frac{1}{n}\sum_{i\in M_\tau}\left(b_{\tau,i}-\mathds{1}\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
\leq & \sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|+\frac{p}{n}
\end{align*}
because $|M_\tau|\leq p$. Apply Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\{f:(a,r)\mapsto \mathds{1}\{r>\mu a+\nu\}:\mu,\nu\in\mathbb{Q}\},\;\;\;\text{and }F\equiv 1.
\]
The subgraph of $\mathcal{F}$ also forms a VC set via similar analysis. Lemma~\ref{lmm:emp} implies that if $t\geq C_1/\sqrt{n}$ for large enough $C_1$
\[
\mathbb{P}\left\{\sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|>t\right\}\leq 4\exp\left(-\left(\frac{\sqrt{n}t}{C_2}-1\right)^2\right)+\mathbb{P}\left\{2>\frac{C_1}{C_2}\right\}.
\]
The second term is 0 for $C_1>2C_2$, and the desired inequality~\eqref{eq:faithful} immediately follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:risk}]
Suppose $\mu_\tau^*, \nu_\tau^*\in \arg\min_{\mu,\nu\in\mathbb{R}}\mathcal{R}(\hat{q}_\tau+\mu A + \nu)$. There exists some finite constant $K$ for which
\[
(\mu_\tau^*, \nu_\tau^*)\in B_K=\{(\mu,\nu): \max (|\mu|,|\nu|)\leq K\}.
\]
Invoke Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\left\{f:(a,r)\mapsto \rho_\tau(r-\mu a-\nu): \mu,\nu\in\mathbb{Q}\right\}.
\]
The subgraph of $\mathcal{F}$ forms a VC class of sets, and on the compact set $B_{K}$, we have $|f|\leq F$ where $F(a,r)=|r|+K|a|+K$ has bounded second moment. By Lemma~\ref{lmm:emp},
\begin{equation}\label{eq:R.concentration}
\sup_{(\mu,\nu)\in B_{2K}}\left|\frac{1}{n}\sum_{i\leq n}\left(\rho_\tau(R_i-\mu A_i-\nu)-\mathbb{E}\rho_\tau (R_i-\mu A_i-\nu)\right)\right|=O_p(1/\sqrt{n}).
\end{equation}
Use continuity of $\rho_\tau$ to deduce existence of some $\delta>0$ for which
\[
\mathbb{E}\rho_\tau(R_1-\mu A_1 -\nu)>\mathbb{E}\rho_\tau(R_1-\mu^*_\tau A_1 -\nu^*_\tau)+2\delta\;\;\;\forall (\mu,\nu)\in\partial B_{2K}.
\]
Use~\eqref{eq:R.concentration} to deduce that with probability $1-o(1)$,
\[
\min_{(\mu,\nu)\in \partial B_{2K}}\frac{1}{n}\sum_{i\leq n}\rho_\tau (R_i-\mu A_i-\nu)>\mathbb{E}\rho_\tau(R_1-\mu^* A_1 -\nu^*)+\delta> \frac{1}{n}\sum_{i\leq n}\rho_\tau (R_i-\mu_\tau^* A_i-\nu_\tau^*).
\]
By convexity of $\rho_\tau$, the minimizers $\hat{\mu}_\tau, \hat{\nu}_\tau$ must appear with $B_{2K}$. Recall that $\hat{\mu}_\tau$ and $\hat{\nu}_\tau$ are obtained by running quantile regression of $R$ against $A$ on the training set, so we have
\begin{equation}\label{eq:basic}
\frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\hat{\mu}_\tau A_i-\hat{\nu}_\tau)\leq \frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\mu_\tau^* A_i-\nu_\tau^*).
\end{equation}
A few applications of the triangle inequality yields
\begin{align*}
& R(\tilde{q}_\tau)-R(\tilde{q}_\tau^*)\\
\leq & \frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\hat{\mu}_\tau A_i-\hat{\nu}_\tau)
-\frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\mu_\tau^* A_i-\nu_\tau^*)\\
& + 2\sup_{(\mu,\nu)\in B_{2K}}\left|\frac{1}{n}\sum_{i\leq n}\left(\rho_\tau(R_i-\mu A_i-\nu)-\mathbb{E}\rho_\tau (R_i-\mu A_i-\nu)\right)\right|\\
\leq & 0+O_p(1/\sqrt{n})
\end{align*}
by~\eqref{eq:basic} and~\eqref{eq:R.concentration} .
\end{proof}
\section{Method and Results}
\label{sec:results}
With samples $(A_i,X_i,Y_i)$ drawn i.i.d. from some joint distribution $F$ on $\mathbb{R}\times \mathbb{R}^{p}\times \mathbb{R}$, consider the problem of estimating the conditional quantile $q_\tau(y \,|\, a,x)$. Let $E_F$ denote the expected value operator under $F$, or $E_F f=\int f(a,x,y)\,dF(a,x,y)$. Similarly define the probability operator under $F$ as $P_F$.
Evaluate the level of fairness of an estimator $\hat{q}_\tau(a,x)$ with
\[
\text{Cov}_F\left(a,\mathds{1}\left\{y> \hat{q}_\tau(a,x)\right\}\right)= E_F(a-E_F a)\left(\mathds{1}\left\{y> \hat{q}_\tau(a,x)\right\}-P_F\left\{y> \hat{q}_\tau(a,x)\right\}\right).
\]
An estimator with a smaller $|\text{Cov}_F(a,\mathds{1}\{y> \hat{q}_\tau\})|$ is considered more fair. This measurement of fairness generalizes the notion of balanced effective quantiles described in section~\ref{sec:fair.def}. Note that when the protected attribute is binary, $\text{Cov}_F(a,\mathds{1}\{y> \hat{q}_\tau\})=0$ is equivalent to $\hat{\tau}_0=\hat{\tau}_1$ for $\hat{\tau}$ defined in~\eqref{eq:effective.quantile}.
From an initial estimator $\hat{q}_\tau$ that is potentially unfair, we propose the following correction procedure.
\vskip10pt
\framebox{
\parbox{\dimexpr\linewidth-10\fboxsep-2\fboxrule}
{\itshape%
On a training set of size $n$, compute $R_i=Y_i-\hat{q}_\tau(A_i,X_i)$ and run quantile regression of $R$ on $A$ at level $\tau$. Obtain regression slope $\hat{\mu}_\tau$ and intercept $\hat{\nu}_\tau$. Define correction $\tilde{q}_\tau(a,x)=\hat{q}_\tau(a,x)+\hat{\mu}_\tau a+\hat{\nu}_\tau$.
}}
\vskip10pt
We show that this estimator $\tilde{q}_\tau$ will
satisfy the following:
\begin{enumerate}
\item Faithful: $P_F\{y>\tilde{q}_\tau\}\approx 1-\tau$
\item Fair: $\text{Cov}_F(a,\mathds{1}\{y>\tilde{q}_\tau\})\approx 0$;
\item Reduced risk: It almost always improves the fit of $\hat{q}_\tau$.
\end{enumerate}
Theorem~\ref{thm:fair} and Theorem~\ref{thm:risk} contain the precise statements of our claims.
\begin{theorem}[Faithfulness and fairness]\label{thm:fair}
Suppose $(A_i,X_i,Y_i)\stackrel{i.i.d.}{\sim} F$, and $A_i-\mathbb{E}A_i$ has finite second moment. Then the corrected estimator $\tilde{q}_\tau$ satisfies
\begin{align}
\label{eq:faithful.rate}
\sup_\tau\bigl|P_F\{y>\tilde{q}_\tau(a,x)\}-(1-\tau)\bigr| = & O_p\left(1/\sqrt{n}\right),\;\;\;\;\;\text{ and }\\
\label{eq:fair.rate}
\sup_\tau\bigl|\text{Cov}_F\left(a,\mathds{1}\{y>\tilde{q}_\tau(a,x)\}\right)\bigr|= & O_p\left(1/\sqrt{n}\right).
\end{align}
Furthermore, there exist positive constants $C,C_1,C_2$ such that $\forall t>C_1/\sqrt{n}$,
\begin{equation}\label{eq:faithful}
\mathbb{P}\left\{\sup_\tau\bigl|P_F\{y>\tilde{q}_\tau(a,x)\}-(1-\tau)\bigr|>t+p/n\right\}\leq C\exp\left(-C_2 nt^2\right).
\end{equation}
Under the stronger assumption that the distribution of $A_i-\mathbb{E}A_i$ is sub-Gaussian, there exist positive constants $C,C_1,C_2,C_3,C_4$ such that $\forall t>C_1/\sqrt{n}$,
\begin{equation}\label{eq:fair}
\mathbb{P}\left\{\sup_\tau\bigl|\text{Cov}_F\left(a,\mathds{1}\{y>\tilde{q}_\tau(a,x)\}\right)\bigr|>t\right\}\leq
C\left(\exp\left(-C_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)+n\exp\left(-C_4 n^2t^2\right)\right).
\end{equation}
\end{theorem}
The following corollary for binary protected attributes is an easy consequence of~\eqref{eq:faithful.rate} and~\eqref{eq:fair.rate}.
\begin{corollary}
\label{cor:fair}
If $A$ is binary, then the correction procedure gives balanced effective quantiles:
\[
\hat{\tau}_0=\tau+O_p(1/\sqrt{n}),\;\;\; \hat{\tau}_1=\tau+O_p(1/\sqrt{n}).
\]
\end{corollary}
\begin{remark}
By modifying the proof of Theorem~\ref{thm:fair} slightly, Corollary~\ref{cor:fair} can be extended to the case where $A$ is categorical with $K$ categories. In this case the correction procedure needs to be adjusted accordingly. Instead of regressing $R$ on $A$, regress $R$ on the span of the indicators $\{A=k\}$ for $k=1,..., K-1$, leaving one category out to avoid collinearity. The corrected estimators will satisfy $\hat{\tau}_k=\tau+O_p(1/\sqrt{n})$ for all categories $k=1,..., K$.
\end{remark}
Define $\mathcal{R}(\cdot)=E_F\rho_\tau (y-\cdot)$ as the risk function, where $\rho_\tau(u)=\tau u\mathds{1}\{u>0\}+(1-\tau) u\mathds{1}\{u\leq 0\}$.
\begin{theorem}[Risk quantification]\label{thm:risk}
The adjustment procedure $\tilde{q}_\tau(A,X)$ satisfies
\[
\mathcal{R}(\tilde{q}_\tau)\leq \inf_{\mu,\nu\in\mathbb{R}}\mathcal{R}(\hat{q}_\tau+\mu A + \nu) +O_p(1/\sqrt{n}).
\]
\end{theorem}
We note that in the different setting where $A$ is a treatment rather than an observational variable,
it is of interest to obtain an unbiased estimate of the treatment effect $\mu_\tau$.
In this case a simple alternative approach is the so-called ``double machine learning'' procedure
by \cite{chernozhukov2016double}; in the quantile regression setting
this would regress the residual onto the transformed attribute $A - \what A$ where
$\what A = \what A(X)$ is a predictive model
of $A$ in terms of $X$.
\section{Full Proofs of Technical Results}\label{sec:lem}
\begin{proof}
To prove the lemma we first transform the problem into bounding the tail of a Rademacher process via a symmetrization technique. Let $\epsilon_i$ be distributed {\it i.i.d.} Rademacher ($\mathbb{P}\{\epsilon_i=1\}=\mathbb{P}\{\epsilon_i=-1\}=1/2$). Write $P_n$ for the empirical (probability) measure that puts mass $n^{-1}$ at each $X_i$. We claim that for all $t>2\sqrt{2}C=:C_1$,
\begin{equation}
\label{eq:symm}
\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\int f dP_n-\int f dP\right|>t\right\}\leq
4\mathbb{P}\left\{\sqrt{n}\sup_{f\in \mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_i f(X_i)\right|>\frac{t}{4}\right\}.
\end{equation}
{\bf Proof of~\eqref{eq:symm}}: Let $\tilde{X}_1,...,\tilde{X}_n$ be independent copies of $X_1,..., X_n$ and let $\tilde{P}_n$ be the corresponding empirical measure. Define events
\[
\mathcal{A}_f = \left\{\sqrt{n}\left|\int f dP_n- \int f dP\right|>t\right\};\;\;\;\text{and }\mathcal{B}_f=\left\{\sqrt{n}\left|\int f d\tilde{P}_n-\int f dP\right|\leq \frac{t}{2}\right\}.
\]
For all $t>C_1$,
\[
\mathbb{P}\mathcal{B}_f= 1-\mathbb{P}\left\{\sqrt{n}\left|\int f d\tilde{P}_n-\int f dP\right|>\frac{t}{2}\right\}
\geq 1-\frac{\text{Var} f(X_1)}{(t/2)^2}
\geq 1-\frac{\int F^2 dP}{(t/2)^2}
\geq \frac{1}{2}.
\]
On the other hand, because $\mathcal{F}$ is countable, we can always find mutually exclusive events $\mathcal{D}_f$ for which
\[
\mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{A}_f=\mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{D}_f = \sum_{f\in\mathcal{F}}\mathbb{P}\mathcal{D}_f.
\]
Since $2\mathbb{P}\mathcal{B}_f\geq 1$ for all $f$, the above is upper bounded by
\[
2\sum_{f\in\mathcal{F}}\mathbb{P}\mathcal{D}_f\mathbb{P} \mathcal{B}_f=2\mathbb{P} \cup_{f\in\mathcal{F}}(\mathcal{D}_f\cap \mathcal{B}_f)\leq 2\mathbb{P}\cup_{f\in\mathcal{F}}(\mathcal{A}_f \cap \mathcal{B}_f),
\]
which is no greater than 2$\mathbb{P}\{\sqrt{n}\sup_f |\int f dP_n-\int f d\tilde{P}_n|>t/2\}$ since
$\mathcal{A}_f\cap \mathcal{B}_f\subset \left\{\sqrt{n}\left|\int f dP_n-\int f d\tilde{P}_n\right|>t/2\right\}$.
Because $\tilde{X}_i$ is an independent copy of $X_i$, by symmetry $f(X_i)-f(\tilde{X}_i)$ and $\epsilon_i (f(X_i)-f(\tilde{X}_i))$ are equal in distribution. Therefore
\begin{align*}
\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\int f dP_n-\int f dP\right|>t\right\}
= & \mathbb{P}\cup _{f\in\mathcal{F}}\mathcal{A}_f \leq 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_i (f(X_i)-f(\tilde{X}_i))\right|>\frac{t}{2}\right\}\\
\leq & 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(X_i)\right|>\frac{t}{4}\right\} + 2\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(\tilde{X}_i)\right|>\frac{t}{4}\right\}\\
= & 4\mathbb{P}\left\{\sqrt{n}\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i\leq n}\epsilon_if(X_i)\right|>\frac{t}{4}\right\}.
\end{align*}
That concludes the proof of~\eqref{eq:symm}.
Denote as $Z_n(f)$ the Rademacher process $n^{-1/2}\sum \epsilon_i f(X_i)$. Let $\mathbb{P}_X$ be the probability measure of $\epsilon$ conditioning on $X$. By independence of $\epsilon$ and $X$, $\epsilon_i$ is still Rademacher under $\mathbb{P}_X$, and it is sub-Gaussian with parameter 1. That implies for all $f,g\in\mathcal{F}$, $Z_n(f)-Z_n(g)\sim \text{subG}\left(\sqrt{\int (f-g)^2 dP_n}\right)$ under $\mathbb{P}_X$. In other words,
\[
\mathbb{P}_X\left\{\left| Z_n(f)-Z_n(g)\right|>2\sqrt{\int (f-g)^2 dP_n}\sqrt{u}\right\}\leq 2e^{-u},\;\;\;\forall u>0.
\]
We have shown that conditioning on $X$, $Z_n(f)$ is a process with sub-Gaussian increments controlled by the $\mathcal{L}^2$ norm with respect to $P_n$. For brevity write $\|f\|$ for $\sqrt{\int f^2 dP_n}$. Apply Theorem 3.5 in~\cite{dirksen2015tail} to deduce that there exists positive constant $C_3$, such that for all $f_0\in\mathcal{F}$,
\begin{equation}
\label{eq:chaining}
\mathbb{P}_X\left\{\sup_{f\in\mathcal{F}}\left|Z_n(f)-Z_n(f_0)\right|\geq C_3\left(\Delta(\mathcal{F},\|\cdot\|)\sqrt{u}+\gamma_2(\mathcal{F},\|\cdot\|)\right)\right\}\leq e^{-u}\;\;\;\forall u\geq 1,
\end{equation}
where $\Delta(\mathcal{F},\|\cdot\|)$ is the diameter of $\mathcal{F}$ under the metric $\|\cdot\|$, and $\gamma_2$ is the generic chaining functional that satisfies
\[
\gamma_2(\mathcal{F},\|\cdot\|)\leq C_4 \int_0^{\Delta(\mathcal{F},\|\cdot\|)} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta
\]
for some constant $C_4$. Here $N(\mathcal{F},\|\cdot\|,\delta)$ stands for the $\delta$-covering number of $\mathcal{F}$ under the metric $\|\cdot\|$.
Because $|f|\leq F$ for all $f\in\mathcal{F}$, we have $\Delta(\mathcal{F},\|\cdot\|)\leq 2\|F\|$, so that
\begin{equation}
\label{eq:c.o.v.}
\int_0^{\Delta(\mathcal{F},\|\cdot\|)} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta
\leq \int_0^{2\|F\|} \sqrt{\log N(\mathcal{F},\|\cdot\|,\delta)}d\delta
= 2\|F\|\int_0^1 \sqrt{\log N(\mathcal{F},\|\cdot\|,2\delta\|F\|)}d\delta
\end{equation}
via change of variables. To bound the covering number, invoke the assumption that $\text{Subgraph}(\mathcal{F})$ is a VC class of sets. By Lemma 19 in~\cite{nolan1987u}, there exist positive constants $C_5$ and $V$ for which the $\mathcal{L}^1(Q)$ covering numbers satisfy
\[
N\left(\mathcal{F},\mathcal{L}^1(Q),\delta \int F dQ\right)\leq (C_5/\delta)^V
\]
for all $0<\delta\leq 1$ and any $Q$ that is a finite measure with finite support on $\mathcal{X}$. Choose $Q$ by $dQ/dP_n=F$. Choose $f_1,..., f_N\in\mathcal{F}$ with $N=N(\mathcal{F},\mathcal{L}^1(Q),\delta \int F dQ)$ and $\min_i \int |f-f_i| dQ \leq \delta \int F dQ$ for each $f\in\mathcal{F}$. If $f_i$ achieves the minimum then
\[
\int \left|f-f_i\right|^2 dP_n \leq \int 2F \left|f-f_i\right| dP_n = 2\int \left|f-f_i\right| dQ\leq 2\delta^2 \int F dQ = 2\delta^2\int F^2 dP_n.
\]
Take square roots on both sides to deduce that
\[
N\left(\mathcal{F},\|\cdot\|, 2\delta \|F\|\right)\leq (C_5/\delta^2)^V.
\]
Plug into~\eqref{eq:c.o.v.} this upper bound on the covering number to deduce that the integral in~\eqref{eq:c.o.v.} converges, and $\gamma_2(\mathcal{F},\|\cdot\|)$ is no greater than a constant multiple of $\|F\|$. Recall that we also have $\Delta(\mathcal{F},\|\cdot\|)\leq 2\|F\|$. From~\eqref{eq:chaining}, there exists positive constant $C_6$ for which
\[
\mathbb{P}_X\left\{\sup_{f\in\mathcal{F}}\left|Z_n(f)-Z_n(f_0)\right|\geq C_6 \|F\|\left(\sqrt{u}+1\right)\right\}\leq e^{-u}\;\;\;\forall u\geq 1.
\]
Take $f_0=0$ so we have $Z_n(f_0)=0$. If the zero function does not belong in $\mathcal{F}$, including it in $\mathcal{F}$ does not disrupt the VC set property, and all previous analysis remain valid for $\mathcal{F}\cup \{0\}$. Letting $u=(t/4C_6 \|F\|-1)^2$ yields
\[
\mathbb{P}_X \left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\}\leq \exp\left(-\left(\frac{t}{4C_6\|F\|}-1\right)^2\right),\;\;\; \forall t\geq 8C_6 \|F\|.
\]
Under $\mathbb{P}$, $\|F\|$ is no longer deterministic. Divide the probability space according to the event $\{t\geq 8C_6 \|F\|\}$:
\begin{align*}
\mathbb{P}\left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\}
\leq &\mathbb{E}\mathds{1}\{t\geq 8C_6 \|F\|\}\mathbb{P}_X \left\{\sup_{f\in\mathcal{F}}|Z_n(f)|>\frac{t}{4}\right\} + \mathbb{P}\{t< 8C_6 \|F\|\}\\
\leq &\mathbb{E}\mathds{1}\{t\geq 8C_6 \|F\|\}\exp\left(-\left(\frac{t}{4C_6\|F\|}-1\right)^2\right)+
\mathbb{P}\{t< 8C_6 \|F\|\}.
\end{align*}
Choose $C_2=4C_6$ and~\eqref{eq:emp.lemma} follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:fair}]
Recall that $\text{Cov}_F(a,\{y>\tilde{q}_\tau\})=\overline{W}_{A-\mathbb{E}A}(\hat{\mu}_\tau,\hat{\nu}_\tau)$. Therefore
\begin{align}
\nonumber& \sup_\tau\left|\text{Cov}_F\left(a,\{y>\tilde{q}_\tau(a,x)\}\right)\right|\\
\nonumber= & \sup_\tau\left|\overline{W}_{A-\mathbb{E}A}(\hat{\mu}_\tau,\hat{\nu}_\tau)\right|\\
\label{eq:triangle}\leq & \sup_{\mu,\nu\in\mathbb{R}}\left|\left(W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right)\right|+ \sup_\tau\left|W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right)\right|.
\end{align}
Use Lemma~\ref{lmm:emp} to control the tail of the first term. Apply Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\left\{f: (a,r)\mapsto (a-\mathbb{E}a)\mathds{1}\{r>\mu a+\nu\}: \mu,\nu\in\mathbb{Q}\right\}.
\]
Note that we are only allowing $\mu,\nu$ to take rational values because Lemma~\ref{lmm:emp} only applies to countable sets of functions. This restriction will not hurt us because the supremum of the $W$ processes over all $\mu,\nu\in\mathbb{R}$ equals the supremum over all $\mu,\nu\in\mathbb{Q}$. Let $F(a,r)=|a-\mathbb{E}a|$ be the envelope function. We need to check that $\text{Subgraph}(\mathcal{F})$ is a VC class of sets.
\begin{align}
\nonumber\text{Subgraph}(\mathcal{F}) =& \left\{\{(a,r,t): (a-\mathbb{E}a)\mathds{1}\{r>\mu a+\nu\}\leq t\} : \mu,\nu\in\mathbb{R}\right\}\\
\label{eq:vc.check}= & \left\{\{(a,r,t): \left(\{r>\mu a+\nu\}\cap \{a-\mathbb{E}a\leq t\}\right)\cup \left(\{r\leq \mu a+\nu\}\cap \{t\geq 0\}\right)\} : \mu,\nu\in\mathbb{Q}\right\}.
\end{align}
Since half spaces in $\mathbb{R}^2$ is of VC dimension 3~\cite[p~221]{alon2004probabilistic}, the set $\{\{r\leq\mu a+\nu\}:\mu,\nu\in\mathbb{Q}\}$ forms a VC class. By the same arguments all four events in~\eqref{eq:vc.check} form VC classes. Deduce that $\text{Subgraph}(\mathcal{F})$ is also a VC class because the VC property is stable under any finitely many union/intersection operations. The assumptions of Lemma~\ref{lmm:emp} are satisfied, which gives that for all $t\geq 2C_1/\sqrt{n}$,
\begin{align*}
&\mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|>\frac{t}{2}\right\}\\
\leq & 4\mathbb{E}\exp\left(-\left(\frac{nt}{2C_2|A-\mathbb{E}A|}-1\right)^2\right)+4\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\}\\
\leq & 4\exp\left(-\left(\frac{\sqrt{n}t}{2C_2 u}-1\right)^2\right)+4\mathbb{P}\left\{|A-\mathbb{E}A|>\sqrt{n}u\right\}+4\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\},\;\;\;\forall u>0.
\end{align*}
Here $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^n$. Under the assumption that $A_i$ has finite second moment, we could pick $u$ to be a large enough constant, and pick $t$ to be a large enough constant multiple of $1/\sqrt{n}$ to make the above arbitrarily small. In other words,
\[
\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|=O_p\left(\frac{1}{\sqrt{n}}\right).
\]
Under the stronger assumption that $A_i-\mathbb{E}A_i$ is sub-Gaussian, we have that $(A_i-\mathbb{E}A_i)^2-\text{Var}(A_i)$ is sub-exponential. Choose $u$ to be a large enough constant and we have
\[
\mathbb{P}\left\{|A-\mathbb{E}A|>\sqrt{n}u\right\}= \mathbb{P}\left\{\frac{1}{\sqrt{n}}\sum_{i\leq n}(A_i-\mathbb{E}A_i)^2>\sqrt{n}u^2\right\}\leq \exp(-C_4(\sqrt{n}u^2-1)).
\]
Similarly if $t>C_1/\sqrt{n}$ for some large enough constant $C_1$,there exists $C_5>0$ such that
\[
\mathbb{P}\left\{2|A-\mathbb{E}A|>nt/2C_2\right\}\leq \exp\left(-C_5(nt^2-1)\right).
\]
Organizing all the terms yields for some positive constants $C,C_1,C_2,C_3$ whose values may have changed from previous lines,
\[
\mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right|>\frac{t}{2}\right\}\leq C\left(\exp\left(-C_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)\right),\;\;\;\forall t>C_1/\sqrt{n}.
\]
For the second term of~\eqref{eq:triangle}, write
\[
W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right) = \frac{1}{n}\sum_{i\leq n}\left(A_i-\mathbb{E}A_i\right)\left\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\right\}.
\]
By the dual form of quantile regression~\cite[p~308]{gutenbrunner1992regression}, there exists regression rank scores $b_\tau\in [0,1]^n$ such that
\[
A^T b=(1-\tau)A^T\mathbbm{1},\;\;\; \mathbbm{1}^T b=(1-\tau)n,\;\;\;\text{and}
\]
\[
b_{\tau,i} =\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\},\;\;\;\forall i\notin M_\tau,
\]
for some $M_\tau\subset[n]$ of size at most $p$. As a result,
\begin{align*}
& \sup_\tau\left|W_{A-\mathbb{E}A}\left(\hat{\mu}_\tau,\hat{\nu}_\tau\right)\right|\\
\leq & \frac{1}{n}\sup_\tau \left|A^T b_\tau-\mathbb{E}A_1 \frac{1}{n}\mathbbm{1}^T b_\tau\right|+\frac{1}{n}\sup_\tau\left|\sum_{i\in M_\tau}(A_i-\mathbb{E}A_i)\left(b_{\tau,i}-\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
= & \frac{1}{n}\sup_\tau\left|(1-\tau)A^T\mathbbm{1}-\mathbb{E}A_1 (1-\tau)n\right|+\frac{1}{n}\sup_\tau\left|\sum_{i\in M_\tau}(A_i-\mathbb{E}A_i)\left(b_{\tau,i}-\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
\leq & \left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|+\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|.
\end{align*}
If $A_i$ has finite second moment, the above is clearly of order $O_p(1/\sqrt{n})$. If we have in addition that $A_i-\mathbb{E}A_i\sim SubG(\sigma)$, then $|\sum_i (A_i-\mathbb{E}A_i)/n|\sim SubG(\sigma/\sqrt{n})$. For all $t>0$,
\[
\mathbb{P}\left\{\left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|>\frac{t}{4}\right\}\leq 2\exp\left(-\frac{nt^2}{32\sigma^2}\right).
\]
We also have
\[
\mathbb{P}\left\{\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|>\frac{t}{4}\right\}\leq n\mathbb{P}\left\{|A_1-\mathbb{E}A_1|>\frac{tn}{4p}\right\}\leq 2\exp\left(-\frac{n^2t^2}{32\sigma^2p^2}+\log n\right).
\]
Hence
\begin{align*}
&\mathbb{P}\left\{\sup_\tau\left|\text{Cov}_F\left(a,\{y>\tilde{q}_\tau(a,x)\}\right)\right|>t\right\}\\
\leq & \mathbb{P}\left\{\sup_{\mu,\nu\in\mathbb{R}}\left|\left(W_{A-\mathbb{E}A}(\mu,\nu)-\overline{W}_{A-\mathbb{E}A}(\mu,\nu)\right)\right|>\frac{t}{2}\right\} + \mathbb{P}\left\{\left|\frac{1}{n}\sum_{i\leq n}(A_i-\mathbb{E}A_i)\right|>\frac{t}{4}\right\} + \mathbb{P}\left\{\frac{p}{n}\max_{i\leq n}|A_i-\mathbb{E}A_i|>\frac{t}{4}\right\}\\
\leq &C\left(\exp\left(-C_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)\right)+ 2\exp\left(-\frac{nt^2}{32\sigma^2}\right)+2\exp\left(-\frac{n^2t^2}{32\sigma^2p^2}+\log n\right)\\
\leq & C\left(\exp\left(-C'_2 nt^2\right)+\exp\left(-C_3 \sqrt{n}\right)+n\exp\left(-C_4 n^2t^2\right)\right).
\end{align*}
That concludes the proof of~\eqref{eq:fair}. The proof of~\eqref{eq:faithful} is similar. Simply note that
\begin{align*}
& \sup_\tau\left|E_F\{y>\tilde{q}_\tau(a,x)\}-(1-\tau)\right|\\
= & \sup_\tau \left|\overline{W}_{\mathds{1}}(\hat{\mu}_\tau,\hat{\nu}_\tau)-(1-\tau)\right|\\
\leq & \sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|+ \sup_\tau \left|\frac{1}{n}\sum_{i\in M_\tau}\left(b_{\tau,i}-\mathds{1}\{R_i>\hat{\mu}_\tau A_i+\hat{\nu}_\tau\}\right)\right|\\
\leq & \sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|+\frac{p}{n}
\end{align*}
because $|M_\tau|\leq p$. Apply Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\{f:(a,r)\mapsto \mathds{1}\{r>\mu a+\nu\}:\mu,\nu\in\mathbb{Q}\},\;\;\;\text{and }F\equiv 1.
\]
Subgraph of $\mathcal{F}$ also forms a VC set via similar analysis. Lemma~\ref{lmm:emp} implies that if $t\geq C_1/\sqrt{n}$ for large enough $C_1$
\[
\mathbb{P}\left\{\sup_{\mu,\nu}\left|W_{\mathds{1}}(\mu,\nu)-\overline{W}_{\mathds{1}}(\mu,\nu)\right|>t\right\}\leq 4\exp\left(-\left(\frac{\sqrt{n}t}{C_2}-1\right)^2\right)+0.
\]
The desired inequality~\eqref{eq:faithful} immediately follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:risk}]
Suppose $\mu_\tau^*, \nu_\tau^*\in \arg\min_{\mu,\nu\in\mathbb{R}}\mathcal{R}(\hat{q}_\tau+\mu A + \nu)$. There exists some finite constant $K$ for which
\[
(\mu_\tau^*, \nu_\tau^*)\in B_K=\{(\mu,\nu): \max (|\mu|,|\nu|)\leq K\}.
\]
Invoke Lemma~\ref{lmm:emp} with
\[
\mathcal{F}=\left\{f:(a,r)\mapsto \rho_\tau(r-\mu a-\nu): \mu,\nu\in\mathbb{Q}\right\}.
\]
Subgraph of $\mathcal{F}$ forms a VC class of sets, and on the compact set $B_{K}$, we have $|f|\leq F$ where $F(a,r)=|r|+R|a|+R$ has bounded second moment. By Lemma~\ref{lmm:emp},
\begin{equation}\label{eq:R.concentration}
\sup_{(\mu,\nu)\in B_{2K}}\left|\frac{1}{n}\sum_{i\leq n}\left(\rho_\tau(R_i-\mu A_i-\nu)-\mathbb{E}\rho_\tau (R_i-\mu A_i-\nu)\right)\right|=O_p(1/\sqrt{n}).
\end{equation}
Use continuity of $\rho_\tau$ to deduce existence of some $\delta>0$ for which
\[
\mathbb{E}\rho_\tau(R_1-\mu A_1 -\nu)>\mathbb{E}\rho_\tau(R_1-\mu^*_\tau A_1 -\nu^*_\tau)+2\delta\;\;\;\forall (\mu,\nu)\in\partial B_{2K}.
\]
Use~\eqref{eq:R.concentration} to deduce that with probability $1-o(1)$,
\[
\min_{(\mu,\nu)\in \partial B_{2K}}\frac{1}{n}\sum_{i\leq n}\rho_\tau (R_i-\mu A_i-\nu)>\mathbb{E}\rho_\tau(R_1-\mu^* A_1 -\nu^*)+\delta> \frac{1}{n}\sum_{i\leq n}\rho_\tau (R_i-\mu_\tau^* A_i-\nu_\tau^*).
\]
By convexity of $\rho_\tau$, the minimizers $\hat{\mu}_\tau, \hat{\nu}_\tau$ must appear with $B_{2K}$. Recall that $\hat{\mu}_\tau$ and $\hat{\nu}_\tau$ are obtained by running quantile regression of $R$ against $A$ on the training set, so we have
\begin{equation}\label{eq:basic}
\frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\hat{\mu}_\tau A_i-\hat{\nu}_\tau)\leq \frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\mu_\tau^* A_i-\nu_\tau^*).
\end{equation}
A few applications of triangle inequality yields
\begin{align*}
& R(\tilde{q}_\tau-R(\tilde{q}_\tau^*))\\
\leq & \frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\hat{\mu}_\tau A_i-\hat{\nu}_\tau)-\frac{1}{n}\sum_{i\leq n}\rho_\tau(R_i-\mu_\tau^* A_i-\nu_\tau^*) + 2\sup_{(\mu,\nu)\in B_{2K}}\left|\frac{1}{n}\sum_{i\leq n}\left(\rho_\tau(R_i-\mu A_i-\nu)-\mathbb{E}\rho_\tau (R_i-\mu A_i-\nu)\right)\right|\\
\leq & 0+O_p(1/\sqrt{n}).
\end{align*}
\end{proof}
\section{Fairness on the training set}
\label{sec:training}
When a set of regression coefficients $\hat{\beta}_\tau$ is obtained by running quantile regression of $Y\in\mathbb{R}^n$ on a design matrix $X\in\mathbb{R}^{n\times p}$, the estimated conditional quantiles on the training set $\hat{q}_\tau(X)=X^T\hat{\beta}_\tau$ are always ``fair" with respect to any binary covariate that enters the regression. Namely, if a binary attribute $X_j$ is included in the quantile regression, then no matter what other attributes are regressed upon, on the training set the outcome $Y$ will lie above the estimated conditional quantile for approximately a proportion $1-\tau$, for each of the two subpopulations $X_{j}=0$ and $X_j=1$. This phenomenon naturally arises from the mathematics behind quantile regression. This section explains this property, and lays some groundwork for the out-of-training-set analysis of the following section.
We claim that for any binary attribute $X_j$, the empirical effective quantiles are balanced:
\begin{equation}
\label{eq:effect.quant}
\mathbb{P}_n\{Y>\hat{q}_\tau(X) \,|\, X_j=0\} \approx \mathbb{P}_n\{Y>\hat{q}_\tau(X) \,|\, X_j=1\} \approx 1-\tau,
\end{equation}
where $\mathbb{P}_n$ denotes the empirical probability measure on the training set.
To see why~\eqref{eq:effect.quant} holds, consider the dual of the quantile regression LP \eqref{eq:dual}.
This optimization has Lagrangian
\[
\mathcal{L}(b,\beta)= -Y^Tb+\beta^T(X^T b-(1-\tau)X^T\mathds{1})
= -\sum_{i\leq n}(Y_i-X_i^T\beta)b_i-(1-\tau)\sum_{i\leq n}X_i^T\beta.
\]
For fixed $\beta$, the vector $b\in [0,1]^n$ minimizing the Lagrangian tends to lie on the ``corners'' of the $n$-dimensional cube, with many of its coordinates taking value either 0 or 1 depending on the sign of $Y_i-X_i^T\beta$. We thus arrive at a characterization for $\hat{b}_\tau$, the solution to the dual program. For $i$ such that $Y_i\neq X_i^T\hat{\beta}_\tau$,
$
\hat{b}_{\tau,i}=\mathds{1}\{Y_i>X_i^T\hat{\beta}_\tau\}.
$
For $i$ such that $Y_i=X_i^T\hat{\beta}_\tau$, the values $\hat{b}_{\tau,i}$ are solutions to the linear system that makes~\eqref{eq:dual} hold. But with $p$ covariates and $n> p$, such equality will typically only occur at most $p$ out of $n$ terms. For large $n$, these only enter the analysis as lower order terms. Excluding these points, the equality constraint in~\eqref{eq:dual} translates to
\begin{equation}\label{eq:effect.quant1}
\sum_{i}X_{ij}\mathds{1}\{Y_i>X_i^T\hat{\beta}_\tau\}=(1-\tau)\sum_{i}X_{ij}\quad\text{for all }j.
\end{equation}
Assuming that the intercept is included as one of the regressors, the above implies that
\begin{equation*}
\frac{1}{n}\sum_{i} \mathds{1}\{Y_i>X_i^T\hat{\beta}_\tau\}=1-\tau,
\end{equation*}
which together with~\eqref{eq:effect.quant1}, implies balanced effective quantiles for binary $X_{\cdot j}$.
In particular, if the protected binary variable $A$ is included in the regression, the resulting model will be fair on the training data,
in the sense that the quantiles for the subpopulations $A=0$ and $A=1$ will be approximately equal, and at the targeted level $\tau$.
This insight gives reason to believe that the quantile regression coefficients, when evaluated on an independent heldout set, should still produce conditional quantile estimates that are what we are calling $\sqrt{n}$-fair. In the following section we establish $\sqrt{n}$-fairness for our proposed adjustment procedure. This requires us to again exploit the connection between the regression coefficients and the fairness measurements formed by the duality of the two linear programs.
| {
"timestamp": "2019-07-23T02:01:02",
"yymm": "1907",
"arxiv_id": "1907.08646",
"language": "en",
"url": "https://arxiv.org/abs/1907.08646",
"abstract": "Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to \"unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the target quantiles are brought into balance, in a statistical sense that we call $\\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a dataset of 200,000 live births, where the objective is to characterize the dependence of the birth weights of the babies on demographic attributes of the birth mother; the protected attribute is the mother's race.",
"subjects": "Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)",
"title": "Fair quantile regression",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846697584034,
"lm_q2_score": 0.7248702702332476,
"lm_q1q2_score": 0.7092019599598406
} |
https://arxiv.org/abs/2010.08023 | Primes in geometric series and finite permutation groups | As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. | \section{Introduction}\label{Intro}
The study of transitive permutation groups of prime degree goes back to the work of Galois on
polynomials of prime degree. It is sometimes asserted that the groups of prime degree are now
completely known, as a consequence of the classification of finite simple groups. This assertion
is true only if one ignores an apparently difficult number-theoretic problem, namely the existence
or otherwise of infinitely many primes of a particular form. The list of such permutation groups
includes several easily described infinite families, three relatively small sporadic examples,
and one other family which will be the subject of this note.
Let $p$ be a prime, and let $q=p^e$, $e\ge 1$, be a prime power. The projective special
linear groups ${\rm L}_n(q)={\rm PSL}_n(q)$ and some closely related groups act doubly transitively,
with degree
\begin{equation}\label{projeqn}
m = \frac{q^n-1}{q-1}=1+q+q^2+\cdots+q^{n-1},
\end{equation}
on the points or hyperplanes of the projective space ${\mathbb P}^{n-1}({\mathbb F}_q)$
for integers $n\ge 2$ and prime powers $q\ge 2$.
\begin{defi}[Projective prime]
If the number $m$ of points and of hyperplanes of the projective space
${\mathbb P}^{n-1}({\mathbb F}_q)$, defined by (\ref{projeqn}), is prime,
then we call it a {\em projective prime}.
\end{defi}
\begin{rema}[$n$ prime]\label{rem:n-prime}
A necessary condition for $m$ in (\ref{projeqn}) to be prime is the primality of the
exponent $n$ since otherwise the polynomial $1+t+t^2+\cdots+t^{n-1}$
would be reducible over $\mathbb Z$.
\end{rema}
The only projective primes with $n=2$ are the Fermat primes $m=2^{2^k}+1$,
while those with $q=2$ are the Mersenne primes, of the form $m=2^n-1$ with $n$ prime.
However, there are many others, such as $m=13$ with $n=q=3$. An interesting case is
the Mersenne prime
$$
m=31=1+2+4+8+16=1+5+25.
$$
Of course, it is an open problem whether there are infinitely many Fermat or Mersenne primes;
at the time of writing, only five Fermat primes (with $k=0,\ldots, 4$) and $51$ Mersenne
primes are known to exist. More generally, the existence of infinitely many projective
primes seems to be an open problem.
As in the case of Mersenne primes, there is plausible heuristic evidence, given in Section~\ref{Heuristic}, to support a conjecture that there are infinitely many projective primes. Indeed, there is much stronger computational
support for this, even in restricted cases such as when $n=3$ and $q$ is prime (see Section~\ref{Comp}). Our aim in
this note is to put forward such evidence, in the hope of inspiring specialists in number
theory to address this problem. Thus, we formulate the following
\begin{conj}[Projective primes]
There are infinitely many projective primes.
\end{conj}
\begin{rema}[Addition of primes]
More than once, some mathematicians (and physicists), among them
very eminent ones, have expressed the opinion that the two famous Goldbach conjectures are completely devoid of
interest since they concern the addition of primes whereas primes are created to be multiplied,
not added. Our note may be considered as a strong case for the interest, in certain contexts,
of the addition of primes and of numbers related to them (such as prime powers). By the way, one of the Goldbach conjectures has already been proved by Harald Helfgott~\cite{Helfgott}.
\end{rema}
\begin{nota}[Primes]
Notation for primes depends on the context. We follow the tradition of denoting a prime number
by the letter $p$ when this number is treated alone. If, however, there are more than one prime
number involved, as is the case, for example, in formula (\ref{projeqn}), then one of these
numbers may be denoted by $m$ or by some other letter.
\end{nota}
\section{Transitive permutation groups of prime degree}\label{Groups}
This section summarises the background in finite permutation groups from which the problem
stated in the Introduction arises. Any reader who is interested only in the problem itself
can safely omit this section. For more details on permutation groups of prime degree,
see~\cite[\S3.5]{DM} or \cite[\S V.21]{Hup}.
An elementary but important fact about transitive groups of prime degree is that they
are all primitive, that is, they leave invariant no non-trivial equivalence relations.
In particular, this means that rational or meromorphic functions of prime degree cannot
be compositions of those of lower degree. Groups of prime degree are also rather rare:
for example, there are $2\,801\,324$ transitive groups of degree $32$ (all but seven of
them imprimitive), and only twelve groups of degree~$31$; similarly there are $315\,842$
groups of degree $40$, but only ten of degree $41$, and six of degree $47$.
The number of transitive groups of degree 32 is computed in \cite{Hulpke}. The database
\cite{Galois-DB} contains a list of all transitive groups of degree $m\le 47$, $m\ne 32$.
The GAP system \cite{GAP} contains a list of all {\sl primitive}\/ groups of degrees
$m\le 2499$, and therefore, in particular, a list of all the groups of prime degrees
up to the same limit.
One of the sections of the memoir by \'Evariste Galois \cite{Galois}\footnote{In his preface
of 16 January 1831, Galois writes that this text is an ``extrait d'un ouvrage que j'ai
eu l'honneur de pr\'esenter \`a l'Acad\'emie il y a un an''. The French word ``ouvrage''
means either a book or just a large piece of work. This larger text was sent to Fourier for
refereeing. Fourier had suddenly died, and the manuscript was never found among his papers.}
is called ``Application to irreducible equations of prime degree''. If we translate the work
of Galois on polynomials and their roots into modern terminology, he showed that the
solvable groups of prime degree $p$ are the subgroups $G$ of the $1$-dimensional
affine group
\[{\rm AGL}_1(p)=\{t\mapsto at+b\mid a, b\in{\mathbb F}_p, a\ne 0\}\]
containing the translation subgroup ${\rm C}_p=\{t\mapsto t+b\mid b\in{\mathbb F}_p\}$. Such
groups $G$ are semidirect products $G={\rm C}_p\rtimes{\rm C}_d$ where ${\rm C}_d$
acts as a subgroup of~${\mathbb F}_p^*$ for some divisor $d$ of $p-1$. There is one
group $G$ for each such $d$, including $G={\rm C}_p$ for $d=1$ and $G={\rm AGL}_1(p)$
for $d=p-1$.
This directs our attention to the nonsolvable groups of prime degree.
Burnside~\cite[\S251]{Bur11} showed that any such group $G$ must be doubly transitive
(as is ${\rm AGL}_1(p)$, unlike its proper subgroups). In fact, in this case
elementary arguments show that a minimal normal subgroup $S$ of $G$ is a nonabelian
simple group, which is also transitive of degree $p$, with trivial centraliser $C_G(S)$
in $G$. Thus $G$ acts faithfully by conjugation on $S$, so
\[S\le G\le{\rm Aut}\,S.\]
This reduces the problem to that of determining the nonabelian simple groups $S$ of
prime degree~$p$, and then studying their automorphism groups for possible subgroups
$G$ of degree~$p$ (the action of~$S$ need not extend to all subgroups of ${\rm Aut}\,S$).
The classification of finite simple groups was announced around 1980, though not
completely proved until over twenty years later. One consequence (see~\cite{Cam81},
for example) was the classification of doubly transitive finite permutation groups.
There are eight families, described
in some detail in~\cite[\S7.7]{DM} and summarised in~\cite[\S7.4]{Cam}\footnote{Note that
${\rm Aut}\,{\rm M}_{22}$, of degree $22$, is omitted from~\cite[p.~252]{DM}. Similarly,
${\rm L}_2(11)$, of degree~$11$, is omitted from the list of groups of prime degree
in~\cite[\S V.21.2]{Hup}, though it is mentioned in II.8.28(6) and in the
Errata in the 2nd printing.}. As far as our problem is concerned, most of them can be
ignored, as their degrees are composite: for example, the symplectic groups
${\rm Sp}_{2n}(2)$ have degrees $2^{n-1}(2^n\pm 1)$, while the unitary and `small'
Ree groups over ${\mathbb F}_q$ have degree $q^3+1$, divisible by $q+1$.
The groups which survive this elimination process are listed in the following theorem:
\begin{theo}[Transitive groups of prime degree]\label{th:groups}
The nonabelian simple permutation groups $S$ of prime degree, together with any
transitive groups $G\le{\rm Aut}\,S$ of degree equal to that of $S$,
are as follows:
\begin{enumerate}
\item[(a)] alternating groups $S={\rm A}_p$ for primes $p\ge 5$, together with the
corresponding symmetric groups ${\rm Aut}({\rm A}_p)={\rm S}_p$;
\item[(b)] $G=S={\rm L}_2(11)$ for $p=11$, acting on the cosets of a subgroup ${\rm A}_5$
{\rm (}two representations, on two conjugacy classes of such subgroups,
equivalent under ${\rm Aut}\,S={\rm PGL}_2(11)${\rm )}, and the Mathieu
groups $S={\rm M}_{11}$ and ${\rm M}_{23}$, acting on Steiner systems with
$p=11$ and $p=23$ points;
\item[(c)] groups $G$ such that
$S={\rm L}_n(q)\le G\le {\rm P\Gamma L}_n(q)\le {\rm Aut}\,({\rm L}_n(q))$,
acting on the points or hyperplanes of the projective space
${\mathbb P}^{n-1}({\mathbb F}_q)$ when the degree $m=(q^n-1)/(q-1)$
is a projective prime, $m\ge 5$.
\end{enumerate}
If we also include the affine groups, where
\begin{enumerate}
\item[(d)] ${\rm C}_p\le G\le {\rm AGL}_1(p)$ for primes $p$,
\end{enumerate}
then we have a complete list of the transitive groups of prime degree.
\end{theo}
\begin{rema}[Commentaries on Theorem \ref{th:groups}]\
\begin{enumerate}
\item The group ${\rm L}_2(11)$ in (b) is one of three cases, known already to Galois,
in which the simple group ${\rm L}_2(p)$ has a non-trivial transitive representation
of degree less that $p+1$, specifically of degree $p=5,7$ or $11$ on the cosets of a
subgroup isomorphic to ${\rm A}_4$, ${\rm S}_4$ or ${\rm A}_5$. The first case appears
in both (a) and (c), via the isomorphisms
${\rm L}_2(5)\cong {\rm A}_5\cong {\rm L}_2(4)$, while the second appears in (c)
via the isomorphism ${\rm L}_2(7)\cong {\rm L}_3(2)$.
The group ${\rm Aut}\,({\rm L}_2(11))={\rm PGL}_2(11)$ does not have a
representation of degree $11$; hence, only the group ${\rm L}_2(11)$ is
a member of our list.
\item For the two Mathieu groups in (b) we have ${\rm Aut}\,({\rm M}_{11})={\rm M}_{11}$
and ${\rm Aut}\,({\rm M}_{23})={\rm M}_{23}$.
\item In (c), we have ${\rm Aut}\,({\rm L}_n(q))={\rm P\Gamma L}_n(q)$ if $n=2$,
but if $n\ge 3$ then ${\rm Aut}\,({\rm L}_n(q))$ contains ${\rm P\Gamma L}_n(q)$
with index~$2$, the `extra automorphism' arising from the point-hyperplane duality
of ${\mathbb P}^{n-1}({\mathbb F}_q)$.
\item In (d), the group ${\rm AGL}_1(p)$ is {\em not}\/ the automorphism group of
${\rm C}_p$ (indeed, ${\rm Aut}\,({\rm C}_p)\cong{\rm C}_{p-1}$). The case (d)
does not correspond to the general scheme of the cases (a), (b), (c) since,
as explained above, the group ${\rm AGL}_1(p)$ is solvable.
\end{enumerate}
\end{rema}
For a given projective prime $m$, the groups $G$ in (c) are easily determined:
they correspond bijectively to the subgroups of
\[{\rm P\Gamma L}_n(q)/{\rm L}_n(q)\cong({\rm PGL}_n(q)/{\rm L}_n(q))\rtimes
{\rm Gal}\,{\mathbb F}_q\cong{\rm C}_d\rtimes{\rm C}_e\]
where $d=\gcd(q-1,n)$ and $q=p^e$ for some prime $p$. In fact, if $m$ is prime then
$n$ is prime (see Remark \ref{rem:n-prime}) and $q\not\equiv 1$ mod~$(n)$,
so $d=1$, the groups ${\rm PGL}_n(q)$ and ${\rm L}_n(q)$ coincide, and
${\rm P\Gamma L}_n(q)/{\rm L}_n(q)\cong{\rm C}_e$. The real problem is to know which
primes are projective, and thus correspond to groups in (c), and in particular whether
or not there are infinitely many of them.
Although this paper concentrates on those cases where ${\rm L}_n(q)$ has prime degree,
there is also interest in cases such as ${\rm L}_5(3)$ where its natural degree $m$ is
a prime power ($11^2$ in this case). For example, Guralnick~\cite{Gur} has shown that
if a nonabelian simple group $S$ has a transitive representation of prime power degree,
then $S$ is an alternating group or ${\rm L}_n(q)$ acting naturally, or ${\rm L}_2(11)$,
${\rm M}_{11}$ or ${\rm M}_{23}$ acting as in Theorem~\ref{th:groups}(b), or the unitary
group ${\rm U}_4(2)\cong {\rm Sp}_4(3)\cong {\rm O}_5(3)$ permuting the $27$ lines on
a cubic surface. In particular, $S$ is doubly transitive in all cases except the last,
where it has rank~3, that is, three orbits on ordered pairs. See also~\cite{EGSS},
where Estes, Guralnick, Schacher and Straus have shown that for each prime $p$
there are only finitely many $e, q, n\ge 3$ such that $p^e=(q^n-1)/(q-1)$.
Another related topic which we will not address here is the Feit--Thompson
Conjecture~\cite{FT62} (see also~\cite[Problem B25]{Guy}), that if $p$ and $q$ are
distinct primes then $(p^q-1)/(p-1)$ does not divide $(q^p-1)/(q-1)$. A proof of this
would significantly shorten the (very long) proof of the theorem~\cite{FT63} that groups
of odd order are solvable.
\section{The Bunyakovsky Conjecture}\label{BunConj}
Viktor Bunyakovsky (1804--1889) was a Russian mathematician and a disciple of Cauchy.
In Russia he is mainly known for the Cauchy--Bunyakovsky inequality
which, in the Western tradition, is named after Cauchy--Schwarz. (As is stated in the
Wikipedia, Bunyakovsky ``\ldots\ is credited with an early discovery of the Cauchy--Schwarz
inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's works on the subject.")
In 1857, Bunyakovsky formulated the following conjecture (see \cite{Bun-1857, Bun-wiki}).
\begin{conj}[Bunyakovsky Conjecture]
The following fairly obvious necessary conditions for a polynomial $f(t)\in{\mathbb Z}[t]$
to have infinitely many prime values for $t\in{\mathbb N}$ are also sufficient:
\begin{itemize}
\item the leading coefficient of $f$ should be positive,
\item $f$ should be irreducible,
\item the integers $f(t)$ for $t\in{\mathbb N}$ should have greatest common divisor $1$.
\end{itemize}
\end{conj}
The last condition is needed in order to avoid examples such as $f(t)=t^2+t+2$, which
satisfies the first two conditions but has even values for all $t\in{\mathbb N}$;
Bunyakovsky gives the surprising example $f(t)=t^9-t^3+2520$, which is irreducible
but has all its values divisible by $504$. His conjecture is a special case of
Schinzel's Hypothesis H~\cite{SS}, which concerns finite sets of polynomials simultaneously
taking prime values.
\begin{rema}[Verification of the coprimality of $f(t)$ for $t\in\mathbb{N}$]
\label{rem:verif-Bun}
The existence of examples like the one above leads to the following question: how to
verify that the greatest common divisor of $f(t)$ for $t\in{\mathbb N}$ is $1$? A~method
(today we would say, an algorithm) proposed by Bunyakovsky is based on the following
observations.
\begin{enumerate}
\item Let $f(t)=c_nt^n+\cdots+c_1t+c_0$. If a prime $p$ divides all the values of
$f(t)$ for $t\in{\mathbb N}$, then $p$ is a divisor of $c_0$. Indeed, substituting
$t=p$ in $f(t)$ and taking the result modulo $p$ we get $c_0 \equiv 0 \mod (p)$.
Thus, we have only a finite number of primes $p$ to test.
\item Let $h(t)$ be a polynomial of degree $k<p$. Then all the values of
$h(t)$, $t\in{\mathbb N}$, are divisible by~$p$ if and only if all the
coefficients of $h$ are divisible by~$p$. Indeed, otherwise, reducing $h(t)$
modulo $p$ we would get a non-zero polynomial of degree less that $p$ which
would have $p$~roots.
\item All the values of the polynomial $t^p-t$ are obviously divisible by $p$.
Let $h(t)$ be the remainder of $f(t)$ on division by $t^p-t$. All that
remains is to determine whether the coefficients of $h(t)$ are all divisible by $p$.
\end{enumerate}
\end{rema}
The conjecture is true for $\deg(f)=1$: this is Dirichlet's Theorem on primes in an
arithmetic progression (see~\cite[\S 5.3.2]{BS} for a proof). However, it has not been
proved for any polynomial of degree greater than $1$, including the case $f(t)=t^2+1$
(see~\cite[\S A1]{Guy},~\cite[\S 2.8]{HW} or~\cite[Ch.~3.IVD]{Rib}); this is sometimes
called Landau's problem, though in fact it goes back to Euler~\cite{Eul}.
In our case we have the advantage
that we are not restricted to a single polynomial: we may consider polynomials
$f(t)=1+t+t^2+\cdots+t^{n-1}$ for any prime $n\ge 3$. (Since we have nothing to add
to the current state of knowledge or ignorance concerning Fermat primes, we will
assume for the rest of this paper that $n\ne 2$.) On the other hand, we require
prime values of $f(t)$ where $t$ is a {\sl prime power\/}, so a proof of the Bunyakovsky
Conjecture for such a polynomial would not necessarily yield infinitely many projective
primes.
Finally, we note that the Bunyakovsky Conjecture has recently arisen in a similar way
in the construction by Amarra, Devillers and Praeger~\cite{ADP} of block-transitive
point-imprimitive $2$-designs with specific parameters.
\section{Heuristic arguments}\label{Heuristic}
In this section we will present some heuristic arguments to support the conjecture that
there are infinitely many projective primes $m=(q^n-1)/(q-1)$. They are based on heuristic
arguments used elsewhere in considering the distribution and number of primes of a given
form. In particular, some of the arguments in this section are adapted from Wagstaff's
treatment~\cite{Wag} of conjectures of Gillies, Lenstra and Pomerance
about Mersenne primes, and its summary in
Prime Pages\footnote{See \url{https://primes.utm.edu/mersenne/heuristic.html}.}.
Of course heuristic arguments, based on assumptions which, although plausible, cannot be
rigorously justified, do not prove anything (in particular, see the warning in Section~\ref{warning}).
However, they may suggest results which one
could attempt to prove by more legitimate means. Authors of classic texts did not disdain
such kind of arguments: see, for example, Sections 2.5 and 22.20 of the book \cite{HW} by Hardy and
Wright, where they present heuristic evidence that there are only finitely many Fermat primes
whereas there are infinitely many prime pairs;
see also the discussion of probabilistic methods in~\cite[Notes on Ch.~8.3]{NZM}
and P\'olya's carefully-qualified defence of heuristic reasoning in number theory
in~\cite{Pol}.
Beside the ``general'' conjecture of infinitely many projective primes we also formulate
a number of ``specific'' (and therefore stronger) conjectures concerning projective
primes of some specific forms. Their plausibility is based mainly on a series of
computational results presented in Sections~\ref{Comp} and \ref{sec:stochastic}.
\subsection{Prime divisors of $m$}\label{sec:small}
We consider firstly the case of any fixed prime $n\ge 3$, and secondly that of any fixed
prime power $q$. In each case, we will need the following lemma in order to give better
estimates for the number of projective primes up to some bound.
\begin{lemm}[Prime divisors of $m$]\label{SmallPrimes}
Let $m=(q^n-1)/(q-1)$ for some integer $q$ and prime $n\ge 3$, and let $r$ be a prime
dividing $m$. Then either $r\equiv 1$ {\rm mod}~$(2n)$
{\rm (}so in particular $r\ge 2n+1${\rm )},
or $r=n$ with $q\equiv 1$ {\rm mod}~$(n)$.
Conversely, if $q\equiv 1$ {\rm mod}~$(n)$ then $m$ is divisible by $n$.
\end{lemm}
\noindent{\sl Proof.} If a prime $r$ divides $m$ then $q^n\equiv 1$ mod~$(r)$.
Since $n$ is prime, it follows that either $n$ divides the order $r-1$ of the
multiplicative group ${\mathbb F}_r^*$, or $q\equiv 1$ mod~$(r)$.
If $n$ divides $r-1$ then $r\equiv 1$ mod~$(n)$. Clearly $m=1+q+\cdots +q^{n-1}$ is odd,
and hence so is $r$, so $r\equiv 1$ mod~$(2n)$ since $n$ is odd and hence $r\ge 2n+1$.
If $q\equiv 1$ mod~$(r)$ then
\[m=1+q+\cdots+q^{n-1}\equiv\underbrace{1+1+\cdots+1}_{n\,\,{\rm times}}\equiv n \;
{\rm mod}~(r).\]
However, $m\equiv 0$ mod~$(r)$, so $n\equiv 0$ mod~$(r)$ and hence $r=n$ since
$n$ is prime. The converse is obvious.
\hfill$\square$
\medskip
Thus $m$ is not divisible by any prime $r\le 2n$, except the prime $r=n$ if
$q\equiv 1$ mod~$(n)$.
\begin{exam}[For Lemma \ref{SmallPrimes}]\label{ex:small-primes}
Let $n=3$. If $q=11\not\equiv 1$ mod~$(3)$ then $m=133=7\cdot 19$; this is divisible
by the primes $r=7$ and $19$, both greater than $2n=6$. However,
if $q=16\equiv 1$ mod~$(3)$ then $m=273=3\cdot7\cdot 13$, divisible by the prime
$r=n=3$ in addition to $r=7$ and $13$. Note that the `large' primes $7, 13$ and $19$
appearing here as divisors of $m$ are all congruent to $1$ mod~$(2n)$.
\end{exam}
\subsection{Fixed $n$, while $q=p\to\infty$}\label{sec:fixed-n}
Let us fix a prime $n\ge 3$, and consider whether $m=(q^n-1)/(q-1)$ is prime.
For simplicity we will restrict $q$ to be prime, rather than a prime power;
therefore, from now on we will denote it by $p$ instead of $q$.
By the Prime Number Theorem (see~\cite[Theorem~6 and Ch.~XXII]{HW} for example),
the number of primes $p$ in the range
$1\le p\le x$ is approximately $x/\ln(x)$ for large $x$. However, if $p\equiv 1$ mod~$(n)$
then $m$ cannot be prime by Lemma~\ref{SmallPrimes}, so we should restrict attention to the
primes $p\not\equiv 1$ mod~$(n)$. Since primes are approximately evenly distributed
between the non-zero congruence classes mod~$(n)$ (see~\cite[\S 5.3.2]{BS}, for example),
the number of primes $p$ we should consider is therefore approximately $(n-2)x/(n-1)\ln(x)$.
Now $1\le m\le (x^n-1)/(x-1)$, and the probability that a randomly-chosen integer $m$
in this range is prime is approximately
\begin{equation}\label{prob1}
\frac{1}{\ln((x^n-1)/(x-1))}\approx\frac{1}{n\ln(x)-\ln(x)}=\frac{1}{(n-1)\ln(x)}.
\end{equation}
However, we know from Lemma~\ref{SmallPrimes} that $m\not\equiv 0$ mod~$(r)$
for each prime $r\le 2n$, including $r=n$ since $p\not\equiv 1$ mod~$(n)$.
For each such $r$, excluding this one congruence class mod~$(r)$ multiplies the
probability of $m$ being prime by $r/(r-1)$. If we regard congruences modulo distinct
primes as statistically independent, then we should multiply the probability
in (\ref{prob1}) by $P(2n)$, where
\begin{equation}\label{prod}
P(y):=\prod_{{\rm prime}\,\,r\le y}\left(1-\frac{1}{r}\right)^{-1}
\end{equation}
for $y\ge 2$ and the product, as indicated, is over all primes $r\le y$.
This gives an approximate probability
\begin{equation}\label{prob7}
\frac{P(2n)}{(n-1)\ln(x)}
\end{equation}
that $m$ is prime. For fixed $n$ this has the form $c_n/\ln(x)$ for a constant
\[c_n := \frac{P(2n)}{(n-1)}.\]
If $n$ is small one can easily calculate $c_n$: for instance $c_3=15/8$ and $c_5=35/32$.
For large $n$ one can approximate $c_n$ by using a theorem of Mertens (see~\cite{Mer},
\cite[\S22.9]{HW} or~\cite[Theorem~8.8(e)]{NZM}) that
\[\prod_{{\rm prime}\,\,r\le y}\left(1-\frac{1}{r}\right)\sim
\frac{\mu}{\ln(y)}\quad{\rm as}\quad y\to\infty,\]
where $\mu:=e^{-\gamma}=0.561459\ldots$\ \label{const:mu}
and $\gamma$ is the Euler--Mascheroni constant $0.577215\ldots$.
This gives an approximate probability
\begin{equation}\label{prob3}
\frac{c_n}{\ln(x)}\sim\frac{e^{\gamma}\ln(2n)}{(n-1)\ln(x)}
\end{equation}
that $m$ is prime.
If we multiply this probability by the approximate number of primes $p\not\equiv 1$ mod~$(n)$ in the range $1\le p\le x$, namely $(n-2)x/(n-1)\ln(x)$, we see that the expected number of primes $m$ arising in this way is approximately
\begin{equation}\label{prob8}
\frac{c_n(n-2)x}{(n-1)(\ln(x))^2}\sim\frac{e^{\gamma}(n-2)\ln(2n)x}{(n-1)^2(\ln(x))^2}\approx\frac{1.781(n-2)\ln(2n)x}{(n-1)^2(\ln(x))^2}.
\end{equation}
Since this number tends to $+\infty$ as $x\to\infty$ for fixed $n$, this suggests
that we should obtain infinitely many projective primes $m$ in this way for any fixed prime $n\ge 3$
(and likewise if we allow $q$ to be an arbitrary prime power).
For each fixed $n$ the estimate in (\ref{prob8}) has the form $C_nx/(\ln(x))^2$ for some constant $C_n$ depending only on $n$. This is analogous to the Hardy--Littlewood estimate $Cx/(\ln(x))^2$ for the number $\pi_2(x)$ of twin prime pairs $p$, $p+2$ with $p\le x$ (see~\cite{PP}), where
\[C=2\cdot\negthinspace\negthinspace\prod_{{\rm prime\;} r\ge 3}\frac{r(r-2)}{(r-1)^2}\approx 1.320323632.\]
\begin{exam}[$n=3$]
If we take $n=3$, so that $c_n=15/8$, then the number of primes $m=1+p+p^2$ for
primes $p\le x$ should be approximately
\begin{equation}\label{n=3estimate}
\frac{15x}{16 \ln(x)^2}
\end{equation}
for large $x$. This estimate is compared with computational evidence in
Section~\ref{primefields} (see Table~\ref{tab:n=3ratios}).
\end{exam}
\subsection{Fixed prime power $q=p^e$ with $e\ge 2$}\label{sec:fixed-q}
Instead, let us now fix $q$ and let $n\to\infty$.
\begin{lemm}\label{e>1}
If $e\ge 2$ then there are no projective primes $m=(q^n-1)/(q-1)$ with $n>e$.
\end{lemm}
\begin{proof}
If $e\ge 2$, so that $q$ is a prime power but not itself a prime, we have
$$
m = \frac{q^n-1}{q-1} =
\frac{(1+p+\cdots+p^{n-1})\,(1+p^n+\cdots+p^{n(e-1)})}{1+p+\cdots+p^{e-1}}.
$$
This is clearly composite if $n>e$ since the two factors in the numerator are
each larger than the denominator.
\end{proof}
Thus, for a fixed $q$ with $e\ge 2$ we can have only a finite number of projective primes $m$.
\begin{rema}[$e=2$]\label{re:e=2}
If $e=2$ and $m$ is prime then $n=2$ (and hence $q=p^2$ is even,
so that $p=2$ and $m=1+q=5$), against our earlier assumption;
thus in dealing with prime powers $q=p^e>p$ (as in Section~\ref{sec:primepowersagain},
where a number of examples are given) we will generally assume that $e\ge 3$.
\end{rema}
Since the ultimate goal of this line of research is to complete the classification
of the permutation groups of prime degree, it would still be of interest to know the
finitely many projective primes arising for each proper prime power $q$.
\subsection{Fixed prime $p$ while $n\to\infty$}\label{sec:fixed-p1}
Let us therefore take $e=1$, so we fix a prime $p$, and consider the primality of
$m=(p^n-1)/(p-1)$ for odd primes $n$ as $n\to\infty$. By Lemma~\ref{SmallPrimes}
we may exclude any primes $n$ dividing $p-1$, since they cannot give prime values
of $m$. Then $m$ is not divisible by any prime $r\le 2n$.
By the Prime Number Theorem, for large $n$ a randomly-chosen integer close to $(p^n-1)/(p-1)$ is prime with probability approximately
\begin{equation}\label{prob}
\frac{1}{\ln((p^n-1)/(p-1))}\approx\frac{1}{n\ln(p)-\ln(p-1)}.
\end{equation}
However, $m$ is not uniformly distributed, since it is coprime to each prime
$r\le 2n$. As before, for each such $r$ this excludes a proportion $1/r$ of the integers
close to $(p^n-1)/(p-1)$, so we should multiply the probability in (\ref{prob}) by
$M(2n)\sim e^{\gamma}\ln(2n)$, giving an approximate probability
\begin{equation}\label{prob2}
\frac{e^{\gamma}\ln(2n)}{n\ln(p)-\ln(p-1)}\sim\frac{e^{\gamma}\ln(n)}{n\ln(p)}
\end{equation}
that $m$ is prime, for large $n$.
(For small $p$, as in Section~\ref{sec:fixed-p2}, this last approximation could induce
significant errors.)
If we choose the prime $n$ uniformly and randomly from the range $p\le n\le x$
for some large $x$ (so that most such $n$ are large as above), then the expected number of
primes $m$ arising is the sum of the probabilities in~(\ref{prob2}), that is
\[\frac{e^{\gamma}}{\ln(p)}\sum_n\frac{\ln(n)}{n}\]
where the sum is over all primes $n$ such that $p\le n\le x$. Now
\[\sum_n\frac{\ln(n)}{n}\approx\ln(x)-\ln(p-1),\]
(see~\cite[Theorem~425]{HW}), so the expected number of primes $m$ is approximately
\begin{equation}\label{qfixed}
\frac{e^{\gamma}(\ln(x)-\ln(p-1))}{\ln(p)}\sim\frac{e^{\gamma}\ln(x)}{\ln(p)}\approx\frac{1.781\ln(x)}{\ln(p)}.
\end{equation}
Since this tends to $+\infty$ with $x$, we may expect to obtain infinitely many
projective primes $m$ from any given prime $p\ge 2$. This estimate is compared with
computational evidence in Section~\ref{sec:fixed-p2} (see Table~\ref{tab:fixed-p}).
\subsection{A warning}\label{warning}
Invoking the independence of congruences modulo different primes in order to make
heuristic estimates, as we did in Section~\ref{sec:fixed-n}, has previously generated
controversy: for instance, Wagstaff discusses this in~\cite{Wag}, citing criticism by
Lenstra in~\cite{Len}. This is best illustrated with P\'olya's discussion in~\cite{Pol}
of the following well-known paradox.
Based on the type of argument used in Section~\ref{sec:fixed-n}, one can attempt a heuristic proof of the Prime Number Theorem. An integer $x$ is prime if and only if $x\not\equiv 0$ mod~$(r)$ for each prime $r\le x$. For each such $r$ this event has probability $(r-1)/r$, so by regarding these events as mutually independent, and by using Mertens's Theorem, one might expect $x$ to be prime with probability
\begin{equation}\label{PNT1}
\prod_{{\rm prime}\,\,r\le x}\left(1-\frac{1}{r}\right)\sim\frac{\mu}{\ln(x)}\quad\hbox{as}\quad x\to\infty,
\end{equation}
where $\mu=e^{-\gamma}=0.561459\ldots$. However, the correct asymptotic probability is $1/\ln(x)$, so this argument underestimates the probability of $x$ being prime (and hence the values of the prime-counting function $\pi(x)$) by a factor of $\mu$. Of course, it is sufficient to eliminate prime factors $r\le x^{1/2}$, rather than $r\le x$, so this alternative approach gives a second estimate
\begin{equation}\label{PNT2}
\prod_{{\rm prime}\,\,r\le x^{1/2}}\left(1-\frac{1}{r}\right)\sim\frac{\mu}{\ln(x^{1/2})}=\frac{2\mu}{\ln(x)}\quad\hbox{as}\quad x\to\infty.
\end{equation}
This overestimates the correct probability by a factor of $2\mu = 1.122918\ldots$, that is, by about $12\%$. If, as suggested by P\'olya in~\cite{Pol}, one takes the product over all primes $r\le x^{\mu}$ then the correct formula is obtained. P\'olya confesses that it is not clear why what he calls this ``trick of the magic $\mu$" works here (Wagstaff~\cite{Wag} calls it a ``fudge factor"), but he goes on to argue that mathematicians should imitate physicists by adapting their theories to fit experimental data when such paradoxes arise. Similar phenomena are discussed by P\'olya~\cite{Pol} in relation to prime pairs and their generalisations, and by Wagstaff~\cite{Wag} in relation to the distribution of divisors of Mersenne numbers.
The great Russian mathematician Andre\u{\i} Kolmogorov used to mention the following episode
(the second author heard it directly from him). Kolmogorov was once present at a talk
given by a prominent Russian physicist. The latter, basing his reasoning on some physical
ideas, introduced the density of a probability distribution on a certain space. Then,
he integrated this density and obtained $\pi$. At this point, Kolmogorov used to say,
I would conclude that we had got a contradiction, and therefore all the reasoning was
wrong. But the conclusion of the physicist was different. Thus, he said, we must divide
the initial formula for the density by $\pi$. It seems that P\'olya would rather line up
with the physicist.
In our case, an appropriate choice of prime factors of $m$ to avoid is also an intricate
matter. For example, for $n=3$, as we will see later, in Section~\ref{primefields}
and Table~\ref{tab:n=3ratios}, formula~(\ref{n=3estimate}) overestimates the number of
projective primes. But we know that,
beside the ``small primes'' 2, 3 and 5, Lemma~\ref{SmallPrimes}
also forbids all primes of the form $r \equiv -1$ mod~$(6)$. However, even if
we adjoin to the product
$$
P(6)=\left( \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)
\left(1-\frac{1}{5}\right) \right)^{-1}
$$
not all such corresponding terms but only $(1-1/11)^{-1}$, we will get $c_3'=33/16$
instead of $c_3=15/8$, and estimate (\ref{n=3estimate}) will be replaced with
$33x/32\ln(x)^2$, which will overestimate the number of projective primes even more
than (\ref{n=3estimate}) does. On the other hand, if we remove the factor $(1-1/5)^{-1}$
from $P(6)$, we will underestimate the desired number.
But this is not yet the end of the story. As we will see in
Section~\ref{sec:empiric} and Figure~\ref{fig:comparison},
the ratio of our estimates to the true number of projective primes grows, that is,
the estimates grow faster than the numbers they are supposed to estimate.
There are even reasons to believe the overestimate grows to infinity
(though slowly). Obviously,
the constant factor does not play any part in this process: the rate of growth of the
estimates depends only on the behavior of the function $x/\ln(x)^2$. Therefore, it is
reasonable to suppose that, for very large $x$, we will have indeed to eliminate 5 from
the set of forbidden primes. Note however that this process of eliminating
or adjoining forbidden primes is not based on any solid theoretical foundation: it is
purely empirical. Therefore, instead of making artificial choices of which primes to
include in the product, it seems at least as reasonable to consider other functions
instead of $x/\ln(x)^2$. This will be done in Section \ref{sec:empiric}.
\section{Primality testing}\label{sec:testing}
Before presenting the experimental results aiming to support our main conjecture
(that there are infinitely many projective primes), let us briefly discuss two problems:
the factorization of integers into prime factors, and the testing of primality. The problems
are, evidently, related to each other, but there is an abyss between their complexities.
\subsection{Integer factorization in modern times}
According to \cite{AGLL}, in 1977, Ronald Rivest, in a letter to
Martin Gardner, estimated that
\begin{quote}
``{\ldots} factoring a 125-digit number
which is the product of two 63-digit
prime numbers would require at least 40 quadrillion years using the best factoring algorithm
known, assuming that $a\cdot b\; {\rm mod}\,(c)$ could be computed in 1~nano\-second, for
125-digit numbers $a$, $b$, and $c$."
\end{quote}
It sounded like a solemn chorus from Purcell's {\sl Dido and Aeneas}\/:
``Never! Never! Never!''
The same year, Gardner \cite{Gardner} launched a challenge: it was proposed to factor
a 129-digit number which was the product of a 64-digit and a 65-digit prime. Apparently
hopeless, whatever the future progress in computer technology would be.
Subsequent years saw spectacular progress in factorization
algorithms. Finally, 17 years later, in 1994, the above 129-digit
number was successfully factored. The project involved some 600 volunteers,
1600 computers, and six months of computation. An account may be found
in \cite{AGLL}; the strange expression ``squeamish ossifrage'' in the title was the
message encrypted using this number by the RSA cryptographic method.
We skip a number of important developments during the next 15 years and go directly to a
milestone of 2009. In December of that year, a 232-digit number was factored: it was a
product two 116-digit primes. This result was the outcome of two years of work by a team
of 13 researchers, and was crowned with a {\$}50\,000 prize. As stated in~\cite{RSA},
\begin{quote}
``The CPU time spent on finding these factors by a collection of parallel computers
amounted approximately to the equivalent of almost 2000 years of computing on a
single-core 2.2 GHz AMD Opteron-based computer."
\end{quote}
Certainly, 2000 years for a 232-digit number as compared to 40\,000\,000\,000\,000\,000
years for a 125-digit one, is incredible progress. A convenient, and machine-independent
measure of an effort in a large-scale computation is GHz-years; in the present case we
have 4400 GHz-years of computing.
Ten more years have passed, and a 240-digit number was factored in November 2019,
and a 250-digit number in February 2020. And here lies the current frontier of the
capability of factoring algorithms. The next challenge, a 260-digit number, still waits
for its turn to be factored.
\subsection{Testing}
The above examples show how difficult, in practical terms, the problem of factorization
can be. However, there exist algorithms which establish whether a given integer is prime
or composite, and this without ever trying to factor it. The most well-known, and the most
used in practice, is the Rabin--Miller algorithm~\cite{Rabin} (see also the compendium
\cite{CLR}). In particular, it is implemented in the Maple command {\em isprime}. Let us
take the above-mentioned 260-digit number (which, we recall, is not yet factored) and see
how this command works. The computation is carried out on a very modest laptop.
\begin{figure}[htbp]
\includegraphics[scale=0.9]{composite-260.eps}
\end{figure}
We see that the correct answer is given, almost literally, ``in no time at all''.
In fact, the Maple time-counter outputs the CPU time within an accuracy of 0.001 seconds.
Therefore, 0. seconds time displayed in the above session means $<0.0005$ seconds, rounded
downwards.
Now consider a harder example, a 6153-digit number we will encounter in
Section~\ref{1stseries}.
\begin{figure}[htbp]
\begin{flushleft}
\hspace*{4mm}
\includegraphics[scale=0.9]{prime-6153.eps}
\end{flushleft}
\end{figure}
This number is prime, and the computation took more than 13 seconds. A good result,
but to perform this testing on a large scale, that is, with large series of numbers,
can turn out to be time-consuming.
\subsection{How the test works}
The Rabin--Miller algorithm is probabilistic. In order to determine whether a given
number $m$ is prime it takes a random element $t\in\mathbb{Z}_m$ and verifies a necessary
primality condition. The condition itself is simple, so we give it here.
The number $m-1$ is even; suppose it is equal to $m-1=(2l+1)\cdot 2^k$. Compute
in $\mathbb{Z}_m$
$$
a_0=t^{2l+1}, \quad \mbox{and then} \quad a_i=a_{i-1}^2 \quad \mbox{for} \quad
i=1,\ldots,k, \quad \mbox{so that} \quad a_k=t^{m-1}.
$$
If one of the following holds then $m$ is composite:
\begin{enumerate}
\item While computing the sequence $a_i$, we come for the first time to $a_i=1$
but the previous number $a_{i-1}\ne -1$. Indeed, in this case the equation
$a^2=1\;{\rm mod}\,(m)$ has, beside two obvious roots 1 and $-1$, a third
root $a_{i-1}$.
\item We get $a_k=t^{m-1}\ne 1$. This contradicts Fermat's little theorem.
\end{enumerate}
Thus, if the test tells us that $m$ is composite then this statement is true, and no
probability is involved. If, however, neither of the two above conditions is satisfied,
we conclude that $m$ is {\em probably prime}. Rabin \cite{Rabin} showed that the
probability of an erroneous answer is bounded by~$1/4$; usually it is much smaller.
For large $m$, in the majority of cases this probability is infinitesimally small.
A dialogue from Gilbert and Sullivan's {\em I am the captain of the Pinafore}\/ comes
to mind: ``What, never? No, never. What, never? Well, hardly ever''.
Nevertheless, in order to be on the safe side, the test is repeated many times with
different (random) values of~$t$. This, by the way, explains why the treatment of a
prime number takes much more time than that of a composite number of the same size.
Notice that raising a number $a$ to a power $a^r$ needs $O(\log r)$ arithmetic operations:
we compute first $a,a^2,a^4,a^8,\ldots$ (taking squares every time), and then multiply
the terms corresponding to the binary expression of the exponent $r$. Notice also that,
in our case, all computations are made modulo $m$, so that the size of the numbers remains
bounded.
\subsection{Polynomial-time algorithms}
The subject of primality testing and factorization has many ramifications. We only mention
very briefly a few of them. We recommend, for an interested reader, a very concise and
clear overview~\cite{Brent} and a more modern and advanced exposition in~\cite{FR}
(especially Chapter~5, ``Primality testing---an overview'').
There are several algorithms for primality testing whose complexity is polynomial in the
size of tested numbers. However, for most of them the estimation of complexity is based
on some as yet unproved hypotheses.
\begin{nota}[Simplified measure of complexity]
Denote $k:=\log m$, and denote $\widetilde{O}(k^s):=O(k^{s+\varepsilon})$ for all $\varepsilon>0$.
\end{nota}
This notation allows one to simplify complexity estimates for algorithms such as, for example,
the Sch\"onhage--Strassen algorithm of multiplication of long integers: we may now write
just $\widetilde{O}(k)$ instead of $O(k\cdot\log k\cdot \log\log k)$.
The complexity of the Rabin--Miller algorithm is $\widetilde{O}(k^2)$: here $O(k)$ is the number
of arithmetic operations, and $\widetilde{O}(k)$ is the complexity of an individual operation.
Four years before Rabin, Miller used the same test but in a deterministic way. Namely,
it suffices to make the test for all $t\le 2\log(m)^2=2k^2$, {\em provided that the
Extended Riemann Hypothesis is true}. Thus, this algorithm, of complexity is $\widetilde{O}(n^4)$,
while being deterministic, is based on an unproved conjecture. Also, the factor $2k^2$
is not innocuous. For $k\sim 6000$, as in the above example, it transforms seconds
into years.
\begin{rema}[Are long computations reliable?]
It is important to note that in a long computation there is a significant probability
of a hardware error. This probability is much greater than that in the Rabin--Miller test.
\end{rema}
In a revolutionary work \cite{Primes-in-P}, an {\em unconditional}\/ polynomial time
algorithm for primality testing was given for the first time. Here `unconditional' means
that the estimate of its complexity does not depend on any unproved statement. After
several improvements its complexity is now established as $\widetilde{O}(k^{15/2})$. It may also
be $\widetilde{O}(k^6)$ if another as yet unproved conjecture is valid. Its theoretical impact
is great but its practical utility is very limited.
Another method is based on the theory of elliptic curves. It is commonly known as the ECPP
algorithm, which means Elliptic Curve Primality Proving. The names we must mention here
are Sh.~Goldwasser, J.~Kilian, A.~Atkin and F.~Morain. This algorithm is probabilistic;
however, it is not of the ``Monte-Carlo type'' but of the ``Las Vegas type''. The latter means
that it always gives the correct answer; it is the computation time which is random.
It is polynomial {\em on average}\/ if certain as yet unproved conjectures are true.
Beside the correct answer, this algorithm also creates a {\em primality certificate}.
A certificate is ``something'' which may be difficult to find but, once found, allows
one to make a verification easily.
In \cite{Brent}, the following example is given. Consider the number
$m=4405^{2638}+2638^{4405}$. It has 15\,071 digits. The {\em proof}\/ of its primality
by the ECPP algorithm was achieved in 5.1~GHz-years. This is a truly remarkable
result if we compare it with other error-free algorithms. Note, however, that the
Rabin--Miller algorithm gives the correct (though unproved) answer in less than
two minutes.
\subsection{A few comments}
Since 1980, when Michael Rabin published his algorithm, not a single case of an erroneous
answer has been observed. Even financiers, in their cryptographic protocols, rely entirely on
this test. However, a mathematical mind resists accepting a ``proof'' which in principle
might be wrong, even if the probability of such an event is infinitesimally small.
What then to do if we have doubts about the validity of the conclusion `prime' given
by the probabilistic test?
In our opinion, the most reasonable way to proceed is to run this test again once or
twice. The test does not repeat exactly the same operations since it chooses different
random elements of $\mathbb{Z}_m$ every time. In this way the probability $\alpha$
of an error, already infinitesimal, will be replaced with $\alpha^2$ or $\alpha^3$.
(We may ask, rather provocatively: how many times can you repeat a two-minute test
if you have 5.1 years at your disposal?)
And what if, by an incredible combination of chances, we take a composite number for
a prime one? Well, let us recall that the aim of our particular study is to collect
evidence that there are infinitely many projective primes. Therefore, one prime less
or one prime more does not change much.
\begin{comment}
\begin{nota}
Denote $k:=\log m$, and denote also $\widetilde{O}(f(k)):=O(f(k)\cdot\gamma(k))$ where $\gamma(k)=o(k)$.
Then the complexity of the Rabin--Miller algorithm is $\widetilde{O}(k^2)$: here $O(k)$ is the number
of arithmetic operations, and $\widetilde{O}(k)$ is the complexity of multiplication and division
of numbers of size $k$ if we use, for example, the Sch\"onhage--Strassen algorithm of
complexity $O(k\cdot\log k\cdot \log\log k)$.
\end{nota}
\subsection{Deterministic polynomial-time algorithms}
The subject of primality testing and factorization has many ramifications. We only mention
very briefly a few of them. We recommend, for an interested reader, a very concise and
clear overview \cite{Brent} and a more modern and advanced exposition in \cite{FR}
(especially Chapter~5, ``Primality testing---an overview'').
Since 1980, when Michael Rabin published his algorithm, not a single case of an erroneous
answer has been observed. Even financiers, in their cryptographic protocols, rely entirely on
this test. However, a mathematical mind resists accepting a ``proof'' which in principle
might be wrong, even if the probability of such an event is infinitesimally small. Does
there exist a way out of this impasse? In fact, four years before Rabin, Miller used
the same test but in a deterministic way. Namely, it suffices to make the test for all
$t\le 2\log(m)^2=2k^2$, {\em provided that the Extended Riemann Hypothesis is true}.
Thus, this algorithm, of complexity is $\widetilde{O}(n^4)$, while being deterministic, is based
on an unproved conjecture. Also, the factor $2k^2$ is not innocuous. For $k\sim 6000$, as
in the above example, it transforms seconds into years.
By the way, in a long computation there is a significant probability of a hardware error.
This probability is much greater than the one in the probabilistic test.
In a revolutionary work \cite{Primes-in-P}, an unconditional polynomial time algorithm
for primality testing was given for the first time. Its complexity is $O(k^{15/2})$
(or $\widetilde{O}(k^6)$ if another as yet unproved conjecture is valid). Its
theoretical impact is great but its practical utility is very limited.
Another line of research is looking for primality certificates. A certificate is
``something'' which may be difficult to find but, once found, permits one to make
a verification easily. Elliptic curves play an important role here. Of course,
the word ``easily'' must be taken with a grain of salt. To give but one example,
consider the number $m=4405^{2638}+2638^{4405}$. It has 15\,071 digits. According
to \cite{Brent}, the {\em proof}\/ of its primality was achieved in 5.1~GHz-years.
The probabilistic test gives the correct (but not proved) answer in less than two
minutes.
\end{comment}
\section{Computational evidence}\label{Comp}
\subsection{Bunyakovsky's conjecture from an experimental perspective}
\label{sec:bun-experim}
The data in favor of the veracity of this conjecture abound. If we take, for example,
$f(t)=t^2+t+1$ and count the number of integers $t\le 10^7$ for which $f(t)$ is prime,
we get 745\,582 solutions. A very ``modest'' particular case of Bunyakovsky's conjecture
is known as Landau's conjecture: it concerns $f(t)=t^2+1$. In this case the number of
$t\le 10^7$ for which $t^2+1$ is prime is 456\,362. There is little doubt that,
at least in these two cases, the conjecture is true. No proof is, however, in view.
The main motivation of this note comes from group theory. Therefore, we will mainly
consider not arbitrary values of $t$ but only prime powers $t=p^e$, $e\ge 1$,
and not arbitrary polynomials $f(t)$ but only those of the form
$$
f(t) = \frac{t^n-1}{t-1} = 1+t+t^2+\cdots+t^{n-1}.
$$
\begin{rema}[Terminological]
While speaking of {\em prime powers}, according to the context we may mean $p^e$ with
$e\ge 1$, that is, including ``pure'' primes, or, sometimes, with $e\ge 2$, in order
to put prime powers in contrast with the pure primes whose exponent is $e=1$.
\end{rema}
\subsection{First series of projective primes}\label{1stseries}
A computer search has revealed
$668$ projective primes with $2\le q\le 2000$ and
$3\le n\le 2000$, including one with $6153$ decimal digits, arising from $q=1201$
and $n=1999$. It is interesting to note that only five pairs $(q,n)$ out of $668$
correspond to prime powers $q=p^e$ with $e\ge 2$, namely,
$$
(q,n) \,=\, (2^3,3),\, (2^7,7),\, (2^9,3),\, (3^3,3),\, (11^3,3).
$$
All the other values of $q$ are ``pure'' primes.
\subsection{Number 31}\label{31}
A computer search of prime degrees up to $10^{12}$ reveals
${\rm L}_3(5)$ and ${\rm L}_5(2)$ as the only pair of groups ${\rm L}_n(q)$ with
the same natural degree in this range; it would be interesting to know whether any
other such pairs exist.
\begin{conj}[Number 31]\label{conj:31}
Beside\/ $31$, there are no other natural degrees common to two different projective
groups ${\rm L}_n(q)$.
\end{conj}
\begin{rema}[Goormaghtigh conjecture]
The Diophantine equation
$$
\frac{x^n-1}{x-1}=\frac{y^k-1}{y-1}
$$
has been studied by many authors (see~\cite[Problem B25]{Guy}, for example).
In 1917, a Belgian engineer and amateur mathematician Ren\'e Goormaghtigh\footnote{See
{\tt https://forvo.com/word/ren\%C3\%A9\_goormaghtigh} to learn how to pronounce this name.}
(1893--1960) conjectured \cite{Goo} that this equation,
for $n\ne k$, $n,k\ge 3$, has only two solutions in $\mathbb N$:
$1+2+4+8+16=1+5+25=31$ and $1+2+4+\cdots+2^{12}=1+90+90^2=8191$.
However, 90 is not a prime power, so that there
is no field with 90 elements. By the way, the number 8191 is prime. Therefore, it is
an instance of Bunyakovsky's conjecture for two different polynomials (and certainly
for many other ones, like $t^2+91$, for example), but it is a projective prime for
only one of them, namely, $1+t+t^2+\cdots+t^{12}$.
In \cite{DLS}, it is proved that for fixed exponents $n\ne k$, $n,k\ge 3$, there can
be only a finite number of solutions. For additional information about this equation
see \cite{Goo-wiki}.
\end{rema}
\subsection{Projective planes over prime fields}\label{primefields}
Let us take only prime values $p$, not taking into account the prime powers $q=p^e$
with $e\ge 2$, let us fix $n=3$ and consider projective primes $m=1+p+p^2$. Our colleague
Jean B\'etr\'ema examined all primes $p\le 10^{11}$ using the package
{\tt Primes.jl} of the language {\tt Julia}. It turns out that {\tt Julia}
is much more efficient than Maple for problems of this sort. We partially reproduce
B\'etr\'ema's results in Table \ref{tab:betrema}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|c|c|c|c}
\hspace*{11mm}Segment & \#(prime $p$) &
\#(prime $m$) & ratio & $\max p$ \\
\hline
\hspace*{9mm} $2,\ldots,\, 10^{10}$ & 455\,052\,511 &
15\,801\,827 & 3.473\% & 9\,999\,999\,491 \\
\hspace*{4mm} $10^{10},\ldots,\, 2\cdot 10^{10}$ & 427\,154\,205 &
13\,882\,936 & 3.250\% & 19\,999\,999\,757 \\
$2\cdot 10^{10},\ldots,\, 3\cdot 10^{10}$ & 417\,799\,210 &
13\,279\,095 & 3.178\% & 29\,999\,999\,921 \\
$3\cdot 10^{10},\ldots,\, 4\cdot 10^{10}$ & 411\,949\,507 &
12\,913\,713 & 3.135\% & 39\,999\,999\,719 \\
$4\cdot 10^{10},\ldots,\, 5\cdot 10^{10}$ & 407\,699\,145 &
12\,645\,233 & 3.102\% & 49\,999\,999\,619 \\
$5\cdot 10^{10},\ldots,\, 6\cdot 10^{10}$ & 404\,383\,577 &
12\,439\,618 & 3.076\% & 59\,999\,999\,429 \\
$6\cdot 10^{10},\ldots,\, 7\cdot 10^{10}$ & 401\,661\,384 &
12\,274\,191 & 3.056\% & 69\,999\,999\,287 \\
$7\cdot 10^{10},\ldots,\, 8\cdot 10^{10}$ & 399\,359\,707 &
12\,136\,112 & 3.039\% & 79\,999\,999\,679 \\
$8\cdot 10^{10},\ldots,\, 9\cdot 10^{10}$ & 397\,369\,745 &
12\,010\,780 & 3.023\% & 89\,999\,999\,981 \\
$9\cdot 10^{10},\ldots,\, 10^{11}$ & 395\,625\,822 &
11\,910\,803 & 3.011\% & 99\,999\,999\,977 \\
\hline
\hspace*{13mm}Total & 4\,118\,054\,813 &
129\,294\,308 & 3.140\% & 99\,999\,999\,977
\end{tabular}
\end{center}
\vspace{2mm}
\caption{The second column gives the number of primes in the corresponding
segment, while the third column gives the number of those primes $p$ which
create a projective prime $m=1+p+p^2$. The proportion of such primes among all the
primes of the second column is given in the fourth column.}
\label{tab:betrema}
\end{table}
We may see from this table that the number of primes $p\le 10^{11}$ which produce a
prime value of $m$ is 129\,294\,308, the largest of them being 99\,999\,999\,977.
The corresponding projective prime is $m=9\,999\,999\,995\,500\,000\,000\,507$.
Such primes $p$ represent approximately 3.140\% of the total number
4\,118\,054\,813 of primes up to $10^{11}$.
Of course, this percentage diminishes together with the growth of the upper limit.
For example, if we count the proportion of such primes up to $10^6$, we get
5.97\%. Nevertheless, it is quite reasonable to conjecture that even in this very
restricted situation there are infinitely many projective primes.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
$x$ & \#(prime $m \mid p\le x$) & estimate (\ref{n=3estimate}) & ratio
& estimate (\ref{eq:rectify}-\ref{eq:C-and-alpha}) & ratio \\
\hline
$1\cdot 10^{10}$ & 15\,801\,827 & $1.7683\times 10^7$ & 1.1190
& $1.5799306\times 10^7$ & 0.999841 \\
$2\cdot 10^{10}$ & 29\,684\,763 & $3.3328\times 10^7$ & 1.1227
& $2.9686686\times 10^7$ & 1.000065 \\
$3\cdot 10^{10}$ & 42\,963\,858 & $4.8096\times 10^7$ & 1.1195
& $4.2969637\times 10^7$ & 1.000135 \\
$4\cdot 10^{10}$ & 55\,877\,571 & $6.2736\times 10^7$ & 1.1227
& $5.5881270\times 10^7$ & 1.000066 \\
$5\cdot 10^{10}$ & 68\,522\,804 & $7.7239\times 10^7$ & 1.1272
& $6.8526763\times 10^7$ & 1.000058 \\
$6\cdot 10^{10}$ & 80\,962\,422 & $9.1332\times 10^7$ & 1.1281
& $8.0965961\times 10^7$ & 1.000044 \\
$7\cdot 10^{10}$ & 93\,236\,613 & $1.0524\times 10^8$ & 1.1287
& $9.3237376\times 10^7$ & 1.000008 \\
$8\cdot 10^{10}$ & 105\,372\,725 & $1.1900\times 10^8$ & 1.1293
& $1.0536780\times 10^8$ & 0.999953 \\
$9\cdot 10^{10}$ & 117\,383\,505 & $1.3262\times 10^8$ & 1.1298
& $1.1737691\times 10^8$ & 0.999944 \\
$10^{11}$ & 129\,294\,308 & $1.4614\times 10^8$ & 1.1303
& $1.2927974\times 10^8$ & 0.999887
\end{tabular}
\end{center}
\vspace{2mm}
\caption{The second column gives the cumulative totals from the second column in
Table~\ref{tab:betrema}, i.e.~the number of projective primes $m$ with $n=3$ arising
from primes $p\le x_i=i\cdot 10^{10}\;(i=1,\ldots, 10)$; the third column gives an
approximation for the estimate for this number from Section~\ref{sec:fixed-n},
while the fourth column gives the ratio of these two numbers. The meaning of the
last two columns is explained in Section~\ref{sec:empiric}.}
\label{tab:n=3ratios}
\end{table}
Table~\ref{tab:n=3ratios} compares the numbers of projective primes $m=1+p+p^2$ for
primes $p\le x_i=i\cdot 10^{10}$, $i=1,\ldots,10$, with the heuristic estimates given
by (\ref{n=3estimate}) in Section~\ref{sec:fixed-n}. It can be seen that the latter
are of the right order of magnitude, but that they consistently over-estimate the
number of such primes by about $12\%$.
In Section~\ref{sec:empiric} we present
another estimate. We do not have any theoretical bases to support it, only empirical
ones, but it approximates the values we need much better than the previous estimate.
The results of this estimate are represented in the two last columns of
Table~\ref{tab:n=3ratios}.
\subsection{An empirical estimate vs.~the theoretical one: rectifying the anomaly}
\label{sec:empiric}
We see that for large $x$ estimate (\ref{n=3estimate}) systematically overestimates the
number of projective primes. Hence, let us instead consider an estimate of the following
form:
\begin{eqnarray}\label{eq:rectify}
y = \frac{C x}{\ln(x)^{\alpha}}
\end{eqnarray}
where the constants $C$ and $\alpha$ are to be found from empirical data.
Denote $z=x/y=\frac{1}{C}\ln(x)^{\alpha}$. Then, taking the logarithm of each side of
this equation we get
$$
\ln(z)=-\ln(C)+\alpha\cdot\ln(\ln(x)).
$$
Thus, in the coordinates
$$
u=\ln(\ln(x)), \qquad v=\ln(z)
$$
equation (\ref{eq:rectify}) takes the form of an equation of a straight line
$$
v = a+\alpha u \quad\mbox{ where }\quad a=-\ln(C).
$$
Our next steps are as follows:
\begin{enumerate}
\item Take a number of pairs $(x_i,y_i)$ where $x_i$ are at our choice while $y_i$
are the numbers of projective primes $m=1+p+p^2$ created from primes $p\le x_i$.
\item Compute the corresponding pairs $(u_i,v_i)$.
\item Put the points $(u_i,v_i)$ on the plane of the coordinates $(u,v)$, in the hope
that they will be reasonably close to a straight line $v=a+\alpha u$.
\item Find the equation of this straight line and thus obtain the values
of $C=e^{-a}$ and of $\alpha$.
\end{enumerate}
The points should be taken with care. As we will see in Section \ref{sec:stochastic},
numbers of projective primes behave rather randomly. Therefore, in order to get a
reasonable estimate, the values of $x_i$ must be large enough, and the spaces between
them must also be large.
The numbers $x_i=i\cdot 10^{10}$, $i=1,\ldots,10$ satisfy both conditions. Taking $x_i$
and $y_i$ from the first two columns of Table~\ref{tab:n=3ratios} we get the results
which are shown in Figure~\ref{fig:empiric-alpha}. We would say that they are even better
than one might hope.
The corresponding constants, found by the method of least squares, are as follows:
\begin{eqnarray}\label{eq:C-and-alpha}
\hspace*{10mm}
a = -0.150383694, \qquad C = e^{-a} = 1.162280117, \qquad \alpha = 2.104419156.
\end{eqnarray}
The estimates given by (\ref{eq:rectify}) with the constants (\ref{eq:C-and-alpha}),
and their ratios to the true number of projective primes are shown in the last two
columns of Table~\ref{tab:n=3ratios}.
\begin{rema}[Overestimate]
If the estimates (\ref{eq:rectify}-\ref{eq:C-and-alpha}) are correct, and it seems
that they are, or at least if they are close to the correct asymptotic, then the
{\em overestimate}\/ of the ratio given by formula (\ref{n=3estimate}) tends to infinity,
though rather slowly: it is proportional to $\ln(x)^{\alpha-2}$.
\end{rema}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.3]{empiric-alpha-10.eps}
\caption{The horizontal axis corresponds to the variable $u=\ln(\ln(x))$, the vertical
one to the variable $v=\ln(x/y)$ where $y$ is meant to count projective primes. The ten
distinguished points correspond to $x_i=i\cdot 10^{10}$, while $y_i$ is the number of
prime $p\le x_i$ such that $m=1+p+p^2$ is prime.}
\label{fig:empiric-alpha}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.3]{comparison.eps}
\caption{Comparison of two estimates. The horizontal scale is logarithmic: an abscissa $k$
corresponds to the number $10^k$. White circles represent the ratios of estimates
(\ref{n=3estimate}) to the true numbers of projective primes $m=1+p+p^2$ obtained from
the primes $p\le 10^k$, $k=3,\ldots,11$. The solid circles represent the similar ratios for
estimates (\ref{eq:rectify}-\ref{eq:C-and-alpha}). Notice that for smaller numbers both
estimates {\em underestimate}\/ the number of projective primes (both black and white
points are below the level 1).}
\label{fig:comparison}
\end{center}
\end{figure}
It would be interesting to understand the nature of the above constants. For example,
$\alpha$ is reasonably close to $1+2\mu=2.1229$ (the constant $\mu$ is defined in
Section~\ref{sec:fixed-n}). It would be tempting to conjecture that they are equal, especially
since $\mu$ appears in several other conjectures related to prime numbers (see, for
example, \cite{Pol}, and also the chapter on the Mersenne primes in \cite{PP}). But no:
experience shows that the estimate of the exponent $\alpha$ diminishes when the
the bound $x$ grows. And let us not forget that for the time being we are unable to
prove even that there are infinitely many projective primes, to say nothing of their
asymptotic behavior.
\subsection{When the exponent $n$ grows}
We considered the prime exponents $n\le 100$ and counted the number of primes $p\le 10^6$
(omitting prime powers) such that $m=(p^n-1)/(p-1)$ is also prime. The results are
presented in Table~\ref{tab:growing-n}: $N$ denotes the number of primes $p$ with the
above property, and $\max p$ is the largest $p\le 10^6$ which, for a given $n$, produces
a prime value of $m$. The total number of primes up to one million is 78\,498. The proportion
of ``good'' primes thus varies (in our table) from 6\% (for $n=3$) to 0.5\% (for $n=83$).
There is a general tendency for this proportion to decrease, but without any apparent
regularity.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|c|c||c|c|c||c|c|c||c|c|c}
$n$ & $N$ & $\max p$ &
$n$ & $N$ & $\max p$ &
$n$ & $N$ & $\max p$ &
$n$ & $N$ & $\max p$ \\
\hline
3 & 4684 & 999\,773 &
19 & 2933 & 999\,067 &
43 & 1119 & 999\,961 &
71 & 848 & 999\,907 \\
5 & 4034 & 999\,653 &
23 & 1150 & 999\,287 &
47 & 1212 & 999\,491 &
73 & 577 & 999\,307 \\
7 & 4436 & 999\,961 &
29 & 1032 & 998\,111 &
53 & 694 & 999\,007 &
79 & 689 & 996\,811 \\
11 & 2243 & 999\,631 &
31 & 1980 & 997\,463 &
59 & 1106 & 999\,953 &
83 & 390 & 993\,557 \\
13 & 2658 & 999\,863 &
37 & 1285 & 999\,269 &
61 & 913 & 999\,763 &
89 & 430 & 995\,339 \\
17 & 2527 & 999\,287 &
41 & 862 & 999\,233 &
67 & 821 & 999\,727 &
97 & 571 & 998\,471
\end{tabular}
\end{center}
\vspace{2mm}
\caption{For a given prime exponent $n\le 100$, the number $N$ shows how many primes
$p\le 10^6$ there are such that $m=(p^n-1)/(p-1)$ is also prime. The column ``$\max p$''
shows the largest such $p$.}\label{tab:growing-n}
\end{table}
We do not present the corresponding projective primes $m$ since their decimal
representations are too long: for example, the number $m$ corresponding to the last
cell of the table, namely, \linebreak
$m=(998\,471^{97}-1)/998\,470$, has 576 digits.
The following conjecture seems quite reasonable:
\begin{conj}[Projective primes for a fixed $n$]
For any fixed prime $n\ge 3$ there are infinitely many prime values $m=1+p+p^2+\ldots+p^{n-1}$,
where $p$ ranges over all prime numbers.
\end{conj}
\subsection{Projective primes with a fixed $p$}\label{sec:fixed-p2}
What if we fix $p$ and allow $n$ to tend to infinity (taking only prime values),
as in Section~\ref{sec:fixed-p1}? Here the computational evidence, presented in Table~\ref{tab:fixed-p}, is less convincing, which is not surprising given the small number of primes $m$ involved.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|c|c|l}
$p$ & estimation (\ref{qfixed}) & true number & exponents $n$ \\
\hline
3 & 13.832 & 12 & $3,7,13,71,103,541,1091,1367,1627,4177,9011,9551$ \\
5 & 8.669 & 11 & $3,7,11,13,47,127,149,181,619,929,3407$ \\
7 & 6.796 & 5 & $5,13,131,149,1699$ \\
11 & 5.136 & 9 & $17,19,73,139,907,1907,2029,4801,5153$ \\
13 & 4.675 & 9 & $5,7,137,283,883,991,1021,1193,3671$ \\
17 & 4.052 & 7 & $5,7,11,47,71,419,4799$ \\
\hline
\end{tabular}
\vspace{2mm}
\caption{The last column gives the list of exponents $n\le 10^4$ such that the number
$m=(p^n-1)/(p-1)$ is prime. The number of such exponents is given in column~3,
while the estimation of this number by formula (\ref{qfixed}) is presented in column~2.}
\label{tab:fixed-p}
\end{center}
\end{table}
For the primes $p=3,5,7,11, 13$ and $17$, the second column of Table~\ref{tab:fixed-p}
gives the estimates based on the first expression in (\ref{qfixed}) for the number of
primes $m=(p^n-1)/(p-1)$ with $n\le x=10^4$. The third column gives the true figures,
found by a computer search, and the relevant exponents $n$ are listed in the fourth column.
We see that the estimation (\ref{qfixed}) of the number of exponents has a reasonably
good correspondence with their actual number. However, we have a feeling that the above
data are not entirely convincing. For example, there are only three exponents in the
table, out of 53, which are greater than 5000. Also, for $p=3$, the next ``good'' exponent,
after the one given in the table, is rather far away: $n=36\,913$. Therefore, in this
case we prefer to formulate not a conjecture but a question:
\begin{ques}[Generalized Mersenne]
Let $p$ be a prime. Do there exist infinitely many values of~$n$ such that the number
$m=(p^n-1)/(p-1)$ is prime?
\end{ques}
\subsection{Prime powers once again}\label{sec:primepowersagain}
We now consider fixed prime powers $q=p^e$ with $e\ge 2$, as $n\to\infty$. By
Remark~\ref{re:e=2}, apart from the example $q=2^2$ and $m=5$ we can ignore the
case $e=2$. An extensive search of prime powers producing projective primes has
given the following results:
\medskip
\noindent
$p=2$: the search for $q=2^e$, $e\ge 2$, producing projective primes,
up to $q\le 10^{60}$, gives eight solutions:
\begin{itemize}
\item There are four solutions for which $m=1+q$\, is a Fermat prime:
$$
(q,n)=(2^2,2),\,(2^4,2),\,(2^8,2),\,(2^{16},2)
$$
(the other known Fermat prime $m=1+2^1=3$ does not correspond to
$e\ge 2$).
\item There are three relatively small solutions: $(q,n)=(2^3,3),(2^7,7),(2^9,3)$.
\item A rather unexpected solution is $(q,n)=(2^{59},59)$. The corresponding
projective prime
$$
m = 1 + 2^{59} + 2^{118} + \cdots + 2^{59\cdot 58}
$$
has 1031 digits.
It is generally believed that there are only five Fermat primes.
However, this example prevents us from conjecturing that there are
only finitely many powers of~2 which yield projective primes.
\end{itemize}
\medskip
\noindent
$p=3$: the search for $q=3^e$, $e\ge 3$, producing projective primes,
up to $q\le 10^{60}$, gives only one solution: $(q,n)=(3^3,3)$, $m=1+27+27^2=757$.
We are inclined to believe that for $p=3$ this solution is unique.
\medskip
\noindent
$q\le 10^{15}$: the total search for all prime powers $q\le 10^{15}$
with $e\ge 3$ producing projective primes gives 337 solutions.
Only eight of them have the exponent $e>3$, namely,
$$
(q,n)=(5^7,7),\,(11^9,3),\,(43^5,5),\,(67^7,7),\,(167^5,5),\,(313^5,5),\,(509^5,5),\,
(859^5,5).
$$
For all the other 329 solutions $q$ is the cube of a prime.
\medskip
\noindent
$q=p^3\le 10^{18}$: the total search for cubes of primes up to
$10^{18}$ reveals 2121 solutions, the largest one being $p=999\,953$,
$q = p^3 = 999\,859\,006\,626\,896\,177$, and
$$
m = 1+q+q^2 = 999\,718\,033\,132\,923\,614\,193\,697\,947\,364\,111\,507.
$$
\medskip
The following conjecture seems to be very plausible:
\begin{conj}[Cubes of primes]
There are infinitely many values of $q=p^3$, with $p$ being prime, such that
$m=1+q+q^2$ is prime.
\end{conj}
Since we did not find any examples where the same prime power $q$ yields more than one
projective prime, we ask:
\begin{ques}[Generalized Mersenne]
Does there exist a proper prime power $q$ such that the number
$m=(q^n-1)/(q-1)$ is prime for more than one value of $n$?
\end{ques}
Of course, Table~\ref{tab:fixed-p} in Section~\ref{sec:fixed-p2} gives a number of examples of this phenomenon where $q=p$ is prime.
\section{Stochastic behavior of projective primes}
\label{sec:stochastic}
In this section, we present a number of observations concerning the stochastic
behavior of the (numbers of) projective primes. Our feeling is that this subject,
while being excitingly interesting, is not yet ready for a profound statistical
analysis. However, we would like to share our observations with the community of
specialists in probabilistic number theory in the hope that they may clarify
certain points of our study.
We deal here exclusively with projective primes of the form $m=1+p+p^2$, where $p$ is prime.
\subsection{Local estimates}
\label{sec:segments}
Let $a,b$ be two integers, $a<b$. Then, according to (\ref{eq:rectify}-\ref{eq:C-and-alpha}),
the primes $p\in[a,b]$ should create, approximately,
\begin{equation}\label{eq:est-by-segm}
C\cdot\left(\frac{b}{\ln(b)^{\alpha}}-\frac{a}{\ln(a)^{\alpha}}\right)
\end{equation}
projective primes of the type $m=1+p+p^2$, where $C = 1.162280117$ and
$\alpha = 2.104419156$. In Figure~\ref{fig:est-by-segm}, left, we
consider the primes $p\le 10^9$. We subdivide this range into $10^4$ segments
$S_i=[(i-1)\cdot 10^5, i\cdot 10^5]\;(i=1, 2, \ldots, 10^4)$ of equal size $10^5$.
Horizontally, we mark the order number $i$ of a segment (from 1 to $10^4$).
For each of these segments, we divide estimate (\ref{eq:est-by-segm}) by the true
number of projective primes $m=1+p+p^2$ for $p\in S_i$; this ratio is the ordinate
of the corresponding point in the picture. The picture thus contains $10^4$ points.
On the right of Figure \ref{fig:est-by-segm}, the construction is similar, but now the
range considered is $p\le 10^{10}$, and the length of each of the $10^4$ segments
is $10^6$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.35]{rand_new_al_4x5.eps}
\hspace{1cm}
\includegraphics[scale=0.35]{rand_new_al_4x6.eps}
\caption{Both pictures contain $10^4$ points. Each point corresponds to a segment
in $\mathbb{N}$ of length $10^5$ (left) or $10^6$ (right). The abscissa of a point
is the order number of the corresponding segment. The ordinate is the ratio of the
estimate (\ref{eq:est-by-segm}) to the true number of projective primes
$m=1+p+p^2$ generated by primes $p$ in this segment.}
\label{fig:est-by-segm}
\end{center}
\end{figure}
We may make the following observations.\
\begin{itemize}
\item There are wild fluctuations in the ratio. Obviously, they are due not to
estimate (\ref{eq:est-by-segm}) itself but to the fluctuations in the numbers
of projective primes $m=1+p+p^2$ with $p$ belonging to the corresponding
segments.
\item Comparing the vertical scales shows that the right-hand band of points is
narrower than the left-hand one. This is
natural since considering larger segments leads to smoothing the fluctuations.
\item The interesting fact is, however, that the variations in both pictures do not
diminish when we let $i$ increase. We may even say that they increase.
\end{itemize}
\subsection{Histogram}\label{sec:histo}
Let us take the right-hand picture of Figure~\ref{fig:est-by-segm}. The minimum value
of the ordinate (i.\,e., of the ratio) in this picture is $r_{\rm min}=0.9217$, the
maximum is $r_{\rm max}=1.1244$. We subdivide the segment $[r_{\rm min},r_{\rm max}]$
into 100 parts and count the number of points whose ordinates belong to each part.
The resulting histogram is shown in Figure~\ref{fig:density}, left.
The mean of this distribution is $46.66$, the standard deviation is $13.66$.
On the right of the same figure we show the density of the normal distribution with
the same parameters. Note that the height of the left picture, which is 300 points out
of 10\,000, corresponds well to the height of the density, which is approximately 0.03.
The resemblance of the two graphs is visible. We leave it to the specialists to use,
if necessary, more sophisticated statistical tools.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.3]{density_experimental.eps}
\hspace{1cm}
\includegraphics[scale=0.3]{density_normal.eps}
\caption{On the left: the histogram of the distribution of heights of the points
in the right-hand picture of Figure \ref{fig:est-by-segm}.
On the right: the density of the normal distribution with the same mean $46.66$
and standard deviation $13.66$.}
\label{fig:density}
\end{center}
\end{figure}
\subsection{Conclusion}
Our main aim in this note has been to give heuristic and computational evidence that
there are infinitely many projective primes, especially in the simplest and apparently
most abundant case, where $n=3$ and $q$ is prime.
We will not pursue these speculations further and will leave the question of more exact
estimates of the number and distribution of projective primes to the community of experts
in probabilistic number theory.
(Our own backgrounds and motivation for this investigation lie in the areas of dessins
d'enfants and permutation groups.)
\bigskip
\paragraph{\bf Acknowledgements} We are greatly indebted to Yuri Bilu who acquainted us
with the Bunyakov\-sky conjecture, which became a crucial point of our study, and to
Peter Cameron, Robert Guralnick and Cheryl Praeger for some very useful comments.
Jean B\'etr\'ema helped us with some computations which were too heavy for our Maple
package on a laptop computer.
Alexander Zvonkin was partially supported by the ANR project {\sc Combin\'e}
(ANR-19-CE48-0011).
| {
"timestamp": "2020-12-08T02:05:23",
"yymm": "2010",
"arxiv_id": "2010.08023",
"language": "en",
"url": "https://arxiv.org/abs/2010.08023",
"abstract": "As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\\rm L}_n(q)$ is prime. We present heuristic arguments and computational evidence to support a conjecture that for each prime $n\\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$.",
"subjects": "Number Theory (math.NT); Group Theory (math.GR)",
"title": "Primes in geometric series and finite permutation groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846691281406,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7092019595029817
} |
https://arxiv.org/abs/1908.08250 | Coloring Hasse diagrams and disjointness graphs of curves | Given a family of curves $\mathcal{C}$ in the plane, its disjointness graph is the graph whose vertices correspond to the elements of $\mathcal{C}$, and two vertices are joined by an edge if and only if the corresponding sets are disjoint. We prove that for every positive integer $r$ and $n$, there exists a family of $n$ curves whose disjointness graph has girth $r$ and chromatic number $\Omega(\frac{1}{r}\log n)$. In the process we slightly improve Bollobás's old result on Hasse diagrams and show that our improved bound is best possible for uniquely generated partial orders. | \section{Introduction}
There are two important, seemingly unrelated, concepts that play important roles in Geometric Graph Theory and in Graph Drawing: {\em Hasse diagrams} and {\em string graphs}.
Hasse diagrams were introduced by Vogt~\cite{V95} at the end of the 19th century for concise representation of partial orders. Today they are widely used in graph drawing algorithms. Let $P$ be a partially ordered set with partial ordering $\prec$. For any $x,y\in P$, we say that $y$ \emph{covers} $x$ if $x\prec y$ and there is no $z\in P$ such that $x\prec z\prec y$. The \emph{Hasse diagram} of $P$ is the directed graph on the elements of $P$, where there is an edge from $x$ to $y$ if and only if $y$ covers $x$. If we disregard the direction of the edges, we obtain the {\em cover graph} of $P$. The graph on $P$ whose two elements are connected by an edge if and only if they are related by $\prec$ is the {\em comparability graph} of $P$. The cover graph is a subgraph of the comparability graph.
The \emph{intersection graph} of a family of sets $\mathcal{C}$ is the graph whose vertices correspond to the elements of $\mathcal{C}$ and two vertices are joined by an edge if and only if the corresponding sets have a nonempty intersection. The \emph{disjointness graph} of $\mathcal{C}$ is the complement of the intersection graph of $\mathcal{C}$. A {\em string}, or \emph{curve}, $\gamma$ is the image of a continuous function $f:[0,1]\rightarrow \mathbb{R}^{2}$. A curve $\gamma$ is \emph{grounded} if one of its endpoints is on the $y$-axis, and $\gamma$ lies in the nonnegative half-plane. See Figure \ref{figure1} for an illustration of a grounded family of curves and its disjointness graph. A {\em string graph} is the intersection graph of curves. The notion was introduced by Benzer~\cite{B59} and Sinden~\cite{Si66} to describe the incidence structures of intervals in chromosomes and metallic layers in printed networks, respectively. The systematic study of string graphs was initiated in \cite{EET76} and \cite{Gr78}.
\smallskip
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node (v1) at (0,6) {};
\node (v2) at (0,-1.5) {};
\draw (v1) edge (v2);
\node at (-0.5,1) {A};
\node at (-0.5,2.5) {B};
\node at (-0.5,3.5) {C};
\node at (-0.5,4.5) {D};
\draw plot[smooth, tension=.7] coordinates {(0,1) (1,1.5) (2,0.5)};
\draw plot[smooth, tension=.7] coordinates {(0,2.5) (1,2) (1.5,2)};
\draw plot[smooth, tension=.7] coordinates {(0,3.5) (1.5,3) (3,1) (2,-0.5) (1,0.5) (2,1.5)};
\draw plot[smooth, tension=.7] coordinates {(0,4.5) (4,2) (4,-1) (1,-0.5) (0.5,1) (1,2.5)};
\node[vertex] (v3) at (6.5,2) {};
\node[vertex] (v4) at (8,2) {};
\node[vertex] (v5) at (9.5,2) {};
\node[vertex] (v6) at (11,2) {};
\draw (v3) edge[bend left] (v4);
\draw (v4) edge[bend left] (v5);
\draw (v5) edge[bend left] (v6);
\node at (6.5,1.5) {A};
\node at (8,1.5) {B};
\node at (9.5,1.5) {C};
\node at (11,1.5) {D};
\end{tikzpicture}
\caption{A family of grounded curves and its disjointness graph. }
\label{figure1}
\end{center}
\end{figure}
The first sign that the above concepts are intimately related was the following simple fact discovered by Golumbic, Rotem, Urrutia~\cite{GoRU83}, and Lov\'asz~\cite{Lo83}: Every comparability graph is the disjointness graph of a collection of curves in the plane. A partial converse of this statement was established in~\cite{FoP12}.
A useful characterization of cover graphs in terms of strings follows directly from Corollary 2.7 of Middendorf and Pfeiffer~\cite{MiP93} and Theorem 1 in~\cite{Si66}. See also~\cite{Kr91} and~\cite{RoW18} (page 2).
\begin{theorem}\label{character} \cite{MiP93}, \cite{Si66} A triangle-free graph is a cover graph of a partially ordered set if and only if it is isomorphic to the disjointness graph of a family of grounded curves.
\end{theorem}
The {\em girth} of a graph $G$ is the length of the shortest cycle in $G$. Obviously, every triangle-free graph has girth at least {\em four}. According to a classical result of Erd\H os~\cite{Er59}, for every $r\ge 3$, there exist graphs with $n$ vertices and girth at least $r$ which have arbitrarily large chromatic numbers. Erd\H os's construction is probabilistic and does not posses any geometric structure.
For geometrically defined graphs, the situation is more complicated. The chromatic number of intersection graphs of axis-parallel rectangles~\cite{AsG60} or chords of a cycle~\cite{Gy85,KoK97,DaM19} and disjointness graphs of segments in the plane~\cite{PaT94,To00} can be bounded from above by a function of their clique numbers. In sharp contrast to this, Pawlik, Kozik, Krawczyk, Laso\'n, Micek, Trotter, and Walczak~\cite{PaKK14} proved that there exist triangle-free intersection graphs of $n$ segments with chromatic number $\Omega(\log \log n)$, which disproved a longstanding conjecture of Erd\H{o}s. In~\cite{PaTT17}, triangle-free disjointness graphs of $n$ curves were constructed, with chromatic number $\Omega(\log n)$, cf.~\cite{MuWW18}. This construction is based on shift graphs, defined by Erd\H os and Hajnal~\cite{ErH64}. It appears to be difficult to extend this method to obtain disjointness graphs of curves with high girth and high chromatic number.
The aim of the present note is to construct such graphs.
\begin{theorem}\label{thm:main1}
For every positive integer $r$ and for every sufficiently large $n$, there exists a family of $n$ curves whose disjointness graph has girth at least $r$ and chromatic number at least $\Omega(\frac{1}{r}\log n)$.
\end{theorem}
This result does not remain true if we are allowed to use only {\em $x$-monotone} curves, that is, if every vertical line meets each curve in at most one point. In this case, the chromatic number of the cover graph is bounded from above by a constant~\cite{PaT94,PaT19}.
In view of Theorem \ref{character}, in order to prove Theorem~\ref{thm:main1}, it is sufficient to establish the following.
\begin{theorem}\label{thm:chi_cover}
For every positive integer $r$ and for every sufficiently large $n$, there exists a poset on $n$ vertices whose cover graph has girth at least $r$ and chromatic number $\Omega(\frac{1}{r}\log n)$.
\end{theorem}
The study of combinatorial properties of cover graphs (Hasse diagrams) is an extensive area of research in the theory of partial orders. Bollob\'as~\cite{Bo77} was the first to show the existence of partial orders (actually, lattices) whose cover graphs have arbitrarily large girth and chromatic number. Alternative constructions were found by Ne\v set\v ril {\em et al.}~\cite{NeR79,KrN91}. Bollob\'as's proof, which gives the best known asymptotic bound, builds on Erd\H{o}s's probabilistic construction~\cite{Er59} mentioned above. It shows that for a fixed girth $r$ and $n\rightarrow\infty$, the chromatic number of a cover graph with $n$ vertices can be as large as $\Omega(\frac{\log n}{\log\log n})$. Our Theorem~\ref{thm:chi_cover} improves on this bound.
It is possible that Theorem~\ref{thm:chi_cover} can be further improved. However, we can show that our bound is tight for an interesting family of cover graphs. A partially ordered set $P$ is called \emph{uniquely generated} if for every comparable pair of vertices $x\prec y$, there exists a unique sequence of vertices $x=v_{1}\prec\dots\prec v_{k}=y$ such that $v_{i+1}$ covers $v_{i}$ for $i=1,\dots,k-1$. Obviously, if there is no chain with $3$ elements in $P$, then $P$ is uniquely generated and its cover graph is bipartite.
\begin{theorem}\label{thm:chi_ug}
(i) If $P$ is a uniquely generated poset on $n$ vertices, then the chromatic number of its cover graph is at most $\lfloor \log_{2} n\rfloor+1$.
(ii) For every integer $r>3$ and for every sufficiently large $n$, there exists a uniquely generated poset on $n$ vertices whose cover graph has girth at least $r$ and chromatic number at least $\Omega(\frac{1}{r}\log n)$.
\end{theorem}
\section{Cover graphs with large chromatic number}
In this section, we prove Theorem \ref{thm:chi_ug}. Note that then Theorem \ref{thm:chi_cover} is an immediate consequence of part (ii) of Theorem \ref{thm:chi_ug}. We omit floors an ceilings for easier readability.
\bigskip
\textit{Proof of Theorem \ref{thm:chi_ug}, part (i).} Let $G$ be the cover graph of $P$, let $<_{P}$ be the partial ordering on $P$, and let $\prec$ be a linear extension of $<_{P}$. For any $x\in P$, let $C(x)$ denote the set of vertices of $P$ covered by $x$.
We prove that the greedy coloring of $G$ with respect to $\prec$ uses at most $1+\lfloor \log_{2} n\rfloor$ colors. Let $v_{1}\prec\dots\prec v_{n}$ be the vertices of $G$. Color them with the elements of $\mathbb{Z}^{+}$, as follows. For $i=1,\dots,n$, if $v_{1},\dots,v_{i-1}$ have already been colored, then color $v_{i}$ with the smallest positive integer $k$ that does not appear among the colors of $C(v_{i})$.
For each vertex $v\in V(G)$, let $T(v)$ denote the set of vertices $u\in V(G)$ such that $u\leq_{P} v$. Note that, as $P$ is uniquely generated, the subgraph of $G$ induced by $T(v)$ is a tree. We claim that if $v$ received color $k$, then $|T(v)|\geq 2^{k-1}$. This clearly implies (i), because if the total number of colors used by our coloring is $K$, then we have $n\geq 2^{K-1}$.
We prove the claim by induction on $k$. For $k=1$, the statement is trivial. Suppose that $k\geq 2$ and that the claim is true for all positive integers smaller than $k$. As $v$ received color $k$, we can find $k-1$ vertices $u_{1},\dots,u_{k-1}\in C(v)$ such that the color of $u_{i}$ is $i$, for $i=1,\dots,k-1$. By the induction hypothesis, we have $|T(u_{i})|\geq 2^{i-1}$. Since the trees $T(u_{1}),\dots,T(u_{k-1})\subset T(v)$ are pairwise disjoint, we obtain $|T(v)|\geq 1+\sum_{i=1}^{k-1}2^{i-1}=2^{k-1}$, as required.\hfill$\Box$
\bigskip
For the proof of part (ii) of Theorem \ref{thm:chi_ug}, we need the following technical lemma.
\begin{lemma}\label{lemma:bipartite}
Let $A$ and $B$ be two $m$-element sets and let $G$ be the random graph on $A\cup B$ in which every $a\in A$ and $b\in B$ are joined by an edge independently with probability $p=\frac{d}{m}$.
Then the probability that there exist $X\subset A$ and $Y\subset B$ such that $|X||Y|\geq 3m^{2}/d$ and there is no edge between $X$ and $Y$ is at most $2^{-m}$.
\end{lemma}
\begin{proof}
Let $N= \frac{3m^{2}}{d}$. For any $X\subset A$ and $Y\subset B$, let $I(X,Y)$ denote the event that there exists no edge between $X$ and $Y$. Obviously, we have $\mathbb{P}(I(X,Y))=(1-p)^{|X||Y|}\leq e^{-p|X||Y|}$. This yields
$$\mathbb{P}\left(\bigcup_{\substack{X\subset A, Y\subset B\\ |X||Y|\geq N}}I(X,Y)\right)\leq \sum_{\substack{X\subset A, Y\subset B\\ |X||Y|\geq N}}e^{-p|X||Y|}\leq 2^{2m}e^{-pN}<2^{-m}.$$
\hfill$\Box$
\end{proof}
\textit{Proof of Theorem \ref{thm:chi_ug}, part (ii).} Assume that $n\geq 2^{10r}$, and let $N=3n$, $k=\frac{\log_{2} N}{10r}$, and $m=\frac{N}{k}$. If $G$ is a graph whose vertex set is a subset of the integers, a \emph{monotone path} in $G$ is a path with vertices $c_{0}<c_{1}<\dots<c_{t}$ and edges $c_{i}c_{i+1}$ for $i=1,\dots,t-1$. A pair of vertices $\{a,b\}$ of $G$ is called \emph{bad}, if there exist two edge-disjoint monotone paths whose endpoints are $a$ and $b$.
Our goal is to construct a graph $G$ on the vertex set $\{1,...,N\}$ satisfying the following three conditions:
\begin{enumerate}
\item $G$ has no independent set of size larger than $7m$,
\item $G$ has at most $\frac{N}{3}$ bad pairs of vertices,
\item the number of cycles in $G$ of length smaller than $r$ is at most $\frac{N}{3}$.
\end{enumerate}
Suppose that such a graph $G$ exists. Let $G'$ denote the graph obtained from $G$ by deleting $\frac{2N}{3}$ vertices: at least one vertex from every bad pair and at least one vertex from every cycle of length smaller than $r$. Then $G'$ has $n$ vertices and girth at least $r$. Condition 1 implies that the chromatic number of $G'$ is at least $\frac{n}{7m}>\frac{1}{10^{3}r}\log_{2}n$. Define a partially ordered set $P$ with partial ordering $<_{P}$ on $V(G')$ in such a way that $a<_{P} b$ if and only if $a<b$ and there exists a monotone path in $G'$ with endpoints $a$ and $b$. Then $P$ meets all the requirements of part (ii) of the theorem. Indeed, as $G'$ has no bad pair of vertices, the cover graph of $P$ is equal to $G'$, and $P$ is uniquely generated.
\smallskip
We construct a graph $G$ with the above three properties, as follows. Divide $\{1,\dots,N\}$ into $k$ intervals of size $m$, denoted by $A_{1},...,A_{k}$. For every $1\leq i<j\leq k$ and for any $x\in A_{i}$, $y\in A_{j}$, join $x$ and $y$ by an edge independently with probability $p_{ij}=\frac{2^{j-i}}{m}$. Denote the resulting graph by $G$.
First, we show that, with probability larger than $\frac{2}{3}$, condition 1 is satisfied: $G$ does not contain an independent set of size larger than $7m$. Let $\mathcal{A}$ denote the event that for every pair $(i,j)$ with $1\leq i<j\leq k$, and for every pair of subsets $X\subset A_{i}$ and $Y\subset A_{j}$ with no edge running between $X$ and $Y$, we have $|X||Y|<3m^{2}2^{i-j}$. By Lemma \ref{lemma:bipartite}, for a fixed pair $(i,j)$ with $1\leq i<j\leq k$, with probability at least $1-2^{-m}$ there exists no $X\subset A_{i}$ and $Y\subset A_{j}$ such that $|X||Y|\geq 3m^{2}2^{i-j}$ and there is no edge between $X$ and $Y$. As there are fewer than $k^{2}$ different pairs $(i,j)$ with $1\leq i<j\leq k$, we have $\mathbb{P}(\mathcal{A})\geq 1-k^{2}2^{-m}>\frac{2}{3}$.
We show that if $\mathcal{A}$ happens, then $G$ has no independent set of size larger than $7m$. Suppose for contradiction that $I\subset V(G)$ is an independent set with $|I|>7m$. For $i=1,...,k$, let $I_{i}=I\cap A_{i}$. Clearly, there exists an index $1\leq h\leq k$ such that $\sum_{i=1}^{h}|I_{i}|\geq 3m$ and $\sum_{i=h+1}^{k}|I_{i}|\geq 3m$. Then we have
\begin{equation}\label{equ:ind}
9m^{2}\leq \left(\sum_{i=1}^{h}|I_{i}|\right)\left(\sum_{i=h+1}^{k}|I_{i}|\right)=\sum_{i=1}^{h}\sum_{j=h+1}^{k}|I_{i}||I_{j}|\leq \sum_{i=1}^{h}\sum_{j=h+1}^{k}3m^{2}2^{i-j},
\end{equation}
where the last inequality holds if $\mathcal{A}$ occurs. However,
$$\sum_{i=1}^{h}\sum_{j=h+1}^{k}2^{i-j}\leq \sum_{l=1}^{k}l2^{-l}<2,$$
which contradicts the left-hand side of (\ref{equ:ind}).
Next, we prove that the probability that $G$ satisfies condition 2 is larger than $\frac{2}{3}$. Let $X$ stand for the number of bad pairs of vertices in $G$, and let $\mathcal{B}$ denote the event that $X \leq \frac{N}{3}$. Let $x\in A_{i}$ and $y\in A_{j}$, where $1\leq i<j\leq k$. Let $x=v_{0},v_{2},\dots,v_{l}=y$ such that $v_{t}\in A_{i_{t}}$ for $t=0,\dots,l$, where $i=i_{0}<\dots<i_{l}=j$. The probability that $v_{0},\dots,v_{l}$ is a monotone path in $G$ is
$$\prod_{t=0}^{l-1}\frac{2^{i_{l+1}-i_{l}}}{m}=\frac{2^{j-i}}{m^l}<\frac{2^{k}}{m^{l}}.$$
There are $\binom{j-i-1}{l-1}m^{l-1}<2^{k}m^{l-1}$ ways to choose the vertices of a monotone path of length $l$ with endpoints $x$ and $y$. Hence, the probability that there exist two edge-disjoint monotone paths with endpoints $x$ and $y$, where one of these paths has length $l$ and the other has length $l'$, is at most
$$(2^{k}m^{l-1})(2^{k}m^{l'-1})\frac{2^{k}}{m^{l}}\frac{2^{k}}{m^{l'}}=\frac{2^{4k}}{m^{2}}.$$
There are fewer than $k^{2}$ ways to choose $(l,l')$, so the probability that $\{x,y\}$ is a bad pair of vertices is less than $\frac{k^{2}2^{4k}}{m^{2}}<\frac{1}{9n}$. Therefore, we have $\mathbb{E}(X)< N^{2}\frac{1}{9N}=\frac{N}{9}$. Applying Markov's inequality, we obtain that $1-\mathbb{P}(\mathcal{B})=\mathbb{P}(X > \frac{N}{3}) < \frac{1}{3}$.
Finally, we show that $G$ satisfies condition 3, with probability larger than $\frac{2}{3}$. Let $Y$ be the number of cycles of length at most $r-1$ in $G$, and let $\mathcal{C}$ denote the event that $Y\leq \frac{N}{3}$. Let $p=n^{-(r-1)/r}$. Note that each pair of vertices in $G$ is joined by an edge with probability at most $\frac{2^{k}}{m}<p$. Then we have
$$\mathbb{E}(Y)<\sum_{l=3}^{r-1}N^{l}p^{l}<rN^{\frac{r-1}{r}}<\frac{N}{9}.$$
Indeed, there are $\frac{(l-1)!}{2}\binom{N}{l}<N^{l}$ possible copies of the cycle of length $l$, and the probability that a fixed copy of such a cycle appears in $G$ is at most $p^{l}$. Applying Markov's inequality, we get $1-\mathbb{P}(\mathcal{C})=\mathbb{P}(Y > \frac{N}{3})<\frac{1}{3}$.
In conclusion, we proved that $\mathbb{P}(\mathcal{A}),\mathbb{P}(\mathcal{B}),\mathbb{P}(\mathcal{C})>\frac{2}{3}$. Thus, the probability that the event $\mathcal{A}\wedge\mathcal{B}\wedge\mathcal{C}$ occurs is nonzero. This means that there exists a graph $G$ satisfying conditions 1,2, and 3, which completes the proof of the theorem.\hfill$\Box$
\section{Acknowledgements}
We are grateful to Bartosz Walczak for valuable discussions. He gave a direct construction proving Theorem~\ref{character} and pointed out where the components of the statement appeared in the literature.
| {
"timestamp": "2019-08-23T02:08:12",
"yymm": "1908",
"arxiv_id": "1908.08250",
"language": "en",
"url": "https://arxiv.org/abs/1908.08250",
"abstract": "Given a family of curves $\\mathcal{C}$ in the plane, its disjointness graph is the graph whose vertices correspond to the elements of $\\mathcal{C}$, and two vertices are joined by an edge if and only if the corresponding sets are disjoint. We prove that for every positive integer $r$ and $n$, there exists a family of $n$ curves whose disjointness graph has girth $r$ and chromatic number $\\Omega(\\frac{1}{r}\\log n)$. In the process we slightly improve Bollobás's old result on Hasse diagrams and show that our improved bound is best possible for uniquely generated partial orders.",
"subjects": "Combinatorics (math.CO)",
"title": "Coloring Hasse diagrams and disjointness graphs of curves",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846684978779,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7092019590461229
} |
https://arxiv.org/abs/2002.12840 | Switching Identities by Probabilistic Means | Switching identities have a long history in potential theory and stochastic analysis. In recent work of Cox and Wang, a switching identity was used to connect an optimal stopping problem and the Skorokhod embedding problem (SEP). Typically switching identies of this form are derived using deep analytic connections. In this paper, we prove the switching identities using a simple probabilistic argument, which furthermore highlights a previously unexplored symmetry between the Root and Rost solutions to the SEP. |
\section{Introduction}
\section{Introduction}
Let $D$ be a rectangle of horizontal length $T$ and let $(0,x),(0,y)$ be points on the left boundary of $D$.
Let $B$ be Brownian motion (started in $x$ or $y$) and write $\sigma$ for the first time at which $(t,B_t)$ leaves the rectangle.
As a particular case of
Hunt's switching identities, we know that
\begin{align} \label{switch}
\mathbb{E}^{x} \left[|B_\sigma - y|\right] = \mathbb{E}^y \left[ |B_\sigma - x| \right].
\end{align}
In this paper, we will be interested in generalisations of this identity, which are of particular relevance in the construction of solutions to the Skorokhod embedding problem.
The Skorokhod embedding problem can be stated as follows: Given measures $\lambda, \mu$ on $\mathbb{R}$, and a Brownian motion $B$ with $B_0 \sim \lambda$, find a stopping time $\tau$ such that
\begin{equation}\label{SEP} B_\tau \sim \mu \text{ and } (B_{t \wedge \tau})_{t \ge 0} \text{ is uniformly integrable.} \tag{$\mathsf{SEP}$} \end{equation}
In this paper we will be interested in a sub-class of solutions to \eqref{SEP} which are given by the first hitting time of a right-barrier (resp.\ left-barrier) $\mathcal R$ which is a closed subset of $\mathbb{R}_+\times \mathbb{R}$ with the property that $(t,x) \in \mathcal{R} \implies (s,x) \in \mathcal{R}$, when $s > t$ (resp. $s<t$).
The most prominent example of such a solution is the Root solution \cite{Ro69}. It establishes the existence of a right-barrier $\mathcal R^{Root}$ such that
\[
\tau^{Root}:=\inf\{t\geq 0 : (t,B_t)\in\mathcal R^{Root}\}
\]
solves \eqref{SEP}.
There is an analogous result for left-barriers by Rost \cite{Ro76}. While Root's original contribution was non-constructive, it was shown in \cite{CoWa13, CoWa12} how to build the Root barrier.
This was done by considering \emph{optimal stopping problems} and establishing the crucial identity
\begin{equation} \label{eq:main_intro}
- \mathbb{E}^\lambda \left[ |B_{\tau^{Root}\wedge T} - y| \right] = \sup_{\sigma \le T} \mathbb{E}^y\left[U_{\mu}(B_\sigma) \mathbbm 1_{\sigma < T} + U_{\lambda}(B_\sigma) \mathbbm 1_{\sigma = T}\right]
\end{equation}
where
the supremum is taken over all stopping times $\sigma \le T$.
Here $U_m$ is the potential function associated with the measure $m$, i.e.\ \[U_m(x) := -\int |y-x| \, m(\mathrm{d} y).\]
From formula \eqref{eq:main_intro} , the barrier region $\mathcal{R}^{Root}$ can be identified with the stopping region of the optimal stopping problem \emph{after a reversal of the time component}. Together with the switching of the starting position between $y$ and $\lambda$, this formula can be seen as a generalisation of \eqref{switch}. Furthermore, the articles \cite{CoWa13, CoWa12} also provide a construction for the Rost barrier similarly based on optimal stopping problems.
The equation \eqref{eq:main_intro} is well understood in potential theoretic terms, and can be seen as a deep consequence of Markov duality, applied to the time-space process $(t,B_t)$. However this connection does not offer much insight into why such equations might arise in these applications. In \cite{CoWa13, CoWa12} this connection was made via viscosity theory and it was noted that a probabilistic explanation has yet to be given.
\medskip\\
{\it The aim of this short paper is to show that \eqref{eq:main_intro} is a natural consequence of some simple probabilistic arguments.} As a bonus, our proof reveals a previously unexplored symmetry between the Root and Rost solutions \cite{Ro76} to \eqref{SEP}.
\\
As a brief taste of the style of argument we will use, and to highlight its fundamental nature, let us first consider equation \eqref{switch}. Introduce a second Brownian motion $W$, independent of $B$, running from right to left and started on the right hand side of the rectangle $D$ at the point $(T,y)$.
For any $s \in [0,T]$ we consider the stopping times
\begin{align*}
\sigma_s &:= \sigma \wedge (T-s)
\\ \tau_s &:= \inf\{t\geq 0 : (T-t, W_t) \not\in D\} \wedge s
\end{align*}
and define
\[
F(s):= \mathbb{E}^{B_0=x, W_0=y}\left[|B_{\sigma_s} - W_{\tau_s}|\right].
\]
Then $F(0)=\mathbb{E}^{x}\left[|B_\sigma - y|\right]$ and $F(T)=\mathbb{E}^y \left[|B_\sigma - x|\right]$. We will show that the function $s\mapsto F(s)$ is constant.
In fact we will first show this for a discrete time version where we replace $B$ by a random walk $X$ and $W$ by a random walk $Y$. For this discrete version, it is not difficult to see that $F(s)=F(s-1)$ (cf. Fig.\ 1 or Section \ref{sec core 1d} for details) so that $F$ is constant. From there, \eqref{switch} follows from an application of Donsker's theorem.
\begin{figure}[H]
\begin{subfigure}{.40\linewidth}
\centering
\input{Prelude1.tex}
\end{subfigure}
\begin{subfigure}{.40\linewidth}
\centering
\input{Prelude2.tex}
\end{subfigure}
\caption{Illustration of $F(s) = F(s-1)$.}
\end{figure}
The Rost optimal stopping problem was subject of investigation in \cite{MC91} by McConnell where it is derived via classical PDE methods and in \cite{DA18} by De Angelis where a probabilistic proof is given relying on stochastic calculus.
Furthermore, the Root optimal stopping problem was also derived by Gassiat, Oberhauser and Zou in \cite{GaObZo19} where a suitable extension for a much wider class of Markov processes is established using classical potential theoretic methods as well as by Cox, Ob{\l}{\'o}j and Touzi in \cite{CoObTo18} where a multi-marginal extension of the problem is found.
\subsection{Overview}
In Section \ref{SSRW case} we shall establish \eqref{eq:main_intro} in the context of simple symmetric random walks (SSRW) on the integer lattice, {as well as a related identity pertaining Rost stopping times.}
Interestingly, the symmetry between Root and Rost cases will be obtained as a consequence of a simple time-reversal principle.
Then in Section \ref{Multidimensional case} we explore extensions to the multidimensional setting.
In Section \ref{Brownian case} we will give some remarks on the passage to continuous time and in Section \ref{conclusions} we will draw some future perspectives.
{
\section{The Root and Rost optimal stopping problems}
}
Recall the notion of Skorokhod embedding problem \eqref{SEP} from the introduction.
This problem was first formulated and solved by Skorokhod \cite{Sk61,Sk65}, and numerous new solutions have been found since.
We refer to the surveys of Hobson \cite{Ho11} and Ob{\l}{\'o}j \cite{Ob04} for an account of many of these solutions.
To guarantee well-posedness, we assume thoroughout that $\lambda,\mu$ have finite first moment and are in convex order.
Let $W$ denote a one-dimensional Brownian motion (in keeping with the introduction, we think of $W$ as a Brownian motion running backwards; the reason for this will become clear in the following section).
Suppose we are given initial and target distributions $\lambda$ and $\mu$, we want to study the Root \cite{Ro69} resp.\ Rost \cite{Ro76} solution to the corresponding \eqref{SEP}.
While Root and Rost solutions are most commonly given as hitting times of so called \emph{barriers}, specific subsets of $\mathbb{R}^2$, keeping the previous notation, we will denote by $D^{Root}$ (resp.\ $D^{Rost}$) the continuation set of the Root (resp.\ Rost) embeddings which can be seen as the complements of the barriers in $\mathbb{R}^2$.
Let us write $\mu^{Root}$ (resp. $\mu^{Rost}$) for the law of the Brownian motion starting with distribution $\lambda$ at the time it leaves $D^{Root}$ (resp. $D^{Rost}$) and $\mu_T^{Root}$ (resp.\ $\mu_T^{Rost}$) for the time it leaves $D^{Root}\cap([0,T)\times\mathbb{R})$ (resp.\ $D^{Rost}\cap([0,T)\times\mathbb{R})$).
Recall that the potential of a measure $m$ is denoted by
$U_{m}(y):=-\int|y-x|\, m(\mathrm{d} x)$,
and for a random variable $Z$ we write $U_Z$ for the potential of the law of $Z$.
Throughout this note we consider \emph{optimal stopping problems}, thus suprema taken over $\tau$ (resp. $\sigma$) will denote suprema over stopping times.
The relations of interest in our article, found in \cite{CoWa13, CoWa12}, are
\begin{align} \label{toprovebrownian}
U_{\mu_{T}^{Root}}(x) &= \Ex \left [ U_{\mu^{Root}}\left(W_{\tau^{*} }\right)\mathbbm 1_{\tau^{*}<T} + U_{\lambda}\left(W_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}=T} \right ] \\\label{toprovesupbrownian}
&= \sup\limits_{ \tau\leq T}\Ex \left [ U_{\mu^{Root}}(W_{\tau})\mathbbm 1_{\tau<T} + U_{\lambda}(W_{\tau})\mathbbm 1_{\tau=T} \right ] ,
\end{align}
where the optimizer is $\tau^*:=\inf\{t \geq 0: (T-t,W_t)\notin D^{Root} \}\wedge T$, and
\begin{align} \label{toproveRostbrownian}
U_{\mu^{Rost}}(x)- U_{\mu_T^{Rost}}(x) &= \Ex \left[ \left(U_{\mu^{Rost}}-U_{\lambda}\right)(W_{\tau_*}) \right]
\\\label{toproveRostsupbrownian}
&= \sup\limits_{\tau\leq T}\Ex \left[ \left(U_{\mu^{Rost}}-U_{\lambda}\right)(W_{\tau}) \right],
\end{align}
where the optimizer is $\tau_*:=\inf\{t \geq 0: (T-t,W_t)\notin D^{Rost} \}\wedge T$.
\\
{
Remark that \eqref{toprovebrownian}-\eqref{toprovesupbrownian} are related to \eqref{eq:main_intro} in the introduction. On the other hand, \eqref{toproveRostbrownian}-\eqref{toproveRostsupbrownian} haven't been stated yet. One of the contributions of the article will be to obtain the latter as a consequence of the former, by means of a previously unexplored symmetry between Root and Rost solutions, as hinted at in the introduction.
In the coming section} we establish the above results \eqref{toprovebrownian}-\eqref{toproveRostsupbrownian} in the context of simple symmetric random walks (SSRW) on the integer lattice.
\section{A common one-dimensional random walk framework}\label{SSRW case}
Consider a set $D \subseteq \mathbb{Z}\times \mathbb{Z}$ satisfying
\begin{itemize}
\item If $(t,m)\in D$, then for all $s<t$ also $(s,m)\in D$.
\end{itemize}
This should be seen as a discretised version of the Root continuation set defined in the introduction.
Likewise a Rost continuation set can be cast in the above form after reflection w.r.t.\ a vertical line.\\
\textbf{Notation:}
Denote by $X,Y$ two mutually independent SSRW on some probability space $(\Omega, \P)$ which are started at possibly random initial positions.
Given $x,y\in \mathbb{Z}$ we write $\P^x$ and $\P_y$ for the conditional distribution given $X_0=x$ and $Y_0=y$ resp.\ Similarly $\P^x_y $ means that we condition on both events simultaneously.
We can consider a probability measure $\lambda$ on $\mathbb{Z}$ a ``starting'' distribution by setting $\P^\lambda := \sum_{x\in \mathbb{Z}} \P^x\lambda(\{x\})$, etc.
Let us then introduce the stopping time
\begin{align*}
\rho^{Root} &= \inf \{t\in \mathbb{N}:(t,X_t)\notin D\},
\end{align*}
where $\mathbb{N} = \{0,1,\dots\}$.
We define by $\mu^{Root}$
the law of $X_{\rho^{Root}}$ under $\P^{\lambda}$,
and assume henceforth that the martingale $\left(X_{\rho^{Root}\wedge t}\right)_{t\in\mathbb{N}}$ is uniformly integrable.
We conveniently drop the dependence of $\mu^{Root}$ on $\lambda$.
Given $T\in\mathbb{N}$ we write $\mu^{Root}_T$ for
the $\P^{\lambda}$-law of $X_{\rho^{Root}\wedge T}$.
Note that this definition is equivalent to $\mu^{Root}_T$ being the law of $X$ started with distribution $\lambda$ at the time it leaves $D^{Root} \cap \left(\{0,\dots, T-1\} \times \mathbb{Z} \right)$.
We will first establish the following identity, which is a discrete-time version of \eqref{toprovebrownian}-\eqref{toprovesupbrownian}
\begin{align} \label{toprove}
U_{\mu_{T}^{Root}}(y)& = \Ey \left [ U_{\mu^{Root}}\left(Y_{\tau^{*} }\right)\mathbbm 1_{\tau^{*}<T} + U_{\lambda}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}=T} \right ] \\\label{toprovesup}
&= \sup\limits_{\tau \leq T} \Ey \left [ U_{\mu^{Root}}(Y_{\tau})\mathbbm 1_{\tau<T} + U_{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T} \right ] ,
\end{align}
where the optimizer is
\( \tau^*:=\inf\{t\in\mathbb{N}: (T-t,Y_t)\notin D \}\wedge T.\)
From here we will derive the discrete-time analogue of \eqref{toproveRostbrownian}-\eqref{toproveRostsupbrownian}.
\subsection{Core argument}\label{sec core 1d}
For convenience of the reader we present here the basis of the argument which we repeatedly use, namely that for $s \in \{1, \dots, T\}$
\begin{equation}\label{eq:trivial}
\Exy\left [ |X_{T-s}-Y_{s}|\right ] = \Exy\left [ |X_{T-(s-1)}-Y_{s-1}|\right ] .
\end{equation}
This represents a formalisation of the discretised argument in the
prelude.
Indeed,
\begin{align*}
\Exy\left [ |X_{T-s}-Y_{s}|\right ]
&= \Exy\left[\Exy\left[ |X_{T-s}-Y_{s}| \big| X_{T-s},Y_{s-1}\right]\right ] \\
&= \Exy\left[\Exy\left[ |(X_{T-s} - Y_{s-1})-(Y_{s}-Y_{s-1})|\big| X_{T-s},Y_{s-1}\right]\right] \\
&= \Exy\left[ |X_{T-s}-Y_{s-1}|+\mathbbm 1_{X_{T-s}=Y_{s-1}}\right ]\\
&= \Exy\left[\Exy\left[ |(X_{T-s}-Y_{s-1})+(X_{T-s+1} -X_{T-s})|\big| X_{T-s},Y_{s-1}\right ]\right ] \\
&= \Exy\left[\Exy\left[ |X_{T-s+1}-Y_{s-1}|\big| X_{T-s},Y_{s-1}\right ]\right ] \\
&= \Exy\left[ |X_{T-(s-1)}-Y_{s-1}|\right ],
\end{align*}
which is best read from the top until the middle equality and then from the bottom until the same equality.
More important than \eqref{eq:trivial} is the reasoning above, especially the appearance of the indicator of the event $\{X_{T-s}=Y_{s-1}\}$, which stems from the fact that $Y_{s-1}$ (resp. $X_{T-s}$) always splits into $Y_{s}\pm 1$ (resp. $X_{T-(s-1)}\pm 1$) with probability $1/2$.
For an illustration of this phenomenon also see Figure \ref{CoreArgumentPlot}.
\begin{figure}
\begin{subfigure}{.40\linewidth}
\centering
\input{CoreArgument1.tex}
\caption{SSRW where $X_{T-s} = Y_{s-1}$}
\end{subfigure}
\begin{subfigure}{.40\linewidth}
\centering
\input{CoreArgument2.tex}
\caption{SSRW where $X_{T-s} \neq Y_{s-1}$}
\end{subfigure}
\caption{Illustrating the appearance of the indicator function in the core argument.}
\label{CoreArgumentPlot}
\end{figure} %
\subsection{The Root case}\label{rootcase}
Let $\tau^* := \min\{t \in \mathbb{N}: (T-t,Y_t) \not\in D\} \wedge T$. We start with a useful observation:
\begin{remark}\label{obspotentials}
The equality $U_{\mu^{Root}}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}<T} =U_{\mu_T^{Root}}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}<T} $ holds.
Indeed, let $z=Y_{\tau^{*}} $ on $\{\tau^{*}<T\}$.
Then $(T-\tau^{*},z)\notin D$ and hence $(X_{T}-z)(X_{\rho^{Root}}-z)\geq 0$ on $\{\rho^{Root}>T\}$ as otherwise $X$ would have left $D$ before $\rho^{Root}$, see also Figure \ref{RootRemarkPlot}.
This implies
\begin{align}
-U_{\mu^{Root}}(z)
&=\Elambda\left[|X_{\rho^{Root}} - z|\right]\nonumber\\
&=\Elambda\left[|X_{\rho^{Root}\wedge T} - z|\mathbbm 1_{\rho^{Root}\leq T} - (X_{\rho^{Root}} - z)\mathbbm 1_{\rho^{Root}> T,\, z>X_T} + (X_{\rho^{Root}} - z)\mathbbm 1_{\rho^{Root}> T,\, z\leq X_T} \right] \nonumber\\
&=\Elambda\left[|X_{\rho^{Root}\wedge T} - z|\mathbbm 1_{\rho^{Root}\leq T} - (X_{\rho^{Root}\wedge T} - z)\mathbbm 1_{\rho^{Root}> T,\, z>X_T} + (X_{\rho^{Root}\wedge T} - z)\mathbbm 1_{\rho^{Root}> T,\, z\leq X_T} \right] \nonumber\\
&=\Elambda\left[|X_{\rho^{Root}\wedge T} - z|\right]=-U_{\mu^{Root}_T}(z).
\end{align}
Accordingly we may replace $\mu^{Root}$ by $\mu^{Root}_T$ in \eqref{toprove} (but we do not do so in \eqref{toprovesup}).
\end{remark}
\begin{figure}
\begin{subfigure}{.40\linewidth}
\centering
\input{SSRWPlot_Remark31_1.tex}
\caption{SSRW where $\tau^* < T$ and $\rho^{Root}\leq T$}
\end{subfigure}
\begin{subfigure}{.40\linewidth}
\centering
\input{SSRWPlot_Remark31_2.tex}
\caption{SSRW where $\tau^* < T$ and $\rho^{Root} > T$}
\end{subfigure}
\caption{Illustrating properties of Root stopping times.}
\label{RootRemarkPlot}
\end{figure}
Given a $Y$-stopping time $\tau\leq T$, we define a stopping time $\sigma(\tau)$ of $X$ as the first time before $T-\tau$ that $X$ leaves $D$, i.e. $\sigma(\tau):= \rho^{Root} \wedge (T-\tau)$.\footnote{Formally, $\tau$ is a stopping time w.r.t.\ the filtration $\mathcal G=(\mathcal G_t)_{t\in\mathbb{N}}$, where $\mathcal G_t=\sigma(\{Y_u:u\leq t\})$. $\sigma(\tau)$ is a stopping time w.r.t.\ the filtration $\mathcal{F}=(\mathcal{F}_s)_{s\leq T} $, where $\mathcal{F}_s=\sigma (\{ X_u, Y_t: u \leq s, t\leq T\})$}
For any $y \in \mathbb{Z}$ we now introduce the crucial interpolating function
\begin{equation}\label{interpolatingfun}
F(s)\;\;:=\;\; F^{\tau^*}(s)\;\;:=\;\;\Elambday\left [| X_{\sigma(\tau^*\wedge s)} - Y_{\tau^*\wedge s} | \right ]\hspace{5pt}\mbox{ for }s\in\{0,\dots,T\}.
\end{equation}
It may help to picture $Y$ evolving ``leftwards'' from the lattice point $(T,y)$ at time zero, so that its exit time $\tau^*$ from $D$ before $T$ is measured as $T-\tau^*$ for the ``rightwards'' process $X$.
\begin{remark}\label{remark:interpolation}
We see that $\sigma(0)=\rho^{Root}\wedge T$, so consequently
\[
F(0)=\Elambda\left[| X_{\rho^{Root}\wedge T} - y|\right]=-U_{\mu^{Root}_{T}}(y).
\]
On the other hand,
$\sigma(\tau^*)=\rho^{Root}\wedge (T-\tau^*)$, so
\begin{align*}
F(T) &=\Elambday\left[|X_{\rho^{Root} \wedge (T-\tau^*)} - Y_{\tau^*} |\right]
\\ &= \Elambday\left[ |X_{\rho^{Root} \wedge (T-\tau^*)} - Y_{\tau^*}|\mathbbm 1_{\tau^*<T} + |X_{\rho^{Root} \wedge (T-\tau^*)} - Y_{\tau^*}|\mathbbm 1_{\tau^*=T} \right]
\\ &= \Elambday\left[ |X_{\rho^{Root} \wedge T} - Y_{\tau^*}|\mathbbm 1_{\tau^*<T} + |X_{0} - Y_{\tau^*}|\mathbbm 1_{\tau^*=T}\right]
\\ &= -\Ey\left[ U_{\mu^{Root}_T}\left(Y_{\tau^*}\right)\mathbbm 1_{\tau^*<T} + U_{\lambda}( Y_{\tau^*})\mathbbm 1_{\tau^*=T}\right] ,
\end{align*}
by independence and by applying the appropriate analogue of the argument in Remark \ref{obspotentials} for the third equality.
\end{remark}
We can now prove \eqref{toprove} and \eqref{toprovesup}; we treat the
cases separately.
\begin{lemma}
\label{constancydiscreteRoot}
The function $F$ is constant. Consequently
\[ U_{\mu_{T}^{Root}}(y) =\Ey \left [ U_{\mu^{Root}}\left(Y_{\tau^{*} }\right)\mathbbm 1_{\tau^{*}<T} + U_{\lambda}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}=T} \right ]\]
\end{lemma}
\begin{proof}
Let $0<s<T$.
Define the stopping times $\tau^*_s:=\tau^*\wedge s$ and $\sigma_s = \sigma(\tau^* \wedge s) = \rho^{Root}\wedge(T-\tau^*_s)$.
Then
\[
F(s) =\Elambday\left[|X_{\sigma_s} - Y_{\tau^*_s}|\right].
\]
Let us first prove that
\begin{equation}\label{eq first to prove}
\Elambday\left[ | X_{\sigma_s} - Y_{\tau^*_s} | \right ]
=\Elambday\left[ | X_{\sigma_s} - Y_{\tau^*_{s-1}} | + \mathbbm 1_{X_{\sigma_s} = Y_{\tau^*_{s-1}},\, \tau^*\geq s} \right ].
\end{equation}
Since
\[
\Elambday\left[| X_{\sigma_s} - Y_{\tau^*_s}| \right ]
=\Elambday\left[| X_{\rho^{Root}\wedge (T-s)} - Y_{s}|\mathbbm 1_{{\tau^*}\geq s} \right ]
+\Elambday\left[| X_{\sigma_s} - Y_{\tau^*_{s-1}}|\mathbbm 1_{{\tau^*}<s} \right ],
\]
and with the appropriate analogue of the core argument \eqref{eq:trivial} (see also Figure \ref{RootCoreArgumentPlot})
\begin{align*}
\Elambday &\left[| X_{\rho^{Root}\wedge (T-s)} - Y_{s}|\mathbbm 1_{{\tau^*}\geq s} \right ]
\\ =&\,\,\Elambday \left[\Elambday\left[ | X_{\rho^{Root}\wedge (T-s)} - Y_{s} | \big|X_{\rho^{Root}\wedge (T-s)}, Y_0,\dots,Y_{s-1} \right]\mathbbm 1_{{\tau^*}\geq s} \right]
\\ =&\,\,\Elambday \left[\Elambday\left[|(X_{\rho^{Root}\wedge (T-s)}-Y_{s-1}) - ( Y_{s}-Y_{s-1})| \big| X_{\rho^{Root}\wedge (T-s)}, Y_0,\dots,Y_{s-1} \right] \mathbbm 1_{{\tau^*}\geq s} \right]
\\ =&\,\,\Elambday \left[\left( |X_{\rho^{Root}\wedge (T-s)}-Y_{s-1}|+ \mathbbm 1_{X_{\rho^{Root}\wedge (T-s)}=Y_{s-1}} \right)\mathbbm 1_{{\tau^*}\geq s} \right]
\\ =&\,\,\Elambday \left[| X_{\sigma_s} - Y_{\tau^*_{s-1}} |\mathbbm 1_{{\tau^*}\geq s} + \mathbbm 1_{X_{\rho^{Root}\wedge (T-s)}=Y_{s-1} \, ,\, {\tau^*}\geq s } \right ],
\end{align*}
clearly \eqref{eq first to prove} follows.
Now let us similarly establish that
\begin{align}
\Elambday\left[| X_{\sigma_{s-1}} - Y_{\tau^*_{s-1}}| \right ]
&=\Elambday\left[| X_{\sigma_s} - Y_{\tau^*_{s-1}}| + \mathbbm 1_{X_{\sigma_s} = Y_{\tau^*_{s-1}} ,\, \tau^*\geq s,\, \rho^{Root}> T-s} \right ]\label{eq second to prove} \\
&=\Elambday\left[| X_{\sigma_s} - Y_{\tau^*_{s-1}}| + \mathbbm 1_{X_{\sigma_s} = Y_{\tau^*_{s-1}} ,\, \tau^*\geq s} \right ] .\label{eq equivalent}
\end{align}
Indeed,
\[
\Elambday\left[| X_{\sigma_{s-1}} - Y_{\tau^*_{s-1}}| \right ]
=\Elambday\left[| X_{T-s+1} - Y_{{s-1}}|\mathbbm 1_{\tau^*\geq s,\, \rho^{Root}> T-s} \right ]
+\Elambday\left[| X_{\sigma_{s}} - Y_{\tau^*_{s-1}}|\mathbbm 1_{\{\tau^*< s\}\cup \{\rho^{Root}\leq T-s\}} \right ],
\]
and again with the appropriate analogue of \eqref{eq:trivial}
\begin{align*}
&\Elambday \left[|X_{T-s+1} - Y_{{s-1}}|\mathbbm 1_{\tau^*\geq s,\, \rho^{Root}> T-s} \right]
\\ &\quad=\Elambday\left[\Elambday \left[|X_{T-s+1} - Y_{s-1}| \big| X_0,\dots,X_{T-s}, Y_0,\dots,Y_{s-1} \right ] \mathbbm 1_{{\tau^*}\geq s,\, \rho^{Root}> T-s}\right ] \\
&\quad=\Elambday\left[\Elambday \left[| (X_{T-s}-Y_{s-1}) + (X_{T-s+1}-X_{T-s})|\big| X_0,\dots,X_{T-s}, Y_0,\dots,Y_{s-1}\right] \mathbbm 1_{{\tau^*}\geq s,\, \rho^{Root}> T-s} \right]
\\ &\quad=\Elambday\left[\left(| X_{T-s} - Y_{{s-1}}| + \mathbbm 1_{X_{T-s}=Y_{s-1}}\right)\mathbbm 1_{{\tau^*}\geq s,\, \rho^{Root}> T-s}\right] \\
&\quad=\Elambday\left[| X_{\sigma_s} - Y_{\tau^*_{s-1}}|\mathbbm 1_{{\tau^*}\geq s,\, \rho^{Root}> T-s} + \mathbbm 1_{X_{\sigma_s}=Y_{\tau^*_{s-1}},\,{\tau^*}\geq s \, ,\, \rho^{Root}> T-s} \right ],
\end{align*}
also \eqref{eq second to prove} follows.
We then see that \eqref{eq equivalent} holds true since on $\{X_{\sigma_s} = Y_{\tau^*_{s-1}}, \tau^*\geq s\}$ we have $(T-(s-1),Y_{s-1})=(T-(s-1),Y_{\tau^*_{s-1}})= (T-s+1,X_{\sigma_s})\in D$, and so by definition of $D$ necessarily $(\rho^{Root}\wedge (T-s),X_{\sigma_s})\in D$, thus $\rho^{Root}\geq T-s+1$ is fulfilled.
The identities \eqref{eq first to prove} and \eqref{eq equivalent} now yield that $F$ is constant.
\end{proof}
\begin{figure}
\begin{subfigure}{.40\linewidth}
\centering
\input{SSRWPlot1.tex}
\caption{SSRW where $\tau^* > s$, $\sigma_s = T-S$ and $X_{\sigma_s} = Y_{s-1}$}
\end{subfigure}
\begin{subfigure}{.40\linewidth}
\centering
\input{SSRWPlot2.tex}
\caption{SSRW where $\tau^* > s$, $\sigma_s < T-S$ and $X_{\sigma_s} \neq Y_{s-1}$}
\end{subfigure}
\caption{Illustration of the core argument in the Root setting.}
\label{RootCoreArgumentPlot}
\end{figure}
The proof of \eqref{toprovesup} follows similar lines.
Given a $Y$-stopping time $\tau$ we consider the interpolating function
\begin{equation}\label{interpolatingfungeneral}
F^{\tau}(s)\;\;:=\;\;\Elambday\left [| X_{\sigma(\tau\wedge s)} - Y_{\tau\wedge s} | \right ]\hspace{5pt}\mbox{ for }s\in\{0,\dots,T\}.
\end{equation}
\begin{lemma}
\label{monotonicitydiscreteRoot}
For every $\{0,\ldots,T\}$-valued stopping time $\tau$ of $Y$, the function $F^{\tau}$ is increasing and
\[ U_{\mu_{T}^{Root}}(y) \geq \Ey \left [ U_{\mu^{Root}}\left(Y_{\tau }\right)\mathbbm 1_{\tau<T} + U_{\lambda}\left(Y_{\tau}\right)\mathbbm 1_{\tau=T} \right ]\]
\end{lemma}
\begin{proof}
Clearly $F^{\tau}(0)= -U_{\mu_{T}^{Root}}(y)$. On the other hand,
\begin{align*}
F^{\tau}(T) &=\Elambday\left[ |X_{\rho^{Root}\wedge (T-\tau)} - Y_{\tau}|\mathbbm 1_{\tau<T} + |X_{0} - Y_{\tau}|\mathbbm 1_{\tau=T} \right] \\
&= -\Elambday\left[ U_{X_{\rho^{Root}\wedge (T-\tau)}}(Y_{\tau})\mathbbm 1_{\tau<T} + U_{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T} \right] \\
&\leq -\Elambday\left [U_{X_{\rho^{Root}}}(Y_{\tau})\mathbbm 1_{\tau<T} + U_{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T}\right ] ,
\end{align*}
where the inequality is a consequence of the potentials $s\mapsto U_{X_{\rho^{Root}\wedge s}}(z)$ being decreasing in $s$ for each $z$ (by Jensen's inequality and optional sampling) and the martingale $\left(X_{\rho^{Root}\wedge t}\right)_{t \in \mathbb{N}}$ being uniformly integrable.
Thus if we show that $F^{\tau}(\cdot)$ is increasing, we can conclude.
Let $0<s<T$.
Define the stopping times $\tau_s:=\tau\wedge s$ and $\sigma_s=\rho^{Root}\wedge(T-\tau_s)$.
Then, analogous to the proof of Lemma \ref{constancydiscreteRoot}, but replacing $\tau^*$ by $\tau$, we get
\begin{align*}
F^{\tau}(s) &=\Elambday\left[|X_{\sigma_s} - Y_{\tau_{s-1}}| + \mathbbm 1_{X_{\sigma_s} = Y_{\tau_{s-1}}, \tau\geq s} \right ]
\\ &\geq\Elambday\left[| X_{\sigma_s} - Y_{\tau_{s-1}}| + \mathbbm 1_{X_{\sigma_s} = Y_{\tau_{s-1}}, \tau\geq s,\,\rho^{Root}\geq T-(s-1)} \right ]
= F^{\tau}(s-1),
\end{align*}
thus $F^{\tau}(\cdot)$ is increasing.
\end{proof}
\begin{remark}
\label{rem generalisation}
The second part of the preceding proof, stating that the function $F^{\tau}$ is increasing, yields after trivial modifications that also
\begin{equation}\label{eq: aux}
s\mapsto F_{\sigma}^{\tau}(s):=\Elambday\left [| X_{\sigma\wedge(T- \tau\wedge s)} - Y_{\tau\wedge s} | \right ],
\end{equation}
is increasing.
We did not use the particular structure of $\rho^{Root}$ there.
\end{remark}
\begin{remark} There may be many other interpolating functions (which must coincide when $\tau=\tau^*$ of course).
For example, if we replace $\sigma(\tau\wedge s)=\rho^{Root}\wedge (T - \tau \wedge s)$ by
\[ \sigma(\tau, s):=\begin{cases}
\rho^{Root} & \text{if } \tau < s\\
\rho^{Root} \wedge {(T-s)} & \text{else }
\end{cases},
\]
and then define
\begin{equation}\label{interpolatingfungeneral_alternative}
\tilde{F}^{\tau}(s)\;\;:=\;\;\Elambday\left [| X_{\sigma(\tau, s)} - Y_{\tau\wedge s} | \right ]\hspace{5pt}\mbox{ for }s\in\{0,\dots,T\},
\end{equation}
we have $\tilde{F}^{\tau}(0)= -U_{\mu_{T}^{Root}}(y)$ and
$\tilde{F}^{\tau}(T)=-\Elambday\left[ U_{X_{\rho^{Root}}}(Y_{\tau})\mathbbm 1_{\tau<T} + U_{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T}\right]$, for each stopping time $\tau\in [0,T]$.
This function can be seen to be increasing for each such $\tau$ and constant for $\tau^*$. \end{remark}
\subsection{The Rost case as a consequence of the Root case}\label{rostcase}
Emboldened by the results in the Root case, we could proceed to establish \eqref{toproveRostbrownian}-\eqref{toproveRostsupbrownian} in a SSRW setting via interpolating functions as well.
It is much more illuminating and elegant, however, to deduce the Rost case from the Root one.
We thus keep the notation as in the previous part.
\begin{proposition} \label{SymmetryProposition}
For each $x,y,T$, any stopping time $\tau$ for $Y$ such that $\Ey[|Y_{\tau}|]<\infty$, and every $\{0,\dots,T\}$-valued stopping time $\sigma$ for $X$, we have
\begin{align}\label{eq: first simmetry}
\Ey\left[|x - Y_{\tau}|-|x - Y_{\tau\wedge T}|\right]&\leq \Exy\left[|X_{\sigma} - Y_{\tau}|-|X_{\sigma}-y| \right].
\end{align}
Suppose furthermore that
\begin{equation}
\tau = \inf \{t\in \mathbb{N}: (T-t,Y_t)\notin D\},\label{eq: geom tau}
\end{equation}
and that
$\sigma=\rho^{Root}\wedge T$. Then there is equality in \eqref{eq: first simmetry}.
\end{proposition}
\begin{proof}
We first prove the inequality
\begin{equation} \label{eq: intermed}
\Ey\left[|x - Y_{\tau}|-|x - Y_{\tau\wedge T}| \right] \leq \Exy\left[|X_{\sigma} - Y_{\tau}|-|X_{\sigma \wedge (T - \tau\wedge T)} - Y_{\tau\wedge T}|\right].
\end{equation}
This follows, on the one hand, by
\[
\Ey\left[\left(|x - Y_{\tau}|-|x - Y_{\tau\wedge T}|\right)\mathbbm 1_{\tau <T} \right] = 0 \leq \Exy\left[ \left( |X_{\sigma} - Y_{\tau}|-|X_{\sigma\wedge (T-\tau)} - Y_{\tau}| \right)\mathbbm 1_{\tau <T} \right],
\]
where the inequality follows by Jensen's inequality and optional sampling.
Similarly we also conclude by Jensen and optional sampling that
\[
\Ey\left[\left(|x - Y_{\tau}| - |x - Y_{\tau\wedge T}|\right)\mathbbm 1_{\tau \geq T} \right] \leq \Exy\left[\left(|X_{\sigma} - Y_{\tau}| - |X_{0} - Y_{ T}| \right) \mathbbm 1_{\tau \geq T} \right].
\]
We furthermore note that considering $F^{\tau}_{\sigma}$ as defined in
\eqref{eq: aux} for the choice $\lambda=\delta_x$ we have that the
r.h.s.\ of \eqref{eq: first simmetry}, resp.\ of \eqref{eq: intermed}, coincides with
$\Exy \left[|X_{\sigma} - Y_{\tau}|\right] - F^{\tau}_{\sigma}(0)$, resp. $\Exy \left[|X_{\sigma} - Y_{\tau}|\right] - F^{\tau}_{\sigma}(T)$.
We can now conclude from Remark \ref{rem generalisation}, stating that $F_{\sigma}^{\tau}$ is increasing, the desired result \eqref{eq: first simmetry}.
In the case $\sigma=\rho^{Root}\wedge T$ and $\tau$ fulfilling \eqref{eq: geom tau}, we obtain that $F_{\rho^{Root}\wedge T}^{\tau} = F^{\tau\wedge T}_{\rho^{Root}\wedge T}=F^{\tau^*}=F $ on $[0,T]$, by \eqref{eq: geom tau}, which by Lemma \ref{constancydiscreteRoot} is constant.
So to conclude we must show that
\[
\Ey\left[|x - Y_{\tau}| - |x - Y_{\tau\wedge T}| \right] = \Exy\left[|X_{\rho^{Root}\wedge T} - Y_{\tau}|\right] - F(T).
\]
We can use the arguments in Remark \ref{obspotentials} resp. \ref{remark:interpolation} to obtain
\[
\Ey\left[\left(|x - Y_{\tau}| - |x - Y_{\tau\wedge T}|\right)\mathbbm 1_{\tau <T} \right] = 0 = \Exy\left[\left(|X_{\rho^{Root}\wedge T} - Y_{\tau}| - |X_{\rho^{Root} \wedge (T-\tau)} - Y_{\tau}| \right)\mathbbm 1_{\tau <T} \right].
\]
Similarly also
\[
\Ey\left[\left(|x - Y_{\tau}| - |x - Y_{\tau\wedge T}|\right)\mathbbm 1_{\tau \geq T} \right] = \Exy\left[\left(|X_{\rho^{Root}\wedge T} - Y_{\tau}| - |X_{0} - Y_{T}| \right)\mathbbm 1_{\tau \geq T}\right],
\]
which concludes the proof.
\end{proof}
\begin{figure}
\begin{subfigure}{.40\linewidth}
\centering
\input{RootToRost1.tex}
\end{subfigure}
\begin{subfigure}{.40\linewidth}
\centering
\input{RootToRost2.tex}
\end{subfigure}
\caption{Illustration of the connection between $D^{Rost}$ and $D$, resp. between $\rho^{Rost}$ and $\rho^{Root}$.}
\label{RootRostPlot}
\end{figure}
A discrete time version of the Rost optimal stopping problem \eqref{toproveRostbrownian}-\eqref{toproveRostsupbrownian} can now be established as a consequence of Proposition \ref{SymmetryProposition}.
A Rost continuation set is a set $D^{Rost}\subset \mathbb{N} \times \mathbb{Z}$ satisfying
\begin{itemize}
\item If $(t,m)\in D^{Rost}$, then for all $s>t$ also $(s,m)\in D^{Rost}$.
\end{itemize}
Given such a set for each fixed $T\in\mathbb{N}$ we may define $D := \{(T-t,m) : (t,m) \in D^{Rost} \}$ which is a Root continuation set to which the previous result is applicable.
Let us introduce
\begin{align}
\rho^{Rost} := \inf \{t\in\mathbb{N}:(T-t,Y_t)\notin D\} = \inf \{t\in\mathbb{N}:(t,Y_t)\notin D^{Rost}\},
\end{align}
and let $\mu^{Rost}$ (resp. $\mu^{Rost}_T$) denote the law of a SSRW started with distribution $\lambda$ and stopped at time $\rho^{Rost}$ (resp. $\rho^{Rost} \wedge T$).
See Figure \ref{RootRostPlot} for an illustration of the connection between Root and Rost continuation sets and the respective hitting times.
We assume uniform integrability of $\left(Y_{\rho_{Rost} \wedge t}\right)_{t \in \mathbb{N}}$.
\begin{corollary}
We have
\begin{align}
U_{\mu^{Rost}}(x)- U_{\mu^{Rost}_T}(x)
&= \Ex \left[\left(U_{\mu^{Rost}} - U_{\lambda}\right)(X_{\sigma_*}) \right] \label{toproveRost}
\\ &= \sup\limits_{\sigma \leq T}\Ex \left[\left(U_{\mu^{Rost}} - U_{\lambda}\right)(X_{\sigma}) \right], \label{toproveRostsup}
\end{align}
where the optimizer is given by
\[
\sigma_*:=\rho^{Root}\wedge T = \inf \{t\in\mathbb{N}:(T-t,X_t)\notin D^{Rost}\}\wedge T.
\]
\end{corollary}
\begin{proof}
For $y \in \mathbb{Z}$ let us first consider $Y_0 = y$, i.e. $\lambda = \delta_{y}$.
Consider Proposition \ref{SymmetryProposition} for the stopping time $\tau = \rho^{Rost}$.
As
\begin{align*}
\Ey\left[|x - Y_{\tau}|-|x - Y_{\tau\wedge T}|\right] &= - \left( U_{\mu^{Rost}}(x) - U_{\mu^{Rost}_T}(x) \right),
\\ \Exy\left[|X_{\sigma} - Y_{\tau}|-|X_{\sigma}-y| \right]&= - \Ex \left[\left(U_{\mu^{Rost}} - U_{\lambda}\right)(X_{\sigma}) \right],
\end{align*}
due to \eqref{eq: first simmetry} we then have
\[
U_{\mu^{Rost}}(x)- U_{\mu^{Rost}_T}(x) \leq \sup\limits_{\sigma \leq T}\Ex \left[\left(U_{\mu^{Rost}} - U_{\lambda}\right)(X_{\sigma}) \right].
\]
To prove \eqref{toproveRost} we note that $\tau = \rho^{Rost}$ satisfies \eqref{eq: geom tau}.
Thus, for $\sigma = \sigma_*$ we have equality in \eqref{eq: first simmetry} which is precisely \eqref{toproveRost} and furthermore also gives \eqref{toproveRostsup}.
As this is true for arbitrary $y \in \mathbb{Z}$, the extension to general $\lambda$ is clear due to identities of the form $\mathbb{E}^x_{\lambda}\left[|X_{\sigma} - Y_{\tau}|\right] = \sum_{y\in\mathbb{Z}} \Exy\left[|X_{\sigma} - Y_{\tau}|\right] \lambda(\{y\})$.
\end{proof}
\section{The multidimensional case}\label{Multidimensional case}
We have established \eqref{toprovebrownian}-\eqref{toproveRostsupbrownian} for the integer lattice in one dimension. We shall extend this to the setting of the $d$-dimensional integer lattice $\mathbb{Z}^d$ for $d$ arbitrary.
Let $Z$ be a SSRW on $\mathbb{Z}^d$ and let
\[z\in\mathbb{Z}^d \mapsto G_n(z)\, :=\,{\mathbb E}^{Z_0=0}[\#\{ t\leq n:Z_t=z\}],\]
denote the expected number of visits to site $z$ of $Z$ started in the origin, prior to $n$.
We then consider the so-called \textit{potential kernel} of the SSRW
\[z\in\mathbb{Z}^d \mapsto a(z)\, :=\,\lim_{n\to\infty} G_n(0)-G_n(z),\]
which is finite in any dimensions and has the desirable property that
\begin{equation}
a(z)=-\mathbbm 1_{z=0}+\frac{1}{2d}\sum_{z'\sim z} a(z'). \label{Laplace}
\end{equation}
Here $z'\sim z$ if $z'$ is an immediate neighbour of $z$ (corresponding to moving away from $z$ along one coordinate only, so there are $2d$ of them).
From this follows that $\left(a(Z_t)\right)_{t\in\mathbb{N}}$ is a (Markovian) submartingale and by induction
\begin{equation}
\mathbb{E}\left[a(Z_{t+n})|Z_t\right]=a(Z_t)+\sum_{\ell=0}^{n-1}\P(Z_{t+\ell}=0|Z_t),\label{supermg}
\end{equation}
which is an identity we will repeatedly use.
In the transient case ($d\geq 3$) we have that $a$ is just the negative of the expected number of visits to a point up to an additive constant.
In the one dimensional case we have $a(\cdot)=|\cdot|$.
We refer to \cite[Chapter 1]{Lawler_intersections} for a review of these concepts/facts.
We first observe that owing to \eqref{Laplace} the core argument \eqref{eq:trivial} in Section \ref{sec core 1d} is still valid, so for $s\in\{1,\dots, T\}$
\[
\Exy\left[a(X_{T-s}-Y_s)\right]= \Exy\left[a(X_{T-s}-Y_{s-1})+\mathbbm 1_{X_{T-s}=Y_{s-1}}\right]=\Exy\left[a(X_{T-(s-1)}-Y_{s-1})\right].
\]
We shall see that all the computations we did using $z\mapsto |z|$ in the one-dimensional case are still valid for the potential kernel $a$. For a measure $\nu$ on $\mathbb{Z}^d$ let
\[A.\nu(y) := - \int a(y - x)\nu(dx).\]
As in the previous section, we denote by $X,Y$ two independent SSRW in $\mathbb{Z}^d$.
\begin{proposition}
Let $\lambda$ be a starting distribution in $\mathbb{Z}^d$ and $D^{Root}$ (resp.\ $D^{Rost}$) be Root-type (resp.\ Rost-type) continuation sets in $\mathbb{Z}^{d+1}$. Denote by $\mu^{Root}$ resp. $\mu_T^{Root}$ the law of a SSRW started with distribution $\lambda$ and stopped upon leaving $D^{Root}$ resp. $D^{Root} \cap \left( \{0, \dots, T-1\} \times \mathbb{Z}^d\right)$ (analogously for $\mu^{Rost}$ and $\mu_T^{Rost}$), and assume that the SSRW stopped when leaving $D^{Root}$ (resp.\ $D^{Rost}$) is uniformly integrable. Then
\begin{align} \label{G1Root}
A.\mu_{T}^{Root}(y)& = \Ey \left [ A.\mu^{Root}\left(Y_{\tau^{*} }\right)\mathbbm 1_{\tau^{*}<T} + A.{\lambda}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}=T} \right ] \\\label{G2Root}
&= \sup\limits_{ \tau\leq T}\Ey \left [ A.{\mu^{Root}}(Y_{\tau})\mathbbm 1_{\tau<T} + A.{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T} \right ] ,
\end{align}
where the optimizer is $\tau^*:=\inf\{t\in\mathbb{N}: (T-t,Y_t)\notin D^{Root} \}\wedge T$, and
\begin{align} \label{G1Rost}
A.{\mu^{Rost}}(x)- A.{\mu_T^{Rost}}(x) &= \Ex \left[ \left(A.{\mu^{Rost}}-A.{\lambda}\right)(X_{\tau_*}) \right]
\\ \label{G2Rost}
&= \sup\limits_{\tau\leq T}\Ex \left[ \left(A.{\mu^{Rost}}-A.{\lambda}\right)(X_{\tau}) \right],
\end{align}
where the optimizer is $\tau_*:=\inf\{t\in\mathbb{N}: (T-t,X_t)\notin D^{Rost} \}\wedge T$.\\
\end{proposition}
\begin{proof}
Let us first prove \eqref{G1Root}.
In analogy to the previous section, we define an interpolating function
\begin{equation}\label{interpolatingfungeneral-G}
F(s)\;\;:=\;\;\Elambday\left[ a( X_{\rho^{Root}\wedge (T- \tau^*\wedge s)} - Y_{\tau^*\wedge s} ) \right ]\hspace{5pt}\mbox{ for }s\in\{0,\dots,T\}.
\end{equation}
Then clearly $F(0) = - A.\mu_{T}^{Root}(y)$ and also
\[
F(T)=\Elambday[a(X_{\rho^{Root}\wedge(T-\tau^*)}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T} ]-\Ey[A.{\lambda}\left(Y_{\tau^{*}}\right)\mathbbm 1_{\tau^{*}=T} ].
\]
If we establish $-\Elambday[a(X_{\rho^{Root}\wedge(T-\tau^*)}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T} ]=\Ey [ A.\mu^{Root}\left(Y_{\tau^{*} }\right)\mathbbm 1_{\tau^{*}<T}]$ then \eqref{G1Root} is implied by $F$ being constant.
Clearly it suffices to show that
\[
\Elambday[a(X_{T-\tau^*}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*} ] =\Elambday[a(X_{\rho^{Root}}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*} ].
\]
Indeed
\begin{align}\label{tower}
\Elambday&\left[a(X_{\rho^{Root}}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*}\right] \nonumber
\\ &= \Elambday\left[\Elambday\left[a(X_{\rho^{Root}}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*} \big|X_0, \dots, X_T, Y_0, \dots, Y_{T-1} \right] \right] \nonumber
\\ &= \Elambday\left[\left( a(X_{T-\tau^*}-Y_{\tau^*})+\sum_{s=T-\tau^*}^{\rho^{Root}-1} \P\left(X_{s}=Y_{\tau^*} \big|X_0, \dots, X_T, Y_0, \dots, Y_{T-1}\right )\right)\mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*} \right] \nonumber
\\ &= \Elambday\left[ a(X_{T-\tau^*}-Y_{\tau^*}) \mathbbm 1_{\tau^{*}<T,\, \rho^{Root}>T-\tau^*}\right],
\end{align}
where the last line holds since, given
$\{X_0, \dots, X_T, Y_0, \dots, Y_{T-1}\}$ on $\{\tau^{*}<T,\, \rho^{Root}>T-\tau^*\}$,
We now prove that $F$ is indeed constant.
First we observe that
\[
F(s)= \Elambday\left[a(X_{\rho^{Root}\wedge (T-\tau^*\wedge s)}-Y_{\tau^*\wedge s})\right]
=\Elambday\left[a(X_{\rho^{Root}\wedge (T-\tau^*\wedge s)}-Y_{\tau^*\wedge (s-1)}) + \mathbbm 1_{X_{\rho^{Root}\wedge(T-s)}= Y_{s-1}, \tau^*\geq s}\right].
\]
To see this we consider the two cases $\{\tau^* < s\}$ and $\{\tau^* \geq s\}$ separately. While the former case is clear, on the latter we apply \eqref{supermg} where we condition on $\{X_0,\dots,X_{T-s},Y_0,\dots,Y_{s-1} \}$.
Analogously but by splitting into $\{\tau^* < s\} \cup \{\rho^{Root}\leq T-s\}$ and $\{\tau^*\geq s, \rho^{Root}>T-s\}$ we obtain
\begin{align*}
F(s-1) &= \Elambday\left[a(X_{\rho^{Root}\wedge (T-\tau^*\wedge (s-1))}-Y_{\tau^*\wedge (s-1)})\right]
\\ &=\Elambday\left[a(X_{\rho^{Root}\wedge (T-\tau^*\wedge s)}-Y_{\tau^*\wedge (s-1)})
+ \mathbbm 1_{X_{\rho^{Root}\wedge(T-s)} = Y_{s-1},\tau^*\geq s, \rho^{Root}>T-s}\right].
\end{align*}
We conclude by observing that the two appearing indicator functions are equal, since on $\{X_{\rho^{Root}\wedge(T-s)} = Y_{s-1}, \tau^*\geq s\}$ we must necessarily have $\rho^{Root}>T-s$.
To show \eqref{G2Root} define the multi dimensional equivalent of \eqref{interpolatingfungeneral}, that is for a $\{0,\dots,T\}$-valued $Y$-stopping time $\tau$ define
\begin{equation}
F(s)\;\;:=\;\;\Elambday\left[ a( X_{\rho^{Root}\wedge (T- \tau\wedge s)} - Y_{\tau \wedge s} ) \right ]\hspace{5pt}\mbox{ for }s\in\{0,\dots,T\}.
\end{equation}
Then clearly $F^\tau(0) = -A.{\mu_T^{Root}}(y)$. Again we can use \eqref{supermg} to show that $F^{\tau}$ is increasing and furthermore
\[
F^\tau(T) \leq - \Ey \left[ A.{\mu^{Root}}(Y_{\tau})\mathbbm 1_{\tau<T} + A.{\lambda}(Y_{\tau})\mathbbm 1_{\tau=T} \right].
\]
The Rost case can be derived from the Root case by analogous arguments as in Section \ref{rostcase}.
A multidimensional version of Proposition \ref{SymmetryProposition} can be proved verbatim replacing the absolute value by the function $a$ and the Jensen arguments by submartingale arguments.
The equality case follows from \eqref{supermg} exploiting the barrier structure as it was done for \eqref{tower}.
\end{proof}
\section{From the Random Walk Setting to the Continuous Case} \label{Brownian case}
While the passage to continuous time is in essence an application of Donsker-type results, we will give a more elaborate explanation using arguments established by Cox and Kinsley in \cite{CoKi19} for the one-dimensional case.
We note that all results and arguments in Section 3 are invariant under uniform scaling of the space-time grid.
Thus for each $N \in \mathbb{N}$ we can consider a rescaled simple symmetric random walk $Y^N$ with space step size $\frac{1}{\sqrt{N}}$ and time step size $\frac{1}{N}$ as it is done in \cite{CoKi19}. The authors discretise an \emph{optimal Skorokhod embedding problem}, an (SEP) featuring the following additional optimisation problem
\begin{equation}
\inf_{\tau \text{ solves (SEP)}} \mathbb{E}[F(B_{\tau}, \tau)]. \tag{OptSEP}
\end{equation}
It is known that for \emph{any} convex (resp. concave) function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, the (OptSEP) with $F(B_{\tau}, \tau) = f(\tau)$ is solved by a Root (resp. Rost) solution, see e.g. \cite{BeCoHu17}.
It is emphasised that the stopping time and the continuation set depend on the measures $\lambda$ and $\mu$ alone and not the specific choice of $f$.
Let $D$ be a Root (resp. Rost) continuation set and consider the corresponding measure $\mu = \mu^{Root}$ (resp. $\mu = \mu^{Rost}$).
Following \cite{CoKi19} we obtain for each $N \in \mathbb{N}$ a discretisation $\mu^N$ of $\mu$ such that $\mu^N \rightarrow \mu$ and moreover $\lambda^N$ and $\mu^N$ are in convex order.
Similarly a discretisation $\lambda^N$ of $\lambda$ can be found such that $\lambda^N \rightarrow \lambda$.
The authors then propose and solve a discretised version of the (OptSEP) for $\lambda^N$ and $\mu^N$.
The optimiser will again be of Root (resp. Rost) form, given as the first time a (scaled) random walk $Y^N$ leaves a Root (resp. Rost) continuation set $\hat D^N$.
Let $D^N$ denote a time-continuous and rescaled completion of the discrete continuation set $\hat D^N$.
In \cite[Chapter 5]{CoKi19} the authors then prove convergence of $D^N$ to $D$.
We note that in the more general setting considered in \cite{CoKi19} a recovery of the initial continuation set $D$ is not guaranteed. However, in the Root case this follows due to \cite{Lo70}.
An analogous uniqueness result for Rost solutions is also true, see e.g. \cite{Gr17} for a generalisation.
By convergence of the continuation sets it is easy to see that for every $T\geq 0$ we have $\mu^N_T \rightarrow \mu_T$.
As in this setting convergence of measures implies uniform convergence of potential functions, $U_{\mu^N_T} \rightarrow U_{\mu_T}$ (see \cite{Ch77} for
details), we have established convergence of the l.h.s of (3.1) to the l.h.s of (2.1).
Let $\left(W_t^{(N)}\right)_{t \geq 0}$ denote the continuous version of the rescaled random walk $Y^N$.
To avoid heavy usage of floor functions, we will assume $T \in I:=\left\{ \frac{m}{2^n}:m,n\in\mathbb{N} \right\}$.
If limits are then taken along the subsequence $\left(Y^{2^n}\right)_{n\in\mathbb{N}}$ (resp. $\left(W^{(2^n)}\right)_{n\in\mathbb{N}}$) there exists an $N_0 \in \mathbb{N}$ such that $T$ will always be a multiple of the step size $\frac{1}{2^n}$ for all $n \geq N_0$.
For arbitrary $T>0$ the results can be recovered via density arguments.
We define the following stopping times
\begin{align} \label{stptimes}
\begin{split}
\hat\tau^{N*} &= \inf\{t \in \mathbb{N} : (NT-t, Y_t^N) \not\in \hat D^N\} \wedge NT,
\\ \bar\tau^{N*} &= \inf\{t >0 : (T-t, W^{(N)}_t) \not\in D^N\} \wedge T,
\\ \tau^{*} &= \inf\{t >0 : (T-t, W_t) \not\in D\} \wedge T,
\end{split}
\end{align}
and the functions
\begin{align*}
G^T(x,t) &:= U_{\mu}(x)\mathbbm 1_{t<T} + U_{\lambda}(x)\mathbbm 1_{t=T},
\\ G^T_N (x,t) &:= U_{\mu^N}(x)\mathbbm 1_{t<T} + U_{\lambda^N}(x)\mathbbm 1_{t=T}.
\end{align*}
The rescaled results of Section 3 then read
\begin{align*}
U_{\mu_T^N}(x) &= \mathbb{E}^x \left[ G^T_N\left(Y^N_{\hat\tau^{N*}}, \frac{\hat\tau^{N*}}{N} \right) \right] \tag{3.1*}
\\ &= \sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[G^T_N \left( Y^N_{\tau}, \frac{\tau}{N} \right) \right]. \tag{3.2*}
\end{align*}
Or, as $\left(Y^N_{\hat\tau^{N*}}, \frac{\hat\tau^{N*}}{N} \right) = \left(W^{(N)}_{\bar\tau^{N*}}, \bar\tau^{N*} \right)$ we consider (3.1*) in $W^{(N)}$-terms
\begin{align*}
U_{\mu_T^N}(x) &= \mathbb{E}^x \left[G^T_N\left(W^{(N)}_{\bar\tau^{N*}}, \bar\tau^{N*}\right) \right]. \tag{3.1**}
\end{align*}
By Lemma 5.5 and 5.6 of \cite{CoKi19} we know
\(
\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right) \stackrel{P}{\rightarrow} \left(W_{\tau^{*}},\tau^{*}\right) \text{ as } N \rightarrow \infty
\).
To see convergence of (3.1**) to (2.1) we need to show
\begin{align*}
&\mathbb{E}^x \left[|G^T_N\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right) - G^T\left(W_{\tau^*},\tau^*\right)|\right]
\\ &\quad \leq \mathbb{E}^x \left[|G^T_N\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)
- G^T\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)|\right]
+ \mathbb{E}^x \left[|G^T\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)
- G^T\left(W_{\tau^*},\tau^*\right)|\right]
\stackrel{N \rightarrow \infty}{\longrightarrow} 0.
\end{align*}
Convergence of the first term is clear due to the fact that uniform convergence of the potential functions implies uniform convergence of $G^T_N$ to $G^T$.
Thus it remains to show convergence of the second term.
Note that $G^T$ is usc, so it suffices to show that
\begin{equation} \label{Gconvergence}
\mathbb{E}^x[G^T(W_{\tau^*}, \tau^*)] \leq \liminf_{N \rightarrow \infty}\, \mathbb{E}^x\left[G^T\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)\right].
\end{equation}
For this, given $\varepsilon>0$ consider the auxiliary function
\[
\tilde G^{\varepsilon}(x,t) := U_{\mu}(x)\mathbbm 1_{t \leq T - \varepsilon} + U_{\lambda}(x) \mathbbm 1_{ T - \varepsilon < t \leq T}.
\]
Then for any random variable $X$ and stopping time $\tau$ we have
\begin{equation*}
\mathbb{E}^x \left[|G^T(X,\tau) - \tilde G^{\varepsilon}(X,\tau)|\right] \leq c \cdot \P\left[ \tau \in (T-\varepsilon, T)\right].
\end{equation*}
Combining this with the fact that $\tilde G^{\varepsilon}$ is lsc and dominating $G^T$ we get
\begin{align*}
\mathbb{E}^x\left[G^T\left(W_{\tau^*}, \tau^*\right)\right]
&\leq \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty}\, \mathbb{E}^x\left[\tilde G^{\varepsilon}\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)\right]
\\ &\leq \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty}\, c \cdot \P\left[ \hat\tau^{N*} \in (T-\varepsilon, T)\right] + \liminf_{N \rightarrow \infty}\, \mathbb{E}^x\left[G^T\left(W^{(N)}_{\bar\tau^{N*}},\bar\tau^{N*}\right)\right].
\end{align*}
Thus we are left to show that
\(
\lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \P\left[ \hat\tau^{N*} \in (T-\varepsilon, T)\right] = 0.
\)
To more easily see the arguments involving specific barrier structures, we consider the following stopping times
\begin{align*}
\bar\rho^N &= \inf\{t >0 : (T-t, W^{(N)}_t) \not\in D^N\} = \inf\{t >0 : (t, W^{(N)}_t) \not\in \tilde D^N\},
\\ \rho &= \inf\{t >0 : (T-t, W_t) \not\in D\} = \inf\{t >0 : (t, W_t) \not\in \tilde D\},
\end{align*}
where $\tilde D^N$ resp. $\tilde D$ is the \emph{Rost} continuation set we obtain by reflecting $D^N$ resp. $D$ along $\left\{\frac{T}{2}\right\}\times \mathbb{R}$.
By \cite[Chapter 5]{CoKi19} we know that $\bar\rho^N \stackrel{P}{\rightarrow} \rho$.
Note that we have $\bar\rho^N\mathbbm 1_{\bar\rho^N<T} = \bar\tau^{N*}\mathbbm 1_{\bar\tau^{N*}<T}$.
For $0 < \tilde T \leq T$ consider
\begin{align*}
x_-:= \sup\{y<x: (\tilde T, y) \in \tilde D\},
\\ x_+:= \inf\{y>x: (\tilde T, y) \in \tilde D\}.
\end{align*}
Since $\tilde D$ is a Rost continuation set and $\rho$ is its Brownian hitting time, we have
\[
\P[\rho = \tilde T] = \P[W_{\tilde T} \in \{x_-,x_+\}] = 0.
\]
So, especially for any $\varepsilon > 0$ we have $\P[\rho = T - \varepsilon] = \P[\rho = T] = 0$.
Altogether we have
\begin{align*}
\lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \P\left[ \bar\tau^{N*} \in (T-\varepsilon, T)\right]
= \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \P\left[ \bar\rho^{N} \in (T-\varepsilon, T)\right]
= \lim_{\varepsilon \searrow 0} \P\left[ \rho \in (T-\varepsilon, T)\right] = 0,
\end{align*}
which concludes the proof of \eqref{Gconvergence}, thus the proof of convergence of (3.1**) to (2.1).
It only remains to show (2.2).
So let $\bar\tau$ be an optimiser of (2.2).
Lemma 5.2 in \cite{CoKi19} then gives a discretisation $\tilde \sigma^N$ of $\bar\tau$ for which $Y^N_{\tilde \sigma^N} \stackrel{a.s.}{\rightarrow} W_{\bar\tau}$ and $\frac{\tilde \sigma^N}{N} \stackrel{P}{\rightarrow} \bar\tau$.
To obtain the other inequality, for $\varepsilon \in I$ define the function
\begin{align*}
\tilde G^{\varepsilon}_{N}(x,t) &:= U_{\mu^N}(x)\mathbbm 1_{t\leq T-\varepsilon} + U_{\lambda^N}(x) \mathbbm 1_{T-\varepsilon<t\leq T},
\end{align*}
and by $\hat\tau^{N*}_{\varepsilon}$ resp. $\tau^{*}_{\varepsilon}$ consider the respective stopping times defined in \eqref{stptimes}, replacing $T$ by $T-\varepsilon$.
Then
\begin{align}
&\sup_{\tau \leq T} \mathbb{E}^x \left[ G^T(W_{\tau}, \tau) \right]
= \mathbb{E}^x \left[G^T\left(W_{\bar\tau}, \bar\tau\right) \right]
= \lim_{\varepsilon \searrow 0} \mathbb{E}^x \left[\tilde G^{\varepsilon}(W_{\bar\tau},\bar\tau)\right] \label{I+II}
\\ &\quad \leq \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \mathbb{E}^x \left[\tilde G^{\varepsilon}_N\left(Y^N_{\tilde \sigma^N}, \frac{\tilde \sigma^N}{N} \right) \right]
\leq \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[ \tilde G^{\varepsilon}_N \left(Y^N_{\tau}, \frac{\tau}{N}\right) \right] \label{III+IV}
\\ &\quad \leq \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \sup_{\frac{\tau}{N} \leq T-\varepsilon} \mathbb{E}^x \left[ G^{T-\varepsilon}_N \left(Y^N_{\tau}, \frac{\tau}{N}\right) \right]
= \lim_{\varepsilon \searrow 0} \liminf_{N \rightarrow \infty} \mathbb{E}^x \left[ G^{T-\varepsilon}_N \left(Y^N_{\hat\tau^{N*}_{\varepsilon}}, \frac{\hat\tau^{N*}_{\varepsilon}}{N}\right) \right] \label{V+VI}
\\ & \quad= \lim_{\varepsilon \searrow 0} \mathbb{E}^x \left[G^{T-\varepsilon} \left(W_{\tau^{*}_{\varepsilon}}, \tau^{*}_{\varepsilon}\right) \right]
= \lim_{\varepsilon \searrow 0} U_{\mu_{T-\varepsilon}}(x)
= U_{\mu_{T}}(x)
= \mathbb{E}^x \left[ G^T \left(W_{\tau^*}, \tau^*\right) \right]. \label{VII+VIII+IX+X}
\end{align}
The fact that $\lim_{\varepsilon \searrow 0} \P\left[ \bar\tau \in (T-\varepsilon, T)\right] = 0$ gives \eqref{I+II} and that $\tilde G^{\varepsilon}$ is l.s.c gives \eqref{III+IV}.
To see \eqref{V+VI} consider the function
\[
H^{\varepsilon}_N(x,t):= U_{\mu^N}(x)\mathbbm 1_{t < T-\varepsilon} + U_{\lambda^N}(x) \mathbbm 1_{T-\varepsilon\leq t\leq T}.
\]
Then $H^{\varepsilon}_N(x,t) \geq G^{\varepsilon}_N(x,t)$ for all $(x,t) \in \mathbb{R} \times [0,T]$ and trivially
\[
\sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[ G^{\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right]
\leq \sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[H^{\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right].
\]
Let $(Z)_{t \geq 0}$ be a martingale, then $\left(H^{\varepsilon}_N\left(Z_{t}, t\right)\right)_{t \in [T-\varepsilon,T]} = \left(U_{\lambda^N}\left(Z_t\right)\right)_{t \in [T-\varepsilon,T]}$ is a supermartingale as $U_{\lambda^N}$ is a concave function.
So for any stopping time $\tau$ we have
\begin{equation} \label{supmartingale}
\mathbb{E}^x \left[ H^{\varepsilon}_N \left(Z_{\tau \wedge (T-\varepsilon)},\tau \wedge (T-\varepsilon) \right) \right] \geq \mathbb{E}^x \left[ H^{\varepsilon}_N \left(Z_{\tau \wedge T},\tau \wedge T \right) \right].
\end{equation}
We see that no optimiser of $\sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[H^{\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right]$ will stop after time $T-\varepsilon$, as this would decrease the value of the objective function.
So we have
\begin{align*}
\sup_{\frac{\tau}{N} \leq T} \mathbb{E}^x \left[H^{\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right]
= \sup_{\frac{\tau}{N} \leq T-\varepsilon} \mathbb{E}^x \left[H^{\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right]
= \sup_{\frac{\tau}{N} \leq T-\varepsilon} \mathbb{E}^x \left[G^{T-\varepsilon}_N\left(Y^N_{\tau}, \frac{\tau}{N}\right) \right].
\end{align*}
As we know that $\hat\tau^{N*}_{\varepsilon}$ is the optimiser of this optimal stopping problem, \eqref{V+VI} follows.
Lastly, \eqref{VII+VIII+IX+X} is due to the convergence result of (3.1*) to (2.1).
To prove convergence of the Rost optimal stopping problem replace the functions $G^T$ and $G^T_N$ above by the following functions
\begin{align*}
G^T(x,t) &= G(x) := U_{\mu}(x) - U_{\lambda}(x),
\\ G^T_N (x,t) &= G_N(x) := U_{\mu^N}(x) - U_{\lambda^N}(x).
\end{align*}
We can now derive our convergence results analogous to the Root case.
\section{Perspectives} \label{conclusions}
We illustrated the elusive connection between Root and Rost's solutions to the (SEP) and optimal stopping problems.
Specialising to the simplest possible setting, this note restricts itself to the case of SSRW and Brownian motion.
In a recent article by Gassiat et.~al.~\cite{GaObZo19} the analytic connection between Root solutions to the (SEP) and solutions to optimal stopping problems was established for a much more general class of Markov processes.
This suggests that our probabilistic arguments would also hold in this generalised setting.
The extension to more general martingales should follow via analogous arguments to the extension made in Chapter~\ref{Multidimensional case} by using the appropriate potential kernel, however for non-martingales some arguments need to be replaced.
\bibliographystyle{plain}
| {
"timestamp": "2021-02-26T02:15:48",
"yymm": "2002",
"arxiv_id": "2002.12840",
"language": "en",
"url": "https://arxiv.org/abs/2002.12840",
"abstract": "Switching identities have a long history in potential theory and stochastic analysis. In recent work of Cox and Wang, a switching identity was used to connect an optimal stopping problem and the Skorokhod embedding problem (SEP). Typically switching identies of this form are derived using deep analytic connections. In this paper, we prove the switching identities using a simple probabilistic argument, which furthermore highlights a previously unexplored symmetry between the Root and Rost solutions to the SEP.",
"subjects": "Probability (math.PR)",
"title": "Switching Identities by Probabilistic Means",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846678676152,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7092019585892643
} |
https://arxiv.org/abs/1406.7651 | A simple construction for a class of $p$-groups with all of their automorphisms central | We exhibit a simple construction, based on elementary linear algebra, for a class of examples of finite $p$-groups of nilpotence class $2$ all of whose automorphisms are central. | \section{Introduction}
In June 2014, Marc van Leeuwen~\cite{MvL} inquired on Mathematics
StackExchange whether there is a group $P$ with an element $a \in P$
such that there is no automorphism of $P$ taking $a$ to its
inverse.
In our answer, we noted that an example was provided by any of the
many constructions in the literature~\cite{HeLie, JoKo, Ea, H79, CaLe,
Ca83, E-group, Mo94, Mo95} of finite $p$-groups of nilpotence class
two in which all automorphisms are central, for $p$ an odd prime. For,
if $P$ is such a group, and $a \in P \setminus Z(P)$, then an image of
$a$ under automorphisms is of the form $a z$, with $z \in Z(P)$. If
$a z = a^{-1}$, then $a^{2} \in Z(P)$, and thus $a \in Z(P)$, as $p$
is odd.
Marc van Leuween commented that ``indeed giving a concrete example is
not so easy''. This made us realize that examples of finite $p$-groups
in which all automorphisms are central, although not conceptually
difficult, usually rely on a fair amount of calculations with
generators and relations. The goal of this paper is to give a class of
examples of such groups for which calculations can be kept to a
minimum, whereas a
central role is played by linear algebra.
The examples are based on one of the cases of~\cite[Section 4]{Ca83}.
They are constructed according to the linear algebra techniques
employed in~\cite{HeLie, DaHe, Ca83, GPS}, as described in~\cite{TAT},
which we review in Section~\ref{sec:prelim}. The examples themselves are
presented in Section~\ref{sec:main}, while in Section~\ref{sec:endo}
we mention an extension to endomorphisms.
\section{Preliminaries}\label{sec:prelim}
Let $P$ be a group. Since the centre $Z(P)$ is a characteristic
subgroup of $P$, there is a natural morphism
$\Aut(P) \to \Aut(P/Z(P))$ whose kernel $\Aut_{c}(P)$ consists of the
central automorphisms of $P$, that is, those automorphisms of $P$ that
take every $a \in P$ to an element of $a Z(P)$.
We review the setup of~\cite{TAT}. Let $V$ be a vector space of
dimension $n+1$ over the field $\text{$\mathbf{F}$} = \GF(p)$, where $p$ is a
prime. Let $W = \Lambda^{2} V$ be the exterior square of $V$. If $f :
V \to W$ is a linear map, we will consider the group $G$ of
the elements of $\GL(V)$ that commute with $f$,
\begin{equation}\label{eq:G}
G = \Set {g \in \GL(V) : (v g) f = (v f) \widehat{g}\text{, for all
$v \in V$}},
\end{equation}
where $\widehat{g}$ is the automorphism of $W$ induced by $g$.
Note that we write maps on the right, so our vectors are row vectors.
Choose now a basis $v_{0}, v_{1}, \dots, v_{n}$ of $V$, and the
corresponding basis $v_{j} \wedge v_{k}$ of $W$, for $j < k$. Write
$f$ in coordinates, that is,
\begin{equation*}
v_{i} f = \sum_{j < k} a_{i,j,k} \cdot v_{j} \wedge v_{k}.
\end{equation*}
If $p$ is odd, we can construct a finite $p$-group $P$ via the
following presentation
\begin{equation}\label{eq:pres}
\begin{aligned}
P
=
\Span{x_{0}, x_{1}, \dots, x_{n}
:\
&\text{$[[x_{i}, x_{j}], x_{k}] = 1$ for all $i, j, k$, }
\\&\text{$x_{i}^{p} = \prod_{j < k} [x_{j}, x_{k}]^{a_{i,j,k}}$
for all $i$,}
\\&\text{$[x_{i}, x_{j}]^{p} = 1$ for all $i, j$}
}.
\end{aligned}
\end{equation}
Note that here the third line of relations follows from the first
two. In fact the first two lines of relations say that commutators
and $p$-th powers of
generators are central, so that we have $1 = [x_{i}^{p}, x_{j}] =
[x_{i}, x_{j}]^{p}$, as $x_{i}$ commutes with $[x_{i}, x_{j}]$.
We have that $P$ is a group of nilpotence class two and order
$\Size{P} = p^{n+1 + \binom{n+1}{2}}$, with $P' = \Phi(P) = Z(P)$ of
order $p^{\binom{n+1}{2}}$, and $P/P'$ of order $p^{n+1}$. Moreover
$P^{p}$ has order $p^{\dim(V f)}$.
If $p = 2$, we appeal to an idea of Zurek~\cite{Zurek}, and modify
\eqref{eq:pres} replacing
$p$-th powers of generators with $4$-th powers.
The presentation thus becomes
\begin{equation*}\label{eq:pres2}
\begin{aligned}
P
=
\Span{x_{0}, x_{1}, \dots, x_{n}
:\
&\text{$[[x_{i}, x_{j}], x_{k}] = 1$ for all $i, j, k$, }
\\&\text{$x_{i}^{4} = \prod_{j < k} [x_{j}, x_{k}]^{a_{i,j,k}}$ for all $i$,}
\\&\text{$[x_{i}, x_{j}]^{2} = 1$ for all $i, j$}
}.
\end{aligned}
\end{equation*}
Here we have $\Size{P} = 2^{2(n+1) +
\binom{n+1}{2}}$, $P'$ has order $2^{\binom{n+1}{2}}$, $P/P'$ has
order $2^{2(n+1)}$, $P^{4}$ has order $2^{\dim(V f)}$, and $P' \le
\Phi(P) = Z(P)$. This time, the relations $[x_{i}, x_{j}]^{2} = 1$
are necessary.
Now it is shown in~\cite[Section~3]{TAT} that
\begin{theorem}\label{theorem}
In the notation above,
\begin{equation*}
\Aut(P)/\Aut_{c}(P) \cong G.
\end{equation*}
\end{theorem}
The point of this, as explained in~\cite{TAT}, is that for an
automorphism $g$ of $P/Z(P) = P/\Phi(P)$ to be induced by an automorphism
of $P$, one needs $g$ to preserve the $p$-th (respectively, $4$-th)
power relations, that is, the linear map $f$.
\section{The examples}\label{sec:main}
We will now construct a class of linear maps $f$, as in the previous
Section, for which the group $G$
of~\eqref{eq:G} is $\Set{1}$. According to Theorem~\ref{theorem}, this
will provide examples of finite $p$-groups $P$ of nilpotence class $2$
with $\Aut(P) = \Aut_{c}(P)$.
Let $V$ be a vector space of dimension $n+1 \ge 4$ over $\text{$\mathbf{F}$} =
\GF(p)$, where $p$ is a prime. Fix a basis $v_{0}, v_{1}, \dots ,
v_{n}$ of $V$, and let
\begin{equation*}
U = \Span{v_{1}, \dots , v_{n}}.
\end{equation*}
On the exterior square $W =
\Lambda^{2} V$, consider a basis which begins with
\begin{equation*}
v_{0} \wedge v_{1}, v_{0} \wedge v_{2}, \dots , v_{0} \wedge v_{n},
\end{equation*}
and continues with the $v_{i} \wedge v_{j}$, for $1 \le i < j \le n$.
We now make our choice for $f$.
\begin{assumption}\label{assump}
Consider the linear map $f : V \to W$ which, with respect to the given
bases, has blockwise matrix
\begin{equation}\label{eq:rels}
\begin{bmatrix}
b & c\\
A & 0
\end{bmatrix},
\end{equation}
where $b$ is a $1 \times n$ vector, $c$ is a $1 \times
\dbinom{n}{2}$ vector,
$A$ is an $n \times n$ matrix, and $0$ is an $n \times \dbinom{n}{2}$ zero
matrix.
Moreover, we take
\begin{itemize}
\item $b, c \ne 0$, and
\item $A$ to be the companion matrix \cite[p.~197]{BAI2} of the
minimal polynomial
$m$ over $\text{$\mathbf{F}$}$
of a primitive element $\alpha$ of $\GF(p^{n})$.
\end{itemize}
\end{assumption}
We collect a few elementary facts about the matrix $A$.
\begin{lemma}\label{lemma:A}\
\begin{enumerate}
\item\label{item:roots} The roots of $m$, i.e.~the eigenvalues of $A$, are
\begin{equation*}
\alpha,
\alpha^{p}, \dots , \alpha^{p^{n-1}}.
\end{equation*}
\item\label{item:order} $A$ has multiplicative order $p^{n}-1$.
\item\label{item:powers} $\text{$\mathbf{F}$}[A]$ is a field of order $p^{n}$, and
$\text{$\mathbf{F}$}[A] = \Set{0} \cup \Set{A^{i} : 0 \le i < p^{n}-1}$.
\item\label{item:simple} $\text{$\mathbf{F}$}^{n}$ is a one-dimensional
$\text{$\mathbf{F}$}[A]$-vector space.
\item\label{item:cent} The centralizer
\begin{equation*}
C_{\End(\text{$\mathbf{F}$}^{n})}(A)
\end{equation*}
of $A$ in $\End(\text{$\mathbf{F}$}^{n})$ is $\text{$\mathbf{F}$}[A]$.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{item:roots} follows from the fact that $\alpha$ is a root of
$m$, and $m$ is irreducible in $\text{$\mathbf{F}$}[x]$, of degree $n$.
\eqref{item:order} follows immediately from the previous point.
\eqref{item:powers} follows from $\text{$\mathbf{F}$}[A] \cong \text{$\mathbf{F}$}[x]/(m)$, and
\eqref{item:order}.
\eqref{item:simple} follows from the fact that $A$ is a companion matrix, and thus $\text{$\mathbf{F}$}^{n}$ is a cyclic $\text{$\mathbf{F}$}[A]$-module.
\eqref{item:cent} now follows from the previous point, as the given
centralizer is the ring of endomorphisms of the $\text{$\mathbf{F}$}[A]$-vector space
$\text{$\mathbf{F}$}^{n}$.
\end{proof}
We now collect a few facts about $f$ and the group $G$ of~\eqref{eq:G}.
\begin{lemma}\label{lemma:omnibus}\
\begin{enumerate}
\item \label{lemma:inj} $f$ is injective.
\item \label{Uf}
$
U f = v_{0} \wedge V = v_{0} \wedge U,
$
and this is a subspace of $W$ of dimension $n$.
\item \label{lemma:u}
If $u \in U$ satisfies $u \wedge V \le V f$, then $u =
0$.
\item \label{lemma:v0}$
\Span{v_{0}} = \Set{x \in V : x \wedge V \le V f}.
$
\item \label{lemma:v0inv} $\Span{v_{0}}$ is left invariant by $G$.
\item \label{lemma:U}$
U = \Set{ x \in V : x f \in v_{0} \wedge V }.
$
\item \label{lemma:Uinv} $U$ is left invariant by $G$.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{lemma:inj} follows from Assumption~\ref{assump}, since $A$ is
invertible, and $c \ne 0$
in~\eqref{eq:rels}.
The formula of~\eqref{Uf} now follows from the shape of the matrix for
$f$ in Assumption~\ref{assump}.
To prove~\eqref{lemma:u}, let $u = c_{1} v_{1} + \dots + c_{n} v_{n}$
satisfy $u \wedge V \le V f$. We will show that $c_{1}
= 0$, but a similar argument yields that all $c_{i}$ have to be
zero. Let us look at the coordinates of $u \wedge v_{2}$ and $u \wedge
v_{3}$ with respect
to $v_{1} \wedge v_{2}$ and $v_{1} \wedge v_{3}$, which yield the $2
\times 2$ matrix
\begin{equation*}
\begin{bmatrix}
c_{1} & 0\\
0 & c_{1}
\end{bmatrix}.
\end{equation*}
By~\eqref{lemma:inj} and~\eqref{Uf}, the dimension of $V f / U f = V
f/(v_{0} \wedge V)$ is
$1$. Thus this matrix must have rank at most $1$, so that
$c_{1}^{2} = 0$.
To prove~\eqref{lemma:v0}, let $0 \ne x \in V$ be such that $x \wedge
V \le V f$. By~\eqref{lemma:u}, $x = c v_{0} + u$, for some $c \ne
0$, and $u \in U$. But then by~\eqref{Uf} $u \wedge V \le V f$, so
that $u = 0$ again by~\eqref{lemma:u}.
\eqref{lemma:v0inv} follows from the previous point.
\eqref{lemma:U} follows from~\eqref{lemma:inj} and~\eqref{Uf}, and
implies~\eqref{lemma:Uinv}, because of~\eqref{lemma:v0inv}.
\end{proof}
(Note that the argument in the proof of~\eqref{lemma:u} fails when $n
= 2$, see~\cite{DaHe}, and this is the reason we have taken $n + 1 \ge
4$.)
We can now state our main result.
\begin{theorem}
With $f$ as in Assumption~\ref{assump}, the group $G$ of~\eqref{eq:G}
is $\Set{1}$.
\end {theorem}
\begin{proof}
\eqref{lemma:v0inv} and~\eqref{lemma:Uinv} of
Lemma~\ref{lemma:omnibus} allow us to write an element
$g \in G$ in matrix form, with
respect to the given basis of $V$, as
\begin{equation*}
g =
\begin{bmatrix}
\gamma & 0\\
0 & \Delta\\
\end{bmatrix}
\end{equation*}
where $\gamma \in \text{$\mathbf{F}$}^{\star}$, and $\Delta \in \GL(n,\text{$\mathbf{F}$})$. By the
definition~\eqref{eq:G}
of $G$, and Assumption~\ref{assump}, we have
\begin{equation}\label{eq:theequality}
\begin{bmatrix}
\gamma & 0\\
0 & \Delta\\
\end{bmatrix}
\begin{bmatrix}
b & c\\
A & 0
\end{bmatrix}
=
\begin{bmatrix}
b & c\\
A & 0
\end{bmatrix}
\begin{bmatrix}
\gamma \Delta & 0\\
0 & \widehat{\Delta}
\end{bmatrix},
\end{equation}
where $\widehat{\Delta}$ is the matrix induced by $\Delta$ on $U
\wedge U$. We will only need consider the following two consequences of~\ref{eq:theequality}:
\begin{equation}\label{eq:a}
\Delta A \Delta^{-1} = \gamma A,
\end{equation}
and
\begin{equation}\label{eq:b}
b \Delta = b.
\end{equation}
To deal with~\eqref{eq:a}, we could appeal to~\cite{Wall}, but prefer
to give a simple direct argument.
As noted in Lemma~\ref{lemma:A}.\eqref{item:roots}, the eigenvalues of $A$ are
\begin{equation}\label{eq:eigenA}
\alpha, \alpha^{p}, \dots, \alpha^{p^{n-1}},
\end{equation}
with $\alpha$ a primitive element, so that those of
$\gamma A$ are
\begin{equation*}
\gamma \alpha, \gamma \alpha^{p}, \dots, \gamma \alpha^{p^{n-1}}.
\end{equation*}
By~\eqref{eq:a} we have $\gamma \alpha = \alpha^{p^{t}}$
for some $t$. If $t > 0$, then
\begin{equation*}
\alpha^{p^{t} - 1} = \gamma \in \GF(p)^{\star},
\end{equation*}
with $p^{t} - 1 > 0$,
so that the order
\begin{equation*}
\frac{p^{n}-1}{p-1} = 1 + p + \dots + p^{n-1}
\end{equation*}
of $\alpha$ in
$\GF(p^{n})^{\star}/\GF(p)^{\star}$ divides $p^{t} - 1 < p^{n-1}$, a
contradiction. Thus $t = 0$ and $\gamma = 1$.
It follows that $\Delta$ is in the centralizer of $A$ in $\GL(n, \text{$\mathbf{F}$})$,
and thus, according to Lemma~\ref{lemma:A}, $\Delta$ is
a power of $A$.
But once more since the eigenvalues of $A$ are as in~\eqref{eq:eigenA}, with
$\alpha$ a primitive element, the only
power of $A$ to have an
eigenvalue $1$ is $1$. Since we have~\eqref{eq:b}, with $b \ne 0$ by
Assumption~\ref{assump}, we obtain that $\Delta = 1$ and thus $G = \{ 1
\}$ as claimed.
\end{proof}
\section{Endomorphisms}\label{sec:endo}
The arguments of the previous Section can be slightly extended to show
that the set
\begin{equation*}
G = \Set {g \in \End(V) : (v g) f = (v f) \hat{g}\text{, for all $v
\in V$}}
\end{equation*}
of the endomorphisms of $V$ that commute with $f$
consists of $0$ and $1$. Now in our examples $P$ the centre
$Z(P)$ is fully invariant, as it equals $\Phi(P)$,
and thus an immediate extension of
Theorem~\ref{theorem} yields that an endomorphism of
$P$ either maps $P$
into $Z(P)$, or is a central automorphism,
so that $P$ is an E-group~\cite{Fau, Mal77, Mal80,
E-group, CFdG}, that is, a
group in which each element commutes with all of its images under
endomorphisms.
To prove this, we proceed as in the previous Section, except that we make no
assumptions on $\gamma$ and $\Delta$. We have
from~\eqref{eq:theequality}
\begin{equation}\label{eq:aa}
\Delta A = \gamma A \Delta.
\end{equation}
If $\gamma = 0$, then $\Delta = 0$. If $\gamma \ne 0$, it follows
from~\ref{eq:aa} that $\ker(\Delta)$ is $A$-invariant, that is, a
$\text{$\mathbf{F}$}[A]$-vector subspace of the one-dimensional $\text{$\mathbf{F}$}[A]$-vector space
$\text{$\mathbf{F}$}^{n}$. Therefore we have either $\ker(\Delta) =
\Set{0}$, that is, $\Delta$ is invertible, and we proceed as above, or
$\ker(\Delta) = \text{$\mathbf{F}$}^{n}$, that is, $\Delta = 0$. Now~\eqref{eq:theequality}
yields $\gamma b = 0$, a contradiction to $\gamma \ne 0$ and $b \ne
0$.
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"timestamp": "2014-07-09T02:09:38",
"yymm": "1406",
"arxiv_id": "1406.7651",
"language": "en",
"url": "https://arxiv.org/abs/1406.7651",
"abstract": "We exhibit a simple construction, based on elementary linear algebra, for a class of examples of finite $p$-groups of nilpotence class $2$ all of whose automorphisms are central.",
"subjects": "Group Theory (math.GR)",
"title": "A simple construction for a class of $p$-groups with all of their automorphisms central",
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https://arxiv.org/abs/0804.4042 | Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes | A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes, and random linear codes. The bound asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. By generalizing the idea of the lower bound, a lower bound on the number of uncorrectable errors for weights larger than half the minimum distance is also obtained, but the generalized lower bound is weak for large weights. The monotone error structure and its related notion larger half and trial set, which are introduced by Helleseth, Kløve, and Levenshtein, are mainly used to derive the bounds. | \section{Introduction}
For a binary linear code, correctable errors we consider here are binary errors correctable by the minimum distance decoding,
which performs a maximum likelihood decoding for binary symmetric channels.
Syndrome decoding is one of the minimum distance decoding.
In syndrome decoding, the correctable errors are coset leaders of the code.
When there are two or more minimum weight vectors in a coset,
we have choices of the coset leader.
If the lexicographically smallest minimum weight vector is taken as the coset leader,
then both the correctable errors and the uncorrectable errors have a monotone structure.
That is, when $\bm{y}$ covers $\bm{x}$ (the support of $\bm{y}$ contains that of $\bm{x}$),
if $\bm{y}$ is correctable, then $\bm{x}$ is also correctable, and
if $\bm{x}$ is uncorrectable, then $\bm{y}$ is also uncorrectable~{\cite{peterson}}.
Using this monotone structure, Z\'emor showed that the residual error probability after maximum likelihood decoding
displays a threshold behavior~\cite{zemor93}.
When uncorrectable (and correctable) errors have the monotone structure, they are characterized by the minimal uncorrectable
(and maximal correctable) errors.
Larger halves of codewords are introduced by Helleseth, Kl\o ve, and Levenshtein~\cite{helleseth05}
to describe the minimal uncorrectable errors.
They also introduced a trial set for a code. It is a set of codewords whose larger halves contain all minimal uncorrectable errors.
Trial sets can be used for a maximum likelihood decoding and for giving an upper bound on the number of uncorrectable errors.
The set of all codewords except for the all-zero codeword and the set of minimal codewords~\cite{ashikhmin98} in the code
are examples of trial sets.
In this paper, we study bounds on the number of correctable/uncorrectable
errors.
There were several works about them.
For the first-order Reed-Muller codes, the exact numbers of correctable errors of weight half the minimum distance and
half the minimum distance plus one were determined~\cite{wu98, yasunaga07}.
For general linear codes, some upper bounds on the number of uncorrectable errors were presented
in~\cite{helleseth05, helleseth97, poltyrev94}.
In this work, we consider lower bounds on the number of uncorrectable errors based on the idea of~\cite{helleseth05}
for general linear codes.
We derive a lower bound on the number of uncorrectable errors of weight half the minimum distance for codes satisfying some condition.
The bound is given in terms of the numbers of codewords with weights $d$ and $d+1$ in a trial set for odd $d$,
where $d$ is the minimum distance of the code.
For the case of even $d$, the bound is given by the number of codewords with weight $d$ in a trial set.
Since the set of all codewords except the all-zero vector is a trial set,
the bound can be evaluated by the numbers of codewords with weights $d$ and $d+1$.
The condition is not too restrictive, and some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes,
and random linear codes satisfy the condition.
For Reed-Muller codes and random linear codes, the lower bound asymptotically coincides with the upper bound of~\cite[Corollary~7]{helleseth05}.
The lower bound can be generalized to a lower bound on the size of the set of larger halves of a trial set,
which is a lower bound on the number of uncorrectable errors.
In the next section, we review some definitions and properties of the monotone error structure, larger halves, and trial sets.
In Section~\ref{sec:halfmd}, a lower bound on the number of uncorrectable errors of weight half the minimum distance is
given for the codes satisfying some condition.
The bound presented in Section~\ref{sec:halfmd} is generalized in Section~\ref{sec:generalize}.
\section{Larger halves and trial sets}
We introduce definitions and properties of larger halves and trial sets.
Let $\mathbb{F}^n$ be the set of all binary vectors of length $n$.
Let $C \subseteq \mathbb{F}^n$ be a binary linear code of length $n$, dimension $k$, and minimum distance $d$.
Then $\mathbb{F}^n$ is partitioned into $2^{n-k}$ cosets $C_1, C_2, \ldots, C_{2^{n-k}}$;
$\mathbb{F}^n = \bigcup_{i=1}^{2^{n-k}}C_i$ and $C_i \cap C_j = \emptyset$ for $i \neq j$,
where each $C_i=\{\bm{v}_i+\bm{c} : \bm{c} \in C\}$ with $\bm{v}_i \in \mathbb{F}^n$.
The vector $\bm{v}_i$ is called a coset leader of the coset $C_i$ if the weight of $\bm{v}_i$ is smallest in $C_i$.
Let $H$ be a parity check matrix of $C$.
The syndrome of a vector $\bm{v} \in \mathbb{F}^n$ is defined as $\bm{v}H^T$.
All vectors having the same syndrome are in the same coset.
Syndrome decoding associates an error vector to each syndrome.
The syndrome decoder presumes that the error vector added to the received vector $\bm{y}$
is the coset leader of the coset which contains $\bm{y}$.
The syndrome decoding function $D : \mathbb{F}^n \rightarrow C$ is defined as
\begin{equation*}
D(\bm{y}) = \bm{y} + \bm{v}_i, \ \ \text{if} \ \bm{y} \in C_i.
\end{equation*}
In this paper, we take as $\bm{v}_i$ the minimum element in $C_i$ with respect to the following total ordering $\preceq$:
\begin{equation}
\bm{x} \preceq \bm{y} \ \ \text{if and only if} \ \
\left\{
\begin{array}{l}
w(\bm{x}) < w(\bm{y}), \ \ \text{or} \\
w(\bm{x}) = w(\bm{y}) \ \text{and} \ v(\bm{x}) \leq v(\bm{y}),
\end{array}
\right.\nonumber
\end{equation}
where $w(\bm{x})$ denotes the Hamming weight of a vector $\bm{x}=(x_1, x_2, \ldots, x_n)$ and
$v(\bm{x})$ denotes the numerical value of $\bm{x}$:
\begin{equation*}
v(\bm{x}) = \sum_{i=1}^{n} x_i 2^{n-i}.
\end{equation*}
We write $\bm{x} \prec \bm{y}$ if $\bm{x} \preceq \bm{y}$ and $\bm{x} \neq \bm{y}$.
Let $E^0(C)$ be the set of all coset leaders of $C$.
In the syndrome decoding, $E^0(C)$ is the set of correctable errors and
$E^1(C) = \mathbb{F}^n \setminus E^0(C)$ is the set of uncorrectable errors.
Since we take the minimum element with respect to $\preceq$ in each coset as its coset leader,
both $E^0(C)$ and $E^1(C)$ have the following well-known monotone structure (see~{\cite[Theorem 3.11]{peterson}}).
Let $\subseteq$ denote a partial ordering called ``covering'' such that
\begin{equation}
\bm{x} \subseteq \bm{y} \ \text{ if and only if} \ S(\bm{x}) \subseteq S(\bm{y}),\nonumber
\end{equation}
where
\begin{equation*}
S(\bm{v})=\{ i : v_i \neq 0\}
\end{equation*}
is the support of $\bm{v}=(v_1,v_2,\ldots,v_n)$.
Consider $\bm{x}$ and $\bm{y}$ with $\bm{x} \subseteq \bm{y}$.
If $\bm{y}$ is a correctable error, then $\bm{x}$ is also correctable.
If $\bm{x}$ is uncorrectable, then $\bm{y}$ is also uncorrectable.
Using this structure, Z\'emor showed that the residual error probability after maximum likelihood decoding
displays a threshold behavior~\cite{zemor93}.
Helleseth, Kl\o ve, and Levenshtein~\cite{helleseth05} studied this structure and introduced {\em larger halves\/} and {\em trial sets\/}.
Since the set of uncorrectable errors $E^1(C)$ has a monotone structure,
$E^1(C)$ can be characterized by {\em minimal uncorrectable errors\/} in $E^1(C)$.
An uncorrectable error $\bm{y} \in E^1(C)$ is minimal if there exists no $\bm{x}$ such that
$\bm{x} \subset \bm{y}$ in $E^1(C)$.
We denote by $M^1(C)$ the set of all minimal uncorrectable errors in $C$.
Larger halves of a codeword $\bm{c} \in C$ are introduced to characterize the minimal uncorrectable errors,
and are defined as minimal vectors $\bm{v}$ with respect to covering such that $\bm{v} + \bm{c} \prec \bm{v}$.
The following condition is a necessary and sufficient condition that $\bm{v} \in \mathbb{F}^n$
is a larger half of $\bm{c} \in C$:
\begin{gather}
\bm{v} \subseteq \bm{c},\label{eq:lhcond1}\\
w(\bm{c}) \leq 2w(\bm{v}) \leq w(\bm{c})+2,\label{eq:lhcond2}\\
l(\bm{v}) \begin{cases}
= l(\bm{c}) & \text{if} \ \ 2w(\bm{v}) = w(\bm{c}),\\
> l(\bm{c}) & \text{if} \ \ 2w(\bm{v}) = w(\bm{c})+2,
\end{cases}\label{eq:lhcond3}
\end{gather}
where
\begin{equation}
l(\bm{x}) = \min S(\bm{x})
\end{equation}
that is, $l(\bm{x})$ is the leftmost non-zero coordinate in the vector $\bm{x}$.
The condition~(\ref{eq:lhcond3}) is not applied if $w(\bm{c})$ is odd.
The proof of equivalence between the definition and the above condition is found in the proof of
\cite[Theorem~1]{helleseth05}.
Let $LH(\bm{c})$ be the set of all larger halves of $\bm{c} \in C$.
For a set $U \subseteq C \setminus \{ \bm{0} \}$, define
\begin{equation*}
LH(U) = \bigcup_{\bm{c} \in U} LH(\bm{c}).
\end{equation*}
When the weight of a codeword $\bm{c}$ is odd, the weight of the vectors in $LH(\bm{c})$ is $(w(\bm{c})+1)/2$.
When the weight of $\bm{c}$ is even, $LH(\bm{c})$ consists of vectors of weights $w(\bm{c})/2$ and $w(\bm{c})/2+1$.
For convenience, let $LH^-(\bm{c})$ and $LH^+(\bm{c})$ denote the sets of larger halves of $\bm{c}$ of weight
$w(\bm{c})/2$ and $w(\bm{c})/2+1$, respectively.
Then $LH(\bm{c}) = LH^-(\bm{c}) \cup LH^+(\bm{c})$.
Also let $LH^-(U) = \bigcup_{\bm{c} \in U}LH^-(\bm{c})$ and $LH^+(U) = \bigcup_{ \bm{c} \in U} LH^+(\bm{c})$
for a subset $U$ of even-weight subcode.
A trial set $T$ for a code $C$ is defined as follows:
\begin{equation*}
T \subseteq C \setminus \{\bm{0}\} \text{ is a trial set for } C \ \text{if} \ M^1(C) \subseteq LH(T)
\end{equation*}
Since every larger half is an uncorrectable error, we have the relation
\begin{equation}
M^1(C) \subseteq LH(T) \subseteq E^1(C).\label{eq:mlhe}
\end{equation}
In the rest of paper, for $\bm{u}, \bm{v} \in \mathbb{F}^n$, we write $\bm{u} \cap \bm{v}$ as the vector in $\mathbb{F}^n$
whose support is ${\rm S}(\bm{u}) \cap{\rm S}(\bm{v})$.
For a set $U \subseteq \mathbb{F}^n$, define
\begin{equation*}
A_i(U)=\{ \bm{v} \in U : w(\bm{v})=i \}.
\end{equation*}
Also we define $M^1_i(C) = A_i(M^1(C))$ and $LH_i(U) = A_i(LH(U))$ for $U \subseteq C \setminus \{ \bm{0} \}$.
\section{A Bound on the Number of Uncorrectable Errors of Weight Half the Minimum Distance}\label{sec:halfmd}
In this section, we derive a lower bound on $|E^1_{\lceil d/2 \rceil}(C)|$.
The bound is given by the number of codewords with weights $d$ and $d+1$ in a trial set.
Since $C \setminus \{ \bm{0} \}$ is a trial set for $C$, the lower bound can be evaluated by the number of codewords of weights $d$ and $d+1$ in $C$.
Since the weight $\lceil d/2 \rceil$ is the minimum weight in $E^1(C)$, every vector in $E^1_{\lceil d/2 \rceil}(C)$ is not covered
by other uncorrectable errors, and thus $M^1_{\lceil d/2 \rceil}(C) = E^1_{\lceil d/2 \rceil}(C)$.
From~(\ref{eq:mlhe}), we have
\begin{equation*}
M^1_{\lceil d/2 \rceil}(C) = LH_{\lceil d/2 \rceil}(T) = E^1_{\lceil d/2 \rceil}(C),
\end{equation*}
where $T$ is a trial set for $C$.
We will give a lower bound on $|E^1_{\lceil d/2 \rceil}(C)|$ by giving a lower bound on $|LH_{\lceil d/2 \rceil}(T)|$.
\subsection{Odd Minimum Weight Case}
When $d$ is odd, $LH_{\lceil d/2 \rceil}(T) = LH(A_d(T)) \cup LH^-(A_{d+1}(T))$.
The next lemma implies that the number of common larger halves among $LH(A_d(T))$ and $LH^-(A_{d+1}(T))$ is small.
\medskip
\begin{lemma}\label{lem:oddcommon}
Let $C$ be a linear code with odd minimum distance $d$.
For every $\bm{c}_1, \bm{c}_1' \in A_d(C)$ and $\bm{c}_2, \bm{c}_2' \in A_{d+1}(C)$, it holds that
$|LH(\bm{c}_1) \cap LH(\bm{c}_1')| = 0$, $|LH(\bm{c}_1) \cap LH^-(\bm{c}_2)| \leq 1$, and $|LH^-(\bm{c}_2) \cap LH^-(\bm{c}_2')| \leq 1$.
\end{lemma}
\medskip
\begin{proof}
For $\bm{c}, \bm{c}' \in C \setminus \{ \bm{0} \}$,
every vector $\bm{v} \in LH(\bm{c}) \cap LH(\bm{c}')$ has the property that $\bm{v} \subseteq \bm{c} \cap \bm{c}'$.
Since every vector in $LH(\bm{c}_1), LH(\bm{c}_1'), LH^-(\bm{c}_2), LH^-(\bm{c}_2')$ has weight $(d+1)/2$,
it is enough to show that $w(\bm{c}_1 \cap \bm{c}_1') < (d+1)/2$, $w(\bm{c}_1 \cap \bm{c}_2) \leq (d+1)/2$,
$w(\bm{c}_2 \cap \bm{c}_2') \leq (d+1)/2$.
We can prove them by using $w(\bm{c} \cap \bm{c}') = (w(\bm{c})+w(\bm{c}')-w(\bm{c}+\bm{c}'))/2$ and $w(\bm{c}+\bm{c}') \geq d$.
\end{proof}
\medskip
From the previous lemma, we give a lower bound on the number of uncorrectable errors of weight half the minimum distance.
The corresponding upper bound is given unconditionally by~\cite[Corollary~7]{helleseth05}.
\medskip
\begin{theorem}\label{th:oddlower}
Let $C$ be a linear code with odd minimum distance $d$ and $T$ be a trial set for $C$.
If
\begin{equation}
\binom{d}{\frac{d+1}{2}} > |A_d(T)|+|A_{d+1}(T)|-1\label{eq:oddcond}
\end{equation}
holds, then
\begin{multline*}
\binom{d}{\frac{d+1}{2}}(|A_d(T)|+|A_{d+1}(T)|)\\ - (2|A_d(T)|+|A_{d+1}(T)|-1)|A_{d+1}(T)|\\
\leq |E^1_{\frac{d+1}{2}}(C)|
\leq \binom{d}{\frac{d+1}{2}}(|A_d(T)|+|A_{d+1}(T)|).
\end{multline*}
\end{theorem}
\medskip
\begin{proof}
From Lemma~\ref{lem:oddcommon}, a codeword $\bm{c} \in A_d(T)$ has at most one common larger half for every $\bm{c}' \in A_{d+1}(T)$
and does not have common larger halves for any $\bm{c}' \in A_{d}(T) \setminus \{ \bm{c} \}$.
Thus at least $|LH(\bm{c})| - |A_{d+1}(T)|$ vectors in $LH(\bm{c})$ does not have common larger halves.
Also, a codeword $\bm{c} \in A_{d+1}(T)$ has at most one common larger half for every $\bm{c}' \in A_{d}(T) \cup \{ A_{d+1}(T) \setminus \{\bm{c}\}\}$,
at least $|LH^-(\bm{c})| - |A_d(T)| - |A_{d+1}(T)| +1$ vectors in $LH^-(\bm{c})$ does not have common larger halves.
For every $\bm{c}_1 \in A_d(T)$ and $\bm{c}_2 \in A_{d+1}(T)$, we have $|LH(\bm{c}_1)|=|LH^-(\bm{c}_2)|=\binom{d}{(d+1)/2}.$
Therefore we have the lower bound $(\binom{d}{(d+1)/2} - |A_{d+1}(T)|)|A_d(T)|
+ (\binom{d}{(d+1)/2} - |A_d(T)| - |A_{d+1}(T)| + 1)|A_{d+1}(T)| \leq |LH_{(d+1)/2}(T)| = |E^1_{(d+1)/2}(C)|$.
The upper bound is obtained from the inequality $|LH_{(d+1)/2}(T)| = |LH(A_d(T)) \cup LH^-(A_{d+1}(T))| \leq |LH(A_d(T))| + |LH^-(A_{d+1}(T))|
\leq \binom{d}{(d+1)/2}|A_d(T)| + \binom{d}{(d+1)/2}|A_{d+1}(T)|$.
\end{proof}
\medskip
The difference between the upper and lower bounds is $(2|A_d(C)|+|A_{d+1}(C)|-1)|A_{d+1}(C)|$.
If the fraction $|A_{d+1}(C)|/\binom{d}{(d+1)/2}$ tends to zero as the code length becomes large,
the lower bound asymptotically coincides with the upper one.
\subsection{Even Minimum Weight Case}
When $d$ is even, $LH_{\lceil d/2 \rceil}(T) = LH^-(A_d(T))$.
The next lemma implies that the number of common larger halves among $LH^-(A_d(T))$ is small.
\medskip
\begin{lemma}\label{lem:evencommon}
Let $C$ be a linear code with even minimum distance $d$.
For every $\bm{c}_1, \bm{c}_2 \in A_d(C)$, it holds that $|LH^-(\bm{c}_1) \cap LH^-(\bm{c}_2)| \leq 1$.
\end{lemma}
\medskip
\begin{proof}
For contradiction, suppose that there exist two distinct vectors in $LH^-(\bm{c}_1) \cap LH^-(\bm{c}_2)$.
Then it holds that $w(\bm{c}_1 \cap \bm{c}_2) \geq d/2+1$, but this leads to the contradiction that
$w(\bm{c}_1 + \bm{c}_2) = w(\bm{c}_1) + w(\bm{c}_2) - 2w(\bm{c}_1 \cap \bm{c}_2) \leq d-1$.
\end{proof}
\medskip
\begin{theorem}\label{th:evenlower}
Let $C$ be a linear code with even minimum distance $d$.
If
\begin{equation}
\frac{1}{2}\binom{d}{\frac{d}{2}} > |A_d(T)|-1\label{eq:evencond}
\end{equation}
holds, then
\begin{multline*}
\frac{1}{2}\binom{d}{\frac{d}{2}}|A_d(T)| - (|A_d(T)|-1) |A_{d}(T)|\\
\leq |E^1_{\frac{d}{2}}(C)|
\leq \frac{1}{2}\binom{d}{\frac{d}{2}}|A_d(T)|.
\end{multline*}
\end{theorem}
\medskip
\begin{proof}
From Lemma~\ref{lem:evencommon}, a codeword $\bm{c} \in A_d(T)$ has at most one common larger half for every
$\bm{c}' \in A_{d}(T) \setminus \{ \bm{c} \}$.
Thus at least $|LH^-(\bm{c})| - |A_{d}(T)| + 1$ vectors in $LH^-(\bm{c})$ does not have common larger halves.
Thus we have the lower bound $(\binom{d}{d/2}/2-|A_d(T)|+1)|A_d(T)| \leq |LH^-(A_d(T))| = |E^1_{d/2}(C)|$.
The upper bound is obtained from the inequality $|E^1_{d/2}(C)| = |LH^-(A_d(C))| \leq \binom{d}{d/2}|A_d(C)|/2$.
\end{proof}
\medskip
The difference between the upper and lower bounds is upper bounded by $|A_{d}(C)|^2$.
If the fraction $|A_{d}(C)|/\binom{d}{d/2}$ tends to zero as the code length becomes large,
the lower bound asymptotically coincides with the upper one.
When we take $C \setminus \{ \bm{0} \}$ as a trial set $T$, the condition for a lower bound can be weaker and
the lower bound can be improved.
\medskip
\begin{theorem}
Let $C$ be a linear code with even minimum distance $d$.
If
\begin{equation*}
\frac{1}{2}\binom{d}{\frac{d}{2}} > \left\lceil \frac{|A_d(C)|-1}{2} \right\rceil
\end{equation*}
holds, then
\begin{multline*}
\frac{1}{2}\binom{d}{\frac{d}{2}}|A_d(C)| - \left\lceil \frac{|A_d(C)|-1}{2} \right\rceil |A_{d}(C)|
\leq |E^1_{\frac{d}{2}}(C)|.
\end{multline*}
\end{theorem}
\medskip
\begin{proof}
From Lemma~\ref{lem:evencommon}, a codeword $\bm{c} \in A_d(C)$ has at most one common larger half for
$\bm{c}' \in A_{d}(C) \setminus \{ \bm{c} \}$.
If $\bm{c}$ and $\bm{c}'$ have the common larger half $\bm{v} \in LH^-(\bm{c}) \cap LH^-(\bm{c}')$, then $\bm{v}$ is
represented as $\bm{v} = \bm{c} \cap \bm{c}'$ and it holds that $l(\bm{c}) = l(\bm{c}')$.
Then the other codeword $\bm{c}+\bm{c}' \in A_d(C)$ does not have common larger halves with $\bm{c}$,
since $l(\bm{c}+\bm{c}') \neq l(\bm{c})$.
Therefore at least $|LH^-(\bm{c})| - \lceil(|A_{d}(C)|-1)/2\rceil$ vectors in $LH^-(\bm{c})$ does not have common larger halves.
Thus we have the lower bound $(\binom{d}{d/2}/2-\lceil (|A_d(C)|-1)/2\rceil)|A_d(C)|$.
\end{proof}
\medskip
In what follows, we see that some BCH codes, Reed-Muller codes, and random linear codes satisfy
the conditions~(\ref{eq:oddcond}) or~(\ref{eq:evencond}).
For an $(n, k)$ linear code $C$, which has code length $n$ and dimension $k$, we choose $C \setminus \{ \bm{0} \}$ as a trial set for $C$.
\subsubsection{Primitive BCH codes}
By using the weight distribution~\cite{desaki97},
we can verify that the $(n, k)$ primitive BCH codes satisfy the condition~(\ref{eq:oddcond})
for $n =127, k \leq 64$ and $n = 63, k \leq 24$.
\subsubsection{Extended Primitive BCH codes}
By using the weight distribution~\cite{desaki97},
we can verify that the $(n, k)$ extended primitive BCH codes satisfy the condition~(\ref{eq:evencond})
for $n =128, k \leq 64$ and $n = 64, k \leq 24$.
\subsubsection{Reed-Muller codes}
For the $r$-th order Reed-Muller code of length $2^m$, the minimum distance is $2^{m-r}$ and
the number of minimum weight codewords $|A_{2^{m-r}}({\rm RM}_{m,r})|$ is presented in Theorem~9 of~\cite[Chapter 13]{macwilliams},
which is upper bounded by $(2^{m+1}-2)^r$.
Then, for a fixed $r$, the condition~(\ref{eq:evencond}) is satisfied except for small $m$.
Table~\ref{tb:rmtrial} shows which parameters meet the condition~(\ref{eq:evencond}).
\begin{table}[t]
\caption{The $r$-th order Reed-Muller code of length $2^m$ satisfying~(\ref{eq:evencond}).}
\label{tb:rmtrial}
\begin{center}
\begin{tabular}{ll}\hline
$r$ & \multicolumn{1}{c}{$m$} \\\hline
1 & $\geq 4$\\
2 & $\geq 6$\\
3 & $\geq 8$\\
4 & $\geq 10$\\
5 & $\geq 11$\\
6 & $\geq 13$\\\hline
\end{tabular}
\end{center}
\end{table}
The fraction $|A_{d}(C)|/\binom{d}{d/2}$ is upper bounded by
\begin{equation*}
\frac{|A_{d}(C)|}{\binom{d}{d/2}} \leq \frac{(2^{m+1}-2)^r}{2^{2^{m-r}}} \leq 2^{(m+1)r - 2^{m-r}}.
\end{equation*}
Thus for a fixed $r$ the fraction tends to zero as $m$ becomes large.
This means the upper and lower bounds in Theorem~\ref{th:evenlower} asymptotically coincide.
\subsubsection{Random Linear Codes}
A random linear code is a code whose generator matrix has equiprobable entries.
That is, first we set a parameter $(n, k)$, and then we choose a generator matrix from all the $2^{nk}$ possible generator matrices
with probability $2^{-nk}$.
It is known that with high probability the minimum distance equals to $n\delta_{\rm GV}$, where $1-H(\delta_{\rm GV})=k/n$ and
$H(x)$ is the binary entropy function of $x$~\cite{gilbert52, varshamov57}.
Also it is known that the weight distribution equals the binomial distribution.
Then, $|A_d(C)| \approx (2^k-1)\binom{n}{d}2^{-n}
\approx \binom{n}{n\delta_{\rm GV}}2^{k-n}
\approx 2^{n(H(\delta_{\rm GV})+k/n-1)} \approx 1,$
where we use the approximation $\binom{n}{n\lambda} \approx 2^{H(\lambda)}$, and
$|A_{d+1}(C)| \approx (2^k-1)\binom{n}{d+1}2^{-n}
\approx \binom{n}{n\delta_{\rm GV}}2^{k-n}(n-d)/(d+1)
\approx 2^{n(H(\delta_{\rm GV})+k/n-1)} \approx 1.
$
Since $\binom{d}{d/2}\approx \sqrt{2/\pi d}2^d \approx 2^{n\delta}$ for even $d$ and
$\binom{d}{(d+1)/2} \approx 1/\sqrt{2\pi(d+1)}2^{d+1} \approx 2^{n\delta}$ for odd $d$,
where $d = n\delta$,
the conditions~(\ref{eq:oddcond}) and~(\ref{eq:evencond}) are satisfied.
Since the fractions $|A_{d+1}(C)|/\binom{d}{(d+1)/2}$ and $|A_{d}(C)|/\binom{d}{d/2}$ tend to zero,
the upper and lower bounds in Theorems~\ref{th:oddlower} and~\ref{th:evenlower} asymptotically coincide.
\subsection*{Remarks}
Note that the condition~(\ref{eq:evencond}) for $T = C \setminus \{ \bm{0} \}$ is a sufficient condition
under which every codewords with weight $d$ is contained in every trial set for $C$ with even minimum distance $d$.
Also, the condition~(\ref{eq:oddcond}) for $T = C \setminus \{ \bm{0} \}$ is a sufficient condition
under which every codewords with weights $d$ and $d+1$ is contained in every trial set for $C$ with odd minimum distance $d$.
When the condition~(\ref{eq:evencond}) holds for $T = C \setminus \{ \bm{0} \}$,
as described in the proof of Theorem~\ref{th:evenlower},
for every $\bm{c} \in A_d(C)$ there exists at least one larger half $\bm{v} \in LH_{\lceil d/2 \rceil}(T)$
that has no common larger half with other codewords in $C$.
Since $M^1_{\lceil d/2 \rceil}(C) = LH_{\lceil d/2 \rceil}(T)$, every larger half of $\bm{c} \in A_d(C)$ is a minimal uncorrectable error.
Every trial set $T$ must satisfy that $M^1(C) \subseteq LH(T)$.
Therefore, every codeword in $A_d(C)$ needs to be contained in every trial set for $C$ in this case.
By a similar argument, we can show that if the condition~(\ref{eq:oddcond}) for $T = C \setminus \{ \bm{0} \}$ holds,
then every codeword in $A_d(C) \cup A_{d+1}(C)$ need to be in every trial set for $C$.
\section{A Generalization of the Bound}\label{sec:generalize}
By generalizing the results in the previous section,
we give a lower bound on the size of $LH_i(C \setminus \{ \bm{0} \})$ for each $i$.
We have the relation $M_i^1(C) \subseteq LH_i(C \setminus \{ \bm{0} \}) \subseteq E_i^1(C)$.
Thus the following lower bound is also a lower bound on the number of uncorrectable errors,
but the bound is weak when $i$ is large.
\medskip
\begin{theorem}\label{th:lhlower}
Let $C$ be a linear code with minimum distance $d$ and $T$ be a trial set for $C$.
Define $B_i = |A_{2i-2}(T)| + |A_{2i-1}(T)| + |A_{2i}(T)|$.
For an integer $i$ with $\lceil d/2 \rceil \leq i \leq \lfloor n/2 \rfloor$, if
\begin{equation*}
\binom{2i-3}{i} > 3 \binom{2i- \lceil \frac{d}{2} \rceil}{i} B_i
\end{equation*}
holds, then
\begin{multline*}
\left(\binom{2i-3}{i} - 3 \binom{2i- \lceil \frac{d}{2} \rceil}{i}B_i\right)B_i
\leq LH_i(T)\\
\leq \binom{2i-3}{i}|A_{2i-2}(T)| + 2\binom{2i-1}{i}(|A_{2i-1}(T)| + |A_{2i}(T)|)
\end{multline*}
\end{theorem}
\medskip
\begin{proof}
First we observe that $LH_i(T) = LH^+(A_{2i-2}(T)) \cup LH(A_{2i-1}(T)) \cup LH^-(A_{2i}(T))$.
We consider the upper bound on the number of common larger halves in $LH_i(T)$.
Let $\bm{c}, \bm{c}'$ be codewords in $A_{2i-2}(T) \cup A_{2i-1}(T) \cup A_{2i}(T)$.
Then $w(\bm{c} \cap \bm{c}') = (w(\bm{c})+w(\bm{c}')-w(\bm{c}+\bm{c}'))/2 \leq (2i+2i-d)/2 = 2i-\lceil d/2 \rceil$.
Therefore the number of common larger halves of weight $i$ between $\bm{c}$ and $\bm{c}'$ is at most $\binom{2i- \lceil d/2 \rceil}{i}$.
For $\bm{c} \in A_{2i-2}(T) \cup A_{2i-1}(T) \cup A_{2i}(T)$, the size of larger halves of $\bm{c}$ with weight $i$ is
at least $\binom{2i-3}{i}$.
Since $\binom{2i-3}{i} > 3 \binom{2i- \lceil d/2 \rceil}{i}B_i$, there is at least
$\binom{2i-3}{i} - 3 \binom{2i- \lceil d/2 \rceil}{i}B_i$ larger halves of $\bm{c}$ with weight $i$ that have no common larger halves.
Thus the lower bound follows.
The upper bound is obtained from the inequality $|LH_i(T)|
\leq |LH^+(A_{2i-2}(T))| + |LH(A_{2i-1}(T))| + |LH^-(A_{2i}(T))|
\leq \binom{2i-3}{i}|A_{2i-2}(T)| + \binom{2i-1}{i}|A_{2i-1}(T)| + \binom{2i-1}{i}|A_{2i}(T)|$.
\end{proof}
\section{Concluding Remarks}
A lower bound on the number of uncorrectable errors of weight half the minimum distance have been derived
for binary linear codes.
The conditions for the bound are not too restrictive, some codes including Reed-Muller codes and random linear codes satisfy the conditions.
A key observation for the results is that an uncorrectable error of weight half the minimum distance
is a larger half of some minimum weight codeword.
The lower bound has been generalized to a lower bound on the size of larger halves of a trial set,
but this bound is not a good lower bound on the number of uncorrectable errors for large weight.
Finding a good lower bound on the number of
uncorrectable error is an interesting future work.
| {
"timestamp": "2008-04-30T10:25:57",
"yymm": "0804",
"arxiv_id": "0804.4042",
"language": "en",
"url": "https://arxiv.org/abs/0804.4042",
"abstract": "A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes, and random linear codes. The bound asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. By generalizing the idea of the lower bound, a lower bound on the number of uncorrectable errors for weights larger than half the minimum distance is also obtained, but the generalized lower bound is weak for large weights. The monotone error structure and its related notion larger half and trial set, which are introduced by Helleseth, Kløve, and Levenshtein, are mainly used to derive the bounds.",
"subjects": "Information Theory (cs.IT)",
"title": "Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846659768267,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.709201957218688
} |
https://arxiv.org/abs/1810.06774 | Intersections of subcomplexes in non-positively curved 2-dimensional complexes | Let $X$ be a contractible $2$-complex which is a union of two contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\cap Z$ contractible as well? In this note, we prove that the inclusion-induced map $\pi _{1}(Y\cap Z)\rightarrow \pi _{1}(Z)$ is injective if $Y$ is $\pi _{1}$-injective subcomplex in a locally CAT(0) 2-complex $X$. In particular, each component in the intersection of two contractible subcomplexes in a CAT(0) 2-complex is contractible. | \section{Introduction}
As a motivation, we consider the following problem.
\begin{problem}
\label{prob1}Let $X$ be a contractible $2$-complex which is a union of two
contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\cap Z$
contractible as well?
\end{problem}
A higher-dimensional version of this problem is already studied by Begle
\cite{b}, which is related to the work of Aronszajn and Borsuk \cite{bb}.
Begle \cite{b} constructs a 3-dimensional contractible simplicial complex
X=Y\cup Z$ whose subcomplexes $Y,Z$ are both contractible but the
intersection $Y\cap Z$ is not simply connected. He left open the question as
to whether or not there are similar counter-examples in dimension two. We
study a more general problem as the following.
\begin{problem}
\label{prob}Let $X$ be a 2-dimensional aspherical simplicial complex (i.e.
the universal cover $\tilde{X}$ is contractible) and $Y$ be any $\pi _{1}
-injective subcomplex. For any subcomplex $Z\subset X,$ is the map $\pi
_{1}(Z\cap Y)\rightarrow \pi _{1}(Z)$ induced by the inclusion injective?
\end{problem}
Note that any subcomplex $Y$ of a contractible $2$-complex $X$ has vanishing
second homology group $H_{2}(Y;\mathbb{Z})=0$, by considering the long exact
sequence of homology groups for the pair $(X,Y)$. When $X$ is contractible
and $Z\subset X$ is also contractible, the triviality of $\pi _{1}(Z\cap Y)$
would imply that each connected component in $Z\cap Y$ is contractible by
the Whitehead theorem. This shows that a positive answer to Problem \re
{prob} gives a positive answer to Problem \ref{prob1}. We define a
subcomplex $Y$ of a 2-dimensional complex $X$ to be \emph{strongly }$\pi
_{1} $\emph{-injective} if for any subcomplex $Z$ of $X,$ the
inclusion-induced map $\pi _{1}(Y\cap Z)\rightarrow \pi _{1}(Z)$ is
injective (cf. Definition \ref{def}). The 2-complex $X$ is said to have
strong $\pi _{1}$-injectivity if any $\pi _{1}$-injective subcomplex $Y$ is
\emph{strongly} $\pi _{1}$-injective.
We will give a positive answer to Problem \ref{prob} for locally $\mathrm{CA
}(0)$ 2-complexes by showing that locally $\mathrm{CAT}(0)$ 2-complexes have
strong $\pi _{1}$-injectivity, as the following.
\begin{theorem}
\label{th1.3}\bigskip Let $X$ be a proper nonpositively curved 2-complex and
$Y$ a $\pi _{1}$-injective subcomplex. For any subcomplex $Z,$ the inclusion
induces an injection $\pi _{1}(Z\cap Y)\rightarrow \pi _{1}(Z).$ In other
words, $X$ has strong $\pi _{1}$-injectivity.
\end{theorem}
\begin{corollary}
Let $X$ be a $\mathrm{CAT}(0)$ 2-complex. For any two contractible
subcomplexes $Y,Z,$ each component in the intersection $Y\cap Z$ is
contractible.
\end{corollary}
Theorem \ref{th1.3} leads to the following observation: when $X$ is a finite
collapsible 2-complex and both $Y$ and $Z$ are contractible, each component
in the intersection $Z\cap Y$ is contractible (cf. Corollary \ref{lem3}).
This is already known by Segev \cite{se} using a different approach.
\textit{Notation: } All complexes are assumed to be connected simplicial
complexes, unless otherwise stated. We use $\pi _{1}(X)$ to denote the
fundamental group of $X$ with a based point in a connected component.
\section{Strong $\protect\pi _{1}$-injectivity}
We first give the following definition.
\begin{definition}
\label{def}A subcomplex $Y$ of a 2-dimensional complex $X$ is \emph{strongly
}$\pi _{1}$\emph{-injective} if for any subcomplex $Z$ of $X,$ the
inclusion-induced map $\pi _{1}(Y\cap Z)\rightarrow \pi _{1}(Z)$ is
injective. The 2-complex $X$ has strong $\pi _{1}$-injectivity if any $\pi
_{1}$-injective subcomplex $Y$ is \emph{strongly} $\pi _{1}$-injective.
\end{definition}
Not every $2$-complex has strong $\pi _{1}$-injectivity. For a simple
counter-example, let $X$ be the sphere $S^2$. Since $S^{2}$ is a union of
two disks with the circle $S^{1}$ as the intersection, the upper disk in the
sphere $S^{2}$ is not strongly $\pi _{1}$-injective.
\begin{lemma}
\label{lemt}Any $\pi _{1}$-injective graph (i.e. $1$-simplicial subcomplex)
is strongly $\pi _{1}$-injective in any $2$-complex.
\end{lemma}
\begin{proof}
Let $X$ be a 2-complex with a $\pi _{1}$-injective $1$-dimensional
subcomplex $Y.$ Shrinking a contractible tree in $Y,$ we see that the
fundamental group of $Y$ is free. For a subcomplex $K\subset Y,$ the
fundamental group of $K$ is still free. If there is a non-nullhomotopic
closed loop in $K,$ the loop represents a nontrivial element in $Y.$ This
implies that the composite $\pi _{1}(K)\rightarrow \pi _{1}(Y)\rightarrow
\pi _{1}(X)$ is injective and thus $Y$ is strongly $\pi _{1}$-injective.
\end{proof}
Next, we study the relation between strong $\pi_1$-injectivity and taking
covering space.
\begin{lemma}
\label{lem1}Let $X$ be a 2-complex with a $\pi _{1}$-injective subcomplex
K. $ Suppose that $p:\tilde{X}\rightarrow X$ is the universal cover$.$ If
p^{-1}(K)$ is strongly $\pi _{1}$-injective in $\tilde{X},$ then the complex
$K$ is strongly $\pi _{1}$-injective in $X.$
\end{lemma}
\begin{proof}
Suppose that there is a subcomplex $Z$ in $X$ such that $\pi _{1}(Z\cap
K)\rightarrow \pi _{1}(Z)$ is not injective. Let $f:S^{1}\rightarrow Z\cap K$
be a map whose homotopic class in $\pi _{1}(Z\cap K)$ is nontrivial, but
trivial in $\pi _{1}(Z).$ The map $f$ has a lifting $f^{\prime
}:S^{1}\rightarrow \tilde{X},$ since $[f]=1\in \pi _{1}(X).$ Moreover,
[f^{\prime }]\neq 1\in \pi _{1}(p^{-1}(K\cap Z),\ast )$ for the base point
in any connected component of $p^{-1}(K\cap Z).$ Since $\pi
_{1}(K)\rightarrow \pi _{1}(X)$ is injective, the complex (each connected
component) $p^{-1}(K)$ is simply connected. By assumption, the induced map
\pi _{1}(p^{-1}(K)\cap p^{-1}(Z))\rightarrow \pi _{1}(p^{-1}(Z))$ is
injective. Therefore, the homotopy class $[f^{\prime }]\neq 1\in \pi
_{1}(p^{-1}(Z)).$ This is a contradiction, since $\pi
_{1}(p^{-1}(Z))\rightarrow \pi _{1}(Z)$ is injective.
\end{proof}
Lemma \ref{lem1} implies that a 2-complex $X$ has strong $\pi _{1}
-injectivity if its universal cover $\tilde{X}$ does.
Let $X$ be a 2-complex and $K$ be a closed triangle (2-simplex). The
2-complex $X\cup K$ obtained by identifying two edges of $K$ with those of
X $ is called an elementary extension of $X,$ while $X$ is called an
elementary collapse of $X\cup K$ (cf. \cite{Bo}). Denote by $e$ the third
edge of $K,$ which is not in $X.$ A 2-complex $X$ is called collapsible if
X $ could be deformed to be a point by finite steps of elementary
extensions, collapses and contracting or adding free edges.
\begin{center}
\begin{equation*}
\FRAME{itbpF}{2.0928in}{1.9571in}{0in}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 2.0928in;height 1.9571in;depth
0in;original-width 6.8476in;original-height 6.4031in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";tempfilename
'PDMVFR00.bmp';tempfile-properties "XPR";}}
\end{equation*}
Figure 1. The elementary extension $X\cup K$
\end{center}
\begin{theorem}
\label{lem2}The 2-complex $X$ has strong $\pi _{1}$-injectivity if and only
if so does an elementary extension $X\cup K.$
\end{theorem}
\begin{proof}
Suppose that the elementary extension $X\cup K$ has strong $\pi _{1}
-injectivity. For any subcomplex $Y$ with injective fundamental group, we
see that
\begin{equation*}
\pi _{1}(Y)\rightarrow \pi _{1}(X)\overset{\cong }{\rightarrow }\pi
_{1}(X\cup K)
\end{equation*
is injective as well. Therefore, for any subcomplex $Z\subset X,$ the map
\pi _{1}(Z\cap Y)\rightarrow \pi _{1}(Z)$ is injective.
Conversely, suppose that $X$ has strong $\pi _{1}$-injectivity. Let
Y\subset X\cup K$ be any $\pi _{1}$-injective subcomplex and $Z\subset X\cup
K$ any subcomplex. We divide the proof into several cases.
\begin{enumerate}
\item[Case 1] $Y\supset K.$
\item[1.1] $Z\supset K.$ For convenience, let $Y\backslash K$ denote the
subcomplex of $Y$ obtained by deleting the interior of $K$ and the open edge
$e.$ We see that
\begin{equation*}
Y\cap Z=(Y\backslash K\cap Z\backslash K)\cup K.
\end{equation*
Note that $Y$ is an elementary extension of $Y\backslash K$ and $Y\cap Z$ is
also an elementary extension of $Y\backslash K\cap Z\backslash K.$
Therefore, we get by the hypothesis on $X$ that
\begin{equation*}
\pi _{1}(Y\cap Z)=\pi _{1}(Y\backslash K\cap Z\backslash K)\hookrightarrow
\pi _{1}(Z\backslash K)=\pi _{1}(Z).
\end{equation*}
\item[1.2] $Z\nsupseteqq K$ but $Z\supset e.$ Let $Z\backslash e$ denote the
complex obtained by removing the interior of $e$ from $Z$. We have that
\begin{equation*}
Y\cap Z=(Y\backslash K\cap Z\backslash e)\cup e.
\end{equation*
If two ends of $e$ are both in the same component of $Y\backslash K\cap
Z\backslash e,$ let $P$ be a path in $Y\backslash K\cap Z\backslash e$
connecting the two ends. Choose a base point $x_{0}\in P.$ Contracting the
path $P,$ we have that (note the injectivity of the first free factor
follows from the hypothesis on $X$)
\begin{equation*}
\pi _{1}(Y\cap Z,x_{0})=\pi _{1}(Y\backslash K\cap Z\backslash e,x_{0})\ast
\mathbb{Z}\hookrightarrow \pi _{1}(Z\backslash e,x_{0})\ast \mathbb{Z}=\pi
_{1}(Z,x_{0}).
\end{equation*
If the two ends of $e$ lie in two different components $Y_{1},Y_{2}$ of
Y\backslash K\cap Z\backslash e,$ choose the path $F$ consisting of the two
attaching edges in $K.$ Note that $F$ is not in $Z$. Since $X$ has strong
\pi _{1}$-injectivity, there is an injection
\begin{equation*}
\pi _{1}(Y\backslash K\cap (Z\backslash e\cup F),x_{0})\hookrightarrow \pi
_{1}(Z\backslash e\cup F,x_{0})
\end{equation*
where the base point $x_{0}\ $is one end of $e.$ Therefore, we have tha
\begin{eqnarray*}
\pi _{1}(Y\cap Z,x_{0}) &=&\pi _{1}(Y_{1})\ast \pi _{1}(Y_{2})=\pi
_{1}((Y\backslash K\cap Z\backslash e)\cup F,x_{0}) \\
&\hookrightarrow &\pi _{1}(Z\backslash e\cup F,x_{0})=\pi _{1}(Z,x_{0}).
\end{eqnarray*}
\item[1.3] $Z\subset X.$ We have that $Y\cap Z=Y\backslash K\cap Z$ and thu
\begin{equation*}
\pi _{1}(Y\cap Z)=\pi _{1}(Y\backslash K\cap Z)\hookrightarrow \pi _{1}(Z).
\end{equation*}
\item[Case 2] $Y\nsupseteqq K$ but $Y\supset e,$ where $e$ is the closed
edge of $K$ not in $X.$
\item[2.1] $Z\supset K.$ We have that $Y\cap Z=(Y\backslash e\cap
Z\backslash K)\cup e.$ Since $\pi _{1}(Y)\rightarrow \pi _{1}(X\cup K)$ is
injective, the path $F$ consisting of the two attaching edges of $K$ does
not lie in $Y.$ If the two ends of $e$ lie in the same component of
Y\backslash e\cap Z\backslash K,$ the edge $e$ is a part of a loop in $Y\cap
Z.$ Then the path $F$ is part of a loop in $Y\backslash e\cap Z\backslash K$
by replacing $e$ with $F$. If the two ends of $e$ lie in different
components, then the edge $e$ will not contribute to the fundamental group.
In any case, we have an injectio
\begin{equation*}
\pi _{1}(Y\cap Z)\hookrightarrow \pi _{1}((Y\backslash e\cup F)\cap
Z\backslash K).
\end{equation*
Since $\pi _{1}(Y)=\pi _{1}(Y\backslash e\cup F)\hookrightarrow \pi
_{1}(X\cup K)=\pi _{1}(X),$ the subcomplex $Y\backslash e\cup F$ is also
\pi _{1}$-injective. Considering that $X$ has strong $\pi _{1}$-injectivity,
there is an injection
\begin{equation*}
\pi _{1}((Y\backslash e\cup F)\cap Z\backslash K)\hookrightarrow \pi
_{1}(Z\backslash K)=\pi _{1}(Z).
\end{equation*
This proves that the inclusion induces an injection $\pi _{1}(Y\cap
Z)\hookrightarrow \pi _{1}(Z).$
\item[2.2] $Z\nsupseteqq K$ but $Z\supset e.$ Since $\pi _{1}(Y)\rightarrow
\pi _{1}(X\cup K)$ is injective, the path $F$ consisting of the two
attaching edges of $K$ does not lie in $Y.$ For the same reason as that of
the case 2.1, we have an injectio
\begin{eqnarray*}
\pi _{1}(Y\cap Z) &=&\pi _{1}((Y\backslash e\cap Z\backslash e)\cup
e)\hookrightarrow \pi _{1}((Y\backslash e\cap Z\backslash e)\cup F) \\
&=&\pi _{1}((Y\backslash e\cup F)\cap (Z\backslash e\cup F)).
\end{eqnarray*
Note that $\pi _{1}(Y\backslash e\cup F)=\pi _{1}(Y)\hookrightarrow \pi
_{1}(X\cup K)=\pi _{1}(X).$ Since $X$ has strong $\pi _{1}$-injectivity, the
inclusion induces an injection $\pi _{1}((Y\backslash e\cup F)\cap
(Z\backslash e\cup F))\hookrightarrow \pi _{1}(Z\backslash e\cup F).$ If
F\nsubseteqq Z,$ we have that $\pi _{1}(Z\backslash e\cup F)=\pi _{1}(Z).$
If $F\subset Z,$ we have that $\pi _{1}(Z\backslash e\cup F)\ast \mathbb{Z
=\pi _{1}(Z).$ In both cases, there is an injection $\pi _{1}(Z\backslash
e\cup F)\hookrightarrow \pi _{1}(Z).$ Therefore, the map $\pi _{1}(Y\cap
Z)\rightarrow \pi _{1}(Z)$ is injective.
\item[2.3] $Z\subset X.$ We have that $Y\cap Z=Y\backslash e\cap Z.$ For the
same reason as that of the case 2.1, the path $F$ is not in $Y$ and there is
an injection $\pi _{1}(Y\backslash e\cup F)\hookrightarrow \pi _{1}(X).$ We
have tha
\begin{equation*}
\pi _{1}(Y\cap Z)=\pi _{1}(Y\backslash e\cap Z)\hookrightarrow \pi
_{1}((Y\backslash e\cup F)\cap Z)\hookrightarrow \pi _{1}(Z).
\end{equation*}
\item[Case 3] $Y\subset X.$
\item[3.1] $Z\supset K.$ We have that $Y\cap Z=Y\cap (Z\backslash K).$ Since
$Z\backslash K$ is a collapse of $Z,$ we get from the hypothesis on $X$ that
\begin{equation*}
\pi _{1}(Y\cap Z)=\pi _{1}(Y\cap (Z\backslash K))\hookrightarrow \pi
_{1}(Z\backslash K)=\pi _{1}(Z).
\end{equation*}
\item[3.2] $Z\nsupseteqq K$ but $Z\supset e.$ In this case, $\pi _{1}(Z)=\pi
_{1}(Z\backslash e)\ast \mathbb{Z}.$ The hypothesis on $X$ implies that $\pi
_{1}(Y\cap Z)=\pi _{1}(Y\cap Z\backslash e)$ injects into $\pi
_{1}(Z\backslash e)$. Therefore, we have an injection
\begin{equation*}
\pi _{1}(Y\cap Z)\hookrightarrow \pi _{1}(Z\backslash e)\hookrightarrow \pi
(Z\backslash e)\ast \mathbb{Z}=\pi _{1}(Z).
\end{equation*}
\item[3.3] $Z\subset X.$ This subcase follows directly from the hypothesis
of $X$.
All the cases are included and the proof is complete
\end{enumerate}
\end{proof}
It is already known by Segev \cite{se} (4.3) that when $X$ is a finite
collapsible 2-complex and both $Y$ and $Z$ are contractible, each connected
component in the intersection $Z\cap Y$ is contractible. This is a special
case of the following.
\begin{corollary}
\label{lem3}A collapsible 2-complex has strong $\pi _{1}$-injectivity. In
particular, each connected component in the intersection $Y\cap Z$ of two
contractible subcomplexes $Y,Z$ in a collapsible 2-complex $X$ is
contractible.
\end{corollary}
\begin{proof}
A collapsible 2-complex is deformed to a point by a finitely many elementary
collapse or extensions. The first part is thus implied by Theorem \ref{lem2
. When $Y$ and $Z$ are contractible, the intersection $Y\cap Z$ is $\pi _{1}
-injective in $Z$ and thus simply connected. A simply connected subcomplex
of a contractible 2-complex is acyclic by the relative homology exact
sequence. Therefore, each connected component in the intersection $Y\cap Z$
is contractible by the Whitehead theorem.
\end{proof}
\section{Non-positively curved complexes}
Recall the notion of non-positively curved complexes from Bridson and
Haefliger \cite{bh} (Chapter II. 1.2). Let $(X,d)$ be a geodesic metric
space. A geodesic triangle $\Delta (x,y,z)$ consists of three vertices
x,y,z\in X$ and three geodesics $[x,y],[y,z],[x,z]$ connecting these
vertices. A comparison triangle $\Delta (\bar{x},\bar{y},\bar{z})$ (or
denoted by $\bar{\Delta}(x,y,z)$) is an Euclidean triangle in the plane
\mathbb{R}^{2}$ with three vertices $\bar{x},\bar{y},\bar{z}$ and edges of
lengths $d(x,y),d(y,z),d(x,z)$ respectively.
\begin{definition}
\label{cat}A geodesic metric space $X$ is CAT(0) if for any geodesic
triangle $\Delta (x,y,z)$ and any two points $p,q\in \Delta (x,y,z),$ we hav
\begin{equation*}
d(p,q)\leq d_{\mathbb{R}^{2}}(\bar{p},\bar{q}),
\end{equation*
where $\bar{p},\bar{q}$ are the corresponding points of $p,q$ in the
comparison triangle $\Delta (\bar{x},\bar{y},\bar{z}).$
\end{definition}
A Euclidean cell is the convex hull of a finite number of points in $\mathbb
R}^{n}$, equipped with the standard Euclidean metric. A Euclidean cell
complex $X$ is a space formed by gluing together Euclidean cell-complexes
via isometries of their faces. It has the piecewise Euclidean path metric.
Precisely, for any $x,y\in X,$ let $x=x_{0},x_{1},\cdots ,x_{n}=y$ be a path
such that each successive $x_{i},x_{i+1}$ is contained in a Euclidean
simplex $S_{i}.$ Define the distance (called path metric) $d_{X}(x,y)=\inf
\sum_{i=0}^{n-1}d_{S_{i}}(x_{i},x_{i+1}),$ where the infimum is taken over
all such paths. Note that a metric space $X$ is proper if any closed ball
B(x,r)\subset X$ is compact.
\begin{definition}
A Euclidean cell complex $X$ is non-positively curved if it is locally
CAT(0), i.e. for every $x\in X$ there exists $r_{x}>0$ such that the ball
B(x,r_{x})$ with the induced metric is a CAT(0).
\end{definition}
Let $X$ be a Euclidean cell complex and $v\in X.$ The (geometric) link
Lk(x,X)$ is the set of unit tangent vectors ("directions") at $x$ in $X.$
Precisely, let $S$ be the set of all geodesics $[x,y]$ with $y$ in a simplex
containing $x.$ Two geodesics are called equivalent if one is contained in
the other. The link $Lk(x,X)$ is the set of equivalence classes of geodesics
in $S.$ If $X$ is one $n$-dimensional Euclidean cell, the link $Lk(x,X)$ is
part of $S^{n-1}$ and thus the topology on $Lk(x,X)$ is defined as the
"angle" topology. In general, the topology on $Lk(x,X)$ is defined as the
path metric coming from each cell.
We will need the following facts about $2$-complexes from \cite{bh}.
\begin{lemma}
\label{key}(1) A finite $\mathrm{CAT}(0)$ 2-complex is collapsible.
(2) (Link condition) A $2$-dimensional Euclidean cell complex $X$ is
non-positively curved if and only if each link $Lk(x,X)$ contains no
injective loops of length less than $2\pi .$
(3) A simply connected non-positively curved complex is $\mathrm{CAT}(0).$
\end{lemma}
\begin{proof}
The first claim (1) is \cite{bh}, 5.34(2), while the second claim (2) is
\cite{bh}, 5.5 and 5.6 in Chapter II.5. The last claim follows the
Cartan-Hadamard theorem (see \cite{bh}, II. 4.1).
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \protect\ref{th1.3}]
Since the universal cover of a non-positively curved complex is $\mathrm{CAT
(0),$ it suffices to prove that a $\mathrm{CAT}(0)$ 2-complex $X$ has strong
$\pi _{1}$-injectivity by Lemma \ref{lem1}. By Corollary \ref{lem3}, any
collapsible 2-complex has strong $\pi _{1}$-injectivity. As a finite
\mathrm{CAT}(0)$ 2-complex is collapsible (see Lemma \ref{key} (1)), it has
strong $\pi _{1}$-injectivity. Suppose that a simply connected subcomplex
Y\subset X$ is not strongly $\pi _{1}$-injective. Let $Z\subset X$ be a
subcomplex such that $\pi _{1}(Y\cap Z)\overset{i}{\rightarrow }\pi _{1}(Z)$
is not injective. Choose $f:S^{1}\rightarrow Y\cap Z$ such that the homotopy
class $[f]\in \ker i$ is not trivial. Since $\text{Im}f$ is compact and any
homotopy $h$ between $f$ and a constant map has compact image in $Z,$ we
could choose finite subcomplexes $Y^{\prime }\subset Y$ containing $\text{Im
f$ and $Z^{\prime }\subset Z$ containing $\text{Im}h.$ The link condition
implies that the subcomplex $Y$ is non-positively curved (cf. Lemma \ref{key}
(2)). Since $Y$ is simply connected, the subcomplex $Y$ is $\mathrm{CAT}(0)$
by Lemma \ref{key} (3). The finite subcomplex $Y^{\prime }$ is contained in
a ball $B_{Y}(x,r)\subset Y$ of sufficiently large radius, for some point
x\in Y$ and sufficient large $r$. When $X$ is proper, the closed ball
B_{Y}(x,r)$ is compact. Since the ball $B_{Y}(x,r)$ is contractible (cf.
\cite{bh}, II.1.4), we may choose a finite contractible subcomplex
Y^{\prime \prime }\subset Y$ containing $Y^{\prime }$ (for example, take
Y^{\prime \prime }=B_{Y}(x,r)$). By the construction, the inclusion-induced
map $\pi _{1}(Y^{\prime \prime }\cap Z^{\prime })\overset{i}{\rightarrow
\pi _{1}(Z^{\prime })$ is not injective. Since both $Y^{\prime \prime }$ and
$Z^{\prime }$ are finite, we may choose a ball $B_{X}(x,r^{\prime })$ of
sufficiently large radius containing $Y^{\prime \prime }$ and $Z^{\prime }.$
Therefore, the subcomplexes $Y^{\prime \prime }$ and $Z^{\prime }$ is
contained in a finite $\mathrm{CAT}(0)$ 2-complex $X^{\prime }.$ The strong
\pi _{1}$-injectivity of $X^{\prime }$ implies that the map $\pi
_{1}(Y^{\prime \prime }\cap Z^{\prime })\overset{i}{\rightarrow }\pi
_{1}(Z^{\prime })$ is injective, which gives a contradiction. This finishes
the proof.
\end{proof}
| {
"timestamp": "2018-10-17T02:05:48",
"yymm": "1810",
"arxiv_id": "1810.06774",
"language": "en",
"url": "https://arxiv.org/abs/1810.06774",
"abstract": "Let $X$ be a contractible $2$-complex which is a union of two contractible subcomplexes $Y$ and $Z.$ Is the intersection $Y\\cap Z$ contractible as well? In this note, we prove that the inclusion-induced map $\\pi _{1}(Y\\cap Z)\\rightarrow \\pi _{1}(Z)$ is injective if $Y$ is $\\pi _{1}$-injective subcomplex in a locally CAT(0) 2-complex $X$. In particular, each component in the intersection of two contractible subcomplexes in a CAT(0) 2-complex is contractible.",
"subjects": "Geometric Topology (math.GT); Algebraic Topology (math.AT)",
"title": "Intersections of subcomplexes in non-positively curved 2-dimensional complexes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465639,
"lm_q2_score": 0.7248702702332476,
"lm_q1q2_score": 0.7092019567618293
} |
https://arxiv.org/abs/1309.0906 | On a Curious Biconditional Involving Divisors of Odd Perfect Numbers | We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality $q<n$ holds unconditionally. | \section{Introduction}
If $J$ is a positive integer, then we write $\sigma(J)$ for the sum of the divisors of $J$. A number $L$ is \emph{perfect} if $\sigma(L)=2L$.
An even perfect number $M$ is said to be given in \emph{Euclidean form} if $$M = (2^p - 1)\cdot{2^{p - 1}}$$
where $p$ and $2^p - 1$ are primes. We call $M_p = 2^p - 1$ the \emph{Mersenne prime} factor of $M$. Currently, there are only $48$ known Mersenne primes \cite{GIMPS}, which correspond to $48$ even perfect numbers.
An odd perfect number $N$ is said to be given in \emph{Eulerian form} if $$N = {q^k}{n^2}$$
where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. We call $q^k$ the \emph{Euler part} of $N$ while $n^2$ is called the \emph{non-Euler part} of $N$.
(We will call $q$ the \emph{Euler prime} factor of $N$.)
It is currently unknown whether there are infinitely many even perfect numbers, or whether any odd perfect numbers exist. It is widely believed that there is an infinite number of even perfect numbers. On the other hand, no examples for an odd perfect number have been found (despite extensive computer searches), nor has a proof for their nonexistence been established.
Ochem and Rao \cite{OchemRao} recently proved that $N > {10}^{1500}$. In a recent preprint, Nielsen \cite{Nielsen1} obtained the lower bound $\omega(N) \geq 10$ for the number of \emph{distinct} prime factors of $N$, improving on his last result $\omega(N) \geq 9$ (see \cite{Nielsen2}).
After testing large numbers $X={t^r}{s^2}$ with $\omega(X) = 8$ for perfection, Sorli conjectured in \cite{Sorli} that $r=\nu_{t}(X)=1$ always holds. (More recently, Beasley \cite{Beasley} points out that Descartes was the first to conjecture $k=\nu_{q}(N)=1$ ``in a letter to Marin Mersenne in 1638, with Frenicle's subsequent observation occurring in 1657".) Dris conjectured in \cite{Dris2} and \cite{Dris} that the components $q^k$ and $n$ are related by the inequality $q^k < n$. This conjecture was made on the basis of the result $I(q^k)<\sqrt[3]{2}<I(n)$.
We denote the abundancy index $I$ of the positive integer $x$ as $I(x) = \sigma(x)/x$.
\section{Preparatory Results}
The following result was communicated to the second author (via e-mail, by Pascal Ochem) on April 17, 2013.
\begin{theorem}\label{Theorem1}
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, then
$$I(n) > {\left(\frac{8}{5}\right)}^{\frac{\ln(4/3)}{\ln(13/9)}} \approx 1.44440557.$$
\end{theorem}
The proof of Theorem \ref{Theorem1} uses the following lemma.
\begin{lemma}\label{LemmaOchemcheckedbyDagal}
Let $x(n) = \ln(I(n^2))/\ln(I(n))$. If $\gcd(a, b) = 1$, then
$$\min(x(a), x(b)) < x(ab) < \max(x(a),x(b)).$$
\end{lemma}
\begin{proof}
First, note that $x(a) \neq x(b)$ (since $\gcd(a , b) = 1$). Without loss of generality, we may assume that $x(a) < x(b)$. Thus, since
$$x(n) = \frac{\ln(I(n^2))}{\ln(I(n))},$$
we have
$$\frac{\ln(I(a^2))}{\ln(I(a))} < \frac{\ln(I(b^2))}{\ln(I(b))}.$$
This implies that
$$\ln(I(a^2))\ln(I(b)) < \ln(I(b^2))\ln(I(a)).$$
Adding $\ln(I(a^2))\ln(I(a))$ to both sides of the last inequality, we get
$$\ln(I(a^2))\bigg(\ln(I(b)) + \ln(I(a))\bigg) < \ln(I(a))\bigg(\ln(I(b^2)) + \ln(I(a^2))\bigg).$$
Using the identity $\ln(X) + \ln(Y) = \ln(XY)$, we can rewrite the last inequality as
$$\ln(I(a^2))\ln(I(ab)) < \ln(I(a))\ln(I((ab)^2))$$
since $I(x)$ is a weakly multiplicative function of $x$ and $\gcd(a, b) = 1$.
It follows that
$$\min(x(a), x(b)) = x(a) = \frac{\ln(I(a^2))}{\ln(I(a))} < \frac{\ln(I((ab)^2))}{\ln(I(ab))} = x(ab).$$
Under the same assumption $x(a) < x(b)$, we can show that $$x(ab) < x(b) = \max(x(a),x(b))$$
by adding $\ln(I(b^2))\ln(I(b))$ ( instead of $\ln(I(a^2))\ln(I(a))$ ) to both sides of the inequality
$$\ln(I(a^2))\ln(I(b)) < \ln(I(b^2))\ln(I(a)).$$
This finishes the proof.
\end{proof}
\begin{remark}\label{RemarkDagal}
From Lemma \ref{LemmaOchemcheckedbyDagal}, we note that $1 < x(n) < 2$ follows from
$$I(n) < I(n^2) < {\left(I(n)\right)}^2$$
and $I(n^2) = {\left(I(n)\right)}^{x(n)}$.
The trivial lower bound for $I(n)$ is
$${\bigg(\frac{8}{5}\bigg)}^{1/2} < I(n).$$
Note that decreasing the denominator in the exponent gives an increase in the lower bound for $I(n)$.
\end{remark}
\begin{remark}\label{ProofTheorem1}
We sketch a proof for Theorem \ref{Theorem1} here, as communicated to the second author by Pascal Ochem.
Suppose that $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form.
We want to obtain a lower bound on $I(n)$. We know that
$$I(n^2) = 2/I(q^k) > 2/(5/4) = 8/5.$$
We need to improve the trivial bound $I(n^2) < {\left(I(n)\right)}^2$.
Let $x(n)$ be such that
$$I(n^2) = {\bigg(I(n)\bigg)}^{x(n)}.$$
That is, $x(n) = \ln(I(n^2))/\ln(I(n))$. We want an upper bound on $x(n)$ for $n$ odd. By Lemma \ref{LemmaOchemcheckedbyDagal}, we consider the component $r^s$ with $r$ prime that maximizes $x(r^s)$.
We have
$$I(r^s) = \frac{r^{s + 1} - 1}{{r^s}(r - 1)} = 1 + \frac{1}{r - 1} - \frac{1}{{r^s}(r - 1)}.$$
Also,
$$I(r^{2s}) = \frac{r^{2s + 1} - 1}{{r^{2s}}(r - 1)} = I(r^s)\bigg(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right)\bigg).$$
So,
$$x(r^s) = \frac{\ln(I(r^{2s}))}{\ln(I(r^s))} = \frac{\ln(I(r^s)) + \ln(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right))}{\ln(I(r^s))},$$
from which it follows that
$$x(r^s) = 1 + \frac{\ln(1 + \left(\frac{1 - r^{-s}}{r^{s + 1} - 1}\right))}{\ln(1 + \frac{1}{r - 1} - \frac{1}{{r^s}(r - 1)})}.$$
We can check that
$$x(r^s) > x(r^t)$$
if $s < t$ and $r \geq 3$. Therefore, $x(r^s)$ is maximized for $s = 1$.
Now,
$$x(r) = 1 + \frac{\ln(1 + (1/(r(r + 1))))}{\ln(1 + (1/r))} = \frac{\ln(1 + (1/r) + (1/r)^2)}{\ln(1 + (1/r))} = \ln(I(r^2))/\ln(I(r)),$$
which is maximized for $r = 3$.
So,
$$x(3) = \ln(I(3^2))/\ln(I(3)) = \ln(13/9)/\ln(4/3) \approx 1.27823.$$
The claim in Theorem \ref{Theorem1} then follows, and the proof is complete.
\end{remark}
The argument in Remark \ref{ProofTheorem1} can be improved to account for the divisibility of $n$ by primes $r$ other than $3$, whereby $x(r) \leq x(5) < x(3)$. We outline an attempt on such an improvement in the next section.
The following claim follows from Acquaah and Konyagin's estimate for the prime factors of an odd perfect number \cite{AcquaahKonyagin}.
\begin{lemma}\label{AcquaahKonyaginLemma}
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, then $q < n\sqrt{3}$.
\end{lemma}
\begin{proof}
First, we observe that we have the estimate $q < n$ when $k > 1$ (see \cite{Dris}).
Now assume that $k = 1$. By Acquaah and Konyagin's estimate for the Euler prime $q$:
$$q < (3N)^{1/3} \Longrightarrow q^3 < 3N = 3{q^k}{n^2} \Longrightarrow q^2 = q^{3-k} < 3n^2.$$
It follows that $q < n\sqrt{3}$.
\end{proof}
Lastly, we have the following result, taken from \cite{DrisMSE}.
\begin{lemma}\label{OPNBiconditionalDris}
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the following biconditionals are true:
$$q^k < n \Longleftrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longleftrightarrow \sigma(q^k) < \sigma(n).$$
\end{lemma}
\begin{proof}
One direction of the biconditional is trivial:
$$q^k < n \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}.$$
This is proved by noting that $I(q^k) < \sqrt[3]{2} < I(n)$, from which the following chain of implications follow:
$$q^k < n \Longrightarrow \sigma(q^k) < \sigma(n)$$
$$q^k < n \Longrightarrow \frac{1}{n} < \frac{1}{q^k}$$
$$\{\sigma(q^k) < \sigma(n)\} \land \{\frac{1}{n} < \frac{1}{q^k}\} \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}$$
Therefore:
$$q^k < n \Longrightarrow \sigma(q^k) < \sigma(n) \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}.$$
For the other direction:
The implication
$$\frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longrightarrow \sigma(q^k) < \sigma(n)$$
can be proved, again by observing that $I(q^k) < \sqrt[3]{2} < I(n)$.
Now, to prove the last implication:
$$\sigma(q^k) < \sigma(n) \Longrightarrow q^k < n$$
we take an indirect approach.
First, we can show that:
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form and
$$I(q^k) + I(n) < \frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k},$$
then $q^k < n \Longleftrightarrow \sigma(q^k) < \sigma(n)$.
Similarly, we can prove that:
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form and
$$\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < I(q^k) + I(n),$$
then $q^k < n \Longleftrightarrow \sigma(n) < \sigma(q^k)$.
Observe that
$$I(q^k) + I(n) = \frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k}$$
if and only if
$$\sigma(q^k) = \sigma(n).$$
Also, observe that if we assume
$$\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < I(q^k) + I(n)$$
then the biconditional
$$q^k < n \Longleftrightarrow \sigma(n) < \sigma(q^k)$$
will contradict $I(q^k) < \sqrt[3]{2} < I(n)$.
Therefore, the following inequality must be true:
$$I(q^k) + I(n) \leq \frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k}.$$
It suffices to consider the case when
$$I(q^k) + I(n) = \frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k}$$
which is true if and only if
$$\sigma(q^k) = \sigma(n).$$
This last equation, together with the inequality $I(q^k) < \sqrt[3]{2} < I(n)$, implies that
$$1 = \frac{\sigma(q^k)}{\sigma(n)} < \frac{q^k}{n}$$
from which it follows that $n < q^k$. Thus, $1/q^k < 1/n$, which then gives, together with the equation $\sigma(n) = \sigma(q^k)$, the inequality
$$\frac{\sigma(n)}{q^k} < \frac{\sigma(q^k)}{n}.$$
But this last inequality, together with $I(q^k) < \sqrt[3]{2} < I(n)$, is known to imply
$$n < q^k.$$
Consequently, if
$$I(q^k) + I(n) = \frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k}$$
then
$$n < q^k \Longleftrightarrow \frac{\sigma(n)}{q^k} < \frac{\sigma(q^k)}{n}$$
We now give a proof for the following biconditional in what follows:
$$q^k < n \Longleftrightarrow \sigma(q^k) < \sigma(n).$$
So we have
$$q^k < n \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}.$$
But since $I(q^k) < I(n)$, we also have
$$\frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longrightarrow \sigma(q^k) < \sigma(n).$$
Now, since $\sigma(q^k)/n < \sigma(n)/q^k$ also implies $q^k < n$, we have
$$\sigma(q^k) < \sigma(n) \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}.$$
But since we have $\sigma(q^k)/n < \sigma(n)/q^k \Longleftrightarrow q^k < n$, then we have
$$\frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longrightarrow q^k < n.$$
Consequently, we obtain
$$q^k < n \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longrightarrow \sigma(q^k) < \sigma(n) \Longrightarrow \frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k} \Longrightarrow q^k < n,$$
and we are done.
\end{proof}
\section{Main Theorem}
We are now ready to prove our main result in this note.
\begin{theorem}\label{TheoremDris}
If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (with smallest prime factor $u$ satisfying $u \geq 5$), then the inequality $q < n$ is true.
\end{theorem}
\begin{proof}
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (with smallest prime factor at least $5$), and suppose to the contrary that $n < q$. By a result in \cite{Dris}, this implies that
$$k = 1.$$
By Lemma \ref{OPNBiconditionalDris}, since $n < q$ and $k = 1$, we have the inequalities
$$n < \sigma(n) \leq q < \sigma(q).$$
Using Theorem \ref{Theorem1} and Lemma \ref{AcquaahKonyaginLemma}, this then gives
$$\frac{1}{\sqrt{3}} < \frac{n}{q} < \frac{\sigma(n)}{q} \leq 1$$
and
$$\sqrt{2} < I(n) \leq \frac{q}{n} < \frac{\sigma(q)}{n}.$$
We now get an upper bound for $\sigma(q)/n$.
Recall the bound
$$\frac{\sigma(q^k)}{q^k} < \frac{q}{q - 1}.$$
Setting $k = 1$, dividing through by $n$ and rearranging, we obtain
$$\frac{\sigma(q)}{n} < \frac{q}{n}\cdot\left(\frac{q}{q - 1}\right) < \sqrt{3}\cdot\left(\frac{n}{n - 1}\right) = \sqrt{3} + \frac{\sqrt{3}}{n - 1} < \sqrt{3}$$
where we have used the fact that
$$\lim_{n\rightarrow\infty}{\frac{\sqrt{3}}{n - 1}} = 0.$$
Consequently,
$$\frac{\sigma(q)}{n} + \frac{\sigma(n)}{q} < 1 + \sqrt{3} \approx 2.732.$$
Again, by Lemma \ref{OPNBiconditionalDris}, we have the lower bound
$$I(q) + I(n) \leq \frac{\sigma(q)}{n} + \frac{\sigma(n)}{q}.$$
Under the assumption that the smallest prime factor $u$ of $N=q{n^2}$ satisfies $u \geq 5$, from Remark \ref{ProofTheorem1}, we get
$$\frac{q + 1}{q} + {\left(\frac{2q}{q + 1}\right)}^{\frac{\log(I(5))}{\log(I(5^2))}} \leq I(q) + {I(n^2)}^{\frac{\log(I(u))}{\log(I(u^2))}} \leq I(q) + I(n)$$
so that
$$\lim_{q\rightarrow\infty}{\left(\frac{q + 1}{q} + {\left(\frac{2q}{q + 1}\right)}^\frac{\log(6/5)}{\log(31/25)}\right)} = 1 + 2^{\frac{\log(6/5)}{\log(31/25)}} \approx 2.799.$$
This is a contradiction.
\end{proof}
\section{Concluding Remarks}
Notice that, in the proof of Theorem \ref{TheoremDris}, if we had the lower bound $u \geq 3$ for the smallest prime factor $u$ of an odd perfect number $N = q^k n^2$, then we will not be able to force a contradiction, because as $q \rightarrow \infty$:
$$2.7199 \approx 1 + 2^{\frac{\log(4/3)}{\log(13/9)}} \leftarrow \frac{q + 1}{q} + {\left(\frac{2q}{q + 1}\right)}^\frac{\log(I(3))}{\log(I(3^2))}$$
$$\leq I(q) + {I(n^2)}^{\frac{\log(I(u))}{\log(I(u^2))}} \leq I(q) + I(n) \leq \frac{\sigma(q)}{n} + \frac{\sigma(n)}{q} < 1 + \sqrt{3} \approx 2.732.$$
Furthermore, when $q = 5$, $3=(q+1)/2 \mid n^2$. It turns out that a necessary condition for $q = 5$ is the Descartes-Frenicle-Sorli conjecture that $k = 1$. This is essentially due to Iannucci (see \cite{Iannucci}). Consequently, we get $5 = q < n$. However, even if $q \ne 5$ but $q \equiv 2 \pmod 3$, then $3 \mid (q+1)/2 \mid n^2$ still holds. It thus remains to consider the case $5 \ne q \equiv 5 \pmod{12}$.
\section*{Acknowledgements}
The second author thanks Carl Pomerance for pointing out the relevance of the paper \cite{AcquaahKonyagin}, and also Peter Acquaah for helpful e-mail exchanges on odd perfect numbers. We would like to thank Severino Gervacio (the M.~Sc.~thesis adviser of the second author) for suggesting that we try to investigate Sorli's conjecture first, and for serving as our research mentor at Far Eastern University. Lastly, we thank the anonymous referee(s) who have made invaluable suggestions for improving the quality of this paper.
\makeatletter
\renewcommand{\@biblabel}[1]{[#1]\hfill}
\makeatother
| {
"timestamp": "2015-06-17T02:06:51",
"yymm": "1309",
"arxiv_id": "1309.0906",
"language": "en",
"url": "https://arxiv.org/abs/1309.0906",
"abstract": "We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality $q<n$ holds unconditionally.",
"subjects": "Number Theory (math.NT)",
"title": "On a Curious Biconditional Involving Divisors of Odd Perfect Numbers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.978384664716301,
"lm_q2_score": 0.7248702702332475,
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https://arxiv.org/abs/1403.5776 | Lyubeznik numbers for nonsingular projective varieties | In this paper, we determine completely the Lyubeznik numbers $\lambda_{i,j}(A)$ of the local ring $A$ at the vertex of the affine cone over a nonsingular projective variety $V$, where $V$ is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of $V$. In particular, we prove that these numbers are intrinsic numerical invariants of $V$, even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties in characteristic zero, which is still open. | \section{Introduction}
Let $A$ be a local ring that admits a surjection from an $m$-dimensional local ring $(R, \mathfrak{m})$ containing its residue field $k$, and let $I \subset R$ be the kernel of the surjection. (All rings in this paper are commutative and have an identity.) For non-negative integers $i$ and $j$, the local cohomology multiplicities, or Lyubeznik numbers, of $A$ are defined to be $\lambda_{i,j}(A) = \dim_k(\Ext^i_R(k, H^{m-j}_I(R)))$, the $i^{th}$ Bass number of $H^{m-j}_I(R)$ with respect to $\mathfrak{m}$. These numbers were defined in \cite{dmodules}. It is proven in \cite[Theorem 3.4(d)]{dmodules} (resp. \cite[Theorem 2.1]{huneke}) that these Bass numbers are all finite when $\ch(k) = 0$ (resp. $\ch(k) = p > 0$), and further, they depend only on $A$, $i$, and $j$, not on the choice of $R$ nor on the choice of surjection $R \rightarrow A$ \cite[Theorem-Definition 4.1]{dmodules}. Some properties of these numbers were subsequently worked out in \cite{walther} and \cite{kawasaki}, and a topological interpretation (in the case $k = \mathbb{C}$) was given in \cite{garcia}.\\
In this paper, we are concerned with the case in which $A$ is the local ring at the vertex of the affine cone over a projective variety. To be precise, let $V$ be a projective variety of dimension $r$ over a field $k$ of characteristic zero. Under an embedding $V \hookrightarrow \mathbb{P}^n_k$, we can write $V = \Proj(k[x_0, \ldots, x_n]/I)$ where $I$ is a homogeneous defining ideal for $V$. Let $\mathfrak{m} = (x_0, \ldots, x_n)$ be the homogeneous maximal ideal of $k[x_0, \ldots, x_n]$, so that $I \subset \mathfrak{m}$. Then $A = (k[x_0, \ldots, x_n]/I)_{\mathfrak{m}}$ is the local ring at the vertex of the affine cone over $V$, and we can define the multiplicities $\lambda_{i,j}(A)$ as in the preceding paragraph. In \cite[p. 133]{survey}, Lyubeznik asked whether $\lambda_{i,j}(A)$ depend only on $V$, $i$, and $j$, and not on the embedding $V \hookrightarrow \mathbb{P}^n_k$ (or, for that matter, on $n$). Zhang settled this question in the affirmative in the case of the ``top'' Lyubeznik number $\lambda_{r+1, r+1}(A)$, in any characteristic, in \cite{zhang}; he went on to give an affirmative answer for all $\lambda_{i,j}(A)$ in the $\ch(k) = p > 0$ case in \cite{zhang2}. In \cite{zhang2}, several preliminary results are established in a characteristic-free setting, but the main line of argument makes crucial use of the Frobenius morphism. This left the characteristic-zero case open for all but the ``top'' Lyubeznik number.\\
In this paper, we determine completely the Lyubeznik numbers $\lambda_{i,j}(A)$, in the case in which $V$ is a \textit{nonsingular} variety, in terms of quantities known already to be embedding-independent. Our proof uses results of Hartshorne and Ogus on algebraic de Rham cohomology, and develops the ideas of \cite[p. 54]{dmodules} and \cite[Remark 2]{garcia}. The characteristic-zero analogue of the result of \cite{zhang2} remains open for singular varieties. Our result on the embedding-independence of the integers $\lambda_{i,j}(A)$ for nonsingular varieties provides further supporting evidence for a positive answer to the question of their embedding-independence for arbitrary varieties.\\
We fix some further notation. $V$, $I$, and $A$ remain as above, except that now and for the remainder of the paper we will assume $V$ is nonsingular. Write $R = (k[x_0, \ldots, x_n])_{\mathfrak{m}}$, a regular local ring, so that $A = R/I$; it is this $R$ which will intervene in the definition of $\lambda_{i,j}(A)$ in terms of Bass numbers. The following quantities will appear repeatedly: $\dim(R) = \hgt(\mathfrak{m}) = n+1$, $\hgt(I) = \codim(V, \mathbb{P}^n_k) = n-r$, and $\dim(A) = r+1$. In particular, since $\dim(R) = n+1$, our definition of the Lyubeznik numbers of $A$ reads $\lambda_{i,j}(A) = \dim_k(\Ext^i_R(k, H^{n+1-j}_I(R)))$.\\
Finally, we remark that, as explained in \cite[$\S 8$]{zhang2}, it is harmless to assume that $k$ is algebraically closed, since Lyubeznik numbers are unaltered under extension of the base field. Under this assumption, we have the following embedding-independent description of $\lambda_{r+1, r+1}$:
\begin{theorem}\label{thm:1.1}
\cite[Theorem 2.7]{zhang} Let $V_1, \ldots, V_s$ be the $r$-dimensional irreducible components of $V$, and let $\Gamma_V$ be the graph on the vertices $V_1, \ldots, V_s$ in which $V_i$ and $V_j$ are joined by an edge if and only if $\dim(V_i \cap V_j) = r-1$. Then $\lambda_{r+1, r+1}(A)$ equals the number of connected components of $\Gamma_V$.
\end{theorem}
Remarks in \cite[$\S 4$]{dmodules} establish that $\lambda_{i,j}(A) = 0$ if either $i$ or $j$ is greater than $r+1$, Theorem \ref{thm:1.1} deals with the case $i=j=r+1$, and our Main Theorem \ref{thm:1.2} below deals with the remaining cases. This furnishes a complete description of all $\lambda_{i,j}(A)$.
\begin{maintheorem}\label{thm:1.2}
Write $\beta_j = \dim_k(H^j_{dR}(V))$, where $H_{dR}$ denotes algebraic de Rham cohomology in the sense of \cite{hartshorne}. \cite[Theorem II.6.2]{hartshorne} implies that all $\beta_j$ are finite. With the above notation and the hypothesis that $V$ is nonsingular, the following hold:
\begin{enumerate}
\item \label{claim:1} $\lambda_{i,j} = 0$ if $i>0$ and $j<r+1$;
\item \label{claim:2} $\lambda_{0,0} = 0, \lambda_{0,1} = \beta_0 - 1$;
\item \label{claim:3} $\lambda_{0,2} = \beta_1, \lambda_{0,j} = \beta_{j-1} - \beta_{j-3}$ for $j = 3, \ldots, r$;
\item \label{claim:4} $\lambda_{0, r+1} = \lambda_{1, r+1} = 0$ (cf. \cite[Theorem \textit{(c)}]{garcia});
\item \label{claim:5} $\lambda_{\ell, r+1} = \lambda_{0, r+2-\ell}$ (determined above) for $\ell = 2, \ldots, r$ (cf. \cite[Remark 1]{garcia}).
\end{enumerate}
\end{maintheorem}
Since algebraic de Rham cohomology is intrinsic to $V$ \cite[Theorem II.1.4]{hartshorne}, the above list, together with the result of \cite{zhang}, immediately implies the following:
\begin{corollary}\label{cor:1.3}
If $V$ is a nonsingular projective variety over a field $k$ of characteristic zero and $\lambda_{i,j}(A)$ is calculated as above, then for all $i$ and $j$, $\lambda_{i,j}(A)$ depends only on $V$, $i$, and $j$, not on $n$ nor on the embedding $V \hookrightarrow \mathbb{P}^n_k$.
\end{corollary}
The rest of the paper consists in a proof of the first three claims of Main Theorem \ref{thm:1.2}. Claim \eqref{claim:1} will follow from known results on the local cohomology $H_I^{n+1-j}(R)$ and from the definition of $\lambda_{i,j}$. To establish Claims \eqref{claim:2} and \eqref{claim:3}, we will first relate local cohomology supported at $I$ to local cohomology of the formal spectrum of the $I$-adic completion of $R$, then we will use a result of Ogus \cite{ogus} to relate this formal local cohomology to local de Rham cohomology. Finally, we will appeal to an exact sequence worked out in \cite{hartshorne} connecting local de Rham cohomology at the vertex of the affine cone over $V$ to the de Rham cohomology of $V$ itself. Claims \eqref{claim:4} and \eqref{claim:5} appeared first in \cite{garcia} and thus are not proven here; see also \cite{blickle} for an alternative proof using the Grothendieck composite-functor spectral sequence.\\
We would like to thank Professor Gennady Lyubeznik for suggesting the problem and for helpful conversations about the results in this paper. We also thank Professor William Messing for profitable conversations about algebraic de Rham cohomology and the hard Lefschetz theorem.
\section{Proof of Claim \eqref{claim:1}}
We need the following lemma on the support of the local cohomology modules under consideration:
\begin{lemma}\label{lem:2.1}
With the above notation, since $V$ is nonsingular, $\Supp(H_I^i(R)) \subset \{\mathfrak{m}\}$ whenever $i \neq \hgt(I) = n-r.$
\end{lemma}
\begin{proof}
Since $V$ is nonsingular, the affine cone over $V$ has an isolated singularity at its vertex, and so all its other local rings are regular local rings. That is, for any prime ideal $\mathfrak{p} \neq \mathfrak{m}$ of the coordinate ring $k[x_0, \ldots, x_n]/I$, the localization $(k[x_0, \ldots, x_n]/I)_{\mathfrak{p}} = (k[x_0, \ldots, x_n])_{\mathfrak{p}}/(I \cdot (k[x_0, \ldots, x_n])_{\mathfrak{p}}) \simeq R_{\mathfrak{p}}/I_{\mathfrak{p}} \simeq A_{\mathfrak{p}}$ is a regular local ring, where we write $I_{\mathfrak{p}} = I R_{\mathfrak{p}}$. We will show that for any prime ideal $\mathfrak{p} \neq \mathfrak{m}$ of $R$ (that is, any non-maximal homogeneous prime ideal of $k[x_0, \ldots, x_n]$), $\mathfrak{p}$ does not belong to the support of $H_I^i(R)$ for any $i \neq \hgt(I)$, \textit{i.e.} the localization $(H^i_I(R))_{\mathfrak{p}}$ is zero. This is clear if $I \not \subset \mathfrak{p}$, so we assume that $I \subset \mathfrak{p}$, in which case $\hgt(I) = \hgt(I_{\mathfrak{p}})$. By the flat base change principle for local cohomology \cite[Theorem 4.3.2]{brodmann}, we have $(H^i_I(R))_{\mathfrak{p}} \simeq H^i_{I_{\mathfrak{p}}}(R_{\mathfrak{p}})$. Since both $R_{\mathfrak{p}}$ and its quotient $R_{\mathfrak{p}}/I_{\mathfrak{p}} \simeq A_{\mathfrak{p}}$ are regular local rings, we conclude \cite[Proposition 2.2.4]{bruns} that $I_{\mathfrak{p}}$ is generated by part of a regular system of parameters of $R_{\mathfrak{p}}$, which must have $\hgt(I_{\mathfrak{p}}) = \hgt(I) = n-r$ elements. But if an ideal is generated by a regular sequence of length $n-r$, local cohomology supported at this ideal cannot be nonzero in any degree other than $n-r$.
\end{proof}
\begin{corollary}\label{cor:2.2}
If $i > \hgt(I)$, $H_I^i(R)$ is an injective $R$-module.
\end{corollary}
\begin{proof}
By \cite[Theorem 3.4(b)]{dmodules}, $\injdim(H_I^i(R)) \leq \dim \Supp(H_I^i(R))$, and the right-hand side is zero if $\Supp(H_I^i(R)) \subset \{\mathfrak{m}\}$.
\end{proof}
Now suppose $j < r+1$. Then $n+1-j > n-r = \hgt(I)$, so by the preceding corollary, $H^{n+1-j}_I(R)$ is an injective $R$-module. This implies that $\Ext^i_R(k, H^{n+1-j}_I(R))$ vanishes for all $i>0$, so that if $i>0$ and $j < r+1$, the dimension of this Ext module (which, by definition, is $\lambda_{i,j}(A)$) is zero, proving Claim \eqref{claim:1}.\qed
\section{Proofs of Claims \eqref{claim:2} and \eqref{claim:3}}
We are now concerned with computing $\lambda_{0,j}(A)$ for $0 \leq j \leq r$. By definition, this is $\dim_k \Ext^0_R(k, H_I^{n+1-j}(R)) = \dim_k \Hom_R(k, H_I^{n+1-j}(R))$, the dimension of the socle of $H^{n+1-j}_I(R)$. As discussed in the previous section, for $j < r+1$, $H^{n+1-j}_I(R)$ is an injective $R$-module supported only at $\mathfrak{m}$. By the structure theory for injective modules over Noetherian rings, $H^{n+1-j}_I(R)$ is thus isomorphic to a direct sum of copies of $E = E(R/\mathfrak{m})$, the injective hull (as $R$-module) of the residue field $k = R/\mathfrak{m}$; by \cite[Theorem 3.4(d)]{dmodules}, the number of copies is finite. Therefore we can write $H^{n+1-j}_I(R) \simeq E^{t_j}$ for some non-negative integer $t_j$. Since the socle of $E$ is one-dimensional, the socle of $H^{n+1-j}_I(R) \simeq E^{t_j}$ is $t_j$-dimensional, from which it follows that $\lambda_{0,j}(A) = t_j$ (for $0 \leq j \leq r$). We therefore turn our attention to the determination of $t_j$.\\
We may complete $R$ at the maximal ideal $\mathfrak{m}$ without changing the $\lambda_{i,j}(A)$ \cite[Lemma 4.2]{dmodules}, so we can, and will, assume that $R$ is the complete regular local ring $k[[x_0, \ldots, x_n]]$. Working over a complete local ring, we have access to the theory of Matlis duality. Let $D$ denote the Matlis dual functor from the category of $R$-modules to itself, so that $D(M) = \Hom_R(M, E)$ for an $R$-module $M$, where $E = E(R/\mathfrak{m})$ is the injective hull mentioned above. We will compute $D(H^{n+1-j}_I(R))$ in two different ways and equate the two answers. On the one hand, $D$ is an exact functor (since $E$ is injective) and $D(E) \simeq R$ \cite[Theorem 10.2.12(i)]{brodmann}, so that
\[
D(H^{n+1-j}_I(R)) \simeq D(E^{t_j}) \simeq (D(E))^{t_j} \simeq R^{t_j}
\]
for each $j$ with $0 \leq j \leq r$. On the other hand, by \cite[Proposition 2.2.3]{ogus}, we have isomorphisms $
D(H_I^{n+1-j}(R)) \simeq H_P^{j}(\hat{X}, \mathcal{O}_{\hat{X}})$ for all $j$. Here $X = \Spec(R)$, $Y \subset X$ is the closed subscheme defined by $I$, $P \in Y$ is the closed point, $\hat{X}$ is the formal completion of $X$ along $Y$ and $\mathcal{O}_{\hat{X}}$ is the structure sheaf of $\hat{X}$.\\
Equating the results of our two calculations of $D(H_I^{n+1-j}(R))$, we see that $H_P^{j}(\hat{X}, \mathcal{O}_{\hat{X}}) \simeq R^{t_j}$. So we have reduced ourselves to the calculation of the $R$-rank of $H_P^{j}(\hat{X}, \mathcal{O}_{\hat{X}})$, for which we will need algebraic de Rham cohomology. We briefly recall the definition of algebraic de Rham cohomology, referring the reader to \cite{hartshorne} for details. If $Y$ is a scheme of finite type over $k$ which admits an embedding as a closed subscheme of a smooth scheme $X$ over $k$, the algebraic de Rham cohomology $H^i_{dR}(Y)$ is by definition the hypercohomology $\mathbb{H}^i(\hat{X}, \hat{\Omega}_{X/k}^{\bullet})$ of the formal completion of the de Rham complex of $X$ along the closed subscheme $Y$. Replacing hypercohomology $\mathbb{H}^i$ with hypercohomology $\mathbb{H}^i_P$ supported in a closed point of $Y$, we get the definition of local algebraic de Rham cohomology $H^i_P(Y)$. In \cite{hartshorne}, it is proven (Theorem II.1.4) that the definition of $H^i_{dR}(Y)$ is independent of the ambient smooth scheme $X$ and of the choice of embedding. \cite[Theorem II.6.1]{hartshorne} establishes that the $H^i_{dR}(Y)$ are finite-dimensional $k$-vector spaces. Analogous embedding-independence and finite-dimensionality statements for $H^i_P(Y)$ are proven in Chapter III of \cite{hartshorne}.\\
We will use the following theorem of Ogus to relate the formal local cohomology obtained above to algebraic de Rham cohomology (we have changed Ogus's notation to match ours):
\begin{theorem}\label{thm:3.1}
\cite[Theorem 2.3]{ogus} Let $k$ be a field of characteristic zero, let $R = k[[x_0, \ldots, x_n]]$, and let $Y$ be a closed subset of $X = \Spec(R)$, defined by an ideal $I$. Let $\hat{X}$ be the formal completion of $X$ along $Y$, and let $P$ be the closed point. Assume $s$ is an integer such that $\Supp(H_I^i(R)) \subset \{P\}$ for all $i > n+1-s$. Then there are natural maps $R \otimes_k H^j_P(Y) \rightarrow H^j_P(\hat{X}, \mathcal{O}_{\hat{X}})$ which are isomorphisms for $j < s$ and injective for $j=s$, where $H^j_P(Y)$ denotes local algebraic de Rham cohomology in the sense defined above.
\end{theorem}
Notice that by Lemma \ref{lem:2.1}, the hypothesis of Ogus's theorem (that the local cohomology be supported at the maximal ideal) holds if we take $s = r + 1$. The conclusion in our case is then that, for $j < r+1 = s$, we have an isomorphism $R \otimes_k H^j_P(Y) \simeq H^j_P(\hat{X}, \mathcal{O}_{\hat{X}})$, where the right-hand side is isomorphic to $R^{t_j}$. Furthermore, by \cite[Theorem III.2.1]{hartshorne}, the local de Rham cohomology $H^j_P(Y)$ is a finite-dimensional $k$-vector space for each $j$. This, together with the isomorphism $R \otimes_k H^j_P(Y) \simeq R^{t_j}$, implies that $\lambda_{0,j}(A) = t_j = \dim_k H^j_P(Y)$, which means we have reduced ourselves further to the calculation of the dimension of this de Rham cohomology space. Ogus indicates a way to compute this dimension in \cite[Remark, p. 354]{ogus}. For the convenience of the reader, we give the full details, showing how the explicit formulas of Claims \eqref{claim:2} and \eqref{claim:3} are obtained.\\
We remark that the calculation $\lambda_{0,j}(A) = \dim_k H^j_P(Y)$ is not new, and explain here how it follows from results in the literature. By the ``Lefschetz principle'' it suffices to assume $k = \mathbb{C}$. Blickle and Bondu prove in \cite[Theorem 1.2(1)]{blickle} that $\lambda_{0,j}(A) = \dim_{\mathbb{C}} H^j_{\{P\}}(Y; \mathbb{C})$, where this last denotes local singular cohomology with complex coefficients. By the local de Rham-Betti comparison theorem \cite[\S IV.3, Remark]{hartshorne}, we have $H^j_{\{P\}}(Y; \mathbb{C}) \simeq H^j_P(Y)$. This argument could have replaced the preceding five paragraphs. However, the proof of Blickle and Bondu (which applies to cases more general than isolated singularities) requires substantial machinery of derived categories and intersection cohomology. We wished to indicate a simpler algebraic proof in our case.\\
Hartshorne in \cite{hartshorne} relates the local de Rham cohomology spaces $H^j_P(Y)$ to global de Rham cohomology of the projective variety $V$ using exact sequences:
\begin{theorem}\label{thm:3.2}
\cite[Proposition III.3.2]{hartshorne} Let $V \subset \mathbb{P}^n_k$ be a projective variety, $C \subset \mathbb{A}^{n+1}_k$ the affine cone over $V$, and $P \in C$ the vertex. Then $H_P^0(C) = 0$ and we have the following two exact sequences, where $H_P^i$ and $H_{dR}^i$ denote local algebraic de Rham cohomology and global algebraic de Rham cohomology, respectively:
\[
0 \rightarrow k \rightarrow H^0_{dR}(V) \rightarrow H_P^1(C) \rightarrow 0
\]
and
\[
0 \rightarrow H^1_{dR}(V) \rightarrow H^2_P(C) \rightarrow H^0_{dR}(V) \rightarrow H^2_{dR}(V) \rightarrow H^3_P(C) \rightarrow H^1_{dR}(V) \rightarrow \cdots
\]
Here the maps $H^i_{dR}(V) \rightarrow H^{i+2}_{dR}(V)$ for $i \geq 0$ are given by cup product with the class of a hyperplane section $\zeta \in H^2_{dR}(V)$.
\end{theorem}
By the ``strong excision theorem'' \cite[Proposition III.3.1]{hartshorne}, the $H^j_P(C)$ of the previous theorem (the de Rham cohomology of $C$ supported at the closed point $P$) and the $H^j_P(Y)$ appearing in Ogus's theorem (local de Rham cohomology of $\Spec(R/I) = \Spec(A)$) are the same, so that $\lambda_{0,j}(A) = \dim_k(H^j_P(C))$.\\
As in the statement of Main Theorem \ref{thm:1.2}, let $\beta_i$ denote the dimension of the $k$-vector space $H^i_{dR}(V)$. Theorem \ref{thm:3.2}, which is true for any projective variety over a field of characteristic zero, is already enough for us to determine $\lambda_{0,0}(A) = 0$ (from the first statement) and $\lambda_{0,1}(A) = \beta_0 - 1$ (from the short exact sequence), proving Claim \eqref{claim:2}. To prove Claim \eqref{claim:3} we will need the hypothesis that $V$ is nonsingular, and also the assumption (justified in the Introduction) that $k$ is algebraically closed. These hypotheses allow us to extract more information from the long exact sequence of Theorem \ref{thm:3.2} by using the hard Lefschetz theorem in the following form:
\begin{theorem}\label{thm:3.3}
(Hard Lefschetz theorem for algebraic de Rham cohomology) If $V$ is a nonsingular projective variety of dimension $r$ over an algebraically closed field $k$ of characteristic zero, then for each $i$ with $0 \leq i \leq r$, the map $H^{r-i}_{dR}(V) \rightarrow H^{r+i}_{dR}(V)$ defined by $i$-fold cup product with the hyperplane section $\zeta$ is an isomorphism.
\end{theorem}
\begin{proof}
The classical hard Lefschetz theorem \cite[p. 122]{griffiths} is the analogue (with $k = \mathbb{C}$) of the preceding assertion with the de Rham cohomology of $V$ replaced by the singular cohomology $H^j(V^{an};\mathbb{C})$ of the associated complex-analytic space. By Hartshorne's comparison theorem \cite[Theorem IV.1.1]{hartshorne}, we have, for all $j$, $H^j_{dR}(V) \simeq H^j(V^{an};\mathbb{C})$ (this only requires that $V$ be a scheme of finite type over $\mathbb{C}$), and this isomorphism carries the de Rham hyperplane class to the Betti hyperplane class, so that the result is true when the base field is $\mathbb{C}$. The statement for an arbitrary algebraically closed base field $k$ with $\ch(k) = 0$ follows from the statement for $k = \mathbb{C}$ by the ``Lefschetz principle''. Take a finite set of generators for the homogeneous defining ideal of $V$, in which only finitely many coefficients from $k$ appear, and consider the field $k'$ obtained by adjoining this finite set of coefficients to $\mathbb{Q}$. $V$ may thus be defined over the field $k'$: if $V'$ is the variety defined over $k'$ by the same homogeneous polynomials as $V$, then $V = V' \times_{k'} k$, and $V'$ is also nonsingular. Since $\mathbb{Q} \subset k' \subset k$ and $k$ is algebraically closed, we may embed $k'$ in $\mathbb{C}$. As $V$ is nonsingular, $H^j_{dR}(V)$ is simply the hypercohomology of the de Rham complex of $V$, which is seen immediately to commute with extensions of the scalar field because formation of the module of K\"{a}hler differentials commutes with such extensions; this reduces the hard Lefschetz theorem for $V$ over $k'$ to the hard Lefschetz theorem for $V$ over $\mathbb{C}$, which we've already seen is true.
\end{proof}
We return to the proof of Claim \eqref{claim:3}. If $j < r$ (so that $H^j_{dR}(V)$ is the source of one of the hard Lefschetz isomorphisms), the map $H^j_{dR}(V) \rightarrow H^{j+2}_{dR}(V)$ defined by cup product with $\zeta$, which occurs in the long exact sequence of Theorem \ref{thm:3.2}, is injective. That long exact sequence hence splits into short exact sequences as follows:
\[
0 \rightarrow H^1_{dR}(V) \rightarrow H^2_P(C) \rightarrow 0
\]
and, for all $j \geq 3$,
\[
0 \rightarrow H^{j-3}_{dR}(V) \rightarrow H^{j-1}_{dR}(V) \rightarrow H^j_P(C) \rightarrow 0.
\]
We see at once from these exact sequences of finite-dimensional $k$-vector spaces that $\lambda_{0,2} = \dim_k H^2_P(C) = \beta_1$ and, for $j \geq 3$, $\lambda_{0,j} = \dim_k H^j_P(C) = \beta_{j-1} - \beta_{j-3}$, proving Claim \eqref{claim:3}.\qed
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"abstract": "In this paper, we determine completely the Lyubeznik numbers $\\lambda_{i,j}(A)$ of the local ring $A$ at the vertex of the affine cone over a nonsingular projective variety $V$, where $V$ is defined over a field of characteristic zero, in terms of the dimensions of the algebraic de Rham cohomology spaces of $V$. In particular, we prove that these numbers are intrinsic numerical invariants of $V$, even though a priori their definition depends on an embedding into projective space. This provides supporting evidence for a positive answer to the question of embedding-independence for arbitrary varieties in characteristic zero, which is still open.",
"subjects": "Commutative Algebra (math.AC); Algebraic Geometry (math.AG)",
"title": "Lyubeznik numbers for nonsingular projective varieties",
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https://arxiv.org/abs/1407.6956 | The $c$-map, Tits Satake subalgebras and the search for $\mathcal{N}=2$ inflaton potentials | In this paper we address the general problem of including inflationary models exhibiting Starobinsky-like potentials into (symmetric) $\mathcal{N}=2$ supergravities. This is done by gauging suitable abelian isometries of the hypermultiplet sector and then truncating the resulting theory to a single scalar field. By using the characteristic properties of the global symmetry groups of the $\mathcal{N}=2$ supergravities we are able to make a general statement on the possible $\alpha$-attractor models which can obtained upon truncation. We find that in symmetric $\mathcal{N}=2$ models group theoretical constraints restrict the allowed values of the parameter $\alpha$ to be $\alpha=1,\,\frac{2}{3},\, \frac{1}{3}$. This confirms and generalizes results recently obtained in the literature. Our analysis heavily relies on the mathematical structure of symmetric $\mathcal{N}=2$ supergravities, in particular on the so called $c$-map connection between Quaternionic Kähler manifolds starting from Special Kähler ones. A general statement on the possible consistent truncations of the gauged models, leading to Starobinsky-like potentials, requires the essential help of Tits Satake universality classes. The paper is mathematically self-contained and aims at presenting the involved mathematical structures to a public not only of physicists but also of mathematicians. To this end the main mathematical structures and the general gauging procedure of $\mathcal{N}=2$ supergravities is reviewed in some detail. | \part{\sc Introducing the subject and motivations}
The present one is a research paper and it contains some new original results. These latter are mostly of mathematical-geometrical character and in our opinion they might be of some interest both for the mathematical scientific community, as well as for that of the theoretical physicists working in the field of supergravity/superstrings. The motivations for the present study is that of analyzing within a general geometric framework some of the recent results \cite{thesearch} relative to the inclusion of candidate inflaton potentials into extended supergravity, the aim being that of singling out general mathematical patterns that eventually can be exported to other examples that include more fields and more multiplets.
\par
From the physicist's viewpoint the obvious ultimate goal is that of singling out possible chains of inclusions of the inflationary models into hierarchically larger and more (super)-symmetric theories finally sheading light on the appropriate place of the inflaton-dynamics, which is revealed by observational cosmology, within a duely unified theory of all interactions.
\par
Notwithstanding the above mentioned specific motivations, the geometrical results presented in this paper, have an intrinsic mathematical interest and moreover admit different applications than those in cosmology within the very framework of supergravities and their gaugings.
For this reason we have chosen to make this paper self-consistent and readable by a wider audience of both mathematicians and physicists working in fields different from that of supergravity cosmological models, by presenting in a systematic way all the mathematical definitions and structures that we utilize in the original part of our work. In the same spirit, in order to provide our reader with orientation, the paper is subdivided into Parts and sections.
\par
The present first part provides a conceptual introduction and the physical motivations in relation to current research.
\par The second part provides the definitions of K\"ahler-Hodge, Special K\"ahler and Quaternionic K\"ahler manifolds. Then introduces the $c$-map \cite{sabarwhal} and illustrates how in Quaternionic K\"ahler manifolds lying in the image of $c$-map, all the quaternionic structures, in particular the HyperK\"ahler two-forms, the $\mathfrak{su}(2)$-connection and the tri-holomorphic moment maps of isometries can all be constructed purely in terms of Special geometry data.
\par
The third part discusses abelian gaugings of hypermultiplet isometries in $\mathcal{N}=2$ supergravity. Using the general mathematical formulae derived in Part Two we discuss generic properties and features of the ensuing scalar potentials.
\par
The fourth part presents concrete examples. In the particular case of the $c$-map of the $S^3$ model we retrieve the results obtained in \cite{thesearch}. Another relevant Special K\"ahler Geometry that we consider corresponds to that of the symmetric space $\mathrm{Sp(6,\mathbb{R})}/\mathrm{SU(3)\times U(1)}$. For this model we provide an in depth, full fledged construction. Analyzing its Quaternionic K\"ahler extension by means of the $c$-map we are able to generalize the results of \cite{thesearch} showing that they fall into a general pattern. Our detailed construction may have applications both in the cosmological perspective of the present paper and in the classification of Black-Hole solutions, possibly also in other contexts.
\par
Part five summarizes the results obtained in the two considered examples and shows that they unveil a deep and universal structure. We briefly summarize the organization of $\mathcal{N}=2$ scalar manifolds that are symmetric spaces into Tits--Satake universality classes
\cite{titsusataku}. Such a concept already proved to be of high value in relation to the construction and classification of supergravity black hole solutions \cite{noinilpotenti}. It is equally effective and precious in relation to the $c$-map and cosmological models. Indeed relying on these structures we are able to show that the inclusion of Starobinsky-like models into extended supergravity theories has a universal character being associated with the gauging of the universal sub-Tits-Satake subalgebra $\mathbb{G}_{\mathrm{subTS}} \, = \, \mathfrak{sl}(2) \times \mathfrak{sl}(2) \times \mathfrak{sl}(2)$. This also leads to the prediction that available values of $\alpha$ for the so named $\alpha$-attractors \cite{alfatrattori} are just $\alpha \, = \, 1,\, \frac{2}{3},\, \frac{1}{3}$. After this conceptual elaboration, part five contains our conclusions and remarks on further perspectives.
\section{\sc Introduction}
The recent observational results on the power-spectrum of the Cosmological Microwave Background radiation \cite{Ade:2013uln},\cite{Ade:2013zuv},\cite{Hinshaw:2012aka},\cite{biceppo} have stirred renewed interest in one-field inflationary cosmological models \cite{Starobinsky:1980te},\cite{lindefund},\cite{guthfund},\cite{steinhardfund}. Indeed the type of cosmology \cite{cosmology},\cite{pietrocosmobook} that seems to be consistent with observations is that based on the simplest scenario of just one scalar field $\phi$ (\textit{the inflaton}) minimally coupled to Einstein Gravity and endowed with a suitable scalar potential $V(\phi)$.
\par
In view of this, several studies have been devoted to the problem of including into supergravity potentials $V(\phi)$ that produce an early inflationary phase and have good cosmological properties, \cite{johndimitri},\cite{Ketov:2010qz},\cite{Ketov:2012jt},\cite{Kallosh:2013hoa},\cite{Kallosh:2013lkr},\cite{Farakos:2013cqa},
\cite{Kallosh:2013maa},\cite{Kallosh:2013daa},\cite{Kallosh:2013tua},\cite{alfatrattori},\cite{Ferrara:2014cca}. A separate investigation was also devoted to determine a list of one field integrable potentials \cite{augustopietrosasha} and to discuss their possible inclusion into $\mathcal{N}=1$ supergravity by means of suitable superpotentials \cite{noiGaugings}. It was shown that such type of inclusion is quite difficult and can be realized only in very few cases \cite{noiGaugings}.
\par
Notwithstanding the difficulties with the inclusion of inflaton potentials by means of a superpotential (F-terms), approximately one year ago, in the seminal paper \cite{minimalsergioKLP} it was instead pointed out that every positive-definite potential $V(\phi)$ can be minimally included into $\mathcal{N}=1$ supergravity as a D-term (see \cite{cfgv}, \cite{castdauriafre} and \cite{primosashapietro} for the complete structure of matter coupled $\mathcal{N}=1$ supergravity). It suffices to introduce a K\"ahler one-fold $\Sigma$ with an abelian group of isometries $\mathrm{G}_{\mathrm{iso}}$ and a K\"ahler potential $\mathcal{K}$ related to the potential $V$ by an appropriate differential relation which allows to interpret this latter as the square of the moment-map of the holomorphic Killing vector generating $\mathrm{G}_{\mathrm{iso}}$. From the mathematical point of view this new vision stimulated the formulation of the concept of $D$-map and its extensive study in \cite{primosashapietro} and \cite{piesashatwo}. The essential mechanism behind the $D$-map is the just the Brout-Englert-Higgs mechanism as it is realized in supergravity. Gauging an isometry of the K\"ahlerian scalar manifold $\mathcal{M}_{K}$ one generates both mass terms for the fermions and a scalar potential that is the square of the moment map $\mathcal{P}_k$ of the corresponding Killing vector. The gauge vector field $A_\mu$ utilized to gauge the considered isometry becomes massive by eating up one of the two scalar fields composing the Wess-Zumino multiplet, while its partner remains in the Lagrangian as a degree of freedom of spin zero, self interacting by means of the $D$-term potential \cite{VanProeyen:1979ks},\cite{Freedman:1976uk}.
\par
Among the positive definite scalar potentials that can be included in supergravity in this way there are the Starobinsky-like potentials \cite{Starobinsky:1980te}:
\begin{equation}\label{starobombolo}
V_{Starobinsky-like} \, = \, \mbox{const} \, \times \, \left( 1\, - \, \exp\left [ - \, \sqrt{\frac{2}{3 \, \alpha}} \, \phi\right]\right)^2\,.
\end{equation}
When $\alpha \, = \, 1$, the potential (\ref{starobombolo}) emerges in the second derivative supergravity dual of a higher derivative $R+R^2$ supergravity model \cite{Whitt:1984pd},
\cite{Cecotti:1987qe},\cite{Cecotti:1987sa},\cite{Ferrara:2013wka},\cite{Ferrara:2013kca}. Due to the high relevance of these potentials in phenomenology, several studies were devoted to the mechanisms for their inclusion in supergravity\footnote{The inclusion of the original Starobinsky $R+R^2$ model in $\mathcal{N}=2$ supergravity was discussed in \cite{thesearch} (see \cite{Ketov:2014qoa} for an earlier discussion) where it is shown to be dual to an $\mathcal{N}=2$ model with two long massive vector multiplets
on a $\mathcal{N}=2$ Minkowski vacuum. The scalaron is subject to a scalar potential of the form (\ref{starobombolo}) with $\alpha=1$. The models considered here are not dual to the $R+R^2$ Starobinsky model and share with it only the scalaron potential, which we shall refer to as the \emph{Starobinsky potential}. If $\alpha\neq 1$ the potential (\ref{starobombolo}) will be called \emph{Starobinsky-like}.}. In their minimal $D$-term realization, the Starobinsky-like potentials were shown to be generated by the gauging of the parabolic subgroups of $\mathrm{SL(2,\mathbb{R})}$ in a theory where the K\"ahler one-fold $\Sigma$ is the homogeneous space $\frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}$, with a value of the curvature directly related to $\alpha$ by:
\begin{equation}\label{kreptosio}
R_{\alpha} \, = \, - \, \frac{2}{3\,\alpha}
\end{equation}
The case $\alpha \, = \, 1$ which is the proper Starobinsky potential corresponds to a realization of the $\frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}$ geometry which is not only K\"ahlerian but actually Special K\"ahlerian. Indeed it is the case of the $S^3$ model.
\par
In more general terms the minimal supergravity approach and the $D$-map from potentials to K\"ahler geometry posed the question of relating the type of generated potential with the global topology of the isometry whose gauging generates it. This issue was thoroughly studied in \cite{pietrosergiosasha1},\cite{pietrosergiosasha2}. Relying on notions developed by Gromov et al \cite{Gromov1985},\cite{Gromov1987}, in \cite{pietrosergiosasha1},\cite{pietrosergiosasha2} it was shown that the global topology of the isometry group can be characterized as, \textit{elliptic, parabolic or hyperbolic} on general K\"ahler manifolds $\Sigma$ of non-positive curvature and that the three cases lead to different distinctive properties of the moment-maps, that either have a fixed-point at finite distance (elliptic case) or a fixed point on the boundary (parabolic) or two fixed point on the boundary (hyperbolic). In the case of constant negative curvature K\"ahler one-folds, \textit{i.e.} of $\Sigma_\alpha \, = \, \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}$ manifolds, it was shown in \cite{pietrosergiosasha1} and \cite{pietrosergiosasha2} that one generates the following three potentials:
\begin{equation}\label{ginolullobrigo}
V(\phi) \, = \, \left \{ \begin{array}{ccccc}
V_{elliptic}& = & \left( \cosh \left[\sqrt{\frac{2}{3 \, \alpha}} \, \phi\right] \, - \, \kappa\right)^2 & \Leftrightarrow & \mbox{from gauging of an elliptic subgroup} \\
V_{hyperbolic}& = & \left( \sinh \left[\sqrt{\frac{2}{3 \, \alpha}} \, \phi\right] \, - \, \kappa\right)^2 & \Leftrightarrow & \mbox{from gauging of a hyperbolic subgroup}\\
V_{parabolic}& = &\left( \exp\left [ - \, \sqrt{\frac{2}{3 \, \alpha}} \, \phi\right]\, - \, \kappa\right)^2 & \Leftrightarrow & \mbox{from gauging of a parabolic subgroup}\\
\end{array}
\right.
\end{equation}
where $\alpha$ parameterizes the curvature of the one-fold $\Sigma_\alpha$ according to eq. (\ref{kreptosio}) and the parameter $\kappa\, = \, \pm1,0$ is interpreted as the coupling constant of a Fayet Iliopoulos term \cite{Fayet:1974jb}. Utilizing instead a flat K\"ahler one-fold $\Sigma_{flat}\, = \, \mathbb{C}$, it was shown in \cite{pietrosergiosasha1} and \cite{pietrosergiosasha2} that the gauging of an elliptic isometry yields a mexican hat Brout-Englert-Higgs potential, while the quadratic potential of chaotic inflation is generated by a translation parabolic gauging.
\par
In view of the above conclusions an obvious and very much relevant question concerns the possible inclusion of the potentials (\ref{ginolullobrigo}) into extended supergravity, in particular $\mathcal{N}=2$. Such a question amounts to asking whether there are consistent one-field truncations of appropriate gauged extended supergravities that produce the considered potentials. Such a question includes two issues:
\begin{enumerate}
\item the choice of a gauging,
\item the existence of an appropriate consistent truncation.
\end{enumerate}
The first of these two issues is relevant for another important problem, that of constructing and classifying de Sitter or Minkowskian, stable or metastable, vacua in supergravity and it was extensively addressed in that context, \cite{mapietoine},\cite{gkp},\cite{uplift},\cite{kklt},
\cite{scrucca},\cite{noscale}. Let us spend some words on the second issue.
\par
The general form of the scalar field equations in supergravity is that of a $\sigma$-model with a potential, namely the following one:
\begin{equation}\label{guberator}
0 \, = \, \square \, \phi^I \, + \, \Gamma^I_{JK} (\phi)\, \partial_\mu \phi^J \, \partial^\mu \, \phi^K \, + \, \frac{\partial}{\partial \phi^I} \, V(\phi)
\end{equation}
where the scalars $\phi^I$ are coordinates of the target Riemaniann manifold $\mathcal{M}_{scalar}$, by $\Gamma^I_{JK} (\phi)$ we have denoted the Levi-Civita connection on the former and $V$ is the potential. The problem of consistent truncation boils down to the following. We consider the embedding of a $\Sigma_\alpha$ K\"ahlerian one-fold into the scalar manifold:
\begin{equation}\label{pullabacca}
\pi \, : \, \Sigma_\alpha \, \mapsto \, \mathcal{M}_{scalar}
\end{equation}
and we require that the pull-back of the field equations (\ref{guberator}) onto the surface $\Sigma_\alpha $:
\begin{equation}\label{guberator2}
0 \, = \, \phi^\star\left[\square \, \phi^I \, + \, \Gamma^I_{JK} (\phi)\, \partial_\mu \phi^J \, \partial^\mu \, \phi^K \, + \, \frac{\partial}{\partial \phi^I} \, V(\phi)\right]
\end{equation}
be consistent, namely that it reproduces always the same equations for all values of $I$. What are the restrictions on the embedding $\pi$ that guarantee such a consistency? This question can be answered in a general form when $\mathcal{M}_{scalar}$ is a symmetric homogeneous space $\mathrm{G/H}$, as it happens most frequently in supergravity models. Considering as usual the symmetric decomposition of the Lie algebra $\mathbb{G}$:
\begin{eqnarray}
\mathbb{G} & =& \mathbb{H} \, \oplus \, \mathbb{K} \nonumber\\
\left [\mathbb{H} \, , \, \mathbb{H}\right] & \subset & \mathbb{H} \nonumber\\
\left [\mathbb{H} \, , \, \mathbb{K}\right] & \subset & \mathbb{K} \nonumber\\
\left [\mathbb{K} \, , \, \mathbb{K}\right] & \subset & \mathbb{H} \label{simmetricoGH}
\end{eqnarray}
the embedding (\ref{pullabacca}) induces a homomorphism:
\begin{equation}\label{girolamo}
\pi \, : \, \mathfrak{sl}(2,\mathbb{R}) \, \mapsto \, \mathbb{G} \quad ; \quad \pi \, : \, \mathbb{O}(2) \, \mapsto \, \mathbb{H}\quad ; \quad \pi \, : \, \mathbb{K}_2 \, \mapsto \, \mathbb{K}
\end{equation}
where:
\begin{equation}\label{birillo2}
\mathfrak{sl}(2,\mathbb{R}) \, =\, \mathbb{O}(2)\, \oplus \, \mathbb{K}_2
\end{equation}
Let us introduce the centralizer algebra of the image of $\mathbb{O}(2)$ in $\mathbb{H}$, namely
\begin{equation}\label{gunther}
\forall \,\mathfrak{g} \, \in \, \mathbb{H} \quad : \quad \mathfrak{g}\, \in \, \mathbb{N}_\pi \quad \mbox{iff} \quad \left[\mathfrak{g}\, , \, \pi\left(\mathbb{O}(2)\right)\right] \, = \,0
\end{equation}
The subspace $\mathbb{K}$ decomposes into irreducible representations of $\pi\left(\mathbb{O}(2)\right)\, \oplus \, \mathbb{N}_\pi$. The embedding $\pi$ leads to a consistent truncation if such a decomposition has the following structure:
\begin{equation}\label{contrallus}
\mathbb{K} \, = \, \underbrace{\left(\mathbf{2}_1\,|\, \mathbf{1}\right)}_{\pi(\mathbb{K}_2 )} \, \oplus_{i=1}^m \, \left(\mathbf{2}_{q_i}\,|\, \mathbf{D}_i\right) \, \oplus \, \left(\mathbf{1}\,|\, \mathbf{D}_0\right)
\end{equation}
where $\mathbf{2}_{q}$ denotes a doublet representation of $\pi\left(\mathbb{O}(2)\right)$ with charge $q$ and all $\mathbf{D}_i$ and $\mathbf{D}_0$ are transitive representations, namely carry a \textit{color} of $\mathbb{N}_\pi$, the only
$\mathbb{N}_\pi$-singlet being the subspace ${\pi(\mathbb{K}_2 )}$ tangent to the embedded one-fold $\Sigma_\alpha$.
In this case setting to zero all the colored fields (those not in ${\pi(\mathbb{K}_2 )}$ ) is consistent because the equation of a colored field cannot receive contribution from two colorless singlets.
\par
In the case of $\mathcal{N}=2$ supergravity the scalar manifold is actually the direct product of two manifolds: a special K\"ahler manifold $\mathcal{M}_{SK}$ that contains the vector multiplet scalars and a Quaternionic K\"ahler $\mathcal{M}_{Q}$ manifold describing the hypermultiplet scalars. Hence one comes to the question whether the one-fold $\Sigma_\alpha$ associated with the inflaton is to be embedded in $\mathcal{M}_{SK}$ or in $\mathcal{M}_{Q}$. The significant advance introduced by \cite{thesearch} is the scenario in which $\Sigma_\alpha$ goes into $\mathcal{M}_{Q}$ and its abelian isometry is gauged by means of vector muliplets assigned to the so named Minimal Coupling Special Geometry. This choice allows for a generic stabilization of the vector multiplet scalars and allows to focus only on the properties of the Quaternionic K\"ahler manifold $\mathcal{M}_{Q}$. In \cite{thesearch} the authors considered the cases where
\begin{equation}\label{g22su2su2}
\mathcal{M}_{Q} \, = \, \frac{\mathrm{G_{(2,2)}}}{\mathrm{SU(2)} \times \mathrm{SU(2)}}\,\,\,\mbox{and}\,\,\,\,\,\, \mathcal{M}_{Q} \, =\,\frac{{\rm SO}(1,4)}{{\rm SO}(4)}\,,
\end{equation}
and it was shown how all the potentials (\ref{ginolullobrigo}) can be included with possible values of $\alpha\, = \, 1,\,\frac{2}{3},\, \frac{1}{3}$ (the value $\alpha=2/3$ could be obtained only for the quaternionic projective space). In the present paper we address the same question for generic homogeneous symmetric Quaternionic K\"ahler manifolds and we eventually arrive at the same conclusion. The three potentials (\ref{ginolullobrigo}) can always be embedded with the same possible values of $\alpha$. Our main weapons in reaching such a conclusion and proving its generality are two:
\begin{description}
\item[a)] The $c$-map from Special K\"ahler manifolds $\mathcal{M}_{\mathcal{SK}}$ to Quaternionic K\"ahler manifolds $\mathcal{M}_{\mathcal{Q}}$
\item[b)] The Tits-Satake projection and the Tits-Satake universality classes.
\end{description}
Combining these two mathematical instruments we are able to look inside the hypermultiplet manifold and to single out its inner core which is the Special K\"ahler STU-model. Working in this framework we were able to derive simple general formulae for the triholomorphic moment maps which besides their present use in cosmology might admit several other interesting and useful applications. Similarly the Tits Satake structure allows for a deep understanding of the universal character of the Starobinsky like potentials and opens the way to their further uplift to higher $\mathcal{N}$ theories in the perspective of finding \textit{microscopic interpretations} of the gaugings that generates them.
\newpage
\part{\sc Mathematical Theory of the $c$-map}
As announced in the introductory part, the aim of this section of the paper is two-fold. On the one hand we want to
review the general geometric structure of homogeneous symmetric Quaternionic K\"ahler manifolds which are in the image of the c-map and to present
in a unified fashion general analytic formulae for the complex structures, $\mathfrak{su}(2)$-connection and tri-holomorphic moment maps.
The ultimate goal is the use of such mathematical instruments in the quest of retrieving \textit{inflaton potentials} inside properly gauged $\mathcal{N}=2$ supergravity theories. On the other hand we aim at a concise, yet comprehensive presentation of Special K\"ahler geometry, Quaternionic K\"ahler geometry and the $c$-map which might be readable by mathematicians, in particular differential geometers and Lie algebrists. Indeed this mathematical subject was mostly developed by theoretical physicists and it is not widely known in the mathematical community. This is rather unfortunate, especially in the light of its relevance to the issue of the classification of nilpotent orbits, which, instead, is largely explored by mathematicians, and is quite relevant both to the classification of black hole solutions and emerges now as quite important in cosmological issues. Unfortunately the mathematical results on nilpotent orbits are obtained within frameworks that make no reference to the very particular structures of special geometry, quaternionic geometry and the magic relations of the $c$-map. Spreading awareness of this sophisticated and beautiful mathematics among mathematicians might be beneficial to both communities and it is the second aim of the following sections. A further, humbler, yet quite important aim, pursued here, is that of establishing some unified and systematic notations that might be utilized in subsequent publications devoted to a systematic exploration of the reach field of \textit{gaugings}, \textit{consistent truncations}, \textit{de Sitter vacua} and \textit{inflaton potentials}.
\section{\sc Special K\"ahler Geometry}
Special K\"ahler geometry in special coordinates was introduced
in 1984--85 by B. de Wit et al.
and E. Cremmer et al. (see pioneering papers in \cite{SKG}), where the coupling of
$\mathcal{N}=2$ vector multiplets to $\mathcal{N}=2$ supergravity was fully determined. The
more intrinsic definition of special K\"ahler geometry in terms of
symplectic bundles is due to Strominger (1990), who
obtained it in connection with the moduli spaces of
Calabi--Yau compactifications, (see \cite{defiskg}). The coordinate-independent description
and derivation of special K\"ahler geometry in the context of $\mathcal{N}=2$
supergravity is due to Castellani, D'Auria, Ferrara \cite{skgintrinsic}
and to D'Auria, Ferrara, Fre' (1991)\cite{D'Auria:1991fj}. Homogenous symmetric special K\"ahler manifolds were classified before by Cremmer and Van Proyen in \cite{ToineCremmerOld}. An early review in modern mathematical language is provided by \cite{mylecture}.
The structure of isometry group for both Special K\"ahler and Quaternionic K\"ahler manifolds was extensively studied in \cite{specHomgeo},\cite{vanderseypen}
\par
Let us summarize the relevant concepts and definitions
\subsection{\sc Hodge--K\"ahler manifolds}
\def{M(k, \IC)}{{M(k, \relax\,\hbox{$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm C}$})}}
Consider a {\sl line bundle} ${\cal L}
{\stackrel{\pi}{\longrightarrow}} {\cal M}$ over a K\"ahler
manifold ${\cal M}$. By definition this is a holomorphic vector
bundle of rank $r=1$. For such bundles the only available Chern
class is the first:
\begin{equation}
c_1 ( {\cal L} ) \, =\, \o{i}{2}
\, {\bar \partial} \,
\left ( \, h^{-1} \, \partial \, h \, \right )\, =
\, \o{i}{2} \,
{\bar \partial} \,\partial \, \mbox{log} \, h
\label{chernclass23}
\end{equation}
where the 1-component real function $h(z,{\bar z})$ is some
hermitian fibre metric on ${\cal L}$. Let $\xi (z)$ be a
holomorphic section of the line bundle ${\cal L}$: noting that
under the action of the operator ${\bar
\partial} \,\partial \, $ the term $\mbox{log} \left ({\bar \xi}({\bar z})
\, \xi (z) \right )$ yields a vanishing contribution, we conclude that
the formula in eq.(\ref{chernclass23}) for the first Chern class can be
re-expressed as follows:
\begin{equation}
c_1 ( {\cal L} ) ~=~\o{i}{2} \,
{\bar \partial} \,\partial \, \mbox{log} \,\parallel \, \xi(z) \, \parallel^2
\label{chernclass24}
\end{equation}
where $\parallel \, \xi(z) \, \parallel^2 ~=~h(z,{\bar z}) \,
{\bar \xi}({\bar z}) \,
\xi (z) $ denotes
the norm of the holomorphic section $\xi (z) $.
\par
Eq.(\ref{chernclass24}) is the starting point for the definition
of Hodge--K\"ahler manifolds. A K\"ahler manifold ${\cal M}$ is a
Hodge manifold if and only if there exists a line bundle ${\cal L}
{\stackrel{\pi}{\longrightarrow}} {\cal M}$ such that its first
Chern class equals the cohomology class of the K\"ahler two-form
$\mathrm{K}$:
\begin{equation}
c_1({\cal L} )~=~\left [ \, \mathrm{K} \, \right ]
\label{chernclass25}
\end{equation}
\par
In local terms this means that there is a holomorphic section $\xi
(z)$ such that we can write
\begin{equation}
\mathrm{K}\, =\, \o{i}{2} \, g_{ij^{\star}} \, dz^{i} \, \wedge
\, d{\bar z}^{j^{\star}} \, = \, \o{i}{2} \, {\bar \partial}
\,\partial \, \mbox{log} \,\parallel \, \xi (z) \,
\parallel^2
\label{chernclass26}
\end{equation}
Recalling the local expression of the K\"ahler metric
in terms of the K\"ahler potential
$ g_{ij^{\star}}\, =\, {\partial}_i \, {\partial}_{j^{\star}}
{\mathcal{K}} (z,{\bar z})$,
it follows from eq.(\ref{chernclass26}) that if the
manifold ${\cal M}$ is a Hodge manifold,
then the exponential of the K\"ahler potential
can be interpreted as the metric
$h(z,{\bar z}) \, = \, \exp \left ( {\cal K} (z,{\bar z})\right )$
on an appropriate line bundle ${\cal L}$.
\par
\subsection{\sc Connection on the line bundle}
On any complex line bundle ${\cal L}$ there is a canonical hermitian connection defined as :
\begin{equation}
\begin{array}{ccccccc}
{\theta}& \equiv & h^{-1} \, \partial \, h = {\o{1}{h}}\, \partial_i h \,
dz^{i} &; &
{\bar \theta}& \equiv & h^{-1} \, {\bar \partial} \, h = {\o{1}{h}} \,
\partial_{i^\star} h \,
d{\bar z}^{i^\star} \cr
\end{array}
\label{canconline}
\end{equation}
For the line-bundle advocated by the Hodge-K\"ahler structure we have
\begin{equation}
\left [ \, {\bar \partial}\,\theta \, \right ] \, = \,
c_1({\cal L}) \, = \, [\mathrm{K}]
\label{curvc1}
\end{equation}
and since the fibre metric $h$ can be identified with the
exponential of the K\"ahler potential we obtain:
\begin{equation}
\begin{array}{ccccccc}
{\theta}& = & \partial \,{\cal K} = \partial_i {\cal K}
dz^{i} & ; &
{\bar \theta}& = & {\bar \partial} \, {\cal K} =
\partial_{i^\star} {\cal K}
d{\bar z}^{i^\star}\cr
\end{array}
\label{curvconline}
\end{equation}
To define special K\"ahler geometry, in addition to the afore-mentioned line--bundle
${\cal L}$ we need a flat holomorphic vector bundle ${\cal SV}
\, \longrightarrow \, {\cal M}$ whose sections play an important role in the construction of the supergravity Lagrangians. For reasons
intrinsic to such constructions the rank of the vector bundle ${\cal SV}$ must be $2\, n_V$ where $n_V$ is the total number of vector fields in the theory. If we have $n$-vector multiplets the total number of vectors is $n_V = n+1$ since, in addition to the vectors of the vector multiplets, we always have the graviphoton sitting in the graviton multiplet. On the other hand the total number of scalars is $2 n$. Suitably paired into $n$-complex fields $z^i$, these scalars span the $n$ complex dimensions of the base manifold ${\cal M}$ to the rank $2n+2$ bundle ${\cal SV}
\, \longrightarrow \, {\cal M}$.
\par
In the sequel we make extensive use of covariant derivatives with
respect to the canonical connection of the line--bundle ${\cal
L}$. Let us review its normalization. As it is well known there
exists a correspondence between line--bundles and
$\mathrm{U(1)}$--bundles. If $\mbox{exp}[f_{\alpha\beta}(z)]$ is
the transition function between two local trivializations of the
line--bundle ${\cal L} {\stackrel{\pi}{\longrightarrow}} {\cal
M}$, the transition function in the corresponding principal
$\mathrm{U(1)}$--bundle ${\cal U} \, \longrightarrow {\cal M}$ is
just $\mbox{exp}[{\rm i}{\rm Im}f_{\alpha\beta}(z)]$ and the
K\"ahler potentials in two different charts are related by: ${\cal
K}_\beta = {\cal K}_\alpha + f_{\alpha\beta} + {\bar
{f}}_{\alpha\beta}$. At the level of connections this
correspondence is formulated by setting: $\mbox{
$\mathrm{U(1)}$--connection} \equiv {\cal Q} \, = \,
\mbox{Im} \theta = -{\o{\rm i}{2}} \left ( \theta - {\bar
\theta} \right)$. If we apply this formula to the case of the
$\mathrm{U(1)}$--bundle ${\cal U} \, \longrightarrow \, {\cal M}$
associated with the line--bundle ${\cal L}$ whose first Chern
class equals the K\"ahler class, we get:
\begin{equation}
{\cal Q} = {\o{\rm i}{2}} \left ( \partial_i {\cal K}
dz^{i} -
\partial_{i^\star} {\cal K}
d{\bar z}^{i^\star} \right )
\label{u1conect}
\end{equation}
Let now
$\Phi (z, \bar z)$ be a section of ${\cal U}^p$. By definition its
covariant derivative is $ \nabla \Phi = (d - i p {\cal Q}) \Phi $
or, in components,
\begin{equation}
\begin{array}{ccccccc}
\nabla_i \Phi &=&
(\partial_i + {1\over 2} p \partial_i {\cal K}) \Phi &; &
\nabla_{i^*}\Phi &=&(\partial_{i^*}-{1\over 2} p \partial_{i^*} {\cal K})
\Phi \cr
\end{array}
\label{scrivo2}
\end{equation}
A covariantly holomorphic section of ${\cal U}$ is defined by the equation:
$ \nabla_{i^*} \Phi = 0 $.
We can easily map each section $\Phi (z, \bar z)$
of ${\cal U}^p$
into a section of the line--bundle ${\cal L}$ by setting:
\begin{equation}
\widetilde{\Phi} = e^{-p {\cal K}/2} \Phi \, .
\label{mappuccia}
\end{equation}
With this position we obtain:
\begin{equation}
\begin{array}{ccccccc}
\nabla_i \widetilde{\Phi}& =&
(\partial_i + p \partial_i {\cal K})
\widetilde{\Phi}& ; &
\nabla_{i^*}\widetilde{\Phi}&=& \partial_{i^*} \widetilde{\Phi}\cr
\end{array}
\end{equation}
Under the map of eq.(\ref{mappuccia}) covariantly holomorphic sections
of ${\cal U}$ flow into holomorphic sections of ${\cal L}$
and viceversa.
\subsection{\sc Special K\"ahler Manifolds}
We are now ready to give the first of two equivalent definitions of special K\"ahler
manifolds:
\begin{definizione}
A Hodge K\"ahler manifold is {\bf Special K\"ahler (of the local type)}
if there exists a completely symmetric holomorphic 3-index section $W_{i
j k}$ of $(T^\star{\cal M})^3 \otimes {\cal L}^2$ (and its
antiholomorphic conjugate $W_{i^* j^* k^*}$) such that the following
identity is satisfied by the Riemann tensor of the Levi--Civita
connection:
\begin{eqnarray}
\partial_{m^*} W_{ijk}& =& 0 \quad \partial_m W_{i^* j^* k^*}
=0 \nonumber \\
\nabla_{[m} W_{i]jk}& =& 0
\quad \nabla_{[m}W_{i^*]j^*k^*}= 0 \nonumber \\
{\cal R}_{i^*j\ell^*k}& =& g_{\ell^*j}g_{ki^*}
+g_{\ell^*k}g_{j i^*} - e^{2 {\cal K}}
W_{i^* \ell^* s^*} W_{t k j} g^{s^*t}
\label{specialone}
\end{eqnarray}
\label{defspecial}
\end{definizione}
In the above equations $\nabla$ denotes the covariant derivative with
respect to both the Levi--Civita and the $\mathrm{U(1)}$ holomorphic connection
of eq.(\ref{u1conect}).
In the case of $W_{ijk}$, the $\mathrm{U(1)}$ weight is $p = 2$.
\par
Out of the $W_{ijk}$ we can construct covariantly holomorphic
sections of weight 2 and - 2 by setting:
\begin{equation}
C_{ijk}\,=\,W_{ijk}\,e^{ {\cal K}} \quad ; \quad
C_{i^\star j^\star k^\star}\,=\,W_{i^\star j^\star k^\star}\,e^{ {\cal K}}
\label{specialissimo}
\end{equation}
The flat bundle mentioned in the previous subsection apparently does not appear in this definition of special geometry.
Yet it is there. It is indeed the essential ingredient in the second definition whose equivalence to the first we shall
shortly provide.
\par
Let ${\cal L} {\stackrel{\pi}{\longrightarrow}} {\cal M}$ denote
the complex line bundle whose first Chern class equals the
cohomology class of the K\"ahler form $\mathrm{K}$ of an
$n$-dimensional Hodge--K\"ahler manifold ${\cal M}$. Let ${\cal
SV} \, \longrightarrow \,{\cal M}$ denote a holomorphic flat
vector bundle of rank $2n+2$ with structural group
$\mathrm{Sp(2n+2,\mathbb{R})}$. Consider tensor bundles of the
type ${\cal H}\,=\,{\cal SV} \otimes {\cal L}$. A typical
holomorphic section of such a bundle will be denoted by ${\Omega}$
and will have the following structure:
\begin{equation}
{\Omega} \, = \, {\twovec{{X}^\Lambda}{{F}_ \Sigma} } \quad
\Lambda,\Sigma =0,1,\dots,n
\label{ololo}
\end{equation}
By definition
the transition functions between two local trivializations
$U_i \subset {\cal M}$ and $U_j \subset {\cal M}$
of the bundle ${\cal H}$ have the following form:
\begin{equation}
{\twovec{X}{ F}}_i=e^{f_{ij}} M_{ij}{\twovec{X}{F}}_j
\end{equation}
where $f_{ij}$ are holomorphic maps $U_i \cap U_j \, \rightarrow
\,\relax\,\hbox{$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm C}$} $
while $M_{ij}$ is a constant $\mathrm{Sp(2n+2,\mathbb{R})}$ matrix. For a consistent
definition of the bundle the transition functions are obviously
subject to the cocycle condition on a triple overlap:
$e^{f_{ij}+f_{jk}+f_{ki}} = 1 $ and $ M_{ij} M_{jk} M_{ki} = 1 $.
\par
Let ${\rm i}\langle\ \vert\ \rangle$ be the compatible
hermitian metric on $\cal H$
\begin{equation}
{\rm i}\langle \Omega \, \vert \, \bar \Omega \rangle \, \equiv \,-
{\rm i} \Omega^T \twomat {0} {\relax{\rm 1\kern-.35em 1}} {-\relax{\rm 1\kern-.35em 1}}{0} {\bar \Omega}
\label{compati}
\end{equation}
\begin{definizione}
We say that a Hodge--K\"ahler manifold ${\cal M}$
is {\bf special K\"ahler} if there exists
a bundle ${\cal H}$ of the type described above such that
for some section $\Omega \, \in \, \Gamma({\cal H},{\cal M})$
the K\"ahler two form is given by:
\begin{equation}
\mathrm{K}= \o{\rm i}{2}
\partial \bar \partial \, \mbox{\rm log} \, \left ({\rm i}\langle \Omega \,
\vert \, \bar \Omega
\rangle \right )=\frac{i}{2}\,g_{i, j^*}\,dz^i\wedge d\bar{z}^{j^*}\,.
\label{compati1} .
\end{equation}
\end{definizione}
From the point of view of local properties, eq.(\ref{compati1})
implies that we have an expression for the K\"ahler potential
in terms of the holomorphic section $\Omega$:
\begin{equation}
{\cal K}\, = \, -\mbox{log}\left ({\rm i}\langle \Omega \,
\vert \, \bar \Omega
\rangle \right )\,
=\, -\mbox{log}\left [ {\rm i} \left ({\bar X}^\Lambda F_\Lambda -
{\bar F}_\Sigma X^\Sigma \right ) \right ]
\label{specpot}
\end{equation}
The relation between the two definitions of special manifolds is
obtained by introducing a non--holomorphic section of the bundle
${\cal H}$ according to:
\begin{equation}
V \, = \, \twovec{L^{\Lambda}}{M_\Sigma} \, \equiv \, e^{{\cal K}/2}\Omega
\,= \, e^{{\cal K}/2} \twovec{X^{\Lambda}}{F_\Sigma}
\label{covholsec}
\end{equation}
so that eq.(\ref{specpot}) becomes:
\begin{equation}
1 \, = \, {\rm i}\langle V \,
\vert \, \bar V
\rangle \,
= \, {\rm i} \left ({\bar L}^\Lambda M_\Lambda -
{\bar M}_\Sigma L^\Sigma \right )
\label{specpotuno}
\end{equation}
Since $V$ is related to a holomorphic section by eq.(\ref{covholsec})
it immediately follows that:
\begin{equation}
\nabla_{i^\star} V \, = \, \left ( \partial_{i^\star} - {\o{1}{2}}
\partial_{i^\star}{\cal K} \right ) \, V \, = \, 0
\label{nonsabeo}
\end{equation}
On the other hand, from eq.(\ref{specpotuno}), defining:
\begin{eqnarray}
U_i & = & \nabla_i V = \left ( \partial_{i} + {\o{1}{2}}
\partial_{i}{\cal K} \right ) \, V \equiv
\twovec{f^{\Lambda}_{i} }{h_{\Sigma\vert i}}\nonumber\\
{\bar U}_{i^\star} & = & \nabla_{i^\star}{\bar V} = \left ( \partial_{i^\star} + {\o{1}{2}}
\partial_{i^\star}{\cal K} \right ) \, {\bar V} \equiv
\twovec{{\bar f}^{\Lambda}_{i^\star} }{{\bar h}_{\Sigma\vert i^\star}}
\label{uvector}
\end{eqnarray}
it follows that:
\begin{equation}
\label{ctensor}
\nabla_i U_j = {\rm i} C_{ijk} \, g^{k\ell^\star} \, {\bar U}_{\ell^\star}
\end{equation}
where $\nabla_i$ denotes the covariant derivative containing both
the Levi--Civita connection on the bundle ${\cal TM}$ and the
canonical connection $\theta$ on the line bundle ${\cal L}$.
In eq.(\ref{ctensor}) the symbol $C_{ijk}$ denotes a covariantly
holomorphic (
$\nabla_{\ell^\star}C_{ijk}=0$) section of the bundle
${\cal TM}^3\otimes{\cal L}^2$ that is totally symmetric in its indices.
This tensor can be identified with the tensor of eq.(\ref{specialissimo})
appearing in eq.(\ref{specialone}).
Alternatively, the set of differential equations:
\begin {eqnarray}
&&\nabla _i V = U_i\\
&& \nabla _i U_j = {\rm i} C_{ijk} g^{k \ell^\star} U_{\ell^\star}\\
&& \nabla _{i^\star} U_j = g_{{i^\star}j} V\\
&&\nabla _{i^\star} V =0 \label{defaltern}
\end{eqnarray}
with V satisfying eq.s (\ref{covholsec}, \ref {specpotuno}) give yet
another definition of special geometry.
In particular it is easy to find eq.(\ref{specialone})
as integrability conditions of(\ref{defaltern})
\subsection{\sc The vector kinetic matrix $\mathcal{N}_{\Lambda\Sigma}$ in special geometry}
\label{scrittaN}
In the construction of supergravity actions another essential item is the complex symmetric matrix $\mathcal{N}_{\Lambda\Sigma}$ whose real and imaginary parts are necessary in order to write the kinetic terms
of the vector fields. The matrix $\mathcal{N}_{\Lambda\Sigma}$ constitutes an integral part of the Special Geometry set up and we provide its general definition in the following lines.
Explicitly $\mathcal{N}_{\Lambda\Sigma}$ which, in relation to its interpretation in the case of Calabi-Yau threefolds, is named
the {\it period matrix}, is defined by means of the following relations:
\begin{equation}
{\bar M}_\Lambda = {{\cal N}}_{\Lambda\Sigma}{\bar L}^\Sigma \quad ;
\quad
h_{\Sigma\vert i} = { {\cal N}}_{\Lambda\Sigma} f^\Sigma_i
\label{etamedia}
\end{equation}
which can be solved introducing the two $(n+1)\times (n+1)$ vectors
\begin{equation}
f^\Lambda_I = \twovec{f^\Lambda_i}{{\bar L}^\Lambda} \quad ; \quad
h_{\Lambda \vert I} = \twovec{h_{\Lambda \vert i}}{{\bar M}_\Lambda}
\label{nuovivec}
\end{equation}
and setting:
\begin{equation}
{{\cal N}}_{\Lambda\Sigma}= h_{\Lambda \vert I} \circ \left (
f^{-1} \right )^I_{\phantom{I} \Sigma}
\label{intriscripen}
\end{equation}
\par
Let us now consider the case where the Special K\"ahler manifold $\mathcal{SK}_n$ of complex dimension $n$ has some isometry group
$\mathrm{U}_{\mathcal{SK}}$. Compatibility with the Special Geometry structure requires the existence of a $2n+2$-dimensional symplectic representation of such a group that we name the $\mathbf{W}$ representation.
In other words that there necessarily exists a symplectic embedding of the isometry group $\mathcal{SK}_n$
\begin{equation}
\mathrm{U}_{\mathcal{SK}} \mapsto \mathrm{Sp(2n+2, \mathbb{R})}
\label{sympembed}
\end{equation}
such that for each element $\xi \in \mathrm{U}_{\mathcal{SK}}$ we have its
representation by means of a suitable real symplectic matrix:
\begin{equation}
\xi \mapsto \Lambda_\xi \equiv \left( \begin{array}{cc}
A_\xi & B_\xi \\
C_\xi & D_\xi \
\end{array} \right)
\label{embeddusmatra}
\end{equation}
satisfying the defining relation (in terms of the symplectic antisymmetric metric $\mathbb{C}$):
\begin{equation}
\Lambda_\xi ^T \, \underbrace{\left( \begin{array}{cc}
\mathbf{0}_{n \times n} & { \mathbf{1}}_{n \times n} \\
-{ \mathbf{1}}_{n \times n} & \mathbf{0}_{n \times n} \
\end{array} \right)}_{ \equiv \, \mathbb{C}} \, \Lambda_\xi = \underbrace{\left( \begin{array}{cc}
\mathbf{0}_{n \times n} & { \mathbf{1}}_{n \times n} \\
-{ \mathbf{1}}_{n \times n} & \mathbf{0}_{n \times n} \
\end{array} \right)}_{\mathbb{C}}
\label{definingsympe}
\end{equation}
which implies the following relations on the $n \times n$ blocks:
\begin{eqnarray}
A^T_\xi \, C_\xi - C^T_\xi \, A_\xi & = & 0 \nonumber\\
A^T_\xi \, D_\xi - C^T_\xi \, B_\xi& = & \mathbf{1}\nonumber\\
B^T_\xi \, C_\xi - D^T_\xi \, A_\xi& = & - \mathbf{1}\nonumber\\
B^T_\xi \, D_\xi - D^T_\xi \, B_\xi & = & 0 \label{symplerele}
\end{eqnarray}
Under an element of the isometry group the symplectic section $\Omega$ of Special Geometry transforms
as follows:
\begin{equation}
\Omega\left( \xi \, \cdot \, z\right) \, = \, \Lambda_\xi \, \Omega\left ( z \right )
\end{equation}
As a consequence of its definition, under the same isometry the matrix ${\cal N}$ transforms by means of a generalized linear fractional
transformation:
\begin{equation}
\mathcal{N}\left(\xi \cdot z,\xi \cdot \bar{z}\right) = \left( C_\xi + D_\xi \, \mathcal{N}(z,\bar{z})\right) \, \left( A_\xi + B_\xi \,\mathcal{N}(z,\bar{z})\right) ^{-1}
\label{Ntransfa}
\end{equation}
\subsection{\sc The holomorphic moment map on K\"ahler manifolds}
The concept of holomorphic moment map applies to all K\"ahler manifolds, not necessarily special. Indeed it can be constructed just in terms of the K\"ahler potential without advocating any further structure. In this subsection we review its properties and definition, as usual in order to fix conventions, normalizations and notations.
\par
Let $g_{i {j^\star}}$ be the K\"ahler metric of a K\"ahler
manifold ${\cal M}$ and let us assume that $g_{i {j^\star}}$ admits
a non trivial group of continuous isometries ${\cal G}$
generated by Killing vectors $k_\mathbf{I}^i$ ($\mathbf{I}=1, \ldots, {\rm dim}
\,{\cal G} )$ that define the infinitesimal variation of the complex
coordinates $z^i$ under the group action:
\begin{equation}
\label{urca1}
z^i \to z^i + \epsilon^\mathbf{I} k_\mathbf{I}^i (z)
\end{equation}
Let $k^i_{\mathbf{I}} (z)$ be a basis of holomorphic Killing vectors for
the metric $g_{i{j^\star}}$. Holomorphicity means the following
differential constraint:
\begin{equation}
\partial_{j^*} k^i_{\mathbf{I}} (z)=0
\leftrightarrow \partial_j k^{i^*}_{\mathbf{I}} (\bar z)=0 \label{holly}
\end{equation}
while the generic Killing equation (suppressing the
gauge index $\mathbf{I}$):
\begin{equation}
\nabla_\mu k_\nu +\nabla_\mu k_\nu=0
\end{equation}
in holomorphic indices reads as follows:
\begin{equation}
\begin{array}{ccccccc}
\nabla_i k_{j} + \nabla_j k_{i} &=&0 & ; &
\nabla_{i^*} k_{j} + \nabla_j k_{i^*} &=& 0
\label{killo}
\end{array}
\end{equation}
where the covariant components are defined as
$k_{j }=g_{j i^*} k^{i^*}$ (and similarly for
$k_{i^*}$).
\par
The vectors $k_{\mathbf{I}}^i$ are generators of infinitesimal
holomorphic coordinate transformations $\delta z^i = \epsilon^\mathbf{I} k^i_{\mathbf{I}} (z)$
which leave the metric invariant. In the same way as the metric is
the derivative of a more fundamental
object, the Killing vectors in a K\"ahler manifold are the
derivatives of suitable prepotentials. Indeed the first of
eq.s (\ref{killo}) is automatically satisfied by holomorphic vectors
and the second equation reduces to the following one:
\begin{equation}
k^i_{\mathbf{I}}=i g^{i j^*} \partial_{j^*} {\cal P}_{\mathbf{I}},
\quad {\cal P}^*_{\mathbf{I}} = {\cal P}_{\mathbf{I}}\label{killo1}
\end{equation}
In other words if we can find a real function ${\cal P}^\mathbf{I}$ such
that the expression $i g^{i j^*} \partial_{j^*}
{\cal P}_{(\mathbf{I})}$ is holomorphic, then eq.(\ref{killo1}) defines a
Killing vector.
\par
The construction of the Killing prepotential can be stated in a more
precise geometrical fashion through the notion of {\it moment map}.
Let us review this construction.
\par
Consider a K\"ahlerian manifold ${\cal M}$ of real dimension $2n$.
Consider an isometry group ${\cal G}$ acting on
${\cal M}$ by means of Killing vector
fields $\overrightarrow{X}$ which are holomorphic
with respect to the complex structure
${ J}$ of ${\cal M}$; then these vector
fields preserve also the K\"ahler 2-form
\begin{equation}
\begin{array}{ccc}
\matrix{
{\cal L}_{\scriptscriptstyle\overrightarrow{X}}g = 0 & \leftrightarrow &
\nabla_{(\mu}X_{\nu)}=0 \cr
{\cal L}_{\scriptscriptstyle\overrightarrow{X}}{ J}= 0 &\null &\null \cr }
\Biggr \} & \Rightarrow & 0={\cal L}_{\scriptscriptstyle\overrightarrow{X}}
K = i_{\scriptscriptstyle\overrightarrow{X}}
dK+d(i_{\scriptscriptstyle\overrightarrow{X}}
K) = d(i_{\scriptscriptstyle\overrightarrow{X}}K) \cr
\end{array}
\label{holkillingvectors}
\end{equation}
Here ${\cal L}_{\scriptscriptstyle\overrightarrow{X}}$ and
$i_{\scriptscriptstyle\overrightarrow{X}}$
denote respectively the Lie derivative along
the vector field $\overrightarrow{X}$ and the contraction
(of forms) with it.
\par
If ${\cal M}$ is simply connected,
$d(i_{\overrightarrow{X}}K)=0$ implies the existence
of a function ${\cal P}_{\overrightarrow{X}}$ such
that
\begin{equation}
-\frac{1}{2}d{\cal P}_{\overrightarrow{X}}=
i_{\scriptscriptstyle\overrightarrow{X}}K
\label{mmap}
\end{equation}
The function ${\cal P}_{\overrightarrow{X}}$ is defined up to a constant,
which can be arranged so as to make it equivariant:
\begin{equation}
\overrightarrow{X} {\cal P}_{\overrightarrow{Y}} =
{\cal P}_{[\overrightarrow{X},\overrightarrow{Y}]}
\label{equivarianza}
\end{equation}
${\cal P}_{\overrightarrow{X}}$ constitutes then a {\it moment map}.
This can be regarded as a map
\begin{equation}
{\cal P}: {\cal M} \, \longrightarrow \,
\mathbb{R} \otimes
{\mathbb{G} }^*
\end{equation}
where ${\mathbb{G}}^*$ denotes the dual of the Lie algebra
${\mathbb{G} }$ of the group ${\cal G}$.
Indeed let $x\in {\mathbb{G} }$ be the Lie algebra element
corresponding to the Killing vector $\overrightarrow{X}$; then, for a given
$m\in {\cal M}$
\begin{equation}
\mu (m)\, : \, x \, \longrightarrow \, {\cal P}_{\overrightarrow{X}}(m) \,
\in \, \mathbb{R}
\end{equation}
is a linear functional on ${\mathbb{G}}$.
If we expand
$\overrightarrow{X} = a^\mathbf{I} k_\mathbf{I}$ in a basis of Killing vectors
$k_\mathbf{I}$ such that
\begin{equation}
[k_\mathbf{I}, k_\mathbf{L}]= f_{\mathbf{I} \mathbf{L}}^{\ \ \mathbf{K}} k_\mathbf{K}
\label{blio}
\end{equation}
we have also
\begin{equation}
{\cal P}_{\overrightarrow{X}}\, = \, a^\mathbf{I} {\cal P}_\mathbf{I}
\end{equation}
In the following we use the
shorthand notation ${\cal L}_\mathbf{I}, i_\mathbf{I}$ for the Lie derivative
and the contraction along the chosen basis of Killing vectors $ k_\mathbf{I}$.
\par
From a geometrical point of view the prepotential,
or moment map, ${\cal P}_\mathbf{I}$ is the Hamiltonian function providing the Poissonian
realization of the Lie algebra on the K\"ahler manifold. This
is just another way of stating the already mentioned
{\it equivariance}.
Indeed the very existence of the closed 2-form $K$ guarantees that
every K\"ahler space is a symplectic manifold and that we can define a
Poisson bracket.
\par
Consider eqs.(\ref{killo1}). To every generator of the abstract Lie algebra
${\mathbb{G}}$ we have associated a function ${\cal P}_\mathbf{I}$ on
${\cal M}$; the Poisson bracket of
${\cal P}_\mathbf{I}$ with ${\cal P}_\mathbf{J}$ is defined as follows:
\begin{equation}
\{{\cal P}_\mathbf{I} , {\cal P}_\mathbf{J}\} \equiv 4\pi K
(\mathbf{I}, \mathbf{J})
\end{equation}
where $K(\mathbf{I}, \mathbf{J})
\equiv K (\vec k_\mathbf{I}, \vec k_\mathbf{J})$ is
the value of $K$ along the pair of Killing vectors.
\par
In reference \cite{D'Auria:1991fj} the following
lemma was proved:
\begin{lemma}
{\it{The following identity is true}}:
\begin{equation}
\{{\cal P}_\mathbf{I}, {\cal
P}_\mathbf{J}\}=f_{\mathbf{I}\mathbf{J}}^{\ \ \mathbf{L}}{\cal
P}_\mathbf{L} + C_{\mathbf{I} \mathbf{J}} \label{brack}
\end{equation}
{\it{where $C_{\mathbf{I} \mathbf{J}}$ is a constant fulfilling the
cocycle condition}}
\begin{equation}
f^{\ \ \mathbf{L}}_{\mathbf{I}\mathbf{M}} C_{\mathbf{L} \mathbf{J}} +
f^{\ \ \mathbf{L}}_{\mathbf{M}\mathbf{J}} C_{\mathbf{L} \mathbf{I}}+
f_{\mathbf{J}\mathbf{I}}^{\ \ \mathbf{L}} C_{\mathbf{L} \mathbf{M}}=0
\label{cocy}
\end{equation}
\end{lemma}
If the Lie algebra ${\mathbb{G}}$ has a trivial second cohomology group
$H^2({\mathbb{G}})=0$, then the cocycle $C_{\mathbf{I} \mathbf{J}}$ is a
coboundary; namely we have
\begin{equation}
C_{\mathbf{I} \mathbf{J}} = f^{\ \ \mathbf{L}}_{\mathbf{I} \mathbf{J}} C_\mathbf{L}
\end{equation}
where $C_\mathbf{L}$ are suitable constants. Hence, assuming
$H^2 (\mathbb{G})= 0$
we can reabsorb $C_\mathbf{L}$ in the definition of ${\cal
P}_\mathbf{I}$:
\begin{equation}
{\cal P}_\mathbf{I} \rightarrow {\cal P}_\mathbf{I}+ C_\mathbf{I}
\end{equation}
and we obtain the stronger equation
\begin{equation}
\{{\cal P}_\mathbf{I}, {\cal P}_\mathbf{J}\} =
f_{\mathbf{I}\mathbf{J}}^{\ \ \mathbf{L}} {\cal P}_\mathbf{L}
\label{2.39}
\end{equation}
Note that $H^2({\mathbb{G}}) = 0$ is true for all semi-simple Lie
algebras.
Using eq.(\ref{brack}), eq.(\ref{2.39})
can be rewritten in components as follows:
\begin{equation}
{i\over 2} g_{ij^*}(k^i_\mathbf{I} k^{j^*}_\mathbf{J} -
k^i_\mathbf{J} k^{j^*}_\mathbf{I})=
{1\over 2} f_{\mathbf{I} \mathbf{J}}^{\ \ \mathbf{L}} {\cal
P}_\mathbf{L}
\label{2.40}
\end{equation}
Equation (\ref{2.40}) is identical with the equivariance condition
in eq.(\ref{equivarianza}).
\par
Finally let us recall the explicit general way of solving eq.(\ref{mmap}) obtaining the real valued function ${\cal P}_\mathbf{I}$ which satisfies eq.(\ref{killo1}). In terms of the K\"ahler potential $\mathcal{K}$ we have:
\begin{equation}\label{sisalvichipuo}
\mathcal{P}_{{\bf I}}{}^x=-\frac{i}{2}\left(k_{{\bf I}}^i\partial_i \mathcal{K}-k_{{\bf I}}^{\bar{\imath}}\partial_{\bar{\imath}} \mathcal{K}\right)+{\rm Im}(f_{{\bf I}})\,,
\end{equation}
where $f_{{\bf I}}=f_{{\bf I}}(z)$ is a holomorphic transformation on the line-bundle, defining a compensating K\"ahler transformation:
\begin{equation}
k_{{\bf I}}^i\partial_i \mathcal{K}+k_{{\bf I}}^{\bar{\imath}}\partial_{\bar{\imath}} \mathcal{K}=-f_{{\bf I}}(z)-\bar{f}_{{\bf I}}(\bar{z})\,.\label{sisalvichipuo2}
\end{equation}
We also have:
\begin{eqnarray}
\mathfrak{T}_{{\bf I}}\cdot \Omega &=&\mathfrak{T}_{{\bf I}}\cdot \Omega+f_{{\bf I}}\,\Omega\,,\label{sisalvichipuo30}\\
\mathfrak{T}_{{\bf I}}\cdot V+i\,{\rm Im}(f_{{\bf I}})\,V
&=& k_{{\bf I}}^i\partial_i V+k_{{\bf I}}^{\bar{\imath}}\partial_{\bar{\imath}} V\,,\label{sisalvichipuo3}
\end{eqnarray}
where $\mathfrak{T}_{{\bf I}}\cdot \Omega$ denotes the symplectic action of the isometry on the section $V$. If $\mathfrak{T}_{{\bf I}}$ is represented by the symplectic matrix $(\mathfrak{T}_{{\bf I}})_\alpha{}^\beta=-(\mathfrak{T}_{{\bf I}})^\beta{}_\alpha$, $\alpha,\,\beta=1,\dots,\,2n+2$:
\begin{equation}\label{cosasimplettica}
\mathfrak{T}_{\mathbf{I}}^T \, \mathbb{C} \, + \, \mathbb{C} \, \mathfrak{T}_{\mathbf{I}} \, = \, 0
\end{equation}
we have $(\mathfrak{T}_{{\bf I}}\cdot V)^\alpha=-\mathfrak{T}_{{\bf I}\,\beta}{}^\alpha\, V^\beta=\mathfrak{T}_{{\bf I}}^\alpha{}_\beta\, V^\beta$. From (\ref{sisalvichipuo3}) and (\ref{sisalvichipuo}) we derive the following useful symplectic-invariant expression for the moment maps:
\begin{equation}
\mathcal{P}_{{\bf I}}{}^x=-\bar{V}^\alpha\,\mathfrak{T}_{{\bf I}\,\alpha}{}^\beta\mathbb{C}_{\beta\gamma}\,V^\gamma\,.
\end{equation}
Eq.s (\ref{sisalvichipuo}), (\ref{sisalvichipuo2}), (\ref{sisalvichipuo3}) generalize the corresponding formulae given in sections 7.1 and 7.2 of \cite{Andrianopoli:1996cm}, where the condition $f_{{\bf I}}=0$ was imposed, to gaugings of non-compact isometries which are associated with non-trivial compensating K\"ahler transformations and/or to gauged (non-compact) isometries whose symplectic action is not diagonal.
\section{\sc Quaternionic geometry}
\label{hypgeosec}
Next we turn our attention to the geometry that pertains to the hypermultiplet sector of an
$\mathcal{N}=2$ supersymmetric theory. Each hypermultiplet contains $4$ real scalar fields
and, at least locally, they can be regarded as the
four components of a quaternion. The locality caveat is, in this
case, very substantial because global quaternionic coordinates can be
constructed only occasionally even on those manifolds that are
denominated quaternionic in the mathematical literature
\cite{hklr}, \cite{gal}. Anyhow, what is important is that, in
the hypermultiplet sector, the scalar manifold $\mathcal{QM}$ has
dimension multiple of four:
\begin{equation}
\mbox{dim}_{\bf R} \, \mathcal{QM} \, = \, 4 \, m \,\equiv \, 4 \, \# \,
\mbox{of hypermultiplets}
\label{quatdim}
\end{equation}
and, in some appropriate sense, it has a quaternionic structure.
\par
We name {\it Hypergeometry} that pertaining to the
hypermultiplet sector, irrespectively whether we deal with global or
local $\mathcal{N}$=2 theories. Yet there are two kinds of hypergeometries.
Supersymmetry requires the existence
of a principal $\mathrm{SU}(2)$--bundle
\begin{equation}
{\cal SU} \, \longrightarrow \, \mathcal{QM} \label{su2bundle}
\end{equation}
The bundle ${\cal SU}$ is
{\bf flat} in the {\it rigid supersymmetry case} while its curvature is
proportional to the K\"ahler forms in the {\it local case}.
\par
These two versions of hypergeometry were already known in mathematics prior to
their use \cite{D'Auria:1991fj},\cite{specHomgeo}, \cite{vanderseypen}, \cite{Andrianopoli:1996cm}, \cite{hklr}, \cite{gal} in the context of $\mathcal{N}=2$
supersymmetry and are identified as:
\begin{eqnarray}
\mbox{rigid hypergeometry} & \equiv & \mbox{HyperK\"ahler geometry.}
\nonumber\\ \mbox{local hypergeometry} & \equiv & \mbox{Quaternionic K\"ahler
geometry} \label{picchio}
\end{eqnarray}
\subsection{\sc Quaternionic K\"ahler, versus HyperK\"ahler manifolds}
Both a Quaternionic K\"ahler or a HyperK\"ahler manifold $\mathcal{QM}$
is a $4 m$-dimensional real manifold endowed with a metric $h$:
\begin{equation}
d s^2 = h_{u v} (q) d q^u \otimes d q^v \quad ; \quad u,v=1,\dots,
4 m \label{qmetrica}
\end{equation}
and three complex structures
\begin{equation}
(J^x) \,:~~ T(\mathcal{QM}) \, \longrightarrow \, T(\mathcal{QM}) \qquad
\quad (x=1,2,3)
\end{equation}
that satisfy the quaternionic algebra
\begin{equation}
J^x J^y = - \delta^{xy} \, \relax{\rm 1\kern-.35em 1} \, + \, \epsilon^{xyz} J^z
\label{quaternionetta}
\label{quatalgebra}
\end{equation}
and respect to which the metric is hermitian:
\begin{equation}
\forall \mbox{\bf X} ,\mbox{\bf Y} \in T\mathcal{QM} \,: \quad h
\left( J^x \mbox{\bf X}, J^x \mbox{\bf Y} \right ) = h \left(
\mbox{\bf X}, \mbox{\bf Y} \right ) \quad \quad
(x=1,2,3)
\label{hermit}
\end{equation}
From eq. (\ref{hermit}) it follows that one can introduce a triplet
of 2-forms
\begin{equation}
\begin{array}{ccccccc}
K^x& = &K^x_{u v} d q^u \wedge d q^v & ; & K^x_{uv} &=& h_{uw}
(J^x)^w_v \cr
\end{array}
\label{iperforme}
\end{equation}
that provides the generalization of the concept of K\"ahler form
occurring in the complex case. The triplet $K^x$ is named the {\it
HyperK\"ahler} form. It is an $\mathrm{SU}(2)$ Lie--algebra valued
2--form in the same way as the K\"ahler form is a $\mathrm{U(1)}$
Lie--algebra valued 2--form. In the complex case the definition of
K\"ahler manifold involves the statement that the K\"ahler 2--form is
closed. At the same time in Hodge--K\"ahler manifolds the K\"ahler 2--form can be
identified with the curvature of a line--bundle which in the case of
rigid supersymmetry is flat. Similar steps can be taken also here and
lead to two possibilities: either HyperK\"ahler or Quaternionic K\"ahler manifolds.
\par
Let us introduce a principal $\mathrm{SU}(2)$--bundle ${\cal SU}$ as
defined in eq. (\ref{su2bundle}). Let $\omega^x$ denote a connection
on such a bundle. To obtain either a HyperK\"ahler or a Quaternionic K\"ahler manifold we must impose the condition that the HyperK\"ahler 2--form
is covariantly closed with respect to the connection $\omega^x$:
\begin{equation}
\nabla K^x \equiv d K^x + \epsilon^{x y z} \omega^y \wedge K^z \,
= \, 0 \label{closkform}
\end{equation}
The only difference between the two kinds of geometries resides in
the structure of the ${\cal SU}$--bundle.
\begin{definizione} A
HyperK\"ahler manifold is a $4 m$--dimensional manifold with the
structure described above and such that the ${\cal SU}$--bundle is
{\bf flat}
\end{definizione}
Defining the ${\cal SU}$--curvature by:
\begin{equation}
\Omega^x \, \equiv \, d \omega^x + {1\over 2} \epsilon^{x y z}
\omega^y \wedge \omega^z \label{su2curv}
\end{equation}
in the HyperK\"ahler case we have:
\begin{equation}
\Omega^x \, = \, 0 \label{piattello}
\end{equation}
Viceversa \begin{definizione} A Quaternionic K\"ahler manifold is a $4
m$--dimensional manifold with the structure described above and such
that the curvature of the ${\cal SU}$--bundle is proportional to the
HyperK\"ahler 2--form \end{definizione} Hence, in the quaternionic
case we can write:
\begin{equation}
\Omega^x \, = \, { {\lambda}}\, K^x \label{piegatello}
\end{equation}
where $\lambda$ is a non vanishing real number.
\par
As a consequence of the above structure the manifold $\mathcal{QM}$ has
a holonomy group of the following type:
\begin{eqnarray}
{\rm Hol}(\mathcal{QM})&=& \mathrm{SU}(2)\otimes \mathrm{H} \quad
(\mbox{Quaternionic K\"ahler}) \nonumber \\ {\rm Hol}(\mathcal{QM})&=& \relax{\rm 1\kern-.35em 1}
\otimes \mathrm{H} \quad (\mbox{HyperK\"ahler}) \nonumber \\ \mathrm{H} &
\subset & \mathrm{Sp (2m,\mathbb{R}) }\label{olonomia}
\end{eqnarray}
In both cases, introducing flat indices $\{A,B,C= 1,2\}
\{\alpha,\beta,\gamma = 1,.., 2m\}$ that run, respectively, in the
fundamental representation of $\mathrm{SU}(2)$ and of
$\mathrm{Sp}(2m,\mathbb{R})$, we can find a vielbein 1-form
\begin{equation}
{\cal U}^{A\alpha} = {\cal U}^{A\alpha}_u (q) d q^u
\label{quatvielbein}
\end{equation}
such that
\begin{equation}
h_{uv} = {\cal U}^{A\alpha}_u {\cal U}^{B\beta}_v
\mathbb{C}_{\alpha\beta}\epsilon_{AB} \label{quatmet}
\end{equation}
where $\mathbb{C}_{\alpha \beta} = - \mathbb{C}_{\beta \alpha}$ and $\epsilon_{AB}
= - \epsilon_{BA}$ are, respectively, the flat $\mathrm{Sp}(2m)$ and
$\mathrm{Sp}(2) \sim \mathrm{SU}(2)$ invariant metrics. The vielbein
${\cal U}^{A\alpha}$ is covariantly closed with respect to the
$\mathrm{SU}(2)$-connection $\omega^z$ and to some
$\mathrm{Sp}(2m,\mathbb{R})$-Lie Algebra valued connection
$\Delta^{\alpha\beta} = \Delta^{\beta \alpha}$:
\begin{eqnarray}
\nabla {\cal U}^{A\alpha}& \equiv & d{\cal U}^{A\alpha} +{i\over 2}
\omega^x (\epsilon \sigma_x\epsilon^{-1})^A_{\phantom{A}B}
\wedge{\cal U}^{B\alpha} \nonumber\\ &+& \Delta^{\alpha\beta} \wedge
{\cal U}^{A\gamma} \mathbb{C}_{\beta\gamma} =0 \label{quattorsion}
\end{eqnarray}
\noindent where $(\sigma^x)_A^{\phantom{A}B}$ are the standard Pauli
matrices. Furthermore ${ \cal U}^{A\alpha}$ satisfies the reality
condition:
\begin{equation}
{\cal U}_{A\alpha} \equiv ({\cal U}^{A\alpha})^* = \epsilon_{AB}
\mathbb{C}_{\alpha\beta} {\cal U}^{B\beta} \label{quatreality}
\end{equation}
Eq.(\ref{quatreality}) defines the rule to lower the symplectic
indices by means of the flat symplectic metrics
$\epsilon_{AB}$ and $\mathbb{C}_{\alpha \beta}$. More specifically we can
write a stronger version of eq. (\ref{quatmet}) \cite{sugkgeom_3}:
\begin{eqnarray}
({\cal U}^{A\alpha}_u {\cal U}^{B\beta}_v + {\cal U}^{A\alpha}_v
{\cal
U}^{B\beta}_u)\mathbb{C}_{\alpha\beta}&=& h_{uv} \epsilon^{AB}\nonumber\\
\label{piuforte}
\end{eqnarray}
\noindent We have also the inverse vielbein ${\cal U}^u_{A\alpha}$
defined by the equation
\begin{equation}
{\cal U}^u_{A\alpha} {\cal U}^{A\alpha}_v = \delta^u_v \label{2.64}
\end{equation}
Flattening a pair of indices of the Riemann tensor ${\cal
R}^{uv}_{\phantom{uv}{ts}}$ we obtain
\begin{equation}
{\cal R}^{uv}_{\phantom{uv}{ts}} {\cal U}^{\alpha A}_u {\cal
U}^{\beta B}_v = -\,{{\rm i}\over 2} \Omega^x_{ts} \epsilon^{AC}
(\sigma_x)_C^{\phantom {C}B} \mathbb{C}^{\alpha \beta}+
\mathbb{R}^{\alpha\beta}_{ts}\epsilon^{AB}
\label{2.65}
\end{equation}
\noindent where $\mathbb{R}^{\alpha\beta}_{ts}$ is the field strength
of the $\mathrm{Sp}(2m) $ connection:
\begin{equation}
d \Delta^{\alpha\beta} + \Delta^{\alpha \gamma} \wedge \Delta^{\delta
\beta} \mathbb{C}_{\gamma \delta} \equiv \mathbb{R}^{\alpha\beta} =
\mathbb{R}^{\alpha \beta}_{ts} dq^t \wedge dq^s \label{2.66}
\end{equation}
Eq. (\ref{2.65}) is the explicit statement that the Levi Civita
connection associated with the metric $h$ has a holonomy group
contained in $\mathrm{SU}(2) \otimes \mathrm{Sp}(2m)$. Consider now
eq.s (\ref{quatalgebra}), (\ref{iperforme}) and (\ref{piegatello}).
We easily deduce the following relation:
\begin{equation}
h^{st} K^x_{us} K^y_{tw} = - \delta^{xy} h_{uw} +
\epsilon^{xyz} K^z_{uw}
\label{universala}
\end{equation}
that holds true both in the HyperK\"ahler and in the quaternionic
case. In the latter case, using eq. (\ref{piegatello}), eq.
(\ref{universala}) can be rewritten as follows:
\begin{equation}
h^{st} \Omega^x_{us} \Omega^y_{tw} = - \lambda^2 \delta^{xy} h_{uw} +
\lambda \epsilon^{xyz} \Omega^z_{uw} \label{2.67}
\end{equation}
Eq.(\ref{2.67}) implies that the intrinsic components of the curvature
2-form $\Omega^x$ yield a representation of the quaternion algebra.
In the HyperK\"ahler case such a representation is provided only by
the HyperK\"ahler form. In the quaternionic case we can write:
\begin{equation}
\Omega^x_{A\alpha, B \beta} \equiv \Omega^x_{uv} {\cal U}^u_{A\alpha}
{\cal U}^v_{B\beta} = - i \lambda \mathbb{C}_{\alpha\beta}
(\sigma_x)_A^{\phantom {A}C}\epsilon _{CB} \label{2.68}
\end{equation}
\noindent Alternatively eq.(\ref{2.68}) can be rewritten in an
intrinsic form as
\begin{equation}
\Omega^x =\,-{\rm i}\, \lambda \mathbb{C}_{\alpha\beta} (\sigma
_x)_A^{\phantom {A}C}\epsilon _{CB} {\cal U}^{\alpha A} \wedge {\cal
U}^{\beta B} \label{2.69}
\end{equation}
\noindent whence we also get:
\begin{equation}
{i\over 2} \Omega^x (\sigma_x)_A^{\phantom{A}B} = \lambda{\cal
U}_{A\alpha} \wedge {\cal U}^{B\alpha} \label{2.70}
\end{equation}
\subsection{\sc The triholomorphic moment map on quaternionic manifolds}
Next, following closely the original derivation of \cite{D'Auria:1991fj,mylecture}
let us turn to a discussion of the triholomorphic isometries of the manifold $\mathcal{QM}$
associated with hypermultiplets.
In $D=4$ supergravity the manifold of hypermultiplet scalars $\mathcal{QM}$ is a Quaternionic
K\"ahler manifold and we can gauge only those of its isometries
that are triholomorphic and that either generate an abelian group $\mathcal{G}$
or are \emph{suitably realized} as isometries also on the special manifold $\widehat{\mathcal{SK}}_n$.
This means that on $\mathcal{QM}$ we have Killing vectors:
\begin{equation}
\vec k_\mathbf{I} = k^u_\mathbf{I} {\vec \partial\over \partial q^u}
\label{2.71}
\end{equation}
\noindent
satisfying the same Lie algebra as the corresponding Killing
vectors on $\widehat{\mathcal{SK}}_n$. In other words
\begin{equation}
{\vec{\mathfrak{K}}}_\mathbf{I} =
\hat{k}^i_\mathbf{I} \vec \partial_i + \hat{k}^{i^*}_\mathbf{I}
\vec\partial_{i^*} + k_\mathbf{I}^u \vec\partial_u
\label{2.72}
\end{equation}
\noindent
is a Killing vector of the block diagonal metric:
\begin{equation}
\mathfrak{g} = \left (
\matrix { \widehat{g}_{ij^\star} & \quad 0 \quad \cr \quad 0
\quad & h_{uv} \cr } \right )
\label{2.73}
\end{equation}
defined on the product manifold\footnote{Following the notations described in the introduction, the Special K\"ahler manifold which describes the interaction of vector multiplets is denoted $\widehat{\mathcal{SK} }$ and all the Special Geometry Structures are endowed with a hat in order to distinguish this Special K\"ahler manifold from the other one which is incapsulated into the Quaternionic K\"ahler manifold $\mathcal{QM}$ describing the hypermultiplets when this latter happens to be in the image of the $c$-map.} $\widehat{\mathcal{SK} }\otimes\mathcal{QM}$.
\par
Let us first focus on the manifold $\mathcal{QM}$.
Triholomorphicity means that the Killing vector fields leave
the HyperK\"ahler structure invariant up to $\mathrm{SU(2)}$
rotations in the $\mathrm{SU(2)}$--bundle defined by eq.(\ref{su2bundle}).
Namely:
\begin{equation}
\begin{array}{ccccccc}
{\cal L}_\mathbf{I} K^x & = &\epsilon^{xyz}K^y
W^z_\mathbf{I} & ; &
{\cal L}_\mathbf{I}\omega^x&=& \nabla W^x_\mathbf{I}
\end{array}
\label{cambicchio}
\end{equation}
where $W^x_\mathbf{I}$ is an $\mathrm{SU(2)}$ compensator associated with the
Killing vector $k^u_\mathbf{I}$. The compensator $W^x_\mathbf{I}$ necessarily
fulfills the cocycle condition:
\begin{equation}
{\cal L}_\mathbf{I} W^{x}_\mathbf{J} - {\cal L}_\mathbf{J} W^x_\mathbf{I} + \epsilon^{xyz}
W^y_\mathbf{I} W^z_\mathbf{J} = f_{\mathbf{I} \mathbf{J}}^{\cdot \cdot \mathbf{L}}
W^x_\mathbf{L}
\label{2.75}
\end{equation}
In the HyperK\"ahler case the $\mathrm{SU(2)}$--bundle is flat and the
compensator can be reabsorbed into the definition of the
HyperK\"ahler forms. In other words we can always find a
map
\begin{equation}
\mathcal{QM} \, \longrightarrow \, L^x_{\phantom{x}y} (q)
\, \in \, \mathrm{SO(3)}
\end{equation}
that trivializes the ${\cal SU}$--bundle globally. Redefining:
\begin{equation}
K^{x\prime} \, = \, L^x_{\phantom{x}y} (q) \, K^y
\label{enfantduparadis}
\end{equation}
the new HyperK\"ahler form obeys the stronger equation:
\begin{equation}
{\cal L}_\mathbf{I} K^{x\prime} \, = \, 0
\label{noncambio}
\end{equation}
On the other hand, in the quaternionic case, the non--triviality of the
${\cal SU}$--bundle forbids to eliminate the $W$--compensator
completely. Due to the identification between HyperK\"ahler
forms and $\mathrm{SU(2)}$ curvatures eq.(\ref{cambicchio}) is rewritten
as:
\begin{equation}
\begin{array}{ccccccc}
{\cal L}_\mathbf{I} \Omega^x& = &\epsilon^{xyz}\Omega^y
W^z_\mathbf{I} & ; &
{\cal L}_\mathbf{I}\omega^x&=& \nabla W^x_\mathbf{I}
\end{array}
\label{cambiacchio}
\end{equation}
In both cases, anyhow, and in full analogy with the case of
K\"ahler manifolds, to each Killing vector
we can associate a triplet ${\cal
P}^x_\mathbf{I} (q)$ of 0-form prepotentials.
Indeed we can set:
\begin{equation}
{\bf i}_\mathbf{I} K^x =
- \nabla {\cal P}^x_\mathbf{I} \equiv -(d {\cal
P}^x_\mathbf{I} + \epsilon^{xyz} \omega^y {\cal P}^z_\mathbf{I})
\label{2.76}
\end{equation}
where $\nabla$ denotes the $\mathrm{SU(2)}$ covariant exterior derivative.
\par
As in the K\"ahler case eq.(\ref{2.76}) defines a moment map:
\begin{equation}
{\cal P}: {\cal M} \, \longrightarrow \,
\mathbb{R}^3 \otimes
{\mathcal{G} }^*
\end{equation}
where ${\mathcal{G}}^*$ denotes the dual of the Lie algebra
${\mathcal{G} }$ of the group ${\cal G}$.
Indeed let $x\in {\mathcal{G} }$ be the Lie algebra element
corresponding to the Killing
vector $\overrightarrow{X}$; then, for a given
$m\in {\cal M}$
\begin{equation}
\mu (m)\, : \, x \, \longrightarrow \, {\cal P}_{\overrightarrow{X}}(m) \,
\in \, \mathbb{R}^3
\end{equation}
is a linear functional on ${\mathcal{G}}$. If we expand
$\overrightarrow{X} = a^\mathbf{I} k_\mathbf{I}$ on a basis of Killing vectors
$k_\mathbf{I}$ such that
\begin{equation}
[k_\mathbf{I}, k_\mathbf{L}]= f_{\mathbf{I} \mathbf{L}}^{\ \ \mathbf{K}} k_\mathbf{K}
\label{blioprime}
\end{equation}
and we also choose a basis ${\bf i}_x \, (x=1,2,3)$ for $\mathbb{R}^3$
we get:
\begin{equation}
{\cal P}_{\overrightarrow{X}}\, = \, a^\mathbf{I} {\cal P}_\mathbf{I}^x \, {\bf i}_x
\end{equation}
Furthermore we need a generalization of the equivariance defined
by eq.(\ref{equivarianza})
\begin{equation}
\overrightarrow{X} \circ {\cal P}_{\overrightarrow{Y}} \,= \,
{\cal P}_{[\overrightarrow{X},\overrightarrow{Y}]}
\label{equivarianzina}
\end{equation}
In the HyperK\"ahler case, the left--hand side of
eq.(\ref{equivarianzina})
is defined as the usual action of a vector field on a $0$--form:
\begin{equation}
\overrightarrow{X} \circ {\cal P}_{\overrightarrow{Y}}\, = \, {\bf i}_{\overrightarrow{X}} \, d
{\cal P}_{\overrightarrow{Y}}\, = \,
X^u \, {\frac{\partial}{\partial q^u}} \, {\cal P}_{\overrightarrow{Y}}\,
\end{equation}
The equivariance condition implies
that we can introduce a triholomorphic Poisson bracket defined
as follows:
\begin{equation}
\{{\cal P}_\mathbf{I}, {\cal P}_\mathbf{J}\}^x \equiv 2 K^x (\mathbf{I},
\mathbf{J})
\label{hykapesce}
\end{equation}
leading to the triholomorphic Poissonian realization of the Lie
algebra:
\begin{equation}
\left \{ {\cal P}_\mathbf{I}, {\cal P}_\mathbf{J} \right \}^x \, = \,
f^{\mathbf{K}}_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}} \, {\cal P}_\mathbf{K}^{x}
\label{hykapescespada}
\end{equation}
which in components reads:
\begin{equation}
K^x_{uv} \, k^u_\mathbf{I} \, k^v_\mathbf{J} \, = \, {\frac{1}{2}} \,
f^{\mathbf{K}}_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}\, {\cal P}_\mathbf{K}^{x}
\label{hykaide}
\end{equation}
In the quaternionic case, instead, the left--hand side of
eq.(\ref{equivarianzina})
is interpreted as follows:
\begin{equation}
\overrightarrow{X} \circ {\cal P}_{\overrightarrow{Y}}\, = \, {\bf i}_{\overrightarrow{X}}\, \nabla
{\cal P}_{\overrightarrow{Y}}\, = \,
X^u \, {\nabla_u} \, {\cal P}_{\overrightarrow{Y}}\,
\end{equation}
where $\nabla$ is the $\mathrm{SU(2)}$--covariant differential.
Correspondingly, the triholomorphic Poisson bracket is defined
as follows:
\begin{equation}
\{{\cal P}_\mathbf{I}, {\cal P}_\mathbf{J}\}^x \equiv 2 K^x (\mathbf{I},
\mathbf{J}) - { {\lambda}} \, \varepsilon^{xyz} \,
{\cal P}_\mathbf{I}^y \, {\cal P}_\mathbf{J}^z
\label{quatpesce}
\end{equation}
and leads to the Poissonian realization of the Lie algebra
\begin{equation}
\left \{ {\cal P}_\mathbf{I}, {\cal P}_\mathbf{J} \right \}^x \, = \,
f^{\mathbf{K}}_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}} \, {\cal P}_\mathbf{K}^{x}
\label{quatpescespada}
\end{equation}
which in components reads:
\begin{equation}
K^x_{uv} \, k^u_\mathbf{I} \, k^v_\mathbf{J} \, - \,
{ \frac{\lambda}{2}} \, \varepsilon^{xyz} \,
{\cal P}_\mathbf{I}^y \, {\cal P}_\mathbf{J}^z\,= \, {\frac{1}{2}} \,
f^{\mathbf{K}}_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}\, {\cal P}_\mathbf{K}^{x}
\label{quatide}
\end{equation}
Eq.(\ref{quatide}), which is the most convenient way of
expressing equivariance in a coordinate basis was originally written
in \cite{D'Auria:1991fj} and has played a fundamental
role in the construction of supersymmetric actions for gauged $\mathcal{N}=2$ supergravity
both in $D=4$ \cite{D'Auria:1991fj,Andrianopoli:1996cm} and in $D=5$ \cite{ceregatta}.
\section{\sc The Quaternionic K\"ahler Geometry in the image of the $c$-map}
The main object of study in the present paper are those Quaternionic K\"ahler manifolds that are in the image of the $c$-map.\footnote{Not all non-compact, homogeneous Quaternionic K\"ahler manifolds which are relevant to supergravity (which are \emph{normal}, i.e. exhibiting a solvable group of isometries having a free and transitive action on it) are in the image of the c-map, the only exception being the quaternionic projective spaces \cite{alek,Cecotti:1988ad,vanderseypen}.} This latter
\begin{equation}\label{cimappo}
\mbox{c-map} \, \, : \,\, \mathcal{SK}_n \, \Longrightarrow \, \mathcal{QM}_{4n+4}
\end{equation}
is a universal construction that starting from an arbitrary Special K\"ahler manifold $\mathcal{SK}_n$ of complex dimension $n$, irrespectively whether it is homogenoeus or not, leads to a unique Quaternionic K\"ahler manifold $\mathcal{QM}_{4n+4}$ of real dimension $4n+4$ which contains $\mathcal{SK}_n$ as a submanifold. The precise modern definition of the $c$-map, originally introduced in \cite{sabarwhal}, is provided below.
\begin{definizione}
Let $\mathcal{SK}_n$ be a special K\"ahler manifold whose complex coordinates we denote by $z^i$ and whose K\"ahler metric we denote by $g_{ij^\star}$. Let moreover $\mathcal{N}_{\Lambda\Sigma}(z,{\bar z})$ be the symmetric period matrix defined by eq.(\ref{intriscripen}), introduce the following set of $4n+4$ coordinates:
\begin{equation}\label{finnico}
\left\{q^u \right\} \, \equiv \, \underbrace{\{U,a\}}_{\mbox{2 real}}\, \bigcup \,\underbrace{\underbrace{\{ z^i\}}_{\mbox{n complex}}}_{\mbox{2n real}} \, \bigcup\, \underbrace{\mathbf{Z} \, = \, \{ Z^\Lambda \, , \, Z_\Sigma \}}_{\mbox{(2n+2) real}}
\end{equation}
Let us further introduce the following $(\mathrm{2n+2})\times(\mathrm{2n+2}) $ matrix ${\cal M}_4^{-1}$:
\begin{eqnarray}
\mathcal{M}_4^{-1} & = &
\left(\begin{array}{c|c}
{\mathrm{Im}}\mathcal{N}\,
+\, {\mathrm{Re}}\mathcal{N} \, { \mathrm{Im}}\mathcal{N}^{-1}\, {\mathrm{Re}}\mathcal{N} & \, -{\mathrm{Re}}\mathcal{N}\,{ \mathrm{Im}}\,\mathcal{N}^{-1}\\
\hline
-\, { \mathrm{Im}}\mathcal{N}^{-1}\,{\mathrm{Re}}\mathcal{N} & { \mathrm{Im}}\mathcal{N}^{-1} \
\end{array}\right) \label{inversem4}
\end{eqnarray}
which depends only on the coordinate of the Special K\"ahler manifold.
The $c$-map image of $\mathcal{SK}_n$ is the unique Quaternionic K\"ahler manifold $\mathcal{QM}_{4n+4}$ whose coordinates are the $q^u$ defined in (\ref{finnico}) and whose metric is given by the following universal formula
\begin{eqnarray}
ds^2_{\mathcal{QM}} &=&\frac{1}{4} \left( d{U}^2+\, 4 g_{ij^\star} \,d{z}^j\, d{{\bar z}}^{j^\star}
+ e^{-2\,U}\,(d{a}+{\bf Z}^T\mathbb{C}d{{\bf
Z}})^2\,-\,2 \, e^{-U}\,d{{\bf
Z}}^T\,\mathcal{M}_4^{-1}\,d{{\bf Z}}\right)
\label{geodaction}
\end{eqnarray}
\end{definizione}
The metric (\ref{geodaction}) has the following positive definite signature
\begin{equation}
\mbox{sign}\left[ds^2_{\mathcal{QM}}\right] \, = \, \left(\underbrace{+,\dots,+}_{4+4\mathrm{n}}\right)
\end{equation}
since the matrix $\mathcal{M}_4^{-1} $ is negative definite.
\par
It is wort mentioning that if we utilize the same $4n+4$ coordinates (\ref{finnico}) and instead of (\ref{geodaction}) we introduce the alternative Lorentzian metric:
\begin{eqnarray}
ds^2_{\mathcal{QM^\star}} &=&\frac{1}{4}\left( d{U}^2+\, 4 g_{ij^\star} \,d{z}^j\, d{{\bar z}}^{j^\star}
+ e^{-2\,U}\,(d{a}+{\bf Z}^T\mathbb{C}d{{\bf
Z}})^2\,+\,2 \, e^{-U}\,d{{\bf
Z}}^T\,\mathcal{M}_4\,d{{\bf Z}}\right)
\label{BHaction}
\end{eqnarray}
that has signature:
\begin{equation}
\mbox{sign}\left[ds^2_{\mathcal{QM}^\star}\right] \, = \, \left(\underbrace{+,\dots,+}_{2\mathrm{n}+2},\underbrace{-,\dots ,-}_{2\mathrm{n}+2}\right)
\end{equation}
we obtain the pseudo-quaternionic manifold $\mathcal{QM}$ which constitutes the target manifold in the $3$-dimensional $\sigma$-model description of $D=4$ supergravity Black-Hole solutions \cite{PietroSashaMarioBH1},\cite{noinilpotenti},\cite{miosasha}. In the case the Special K\"ahler pre-image is a symmetric space $\mathrm{U}_{\mathcal{SK}}/\mathrm{H}_{\mathcal{SK}}$, both $\mathcal{QM}$ and $\mathcal{QM}^\star$ turn out to be symmetric spaces as well, $\mathrm{U}_{Q}/\mathrm{H}_{Q}$ and $\mathrm{U}_{Q}/\mathrm{H}_{Q}^\star$, the numerator group being the same. We will come back to the issue of symmetric homogeneous Quaternionic K\"ahler manifolds in section \ref{omosymmetro}
\subsection{\sc The HyperK\"ahler two-forms and the $\mathfrak{su}(2)$-connection}
The reason why we state that $\mathcal{QM}_{4n+4}$ is Quaternionic K\"ahler is that, by utilizing only the identities of Special K\"ahler Geometry we can construct the three complex structures $J_u^{x|v}$ satisfying the quaternionic algebra (\ref{quaternionetta}) the corresponding HyperK\"ahler two-forms $K^x$ and the $\mathfrak{su}(2)$ connection $\omega^x$ with respect to which they are covariantly constant.
\par
The construction is extremely beautiful and it is the following one.
\par
Consider the K\"ahler connection $\mathcal{Q}$ defined by eq. (\ref{u1conect}) and furthermore introduce the following differential form:
\begin{equation}
\label{Phidiffe}
\Phi \, = \, da + \mathbf{Z}^T \, \mathbb{C}\, \mathrm{d}\mathbf{Z}
\end{equation}
Next define the two dimensional representation of both the $\mathfrak{su}(2)$ connection and of the HyperK\"ahler $2$-forms as it follows:
\begin{eqnarray}
\omega &=& \frac{\rm i}{\sqrt{2}}\,\sum_{x=1}^3 \, \omega^x \, \gamma_x \label{cunnettasu2}\\
\mathbf{K} &=& \frac{\rm i}{\sqrt{2}}\,\sum_{x=1}^3 \, K^x \, \sigma_x \label{HypKalmatra}
\end{eqnarray}
where $\gamma_x$ denotes a basis of $2\times 2$ euclidian $\gamma$-matrices for which we utilize the following basis
which is convenient in the explicit calculations we perform in Part Four\footnote{The chosen $\gamma$-matrices are a permutation of the standard pauli matrices divided by $\sqrt{2}$ and multiplied by $\frac{\rm i}{2}$ can be used as a basis of anti-hermitian generators for the $\mathfrak{su}(2)$ algebra in the fundamental defining representation.}:
\begin{eqnarray}
\gamma_1&=& \left(
\begin{array}{ll}
\frac{1}{\sqrt{2}} & 0 \\
0 & -\frac{1}{\sqrt{2}}
\end{array}
\right) \nonumber\\
\gamma_2&=& \left(
\begin{array}{ll}
0 & -\frac{i}{\sqrt{2}} \\
\frac{i}{\sqrt{2}} & 0
\end{array}
\right)\nonumber\\
\gamma_3 &=& \left(
\begin{array}{ll}
0 & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & 0
\end{array}
\right) \label{gammini}
\end{eqnarray}
These $\gamma$-matrices satisfy the following Clifford algebra:
\begin{equation}\label{cliffordus}
\left\{ \gamma_x \, , \, \gamma_y \right \} \, = \, \delta^{xy} \, \mathbf{1}_{2 \times 2}
\end{equation}
and $\frac{\rm i}{2} \, \gamma_x$ provide a basis of generators of the $\mathfrak{su}(2)$ algebra.
\par
Having fixed these conventions the expression of the quaternionic $\mathfrak{su}(2)$-connection in terms of Special Geometry structures is encoded in the following expression for the $2\times 2$-matrix valued $1$-form $\omega$. Explicitly we have:
\begin{equation}\label{omegaSu2}
\omega \, = \, \left( \begin{array}{cc}
-\frac{\rm i}{2} \, \mathcal{Q} \, - \, \frac{\rm i}{4} \,e^{-U} \, \Phi & e^{-\frac{U}{2}} \, V^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\\
- \, e^{-\frac{U}{2}} \, \overline{V}^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z} & \frac{\rm i}{2} \, \mathcal{Q} \, + \, \frac{\rm i}{4} \,e^{-U} \, \Phi
\end{array}
\right)
\end{equation}
where $V$ and $\overline{V}$ denote the covariantly holomorphic sections of Special geometry defined in eq.s (\ref{covholsec}).
The curvature of this connection is obtained from a straight-forward calculation:
\begin{eqnarray}
\label{K2per2}
\mathbf{K} &\equiv& d\omega \, + \, \omega \, \wedge \, \omega \nonumber \\
\null &=& \left(\begin{array}{cc}
\mathfrak{u} & \mathfrak{v} \\
- \,\overline{\mathfrak{v}}& -\,\mathfrak{u}
\end{array}
\right)
\end{eqnarray}
the independent $2$-form matrix elements being given by the following explicit formulae:
\begin{eqnarray}
\label{uvvb}
\mathfrak{u} &=& -{\rm i} \frac{1}{2} \, K \, -\frac{1}{8} dS \,\wedge \, d\bar{S}\, - \, e^{-U} \, V^T \, \mathbb{C} \,\mathrm{d}\mathbf{Z}\, \wedge\, \bar{V}^T \, \mathbb{C} \,\mathrm{d}\mathbf{Z} \, - \, \frac{1}{4} \, e^{-U} \,\mathrm{d}\mathbf{Z}^T \, \wedge \, \mathbb{C} \, \mathrm{d}\mathbf{Z} \nonumber\\
\mathfrak{v }&=& e^{-\frac{U}{2}} \left( \, DV^T \, \wedge \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\, - \, \frac{1}{2} dS \, \wedge \,
V^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\right)\nonumber\\
\overline{\mathfrak{v }} &=& e^{-\frac{U}{2}} \left( \, D\overline{V}^T \, \wedge \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\, - \, \frac{1}{2} d\overline{S} \, \wedge \,
\overline{V}^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\right)
\end{eqnarray}
where
\begin{equation}\label{kalleforma}
K \, = \, \frac{ {\rm i}}{2} \, g_{ij^\star} \, dz^i \, \wedge \, d\bar{z}^{j^\star}
\end{equation}
is the K\"ahler $2$-form of the Special K\"ahler submanifold and where we have used the following short hand notations:
\begin{eqnarray}
dS &=& dU \, + \, {\rm i} \, e^{-U}\, \left(da \, + \, \mathbf{Z}^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\right) \label{firbone1} \\
d\overline{S} &=& dU \, - \, {\rm i} \, e^{-U}\, \left(da \, + \, \mathbf{Z}^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\right) \label{firbone2} \\
DV &=& dz^i \, \nabla_i V \label{firbone3} \\
D\overline{V} &=& d\bar{z}^{i^\star} \, \nabla_{i^\star} V \label{firbone4}
\end{eqnarray}
The three HyperK\"ahler forms $K^x$ are easily extracted from eq.s (\ref{K2per2}-\ref{uvvb}) by collecting the coefficients of the $\gamma$-matrix expansion and we need not to write their form which is immediately deduced. The relevant thing is that the components of $K^x$ with an index raised through multiplication with the inverse of the quaternionic metric $h^{uv}$ exactly satisfy the algebra of quaternionic complex structures (\ref{quatalgebra}). Explicitly we have:
\begin{eqnarray}
K^x &=& - \, {\rm i} \, 4 \sqrt{2} \, \mbox{Tr} \, \left( \gamma^x \, \mathbf{K}\right) \, \equiv \, K^x_{uv} \, dq^u \,\wedge \, dq^v \nonumber\\
J^{x|s}_u &=& K^x_{uv}\, h^{vs} \nonumber \\
J^{x|s}_u \, J^{y|v}_s &=& - \delta^{ xy} \, \delta^v_u \, + \, \epsilon^{xyz} \, J^{z|v}_u \label{quatKvera}
\end{eqnarray}
The above formulae are not only the general proof that the Riemaniann manifold $\mathcal{QM}$ defined by the metric (\ref{geodaction}) is indeed a Quaternionic K\"ahler manifold, but, what is most relevant, they also provide an algorithm to write in terms of Special Geometry structures the tri-holomorphic moment map of the principal isometries possessed by $\mathcal{QM}$.
\subsection{\sc Isometries of $\mathcal{QM}$ in the image of the $c$-map and their tri-holomorphic moment maps}
\label{triholoformul}
Let us now consider the isometries of the metric (\ref{geodaction}). There are three type of isometries:
\begin{description}
\item[a)] The isometries of the $(2n+3)$--dimensional Heisenberg algebra $\mathrm{\mathbb{H}eis}$ which is always present and is universal for any $(4n+4)$--dimensional Quaternionic K\"ahler manifold in the image of the $c$-map. We describe it below.
\item[b)] All the isometries of the pre-image Special K\"ahler manifold $\mathcal{SK}_n$ that are promoted to isometries of the image manifold in a way described below.
\item[c)] The additional $2n+4$ isometries that occur only when $\mathcal{SK}_n$ is a symmetric space and such, as a consequence, is also the $c$-map image $\mathcal{QM}_{4n+4}$. We will discuss these isometries
in section (\ref{omosymmetro}).
\end{description}
For the first two types of isometries a) and b) we are able to write general expressions for the tri-holomorphic moment maps that utilize only the structures of Special Geometry. In the case that the additional isometries c) do exist we have another universal formula which can be used for all generators of the isometry algebra $\mathbb{U}_\mathcal{Q}$ and which relies on the identification of the generators of the $\mathfrak{su}(2) \subset \mathbb{H}$ subalgebra with the three complex structures. We will illustrate the details of such an identification while discussing the example of the $S^3$-model.
\par
First of all let us fix the notation writing the general form of a Killing vector. This a tangent vector:
\begin{eqnarray}\label{killingus}
\vec{\mathbf{k}}& = & k^u (q) \, {\partial}_u \nonumber\\
&=& k^\diamond \, \frac{\partial}{\partial U} \, + \, k^i \, \frac{\partial}{\partial z^i}\, + \, k^{i^\star} \, \frac{\partial}{\partial \bar{z}^{i^\star}}\, + \, k^\bullet \, \frac{\partial}{\partial a}\, + \, k^\alpha \, \frac{\partial}{\partial \mathbf{Z}^\alpha}\nonumber\\
& \equiv & k^\diamond \, \partial_\diamond + \, k^i \, \partial_i \, + \, k^{i^\star} \,\partial_{i^\star} + \, k^\bullet \, \partial_\bullet \, + \, k^\alpha \, \partial_\alpha
\end{eqnarray}
with respect to which the Lie derivative of the metric element (\ref{geodaction}) vanishes:
\begin{equation}\label{isoverissima}
\ell_{\vec{\mathbf{k}}}\, ds^2_{\mathcal{QM}} \, = \,0
\end{equation}
\subsubsection{\sc Tri-holomorphic moment maps for the Heisenberg algebra translations}
First let us consider the isometries associated with the Heisenberg algebra. The transformation:
\begin{equation}\label{infinoLam}
Z^\alpha \, \mapsto \, Z^\alpha \, + \, \Lambda^\alpha \quad ; \quad a \, \mapsto \, a \, - \, \Lambda^T \, \mathbb{C} \, \mathbf{Z}
\end{equation}
where $\Lambda^\alpha$ is an arbitrary set of $2n+2$ real infinitesimal parameters is an infinitesimal isometry for the metric $ ds^2_{\mathcal{QM}}$ in (\ref{geodaction}). It corresponds to the following Killing vector:
\begin{eqnarray}\label{KillusW}
\overrightarrow{\mathbf{k}}_{[\Lambda]} & = & \Lambda^\alpha \, \overrightarrow{\mathbf{k}}_\alpha \nonumber\\
& = & \Lambda^\alpha \, \partial_\alpha \, - \, \Lambda^T \, \mathbb{C} \, \mathbf{Z} \, \partial_\bullet
\end{eqnarray}
whose components are immediately deduced by comparison of eq.(\ref{KillusW}) with eq.(\ref{killingus}).
\par
We are interested in determining the expression of the tri-holomorphic moment map $\mathfrak{P}_{[\Lambda]}$ which satisfies the defining equation:
\begin{eqnarray}
\mathbf{i}_{[\Lambda]}\, \mathbf{K} \, \equiv \,
\left(\begin{array}{cc}
\mathbf{i}_{[\Lambda]}\,\mathfrak{u} & \mathbf{i}_{[\Lambda]}\,\mathfrak{v} \\
- \,\mathbf{i}_{[\Lambda]}\,\overline{\mathfrak{v}}& -\,\mathbf{i}_{[\Lambda]}\,\mathfrak{u}
\end{array}
\right) &=& \mathrm{d} \mathfrak{P}_{[\Lambda]} \, + \, \left[ \omega \, , \, \mathfrak{P}_{[\Lambda]}\right ]
\label{pullusLam}
\end{eqnarray}
The general solution to this problem is
\begin{eqnarray}
\label{triholoHeis}
\mathfrak{P}_{[\Lambda]}&=&\left(\begin{array}{cc}
- \, \frac{\rm i}{4}\, e^{-U}\, \Lambda^T \, \mathbb{C} \, \mathbf{Z} &\frac{1}{2} \, e^{-\frac{U}{2}} \, \Lambda^T \, C \, V \\
- \,\frac{1}{2} \, e^{-\frac{U}{2}} \, \Lambda^T \, C \, \overline{V} & \, \frac{\rm i}{4}\, e^{-U}\, \Lambda^T \, \mathbb{C} \, \mathbf{Z}
\end{array}
\right) \nonumber\\
\end{eqnarray}
\subsubsection{\sc Tri-holomorphic moment map for the Heisenberg algebra central charge}
Consider next the isometry associated with the Heisenberg algebra central charge. The transformation:
\begin{equation}\label{infinoZeta}
a \, \mapsto \, a \, + \, \varepsilon
\end{equation}
where $\varepsilon$ is an arbitrary real small parameter is an infinitesimal isometry for the metric $ ds^2_{\mathcal{QM}}$ in (\ref{geodaction}). It corresponds to the following Killing vector:
\begin{eqnarray}\label{KillusZ}
\varepsilon \, \overrightarrow{\mathbf{k}}_{[\bullet]} & = & \varepsilon \, \partial_\bullet
\end{eqnarray}
whose components are immediately deduced by comparison of eq.(\ref{KillusZ}) with eq.(\ref{killingus}).
\par
We are interested in determining the expression of the tri-holomorphic moment map $\mathfrak{P}_{[\bullet]}$ which satisfies the defining equation analogous to eq.(\ref{pullusLam}):
\begin{eqnarray}
\mathbf{i}_{[\bullet]}\, \mathbf{K} &=& \mathrm{d} \mathfrak{P}_{[\bullet]} \, + \, \left[ \omega \, , \, \mathfrak{P}_{[\bullet]}\right ]
\label{pullusBull}
\end{eqnarray}
The solution of this problem is even simpler than in the previous case. Explicitly we obtain:
\begin{eqnarray}
\label{triholoHeisZ}
\mathfrak{P}_{[\bullet]}&=&\left(\begin{array}{cc}
- \, \frac{\rm i}{8}\, e^{-U} &0 \\
0 & \, \frac{\rm i}{8}\, e^{-U}
\end{array}
\right) \nonumber\\
\end{eqnarray}
The explicit expression of the moment maps and Killing vectors associated with the Heisenberg isometries was used in the gauging of abelian subalgebras of the Heisenberg algebra, which is relevant to the description of compactifications of
Type II superstring on a generalized Calabi-Yau manifold \cite{heisenberg}.
\subsubsection{\sc Tri-holomorphic moment map for the extension of $\mathcal{SK}_n$ holomorphic isometries}
Next we consider the question how to write the moment map associated with those isometries that where already present in the original
Special K\"ahler manifold $\mathcal{SK}_n$ which we $c$-mapped to a Quaternionic K\"ahler manifold.
\par
Suppose that ${\mathcal{SK}}_n$ has a certain number of holomorphic Killing vectors $k_{\mathbf{I}}^i(z)$ satisfying eq.s (\ref{killo},\ref{killo1},\ref{holkillingvectors}) necessarily closing some Lie algebra $\mathfrak{g}_{\mathcal{SK}}$ among themselves. Their holomorphic momentum-map is provided by eq.(\ref{sisalvichipuo}). Necessarily every isometry of a special K\"ahler manifold has a linear symplectic $(2n+2)$-dimensional realization on the holomorphic section $\Omega(z)$ up to an overall holomorphic factor. This means that for each holomorphic Killing vector we have (see Eq. (\ref{sisalvichipuo30})):
\begin{equation}\label{bellacosa}
k_{\mathbf{I}}^i(z)\, \partial_i \, \Omega(z) \, = \, \exp\left[f_{\mathbf{I}}(z)\right] \, \mathfrak{T}_{\mathbf{I}} \, \Omega(z) \,.
\end{equation}
where $f_{\mathbf{I}}(z)$ the holomorphic K\"ahler compensator.
Then it can be easily checked that the transformation:
\begin{equation}\label{kQupKil}
z^i \, \mapsto \, z^i \, + \, k^i_{\mathbf{I}}(z) \quad ; \quad \mathbf{Z} \, \mapsto \, \mathbf{Z} \, + \, \mathfrak{T}_{\mathbf{I}} \, \mathbf{Z}
\end{equation}
is an infinitesimal isometry of the metric (\ref{geodaction}) corresponding to the Killing vector:
\begin{equation}\label{promossoKil}
\vec{\mathbf{k}}_{\mathbf{I}} \, = \, k^i_{\mathbf{I}}(z) \, \partial_i \, + \, k^{i^\star}_{\mathbf{I}}(\bar{z}) \,\partial_{i^\star} \, + \, \left(\mathfrak{T}_{\mathbf{I}}\right)^\alpha_{\phantom{\alpha}\beta} \, \mathbf{Z}^\beta \, \partial_\alpha
\end{equation}
Also in this case we are interested in determining the expression of the tri-holomorphic moment map $\mathfrak{P}_{[\mathbf{I}]}$ satisfying the defining equation:
\begin{eqnarray}
\mathbf{i}_{\vec{\mathbf{k}}_{\mathbf{I}}}\, \mathbf{K} &=& \mathrm{d} \mathfrak{P}_{[\mathbf{I}]} \, + \, \left[ \omega \, , \,
\mathfrak{P}_{[\mathbf{I}]}\right ]
\label{pullusBull2}
\end{eqnarray}
The solution is given by the expression below:
\begin{eqnarray}
\label{triholoSK}
\mathfrak{P}_{[\mathbf{I}]}&=&\left(\begin{array}{cc}
\frac{\rm i}{4} \left(\mathcal{P}_{\mathbf{I}} \, + \, \frac{1}{2} \, e^{-U} \, \mathbf{Z}^T \, \mathbb{C} \, \mathfrak{T}_{\mathbf{I}} \, \mathbf{Z} \right)& - \, \frac{1}{2} \, e^{-U/2} \, V^T \, \mathbb{C} \, \mathfrak{T}_{\mathbf{I}} \, \mathbf{Z} \\
\, \frac{1}{2} \, \, e^{-U/2} \,\overline{V}^T \, \mathbb{C} \, \mathfrak{T}_{\mathbf{I}} \, \mathbf{Z} & \, - \frac{\rm i}{4} \left(\mathcal{P}_{\mathbf{I}} \, + \, \frac{1}{2} \, e^{-U} \, \mathbf{Z}^T \, \mathbb{C} \, \mathfrak{T}_{\mathbf{I}} \, \mathbf{Z} \right)
\end{array}
\right) \nonumber\\
\end{eqnarray}
where $\mathcal{P}_{\mathbf{I}}$ is the moment map of the same Killing vector in pure Special Geometry.
\subsection{\sc Homogeneous Symmetric Special Quaternionic K\"ahler manifolds}
\label{omosymmetro}
When the Special K\"ahler manifold $\mathcal{SK}_n$ is a symmetric coset space, it turns out that the metric (\ref{geodaction}) is actually the symmetric metric on an enlarged symmetric coset manifold
\begin{equation}\label{qcosetto}
\mathcal{QM}_{4n+4} \, = \, \frac{\mathrm{U}_Q}{\mathrm{H}_Q} \, \supset \, \frac{\mathrm{U}_{\mathcal{SK}}}{\mathrm{H}_{\mathcal{SK}}}
\end{equation}
\par
Naming $\Lambda[\mathfrak{g}]$ the $\mathbf{W}$-representation of any finite element of the $\mathfrak{g}\in\mathrm{U}_{\mathcal{SK}}$ group, we have that the matrix $\mathcal{M}_4(z,\bar{z})$ transforms as follows:
\begin{equation}\label{traduco}
\mathcal{M}_4\left( \mathfrak{g}\cdot z,\mathfrak{g}\cdot \bar{z} \right)\, = \, \Lambda[\mathfrak{g}] \, \mathcal{M}_4\left(z,\bar{z}\right ) ]\, \Lambda^T[\mathfrak{g}]
\end{equation}
where $\mathfrak{g}\cdot z$ denotes the non linear action of $\mathrm{U}_{\mathcal{SK}}$ on the scalar fields. Since the space $\frac{\mathrm{U}_{\mathcal{SK}}}{\mathrm{H}_{\mathcal{SK}}}$ is homogeneous, choosing any reference point $z_0$ all the others can be reached by a suitable group element $\mathfrak{g}_z$ such that $\mathfrak{g}_z\cdot z_0 \, = \, z$ and we can write:
\begin{equation}\label{turnaconto}
\mathcal{M}_4^{-1}(z,\bar{z}) \, = \, \Lambda^T[\mathfrak{g}_z^{-1}] \, \mathcal{M}_4^{-1}(z_0,\bar{z}_0) ]\, \Lambda[\mathfrak{g}^{-1}_z]
\end{equation}
This allows to introduce a set of $4n+4$ vielbein defined in the following way:
\begin{equation}\label{filibaine}
E^I_{\mathcal{QM}} \, = \, \frac{1}{2} \, \left\{ dU \, , \, \underbrace{e^i(z)}_{2\,n} \, , \, e^{-U} \,\left(d{a}+{\bf Z}^T\mathbb{C}d{{\bf
Z}}\right) \, , \, \underbrace{e^{-\frac {U} {2}}\, \Lambda[\mathfrak{g}_z^{-1}] \, \mathrm{d}\mathbf{Z}}_{2n+2} \right\}
\end{equation}
and rewrite the metric (\ref{geodaction}) as it follows:
\begin{equation}\label{cornish}
ds^2_{\mathcal{QM}} \, = \, E^I_{\mathcal{QM}} \, \mathfrak{q}_{IJ} \, E^J_{\mathcal{QM}}
\end{equation}
where the quadratic symmetric constant tensor $\mathfrak{q}_{IJ}$ has the following form:
\begin{equation}\label{quadrotta}
\mathfrak{q}_{IJ} \, = \, \left( \begin{array}{c|c|c|c}
1 & 0 & 0 & 0 \\
\hline
0 & \delta_{ij} & 0 & 0 \\
\hline
0 & 0 & 1 & 0 \\
\hline
0 & 0 & 0 & -\, 2\, \mathcal{M}_4^{-1}(z_0,\bar{z}_0)
\end{array}
\right)
\end{equation}
The above defined vielbein are endowed with a very special property namely they identically satisfy a set of Maurer Cartan equations:
\begin{equation}\label{MCSolv}
dE^I_{\mathcal{QM}} \, - \, \frac{1}{2} f^I_{\phantom{I}JK} \, E^J_{\mathcal{QM}} \, \wedge \, E^K_{\mathcal{QM}} \, = \, 0
\end{equation}
where $f^I_{\phantom{I}JK}$ are the structure constants of a solvable Lie algebra $\mathfrak{A}$ which can be identified as follows:
\begin{equation}\label{solvableGHalg}
\mathfrak{A} \, = \, Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}} \right)
\end{equation}
In the above equation $Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{\mathcal{Q}}} \right)$ denotes the Lie algebra of the solvable group manifold metrically equivalent to the non-comapact coset manifold $\frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}}$ according to a well developed mathematical theory extensively used in supergravity theories\cite{solvableparam}. In the case ${\mathrm{U}_{\mathcal{SK}}}$ is a \textit{maximally split} real form of a complex Lie algebra, then also ${\mathrm{U}_{\mathcal{Q}}}$ is maximally split and we have:
\begin{equation}\label{solvableGHalg2}
Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}} \right) \, =\, \mbox{Bor}\left ( \mathbb{U}_\mathcal{Q} \right)
\end{equation}
where $\mbox{Bor}\left ( \mathbb{U}_\mathcal{Q} \right)$ denotes the \textit{Borel subalgebra} of the semi-simple Lie algebra $\mathbb{G}$, generated by its Cartan generators and by the step operators associated with all positive roots.
\par
According to the general mathematical theory mentioned above, the very fact that the vielbein (\ref{filibaine}) satisfies the Maurer-Cartan equations of $Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}} \right)$ implies that the metric (\ref{cornish}) is the symmetric metric on the coset manifold $\frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}}$ which therefore admits continuous isometries associated with all the generators of the Lie algebra $\mathbb{U}_\mathcal{Q}$. This latter admits the following general decomposition:
\begin{equation}
\mbox{adj}(\mathbb{U}_{\mathcal{Q}}) =
\mbox{adj}(\mathbb{U}_{\mathcal{SK}})\oplus\mbox{adj}(\mathrm{SL(2,\mathbb{R})_E})\oplus
\mathbf{W}_{(2,\mathbf{W})}
\label{gendecompo}
\end{equation}
where $\mathbf{W}$ is the {\bf symplectic} representation of
$\mathbb{U}_{\mathcal{SK}}$ in which the symplectic section of Special Geometry transforms and which was used to construct the vielbein (\ref{cornish}). Denoting the generators of
$\mathbb{U}_{\mathcal{SK}}$ by $T^a$, the generators of
$\mathrm{SL(2,\mathbb{R})_E}$ by $\mathrm{L^x}$ and denoting by
$\mathbf{W}^{i\alpha}$ the generators in $\mathbf{W}_{(2,\mathbf{W})}$, the commutation
relations that correspond to the decomposition (\ref{gendecompo})
have the following general form \cite{MarioPietroKsenyaKM}:
\begin{eqnarray}
\nonumber && [T^a,T^b] = f^{ab}_{\phantom{ab}c} \, T^c \\
\nonumber && [L^x_E,L^y_E] = f^{xy}_{\phantom{xy}z} \, L^z , \\
&&\nonumber [T^a,\mathbf{W}^{i\alpha}] = (\Lambda^a)^\alpha_{\,\,\,\beta} \, \mathbf{W}^{i\beta},
\\ \nonumber && [L^x_E, \mathbf{W}^{i\alpha}] = (\lambda^x)^i_{\,\, j}\, \mathbf{W}^{j\alpha}, \\
&&[\mathbf{W}^{i\alpha},\mathbf{W}^{j\beta}] = \epsilon^{ij}\, (K_a)^{\alpha\beta}\, T^a + \,
\mathbb{C}^{\alpha\beta}\, k_x^{ij}\, L^x_E \label{genGD3pre}
\end{eqnarray}
where the $2 \times 2$ matrices $(\lambda^x)^i_j$, are the canonical generators of $\mathrm{SL(2,\mathbb{R})}$
in the fundamental, defining representation:
\begin{equation}
\lambda^3 = \left(\begin{array}{cc}
\ft 12 & 0 \\
0 & -\ft 12 \
\end{array} \right) \quad ; \quad \lambda^1 = \left(\begin{array}{cc}
0 & \ft 12 \\
\ft 12 & 0\
\end{array} \right) \quad ; \quad \lambda^2 = \left(\begin{array}{cc}
0 & \ft 12 \\
-\ft 12 & 0\
\end{array} \right)
\label{lambdax}
\end{equation}
while $\Lambda^a$ are the generators
of $\mathbb{U}_{\mathcal{SK}}$ in the symplectic representation $\mathbf{W}$. By
\begin{equation}
\mathbb{C}^{\alpha\beta} \equiv \left( \begin{array}{c|c}
\mathbf{0}_{(n+1)\times (n+1)} & \mathbf{1}_{(n+1)\times (n+1)} \\
\hline
-\mathbf{1}_{(n+1)\times (n+1)} & \mathbf{0}_{(n+1)\times (n+1)} \
\end{array}\right)
\label{omegamatra}
\end{equation}
we denote the antisymmetric symplectic metric in $2n+2$ dimensions, $n$
being the complex dimension of the Special K\"ahler manifold $\frac{\mathbb{U}_{\mathcal{SK}}}{\mathbb{H}_{\mathcal{SK}}}$. The symplectic character
of the representation $\mathbf{W}$ is asserted by the identity:
\begin{equation}
\Lambda^a\, \mathbb{C} + \mathbb{C}\, \left( \Lambda^a \right )^T = 0
\label{Lamsymp}
\end{equation}
The fundamental doublet representation of $\mathrm{SL(2,\mathbb{R})_E}$
is also symplectic and we have denoted by $\epsilon^{ij}= \left( \begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right) $ the
$2$-dimensional symplectic metric, so that:
\begin{equation}
\lambda^x\, \epsilon + \epsilon\, \left( \lambda^x \right )^T = 0,
\label{lamsymp}
\end{equation}
The matrices
$\left(K_a\right)^{\alpha\beta}=\left(K_a\right)^{\beta\alpha}$ and
$\left(k_x\right)^{ij}=\left(k_y\right)^{ji}$ are just symmetric matrices
in one-to-one correspondence with the generators of $\mathbb{U}_{\mathcal{Q}}$ and
$\mathrm{SL(2,\mathbb{R})}$, respectively. Implementing Jacobi
identities we find the following relations:
\begin{eqnarray}
&& \nonumber K_a\Lambda^c +
\Lambda^c K_a = f^{bc}_{\phantom{bc}a}K_b, \quad k_x\lambda^y + \lambda^y k_x
= f^{yz}_{\phantom{yz}x}k_z,
\label{jacobrele}
\end{eqnarray}
which admit the unique solution:
\begin{equation}
K_a = c_1 \, \mathbf{g}_{ab} \,\Lambda^b\mathbb{C}, \quad ; \quad k_x
= c_2 \, \mathbf{g}_{xy} \, \lambda^y \epsilon
\label{uniquesolutK&k}
\end{equation}
where $\mathbf{g}_{ab}$, $\mathbf{g}_{xy}$ are the Cartan-Killing metrics
on the algebras $\mathbb{U}_{\mathcal{SK}}$ and $\mathrm{SL(2,\mathbb{R})}$, respectively
and $c_1$ and $c_2$ are two arbitrary constants. These latter
can always be reabsorbed into the normalization of the generators
$\mathbf{W}^{i\alpha}$ and correspondingly set to one. Hence the algebra
(\ref{genGD3pre}) can always be put into the following elegant form:
\begin{eqnarray}
&& [T^a,T^b] = f^{ab}_{\phantom{ab}c} \, T^c \nonumber\\
&& [L^x,L^y] = f^{xy}_{\phantom{xy}z} \, L^z , \nonumber\\
&&
[T^a,\mathbf{W}^{i\alpha}] = (\Lambda^a)^\alpha_{\,\,\,\beta} \, \mathbf{W}^{i\beta},
\nonumber\\
&& [L^x, \mathbf{W}^{i\alpha}] = (\lambda^x)^i_{\,\, j}\, \mathbf{W}^{j\alpha}, \nonumber \\
&&[\mathbf{W}^{i\alpha},\mathbf{W}^{j\beta}] =
\epsilon^{ij}\, (\Lambda_a)^{\alpha\beta}\, T^a + \, \mathbb{C}^{\alpha\beta}\, \lambda_x^{ij}\, L^x
\label{genGD3}
\end{eqnarray}
where we have used the convention that symplectic indices are raised
and lowered with the symplectic metric, while adjoint representation
indices are raised and lowered with the Cartan-Killing metric.
\par
For the reader's convenience the list of Symmetric Special manifolds and of their Quaternionic K\"ahler counterparts in the image of the c-map is recalled in table \ref{homomodels} which reproduces the results of \cite{ToineCremmerOld}, according to which there is a short list of Symmetric Homogeneous Special manifolds comprising five discrete cases and two infinite series.
\begin{table}
\begin{center}
{\small
\begin{tabular}{||c|c||c||}
\hline
$\mathcal{SK}_n$ & $\mathcal{QM}_{4n+4}$ & $\mbox{dim} \, \mathcal{SK}_n \, = \, $ \\
Special K\"ahler manifold & Quaternionic K\"ahler manifold & $n$ \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}$ & $ \frac{\mathrm{G_{2(2)}}}{\mathrm{SU(2)\times SU(2)}}$ & $n=1$\\
\null & \null & \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{Sp(6,R)}}{\mathrm{SU(3)\times U(1)}}$ & $ \frac{\mathrm{F_{4(4)}}}{\mathrm{USp(6)\times SU(2)}}$ &$n=6$\\
\null & \null & \\
\null & \null &\null \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{SU(3,3)}}{\mathrm{SU(3)\times SU(3) \times U(1)}}$ & $ \frac{\mathrm{E_{6(2)}}}{\mathrm{SU(6)\times SU(2)}}$ &$n=9$\\
\null & \null & \\
\null & \null &\null \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{SO^\star(12)}}{\mathrm{SU(6)\times U(1)}}$ & $ \frac{\mathrm{E_{7(-5)}}}{\mathrm{SO(12)\times SU(2)}}$ & $n=15$ \\
\null & \null & \\
\null & \null &\null \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{E_{7(-25)}}}{\mathrm{E_{6(-78)} \times U(1)}}$ & $ \frac{\mathrm{E_{8(-24)}}}{\mathrm{E_{7(-133)}\times SU(2)}}$ & $n=27$ \\
\null & \null & \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,2+p)}}{\mathrm{SO(2)\times SO(2+p)}}$ & $ \frac{\mathrm{SO(4,4+p)}}{\mathrm{SO(4)\times SO(4+p)}}$ & $n=3+p$ \\
\null & \null & \\
\hline
\null & \null &\null \\
$ \frac{\mathrm{SU(p+1,1)}}{\mathrm{SU(p+1)\times U(1)}}$ & $ \frac{\mathrm{SU(p+2,2)}}{\mathrm{SU(p+2)\times SU(2)}}$ & $n=p+1$ \\
\null & \null &\null \\
\hline
\end{tabular}
}
\caption{List of special K\"ahler symmetric spaces with their Quaternionic K\"ahler c-map images. The number $n$ denotes the complex dimension of the Special K\"ahler preimage. On the other hand $4n+4$ is the real dimension of the Quaternionic K\"ahler c-map image. \label{homomodels}}
\end{center}
\end{table}
\par
Inspecting eq.s (\ref{genGD3}) we immediately realize that the Lie Algebra $\mathbb{U}_{Q}$ contains two universal Heisenberg subalgebras of dimension $(2n+3)$, namely:
\begin{eqnarray}
\mathbb{U}_{\mathcal{Q}} \, \supset \, \mathbb{H}\mathrm{eis}_1 \, &=& \mbox{span}_\mathbb{R}\, \left\{\mathbf{W}^{1\alpha} \, , \, \mathbb{Z}_1 \right\} \quad ; \quad \mathbb{Z}_1 \, =\, L_+ \, \equiv \, L^1\, + \, L^2 \nonumber\\
\null &\null & \left[\mathbf{W}^{1\alpha}\, , \, \mathbf{\mathbf{W}}^{1\beta} \right]\, = \, -\, \frac{1}{2} \, \mathbb{C}^{\alpha\beta} \, \mathbb{Z}_1 \quad ; \quad \left[\mathbb{Z}_1\, , \, \mathbf{W}^{1\beta} \right]\, = \,0 \label{Heinber1}\\
\mathbb{U}_{\mathcal{Q}} \, \supset \, \mathbb{H}\mathrm{eis}_2 \, &=& \mbox{span}_\mathbb{R}\, \left\{\mathbf{W}^{2\alpha} \, , \, \mathbb{Z}_2 \right\} \quad ; \quad \mathbb{Z}_2 \, =\, L_- \, \equiv \, L^1\, - \, L^2 \nonumber\\
\null &\null & \left[\mathbf{W}^{2\alpha}\, , \, \mathbf{W}^{2\beta} \right]\, = \, -\, \frac{1}{2} \, \mathbb{C}^{\alpha\beta} \, \mathbb{Z}_2 \quad ; \quad \left[\mathbb{Z}_2\, , \, \mathbf{W}^{2\beta} \right]\, = \,0 \label{Heinber2}
\end{eqnarray}
The first of these Heisenberg subalgebras of isometries is the universal one that exists for all Quaternionic K\"ahler manifolds $\mathcal{QM}_{4n+4}$ lying in the image of the $c$-map, irrespectively whether the pre-image Special K\"ahler manifold $\mathcal{SK}_n$ is a symmetric space or not. The tri-holomorphic moment map of these isometries was presented in eq.s(\ref{triholoHeis}) and (\ref{triholoHeisZ}). The second Heisenberg algebra exists only in the case when the Quaternionic K\"ahler manifold $\mathcal{QM}_{4n+4}$ is a symmetric space.
\par
From this discussion we also realize that the central charge $\mathbb{Z}_1$ is just the $L_+$ generator of a universal $\mathfrak{sl}(2,R)_E$ Lie algebra that exists only in the symmetric space case and which was named the Ehlers algebra in the context of dimensional reduction analysis from $D=4$ to $D=3$ \cite{PietroSashaMarioBH1}. When $\mathfrak{sl}(2,R)_E$ does exist we can introduce the universal compact generator:
\begin{equation}\label{ruotogrande}
\mathfrak{S}\, \equiv\, L_+ \, - \, L_- \,= \, 2 \, \lambda^2
\end{equation}
which rotates the two sets of Heisenberg translations one into the other:
\begin{equation}\label{tabarro}
\left[ \mathfrak{S}\, , \, \mathbf{W}^{i\alpha}\right ] \, = \, \epsilon^{ij} \, \mathbf{W}^{j\alpha}
\end{equation}
As we shall see, the gauging of this generator is a rather essential ingredient in the inclusion of one-field cosmological models into gauged $\mathcal{N}=2$ supergravity.
\paragraph{The embedding tensor formulation of the gauging.} It is useful to encode the choice of the gauge algebra in an \emph{embedding tensor} $\theta_{\Lambda}{}^{\mathcal{A}}$ \cite{Cordaro:1998tx,Nicolai:2001sv,deWit:2002vt,deWit:2005ub} though which the gauge generators are expressed in terms of the global symmetry ones. If we denote by $\{t_{\mathcal{A}}\}\equiv \{T^a,\,L^x,\,{\bf W}^{i\,\alpha}\}$ the generators of $\mathbb{U}_{\mathcal{Q}}$ and by $X_{\Lambda}$
the gauge generators, since we are gauging only (abelian) isometries of the quaternionic manifold, we can write:
\begin{equation}
X_{\Lambda}=\theta_{\Lambda}{}^{\mathcal{A}}\,t_{\mathcal{A}}\,,
\end{equation}
where the index $\Lambda$ runs over all the vector fields. Of these only a subset will actually gauge the chosen isometries. This subset will be labelled by boldface latin indices ${\bf I},\,{\bf J},\dots$.
The only condition on this tensor originates from the structure of the algebra:
\begin{equation}
[X_{\Lambda},\,X_{\Sigma}]=0\,\,\Rightarrow\,\,\,\,\,\theta_{\Lambda}{}^{\mathcal{A}}\theta_{\Lambda}{}^{\mathcal{B}}\,f_{\mathcal{A}\mathcal{B}}{}^\mathcal{C}=0\,,
\end{equation}
where $f_{\mathcal{A}\mathcal{B}}{}^\mathcal{C}$ are the structure constants of $\mathbb{U}_{\mathcal{Q}}$. The triholomorphic moment maps $\mathcal{P}^x_{\Lambda}$ and the Killing vectors $k_\Lambda^u$ can then be expressed in the following way:
\begin{equation}
\mathcal{P}^x_{\Lambda}=\theta_{\Lambda}{}^{\mathcal{A}}\,\mathcal{P}^x_{\mathcal{A}}\,\,;\,\,\,\,k_\Lambda^u=\theta_{\Lambda}{}^{\mathcal{A}}\,k_{\mathcal{A}}^u\,,
\end{equation}
where $\mathcal{P}^x_{\mathcal{A}},\,k_{\mathcal{A}}^u$ are the \emph{intrinsic} moment maps and Killing vectors associated with the quaternionic isometries.
In order to make the analysis independent of the initial symplectic frame of the vector multiplet sector, it is useful to describe the gauge algebra generators by the (redundant) symplectic notation $X_M=(X_\Lambda,\,X^\Lambda)=\theta_{M}{}^{\mathcal{A}}\,t_{\mathcal{A}}$, see \cite{deWit:2005ub}. The tensor $\theta_{M}{}^{\mathcal{A}}$ should then satisfy the locality constraint:
\begin{equation}
\theta_{\Lambda}{}^{\mathcal{A}}\theta^{\Lambda\,\mathcal{B}}-\theta_{\Lambda}{}^{\mathcal{B}}\theta^{\Lambda\,\mathcal{A}}=0\,,
\end{equation}
which guarantees that the tensor can be rotated, by means of a symplectic transformation, to an \emph{electric frame} in which $\theta^{\Lambda\,\mathcal{A}}=0$. For the restricted kind of gauging that we shall be dealing with, by extending the arguments given in \cite{thesearch}, we can work in the electric frame to start with, with no loss of generality.
\subsubsection{\sc The tri-holomorphic moment map in homogeneous symmetric Quaternionic K\"ahler manifolds}
\label{MezhduAlgGeom}
In the case the Quaternionic K\"ahler manifold $\mathcal{QM}_{4n+4}$ is a homogeneous symmetric space $\frac{\mathrm{U}_\mathcal{Q}}{\mathrm{H}_\mathcal{Q}}$, the tri-holomorphic moment map associated with any generator of $\mathfrak{t} \, \in \, \mathbb{U}_\mathcal{Q}$ of the isometry Lie algebra can be easily constructed by means of the formula:
\begin{equation}\label{generMapformula}
\mathcal{P}_\mathfrak{t}^x \, = \, \mbox{Tr}_{[{\mathbf{fun}}]}\, \left( J^x \, \mathbb{L}_{Solv}^{-1} \, \mathfrak{t} \, \mathbb{L}_{Solv} \right)
\end{equation}
where:
\begin{description}
\item[a)] $J^x$ are the three generators of the $\mathfrak{su}(2)$ factor in the isotropy subalgebra $\mathbb{H}\, = \, \mathfrak{su}(2) \, \oplus \, \mathbb{H}^\prime$, satisfying the quaternionic algebra (\ref{quatKvera}). They should be normalized in such a way as to realize the following condition. Naming:
\begin{equation}\label{MaurAmiCartan}
\Xi \, = \, \mathbb{L}_{Solv}^{-1}(q) \, \mathrm{d} \mathbb{L}_{Solv}(q)
\end{equation}
the Maurer Cartan differential one-form its projection on $J^x$ should precisely yield the $\mathfrak{su}(2)$ one-form defined in eq. (\ref{omegaSu2}):
\begin{equation}\label{omegacorrusco}
\omega \, = \, - \, \frac{{\rm i}}{ \sqrt{2} N_f } \, \sum_{x=1}^3 \,\mbox{Tr}_{[\mathbf{fun}]}\, \left( J^x \, \Xi\right) \, \gamma_x\, = \, \left( \begin{array}{cc}
-\frac{\rm i}{2} \, \mathcal{Q} \, - \, \frac{\rm i}{4} \,e^{-U} \, \Phi & e^{-\frac{U}{2}} \, V^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z}\\
- \, e^{-\frac{U}{2}} \, \overline{V}^T \, \mathbb{C} \, \mathrm{d}\mathbf{Z} & \frac{\rm i}{2} \, \mathcal{Q} \, + \, \frac{\rm i}{4} \,e^{-U} \, \Phi
\end{array}
\right)
\end{equation}
In the above equation, which provides the precise link between the $c$-map description and the coset manifold description of the same geometry, $N_f \, = \, \mbox{dim}\,\mathbf{fun}$ denotes the dimension of the fundamental representation of $\mathbb{U}_\mathcal{Q}$.
\item[b)] The solvable coset representative $\mathbb{L}_{Solv}(q)$ is obtained by exponentiation of the Solvable Lie algebra:
\begin{equation}\label{solvoexpo}
\mathbb{L}_{Solv}(q) \, \simeq \, \exp \left[ q \, \cdot \, Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathcal{H}_\mathcal{Q}}\right) \right]
\end{equation}
but the detailed exponentiation rule has to be determined in such a way that projecting the same Maurer Cartan form (\ref{MaurAmiCartan}) along an appropriate basis of generators $T_{I|Solv}$ of the solvable Lie algebra $Solv\left( \frac{\mathrm{U}_\mathcal{Q}}{\mathcal{H}_\mathcal{Q}}\right)$ we precisely obtain the vielbein $E^I_{QM}$ defined in eq.(\ref{filibaine}). This summarized in the following general equations:
\begin{eqnarray}\label{sopore}
E^I_{\mathcal{QM}} & = & \mbox{Tr}_{\mathbf{[fun]}} \left( T^I_{Solv} \, \Xi \right) \nonumber\\
\delta^{I}_{J} & = & \mbox{Tr}_{\mathbf{[fun]}} \left( T^I_{Solv} \, T_{I|Solv} \right) \nonumber\\
\Xi & = & E^I_{\mathcal{QM}} \, T_{I|Solv}
\end{eqnarray}
\end{description}
In eq.(\ref{sopore}) by $T^I_{Solv}$ we have denoted the conjugate (with respect to the trace) of the solvable Lie algebra generators.
\par
A general comment is in order. The precise calibration of the basis of the solvable generators $T^I_{Solv}$ and of their exponentiation outlined in eq.(\ref{solvoexpo}) which allows the identification (\ref{sopore}) is a necessary and quite laborious task in order to establish the bridge between the general $c$-map description of the quaternionic geometry and its actual realization in each symmetric coset model. This is also an unavoidable step in order to give a precise meaning to the very handy formula (\ref{generMapformula}) for the tri-holomorphic map. It should also be noted that although (\ref{generMapformula}) covers all the cases, the result of such a purely algebraic calculation is difficult to be guessed a priori. Hence educated guesses on the choice of generators whose gauging produces a priori determined features are difficult to be inferred from (\ref{generMapformula}). The analytic structure of the tri-holomorphic moment map instead is much clearer in the $c$-map framework of formulae (\ref{triholoHeis},\ref{triholoHeisZ},\ref{triholoSK}). The use of both languages and the construction of the precise bridge between them in each model is therefore an essential ingredient to understand the nature and the properties of candidate gaugings in whatever physical application.
\newpage
\part{\sc Abelian Gaugings and General Properties of their Potentials in the $c$-map Framework}
\label{grandiscussia}
As we stressed in the introduction the inclusion into $\mathcal{N}=2$ supergravity obtained in \cite{thesearch} of inflaton potentials such as the Starobinsky potential\footnote{Just as in \cite{thesearch} we mention scalar fields that typically have non canonical kinetic terms}
\begin{equation}\label{starotto}
V_{Starobinsky}(\phi) \, \equiv \, \left(1 \, - \, \exp\left[-\phi\right]\right)^2
\end{equation}
is not occasional and limited to the case of hypermultiplets lying in $\frac{\mathrm{G_{(2,2)}}}{\mathrm{SU(2)} \times \mathrm{SU(2)}}$, rather it follows a general pattern that can be uncovered and relies on the properties of the $c$-map. In this way the mechanisms of the \cite{thesearch} can be generalized to
larger Quaternionic K\"ahler manifolds opening a quite interesting new playground for the search of inflaton potentials that can be classified and understood in their geometrical origin.
\par
Let us schematically summarize the main ingredients of the approach pioneered in \cite{thesearch} whose generalization we pursue in this paper:
\begin{description}
\item[A)] The inflaton field $\phi$ is assumed to belong to the hypermultiplet Quaternionic K\"ahler manifold $\mathcal{QM}$.
\item[B)] In analogy with the construction in \cite{thesearch}, we require the graviphoton not to be minimally coupled to any other field. This condition originally followed from the general argument that in the dual to the $R+R^2$ supergravity the central charge is gauged. This will amount to a constraint on the form of the embedding tensor $\theta$ defining the gauge algebra.
\item[C)] The inflaton potential is generated by the gauging of an abelian subalgebra $\mathcal{A} \subset \mbox{iso} \left[\mathcal{QM}\right]$ of the isometry algebra of the hypermultipet manifold.
\item[D)] Since $\mathcal{A}$ is abelian it is not required to have any action on the vector multiplet scalars $\omega^i$ which are inert. Actually it is quite desirable that the potential $V_{gauging}$ generated by the gauging allows to fix all the $\omega^i$ to their values at some reference point, say $\omega^i \, = \, 0$:
\begin{equation}\label{fixing}
\left. \frac{\partial}{\partial \omega^i} \, V_{gauging} \right|_{\omega^i \, = \, 0} \, = \, 0
\end{equation}
As shown in \cite{thesearch}, one can generically guarantee the fixing conditions (\ref{fixing}) if the Special K\"ahler Geometry of the vector multiplets is chosen to be that of the so named Minimal Coupling, defined below in eq.s (\ref{minicup1}-\ref{minicup3}).
\item[E)] With the above choice of the vector multiplet geometry, after fixing the scalars $\omega^i$ the effective potential reduces to a sum of squares of the tri-holomorphic moment maps $P_{\mathcal{A}}^x$ which still depend on the variables $\left\{Z,U,a,z^i,{\bar z}^{i^\star}\right\}$. In order to approach effective potentials recognizable also as $\mathcal{N}=1$ supergravity potentials one would like to be able to fix all the Heisenberg fields $\mathbf{Z}$ (and possibly also the other fields $U$ and $a$) to zero, remaining only with the complex fields $\left ( z^i,{\bar z}^{i^\star}\right)$ of the inner Special K\"ahler manifold. Looking at the general form (\ref{triholoHeis}) of the tri-holomorphic moment map for the Heisenberg algebra generators and (\ref{triholoSK}) for the tri-holomorphic moment map of the inner Special K\"ahler isometries we immediately realize that, gauging these isometries \emph{separately}, the condition:
\begin{equation}\label{secondfixing}
\left. \frac{\partial}{\partial \mathbf{Z}^\alpha} \sum_{\mathfrak{t} \,\in \, \mathcal{A}} \, \left(\mathcal{P}_{\mathfrak{t}}\right)^2 \right|_{\mathbf{Z}\, = \, 0} \, = \, 0
\end{equation}
is always satisfied. A gauge generator which is a combination of a translation in the Heisenberg algebra and a Special K\"ahler isometry, yields in general a scalar potential exhibiting linear terms in ${\bf Z}$, so that (\ref{secondfixing}) provides a non-trivial constraint.\par
The definition of the locus $\mathcal{L}$ involves setting to zero a certain number of fields $\phi^r$ belonging to $\mathcal{SK}_n$ so that we should also realize the consistency condition:
\begin{equation}\label{thirdfixing}
\left. \frac{\partial}{\partial \phi^r} \sum_{\mathfrak{t} \,\in \, \mathcal{A}} \, \left(\mathcal{P}_{\mathfrak{t}}\right)^2 \right|_{\begin{array}{ccc}\mathbf{Z}& = & 0\\
\phi^r & = & 0 \\
\end{array}} \, = \, 0\,.
\end{equation}
As mentioned earlier, the gauging yielding Starobinsky-like potentials need also involve the compact generator $\mathfrak{S}$. As we shall show in the following, if the gauged isometry is a combination of $\mathfrak{S}$ and an $\mathcal{SK}_n$ isometry, (\ref{secondfixing}) poses no constraint on the gauging.
\item[F)] A favorite, though not mandatory, choice corresponds to looking for abelian generators of $\mbox{iso}\left[\mathcal{SK}_n\right]$ such that the locus which satisfies conditions (\ref{thirdfixing}) is defined by setting to zero all the axions $p_r$, namely all the fields associated with nilpotent generators of the solvable Lie algebra of $\mathcal{SK}_n$. The inclusion of the Starobinsky potential in supergravity was obtained in \cite{thesearch} precisely in this way. In section \ref{generalonuovo} we show a generalization of the same mechanism in the case of a bigger manifold $\mathcal{QM}_{4n+4}$, obtaining what can be denominated a multi Starobinsky model.
\item[G)] \textbf{The $U$-problem}. If we use only the type of isometries yielding the tri-holomorphic moment maps (\ref{triholoHeis}), (\ref{triholoHeisZ}) and (\ref{triholoSK}) we face a serious problem with the fields $U$. It appears only through exponentials all of the same sign ($\exp[ - 2 \, U]$ or $\exp[ - \, U]$ in front of perfect squares. Hence the field $U$ cannot be stabilized unless all such squares are zero which means no residual potential. To overcome such a problem one should have moment maps with the opposite sign of $U$ in the exponential and this can happen only by introducing in the gauging either $L^E_-$ or generators $\mathbf{W}^{2,\alpha}$ this means that such generators should exist, namely the manifold $\mathcal{QM}_{4n+4}$ should be a symmetric space. In \cite{thesearch} the $U$-problem was solved by adding to a parabolic generator of a $\mathcal{SK}_n$-isometry the universal compact generator (\ref{ruotogrande}). As we have emphasized the Ehlers subalgebra exists in all symmetric spaces and so does the compact generator (\ref{ruotogrande}). This implies that the mechanism leading to the inclusion of the Starobinsky model found in \cite{thesearch} is actually rather universal and can be generalized in several ways.
\end{description}
The above discussion provides a framework for the search of other inflaton potentials.
\section{\sc Minimal Coupling Special Geometry}
\label{minicoup}
In this section we shortly describe the structure of the Minimal Coupling Special K\"ahler manifold $\mathcal{MSK}_{p+1}$, mostly in order to fix our conventions and to establish our notations. As announced in the introduction, this kind of Special Geometry is our favorite choice for the vector multiplet sector of the $\mathcal{N}=2$ lagrangian which allows us to construct an entire class of theories where the vector multiplet scalars can be stabilized and the effective potential of an abelian gauging is reduced only to the hypermultiplet sector. In view of such a use of $\mathcal{MSK}_{p+1}$, all items of its Special Geometry will be denoted with a hat, and its complex coordinates will be named $\omega_i$ rather than $z^i$. However it is clear that $\mathcal{MSK}_{p+1}$ might also be used as $c$-map preimage of a Quaternionic K\"ahler manifold describing hypermultiplets.
\par
As a manifold $\mathcal{MSK}_{p+1}$ is the following coset:
\begin{equation}\label{minicup1}
\mathcal{MSK}_{p+1} \, = \, \frac{\mathrm{SU(1,p+1)}}{\mathrm{U(1)} \times \mathrm{SU(p+1)}}
\end{equation}
In terms of the complex coordinates $\omega^i$ a convenient choice of the $(2 \,p \,+\, 4)$-dimensional holomorphic symplect section is the following one:
\begin{equation}\label{minicup2}
\widehat{\Omega} \, = \, \left( \begin{array}{c}
\widehat{X}^\Lambda \\
\hline
\widehat{F}_\Sigma
\end{array}
\right) \, = \,\left( \begin{array}{c}
1 \\
\omega^i\\
\hline
-{\rm i}\\
{\rm i}\, \omega^i
\end{array}
\right) \quad ; \quad (i\,= \, 1, \dots \, p+1 )
\end{equation}
which leads to the following K\"ahler potential:
\begin{equation}\label{minicup3}
\widehat{\mathcal{K}} \, = \, - \, \log \,\left[ - \, {\rm i} \widehat{\Omega} \, \widehat{\mathbb{C}} \, \widehat{{\overline{\Omega}}}\right ] \, = \, - \, \log \, \left[\,2 \,\left( 1 \, - \, \omega\, \cdot \,\bar{\omega}\right)\right]
\end{equation}
and to the following K\"ahler metric:
\begin{equation}\label{minicup4}
\widehat{g}_{ij^\star} \, = \, \partial_i\,\partial_{j^\star} \,\widehat{\mathcal{K}}\, = \, \frac{1}{\left(1-\omega\cdot\ \bar{\omega} \right)^2} \, \left( \delta^{ij}\, \left(1-\omega\cdot\ \bar{\omega} \right)\, + \, \bar{\omega}^i \, \omega^j\right)
\end{equation}
Defining the K\"ahler covariant derivatives of the covariantly holomorphic sections as in eq.s (\ref{uvector}) we obtain three results that are very important for the discussion of reduced scalar potentials in the present paper.
Firstly we get:
\begin{equation}\label{minicup4barra}
\nabla_i \, \widehat{U}_j \, \equiv \, \nabla_i \, \nabla_j \widehat{V} \, = \,0
\end{equation}
which compared with eq.(\ref{defaltern}) implies the vanishing of the three-index symmetric tensor $\widehat{C}_{ijk}$. This unique property of the special K\"ahler manifold $\mathcal{MSK}_{p+1}$ defined by eq.(\ref{minicup1}) is the reason why it has been named the Minimal Coupling Special Geometry, the interpretation of the tensor $C_{ijk}$ in phenomenological applications being that of Yukawa couplings of the gauginos. In ref. \cite{thesearch} it was shown that the vanishing of $\widehat{C}_{ijk}$ guarantees the consistency (see eq. (3.10) of the quoted reference) of the truncation of the classical supergravity theory to the hypermultiplet quaternionic scalars by fixing the vector multiplet scalars to the origin of their manifold:
\begin{equation}\label{golubchika}
\omega^i \, = \, 0
\end{equation}
Secondly we evaluate the the covariantly symplectic holomorphic section in the origin of the manifold and we obtain:
\begin{equation}\label{miniculpan1}
\left. \widehat{V} \right|_{\omega\, =\,0} \, = \,\frac{1}{\sqrt{2}}\,\left\{\begin{array}{c|c||c|c}
1 & 0 & -\,{\rm i} & 0
\end{array}
\right\}
\end{equation}
In the same point we have:
\begin{equation}\label{fantamini}
\left.\left(\widehat{g}^{ij^\star}\,\nabla_i \widehat{V}^\alpha \,\nabla_{j^\star} \widehat{\overline{V}}\right)\right|_{\omega\, =\,0} \, = \, \frac{1}{2} \, \left(\begin{array}{c|c||c|c}
0 & 0 & 0 & 0 \\
\hline
0 & \mathbf{1}_{(p+1)\times(p+1)} & 0 & {\rm i} \,\mathbf{1}_{(p+1)\times(p+1)} \\
\hline
\hline
0 & 0 & 0 & 0 \\
\hline
0 & - \,{\rm i} \,\mathbf{1}_{(p+1)\times(p+1)} & 0 & \mathbf{1}_{(p+1)\times(p+1)}
\end{array}
\right)
\end{equation}
\subsection{\sc Gauging abelian isometries of the hypermultiplets}
Relying on these results we see that if the hypermultiplet Quaternionic manifold $\mathcal{QM}_{4m}$ possesses a $p+1$-dimensional abelian Lie algebra of isometries, we can always gauge them by using, for the vector multiplets, the Special K\"ahler manifold
$\mathcal{MSK}_{p+1}$ introducing also the following embedding tensor:
\begin{equation}\label{gioisco}
\theta_{M}^{I} \, \equiv \, \left\{\theta_{\Lambda}^I\, , \, \theta^{\Sigma|I }\right\} \, = \, \left \{\theta_0{}^{I} \, = \,0 \, , \, \theta_J{}^{I} \, = \, \delta^I_J \, , \, \theta^{\Sigma|I }\, = \, 0\right\}\,.
\end{equation}
Notice that the choice of setting $\theta_0{}^{I} \, = \,0$ follows from the requirement $B)$ that the graviphoton should not be gauged. This indeed amounts to
requiring:
\begin{equation}
\left. \widehat{V} \right|_{\omega\, =\,0}^M\,\theta_M{}^{\mathcal{A}}=0\,\,\,\Rightarrow\,\,\,\,\,\,\theta_0{}^{I} \, = \,0\,.
\end{equation}
In such a theory the scalar potential has the following general form:
\begin{equation}\label{cornettoalgida}
\mathcal{V}_{scalar}(\omega, \bar{\omega},q) \, = \, 4 \, k_I^u k_J^v \, h_{uv} \, \widehat{V}^I \, \widehat{\overline{V}}{}^J
\, + \,
\left(\widehat{g}^{ij^\star}\,\nabla_i \widehat{V}^I \,\nabla_{j^\star} \widehat{\overline{V}}{}^J \, - \, 3 \,
\widehat{V}^I \,\widehat{\overline{V}}{}^J \right) \, \mathcal{P}^x_I \, \mathcal{P}^x_J
\end{equation}
setting $\omega^i \, = \, 0$ is a consistent truncation and the reduced potential takes the following universal general form
which is positive definite by construction:
\begin{equation}\label{gartolini}
\mathcal{V}_{scalar}(0, 0,q) \, = \, \sum_{I=1}^{p+1} \,\mathcal{P}^x_I(q) \, \mathcal{P}^x_I(q)
\end{equation}
In the next section \ref{starobin1} we reconsider the derivation of the Starobinsky potential obtained in \cite{thesearch} from a parabolic gauging as a master example that can be generalized to bigger manifolds.
\subsection{\sc The Starobinsky potential}
\label{starobin1}
Last year a great deal of activity was devoted to the inclusion of phenomenologically interesting inflaton potentials into $\mathcal{N}=1$ supergravity as we recalled in the introduction. A first wave of investigations considered the possible generation of potentials by means of suitably chosen superpotentials, subsequently, after an important new viewpoint was introduced in \cite{minimalsergioKLP} and was subsequently developed in
\cite{primosashapietro},\cite{piesashatwo},\cite{Ferrara:2013wka},\cite{Ferrara:2013kca},
\cite{pietrosergiosasha1},\cite{pietrosergiosasha2}, it became clear that positive definite inflaton potentials can be generated by the gauging of some isometry of the K\"ahler manifold of scalar multiplets. Such potentials have the form of squares of K\"ahler moment maps. In \cite{pietrosergiosasha1} this mechanism was applied to the case of constant curvature one-dimensional K\"ahler manifolds and it was shown that Starobinsky-like potentials \cite{Starobinsky:1980te} emerge from the moment map of a parabolic isometry in $\mathrm{SL(2,\mathbb{R})} \simeq \mathrm{SU(1,1)}$ with the addition of a Fayet Iliopoulos term. In particular the standard Starobinsky model that is dual to an $R+R^2$ supergravity emerges from gauging the parabolic shift isometry of an $\frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}$ manifold with K\"ahler potential $\mathcal{K}\, = \, - 3\, \log (z-{\bar z})$ which is precisely the Special K\"ahler manifold $S^3$. Let us now consider eq.(\ref{triholoSK}) and we can learn an important lesson. If in the $c$-map image of some $\mathcal{SK}$ Special K\"ahler manifold, for instance the $S^3$ model, we gauge, according to the scheme discussed in section \ref{grandiscussia}, some nilpotent Lie algebra element $\mathfrak{N}_+\, \in \, \mathbb{U}_{\mathcal{SK}} \, \subset \mathbb{U}_\mathcal{Q}$ identical with the parabolic shift generator that we would have gauged in $\mathcal{N}=1$ supergravity, (for instance the generator $L_+\, \in \, \mathfrak{sl}(2,\mathbb{R})$ in the case of the $S^3$ model), we obtain a moment map that contains precisely the $\mathcal{P}_\mathbf{I}$ of the $\mathcal{N}=1$ case, modified by $\mathbf{Z}$ dependent terms. In case the $\mathbf{Z}$ can be stabilized to zero the remaining effective potential is that of the corresponding $\mathcal{N}=1$ theory, apart from the Fayet Iliopoulos term. There are two remaining problems. The generation of a Fayet Iliopoulos term and the stabilization of the $U$ field. They are solved in one stroke by modifying the parabolic generator of the inner Special K\"ahler isometry with the addition of the universal Ehlers rotation (\ref{ruotogrande}).
\par
Let us see how this works.
\par
With reference to eq.s (\ref{generillini}) let us consider the following generator:
\begin{equation}\label{costianovo}
\mathfrak{p}\, = \, \mathfrak{N}_+ \, + \, \kappa \, \mathfrak{S}
\end{equation}
where $\mathfrak{N}_+$ is the previously mentioned nilpotent element of the Special K\"ahler subalgebra ($\mathfrak{N}_+^r \, = \, 0$, for some positive integer $r$) and $\kappa$ is a parameter. Let us then calculate the tri-holomorphic moment map $\mathcal{P}^x_\mathfrak{p}$ according to formula (\ref{fisterone}).
\par
Because of the linearity of the momentum map in Lie algebra elements we have:
\begin{eqnarray}
\mathfrak{P}_\mathfrak{p} &=& \mathfrak{P}_{\mathfrak{N}_+} \, + \, \mathfrak{P}_{\mathfrak{S}}\nonumber \\
\mathfrak{P}_{\mathfrak{N}_+} &=& \left( \begin{array}{c|c}
\frac{\rm i}{4} \, \mathcal{P}_{\mathfrak{N}_+} \, + \, \mathcal{O} \, \left( \mathbf{Z}^2\right)& \mathcal{O} \, \left( \mathbf{Z}\right) \\
\hline
\mathcal{O} \, \left( \mathbf{Z}\right) & \, - \, \frac{\rm i}{4} \, \mathcal{P}_{\mathfrak{N}_+}\, - \, \mathcal{O} \, \left( \mathbf{Z}^2\right)
\end{array}
\right) \nonumber\\
\mathfrak{P}_{\mathfrak{S}} &=& \left( \begin{array}{c|c}
\frac{\rm i}{8} \, e^{-U} \,\left(1 + a^2 \, + \, e^{2U}\right)\, + \, \mathcal{O} \, \left( \mathbf{Z}^2\right)& \mathcal{O} \, \left( \mathbf{Z}\right) \\
\hline
\mathcal{O} \, \left( \mathbf{Z}\right) & \, - \, \frac{\rm i}{8} \, e^{-U} \,\left(1 + a^2 \, + \, e^{2U}\right)\, - \, \mathcal{O} \, \left( \mathbf{Z}^2\right)
\end{array}
\right) \nonumber\\
\label{gomorratano}
\end{eqnarray}
where $\mathcal{P}_{\mathfrak{N}_+}$ is the K\"ahlerian moment map of the Killing vector associated with the generator $\mathfrak{N}_+$ as defined in eq.(\ref{sisalvichipuo}).
It is evident by the above completely universal formulae that the potential:
\begin{equation}\label{potentus}
V_{gauging} \, = \, \mbox{const} \, \mbox{Tr} \left[ \, \mathfrak{P}_\mathfrak{p}\, \cdot \, \mathfrak{P}_\mathfrak{p} \right]
\end{equation}
possesses the following universal property:
\begin{equation}\label{Zetaseneva}
\left. \frac{\partial}{\partial \mathbf{Z}^\alpha} \, V_{gauging}\right|_{\mathbf{Z}=0} \, = \,0
\end{equation}
allowing for a consistent truncation of the Heisenberg fields. After such truncation we find:
\begin{equation}\label{Vzero}
V_{eff}(U,a, z,\bar z) \, =\, \left.V_{gauging}\right|_{\mathbf{Z}=0} \, = \, \mbox{const} \, \times \, \left[ \mathcal{P}_{\mathfrak{N}_+} \, + \, \frac{\kappa}{2} \, e^{-U} \left(1+a^2+e^{2U}\right)\right]^2
\end{equation}
From equation (\ref{Vzero}) we further learn that we can consistently truncate the fields $a$ and $U$ setting them to zero since
\begin{equation}\label{ciurlonelmanicozzo}
\left. \frac{\partial}{\partial U} \, V_{eff}\right|_{U=a=0} \, = \,0 \quad ; \quad \left. \frac{\partial}{\partial a} \, V_{eff}\right|_{U=a=0} \, = \,0
\end{equation}
We find:
\begin{equation}\label{curiosone}
V_{infl}(z,\bar z) \, \equiv \, V_{eff}(0,0, z,\bar z) \, = \, \left( \, \mathcal{P}_{\mathfrak{N}_+} \, + \, \kappa \right)^2
\end{equation}
which clearly shows how the universal generator $\mathfrak{S}$ provides, after stabilization of the $U$ field, the mechanism that generates the Fayet Iliopoulos term \cite{Fayet:1974jb} essential for inflation.
\newpage
\part{\sc Examples}
As an illustration of the general patterns and mechanisms described in the previous pages we consider two examples of Quaternionic K\"ahler manifolds $\mathcal{QM}_{4n+4}$ obtained from the $c$-map of two homogeneous symmetric Special K\"ahler manifolds $\mathcal{SK}_n$.
\begin{enumerate}
\item The manifold $\frac{\mathrm{G_{(2,2)}}}{\mathrm{SU(2)} \times \mathrm{SU(2)}}$ which is the $c$-map image of the Special K\"ahler manifold $\frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}$ with cubic embedding of $\mathrm{SU(1,1)}$ in $\mathrm{Sp(4,\mathbb{R})}$. In this case $n=1$ and the corresponding coupling of one vector multiplet to supergravity is usually named the $S^3$ model in the literature.
\item The manifold $\frac{\mathrm{F_{(4,4)}}}{\mathrm{SU(2)} \times \mathrm{USp(6)}}$ which is the $c$-map image of the Special K\"ahler manifold $\frac{\mathrm{Sp(6,\mathbb{R})}}{\mathrm{SU(3) \times U(1)}}$ . In this case $n=6$.
\end{enumerate}
For these two models we provide a full fledged construction of all the geometrical items and in particular we realize the bridge between the algebraic description and the analytic one advocated at the end of section \ref{MezhduAlgGeom}. This allows us to discuss a couple of examples of gaugings. In particular in the case of the of the first model which we utilized as a calibration device for our general formulae we retrieve the inclusion of the Starobinsky model first demonstrated in \cite{thesearch}.
\par
The detailed construction of the second model, which we plan to utilize in future publications for an extensive and possibly exhaustive analysis of gaugings in larger hypermultiplet spaces, is utilized in the present paper to provide an example of generalization of the results of \cite{thesearch} by means of the inclusion of a multi Starobinsky model.
\section{\sc The $S^3$ model and its quaternionic image $\frac{\mathrm{G_{(2,2)}}}{\mathrm{SU(2)} \times \mathrm{SU(2)}}$}
In this which is the simplest example $n=1$, namely the Special K\"ahler manifold has complex dimension $1$ and it can be identified with the time honored Poincar\'e Lobachevsky plane:
\begin{equation}\label{tripini}
\mathcal{SK}_{1} \, = \, \frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}
\end{equation}
\subsection{\sc The special K\"ahler structure of $S^3$}
The corresponding K\"ahler potential is:
\begin{equation}\label{kelero1}
\mathcal{K} \, = \, - \, \log \, \left[ (z \, - \, \bar{z})^3\right ]
\end{equation}
which leads to the K\"ahler metric:
\begin{equation}\label{kelero2}
g_{z\bar{z}} \, = \, \frac{3}{4} \, \frac{1}{(z \, - \, \bar{z})^2}
\end{equation}
Setting:
\begin{equation}\label{realcordo}
z \, = \, {\rm i} \, \exp[h] \, + \, y
\end{equation}
we get:
\begin{equation}\label{metrullone}
g_{z\bar{z}} \, = \, \frac{3}{2} \, \left(dh^2 \, + \, \exp[-2h] \, dy^2\right)
\end{equation}
In the notations of \cite{PietroSashaMarioBH1} the holomorphic symplectic section governing this special geometry is given
by the following four component vector:
\begin{equation}\label{seziona}
\Omega \, = \,\left\{-\sqrt{3}z^2,z^3,\sqrt{3} z,1\right\}
\end{equation}
In this case the $\mathbf{W}$-representation is the spin $j\, = \, \frac{3}{2}$ of the $\mathrm{SL(2,\mathbb{R})} \sim \mathrm{SU(1,1)}$ group that happens to be
four dimensional symplectic:
\begin{equation}
\label{frilli}
\mathrm{SL(2,\mathbb{R})} \, \ni \,\left(\begin{array}{ll}
a & b \\
c & d
\end{array} \right) \, \Longrightarrow \, \left(
\begin{array}{llll}
d a^2+2 b c a & -\sqrt{3} a^2
c & -c b^2-2 a d b &
-\sqrt{3} b^2 d \\
-\sqrt{3} a^2 b & a^3 &
\sqrt{3} a b^2 & b^3 \\
-b c^2-2 a d c & \sqrt{3} a
c^2 & a d^2+2 b c d &
\sqrt{3} b d^2 \\
-\sqrt{3} c^2 d & c^3 &
\sqrt{3} c d^2 & d^3
\end{array}
\right) \, \in \, \mathrm{Sp(4,\mathbb{R})}
\end{equation}
the preserved symplectic metric being the following one:
\begin{equation}\label{goriaci}
\mathbb{C} \, = \, \left(
\begin{array}{llll}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0
\end{array}
\right)
\end{equation}
According to the general rule the K\"ahler potential (\ref{kelero1}) is retrieved by setting:
\begin{equation}\label{fagano}
\mathcal{K}(z,{\bar z}) \, = \, - \log \left[ - {\rm i} \Omega \, \mathbb{C} \, \
\overline{\Omega} \right ]
\end{equation}
\subsection{\sc The matrix $\mathcal{M}_4^{-1}$ and the $c$-map}
For the $S^3$ model the matrix $\mathcal{M}_4$ and its inverse have the following explicit appearance:
\begin{equation}\label{m4diretto}
\mathcal{M}_4 \, = \, \left(
\begin{array}{llll}
\frac{4 i z {\bar z} \left(z^2+4 {\bar z}
z+{\bar z}^2\right)}{(z-{\bar z})^3} & -\frac{4 i \sqrt{3} z^2
{\bar z}^2 (z+{\bar z})}{(z-{\bar z})^3} & -\frac{i
(z+{\bar z}) \left(z^2+10 {\bar z}
z+{\bar z}^2\right)}{(z-{\bar z})^3} & -\frac{2 i \sqrt{3}
(z+{\bar z})^2}{(z-{\bar z})^3} \\
-\frac{4 i \sqrt{3} z^2 {\bar z}^2 (z+{\bar z})}{(z-{\bar z})^3} &
\frac{8 i z^3 {\bar z}^3}{(z-{\bar z})^3} & \frac{2 i \sqrt{3} z
{\bar z} (z+{\bar z})^2}{(z-{\bar z})^3} & \frac{i
(z+{\bar z})^3}{(z-{\bar z})^3} \\
-\frac{i (z+{\bar z}) \left(z^2+10 {\bar z}
z+{\bar z}^2\right)}{(z-{\bar z})^3} & \frac{2 i \sqrt{3} z
{\bar z} (z+{\bar z})^2}{(z-{\bar z})^3} & \frac{4 i \left(z^2+4
{\bar z} z+{\bar z}^2\right)}{(z-{\bar z})^3} & \frac{4 i
\sqrt{3} (z+{\bar z})}{(z-{\bar z})^3} \\
-\frac{2 i \sqrt{3} (z+{\bar z})^2}{(z-{\bar z})^3} & \frac{i
(z+{\bar z})^3}{(z-{\bar z})^3} & \frac{4 i \sqrt{3}
(z+{\bar z})}{(z-{\bar z})^3} & \frac{8 i}{(z-{\bar z})^3}
\end{array}
\right)
\end{equation}
its inverse being:
\begin{equation}\label{m4inverso}
\mathcal{M}_4^{-1} \, = \,\left(
\begin{array}{llll}
\frac{4 i \left(z^2+4 {\bar z}
z+{\bar z}^2\right)}{(z-{\bar z})^3} & \frac{4 i \sqrt{3}
(z+{\bar z})}{(z-{\bar z})^3} & \frac{i \left(z^3+11 {\bar z}
z^2+11 {\bar z}^2 z+{\bar z}^3\right)}{(z-{\bar z})^3} &
-\frac{2 i \sqrt{3} z {\bar z} (z+{\bar z})^2}{(z-{\bar z})^3}
\\
\frac{4 i \sqrt{3} (z+{\bar z})}{(z-{\bar z})^3} & \frac{8
i}{(z-{\bar z})^3} & \frac{2 i \sqrt{3}
(z+{\bar z})^2}{(z-{\bar z})^3} & -\frac{i
(z+{\bar z})^3}{(z-{\bar z})^3} \\
\frac{i \left(z^3+11 {\bar z} z^2+11 {\bar z}^2
z+{\bar z}^3\right)}{(z-{\bar z})^3} & \frac{2 i \sqrt{3}
(z+{\bar z})^2}{(z-{\bar z})^3} & \frac{4 i z {\bar z}
\left(z^2+4 {\bar z} z+{\bar z}^2\right)}{(z-{\bar z})^3} &
-\frac{4 i \sqrt{3} z^2 {\bar z}^2 (z+{\bar z})}{(z-{\bar z})^3}
\\
-\frac{2 i \sqrt{3} z {\bar z} (z+{\bar z})^2}{(z-{\bar z})^3} &
-\frac{i (z+{\bar z})^3}{(z-{\bar z})^3} & -\frac{4 i \sqrt{3}
z^2 {\bar z}^2 (z+{\bar z})}{(z-{\bar z})^3} & \frac{8 i z^3
{\bar z}^3}{(z-{\bar z})^3}
\end{array}
\right)
\end{equation}
Furthermore, in this case a convenient reference point is given by $z_0 \, = \, {\rm i}$ that can be mapped into any point of the upper complex plane by means of the element:
\begin{equation}\label{trasluco}
\mathfrak{g}_z \, = \, \ \left(
\begin{array}{ll}
e^{h/2} & e^{-h/2} y \\
0 & e^{-h/2}
\end{array}
\right) \, \in \, \mathrm{SL(2,\mathbb{R})}
\end{equation}
acting by means of fractional linear transformations. The explicit form of the $\Lambda(\mathfrak{g})$ matrix in the $\mathbf{W}$-representation was given in eq.(\ref{frilli}). This provides us with all the necessary information in order to write down the explicit form of the $E^I_{\mathcal{QM}} $ vielbein for the $S^3$ case.
\subsection{\sc The vielbein and the borellian Maurer Cartan equations}
They are the following ones:
\begin{equation}\label{EBHfilo}
E^I_{\mathcal{QM}} \, = \, \frac{1}{2} \left(
\begin{array}{l}
\mathrm{dU} \\
\sqrt{3} \mathrm{dh} \\
\sqrt{3} \mathrm{dy} e^{-h} \\
e^{-U} \left(\mathrm{da}+\mathrm{dZ}_3 Z_1+\mathrm{dZ}_4 Z_2-\mathrm{dZ}_1 Z_3-\mathrm{dZ}_2
Z_4\right) \\
\sqrt{2} e^{-\frac{h}{2}-\frac{U}{2}}
\left(\mathrm{dZ}_1+y \left(2
\mathrm{dZ}_3-\sqrt{3} y
\mathrm{dZ}_4\right)\right) \\
\sqrt{2} e^{-\frac{3 h}{2}-\frac{U}{2}}
\left(\left(\sqrt{3} \mathrm{dZ}_3-y
\mathrm{dZ}_4\right) y^2+\sqrt{3} \mathrm{dZ}_1
y+\mathrm{dZ}_2\right)\\
\sqrt{2} e^{\frac{h-U}{2}}
\left(\mathrm{dZ}_3-\sqrt{3} y
\mathrm{dZ}_4\right) \\
\sqrt{2} e^{\frac{3 h}{2}-\frac{U}{2}}
\mathrm{dZ}_4
\end{array}
\right)
\end{equation}
Furthermore we find $\mathcal{M}_4^{-1}({\rm i},-{\rm i}) \, = \, - \, \mathbf{1}_{4\times4}$ so that the quadratic form (\ref{quadrotta}) is just:
\begin{equation}\label{doremito}
\mathfrak{q}_{AB} \, = \, \mbox{diag} \,\left( 1,1,1,1,1,1,1,1\right)
\end{equation}
The next step consists of calculating the geometry of the space described by the above vielbein and flat metric (\ref{doremito}). To this effect we have first to calculate the contorsion, namely the exterior derivatives of the vielbein and then using such a result the spin connection $\omega^{IJ}$, finally the curvature two-form from which we extract the Riemann and the Ricci tensor.
\par
Addressing the first step, namely the contorsion, we have the first important surprise. The exterior derivatives of the vielbein are expressed in terms of wedge-quadratic products of the same vielbein with constant numerical coefficients. This means that the above constructed vielbein satisfy a set of Maurer Cartan equations describing a Lie algebra, namely\footnote{Note that here, for simplicity we have dropped the suffix $\mathcal{SK}$. This is done for simplicity since there is no risk of confusion.}:
\begin{equation}\label{glorioso}
dE^I \, - \, \frac{1}{2} \, f_{JK}^{\phantom{JK}I} \, E^I \, \wedge \, E^J \, = \, 0
\end{equation}
the tensor $f_{BC}^{\phantom{BC}A}$ being the structure constants of such a Lie algebra. Explicitly for the $S^3$ model we get:
\begin{equation}\label{primocartano}
\begin{array}{l}
0 \, = \, dE^{1} \\
0 \, = \, dE^{2} \\
0 \, = \, dE^{3}\, + \, 2\,\frac{E^{2}\wedge
E^{3}}{\sqrt{3}} \\
0 \, = \, dE^{4}\, +\, 2 \, E^{1}\wedge
E^{4}\, -\, 2 \, E^{5}\wedge
E^{7}\, -\, 2 \, E^{6}\wedge E^{8}
\\
0 \, = \, dE^{5}+ E^{1}\wedge
E^{5}\, +\, \frac{E^{2}\wedge
E^{5}}{ \sqrt{3}}-\frac{4 \,
E^{3}\wedge E^{7}}{\sqrt{3}} \\
0 \, = \, dE^{6}+ E^{1}\wedge
E^{6}+ \sqrt{3}
E^{2}\wedge
E^{6}\, -\, 2\, E^{3}\wedge E^{5}
\\
0 \, = \, dE^{7}+ E^{1}\wedge
E^{7}-\frac{E^{2}\wedge
E^{7}}{\sqrt{3}}\, +\, 2 \, E^{3}\wedge
E^{8} \\
0 \, = \, dE^{8}+E^{1}\wedge
E^{8}- \sqrt{3}
E^{2}\wedge E^{8}
\end{array}
\end{equation}
Hence it arises the following question: which Lie algebra is described by such Maurer Cartan equations?
Utilizing the standard method of diagonalizing the adjoint action of the two commuting generators $H_{1,2}$ dual to $E^{1,2}$ we find that the eigenvalues are just the positive roots of $\mathfrak{g}_{2,2}$:
\begin{equation}
\label{g2rootsystem}
\begin{array}{rclcrcl}
\alpha_1&=&(1,0)&;&\alpha_2&=&\frac{\sqrt{3}}{2}\,(-\sqrt{3},1)\\
\alpha_3 \, =\, \alpha_1+\alpha_2&=&\frac{1}{2}\,(-1,\sqrt{3}) &;&
\alpha_4 \, = \, 2\,\alpha_1+\alpha_2 &=&
\frac{1}{2}\,(1,\sqrt{3}) \\
\alpha_5 \, = \, 3\,\alpha_1+\alpha_2&=&\frac{\sqrt{3}}{2}\,(\sqrt{3},1)&;&\alpha_6 \, = \, 3\,\alpha_1+2\,\alpha_2 &=&
(0,\sqrt{3})\
\end{array}
\end{equation}
As it is well known the complex Lie algebra $\mathfrak{g}_2(\mathbb{C})$ has rank two and it is defined by the $2\times 2$ Cartan matrix encoded in the following Dynkin diagram:
\begin{center}
\begin{picture}(110,30)
\put (-60,20){$\mathfrak{g}_2$}
\put (10,23){\circle {10}}
\put (15,25){\line (1,0){20}}
\put (15,23){\line (1,0){20}}
\put (20.5,19){{\LARGE$>$}}
\put (15,20.5){\line (1,0){20}}
\put (40,23){\circle {10}}
\put (65,21){$=\quad\quad\left (\begin{array}{cc}
2 & -3\\
-1 & 2
\end{array} \right)$}
\end{picture}
\end{center}
The real form $\mathfrak{g}_{2,2}$ is the maximally split form of the above complex Lie algebra.
With a little bit of more work we can put eq.s(\ref{primocartano}) into the standard Cartan Weyl form for the Borel subalgebra of $\mathfrak{g}_{2,2}$, composed by the Cartan generators and by all the positive root step operators. Naming $T_J$ the generators dual to the vielbein $E^I$ such that $E^I(T_J) \, = \delta^I_J$, we find that the appropriate identifications are the following ones:
\begin{equation}\label{interpretaziag22}
\begin{array}{l}
T_2 \, = \, 2 \, \frac{\mathcal{H}_1}{\sqrt{3}} \\
T_1 \, = \, 2 \, \frac{\mathcal{H}_2}{\sqrt{3}} \\
T_3 \, = \, 2 \, E^{\alpha _1} \\
T_4 \, = \, 2 \, E^{\alpha _6} \\
T_8 \, = \, 2 \, E^{\alpha _2} \\
T_7 \, = \, 2 \, E^{\alpha _3 } \\
T_5 \, = \, 2 \, E^{\alpha _4} \\
T_6 \, = \, 2 \, E^{\alpha _5}
\end{array}
\end{equation}
We conclude that the manifold on which the metric (\ref{geodaction}) is constructed is homeomorphic to the solvable group-manifold $\mbox{Bor}(\mathfrak{g}_{2,2})$.
\subsection{\sc The spin connection}
Next, calculating the Levi-Civita spin connection from its definition, namely the vanishing torsion condition:
\begin{equation}\label{surcallo}
0 \, = \, dE^I \, + \, \omega^{IJ} \, \wedge \, E^J
\end{equation}
we find the following result:
\begin{equation}\label{ristolone}
\omega^{IJ} \, = \, \left(
\begin{array}{llllllll}
0 & 0 & 0 & E^{4} &
\frac{E^{5}}{2} &
\frac{E^{6}}{2} &
\frac{E^{7}}{2} &
\frac{E^{8}}{2} \\
0 & 0 & \frac{E^{3}}{\sqrt{3}} & 0 &
\frac{E^{5}}{2 \sqrt{3}} & \frac{1}{2}
\sqrt{3} E^{6} & -\frac{E^{7}}{2
\sqrt{3}} & -\frac{1}{2} \sqrt{3} E^{8}
\\
0 & -\frac{E^{3}}{\sqrt{3}} & 0 & 0 &
-\frac{E^{6}}{2}-\frac{E^{7}}{\sqrt{3}} & -\frac{E^{5}}{2} &
\frac{E^{8}}{2}-\frac{E^{5}}{\sqrt{3}} & \frac{E^{7}}{2} \\
-E^{4} & 0 & 0 & 0 &
\frac{E^{7}}{2} &
\frac{E^{8}}{2} &
-\frac{E^{5}}{2} &
-\frac{E^{6}}{2} \\
-\frac{E^{5}}{2} & -\frac{E^{5}}{2
\sqrt{3}} &
\frac{E^{6}}{2}+\frac{E^{7}}{\sqrt{3}} & -\frac{E^{7}}{2} & 0 &
\frac{E^{3}}{2} &
-\frac{E^{3}}{\sqrt{3}}-\frac{E^4}{2} & 0 \\
-\frac{E^{6}}{2} & -\frac{1}{2} \sqrt{3} \,
E^{6} & \frac{E^{5}}{2} &
-\frac{E^{8}}{2} &
-\frac{E^{3}}{2} & 0 & 0 &
-\frac{E^{4}}{2} \\
-\frac{E^{7}}{2} & \frac{E^{7}}{2
\sqrt{3}} &
\frac{E^{5}}{\sqrt{3}}-\frac{E^{
8}}{2} & \frac{E^{5}}{2} &
\frac{E^{3}}{\sqrt{3}}+\frac{E^{
4}}{2} & 0 & 0 & \frac{E^{3}}{2} \\
-\frac{E^{8}}{2} & \frac{1}{2} \sqrt{3}
E^{8} & -\frac{E^{7}}{2} &
\frac{E^{6}}{2} & 0 &
\frac{E^{4}}{2} &
-\frac{E^{3}}{2} & 0
\end{array}
\right)
\end{equation}
which can be decomposed in the way we now describe.
\subsection{\sc Holonomy algebra and decompostion of the spin connection}
Let us introduce two triplets $J^x_{[I]}$ and $J^x_{[II]}$ of $8 \times 8$ matrices that can be read off explicitly as the coefficients of $\alpha_x$ and $\beta_x$ in the following linear combinations:
\begin{eqnarray}\label{gringo1}
&\sum_{x=1}^3 \, \alpha_x \, J^x_{[I]} \, = \, & \nonumber\\
& \left(
\begin{array}{llllllll}
0 & 0 & 0 & -\frac{\alpha _1}{2} & -\frac{1}{4}
\sqrt{3} \alpha _3 & -\frac{\alpha _2}{4} &
\frac{\sqrt{3} \alpha _2}{4} & -\frac{\alpha
_3}{4} \\
0 & 0 & \frac{\alpha _1}{2} & 0 & -\frac{\alpha
_3}{4} & -\frac{1}{4} \sqrt{3} \alpha _2 &
-\frac{\alpha _2}{4} & \frac{\sqrt{3} \alpha
_3}{4} \\
0 & -\frac{\alpha _1}{2} & 0 & 0 & -\frac{\alpha
_2}{4} & \frac{\sqrt{3} \alpha _3}{4} &
\frac{\alpha _3}{4} & \frac{\sqrt{3} \alpha
_2}{4} \\
\frac{\alpha _1}{2} & 0 & 0 & 0 & \frac{\sqrt{3}
\alpha _2}{4} & -\frac{\alpha _3}{4} &
\frac{\sqrt{3} \alpha _3}{4} & \frac{\alpha
_2}{4} \\
\frac{\sqrt{3} \alpha _3}{4} & \frac{\alpha
_3}{4} & \frac{\alpha _2}{4} & -\frac{1}{4}
\sqrt{3} \alpha _2 & 0 & \frac{\sqrt{3} \alpha
_1}{4} & -\frac{\alpha _1}{4} & 0 \\
\frac{\alpha _2}{4} & \frac{\sqrt{3} \alpha
_2}{4} & -\frac{1}{4} \sqrt{3} \alpha _3 &
\frac{\alpha _3}{4} & -\frac{1}{4} \sqrt{3}
\alpha _1 & 0 & 0 & \frac{\alpha _1}{4} \\
-\frac{1}{4} \sqrt{3} \alpha _2 & \frac{\alpha
_2}{4} & -\frac{\alpha _3}{4} & -\frac{1}{4}
\sqrt{3} \alpha _3 & \frac{\alpha _1}{4} & 0 &
0 & \frac{\sqrt{3} \alpha _1}{4} \\
\frac{\alpha _3}{4} & -\frac{1}{4} \sqrt{3}
\alpha _3 & -\frac{1}{4} \sqrt{3} \alpha _2 &
-\frac{\alpha _2}{4} & 0 & -\frac{\alpha
_1}{4} & -\frac{1}{4} \sqrt{3} \alpha _1 & 0
\end{array}
\right) & \nonumber\\
\end{eqnarray}
\begin{eqnarray}\label{gringo2}
&\sum_{x=1}^3 \, \beta_x \, J^x_{[II]} \, = \, & \nonumber\\
& \left(
\begin{array}{llllllll}
0 & 0 & 0 & \frac{3 \beta _1}{2} & -\frac{1}{4}
\sqrt{3} \beta _3 & -\frac{3 \beta _2}{4} &
-\frac{1}{4} \sqrt{3} \beta _2 & \frac{3 \beta
_3}{4} \\
0 & 0 & \frac{\beta _1}{2} & 0 & -\frac{\beta
_3}{4} & -\frac{3}{4} \sqrt{3} \beta _2 &
\frac{\beta _2}{4} & -\frac{3}{4} \sqrt{3}
\beta _3 \\
0 & -\frac{\beta _1}{2} & 0 & 0 & \frac{5 \beta
_2}{4} & \frac{\sqrt{3} \beta _3}{4} & \frac{5
\beta _3}{4} & -\frac{1}{4} \sqrt{3} \beta _2
\\
-\frac{3 \beta _1}{2} & 0 & 0 & 0 & -\frac{1}{4}
\sqrt{3} \beta _2 & \frac{3 \beta _3}{4} &
\frac{\sqrt{3} \beta _3}{4} & \frac{3 \beta
_2}{4} \\
\frac{\sqrt{3} \beta _3}{4} & \frac{\beta _3}{4}
& -\frac{5 \beta _2}{4} & \frac{\sqrt{3} \beta
_2}{4} & 0 & \frac{\sqrt{3} \beta _1}{4} &
-\frac{5 \beta _1}{4} & 0 \\
\frac{3 \beta _2}{4} & \frac{3 \sqrt{3} \beta
_2}{4} & -\frac{1}{4} \sqrt{3} \beta _3 &
-\frac{3 \beta _3}{4} & -\frac{1}{4} \sqrt{3}
\beta _1 & 0 & 0 & -\frac{3 \beta _1}{4} \\
\frac{\sqrt{3} \beta _2}{4} & -\frac{\beta
_2}{4} & -\frac{5 \beta _3}{4} & -\frac{1}{4}
\sqrt{3} \beta _3 & \frac{5 \beta _1}{4} & 0 &
0 & \frac{\sqrt{3} \beta _1}{4} \\
-\frac{3 \beta _3}{4} & \frac{3 \sqrt{3} \beta
_3}{4} & \frac{\sqrt{3} \beta _2}{4} &
-\frac{3 \beta _2}{4} & 0 & \frac{3 \beta
_1}{4} & -\frac{1}{4} \sqrt{3} \beta _1 & 0
\end{array}
\right) & \nonumber\\
\end{eqnarray}
Both triplets form an 8-dimensional representation of the $\mathfrak{su}(2)$ Lie algebra and the two triplets commute with each other:
\begin{eqnarray}
\left[ \,J^x_{[I]} \, , \, J^y_{[I]} \right ] &=& \epsilon^{xyz} \, J^y_{[I]} \nonumber\\
\left[ \,J^x_{[II]} \, , \, J^y_{[II]} \right ] &=& \epsilon^{xyz} \, J^y_{[II]} \nonumber\\
\left[ \,J^x_{[I]} \, , \, J^y_{[II]} \right ] &=&0 \label{guglielmotell}
\end{eqnarray}
Furthermore all matrices are antisymmetric so that the two Lie algebras $\mathfrak{su}_{\mathrm{I}}(2)$ and $\mathfrak{su}_{\mathrm{II}}(2)$ are both subalgebras of $\mathfrak{so}(8)$.
The distinction between these two representations becomes clear when we calculate the Casimir operator for both of them. We obtain:
\begin{equation}\label{gaglioffo}
\sum_{x=1}^3 \, J^x_{[I]}\cdot J^x_{[I]} \, = \, - \, \frac{3}{4} \, \mathbf{1} \quad ; \quad \sum_{x=1}^3 \, J^x_{[II]}\cdot J^x_{[II]} \, = \, - \, \frac{15}{4} \, \mathbf{1}
\end{equation}
Hence the first $\mathfrak{su}_{\mathrm{I}}(2)$ Lie algebra in realized on the considered eight--dimensional space in the $j=\frac{1}{2}$ representation, while the second
$\mathfrak{su}_{\mathrm{II}}(2)$ Lie algebra in realized on the same space in the $j=\frac{3}{2}$. In other words, with respect to both subalgebras of $\mathfrak{so}(8)$, the fundamental representation decomposes as follows:
\begin{equation}\label{cirimello}
\mathbf{8} \, \stackrel{\mathfrak{su}_{\mathrm{I}}(2) \oplus \mathfrak{su}_{\mathrm{II}}(2) \subset \mathfrak{so}(8)}{\Longrightarrow} \, (\mathbf{2},\mathbf{4})
\end{equation}
By direct calculation we verify that the spin connection displayed in equation (\ref{ristolone}) has the following structure:
\begin{equation}\label{decompospincon}
\omega \, = \, \omega^{[I]}_x \, J^x_{[I]} \, \oplus \, \omega^{[II]}_x \, J^x_{[II]}
\end{equation}
where:
\begin{equation}\label{cucurucu}
\omega^{[I]}_x \, = \, \left(
\begin{array}{l}
\sqrt{3} \mathrm{E}^3- \mathrm{E}^4 \\
\sqrt{3} \mathrm{E}^7- \mathrm{E}^6 \\
- \mathrm{E}^8- \sqrt{3} \mathrm{E}^5
\end{array}
\right) \quad ; \quad \omega^{[II]}_x \, = \, \left(
\begin{array}{l}
\frac{\mathrm{E}^4}{2}+\frac{\mathrm{E}^3}{2
\sqrt{3}} \\
-\frac{\mathrm{E}^7}{2
\sqrt{3}}-\frac{\mathrm{E}^6}{2} \\
\frac{\mathrm{E}^8}{2}-\frac{\mathrm{E}^5}{2
\sqrt{3}}
\end{array}
\right)
\end{equation}
This structure clearly demonstrates the reduced holonomy of the Quaternionic K\"ahler manifold. Indeed, according to eq.(\ref{cirimello}) the vielbein transforms in the doublet of $\mathfrak{su}_{\mathrm{I}}(2)$ tensored with the fundamental representation of $\mathfrak{sp}(4,\mathbb{R})$. In the present case the symplectic $4 \times 4$ matrices are actually reduced to the subalgebra $\mathfrak{su}_{\mathrm{II}}(2) \subset \mathfrak{sp}(4,\mathbb{R})$ with respect to which the fundamental of $\mathfrak{sp}(4,\mathbb{R})$ remains irreducible and coincides with the $j=\frac{3}{2}$ representation of $\mathfrak{su}_{\mathrm{II}}(2)$. The above discussion can be summarized by the statement:
\begin{equation}\label{ollonomio}
\mathfrak{so}(8) \, \subset \, \mathfrak{su}(2) \, \oplus \, \mathfrak{usp}(4) \, \subset \, \mathrm{Hol} \, = \, \mathfrak{su}_{\mathrm{I}}(2) \, \oplus \, \mathfrak{su}_{\mathrm{II}}(2)
\end{equation}
by definition the holonomy algebra being the Lie algebra in which the Levi-Civita spin connection takes values.
\subsection{\sc Structure of the isotropy subalgebra $\mathbb{H}$}
It remains to single out the structure of the denominator subalgebra ${\mathbb{H}} \, \subset \, \mathbb{U} \, \equiv \, {\mathfrak{g}}_{2,2}$ in the orthogonal decomposition:
\begin{equation}\label{ortogonallodecompo}
\mathbb{U} \, = \, \mathbb{H} \, \oplus \, \mathbb{K} \quad ; \quad \left \{ \begin{array}{ccc}
\left[ \mathbb{H} \, , \, \mathbb{H} \right] & \subset & \mathbb{H} \\
\left[ \mathbb{H} \, , \, \mathbb{K} \right] & \subset & \mathbb{K} \\
\left[ \mathbb{K} \, , \, \mathbb{K} \right] & \subset & \mathbb{H} \\
\end{array}\right.
\end{equation}
Since our quaternionic K\"ahler manifold is a symmetric space it follows that Lie algebra $\mathbb{H}$ must be isomorphic with the holonomy algebra $\mathrm{Hol} \, = \,\mathfrak{su}_{\mathrm{I}}(2) \oplus \mathfrak{su}_{\mathrm{II}}(2)$ that we have calculated in the previous subsection.
By definition the Lie algebra $\mathbb{H}$ is the maximal compact subalgebra which for maximal split algebras has a universal definition in terms of the step operators associated with the positive roots $E^\alpha$ and their conjugates $E^{-\alpha}$. In the case of $\mathfrak{g}_{2,2}$ which has six positive roots we can write:
\begin{equation}\label{Hfavola}
\mathbb{H} \, \equiv \,\mbox{span}_{\mathbb{R}} \left \{ E^{\alpha _1} - E^{\alpha _1}\, , \, E^{\alpha _2} - E^{\alpha _2} \, , \, E^{\alpha _3} - E^{\alpha _3} \, , \, E^{\alpha _4} - E^{\alpha _4}\, , \, E^{\alpha _5} - E^{\alpha _5} \, , \, E^{\alpha _6} - E^{\alpha _6} \right \}
\end{equation}
The structure of (\ref{Hfavola}) is the following:
\begin{equation}\label{fattucco}
\mathbb{H} \, = \, \mathfrak{su}_{\mathrm{I}}(2) \, \oplus \, \mathfrak{su}_{\mathrm{II}}(2)
\end{equation}
where the generators of the two subalgebras are:
\begin{equation}\label{friccouno}
j^x_{[I]} \, = \,\left(
\begin{array}{l}
\frac{-3 \mathrm{E}^{-\alpha_1}+3
\mathrm{E}^{\alpha_1}+\sqrt{3}
\left(\mathrm{E}^{\alpha_6}-\mathrm{E}^{-\alpha_6}\right)}{6 \sqrt{2}} \\
\frac{3 \mathrm{E}^{-\alpha_3}-3
\mathrm{E}^{\alpha_3}+\sqrt{3}
\left(\mathrm{E}^{-\alpha_5}-\mathrm{E}^{\alpha_5}\right)}{6 \sqrt{2}} \\
\frac{\sqrt{3} \left(\mathrm{E}^{\alpha_2}-\mathrm{E}^{-\alpha_2}\right)+3
\left(\mathrm{E}^{-\alpha_4}-\mathrm{E}^{\alpha_4}\right)}{6 \sqrt{2}}
\end{array}
\right)
\end{equation}
and
\begin{equation}\label{friccodue}
j^x_{[II]} \, = \,\left(
\begin{array}{l}
\frac{-\mathrm{E}^{-\alpha_1}+\mathrm{E}^{\alpha_1}+\sqrt{3} \left(\mathrm{E}^{-\alpha_6}-\mathrm{E}^{\alpha_6}\right)}{2 \sqrt{2}}
\\
\frac{-\mathrm{E}^{-\alpha_3}+\mathrm{E}^{\alpha_3}+\sqrt{3} \left(\mathrm{E}^{-\alpha_5}-\mathrm{E}^{\alpha_5}\right)}{2 \sqrt{2}}
\\
\frac{\sqrt{3} \left(\mathrm{E}^{-\alpha_2}-\mathrm{E}^{\alpha_2}\right)+\mathrm{E}^{-\alpha_4}-\mathrm{E}^{\alpha_4}}{2 \sqrt{2}}
\end{array}
\right)
\end{equation}
and satisfy among themselves the same relations (\ref{guglielmotell}) as their homologous generators $ J^x_{[I]}$ and $ J^x_{[II]}$.
In eq.(\ref{fattucco}) we have used the same notation as in eq.(\ref{ollonomio}) using the obligatory homomorphism between the the holonomy algebra $\mathrm{Hol}$ and the isotropy subalgebra $\mathbb{H}$. The precise correspondence between generators of one algebra and generators of the other will be establishe in the next subsection by means of the use of the coset representative.
\subsection{\sc The coset representative}
The next step in the development of the coset approach is the construction of the solvable coset representative $\mathbb{L}_{Solv}(\phi)$, advocated in eq.s(\ref{solvoexpo}-\ref{sopore} ), namely a coordinate dependent element of the Borel group of $\mathfrak{g}_{(2,2)}$ such that the Maurer Cartan form
\begin{equation}\label{rupiaindu}
\Xi \, = \, \mathbb{L}_{Solv}(\phi)^{-1} \, \mathrm{d}\mathbb{L}_{Solv}(\phi)
\end{equation}
projected along the Borel algebra generators, as given in eq.(\ref{interpretaziag22}), reproduces the vielbein of eq.(\ref{EBHfilo}).
The appropriate coset representative is obtained by exponentiating the Borel Lie algebra and the precise recipe is provided below.
First define:
\begin{eqnarray}\label{generillini}
\mathbf{L}^E_0 & = & \frac{1}{\sqrt{3}} \,\mathcal{H}_2 \quad ; \quad \mathbf{L}^E_+ \, = \, -\sqrt{\frac{2}{3}}\, E^{\alpha_6} \nonumber\\
\mathbf{L}_0 & = &\,\mathcal{H}_1 \quad ; \quad \mathbf{L}_+ \, = \, \sqrt{2} E^{\alpha_1} \nonumber\\
\mathbf{W}^I & = & \sqrt{\frac{2}{3}} \, \left \{ E^{\alpha_4} \, , \, E^{\alpha_5} \, , \, - \, E^{\alpha_3}\, , \, - \, E^{\alpha_2} \, , \, \right \}
\end{eqnarray}
and then set:
\begin{eqnarray}\label{fornarina}
\mathbb{L}& = & \exp\left [a \mathbf{L}^E_+\right]\,\cdot \, \exp\left [\sqrt{2} \left(Z_1 \, \mathbf{W}^1 \, + \, Z_3 \, \mathbf{W}^3\right )\right]\,\cdot \, \nonumber\\
&&\,\cdot \,\exp\left [\sqrt{2} \left(Z_1 \, \mathbf{W}^1 \, + \, Z_3 \, \mathbf{W}^3\right )\right]\,\cdot \,
\exp\left [y \mathbf{L}_+\right]\,\cdot \,\exp\left [h \mathbf{L}_0\right]\,\cdot \,\exp\left [U \mathbf{L}^E_0\right]
\end{eqnarray}
By explicit evaluation we obtain the result displayed in the appendix in formulae (\ref{fornoalegna}) and (\ref{cosettuspresento})
and we verify that, if we set:
\begin{equation}\label{sequenzus}
\mathfrak{T}_I \, = \, \left\{ \mathbf{L}^E_0 \, , \, \mathbf{L}_0 \, , \, \mathbf{L}^E_+ \, , \, \mathbf{L}_+ \, , \,\mathbf{W}_1\, , \,\mathbf{W}_2\, , \,\mathbf{W}_3\, , \,\mathbf{W}_4\right\}
\end{equation}
upon substitution of (\ref{cosettuspresento}) into the Maurer Cartan form (\ref{rupiaindu})
we obtain:
\begin{equation}\label{rupiamoresca}
\mathbb{L}_{Solv}(\phi)^{-1} \, \mathrm{d}\mathbb{L}_{Solv}(\phi) \, = \, \sum_{I=1}^8 \, \mathfrak{T}_I \, E^I_{\mathcal{QM}}
\end{equation}
the forms $E^I_{\mathcal{QM}}$ being given in equation (\ref{EBHfilo}). Alternatively we can also write:
\begin{equation}\label{curlandico}
\mathbb{L}_{Solv}(\phi)^{-1} \, \mathrm{d}\mathbb{L}_{Solv}(\phi) \, = \, \sum_{x=1}^3 \, \left (\omega^{[I]}_x \, j^x_{[I]}\, \oplus \, \omega^{[II]}_x \, j^x_{[II]}\right) \, \oplus \, \sum_{I}^8 \, \mathbf{T}_I \, E^I_{\mathcal{SQ}}
\end{equation}
In the above equation $\omega^{[I]}_x$ and $\omega^{[II]}_x$ are the components of the spin connections given in eq. (\ref{cucurucu}),
$j^x_{[I]}$ and $j^x_{[II]}$ are the generators of $\mathbb{H}$ defined in eq. (\ref{friccouno},\ref{friccodue}) and $\mathbf{T}_I $ denotes a suitable base of generators in the $\mathbb{K}$ subspace of $\mathfrak{g}_{(2,2)}$ defined as:
\begin{eqnarray}\label{cofimus}
\mathbb{K} & \equiv & ,\mbox{span}_{\mathbb{R}} \left \{ \mathcal{H}_1 \, , \, \mathcal{H}_2 \, , \, E^{\alpha _1} + E^{\alpha _1}\, , \, E^{\alpha _2} + E^{\alpha _2} \, , \, E^{\alpha _3} + E^{\alpha _3} \right. \nonumber\\
& &\left. E^{\alpha _4} + E^{\alpha _4}\, , \, E^{\alpha _5} + E^{\alpha _5} \, , \, E^{\alpha _6} + E^{\alpha _6} \right \}
\end{eqnarray}
The precise form of the generators $\mathbf{T}_I $ is not relevant to our purposes and we omit it. The key point is instead the identification of the generators $j^x_{[I]}$ of $\mathbb{H}$ with generators $J^x_{[I]}$ of the holonomy algebra. This provides us with the knowledge of the quaternionic complex structures within the algebra $\mathbb{U}_\mathcal{Q}$ and allows to calculate the tri-holomorphic moment map of any generator $\mathbf{t}\, \in \, \mathbb{U}_\mathcal{Q}$ by means of the formula (\ref{generMapformula}) which in our case reads:
\begin{equation}\label{fisterone}
\mathcal{P}_{\mathbf{t}}^x \, = \,\frac{1}{2} \mbox{Tr}_{\mathbf{7}} \left ( j^x_{[I]} \, \mathbb{L}_{Solv}^{-1} \, \mathbf{t} \, \mathbb{L}_{Solv} \right )
\end{equation}
having denoted by $\mathbf{7}$ the $7$-dimensional fundamental representation of $\mathfrak{g}_{(2,2)}$.
\paragraph{The Starobinsky potential}
As an immediate application of eq.(\ref{fisterone}) one can retrieve the results of \cite{thesearch} on the inclusion of the Starobinsky potential into supergravity. In section \ref{starobin1} we presented a general discussion of the gaugings of nilpotent generators in the Special K\"ahler subalgebra $\mathbb{U}_{\mathcal{SK}} \, \subset \, \mathbb{U}_\mathcal{Q}$. In the present case where $\mathbb{U}_{\mathcal{SK}} \, = \, \mathfrak{sl}(2,\mathbb{R})$ the only available nilpotent operator is $L_+$ and from the general formula (\ref{sisalvichipuo}) applied to the case where the metric is given by (\ref{kelero2}) and the complex coordinate is parameterized as in eq.(\ref{realcordo}) we find:
\begin{equation}\label{cubitagorio}
\mathcal{P}_{L_+} \, = \, \mbox{const} \, \times \, \exp[-h] \, = \, \mbox{const} \, \times \, \left(\mbox{Im} \,z\right)^{-1}
\end{equation}
This result inserted into the general formula (\ref{curiosone}) yields
\begin{equation}\label{starobbo}
V(h) \, = \, \mbox{const} \, \times \, \left(\exp[-h] \, + \, \kappa \right)^2
\end{equation}
which is indeed the Starobinsky potential, since, once expressed in terms of $h$, the K\"ahler potential is exactly $\mathcal{K}\, = \, 3 \, h$.
The same result is directly obtained with precise coefficients by inserting in eq.(\ref{fisterone}) the $7$-dimensional image of $L_+$ in the fundamental representation of $\mathfrak{g}_{(2,2)}$.
\newpage
\section{\sc The $\mathrm{Sp(6,\mathbb{R})}/\mathrm{SU(3)\times U(1)}$ - model and its c-map image.}
Next we consider the Special K\"ahler manifold
\begin{equation}\label{gurto}
\mathcal{M}_{\mathop{\rm {}Sp} 6} \, = \, \frac{\mathop{\rm {}Sp}(6,\mathbb{R})}{\mathrm{SU(3)\times U(1)}}
\end{equation}
and its c-map image which is the following quaternionic manifold:
\begin{equation}\label{quatergurto}
\mbox{$c$-map} \quad : \quad \mathcal{M}_{\mathop{\rm {}Sp} 6} \, \mapsto \, \mathcal{QM}_{F4} \, \equiv \, \frac{\mathrm{F_{(4,4)}}}{\mathrm{SU(2)} \times \mathrm{USp(6)}}
\end{equation}
$\mathcal{M}_{\mathop{\rm {}Sp} 6}$ belongs to the magic square of exceptional special K\"ahler manifolds whose quaternionic $c$-map is a homogeneous symmetric space having, as it is evident from (\ref{quatergurto}), an exceptional Lie group as isometry group.
\par
We begin by illustrating some general properties of this remarkable manifold. First of all, in order to discuss them adequately we need to choose a basis for the $\mathfrak{sp}(6,\mathbb{R})$ Lie algebra. Since we are not interested in solving Lax equations we do not choose the basis where the matrices of the Borel subalgebra are upper triangular. We rather use the basis where the symplectic preserved metric is the standard one, namely:
\begin{equation}\label{Cmatra}
\mathbb{C} \, = \, \left(
\begin{array}{llllll}
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0
\end{array}
\right)
\end{equation}
This traditional choice allows to describe in a simple way other aspects of the manifold geometry that are more relevant to our present purposes.
\par
According to the above choice, an element of the $\mathop{\rm {}Sp}(6,\mathbb{R})$ group and an element of the $\mathfrak{sp}(6,\mathbb{R})$ Lie-algebra are matrices respectively fulfilling the following two constraints:
\begin{equation}\label{giluro}
\left( \begin{array}{c|c}
A & B \\
\hline
C & D
\end{array}
\right)^T \, \mathbb{C} \, \left( \begin{array}{c|c}
A & B \\
\hline
C & D
\end{array}
\right) \, = \, \mathbb{C} \quad ; \quad \left( \begin{array}{c|c}
\mathbf{A} & \mathbf{B} \\
\hline
\mathbf{C} & \mathbf{D}
\end{array}
\right)^T \, \mathbb{C} + \mathbb{C} \left( \begin{array}{c|c}
\mathbf{A} & \mathbf{B} \\
\hline
\mathbf{C} & \mathbf{D}
\end{array}
\right) \, = \, 0
\end{equation}
where $A,B,C,D$, $\mathbf{A,B,C,D}$ are $3\times 3$ blocks. By means of the so called Cayley transformation
\begin{equation}\label{caylus}
\mathcal{C} \, = \, \left(
\begin{array}{llllll}
\frac{1}{\sqrt{2}} & 0 & 0 & \frac{i}{\sqrt{2}} & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & 0 & 0 & \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & \frac{i}{\sqrt{2}} \\
\frac{1}{\sqrt{2}} & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 & 0 \\
0 & \frac{1}{\sqrt{2}} & 0 & 0 & -\frac{i}{\sqrt{2}} & 0 \\
0 & 0 & \frac{1}{\sqrt{2}} & 0 & 0 & -\frac{i}{\sqrt{2}}
\end{array}
\right)
\end{equation}
a real element of the symplectic group (or algebra) can be mapped into a matrix that is simultaneously symplectic and pseudounitary:
\begin{equation}\label{grillo}
\mathcal{S} \, = \, \mathcal{C}^\dagger \, \, \left( \begin{array}{c|c}
{A} & {B} \\
\hline
{C} & {D}
\end{array}
\right) \,\mathcal{C} \, = \, \left(\begin{array}{cc}
U_0 & U_1^\star \\
U_1 & U_0 ^\star
\end{array}
\right) \,\, \in \, \mathop{\rm {}Sp}(6,\mathbb{C}) \bigcap \mathrm{SU(3,3)}
\end{equation}
The diagonal blocks $U_0 \in \mathrm{U(3)}$ span the $\mathrm{H}$-subgroup of the coset (\ref{gurto}). This allows to introduce a set projective coordinates that parameterize the points of the manifold (\ref{gurto}) and have a nice fractional linear transformation under the action of the group $\mathop{\rm {}Sp}(6,\mathbb{R})$. Given any coset parameterization
\begin{equation}\label{cosetpar}
\left( \begin{array}{c|c}
{A}(\phi) & {B}(\phi) \\
\hline
{C}(\phi) & {D}(\phi)
\end{array}
\right) \, \in \, \mathop{\rm {}Sp}(6,\mathbb{R})
\end{equation}
namely a family of symplectic group elements depending on 12 parameters $\phi^i$ such that each different choice of the $\phi^i$ provides a representative of a different equivalence class in (\ref{gurto}), we can construct the following, \textit{symmetric complex matrix}:
\begin{equation}\label{Zmatra}
Z(\phi) \, \equiv \, \left(A(\phi) \, - \, {\rm i} \,B(\phi) \right) \, \left( C(\phi) \, - \, {\rm i} D(\phi) \right)^{-1}
\end{equation}
which has a very simple transformation under the action of the symplectic group. Let us consider the action of any element of $\mathop{\rm {}Sp}(6,\mathbb{R})$ on the coset representative. We have:
\begin{equation}\label{canalito}
\underbrace{\left( \begin{array}{c|c}
{\hat{A}} & {\hat{B}} \\
\hline
{\hat{C}} & {\hat{D}}
\end{array}
\right)}_{= \mathfrak{g}\in \,\mathop{\rm {}Sp}(6,\mathbb{R})} \left( \begin{array}{c|c}
{A}(\phi) & {B}(\phi) \\
\hline
{C}(\phi) & {D}(\phi)
\end{array}
\right) \, = \, \left( \begin{array}{c|c}
{A}(\phi^\prime) & {B}(\phi^\prime) \\
\hline
{C}(\phi^\prime) & {D}(\phi^\prime)
\end{array}
\right) \, \mathrm{H}(\phi, \mathfrak{g})
\end{equation}
where $\phi^\prime$ is the label of a new equivalence class and $\mathrm{H}(\phi, \mathfrak{g}) \, \in \, \mathrm{U(3)}$ is a suitable $\mathrm{H}$-compensator. Calculating the matrix $Z(\phi^\prime)$ according to the definition (\ref{Zmatra}) we find that it is related to $Z(\phi)$ by a simple linear fractional transformation (generalized to matrices):
\begin{equation}\label{gourmet}
Z(\phi^\prime) \, = \, \left( A Z(\phi) \, + \, B\right) \, \left( C Z(\phi) \, + \, D\right)^{-1}
\end{equation}
Formula (\ref{gourmet}) is of crucial relevance and requires several comments. From a mathematical point of view, (\ref{gourmet}) is the well known generalization of the action of the $\mathrm{SL(2,\mathbb{R})}\, \simeq \, \mathop{\rm {}Sp}(2,\mathbb{R})$ group on the upper complex plane of Poincar\'e-Lobachevsky. The complex numbers $z$ with positive imaginary parts (${\rm Im} z > 0$) are replaced by the complex symmetric matrices $Z_{ij}$ whose imaginary part is positive definite. Such matrices constitute the so named \textbf{upper Siegel plane}, which indeed is homeomorphic to the coset $\mathop{\rm {}Sp}(2n,\mathbb{R})/\mathrm{U(n)}$. From the physical point of view (\ref{gourmet}) is just identical to the Gaillard-Zumino formula for the construction of the kinetic matrix $\mathcal{N}_{\Lambda\Sigma}$ which appears in the lagrangian of the vector fields in $\mathcal{N}=2$ supergravity and is rooted in the structure of special K\"ahler geometry. Indeed for any special K\"ahler manifold $\mathcal{M}_{n}$ of complex dimension $n$ that is also a symmetric space $\mathrm{G/H}$, there exists a so named $\mathbf{W}$-representation of $\mathrm{G}$, which is symplectic, has dimension $2n+2$ and hosts the electric and magnetic field strengths of the model. Such a representation defines a symplectic embedding:
\begin{equation}\label{gongolini}
\mathrm{G} \, \rightarrow \, \mathop{\rm {}Sp}\left( 2n+2,\mathbb{R}\right )
\end{equation}
which associates to any coset representative $\mathfrak{g}(\phi) \in \mathrm{G/H}$ its corresponding symplectic $(2n+2) \times (2n +2)$
representation $\left( \begin{array}{c|c}
{A}(\phi) & {B}(\phi) \\
\hline
{C}(\phi) & {D}(\phi)
\end{array}
\right)$. From this latter, utilizing the recipe provided by formula (\ref{gourmet}) we obtain an $(n+1) \times (n+1)$ complex symmetric matrix to be identified with the appropriate $\mathcal{N}$ kinetic matrix largely discussed and utilized in section \ref{scrittaN}.
\par
The peculiarity of the $\mathcal{N}=2$ model under investigation is that the original isometry group $\mathrm{G}$ is already symplectic so that we can utilize the Gaillard-Zumino formula (\ref{gourmet}) in the fundamental $6$ dimensional representation in order to construct a Siegel parametrization of the coset in terms of a symmetric complex $ 3 \times 3$ matrix $Z$. The $\mathbf{W}$-representation is the $\mathbf{14}^\prime$ and this defines the embedding:
\begin{equation}\label{gallettoalladiavola}
\mathop{\rm {}Sp}(6,\mathbb{R}) \mapsto \mathop{\rm {}Sp}(14,\mathbb{R})
\end{equation}
from which we can construct the $ 7\times 7 $ kinetic matrix $\mathcal{N}(Z)$.
\paragraph{\sc The transitive action of $\mathop{\rm {}Sp}(6,\mathbb{R})$ on the upper Siegel plane.} Before proceeding with the actual construction of the Lie algebra let us comment on the transitive action of the symplectic group on the Siegel plane. Focusing on the the formula (\ref{gourmet}), consider the $\mathop{\rm {}Sp}(6,\mathbb{R})$ parabolic subgroup composed by the following matrices:
\begin{equation}\label{caripollo}
\mathfrak{g}(B) \, = \, \left(\begin{array}{c|c}
\mathbf{1}_{3\times 3} & B \\
\hline
\mathbf{0}_{3\times 3} & \mathbf{1}_{3\times 3}
\end{array}
\right)
\end{equation}
where $B$ is symmetric and real. By means of such a subgroup we can always map a generic $Z$ matric into one that has vanishing real part $\mbox{Re} Z \, = \,0$.
Next consider the action on the residual imaginary part of $Z$ of the $\mathrm{GL(3,\mathbb{R})} \subset \mathop{\rm {}Sp}(6,\mathbb{R})$ subgroup
composed by the matrices:
\begin{equation}\label{caripollo2}
\mathfrak{g}(B) \, = \, \left(\begin{array}{c|c}
\mathcal{A} & \mathbf{0}_{3\times 3} \\
\hline
\mathbf{0}_{3\times 3} & \left(\mathcal{A}^T\right) ^{-1}
\end{array}
\right) \quad ; \quad \mathcal{A} \, \in \, \mathrm{GL(3,\mathbb{R})}
\end{equation}
We obtain:
\begin{equation}\label{struzzo}
\mbox{Im} Z \, \mapsto \, \mathcal{A} \, \mbox{Im} Z \, \mathcal{A}^T
\end{equation}
Choosing $\mathcal{A} \, = \, (\mbox{Im} Z)^{\frac 12}$, which is always possible since $\mbox{Im} Z$ is positive definite we can reduce the imaginary part to the identity matrix. This shows the transitive action of the symplectic group on the Siegel plane and also provides a nice coset parameterization of the coset manifold. Indeed we can introduce the following matrix:
\begin{equation}\label{godereccio}
\mathfrak{g}(Z) \, \equiv \, \left (\begin{array}{c|c}
\left(\mbox{Im} Z\right)^{\frac 12} &\mbox{Re} Z \, \left(\mbox{Im} Z\right)^{-\frac 12} \\
\hline
\mathbf{0} & \left(\mbox{Im} Z\right)^{-\frac 12}
\end{array}
\right)
\end{equation}
which maps the origin of the manifold ${\rm i} \mathbf{1}_{3\times 3}$ in the complex symmetric matrix $Z$.
\subsection{\sc The $\mathfrak{sp}(6,\mathbb{R})$ Lie algebra}
From the point of view of the Dynkin classification the Lie algebra $\mathfrak{sp}(6,\mathbb{R})$ is the maximally split real section of the complex Lie algebra $C_3$ whose Dynkin diagram is displayed in fig.\ref{C3dynk}.
\begin{figure}
\centering
\begin{picture}(90,50)
\put (-70,35){$C_3$} \put
(10,35){\circle {10}} \put (7,20){$\alpha_1$} \put (15,35){\line
(1,0){20}} \put (40,35){\circle {10}} \put (37,20){$\alpha_2$}\put (45,38){\line (1,0){20}}
\put (55,35){\line (1,1){10}} \put (55,35){\line (1,-1){10}}\put (45,33){\line (1,0){20}} \put
(70,35){\circle {10}} \put (67,20){$\alpha_{3}$}
\end{picture}
\caption{The Dynkin diagram of $C_{3}$. \label{C3dynk}}
\end{figure}
The root system is composed of $18$-roots whose subset of $9$ positive ones is displayed here below:
\begin{equation}\label{cornette}
\left[\begin{array}{c}
\alpha_1 \\
\alpha_2 \\
\alpha_3\\
\alpha_4 \\
\alpha_5 \\
\alpha_6 \\
\alpha_7 \\
\alpha_8 \\
\alpha_9
\end{array}
\right] \, = \, \left[
\begin{array}{ll}
\alpha _1 & \{1,-1,0\} \\
\alpha _2 & \{0,1,-1\} \\
\alpha _3 & \{0,0,2\} \\
\alpha _1+\alpha _2 &
\{1,0,-1\} \\
\alpha _2+\alpha _3 &
\{0,1,1\} \\
\alpha _1+\alpha _2+\alpha _3
& \{1,0,1\} \\
2 \alpha _2+\alpha _3 &
\{0,2,0\} \\
\alpha _1+2 \alpha _2+\alpha
_3 & \{1,1,0\} \\
2 \alpha _1+2 \alpha
_2+\alpha _3 & \{2,0,0\}
\end{array}
\right]
\end{equation}
The simple roots are the first three. Of the remaining $6$ we have provided both their expression in terms of the simple roots and their realization as three-vectors in $\mathbb{R}^3$. Such a realization is spelled out also for the simple roots.
Next we present the basis of $6\times6$ matrices that fulfill the standard commutation relations of the Lie Algebra in the Cartan Weyl basis.
\paragraph{Cartan Generators.} The Cartan generators are named $\mathcal{H}^i$ and can be easily read-off from the following formula:
\begin{equation}\label{cartolini}
\sum_{i=1}^{3} \, h_i \, \mathcal{H}^i \, = \,\left(
\begin{array}{llllll}
h_1 & 0 & 0 & 0 & 0 & 0 \\
0 & h_2 & 0 & 0 & 0 & 0 \\
0 & 0 & h_3 & 0 & 0 & 0 \\
0 & 0 & 0 & -h_1 & 0 & 0 \\
0 & 0 & 0 & 0 & -h_2 & 0 \\
0 & 0 & 0 & 0 & 0 & -h_3
\end{array}
\right)
\end{equation}
by collecting the coefficient of the parameter $h_i$.
\paragraph{\sc Positive Root Step Operators.} The step operator associated with the positive root $\alpha_i$ is named $\mathcal{E}^{\alpha_i}$ and can be easily read-off from the following formula:
\begin{equation}\label{steppinisu}
\sum_{i=1}^{9} \, a_i \, \mathcal{E}^{\alpha_i} \, = \,\left(
\begin{array}{llllll}
0 & a_1 & a_4 & \sqrt{2} a_9
& a_8 & a_6 \\
0 & 0 & a_2 & a_8 & \sqrt{2}
a_7 & a_5 \\
0 & 0 & 0 & a_6 & a_5 &
\sqrt{2} a_3 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -a_1 & 0 & 0 \\
0 & 0 & 0 & -a_4 & -a_2 & 0
\end{array}
\right)
\end{equation}
by collecting the coefficient of the parameter $a_i$.
\paragraph{\sc Negative Root Step Operators.} The step operator associated with the negative root $-\alpha_i$ is named $\mathcal{E}^{-\alpha_i}$ and can be easily read-off from the following formula:
\begin{equation}\label{steppinigiu}
\sum_{i=1}^{9} \, b_i \, \mathcal{E}^{-\alpha_i} \, = \,\left(
\begin{array}{llllll}
0 & 0 & 0 & 0 & 0 & 0 \\
b_1 & 0 & 0 & 0 & 0 & 0 \\
b_4 & b_2 & 0 & 0 & 0 & 0 \\
\sqrt{2} b_9 & b_8 & b_6 & 0
& -b_1 & -b_4 \\
b_8 & \sqrt{2} b_7 & b_5 & 0
& 0 & -b_2 \\
b_6 & b_5 & \sqrt{2} b_3 & 0
& 0 & 0
\end{array}
\right)
\end{equation}
by collecting the coefficient of the parameter $b_i$.
\subsection{\sc The representation $14^\prime$}
\label{seziona14}
The $\mathbf{14}^\prime$ representation of $\mathfrak{sp}(6,\mathbb{R})$ which plays the role $\mathbf{W}$-representation for the special manifold under consideration is defined as the representation obeyed by the three-times antisymmetric tensors with vanishing $\mathbb{C}$-traces, namely:
\begin{equation}\label{goliutico}
\underbrace{t_{ABC}}_{\mbox{antisymmetric in} A,B,C} \times \quad \mathbb{C}^{BC} \, = \,0
\end{equation}
The generators are constructed in the appendix, subsection \ref{seziona14}, and displayed in eq.s (\ref{cartolini14}), (\ref{steppinisu14}) and (\ref{steppinigiu14}).
\subsection{\sc The holomorphic symplectic section and its transformation in the $14^\prime$}
\label{holoseziona}
In order to construct the special geometry of the manifold (\ref{gurto}) we need to introduce the holomorphic symplectic section that, by definition, should transform in the $14^\prime$ representation of $\mathop{\rm {}Sp}(6,\mathbb{R})$. To this effect, we choose as special coordinates the components of the symmetric complex matrix defined by eq.(\ref{gourmet}) and we choose a lexicographic order to enumerate its independent components, namely we set:
\begin{equation}\label{specoord}
Z \, = \, \left(
\begin{array}{lll}
z_1 & z_2 & z_3 \\
z_2 & z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right)
\end{equation}
Next we introduce the holomorphic prepotential defined by:
\begin{eqnarray}
\mathcal{F} & \equiv & Z_{a,i} \, Z_{b,j} \, Z_{c,k} \, \epsilon^{abc} \,\epsilon^{ijk}\nonumber \\
\null &=& -6 \left(z_6 z_2^2-2 z_3 z_5
z_2+z_3^2 z_4+z_1
\left(z_5^2-z_4
z_6\right)\right)
\end{eqnarray}
and we can introduce a first ansatz for the symplectic section by writing:
\begin{eqnarray}
\widetilde{\Omega} &=& \left \{1, \, z^I , \, \mathcal{F} , \, \frac{\partial \mathcal{F} }{\partial z^J} \right\} \nonumber \\
\null &=& \left\{1,z_1,z_2,z_3,z_4,z_5,z
_6,-6 \left(z_6 z_2^2-2 z_3
z_5 z_2+z_3^2 z_4+z_1
\left(z_5^2-z_4
z_6\right)\right), \right.\nonumber\\
&&\left.6 z_4
z_6-6 z_5^2,12 \left(z_3
z_5-z_2 z_6\right),12
\left(z_2 z_5-z_3
z_4\right),6 z_1 z_6-6
z_3^2,12 \left(z_2 z_3-z_1
z_5\right),6 z_1 z_4-6
z_2^2\right\}\nonumber\\
\label{preomega}
\end{eqnarray}
In order to match the transformation of this holomorphic section with the transformations of the $\mathbf{14}^\prime$ representation as we defined it in subsection \ref{holoseziona} we still need a change of basis. Consider the following matrix
{\scriptsize
\begin{equation}\label{paola}
\mathfrak{S}\, = \, \left(
\begin{array}{llllllllllllll}
0 & 0 & 0 & 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3 \sqrt{2}} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{3 \sqrt{2}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3 \sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{6} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\frac{1}{6} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{6} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{3 \sqrt{2}} \\
0 & 0 & 0 & 0 & -\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right)
\end{equation}
}
and define:
\begin{equation}\label{gargamelle}
\Omega\left( Z\right) \, = \, \mathfrak{S}\, \widetilde{\Omega}(Z) \, = \, \left(
\begin{array}{l}
\sqrt{2} z_6 \\
\sqrt{2} \left(z_1 z_6-z_3^2\right) \\
\sqrt{2} \left(z_5^2-z_4 z_6\right) \\
-\sqrt{2} \left(z_6 z_2^2-2 z_3 z_5 z_2+z_3^2 z_4+z_1 \left(z_5^2-z_4 z_6\right)\right) \\
2 z_2 z_3-2 z_1 z_5 \\
2 z_3 z_4-2 z_2 z_5 \\
2 z_3 z_5-2 z_2 z_6 \\
\sqrt{2} \left(z_1 z_4-z_2^2\right) \\
-\sqrt{2} z_4 \\
\sqrt{2} z_1 \\
\sqrt{2} \\
-2 z_5 \\
2 z_3 \\
-2 z_2
\end{array}
\right)
\end{equation}
Naming $\mathcal{D}_{14}\left[g\right]$ the 14-dimensional representation of a finite element $g , \in \, \mathop{\rm {}Sp}(6,\mathbb{R})$ of the symplectic group that corresponds to the representation of the algebra as we constructed it above, the holomorphic symplectic section (\ref{gargamelle}) transforms in the following way:
\begin{equation}\label{buonotrasformo}
\Omega \left[ (A\,Z \, + \, B) \, (C \, Z \, + \, D )^{-1} \right] \, = \, \frac{1}{\mbox{Det}\left(C \, Z \, + \, D\right)} \, \mathcal{D}_{14}\left[\left(\begin{array}{c|c}
A & B \\
\hline
C & D
\end{array}\right)
\right] \, \Omega[Z]
\end{equation}
The formula (\ref{buonotrasformo}) can be in particular applied to the case where the original $Z$ is the origin of the coset manifold:
$Z_0 \, = \, {\rm i} \, \mathbf{1}_{3\times 3}$. In that case, recalling eq. (\ref{godereccio}) we find:
\begin{equation}\label{frittellaprima}
\Omega[Z_0] \, = \, \left\{i \sqrt{2},-\sqrt{2},\sqrt{2},-i
\sqrt{2},0,0,0,-\sqrt{2},-i \sqrt{2},i
\sqrt{2},\sqrt{2},0,0,0\right\}
\end{equation}
and
\begin{equation}\label{caviccio}
\Omega[Z_0] \, = \, \sqrt{\mbox{Det}\left[\mbox{Im} \, Z\right] } \, \times \, \mathcal{D}_{14}\left[ \left (\begin{array}{c|c}
\left(\mbox{Im} Z\right)^{\frac 12} &\mbox{Re} Z \, \left(\mbox{Im} Z\right)^{-\frac 12} \\
\hline
\mathbf{0} & \left(\mbox{Im} Z\right)^{-\frac 12}
\end{array}
\right)\right] \, \cdot \, \Omega[Z_0]
\end{equation}
\subsection{\sc The K\"ahler potential and the metric}
Provided with this information we can now write the explicit form of the K\"ahler potential and of the K\"ahler metric for the manifold (\ref{gurto}) according to the rules of special K\"ahler geometry. We have:
\begin{eqnarray}
\mathcal{K} & \equiv & - \log \left( {\rm i} \Omega[Z] \, \mathbb{C}_{14} \, \overline{\Omega}[\bar{Z}]\right) \nonumber\\
\null &=& -\log \left(2 i \left(-z_6
z_2^2+{\bar z}_6 z_2^2+2 z_6
{\bar z}_2 z_2-2 z_5 {\bar z}_3
z_2+2 {\bar z}_3 {\bar z}_5 z_2-2
{\bar z}_2 {\bar z}_6 z_2-z_6
{\bar z}_2^2-z_4
{\bar z}_3^2+{\bar z}_1
{\bar z}_5^2+\right.\right. \nonumber\\
&&\left.\left. z_5^2 {\bar z}_1-z_4
z_6 {\bar z}_1+2 z_5 {\bar z}_2
{\bar z}_3+{\bar z}_3^2
{\bar z}_4+z_6 {\bar z}_1
{\bar z}_4 +z_3^2
\left({\bar z}_4-z_4\right)-2 z_5
{\bar z}_1 {\bar z}_5-2
{\bar z}_2 {\bar z}_3
{\bar z}_5 \right.\right.\nonumber\\
&&\left.\left.+2 z_3 \left(-z_5
{\bar z}_2+{\bar z}_5
{\bar z}_2+z_4
{\bar z}_3-{\bar z}_3
{\bar z}_4+z_2
\left(z_5-{\bar z}_5\right)\right)+
{\bar z}_2^2 {\bar z}_6+z_4
{\bar z}_1 {\bar z}_6-{\bar z}_1
{\bar z}_4 {\bar z}_6 \right.\right.\nonumber\\
&&\left.\left. -z_1
\left(z_5^2-2 {\bar z}_5
z_5+{\bar z}_5^2+z_6
{\bar z}_4-{\bar z}_4
{\bar z}_6+z_4
\left({\bar z}_6-z_6\right)\right)\right)\right) \nonumber\\
\end{eqnarray}
and the line element on the manifold, in terms of the special coordinates $z_i$ takes the standard form:
\begin{equation}\label{gumilevo}
ds_K^2 \, = \, \frac{\partial}{\partial z^i} \, \frac{\partial}{\partial \bar{z}^{j}} \, \mathcal{K} \, dz^i \otimes d\bar{z}^j
\end{equation}
The explicit form of $ds_K^2$ in terms of the special coordinate $z^i$ can be worked out by simple derivatives, yet its explicit form is quite lengthy and so much involved that we think it better not to display it. For the purposes that we pursue we rather prefer to write the form of the metric in terms of solvable real coordinates.
\subsubsection{\sc The solvable parametrization}
The transition to a solvable parametrization of the coset is rather simple. Let us define the solvable coset representative as the product of the exponentials of all the generators of the Borel subalgebra of $\mathfrak{sp}(6,\mathbb{R})$:
\begin{eqnarray}
& \mathbb{L}(h,p)\, = \, \prod_{i=1}^9 \, \exp\left [ p_{10-i} \, \mathcal{E}^{\alpha_{10-i}}\right ] \, \prod_{j=3}^3 \exp\left [ h_{j} \, \mathcal{H}^{j}\right ] \, = \, & \nonumber\\
& \mbox{\scriptsize $\left(
\begin{array}{l|l|l|l|l|l}
e^{h_1} & e^{h_2} p_1 & e^{h_3} p_4 &
\begin{array}{l}
e^{-h_1} \left(\sqrt{2} p_1 p_2 p_3
p_4 \right.\\
\left. +\left(p_1 p_2-p_4\right) p_6\right.\\
\left.-p_1
p_8+\sqrt{2} p_9 \right)
\end{array}
& \begin{array}{c}
e^{-h_2}
\left(-\sqrt{2} p_2 p_3 p_4\right. \\
\left. -p_2
p_6+p_8\right)
\end{array}
& e^{-h_3}
\left(\sqrt{2} p_3 p_4+p_6\right) \\
\hline
0 & e^{h_2} & e^{h_3} p_2 &
\begin{array}{c}
e^{-h_1}
\left(\left(p_1 p_2-p_4\right)
p_5\right. \\
\left.-\sqrt{2} p_1 p_7+p_8\right)
\end{array}
&
\begin{array}{c}
e^{-h_2} \left(\sqrt{2} p_7 \right. \\
\left. -p_2
p_5\right)
\end{array}
& e^{-h_3} p_5
\\
\hline
0 & 0 & e^{h_3} & e^{-h_1}
\left(\sqrt{2} p_1 p_2 p_3-p_1
p_5+p_6\right) & e^{-h_2}
\left(p_5-\sqrt{2} p_2 p_3\right) &
\sqrt{2} e^{-h_3} p_3 \\
\hline
0 & 0 & 0 & e^{-h_1} & 0 & 0 \\
\hline
0 & 0 & 0 & -e^{-h_1} p_1 & e^{-h_2} &
0 \\
\hline
0 & 0 & 0 & e^{-h_1} \left(p_1
p_2-p_4\right) & -e^{-h_2} p_2 &
e^{-h_3}
\end{array}
\right) $}& \nonumber\\
\label{baltico}
\end{eqnarray}
The real coordinates of the manifold are now the $12$ parameters:
\begin{equation}\label{coordine}
\mbox{coordinates} \, \equiv \, \left\{ h_1, \dots , h_3, \, p_1 , \, \dots \, p_9 \right \}
\end{equation}
Extracting the complex matrix $Z$ from the symplectic matrix $\mathbb{L}(h,p)$ we find:
\begin{eqnarray}\label{zph}
& Z(h,p) \, = \, & \nonumber\\
& \mbox{\scriptsize $\left(
\begin{array}{lll}
i e^{2 h_2} p_1^2+i e^{2
h_1}+\left(\sqrt{2} p_3+i e^{2
h_3}\right) p_4^2+\sqrt{2} p_9 & i
e^{2 h_2} p_1+i e^{2 h_3} p_2
p_4+p_8 & \left(\sqrt{2} p_3+i e^{2
h_3}\right) p_4+p_6 \\
i e^{2 h_2} p_1+i e^{2 h_3} p_2
p_4+p_8 & i e^{2 h_3} p_2^2+i e^{2
h_2}+\sqrt{2} p_7 & i e^{2 h_3}
p_2+p_5 \\
\left(\sqrt{2} p_3+i e^{2 h_3}\right)
p_4+p_6 & i e^{2 h_3} p_2+p_5 &
\sqrt{2} p_3+i e^{2 h_3}
\end{array}
\right)$} &
\end{eqnarray}
which defines the coordinate transformation from the special to the solvable coordinates:
\begin{equation}\label{carlinus}
\left(
\begin{array}{l}
z_1 \\
z_2 \\
z_3 \\
z_4 \\
z_5 \\
z_6
\end{array}
\right)\, = \, \left(
\begin{array}{l}
\sqrt{2} p_3 p_4^2+i \left(e^{2 h_2}
p_1^2+e^{2 h_1}+e^{2 h_3}
p_4^2\right)+\sqrt{2} p_9 \\
i \left(e^{2 h_2} p_1+e^{2 h_3} p_2
p_4\right)+p_8 \\
i e^{2 h_3} p_4+\sqrt{2} p_3 p_4+p_6
\\
i \left(e^{2 h_3} p_2^2+e^{2
h_2}\right)+\sqrt{2} p_7 \\
i e^{2 h_3} p_2+p_5 \\
\sqrt{2} p_3+i e^{2 h_3}
\end{array}
\right)
\end{equation}
Inserting such a coordinate transformation into the K\"ahler metric (\ref{gumilevo}) we obtain its form in terms of the real coordinates
(\ref{coordine}). For the explicit form of the metric, we refer the reader to the appendix, eq. (\ref{formidabile}).
The complete metric is quite formidable (\ref{formidabile}) since it contains a total of 100 terms. It has however quite simple properties when we sit in the neighborhood of the coset origin, in particular at $p_i \sim 0$. In this case it drastically simplifies and becomes diagonal:
\begin{eqnarray}\label{pippa}
&&ds_K^2 \, \stackrel{p_i \rightarrow 0}{\Longrightarrow} \, \mathrm{dh}_1^2+\mathrm{dh}_2^2+\mathrm{dh}_3
^2+\frac{1}{2} e^{2 h_2-2 h_1}
\mathrm{dp}_1^2+\frac{1}{2} e^{2 h_3-2
h_2} \mathrm{dp}_2^2+\frac{1}{2} e^{-4
h_3} \mathrm{dp}_3^2+\frac{1}{2} e^{2
h_3-2 h_1} \mathrm{dp}_4^2\nonumber\\
&&+\frac{1}{2}
e^{-2 h_2-2 h_3}
\mathrm{dp}_5^2+\frac{1}{2} e^{-2
h_1-2 h_3} \mathrm{dp}_6^2+\frac{1}{2}
e^{-4 h_2} \mathrm{dp}_7^2+\frac{1}{2}
e^{-2 h_1-2 h_2}
\mathrm{dp}_8^2+\frac{1}{2} e^{-4 h_1}
\mathrm{dp}_9^2
\end{eqnarray}
which shows that it is positive definite as it should be. It is also interesting to note that if the truncation to the Cartan is permitted by the potential, then we just have three dilatons with canonical kinetic terms.
\subsection{\sc The quartic invariant in the $14^\prime$}
Of crucial relevance for the analysis of Black Hole charges and in general for the classification of orbits in the $\mathbf{W}$-representation is the quartic symplectic invariant. Given a $14$-vector
\begin{equation}\label{gongolando}
\mathcal{Q} \, = \, \left\{ q_1,\, q_2 \, \dots ,\,q_{14}\right\}
\end{equation}
the standard form of this invariant can be expressed in the following manifestly ${\rm Sp}(6,\mathbb{R})$-invariant form (see for instance \cite{Ferrara:2013zga})
\begin{eqnarray}
\mathfrak{J}_4(\mathcal{Q}) &=& -\frac{n_V(2n_V+1)}{6d}\,(\Lambda_a)_{\alpha\beta}\,(\Lambda^a)_{\gamma\delta}\,\mathcal{Q}^\alpha\,\mathcal{Q}^\beta\,\mathcal{Q}^\gamma\,\mathcal{Q}^\delta\,,
\end{eqnarray}
where in our case $n_V=7$ and $d={\rm dim}{\rm Sp}(6,\mathbb{R})=21$, the symplectic indices are raised and lowered by $\mathbb{C}_14^{\alpha\beta}$ and $\mathbb{C}_{14\,\alpha\beta}$ and the index $a$ is raised by the inverse of $\eta_{ab}\equiv
{\rm Tr}(\Lambda_a\,\Lambda_b)$.
The explicit form of $\mathfrak{J}_4(\mathcal{Q})$ reads:
\begin{eqnarray}
\mathfrak{J}_4(\mathcal{Q}) &=& -2 q_1 q_9 q_5^2+2 q_3 q_{11} q_5^2-2 \sqrt{2} q_6 q_7 q_{11} q_5-2 q_1 q_8 q_{12}
q_5+2 q_2 q_9 q_{12} q_5-2 q_3 q_{10} q_{12} q_5\nonumber\\
&&+2 q_4 q_{11} q_{12} q_5-2
\sqrt{2} q_7 q_9 q_{13} q_5 +2 \sqrt{2} q_1 q_6 q_{14} q_5+2 \sqrt{2} q_3 q_{13}
q_{14} q_5+q_1^2 q_8^2\nonumber\\
&&+q_2^2 q_9^2+q_3^2 q_{10}^2+q_4^2 q_{11}^2+2 q_2 q_8
q_{12}^2-2 q_4 q_{10} q_{12}^2-2 q_3 q_8 q_{13}^2+2 q_4 q_9 q_{13}^2-2 q_2 q_3
q_{14}^2\nonumber\\
&&-2 q_1 q_4 q_{14}^2+2 q_1 q_2 q_8 q_9+2 q_1 q_6^2 q_{10}+2 q_1 q_3 q_8
q_{10}-2 q_7^2 q_9 q_{10}-2 q_2 q_3 q_9 q_{10}\nonumber\\
&&-4 q_1 q_4 q_9 q_{10}-2 q_2 q_6^2
q_{11}-2 q_7^2 q_8 q_{11}-4 q_2 q_3 q_8 q_{11}\nonumber\\
&&-2 q_1 q_4 q_8 q_{11}+2 q_2 q_4
q_9 q_{11}+2 q_3 q_4 q_{10} q_{11}\nonumber\\
&&+2 \sqrt{2} q_6 q_7 q_{10} q_{12}-2 q_1 q_6
q_8 q_{13}-2 q_2 q_6 q_9 q_{13}+2 q_3 q_6 q_{10} q_{13}\nonumber\\
&&+2 q_4 q_6 q_{11}
q_{13}-2 \sqrt{2} q_7 q_8 q_{12} q_{13}+2 q_1 q_7 q_8 q_{14}\nonumber\\
&&+2 q_2 q_7 q_9
q_{14}+2 q_3 q_7 q_{10} q_{14}+2 q_4 q_7 q_{11} q_{14}-2 \sqrt{2} q_2 q_6 q_{12}
q_{14}+2 \sqrt{2} q_4 q_{12} q_{13} q_{14}\nonumber\\
\label{I4inv}
\end{eqnarray}
\subsection{\sc Truncation to the $STU$-model}
\label{truncazia}
Next we analyze how the $STU$-model is embedded into the $\mathop{\rm {}Sp}(6,\mathbb{R})$-model.
At the level of the special coordinates the truncation to the $STU$-model is very simply done. It suffices to
set to zero the complex coordinates $z_2,z_3,z_5$ keeping only $z_1,z_4,z_6$ that can be identified with the fields $S,T,U$. When we do so the symplectic section reduces as follows:
\begin{equation}\label{stuSymsec}
\Omega\left[Z \left(
\begin{array}{ccc}
z_1 & 0 & 0 \\
0 & z_4 & 0 \\
0 & 0 & z_6 \\
\end{array}
\right)
\right] \, = \, \left(
\begin{array}{l}
\sqrt{2} z_6 \\
\sqrt{2} z_1 z_6 \\
-\sqrt{2} z_4 z_6 \\
\sqrt{2} z_1 z_4 z_6 \\
0 \\
0 \\
0 \\
\sqrt{2} z_1 z_4 \\
-\sqrt{2} z_4 \\
\sqrt{2} z_1 \\
\sqrt{2} \\
0 \\
0 \\
0
\end{array}
\right)
\end{equation}
and the K\"ahler potential reduces to:
\begin{equation}\label{stuKalpot}
\mathcal{K} \, \rightarrow\, -\log \left[2 i \left(z_1-{\bar z}_1\right) \left(z_4-{\bar z}_4\right)
\left(z_6-{\bar z}_6\right)\right]
\end{equation}
which yields three copies of the Poincar\'e metric, one for each of the three $\frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}$
submanifolds.
\par
The result (\ref{stuSymsec}) is in agreement with the decomposition of the $\mathbf{14}^\prime$ of $\mathfrak{sp}(6,\mathbb{R})$ with respect to the three subalgebras $\mathfrak{sl}(2)$:
\begin{equation}\label{decumpo}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2) \times \mathfrak{sl}(2) \times \mathfrak{sl}(2)}{\Longrightarrow} \, \left(\mathbf{2,2,2}\right)
\oplus \left(\mathbf{2,1,1}\right) \oplus \left(\mathbf{1,2,1}\right) \oplus \left(\mathbf{1,1,2}\right)
\end{equation}
From (\ref{stuSymsec}) we also learn that the directions $\{1,2,3,4,8,9,10,11\}$ of the $\mathbf{14}^\prime$ vector space span the representation $\left(\mathbf{2,2,2}\right)$, while the directions $\{5,6,7,12,13,14\}$ of the same space span the representations $\left(\mathbf{2,1,1}\right) \oplus \left(\mathbf{1,2,1}\right) \oplus \left(\mathbf{1,1,2}\right)$.
The adjoint representation of $\mathfrak{sp}(6,\mathbb{R})$ decomposes instead in the following way:
\begin{eqnarray}
\mbox{adj} \,\left[\mathfrak{sp}(6,\mathbb{R})\right] & \stackrel{\mathfrak{sl}(2) \times \mathfrak{sl}(2) \times \mathfrak{sl}(2)}{\Longrightarrow} &
\left(\mathbf{3,1,1}\right) \oplus \left(\mathbf{1,3,1}\right) \oplus \left(\mathbf{1,1,3}\right) \nonumber\\
&&
\oplus \left(\mathbf{2,2,1}\right) \oplus \left(\mathbf{2,1,2}\right) \oplus \left(\mathbf{1,2,2}\right) \label{adjointo}
\end{eqnarray}
as it is evident by a quick inspection of the roots (\ref{cornette}). In terms of the Cartan-Weyl basis the three
$\mathrm{\mathfrak{sl}(2,\mathbb{R})}$ subalgebra contains the three Cartan generators $\mathcal{H}_i$ and the step operators
$\mathcal{E}^{\pm \alpha_3}$, $\mathcal{E}^{\pm \alpha_7}$ , $\mathcal{E}^{\pm \alpha_9}$. The remaining 12 step operators
span the representation $\left(\mathbf{2,2,1}\right) \oplus \left(\mathbf{2,1,2}\right) \oplus \left(\mathbf{1,2,2}\right)$, namely:
\begin{equation}\label{spannone}
\left(\mathbf{2,2,1}\right) \oplus \left(\mathbf{2,1,2}\right) \oplus \left(\mathbf{1,2,2}\right) \, = \, \mbox{span} \left[
\mathcal{E}^{\pm \alpha_1}, \mathcal{E}^{\pm \alpha_2} , \mathcal{E}^{\pm \alpha_4} ,\mathcal{E}^{\pm \alpha_5}, \mathcal{E}^{\pm \alpha_6} , \mathcal{E}^{\pm \alpha_8}
\right]
\end{equation}
The explicit form of an $\mathfrak{sp}(6,\mathbb{R})$ Lie algebra element reduced to the $\mathfrak{sl}(2)^3$ subalgebra is the following one:
\begin{equation}\label{pagnocco}
\left(
\begin{array}{llllll}
h_1 & 0 & 0 & b_1 & 0 & 0 \\
0 & h_2 & 0 & 0 & b_2 & 0 \\
0 & 0 & h_3 & 0 & 0 & b_3 \\
c_1 & 0 & 0 & -h_1 & 0 & 0 \\
0 & c_2 & 0 & 0 & -h_2 & 0 \\
0 & 0 & c_3 & 0 & 0 & -h_3
\end{array}
\right) \, \in \, \mathfrak{sl}(2) \otimes \mathfrak{sl}(2) \otimes \mathfrak{sl}(2) \, \subset \, \mathfrak{sp}(6,\mathbb{R})
\end{equation}
\subsection{\sc Reduction of the charge vector to the $(2,2,2)$}
In order to study the orbits of the charge vectors in the $\mathbf{14}^\prime$ our first step consists of reducing it to normal form, namely to the $(2,2,2)$ representation. We claim that for generic charge vectors this is always possible by means of $\mathop{\rm {}Sp}(6,\mathbb{R})$ rotations generated by elements of the $\left(\mathbf{2,2,1}\right) \oplus \left(\mathbf{2,1,2}\right) \oplus \left(\mathbf{1,2,2}\right)$ subspace. To show this let us consider the six dimensional compact Lie algebra element:
\begin{eqnarray}\label{cappotto}
\mathbb{K}_\psi & = & \psi_1 \,\left(\mathcal{E}^{\alpha_1} -\mathcal{E}^{-\alpha_1}\right) +\psi_2 \, \left(\mathcal{E}^{\alpha_2} -\mathcal{E}^{-\alpha_2}\right) +\psi_3 \, \left(\mathcal{E}^{\alpha_4} -\mathcal{E}^{-\alpha_4}\right)\nonumber\\
&&\psi_4 \,\left(\mathcal{E}^{\alpha_5} -\mathcal{E}^{-\alpha_5}\right) +\psi_5 \, \left(\mathcal{E}^{\alpha_6} -\mathcal{E}^{-\alpha_6}\right) +\psi_6 \, \left(\mathcal{E}^{\alpha_8} -\mathcal{E}^{-\alpha_8}\right)
\end{eqnarray}
and a generic charge vector that has components only in the $\mathbf{(2,2,2)}$ subspace.
\begin{equation}\label{toroidallo}
\mathcal{Q}_{2,2,2} \, = \, \left\{\Theta _1,\Theta _2,\Theta _3,\Theta _4,0,0,0,\Theta _5,\Theta _6,\Theta
_7,\Theta _8,0,0,0\right\}
\end{equation}
If we apply the $\mathbf{14}^\prime$ representation of $\mathbb{K}_\psi$ to the charge vector $\mathcal{Q}_N$ we obtain:
\begin{equation}\label{gorgonzola}
\mathcal{D}_{14}\left( \mathbb{K}_\psi\right) \,\mathcal{Q}_{2,2,2} \, = \, \left(
\begin{array}{l}
0 \\
0 \\
0 \\
0 \\
-\sqrt{2} \Theta _2 \psi _2+\sqrt{2} \Theta _5 \psi _2-\sqrt{2} \Theta _4 \psi
_4-\sqrt{2} \Theta _7 \psi _4 \\
-\sqrt{2} \Theta _3 \psi _3-\sqrt{2} \Theta _5 \psi _3+\sqrt{2} \Theta _4 \psi
_5-\sqrt{2} \Theta _6 \psi _5 \\
\sqrt{2} \Theta _2 \psi _1+\sqrt{2} \Theta _3 \psi _1-\sqrt{2} \Theta _1 \psi
_6-\sqrt{2} \Theta _4 \psi _6 \\
0 \\
0 \\
0 \\
0 \\
-\sqrt{2} \Theta _1 \psi _2-\sqrt{2} \Theta _6 \psi _2+\sqrt{2} \Theta _3 \psi
_4-\sqrt{2} \Theta _8 \psi _4 \\
\sqrt{2} \Theta _1 \psi _3-\sqrt{2} \Theta _7 \psi _3+\sqrt{2} \Theta _2 \psi
_5+\sqrt{2} \Theta _8 \psi _5 \\
\sqrt{2} \Theta _6 \psi _1+\sqrt{2} \Theta _7 \psi _1-\sqrt{2} \Theta _5 \psi
_6-\sqrt{2} \Theta _8 \psi _6
\end{array}
\right)
\end{equation}
which clearly shows that the six parameters $\psi_{1,\dots,6}$ are sufficient to generate arbitrary components $\{5,6,7,12,13,14\}$ of the charge vector starting from vanishing ones. Reverting the path this means that by means of the same rotations, apart from singular orbits that deserve a separate study we can always fix the gauge where the six components $\{5,6,7,12,13,14\}$ vanish.
\subsubsection{\sc Further reduction to normal form of the charge vector}
Once the charge vector is reduced to $(2,2,2)$ representation, we can further act on it with the $\mathrm{SL(2,\mathbb{R})}^3$ group in order to further reduce its components. By using the three parameters of the abelian translation group $\mathbb{R}^3$ contained in $\mathrm{SL(2,\mathbb{R})}^3$ we can put to zero three of the eight charges and a possible normal form of the charge vector is the following one:
\begin{equation}\label{normaldata}
\mathcal{Q}_{N} \, = \, \left\{0,P_1,P_2,P_3,0,0,0,P_4,0,0,P_5,0,0,0\right\}
\end{equation}
The corresponding quartic invariant is:
\begin{equation}\label{normaldata2}
\mathfrak{J}_4 \left(\mathcal{Q}_{N}\right)\, = \, P_3^2 P_5^2-4 P_1 P_2 P_4 P_5
\end{equation}
\section{\sc The $\frac{\mathrm{F_{(4,4)}}}{\mathrm{SU(2)} \times \mathrm{USp(6)}}$ quaternionic K\"ahler manifold}
Let us now come to the $c$-map image of the Special K\"ahler manifold (\ref{gurto}), namely to the quaternionic K\"ahler manifold (\ref{quatergurto}). The $F_{(4,4)}$ Lie algebra has rank four and its structure is codified in the Dynkin diagram presented in fig.\ref{F4dynk}.
\begin{figure}
\centering
\begin{picture}(90,50)
\put (-70,35){$F_4$} \put
(10,35){\circle {10}} \put (7,20){$\beta_4$} \put (15,35){\line
(1,0){20}} \put (40,35){\circle {10}} \put (37,20){$\beta_3$}\put (45,38){\line (1,0){20}}
\put (55,35){\line (1,1){10}} \put (55,35){\line (1,-1){10}}\put (45,33){\line (1,0){20}} \put
(70,35){\circle {10}} \put (67,20){$\beta_{2}$} \put
(75,35){\line (1,0){20}} \put (100,35){\circle {10}} \put (100,35){\circle {9}}\put (100,35){\circle {8}}\put (100,35){\circle {10}}\put (100,35){\circle {7}}
\put (100,35){\circle {6}}\put (100,35){\circle {5}}\put (100,35){\circle {4}}\put (100,35){\circle {3}}\put (100,35){\circle {2}}\put (100,35){\circle {1}}\put
(97,20){$\beta_{1}$}
\end{picture}
$$ \begin{array}{l}\psi \, = \, \beta_{24} \, = \, 2\beta_1 +3\beta_2 +4\beta_3+2\beta_4 \\(\psi \, , \,\beta_1) = 2
\quad; \quad (\psi \, , \, \beta_i ) = 0 \quad i \ne 1 \
\end{array} $$
\vskip 1cm \caption{The Dynkin diagram of $F_{4(4)}$. The only root which is not orthogonal
to the highest root is $\beta_V = \beta_1$. The root $\beta_V = \beta_1 $
is the highest weight of the $\mathbf{W}$-representation of $\mathfrak{sp}(6,\mathbb{R})$ \label{F4dynk}}
\end{figure}
The complete set of positive roots contains $24$ elements that are listed below:
\begin{equation}\label{radicioneF4}
\begin{array}{lllll}
\beta _1 & = & \beta _1 & = & \{-1,-1,-1,1\} \\
\beta _2 & = & \beta _2 & = & \{0,0,2,0\} \\
\beta _3 & = & \beta _3 & = & \{0,1,-1,0\} \\
\beta _4 & = & \beta _4 & = & \{1,-1,0,0\} \\
\beta _5 & = & \beta _1+\beta _2 & = & \{-1,-1,1,1\} \\
\beta _6 & = & \beta _2+\beta _3 & = & \{0,1,1,0\} \\
\beta _7 & = & \beta _3+\beta _4 & = & \{1,0,-1,0\} \\
\beta _8 & = & \beta _1+\beta _2+\beta _3 & = &
\{-1,0,0,1\} \\
\beta _9 & = & \beta _2+2 \beta _3 & = & \{0,2,0,0\} \\
\beta _{10} & = & \beta _2+\beta _3+\beta _4 & = &
\{1,0,1,0\} \\
\beta _{11} & = & \beta _1+\beta _2+2 \beta _3 & = &
\{-1,1,-1,1\} \\
\beta _{12} & = & \beta _1+\beta _2+\beta _3+\beta _4 & = &
\{0,-1,0,1\} \\
\beta _{13} & = & \beta _2+2 \beta _3+\beta _4 & = &
\{1,1,0,0\} \\
\beta _{14} & = & \beta _1+2 \beta _2+2 \beta _3 & = &
\{-1,1,1,1\} \\
\beta _{15} & = & \beta _1+\beta _2+2 \beta _3+\beta _4 & =
& \{0,0,-1,1\} \\
\beta _{16} & = & \beta _2+2 \beta _3+2 \beta _4 & = &
\{2,0,0,0\} \\
\beta _{17} & = & \beta _1+2 \beta _2+2 \beta _3+\beta _4 &
= & \{0,0,1,1\} \\
\beta _{18} & = & \beta _1+\beta _2+2 \beta _3+2 \beta _4 &
= & \{1,-1,-1,1\} \\
\beta _{19} & = & \beta _1+2 \beta _2+3 \beta _3+\beta _4 &
= & \{0,1,0,1\} \\
\beta _{20} & = & \beta _1+2 \beta _2+2 \beta _3+2 \beta _4
& = & \{1,-1,1,1\} \\
\beta _{21} & = & \beta _1+2 \beta _2+3 \beta _3+2 \beta _4
& = & \{1,0,0,1\} \\
\beta _{22} & = & \beta _1+2 \beta _2+4 \beta _3+2 \beta _4
& = & \{1,1,-1,1\} \\
\beta _{23} & = & \beta _1+3 \beta _2+4 \beta _3+2 \beta _4
& = & \{1,1,1,1\} \\
\beta _{24} & = & 2 \beta _1+3 \beta _2+4 \beta _3+2 \beta
_4 & = & \{0,0,0,2\}
\end{array}
\end{equation}
In eq.(\ref{radicioneF4}) the first column is the name of the root, the second column gives its decomposition in terms of simple roots, while the last column provides the component of the root vector in $\mathbb{R}^4$.
\par
The standard Cartan-Weyl form of the Lie algebra is as follows:
\begin{eqnarray}
\left[ \mathcal{H}_i \, , \, E^{\pm\beta}\right] &=& \pm \, \beta^i \, E^{\pm\beta_I} \, \label{weylus1} \\
\left[ E^{\beta} \, , \, E^{-\,\beta}\right] &=& \, \beta \, \cdot \, \mathcal{H} \, \label{weylus2} \\
\left[ E^{\beta}\, , \, E^{\gamma}\right] &=& \left\{ \begin{array}{lc}N_{\beta \gamma} \, E^{\beta \, + \, \gamma}& \mbox{if $\beta+\gamma$ is a root}\\
0 &\mbox{if $\beta+\gamma$ is not a root}\end{array} \right. \label{weylus3}
\end{eqnarray}
where $N_{\beta \gamma}$ are numbers that can be predicted from Lie algebra theory. They are irrelevant, since they follows from commutators, when one has an explicit matrix realization of all the Cartan generators $\mathcal{H}_i$ and of the step operators $E^{\pm\beta}$. The fundamental representation of $F_{(4,4)}$ is $26$-dimensional and real. We have constructed in a MATHEMATICA code all the $26 \times 26$ matrix representations of the $52$ generators of the Cartan Weyl basis. Obviously we can not present them here because they are too big. However all the rest of the construction can be easily presented in terms of these Weyl generators sand this is what we presently do.
\subsection{\sc The maximal compact subalgebra $\mathbb{H} \, = \, \mathfrak{su}(2) \oplus \mathfrak{usp}(6)$}
The maximal compact subalgebra $\mathbb{H}$ of a maximally split simple Lie algebra such as $F_{(4,4)}$, is just the real span of all the independent compact generators $E^{\beta_i} \, - \, E^{- \beta_i}$. In our case we have $24$ positive roots and we can write:
\begin{equation}\label{algebraH}
\mathbb{H} \, = \, \mbox{span}_\mathbb{R} \, \left\{ H_1\, , \, H_2 \, , \, \dots \, , \, H_{24} \right\}
\end{equation}
where we have defined:
\begin{equation}\label{Higene}
H_i \, = \, E^{\beta_i} \, - \, E^{- \beta_i}
\end{equation}
the positive roots being numbered as in eq.(\ref{radicioneF4}). We know from theory that this maximal compact subalgebra has the structure:
\begin{equation}\label{furtivo}
\mathbb{H} \, = \, \mathfrak{su}(2) \oplus \mathfrak{usp}(6)
\end{equation}
It is important to derive an explicit basis of generators satisfying the standard commutation relations of the two simple factors in eq.(\ref{furtivo}) for holonomy calculations of the coset manifold. Particularly important are the three generators $J^x$ of the $\mathfrak{su}(2)$ subalgebra since they will act as quaternionic complex structures in the calculation of the tri-holomorphic moment map. By means of standard techniques of diagonalization of the adjoint action of generators we have retrieved the required basis rearrangement.
\subsubsection{\sc The $\mathfrak{su}(2)$ Lie algebra} The three generators $J^x$ have tho following explicit form:
\begin{eqnarray}
J^1 &=& \frac{H_1-H_{14}+H_{20}-H_{22}}{4 \sqrt{2}} \nonumber\\
J^2 &=& \frac{H_5+H_{11}-H_{18}+H_{23}}{4 \sqrt{2}}\nonumber \\
J^3 &=& -\frac{H_2-H_9+H_{16}+H_{24}}{4 \sqrt{2}}\label{su2generati}
\end{eqnarray}
and close the standard commutation relations:
\begin{equation}\label{stundasu2}
\left[ J^x \, , \, J^y \right] \, = \, \epsilon^{xyz} \, J^y
\end{equation}
\subsubsection{\sc The $\mathfrak{usp}(6)$ Lie algebra} The $21$ generators of the $\mathfrak{usp}(6)$ Lie algebra are given by the following combinations.
First we have three mutually commuting generators (the compact Cartan generators):
\begin{equation}\label{cartacantabene}
\left[ \mathcal{L}^i \, , \, \mathcal{L}^j \right]\, = \, 0
\end{equation}
that are given by the following combinations:
\begin{equation}\label{compacarta}
\begin{array}{lll}
\mathcal{L}^1 & = &
-\frac{H_2}{2}-\frac{H_9}{2}+\frac{H_{16
}}{2}-\frac{H_{24}}{2} \\
\mathcal{L}^2 & = &
-\frac{H_2}{2}+\frac{H_9}{2}+\frac{H_{16
}}{2}+\frac{H_{24}}{2} \\
\mathcal{L}^3 & = &
\frac{H_2}{2}+\frac{H_9}{2}+\frac{H_{16}
}{2}-\frac{H_{24}}{2}
\end{array}
\end{equation}
Secondly we have $9$ pairs of generators $\left\{X_{i}\, , \, Y_{i}\right\}$ which are in correspondence with the $9$ positive roots of the $\mathfrak{sp}(6,C)$ Lie algebra (see eq.(\ref{cornette}). Explicitly we have:
\begin{equation}\label{coppietteXY}
\begin{array}{lllllll}
X_1 & = & H_{10} & ; &
Y_1 & = & H_7 \\
X_2 & = & H_4 & ; & Y_2 &
= & -H_{13} \\
X_3 & = & H_6 & ; & Y_3 &
= & -H_3 \\
X_4 & = &
-H_1+H_{14}+H_{20}-H_{22} & ; &
Y_4 & = &
-H_5-H_{11}-H_{18}+H_{23} \\
X_5 & = & H_{21} & ; &
Y_5 & = & -H_8 \\
X_6 & = & H_1+H_{14}+H_{20}+H_{22}
& ; & Y_6 & = &
H_5-H_{11}-H_{18}-H_{23} \\
X_7 & = &
-H_1-H_{14}+H_{20}+H_{22} & ; &
Y_7 & = &
H_5-H_{11}+H_{18}+H_{23} \\
X_8 & = & H_{17} & ; &
Y_8 & = & H_{15} \\
X_9 & = & H_{12} & ; &
Y_9 & = & H_{19}
\end{array}
\end{equation}
The commutation relations with the compact Cartan generators are as follows:
\begin{equation}\label{cirocondo}
\left[ \mathcal{L}^i \, , \, X_I \right] \, = \, \alpha_I^i \, Y_I \quad ; \quad \left[ \mathcal{L}^i \, , \, Y_I \right] \, = \, - \, \alpha_I^i \, X_I
\end{equation}
where $\alpha_I$ are the roots of eq.(\ref{cornette}). The remaining commutation relations mix the $Y$ and the $X$ among themselves and reproduce the Cartan generators.
\subsection{\sc The subalgebra $\mathfrak{sl}(2,\mathbb{R})_E \, \oplus \, \mathfrak{sp}(6,\mathbb{R})$ and the $\mathbf{W}$-generators}
Of great relevance in all applications of the (pseudo)-quaternionic geometry either in the construction of Black-Hole solutions or in the quest of inflaton potentials by means of the gauging of hypermultiplet isometries is the identification of the subalgebra:
\begin{equation}\label{cannula}
\mathfrak{sl}(2,\mathbb{R})_E \, \oplus \, \mathfrak{sp}(6,\mathbb{R}) \, \subset \, \mathfrak{f}_{(4,4)}
\end{equation}
and the recasting of $\mathfrak{f}_{(4,4)}$ in the general form \ref{genGD3} by means of the identification of the $\mathbf{W}$-generators.
\par
To this effect a very powerful tool is provided by the comparison of the $\mathfrak{f}_{(4,4)}$ root system displayed in eq.(\ref{radicioneF4}) with the $\mathfrak{sp}(6,\mathbb{R})$ root system displayed in eq.(\ref{cornette}). The step operators associated with the highest (lowest) root $\pm\beta_{24}$ are the only ones that have a grading $\pm 2$ with respect to the fourth Cartan generator $\mathcal{H}_4$. These three operators close among themselves the Lie algebra $\mathfrak{sl}(2,\mathbb{R})_E $. There are $9$ roots that have grading zero with respect to $\mathcal{H}_4$. Projected onto the plane $\mathcal{H}_4\, = \,0$ these $9$ roots form, together with their negatives, a $\mathfrak{sp}(6,\mathbb{R})$ root system. Correspondingly the $\mathfrak{sp}(6,\mathbb{R})$ subalgebra is generated by the step operators associated with these $9$ roots (and with their negaives) plus the first $3$ Cartan generators. Finally there are $14$ positive roots $\beta$ that have have grading $1$ with respect to $\mathcal{H}_4$. The step operators associated with these $14$ roots form the $\mathbf{W}$-generators with index $1$ of $\mathrm{SL(2,\mathbb{R})}_E$. Their partners with index $2$ are provided by the corresponding negative root step operators.
\par
It is quite important to arrange the generators $\mathbf{W}$ in such a way that under any element $\mathfrak{g }\, \in \, \mathfrak{sp}(6,\mathbb{R}) \, \subset \, \mathfrak{f}_{(4,4)} $ they transform exactly with the $\mathcal{D}_{14}(\mathfrak{g})$ matrices defined in eq.s(\ref{cartolini14}), (\ref{steppinisu14}) and (\ref{steppinigiu14}).
\par
The precise definition of all the generators that satisfy the specified requirements is given below.
\subsubsection{\sc The Ehlers subalgebra $\mathfrak{sl}(2,\mathbb{R})_E$.} The standard commutation relations:
\begin{eqnarray}
\left[ L_0^E \, , \, L^E_\pm\right] &=& \pm \, L^E_\pm \\
\left[ L^E_+\, , \, L^E_- \right] &=& 2 \, L^E_0 \label{EhlersF4}
\end{eqnarray}
are satisfied by the following generators:
\begin{eqnarray}
L_0^E&=& \frac{1}{2} \, \mathcal{H}_4 \nonumber \\
L_+^E &=& \frac{1}{\sqrt{2}} \, \mathcal{E}^{\beta_{24}}\nonumber \\
L_- ^E&=& \frac{1}{\sqrt{2}} \, \mathcal{E}^{- \,\beta_{24}} \label{EhlersF4BIS}
\end{eqnarray}
\subsubsection{\sc The subalgebra $\mathfrak{sp}(6,\mathbb{R})$.} The Cartan generators are the following ones:
\begin{eqnarray}
\mathcal{H}_1 &=& \mathcal{H}_1 \nonumber\\
\mathcal{H}_2 &=& \mathcal{H}_2 \nonumber\\
\mathcal{H}_3 &=& \mathcal{H}_3 \label{samicarti}
\end{eqnarray}
while the step operators are identified as follows
\begin{equation}
\begin{array}{lll}
\mathcal{E}^{\pm \alpha _1} & = & \mathcal{E}^{\pm\beta_{4}} \\
\mathcal{E}^{\pm \alpha _2} & = & \mathcal{E}^{\pm\beta_{3}} \\
\mathcal{E}^{\pm \alpha _3} & = & \mathcal{E}^{\pm\beta_{2}} \\
\mathcal{E}^{\pm \alpha _4} & = & \mathcal{E}^{\pm\beta_{7}} \\
\mathcal{E}^{\pm \alpha _5} & = & -\mathcal{E}^{\pm\beta_{6}}\\
\mathcal{E}^{\pm \alpha _6} & = & \mathcal{E}^{\pm\beta_{10}}\\
\mathcal{E}^{\pm \alpha _7} & = & -\mathcal{E}^{\pm\beta_{9}}\\
\mathcal{E}^{\pm \alpha _8} & = & \mathcal{E}^{\pm\beta_{13}}\\
\mathcal{E}^{\pm \alpha _9} & = & \mathcal{E}^{\pm\beta_{16}}
\end{array}\label{fognato}
\end{equation}
We would like to attract the attention of the reader on the two minus signs introduced in the identifications (\ref{fognato}). Together with the other minus signs that appear below in the identification of the $W$-generators these signs are essential in order for the transformations of the $W$.s to be identical with those given by the previously defined $\mathcal{D}_{14}(\mathfrak{g})$ matrices.
\subsubsection{\sc The $\mathbf{W}$-generators} Casting the $\mathfrak{f}_{(4,4)}$ Lie algebra in the general form (\ref{genGD3}) is completed by the identification of the $\mathbf{W}$-generators. We find:
\begin{equation}\label{widentifio}
\begin{array}{lll}
\mathbf{W}^{1,1} & = & \mathcal{E}^{ \beta_{5}} \\
\mathbf{W}^{1,2} & = & \mathcal{E}^{ \beta_{20}} \\
\mathbf{W}^{1,3} & = & \mathcal{E}^{ \beta_{14}} \\
\mathbf{W}^{1,4} & = & -\mathcal{E}^{ \beta_{23}} \\
\mathbf{W}^{1,5} & = & \mathcal{E}^{ \beta_{21}} \\
\mathbf{W}^{1,6} & = & \mathcal{E}^{ \beta_{19}} \\
\mathbf{W}^{1,7} & = & -\mathcal{E}^{ \beta_{17}} \\
\mathbf{W}^{1,8} & = & -\mathcal{E}^{ \beta_{22}} \\
\mathbf{W}^{1,9} & = & -\mathcal{E}^{ \beta_{11}} \\
\mathbf{W}^{1,10} & = & -\mathcal{E}^{ \beta_{18}} \\
\mathbf{W}^{1,11} & = & -\mathcal{E}^{ \beta_{1}} \\
\mathbf{W}^{1,12} & = & -\mathcal{E}^{ \beta_{8}} \\
\mathbf{W}^{1,13} & = & -\mathcal{E}^{ \beta_{12}} \\
\mathbf{W}^{1,14} & = & -\mathcal{E}^{ \beta_{15}}
\end{array}
\end{equation}
and for all $\mathfrak{g }\, \in \, \mathfrak{sp}(6,\mathbb{R}) \, \subset \, \mathfrak{f}_{(4,4)} $ we have:
\begin{equation}\label{ciucciatiquesta}
\left[ \mathfrak{g } \, , \, \mathbf{W}^{1,\alpha} \right] \, = \, \mathcal{D}_{14}(\mathfrak{g})^\alpha_{\phantom{\alpha}\gamma} \, \mathbf{W}^{1,\gamma}
\end{equation}
The generators $\mathbf{W}^{2,\alpha}$ are then easily obtained from by means of a rotation with the unique compact generator of the Ehlers subalgebra introduced in eq.(\ref{ruotogrande}):
\begin{equation}\label{fungatorepazzo}
\left[ \mathfrak{S} \, , \, \mathbf{W}^{1,\alpha} \right] \, = \,\mathbf{W}^{2,\alpha}
\end{equation}
\subsection{\sc The solvable coset representative}
The precise constructions of the previous sections enable us to introduce the solvable coset representative $\mathbb{L}_{Solv}\left(a,U,h,p,Z\right)$ of the manifold (\ref{quatergurto}) such that the Maurer Cartan form:
\begin{equation}\label{maurocartaneggia}
\Xi \, \equiv \, \mathbb{L}_{Solv}^{-1} \, \mathrm{d} \mathbb{L}_{Solv}
\end{equation}
decomposed along the generators of the Borel Lie algebra:
\begin{eqnarray}\label{solubilino}
\Xi & = & E^I_{\mathcal{QM}} \, T_I\nonumber\\
T_I & = & \left\{ \underbrace{L_0^E \, , \, L^E_+ }_{2 \,\hookrightarrow\, Solv\left[\mathfrak{sl}(2)\right]} \, , \, \underbrace{\mathcal{H}^i \, , \, \mathcal{E}^{\alpha_i}}_{12 \, \hookrightarrow \, Solv\left[\mathfrak{sp}(6)\right]} \, ,\, \underbrace{\mathbf{W}^{1\alpha}}_{14 \, \hookrightarrow \, \mathbb{H}\mathrm{eis}}\right\}
\end{eqnarray}
provides the vielbein $E^I_{\mathcal{QM}}$ mentioned in eq.(\ref{filibaine}) and by squaring the metric (\ref{cornish}).
\par
In full analogy with eq.s (\ref{fornarina}) and (\ref{baltico}) we write:
\begin{eqnarray}\label{SolvCosF4}
\mathbb{L}_{Solv} & = & \exp \left[ a \, L^E_+\right] \, \cdot \, \exp\left[\sum_{j=1}^7 \, \mathbf{Z}_{2j-1} \, \mathbf{W}^{1,2j-1}\right] \, \cdot \, \exp\left[\sum_{j=1}^7 \, \mathbf{Z}_{2j} \, \mathbf{W}^{1,2j}\right] \, \times \nonumber \\
&& \times \prod_{i=1}^9 \, \exp\left [ p_{10-i} \, \mathcal{E}^{\alpha_{10-i}}\right ] \, \cdot \, \prod_{j=3}^3 \exp\left [ h_{j} \, \mathcal{H}^{j}\right ] \, \cdot \, \exp\left[U \, L^E_0 \right]
\end{eqnarray}
The explicit expression of $\mathbb{L}_{Solv}$ in the fundamental $26$-dimensional representation is obviously very large but it can be dealt with by means of an appropriate MATHEMATICA code.
\par
We are finally in the position of calculating the tri-holomorphic moment map of any element $ \mathfrak{t}\, \in \, \mathfrak{f}_{(4,4)}$
of the isometry Lie algebra of $\mathcal{QM}$ through the formula:
\begin{equation}\label{cordiglierAndina}
\mathcal{P}^x_\mathfrak{t} \, = \, \mbox{Tr}_{\mathbf{26}} \, \left( J^x \, \mathbb{L}_{Solv}^{-1} \, \mathfrak{t} \, \mathbb{L}_{Solv} \right)
\end{equation}
\subsection{\sc The example of the inclusion of multi Starobinsky models}
\label{generalonuovo}
In section \ref{truncazia} we studied the truncation of the $\mathfrak{sp}(6,\mathbb{R})$ model to the STU model. There we showed that setting to zero the three complex coordinates $z_2,z_3,z_5$, the remaining ones $z_1,z_4,z_6$ span the STU model, namely they parameterize three copies of the Lobachevsky-Poincar\'e hyperbolic plane. Inspecting eq. (\ref{carlinus}) we also see that the three coordinates $z_1,z_4,z_6$ are the only ones that survive when all the axions $p_i$ are set to zero. We also recall from sect. \ref{truncazia} that the three parabolic generators of the three $\mathrm{SL(2,\mathbb{\mathbb{R}})}$ groups spanning the STU model are $\mathcal{E}^{\alpha_3},\, \mathcal{E}^{\alpha_7},\,\mathcal{E}^{\alpha_9}$ whose identification with $\mathfrak{f}_{(4,4)}$ generators is provided by eq.(\ref{fognato}). Correspondingly we introduce the following generator:
\begin{equation}\label{carnevaleSTU}
\mathfrak{t}_{STU}\, = \, \beta_3 \, \mathcal{E}^{\alpha_3}\, + \, \beta_2 \, \mathcal{E}^{\alpha_7}\, + \, \beta_1 \, \mathcal{E}^{\alpha_9}\, - \, \kappa \, \mathfrak{S}
\end{equation}
and we calculate its tri-holomorphic moment map, by means of eq.(\ref{cordiglierAndina}). Defining the potential:
\begin{equation}\label{gogamigoga}
V_{STU} \, = \, \sum_{x=1}^3 \, \left(\mathcal{P}^x_{\mathfrak{t}_{STU}}\right)^2
\end{equation}
We can verify that:
\begin{eqnarray}
\left. \frac{\partial}{\partial \mathbf{Z}^\alpha} \, V_{STU}\right|_{\mathbf{Z}=U=a =0} & = &0 \nonumber\\
\left. \frac{\partial}{\partial U} \, V_{STU}\right|_{\mathbf{Z}=U=a =0} & = &0 \nonumber\\
\left. \frac{\partial}{\partial a} \, V_{STU}\right|_{\mathbf{Z}=U=a =0} & = &0 \label{UaZetaseneva}
\end{eqnarray}
Hence we can consistently truncate $U$, $a$ and the Heisenberg fields $\mathbf{Z}$. We find:
\begin{equation}\label{multistarobbo}
\left. V_{STU}\right|_{\mathbf{Z}=U=a =0} \, = \, \frac{9}{4} \left( \, 2\kappa \, - \, \sqrt{2} \sum_{i=1}^3
\beta_i \,e^{-2 h_i} \right)^2
\end{equation}
The above potential can be named a multi-Starobinsky model with three independent dilatons.
\par
First of all let us note that in the above model the absolute value of $\beta_i$ is irrelevant since we can always reabsorb it by a constant shift $h_i \to h_i \, - \, \log|\beta_i|$. The only relevant thing are the signs of $\beta_i$ including in this notion also zero, namely $\beta_i $ can be $\pm 1$ or $0$. Secondly we observe that when all the non vanishing $\beta_i$ have the same sign we can make a consistent one field truncation to
\begin{equation}\label{correlo}
h_i \, = \, h \quad ; \quad \mbox{for all $i$ such that $\beta_i \ne 0$}
\end{equation}
After this truncation the potential (\ref{multistarobbo}) becomes the following:
\begin{equation}\label{riducione}
V_{eff} \, = \, \frac{9}{4} \left(\, 2\kappa \, - \, \sqrt{2} \, q\, e^{-2 h}\right)^2
\end{equation}
where $q$ is the number of equal sign non zero $\beta_i$, which obviously can take only three values $q=1,2,3$. In order to compare this result with the definition of $\alpha$-attractors introduced in \cite{alfatrattori}, we just have to compare the potential (\ref{riducione}) with the normalization of the scalar kinetic terms in the lagrangian:
\begin{equation}\label{kinescalo}
\mathcal{L} \, = \, \dots \, + \, \frac{1}{4} \, (\partial U)^2 \, + \, (\partial h_1)^2\, + \, (\partial h_2)^2 \, + \, (\partial h_3)^2 \, + \, \dots
\end{equation}
which follows from eq.s(\ref{geodaction}, \ref{formidabile}). Renaming $h\, = \, \frac{1}{\sqrt{2 \, q}} \, \phi$, so that the new field $\phi$ has canonical kinetic term $\frac{1}{2} \, (\partial \phi)^2$, we obtain a potential:
\begin{equation}\label{bambolone}
V_{eff} \, = \, \mbox{const} \, \times \, \left( 2\kappa \, - \, \sqrt{2} \, q\, \exp\left [ - \, \sqrt{\frac{2}{q}} \, \phi\right]\right)^2
\end{equation}
which, in the notation of \cite{alfatrattori}, corresponds to $\alpha = \frac{q}{3} $, namely to:
\begin{equation}\label{valoretti}
\alpha \, = \, 1 \, , \, \frac{2}{3} \, , \, \frac{1}{3}
\end{equation}
The above result has been obtained by gauging only one generator, namely (\ref{carnevaleSTU}). Correspondingly we have generated Starobinsky-like models with only one massive vector that is the gauge vector associated with the gauged generator. There is another way of obtaining the same potential but with $q$-massive vectors (one for each constituent Starobinsky model with $q=\frac{1}{3}$). This is very simply understood remarking that the $\mathfrak{f}_{(4,4)}$ algebra contains an $\mathfrak{sl}(2,\mathbb{R})^4$ subalgebra singled out as follows:
\begin{equation}\label{governolato}
\mathfrak{f}_{(4,4)} \, \supset \, \mathfrak{sl}(2,\mathbb{R})_E \, \oplus \, \underbrace{\mathfrak{sl}(2,\mathbb{R})_S \, \oplus \, \mathfrak{sl}(2,\mathbb{R})_T\, \oplus \, \mathfrak{sl}(2,\mathbb{R})_U}_{\subset \, \mathfrak{sp}(6,\mathbb{R})}
\end{equation}
where $\mathfrak{sl}(2,\mathbb{R})_S \, \oplus \, \mathfrak{sl}(2,\mathbb{R})_T\, \oplus \, \mathfrak{sl}(2,\mathbb{R})_U$ describes the STU model embedded in the K\"ahler manifold (\ref{gurto}). These four $\mathfrak{sl}(2,\mathbb{R})$ algebras are completely symmetric among themselves and the gauging of their generators produce identical results. So we can introduce the abelian gauge algebra spanned by the following three commuting generators:
\begin{eqnarray}
\mathfrak{t}_{S} &=&\beta_3 \, \mathcal{E}^{\alpha_3}\, - \, \kappa_3 \, \mathfrak{S} \nonumber \\
\mathfrak{t}_{T} &=& \beta_2 \, \mathcal{E}^{\alpha_7}\, - \, \kappa_2 \, \mathfrak{S}\nonumber\\
\mathfrak{t}_{U} &=& \, \beta_1 \, \mathcal{E}^{\alpha_9}\, - \, \kappa_1\, \mathfrak{S}
\end{eqnarray}
Gauging with three separate vectors each of the above generators we obtain a new potential:
\begin{equation}\label{STUnew}
\widehat{V}_{STU} \, = \, \sum_{x=1}^3 \, \left(\mathcal{P}^x_{\mathfrak{t}_{S}}\right)^2 \, + \, \sum_{x=1}^3 \, \left(\mathcal{P}^x_{\mathfrak{t}_{T}}\right)^2 \, + \, \sum_{x=1}^3 \, \left(\mathcal{P}^x_{\mathfrak{t}_{U}}\right)^2 \,
\end{equation}
that has the same property as the potential (\ref{gogamigoga}), namely it allows us to truncate consistently to zero all the axions $p_i$, all the Heisenberg fields $\mathbf{Z}^\alpha$ and the Taub NUT field $a$. The reduced potential after such a truncation has the form:
\begin{equation}\label{ciurlone}
\widehat{V}_{red}\, = \, \frac{9}{4} \, \sum_{i=1}^3 \, \left( \, 2\kappa_i \, - \, \sqrt{2}
e^{-2 h_i} \beta_i \right)^2
\end{equation}
As we already remarked before, the absolute value of the $\beta_i$ parameters is irrelevant: what matters is only the relative signs of the $\beta_i$ with respect to the sign of their corresponding $\kappa_i$. If for all non vanishing $\beta_i$ we have $\frac{\beta_i}{\kappa_i} \, = 1$, then we can consistently perform the same truncation (\ref{correlo}) as before and we reobtain the potentials (\ref{bambolone}) with the same spectrum of $\alpha$-values (\ref{valoretti}). The difference with the previous case is, as we emphasized at the beginning o this discussion, that now the number of massive fields is $q$, namely as many as the elementary non trivial constituent Starobinsky-like models.
\subsection{\sc Nilpotent gaugings and truncations}
\label{orbitando}
Let us now put the above obtained results in the general framework discussed in sect.\ref{starobin1}. The issue is the classification of orbits of nilpotent operators and the question whether for each of these orbits we can find a consistent one-field reduction that produces a Starobinsky-like model with an appropriate value of $\alpha$.
\par
To answer this question we have followed the algorithm described in the second paper of \cite{noinilpotenti}. According to a general mathematical set, up to conjugation, every nilpotent orbit is associated with a standard triple $\left \{ x,y,h \right \}$ satisfying the standard commutation relations of the $\mathfrak{sl}(2)$ Lie algebra, namely:
\begin{equation}\label{basictriple}
\left [ h \, , \, x\right ] \, = \, x \quad ; \quad \left [ h \, , \, y\right ] \, = -\, y
\quad ; \quad \left [ x \, , \, y \right ] \, = \, 2 \, h
\end{equation}
Interesting for us is the classification of nilpotent orbits in the K\"ahler subalgebra $\mathfrak{sp}(6,\mathbb{R})$ and, according to the above mathematical theory, this is just the classification of embeddings of an $\mathfrak{sl}(2)$ Lie algebra in the ambient one, modulo conjugation by the full group $\mathrm{Sp(6,\mathbb{R})}$.
The second relevant point emphasized in \cite{noinilpotenti} is that embeddings of subalgebras $\mathfrak{h} \subset \mathfrak{g}$ are characterized by the branching law of any representation of $\mathfrak{g}$ into irreducible representations of $\mathfrak{h}$. Clearly two embeddings might be conjugate only if their branching laws are identical. Embeddings with different branching laws necessarily belong to different orbits. In the case of the $\mathfrak{sl}(2) \sim \mathfrak{so}(1,2)$ Lie algebra, irreducible representations are uniquely identified by their spin $j$, so that the branching law is expressed by listing the angular momenta $\left\{ j_1 , j_2 , \dots j_n\right \}$ of the irreducible blocks into which any representation of the original algebra, for instance the fundamental, decomposes with respect to the embedded subalgebra. The dimensions of each irreducible module is $2j+1$ so that an a priori constraint on the labels $\left\{ j_1 , j_2 , \dots j_n\right \}$ characterizing an irreducible orbit of $\mathfrak{sp}(6,\mathbb{R})$ is the summation rule:
\begin{equation}\label{summarulla}
\sum_{i=1}^{n} (2 j_i +1) \, = \, 6\, = \, \mbox{dimension of the fundamental representation}
\end{equation}
Therefore we have considered all possible partitions of the number $6$ into integers and for each partition we have constructed a candidate $h$ element in the Cartan subalgebra of $\mathfrak{sp}(6,\mathbb{R})$ containing as eigenvalues all the $J_3$ values of the corresponding $\left\{ j_1 , j_2 , \dots j_n\right \}$ representation. To clarify what we mean by this it suffices to consider the example of the first partition $6=6$. In this case the $6$ dimensional representation of $\mathfrak{sl}(2)$ is the $j \, = \, \frac{5}{2}$ and the $6$ eigenvalues are $\pm \frac{5}{2}$, $\pm \frac{3}{2}$, $\pm \frac{1}{2}$. Having so fixed the so named central element $h$ of the candidate standard triplet we have tried to construct the corresponding $x$ and $y$. Imposing the standard commutation relations (\ref{basictriple}) one obtains quadratic equations on the coefficients of the linear combinations expressing the candidate $x$ and $y$ that may have or may not have solutions. If the solutions exist, then the corresponding standard triple is found, the orbit exists and we have constructed one representative $x$.
\par
Next, given the existing orbits and the corresponding standard triples, for each of them we have constructed a Lobachevsky complex plane immersed in the Special K\"ahler manifold $\mathcal{M}_{Sp6}$ defined by eq.(\ref{gurto}). The construction is very simple. One calculates the group element $\mathfrak{g}(\lambda,\psi) \, \in \, \mathfrak{sp}(6,\mathbb{R})$ defined below:
\begin{equation}\label{lambopsi}
\mathfrak{g}(\lambda,\psi) \, = \, \exp\left[ \psi \, x\right] \, \cdot \, \, \exp\left[ \lambda \, h\right] \, = \, \left (\begin{array}{c|c}
\mathbf{A}(\lambda,\psi) & \mathbf{B}(\lambda,\psi) \\
\hline
\mathbf{C}(\lambda,\psi) & \mathbf{D}(\lambda,\psi)
\end{array}
\right)
\end{equation}
and using equation (\ref{Zmatra}), we write:
\begin{eqnarray}\label{embeddone}
Z(\lambda\, , \, \psi) & = & \left( \mathbf{A}(\lambda,\psi) \, - \, {\rm i} \mathbf{B}(\lambda,\psi)\right) \, \cdot \, \left( \mathbf{C}(\lambda,\psi) \, - \, {\rm i} \mathbf{D}(\lambda,\psi)\right)^{-1} \, \nonumber\\
& \equiv & \left( \begin{array}{ccc}
z_1(\lambda,\psi) & z_2(\lambda,\psi) & z_3(\lambda,\psi) \\
z_2(\lambda,\psi) & z_4(\lambda,\psi) & z_5(\lambda,\psi) \\
z_3(\lambda,\psi) & z_5(\lambda,\psi) & z_6(\lambda,\psi)
\end{array}
\right)
\end{eqnarray}
which defines the explicit embedding:
\begin{equation}\label{embeddus}
\phi \, : \, \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}} \, \rightarrow \, \frac{\mathrm{Sp(6,\mathbb{R})}}{\mathrm{SU(3)} \times \mathrm{U(1)}} \, \equiv \, \mathcal{M}_{Sp6}
\end{equation}
of the Lobachevsky plane in $\mathcal{M}_{Sp6}$. Indeed from (\ref{embeddone}) we read off the parameterization of the complex coordinates $z_i$ ($i=1,\dots ,6$) as functions of $\lambda \, = \, \log \mbox{Im}\, w$ and $\psi \, = \, \mbox{Re}\,w$, the complex variable $w$ being the local variable over the embedded Poincar\'e-Lobachevsky plane.
\par
The question is whether the field equations of the scalar fields:
\begin{equation}\label{finocchione}
\partial_i \, \partial_{j^\star} \,\mathcal{K}\, \partial^\mu\partial_\mu \,\bar{z}^{j^\star} \, + \, \partial_i \, \partial_{j^\star}\,\partial_{k^\star} \, \mathcal{K} \, \partial_\mu z^{j^\star} \, \partial_\mu z^{k^\star}\, - \, \frac{1}{4}\,\partial_i \, V_{gauging}\left(z\, ,\, {\bar z}\right) \, = \, 0
\end{equation}
admit first a consistent reduction to the complex scalar field $w$ and then a consistent truncation to a vanishing axion $\psi \, = \, 0$. Consistency of the truncation can be verified or disproved in the following simple way. The pull-back on the immersed surface $\phi^\star \left(\frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\right) \, \subset \, \mathcal{M}_{Sp6}$ of the twelve field equations (\ref{finocchione}) (six complex equations) should be consistent among themselves and be identical with the two field equations obtained from the variation of the pull-back $\phi^\star(\mathcal{L})$ on the immersed surface of the Lagrangian $\mathcal{L}$ from which eq.s (\ref{finocchione}) derive, namely:
\begin{equation}\label{lagrandona}
\mathcal{L} \, = \, 4 \, \partial_i \, \partial_{j^\star} \, \mathcal{K}\, \partial_\mu \,{z}^{i} \,\partial^\mu \,\bar{z}^{j^\star} \, - \, V_{gauging}\left(z\, ,\, {\bar z}\right)
\end{equation}
In other words, defining $ w\, = \, {\rm i} \, e^{\lambda} \, + \, \psi $, the truncation is consistent if the following diagram is commutative:
\begin{eqnarray}
&&\begin{array}{ccc}
\mathcal{L}(z,{\bar z}) & \stackrel{\phi^\star}{\Longrightarrow} & \phi^\star \mathcal{L}(w,{\bar w}) \\
\downarrow & \null & \downarrow \\
\partial^\mu \, \frac{\partial \mathcal{L}}{\partial (\partial_\mu z)} \, - \, \frac{\partial \mathcal{L}}{\partial z} & \stackrel{\phi^\star}{\Longrightarrow} & \partial^\mu \, \frac{\partial \phi^\star \mathcal{L}}{\partial (\partial_\mu w)} \, - \, \frac{\partial \phi^\star \mathcal{L}}{\partial w}
\end{array} \label{coomodiag}
\end{eqnarray}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Partition&J.s&Orbit Name&One field reduction \\
\hline
6=6 & $\left(\frac{5}{2}\right)$ & $\mathfrak{O}_1$ & NO \\
6=5+1 & $\left(2,0\right)$ & Orbit does not exist & NO \\
6=4+2 & $\left(\frac{3}{2},\frac{1}{2}\right)$ & $\mathfrak{O}_2$ & NO \\
6=3+3 & $\left(1,1\right)$ & $\mathfrak{O}_3$ & NO \\
6=3+2+1 & $\left(1,\frac{1}{2},0\right)$ &Orbit does not exist & NO \\
6=3+1+1+1 & $\left(1,0,0,0\right)$ & Orbit does not exist & NO \\
6=2+2+2 & $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ & $\mathfrak{O}_4$ & YES \\
6=2+2+1+1 & $\left(\frac{1}{2},\frac{1}{2},0,0\right)$ & $\mathfrak{O}_5$ & YES \\
6=2+1+1+1+1 & $\left(\frac{1}{2},0,0,0,0\right)$ & $\mathfrak{O}_6$ & YES \\
\hline
\end{tabular}
\end{center}
In the above table we have summarized the results of this simple investigation. There is a total of six orbits (up to possible further splitting in Weyl group orbits which we have not analyzed) and for each of them the corresponding immersion formulae in the $\mathcal{M}_{\mathrm{Sp6}}$ manifolds are those described below.
\paragraph{Orbit $\mathfrak{O}_1$: ($j=\frac{5}{2}$).}
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
-6 \psi ^5+10 i e^{\lambda } \psi ^4+5 i e^{3 \lambda } \psi ^2+i
e^{5 \lambda } & \sqrt{5} \psi \left(3 \psi ^3-4 i e^{\lambda
} \psi ^2-i e^{3 \lambda }\right) & i \sqrt{10} \left(i \psi
+e^{\lambda }\right) \psi ^2 \\
\sqrt{5} \psi \left(3 \psi ^3-4 i e^{\lambda } \psi ^2-i e^{3
\lambda }\right) & i \left(8 i \psi ^3+8 e^{\lambda } \psi
^2+e^{3 \lambda }\right) & \sqrt{2} \psi \left(3 \psi -2 i
e^{\lambda }\right) \\
i \sqrt{10} \left(i \psi +e^{\lambda }\right) \psi ^2 & \sqrt{2}
\psi \left(3 \psi -2 i e^{\lambda }\right) & i e^{\lambda }-3
\psi
\end{array}
\right) \nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, + \, \psi \label{immerO1}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittaprima}
\phi^\star \mathcal{L} \, = \, 35 \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, \frac{1}{4}\, g^2 \, \left(3\,e^{-\lambda} \, - \, \kappa\right)^2
\end{equation}
The pull-backs of the scalar field equations are inconsistent among themselves and differ from the equations derived from the pull-back of the lagrangian (\ref{orbittaprima}), hence the truncation is not consistent. No Starobinsky--like model can be obtained from this orbit.
\par
One might wonder whether the inconsistency is due to the particularly chosen coset representative (\ref{lambopsi}) and to the explicit form of the embedding (\ref{immerO1}) which turns out to be non-holomorphic. To clarify such a doubt and show that the inconsistency of the equations is an intrinsic property of the orbit, we have addressed the problem from a different view point which leads to a perfectly holomorphic embedding of the Lobachevsky plane associated with the considered orbit into the target Special K\"ahler manifold (\ref{gurto}).
\par
The argument is the following one. Having fixed the embedding $\mathfrak{sl}(2,\mathbb{R})\mapsto \mathfrak{sp}(6,\mathbb{R})$ at the level of the fundamental representation $\mathbf{6}$ it is fixed also in all other representations and we can wonder what is the branching rule of the $\mathbf{W}$-representation $\mathbf{14}^\prime$ such an embedding. By direct evaluation of the Casimir we obtain the following branching:
\begin{equation}\label{frikandello1}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, \left( j\, =\, \frac{9}{2}\right) \, \oplus \, \left( j\, =\, \frac{3}{2}\right)
\end{equation}
This means that the symplectic section (\ref{gargamelle}) splits into the sum of two vectors, one lying in the 10-dimensional space of the first representation, the other in the $4$-dimensional space of the second representation. Imposing the vanishing of the lowest spin representation introduces a set of $4$ holomorphic constraints on the six coordinates $z_i$. By construction these constraints are $\mathfrak{sl}(2,\mathbb{R})$ invariant: therefore the sought for Lobachevsky plane certainly lies in the complex two-folds defined by the vanishing of these constraints. With a little bit of work one can further eliminate one of the two remaining complex coordinates in such a way that the ten entries of the $ \left( j\, =\, \frac{9}{2}\right)$ representation correspond to all the powers $w^r$, with $r=0, 1, \dots , 9$ of a complex parameter $w$. Because of this very property $w$ can be interpreted as the local coordinate of the sought for Lobachevsky plane embedded in the K\"ahler manifold (\ref{gurto}) according to the specified orbit. Indeed if $w$ transforms by fractional linear transformation under some algebra $\mathfrak{sl}(2)$, then the $2j+1$ first powers of $w$ provide a basis for the $j$-representation of that $\mathfrak{sl}(2)$. Viceversa, if a vector, which is known to transform in the $j$-represenation of a given $\mathfrak{sl}(2)$ (up to an overall function of $w$), is made by linear combinations of the first $2j+1$ powers of a coordinate $w$, then that $w$ is the local coordinate on a Lobachevsky plane transitive under the action of that very $\mathfrak{sl}(2)$.
\par
In our case the four holomorphic constraints that express the vanishing of the $j=\frac{3}{2}$ representation inside the $14^\prime$ are the following ones:
\begin{eqnarray}
\sqrt{\frac{2}{7}} \left(\sqrt{5} \left(z_4
z_6-z_5^2\right)-2 z_2\right) &=& 0 \nonumber \\
\frac{8 z_3-\sqrt{10} z_4}{\sqrt{21}}&=& 0 \nonumber\\
\sqrt{\frac{2}{7}} \left(\sqrt{5} z_1+2 z_3 z_5-2 z_2
z_6\right) &=& 0 \nonumber \\
\frac{-\sqrt{10} z_3^2+8 z_4 z_3-8 z_2 z_5+\sqrt{10}
z_1 z_6}{\sqrt{21}}&=& 0 \label{bamboccione}
\end{eqnarray}
The explicit form of (\ref{bamboccione}) obviously depends on the standard triple chosen as representative of the orbit, yet for whatever representative the four constraints are holomorphic. The next point consists in solving (\ref{bamboccione}) in terms of a parameter $w$ so that the complementary set of ten polynomials of the $z_i$ spanning the $j=\frac{9}{2}$ representation provide all the powers of $w$ from $0$ to $9$.
The requested solution is given by:
\begin{equation}\label{sicumerolo}
z_1\to \frac{3 w^5}{16}, \, \, z_2\to \frac{3\sqrt{5} w^4}{16},\, \, z_3\to \frac{1}{4}
\sqrt{\frac{5}{2}} w^3,\, \, z_4\to w^3,\, \,z_5\to \frac{3w^2}{2 \sqrt{2}},\, \, z_6\to \frac{3 w}{2}
\end{equation}
Implementing the transformation (\ref{sicumerolo}) in the symplectic section (\ref{gargamelle}) one finds:
\begin{equation}\label{novemezzisezia}
\Omega[Z] \, \stackrel{\phi}{\Longrightarrow} \, \Omega_{\frac{9}{2}}[w] \, = \, \left(
\begin{array}{l}
\frac{3 w}{\sqrt{2}} \\
\frac{w^6}{4 \sqrt{2}} \\
-\frac{3 w^4}{4 \sqrt{2}} \\
\frac{w^9}{256 \sqrt{2}} \\
-\frac{3 w^7}{32 \sqrt{2}} \\
-\frac{1}{16} \sqrt{\frac{5}{2}} w^6 \\
-\frac{3}{16} \sqrt{5} w^5 \\
\frac{3 w^8}{128 \sqrt{2}} \\
-\sqrt{2} w^3 \\
\frac{3 w^5}{8 \sqrt{2}} \\
\sqrt{2} \\
-\frac{3 w^2}{\sqrt{2}} \\
\frac{1}{2} \sqrt{\frac{5}{2}} w^3 \\
-\frac{3}{8} \sqrt{5} w^4
\end{array}\right)
\end{equation}
which as requested contains all the powers of $w$ and has vanishing projection on the $j=\frac{3}{2}$ representation. Calculating the K\"ahler potential from such a section we obtain:
\begin{equation}\label{furutto}
\mathcal{K}_{\frac{9}{2}} \, = \, - \, \log \,\left(\bar{\Omega}_{\frac{9}{2}}[\bar{w}]\,\mathbb{C}_{14} \, \Omega_{\frac{9}{2}}[w] \right) \, = \, \log\left ( -\frac{\rm i}{256} \,(w-\bar{w})^9\right)
\end{equation}
Now the question of consistency can be readdressed in the present context. Implementing the substitution (\ref{sicumerolo}) in the six complex equations (\ref{finocchione}) (with for instance vanishing potential) do we obtain six consistent equations or not? The answer is no. The six equations (\ref{finocchione}) are inconsistent and this confirms in a holomorphic set up the same result we had previously obtained in the direct approach of eq.s (\ref{lambopsi}-\ref{embeddone}). Hence the $\mathfrak{sl}(2)$ embedding of orbit $\mathfrak{O}_1$ leads to inconsistent truncations and has to be excluded.
\paragraph{Orbit $\mathfrak{O}_2$: ($j_1=\frac{3}{2}, \, j_2=\frac{1}{2}$).}
For the second orbit, the direct approach (\ref{lambopsi}-\ref{embeddone}) leads to:
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
\left(e^{\lambda }-i \psi \right)^2 \left(2 \psi -i e^{\lambda
}\right) & 0 & \sqrt{3} \psi \left(\psi +i e^{\lambda }\right)
\\
0 & -\psi -i e^{\lambda } & 0 \\
\sqrt{3} \psi \left(\psi +i e^{\lambda }\right) & 0 & -2 \psi -i
e^{\lambda }
\end{array}
\right) \nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, + \, \psi \label{immerO2}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittaseconda}
\phi^\star \mathcal{L} \, = \, 11 \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, \frac{1}{4}\, g^2 \, \left(3\,e^{-\lambda} \, - \, \kappa\right)^2
\end{equation}
Also in this case the pull-back of the scalar field equations yields an inconsistent set and there is no truncation. No Starobinsky--like model can be obtained from this orbit. In a similar way to the previous case we can discuss the same issue in a holomorphic set up. The branching rule of the $\mathbf{14}^\prime$ representation in the considered embedding is the following one:
\begin{equation}\label{frikandello2}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, \left( j\, =\, \frac{5}{2}\right) \, \oplus \, \left( j\, =\, \frac{3}{2}\right) \, \oplus \, \left( j\, =\, \frac{3}{2}\right)
\end{equation}
and we can impose holomorphic constraints that suppress the two lowest spin representations $\left( j\, =\, \frac{3}{2}\right)$ leaving only the top one $\left( j\, =\, \frac{5}{2}\right)$ spanned by the powers of a parameter $w$ from $0$ to $5$.
Such a holomorphic embedding is given:
\begin{equation}\label{frikandellus2}
z_1 \frac{2 w^3}{3^{3/4}}, \, \, z_2\to 0,\, \, z_3\to w^2,z_4\to
\frac{w}{\sqrt[4]{3}},\, \, z_5\to 0,\, \, z_6\to \frac{2
w}{\sqrt[4]{3}}
\end{equation}
Substitution of eq.s (\ref{frikandellus2}) into the field equations (\ref{finocchione}) confirms that their pull-back on this surface is inconsistent.
\paragraph{Orbit $\mathfrak{O}_3$: ($j_1=1, \, j_2=1$).}
For the third orbit, the direct approach (\ref{lambopsi}-\ref{embeddone}) leads to
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
-i e^{2 \lambda } & -\psi ^2 & -\sqrt{2} \psi \\
-\psi ^2 & -i \left(2 \psi ^2+e^{2 \lambda }\right) & -i \sqrt{2}
\psi \\
-\sqrt{2} \psi & -i \sqrt{2} \psi & -i
\end{array}
\right) \nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, + \, \psi \label{immerO3}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittaterza}
\phi^\star \mathcal{L} \, = \, 8 \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, \frac{1}{4}\, g^2 \, \kappa^2
\end{equation}
while the pull-back of the scalar field equations is an inconsistent set. Hence this truncation is not consistent and no Starobinsky--like model can be obtained from this orbit. As in the previous two cases we can confirm the same result in a holomorphic set up, yet we consider it useless to repeat once more the same type of calculations. What is relevant to mention in view of our subsequent considerations
is the branching rule of the $\mathbf{14}^\prime$ representation under this forbidden embedding leading to inconsistent field equations::
\begin{equation}\label{frikandello3}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, \left( j\, =\, 2\right) \, \oplus \, \left( j\, =\, 2\right) \,
\oplus \, 4 \, \times \, \left( j\, =\, 0\right)
\end{equation}
\paragraph{Orbit $\mathfrak{O}_4$: ($j_1=\frac{1}{2}, \, j_2=\frac{1}{2},j_3=\frac{1}{2}$).} For the fourth orbit, the direct approach (\ref{lambopsi}-\ref{embeddone}) leads to
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
i e^{\lambda }-\psi & 0 & 0 \\
0 & i e^{\lambda }-\psi & 0 \\
0 & 0 & i e^{\lambda }-\psi
\end{array}
\right)
\nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, - \, \psi \label{immerO4}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittaquarta}
\phi^\star \mathcal{L} \, = \, 3 \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, g^2 \, \frac{1}{4} \,\left( 3 \, e^{-\lambda}\, - \, 2\, \kappa\right)^2
\end{equation}
The pull-back of the scalar field equations produces equations consistent among themselves which coincide with the equations derived from the pull-back of the lagrangian (\ref{orbittaquarta}), hence the truncation is consistent. We reobtain the Starobinsky model discussed in the previous section with $q=3$ and hence with $\alpha \, = \, 1$. In this case the consistent truncation is already produced form holomorphic constraints. Indeed equation (\ref{immerO4}) can be summarized as:
\begin{equation}\label{holomorph4}
z_2 \, = \, z_3 \, = \, z_5 \, = \, 0 \quad ; \quad z_1 \, = \, z_4 \, = \, z_6 \, = \, w
\end{equation}
It is interesting and important for our future consideration to mention the branching rule of the $\mathbf{14}^\prime$ representation under this $\mathfrak{sl}(2)$ subalgebra:
\begin{equation}\label{frikandello4}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, \left( j\, =\, \frac{3}{2}\right) \,
\oplus \, 5 \, \times \, \left( j\, =\, \frac{1}{2}\right)
\end{equation}
and the constraints (\ref{holomorph4}) precisely are the conditions under which the five representations $\left( j\, =\, \frac{1}{2}\right)$ vanish and we are left with the representation $\left( j\, =\, \frac{3}{2}\right)$ duely spanned by the powers $1,w,w^2,w^3$.
\paragraph{Orbit $\mathfrak{O}_5$: ($j_1=\frac{1}{2}, \, j_2=\frac{1}{2},j_3=0$).} For the fifth orbit, the direct approach (\ref{lambopsi}-\ref{embeddone}) leads to
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
i e^{\lambda }-\psi & 0 & 0 \\
0 & i e^{\lambda }-\psi & 0 \\
0 & 0 & i
\end{array}
\right) \nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, - \, \psi \label{immerO5}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittaquinta}
\phi^\star \mathcal{L} \, = \, 2 \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, g^2 \, \left(e^{-\lambda} \, - \, \kappa\right)^2
\end{equation}
The pull-back of the scalar field equations yields a consistent system identical with the field equations derived from the pull-back of the lagrangian (\ref{orbittasesta}), hence the truncation is consistent. We reobtain the Starobinsky--like model discussed in the previous section with $q=2$ and hence with $\alpha \, = \, \frac{2}{3}$.
\par
In this, as in the previous case, the consistent truncation is produced from holomorphic constraints. Indeed equation (\ref{immerO4}) can be summarized as:
\begin{equation}
z_2 \, = \, z_3 \, = \, z_5 \, = \, 0 \quad ; \quad z_1 \, = \, z_4 \, = \, w \quad ; \quad z_6 \, = \, {\rm i} \label{belgiuro}
\end{equation}
In this case the branching rule of the $\mathbf{14}^\prime$ representation under the considered $\mathfrak{sl}(2)$ subalgebra is the following one:
\begin{equation}\label{frikandello5}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, \left( j\, =\, 1\right) \,
\oplus \,\left( j\, =\, 1\right)\, \oplus \, 2 \, \times \, \left( j\, =\, \frac{1}{2}\right) \, + \, 4 \, \times \, \left( j\, =\, 0\right)
\end{equation}
and the constraint (\ref{belgiuro}) guarantees that the singlets and the $\left( j\, =\, \frac{1}{2}\right)$ representations are all set to zero.s
\paragraph{Orbit $\mathfrak{O}_6$: ($j_1=\frac{1}{2}, \, j_2=0,\, j_3=0$).} For the sixth orbit, the direct approach (\ref{lambopsi}-\ref{embeddone}) leads to
\begin{eqnarray}
&& \left( \begin{array}{ccc}
z_1& z_2 & z_3\\
z_2& z_4 & z_5 \\
z_3 & z_5 & z_6
\end{array}
\right) = \, \nonumber\\
&& \left(
\begin{array}{lll}
\psi +i e^{\lambda } & 0 & 0 \\
0 & i & 0 \\
0 & 0 & i
\end{array}
\right) \nonumber\\
&& w \,=\, {\rm i} \, e^\lambda \, + \, \psi \label{immerO6}
\end{eqnarray}
The pull-back of the lagrangian is the following one:
\begin{equation}\label{orbittasesta}
\phi^\star \mathcal{L} \, = \, \left( \partial^\mu\psi\, \partial^\mu \psi \, e^{-2\lambda} \, + \, \partial^\mu\lambda\, \partial^\mu \lambda\right) \, - \, \frac{1}{4}\, g^2 \, \left( e^{\lambda}\, + \, \kappa\right)^2
\end{equation}
The pull-back of the scalar field equations yields a consistent system coinciding with the equations derived from the pull-back of the lagrangian (\ref{orbittasesta}). So we have a consistent truncation and we reobtain the Starobinsky--like model discussed in the previous section with $q=1$. It corresponds to $\alpha \, = \, \frac{1}{3}$. Equation (\ref{immerO6}) can be summarized as:
\begin{equation}\label{holomorph6}
z_2 \, = \, z_3 \, = \, z_5 \, = \, 0 \quad ; \quad z_1 \, = \, w \quad ; \quad z_4 \, = \, z_6 \, = \, {\rm i}
\end{equation}
The branching of the $\mathbf{14}^\prime$ dimensional representation under this $\mathfrak{sl}(2)$ subalgebra is the following one:
\begin{equation}\label{frikandello6}
\mathbf{14}^\prime \, \stackrel{\mathfrak{sl}(2,\mathbb{R})}{\longrightarrow} \, 5 \, \times \,\left( j\, =\, \frac{1}{2}\right) \,
\oplus \, 4 \, \times \, \left( j\, =\, 0\right)
\end{equation}
\subsubsection{\sc Conclusion of the above discussion}
This concludes our preliminary study of the orbits and shows that the embedded Starobinsky-like models described in section \ref{generalonuovo} exhaust the list of possible embeddings, the values of $\alpha\, = \, 1, \frac{2}{3} ,\frac{1}{3}$ being, apparently the only admissible ones. Next let us observe that the branching rules of the $\mathbf{14}^\prime$ dimensional representation which lead to consistent truncations, namely, (\ref{frikandello4},\ref{frikandello5},\ref{frikandello6}) are the only possible ones that we can obtain by embedding:
\begin{equation}\label{grullonevero}
\mathfrak{sl}(2) \, \mapsto \, \mathfrak{sl}(2) \, \times \, \mathfrak{sl}(2) \,\times \, \mathfrak{sl}(2)
\end{equation}
if the considered $\mathbf{14}^\prime$ representation of $\mathfrak{sl}(2) \, \times \, \mathfrak{sl}(2) \,\times \, \mathfrak{sl}(2)$ is the following one:
\begin{equation}\label{cannoleccio}
\mathbf{14}^\prime \, = \, \left(\frac{1}{2} \, , \, \frac{1}{2} \, , \, \frac{1}{2}\right)\, \oplus \, \left(\frac{1}{2},0,0\right) \, \oplus \,
\left(0, \frac{1}{2},0\right) \, \oplus \, \left(0, 0, \frac{1}{2}\right)
\end{equation}
This has a profound meaning. It implies that the only consistent truncations occur when the $\mathfrak{sl}(2)$ Lie algebra is embedded in the \textit{sub-Tits-Satake} Lie algebra, which as we discuss in the conclusive part is universal for all $\mathcal{N}=2$ models. This allows us to make the bold statement that the only values of $\alpha$ one can obtain form the gauging of hypermultiplet isometries in any supergravity theory based on symmetric manifolds is just $\alpha \, = \, 1, \frac{2}{3} , \, \frac{1}{3}$.
\newpage
\part{\sc Conclusions and perspectives}
\par
Considering the structure of the $c$-map and the results relative to the inclusion of Starobinsky like potentials that we have concretely obtained in the case of the $\mathrm{Sp(6,\mathbb{R})}$-model, we interpret them within the framework provided by Tits-Satake subalgebras and Tits Satake universality classes. This allows us to advocate that the mechanism underlying the generation of such cosmological potentials is universal and the prediction on the possible values of $\alpha$ equally general. Actually the analysis we are going to present in the next section supports the following general conclusion: the cosmological potentials classified in \cite{pietrosergiosasha1}, that follow from the gauging of a generator respectively elliptic, hyperbolic or parabolic of a constant curvature K\"ahler surface admit a universal uplifting to all $\mathcal{N}=2$ models based on symmetric spaces for the hypermultiplets, provided the curvature of the K\"ahler surface is duely quantized. Indeed there exist consistent one-field truncation only if the gauging occurs inside the universal \textit{sub-Tits-Satake} Lie algebra.
\section{\sc The Tits Satake projection}
In most cases of lower supersymmetry, neither the algebra $\mathbb{U}_{\mathcal{SK}}$ nor the
algebra $\mathbb{U}_{\mathcal{QM}}$ are \textbf{maximally split}. In short
this means that the non-compact rank $r_{nc} < r $ is less than
the rank of $\mathbb{U}$, namely not all the Cartan generators are
non-compact. Rigorously $r_{nc}$ is defined as follows:
\begin{equation}
r_{nc}\, = \, \mbox{rank} \left( \mathrm{U/H}\right) \, \equiv \, \mbox{dim} \,
\mathcal{H}^{n.c.} \quad ; \quad \mathcal{H}^{n.c.} \, \equiv \,
\mbox{CSA}_{\mathbb{U}(\mathbb{C})} \, \bigcap \, \mathbb{K}
\label{rncdefi}
\end{equation}
When this happens it means that the structure of both black-hole-like and cosmological-like solutions of supergravity is effectively determined by a
\textit{maximally split subalgebra} $\mathbb{U}^{TS} \subset
\mathbb{U}$ named the \textit{Tits Satake} subalgebra of
$\mathbb{U}$, whose rank is equal to $r_{nc}$. Effectively determined
does not mean that solutions of the big system
coincide with those
of the smaller system rather it means that the former can be obtained from the
latter by means of rotations of a compact subgroup $\mathrm{G_{paint} \subset U}$ of the big group
which we name the \textit{paint group}, for whose precise definition we refer the reader to \cite{titsusataku}.
Here we just emphasize few important facts, relevant for our goals. To this effect we recall that the Tits Satake algebra is obtained from the
original algebra via a projection of the root system of $\mathbb{U}$ onto the subspace orthogonal to the compact part of the Cartan subalgebra
of $\mathbb{U}^{TS}$:
\begin{equation}
\Pi^{TS} \quad ; \quad \Delta_\mathbb{U} \,\mapsto \,
\overline{\Delta}_{\mathbb{U}^{TS}}
\label{Tsproj}
\end{equation}
In euclidian geometry $\overline{\Delta}_{\mathbb{U}^{TS}}$ is just a collection of vectors in $r_{nc}$ dimensions; a priori there is no reason why it should be the root system of another Lie algebra. Yet in almost all cases, $\overline{\Delta}_{\mathbb{U}^{TS}}$ turns out to be a Lie algebra root system and the maximal split Lie algebra corresponding to it, $\mathbb{U}^{TS}$, is, by definition, the Tits Satake subalgebra of the original non maximally split Lie algebra: $\mathbb{U}^{TS} \subset \mathbb{U}$. Such algebras $\mathbb{U}$ are called \textit{non-exotic}. The \textit{exotic} non compact algebras are those for which the system $\overline{\Delta}_{\mathbb{U}^{TS}}$ is not an admissible root system. In such cases there is no Tits Satake subalgebra $\mathbb{U}^{TS}$. Exotic subalgebra are very few and in supergravity they appear only in three instances that display additional peculiarities relevant for the black-hole and cosmological solutions. As for $\mathcal{N}=2$ theories the only exotic homogeneous symmetric Special K\"ahler manifolds are those of the Minimal Coupling series discussed in section \ref{minicoup}. Exotic are also the Quaternionic K\"ahler manifolds in the $c$-map image of the former.
\par
For the non exotic models we have that the decomposition (\ref{gendecompo}) commutes with the projection,
namely:
\begin{equation}
\begin{array}{rcl}
\mbox{adj}(\mathbb{U}_{\mathcal{QM}}) &=&
\mbox{adj}(\mathbb{U}_{\mathcal{SK}})\oplus\mbox{adj}({\mathfrak{sl}(2,\mathbb{R})_E})\oplus
\mathbf{W}_{(2,W)} \\
\null &\Downarrow & \null\\
\mbox{adj}(\mathbb{U}^{TS}_{\mathcal{QM}}) &=&
\mbox{adj}(\mathbb{U}^{TS}_{\mathcal{SK}})\oplus\mbox{adj}({\mathfrak{sl}(2,\mathbb{R})_E})\oplus
\mathbf{W}_{(2,W^{TS})} \\
\end{array}
\label{gendecompo2}
\end{equation}
In other words the projection leaves the $A_1$ Ehlers subalgebra untouched and has a non trivial effect only
on the algebra $\mathbb{U}_{\mathcal{SK}}$. Furthermore the image under the projection of the highest root
of $\mathbb{U}$ is the highest root of $\mathbb{U}^{TS}$:
\begin{equation}
\Pi^{TS} \quad : \quad \psi \, \rightarrow \, \psi^{TS}
\label{Pionpsi}
\end{equation}
The reason why the Tits Satake projection is relevant to us is that the classification of nilpotent orbits (standard triples) and hence of abelian gaugings depends only on the Tits Satake subalgebra and therefore is universal for all members of the same Tits Satake universality class. By this name we mean all algebras that share the same Tits Satake projection. A similar property was extensively used in \cite{noinilpotenti} in the discussion of extremal black-hole solutions. Indeed the classification of extremal black-holes also boils down to the classification of nilpotent orbits, so that the mathematical problem at stake is just the same.
\par
Having clarified these points we can proceed with the classification of homogeneous symmetric spaces relevant to $\mathcal{N}=2$ supergravity either in the vector multiplet sector (Special K\"ahler) or in the hypermultiplet sector (Quaternionic K\"ahler). These spaces are listed in table \ref{homomodelTS}.
\begin{table}
\begin{center}
{\tiny
\begin{tabular}{|l|c|c||c|c||c|c||c||}
\hline
\null & TS & TS & coset &coset & Paint & subP & susy\\
$\#$ & $\mathcal{SK}_{TS}$ & $\mathcal{QM}_{TS}$ & $\mathcal{SK}$ & $\mathcal{QM}$ & Group & Group & \\
\hline
\hline
1 &\null & \null & \null & \null &\null & \null & \null \\
\null & $ \frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}$ & $ \frac{\mathrm{G_{2(2)}}}{\mathrm{SU(2)\times SU(2)}}$ & $ \frac{\mathrm{SU(1,1)}}{\mathrm{U(1)}}$ & $ \frac{\mathrm{G_{2(2)}}}{\mathrm{SU(2)\times SU(2)}}$ & $1$ & $1$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & n=1 \\
\hline
\hline
2 &\null & \null & \null & \null &\null & \null & \null \\
\null & \null & \null & $ \frac{\mathrm{Sp(6,R)}}{\mathrm{SU(3)\times U(1)}}$ & $ \frac{\mathrm{F_{4(4)}}}{\mathrm{USp(6)\times SU(2)}}$ & $1$ & $1$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & $n=6$ \\
\cline{1-1} \cline{4-8}
3 &\null & \null & \null & \null &\null & \null & \null \\
\null & \null & \null & $ \frac{\mathrm{SU(3,3)}}{\mathrm{SU(3)\times SU(3) \times U(1)}}$ & $ \frac{\mathrm{E_{6(2)}}}{\mathrm{SU(6)\times SU(2)}}$ & $\mathrm{SO(2)\times SO(2)}$ & $1$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & $n=9$ \\
\cline{1-1} \cline{4-8}
4 &\null & \null & \null & \null &\null & \null & $\mathcal{N}=6$ \\
\null & $ \frac{\mathrm{Sp(6,R)}}{\mathrm{SU(3)\times U(1)}}$ & $ \frac{\mathrm{F_{4(4)}}}{\mathrm{Sp(6,R)\times SL(2,R)}}$ & $ \frac{\mathrm{SO^\star(12)}}{\mathrm{SU(6)\times U(1)}}$ & $ \frac{\mathrm{E_{7(-5)}}}{\mathrm{SO(12)\times SU(2)}}$ & $\mathrm{SO(3)\times SO(3)}$ & $\mathrm{SO(3)_{d}}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &$\mathrm{\times SO(3)}$ & \null & n=15 \\
\cline{1-1} \cline{4-8}
5 &\null & \null & \null & \null &\null & \null & \null \\
\null & \null & \null & $ \frac{\mathrm{E_{7(-25)}}}{\mathrm{E_{6(-78)} \times U(1)}}$ & $ \frac{\mathrm{E_{8(-24)}}}{\mathrm{E_{7(-133)}\times SU(2)}}$ & $\mathrm{SO(8)}$ & $G_{2(2)}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & $n=27$ \\
\hline
\hline
\hline
6 & \null \null & \null & \null & \null &\null & \null & \null \\
\null & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,1)}}{\mathrm{SO(2)}}$ & $ \frac{\mathrm{SO(4,3)}}{\mathrm{SO(4)\times SO(3)}}$ & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,1)}}{\mathrm{SO(2)}}$ & $ \frac{\mathrm{SO(4,3)}}{\mathrm{SO(4)\times SO(3)}}$ & $\mathrm{1}$ & $\mathrm{1}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & n=2 \\
\hline
7 & \null \null & \null & \null & \null &\null & \null & \null \\
\null & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,2)}}{\mathrm{SO(2)\times SO(2)}}$ & $ \frac{\mathrm{SO(4,4)}}{\mathrm{SO(4)\times SO(4)}}$ & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,2)}}{\mathrm{SO(2)\times SO(2)}}$ & $ \frac{\mathrm{SO(4,4)}}{\mathrm{SO(4)\times SO(4)}}$ & $\mathrm{1}$ & $\mathrm{1}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & n=3 \\
\hline
8 & \null \null & \null & \null & \null &\null & \null & \null \\
\null & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,3)}}{\mathrm{SO(2)\times SO(3)}}$ & $ \frac{\mathrm{SO(4,5)}}{\mathrm{SO(4)\times SO(5)}}$ & $ \frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\times\frac{\mathrm{SO(2,2+p)}}{\mathrm{SO(2)\times SO(2+p)}}$ & $ \frac{\mathrm{SO(4,4+p)}}{\mathrm{SO(4)\times SO(4+p)}}$ & $\mathrm{SO(p)}$ & $\mathrm{SO(p-1)}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null & \null & n=3+p \\
\hline
\hline
\hline
$\mathrm{exot}$&\null & \null & \null & \null &\null & \null & \null \\
\null \null & $ bc_1$ & $ bc_2$ & $ \frac{\mathrm{SU(p+1,1)}}{\mathrm{SU(p+1)\times U(1)}}$ & $ \frac{\mathrm{SU(p+2,2)}}{\mathrm{SU(p+1,1)\times SL(2,R)_{h^\star}}}$ & $\mathrm{U(1)\times U(1) \times U(p)}$ & $\mathrm{U(p-1)}$ & $\mathcal{N}=2$ \\
\null &\null & \null & \null & \null &\null& \null & n=p+1 \\
\hline
\end{tabular}
}
\caption{The first eight rows of this table list the \textit{non-exotic} homogenous symmetric Special K\"ahler manifolds. They are displayed together with their Quaternionic K\"ahler $c$-map images and are organized in five Tits Satake universality classes. Non exotic means that the Tits Satake projection of the root system is a standard Lie Algebra root system. Of the five universality classes three contain only one maximally split element, one contains four elements filling the Tits magic square, while the last class contains an infinite number of elements. Within each class the models are distinguished by the different structure of the Paint Group and of its subPaint subgroup. The Paint group is a $c$-map invariant. It is the same in the Special K\"ahler and in the Quaternionic K\"ahler case. The last line displays the unique family of exotic Special K\"ahler symmetric spaces for which the Tits Satake projection of the root system is not a root system. They correspond to the Minimal Coupling Models discussed in the main text. Their $c$-map images of the exotic models are also exotic. Notwithstanding this anomaly the concept of Paint Group, according to its definition as group of external automorphisms of the solvable Lie algebra generating the non compact coset manifold still exists. The Paint group is the same for the K\"ahler and the quaternionic K\"ahler case as in the non exotic cases. \label{homomodelTS}}
\end{center}
\end{table}
In table \ref{homomodelTS} we have also listed the Paint groups and the subpaint groups. These latter are always compact and their different structures is what distinguishes the different elements belonging to the same class. As it was shown in \cite{titsusataku}, these groups are dimensional reduction invariant and therefore $c$-map invariant, namely they are the same in $\mathrm{U}_{\mathcal{SK}}$ and in $\mathrm{U}_{\mathcal{QM}}$. Hence the representation $\mathbf{W}$ which, as we have seen, hosts the symplectic section of Special Geometry and regulates, by means of its branching, the existence or non existence of consistent truncations, can be decomposed with respect to the Tits Satake subalgebra and the Paint group revealing a regularity structure inside each Tits Satake universality class which is what allows us to draw general conclusions and make universal predictions. In the case of black-holes the same Tits-Satake decomposition of the $\mathbf{W}$-representation is at the heart of the classification of \textit{charge orbits} of the hole, as we extensively discussed in \cite{noinilpotenti}.
\section{\sc Tits Satake Universality classes and the embedding of Starobinsky-like models}
In the present section we consider the decomposition of the $\mathbf{W}$-representations with respect to Tits-Satake subalgebras and Paint groups for all the non-exotic models.
\par
In \cite{titsusataku} the \textit{paint algebra} was defined as the algebra of external automorphisms of the solvable Lie algebra $\mathop{\rm Solv}\nolimits_\mathcal{M}$ generating the non-compact symmetric space: $\mathcal{M}\, = \, \mathrm{U/H}$, namely
\begin{equation}
\mathbb{G}_{\mathrm{paint}} \, = \, \mathrm{Aut}_{\mathrm{Ext}} \, \left[
\mathop{\rm Solv}\nolimits_\mathcal{M}\right]. \label{pittureFuori}
\end{equation}
where:
\begin{equation}
\mathrm{Aut}_{\mathrm{Ext}} \, \left[ \mathop{\rm Solv}\nolimits_\mathcal{M}\right] \,
\equiv \, \frac{\mathrm{Aut} \, \left[
\mathop{\rm Solv}\nolimits_\mathcal{M}\right]}{\mathop{\rm Solv}\nolimits_\mathcal{M}},
\label{outerauto}
\end{equation}
Given the paint algebra $\mathbb{G}_{\mathrm{paint}} \, \subset \, \mathbb{U}$ and the Tits Satake subalgebra $\mathbb{G}_{\mathrm{TS}}\, \subset \, \mathbb{U}$, whose construction we have briefly recalled above, following \cite{titsusataku}
one introduces the \textit{sub Tits Satake} and \textit{sub paint} algebras as the centralizers of the paint algebra and of the Tits Satake algebra, respectively. In other words we have:
\begin{equation}\label{subTs}
\mathfrak{s} \, \in \, \mathbb{G}_{\mathrm{subTS}} \, \subset \, \mathbb{G}_{\mathrm{TS}}\, \subset \, \mathbb{U} \quad \Leftrightarrow \, \quad \left[ \mathfrak{s} \, , \, \mathbb{G}_{\mathrm{paint}}\right] \, = \, 0
\end{equation}
and
\begin{equation}\label{subpaint}
\mathfrak{t} \, \in \, \mathbb{G}_{\mathrm{subpaint}} \, \subset \, \mathbb{G}_{\mathrm{paint}}\, \subset \, \mathbb{U} \quad \Leftrightarrow \, \quad \left[ \mathfrak{t} \, , \, \mathbb{G}_{\mathrm{TS}}\right] \, = \, 0
\end{equation}
A very important property of the paint and subpaint algebras is that they are conserved int the $c$-map, namely they are the same for $\mathbb{U}_{\mathcal{SK}}$ and $\mathbb{U}_{\mathcal{QM}}$.
\par
In the next lines we analyze the decomposition of the $\mathbf{W}$-representations with respect to these subalgebras for each Tits Satake universality class of non maximally split models. In the case of maximally split models there is no paint algebra and there is nothing with respect to which to decompose.
\subsection{\sc Universality class $\mathfrak{sp}(6,\mathbb{R})\Rightarrow \mathfrak{f}_{4(4)}$}
In this case the sub Tits Satake Lie algebra is
\begin{equation}\label{cuffio}
\mathbb{G}_{\mathrm{subTS}} \, = \, \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R}) \subset \mathfrak{sp}(6,\mathbb{R}) \, = \, \mathbb{G}_{\mathrm{TS}}
\end{equation}
and the $\mathbf{W}$-representation of the maximally split model decomposes as follows:
\begin{equation}\label{scompo14}
\mathbf{14}^\prime \, \stackrel{\mathbb{G}_{\mathrm{subTS}}}{\Longrightarrow} \, (\mathbf{2},\mathbf{1},\mathbf{1}) \oplus (\mathbf{1},\mathbf{2},\mathbf{1}) \oplus (\mathbf{1},\mathbf{1},\mathbf{2}) \oplus (\mathbf{2},\mathbf{2},\mathbf{2})
\end{equation}
This decomposition combines in the following way with the paint group representations in the various models belonging to the same universality class.
\subsubsection{\sc $\mathfrak{su}(3,3)$ model}
For this case the paint algebra is
\begin{equation}\label{furadino}
\mathbb{G}_{\mathrm{paint}} \, = \, \mathfrak{so}(2) \oplus \mathfrak{so}(2)
\end{equation}
and the $\mathbf{W}$-representation is the $\mathbf{20}$ dimensional of $\mathfrak{su}(3,3)$ corresponding to an antisymmetric tensor with a reality condition of the form:
\begin{equation}\label{realata}
t_{\alpha\beta\gamma}^\star = \frac{1}{3!} \, \epsilon_{\alpha\beta\gamma\delta\eta\theta} \, t_{\delta\eta\theta}
\end{equation}
The decomposition of this representation with respect to the Lie algebra $ \mathbb{G}_{\mathrm{paint}}\oplus {\mathbb{G}_{\mathrm{subTS}}}$ is the following one:
\begin{equation}\label{ruppatoA}
\mathbf{20} \, \stackrel{\mathbb{G}_{\mathrm{paint}} \oplus \mathbb{G}_{\mathrm{subTS}}}{\Longrightarrow} \, (2,q_1|\mathbf{2},\mathbf{1},\mathbf{1}) \oplus (2,q_2|\mathbf{1},\mathbf{2},\mathbf{1}) \oplus (2,q_3|\mathbf{1},\mathbf{1},\mathbf{2}) \oplus (1,0|\mathbf{2},\mathbf{2},\mathbf{2})
\end{equation}
where $(2,q)$ means a doublet of $\mathfrak{so}(2)\oplus\mathfrak{so}(2)$ with a
certain grading $q$ with respect to the generators, while $(1,0)$
means the singlet that has $0$ grading with respect to both
generators. The subpaint algebra in this case is $
\mathbb{G}_{\mathrm{subpaint}}\, = \,0$ and the decomposition of
the same $\mathbf{W}$-representation with respect to
$\mathbb{G}_{\mathrm{subpaint}} \oplus \mathbb{G}_{\mathrm{TS}}$
is:
\begin{equation}\label{ruppatoB}
\mathbf{20} \, \stackrel{\mathbb{G}_{\mathrm{subpaint}} \oplus \mathbb{G}_{\mathrm{TS}}}{\Longrightarrow} \, \mathbf{6} \, \oplus \, \mathbf{14}
\end{equation}
This follows from the decomposition of the $\mathbf{6}$ of $\mathfrak{sp}(6,\mathbf{R})$ with respect to the sub Tits Satake algebra (\ref{cuffio}):
\begin{equation}\label{seifracco}
\mathbf{6} \, \stackrel{\mathbb{G}_{\mathrm{subTS}}}{\Longrightarrow} \, (\mathbf{2},\mathbf{1},\mathbf{1}) \oplus (\mathbf{1},\mathbf{2},\mathbf{1}) \oplus (\mathbf{1},\mathbf{1},\mathbf{2})
\end{equation}
\subsubsection{\sc $\mathfrak{so}^\star(12)$ model}
For this case the paint algebra is
\begin{equation}\label{furadinotwo}
\mathbb{G}_{\mathrm{paint}} \, = \, \mathfrak{so}(3) \oplus \mathfrak{so}(3) \oplus \mathfrak{so}(3)
\end{equation}
and the $\mathbf{W}$-representation is the $\mathbf{32}_s $
dimensional spinorial representation of $\mathfrak{so}^\star(12)$. The
decomposition of this representation with respect to the Lie
algebra $ \mathbb{G}_{\mathrm{paint}}\oplus
{\mathbb{G}_{\mathrm{subTS}}}$ is the following one:
\begin{equation}\label{ruppatoAtwo}
\mathbf{32}_s \, \stackrel{\mathbb{G}_{\mathrm{paint}} \oplus \mathbb{G}_{\mathrm{subTS}}}{\Longrightarrow} \, (\underline{2},\underline{2},\underline{1}|\mathbf{2},\mathbf{1},\mathbf{1}) \oplus (\underline{2},\underline{1},\underline{2}|\mathbf{1},\mathbf{2},\mathbf{1}) \oplus (\underline{1},\underline{1},\underline{2}|\mathbf{1},\mathbf{1},\mathbf{2}) \oplus (\underline{1},\underline{1},\underline{1}|\mathbf{2},\mathbf{2},\mathbf{2})
\end{equation}
where $\underline{2}$ means the doublet spinor representation of
$\mathfrak{so}(3)$. The subpaint algebra in this case is $
\mathbb{G}_{\mathrm{paint}}\, = \,\mathfrak{so}(3)_{\mathrm{diag}}$ and the
decomposition of the same $\mathbf{W}$-representation with respect
to $\mathbb{G}_{\mathrm{subpaint}} \oplus
\mathbb{G}_{\mathrm{TS}}$ is:
\begin{equation}\label{ruppatoBtwo}
\mathbf{32}_s \, \stackrel{\mathbb{G}_{\mathrm{TS}}\oplus \mathbb{G}_{\mathrm{subpaint}}}{\Longrightarrow} \, (\mathbf{6}|\underline{3}) \, \oplus \, (\mathbf{14}^\prime|\underline{1})
\end{equation}
This follows from the decomposition of the product $\underline{2} \times \underline{2}$ of $\mathfrak{so}(3)_{\mathrm{diag}}$ times the Tits Satake algebra (\ref{cuffio}):
\begin{equation}\label{seifraccotwo}
\underline{2} \times \underline{2} \, = \, \underline{3} \, \oplus \, \underline{1}
\end{equation}
\subsubsection{\sc $\mathfrak{e}_{7(-25)}$ model}
For this case the paint algebra is
\begin{equation}\label{furadinothree}
\mathbb{G}_{\mathrm{paint}} \, = \, \mathfrak{so}(8)
\end{equation}
and the $\mathbf{W}$-representation is the fundamental $\mathbf{56} $ dimensional representation of $\mathfrak{e}_{7(-25)}$
The decomposition of this representation with respect to the Lie algebra $ \mathbb{G}_{\mathrm{paint}}\oplus {\mathbb{G}_{\mathrm{subTS}}}$ is the following one:
\begin{equation}\label{ruppatoAthree}
\mathbf{56} \, \stackrel{\mathbb{G}_{\mathrm{paint}} \oplus \mathbb{G}_{\mathrm{subTS}}}{\Longrightarrow} \, (\mathbf{8}_v|\mathbf{2},\mathbf{1},\mathbf{1}) \oplus (\mathbf{8}_s|\mathbf{1},\mathbf{2},\mathbf{1}) \oplus (\mathbf{8}_c|\mathbf{1},\mathbf{1},\mathbf{2}) \oplus (\mathbf{1}|\mathbf{2},\mathbf{2},\mathbf{2})
\end{equation}
where $\mathbf{8}_{v,s,c}$ are the three inequivalent eight-dimensional representations of $\mathfrak{so}(8)$, the vector, the spinor and the conjugate spinor. The subpaint algebra in this case is $ \mathbb{G}_{\mathrm{paint}}\, = \,\mathfrak{g}_{2(-14)}$ with respect to which all three $8$-dimensional representations of $\mathfrak{so}(8)$ branch as follows:
\begin{equation}\label{seifraccothree}
\mathbf{8}_{v,s,c} \, \stackrel{\mathfrak{g}_{2(-14)}}{\Longrightarrow} \, \mathbf{7 }\, \oplus \, \mathbf{1}
\end{equation}
In view of this the decomposition of the same $\mathbf{W}$-representation with respect to $\mathbb{G}_{\mathrm{subpaint}} \oplus \mathbb{G}_{\mathrm{TS}}$ is:
\begin{equation}\label{ruppatoBthree}
\mathbf{56} \, \stackrel{\mathbb{G}_{\mathrm{TS}} \oplus \mathbb{G}_{\mathrm{subpaint}}}{\Longrightarrow} \, (\mathbf{6}|\mathbf{7}) \, \oplus \, (\mathbf{14}^\prime|\mathbf{1})
\end{equation}
\subsection{\sc Universality class $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(2,3) \Rightarrow \mathfrak{so}(4,5)$}
This case corresponds to one of the possible infinite families of $\mathcal{N}=2$ theories with a symmetric homogeneous special K\"ahler manifold and a number of vector multiplets larger than three ($n=3+p$). The other infinite family corresponds instead to one of the three exotic models.
\par
The generic element of this infinite class corresponds to the following algebras:
\begin{eqnarray}
\mathbb{U}_{\mathcal{SK}} &=& \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(2,2+p)\nonumber\\
\mathbb{U}_{\mathcal{QM}} &=& \mathfrak{so}(4,4+p) \label{feldane}
\end{eqnarray}
In this case the sub Tits Satake algebra is:
\begin{equation}\label{casilino1}
\mathbb{G}_{\mathrm{subTS}} \, = \, \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{sl}(2,\mathbb{R}) \, \simeq \, \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(2,2) \, \, \subset \, \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(2,3) \, = \, \mathbb{G}_{\mathrm{TS}}
\end{equation}
an the paint and subpaint algebras are as follows:
\begin{eqnarray}
\mathbb{G}_{\mathrm{paint}} &=& \mathfrak{so}(p) \nonumber\\
\mathbb{G}_{\mathrm{subpaint}} &=& \mathfrak{so}(p-1) \label{felanina1}
\end{eqnarray}
The symplectic $\mathbf{W}$ representation of $\mathbb{U}_{\mathcal{SK}}$ is the tensor product of the fundamental representation of $\mathfrak{sl}(2)$ with the fundamental vector representation of $\mathfrak{so}(2,2+p)$, namely
\begin{equation}\label{erast1}
\mathbf{W} \, = \, \left( \mathbf{2 | 4}+p\right) \quad ; \quad \mbox{dim} \,\mathbf{W} \, = \, 8+2p
\end{equation}
The decomposition of this representation with respect to $\mathbb{G}_{\mathrm{subTS}} \oplus \mathbb{G}_{\mathrm{subpaint}}$ is the following one:
\begin{equation}\label{decompusOne}
\mathbf{W} \, \stackrel{\mathbb{G}_{\mathrm{subTS}} \oplus \mathbb{G}_{\mathbf{subpaint}}}{\Longrightarrow} \, \left(\mathbf{2,2,2|1}\right) \oplus \left(\mathbf{2,1,1|1}\right) \oplus \left(\mathbf{2,1,1}|p-1\right)
\end{equation}
where $\mathbf{2,2,2}$ denotes the tensor product of the three fundamental representations of $\mathfrak{sl}(2,\mathbb{R})^3$. Similarly $\mathbf{2,1,1}$ denotes the doublet of the first $\mathfrak{sl}(2,\mathbb{R})$ tensored with the singlets of the following two $\mathfrak{sl}(2,\mathbb{R})$ algebras. The representations appearing in (\ref{decompusOne}) can be grouped in order to reconstruct full representations either of the complete Tits Satake or of the complete paint algebras. In this way one obtains:
\begin{eqnarray}
\mathbf{W} & \stackrel{\mathbb{G}_{\mathrm{subTS}} \oplus \mathbb{G}_{\mathbf{paint}}}{\Longrightarrow} & \left(\mathbf{2,2,2|1}\right) \oplus \left(\mathbf{2,1,1}|p+1\right)\nonumber\\
\mathbf{W} & \stackrel{\mathbb{G}_{\mathrm{TS}} \oplus \mathbb{G}_{\mathbf{subpaint}}}{\Longrightarrow} & \left(\mathbf{2,5|1}\right) \oplus \left(\mathbf{2,1}|p-1\right)
\end{eqnarray}
\subsection{\sc $\mathbf{W}$-representations of the maximally split non exotic models}
In the previous subsections we have analyzed the Tits-Satake decomposition of the $\mathbf{W}$-representation for all those models that are non maximally split. The remaining models are the maximally split ones for which there is no paint algebra and the Tits Satake projection is the identity map. There are essentially four type of models:
\begin{enumerate}
\item The $\mathrm{SU(1,1)}$ non exotic model where the $\mathbf{W}$-representation is the $j=\ft 32$ of $\mathfrak{so}(1,2)\sim \mathfrak{su}(1,1)$
\item The $\mathrm{Sp(6,\mathbb{R})}$ model where the $\mathbf{W}$-representation is the $\mathbf{14}^\prime$ (antisymmetric symplectic traceless three-tensor).
\item The models $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(q,q)$ where the $\mathbf{W}$-representation is the $\left(\mathrm{2,2q}\right)$, namely the tensor product of the two fundamentals.
\item The models $\mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{so}(q,q+1)$ where the $\mathbf{W}$-representation is the $\left(\mathrm{2,2q+1}\right)$, namely the tensor product of the two fundamentals.
\end{enumerate}
Therefore, for the above maximally split models, we need the classification of $\mathrm{U_{\mathcal{SK}}}$ orbits in the mentioned $\mathbf{W}$-representations. Actually such orbits are sufficient also for the non maximally split models. Indeed each of the above $4$-models correspond to one Tits Satake universality class
and, within each universality class, the only relevant part of the $\mathbf{W}$-representation is the subpaint group singlet which is universal for all members of the class. This is precisely what we verified in the previous subsections.
\par
For instance for all members of the universality class of $\mathrm{Sp(6,\mathbb{R})}$, the $\mathbf{W}$-representation splits as follows with respect to the subalgebra $\mathfrak{sp}(6,\mathbb{R})\oplus \mathbb{G}_{\mathrm{subpaint}}$:
\begin{equation}\label{gribochky}
\mathbf{W} \, \stackrel{\mathfrak{sp}(6,\mathbb{R})\oplus \mathbb{G}_{\mathrm{subpaint}}}{\Longrightarrow} \, \left( \mathbf{6}\, | \, \mathcal{D}_{\mathrm{subpaint}}\right) \, + \, \left( \mathbf{14}^\prime \, | \, \mathbf{1}_{\mathrm{subpaint}}\right)
\end{equation}
where the representation $\mathcal{D}_{\mathrm{subpaint}}$ is the following one for the three non-maximally split members of the class:
\begin{equation}\label{reppisubpitturi}
\mathcal{D}_{\mathrm{subpaint}} \, = \, \left\{ \begin{array}{ccccc}
\mathbf{1} & \mbox{of} & \mathbf{1} & \mbox{for the} & \mathfrak{su}(3,3)-\mbox{model} \\
\mathbf{3} & \mbox{of} & \mathfrak{so}(3) & \mbox{for the} & \mathfrak{so}^\star(12)-\mbox{model} \\
\mathbf{7} & \mbox{of} & \mathfrak{g}_{2(-14)} & \mbox{for the} & \mathfrak{e}_{7(-25)}-\mbox{model}\\
\end{array}\right.
\end{equation}
Clearly the condition:
\begin{equation}\label{condosputta}
\left( \mathbf{6}\, | \, \mathcal{D}_{\mathrm{subpaint}}\right)\, = \, 0
\end{equation}
imposed on a vector in the $\mathbf{W}$-representation breaks the group $\mathrm{U}_{\mathcal{SK}}$ to its Tits Satake subgroup. The key point is that each $\mathbf{W}$-orbit of the big group $\mathrm{U}_{\mathcal{SK}}$ crosses the locus (\ref{condosputta}) so that the classification of $\mathrm{Sp(6,\mathbb{R})}$ orbits in the $\mathbf{14}^\prime$-representation exhausts the classification of $\mathbf{W}$-orbits for all members of the universality class.
\par
In order to prove that the gauge (\ref{condosputta}) is always reachable it suffices to show that the representation $\left( \mathbf{6}\, | \,\mathcal{D}_{\mathrm{subpaint}}\right )$ always appears at least once in the decomposition of the Lie algebra $\mathbb{U}_{\mathcal{SK}}$ with respect to the subalgebra $\mathfrak{sp}(6,\mathbb{R})\oplus \mathbb{G}_{\mathrm{subpaint}}$. The corresponding parameters of the big group can be used to set to zero the projection of the $\mathbf{W}$-vector onto $\left( \mathbf{6}\, | \,\mathcal{D}_{\mathrm{subpaint}}\right )$.
\par
The required condition is easily verified since we have:
\begin{eqnarray}
\underbrace{\mbox{adj}\, \mathfrak{su}(3,3)}_{\mathbf{35}} &\stackrel{\mathfrak{sp}(6,\mathbb{R})}{\Longrightarrow} \,& \underbrace{\mbox{adj}\, \mathfrak{sp}(6,\mathbb{R})}_{\mathbf{21}} \, \oplus \, \mathbf{6} \, \oplus \, \mathbf{6} \, \oplus \, \mathbf{1} \, \oplus \, \mathbf{1}\nonumber \\
\underbrace{\mbox{adj}\, \mathfrak{so}^\star(12)}_{\mathbf{66}} &\stackrel{\mathfrak{sp}(6,\mathbb{R})\oplus \mathfrak{so}(3)}{\Longrightarrow} \,& \underbrace{\mbox{adj}\, \mathfrak{sp}(6,\mathbb{R})}_{\mathbf{21}} \, \oplus \, \underbrace{\mbox{adj}\, \mathfrak{so}(3)}_{\mathbf{3}}\,
\oplus \, \left(\mathbf{6} , \mathbf{3}\right)\, \oplus \, \left(\mathbf{6} , \mathbf{3}\right)\oplus \, \left(\mathbf{1},\mathbf{3}\right)\,\oplus \, \left(\mathbf{1},\mathbf{3}\right)\,\nonumber \\
\underbrace{\mbox{adj}\, \mathfrak{e}_{7(-25)}}_{\mathbf{133}} &\stackrel{\mathfrak{sp}(6,\mathbb{R})\oplus \mathfrak{g}_{2(-14)}}{\Longrightarrow} \,& \underbrace{\mbox{adj}\, \mathfrak{sp}(6,\mathbb{R})}_{\mathbf{21}} \, \oplus \, \underbrace{\mbox{adj}\, \mathfrak{g}_{2(-14)}}_{\mathbf{14}}\,
\oplus \, \left(\mathbf{6} , \mathbf{7}\right)\, \oplus \, \left(\mathbf{6} , \mathbf{7}\right)\oplus \, \left(\mathbf{1},\mathbf{7}\right)\,\oplus \, \left(\mathbf{1},\mathbf{7}\right)\,\nonumber \\
\end{eqnarray}
The reader cannot avoid being impressed by the striking similarity of the above decompositions which encode the very essence of Tits Satake universality. Indeed the representations of the common Tits Satake subalgebra appearing in the decomposition of the adjoint are the same for all members of the class. They are simply uniformly assigned to the fundamental representation of the subpaint algebra which is different in the three cases. The representation $\left( \mathbf{6}\, | \,\mathcal{D}_{\mathrm{subpaint}}\right )$ appears twice in these decompositions and can be used to reach the gauge (\ref{condosputta}) as we claimed above.
\paragraph{\sc Holomorphic consistent truncations} The next point to remark is that the condition (\ref{condosputta}) has another important interpretation if applied to the holomorphic section of special geometry. The key point is the following numerical identity valid for all members of the universality class:
\begin{equation}\label{curiosona}
\mbox{dim} \, \frac{\mathrm{U}_{\mathcal{SK}}}{\mathrm{H}_{\mathcal{SK}}}\, = \,
\mbox{dim} \, \frac{\mathrm{U}^{TS}_{\mathcal{SK}}}{\mathrm{H}^{TS}_{\mathcal{SK}}}\, \oplus \, 6 \, \times \, \mbox{dim}\, \mathcal{D}_{\mathrm{subpaint}}
\end{equation}
This means that if we decompose the symplectic section of the big group according to the Tits-Satake subalgebra and we impose on it the condition (\ref{condosputta}) we just obtain the right number of holomorphic constraints to project onto the submanifold $\frac{\mathrm{U}_{\mathcal{SK}}}{\mathrm{H}_{\mathcal{SK}}}$. At the level of field equations this is certainly a consistent truncation, since we project onto the singlets of the subpaint group.
\par
On the other hand if we decompose the $\mathbf{W}$-representation with respect to the sub-Tits-Satake subalgebra $\mathfrak{sl}(2)\times \mathfrak{sl}(2) \times \mathfrak{sl}(2)$ we have the branching rule:
\begin{equation}\label{fillopona}
\mathbf{W} \, \rightarrow \, \left(\mathcal{D}_1 |\mathbf{2},\mathbf{1},\mathbf{1}\right)\, \oplus \, \left(\mathcal{D}_2 |\mathbf{1},\mathbf{2},\mathbf{1}\right)\, \oplus \,\left(\mathcal{D}_3 |\mathbf{1},\mathbf{1},\mathbf{2}\right) \, \oplus \, \left(\mathbf{1} |\mathbf{2},\mathbf{2},\mathbf{2}\right)
\end{equation}
where $\mathcal{D}_{1,2,3}$ are three suitable representations of the Paint Group. Imposing on the symplectic section of the big model
the constraints:
\begin{eqnarray}
\left(\mathcal{D}_1 |\mathbf{2},\mathbf{1},\mathbf{1}\right)&=& 0 \nonumber\\
\left(\mathcal{D}_2 |\mathbf{1},\mathbf{2},\mathbf{1}\right) &=& 0 \nonumber\\
\left(\mathcal{D}_3 |\mathbf{1},\mathbf{1},\mathbf{2}\right) &=& 0
\end{eqnarray}
yields precisely the correct number of holomorphic constraints that restrict the considered Special K\"ahler manifold to the Special K\"ahler manifold of the STU-model namely to $\left(\frac{\mathrm{SL(2,\mathbb{R})}}{\mathrm{SO(2)}}\right)^3$. This follows from the numerical identity true for all members of the universality class:
\begin{equation}\label{guttallaxa}
\mbox{dim} \, \frac{\mathrm{U}_{\mathcal{SK}}}{\mathrm{H}_{\mathcal{SK}}}\, = \, \sum_{i=1}^3 \, 2 \, \times \, \mbox{dim} \, \mathcal{D}_i \, + \, 6
\end{equation}
The reason why the truncation to the STU-model is always a consistent truncation at the level of field equations is obvious in this set up. It corresponds to the truncation to the Paint Group singlets.
\paragraph{\sc $\mathbf{W}$-representations for the remaining models}
For the models of type $\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+p)$ having $\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+1)$ as Tits Satake subalgebra and $\mathfrak{so}(p-1)$ as subpaint algebra the decomposition of the $\mathbf{W}$-representation is the following one:
\begin{equation}\label{funghifritti}
\mathbf{W} \, = \, \left(\mathbf{2,2q+p}\right) \, \stackrel{\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+1)\oplus \mathfrak{so}(p-1)}{\Longrightarrow} \, \left(\mathbf{2,2q+1}|\mathbf{1}\right) \, \oplus \, \left(\mathbf{2,1}|\mathbf{p-1}\right)
\end{equation}
and the question is whether each $\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+p)$ orbit in the $\left(\mathbf{2,2q+p}\right)$ representation intersects the $\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+1)\oplus \mathfrak{so}(p-1)$-invariant locus:
\begin{equation}\label{fittone}
\left(\mathbf{2,1}|\mathbf{p-1}\right) \, = \,0
\end{equation}
The answer is yes since we always have enough parameters in the coset
\begin{equation}\label{ciabatta}
\frac{\mathrm{SL(2,\mathbb{R})}\times \mathrm{SO(q,q+p)}}{\mathrm{SL(2,\mathbb{R})\times SO(q,q+1)\times SO(p-1})}
\end{equation}
to reach the desired gauge (\ref{fittone}). Indeed let us observe the decomposition:
\begin{equation}\label{fruttodimare}
\mbox{adj}\, \left[\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(q,q+p)\right] \, = \, \mbox{adj}\, \left[\mathfrak{sl}(2,\mathbb{R})\right] \oplus \mbox{adj}\, \left[\mathfrak{so}(q,q+1)\right] \, \oplus \mbox{adj}\, \left[\mathfrak{so}(p-1)\right] \, \oplus \, \left(\mathbf{1,2q+1 | p-1}\right)
\end{equation}
The $2q+1$ vectors of $\mathfrak{so}(p-1)$ appearing in (\ref{fruttodimare}) are certainly sufficient to set to zero the $2$ vectors of $\mathfrak{so}(p-1)$ appearing in $\mathbf{W}$.
\par
Relevant for the case of $\mathcal{N}\,=2\,$ supersymmetry is the value $q=2$ and in this case the sub-Tits-Satake Lie algebra is :
\begin{equation}\label{grilloparlante}
\mathbb{G}_{\mathrm{subTS}} \, = \, \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(2,2) \, = \, \mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})
\end{equation}
namely it is once again the Lie algebra of the STU-model. Reduction to the STU-model is consistent for the same reason as in the other universality classes: it corresponds to truncation to Paint Group singlets.
\subsection{\sc Gaugings with consistent one-field truncations}
On the basis of the analysis presented in the previous section we arrive at the following conclusion. By gauging a nilpotent element of the isometry subalgebra of $\mathcal{SK}$ inside $\mathcal{QM}$ we generate a potential. The structure of the theory depends on the nilpotent orbit, namely on the embedding of an $\mathfrak{sl}(2)$ Lie algebra in $\mathbb{U}_{\mathcal{SK}}$ and there are many ways of doing this (the orbits), yet the gauged theory will admit a one-field truncation if and only if the $\mathfrak{sl}(2)$ is embedded into the sub Tits Satake Lie algebra:
\begin{equation}\label{gongolato}
\mathfrak{sl}(2) \, \mapsto \, \mathbb{G}_{\mathrm{subTS}}\, \subset \, \mathbb{U}_{\mathcal{SK}}
\end{equation}
There are only three different embeddings of $\mathfrak{sl}(2)$ into $\left( \mathfrak{sl}(2) \right)^3$ and these correspond to the three admissible values $\alpha\, = \, 1 , \, \frac{2}{3}, \, \frac{1}{3}$ in the Starobinsky-like model.
\section{\sc Conclusions}
In this paper we have analyzed in detail the structure of the $c$-map from Special K\"ahler manifolds to Quaternionic K\"ahler manifolds in connection with the abelian gaugings of hypermultiplet isometries in $\mathcal{N}=2$ supergravity and the generation of effective one field potentials that might describe inflaton dynamics.
\par
The motivations of such a study have been put forward in the introduction. Here we try to summarize the main results we have obtained and the perspectives for future investigations that have emerged.
\paragraph{\sc Results}
\begin{description}
\item[I] The $c$-map description of Quaternionic K\"ahler manifolds $\mathcal{QM}_{4n+4}$ allows to distribute the isometries of
$\mathcal{QM}_{4n+4}$ into three classes:
\begin{description}
\item[Ia)] The isometries associated with the Heisenberg algebra which exists in all cases, even when the internal Special K\"ahler manifold $\mathcal{SK}_{n}$ has no continuous symmetries as it might happen when we deal the moduli space of a Calabi-Yau three-fold and the Heisenberg fields are the Ramond--Ramond scalars of a superstring compactification. The gauging of such perturbative isometries produces one field potentials that are of the pure exponential type.
\item[Ib)] The isometries of the inner Special K\"ahler manifold $\mathcal{SK}_{n}$ that can always be promoted to isometries of the full Quaternionic manifold $\mathcal{QM}_{4n+4}$. Gauging these isometries one obtains a potential that at $\mathbf{Z}^\alpha \, =\, a \, = \, U \, = \, 0$ is identical with the potential obtained from the gauging of the same K\"ahler isometries in $\mathcal{N}=1$ supersymmetry. Hence one would like to be able to stabilize these fields. As for $\mathbf{Z}^\alpha \, =\, a \, = \, 0$ there is no problem. We can always truncate them. The field $U$ instead, that in superstring interpretations of the hypermultiplets can be identified with the coupling constant dilaton field, appears through exponentials all of the same sign in these perturbative gaugings and cannot be stabilized.
\item[Ic)] The non perturbative isometries that mix the Heisenberg symmetries with the K\"ahler symmetries. These exist only when $\mathcal{QM}_{4n+4}$ is a symmetric homogeneous space and the Lie algebra of the full isometry group takes the universal form (\ref{genGD3}). Gauging a compact non perturbative generator appears to be the only way of introducing exponentials with opposite sign of the field $U$ allowing for its stabilization.
\end{description}
\item[II] The Starobinsky-like potential:
\begin{equation}\label{bambolone2}
V_{Starobinsky} \, = \, \mbox{const} \, \times \, \left( 1\, - \, \exp\left [ - \, \sqrt{\frac{2}{3 \, \alpha}} \, \phi\right]\right)^2
\end{equation}
can be obtained universally from all homogeneous symmetric Quaternionic K\"ahler manifolds by means of an admissible embedding of a $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra into the Special K\"ahler subalgebra $\mathbb{U}_\mathcal{SK}\,\subset \, \mathbb{U}_\mathcal{Q}$. The problem of embedding of $\mathfrak{sl}(2,\mathbb{R})$ algebras is the same as the problem of classifying standard triples and nilpotent algebras, yet the embedding must also be admissible, in the sense that it should allow for consistent one-field truncations. In the case of $\mathcal{SK} \, = \, \mathfrak{sp}(6,\mathbb{R})$ we exhausted the analysis of orbits and showed that admissible subalgebras correspond to embeddings of $\mathfrak{sl}(2)$ into the maximal subalgebra $\left(\mathfrak{sl}(2)\right)^3$ yielding $\alpha \, = \, 1, \, \frac{2}{3}, \, \frac{1}{3}$. By means of arguments based on Tits Satake universality classes, we have advocated that this result is general for all supergravity theories where the hypermultiplets are described by a homogeneous symmetric space.
\item[III] The above results rely on the use of the minimal coupling Special Geometry for the description of the vector multiplets. Staying within such a framework one can gauge by different vector fields the generators of a maximal set of mutually commuting $\mathfrak{sl}(2,\mathbb{R})$ Lie algebras and obtain as many massive fields. The number of massive vector fields appears therefore to be equal to the rank of the Tits Satake subalgebra $\mathbb{G}_{\mathrm{subTS}} \, \subset \, \mathbb{U}_\mathcal{SK}\,\subset \, \mathbb{U}_\mathcal{Q}$. The result of \cite{thesearch} where it was found that the maximal number of massive vector fields in this sort of gauging is one is confirmed by this general rule. In \cite{thesearch} we have $\mathbb{G}_{\mathrm{subTS}}\, =\, \mathfrak{sl}(2,\mathbb{R}) \, = \, \mathbb{U}_\mathcal{SK}\,\subset \, \mathbb{U}_\mathcal{Q} \, \equiv \, \mathfrak{g}_{(2,2)}$ and $\mbox{rank}\,\mathfrak{sl}(2,\mathbb{R})\, = \, 1$. In all other cases the rank of the sub Tits Satake Lie algebra is three.
\end{description}
\paragraph{\sc Perspectives and Generalizations}
In order to improve our understanding of the possible embedding of inflaton dynamics within larger unified theories, we consider necessary and feasible to explore the following directions.
\begin{description}
\item[A)] Enlarge the scope of gaugings of hypermultiplet isometries to non abelian algebras and consider the attractive situation where the Special K\"ahler manifold describing the vector multiplets and that, whose $c$-map provides the Quaternionic K\"ahler manifold description of the hypermultiplets are just the same or at least belong to the same class of homogeneous Special K\"ahler spaces.
\item[B] Utilize the above $\mathcal{N}=2$ set up to promote the embedding of inflaton dynamics from $\mathcal{N}=2$ to higher $\mathcal{N}$, in particular to $\mathcal{N}=3$ and $\mathcal{N}=4$.
\item[C] Consider such scenarios as superstring compactification on $T^2 \times K3$ and try to interpret the gaugings that produce inflaton dynamics in terms of fluxes.
\end{description}
We plan to address such questions in forthcoming publications.
\par
\section*{Acknowledgements}
We are grateful to our friends Sergio Ferrara and Antoine Van Proyen for useful discussions during the completion of this work.
The work of A.S. was supported in part by the RFBR Grants
No. 13-02-91330-NNIO-a, No. 13-02-90602-Arm-a and by the
Heisenberg-Landau program.
\newpage
| {
"timestamp": "2014-07-28T02:10:54",
"yymm": "1407",
"arxiv_id": "1407.6956",
"language": "en",
"url": "https://arxiv.org/abs/1407.6956",
"abstract": "In this paper we address the general problem of including inflationary models exhibiting Starobinsky-like potentials into (symmetric) $\\mathcal{N}=2$ supergravities. This is done by gauging suitable abelian isometries of the hypermultiplet sector and then truncating the resulting theory to a single scalar field. By using the characteristic properties of the global symmetry groups of the $\\mathcal{N}=2$ supergravities we are able to make a general statement on the possible $\\alpha$-attractor models which can obtained upon truncation. We find that in symmetric $\\mathcal{N}=2$ models group theoretical constraints restrict the allowed values of the parameter $\\alpha$ to be $\\alpha=1,\\,\\frac{2}{3},\\, \\frac{1}{3}$. This confirms and generalizes results recently obtained in the literature. Our analysis heavily relies on the mathematical structure of symmetric $\\mathcal{N}=2$ supergravities, in particular on the so called $c$-map connection between Quaternionic Kähler manifolds starting from Special Kähler ones. A general statement on the possible consistent truncations of the gauged models, leading to Starobinsky-like potentials, requires the essential help of Tits Satake universality classes. The paper is mathematically self-contained and aims at presenting the involved mathematical structures to a public not only of physicists but also of mathematicians. To this end the main mathematical structures and the general gauging procedure of $\\mathcal{N}=2$ supergravities is reviewed in some detail.",
"subjects": "High Energy Physics - Theory (hep-th)",
"title": "The $c$-map, Tits Satake subalgebras and the search for $\\mathcal{N}=2$ inflaton potentials",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860382,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7092019558481115
} |
https://arxiv.org/abs/1211.5218 | Multi-frequency Calderon-Zygmund analysis and connexion to Bochner-Riesz multipliers | In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderon-Zygmund analysis introduced in [9]. We set a definition of general multi-frequency Calderon-Zygmund operator. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [9]. Involving a new kind of maximal sharp function, weighted estimates are obtained. | \section{Notations and preliminaries}
Let us consider the Euclidean space $\rn$ equipped with the Lebesgue measure $dx$ and its Euclidean distance $|x-y|$. Given a ball $Q\subset\rn$ we denote its center by $c(Q)$ and its radius by $r_Q$. For any $\lambda>1$, we denote by $\lambda\,Q:=B(c(Q),\lambda r_Q)$.
We write $L^p$ for $L^p(\rn,\re)$ or $L^p(\rn,{\mathbb C})$. For a subset $E\subset \rn$ of finite and non-vanishing measure and $f$ a locally integrable function, the average
of $f$ on $E$ is defined by $$ \aver{E} f dx := \frac{1}{|E|}\int_E f(x) dx.$$
Let us denote by ${\mathcal Q}$ the collection of all balls in $\rn$. We write $\M$ for the maximal Hardy-Littlewood function:
$$
{\mathcal M} f(x)= \sup_{{\genfrac{}{}{0pt}{}{Q\in{\mathcal Q}}{x\in Q}}} \aver{Q}|f|dx.$$
For $p\in (1,\infty)$, we set $\M_p f(x)=\M(|f|^p)(x)^{1/p}$. The Fourier transform will be denoted by $\mathcal F$ as an operator and we make use of the other usual notation ${\mathcal F}(f)=\widehat{f}$ too.
\mb
In the current work, we aim to develop a multi-frequency analysis, based on the following lemma:
\begin{lemma}[\cite{BE}] \label{lem} Let $\Theta\subset \R^n$ be a finite collection of frequencies and $Q$ be a ball. For every function $\phi$ belonging to the subspace of $L^2(3Q)$, spanned by $(e^{i\xi\cdot})_{\xi\in \Theta}$, we have for $p\in[1,2]$
\begin{equation} \|\phi\|_{L^\infty(Q)} \lesssim (\sharp \Theta)^{\frac{1}{p}} \left(\aver{3Q} |\phi|^p dx \right)^{\frac{1}{p}}. \label{aze} \end{equation}
\end{lemma}
\begin{remark} In \cite{BE}, this lemma is stated and proved in a one-dimensional setting. However, the proof only relies on the additive group structure of the ambient space by using translation operators. So the exact same proof can be extended to a multi-dimensional setting.
\end{remark}
\begin{remark} \label{rem:lem} The question of extending the previous lemma for $p\in(2,\infty)$ is still open in such a general situation. Of course, (\ref{aze}) is true for $p=\infty$ and so it would be reasonable to expect the result for intermediate exponents $p \in(2,\infty)$. Unfortunately, the well-known interpolation theory does not apply here. \\
However, in some specific situations, we may extend this lemma for $p\geq 2$. Indeed, if $p=2k$ is an even integer then applying (\ref{aze}) with $p=2$ and $\Theta^k:=\{\theta_{i_1}+...+\theta_{i_k},\ \theta_i\in \Theta\}$ to $\phi^k$ yields
\begin{align*}
\|\phi \|_{L^\infty(Q)} & \lesssim \|\phi^k \|_{L^\infty(Q)}^{\frac{1}{k}} \\
& \lesssim (\sharp \Theta^k )^{\frac{1}{2k}} \left(\aver{3Q} |\phi|^{2k} dx \right)^{\frac{1}{2k}} \\
& \simeq (\sharp \Theta^k )^{\frac{1}{p}} \left(\aver{3Q} |\phi|^{p} dx \right)^{\frac{1}{p}}.
\end{align*}
By this way, we see that an extension of (\ref{aze}) for $p\geq 2$ may be related to sharp combinatorial arguments, to estimate $\sharp \Theta^k$ (a trivial bound is $\sharp \Theta^k \leq (\sharp \Theta)^{k}$ which does not improve (\ref{aze})).
\end{remark}
We aim to obtain weighted estimates, involving Muckenhoupt's weights.
\begin{dfn} \label{def:1} A weight $\omega$ is a non-negative locally integrable function.
We say that a weight $\omega\in {\mathbb A}_p$, $1<p<\infty$, if there exists a positive constant $C$ such that for every ball $Q$,
$$\bigg(\aver{Q} \omega\,dx\bigg)\, \bigg(\aver{Q} \omega^{1-p'}\,dx\bigg)^{p-1}\le C.
$$
For $p=1$, we say that $\omega\in {\mathbb A}_1$ if there is a positive constant $C$ such
that for every ball $Q$,
$$
\aver{Q} \omega \,dx
\le
C\, \omega (y),
\qquad \mbox{for a.e. }y\in Q.
$$
We write ${\mathbb A}_\infty=\cup_{p\ge 1} {\mathbb A}_p$.
\end{dfn}
We just recall that for $p\in(1,\infty)$, the maximal function $\M$ is bounded on $L^p(\omega)$ if and only if $\omega \in {\mathbb A}_p$.
We also need to introduce the reverse H\"older classes.
\begin{dfn} \label{def:2} A weight $\omega \in RH_p$, $1<p<\infty$, if there is a constant $C$ such that for every ball $Q$,
$$
\bigg( \aver{Q} \omega^p\,dx \bigg)^{1/p} \leq C \bigg(\aver{Q} \omega\,dx\bigg).
$$
It is well known that ${\mathbb A}_\infty=\cup_{r>1} RH_r$. Thus, for $p=1$ it is understood that $RH_1={\mathbb A}_\infty$.
\end{dfn}
\subsection{Examples of multi-frequency Calder\'on-Zygmund operators}
Let us detail particular situations where such multi-frequency operators appear.
\subsubsection*{The multi-frequency Hilbert transform}
As explained in the introduction, an example of such multi-frequency operators in the $1$-dimensional setting is the multi-frequency Hilbert transform. In $\R$, consider an arbitrary collection of frequencies $\Theta:=(\xi_1,...,\xi_N)$ (assumed to be indexed by the increasing order $\xi_1< \xi_2< \cdots < \xi_N$). The associated multi-frequency Hilbert transform is the Fourier multiplier corresponding to the symbol
$$ m(\xi) = \left\{ \begin{array}{ll}
-1, & \xi<\xi_1 \\
(-1)^{j+1}, & \xi_j<\xi<\xi_{j+1} \\
(-1)^{N+1}, & \xi>\xi_N.
\end{array} \right. $$
Associated to $\Theta$, we have a collection of disjoint intervals $\Delta:=\{ (-\infty,\xi_1), (\xi_1,\xi_2),...,(\xi_N,\infty)\}$. It is well-known by Rubio de Francia's work \cite{Rubio} that for $q\in(1,2]$, the functional
\begin{equation} f \rightarrow \left( \sum_{\omega\in\Delta} \left|{\mathcal F}^{-1}[{\bf 1}_{\omega} {\mathcal F}f]\right|^q \right)^{\frac{1}{q}} \label{eq:f} \end{equation}
is bounded on $L^p$ for $p\in(q',\infty)$.
The boundedness of the multi-frequency Hilbert transform is closely related to the understanding of (\ref{eq:f}) for $q\to 1$.
\mb
We point out that in Rubio de Francia's result, the obtained estimates do not depend on the collection of intervals $\Delta$. More precisely, excepted the end-point $p=q'$, the range $(q',\infty)$ is optimal for a uniform (with respect to the collection $\Delta$) $L^p$-boundedness of (\ref{eq:f}). So it is natural that for $q\to 1$ things are more difficult, which is illustrated by our multi-frequency Calder\'on-Zygmund analysis. Indeed, for example if one considers the particular case $\Theta:=(1,...,N)$, then following the notations of Remark \ref{rem:lem}, we have $\Theta^k =\{k,...,kN\}$ and so $\sharp \Theta^k = k(N-1)+1 \simeq k N$. Hence, in this situation we have observed (see Remark \ref{rem:lem}) that we can extend Lemma \ref{lem} to exponents $p\in[1,\infty]$ (the implicit constant appearing in (\ref{aze}) is only depending on $p$). By this way, Theorem \ref{thm:weight} can be improved and we obtain a better exponent
$$ \gamma=\frac{tp}{s^2}+\left|\frac{1}{2}-\frac{1}{s}\right|.$$
Consequently, it seems that for the $L^p$-boundedness of the multi-frequency Hilbert transform, the collection $\Theta$ could play an important role (which was not the case for the $\ell^q$-functional (\ref{eq:f}) with $q'<p$).
\subsubsection*{Multi-frequency operators coming from a covering of the frequency space}
Let $(Q_j)_{j=1,...,N}$ be a family of disjoint cubes and $\phi_j$ a smooth function with $\widehat{\phi_j}$ supported and adapted to $Q_j$. Then consider the linear operator given by
$$ T(f) = \sum_{j=1}^N \phi_j \ast f.$$
It is easy to check that $T$ is a multi-frequency Calder\'on-Zygmund operator, associated to the collection $\Theta:=(\xi_1,...,\xi_N)$ where for every $j$, $\xi_j:=c(Q_j)$ is the center of the ball $Q_j$. With $r_j$ the radius of $Q_j$, we have the regularity estimate
$$ \sum_{j=1}^N \left|\nabla_{(x,y)}\ e^{i\xi_j\cdot(x-y)} \phi_j(x-y)\right| \lesssim |x-y|^{-n-1} \sum_{j=1}^N \frac{(r_j |x-y|)^{n+1}}{(1+r_j|x-y|)^{M}},$$
for every integer $M>0$.
So boundedness of $T$ (Theorem \ref{thm:boundedness}) yields the inequality
\begin{equation} \left\| \sum_{j=1}^N \phi_j \ast f \right\|_{L^p} \lesssim C(r_1,...,r_N) N^{\left|\frac{1}{p}-\frac{1}{2}\right|} \|f\|_{L^p}, \label{eq:ex} \end{equation}
with
$$ C(r_1,...,r_N):= \sup_{t>0} \sum_{j=1}^N \frac{(r_j t)^{n+1}}{(1+r_jt)^{M}}.$$
Let us examine some particular situations:
\begin{itemize}
\item If the cubes $(Q_j)_j$ have an equal side-length, then as for Proposition \ref{prop}, simple arguments imply (\ref{eq:ex}) for $p\in[1,\infty]$ without the constant $C(r_1,...,r_N)$.
\item If the collection $(Q_j)_j$ is dyadic: it exists a point $\xi_0$, $d(Q_j,\xi_0) \simeq r_{Q_j} \simeq 2^j$ then Littlewood-Paley theory implies (\ref{eq:ex}) without the factor $N^{|\frac{1}{p}-\frac{1}{2}|}$ (in this case $C(r_1,...,r_N)\simeq 1$).
\item If the cubes $(Q_j)$ have only the dyadic scale: $r_{Q_j} \simeq 2^j$ (but no assumptions on the centers of the balls) then Littlewood-Paley theory cannot be used. However, our previous results can be applied in this situation and so (\ref{eq:ex}) holds and $C(r_1,...,r_N) \simeq 1$.
\end{itemize}
We aim to use the new multi-frequency Calder\'on-Zygmund analysis to extend these inequalities with replacing the convolution operators by more general Calder\'on-Zygmund operators, still satisfying some orthogonality properties.
\subsubsection*{Multi-frequency operators coming from variation norm estimates}
As explained in the introduction, the multi-frequency Calder\'on-Zygmund analysis has been first developed for proving a variation norm variant of a Bourgain's maximal inequality.
So our results can be adapted in such a framework. For example, in \cite{GMS} Grafakos, Martell and Soria have studied maximal inequalities of the form
$$ \left\| \sup_{j=1,...,N} \left| T( e^{i\theta_j\cdot} f) \right| \right\|_{L^p} \lesssim \|f\|_{L^p}$$
where $(\theta_j)_{j=1,...,N}$ is a collection of frequencies and $T$ a fixed Calder\'on-Zygmund operator.
We can ask the same question, for a variation norm variant: for $q\in[1,\infty)$ consider
$$ \left(\sum_{j=1}^N \left| T( e^{i\theta_j\cdot} f) \right|^q \right)^{\frac{1}{q}}$$
and study its boundedness on $L^p$, with a sharp control of the behaviour with respect to $N$. By a linearization argument (involving Rademacher's functions), this $\ell^q$-functional can be realized as an average of modulated Calder\'on-Zygmund operators, associated to the collection $\Theta:=(\theta_j)_j$.
\section{Unweighted estimates for multi-frequency Calder\'on-Zygmund operators} \label{sec1}
In this section, we aim to prove the weak $L^1$-estimate for a multi-frequency Calder\'on-Zygmund operator, then Theorem \ref{thm:boundedness} will easily follow from interpolation and duality.
\begin{proposition} \label{prop:bound} Let $\Theta=(\xi_1,...,\xi_N)$ be a collection of $N$ frequencies as above and $T$ be a Calder\'on-Zygmund operator relatively to $\Theta$. Then $T$ is of weak type $(1,1)$ with (uniformly with respect to $N$)
$$ \|T\|_{L^1\to L^{1,\infty}} \lesssim N^{\frac{1}{2}}.$$
\end{proposition}
\begin{proof} Consider $f$ a function in $L^1$ and $\lambda>0$, we use the Calder\'on-Zygmund decomposition\footnote{In \cite{NOT}, the multi-frequency Calder\'on-Zygmund decomposition is only described in $\R$. The proof is a combination of Lemma \ref{lem} and the usual Calder\'on-Zygmund decomposition. Since both of them can be extended in a multi-dimensional framework, the multi-frequency Calder\'on-Zygmund decomposition performed in \cite{NOT} still holds in $\R^n$.} of \cite{NOT} related to the collection of frequencies $\Theta$. So the function $f$ can be decomposed $f = g + \sum_{J\in {\bf J}} b_J$ with the following properties:
\begin{itemize}
\item ${\bf J}$ is a collection of balls and $(3J)_{J\in{\bf J}}$ has a bounded overlap;
\item for each $J\in{\bf J}$, $b_J$ is supported in $3J$;
\item we have
\begin{equation} \sum_{J \in{\bf J}} |J| \lesssim \sqrt{N} \|f\|_{L^1} \lambda^{-1}; \label{eq:sumJ} \end{equation}
\item the ``good part'' $g$ satisfies
\begin{equation} \|g\|_{L^2}^2 \lesssim \|f\|_{L^1} \sqrt{N} \lambda; \label{eq:g} \end{equation}
\item the cubes $J$ satisfy
\begin{equation} \|f\|_{L^1(J)} \lesssim |J|\lambda N^{-\frac{1}{2}} , \quad \|f-b_J\|_{L^2(J)} \lesssim \sqrt{|J|} \lambda; \label{eq:b} \end{equation}
\item we have cancellation for all the frequencies of $\Theta$: for all $j=1,...,N$ and $J\in{\bf J}$, $\widehat{b_J}(\xi_j)=0$.
\end{itemize}
We aim to estimate the measure of the level-set
$$ \Upsilon_\lambda:=\left\{x, |T(f)(x)|>\lambda\right\}.$$
With $b=\sum_J b_J$, we have
\begin{align*}
|\Upsilon_\lambda| & \leq \left|\left\{x, |T(g)(x)|>\lambda/2\right\}\right| + \left|\left\{x, |T(b)(x)|>\lambda/2\right\}\right| \\
& \lesssim \lambda^{-2} \|T(g)\|_{L^2}^2 + \left|\left\{x, |T(b)(x)|>\lambda/2\right\}\right| \\
& \lesssim \lambda^{-1}\sqrt{N}\|f\|_{L^1} + \left|\left\{x, |T(b)(x)|>\lambda/2\right\}\right|,
\end{align*}
where we used the $L^2$-boundedness of $T$. So it remains us to study the last term.
Since (\ref{eq:sumJ}), we get
$$ \left| \bigcup_{J\in {\bf J}} 4J \right| \lesssim \sum_{J} |J| \lesssim \sqrt{N} \|f\|_{L^1} \lambda^{-1}.$$
Consequently, it only remains to estimate the measure of the set
$$ O_\lambda:= \left\{x \in \left(\bigcup_{J\in {\bf J}} 4J\right)^{c}, \quad |T(b)(x)|>\lambda/2\right\}.$$
Since
\begin{equation}
|O_\lambda| \lesssim \lambda^{-1} \sum_{J} \|T(b_J)\|_{L^1((2J)^c)}, \label{eq:Olambda}
\end{equation}
it is sufficient to estimate the $L^1$-norms.
Consider $K$ the kernel of $T$ and a point $x_0\in \left(\bigcup_{J\in {\bf J}} 4J\right)^{c}$.
Then, we can use the integral representation and we have
$$ T(b)(x_0) = \int K(x_0,y) b(y) dy = \sum_{J} \int_{3J} K(x_0,y) b_J(y) dy.$$
To each $J$, we aim to take advantage of the cancellation properties of $b_J$, so we subtract the projection of $\left[y\to K(x_0,y)\right]$ on the space, spanned by $(e^{iy\cdot\eta})_{\eta\in \Theta}$. So we have
\begin{align*}
T(b)(x_0) & = \sum_{J}\sum_{j=1}^N \int_{3J} \left[K_j(x_0,y) - e^{i\xi_j \cdot c(J)} K_j(x_0,c(J))e^{-i\xi_j \cdot y}\right] b_J(y) dy \\
& = \sum_{J}\sum_{j=1}^N \int_{3J} \left[\widetilde{K}_{j}(x_0,y) - \widetilde{K}_j(x_0,c(J))\right]e^{i\xi_j\cdot (x_0-y)} b_J(y) dy
\end{align*}
where $c(J)$ is the center of $J$ and $\widetilde{K}_j(x,y):=K_j(x,y)e^{-i\xi_j \cdot (x-y)}$.
We then write
$$ T_j(b)(x_0) := \int \left[\widetilde{K}_{j}(x_0,y) - \widetilde{K}_j(x_0,c(J))\right]e^{i\xi_j \cdot (x_0-y)} b(y) dy.$$
such that $T(b) =\sum_j T_j(b)$. Due to the regularity assumption on $K$ (and so on $\widetilde{K}_j$), it comes for $y\in J$ and $x_0\in (2J)^c$
\begin{equation}
\sum_{j=1}^N \left|\widetilde{K}_{j}(x_0,y) - \widetilde{K}_j(x_0,c(J))\right| \lesssim \frac{r_J}{|x_0-y|^{n+1}}. \label{eq:kernelbis} \end{equation}
So we have
$$ \|T(b_J)\|_{L^1((2J)^c)} \lesssim \iint_{|x-y|\geq r_J} \frac{r_J}{|x-y|^{n+1}} |b_J(y)| dxdy \lesssim \|b_J\|_{L^1} \lesssim |J| \lambda.$$
Finally, we obtain with (\ref{eq:Olambda}) that
$$ |O_\lambda| \lesssim \sum_{J} |J| \lesssim \sqrt{N} \|f\|_{L^1} \lambda^{-1}, $$
which concludes the proof.
\end{proof}
\begin{remark}
Following \cite{NOT}, the bound of order $N^{\frac{1}{2}}$ is optimal for the multi-frequency decomposition and for the weak-$L^1$ estimate.
\end{remark}
\section{Weighted estimates for multi-frequency Calder\'on-Zygmund operators} \label{sec2}
Aiming to obtain weighted estimates on such multi-frequency operators (using {\it Good-lambda inequalities}), we also have to define a suitable maximal sharp function, associated to a collection of frequencies.
\begin{dfn}[Maximal sharp function] \label{def:max} Let $\Theta$ be a collection of $N$ frequencies and $s\in[1,\infty)$. Consider a ball $Q$, we denote by $\pr_{\Theta,Q}$ the projection operator (in the $L^s$-sense) on the subspace of $L^s(3Q)$, spanned by $(\exp {i\xi\cdot})_{\xi\in \Theta}$. Let us specify this projection operator: consider $E$ the finite dimensional sub-space of $L^s(3Q)$, spanned by $(e^{i\xi\cdot})_{\xi\in \Theta}$ and equipped with the $L^s(3Q)$-norm.
Since $E$ is of finite dimension, then for every $f\in L^s(Q)$ there exists $v:=\pr_{\Theta,Q}(f)\in E$ such that
$$ \|f-v\|_{L^s(3Q)} = \inf_{\phi\in E} \|f-\phi\|_{L^s(3Q)}.$$
This projection operator may depend on $s$, which is not important for our purpose so this is implicit in the notation and we forget it.
\mb
Since $0\in E$, we obviously have
\begin{equation} \label{eq:projection}\|\pr_{\Theta,Q}(f)\|_{L^s(3Q)} \leq 2\|f\|_{L^s(Q)}. \end{equation} Then, we may define the maximal sharp function
$$ \M_{s,\Theta}^\sharp (f)(x_0) := \sup_{x_0\in Q} \left(\aver{Q} \left|f-\pr_{\Theta,Q}(f{\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}}.$$
\end{dfn}
Note that the usual sharp maximal function is the one obtained for $\Theta:=\{0\}$ and in this situation it is well-known that the maximal sharp function satisfies a so-called Fefferman-Stein inequality (see \cite{FS}). We first prove an equivalent property for this generalised maximal sharp function:
\begin{proposition} \label{prop:FeffermanStein} Let $s\in(1,\infty)$, $t\in[1,\infty)$ and $p \in(s,\infty)$ be fixed. Then for every function $f\in L^s$ and every weight $\omega\in RH_{t'}$, we have for every $p\geq s$
$$ \|f\|_{L^p(\omega)} \lesssim N^{\frac{tp}{s}\max\{\frac{1}{2},\frac{1}{s}\}} \left\| \M_{s,\Theta}^\sharp(f) \right\|_{L^p(\omega)}.$$
\end{proposition}
The proof relies on a {\it Good-lambda inequality} and Lemma \ref{lem}.
\begin{proof} We make use on the abstract theory developed in \cite{AM} by Auscher and Martell.
We also follow notations of \cite[Theorem 3.1]{AM}. Indeed, for each ball $Q\subset \rn$ we have the following
$$ F(x) := |f(x)|^s \lesssim \left|f(x)-\pr_{\Theta,Q}(f{\bf 1}_Q)(x) \right|^s + \left|\pr_{\Theta,Q}(f{\bf 1}_Q)(x) \right|^s := G_Q(x)+H_Q(x).
$$
By definition, it comes
$$ \aver{Q} G_Q dx \leq \inf _{Q} \M_{s,\Theta}^\sharp (f)^s$$
and following Lemma \ref{lem} (with (\ref{eq:projection}))
\begin{align*}
\sup_{x\in Q} H_Q & = \|\pr_{\Theta,Q}(f {\bf 1}_Q)\|_{L^\infty(Q)}^s \lesssim N^{s\max\{\frac{1}{2},\frac{1}{s}\}} \left( \aver{3Q} |\pr_{\Theta,Q}(f {\bf 1}_Q)|^s dx\right) \\
& \lesssim N^{s\max\{\frac{1}{2},\frac{1}{s}\}} \left( \aver{Q} |f|^s dx\right) \lesssim N^{s\max\{\frac{1}{2},\frac{1}{s}\}} \inf_Q {\mathcal M} F.
\end{align*}
So we can apply \cite[Theorem 3.1]{AM} (with $q=\infty$ and $a\simeq N^{s\max\{\frac{1}{2},\frac{1}{s}\}}$) and by checking the behaviour of the constants with respect to ``$a$" in its proof, we obtain for every $p\geq 1$
$$ \left\| {\mathcal M}_s(f)^s \right\|_{L^p(\omega)} \lesssim N^{spt \max\{\frac{1}{2},\frac{1}{s}\}} \left\| \M_{s,\Theta}^\sharp (f)^s \right\|_{L^p(\omega)},$$
which yields the desired result.
\end{proof}
Then, we evaluate a multi-frequency Calder\'on-Zygmund operator via this new maximal sharp function.
\begin{proposition} \label{prop:maximal} Let $T$ be a Calder\'on-Zygmund operator relatively to $\Theta$ and $s\in(1,\infty)$. Then, we have the following pointwise estimate:
$$ \M_{s,\Theta}^\sharp (T(f)) \lesssim N^{|\frac{1}{s}-\frac{1}{2}|} {\mathcal M}_s(f) .$$
\end{proposition}
\begin{proof} We follow the well-known proof for usual Calder\'on-Zygmund operators and adapt the arguments to the current situation. So consider a point $x_0$ and a ball $Q\subset \rn$ containing $x_0$, we have to estimate
$$ \left( \aver{Q} \left|T(f)-\pr_{\Theta,Q}(T(f){\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}}.$$
We split the function into a local part $f_0$ and an off-diagonal part $f_\infty$:
$$f=f_0+f_\infty:= f{\bf 1}_{10 Q} + f {\bf 1}_{(10 Q)^c}.$$
By definition of the projection operator, we know that
\begin{align*}
\left( \aver{Q} \left|T(f)-\pr_{\Theta,Q}(T(f){\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}} & \leq \left( \aver{Q} \left|T(f)-\pr_{\Theta,Q}(T(f_\infty){\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}} \\
& \leq \left( \aver{Q} \left|T(f_0) \right|^s dx \right)^{\frac{1}{s}} + \left( \aver{Q} \left|T(f_\infty)-\pr_{\Theta,Q}(T(f_\infty){\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}}.
\end{align*}
For the local part, we use boundedness in $L^s$ of the operator $T$ (Proposition \ref{prop:bound}), hence
\begin{align*}
\left( \aver{Q} \left|T(f_0)\right|^s dx \right)^{\frac{1}{s}} & \lesssim |Q|^{-\frac{1}{s}} \|T(f_0)\|_{L^s(Q)} \lesssim N^{(\frac{1}{2}-\frac{1}{s})} \left(|Q|^{-\frac{1}{s}} \|f_0\|_{L^s}\right) \\
& \lesssim N^{|\frac{1}{2}-\frac{1}{s}|} {\mathcal M}_s(f)(x_0).
\end{align*}
Then let us focus on the second part, involving $f_\infty$. \\
We use the decomposition (with an integral representation) since we are in the off-diagonal case: for $x\in Q$
$$ T(f_\infty)(x) = \sum_{j=1}^N \int K_j(x,y) f_\infty(y) dy.$$
Consider the following function, defined on $3Q$ by (where $c(Q)$ is the center of $Q$)
$$ \Phi:= x\in 3Q \rightarrow \sum_{j=1}^N \int e^{i\xi_j\cdot (x-c(Q))} K_j(c(Q),y) f_\infty(y) dy.$$
So $\Phi \in E$ (see Definition \ref{def:max}) and hence
\begin{equation} \left( \aver{Q} \left|T(f_\infty)-\pr_{\Theta,Q}(T(f_\infty){\bf 1}_Q) \right|^s dx \right)^{\frac{1}{s}} \leq \left( \aver{Q} \left|T(f_\infty)-\Phi \right|^s dx \right)^{\frac{1}{s}}. \label{eq::} \end{equation}
If we set $\widetilde{K}_j(x,z):=K_j(x,z)e^{-i\xi_j\cdot (x-z)}$, then
$$ T(f_\infty)(x) -\Phi(x) = \sum_j \int \left[\widetilde{K}_{j}(x,y) - \widetilde{K}_j(c(Q),y)\right] e^{i\xi_j(x-y)} f_\infty(y) dy.$$
From the regularity assumption on the kernels $K_j$'s, we have for $y\in (10 Q)^c$
\begin{equation} \sum_{j} \left| \widetilde{K}_{j}(x,y) - \widetilde{K}_j(c(Q),y) \right| \lesssim r_Q \sup_{z\in Q} \sum_j \left| \nabla_{x} \widetilde{K}_{j}(z,y)\right| \lesssim r_Q^{-n}\left(1+\frac{d(y,Q)}{r_Q}\right)^{-n-1}. \label{eq:kernel} \end{equation}
We also have (since $y\in (10 Q)^c$ and $x,c(Q)\in Q$)
\begin{align*}
\left| T(f_\infty)(x) -\Phi(x)\right| & \lesssim \int_{|z|\geq 10 r_Q} r_Q^{-n}\left(1+\frac{|x-c(Q)-z|}{r_Q}\right)^{-n-1} |f(c(Q)+z)| dz \\
& \lesssim \int_{|z|\geq 5 r_Q} r_Q^{-n}\left(1+\frac{|z|}{r_Q}\right)^{-n-1} |f(x_0+z)| dz \\
& \lesssim {\mathcal M}(f)(x_0),
\end{align*}
which concludes the proof.
\end{proof}
We obtain the following corollary:
\begin{corollary} \label{cor} Let $\Theta$ be a collection of $N$ frequencies. For $p\in (2,\infty)$, $s\in[2,p)$ and $t\in(1,\infty)$, a multi-frequency Calder\'on-Zygmund operator $T$ is bounded on $L^p(\omega)$ for every weight $\omega \in RH_{t'} \cap {\mathbb A}_{\frac{p}{s}}$ with
$$ \|T\|_{L^p(\omega) \to L^p(\omega)} \lesssim N^{\frac{tp}{2s}+\left(\frac{1}{2}-\frac{1}{s}\right)}.$$
\end{corollary}
\begin{proof} Using Propositions \ref{prop:FeffermanStein} and \ref{prop:maximal}, it follows that for $p>s\geq 2$ (assuming $\omega \in {\mathbb A}_{\frac{p}{s}}$)
\begin{align*} \|T(f) \|_{L^p(\omega)} & \lesssim N^{\frac{tp}{2s}} \left\| \M_{s,\Theta}^\sharp [T(f)] \right\|_{L^p(\omega)} \\
& \lesssim N^{\frac{tp}{2s}+\left(\frac{1}{2}-\frac{1}{s}\right)} \left\| {\mathcal M}_s(f) \right\|_{L^p(\omega)} \\
& \lesssim N^{\frac{tp}{2s}+\left(\frac{1}{2}-\frac{1}{s}\right)} \left\| f \right\|_{L^p(\omega)},
\end{align*}
where we used weighted boundedness of the maximal function since $\omega \in {\mathbb A}_{\frac{p}{s}}$.
\end{proof}
\mb As explained in the introduction, this estimate is only interesting when the exponent $\frac{tp}{2s}+\left(\frac{1}{2}-\frac{1}{s}\right)$ is lower than $1$.
\section{Connexion to Bochner-Riesz multipliers} \label{sec3}
In this section, we aim to describe how such arguments could be applied to generalized Bochner-Riesz multipliers. Weighted estimates for Bochner-Riesz multipliers has been initiated in \cite{Vargas, Christ, CDL}. We first emphasize that we do not pretend to obtain new weighted estimates for Bochner-Riesz multipliers. But we only want to describe here a new point of view and a new approach for such estimates, which will be the subject of a future investigation. Such an application is a great motivation for pursuing the study of a multi-frequency Calder\'on-Zygmund analysis.
\mb
Consider also $\Omega$ a bounded open subset of $\rn$ such that its boundary $\Gamma:=\overline{\Omega}\setminus \Omega$ is an hyper-manifold of Hausdorff dimension $n-1$. For $\delta>0$, we then define the generalized Bochner-Riesz multiplier, given by
$$ R_{\Omega,\delta}(f) (x) := \int_\Omega e^{ix\cdot \xi} \widehat{f}(\xi) m_\delta d\xi,$$
where $m_\delta$ is a smooth symbol supported in $\overline{\Omega}$ and satisfying in $\Omega$
$$ | \partial^\alpha m_\delta(\xi)| \lesssim d(\xi,\Gamma)^{\delta-|\alpha|}.$$
\mb
We first use a Whitney covering $(O_i)_i$ of $\Omega$. That is a collection of sub-balls such that
\begin{itemize}
\item the collection $(O_i)_i$ covers $\Omega$ and has a bounded overlap;
\item the radius $r_{O_i}$ is equivalent to $d(O_i,\Gamma)$.
\end{itemize}
Associated to this collection, we build a partition of the unity $(\chi_i)_i$ of smooth functions such that $\chi_i$ is supported on $O_i$ with
$$ \sum_i \chi_i(\xi) = {\bf 1}_{\Omega}(\xi)$$
and $\|\partial^\alpha \chi_i\|_\infty \lesssim r_{O_i}^{-|\alpha|}$. \\
Then, $R_{\delta}$ may be written as
$$ R_{\delta}(f) (x) = \sum_{j=-\infty}^\infty T_j(f)(x),$$
with
\begin{align}
T_j(f)(x) & :=\sum_{\genfrac{}{}{0pt}{}{l,}{2^j \leq r_{O_l}< 2^{j+1}}} \int_\Omega e^{ix\cdot \xi} \widehat{f}(\xi) m_\delta(\xi) \chi_l(\xi) d\xi \nonumber \\
& =2^{j\delta} U_j(f)(x), \label{eq:dec}
\end{align}
where we set
$$ U_j(f)(x):= \sum_{\genfrac{}{}{0pt}{}{l,}{2^j \leq r_{O_l}< 2^{j+1}}} \int_\Omega e^{ix\cdot \xi} \widehat{f}(\xi) (2^{-j\delta} m_\delta(\xi)) \chi_l(\xi) d\xi.$$
\gb
{\bf Observation :} The main idea is to observe that the operator $U_j$ is a multi-frequency Calder\'on-Zygmund operator associated to the collection
$$ \Theta_j :=\{ c(O_l), \ 2^j \leq r_{O_l}< 2^{j+1} \} \qquad \textrm{with} \qquad \sharp \Theta_j \simeq 2^{-j(n-1)}.$$
However, these operators have specific properties, one of them is that the considered balls have equivalent radius, which means that these operators have only one scale $2^{j}$. For example, this observation allows us to easily prove some boundedness:
\begin{proposition} \label{prop} Uniformly with $j\lesssim 0$, the multiplier $U_j$ is a convolution operation with a kernel $K_j$ satisfying
$$ \|K_j\|_{L^1} \lesssim 2^{-j \frac{n-1}{2}}.$$
Hence, it follows that $U_j$ is bounded on Lebesgue space $L^p$ for every $p\in[1,\infty]$. Moreover for every $s\in[1,2]$, $p\in(s,\infty)$ and every weight $\omega\in {\mathbb A}_{\frac{p}{s}}$, $U_j$ is bounded on $L^p(\omega)$ with
$$ \| U_j\|_{L^p(\omega) \rightarrow L^p(\omega)} \lesssim 2^{-j\frac{n-1}{s}}.$$
\end{proposition}
\begin{proof} The operator $U_j$ is a Fourier multiplier, associated to the symbol
$$\sigma_j(\xi):=\sum_{\genfrac{}{}{0pt}{}{l,}{2^j \leq r_{O_l}< 2^{j+1}}} (2^{-j\delta} m_\delta(\xi)) \chi_l(\xi).$$
Since the considered balls $(O_l)_l$ are almost disjoint, it comes that
$$ \|\sigma_j\|_{L^2} \lesssim |\{\xi, d(\xi,\partial \Omega)\simeq 2^j \}|^{\frac{1}{2}} \lesssim 2^{\frac{j}{2}}.$$
Moreover, using regularity assumptions on $m_\delta$, we deduce that for every $\alpha$
$$ \|\partial^\alpha \sigma_j\|_{L^2} \lesssim 2^{-j|\alpha|} |\{\xi, d(\xi,\partial \Omega) \simeq 2^j \}|^{\frac{1}{2}} \lesssim 2^{j(\frac{1}{2}-|\alpha|)}.$$
So with $K_j:= \mathcal{F}(\sigma_j)$, it follows that for any integer $M$
\begin{equation} \left\| (1+2^{j}|\cdot|)^{M} K_j\right\|_{L^2} \lesssim 2^{\frac{j}{2}}. \label{eq:L^2} \end{equation}
Hence
$$ \left\| K_j\right\|_{L^1} \lesssim 2^{-j\frac{n-1}{2}}.$$
Using Minkowski inequality, we deduce that for every $p\in[1,\infty]$
$$ \| U_j\|_{L^p \to L^p} \lesssim \|K_j\|_{L^1} \lesssim 2^{-j\frac{n-1}{2}}.$$
Let us now focus on the second claim about weighted estimates. Using integrations by parts for computing the kernel $K_j$, it comes for any integer $M$
\begin{equation} \left\| (1+2^{j}|\cdot|)^{M} K_j\right\|_{L^\infty} \lesssim 2^{j}. \end{equation}
By interpolation with (\ref{eq:L^2}), for $s\in[1,2]$ we get
\begin{equation} \left\| (1+2^{j}|\cdot|)^{M} K_j\right\|_{L^{s'}} \lesssim 2^{\frac{j}{s}}, \end{equation}
which gives
$$ U_j(f) \lesssim 2^{-j\frac{n-1}{s}} \M_{s}(f).$$
Hence, for every $p> s$ and every weight $\omega \in {\mathbb A}_{\frac{p}{s}}$
$$ \|U_j\|_{L^p(\omega) \to L^p(\omega)} \lesssim 2^{-j\frac{n-1}{s}}.$$
\end{proof}
\mb
In this context, $\sharp \Theta_j \simeq 2^{-j(n-1)}$, so the constant $2^{-j \frac{n-1}{s}}$ is equivalent to $(\sharp \Theta_j)^{\frac{1}{s}}$ and this is a better constant than the one obtained in Corollary \ref{cor} (for a subclass of ${\mathbb A}_{\frac{p}{s}}$ weights). \\
So improving these ``easy bounds'' means to obtain inequalities such as
$$ \| U_j\|_{L^p(\omega) \rightarrow L^p(\omega)} \lesssim (\sharp \Theta_j)^{\gamma}$$
for some better exponent $\gamma<\frac{1}{s}$.
\gb
Let us finish by suggesting how could we get improvements of our approach to get interesting results for Bochner-Riesz multipliers:
\mb
{\bf Question :} The general approach, developed in the previous section, only allows to get an exponent
$$ \gamma=\frac{tp}{2s}+\left(\frac{1}{2}-\frac{1}{s}\right)$$
(with some $s\in[2,p)$) which is bigger than $\frac{1}{2}$ (since $p>s\geq 2$ and $t>1$).
So to improve this exponent $\gamma$, two things seem to be crucial:
\begin{itemize}
\item to extend the use of Lemma \ref{lem} for $p\geq 2$ which would allow us to get an exponent $\frac{tp}{s^2}$ instead of $\frac{tp}{2s}$;
\item to use the geometry of the boundary $\Gamma$ to get better exponents, even for the unweighted estimates. Indeed, for example for the unit ball (using its non vanishing curvature), we know that (see \cite{Lee, survey})
$$ \|U_j\|_{L^p\to L^p} \lesssim 2^{-j\delta(p)}$$
with if $n=2$
$$ \delta(p):= \max\left\{ 2\left|\frac{1}{2}-\frac{1}{p}\right|-\frac{1}{2},0 \right\}.$$
and if $n\geq 3$ and $p\geq \frac{2(n+2)}{n}$ or $p\leq \frac{2(n+2)}{n+4}$
$$ \delta(p):= \max\left\{ n\left|\frac{1}{2}-\frac{1}{p}\right|-\frac{1}{2},0 \right\}.$$
\end{itemize}
| {
"timestamp": "2013-05-02T02:01:05",
"yymm": "1211",
"arxiv_id": "1211.5218",
"language": "en",
"url": "https://arxiv.org/abs/1211.5218",
"abstract": "In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderon-Zygmund analysis introduced in [9]. We set a definition of general multi-frequency Calderon-Zygmund operator. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [9]. Involving a new kind of maximal sharp function, weighted estimates are obtained.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "Multi-frequency Calderon-Zygmund analysis and connexion to Bochner-Riesz multipliers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846703886661,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7092019546015942
} |
https://arxiv.org/abs/2104.05870 | Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method | We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of the this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented. | \section{Introduction}
The aim of this paper is to compute viscosity solutions to a large class of Hamilton-Jacobi equations possibly involving nonconvex Hamiltonians.
The key ingredient for us to reach this achievement is the use of a new Carleman estimate and the convexification method.
This method is only applicable when
a viscosity term is added to the Hamilton-Jacobi equation under consideration. The idea of adding the viscosity term and passing to the limit to obtain viscosity solutions is due to the seminal works \cite{CrandallLions83,CrandallEvansLions84}.
Let $R > 0$ and $\Omega = (-R, R)^d$ where $d \geq 1$ is the spatial dimension.
Let $F: \overline \Omega \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}$ and $f: \overline \Omega \to \mathbb{R}$ be functions of the class $C^2$.
In this paper, we propose a globally convergent numerical method to solve the Hamilton-Jacobi equation
\begin{equation}
F({\bf x}, u({\bf x}), \nabla u({\bf x})) = 0 \quad \mbox{for all } {\bf x} \in \Omega
\label{HJ}
\end{equation}
with the Dirichlet boundary condition
\begin{equation}
u({\bf x}) = f({\bf x}) \quad \mbox{for all } {\bf x} \in \partial \Omega.
\label{dir}
\end{equation}
The smoothness condition imposing on $F$ and $f$ is for the simplicity. We need it when analytically establishing the convergence of the proposed method.
However, this technical condition can be relaxed in our numerical study.
In this paper, we are interested in computing the viscosity solution to \eqref{HJ}--\eqref{dir}.
We only deal with the case that the Dirichlet boundary condition holds in the classical sense in this paper.
We refer the readers to \cite{CrandallLions83, CrandallEvansLions84, Lions, Barles, BCD,Tran19} and the references therein for the theory of viscosity solutions to \eqref{HJ}--\eqref{dir}.
It is worth mentioning that a number of different extremely efficient and fast numerical approaches and techniques (many of which are of high orders) have been developed for Hamilton-Jacobi equations.
For finite difference monotone and consistent schemes of first-order equations and applications, see \cite{CL-rate, Sou1, BS-num, Sethian, OsFe} for details and recent developments.
If $F=F({\bf x},s,{\bf p})$ is convex in ${\bf p}$ and satisfies some appropriate conditions, it is possible to construct some semi-Lagrangian approximations by the discretization of the Dynamical Programming Principle associated to the problem, see \cite{FaFe1, FaFe2} and the references therein.
For a non-exhaustive list of results along these directions, see \cite{OsherSethian, OsherShu, Tsitsiklis, Abgrall, Abgrall2, BrysonLevy, SethianVladimirsky, TsaiChengOsherZhao, KaoOsherQian, QianZhangZhao, CamilliCDGomes, CockburnMerevQian, CagnettiGomesTran, ObermanSalvador, LiQian, GallistlSprekelerSuli}.
Another approach to solve \eqref{HJ}--\eqref{dir} is based on optimization \cite{HornBrooks, Szeliski, LeclercBobick, DanielDurou}.
However, due to the nonlinearity of the function $F$ in \eqref{HJ}, the least squares cost functional is nonconvex and might have multiple local minima and ravines.
Hence, the methods based on optimization only provide reliable numerical solutions if good initial guesses of the true solutions are given.
\smallskip
Unlike the mentioned optimization approach, we propose to use the convexification method, which
does not rely on the above assumptions.
This method is globally convergent in the sense that
\begin{enumerate}
\item it delivers good approximation of the true solution without knowing any advance knowledge of the true solution even when the given data is noisy;
\item the claim in \# 1 above is rigorously proved and numerically verified.
\end{enumerate}
To the best of our knowledge, our method is new in the context of viscosity solutions to Hamilton-Jacobi equations.
It has two major advantages in solving \eqref{HJ}--\eqref{dir} numerically.
Firstly, it works for some quite general $F({\bf x},u,\nabla u)$, which might be nonconvex in $\nabla u$, and it does not require a lot of specific structures on $F$.
In particular, $F$ might be dependent on $u$ and $\nabla u$ in a rather complicated way.
Secondly, it is quite stable and robust even with some noise level on the boundary data, which occurs naturally in applications.
The main idea of the convexification method is to employ a suitable Carleman
weight function to convexify the mismatch functional derived from the given
boundary value problem. Several versions of the convexification method have
been developed since it was first introduced in \cite%
{KlibanovIoussoupova:SMA1995} for a coefficient inverse problem for a hyperbolic equation. We cite here \cite%
{Klibanov:ip2015,Klibanov:sjma1997, Klibanov:IPI2019,Klibanov:nw1997,KlibanovNik:ra2017, KhoaKlibanovLoc:SIAMImaging2020, VoKlibanovNguyen:IP2020, SmirnovKlibanovNguyen:IPI2020, Klibanov:ip2020} and references therein
for some important works in this area and their real-world applications in bio-medical imaging, non-destructed testing,
travel time tomography, identifying anti-personnel explosive devices buried under the ground, {\it etc.}
The crucial mathematical ingredients that
guarantee the strict convexity of this functional are the
Carleman estimates.
The original idea of applying Carleman estimates to prove the uniqueness for a large class of important nonlinear mathematical problems was first published in \cite{BukhgeimKlibanov:smd1981}.
It was discovered later in \cite{KlibanovIoussoupova:SMA1995, KlibanovLiBook}, that the idea of \cite%
{BukhgeimKlibanov:smd1981} can be successfully modified to develop globally
convergent numerical methods for coefficient inverse problems using the
convexification.
In this paper, it is the first time we use a Carleman weight function to numerically solve Hamilton-Jacobi equations.
One of the strengths of the convexification method is that it does not require the convexity of $F({\bf x}, s, {\bf p})$ with respect to ${\bf p}$. Still, it has a drawback. The theory of the convexification method requires
an additional information about $u_z$ on a part of $\partial \Omega$. In this paper, that part is
\begin{equation}
\Gamma^+ = \big\{
{\bf x} = (x_1, x_2, \dots, x_{d - 1}, z = R): |x_i| \leq R, 1 \leq i \leq d - 1
\big\}
\subset \partial \Omega.
\end{equation}
For ${\bf x}\in \mathbb{R}^d$, write ${\bf x} = (x_1, x_2, \dots, x_{d - 1}, z)$.
More precisely, we impose the following condition.
\begin{Assumption}
Write $\nabla u = (u_{x_1}, \dots, u_{x_{d - 1}}, u_z)$. We assume that $u_z$ on $\Gamma^+$ is known.
\label{assumption 1}
\end{Assumption}
In general, the additional knowledge of $u_z$ on $\Gamma^+$ in Assumption \ref{assumption 1} makes the problem of computing solutions to Hamilton-Jacobi equations with the Dirichlet data on $\partial \Omega$ and the Neumann data on $\Gamma^+$ over-determined.
However, in many real-world circumstances, we are able to compute $u_z$ on $\Gamma^+$ from the knowledge of $u$ on $\Gamma^+$ without further measurement.
We provide here a classical example arising from the traveling time problem. Denote by $c({\bf x})$, ${\bf x} \in \overline \Omega$, the velocity of the light at the point ${\bf x}$.
Here, $c \in C(\overline \Omega, (0,\infty))$ is a given function.
Let $u({\bf x})$ be the minimal time for the light to travel from $\partial \Omega$ to ${\bf x} \in \Omega$.
This function is governed by the boundary value problem for the eikonal equation
\begin{equation}
\left\{
\begin{array}{rcll}
c({\bf x})^2|\nabla u({\bf x})|^2 &=& 1 &{\bf x} \in \Omega,\\
u({\bf x}) &=& 0 &{\bf x} \in \partial \Omega.
\end{array}
\right.
\label{eik}
\end{equation}
Since $u = 0$ on $\partial \Omega$, in particular, $u = 0$ on $\Gamma^+.$
Hence $u_{x_i} = 0$ on $\Gamma^+$ for all $1 \leq i \leq d - 1$.
The function $u_z$ on $\Gamma^+$ is given by
\[
u_z({\bf x}) = -\frac{1}{c({\bf x})} \quad \mbox{for all } {\bf x} \in \Gamma^+.
\]
Above, the case $u_z({\bf x}) = \frac{1}{c({\bf x})}$ is negligible since in reality, $u \geq 0$ on $\overline \Omega$ and therefore, its partial derivative with respect to $z$ on $\Gamma^+$ is non-positive.
\begin{remark}[Reducing Assumption \ref{assumption 1}]
We have the following points.
\begin{itemize}
\item[1.] It follows from the example above that in general, since $u$, and; therefore, $u_{x_1},$ $\dots,$ $u_{x_{d - 1}}$ on $\Gamma^+$ are known, to verify Assumption \ref{assumption 1}, we simply solve the equation
\begin{equation}
F({\bf x}, u, u_{x_1}, \dots, u_{x_{d - 1}}, u_z) = 0
\label{1.5}
\end{equation}
for $u_z$.
So, a condition on $F$ such that Assumption \ref{assumption 1} holds true is that \eqref{1.5} is uniquely solvable for $u_z$. For example,
\[
\frac{\partial }{\partial z}F({\bf x}, u, u_{x_1}, \dots, u_{x_{d - 1}}, u_z) \not = 0 \quad \mbox{for all } {\bf x} \in \Gamma^+.
\]
\item[2.] In many important Hamilton-Jacobi equations; for e.g., the eikonal equation $F(x, u, \nabla u) = c({\bf x})^2|\nabla u|^2 - 1$ in \eqref{eik} or $F({\bf x}, u, \nabla u) = |u_{x_1}| - |u_{z}| - g$ (the case $d = 2$), for some function $g$, equation \eqref{1.5} only provides $|u_z|$ rather than $u_z$ on $\Gamma^+$. In this case, the sign of $u_z$ on $\Gamma^+$ is required.
\item[3.]We provide here an example in which the sign of $u_{z}$ is known.
Let $d=3$ and $\mathbf{x}_{0}$ be a point in $\mathbb{R}^{3}\setminus \Omega
$. In travel time tomography, the Hamilton-Jacobi equation that describes
the the travel time of light traveling from the source $\mathbf{x}_{0}$ to
a point $\mathbf{x}\in \mathbb{R}^{3},$ has the form $c^{2}(\mathbf{x}%
)|\nabla _{\mathbf{x}}u(\mathbf{x},\mathbf{x}_{0})|^{2}=1$ for all $\mathbf{x%
}\in \mathbb{R}^{3}\setminus \{\mathbf{x}_{0}\}$ with $u(\mathbf{x}_{0},%
\mathbf{x}_{0})=0.$ Here $c(\mathbf{x})$ is the speed of the light at $%
\mathbf{x}$. It was proved in \cite[Lemma 4.1]{Klibanov:IPI2019} that if $c(%
\mathbf{x})$, $\mathbf{x}\in \Omega $ is an increasing function with respect
to $z$ and the source $x_{0}\in \left\{ z<-R\right\} ,$
then the function $u(x,x_{0})$ is strictly increasing in the $z-$%
direction for $x\in \Omega $\ implying $u_{z}>0$\
on $\Gamma ^{+}.$ This result can be extended to all dimensions
by repeating the proof in \cite[Lemma 4.1]{Klibanov:IPI2019}.
\item[4.] In the case when Assumption \ref{assumption 1} cannot be verified or even when it might not hold true, for e.g.,
\[
F({\bf x}, u, \nabla u) = 10 u + \min\{|\nabla u|, ||\nabla u| - 8| + 6\} - g
\] or
\[ F({\bf x}, u, \nabla u) =u+ |\nabla u| - V\cdot \nabla u
\]
for some function $g$ and vector valued function $V$, the convexification method still provides good numerical solutions, see Test 4 and Test 5 in Section \ref{sec:num}. However, the rigorous theorem that guarantees the efficiency of this method is missing in this paper.
\end{itemize}
\label{rm 1}
\end{remark}
\medskip
Let us give a brief description of the main results in the paper.
We consider the vanishing viscosity process (equations \eqref{2.1} and \eqref{4.1}) and aim at computing $u^{\epsilon_0}$ for $\epsilon_0>0$ sufficiently small, which is a good approximation of $u$, the viscosity solution to \eqref{HJ}--\eqref{dir}.
The convexification is developed to compute this $u^{\epsilon_0}$.
Firstly, we obtain a new Carleman inequality in Theorem \ref{thm Car est}:
For $\beta > 1$, $r > R + 1$, and $b > R + r$, we can find two numbers $\lambda_0 = \lambda_0(\beta, r, R, b, d)>0$, $C = C(r, R, b, d)>0$ such that for all $\lambda > \lambda_0$
and for all $u \in C^2(\overline \Omega)$ with $u = 0$ on $\partial \Omega$ and $u_z = 0$ on $\Gamma^+$, we have
\begin{multline*}
\int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2 d{\bf x}
\geq C\lambda^3 \beta^2 (\beta - 1) b^{-3\beta} (-R + r)^{2\beta} \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}
\\
+ C \lambda (\beta - 1)b^{-\beta}\int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\end{multline*}
We then use this Carleman estimate to show in Theorem \ref{thm convex} that the functional
\begin{equation}
J_{\lambda, \beta, \eta}(u) = \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} \big|-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u)\big|^2 \,d{\bf x}
+ \eta \|u\|^2_{H^p(\Omega)}
\label{Jdef}
\end{equation}
is strictly convex for $u\in H \cap \overline{B(M)}$.
Here, $p > \lceil d/2 \rceil + 2$ is such that $H^p \hookrightarrow C^2(\overline \Omega)$, and
\[
H = \{u \in H^p(\Omega): u|_{\partial \Omega} = f \mbox{ and } u_z|_{\Gamma^+} = g\}, \quad
{B(M)} = \{u \in H^p(\Omega): \|u\|_{H^p(\Omega)} < M\}.
\]
Then, we use a gradient descent method (Theorem \ref{thm 4.2}) to compute the minimizer $u_{\rm min}$ of this functional.
Assuming that $u_{\rm min} \in H \cap B(M/3)$, we can start the gradient descent method at $u^{(0)} \in H \cap B(M/3)$ and iterate
\begin{equation*}
u^{(k)} = u^{(k - 1)} - \kappa DJ_{\lambda, \beta, \eta}(u^{(k-1)}) \quad \text{ for } k \in \mathbb{N}.
\end{equation*}
Here, $ \kappa \in (0, \kappa_0)$ where $ \kappa_0 \in (0, 1)$ depends only on $\lambda,$ $\beta$, $R,$ $r,$ $b$, $d$, $M$ and $\epsilon_0$.
We are able to obtain
\begin{equation*}
\big\|u^{(k)} - u_{\rm min}\big\|_{H^p(\Omega)} \leq \theta^{k/2} \big\|u^{(0)} - u_{\rm min} \big\|_{H^p(\Omega)} \quad \text{ for } k \in \mathbb{N},
\end{equation*}
for some $\theta \in (0, 1)$ depending only on $ \kappa,$ $\lambda,$ $\beta$, $R,$ $r,$ $b$, $d$, $M$, $F$ and $\epsilon_0$.
Then, in Theorem \ref{thm 4.3}, we show that, even if there is a noise of size $\delta>0$, we still have a nice bound
\begin{equation*}
\|u_{\min}^\delta - u^{\epsilon_0}\|^2_{H^1(\Omega)} \leq C(\eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + \delta^2).
\end{equation*}
Here, $u^{\delta}_{\min}$ is the minimizer of $J_{\lambda, \beta, \eta}(u)$ with noisy data $u|_{\partial \Omega} = f^\delta$, $u_z|_{\Gamma^+} = g^\delta$.
Here, by saying that $\delta$ is the noise level, there exists an ``error" function $\mathcal E$ satisfying
$\|\mathcal E\|_{H^p(\Omega)} \leq \delta$, $\mathcal E|_{\partial \Omega} = f^\delta - f$, $\mathcal E_z|_{\Gamma^+} = g^\delta - g$.
Combining Theorem \ref{thm 4.2} and Theorem \ref{thm 4.3}, we have for each $k \geq 1$,
\[
\|u^{(k)} - u^{\epsilon_0}\|_{H^1(\Omega)} \leq C(\sqrt{\eta} \|u^{\epsilon_0}\|_{H^p(\Omega)} + \delta) +
\theta^{k/2} \|u^{(0)} - u^\delta_{\min}\|_{H^p(\Omega)}.
\]
This inequality shows the stability of our method with respect to noise. If $\theta^{k}$ and $\eta$ are $O(\delta^2)$ as $\delta$ tends to $0$, then the convergence rate is Lipschitz.
Finally, in Section \ref{sec:num}, we implement the convexification method based on the finite difference method
and obtain interesting numerical results in two dimensions.
\medskip
We now address a bit further some state of the art numerical methods in solving \eqref{HJ}--\eqref{dir} in the literature.
If $F$ is generically convex in $\nabla u$, there have been extremely powerful approaches to compute the solutions such as monotone numerical Hamiltonian based finite difference methods (see \cite{OsherSethian,Sethian, SethianVladimirsky, TsaiChengOsherZhao, QianZhangZhao} and the references therein).
When $F$ is nonconvex in $\nabla u$, the Lax--Friedrichs schemes (\cite{OsherShu, Abgrall, OsFe}) and the Lax--Friedrichs sweeping algorithm (\cite{KaoOsherQian,LiQian}), in which numerical viscosity terms appear naturally, are very efficient and accurate.
Moreover, all the mentioned methods have very quick running times with not too many iterations.
Similar to the Lax--Friedrichs schemes, the addition of a viscosity term is natural in our approach as we deal with general $F$, which is possibly nonconvex in $\nabla u$.
\medskip
The paper is organized as follows.
In Section \ref{sec:prelim}, we give some preliminaries about viscosity solutions to Hamilton-Jacobi equations, which are rather well-known in the literature.
We state and prove a Carleman estimate in Theorem \ref{thm Car est} in Section \ref{sec:Carleman}.
Section \ref{sec conv} is devoted to the theoretical results of the convexification (Theorems \ref{thm convex}--\ref{thm 4.3}), which is our main focus in this current paper.
Then, in Section \ref{sec:num}, we implement the convexification method based on the finite difference method
and obtain interesting numerical results in two dimensions.
\section{Some preliminaries about viscosity solutions to Hamilton-Jacobi equations} \label{sec:prelim}
\begin{Definition}[Viscosity solutions of \eqref{HJ}--\eqref{dir}]
Let $u \in C(\overline \Omega)$.
\begin{itemize}
\item[(a)] We say that $u$ is a viscosity subsolution to \eqref{HJ}--\eqref{dir} if for any test function $\varphi \in C^1(\overline \Omega)$ such that $u-\varphi$ has a strict maximum at ${\bf x}_0 \in \overline \Omega$, then
\[
F({\bf x}_0, u({\bf x}_0), \nabla \varphi({\bf x}_0)) \leq 0 \quad \text{ if } {\bf x}_0 \in\Omega,
\]
or
\[
\min\left\{F({\bf x}_0, u({\bf x}_0), \nabla\varphi({\bf x}_0)), u({\bf x}_0) - f({\bf x}_0) \right\} \leq 0 \quad \text{ if } {\bf x}_0 \in\partial\Omega.
\]
\item[(b)] We say that $u$ is a viscosity supersolution to \eqref{HJ}--\eqref{dir} if for any test function $\varphi \in C^1(\overline \Omega)$ such that $u-\varphi$ has a strict minimum at ${\bf x}_0 \in \overline \Omega$, then
\[
F({\bf x}_0, u({\bf x}_0), \nabla \varphi({\bf x}_0)) \geq 0 \quad \text{ if } {\bf x}_0 \in\Omega,
\]
or
\[
\max\left\{F({\bf x}_0, u({\bf x}_0), \nabla \varphi({\bf x}_0)), u({\bf x}_0) - f({\bf x}_0) \right\} \geq 0 \quad \text{ if } {\bf x}_0 \in\partial\Omega.
\]
\item[(c)] We say that $u$ is a viscosity solution to \eqref{HJ}--\eqref{dir} if it is both its viscosity subsolution and its viscosity supersolution.
\end{itemize}
\end{Definition}
It is worth noting that the Dirichlet boundary condition might not hold in the classical sense for viscosity solutions to \eqref{HJ}--\eqref{dir} (see \cite[Appendix E]{Tran19} for example).
In the following, we impose some compatibility conditions to make sure that $u=f$ on $\partial \Omega$ classically.
\medskip
We write $F = F({\bf x}, s, {\bf p})$. Let $\nabla_{{\bf x}} F$ and $\nabla_{{\bf p}} F$ the gradient vector of $F$ with respect to the first and the third variables, respectively, and $\partial_s F=F_s$ the partial derivative of $F$ with respect to the second variable.
Here is one set of conditions that often occurs in the literature.
\begin{Assumption}
1. There exists $\alpha>0$ such that
\[
\alpha \leq F_s({\bf x},s,{\bf p}) \leq \frac{1}{\alpha} \quad \text{ for all } ({\bf x},s,{\bf p}) \in \overline \Omega \times \mathbb{R} \times \mathbb{R}^d.
\]
2. $F$ is coercive in ${\bf p}$ in the sense that
\[
\lim_{|{\bf p}| \to \infty} \inf_{({\bf x},s) \in \overline \Omega \times \mathbb{R}} \left ( F({\bf x},s,{\bf p})^2 + \partial_s F({\bf x},s,{\bf p}) |{\bf p}|^2 + \nabla_{{\bf x}}F({\bf x},s,{\bf p})\cdot {\bf p} \right) = +\infty.
\]
3. There exists $\phi \in C^2(\overline \Omega)$ such that $\phi = f$ on $\partial \Omega$, and
\[
F({\bf x},\phi({\bf x}),\nabla \phi({\bf x})) < 0 \quad \text{ in } \Omega.
\]
\label{assumption 2}
\end{Assumption}
\begin{Theorem}
Suppose that Assumption \ref{assumption 2} holds.
Then, problem \eqref{HJ}--\eqref{dir} has a unique viscosity solution, denoted by $u$.
Moreover, for $\epsilon > 0,$ the following problem
\begin{equation}
\left\{
\begin{array}{rcll}
-\epsilon \Delta u^{\epsilon} + F({\bf x}, u^\epsilon, \nabla u^\epsilon) &=& 0 &{\bf x} \in \Omega,\\
u^{\epsilon} &=& f &{\bf x} \in \partial\Omega.
\end{array}
\right.
\label{2.1}
\end{equation}
has a unique solution in $u^\epsilon \in C^2(\overline \Omega)$
and
\begin{equation}
\lim_{\epsilon \to 0^+} \|u^\epsilon - u\|_{L^\infty(\Omega)} = 0.
\label{vanishing epsilon}
\end{equation}
\label{Theorem vis}
\end{Theorem}
Theorem \ref{Theorem vis} holds true under other appropriate sets of assumptions too.
We here just give one prototypical set of conditions, Assumption \ref{assumption 2}, for demonstration.
See \cite{CrandallLions83, CrandallEvansLions84, Lions, Tran19} for its proof.
For other set of appropriate conditions on $F$, see references listed in Section \ref{sec:num} in each numerical test.
\begin{remark}
In fact, under Assumption \ref{assumption 2}, we are able to quantify the rate of convergence of $u^\epsilon$ to $u$ in $L^\infty(\Omega)$. There exists $C>0$ independent of $\epsilon \in (0,1)$ such that
\[
\|u^\epsilon - u\|_{L^\infty(\Omega)} \leq C \epsilon^{1/2}.
\]
See \cite{CL-rate, CagnettiGomesTran-2, Tran19} for more details.
\end{remark}
\section{A Carleman estimate} \label{sec:Carleman}
We prove a Carleman estimate that plays an important role in our proof for the convergence of the convexification method.
In the proof of the Carleman estimate, we will need the notation
\begin{equation}
\Gamma^- = \big\{
{\bf x} = (x_1, x_2, \dots, x_{d - 1}, z = -R): |x_i| \leq R, 1 \leq i \leq d - 1
\big\}
\subset \partial \Omega.
\end{equation}
\begin{Theorem}[Carleman estimate]
For $\beta > 1$, $r > R + 1$, and $b > R + r$, we can find a number $\lambda_0 = \lambda_0(\beta, r, R, b, d)$ such that for all $\lambda > \lambda_0$
and
for all $u \in C^2(\overline \Omega)$ with $u = 0$ on $\partial \Omega$ and $u_z = 0$ on $\Gamma^+$, we have
\begin{multline}
\int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2 d{\bf x}
\geq C\lambda^3 \beta^2 (\beta - 1) b^{-3\beta} (-R + r)^{2\beta} \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}
\\
+ C \lambda (\beta - 1)b^{-\beta}\int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\label{Car est}
\end{multline}
where $C = C(r, R, b, d)$ is a constant.
Here, $\lambda_0$ and $C$ depend only on the listed parameters.
\label{thm Car est}
\end{Theorem}
\proof
In the proof below, $C_1,$ $C_2$ and $C_3$ are constants depending only on $r$, $R$, $b$, and $d$. We split the proof into several steps.
\noindent{\it Step 1.}
Define the function
\begin{equation}
v = e^{\lambda (\frac{z + r}{b})^\beta} u.
\end{equation}
For ${\bf x} = (x_1, \dots, x_{d- 1}, z)$, $i = 1, \dots, d - 1$, we have
\begin{align*}
u &= e^{-\lambda (\frac{z + r}{b})^\beta} v,
&u_{x_i} &=e^{-\lambda (\frac{z + r}{b})^\beta} v_{x_i}, \\
u_{x_i x_i} &=e^{-\lambda (\frac{z + r}{b})^\beta} v_{x_i x_i},
&u_z &= e^{-\lambda (\frac{z + r}{b})^\beta}\Big(-\lambda \beta b^{-\beta} (z + r)^{\beta-1} v + v_z\Big),\\
\end{align*}
and
\[
u_{zz} = e^{-\lambda (\frac{z + r}{b})^\beta}\Big(\Big( \lambda^2 \beta^2 b^{-2\beta} (z + r)^{2\beta - 2} -\lambda \beta (\beta-1) b^{-\beta}(z + r)^{\beta - 2}\Big) v
- 2\lambda \beta b^{-\beta} (z + r)^{\beta - 1} v_z + v_{zz} \Big).
\]
Therefore, using the inequality $(a - b + c)^2 \geq -2ba - 2bc$, we have
\begin{align}
&e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2 \nonumber\\
=\ & \Big[ \big(\lambda^2 \beta^2 b^{-2\beta} (z + r)^{2\beta - 2} -\lambda \beta (\beta-1) b^{-\beta}(z + r)^{\beta - 2}\big) v
- 2\lambda \beta b^{-\beta} (z + r)^{\beta - 1} v_z + \Delta v \Big]^2 \nonumber
\\
\geq \ &- 4\lambda^2 \beta^2 b^{-2\beta} (z + r)^{2\beta - 3} \big(\lambda \beta b^{-\beta} (z + r)^{\beta } -(\beta-1)\big) v_z v
-4\lambda \beta b^{-\beta} (z + r)^{\beta - 1} v_z \Delta v.
\label{3.3}
\end{align}
Dividing both sides of \eqref{3.3} by $2\lambda \beta b^{-\beta}(z + r)^{\beta - 1}$ and integrating the resulting equation, we have
\begin{equation}
\int_{\Omega}\frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{2\lambda \beta b^{-\beta} (z + r)^{\beta - 1}} \,d{\bf x}
\geq I_1 + I_2,
\label{step1}
\end{equation}
where
\begin{equation}
I_1 = -2\lambda \beta\int_{\Omega} b^{-\beta}(z + r)^{\beta - 2} \big(\lambda \beta b^{-\beta} (z + r)^{\beta } -(\beta-1)\big) v_z v \,d{\bf x}
\label{I1}
\end{equation}
and
\begin{equation}
I_2 = -2\int_{\Omega} v_z \Delta v\,d{\bf x}.
\label{I2}
\end{equation}
\noindent{\it Step 2.} In this step, we estimate $I_2$.
Since $u = 0$ on $\partial \Omega$, $v = 0$ on $\partial \Omega$.
Since $u = u_z = 0$ on $\Gamma^+$, $v_z = 0$ on $\Gamma^+$.
Hence, $v_z = 0$ on $\partial \Omega \setminus \Gamma^-$.
Using integration by parts, we have
\begin{align*}
I_2 &= -2 \int_{\Omega} {\rm div}( v_z \nabla v)\,d{\bf x}
+ 2 \int_{\Omega} \nabla v_z\cdot \nabla v\,d{\bf x}
= -2\int_{\Gamma^+} |v_z|^2\,d\sigma
+ 2\int_{\Gamma^-} |v_z|^2\,d\sigma
+ \int_{\Omega} (|\nabla v|^2)_z \,d{\bf x}
\\
&
=
2\int_{\Gamma^-} |v_z|^2\,d\sigma
+ \int_{\Gamma^+} |\nabla v|^2 \,d\sigma
- \int_{\Gamma^-} |\nabla v|^2 \,d\sigma.
\end{align*}
Since $v_{x_i} = 0$ on $\Gamma^+ \cup \Gamma^-$, $i = 1, \dots, d - 1$, $|\nabla v| = |v_z|$ on this set.
Using the fact that $v_z = 0$ on $\partial \Omega \setminus \Gamma^-$ again, we have
\begin{equation}
I_2= \int_{\Gamma^+} |v_z|^2\,d\sigma
+ \int_{\Gamma^-} |v_z|^2\,d\sigma
\geq
0.
\label{3.6}
\end{equation}
\noindent{\it Step 3.} We estimate $I_1.$ It follows from \eqref{I1} that
\begin{align*}
I_1 &= -\lambda \beta\int_{\Omega} b^{-\beta}(z + r)^{\beta - 2} \big(\lambda \beta b^{-\beta} (z + r)^{\beta } - (\beta-1)\big) (|v|^2)_z \,d{\bf x}
\\
& =-\lambda \beta\int_{\Omega} \big[ b^{-\beta}(z + r)^{\beta - 2} \big(\lambda \beta b^{-\beta} (z + r)^{\beta } -(\beta-1)\big) |v|^2\big]_z \,d{\bf x}
\\
&\hspace{6cm}
+ \lambda \beta\int_{\Omega} \big[ b^{-\beta}(z + r)^{\beta - 2} \big(\lambda \beta b^{-\beta} (z + r)^{\beta } -(\beta-1)\big)\big]_z |v|^2 \,d{\bf x}
\\
&= \lambda \beta \int_{\Omega} \big[2\lambda \beta (\beta - 1) b^{-2\beta}(z + r)^{2\beta - 3}
- (\beta-1)(\beta - 2) b^{-\beta}(z + r)^{\beta - 3}
\big]|v|^2\,d{\bf x}.
\end{align*}
Now, for fixed $\beta > 1$, we can find $\lambda_0 \geq 1$, only depending on $\beta$, $r$ and $R$, sufficiently large such that for all $\lambda > \lambda_0$,
\begin{equation}
I_1 \geq C_1\lambda^2 \beta^2 (\beta - 1) b^{-2\beta} \int_{\Omega} (z + r)^{2\beta} |v|^2 d{\bf x}.
\label{3.7}
\end{equation}
\noindent{\it Step 3}.
Combining \eqref{step1}, \eqref{3.6} and \eqref{3.7}, we have
\begin{equation}
\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{2\lambda \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}\geq
C_1\lambda^2 \beta^2 (\beta - 1) b^{-2\beta} \int_{\Omega} (z + r)^{2\beta} |v|^2 d{\bf x}.
\label{3.8}
\end{equation}
Recall that $v = e^{\lambda (\frac{z + r}{b})^\beta} u$.
It follows from \eqref{3.8} that
\begin{equation*}
\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{2\lambda \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq
C_1\lambda^2 \beta^2 (\beta - 1) b^{-2\beta} \int_{\Omega} (z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}.
\end{equation*}
or equivalently,
\begin{equation}
\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{ \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq
C_1\lambda^3 \beta^2 (\beta - 1) b^{-2\beta} \int_{\Omega} (z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}.
\label{3.9}
\end{equation}
\noindent {\it Step 4.}
We estimate the term $\displaystyle \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} u\Delta u \,d{\bf x}$.
Using integration by parts, since $u = 0$ on $\partial \Omega,$ we have
\begin{align*}
-\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} u\Delta u \,d{\bf x}
&= -\int_{\Omega} {\rm div}\big[ e^{2\lambda (\frac{z + r}{b})^\beta} u\nabla u\big] \,d{\bf x}
+ \int_{\Omega} \nabla(e^{2\lambda (\frac{z + r}{b})^\beta} u) \cdot \nabla u \,d{\bf x}
\\
&\hspace{-3cm}= \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta}\big[2\lambda \beta b^{-\beta}(z + r)^{\beta-1}u + u_z\big]u_z\,d{\bf x}
+ \sum_{i = 1}^{d - 1}\int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|u_{x_i}|^2\,d{\bf x}
\\
&\hspace{-3cm}
= \lambda \beta \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} b^{-\beta} (z + r)^{\beta-1} (|u|^2)_z\,d{\bf x}
+ \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}
\\
&\hspace{-3cm}
= \lambda \beta \int_{\Omega} \Big[e^{2\lambda (\frac{z + r}{b})^\beta} b^{-\beta}(z + r)^{\beta-1} |u|^2\Big]_z\,d{\bf x}
-\lambda \beta \int_{\Omega} \Big[e^{2\lambda (\frac{z + r}{b})^\beta} b^{-\beta} (z + r)^{\beta-1}\Big]_z |u|^2\,d{\bf x}
\\
&\hspace{6cm}
+ \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta b^{-\beta}}|\nabla u|^2\,d{\bf x}.
\end{align*}
Due to the fact that $u = 0$ on $\partial \Omega$, the first integral in the last row above vanishes.
We have
\begin{multline*}
-\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} u\Delta u \,d{\bf x} =
-\lambda \beta \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} \Big(\lambda \beta b^{-2\beta} (z + r)^{2\beta - 2} + (\beta - 1)b^{-\beta} (z + r)^{-2+\beta}\Big) |u|^2\,d{\bf x}
\\
+ \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta b^{-\beta}}|\nabla u|^2\,d{\bf x}.
\end{multline*}
Hence, for $\lambda > \lambda_0$,
\begin{equation}
-\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} u\Delta u \,d{\bf x}
\geq
-C_2\lambda^2 \beta^2 b^{-2\beta} \int_{\Omega}
(z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta}
|u|^2\,d{\bf x}
+ \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\label{3.12}
\end{equation}
\noindent{\it Step 5.} We complete the proof in this step. Using the inequality $x^2 + y^2 \geq -2xy$, we have
\begin{multline}
\lambda^{5/2} \beta b^{-\beta} \int_{\Omega} (z + r)^{\beta-1} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}
+ \int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{4\lambda^{1/2} \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\\
\geq -\lambda\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} u\Delta u \,d{\bf x}.
\label{3.1222}
\end{multline}
Combining \eqref{3.12} and \eqref{3.1222}, we have
\begin{multline*}
\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{4\lambda^{1/2} \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq
- \lambda^{5/2} \beta b^{-\beta} \int_{\Omega} (z + r)^{\beta-1} e^{2\lambda (\frac{z + r}{b})^\beta} |u|^2 d{\bf x}
\\
-C_2\lambda^3 \beta^2 b^{-2\beta} \int_{\Omega}
(z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta}
|u|^2\,d{\bf x}
+ \lambda \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x},
\end{multline*}
which implies
\begin{multline}
\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{4\lambda^{1/2} \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq
-C_3\lambda^3 \beta^2 b^{-2\beta} \int_{\Omega}
(z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta}
|u|^2\,d{\bf x}
\\
+ \lambda \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\label{3.1414}
\end{multline}
Multiply $\frac{C_1(\beta - 1)}{2C_3}$ to both sides of \eqref{3.1414} where $C_1$ is the constant in \eqref{3.9}, we yield
\begin{multline}
\frac{C_1(\beta - 1)}{2C_3}\int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{4\lambda^{1/2} \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq
- \frac{C_1\lambda^3 \beta^2(\beta - 1) b^{-2\beta} }{2} \int_{\Omega}
(z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta}
|u|^2\,d{\bf x}
\\
+ \frac{C_1 \lambda (\beta - 1)}{2C_3} \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\label{3.13}
\end{multline}
Adding \eqref{3.9} and \eqref{3.13},
we have
\begin{multline}
\Big(1 + \frac{C_1(\beta - 1)}{2C_3\lambda^{1/2}}\Big) \int_\Omega \frac{e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta u|^2}{ \beta b^{-\beta} (z + r)^{\beta - 1}} d{\bf x}
\geq \frac{C_1\lambda^3 \beta^2(\beta - 1) b^{-2\beta} }{2} \int_{\Omega}
(z + r)^{2\beta} e^{2\lambda (\frac{z + r}{b})^\beta}
|u|^2\,d{\bf x}
\\
+ \frac{C_1 \lambda (\beta - 1)}{2C_3} \int_\Omega e^{2\lambda (\frac{z + r}{b})^\beta}|\nabla u|^2\,d{\bf x}.
\label{3.14}
\end{multline}
The desired Carleman estimate \eqref{Car est} follows from \eqref{3.14}.
\QEDB
\begin{remark}
In our previous publications, see e.g. \cite{NguyenLiKlibanov:IPI2019, LeNguyen:2020}, the number $\beta$ must be large. The reason for \eqref{Car est} holds true when $\beta > 1$ is our trick of dividing both sides of \eqref{3.3} by $2\lambda \beta b^{-\beta}(z + r)^{\beta - 1}$ so that the corresponding $I_2$, defined in \eqref{I2}, is nonnegative.
This trick was used in \cite[Theorem 3.1]{LeNguyen:2020}. However, since the Carleman weight in \cite[Theorem 3.1]{LeNguyen:2020} is different from the one in \eqref{Car est}, the parameter $\beta$ in \cite[Theorem 3.1]{LeNguyen:2020} must be large.
In the case when the principal differential operator in \eqref{Car est} is replaced by the general elliptic operator, this trick is not applicable.
\end{remark}
\section{The convexification method to compute the viscosity solution to Hamilton-Jacobi equations} \label{sec conv}
In this section, we propose to use the convexification method to solve \eqref{HJ} together with the boundary condition \eqref{dir} supposing that Assumption \ref{assumption 1} holds true.
That means we know the boundary value $u = f$ on $\partial \Omega$ and the function $u_z|_{\Gamma^+}$ can be computed, say $u_z = g$ on $\Gamma^+.$
Due to Theorem \ref{Theorem vis}, it is natural to try to approximate the solution $u$ by a function $u^{\epsilon_0}$ that satisfies
\begin{equation}
\left\{
\begin{array}{rcll}
-\epsilon_0 \Delta u^{\epsilon_0} + F({\bf x}, u^{\epsilon_0}, \nabla u^{\epsilon_0}) &=& 0 &{\bf x} \in \Omega,\\
u^{\epsilon_0} &=& f({\bf x}) &{\bf x} \in \partial\Omega,\\
u^{\epsilon_0}_z &=& g({\bf x}) &{\bf x} \in \Gamma^+.
\end{array}
\right.
\label{4.1}
\end{equation}
\begin{remark}
1. In general, \eqref{4.1} is over-determined.
It might have no solution; especially when the boundary data contains noise.
However, the convexification method can deliver a function that ``most fits" \eqref{4.1} and show that this function is an approximation of the true solution to \eqref{HJ}--\eqref{dir} when the given boundary data are noiseless.
Again, due to Assumption \ref{assumption 1}, the Neumann condition imposed in \eqref{4.1} makes sense.
2. On the other hand, \eqref{4.1} is not over-determined in the sense that
\begin{equation}
\left\{
\begin{array}{rcll}
-\epsilon_0 \Delta u^{\epsilon_0} + F({\bf x}, u^{\epsilon_0}, \nabla u^{\epsilon_0}) &=& 0 &{\bf x} \in \Omega,\\
u^{\epsilon_0} &=& f({\bf x}) &{\bf x} \in \partial\Omega,\\
\end{array}
\right.
\label{4.2222}
\end{equation}
might not be uniquely solvable if we do not impose Assumption \ref{assumption 2} or other sets of appropriate conditions.
For example, when $F({\bf x}, u^{\epsilon_0}, \nabla u^{\epsilon_0}) = -u^{\epsilon_0}$, $f=0$, and $\frac{1}{\epsilon_0}$ is an eigenvalue of $-\Delta$, \eqref{4.2222} has multiple solutions.
Therefore, imposing the additional Neumann boundary condition on $\Gamma^+$, in the general case, is necessary.
\end{remark}
Let $p > \lceil d/2 \rceil + 2$ such that $H^p(\Omega) \hookrightarrow C^2(\overline \Omega).$
Define
\[
H = \{
u \in H^p(\Omega): u|_{\partial \Omega} = f \mbox{ and } u_z|_{\Gamma^+} = g
\}.
\]
Clearly, $H$ is a closed subset of $H^p(\Omega).$
We will also need the following set of test functions
\[
H_0 = \{
u \in H^p(\Omega): u|_{\partial \Omega} = 0 \mbox{ and } u_z|_{\Gamma^+} = 0
\}.
\]
Let $M > 0$ be chosen later, and set
\[
{B(M)} = \{u \in H^p(\Omega): \|u\|_{H^p(\Omega)} < M\}.
\]
We assume that $H \cap \overline{B(M)} \not = \emptyset$.
For each $\lambda > 1$, $\beta > 1$ and $\eta > 0$, introduce the functional
\begin{equation}
J_{\lambda, \beta, \eta}(u) = \int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} \big|-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u)\big|^2 \,d{\bf x}
+ \eta \|u\|^2_{H^p(\Omega)}
\quad u \in H \cap \overline{B(M)}
\label{Jdef}
\end{equation}
where $\epsilon_0$ is a fixed small positive number.
\medskip
The convexification theorem is to prove that for each $\beta > 1$, there is a number $\lambda_0 > 1$ such that for all $\lambda \geq \lambda_0$ and for all $\eta > 0$, the function $J_{\lambda, \beta, \eta}$ is strictly convex in $H \cap \overline{B(M)}.$
The word ``convexification" is suggested by the fact that the Carleman weight function $e^{2\lambda (\frac{z + r}{b})^\beta}$ convexifies this functional.
The convexification theorem is stated below.
\begin{Theorem}[The convexification theorem]
1. For all $\lambda, \beta >1$, and $\eta > 0$, for all $u \in H^p(\Omega)$, $h\in H_0$, we have \begin{equation}
\lim_{H_0 \in h \to 0} \frac{|J_{\lambda, \beta, \eta}(u + h) - J_{\lambda, \beta, \eta}(u) -
DJ_{\lambda, \beta, \eta}(u)h |}{\|h\|_{H^p(\Omega)}} = 0
\label{4.2}
\end{equation}
where
\begin{multline}
DJ_{\lambda, \beta, \eta}(u)h =
2\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta}\Big[-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u)\Big]
\Big[-\epsilon_0 \Delta h + \partial_s F({\bf x}, u, \nabla u) h
\\
+ \nabla_{{\bf p}} F({\bf x}, u, \nabla u) \cdot \nabla h\Big]\,d{\bf x}
+ 2\eta\langle u, h\rangle_{H^p(\Omega)}.
\label{DJ}
\end{multline}
2. Let $M$ be an arbitrarily large number.
For each $\beta> 1$,
$\lambda >\lambda_0 = \lambda_0(\epsilon_0, M, b, d, r, F, \beta) > 1$, $\eta > 0$, $u \in H\cap \overline{B(M)}$ and $v \in H \cap \overline{B(M)}$, we have
\begin{equation}
J_{\lambda, \beta, \eta}(v) - J_{\lambda, \beta, \eta}(u) - DJ_{\lambda, \beta, \eta}(u)h \geq C \|v - u\|^2_{H^1(\Omega)} + \eta \|v - u\|^2_{H^p(\Omega)},
\label{convex}
\end{equation}
and
\begin{equation}
\langle DJ_{\lambda, \beta, \eta}(v) - DJ_{\lambda, \beta, \eta}(u),(v - u)\rangle_{H^p(\Omega)} \geq C \|v - u\|^2_{H^1(\Omega)} + \eta \|v - u\|^2_{H^p(\Omega)}.
\label{convex1}
\end{equation}
Here, the constant $C$ depends only on $\lambda,$ $\beta$, $R,$ $r,$ $b$, $d$, $M$, $F$ and $\epsilon_0$.
As a result, the functional $J_{\lambda, \beta, \eta}$ has a unique minimizer in $\overline{B(M)}$.
\label{thm convex}
\end{Theorem}
The key point for us to successfully establish the inequalities \eqref{convex} and \eqref{convex1} is the presence of the Carleman weight function $e^{2\lambda (\frac{z + r}{b})^\beta}$ and the use of the Carleman estimate \eqref{Car est}.
A direct consequence of the inequalities \eqref{convex} and \eqref{convex1} is the strict convexity of $J_{\lambda, \beta, \eta}$ in $H \cap \overline{B(M)}.$
It is worth mentioning that $C$ in \eqref{convex}--\eqref{convex1} depends on the viscosity coefficient $\epsilon_0$, and we need $\lambda >\lambda_0 = \lambda_0(\epsilon_0, M, b, d, r, F, \beta) > 1$ so that $J_{\lambda, \beta, \eta}$ is convex in $H \cap \overline{B(M)}.$
On the other hand, $D^2 J_{\lambda, \beta, \eta} \geq \eta Id$ in $H \cap \overline{B(M)}$, which means that the uniform convexity of $J_{\lambda, \beta, \eta}$ only depends on $\eta$, not $\epsilon_0$.
The proof of Theorem \ref{thm convex} is similar to that of the convexification theorem in \cite{KlibanovNik:ra2017}, which is originally designed to solve highly nonlinear and severely ill-posed inverse problems. Since this is the first time the convexification method is employed in the area of numerical methods for Hamilton-Jacobi equations, we present the proof here for the reader's convenience.
\begin{remark}
For all $u \in H^p(\Omega)$, since $DJ_{\lambda, \beta, \eta}(u)$ is a bounded linear map from $H_0$ into $\mathbb{R}$, by the Riesz theorem there exists uniquely the function $J'_{\lambda, \beta, \nu}(u) \in H_0$ such that
\begin{equation*}
\langle J'_{\lambda, \beta, \nu}(u), h \rangle_{H^p(\Omega)} = DJ_{\lambda, \beta, \eta}(u) h
\end{equation*}
for all $h \in H_0.$
\label{rem 4.2}
\end{remark}
\proof[Proof of Theorem \ref{thm convex}]
Since $F$ is in the class $C^2(\overline \Omega \times \mathbb{R} \times \mathbb{R}^d)$, for all $u, h \in H^p(\Omega)$ we can write
\begin{equation}
F({\bf x}, u + h, \nabla (u + h)) = F({\bf x}, u, \nabla u) + \partial_s F({\bf x}, u, \nabla u) h + \nabla_{{\bf p}} F({\bf x}, u, \nabla u) \cdot \nabla h
+ O(|h|^2) + O(|\nabla h|^2).
\label{4.4444}
\end{equation}
Here, $O(s)$ is the quantity satisfying $|O(s)| \leq C|s|$ where $C$ is a constant that might depend on an upper bound of $\|u\|_{C^1(\overline \Omega)}$ and the function $F$.
Using \eqref{Jdef} and \eqref{4.4444}, we obtain
\begin{align*}
J_{\lambda, \beta, \eta}(&u + h) - J_{\lambda, \beta, \eta}(u) - 2\eta \langle u, h\rangle_{H^p(\Omega)} - \eta\|h\|^2_{H^p(\Omega)}
\\
&= \int_{\Omega} e^{2\lambda(\frac{z + r}{b})^\beta} \Big(|-\epsilon_0 \Delta(u+h) + F({\bf x}, u + h, \nabla (u + h))|^2 - |-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u)|^2\Big)\,d{\bf x}
\\
&= \int_{\Omega} e^{2\lambda(\frac{z + r}{b})^\beta} \big(-\epsilon_0 \Delta h + F({\bf x}, u + h, \nabla (u + h)) - F({\bf x}, u, \nabla u)\big)\big(-\epsilon_0 \Delta h - 2 \epsilon_0 \Delta u
\\
&\hspace{8cm}
+ F({\bf x}, u + h, \nabla (u + h)) + F({\bf x}, u, \nabla u)\big)\,d{\bf x} \\
&= \int_{\Omega} e^{2\lambda(\frac{z + r}{b})^\beta} \Big[-\epsilon_0 \Delta h + \partial_s F({\bf x}, u, \nabla u) h + \nabla_{{\bf p}} F({\bf x}, u, \nabla u) \cdot \nabla h + O(|h|^2) + O(|\nabla h|^2)\Big]
\\ &\hspace{1cm} \Big[
2\big(-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u)\big)
-\epsilon_0 \Delta h
+ \partial_s F({\bf x}, u, \nabla u) h + \nabla_{{\bf p}} F({\bf x}, u, \nabla u) \cdot \nabla h
\\
&\hspace{10cm}+ O(|h|^2) + O(|\nabla h|^2)
\Big]\,d{\bf x}.
\end{align*}
Hence,
\begin{multline}
J_{\lambda, \beta, \eta}(u + h) - J_{\lambda, \beta, \eta}(u) - 2\eta\langle u, h\rangle_{H^p(\Omega)} - \eta\|h\|^2_{H^p(\Omega)}
- 2\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} \Big[ -\epsilon_0 \Delta u + F({\bf x}, u, \nabla u) \Big]
\\
\Big[-\epsilon_0 \Delta h
+ \partial_s F({\bf x}, u, \nabla u) h
+ \nabla_{{\bf p}} F({\bf x}, u, \nabla u) \cdot \nabla h\Big]\,d{\bf x}
\\
=2\int_{\Omega} e^{2\lambda (\frac{z + r}{b})^\beta} \Big[\epsilon_0^2|\Delta h|^2 + O(|h|^2) + O(|\nabla h|^2)\Big]\,d{\bf x}.
\label{4.4}
\end{multline}
Defining $DJ_{\lambda, \beta, \eta}$ as in \eqref{DJ} and using \eqref{4.4}, we have
\begin{multline*}
\lim_{H_0 \ni h \to 0} \frac{|J_{\lambda, \beta, \eta}(u + h) - J_{\lambda, \beta, \eta}(u) - DJ_{\lambda, \beta, \eta}(u)h|}{\|h\|_{H^p(\Omega)}}
\\
\leq
\lim_{H_0 \ni h \to 0} \frac{|J_{\lambda, \beta, \eta}(u + h) - J_{\lambda, \beta, \eta}(u) - DJ_{\lambda, \beta, \eta}(u)h|}{\|h\|_{H^2(\Omega)}} = 0.
\end{multline*}
We have proved part 1 of this theorem.
We next prove part 2.
\medskip
For any $u$ and $v$ in $H \cap \overline{B(M)}$, set $h = v - u \in H_0$. It follows from \eqref{4.4} that
\begin{align}
J_{\lambda, \beta, \eta}(v) - J_{\lambda, \beta, \eta}(u) - & DJ_{\lambda, \beta, \eta}(u)(v - u)
= J_{\lambda, \beta, \eta}(u + h) - J_{\lambda, \beta, \eta}(u) - DJ_{\lambda, \beta, \eta}(u)h \nonumber\\
&\geq \int_{\Omega}e^{2\lambda (\frac{z + r}{b})^\beta}\Big[ \epsilon_0^2|\Delta h|^2 - O(|h|^2) - O(|\nabla h|^2)\Big]\,d{\bf x} + \eta \|h\|_{H^p(\Omega)}^2. \label{4.6}
\end{align}
Note that $h|_{\partial \Omega}=0$ and $h_z|_{\Gamma^+} = 0$.
Applying Theorem \ref{thm Car est}, for each $\beta > 1$, we can find $\lambda_0$ depending on $R$, $M$, $F$, $\beta,$ $b$ and $d$ such that for all $\lambda > \lambda_0$,
\begin{equation}
\int_{\Omega}e^{2\lambda (\frac{z + r}{b})^\beta} |\Delta h|^2\,d{\bf x} \geq C\lambda \int_{\Omega}e^{2\lambda (\frac{z + r}{b})^\beta}( |\nabla h|^2 + \lambda^2 |h|^2)\,d{\bf x}.
\label{4.7}
\end{equation}
Here, the constant $C$ is allowed to depend on $\beta.$
We now choose $\lambda$ such that $\epsilon_0^2 \lambda$ is sufficiently large.
Combining \eqref{4.6} and \eqref{4.7}, we have
\begin{equation*}
J_{\lambda, \beta, \eta}(v) - J_{\lambda, \beta, \eta}(u) - DJ_{\lambda, \beta, \eta}(u)(v - u)\geq C \|h\|^2_{H^1(\Omega)} + \eta \|h\|^2_{H^p(\Omega)}.
\end{equation*}
We have proved \eqref{convex}.
Here, $C$ depends on $\lambda,$ $\beta,$ $\epsilon_0$, $R$, $M$, $b$ and $d$.
Interchanging the roles of $u$ and $v$ in \eqref{convex} and adding the resulting estimate to \eqref{convex}, we obtain \eqref{convex1}.
\medskip
We next prove that if \eqref{convex1} holds true, $J_{\lambda, \beta, \eta}$ has a unique minimizer.
The existence of the minimizer is obvious. It follows from the fact that $H \cap \overline{B(M)}$ is convex in $H^p(\Omega)$ and the compact embedding of $H^p(\Omega)$ to $H^2(\Omega).$
An alternative way to obtain the existence of the minimizer is to argue similarly to the proofs of Theorem 2.1 in \cite{KlibanovNik:ra2017} or Theorem 4.1 in \cite{KlibanovLeNguyenARL:preprint2021}.
We now prove the uniqueness.
Let $u_1$ and $u_2$ be two local minimizers of $J_{\lambda, \beta, \eta}$ on $H \cap \overline{B(M)}$.
It is clear that, see \cite[Lemma 2]{KlibanovNik:ra2017},
\[
DJ_{\lambda, \beta, \eta}(u_1)(u_2 - u_1) \geq 0 \quad
\mbox{and} \quad
DJ_{\lambda, \beta, \eta}(u_2)(u_1 - u_2) \geq 0.
\]
Thus,
\begin{equation}
(DJ_{\lambda, \beta, \eta}(u_1) - DJ_{\lambda, \beta, \eta}(u_2))(u_1 - u_2) \leq 0
\label{4.12}
\end{equation}
Combining \eqref{convex1} for $u_1$ and $u_2$ and \eqref{4.12}, we have
\[
C\|u_1 - u_2\|^2_{H^1(\Omega)} + \eta \|u_1 - u_2\|^2_{H^p(\Omega)} \leq 0.
\]
The proof is complete.
\QEDB
\begin{Theorem}[The convergence of the gradient descent method]
Let $\lambda, \beta$ and $\eta$ as in part 2 of Theorem \ref{thm convex} and let $J'_{\lambda, \beta, \eta}$ be as in Remark \ref{rem 4.2}.
Let $u^{(0)}$ be any function in $H \cap B(M/3)$.
For $k \in \mathbb{N}$, define
\begin{equation}
u^{(k)} = u^{(k - 1)} - \kappa J_{\lambda, \beta, \eta}'(u^{(k-1)})
\label{gdm}
\end{equation}
for all $ \kappa \in (0, \kappa_0)$ where $ \kappa_0 \in (0, 1)$ is a number that depends only on $\lambda,$ $\beta$, $R,$ $r,$ $b$, $d$, $M$ and $\epsilon_0$.
Then, if the minimizer $u_{\rm min}$ of $J_{\lambda, \beta, \eta}$ is in $H \cap B(M/3)$ then
there is a number $\theta \in (0, 1)$ depending only on $ \kappa,$ $\lambda,$ $\beta$, $R,$ $r,$ $b$, $d$, $M$, $F$ and $\epsilon_0$ such that for all $k \geq 1$
\begin{equation}
\big\|u^{(k)} - u_{\rm min}\big\|_{H^p(\Omega)} \leq \theta^{k/2} \big\|u^{(0)} - u_{\rm min} \big\|_{H^p(\Omega)}.
\label{4.1313}
\end{equation}
\label{thm 4.2}
\end{Theorem}
Theorem \ref{thm 4.2} and the estimate \eqref{4.1313} guarantee that the minimizer of $J_{\lambda, \beta, \eta}$ can be found by the popular gradient descent method.
The success is due to the hypothesis that the desired minimizer is in the interior of $B(M/3)$.
We do not experience any difficulty due to not checking this condition in the numerical study.
However, to be more rigorous,
in general, if this condition cannot be verified, one can use the projected gradient method as in \cite{KlibanovNik:ra2017, KhoaKlibanovLoc:SIAMImaging2020} to find the minimizer. In this paper, we choose the gradient descent method because the implementation of the projection in the projected gradient method is more complicated while there are many ready-to-use packages; for e.g., the optimization toolbox of Matlab, for the gradient descent method.
In other words, Theorem \ref{thm 4.2} significantly reduces our efforts in implementation.
Although the proof of Theorem \ref{thm 4.2} is similar to the proofs of \cite[Theorem 6]{Klibanov:2ndSAR2021} in 1D case and \cite[Theorem 2.2]{LeNguyen:preprint2021} in higher dimensions, we briefly present the proof of Theorem \ref{thm 4.2} here for the convenience of the reader.
\proof[Proof of Theorem \ref{thm 4.2}]
Since $F$ is a function in the class $C^2(\overline \Omega \times \mathbb{R} \times \mathbb{R}^d)$, it is obvious that $\nabla_{{\bf x}}F$, $\partial_s F$, $\nabla_{{\bf p}} F$ are all Lipschitz continuous in any bounded subdomain of $\overline \Omega \times \mathbb{R} \times \mathbb{R}^{d}$.
As a result, $DJ_{\lambda, \beta, \eta}$, see \eqref{DJ} for its definition, is Lipschitz continuous on $H \cap \overline{B(M)}$.
Hence, there is a positive number $L$ such that
\begin{equation}
\|J_{\lambda, \beta, \eta}'(u_1) - J_{\lambda, \beta, \eta}'(u_2)\|_{H^p(\Omega)} \leq L \|u_1 - u_2\|_{H^p(\Omega)}
\label{4.13}
\end{equation}
for all $u_1, u_2 \in H \cap B(M).$
We claim that for any $k \geq 0$,
\begin{equation}
\|u^{(k)} - u_{\rm min}\|_{H^p(\Omega)} < \frac{2M}{3}.
\label{bbb}
\end{equation}
This is true when $k = 0$. Assume \eqref{bbb} is true for some $k$, we will prove that \eqref{bbb} holds true for $k + 1.$ Due to \eqref{bbb}, $u^{(k)} \in H \cap B(M)$.
Since $u_{\rm min}$ is in $H \cap B(M/3)$, $J_{\lambda, \beta, \eta}'(u_{\rm min}) = 0$.
Using \eqref{convex1} and \eqref{4.13}, we have
\begin{align*}
\big\|u^{(k+1)} &- u_{\rm min}\big\|_{H^p(\Omega)}^2
= \big\|u^{(k)} - u_{\rm min} - \kappa [J_{\lambda, \beta, \eta}'(u^{(k)}) - J_{\lambda, \beta, \eta}'(u_{\rm min})] \big\|_{H^p(\Omega)}^2
\\
&= \big\|u^{(k )} - u_{\rm min}\big\|_{H^p(\Omega)}^2
- 2 \kappa \big\langle J_{\lambda, \beta, \eta}'(u^{(k)}) - J_{\lambda, \beta, \eta}'(u_{\rm min}), u^{(k )} - u_{\rm min}\big\rangle_{H^p(\Omega)}
\\
&\hspace{7cm}
+ \kappa^2\big\|J_{\lambda, \beta, \eta}'(u^{(k)}) - J_{\lambda, \beta, \eta}'(u_{\rm min})\big\|_{H^p(\Omega)}^2
\\
&\leq (1 - 2 \kappa\eta + \kappa^2 L^2)\big\|u^{(k)} - u_{\rm min} \big\|_{H^p(\Omega)}^2.
\end{align*}
Choosing $ \kappa \in (0, \kappa_0)$ where $ \kappa_0 = \eta L^{-2}$, we have $\theta = 1 - 2 \kappa\eta + \kappa^2 L^2 \in (0, 1)$ and
\begin{equation}
\big\|u^{(k+1)} - u_{\rm min}\big\|_{H^p(\Omega)} \leq \theta^{1/2} \big\|u^{(k)} - u_{\rm min} \big\|_{H^p(\Omega)} < \big\|u^{(k)} - u_{\rm min} \big\|_{H^p(\Omega)} < \frac{2M}{3}.
\label{4.15}
\end{equation}
We have proved \eqref{bbb}. The estimate \eqref{4.1313} follows from \eqref{4.15}.
\QEDB
\medskip
Consider the case when the boundary data $f = u|_{\partial \Omega}$ contains noise with the noise level $\delta > 0$.
Since the knowledge of $g = u_z|_{\Gamma^+}$ can be computed from the knowledge of $u$ on this set, see Assumption \ref{assumption 1} and Remark \ref{rm 1}, the Neumann data $u_z|_{\Gamma^+}$ also contains noise. We assume that the noise level is still $\delta$.
Denote by $f^{\delta}$ and $g^{\delta}$ the noisy boundary data and denote the corresponding noiseless data $f^*$ and $g^*$.
Here, by saying that $\delta$ is the noise level, there exists an ``error" function $\mathcal E$ satisfying
\[
\left\{
\begin{array}{ll}
\|\mathcal E\|_{H^p(\Omega)} \leq \delta,\\
\mathcal E|_{\partial \Omega} = f^\delta - f^*,\\
\mathcal E_z|_{\Gamma^+} = g^\delta - g^*.
\end{array}
\right.
\]
The following theorem guarantees that the minimizer of $J_{\lambda, \beta, \eta}$ subject to the boundary constraints defined by the noisy data is an approximation of the true solution to \eqref{4.1}.
\begin{Theorem}
Assume that the set
\[
H^{\delta} = \{
u \in H^p(\Omega): u|_{\partial \Omega} = f^\delta \mbox{ and } u_z|_{\Gamma^+} = g^\delta
\}
\] is nonempty
and let $u^{\delta}_{\min}$ be the minimizer of $J_{\lambda, \beta, \eta}$ in $H^{\delta}.$
Assume further that Problem \eqref{4.1}, in which the Dirichlet and Neumann data $f$ and $g$ are replaced by $f^*$ and $g^*$, respectively, has the unique solution $u^{\epsilon_0} \in B(M - \delta)$.
Then, for all $\lambda$ and $\beta$ such that \eqref{convex} holds true, we have
\begin{equation}
\|u_{\min}^\delta - u^{\epsilon_0}\|^2_{H^1(\Omega)} \leq C(\eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + \delta^2).
\label{4.1717}
\end{equation}
\label{thm 4.3}
\end{Theorem}
\proof
Define $u = u^{\epsilon_0} + \mathcal{E}.$
We have $u \in H^{\delta} \cap B(M).$
Since $u^\delta_{\min}$ is the minimizer of $J_{\lambda, \beta, \eta}$ in $H^{\delta}$, by \cite[Lemma 2]{KlibanovNik:ra2017},
\[\langle J_{\lambda, \beta, \eta}'(u^\delta_{\min}), u^\delta_{\min} - u^{\epsilon_0} - \mathcal{E} \rangle_{H^p(\Omega)}\leq 0.\]
Applying \eqref{convex} for $u$ and $u_{\min}$ gives
\begin{align}
J_{\lambda, \beta, \eta}(u^{\epsilon_0} + \mathcal{E})
& \geq J_{\lambda, \beta, \eta}(u^{\epsilon_0} + \mathcal{E}) - J_{\lambda, \beta, \eta}(u^\delta_{\rm min}) - \langle J_{\lambda, \beta, \eta}'(u^{\delta}_{\rm min}), u^{\epsilon_0} + \mathcal{E} - u^{\delta}_{\rm min} \rangle_{H^p(\Omega)}
\nonumber
\\
&\geq C \|u^{\epsilon_0} + \mathcal{E} - u^\delta_{\min}\|^2_{H^1(\Omega)} + \eta \|u^{\epsilon_0} + \mathcal{E} - u^\delta_{\min}\|^2_{H^p(\Omega)}
\nonumber
\\
&\geq C \|u^{\epsilon_0} + \mathcal{E} - u^\delta_{\min}\|^2_{H^1(\Omega)}. \label{4.17}
\end{align}
Applying \eqref{4.2}, since $u^{\epsilon_0}$ solves \eqref{4.1}, we have
\begin{align}
J_{\lambda, \beta, \eta}(u^{\epsilon_0} + \mathcal{E})
&= J_{\lambda, \beta, \eta}(u^{\epsilon_0}) + J_{\lambda, \beta, \eta}'(u^{\epsilon_0})(\mathcal E) + C\delta^2
= \eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + 2\eta\langle u^{\epsilon_0}, \mathcal E \rangle_{H^p(\Omega)} + C\delta^2 \nonumber
\\
&\leq \eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + \eta^2\|u^{\epsilon_0}\|_{H^p(\Omega)}^2 + C\delta^2
\leq 2\eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + C\delta^2.
\label{4.18}
\end{align}
Using the inequality $(a - b)^2 \geq \frac{1}{2}a^2 - b^2$ and combining \eqref{4.17} and \eqref{4.18}, we have
\begin{align*}
\frac{1}{2}\|u^{\epsilon_0} - u^\delta_{\min}\|^2_{H^1(\Omega)} - \|\mathcal{E}\|^2_{H^1(\Omega)}
\leq \|u^{\epsilon_0} + \mathcal{E} - u^\delta_{\min}\|^2_{H^1(\Omega)}
\leq C(\eta \|u^{\epsilon_0}\|^2_{H^p(\Omega)} + \delta^2)
\end{align*}
The estimate \eqref{4.1717} follows.
\QEDB
\medskip
Combining Theorem \ref{thm 4.2} and Theorem \ref{thm 4.3}, we have for each $k \geq 1$,
\[
\|u^{(k)} - u^{\epsilon_0}\|_{H^1(\Omega)} \leq C(\sqrt{\eta} \|u^{\epsilon_0}\|_{H^p(\Omega)} + \delta) +
\theta^{k/2} \|u^{(0)} - u^\delta_{\min}\|_{H^p(\Omega)}
\]
for some constant $C > 0$ and $\theta \in (0, 1)$ where $u^{(k)}$ is the minimizing sequence defined in Theorem \ref{thm 4.2}. This inequality shows the stability of our method with respect to noise. If $\theta^{k}$ and $\eta$ are $O(\delta^2)$, then the convergence rate is Lipschitz.
\begin{remark}
In the case when the function $F$ is such that the comparison principle for $-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u) = 0$ holds true; for e.g., $F$ is strictly increasing with respect to its second variable (Assumption \ref{assumption 2}), one can prove the stability of $u$ with respect to noise without imposing Assumption \ref{assumption 1} by the fact that $u - \delta$ and $u+\delta$ are a subsolution and supersolution of $-\epsilon_0 \Delta u + F({\bf x}, u, \nabla u) = 0$, respectively. The main reason for us to successfully establish the stability without assuming the comparison principle is due to the presence of the Neumann data in Assumption \ref{assumption 1}.
\label{rm 4.2}
\end{remark}
In the next section, we present the numerical implementation and some interesting numerical results.
\section{Numerical study} \label{sec:num}
We implement the convexification method based on the finite difference method. For the simplicity, in this section, we only consider the case $d = 2.$
On $\overline{\Omega} = [-R, R]^2$, we arrange a uniform grid of points
\begin{equation}
\mathcal{G} = \big\{{\bf x}_{ij} = (x_i, z_j): x_i = -R + (i - 1) \delta_x, z_j = -R + (j - 1)\delta_z), 1 \leq i, j \leq N\big\}
\label{5.1}
\end{equation}
where $\delta_x = \delta_z = h = 2R/(N - 1)$.
In our computation, $R = 1$ and $N = 50.$
The finite difference version of the objective function $J_{\lambda, \beta, \eta}(u)$ is given by
\begin{multline}
J_{\lambda, \beta, \eta}(u) = h^2 \sum_{i, j = 2}^{N-1} e^{2\lambda \big|\frac{z_j + r)}{b}\big|^\beta}
\Big|
-\epsilon_0 \Delta^h u(x_i, z_j)
+ F((x_i, z_j), u(x_i, z_j), \partial_x^h u(x_i, z_j), \partial_z^h u(x_i, z_j))
\Big|^2
\\
+ \eta h^2 \Big(\sum_{i, j = 1}^{N}|u(x_i, z_j)|^2 +\sum_{i, j = 2}^{N-1} \big( |\partial_x^h u(x_i, z_j)|^2 + |\partial_z^h u(x_i, z_j)|^2 + |\Delta^h u(x_i, z_j)|^2\big)\Big).
\end{multline}
where
\begin{align*}
\Delta^h u(x_i, z_j) &= \frac{u(x_i, z_{j-1}) + u(x_i, z_{j + 1}) + u(x_{i-1}, z_j) + u(x_{i+1}, z_{j}) -4u(x_i, z_j) }{h^2}\\
\partial_x^h u(x_i, z_j) &=\frac{u(x_{i+1}, z_{j}) - u(x_{i - 1}, z_{j})}{2h}\\
\partial_z^h u(x_i, z_j) &= \frac{u(x_i, z_{j+1}) - u(x_i, z_{j-1})}{2h}.
\end{align*}
\begin{remark}
1. In our computation $R = 1$, $N = 50$, $\epsilon_0 = 10^{-3}$, $\lambda = 2,$ $\beta = 8$, $b = 10$, $r = 1.2$ and $\eta = 10^{-4}$. Although in the theoretical part, the parameters $\lambda$ and $\beta$ should be large, these values are already good for the numerical part. For the simplicity, we use this set of parameters for all tests below.
2. In theory $p > 3$ when $d = 2$. However, in numerical study, we can reduce the norm in the regularization term to $p = 2$ to simplify the implementation and to improve the speed of computation. We do not experience any difficulty with this small change.
\end{remark}
In our implementation, instead of writing the computational code for the gradient descent method, we use the optimization toolbox of Matlab, in which the gradient descent method is coded.
More precisely, we use the command ``fmincon" to minimize the functional $J_{\lambda, \beta, \eta}$ subject to the boundary constraint in \eqref{4.1}.
The command ``fmincon" requires an initial solution, that is the function $u^{(0)}$ in Theorem \ref{thm 4.2}.
The function $u^{(0)}$ is naturally assigned to be the zero function.
This function $u^{(0)}$ does not satisfy the Dirichlet boundary condition on $\partial \Omega$ and Neumann condition on $\Gamma^+$. However, the command ``fmincon" corrects this error automatically.
We present here five (5) numerical tests in which the given boundary data are noisy with noise level $\delta = 0\%$, $5\%$, and $10\%$, respectively.
In each test, $u_{\rm true}$ and $u_{\rm comp}$ denote the true and computed viscosity solutions, respectively.
Given functions $f,g$, the noisy versions of $f, g$ are given by
\[
f^\delta = f(1 + \delta \cdot \mbox{rand}) \quad \text{ and } \quad
g^\delta = g(1 + \delta \cdot \mbox{rand}),
\]
where rand is the function that generates uniformly distributed random numbers in $[-1, 1].$
The relative computed error is defined as
\[
{\rm err}(\delta) = \frac{\|u_{\rm comp} - u_{\rm true}\|_{L^{\infty}(\Omega)}}{\|u_{\rm true}\|_{L^{\infty}(\Omega)}} \quad \text{ for } \delta > 0.
\]
{\it Test 1.} In this test, we find the viscosity solution to \eqref{HJ}--\eqref{dir} where
\begin{equation}
F({\bf x}, s, {\bf p}) = \frac{1}{150} s + |{\bf p}| + \frac{1}{150} (x^2 + z^2) - 2\sqrt{x^2 + z^2} \quad \mbox{for all } {\bf x} \in \Omega, s \in \mathbb{R}, {\bf p} \in \mathbb{R}^2
\label{F1}
\end{equation} and the boundary data are given by
\begin{equation}
u({\bf x}) = f({\bf x}) = -(x^2 + z^2)\quad \mbox{for all } {\bf x} = (x, z) \in \partial\Omega
\label{f1}
\end{equation}
and
\begin{equation}
u_z({\bf x}) = g({\bf x}) = -2z \quad \mbox{for all } {\bf x} = (x, z) \in \Gamma^+.
\label{g1}
\end{equation}
The true solution is $u_{\rm true}(x, z) = -(x^2 + z^2)$, which is smooth in $\Omega$.
We are here in a standard setting, and the convergence of $u^\epsilon$, the solution to \eqref{2.1}, to the solution $u_{\rm true}$ to \eqref{HJ}--\eqref{dir} is guaranteed in \cite{CrandallLions83, CrandallEvansLions84, Lions, Barles, BCD,Tran19}.
\begin{figure}[h!]
\subfloat[The true solution $u_{\rm true} = -(x^2 + z^2)$.]{\includegraphics[width=.3\textwidth]{u_trueModel1}}
\subfloat[The solution $u_{\rm comp}$, computed from noiseless boundary data.]{\includegraphics[width=.3\textwidth]{up2Model1Noise0}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 5\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model1Noise5}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 10\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model1Noise10}}
\subfloat[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 0\%$.}]{\includegraphics[width=.3\textwidth]{u_diff1Noise0}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 5\%$.}]{\includegraphics[width=.3\textwidth]{u_diff1Noise5}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 10\%$.}]{\includegraphics[width=.3\textwidth]{u_diff1Noise10}}
\caption{\label{fig 1} Test 1. The true and computed viscosity solutions with $\delta$ is $0\%$, 5\%, and 10\% noisy boundary Dirichlet data on $\partial \Omega$ and Neumann data on $\Gamma^+$. }
\end{figure}
We show in Figure \ref{fig 1} the graphs of this true solution and that of the computed ones from noiseless and noisy boundary data.
It is evident that we successfully obtained computed viscosity solutions to \eqref{HJ}--\eqref{dir}. By adding high level noise into the boundary data, we have numerically shown that the convexification is stable.
{It is evident from Table \ref{tab1} that the relative computed error is about the noise level,
which clearly illustrates the Lipschitz stability in Theorem \ref{thm 4.3}.}
In this test, the true solution is smooth.
The function $F({\bf x}, s, {\bf p})$ is strictly increasing with respect to $s$.
\begin{table}[h!]
\caption{\label{tab1} {Test 1. The performance of the convexification method. The computational time is the time for a Precisions Workstations T7810 with 24 cores to compute the solution $u_{\rm comp}$. In this table,
the relative $L^\infty(\Omega)$ error is $\|u_{\rm comp} - u_{\rm true}\|_{L^\infty(\Omega)}/\|u_{\rm true}\|_{L^\infty(\Omega)}$}.}
{
\centering
\begin{tabular}{c c c c }
\hline\hline
Noise level &computational time& number of iterations & relative $ L^\infty(\Omega)$ error\\
\hline
0\%&23.47 minutes&279 &$0.24\%$ \\
5\%& 24.40 minutes &292&4.51\%\\
10\%& 27.35 minutes &329&9.95\%\\
\hline
\end{tabular}
}
\end{table}
\medskip
{\it Test 2.} We now solve the {\it eikonal} equation of which the function $F$ in \eqref{HJ} is not in the class $C^1$.
In this test, the function $F$ is given by
\begin{equation}
F({\bf x}, s, {\bf p}) = |{\bf p}| - \sqrt{2}
\quad \mbox{for all } {\bf x} \in \Omega, s \in \mathbb{R}, {\bf p} \in \mathbb{R}^2
\label{F2}
\end{equation}
and the boundary data are
\begin{equation}
u({\bf x}) = f({\bf x}) = -(|x| + |z|)
\quad \mbox{for all } {\bf x} = (x, z) \in \partial \Omega
\label{f2}
\end{equation}
and
\begin{equation}
u_z({\bf x}) = g({\bf x}) = \left\{
\begin{array}{rl}
1 &z < 0,\\
-1 &z > 0,
\end{array}
\right.
\quad \mbox{for all } {\bf x} = (x, z) \in \Gamma^+.
\label{g2}
\end{equation}
We claim that the true solution to \eqref{HJ}--\eqref{dir} is $u_{\rm true}({\bf x}) = -(|x| + |z|)$ for all ${\bf x} = (x, z) \in \Omega.$
Intuitively, this claim holds as the graph of $u_{\rm true}$ only has corners from above and $F$ is convex in ${\bf p}$.
Let us provide a rigorous verification here.
If $x \neq 0$ and $z \neq 0$, then $u_{\rm true}$ is differentiable at ${\bf x}=(x,z)$, and
\[
\nabla u_{\rm true}({\bf x}) =\left(-\frac{x}{|x|}, - \frac{z}{|z|} \right ) \quad \Rightarrow \quad |\nabla u_{\rm true}({\bf x}) |=\sqrt{2}.
\]
If $x=0$ or $z=0$, then $u_{\rm true}$ is not differentiable at ${\bf x}=(x,z)$.
We can only find smooth test functions that touch $u_{\rm true}$ from above at ${\bf x}$, and we cannot find smooth test functions that touch $u_{\rm true}$ from below at ${\bf x}$.
Let $\phi$ be a smooth test function that touches $u_{\rm true}$ from above at ${\bf x}$.
Without loss of generality, we only need to consider the case $x=0$.
If $z \neq 0$, then we have that
\[
\phi_x({\bf x}) \in [-1,1], \ \phi_z({\bf x})=-\frac{z}{|z|}.
\]
If $z = 0$, then we have that
\[
\phi_x({\bf x}) \in [-1,1], \ \phi_z({\bf x}) \in [-1,1].
\]
In both cases,
\[
|\nabla \phi({\bf x}) | \leq \sqrt{2}.
\]
Thus, the subsolution test holds for $u_{\rm true}$ at ${\bf x}$.
We conclude that $u_{\rm true}$ is a viscosity solution to \eqref{HJ}--\eqref{dir}.
This true solution and its computed versions $u_{\rm comp}$ from noisy boundary data are displayed in Figure \ref{fig 2}.
The convergence of $u^\epsilon$, the solution to \eqref{2.1}, to the solution $u_{\rm true}$ to \eqref{HJ}--\eqref{dir} is guaranteed in \cite{FlemingSouganidis, Tran11} with convergence rate $O(\epsilon^{1/2})$.
\begin{figure}[h!]
\subfloat[The true solution $u_{\rm true} = -(|x| + |z|)$.]{\includegraphics[width=.3\textwidth]{u_trueModel2}}
\subfloat[The solution $u_{\rm comp}$, computed from noiseless boundary data.]{\includegraphics[width=.3\textwidth]{up2Model2Noise0}}
\subfloat[The solution $u_{\rm comp}$, computed from 5\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model2Noise5}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 10\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model2Noise10}}
\subfloat[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 0\%$.}]{\includegraphics[width=.3\textwidth]{u_diff2Noise0}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 5\%$.}]{\includegraphics[width=.3\textwidth]{u_diff2Noise5}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 10\%$.}]{\includegraphics[width=.3\textwidth]{u_diff2Noise10}}
\caption{\label{fig 2} Test 2. The true and computed viscosity solutions with $\delta$ is $0\%$, 5\%, and 10\% noisy boundary Dirichlet data on $\partial \Omega$ and Neumann data on $\Gamma^+$. }
\end{figure}
This test is more challenging than Test 1.
In this test, the functions $F$ and $f$ are both not smooth.
So, the smoothness condition for the theoretical part does not satisfy.
However, the convexification method still provides reliable solutions even when the given boundary is noisy with the noise level $\delta = 10\%.$
This shows the robustness of the convexification method.
{The relative errors are acceptable, see Table \ref{tab2}.}
\begin{table}[h!]
\caption{\label{tab2} {Test 2. The performance of the convexification method. The computational time is the time for a Precisions Workstations T7810 with 24 cores to compute the solution $u_{\rm comp}$. In this table,
the relative $L^\infty(\Omega)$ error is $\|u_{\rm comp} - u_{\rm true}\|_{L^\infty(\Omega)}/\|u_{\rm true}\|_{L^\infty(\Omega)}$.}}
\centering
{
\begin{tabular}{c c c c }
\hline\hline
Noise level &computational time& number of iterations & relative $ L^\infty(\Omega)$ error\\
\hline
0\%& 48 minutes&581&3.75\%\\
5\%& 89 minutes&1071& 5.23\%\\
10\%& 64 minutes & 766 &12.22 \%\\
\hline
\end{tabular}
}
\end{table}
\medskip
{\it Test 3.}
We next consider a more complicated Hamilton-Jacobi equation. Unlike the previous two tests, the function $F$ in this test is not convex and; more interestingly, not coercive and not continuous. It is given by
\begin{multline}
F({\bf x}, s, {\bf p}) =20 s + |p_1| - |p_2|
\\
- \Big(20\big(-|x - 0.5| + e^{\sin(\pi(x^2 + z^2))}\big) + G(x, z) - \Big|2\pi z \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))}\Big| \Big)
\label{5.9999}
\end{multline}
for all ${\bf x} = (x, z) \in \Omega, s \in \mathbb{R}, {\bf p}= (p_1, p_2) \in \mathbb{R}^2$ where
\[
G(x, z) =
\left\{
\begin{array}{ll}
\big|1 + 2\pi x \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))}\big| &x < 0.5,\\
\big|-1 + 2\pi x \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))}\big| &x > 0.5.
\end{array}
\right.
\]
Note that $G$ and $F$ are discontinuous at $x=0.5$ in general.
The boundary data are given by
\begin{equation}
u({\bf x}) = f({\bf x}) = - |x - 0.5| + e^{\sin(\pi(x^2 + z^2))} \quad \mbox{for all } {\bf x} = (x, z) \in \partial \Omega
\end{equation}
and
\begin{equation}
u_z({\bf x}) = g({\bf x}) = 2\pi z \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))} \quad \mbox{for all } {\bf x} = (x, z) \in \Gamma^+.
\end{equation}
The function $u_{\rm true}(x, z) = - |x - 0.5| + e^{\sin(\pi(x^2 + z^2))} $ is the true viscosity solution for this test.
Indeed, if $x \neq 0.5$, then $u_{\rm true}$ is differentiable at ${\bf x}=(x,z)$, and
\[
\nabla u_{\rm true}({\bf x}) =\left(-\frac{x-0.5}{|x-0.5|}+ 2\pi x \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))}, 2\pi z \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))} \right ),
\]
which gives that $F({\bf x},u_{\rm true}({\bf x}),\nabla u_{\rm true}({\bf x}) )=0$.
If $x=0.5$, then $u_{\rm true}$ is not differentiable at ${\bf x}=(x,z)$.
We can only find smooth test functions that touch $u_{\rm true}$ from above at ${\bf x}$, and we cannot find smooth test functions that touch $u_{\rm true}$ from below at ${\bf x}$.
Let $\phi$ be a smooth test function that touches $u_{\rm true}$ from above at ${\bf x}$.
Then,
\[
\phi_x({\bf x}) \in 2\pi x \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))}+[-1,1], \ \phi_z({\bf x})=2\pi z \cos(\pi(x^2 + z^2)) e^{\sin(\pi(x^2 + z^2))},
\]
which yields that $F_*({\bf x},u_{\rm true}({\bf x}),\nabla \phi({\bf x}) ) \leq 0$.
Here, $F_*$ is the lower semicontinuous envelope of $F$.
Therefore, the subsolution test holds for $u_{\rm true}$ at ${\bf x}$.
We imply that $u_{\rm true}$ is a viscosity solution to \eqref{HJ}--\eqref{dir}.
\begin{figure}[h!]
\subfloat[The true solution $u_{\rm true}(x, z) = - |x - 0.5| + e^{\sin(\pi(x^2 + z^2))} $.]{\includegraphics[width=.3\textwidth]{u_trueModel3}}
\subfloat[The solution $u_{\rm comp}$, computed from noiseless boundary data.]{\includegraphics[width=.3\textwidth]{up2Model3Noise0}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 5\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model3Noise5}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 10\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model3Noise10}}
\subfloat[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 0\%$.}]{\includegraphics[width=.3\textwidth]{u_diff3Noise0}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 5\%$.}]{\includegraphics[width=.3\textwidth]{u_diff3Noise5}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 10\%$.}]{\includegraphics[width=.3\textwidth]{u_diff3Noise10}}
\caption{\label{fig 3} Test 3. The true and computed viscosity solutions with $\delta$ is $0\%$, 5\%, and 10\% noisy boundary Dirichlet data on $\partial \Omega$ and Neumann data on $\Gamma^+$.}
\end{figure}
The numerical results are given in Figure \ref{fig 3}.
As mentioned, this test is interesting since the function $F$, see \eqref{5.9999}, is nonconvex, noncoercive, and discontinuous.
{Solving the Hamilton-Jacobi equation with this Hamiltonian is challenging.}
Some existing methods might not be applicable.
In contrast, the numerical results in Figure \ref{fig 3} are out of expectation.
The errors of computation are small, see Table \ref{tab3}, although the solution has complicated structure.
This kind of nonconvex Hamiltonian occurs in the context of two-player zero-sum differential games (see \cite{BCD, Tran19}).
\begin{table}[h!]
\caption{\label{tab3}{Test 3. The performance of the convexification method. The computational time is the time for a Precisions Workstations T7810 with 24 cores to compute the solution $u_{\rm comp}$. In this table,
the relative $L^\infty(\Omega)$ error is $\|u_{\rm comp} - u_{\rm true}\|_{L^\infty(\Omega)}/\|u_{\rm true}\|_{L^\infty(\Omega)}$.}}
{
\centering
\begin{tabular}{c c c c }
\hline\hline
Noise level &computational time& number of iterations & relative $ L^\infty(\Omega)$ error\\
\hline
0\%& 10.72 minutes & 129 & 1.32\%\\
5\%& 11.12 minutes &131 & 2.04\%\\
10\%& 8.43 minutes & 101 & 3.97\%\\
\hline
\end{tabular}
}
\end{table}
\begin{remark}
In Tests 1, 2 and 3 above, we are in the context that the knowledge of $u_z$ on $\Gamma^+$ can be computed from the knowledge of $u$ and the form of the Hamilton-Jacobi equation, see Assumption \ref{assumption 1} and Remark \ref{rm 1}.
However, if the given Hamilton-Jacobi equation is rather complicated as in Tests 4 and 5 below, solving $u_z|_{\Gamma^+}$ from $u|_{\Gamma^+}$ is impossible.
In this case, we minimize $J_{\lambda, \beta, \eta}$ on the set $\{u \in H^p(\Omega): u({\bf x}) = f({\bf x}) \mbox{ for } {\bf x} \in \partial \Omega\}$.
The convexification method still gives us out of expectation numerical results in these two tests.
However, the proofs of the convexification theorem and the convergence of the numerical scheme for the problem with only Dirichlet boundary condition \eqref{HJ}--\eqref{dir} are extremely challenging, and they are out of the scope of this paper.
\end{remark}
\medskip
{\it Test 4.} We next consider the G-equation, which arises from instantaneous flame position.
We solve \eqref{HJ}--\eqref{dir} when
\begin{equation}
F({\bf x}, s, {\bf p}) = s + |{\bf p}| - x p_1
\label{F4}
\end{equation}
for ${\bf x} \in \Omega, s \in \mathbb{R}, {\bf p} = (p_1, p_2)\in \mathbb{R}^2$
and the boundary data is given by
\begin{equation}
f({\bf x}) = -|x| - 1 \quad {\bf x} = (x, z) \in \partial \Omega.
\label{f4}
\end{equation}
The function $u_{\rm true}(x, z) = -|x| - 1$ is the true viscosity solution for this test.
The verification of this is similar to that of Tests 2 and 3, and is hence omitted here.
The numerical results are given in Figure \ref{fig 5}. Relative errors are $0.91\%$, $4.97\%$ and $9.99\%$ when $\delta$ is 0\%, 5\% and $10\%$ respectively {(see Table \ref{tab4})}.
The G-equation is quite popular in the combustion science literature.
We refer the readers to \cite{CarNoSou, XinYu, LiuXinYu} for some recent important mathematical developments.
\begin{figure}[h!]
\subfloat[The true solution $u_{\rm true} = -|x| - 1$.]{\includegraphics[width=.3\textwidth]{u_trueModel5}}
\subfloat[The solution $u_{\rm comp}$, computed from noiseless boundary data.]{\includegraphics[width=.3\textwidth]{up2Model5Noise0}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 5\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model5Noise5}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 10\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model5Noise10}}
\subfloat[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 0\%$.}]{\includegraphics[width=.3\textwidth]{u_diff5Noise0}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 5\%$.}]{\includegraphics[width=.3\textwidth]{u_diff5Noise5}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 10\%$.}]{\includegraphics[width=.3\textwidth]{u_diff5Noise10}}
\caption{\label{fig 5} Test 4. The true and computed viscosity solutions with $\delta$ is $0\%$, 5\% and 10\% noisy boundary Dirichlet data on $\partial \Omega$.}
\end{figure}
\begin{table}[h!]
\caption{\label{tab4}{Test 4. The performance of the convexification method. The computational time is the time for a Precisions Workstations T7810 with 24 cores to compute the solution $u_{\rm comp}$.
The relative $L^\infty(\Omega)$ error is $\|u_{\rm comp} - u_{\rm true}\|_{L^\infty(\Omega)}/\|u_{\rm true}\|_{L^\infty(\Omega)}$.}}
{
\centering
\begin{tabular}{c c c c }
\hline\hline
Noise level &computational time& number of iterations & relative $ L^\infty(\Omega)$ error\\
\hline
0\%& 22.62 minutes & 280 &0.91\% \\
5\%& 24.99 minutes & 311 & 4.97\%\\
10\%&26.50 minutes & 322& 9.99\%\\
\hline
\end{tabular}
}
\end{table}
\medskip
{\it Test 5.} We finally consider a quite complicated form of the function $F$.
For ${\bf x} = (x, z) \in \Omega$, let
\[
G(x, z) = \left\{
\begin{array}{ll}
1 + 2\pi \cos(2\pi(x + z)) &x > 0,\\
-1 + 2\pi \cos(2\pi(x + z)) &x < 0.
\end{array}
\right.
\]
We solve \eqref{HJ}--\eqref{dir} when
\begin{multline}
F({\bf x}, s, {\bf p}) = 15s + \min \left\{ |{\bf p}|, ||{\bf p}|-10|+6 \right\}
- \Big[15(-|x| + \sin(2\pi (x + z)))
\\
+\min\left\{ |\sqrt{|G(x, z)|^2 + 4\pi^2\cos(2\pi (x + z))}|, \big|\sqrt{|G(x, z)|^2 + 4\pi^2\cos(2\pi (x + z))}-10\big|+6 \right\} \Big]
\label{F6}
\end{multline}
for ${\bf x} \in \Omega, s \in \mathbb{R}, {\bf p} \in \mathbb{R}^2$,
and the Dirichlet boundary data is given by
\begin{equation}
f({\bf x}) = -|x| + \sin(2\pi (x + z)) \quad \text{ for all } {\bf x} = (x, z) \in \partial \Omega.
\label{f6}
\end{equation}
It it worth mentioning that $F, G$ are not continuous in general at $x=0$.
The function $u_{\rm true}(x, z) = -|x| + \sin(2\pi (x + z))$ is the true viscosity solution for this test.
Its graph and the graphs of the computed solutions are displayed in Figure \ref{fig 6}.
Let us give a rigorous verification here.
If $x \neq 0$, then $u_{\rm true}$ is differentiable at $(x, z)$, and
\[
\nabla u_{\rm true}(x, z) = \left(-\frac{x}{|x|}+2\pi \cos(2\pi(x+z)), 2\pi \cos(2\pi(x+z)) \right),
\]
which implies that $F({\bf x},u({\bf x}),\nabla u_{\rm true}({\bf x}))=0$.
If $x=0$, then $u_{\rm true}$ is not differentiable at ${\bf x}=(x,z)$.
We can only find smooth test functions that touch $u_{\rm true}$ from above at ${\bf x}$, and we cannot find smooth test functions that touch $u_{\rm true}$ from below at ${\bf x}$.
Let $\phi$ be a smooth test function that touches $u_{\rm true}$ from above at ${\bf x}$.
Then,
\[
\phi_x({\bf x}) \in 2\pi \cos(2\pi z)+[-1,1], \ \phi_z({\bf x})=2\pi \cos(2\pi z),
\]
which yields that
\[
|\nabla \phi({\bf x})|^2 \leq 1 + 4 \pi + 4 \pi^2 \quad \Rightarrow \quad |\nabla \phi({\bf x})| \leq 1 + 2\pi <8.
\]
Therefore,
\begin{align*}
F_*({\bf x},u_{\rm true}({\bf x}),\nabla \phi({\bf x}) ) &= 10 \sin(2\pi z) + |\nabla \phi({\bf x})| - 10 \sin(2\pi z) - \sqrt{|G|^*(0, z) + 4\pi^2\cos(2\pi z)}\\
&= |\nabla \phi({\bf x})| - \sqrt{1 + 2\pi |\cos(2\pi z)|+4\pi^2} \leq 0.
\end{align*}
Here, $F_*$ is the lower semicontinuous envelope of $F$, and $|G|^*$ is the upper semicontinuous envelope of $|G|$.
Thus, the subsolution test holds for $u_{\rm true}$ at ${\bf x}$.
We imply that $u_{\rm true}$ is a viscosity solution to \eqref{HJ}--\eqref{dir}.
It is evidently clear that the convexification method delivers satisfactory numerical results.
{The relative errors are provided in Table \ref{tab5}}.
This kind of Hamiltonian was considered in \cite{QianTranYu} in the context of the periodic homogenization theory.
\begin{figure}[h!]
\subfloat[The true solution $u_{\rm true} = -|x| + \sin(2\pi (x + z))$.]{\includegraphics[width=.3\textwidth]{u_trueModel6}}
\subfloat[The solution $u_{\rm comp}$, computed from noiseless boundary data.]{\includegraphics[width=.3\textwidth]{up2Model6Noise0}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 5\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model6Noise5}}
\quad
\subfloat[The solution $u_{\rm comp}$, computed from 10\% noisy boundary data.]{\includegraphics[width=.3\textwidth]{up2Model6Noise10}}
\subfloat[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 0\%$.}]{\includegraphics[width=.3\textwidth]{u_diff6Noise0}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 5\%$.}]{\includegraphics[width=.3\textwidth]{u_diff6Noise5}}
\quad
\subfloat[[{The relative error $\frac{|u_{\rm comp} - u_{\rm true}|}{\|u_{\rm true}\|_{L^{\infty}}}$, $\delta = 10\%$.}]{\includegraphics[width=.3\textwidth]{u_diff6Noise10}}
\caption{\label{fig 6} Test 5. The true and computed viscosity solutions with $\delta$ is $0\%$, 5\%, and 10\% noisy boundary Dirichlet data on $\partial \Omega$. {The relative errors with these noise levels are given in Table \ref{tab5}.}}
\end{figure}
\begin{table}[h!]
\caption{\label{tab5}{Test 5. The performance of the convexification method. The computational time is the time for a Precisions Workstations T7810 with 24 cores to compute the solution $u_{\rm comp}$. In this table,
the relative $L^\infty(\Omega)$ error is $\|u_{\rm comp} - u_{\rm true}\|_{L^\infty(\Omega)}/\|u_{\rm true}\|_{L^\infty(\Omega)}$.}}
{
\centering
\begin{tabular}{c c c c }
\hline\hline
Noise level &computational time& number of iterations & relative $ L^\infty(\Omega)$ error\\
\hline
0\%& 11.83 minutes &144 & 1.78\%\\
5\%& 10.06 minutes&121 &4.97\%\\
10\%& 11.54 minutes&143 &9.77\%\\
\hline
\end{tabular}
}
\end{table}
\begin{remark}
It follows from tests 3 and 5 that the convexification method is effective in the interesting case when the function $F$ is nonconvex in ${\bf p}.$
The non-convexity is illustrated in Figure \ref{fig nonconvex}.
This numerically confirms the strength of our method in solving Hamilton-Jacobi equations.
We refer the readers to \cite{KaoOsherQian, LiQian} for some other examples dealing with non-convex, discontinuous Hamiltonians via the Lax-Friedrichs sweeping method.
\end{remark}
\begin{figure}[h!]
\begin{center}
\subfloat[${\bf p} \mapsto |p_1| - |p_2|$.]{\includegraphics[width=.35\textwidth]{nonconvex3}}
\quad
\subfloat[${\bf p} \mapsto \min \left\{ |{\bf p}|, ||{\bf p}|-10|+6 \right\}$]{\includegraphics[width=.35\textwidth]{nonconvex6}}
\caption{\label{fig nonconvex} The dependence of the function $F({\bf x}, s, {\bf p})$ on the third variable ${\bf p}$ in Test 3 (a) and Test 5 (b). It is evident that in those tests, the function $F$ is nonconvex in ${\bf p}.$}
\end{center}
\end{figure}
{
\begin{remark}
The relative errors in computation are displayed in the last rows of Figures \ref{fig 1}--\ref{fig 6}.
It is evident from those figures that the errors in computation occur at $\partial \Omega$ where the noise is added and at the places where the true solution $u_{\rm true}$ is not differentiable.
This again reflects the strength of the convexification method.
\end{remark}
}
{
It has been shown both analytically and numerically that the convexification method is robust in solving a large class of Hamilton-Jacobi equations.
The strengths of the convexification method involves the facts
\begin{enumerate}
\item that it does not require any special structure of the Hamiltonian; especially, the convexity condition of the Hamiltonian with respect to the third variable is relaxed;
\item that it yields the satisfactory numerical solutions even when the given boundary is noisy.
\end{enumerate}
However, the convexification method has a drawback.
It is time consuming in comparison to the well-known methods for nonconvex Hamiltonians; for e.g., the Lax--Friedrichs schemes (\cite{OsherShu, Abgrall, OsFe}) and the Lax--Friedrichs sweeping algorithm (\cite{KaoOsherQian,LiQian}).
In our tests, it takes from 10 minutes to 90 minutes, depending on the forms of the given Hamiltonians, for a Workstations T7810 with 24 cores to compute the solutions (see Tables \ref{tab1}--\ref{tab5}).
The slow computational time is acceptable in the sense that we consider the computational program as a ``proof of concept" to numerically confirm the analysis of the convexification method.
The convexification method is the first generalization of the numerical method based on Carleman estimates to solve Hamilton-Jacobi equations.
We expect to improve the computational time in the next generation.
The next generation will be developed based on the fixed point iterative scheme similar to the ones in \cite{LeetalPreprint2021, LeNguyen:2020, NguyenKlibanov:preprint2021} for quasi-linear elliptic and hyperbolic equations. The rate of convergence in those papers is $O(\theta^n)$ as $n \to \infty$ for some $\theta \in (0, 1).$
Hence, the success in reducing computational time is very promising.
}
\section{Concluding remarks}
In this paper, we introduce a new method to solve highly nonlinear Hamilton-Jacobi equations in a rectangular domain.
This method is called the convexification.
The key idea of the convexification is to involve a Carleman weight function into a cost functional defined directly from the equation under consideration.
Using a Carleman estimate, we established some important theoretical results.
The first theorem guarantees that this cost functional is strictly convex and has a unique solution.
Then, we proved in the second theorem that the gradient descent method with sufficiently small step size delivers a sequence converging to the unique minimizer.
Then, in the third theorem, we prove that the minimizer above converges to the solution in the vanishing viscosity process, a good approximation of the viscosity solution to the Hamilton-Jacobi equation, as the noise contained in the boundary data tends to $0$.
The rate of the convergence is Lipschitz.
All theoretical results are valid in the framework that we know the value of the solution on the boundary of the domain and its normal derivative in a part of this boundary.
We have pointed out that this framework is acceptable in some real-world applications.
We have shown some interesting numerical tests in 2D.
These tests confirm the convergence of the convexification method even when the Hamiltonian is not convex or discontinuous.
Moreover, these numerical results are out of expectation even when the solved equations are not in the framework above.
\medskip
As of now, the convexification method is more time consuming in comparison to the well-known methods as addressed by the end of Section \ref{sec:num}.
We expect to improve the computational time in the next generation.
Besides, we also intend to study Hamilton-Jacobi equations with other types of boundary conditions in the near future.
\section*{Acknowledgement} The works of MVK and LHN were supported by US Army Research Laboratory and US Army Research
Office grant W911NF-19-1-0044.
The work of HT is supported in part by NSF grant DMS-1664424 and NSF CAREER grant DMS-1843320.
| {
"timestamp": "2021-11-08T02:05:06",
"yymm": "2104",
"arxiv_id": "2104.05870",
"language": "en",
"url": "https://arxiv.org/abs/2104.05870",
"abstract": "We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of the this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846691281406,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7092019536878768
} |
https://arxiv.org/abs/1307.6477 | On construction and analysis of sparse random matrices and expander graphs with applications to compressed sensing | We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency matrices of lossless expander graphs. We present tail bounds on the probability that the cardinality of the set of neighbors for these graphs will be less than the expected value. The bounds are derived through the analysis of collisions in unions of sets using a {\em dyadic splitting} technique. This analysis led to the derivation of better constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse non mean-zero measurement matrices. | \section{Introduction}
\section{Introduction}\label{sec:intro}
Sparse matrices are particularly useful in applied and computational mathematics because of their low storage complexity and fast implementation as compared to dense matrices. Of late, significant progress has been made to incorporate sparse matrices in compressed sensing, with \cite{berinde2008combining,berinde2008sparse,jafarpour2009efficient,xu2007further} giving both theoretical performance guarantees and also exhibiting numerical results that shows sparse matrices coming from expander graphs can be as good sensing matrices as their dense counterparts. In fact, Blanchard and Tanner \cite{blanchard2012gpu} recently demonstrated in a GPU implementation how well these type of matrices do compared to dense Gaussian and Discrete Cosine Transform matrices even with very small fixed number of nonzeros per column (as considered here).
In this manuscript we consider random sparse matrices that are adjacency matrices of lossless expander graphs. Expander graphs are highly connected graphs with very sparse adjacency matrices, a precise definition of a lossless expander graph is given in Definition \ref{def:expander}.
\begin{definition}
\label{def:expander}
$G \left(U,V,E\right)$ is a lossless $(k,d,\epsilon)$-expander if it is a bipartite graph with $|U| = N$ left vertices, $|V| = n$ right vertices and has a regular left degree $d$, such that any $X \subset U$ with $|X| \leq k$ has a set of neighbors $\Gamma(X) \subset V$ with $|\Gamma(X)| \geq \left(1 - \epsilon \right) d |X|$ neighbors.
\end{definition}
Note that these graphs are {\em lossless} because $\epsilon\ll1$, they are also referred to as {\em unbalanced expanders} in the literature because $n\ll N$ and a $(k,d,\epsilon)$-lossless expander graph has an {\em expansion} of $\left(1 - \epsilon \right) d$. Such graphs have been well studied in theoretical computer science and mathematics and have many applications. Probabilistic constructions of such graphs using random left-regular bipartite graphs with optimal parameters exist but deterministic constructions only achieve sub-optimal parameters, see \cite{capalbo2002randomness} or \cite{hoory2006expander} for a more detailed survey.
Using a novel technique of {\em dyadic splitting of sets}, this work derives quantitative guarantees on the probabilistic construction of these graphs in the form of a bound on the tail probability of the size of the {\em set of neighbors}, $\Gamma(X)$ for a given $X \subset U$, of a randomly generated left-degree bipartite graph. Moreover, this tail bound proves a bound on the tail probability of the {\em expansion} of the graph, $|\Gamma(X)|/|X|$. In addition, we derive the first phase transitions showing regions in parameter space that depicting when a left-regular bipartite graph with a given set of parameters is guaranteed to be a lossless expander with high probability. Similar results in terms of the adjacency matrices of these graphs is also presented. Another contribution of this work is the derivation of sampling theorems comparing performance guarantees for some of the algorithms proposed for compressed sensing using such sparse matrices as well as the more traditional $\ell_1$ minimization compressed sensing formulation. It also provides phase transitions of $\ell_1$ minimization performance guarantees for such sparse matrices compared to what $\ell_2$ restricted isometry constants ($\mathrm{RIC}_2$) analysis yields for Gaussian matrices.
\section{Tail Bound}\label{sec:tail}
Our main result is the presentation of formulae for the expected cardinality of the {\em set of neighbors} of $(k,d,\epsilon)$-lossless expander graphs and the sparse non-mean zero matrices from these graphs. Based on this, we present a tail bound on the probability that this cardinality will be less than the expected value. We start by defining the class of matrices we consider and a key concept of a {\em set of neighbors} used in our derivation.
\begin{definition}
\label{def:1set_expansion}
Let $A$ be an $n\times N$ matrix with $d$ nonzeros in each column. We refer to $A$ as a random a) sparse expander (SE) if every nonzero has value $1$ and b) sparse signed expander (SSE) if every nonzero has value from $\{-1,1\}$.
\end{definition}
The support set of the $d$ nonzeros per column of these matrices are drawn uniformly at random and independently for each column. An SE matrix is an adjacency matrix of $(k,d,\epsilon)$-lossless expander graph while an SSE matrix have random sign patterns in the nonzeros of an adjacency matrix of a $(k,d,\epsilon)$-lossless expander graph. If $A$ is either an SE or SSE it will have only $d$ nonzeros per column and since we fix $d\ll n$, $A$ is therefore extremely sparse.
We formally define the {\em set of neighbors} in both graph theory and linear algebra notation to aid translation between the terminology of the two communities. Denote $A_S$ as a submatrix of $A$ composed of columns of $A$ indexed by the set $S$ with $|S|=s$.
\begin{definition}
\label{def:neighbours}
Consider a bipartite graph $G(U,V,E)$ where $E$ is the set of edges and $e_{ij}=(x_i,y_j)$ is the edge that connects vertex $x_i$ to vertex $y_j$. For a given subset of left vertices $S\subset U$ its set of neighbors $\Gamma(S) \subset V$ is defined as $\Gamma(S) := \{y_j|x_i\in S \mbox{ and } e_{ij}\in E\}$. In terms of the adjacency matrix, $A$, of $G(U,V,E)$ the set of neighbors of $A_S$ denoted by $A_s$, is the set of rows with at least one nonzero.
\end{definition}
Henceforth, we will only use the linear algebra notation $A_s$ which is equivalent to $\Gamma(S)$. Note that $\left|A_s\right|$ is a random variable depending on the draw of the set of columns, $S$, for each fixed $A$. Therefore, we can ask what is the probability that $\left|A_s\right|$ is not greater than $a_s$, in particular where $a_s$ is smaller than the expected value of $\left|A_s\right|$. This is the question that Theorem \ref{thm:prob_bound_1set_expansion} attempts to answers.
\begin{theorem}[Theorem 1.6, \cite{bah2012vanishingly}]
\label{thm:prob_bound_1set_expansion}
For fixed $s,n,N,d$ and $d\leq a_s <\infty$, let an $n\times N$ matrix, $A$ be drawn from either of the classes of matrices defined in Definition \ref{def:1set_expansion}, then
\begin{equation}
\label{eq:prob_bound_1set_expansion}
\hbox{Prob}\left(\left|A_s\right| \leq a_s\right) < p_{max}(s,d) \cdot e^{\left[n\cdot\Psi\left(a_s,\ldots,a_1\right)\right]}
\end{equation}
where $p_{max}(s,d) = \frac{2}{25\sqrt{2\pi s^3d^3}}$, and for random variables $a_s, \ldots, a_2$ and $a_1:=d$, $\Psi\left(a_s,\ldots,a_1\right)$ is given by
\begin{multline*}
\frac{1}{n} \bigg{[}3s\log\left(5d \right) + \sum_{i=1}^{\lceil s/2\rceil} \frac{s}{2i}\left(\left(n-a_i\right) \cdot \hbox{H}\left(\frac{a_{2i}-a_i}{n-a_i}\right) \right. \\ \left. + a_i\cdot \hbox{H}\left(\frac{a_{2i}-a_i}{a_i}\right) - n\cdot \hbox{H}\left(\frac{a_i}{n}\right) \right) \bigg{]},
\end{multline*}
where $\hbox{H}(\cdot)$ is the Shannon entropy function of base $e$ logarithm. Consequently:
\begin{enumerate}
\item if no restriction is imposed on $a_s$ then the $a_i$ for $i>1$ take on the expected values of $\left|A_s\right|$, which are given by $ \hat{a}_{2i} = \hat{a}_{i}\left(2 - \frac{\hat{a}_{i}}{n}\right)$ for $i=1,2,4,\ldots,\lceil s/2\rceil$;
\item else if $a_{s}$ is restricted to be less than $\hat{a}_{s}$, then the $a_i$ for $i>1$ are the unique solutions to the following polynomial system $ a_{2i}^3 - 2a_ia_{2i}^2 + 2a_i^2a_{2i} - a_i^2a_{4i} = 0$ for $i = 1, 2,\ldots,\lceil s/4\rceil$ with $a_{2i}\ge a_i$ for each $i$.
\end{enumerate}
\end{theorem}
Theorem \ref{thm:prob_bound_1set_expansion} gives a bound on the probability that the cardinality of a union of $k$ sets each with $d$ elements is less than $a_k$. Figure \ref{fig:cardinalities1} shows plots of values of $\left|A_k\right|$ (size of set of neighbors) for different $k$ taken over 500 realizations (in blue), superimposed on these plots is the empirical mean values of $\left|A_k\right|$ over the 500 runs (in red) and the $\hat{a}_k$ in green.
\vspace{-2mm}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah02.eps}
\caption{For fixed $d=8$ and $n=2^{10}$, over $500$ realizations, plots (in blue) of the cardinalities of the index sets of nonzeros in a given number of set sizes, $k$. The dotted red curve is mean of the simulations and the green squares are the $\hat{a}_k$.}
\label{fig:cardinalities1}
\end{figure}
\vspace{-2mm}
Furthermore, simulations illustrate that the $\hat{a}_k$ are the expected values of the cardinalities of the union of $k$ sets, $\left|A_k\right|$, as shown in Figure \ref{fig:cardinalities2}, where we show the relative error between $\hat{a}_k$ and the empirical mean values of the $\left|A_k\right|$, denoted by $\bar{a}_k$, realized over $500$ runs, to be less than $10^{-3}$.
\vspace{-2mm}
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah04.eps}
\caption{For fixed $d=8$ and $n=2^{10}$, over $500$ realizations, plots of the relative error between the mean values of $a_k$ (referred to as $\bar{a}_k$) and the $\hat{a}_k$.}
\label{fig:cardinalities2}
\end{figure}
\vspace{-2mm}
\section{Sampling Theorems}\label{sec:sample}
We now use Theorem \ref{thm:prob_bound_1set_expansion} with the $\ell_1$-norm restricted isometry property (RIP-1), introduced by Berinde et. al. in \cite{berinde2008combining}, to deduce the corollaries that follow which are about the probabilistic construction of expander graphs, the matrices we consider, and sampling theorems of some selected compressed sensing algorithms. Firstly, using only the expansion property of these graphs we can draw the following corollary from Theorem \ref{thm:prob_bound_1set_expansion}.
\begin{corollary}
\label{cor:prob_bound_1set_expander}
For fixed $s,n,N,d$ and $0<\epsilon<1/2$, let an $n\times N$ matrix, $A$ be drawn from the class of matrices defined in Definition \ref{def:1set_expansion}, then
\begin{equation*}
\label{eq:prob_bound_1set_expander}
\hbox{Prob}\left(\mathop{\|A_Sx\|_1} \leq (1-2\epsilon)d\|x\|_1\right) < p_{max}(s,d) \cdot e^{\left[n\cdot\Psi\left(s,d,\epsilon\right)\right]},
\end{equation*}
where $\Psi\left(s,d,\epsilon\right) = \Psi\left(a_s,\ldots,a_1\right)$ with $a_s = (1-\epsilon)ds$.
\end{corollary}
Theorem \ref{thm:prob_bound_1set_expansion} and Corollary \ref{cor:prob_bound_1set_expander} allow us to calculate $s,n,N,d,\epsilon$ where
the probability of the probabilistic constructions in Definition \ref{def:1set_expansion} not being a $(s,d,\epsilon)$-lossless expander is exponentially small. Using Corollary \ref{cor:prob_bound_1set_expander} and the RIP-1 results in \cite{berinde2008combining} we derived a bound for the probability that a random draw of a matrix with $d ~1$s or $\pm 1$s in each column fails to satisfy the lower bound of the RIP-1 constant ($\mathrm{RIC}_1$) and hence fails to come from the class of matrices given in Definition \ref{def:1set_expansion}, for details see \cite{bah2012vanishingly}. From this bound we deduce the following corollary which is a sampling theorem on the existence of lossless expander graphs.
\begin{corollary}
\label{cor:prob_expander_existence2}
Consider $0<\epsilon<1/2$ and $d$ fixed. If $A$ is drawn from the class of matrices in Definition \ref{def:1set_expansion} and any $k$-sparse $x$ with $(k,n,N) \rightarrow \infty$ while $k/n \rightarrow \rho \in (0,1)$ and $n/N \rightarrow \delta \in (0,1)$ then for $\rho < (1-\gamma)\rho^{exp}(\delta;d,\epsilon)$ and $\gamma>0$
\begin{equation}
\label{eq:prob_expander_existence2}
\hbox{Prob}\left(\|Ax\|_1 \geq (1-2\epsilon)d\|x\|_1\right) \rightarrow 1
\end{equation}
exponentially in $n$, where $\rho^{exp}(\delta;d,\epsilon)$ is the largest limiting value of $k/n$ for which $\hbox{H}\left(\frac{k}{N}\right) + \frac{n}{N}\Psi\left(k,d,\epsilon\right) = 0.$
\end{corollary}
For each fixed $0<\epsilon<1/2$ and each fixed $d$, $\rho^{exp}(\delta;d,\epsilon)$ in Corollary \ref{cor:prob_expander_existence2} is a function of $\delta$ and a phase transition function in the $(\delta,\rho)$ plane. Below the curve of $\rho^{exp}(\delta;d,\epsilon)$ the probability in \eqref{eq:prob_expander_existence2} goes to one exponentially in $n$ as the problem size grows. That is if $A$ is drawn at random with $d ~1$s or $d ~\pm 1$s in each column and having parameters $(k,n,N)$ that fall below the curve of $\rho^{exp}(\delta;d,\epsilon)$ then we say it is from the class of matrices in Definition \ref{def:1set_expansion} with probability approaching one exponentially in $n$. In terms of $|\Gamma(X)|$ for $X\subset U$ and $|X|\leq k$, Corollary \ref{cor:prob_expander_existence2} say that the probability $|\Gamma(X)| \geq (1-\epsilon) dk$ goes to one exponentially in $n$ if the parameters of our graph lies in the region below $\rho^{exp}(\delta;d,\epsilon)$. This implies that if we draw a random bipartite graphs that has parameters in the region below the curve of $\rho^{exp}(\delta;d,\epsilon)$ then with probability approaching one exponentially in $n$ that graph is a $(k,d,\epsilon)$-lossless expander.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah07.eps}
\caption{Phase transition plots of $\rho^{exp}(\delta;d,\epsilon)$ for fixed $\epsilon=1/6$ and $n = 2^{10}$ with $d$ varied.}
\label{fig:phase_transition1}
\end{figure}
Figure \ref{fig:phase_transition1} shows a plot of what $\rho^{exp}(\delta;d,\epsilon)$ converge to for different values of $d$ with $\epsilon$ and $n$ fixed. It is interesting to note how increasing $d$ increases the phase transition up to a point then it decreases the phase transition. Essentially beyond $d=16$ there is inconsequential gain in increasing $d$. This vindicates the use of small $d$ in most of the numerical simulations involving the class of matrices considered here. Note the vanishing sparsity as the problem size $(k,n,N)$ grows while $d$ is fixed to a small value of $8$.
Corollary \ref{cor:prob_expander_existence2} can also be arrived at based on similar probabilistic constructions of expander graphs first proven by Pinsker in \cite{pinsker1973complexity} with more recent proofs in \cite{berinde2009advances,capalbo2002randomness}. To put our results in perspective, we compare them to the phase transitions derived from the constants from the construction in \cite{berinde2009advances}, shown in Figure \ref{fig:phase_transition2}.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah09.eps}
\caption{A comparison of $\rho^{exp}$ in Corollary \ref{cor:prob_expander_existence2} to $\rho_{bi}^{exp}$ derived from the alternative construction proven in \cite{berinde2009advances}.}
\label{fig:phase_transition2}
\end{figure}
Furthermore, for moderate values of $\epsilon$ this allows us to make quantitative sampling theorems for some compressed sensing reconstruction algorithms. As usual in compressed sensing, in addition to $\ell_1$-minimization quite a few {\em combinatorial} greedy algorithms have been proposed for these sparse non-mean zero matrices. These algorithms iteratively locates and eliminate large (in magnitude) components of the vector, \cite{berinde2008combining}. They include Sequential Sparse Matching Pursuit (SSMP), see \cite{berinde2009sequential}; and Expander Recovery (ER), see \cite{jafarpour2009efficient}. Besides, theoretical guarantees have been given for $\ell_1$ recovery and some of the greedy algorithms including SSMP and ER. Base on these theoretical guarantees, we derived sampling theorems and present here phase transition curves which are plots of phase transition functions $\rho^{alg}(\delta;d,\epsilon)$ of algorithms such that for $k/n \rightarrow \rho < (1-\gamma)\rho^{alg}(\delta;d,\epsilon), ~\gamma>0$, a given algorithm is guaranteed to recovery all $k$-sparse signals with overwhelming probability approaching one exponentially in $n$.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah11.eps}
\caption{Phase transition curves $\rho^{alg}\left(\delta;d,\epsilon\right)$ computed over finite values of $\delta \in (0,1)$ with $d$ fixed and the different $\epsilon$ values for each algorithm - 1/4, 1/6 and 1/16 for ER, $\ell_1$ and SSMP respectively.}
\label{fig:pt_alg_compare1}
\end{figure}
Figure \ref{fig:pt_alg_compare1} compares the phase transition of thee above mentioned algorithms. Remarkably, for ER recovery is guaranteed for a larger portion of the $(\delta,\rho)$ plane than is guaranteed by the theory for $\ell_1$-minimization using sparse matrices; however, $\ell_1$-minimization has a larger recovery region than does SSMP. Figure \ref{fig:pt_alg_compare2} shows a comparison of the phase transition of $\ell_1$-minimization as presented by Blanchard et. al. in \cite{blanchard2011phase} for dense Gaussian matrices based on $\mathrm{RIC}_2$ analysis and the phase transition we derived here for the sparse binary matrices coming from lossless expander based on $\mathrm{RIC}_1$ analysis. This shows a significant difference between the two with sparse matrices having better performance guarantees.
However, these improved recovery guarantees are likely more due to the closer match of the method of analysis than to the efficacy of sparse matrices over dense matrices.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{bah12.eps}
\caption{Phase transition plots of $\ell_1$, $\rho^{\ell_1}_G\left(\delta\right)$, for Gaussian matrices derived using $\mathrm{RIC}_2$ and $\rho^{\ell_1}_E\left(\delta;d,\epsilon\right)$ for adjacency matrices of expander graphs with $n=1024$, $d=8$, and $\epsilon=1/6$.}
\label{fig:pt_alg_compare2}
\end{figure}
\section{Sketch of Main Proof}\label{sec:proof}
Due to space constraints the details of the proofs are skipped and the interested reader is referred to \cite{bah2012vanishingly}. It is however important to briefly describe the key innovations in the derivation of the main result, Theorem \ref{thm:prob_bound_1set_expansion}.
For one fixed set of columns of $A$, denoted $A_S$, the probability in \eqref{eq:prob_bound_1set_expansion} can be understood as the cardinality of the unions of nonzeros in the columns. Our analysis of this probability follows from a nested unions of subsets using a {\em dyadic splitting} technique. Given a starting set of columns we recursively split the number of columns from this set and the resulting sets into two sets of cardinality of the ceiling and floor of the cardinality of their union until a level when the cardinalities are at most two. Resulting from this type of splitting is a regular binary tree where the size of each child is either the ceiling or the floor of the size of it's parent set. The probability of interest becomes a product of the probabilities involving all the children from the dyadic splitting of $A_s$. The proof therefore reduces to upper bounding this product.
Furthermore, in the binary tree resulting from our dyadic splitting scheme the number of columns in the two children of a parent node is the ceiling and the floor of half of the number of columns of the parent node. At each level of the split the number of columns of the children of that level differ by one. The enumeration of these two quantities at each level of the splitting process is necessary in the computation of the bound in \eqref{eq:prob_bound_1set_expansion}. This led to another novel technical result in our derivation, i.e. {\em dyadic splitting lemma} (Lemma 2.5 in \cite{bah2012vanishingly}).
\section{Conclusions}\label{sec:conclusion}
This work derived bounds on the tail probability of the cardinality of the {\em set of neighbours} of expander graphs resulting into better order constants than the standard probabilistic construction. Using this bound and $\mathrm{RIC}_1$ analysis, we deduce sampling theorems for the existence of expander graphs and their adjacency matrices. The derivation of the tail bound used a novel technique of {\em dyadic set splitting}. We also compared quantitatively, performance guarantees of compressed sensing algorithms which show greater phase transitions for ER than $\ell_1$-minimization which in turn is greater than SSMP. A comparison of $\ell_1$-minimization for dense and sparse matrices shows a higher phase transition for sparse matrices.
| {
"timestamp": "2013-07-25T02:08:22",
"yymm": "1307",
"arxiv_id": "1307.6477",
"language": "en",
"url": "https://arxiv.org/abs/1307.6477",
"abstract": "We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency matrices of lossless expander graphs. We present tail bounds on the probability that the cardinality of the set of neighbors for these graphs will be less than the expected value. The bounds are derived through the analysis of collisions in unions of sets using a {\\em dyadic splitting} technique. This analysis led to the derivation of better constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse non mean-zero measurement matrices.",
"subjects": "Information Theory (cs.IT)",
"title": "On construction and analysis of sparse random matrices and expander graphs with applications to compressed sensing",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9783846672373524,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7092019523173007
} |
https://arxiv.org/abs/1202.3183 | Zeta functions for function fields | We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group, maximal parabolic subgroup). Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given. We end this paper with an explanation on why these zetas are non-abelian in nature, using our up-coming works on 'parabolic reduction, stability and the mass'.The constructions and results were announced in our paper on 'Counting Bundles'arXiv:1202.0869. | \section{Pure Non-Abelian Zeta Functions}
Non-Abelian zeta functions for function fields were introduced in [W1] about 10 years ago.
However, due to the lack of the Riemann Hypothesis, we have faced some essential difficulties.
Recently, with an old paper of Drinfeld ([D]) on counting rank two cuspidal $\mathbb Q_l$-representations
for function fields, we realize that our old
definition of zeta should be altered: instead of counting rational semi-stable bundles of all degrees as done in [W],
only these of degree zero and more generally degrees of multiples of the rank should be counted.
This then leads to the definition of {\it pure} high rank zeta functions of this paper. This purity proves to be very essential: We expect that the Riemann Hypothesis holds for pure zetas. Indeed, in this direction, we now have the work of Yoshida [Y] for the RH in rank two and of mine [W4] for elliptic curves.
\subsection{Counting Semi-Stable Bundles}
Let $X$ be an irreducible, reduced, regular projective curve of genus $g$ defined over $\mathbb F_q$. Denote by $\mathcal M_{X,r}(d)$ the moduli space of rank $r$ semi-stable bundles of degree $d$ consisting of the Seshadri Jordan-H\"older equivalence classes of $\mathbb F_q$-rational semi-stable bundles on $X$. For our own purpose, we consider $\mathcal M_{X,r}(d)$ in the sense of the {\it fat} moduli, meaning that ordinary moduli spaces equipped with an additional structure at Seshadri class $[\mathcal E]$
defined by the collection of semi-stable bundles in $[\mathcal E]$, namely, the set $\big\{\mathcal E:\mathcal E\in [\mathcal E]\big\}$, is added at the point $[\mathcal E]$. For our own convenience,
$\mathcal M_{X,r}(d)$ eqiupped with such a structure is called a {\it fat moduli space} and denoted as ${\bf M}_{X,r}(d)$.
A natural question is to count these
$\mathbb F_q$-rational semi-stable bundles $\mathcal E$ on $X$. For this purpose, two naive invariants, namely, the automorphism group $\mathrm{Aut}(\mathcal E)$ and its global sections $h^0(X,\mathcal E)$, will be used.
This then leads to the refined Brill-Noether loci $$W_{X,r}^i(d):=\Big\{[\mathcal E]\in {\bf M}_{X,r}(d):
\min_{\mathcal E\in[\mathcal E]}\{h^0(X,\mathcal E)\}\geq i\Big\}$$ and $$[\mathcal E]^j:=\{\mathcal E\in[\mathcal E]: \dim_{\mathbb F_q}\mathrm{Aut}\,{\mathcal E}\geq j\}.$$ Recall that there exist natural
isomorphisms $${\bf M}_{X,r}(d)\to {\bf M}_{X,r}(d+rm),\qquad
\mathcal E\mapsto A^m\otimes\mathcal E$$ and $${\bf M}_{X,r}(d)\to {\bf M}_{X,r}(-d+r(2g-2)),\qquad
\mathcal E\mapsto K_X\otimes\mathcal E^\vee,$$
where $A$ is an Artin line bundle of degree one on $X/\mathbb F_q$ and $K_X$ denotes the dualizing bundle of $X/\mathbb F_q$. So we only need to count ${\bf M}_{X,r}(d_0)$ for $d_0= 0, 1,\dots, r(g-1).$ Accordingly, we introduce
$$\alpha_{X,r}(d):=\sum_{\mathcal E\in {\bf M}_{X,r}(d)}\frac{q^{h^0(X,\mathcal E)}-1}{\mathrm{Aut}(\mathcal E)},\qquad \beta_{X,r}(d):=\sum_{\mathcal E\in {\bf M}_{X,r}(d)}\frac{1}{\mathrm{Aut}(\mathcal E)}$$ with $\beta$ a classical invariant ([HN]).
So to count bundles, the problem becomes how to control $\alpha_{X,r}(d_0)$'s with $d_0$ ranging as above, and
$\beta_{X,r}(d)$ with $d=0,1,\dots r-1$. For $\alpha$, two general principles can be used for counting semi-stable bundles, namely,
\noindent
(i) the vanishing theorem claiming that, for semi-stable $\mathcal E$,
$$h^1(X,\mathcal E)=0\qquad\mathrm{if}\qquad d(\mathrm E)\geq r(2g-2)+1;$$
\noindent
(ii) the Clifford lemma claiming that, for semi-stable $\mathcal E$,
$$h^0(X,\mathcal E)\leq r+\frac{d}{2}\qquad\mathrm{if}\qquad 0\leq \mu(\mathcal E)\leq 2g-2.$$
By contrasting, the invariant $\beta$ has already been understood, thanks to the high profile works of Harder-Narasimhan ([HN]),
Desale-Ramanan ([DR]), Atiyah-Bott ([AB]), Witten ([Wi]) and Zagier ([Z]).
To state it, let $$\zeta_X(s):=\frac{\prod_{i=1}^{2g}(1-\omega_iq^{-s})}{(1-q^{-s})(1-qq^{-s})}$$ be the Artin zeta function of $X/\mathbb F_q$,
$$v_n(q):=\frac{\prod_{i=1}^{2g}(1-\omega_i)}{q-1}q^{(r^2-1)(g-1)}\zeta_X(2)\cdots\zeta_X(r)$$ and for a partition $r=n_1+\cdots+n_k$ of $r$, set
$$c_{r,d}(t):=\prod_{i=1}^{s-1}\frac{t^{(n_i+n_{i+1})\{n_1+\cdots+n_i)d/n\}}}{1-t^{n_i+n_{i+1}}}.$$
\begin{thm} ([HN], [DS], in particular, [Z, Thm 2]) For any pair $(r,d)$, we have
$$\beta_{X,r}(d)=\sum_{n_1,\dots,n_s>0,\ \sum n_i=r}q^{(g-1)\sum_{i<j}n_in_j}c_{r,d}(q)\cdot \prod_{i=1}^s v_n(q).$$
\end{thm}
\subsection{Pure Non-Abelian Zeta Functions}
Practically, the difficulty of counting semi-stable bundles comes form the fact that direct summands of the associated
Jordan-H\"older graded bundle, or equivalently, the Jordan-H\"older filtrations, of an
$\mathbb F_q$-rational semi-stable bundle in general would not be defined over $X/\mathbb F_q$, but
rather its scalar extension $X_n/\mathbb F_{q^n}$. Theoretically, this is the junction point where the abelian and non-abelian ingredients of curves interact. For examples, torsions of Jacobians, Weierastrass points and stable but not absolutely stable bundles are closely related and hence get into the picture naturally. Good examples may be found in [W1].
To uniformly study $\alpha$ and $\beta$'s, we, in [W], introduce the non-abelian zeta functions with the hope that
the Riemann Hypothesis would hold for them. Unfortunately, examples shows that there are zeros off the central line
for these old zetas. (For details, see the examples below.)
This, in practice, has prevented any further studies for such zetas. However, during my visit to IHES in September 2011,
we got to know the work of Drinfeld ([D]). Learning from it, we now know where the problem lies for old zetas:
We should {\it count only the pure part, instead of counting all}.
\vskip 0.30cm
\noindent
{\bf Main Definition 1.} {\it For an irreducible, reduced, regular projective curve $X$ of genus $g$ defined over finite field $\mathbb F_q$, define its rank r pure non-abelian zeta function by
$$\begin{aligned}\zeta_{X,r}(s):=&\sum_{m=0}^\infty
\sum_{V\in {\bf M}_{X,r}(d), d=rm}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot (q^{-s})^{d(V)},\\\widehat\zeta_{X,r}(s):=&\sum_{m=0}^\infty
\sum_{V\in {\bf M}_{X,r}(d), d=rm}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot (q^{-s})^{\chi(X,V)}.\end{aligned}$$ As usual, set
$$\qquad Z_{X,r}(t):=\zeta_{X,r}(s)\qquad\mathrm{and}\qquad \widehat Z_{X,r}(t):=\widehat\zeta_{X,r}(s)\qquad\mathrm{with}\qquad t:=q^{-s}.$$}
By [W1, Prop 1.2.1], when the rank is one,
$$\zeta_{X,1}(s)=\zeta_{X}(s)$$ is the Artin zeta function.
Moreover
$$\begin{aligned}\widehat Z_{X,r}(t)
=&\sum_{m=0}^{2g-2}
\sum_{V\in {\bf M}_{X,r}(d), d=rm}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{\chi(C,V)}\\
&+\sum_{m\,>\, 2g-2}
\sum_{V\in {\bf M}_{X,r}(d), d=rm}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{\chi(C,V)}\\
=&\Big(\sum_{m=0}^{(g-1)-1}
\sum_{V\in {\bf M}_{X,r}(d), d=rm, r[(2g-2)-m]}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{\chi(C,V)}\\
&\qquad+\sum_{V\in {\bf M}_{X,r}(d), d=r(g-1)}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{\chi(C,V)}\Big)\\
&+\sum_{m\,>\, 2g-2}
\sum_{V\in {\bf M}_{X,r}(0)}\frac{q^{r[m-(g-1)]}-1}{\#\mathrm{Aut}(V)}\cdot t^{r[m-(g-1)]}\\
&\hskip 1.0cm\mathrm{(by\ the\ Vanishing\ Thm\ and\ the\ Riemann-Roch\ Thm)}\\
=&\Big[\sum_{m=0}^{(g-1)-1}
\Big(\sum_{V\in {\bf M}_{X,r}(rm)}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{r[m-(g-1)]}\\
&\qquad+\sum_{W\in {\bf M}_{X,r}(rm)}\frac{q^{h^0(C,W)-r[m-(g-1)]}-1}{\#\mathrm{Aut}(W)}\cdot t^{r[(g-1)-m]}\Big)\\
&\qquad+\sum_{V\in {\bf M}_{X,r}(r(g-1))}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{0}\Big]\\&+\sum_{V\in {\bf M}_{X,r}(0)}\frac{1}{\#\mathrm{Aut}(V)}\cdot\Big(\frac{(qt)^{rg}}{1-(qt)^r}-\frac{t^{rg}}{1-t^r}\Big)\\
&\hskip 7.5cm\mathrm{(by\ the\ duality)}\\
=&\Big[\sum_{m=0}^{(g-1)-1}
\Big(\sum_{V\in {\bf M}_{X,r}(rm)}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{r[m-(g-1)]}\\
&+\sum_{W\in {\bf M}_{X,r}(rm)}\frac{q^{h^0(C,W)}-1}{\#\mathrm{Aut}(W)}\cdot ({qt})^{r[(g-1)-m]}
\\
&+\sum_{W\in {\bf M}_{X,r}(rm)}\frac{q^{-r[m-(g-1)]}-1}{\#\mathrm{Aut}(W)}\cdot t^{r[(g-1)-m]}\Big)+\sum_{V\in {\bf M}_{X,r}(r(g-1))}\frac{q^{h^0(C,V)}-1}{\#\mathrm{Aut}(V)}\cdot t^{0}\Big]\\
&+\sum_{V\in {\bf M}_{X,r}(0)}\frac{1}{\#\mathrm{Aut}(V)}\cdot\Big(\frac{(qt)^{rg}}{1-(qt)^r}-\frac{t^{rg}}{1-t^r}\Big)\\
=&\Big[\sum_{m=0}^{(g-1)-1}
\alpha_{X,r}(rm)\cdot \Big(t^{r[m-(g-1)]}+ \Big(\frac{1}{qt}\Big)^{r[m-(g-1)]}\Big)+\alpha_{X,r}(r(g-1))\Big]\\
&\qquad+\beta_{X,r}(0)\cdot \Big(\frac{(qt)^r}{1-(qt)^r}-\frac{t^r}{1-t^r}\Big)
\end{aligned}$$
Consequently,
$$\widehat Z_{X,r}(\frac{1}{qt})=\widehat Z_{X,r}(t),$$
and, if we introduce $T:=t^r$ and $Q:=q^r$,
$$\begin{aligned}Z_{X,r}&(t)
=\sum_{m=0}^{(g-1)-1}\alpha_{X,r}(mr)
\cdot \Big(T^{m}+ Q^{(g-1)-m}\cdot T^{2(g-1)-m}\Big)\\
&+\alpha_{X,r}\big(r(g-1)\big)\cdot T^{g-1}+(Q-1)\beta_{X,r}(0)\cdot \frac{T^{g}}{(1-T)(1-QT)}.
\end{aligned}\eqno(1)$$
This then completes the proof of the following
\begin{thm} ({\bf Zeta Facts}) (i) $\zeta_{X,1}(s)=\zeta_X(s)$, the Artin zeta function for $X/\mathbb F_q$;
\noindent
(ii) ({\bf Rationality}) There exists a degree $2g$ polynomial $P_{X,r}(T)\in\mathbb Q[T]$ of $T$
such that
$$Z_{X,r}(t)=\frac{P_{X,r}(T)}{(1-T)(1-QT)}\quad\mathrm{with}\quad T=t^r,\ Q=q^r;$$
\noindent
(iii) ({\bf Functional equation}) $$\widehat Z_{X,r}(\frac{1}{qt})=\widehat Z_{X,r}(t),$$
\end{thm}
These non-abelian zetas give a systematical treatment of invariants $\alpha$'s and $\beta$'s in counting semi-stable bundles.
With $\beta_{X,r}(0)$ known, we expect the uniform control of $$\alpha_{X,r}(0), \alpha_{X,r}(0),\dots, \alpha_{X,r}\big(r(g-1)\big)$$ through the following
\noindent
{\bf Riemann Hypothesis.} {\it Let $P_{X,r}(T)=P_{X,r}(0)\cdot\prod_{i=1}^{2g}(1-\omega_{X,r}(i)T)$, then $$|\omega_{X,r}(i)|=Q^{\frac{1}{2}},\qquad\forall i,\ 1\leq i\leq 2g.$$}
\noindent
{\bf Examples.} (i) ([W4]) {\it Rank 2 Zeta for Elliptic Curves} Let $E$ be an elliptic curve defined over $\mathbb F_q$ with $N$ the number of $\mathbb F_q$-rational points. For rank two pure zeta, it suffices to calculate
$\alpha_{E,2}(0)$ and $\beta_{E,2}(0)$. By Thm 1, $$\beta_{E,2}(0)=\frac{N}{q-1}\Big(1+\frac{N}{q^2-1}\Big).$$
On the other hand, by the classification of Atiyah ([A]), over $\overline{\mathbb F_q}$, the graded bundle associated to a Jordan-H\"older filtration of a semi-stable bundle $V\otimes_{\mathbb F_q}\overline{\mathbb F_q}$ is of the form $\mathrm{Gr}(V\otimes_{\mathbb F_q}\overline{\mathbb F_q})=L_1\oplus L_2$ with $L_i$ degree zero line bundles, which may not be defined over $\mathbb F_q$. Consequently, for $\mathbb F_q$-rational semi-stable bundles $V$ of rank two, $h^0(E,V)\not=0$ if and only if $V=\mathcal O_E\oplus L$ or $V=I_2$ with $L$ a $\mathbb F_q$-rational line bundle of degree 0 and $I_2$ the only non-trivial extension of $\mathcal O_E$ by itself. Thus $\alpha_{E,2}(0)$ is given by $$\begin{aligned}&\Big(\frac{q^{h^0(E,\mathcal O_E\oplus \mathcal O_E)}-1}{\#\mathrm{Aut}(\mathcal O_E\oplus \mathcal O_E)}+\frac{q^{h^0(E,I_2)}-1}{\#\mathrm{Aut} (I_2)}\Big)+\sum_{L\in\mathrm{Pic}^0(E), L\not=\mathcal O_E}\frac{q^{h^0(E,\mathcal O_E\oplus L)}-1}{\#\mathrm{Aut}(\mathcal O_E\oplus L)}\\
=&\Big(\frac{q^2-1}{(q^2-1)(q^2-q)}+\frac{q-1}{(q-1)q}\Big)+(N-1)\frac{q-1}{(q-1)^2}=\frac{N}{q-1}.\end{aligned}$$ Thus, $$Z_{E,2}(t)=\alpha_{E,2}(0)\cdot\frac{1+(N-2)T+QT^2}{(1-T)(1-QT)},$$
the Riemann Hypothesis holds since $$\Delta=(N-2)^2-4Q=(N-2-2q)(N-2+2q)<0$$ using Hasse's theorem for the Riemann Hypothesis of elliptic curves, namely $$N\leq 2\sqrt q.$$
\noindent
(ii) {\it Rank Two Bundles on Genus Two Curves} Let $X$ be a genus 2 curve. For rank two zeta,
$$\begin{aligned}Z_{X,2}(t)
=&\alpha_{X,2}(0)
\cdot \Big(1+ Q\cdot T^{2}\Big)+\alpha_{X,r}\big(2\big)\cdot T\\
&+(Q-1)\beta_{X,2}(0)\cdot \frac{T^{2}}{(1-T)(1-QT)}.
\end{aligned}$$
Thus $$\begin{aligned}P_{X,2}(t)=&\alpha_{C,2}(0)\Big(1+Q^2T^4\Big)+\Big(\alpha_{X,2}(2)-
\alpha_{X,2}(0)(Q+1)\Big)\Big(T+QT^3\Big)\\
&+
\Big(2Q\,\alpha_{X,2}(0)-(Q+1)\alpha_{X,2}(2)+\beta_{X,2}(0)(Q-1)\Big)T^2.\end{aligned}$$
So we need to consider the spaces ${\bf M}_{X,2}(0)$, ${\bf M}_{X,2}(2)$ (and ${\bf M}_{X,2}(4)$). By the Clifford lemma,
$$h^0(X,V)=\begin{cases}0,1,2,&$if$\ \ V\in {\bf M}_{X,2}(0);\\
0,1,2,& $if$\ \ V\in {\bf M}_{X,2}(2);\\
2,3,4,&$if$\ \ V\in {\bf M}_{X,2}(4).\end{cases}$$ Consequently,
$$\begin{aligned}\alpha_{X,2}(0)=&\sum_{V\in W^1_{X,2}(0)}\frac{q-1}{\#\mathrm{Aut}(V)}+\frac{q^2-1}{(q^2-1)(q^2-q)},\\
\alpha_{X,2}(2)=&\sum_{V\in W^1_{X,2}(0)}\frac{q-1}{\#\mathrm{Aut}(V)}+\sum_{V\in W^2_{X,2}(0)}\frac{q^2-1}{\#\mathrm{Aut}(V)},\end{aligned}$$ note that $h^0(X,V)=2$ and $d(V)=0$ iff $V=\mathcal O_X\oplus\mathcal O_X$. Moreover, the Riemann Hypothesis now is equivalent to the conditions that
$$A^2<4Q, \qquad B^2<4Q$$ with real constants $A,\ B$ defined by
$$P_{X,2}(T)=\alpha_{X,2}(0)\cdot (1-AT+QT^2)(1-BT+QT^2).$$
That is to say, $$\begin{aligned}A+B=&(Q+1)-\alpha_{X,2}'(2),\\
AB=&(Q-1)\beta_{X,2}'(0)-(Q+1)\alpha_{X,2}'(2),\end{aligned}$$
where $$\alpha_{X,r}'(d):=\frac{\alpha_{X,r}(d)}{\alpha_{X,r}(0)},\qquad \beta_{X,r}'(0):=\frac{\beta_{X,r}(0)}{\alpha_{X,r}(0)}.$$
While the above does give a good control of $\alpha'$'s and $\beta'$, it looks a bit clumsy.
A much better way is to set $$Z_{X,r}(t)=\alpha_{X,r}(0)\cdot\exp\Big(\sum_{m=1}^\infty N_{X,r}(m)\frac{T^m}{m}\Big).$$
Then $$N_{X,r}(m)=1+Q^m-\sum_{i=1}^{2g}\omega_{X,r}(i)^m$$
and the Riemann Hypothesis gives a much elegant control of $N_{X,r}(m)$'s.
We expect that $N_{X,r}(m)$'s measure rank $r$ stable bundles over $X/\mathbb F_{q^m}$. This is certainly the case in rank one through Weil's counting zeta,
and in rank two for elliptic curves, as indicated in the above example.
\subsection{Why Purity}
Next, we explain why purity is introduced for our study of zeta functions. Simply put,
this is due to the Riemann Hypothesis.
For this purpose, let $E$ be an irreducible, reduced regular elliptic curve defined over $\mathbb F_q$. We will concentrate on ranks two and three. Due to isomorphisms $${\bf M}_{E,r}(d)\to {\bf M}_{E,r}(d+rm),\qquad
\mathcal E\mapsto A^m\otimes\mathcal E$$ and $${\bf M}_{E,r}(d)\to {\bf M}_{E,r}(-d),\qquad
\mathcal E\mapsto K_E\otimes\mathcal E^\vee,$$
among all invariants $\alpha$ and $\beta$'s, for rank two and three,
it suffices to understand $\alpha_{E,r}(d)$ and $\beta_{E,r}(d)$ for $d=0,1$.
\subsubsection{Rank Two}
From Ex(i) in \S1.2,
$$\beta_{E,2}(0)=\frac{N}{q-1}\Big(1+\frac{N}{q^2-1}\Big)\qquad\mathrm{and}\quad
\alpha_{E,2}(0)=\frac{N}{q-1}.$$
On the other hand, $\alpha_{E,2}(1)$ and $\beta_{E,2}(1)$ are easy to calculate.
Indeed, all semi-stable bundles of rank 2 and degree 1 are stable. Thus, by the classification of Atiyah ([A]),
${\bf M}_{E,2}(1)$ via the determinant line bundle map is isomorphic to $\mathrm{Pic}^1(E)$.
So $$\beta_{E,2}(1)=\frac{N}{q-1}.$$ Moreover, by the vanishing theorem, $h^0(E,\mathcal E)=1$.
Thus $$\alpha_{E,2}(1)=N.$$
As such, from [W1, \S1.2.2], we know that
the original zeta function $\zeta$ of counting all degree semi-stable bundles defined as
$$\sum_{V\in{\bf M}_{E,2}(d)}\frac{q^{h^0(E,V)}-1}{\#\mathrm{Aut}\,V}=:\zeta_{E,2}(s)+\zeta_{E,2}^1(s)$$
is given by
$$\begin{aligned}\Big(\alpha_{E,2}(0)+&\beta_{E,2}(0)\cdot\frac{(q^2-1)t^2}{(1-t^2)(1-q^2t^2)}\Big)+\beta_{E,2}(1)\Big(\frac{qt}{1-q^2t^2}-\frac{t}{1-t^2}\Big)\\
=&\frac{N}{q-1}\cdot\frac{1+(q-1)t+(N-1)t^2+(q-1)qt^3+q^2t^4}{(1-t^2)(1-q^2t^2)}.\end{aligned}$$
Here $$\zeta_{E,2}^1(s):=\sum_{V\in{\bf M}_{E,2}(2m+1),\ m\geq 0}\frac{q^{h^0(E,V)}-1}{\#\mathrm{Aut}\,V}.$$
Note that for the polynomial appeared in the numerator
$$P_2(t):=1+(q-1)t+(N-1)t^2+(q-1)qt^3+q^2t^4,$$ by the functional equation ([W1]), we have the
factorization $$P_2(t)=(qt^2+A_+t+1)(qt^2+A_-t+1)$$ in $\mathbb R[t]$. Assume, as we may, that $|A_+|>|A_-|$. By Hasse's theorem for Artin zeta functions,
the coefficients of $t^2$ in $P_2(t)$ is $N-1$, which is of the same order as
$q-1$, the coefficient of $t$. Consequently, $$A_+^2-4q>0,\qquad\mathrm{while}\qquad A_-^2-4q<0.$$
So there is no RH for the zeta defined by counting semi-stable bundles of all degrees.
\subsubsection{Rank Three}
We already saw that for pure rank two zeta functions, the RH holds. In fact, one can show that this patten persists ([W4]). This then leads to the problem of whether partial zeta functions defined by counting semi-stable bundles of other types of degrees satisfy the RH. Here, we use an example in rank 3 to indicate that another seemly natural choice does not work neither.
Introduce then the function
$$\zeta_{E,3}^{12}(s):=\sum_{\substack{V\in{\bf M}_{E,2}(d)\\
d\equiv 1,\,2\,(\mathrm{mod}\,3)}}\frac{q^{h^0(E,V)}-1}{\#\mathrm{Aut}\,V}.$$
(The reason for taking both 1 and 2, not just a single one, 1 or 2, in the congruence classes is that otherwise
the functional equation does not hold.)
By the vanishing theorem and the fact that all rank three semi-stable bundles of degree 1 or 2 are stable,
one checks that
$$\zeta_{E,3}^{12}(s)=N\cdot \Big(\frac{qt+q^2t^2}{1-q^3t^3}-\frac{t+t^2}{1-t^3}\Big)=\frac{P_{E,3}(t)}{(1-t^3)(1-q^3t^3)}$$ where $$P_{E,3}^{12}(t)=(q-1)t\Big[q^2t^4+q(q-1)t^3+(q+1)t+1\Big].$$
The polynomial $$q^2t^4+q(q-1)t^3+(q+1)t+1$$ does not satisfy the Riemann Hypothesis.
\section{Group Zeta Functions}
\subsection{Number Fields versus Function Fields}
For number fields, we have yet another type of zeta functions defined for pairs consisting of (reductive group, maximal parabolic subgroup)'s ([W2,3]). We will introduce such zeta functions for function fields next.
For this purpose, we first examine analogue between function fields and number fields in our setting.
To be more precise, we will analysis
Zagier's formula for counting semi-stable bundles
over curves on finite fields and our own volume formula for semi-stable lattices over number fields.
Set then $$\widehat\zeta_{F}(1):=\begin{cases}\mathrm{Res}_{s=1}\widehat\zeta_F(s), & F\ \mathrm{number\ field};\\
\mathrm{Res}_{s=1}\widehat\zeta_F(s)\cdot\log q, & F\ \mathrm{function\ field}.\end{cases}$$
And denote by $\mathcal M_{\mathbb Q,r}[1]$ the moduli space of rank $r$
semi-stable lattices of volume 1.
\begin{thm} (i) (Reformulation of [Z, Thm 2]) For an irreducible, reduced, regular projective curve $X/\mathbb F_q$ of genus $g$,
$$\frac{\beta_{X,r}(0)}{q^{-(g-1)\cdot\frac{r^2-r}{2}}}=\sum_{\substack{n_1,\dots,n_s>0,\\ n_1+\cdots+n_k=r}}
\frac{(-1)^{k-1}}{\prod_{j=1}^{k-1}(q^{n_j+n_{j+1}}-1)}\prod_{j=1}^k \prod_{i=1}^{n_j}\widehat\zeta_X(i);$$
\noindent
(ii) ([W2, \S4.8]) For a number field $F$, $$\frac{1}{r}\cdot \mathrm{Vol}\Big(\mathcal M_{\mathbb Q,r}[1]\Big)=\sum_{\substack{n_1,\dots,n_s>0,\\ n_1+\cdots+n_k=r}}
\frac{(-1)^{k-1}}{\prod_{j=1}^{k-1}(n_j+n_{j+1})}\prod_{j=1}^k \prod_{i=1}^{n_j}\widehat\zeta(i).$$
\end{thm}
Put Zagier's result in our form as above, the hidden parallel structures in these two worlds becomes crystal
clear. That is to say, for the mass of moduli space of semi-stable objects,
when shift from number fields to function fields, the integers $n_j+n_{j+1}$ should be replaces by
$q^{(n_j+n_{j+1})}-1$. This then would suggest that, more generally, when defining group
zeta functions associated to $(G,P)$ with $G$ reductive and $P$ maximal parabolic for function fields, based on these for number fields investigated in [W2,3],
we should replace the rational factor $\langle w\lambda-\rho,\alpha^\vee\rangle$ by
$q^{\langle w\lambda-\rho,\alpha^\vee\rangle}-1$.
In reality, even this is the direction we would go, this is not exactly the path we really pave. As a matter of fact, when shifting from number fields to function fields, {\it the rational factor $\langle w\lambda-\rho,\alpha^\vee\rangle$ should be replaced by
$1-q^{-\langle w\lambda-\rho,\alpha^\vee\rangle}$, instead of $q^{\langle w\lambda-\rho,\alpha^\vee\rangle}-1$.}
\subsection{Definitions}
Let $X$ be an irreducible, reduced, regular projective curve of genus $g$ defined on $\mathbb F_q$. Denote by $F$ its function field. Let $G$ be a split connected reductive group with $B$ a fixed Borel over $F$.
Denote by $\Sigma(G):=\Sigma:=$
$$\Big(V,\langle\cdot,\cdot\rangle,\Delta=\{\alpha_1,\dots,\alpha_n\},\Lambda:=\{\lambda_1,\dots,\lambda_n\},\Phi=\Phi^+\cup\Phi^-,W\Big)$$ the associated root system with the Weyl vector $\rho:=\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha$. For $w\in W$, set
$\Phi_w:=\Phi^+\,\cap\, w^{-1}\Phi^-$, and
for $\alpha\in\Phi$, denote its coroot by $\alpha^\vee:=\frac{2}{\langle\alpha,\alpha\rangle}\cdot\alpha$.
From Lie theory, (see e.g., [H]), there is a well-known one-to-one correspondence between standard parabolic subgroups of $G$
and subsets of $\Delta$. Consequently, for a maximal standard parabolic subgroup $P$, there exists a unique $p=p(P)$ such that the subset of $\Delta$ above for $P$ is given by $$\Delta_p=:\Delta\backslash\{\alpha_p\}:=\{\beta_{P,1},\dots,\beta_{P,n-1}\}.$$
For such $p$, let $\Phi_p$ be the corresponding root system. Then $\Phi_p$ is normal to the fundamental weight $\lambda_p$, since $\langle\alpha_i,\lambda_j\rangle=\delta_{ij}$. This similarly leads to positive $\Phi^+_p:=\Phi^+\,\cap\,\Phi_p$, $\Phi_p^-,\,\rho_p$, and $W_p$.
Set $$c_p:=2\langle\lambda_p-\rho_p,\alpha_p^\vee\rangle.$$
Moreover, for $\lambda \in V$, introduce a specific coordinate system via $$\lambda=\sum_{j=1}^n(1+s_j)\lambda_j=\rho+\sum_{j=1}^ns_j\lambda_j.$$
\vskip 0.30cm
\noindent
{\bf Main Definition 2.} {\it (i) The period of $G$ for $X$ is defined by
$$\omega_X^G(\lambda):=\sum_{w\in W}\frac{1}{\prod_{\alpha\in\Delta}(1-q^{-\langle w\lambda-\rho,\alpha^\vee\rangle})}\prod_{\alpha\in\Phi_w}\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}$$ where $\widehat\zeta_X$ denotes the complete
Artin zeta function of $X$;
\noindent
(ii) The period of $(G,P)$ for $X$ is defined by
$$\begin{aligned}\omega^{(G,P)}_X(s):=&\mathrm{Res}_{\langle\lambda-\rho,\beta_{P,n-1}^\vee\rangle=0}\cdots
\mathrm{Res}_{\langle\lambda-\rho,\beta_{P,2}^\vee\rangle=0}\mathrm{Res}_{\langle\lambda-\rho,\beta_{P,1}^\vee\rangle=0}\,\omega_X^G(\lambda)\\
=&\mathrm{Res}_{s_n=0}\cdots \mathrm{Res}_{s_{p+1}=0}\mathrm{Res}_{s_{p-1}=0}\cdots
\mathrm{Res}_{s_1=0}\,\omega_X^G(\lambda)\end{aligned}$$ with $s=s_p$.
}
It is clear that there exists a minimal number $I(G,P)$ and factors $$\widehat\zeta_X\Big(a_1^{(G,P)}s+b_1^{(G,P)}\Big),\ \widehat\zeta_X\Big(a_2^{(G,P)}s+b_2^{(G,P)}\Big),\ \cdots,\
\widehat\zeta_X\Big(a_{I(G,P)}^{(G,P)}s+b_{I(G,P)}^{(G,P)}\Big),$$
such that there are no zeta factors appeared in the denominators of
all terms of the product $\Big[\prod_{i=1}^{I(G,P)}\widehat\zeta_X\Big(a_i^{(G,P)}s+b_i^{(G,P)}\Big)\Big]\cdot \omega_{\mathbb Q}^{(G,P)}(s).$
\vskip 0.30cm
\noindent
{\bf Main Definition 3.} {\it The zeta function of $X$ associated to $(G,P)$
is defined by $${\widehat\zeta}_X^{(G,P)}(s):=\Big[\prod_{i=1}^{I(G,P)}\widehat\zeta_X\Big(a_i^{(G,P)}s+b_i^{(G,P)}\Big)\Big]\cdot \omega_{X}^{(G,P)}(s).$$}
\begin{thm} (Functional Equation) We have
$${\widehat\zeta}_X^{(G,P)}(-c_p-s)={\widehat\zeta}_X^{(G,P)}(s).$$
\end{thm}
\subsection{Proof of the Functional Equation}
Using the Lie structures exposed, next, we give a proof of the functional equation for the group zetas of function fields,
following [Ko], in which the group zetas for the field $\mathbb Q$ of rational numbers is treated.
\subsubsection{Lie Structures}
For $w\in W$, denote by $l(w):=|\Phi_w|$ the length of $w$. Write the longest element of $W$ as $w_0$. Then, $$w_0^2=id,\quad w_0\Delta=\Delta\qquad\mathrm{and}\quad w_0\Phi^+=\Phi^-.$$ Similarly, for a fixed $p$, denote by $w_p$ the longest element of $W_p$.
Now, for $w\in W$, introduce the subset $\frak{W}_p$ of $W$
by $$\frak{W}_p:=\{w\in W:w\Delta_p\subset\Delta\cup\Phi^-\}.$$ One checks that $\mathrm{id},w_0,w_p\in\frak{W}_p$.
For each $\alpha\in \Phi$, define its height by $\mathrm{ht}\,\alpha^\vee:=\langle\rho,\alpha^\vee\rangle$.
For $w\in\frak{W}_p$ and $(k,h)\in\mathbb Z^2$, set
$$\begin{aligned}N_{p,w}(k,h):=&\#\{\alpha\in w^{-1}\Phi^-:\langle\lambda_p,\alpha^\vee\rangle=k,\mathrm{ht}\,\alpha^\vee=h\},\\
N_{p}(k,h):=&\#\{\alpha\in \Phi:\langle\lambda_p,\alpha^\vee\rangle=k,\mathrm{ht}\,\alpha^\vee=h\},\\
M_p(k,h):=&\max_{w\in \frak{W}_p}\{N_{p,w}(k,h-1)-N_{p,w}(k,h)\},\qquad\mathrm{and}\\
\widetilde M_p(k,h):=&\max_{w\in \frak{W}_p}\{\delta(N_{p,w}(k,h-1)-N_{p,w}(k,h))\},\end{aligned}$$
where $\delta(a)=a$ if $a>0$ and 0 otherwise.
\begin{lem} The following relations hold.
\noindent
(i) If $h\geq 1$, $M_p(k,h)=\widetilde M_p(k,h)$;
\noindent
(ii) $N_p(k,kc_p-h)-M_p(k,kc_p-h+1)=N_p(k,h-1)-M_p(k,h)$;
\noindent
(iii) $c_p\lambda_p-w_p\rho=\rho$.
\end{lem}
They are various lemmas of [Ko]. More precisely, (i), (ii) and (iii) correspond to
Lem. 5.4 (1), (2) and Lem 4.1, respectively.
\subsubsection{A local decomposition}
Write by $$\omega_{X}^G(\lambda)=:\sum_{w\in W}\omega_w^G(\lambda)$$ where
$$\omega_w^G(\lambda):=\Big(\prod_{\alpha\in\Delta}\frac{1}{1-q^{-\langle w\lambda-\rho,\alpha^\vee\rangle}}\Big)\Big(\prod_{\alpha\in\Phi_w}\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big).$$
Since $$\langle w\lambda,\lambda'\rangle=\langle \lambda,w^{-1}\lambda'\rangle,\quad \ w\alpha^\vee=(w\alpha)^\vee,$$ we have, for each $w\in W$, locally,
$$\begin{aligned}\omega_w^G&(\lambda)=
\Big(\prod_{\alpha\in\Delta}\frac{1}{1-q^{-\langle w\lambda-\rho,\alpha^\vee\rangle}}\Big)
\Big[\prod_{\alpha\in\Phi_w\,\cap\, \Delta_p}\frac{1}{1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}}
\Big]\\
\times&\Big(\Big[\prod_{\alpha\in\Phi_w\,\cap\, \Delta_p}\big(1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}\big)\cdot\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big]\Big[\prod_{\alpha\in\Phi_w\backslash\Delta_p}\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big]\Big)\\
=&\Big[\Big(\prod_{\alpha\in (w^{-1}\Delta\cup \Phi_w)\,\cap\,\Delta_p}\frac{1}{1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}}\Big)
\Big(\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}}
\Big)\Big]\\
\times&\Big(\Big[\prod_{\alpha\in\Phi_w\,\cap\, \Delta_p}\big(1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}\big)\cdot\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big]\Big[\prod_{\alpha\in\Phi_w\backslash\Delta_p}\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big]\Big).\end{aligned}
$$
\subsubsection{Taking residues}
Next, for $\omega_w^G(\lambda)$, we take the residues at $s_k=0$ for $k\not=p$ and put $s_p=s$. Recall that $$\Big[\alpha\in\Delta\quad\Leftrightarrow\quad\langle\rho,\alpha^\vee\rangle=1\Big]\qquad\Rightarrow\qquad\langle\lambda-\rho,\alpha^\vee\rangle=\sum_{k=1}^na_ks_k.$$
Consequently, for each of four products appeared in the latest expression for $\omega_w^G(\lambda)$, (after taking the residue), we have
(i) For the first term, $$\prod_{\alpha\in (w^{-1}\Delta\cup \Phi_w)\,\cap\,\Delta_p}\frac{1}{1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}}=\prod_{\alpha\in (w^{-1}\Delta\cup \Phi_w)\,\cap\,\Delta_p}\frac{1}{1-q^{-s_k}};$$
(ii) For the second term,
$$\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda-\rho,\alpha^\vee\rangle}}
\Big|_{s_k=0, k\not=p;s_p=s}=\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}.$$ Since
$\alpha\in (w^{-1}\Delta)\backslash \Delta_p$, $\mathrm{ht}\,\alpha^\vee\not=1$ or $\langle \lambda_p,\alpha^\vee\rangle\not=0$. Thus the denominator do not vanish identically.
(iii) In the third term, for $\alpha_k\in\Phi_w\,\cap\, \Delta_p$, we have
$$\Big(1-q^{-\langle \lambda-\rho,\alpha_k^\vee\rangle}\Big)\cdot\frac{\widehat\zeta_X(\langle\lambda,\alpha_k^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha_k^\vee\rangle+1)}=\big(1-q^{-s_k}\Big)\cdot\frac{\widehat\zeta_X(s_k+1)}{\widehat\zeta_X(s_k+2)}=\frac{\widehat\zeta_X(1)}{\widehat\zeta_X(2)}+o(s_k)$$ as $s_k\to 0$, where
$\widehat\zeta_X(1):=\mathrm{Res}_{s=1}\widehat\zeta_X(s)$.
(iv) In the forth term, for $
\alpha\in\Phi_w\backslash\Delta_p$, we have
$$\frac{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle)}
{\widehat\zeta_X(\langle\lambda,\alpha^\vee\rangle+1)}\Big|_{s_k=0,k\not=p;s_p=s}=
\frac{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}.$$
Consequently, when taking the residues, all terms $\omega_w^G(\lambda)$ vanish
except for the $w$'s satisfying $\Delta_p\subset w^{-1}\Delta\cup \Phi_w$, i.e., $w\in\frak W_p$. Moreover, for $w\in\frak W_p$,
$$\begin{aligned}\mathrm{Res}_{s_k=0,k\not=p}\omega_w^G(\lambda)=&\Big(\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\\
&\times\Big(\prod_{\alpha\in\Phi_w\,\cap\, \Delta_p}
\frac{\widehat\zeta_X(1)}{\widehat\zeta_X(2)}\Big)\Big(\prod_{\alpha\in\Phi_w\backslash\Delta_p}\frac{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\Big).\end{aligned}$$
Therefore,
$$\begin{aligned}\omega_X^{(G,P)}(s)=&\sum_{w\in \frak W_p}\Big(
\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\\
&\times\Big(\prod_{\alpha\in\Phi_w\,\cap\, \Delta_p}
\frac{\widehat\zeta_X(1)}{\widehat\zeta_X(2)}\Big)\Big(\prod_{\alpha\in\Phi_w\backslash\Delta_p}\frac{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\Big)\\
&\hskip 1.0cm(\mathrm{since}\quad\Delta_p\subset w^{-1}\Delta\cup \Phi_w\ \Leftrightarrow\ \Delta_p\subset w^{-1}(\Delta\cup \Phi^-))\\
=&\sum_{w\in \frak W_p}\Big(
\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\\
&\times\Big(\prod_{\alpha\in\Phi_w\backslash \Delta_p}
\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\Big)
\Big(\prod_{\alpha\in\Phi_w}\frac{1}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\Big)\\
=&\sum_{w\in \frak W_p}\Big(
\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\cdot \widehat\zeta_{p,w}(s)\\
\end{aligned}\eqno(2)
$$
where, for $w\in\frak W_p$, we let
$$\begin{aligned}\widehat\zeta_{p,w}(s):=&\Big(\prod_{\alpha\in\Phi_w\backslash \Delta_p}
\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\Big)\\
&\times
\Big(\prod_{\alpha\in\Phi_w}\frac{1}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\Big).\end{aligned}$$
\subsubsection{Minimal number of factors}
With the above decomposition, we are ready to find out the minimal number of factors used in the normalization process appeared in Main Definition 3. With the expression for $\omega_X^{(G,P)}(s)$ in (2), we concentrate the zeta factors in $\widehat\zeta_{p,w}(s)$ for $w\in\frak W_p$.
By definition,
$$\begin{aligned}\prod_{\alpha\in\Phi_w\backslash \Delta_p}&
\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\\
=&\widehat\zeta_X(s+1)^{N_{p,w}(1,1)}\prod_{\alpha\in\Phi_w\backslash \Delta}
\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\\
=&\widehat\zeta_X(s+1)^{N_{p,w}(1,1)}\prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{N_{p,w}(k,h)},\end{aligned}$$ and
$$\begin{aligned}
\prod_{\alpha\in\Phi_w}&\frac{1}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\\
=&\prod_{k=0}^\infty\prod_{h=1}^\infty \widehat\zeta_X(k s+h+1)^{-N_{p,w}(k,h)}\\
=&\prod_{k=0}^\infty\prod_{h=1}^\infty \widehat\zeta_X(k s+h+1)^{-N_{p,w}(k,h-1)}.\end{aligned}$$
Hence
$$\widehat\zeta_{p,w}(s)=\widehat\zeta_X(s+1)^{N_{p,w}(1,1)}\prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{N_{p,w}(k,h)-N_{p,w}(k,h-1)}.$$
Therefore,
\hskip 2.0cm$\widehat\zeta_X(k s+h)$ {\it appears in the denominator of} $\omega_X^{(G,P)}(s)$\\
\hskip 6.0cm$\Updownarrow$\\
\hskip 4.0cm $N_{p,w}(k,h)-N_{p,w}(k,h-1)<0$.
Consequently, $$\prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{\widetilde M_{p}(k,h)}=\prod_{i=1}^{I(G,P)}\widehat\zeta_X\Big(a_i^{(G,P)}s+b_i^{(G,P)}\Big)$$
is exactly the minimal zeta factors appeared in the normalization process in defining
$\widehat\zeta_X^{(G,P)}(s)$.
Thus, by Lem.\,5(i), we have proved the following
\begin{thm} The zeta function for $X$ associated to $(G,P)$ is given by
$$\widehat \zeta_X^{(G,P)}(s)=\omega_X^{(G,P)}(s)\cdot \prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{M_{p}(k,h)}.\eqno(3)$$
\end{thm}
\subsubsection{A global decomposition}
The factor
$$\prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{M_{p}(k,h)}$$
appeared in (3) proves to be a bit hard. To overcome this, we go back to
the expression of $\omega_X^{(G,P)}(s)$ in (2). Introduce the \lq overdone' maximal factor
$$\begin{aligned}M_X^{(G,P)}(s):=M_p(s):=&\prod_{\alpha\in\Phi^+}\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)\\
=&\prod_{\alpha\in\Phi^-}\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\end{aligned}.$$ Obviously,
being maximal, $M_p(s)$ does clear up all the zeta factors in the denominators of terms of (2).
Moreover, by definition,
$$M_p(s)=\prod_{k=0}^\infty\prod_{h=1}^\infty \widehat\zeta_X(k s+h+1)^{N_{p}(k,h)}
=\prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{N_{p}(k,h-1)}.$$ Here, in the last step, we have used the functional equation for Artin zetas. This then leads to the global decomposition
$$\widehat \zeta_X^{(G,P)}(s)=\frac{\Omega_X^{(G,P)}(s)}{D_X^{(G,P)}(s)}$$
where we have set
$$\begin{aligned}\Omega_X^{(G,P)}(s):=&M_X^{(G,P)}(s)\cdot \omega_X^{(G,P)}(s),\quad\mathrm{and}\\
D_X^{(G,P)}(s):=& \prod_{k=0}^\infty\prod_{h=2}^\infty \widehat\zeta_X(k s+h)^{-M_{p}(k,h)+N_{p,w}(k,h-1)}.\end{aligned}$$
As such, then the functional equation of our zeta functions is equivalent to the following
\begin{prop}
$$D_X^{(G,P)}(-c_P-s)=D_X^{(G,P)}(s),\qquad\mathrm{and}\qquad \Omega_X^{(G,P)}(-c_P-s)=\Omega_X^{(G,P)}(s).$$
\end{prop}
\subsubsection{Functional Equation for $D_X^{(G,P)}(s)$}
This is rather easy.
Decompose $D$ according to whether it consists of special values of zetas or not to get $$D_X^{(G,P)}(s):=D_p^0\cdot
D_p^1(s)$$ where
$$\begin{aligned}D^0_p:=&\prod_{h=2}^\infty\widehat\zeta_X(h)^{N_p(0,h-1)-M_p(0,h)},\\
D_p^1(s):=&\prod_{k=1}^\infty\prod_{h=2}^\infty\widehat\zeta_X(ks+h)^{N_p(k,h-1)-M_p(k,h)}.\end{aligned}$$
It suffices to show that
$$D_p^1(-c_p-s)=D_p^1(s).$$
Since $N_{p,w}(k,h-1)=0$ and $M_p(k,h)=0$ for $k\geq 1$ and $h\leq 1$, we have
$$D_p^1(s)=\prod_{k=1}^\infty\prod_{h=-\infty}^\infty\widehat\zeta_X(ks+h)^{N_p(k,h-1)-M_p(k,h)}.$$ Consequently,
$$\begin{aligned}D_p^1(-c_p-s)=&\prod_{k=1}^\infty\prod_{h=-\infty}^\infty\widehat\zeta_X(-kc_p-ks+h)^{N_p(k,h-1)-M_p(k,h)}\\
=&\prod_{k=1}^\infty\prod_{h=-\infty}^\infty\widehat\zeta_X(ks+kc_p-h+1)^{N_p(k,h-1)-M_p(k,h)}\\
&\qquad(\mathrm{by\ the\ functional\ equation}\ \widehat\zeta_X(1-s)=\widehat\zeta_X(s))\\
=&\prod_{k=1}^\infty\prod_{h=-\infty}^\infty\widehat\zeta_X(ks+h)^{N_p(k,kc_p-h)-M_p(k,kc_p-h+1)}\\
=&\prod_{k=1}^\infty\prod_{h=-\infty}^\infty\widehat\zeta_X(ks+h)^{N_p(k,h-1)-M_p(k,h)}\ (\mathrm{by\ Lem\,5(ii)})\\
=&D_p^1(s).\end{aligned}$$
\subsubsection{Involution Structure on $\frak W_p$}
We are left with the proof of the functional equation for $\Omega_X^{(G,P)}(s)$. For this,
we use an involution structure on $\frak W_p$ given by $w\mapsto w_0ww_p$.
Set then $$\begin{aligned}
f_{p,w}(s):=&\prod_{\alpha\in(w^{-1}\Delta\backslash\Delta_p)}\frac{1}{1-q^{-\langle\lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}},\\
g_{p,w}(s):=&\prod_{\alpha\in(w^{-1}\Phi^-)\backslash\Delta_p}\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee).\end{aligned}$$
Similarly as in [Ko], we have the following
\begin{prop}
(i) ({\bf Involution Structure})
$$f_{p,w}(-c_p-s)=f_{p,w_0ww_p}(s),\qquad g_{p,w}(-c_p-s)=g_{p,w_0ww_p}(s);$$
\noindent
(ii) $$
\Omega_X^{(G,P)}(s)=
\sum_{w\in \frak W_p} f_{p,w}(s)\cdot g_{p,w}(s).$$
\end{prop}
\noindent
{\it Proof.} (i) For a fixed subset $A\subset\Phi, w\in W$, set
$$S_{p,A}(s;w):=\big\{\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee:\alpha\in(w^{-1}A)\backslash \Delta_p\big\}.$$
Then, note that, for $A=\Delta$ or $\Phi^-$,
$w_0A=-A$ and
$$-w_p(w^{-1}A\backslash\Delta_p)=(w_pw^{-1}(-A))\backslash (w_p(-\Delta_p))=(w_pw^{-1}w_0A)\backslash \Delta_p.$$ So, we have $$\begin{aligned}f_{p,w}(s)=&\prod_{as+b\in S_{p,\Delta}(s;w)}\frac{1}{1-q^{-as-b+1}},\\
g_{p,w}(s)=&\prod_{as+b\in S_{p,\Phi^-}(s;w)}\frac{1}{\widehat\zeta_X(as+b)},\end{aligned}$$
Moreover,
$$\begin{aligned}S_{p,A}(-c_p-s;w)=&\{\langle\lambda_p,\alpha^\vee\rangle(-c_p-s)+\mathrm{ht}\,\alpha^\vee:\alpha\in (w^{-1}A)\backslash \Delta_p\}\\
=&\{\langle\lambda_p,-w_p\alpha^\vee\rangle s+\langle c_p\lambda_p-w_p\rho,-w_p\alpha^\vee\rangle
:\alpha\in (w^{-1}A)\backslash \Delta_p\}\\
=&\{\langle\lambda_p,\beta^\vee\rangle s+\langle \rho,\beta^\vee\rangle
:\beta\in (w_pw^{-1}w_0A)\backslash \Delta_p\}\\
&\hskip 4.0cm (\mathrm{by\ Lem\, 5(iii)})\\
=&S_{p,A}(s;w_0ww_p).\end{aligned}$$
\noindent
(ii) In [Ko], the following Lie structures are exposed.
$$\begin{aligned}(a)\ \Phi^-\backslash (-\Phi_w)&=\Phi^-\backslash (\Phi^-\,\cap\, w^{-1}\Phi^+)=\Phi^-\backslash w^{-1}\Phi^+=\Phi^-\,\cap\, w^{-1} \Phi^-,\\
(b)\ \ (\Phi_w\backslash\Delta_p)\cup&(\Phi^-\,\cap\, w^{-1}\Phi^-)=
((\Phi^+\cap w^{-1}\Phi^-)\backslash\Delta_p)\cup(\Phi^-\,\cap\, w^{-1}\Phi^-)\\
=& ((\Phi^+\cap w^{-1}\Phi^-)\cup(\Phi^-\,\cap\, w^{-1}\Phi^-))\backslash\Delta_p
=w^{-1}\Phi^-\,\backslash\,\Delta_p.\end{aligned}$$
Consequently,
$$\begin{aligned}\Omega_X^{(G,P)}(s)\buildrel {(a)}\over=&\sum_{w\in W,\Delta_p\subset w^{-1}(\Delta\cup\Phi^-)}\Big(\prod_{\alpha\in(w^{-1}\Delta)\backslash\Delta_p}\frac{1}{1-q^{-\langle\lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\\
&\hskip 1.5cm\times\Big(\prod_{\alpha\in(\Phi_w\backslash\Delta_p)\cup(\Phi^-\,\cap\, w^{-1}\Phi^-)}\widehat\zeta_X\big(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee\big)\Big)\\
\buildrel {(b)}\over=&\sum_{w\in \frak W_p} \Big(\prod_{\alpha\in(w^{-1}\Delta\backslash\Delta_p)}\frac{1}{1-q^{-\langle\lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}\Big)\\
&\hskip 1.5cm\times\Big(\prod_{\alpha\in(w^{-1}\Phi^-)\backslash\Delta_p}\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\Big).\end{aligned}$$
\noindent
{\it Proof for Thm 4.}
With Prop. 8, the functional equation $$\Omega_X^{(G,P)}(c_P-s)=\Omega_X^{(G,P)}(s)$$
is a direct consequence of the fact that $w_0,w_p\in\frak W_p$ so $w\mapsto w_0ww_p$ induces an involution structure of $\frak W_p$. This then completes the proof of Thm 4 as well.
\vskip 0.30cm
\noindent
{\it Remark.} The functional equation for $\widehat\zeta_X^{(G,P)}(s)$ can be more directly proved using the involution structure $w\mapsto w_0w w_p$ on $\frak{W}_p$
directly, based on the relation ([KKS, Cor 8.7])
$$M_p(k,h)=N_{p,w_0}(k,h-1)-N_{p,w_0}(k,h).$$
\section{Counting Bundles}
\subsection{Uniformity and the Riemann Hypothesis}
Recall that, by Thm 2, or better, the equality (1), we have
$$\begin{aligned}\zeta_{X,r}&(s)
=\sum_{m=0}^{(g-1)-1}\alpha_{X,r}(mr)
\cdot \Big((q^{-rs})^{m}+ (q^r)^{(g-1)-m}\cdot (q^{-rs})^{2(g-1)-m}\Big)\\
&+\alpha_{X,r}\big(r(g-1)\big)\cdot (q^{-rs})^{g-1}+(q^r-1)\beta_{X,r}(0)\cdot \frac{(q^{-rs})^{g}}{(1-q^{-rs})(1-q^rq^{-rs})}.
\end{aligned}$$
Thus, the following conjecture, motivated by our works on zetas for number fields ([W2,3]),
counts semi-stable bundles decisively.
\begin{conj} (1) ({\bf Uniformity}) There are universal constants $a_{F,r}, b_{F,r}$ and rational functions $c_{F,r}(q)$ depending on $F$ and $r$ such that
$$\widehat\zeta_{F,r}(s)=c_{F,r}(q)\cdot \widehat\zeta_X^{(SL_r,P_{r-1,1})}(a_{F,r}\cdot s+b_{F,r}).$$
\noindent
(2) ({\bf The Riemann Hypothesis})
$${\widehat\zeta}_{F,r}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{1}{2}.$$
\end{conj}
That is to say, all weighted counts on semi-stables via the invariants $\alpha$'s and $\beta$'s can be read from Artin's zeta functions defined using only line bundles, while the Riemann Hypothesis
gives an effective control of the invariants $\alpha$'s and $\beta$'s.
We have the following supportive evidences.
\begin{thm} (i) ({\bf Uniformity}, [W4]) For elliptic curves, the uniformity holds when $r=1,\,2,\,3,\,4,\,5.$
\noindent
(ii) ({\bf Riemann Hypothesis}) The Riemann Hypothesis holds for
\noindent
(a) (Weil) $\widehat\zeta_X(s)$;
\noindent
(b) ([Y]) $\widehat\zeta_X^{(SL_2,P_{1,1})}(s)$;
\noindent
(c) ([W4]) $\widehat\zeta_{E,r}(s)$ for $r=2,3,4,5$ with $E$ an elliptic curve.
\end{thm}
\subsection{Parabolic Reduction, Stability and the Mass}
To end this paper, we explain the reasons why $\zeta_{X,r}(s)$ are {\it non-abelian} zeta functions of $X$, despite the uniformity claiming that, up to certain rational function factors,
$\zeta_{X,r}(s)$ can be read from {\it abelian} Artin zetas.
The central reason is certainly that $\zeta_{X,r}(s)$'s are defined using moduli spaces of semi-stable bundles, highly non-commutative objects associated to $X$. Furthermore, even assuming the uniformity, from the equation (2), we can still detect where the non-abelian structure lies on. More precisely, in each term $\omega_w^{G}(s),\ w\in\frak W_p$, while, for the zeta factor part
$$\begin{aligned}\widehat\zeta_{p,w}(s):=&\Big(\prod_{\alpha\in\Phi_w\backslash \Delta_p}
\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee)\Big)\\
&\times
\Big(\prod_{\alpha\in\Phi_w}\frac{1}
{\widehat\zeta_X(\langle\lambda_p,\alpha^\vee\rangle s+\mathrm{ht}\,\alpha^\vee+1)}\Big),\end{aligned}$$ there are involved only Artin zetas which are abelian, the non-abelian structure, through the group structure,
is naturally reflected via the
rational function factors $$\prod_{\alpha\in (w^{-1}\Delta)\backslash \Delta_p}\frac{1}{1-q^{-\langle \lambda_p,\alpha^\vee\rangle s-\mathrm{ht}\,\alpha^\vee+1}}.$$
To properly understand this, let us examine the so-called parabolic reduction structure appeared in the mass formula for function fields, and similarly, the volume formula for number fields, respectively.
As usual, let $\mathcal D_{\mathbb Q,r}$ be the volume of fundamental domain of $SL_r(\mathbb Z)$, and $\mathcal M_{\mathbb Q,r}[1]$ the moduli space of rank $r$
semi-stable lattices of volume 1. (For background materials, please refer to [W2].)
Then we have
\begin{thm} (i) (Siegel) $$\mathrm{Vol}\Big(\mathcal D_{\mathbb Q,r}\Big)=r\cdot\prod_{i=1}^r\widehat\zeta(i);$$
\noindent
(ii)
(Reformulation of [KS, \S16])
$$\mathrm{Vol}\Big(\mathcal D_{\mathbb Q,r}\Big)=\sum_{\substack{n_1,\dots,n_k\geq 1,\\ n_1+\cdots+n_k=r}}\frac{\prod_{j=1}^k\mathrm{Vol}\Big(\mathcal M_{\mathbb Q,n_j}[1]\Big)}{n_1(n_1+n_2)\cdots(n_1+\cdots+n_k)\cdots(n_k-1+n_k)n_k};$$
\noindent
(iii) (Weng [W2, \S4.8]) $$\frac{1}{r}\cdot \mathrm{Vol}\Big(\mathcal M_{\mathbb Q,r}[1]\Big)=\sum_{\substack{n_1,\dots,n_s>0,\\ n_1+\cdots+n_k=r}}
\frac{(-1)^{k-1}}{\prod_{j=1}^{k-1}(n_j+n_{j+1})}\prod_{j=1}^k \mathrm{Vol}\Big(\mathcal D_{\mathbb Q,n_j}\Big).$$
\end{thm}
\noindent
{\it Remarks.} (1) Siegel's formula claims that the volume of {\it non-abelian} fundamental domain can be
measured using special values of the {\it abelian} zeta;
\noindent
(2) Even the roots for [KS, \S16] and [W2] are very much different: the former uses arithmetic truncation of Harder-Narasimhan filtration, and the later
uses analytic truncation and Eisenstein series, they share a common origin, as we observed, namely,
the {\it parabolic reduction structure};
\noindent
(3) The part of non-abelian group structure and the part of the abelian zeta are well-organized so that they fit into a uniform theory naturally. For example, roughly, we see that fundamental domains consists of an essential part coming from stable lattices and boundary parts coming from tubular neighborhoods of cusps associated to proper parabolic subgroups.
Motivated by this, more generally, for a split reductive group $G$ defined over a number field $F$, $B$ a fixed Borel ...
denote by $G(\mathbb A)^{\mathrm{ss}}$ the adelic elements of $G$ corresponding to semi-stable principle $G$-lattices ([G]). Write $\mathbb K_G$ for the associated maximal compact subgroup.
Also for a standard parabolic subgroup $P$, write its Levi decomposition as $P=UM$ with $U$ the unipotent radical and $M$ its Levi factor. Denote the corresponding simple decomposition of $M$ as
$\prod_iM_i$ with $M_i$'s the simple factors of $M$.
Introduce invariants $$\nu_P:=\prod_i\mathrm{Vol}\Big(\mathbb K_{M_i} Z_{M_i^1(\mathbb A)}\big\backslash M_i^1(\mathbb A)\big/M_i(F)\Big)$$
and $$\mu_P:=\prod_i\mathrm{Vol}\Big(\mathbb K_{M_i} Z_{M_i^1(\mathbb A)}\big\backslash M_i^1(\mathbb A)^{\mathrm{ss}}\big/M_i(F)\Big).$$ In parallel, we have similar constructions for function fields $F=\mathbb F_q(X)$. Based on all this, then we have the following
\begin{conj} ({\bf Parabolic Reduction}) Let $G/F$ be a split reductive group with $B/F$ a fixed Borel. Then, for each standard parabolic subgroup $P$ of $G$, there exist constants
$c_P\in\mathbb Q,\ e_P\in\mathbb Q_{>0}$ such that
$$\nu_G=\sum_{P}\,c_P\cdot\nu_P,\qquad
\mu_G=\sum_{P}\,\mathrm{sgn}(P)\,e_P\cdot\nu_P,$$ where $P$ runs over all standard parabolic subgroups of $G$, and $\mathrm{sgn}(P)$ denotes the sign of $P$.
\end{conj}
The exact values of $e_P$'s can be written out in terms of the associated root system.
Indeed,
if $$W_0:=\Big\{w\in W:\{\alpha\in\Delta:w\alpha\in \Delta\cup \Phi^- \}=\Delta\Big\},$$
then there is a natural one-to-one correspondence between $W_0$ and the set of subsets of $\Delta$, and hence to the set of standard parabolic subgroups of $G$. Thus we will write
$$W_0:=\Big\{w_P:\ P\ \mathrm{standard\ parabolic\ subgroup}\Big\},$$
and, for $w=w_P\in W_0$, write $J_P\subset\Delta$ the corresponding subset.
\begin{conj} ({\bf Parabolic Reduction, Stability \& the Mass}, [W5]) Let $G$ be a split type reductive group
with $P$ its maximal parabolic subgroup.
\noindent
(1) Over a number field $F$,
\noindent
(i) The volume of moduli space of semi-stable principal lattices is given by
$$\nu_G=\mathrm{Res}_{s=-c_P}\widehat\zeta_F^{(G,P)}(s)=\mathrm{Res}_{\lambda=\rho}\omega_F^{G}(\lambda);$$
\noindent
(ii) We have the following formula
$$ \mu_G=\sum_{P}\frac{(-1)^{\mathrm{rank}(P)}}{\prod_{\alpha\in\Delta\backslash w_JJ_P}(1-\langle w_J\rho,\alpha^\vee\rangle)}\cdot\nu_P;$$
\noindent
(2) Over an irreducible reduced regular projective curve $X$,
\noindent
(i) The mass of moduli space of semi-stable principal bundles is given by
$$\log q\cdot \nu_G=\mathrm{Res}_{s=-c_P}\widehat\zeta_X^{(G,P)}(s)=\mathrm{Res}_{\lambda=\rho}\omega_X^{G}(\lambda);$$
\noindent
(ii) We have the following formula
$$\mu_G=\sum_{P}\frac{(-1)^{\mathrm{rank}(P)}}{\prod_{\alpha\in\Delta\backslash w_JJ_P}(1-q^{\langle w_J\rho,\alpha^\vee\rangle-1})}\cdot\nu_P;$$
\end{conj}
\noindent
{\it Remarks.} (1) We expect that $c_P>0$ for and only for number fields.
\noindent
(2) Calculations in [Ad] for lower ranks groups indicates that, for number fields, $\frac{1}{c_P}\in\mathbb Z_{>0}$. It would be very interesting to find a close formula for them.
\vskip 0.30cm
All this indicates that non-commutative group structures are naturally embedded into our pure high rank zeta functions.
\vskip 0.70cm
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\vskip 0.40cm
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\vskip 0.10cm
\noindent
[Z] D. Zagier, Elementary aspects of the Verlinde formula and the Harder-Narasimhan-Atiyah-Bott formula, in {\it Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry}, 445-462 (1996)
\vskip 0.30cm
\noindent
Lin WENG\footnote{{\bf Acknowledgement.}
We would like to thank IHES for providing us an excellent working environment during our September visit in 2011. Special thanks also due to Kontsevich and Yoshida for sharing with us their works, which motivate
our current works,
to Lafforgue for enlightening us the interchangeability between taking residues and taking integrations over moduli spaces of semi-stable lattices, which leads to the uniformity conjecture, and to Deninger and Hida for constant encouragements.
This work is partially supported by JSPS.}
\noindent
Institute for Fundamental Research, The $L$ Academy {\it and}
\noindent
Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395,
JAPAN
\noindent
E-Mail: weng@math.kyushu-u.ac.jp
\end{document} | {
"timestamp": "2012-02-21T02:00:32",
"yymm": "1202",
"arxiv_id": "1202.3183",
"language": "en",
"url": "https://arxiv.org/abs/1202.3183",
"abstract": "We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group, maximal parabolic subgroup). Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given. We end this paper with an explanation on why these zetas are non-abelian in nature, using our up-coming works on 'parabolic reduction, stability and the mass'.The constructions and results were announced in our paper on 'Counting Bundles'arXiv:1202.0869.",
"subjects": "Algebraic Geometry (math.AG)",
"title": "Zeta functions for function fields",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846653465638,
"lm_q2_score": 0.72487026428967,
"lm_q1q2_score": 0.7092019509467241
} |
https://arxiv.org/abs/1804.05985 | Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem | We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244. | \section{Introduction}
\subsection{The Integer Chebyshev Problem}
The supremum norm of a polynomial $p$ over an interval $I$ is defined as
\begin{equation}
\|p(x)\|_I=\sup_{x\in I}|p(x)|
\end{equation}
Let $\mathbb{Z}_n[x]$ denote the polynomials of degree at most $n$ with integer coefficients. The integer Chebyshev problem is the problem of finding a polynomial in $\mathbb{Z}_n[x]$ of minimal supremum norm on the interval $I$, normalized by the degree of the polynomial; most commonly, and in this paper, the case $I=[0,1]$ is considered.
More precisely, we have the following definition
\begin{definition}
For $n>0$, define
\begin{equation}
t_{\mathbb{Z},n}(I) = \min_{\substack{p\in \mathbb{Z}_n[x] \\ p \neq 0}}\|p(x)\|^{1/n}_I
\end{equation}
Any degree $n$ polynomial $p\in \mathbb{Z}_n[x]$ such that $\|p(x)\|^{1/n}_I=t_{\mathbb{Z},n}(I)$ is called an integer Chebyshev polynomial of degree $n$. Then the value
\begin{equation}
t_{\mathbb{Z}}(I)=\lim_{n\rightarrow \infty} t_{\mathbb{Z},n}(I)
\end{equation}
is called the integer Chebyshev constant, or integer transfinite diameter for the interval $I$.
\end{definition}
We see that the limit in the definition exists, since
\begin{equation} \label{eq:4}
t_{\mathbb{Z},n+m}(I)^{n+m}\leq t_{\mathbb{Z},n}(I)^nt_{\mathbb{Z},m}(I)^m.
\end{equation}
This follows from the fact that if $p(x)$ and $q(x)$ are integer Chebyshev polynomials of degree $n$ and $m$ respectively, then $t_{\mathbb{Z},n+m}(I)\leq \|p(x)q(x)\|^\frac{1}{n+m}_I$.
We note here that for intervals of size greater than 4, the problem as stated here is solved, with the integer Chebyshev constant for the interval as 1, and the polynomial identically 1 on that interval.
Let $p(x) = a_n x^n + \dots + a_0$ be a polynomial with integer coefficients, and all of whose conjugates are in an interval $I$.
Then it is relatively easy to show, by means of resultants, that $t_{\mathbb Z}(I) \geq \frac{1}{a_n^{1/n}}$.
See for example \cite{b02}.
By considering the numerator of the iterates of the function
\[ u(x) = \frac{z (1-z)}{1-3 z (1-z)} \]
we get an infinite family of polynomials, known as the Gorshkov-Wirsing polynomials, all of whose conjugates are in $[0,1]$.
This is used to show that $t_{\mathbb Z}([0,1]) \geq \frac{1}{2.3768417062} \approx 0.4207263$.
Here this constant is explicitly computable to an arbitrary number of digits.
See \cite{g59, m94} for details.
In \cite{be95} it was shown that the lower bound coming from this infinite family is in fact not best possible.
That is, there exists an $\epsilon > 0$ such that $t_{\mathbb Z}([0,1]) \geq 0.4207263\dots + \epsilon$.
At the time no non-trivial lower bound for $\epsilon$ was determined.
Pritsker showed in \cite{p05}, by means of weighted potential theory, that
$t_{\mathbb Z}([0,1]) \geq 0.4213$.
Generalizations of these Gorshkov-Wirsing polynomails were considered in \cite{h11}.
Given the submultiplicative nature of $t_{\mathbb Z, n}(I)$ we have
$t_{\mathbb Z}(I) \leq t_{\mathbb Z, n}(I)$ for all $n$.
This gives a simple method to find an upper bound for $t_{\mathbb Z}(I)$; find large degree polynomials with
small supremum norm.
In \cite{be95} a set of $9$ polynomials $p_i(x)$ and exponents $a_i$ were found such that the resulting
polynomial $P(x) = p_1(x)^{a_1} \dots p_9(x)^{a_9}$ had small supremum norm.
This was used to show that $t_{\mathbb Z}([0,1]) \leq \frac{1}{2.3605\dots} \approx 0.42364$.
This was done computationally by means of a simplex search on a very large grid of points in $[0,1]$.
The upper bound was improved by Habsieger and Salvy in \cite{hs97} to give $t_{\mathbb Z}([0,1]) \leq 0.4234794$.
In addition, Habsieger and Salvy explicitly computed integer Chebyshev polynomials of degree $n$ for all $n \leq 75$.
Wu in \cite{w03} extended this list of integer Chebshev polynomials up to degree 100.
In \cite{p05,p99} Pritsker introduced techniques from weighted potential theory to help improve
lower bounds for the multiplicity of factors in an integer Chebyshev polynomial of degree $n$.
With this he improved the upper bound to $t_{\mathbb Z}([0,1]) \leq 0.4232$.
Flammang in \cite{f09}, by means of auxiliary functions constructed with $28$ polynomials,
improved this upper bound to 0.42291334.
In the 2009 PhD thesis of Meichsner \cite{m09}, extensive computational work was done on this problem.
An upper bound of $1/2.36482727 \approx 0.42286386$ was found.
In addition, Meichsner extended the collection of known integer Chebyshev polynomials completely up to degree
$145$, and also for a large collection of polynomials up to degree $230$.
It is upon this result that we improve.
Later Flammang, in \cite{f14} improved these upper bounds again to 0.422685
by finding a good lower bound in the absolute length of a polynomial.
This last bound is the best known bound.
We unfortunately were not able to improve upon this bound in this paper.
There have been many variations of this problem considered.
For example, the integer Chebyshev problem on sets other than $[0,1]$ has been
considered in \cite{ap07, frs97, frs06}.
The monic integer Chebyshev polynomial, denoted $t_M(I)$, was introduced in \cite{bpp03}.
What is interesting in the monic case is that often exact value can be computed.
In \cite{hs06} it was shown, for all $a$, that $t_M([a, a+1]) = 1/2$.
It was shown that $t_M([0,x])$ is continuous in $x$ for $x > 0$ in \cite{h08b}.
The multivariate case has been looked at in \cite{bp08, m09}.
Applications of this to the leading coefficient in Schur's problem were studied in \cite{p12}.
In this paper, we will apply semi-infinite programming techniques to the problem of finding integer Chebyshev polynomials of degree $n$ on $[0,1]$. We prove 16 polynomials on $[0,1]$ to be integer Chebyshev polynomials of degree 147, 149, 152, 153, 154, 158, 175, 191, 194, 198, 202, 236, 238, 239, 241 and 244 respectively. These new polynomials were not previously known to be integer Chebyshev polynomials, and the previous largest-known integer Chebyshev polynomial was of degree 230. There still remain 20 integer Chebyshev polynomials of degree less than 244 that are unknown.
\subsection{Semi-Infinite Programming}
A semi-infinite programming (SIP) problem is an optimization problem that, in primal form, can be formulated as
\begin{equation*}
\begin{array}{llll}
(P) &\displaystyle\min_{x\in D} f(x), & D = \{x\in \mathbb{R}: g(x,t) \geq 0, &\forall t\in T\}\\
\end{array}
\end{equation*}
where $x\in \mathbb{R}^n$ and $T$ is an infinite set. Thus the optimization problem is over a finite number of variables subject to an infinite number of constraints. In a linear semi-infinite programming (LSIP) problem, the objective function $f$ and constraint functions $g(x,t)$, $t\in T$ are affine in the variable $x$. For a general reference on linear semi-infinite programming, see \cite{gl98}.
One of the difficulties of solving LSIP problems is that in order to check the feasibility of a solution $\bar{x}$, we must be able to check whether $\bar{x}\in D$, that is, whether the solution to
\begin{equation*}
(Q) \quad \text{min} \quad g(\bar{x},t) \ \forall t\in T
\end{equation*}
is greater than 0. This is known as the lower-level problem. The problem $(Q)$ is, in general, a non-linear global optimization problem, so may be very difficult to solve. Handling this part is often left to the specific application.
The cutting plane algorithm used in this paper to solve the LSIP problems that will arise is in the class of discretization algorithms. The common theme of this class is to replace the set $T$ by a finite subset $T_k \subset T$, so that we can solve the resulting LP, denoted by $P(T_k)$, with finite constraints, using the simplex method. These algorithms are covered in a general setting in chapter 10 of \cite{gl98}.
The cutting plane algorithm is an iterative algorithm, which produces the next discretized set $T_{k+1}$ by using the solution to the discretized problem on iteration $k$. In particular, given a solution $\overline{x}$ to $P(T_k)$, we solve the lower level problem $(Q)$ to determine a set of values $T'={t\in T : g(\overline{x},t) > 0}$. We then set $T_{k+1}=T\cup T'$ and iterate. Termination occurs when $\overline{x}$ is feasible to within some tolerance $\epsilon$. The constraints given by $t\in T'$ are called cutting planes, as they separate the current solution from some set containing all feasible solutions. This algorithm will be further described in the context it is used in Section 2.
\section{Integer Chebyshev Polynomials of Degree $n$}
The best known methods for finding an integer Chebyshev polynomial $p_n$ for given $n$ use 3 steps:
\begin{enumerate}
\item Determine an initial upper bound for $t_{\mathbb{Z},n}[0,1]$. The typical way to do this is to make use of Equation \eqref{eq:4}, and take
\begin{equation}
c_n=\displaystyle\min_{k\in \{1..n-1\}} \|p_{k}(x)p_{n-k}(x)\|_{[0,1]}
\end{equation}
where $p_i$ is an integer Chebyshev polynomial for each $i\in \{1,..,n-1\}$.
\item Using a set of known factors of integer Chebyshev polynomials of degree $k < n$, determine which are necessary factors of an integer Chebyshev polynomial of degree $n$ with supremum norm less than $c_n$. At the end of this step, we refer to the product of the known factors as $F(x)$ and the product of the unknown factors as $G(x)$.
\item Compute $G(x)$.
\end{enumerate}
Step 3 begins where the techniques of Step 2 fail to produce additional factors, and consists of performing some type of exhaustive search of the possible factors $G(x)$ and choosing the one that minimizes $\|G(x)F(x)\|_{[0,1]}$. Thus $p_n(x)=G(x)F(x)$ is an integer Chebyshev polynomial of degree $n$.
We note the following lemmas, whose proofs can be found in \cite{hs97}:
\begin{lemma}
If $p_n(x)$ is an integer Chebyshev polynomial of degree $n$ for the interval $[0,1/4]$, then $p_n(x(1-x))$ is an integer Chebyshev polynomial of degree $2n$ for the interval $[0,1]$.
\end{lemma}
It follows that there is a symmetric integer Chebyshev polynomial of degree $n$ on $[0,1]$ for all even $n$. We also have
\begin{lemma}
If $n$ is odd, then there exists a symmetric integer Chebyshev polynomial of degree $n$ of the form $(2x-1)p(x(1-x))$
\end{lemma}
These results allow us to work on the interval $[0,1/4]$ instead of $[0,1]$. This proves useful, for example in Step 3 above, by reducing the degree of the unknown factor $G$ before the exhaustive search.
In attempting to find an integer Chebyshev polynomial $p_n(x)$ of degree $n$, we first use methods to determine necessary factors (and their multiplicities) of $p_n(x)$, given that $\|p_n(x)\|\leq c_n$. For all current methods, once we are no longer able produce new factors, we are left with known factors $F$ where $\deg(F) < \deg(p_n)$, if $n$ is sufficiently large.
Although the focus of this paper is on improvements to the search for the missing factor $G(x)$, we will briefly describe some of the basic methods that allow for determining the necessary factors $F$. In particular, a simple method used in \cite{hs97} and others is as follows. Given an upper bound $c_n$ on $t_{\mathbb{Z}}[0,1]$, we have
$$|p_n(x)|=|G(x)|\cdot |F(x)|\leq c_n$$
for all $x \in [0,1]$. We then may be able to prove that a factor of the form $ax-b$ $(a,b\in \mathbb{Z})$ must divide $G(x)$. If
$$c_n \leq \frac{|F(b/a)|}{a^g},$$
where $g=\deg(G)$, then $a^g|G(b/a)|<1$. But $a^gG(b/a)$ is the resultant of $G(x)$ and $ax-b$, and since both have integer coefficients, the resultant is also integer. Thus we conclude that $a^g|G(b/a)|=0$ and so $ax-b$ divides $G(x)$. We then update $F(x)$ to $F(x)\cdot (ax-b)$ and $G(x)$ to $G(x)/(ax-b)$, so $G(x)$ is still the remaining unknown factor.
Similar techniques can be used for algebraic numbers whose conjugates all lie
within the interval.
This technique can easily be extended to multiple factors by using Markov's bound on $m$th derivatives, as done in \cite{hs97}. More sophisticated techniques have been developed, ranging from Lagrange interpolation in \cite{hs97} to generalized Muntz-Legendre polynomials in \cite{w03} to applications of the simplex method in \cite{m09}. An overview of these methods as well as the most recent and most effective techniques can be found in \cite{m09}.
For the remainder of this section, we use the results of Lemmas 1 and 2 to work instead on the interval $[0,1/4]$ with the transformation $x\mapsto \frac{1-\sqrt{1-4x}}{2}$. We will use $G(x)$ and $F(x)$ to refer to the unknown and known factors respectively on the interval $[0,1/4]$. This transformation halves the degree of $G(x)$, resulting in nearly half as many coefficients to determine.
The algorithm presented here requires the known factors $F(x)$ as input, since we are only concerned with computing $G(x)$. The current best methods for finding $F$ can be found in \cite{m09}, where an implementation in Maple is provided. All results of this section are based on the known factors $F$ provided by that implementation.
\subsection{Formulation as an Integer Semi-Infinite Program}
We can formulate our problem as the following semi-infinite program (SIP) with integer variables $a_i$ and continuous variable $c$:
\begin{equation*}
\begin{array}{llll}
(IP)&\text{minimize} & c &\\
& \text{subject to}& -c \leq \left|F(x)\right| \displaystyle\sum_{i=0}^g a_ix^i \leq c & \forall x\in [0,1/4] \\
& & a_i \in \mathbb{Z} & \forall i\in \{0,\dots,g\}
\end{array}
\end{equation*}
where $\sum_{i=0}^g a_ix^i=G(x)$, $g=\deg(G)$. We note that the constraints are linear in the coefficients $a_i$, so this is a linear mixed-integer SIP. To handle the integrality constraints, we propose a branch and bound algorithm similar to those used for solving integer linear programs.
\subsection{Branch and Bound}
Branch and bound algorithms for minimizing an arbitrary objective function work by producing a tree of search nodes and maintaining a record of the best known solution; in our case, this will be $G^*(x)$, where $\|G^*(x)F(x)\|=c^*$
The root node of the search tree represents the optimization problem over the full solution space, while each branch involves the partition of the search space into two or more components. By finding a lower bound on the objective value of each node, we can eliminate those nodes whose lower bound is greater than $c^*$. We call this "cutting" or "pruning" the branch.
Branch and bound methods to solve integer programs find lower bounds by solving a relaxed linear program, and branch by adding constraints to the integer variables to produce new nodes partitioning the search space. Our approach here is similar, but the relaxed problem is instead a linear SIP.
Each node $N$ is defined by the set of constraints $C$ that have been added to the variables during branching. In order to get a lower bound on the solutions obtainable from $N$, we solve the SIP $(R)$ given by the relaxation of the integrality constraints of $(IP)$ together with the constraints $C$, to get a solution $(\bar{a}, \bar{c})$ with $\bar{c}\leq \nu(IP)$, where $\nu(IP)$ is the optimal solution to $(IP)$. We can find candidates for new best solutions at very low cost by rounding $\bar{a}$ to get $a^*$. If
\begin{equation}
\left\|\left( \sum_{i=0}^g a_i^* x^i \right)F(x)\right\|\leq c^*
\end{equation}
then we replace $G^*(x)$ by $\sum_{i=0}^g a_i^* x^i$. At the end of the algorithm, we output $G^*(x)$, where $G^*(x)F(x)$ is the integer Chebyshev polynomial of degree $n$.
In order to fully specify the branch and bound algorithm, we must describe three more procedures:
\begin{enumerate}
\item A lower bounding method for each node in the branch and bound tree.
\item A branching protocol.
\item A policy for deciding which node to process next.
\end{enumerate}
\subsubsection{Lower bounding at each node}
To find a lower bound at node $N$, we use a cutting plane algorithm as described in the introduction to solve the LSIP. We use an iterative process beginning with a finite subset $T_0 \subset [0,1]$, and compute a solution $(\bar{a}, \bar{c})$ to the LP given by only considering constraints defined by $x\in T_0$ in the SIP $R$ of $N$. We then let
$$f(x)=\left(\sum_{i=0}^g a_i^* x^i\right)F(x),$$
and set
$$T_1=T_0 \cup \{x\in [0,1] : f(x) > \bar{c}, f'(x)=0 \}.$$
We iterate in this fashion until no such values of $x$ can be found or the changes to $\bar{c}$ between successive iterations is sufficiently small. A lower bound for the node $N$ is then provided by the value of $\bar{c}$ from the last iteration.
\subsubsection{Branching protocol}
Before detailing the branching technique, we first note that we need only consider polynomials $G(x)$ such that $a_g \geq 1$, since we are searching for a degree $g$ polynomial and negation does not change the supremum norm. Experimentally, combining this constraint with constraints on the variable $a_0$ (e.g. $a_0=1$) produced the largest likelihood of branch cutting occurring near the start of the algorithm. This motivates our branching protocol: given a set $C$ of constraints for the node $N$, we branch on the coefficient $a_i$ with the smallest index $i$ where $C$ does not already contain an equality constraint for $a_i$. If there are no constraints for $a_i$ in $C$, then we produce four new nodes $N_1, N_2, N_3, N_4$ with constraint sets
\begin{itemize}
\item $C_1 = C \cup \{a_i = \lceil \overline{a}_i \rceil \}$
\item $C_2 = C \cup \{a_i = \lfloor \overline{a}_i \rfloor \}$
\item $C_3 = C \cup \{a_i \geq \lceil \overline{a}_i \rceil +1 \}$
\item $C_4 = C \cup \{a_i \leq \lfloor \overline{a}_i \rfloor -1 \}$
\end{itemize}
respectively.
If there are already inequality constraints on $a_i$ in $C$, then we produce nodes with constraint sets similarly to the above, only where the new constraints do not conflict with the existing constraints in $C$, always producing at least 2 equality constraints.
This branching is a trade off between branching with equality on all possible values of $a_i$, and binary branching using only inequalities. This method avoids computing bounds on the value of $a_i$ while also reducing the total number of nodes created compared to binary branching.
\subsubsection{Selecting the next node to process}
The selection of the next node to process is determined by the data structure in which the nodes are placed. We used a priority queue whereby nodes with smaller $\overline{c}$ are processed first, yielding a "best first search". The heuristic justification of this choice is that since we are searching for nodes with smaller lower bounds first, we will not spend time on a branch which has a better chance of being cut.
\subsubsection{The algorithm}
Algorithm \ref{alg:bnb} is a pseudo-code description of the branch and bound algorithm, given procedures for finding lower bounds and determining new nodes by branching, as described above
\begin{algorithm}
\caption{Branch and Bound}
\begin{algorithmic}[1]
\Procedure{BranchAndBound}{$n,F,c_0$}
\State $UpperBound \gets c_0$
\State $BestSol \gets \text{empty}$
\State $PriQueue \gets (N_0, 0)$ \Comment{$N_0$ has no variable constraints}
\While{$PriQueue \ \text{not empty}$}
\State $(N,\overline{a}) \gets \text{GetNode}(PriQueue)$
\State $a^* \gets \text{Round}(\overline{a})$
\State $c^* \gets \left\|\left( \sum a_i^* x^i \right)F(x)\right\|$
\If {$c^* \leq UpperBound$}
\State $UpperBound \gets c^*$
\State $BestSol \gets a^*$
\EndIf
\State $(N_1,N_2,N_3,N_4) \gets \text{Branch}(N)$
\For{$i$ from 1 to 4}
\State $(\overline{a},\overline{c}) \gets \text{LowerBound}(N_i)$
\If{$\overline{c} < UpperBound$}
\State $\text{AddNode}(PriQueue, \text{node}=(N_i, \overline{a}), \text{priority}=\overline{c})$
\EndIf
\EndFor
\EndWhile
\EndProcedure
\end{algorithmic}
\label{alg:bnb}
\end{algorithm}
We note that this algorithm is similar in spirit to the technique used in \cite{hs97} for computing $G(x)$, although described in a different framework. Improvements in our algorithm come from eliminating the need to find bounds on each coefficient and that the branch and bound algorithm is more likely to find good solutions sooner by use of the priority queue.
\subsection{Resultant Search}
We describe here the method used in \cite{m09} to find $G(x)$. This method produced the best results before this paper, and will be used in combination with our branch and bound algorithm to find new integer Chebyshev polynomials.
The main idea is to write the coefficients of $G(x)$ in terms of the resultants of $G(x)$ with $g+1$ linear polynomials, and to perform a search on these resultants, which are also integers, instead of the coefficients themselves. Then we can determine a set of congruences the resultants must satisfy, greatly reducing the number of possible combinations of coefficients.
Consider the $g+1$ linear polynomials $v_ix-w_i$. Then the resultants $r_i=v_i^gG(w_i/v_i)$ for each $i$ give a system of equations of the resultants in terms of the coefficients $a_k$ of $G(x)$. We can solve this system for the variables $a_k$ to get
$$\frac{1}{m_k}\sum_{i=1}^{g+1}t_{k,i}r_i=a_k$$
with $m_k,t_{k,i}\in \mathbb{Z}$. Then
$$\sum_{i=1}^{g+1}t_{k,i}r_i \equiv 0 \ (\text{mod}\ m_k).$$
Letting $M=\text{lcm}(m_1, m_2, ...,m_{g+1})$ gives
$$\sum_{i=1}^{g+1}\frac{M}{m_k}t_{k,i}r_i \equiv 0 \ (\text{mod} \ M).$$
We write this in matrix form as
$$S\mathbf{r}=
\begin{bmatrix}
s_{1,1} & s_{1,2} & \dots & s_{1,g+1} \\
s_{2,1} & s_{2,2} & \dots & s_{2,g+1} \\
\vdots & \vdots & \ddots & \vdots \\
s_{g+1,1} & s_{g+1,2} & \dots & s_{g+1,g+1}
\end{bmatrix}
\begin{bmatrix}
r_1 \\ r_2 \\ \vdots \\ r_{g+1}
\end{bmatrix}
\equiv
\begin{bmatrix}
0 \\ 0 \\ \vdots \\ 0
\end{bmatrix}
\ (\text{mod} \ M).
$$
Using a form of Gaussian elimination with the Euclidean algorithm, we can reduce the matrix $S$ to an upper triangular matrix $S'$. Given bounds on each $r_i$, we can then try all possible values of $\mathbf{r}$ by back substitution in $S'$, solving the linear congruence in one variable at each step. We find bounds on the $r_i$ by noting that $r_i=v_i^g|G(w_i/v_i)|\leq v_i^gc_n/|F(w_i/v_i)|$ or by solving the SIP
$$
\begin{array}{llll}
&\text{minimize} & r_i=v_i^g|G(w_i/v_i)| &\\
&\text{subject to} &\left| G(x) \right| \left|F(x)\right| \leq c_n &\forall x\in [0,1/4]\\
\end{array}
$$
with the coefficients of $G(x)$ as variables, using some subset of $[0,1/4]$ for the constraints (around 200 constraints), whichever gives a better bound.
For each valid $\mathbf{r}$, we can compute $G(x)$ and find the $\mathbf{r}$ giving $G(x)$ that minimizes $\|G(x)F(x)\|$. Then $G(x)F(x)$ is an integer Chebyshev polynomial of degree $n$ on $[0,1/4]$, which we can then transform to find the symmetric integer Chebyshev polynomial on $[0,1]$.
\subsection{Combining Methods}
Along with the other methods given in \cite{m09}, the resultant search method allowed the computation of all integer Chebyshev polynomials up to degree 145, and many up to degree 230, and included all integer Chebyshev polynomials for which the degree of the missing factor $G(x)$ on the interval $[0,1/4]$ was less than 15. Unfortunately, the exponential nature of the branch and bound method prevents scaling to the same extent. For example, in step 3 of finding the integer Chebyshev polynomial of degree 120, $G(x)$ has degree 12. The resultant search method completes in 3 minutes, while branch and bound takes over 8 hours. However, fast progress by the branch and bound algorithm in the early stages motivates a combination of the two methods wherein we start off branching before doing a resultant search on the remaining variables.
More precisely, we observed that large amounts of branch cutting occurred in the early stages of the branch and bound algorithm. This was especially true when the signs of the early variables did not match with the signs of the variables in the minimizing factor. For example, if $a_0=1$ for the factor $G^*(x)$ minimizing $\|G(x)F(x)\|$, then the node with constraint $a_0=-1$ would always produce a lower bound greater than $c_n$. Such asymmetry does not exist when searching the resultants.
We exploit this early effectiveness of branching by combining the two methods; we first start with branch and bound, and branch until a specified variable is reached. Given the determined coefficients, we then do a resultant search to find the remaining coefficients that minimize $\|G(x)F(x)\|$ for each node.
Suppose that we branch until coefficient $j$. Let $\overline{a}_0,\overline{a}_1,...,\overline{a}_j$ be the known coefficients and let $a_{j+1},a_{j+2},...,a_g$ be the unknown coefficients. Then the resultant of $G(x)$ with $v_ix-w_i$ is
$$r_i=v_i^g\left( \sum_{k=0}^j\overline{a}_k(w_i/v_i)^k+ \sum_{k=j+1}^ga_k(w_i/v_i)^k \right).$$
Rearranging, we have
$$\overline{r}_i=r_i-v_i^g\sum_{k=0}^j\overline{a}_k(w_i/v_i)^k=v_i^g\sum_{k=j+1}^ga_k(w_i/v_i)^k$$
with $\overline{r}_i\in \mathbb{Z}$. We can use the system of equations between the $\overline{r_i}$ and the $a_k$ given in this way for the resultant search, shifting the bounds on $r_i$ by $\sum_{k=0}^j\overline{a}_k(w_i/v_i)^k$ to get bounds on $\overline{r_i}$.
In practice, branching until 11 variables remained proved effective. Branching later resulted in set up computations for the resultant search being wasted on a small search, and branching earlier often made the resultant search too large to be feasible in our implementation.
A running time comparison on a select few examples between the combined method and the branch and bound and resultant search alone is given in Table \ref{table:time}. As can be seen, the combined method is not unilaterally better than the resultant search alone, particularly for smaller $n$. However, for larger $n$, we can see that there are instances where the combined method will finish quite quickly, while the resultant search alone does not finish within any reasonable timeframe.
\begin{table}
\begin{center}
\begin{tabular}{ l | c | c | c }
$n$ & Branch and Bound & Resultant Search & Combined Method \\
\hline
120 & 8.1 hours & 182 seconds & 1560 seconds \\
145 & - & 17.5 hours & 24.0 hours\\
154 & - & - & 15.7 hours \\
199 & - & 87.1 hours & 39.9 hours \\
\hline
\end{tabular}
\caption{Time for algorithms to compute integer Chebyshev polynomial of degree $n$. An entry of "-" indicates the program did not terminate within 7 days. Tests were run on AMD Opteron 6174 2.2 GHz processors.}
\label{table:time}
\end{center}
\end{table}
We note two additional advantages of the combined method that are inherited from the branch and bound algorithm. The "best-first" search of the branch and bound method applies here as well, and the variable sets chosen for the first few searches will be the variable sets indicated to be most promising by the LP solutions at all active nodes. Also, as the vast majority of the computations happen within nodes and not between them, the branching portion of the algorithm is very simple to parallelize. Furthermore, it parallelizes nearly perfectly once the number of active nodes exceeds the number of cores available. Future research could implement a parallelized version of the algorithm that could help find integer Chebyshev polynomials that have proven elusive thus far.
\subsection{Results}
Our implementation in Maple of the combined branch and bound and resultant search method, as well as additional data files, can be found at \cite{kghareweb}.
Using the combined branch and bound and resultant search algorithm, we were able to find 16 more integer Chebyshev polynomials not given in \cite{m09}, listed in Table \ref{table:ICPs}. The polynomials $h_i$ are given in Appendix A. The missing factors found for these all had degrees in the range 14-16, except for the degree 202 integer Chebyshev polynomial, which had a degree 17 missing factor.
\begin{table}
\begin{center}
\begin{tabular}{ l | c | r }
$n$ & $p_n $ & $t_{\mathbb{Z}, n}[0,1]$ \\
\hline
147 & $h_1 ^{48}h_2 ^{17}h_3 ^6h_5 ^2h_{10}h_{14}$ & 0.42591455 \\
149 & $h_1 ^{47}h_2 ^{17}h_3 ^6h_5 ^3h_{10} h_{14} $ & 0.42578804\\
152 & $h_1 ^{47}h_2 ^{16}h_3 ^6h_5 ^2h_{10} h_{23} $ & 0.42577465\\
153 & $h_1 ^{48}h_2 ^{19}h_3 ^5h_4 h_5 ^2h_7 h_{10} h_{14} $ & 0.42547485\\
154 & $h_1 ^{49}h_2 ^{18}h_3 ^6h_5 ^3h_{10} h_{14} $ & 0.42548736\\
158 & $h_1 ^{51}h_2 ^{18}h_3 ^6h_5 ^3h_{10} h_{14} $ & 0.42536299\\
175 & $h_1 ^{56}h_2 ^{23}h_3 ^6h_4 h_5 ^2h_7 h_{10} h_{14} $ & 0.42542222\\
191 & $h_1 ^{60}h_2 ^{21}h_3 ^8h_5 ^3h_{10} h_{14} h_{15} $ & 0.42512849\\
194 & $h_1 ^{61}h_2 ^{22}h_3 ^8h_5 ^3h_{10} h_{14} h_{15} $ & 0.42517829\\
198 & $h_1 ^{63}h_2 ^{22}h_3 ^8h_5 ^3h_{10} h_{14} h_{15} $ & 0.42505003\\
202 & $h_1 ^{64}h_2 ^{24}h_3 ^9h_4 h_5 ^3h_6 h_{10} h_{14} $ & 0.42514131\\
236 & $h_1 ^{75}h_2 ^{28}h_3 ^9h_4 h_5 ^4h_{10} h_{14} h_{15} $ & 0.42434377\\
238 & $h_1 ^{76}h_2 ^{28}h_3 ^9h_4 h_5 ^4h_{10} h_{14} h_{15} $ & 0.42468031\\
239 & $h_1 ^{76}h_2 ^{27}h_3 ^{10}h_5 ^3h_{10} h_{14} h_{21} $ & 0.42461390\\
241 & $h_1 ^{77}h_2 ^{27}h_3 ^{10}h_5 ^3h_{10} h_{14} h_{21} $ & 0.42448242\\
244 & $h_1 ^{78}h_2 ^{30}h_3 ^{11}h_4 h_5 ^4h_6 h_{10} h_{14}$ & 0.42456112\\
\hline
\end{tabular}
\caption{New integer Chebyshev polynomials for the interval $[0,1]$. The polynomials $h_i$ are given in Appendix A.}
\label{table:ICPs}
\end{center}
\end{table}
\section{Conclusion}
In this paper, we presented an improved method for finding integer Chebyshev polynomials of degree $n$. We implemented additions to the standard semi-infinite programming methods, including a branch and bound technique combined with previous methods. These efforts yielded 16 new integer Chebyshev polynomials of degree ranging from 147 to 244. There remain 20 integer Chebyshev polynomials of degree less than 244 that are not known.
In the search for integer Chebyshev polynomials, any improvement on either computing the necessary degree of known factors or on the exhaustive search to find remaining factors will likely yield improved results. As noted previously, the branch and bound algorithm naturally parallelizes, and this would be a good way to continue to find integer Chebyshev polynomials, although there are limitations compared to an algorithmic improvement.
| {
"timestamp": "2018-10-29T01:06:30",
"yymm": "1804",
"arxiv_id": "1804.05985",
"language": "en",
"url": "https://arxiv.org/abs/1804.05985",
"abstract": "We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.",
"subjects": "Number Theory (math.NT)",
"title": "Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860382,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7092019500330067
} |
https://arxiv.org/abs/2003.07852 | String topology of finite groups of Lie type | We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration.We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings. | \section{Introduction}
For large primes $\ell$,
the mod $\ell$ cohomology ring of a finite group of Lie type
${\mathbf G}({\mathbb {F}}_q)$ over a finite field ${\mathbb {F}}_q$ of characteristic $p\neq \ell$
has been known since Quillen
\cite[\S2]{quillen71icm}: it is the tensor product of
a polynomial algebra and an exterior algebra on
generators of known degrees.
When also $q\equiv 1$ mod $\ell$,
this ring can be observed to coincide with the mod $\ell$ cohomology
ring of the free loop space
$LB {\mathbf G}(\mathbb{C}) = \operatorname{map}(S^1,B{\mathbf G}(\mathbb{C}))$,
or, equivalently, with that of the classifying space $BL{\mathbf G}(\mathbb{C})$
of the loop group $L{\mathbf G}(\mathbb{C})$ studied
e.g.\ in \cite{PS86}.
Here ${\mathbf G}$ is the underlying reductive group scheme over ${\mathbb {Z}}$
and ${\mathbf G}(\mathbb{C})$ denotes the complex points of ${\mathbf G}$ with the analytic
topology; the two spaces $LB {\mathbf G}(\mathbb{C})$ and $BL{\mathbf G}(\mathbb{C})$ are homotopy
equivalent.
The isomorphism
between $H^\ast(B{\mathbf G}({\mathbb {F}}_q);{\mathbb {F}}_\ell)$ and $H^\ast(LB{\mathbf G}(\mathbb{C});{\mathbb {F}}_\ell)$
arises as a result of the collapse of two spectral sequences
with abstractly isomorphic $E_2$ pages, and comes without a satisfying structural explanation.
(In both cases, the cohomology
is isomorphic to ${\mathbb {F}}_\ell[x_1, \ldots,x_r] \otimes
\wedge_{{\mathbb {F}}_\ell}(y_1,\ldots,y_r) $ with $|x_i|=2d_i$ and $|y_i|=2d_i-1$, for $d_i$'s the fundamental
degrees of the root system of ${\mathbf G}$ \cite[\S3.7,Table~1]{humphreys90}.)
When $\ell$ is small, more specifically when $\ell$ is a
torsion prime for ${\mathbf G}$, both
$H^*(B{\mathbf G}({\mathbb {F}}_q);{\mathbb {F}}_\ell)$ and $H^*(L B{\mathbf G}(\mathbb{C});{\mathbb {F}}_\ell)$
remain highly interesting but
become very hard to compute, and are in general unknown.
Calculations have revealed isomorphic rings, even in the presence of $\ell$--torsion in ${\mathbf G}$, see
\cite{quillen72}, \cite{quillen71spin}, \cite{FP78},
\cite{kleinerman82}, \cite{MT91}, \cite{KK93}.
Indeed, in an unpublished note \cite{tezuka98}, Tezuka asked
if they \emph{always} agree, as long as $q \equiv 1$ mod $\ell$
(or $1$ mod $4$ in the case $\ell =2$).
Further calculations
supporting this ``Tezuka conjecture'' have been worked out in
\cite{KMT00}, \cite{KK10}, \cite{KMN06}, \cite{KTY12},
\cite{Kameko-spin}, yet again without any structural explanation why it
might be true. The underlying spaces are certainly not homotopy
equivalent in any sense, as even the most basic cases show: For ${\mathbf {T}}$ a
one-dimensional torus, $B{\mathbf {T}}({\mathbb {F}}_q) \simeq B{\mathbb {Z}}/(q-1)$
is a rationally trivial space depending heavily on
$q$, whereas $L B{\mathbf {T}}(\mathbb{C}) \simeq S^1 \times {\mathbb{C}P}^\infty$
is rationally nontrivial and independent of $q$.
\smallskip
The goal of this paper is to use string topology
to produce a general structural relationship between
$H^*(LB{\mathbf G}(\mathbb{C});{\mathbb {F}}_\ell)$ and $H^*(B{\mathbf G}({\mathbb {F}}_q);{\mathbb {F}}_\ell)$:
we show that we can equip the former cohomology groups with a
Chas--Sullivan-type string product so that the latter cohomology
groups become a module over the former. The non-vanishing
of a single fundamental class, defined in an elementary fashion,
implies that the module structure is free of rank $1$, which
in turn suffices to guarantee that the two cohomology rings (with
respect to the cup product) are isomorphic up to a specified filtration.
We prove the non-vanishing of the fundamental class for
all non-exceptional groups, by case-by-case methods, as long as $q$ satisfies the aforementioned congruence
condition, or alternatively allowing a modification of
${\mathbf G}$ described below. In this setup we know of no case where the fundamental
class vanishes, raising the question whether it is always
nonzero.
\medskip
We now proceed to describe our results in more detail.
Our structural theorems are most naturally formulated in the
generality of connected $\ell$--compact groups, which we briefly
recall (see e.g.\ \cite{grodal10} for a survey).
A connected $\ell$--compact group is a pointed simply connected space $BG$
which is
local with respect to homology with coefficients in ${\mathbb {F}}_\ell$
\cite{bousfield75} (or here equivalently
Bousfield--Kan $\ell$--complete \cite[\S11]{DW94}),
and whose based loop space $G = \Omega BG$ has
finite mod $\ell$ cohomology. Examples include
${\mathbb {F}}_\ell$--localizations of classifying spaces of
connected compact Lie groups, noting that
${\mathbb {F}}_\ell$--localization does not change the mod $\ell$
homology.
The dimension of the $\ell$--compact group $BG$ is
defined as the degree of the
top nontrivial mod $\ell$ homology group of $G$.
Connected $\ell$--compact groups and their
automorphisms have been classified: connected $\ell$--compact groups are in
one-to-one correspondence with root data $\mathbb{D}$ over the $\ell$--adic
integers ${\mathbb {Z}}_\ell$, and $\mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$,
where $\mathbb{D}_G$ denotes the root datum corresponding to $BG$
\cite[Thm.~1.2]{AG09};
in other words, the classification is wholly analogous to that of
compact connected Lie groups, but with
${\mathbb {Z}}$--root data replaced with ${\mathbb {Z}}_\ell$--root data.
Here $\mathrm{Out}(BG)$ is the group of free homotopy classes of
self-homotopy equivalences of $BG$;
for the definition of a ${\mathbb {Z}}_\ell$--root datum $\mathbb{D}$ and
its outer automorphism group
$\mathrm{Out}(\mathbb{D})$,
see Subsection~\ref{subsec:rootdata}.
Any compact
connected Lie group $K$ has an associated $\ell$--compact group,
obtained as the ${\mathbb {F}}_\ell$--homology localization $BK\hat{{}_\ell}$, a process which on the level of root data
corresponds to tensoring with the $\ell$--adic integers ${\mathbb {Z}}_\ell$.
For ${\mathbf G}$ a connected reductive algebraic group and $K$ a maximal
compact subgroup of the complex algebraic group ${\mathbf G}(\mathbb{C})$, we have a homotopy equivalence
$BK \xrightarrow{\simeq} B{\mathbf G}(\mathbb{C})$ (see e.g.\ \cite[\S8.1]{AGMV08}), so
$B{\mathbf G}(\mathbb{C})\hat{{}_\ell}$ is a connected $\ell$--compact group as well.
\smallskip
Fundamental to our construction of the module structure
on the cohomology $H^*(B{\mathbf G}({\mathbb {F}}_q);{\mathbb {F}}_\ell)$ of a finite group of Lie type
is that up to homotopy equivalence,
the space $B{\mathbf G}({\mathbb {F}}_q)\hat{{}_\ell}$ may be realized as a space of paths
in the $\ell$--compact group $BG = B{\mathbf G}(\mathbb{C})\hat{{}_\ell}$, as we will
now explain. Let us start by recalling the definition of a
general finite group
of Lie type. Let ${\mathbf G}$ be a connected split reductive algebraic group scheme
over ${\mathbb {Z}}$ with $\bar {\mathbb {F}}_p$--rational points ${\mathbf G}(\bar {\mathbb {F}}_p)$, and
let $\sigma$ be a Steinberg endomorphism, i.e., an endomorphism of
${\mathbf G}(\bar {\mathbb {F}}_p)$, as an algebraic group over $\bar {\mathbb {F}}_p$, which raised to some power
becomes a standard Frobenius map $\psi^q\colon\thinspace {\mathbf G}(\bar{\mathbb {F}}_p) \to {\mathbf G}(\bar{\mathbb {F}}_p)$
induced by the $q$--th power map on $\bar {\mathbb {F}}_p$.
A finite group of Lie type is a group (necessarily finite)
which arises as the fixed points
${\mathbf G}(\bar {\mathbb {F}}_p)^\sigma$ for some such ${\mathbf G}$ and $\sigma$;
important examples are of course given by
the ``untwisted case'' where $\sigma =
\psi^q$ and ${\mathbf G}(\bar {\mathbb {F}}_p)^{\psi^q} = {\mathbf G}( {\mathbb {F}}_q)$.
The classical groups
$\mathrm{GL}_n({\mathbb {F}}_q)$, $\mathrm{Sp}_n({\mathbb {F}}_q)$, etc.\ are
examples of finite groups of Lie type;
see e.g.\ \cite[\S22.1]{MT11} for more information.
By a theorem of Friedlander--Mislin \cite[Thm.~1.4]{FM84}
(generalizing work of Quillen \cite{quillen72}), there is a homotopy equivalence
\begin{equation}\label{lie}
BG = B{\mathbf G}(\mathbb{C})\hat{{}_\ell} \xleftarrow{\ \simeq\ } (B{\mathbf G}(\bar {\mathbb {F}}_p))\hat{{}_\ell}
\end{equation}
for $\ell \neq p$
relating characteristic $p$ to characteristic $0$,
as long as we apply
${\mathbb {F}}_\ell$--localization.
Combining this with another theorem of Quillen and Friedlander,
we obtain homotopy equivalences
\begin{equation}\label{lietype}
(B{\mathbf G}(\bar {\mathbb {F}}_p)^\sigma)\hat{{}_\ell}
\xrightarrow{\ \simeq\ }
(B{\mathbf G}(\bar{\mathbb {F}}_p)\hat{{}_\ell})^{h\sigma}
\xrightarrow{\ \simeq\ }
BG^{h\sigma}
\end{equation}
for $\sigma$ a Steinberg endomorphism, relating actual fixed points to homotopy fixed points, where we have continued to write $\sigma$ for the self-equivalences
of $B{\mathbf G}(\bar{\mathbb {F}}_p)\hat{{}_\ell}$ and $BG$ induced by $\sigma$.
See \cite[Thm.~2.9]{friedlander76},
\cite[Thm.~12.2]{FriedlanderEtaleHomotopy}, and also \cite[Thm.~3.1]{BrotoMoellerOliver}.
In particular,
in this picture the Frobenius map $\psi^q$ of ${\mathbf G}(\bar{\mathbb {F}}_p)$ gives rise to the ``unstable Adams operation'' self-equivalence
$\psi^q$ of $BG$ which on the root datum corresponds to multiplication
by $q \in {\mathbb {Z}}_\ell^\times$, a central element of $\mathrm{Out}(\mathbb{D}_G)$.
Here, and throughout, by the homotopy fixed point space $X^{h\sigma}$ of a
self-map $\sigma\colon\thinspace X \to X$ we mean
the space
$X^{h\sigma} = \{ \alpha\colon\thinspace I \to X\ |\ \sigma\alpha(1) = \alpha(0)\}
$,
a subspace of the mapping space $X^I$. It also identifies with the homotopy pullback of the diagram $X \xrightarrow{\
\Delta\ } X \times X \xleftarrow{\ (\sigma,1)\ } X$ as well as the
homotopy fixed points $X^{h\mathbb{N}_0}$ e.g.\ defined by the universal property of a
right adjoint,
letting the monoid $\mathbb{N}_0$ act on $X$ via $\sigma$; it also agrees with
the homotopy fixed points
$X^{h{\mathbb {Z}}}$ of the ${\mathbb {Z}}$--action defined by $\sigma$ when $\sigma$ is a homeomorphism.
In particular the
homotopy type of $X^{h\sigma}$ only depends on the free homotopy class
of $\sigma$.
See e.g.\ \cite[\S4]{BrotoMoellerOliver}
and \cite[Ch.~XI\S8]{bk} for details.
With the above dictionary in place,
the rest of the paper will be formulated in the context of
homotopy fixed points on $\ell$--compact groups.
Via \eqref{lie} and \eqref{lietype}, our results then imply
results about finite groups of Lie type in any characteristic $p \neq
\ell$. The space $BG^{h\sigma}$ is also quite interesting
for general $\ell$--compact groups
$BG$ and self-maps $\sigma$
not coming from algebraic groups and Steinberg endomorphisms,
and is often known to be an ``exotic'' $\ell$--local
finite group, in the sense of \cite{BLO03jams}, although
such results so far build on
case-by-case considerations; see \cite[Thm.~4.5]{LO02} \cite[Thm.~A]{BM07}.
We emphasise that the group $\mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$
can be described explicitly, just as in the Lie group
case, with the main structural difference being that ${\mathbb {Z}}_\ell$ has a
lot more units than ${\mathbb {Z}}$.
The calculation of $\mathrm{Out}(\mathbb{D})$ is reduced to
the case where $\mathbb{D}$ is simple and simply connected
in \cite[\S8.4]{AG09}, and the results in these cases are tabulated in
\cite[Thm.~13.1]{AGMV08}. See also \cite{JMO92, JMO95} for earlier
work.
\smallskip
Returning to string topology, Chas and Sullivan
\cite{ChasSullivan} and later authors, see e.g.\ \cite{Sullivan04, CG04},
observed that, for $X$ a closed oriented manifold, $H^*(LX)$ and
$H_*(LX)$ carry additional ``string'' products and coproducts.
Roughly speaking, these structures are constructed by reversing
the direction of one of the two maps in the diagram
\begin{equation}\label{eq:stringtop101}
LX \times LX = \operatorname{map}(S^1 \amalg S^1,X) \xot{\quad} \operatorname{map}(S^1 \vee S^1,
X) \xto{\quad} \operatorname{map}(S^1,X) = LX
\end{equation}
by an umkehr map construction; here the first map is induced by joining the circles at the basepoint and the
second map is induced by the pinch map. Furthermore,
versions of these constructions, letting $X$ be an orbifold,
Borel construction, Gorenstein space, stack, or classifying space have been considered by a number of authors, see e.g.\
\cite{LUX08, FelixThomas, BGNX07, BGNX12, GS08, GW08, ChataurMenichi,HL15}.
In particular, Chataur and Menichi showed in \cite{ChataurMenichi} that
the shifted cohomology $\mathbb{H}^\ast(LBG) = H^{*+d}(LBG;\,{\mathbb {F}}_\ell)$ can
be endowed with a string topological product, which should be thought
of as mixing the cup product
on $H^*(BG)$ with a dual of the Pontryagin product on $H_*(G)$, by
choosing an umkehr map for the right-hand map in \eqref{eq:stringtop101}.
The group theoretic significance of these string topological
structures has not been clear so far, however.
\smallskip
In this paper, we show that $H^\ast(BG^{h\sigma};\,{\mathbb {F}}_\ell)$
can be endowed with a module structure over
$\mathbb{H}^\ast(LBG)$, which via
\eqref{lie} and \eqref{lietype} provides the cohomology groups of
finite groups of Lie type with a large amount of useful new structure.
As a part of our work, we give a new construction of a
string product on $\mathbb{H}^\ast (LBG)$;
see Definition~\ref{def:stringprodandmod} and Remark~\ref{rk:summary}.
The new construction has the advantage of providing a product
that is manifestly unital and associative,
and it is needed for the construction of the
product structures on the level of spectral sequences in
Theorem~\ref{thm:mainresult} below.
The product constructed here should agree with the product constructed by
Chataur and Menichi \cite{ChataurMenichi}
(in the sign-corrected form of \cite[\S7]{KM16}),
although our work is logically independent of \cite{ChataurMenichi}
and we will not verify the relationship here.
We can now state our main result on the module structure precisely:
\begin{Th}
\label{thm:mainresult}
Let $\ell$ be a prime, let $BG$ be a connected $\ell$--compact group of
dimension $d$, and let $\sigma\colon\thinspace BG \to BG$ be a self-map of $BG$.
Set $G = \Omega BG$, $H^\ast(-) = H^\ast(-;\,{\mathbb {F}}_\ell)$, $\mathbb{H}^\ast = H^{\ast+d}$, and $\mathbb{E}^{\ast,\ast}_r = E^{\ast,\ast+d}_r$.
\begin{enumerate}[(i)] %
\item \label{it:mainresult-modstr}
The cohomology groups $H^\ast(BG^{h\sigma})$ admit a module structure
over $\mathbb{H}^\ast(LBG)$ equipped with the string product of
Definition~\ref{def:stringprodandmod}.
\item \label{it:mainresult-ringss} The Serre spectral sequence of the evaluation
fibration $LBG\to BG$, $\omega \mapsto \omega(1)$,
shifted down by $d$ degrees,
is a strongly convergent
spectral sequence of algebras
\begin{equation}
\label{ss:alg}
\mathbb{E}^{\ast,\ast}_2(LBG)
\cong
H^\ast(BG) \otimes \mathbb{H}^\ast (G)
\Longrightarrow \mathbb{H}^\ast(LBG).
\end{equation}
Here
$H^\ast(BG)$ is equipped with the cup product
and the product on $\mathbb{H}^\ast (G)$ is a dual of the
Pontryagin product on $H_\ast(G)$ (see Theorem~\ref{thm:pontryaginproduct}).
\item \label{it:mainresult-modss}
The Serre spectral sequence
\begin{equation}
\label{ss:mod}
E^{\ast,\ast}_2(BG^{h\sigma})
\cong
H^\ast(BG) \otimes H^\ast (G) \Longrightarrow H^\ast(BG^{h\sigma})
\end{equation}
of the fibration $BG^{h\sigma} \to BG$, $\alpha \mapsto \alpha(1)$,
with fibre homotopy equivalent to $G$,
is a module spectral sequence over
the spectral sequence \eqref{ss:alg}
and converges to $H^\ast(BG^{h\sigma})$ as a module over
$\mathbb{H}^\ast(LBG)$. On the $E_2$--page, the module structure
is free of rank $1$ on
a generator of
$E_2^{0,d} \cong H^d(G) \cong {\mathbb {F}}_\ell$.
\end{enumerate}
\end{Th}
The construction of the module structure is given in
Section~\ref{sec:products}, and in overview form in Remark~\ref{rk:summary}; the
properties of the spectral sequence are proven in
Section~\ref{sec:spectralsequences}; and everything is put together to
give Theorem~\ref{thm:mainresult} in the
beginning of Section~\ref{sec:tezuka}.
More general coefficient
rings than ${\mathbb {F}}_\ell$ are possible; see Remark~\ref{rk:generalcoefficients}.
In the case where $H^*(BG)$ is a polynomial ring and $\sigma$ acts as the identity on $H^*(BG)$, the spectral sequences collapse at the $E_2$--pages, providing a
structured isomorphism between the $E_\infty$--pages, and hence a
structured isomorphism between $H^*(LBG)$ and $H^*(BG^{h\sigma})$. See
Remark~\ref{rk:polysscollapse}.
In general, the spectral sequences of
Theorem~\ref{thm:mainresult}(\ref{it:mainresult-ringss}),(\ref{it:mainresult-modss})
are unknown, and have differentials. For example, our
product on $\mathbb{H}^*(LBG)$ is expected to always be commutative, see
Remark~\ref{rk:cmcomp}, whereas the Pontryagin
product on $H_*(G)$ is usually non-commutative in the presence of
torsion, see Remark~\ref{rk:noncomm}, forcing nontrivial differentials to appear in (\ref{it:mainresult-ringss}). Nevertheless, we show that the question whether the $E_\infty$--pages
are isomorphic is equivalent to a single class
being nonzero.
To state this precisely, consider the homotopy fibre sequence
\begin{equation}\label{eq:fibseq}
G \xrightarrow{\ i\ } BG^{h\sigma} \xto{\quad} BG
\end{equation}
associated to the evaluation fibration
$BG^{h\sigma} \to BG$, $\alpha \mapsto \alpha(1)$.
\begin{defn}[{$[G]$--fundamental class}] \label{defn:fundclass} For $\sigma\colon\thinspace BG \to BG$ a
self-map, consider the homomorphism $i_*\colon\thinspace H_d(G) \to H_d(BG^{h\sigma})$
induced by \eqref{eq:fibseq}, where $d$ is the top nontrivial
degree of $H_*(G)$. We have a canonical class $i_*([G])$, given by evaluating $i_*$ on a generator $[G] \in
H_d(G)\cong {\mathbb {F}}_\ell$, well
defined up to a unit in ${\mathbb {F}}_\ell$.
If this class $i_*([G])$ is nontrivial, we call it
the {\em $[G]$--fundamental class} of $BG^{h\sigma}$.
%
\end{defn}
In other words, $BG^{h\sigma}$ has a $[G]$--fundamental
class if the map $i_\ast\colon\thinspace H_d(G) \to H_d(BG^{h\sigma})$ is nontrivial, or, formulated dually, if $i^*\colon\thinspace H^d(BG^{h\sigma}) \to H^d(G)$ is nontrivial.
\begin{Th}
\label{thm:strtoptezukacrit}
With the setup of Theorem~\ref{thm:mainresult},
the module $H^\ast(BG^{h\sigma})$ is free of rank 1 over $\mathbb{H}^\ast (LBG)$
if and only if $BG^{h\sigma}$ has a $[G]$--fundamental class.
Indeed, an element $x \in H^\ast (BG^{h\sigma})$ generates $H^*(BG^{h\sigma})$ as a
free rank 1 module over $\mathbb{H}^\ast(LBG)$ if and only if $x$ has degree $d$
and $i^\ast (x) \neq 0 \in H^d (G)$.
\end{Th}
The last condition can
alternatively be phrased as saying that $x \in H^d(BG^{h\sigma})$
generates $H^\ast(BG^{h\sigma})$ as a free of rank 1 module over
$\mathbb{H}^\ast(LBG)$ if and only if it evaluates
nontrivially against $i_\ast [G] \in H_d(BG^{h\sigma})$ for a generator $[G] \in H_d(G) \cong {\mathbb {F}}_\ell$.
Note also that the notion of a $[G]$--fundamental class depends on the map
$BG^{h\sigma} \to BG$ and not just the space $BG^{h\sigma}$.
We will return to this point later
in Definition~\ref{def:ghmuetfunclass}.
\smallskip
The existence of a fundamental class implies
a strong link between the $H^*(BG)$--module structures
on $H^\ast(LBG)$ and $H^\ast(BG^{h\sigma})$,
and similarly for the ring structures.
\begin{Th}
\label{thrm:cup-products}
With the setup of Theorem~\ref{thm:mainresult}, suppose that
$BG^{h\sigma}$ has a $[G]$--fundamental class (Definition~\ref{defn:fundclass}),
so that by Theorem~\ref{thm:strtoptezukacrit} we may find
an $x\in H^d(BG^{h\sigma})$ such that the map
\begin{equation}
\label{map:circx}
- \circ x \colon\thinspace H^\ast(LBG) \xrightarrow{\ \cong\ } H^\ast(BG^{h\sigma})
\end{equation}
given by module multiplication with $x$ is an isomorphism of graded
vector spaces. Then the following holds.
\begin{enumerate}[(i)]
\item \label{it:modiso}
The isomorphism \eqref{map:circx}
is an isomorphism of $H^\ast(BG)$--modules,
where the source and target are given
the $H^\ast(BG)$--module structures induced by the fibrations
$LBG \to BG$, $\omega \mapsto \omega(1)$ and
$BG^{h\sigma} \to BG$, $\alpha \mapsto \alpha(1)$
respectively. In particular, the induced map $H^*(BG) \to H^*(BG^{h\sigma})$ is injective.
\item \label{it:gralgiso} If $1\circ x =1$
(as may be arranged by replacing $x$ by a multiple if necessary), the isomorphism \eqref{map:circx} induces
an algebra isomorphism
\[
\gr H^\ast(LBG) \xrightarrow{\ \cong\ } \gr H^\ast(BG^{h\sigma})
\]
between the associated graded algebras of $H^\ast(LBG)$ and
$H^\ast(BG^{h\sigma})$ corresponding to the Serre spectral
sequences of the
aforementioned fibrations $LBG\to BG$ and $BG^{h\sigma} \to BG$,
respectively.
\end{enumerate}
\end{Th}
Notice in particular that (\ref{it:modiso}) implies that $\sigma$
necessarily has to act as the identity on $H^*(BG)$ for a
$[G]$--fundamental class to exist, and thus $\sigma$ will have to be a
self-equivalence, as $BG$ is an ${\mathbb {F}}_\ell$--local space.
Also note that without further assumptions,
(\ref{it:gralgiso}) cannot be improved to
an algebra isomorphism of the underlying algebras, again by the
example of a torus: For $BG =
(BS^1)\hat{{}_2}$ and $\sigma = \psi^3$,
the map $i\colon\thinspace G \to BG^{h\sigma}$
identifies with the map $(S^1)\hat{{}_2} \to B{\mathbb {Z}}/2$
arising from the fibration
$B{\mathbb {Z}}/2 \to (BS^1)\hat{{}_2} \xrightarrow{2} (BS^1)\hat{{}_2}$,
so $BG^{h\sigma}$ has an $[(S^1)\hat{{}_2}]$--fundamental class.
However, the cohomology rings of
$L(BS^1)\hat{{}_2} \simeq (S^1 \times BS^1)\hat{{}_2}$ and
$((BS^1)\hat{{}_2})^{h\sigma}\simeq B{\mathbb {Z}}/2$
are not isomorphic. It may be that
this problem is limited to the case where $\ell =2$ and
$q \equiv 3\ \textrm{mod}\ 4$, as suggested by Tezuka's question,
though we do not know of a general result to that effect.
\medskip
We now proceed to describe when we know that a fundamental class
exist; our main result in this regard is Theorem~\ref{thm:examples} below.
Let us assume that $\sigma$ is of the form $\sigma = \tau \psi^q$
for $\tau \in
\mathrm{Out}(\mathbb{D}_G)$ of finite order and $q \in
{\mathbb {Z}}_\ell^\times$.
We write
\begin{equation}\label{eq:lietypegrp}
\BtGq = BG^{h(\tau\psi^q)}
\end{equation}
for short, and abbreviate further to $BG(q)$ when $\tau =1$;
by~\eqref{lietype},
this notation
agrees with standard Lie theoretic notation, as in e.g.\
\cite[Table~22.1]{MT11}, when
the latter makes sense.
All possibilities for the ``twistings'' $\tau$
are tabulated in \cite[Thm.~13.1]{AGMV08};
see
Remark~\ref{rk:twistingclassification}
for an elaboration of the relationship to the Lie case, where we e.g.\
pick up some ``very twisted'' groups as $q \in {\mathbb {Z}}_\ell^\times$.
To state Theorem~\ref{thm:examples},
we first explain how to
use the theory of $\ell$--compact groups to reduce the
study of $\BtGq$ to the case where $\tau =1$
and $q$ is congruent to $1$ modulo $\ell$,
at least when $\tau$ has order prime to $\ell$. The following
result is a small generalization of a result of Broto--M{\o}ller
\cite[Thms.~B and E(1)]{BM07}, proved in Appendix~\ref{appendix:pcg} as
Theorem~\ref{remove-twisting} via a
general argument.
\newcommand{\untwistingThmStatement}{%
Suppose $BG$ is a connected $\ell$--compact
group, $q \in
{\mathbb {Z}}_\ell^\times$, and $\tau \in \mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$
is of finite order prime to $\ell$.
Let $e$ be the multiplicative order of $q$
mod $\ell$, and write $q =
\zeta_e q'$ in ${\mathbb {Z}}_\ell^\times$, for $\zeta_e$ a primitive $e$-th root of unity and
$q' \equiv 1$ mod $\ell$. Set $\tau_e = \tau \psi^{\zeta_e}$
and $\bbtau_e = \langle \tau_e\rangle$.
Then the finite group $\bbtau_e$ has a
canonical action on $BG$
such that the homotopy fixed points $BG^{h\bbtau_e}$ are %
a connected $\ell$--compact group and there exists
a homotopy equivalence
\begin{equation*}%
\BtGq \xrightarrow{\ \simeq\ } (BG^{h\bbtau_e})(q').
\end{equation*}%
}
\begin{thrm}[Untwisting theorem]\label{thm:untwisting-intro}
\untwistingThmStatement
\end{thrm}
The ``untwisting'' procedure above is analogous to the
$\Phi_e$--theory of finite group of Lie
type of Brou\'e--Malle--Michel \cite{BM92,malle98}, but has the
advantage that the unwisted group $BG^{h\bbtau_e}$ is again an $\ell$--compact
group, rather than a mythical ``Spets'' (though in general {\em not} the
$\ell$--completion of a compact Lie group, even if $BG$ is);
see Appendix~\ref{appendix:pcg}.
With the notation of Theorem~\ref{thm:untwisting-intro},
we can now make the following definition.
\begin{defn}[{$[G^{h\bbtau_e}]$}--fundamental class]
\label{def:ghmuetfunclass}
We say that $\BtGq$ has a {\em $[G^{h\bbtau_e}]$--fundamental
class} if
$$\BtGq \simeq (BG^{h\bbtau_e})(q') \xto{\quad} BG^{h\bbtau_e}$$
has a fundamental class in the sense of
Definition~\ref{defn:fundclass}.
\end{defn}
We are now ready to state our main existence result.
\begin{Th}
\label{thm:examples}
Suppose $BG$ is a simply connected
$\ell$--compact group, $q\in {\mathbb {Z}}_\ell^\times$,
and $\tau \in\mathrm{Out}(BG)\cong \mathrm{Out}(\mathbb{D}_G)$
is an element of finite order prime to $\ell$. Then
$\BtGq$ has a
$[G^{h\bbtau_e}]$--fundamental class,
and Theorems~\ref{thm:strtoptezukacrit} and
\ref{thrm:cup-products} thus provide structured isomorphisms between
$H^*(\BtGq)$ and $H^*(LBG^{h\bbtau_e})$, except possibly in the following 8 cases: when $\ell =5$ and
$G$ contains an $E_8$--summand; when $\ell=3$ and $G$ contains an $F_4$
or $E_i$--summand, $i=6,7,8$; and when $\ell=2$ and $G$ contains an
$E_i$--summand, $i=6,7,8$.
\end{Th}
As mentioned, we suspect that the restrictions on $BG$ in
the theorem may be superfluous,
and that the theorem in fact holds for any connected
$\ell$--compact group. Theorem~\ref{thm:examples2} provides a more elaborate
statement which also applies to some non-simply connected cases.
Theorem~\ref{thm:examples} can be seen as generalizing a number of
previous results in the literature where both $H^*(BG(q))$ and $H^*(LBG)$
were calculated to various degrees of precision and amount of
structure, and subsequently observed to coincide. See e.g.\
\cite{kleinerman82, milgram96, KMT00, Kameko-pb, grbic06, KK10, KMN06,KTY12}
for previous work on polynomial cases, and
\cite{Kameko-spin} for the spin groups, where an isomorphism of graded
abelian groups
was established---it was noticing this last
paper in an arXiv listing which originally alerted
us to Tezuka's question.
Recall that for
$BG^{h\sigma}$
to have a $[G]$--fundamental class,
it is necessary for the endomorphism $\sigma$
to induce the identity map on
$H^*(BG;{\mathbb {F}}_\ell)$,
and in particular
for $\sigma$ to be an self-homotopy equivalence.
For general automorphisms, we prove the following result,
which gives some information
also in the cases
excluded in Theorem~\ref{thm:examples}.
\begin{Th}
\label{thm:tezukasubgrp}
The subset of $[\sigma] \in \mathrm{Out}(BG)$ for which
$BG^{h\sigma}$ has a $[G]$--fundamental class
is an uncountable subgroup of
$\{ [\sigma] \in \mathrm{Out}(BG) \mid \sigma^* = \mathrm{id} \in \mathrm{Aut}(H^\ast BG)\}$.
Moreover, this subgroup is
closed in the topology induced on $\mathrm{Out}(BG)$ from $\mathrm{Out}(\mathbb{D}_G)$
when $BG$ is semisimple.
\end{Th}
Theorem~\ref{thm:tezukaconjsubgrp2} and Corollary~\ref{cor:psiqsubgrp} provide slightly more precise
statements.
We do not know an example of connected $\ell$--compact group for which the subgroup in
Theorem~\ref{thm:tezukasubgrp} is proper (even in the non-semisimple case).
\smallskip
We would like to end
this introduction with some questions about possible extensions and
refinements of our results:
\begin{Qu}[Existence of a fundamental class] \label{qu:fundclass} Let $BG$ be a connected
$\ell$--compact group.
\begin{enumerate}
\item Does a $[G]$--fundamental class always exist for $BG^{h\sigma}$
if $\sigma^*$ acts as the identity on $H^*(BG)$? (Notice that
the condition on $\sigma^\ast$
is necessary by Theorem~\ref{thm:tezukasubgrp}.)
\item Does a $[G^{h\bbtau_e}]$--fundamental class always exist for
$\BtGq$ when $\ell \nmid |\tau|$?
(A positive answer implies, for example, that $\psi^q$
induces the identity map on $H^*(B{\mathbf G}(\bar {\mathbb {F}}_q))$, that $H^*(B{\mathbf G}(\bar
{\mathbb {F}}_q)) \to H^*(B{\mathbf G}({\mathbb {F}}_q))$ is injective,
and that every elementary abelian $\ell$--subgroup of ${\mathbf G}(\bar {\mathbb {F}}_q)$ is
conjugate to a subgroup
in ${\mathbf G}({\mathbb {F}}_q)$ whenever ${\mathbf G}$
is a connected split reductive group scheme over the integers
and $q \equiv 1\ \mathrm{mod}\ \ell$; whether or not these
properties hold does not seem to be known in general.)
\end{enumerate}
\end{Qu}
The
calculational literature often work with the explicit or implicit
assumption that $G$ is simply connected or simple, though it is not
clear if this is necessary. (Indeed, Tezuka in his original question \cite{tezuka98}
seems implicitly to be in the simply connected case.)
The second question which we would like to raise is to which extent the isomorphism \eqref{map:circx}
can be extended to respect the multiplicative and
Steenrod algebra structure.
\begin{Qu}[Multiplicative and Steenrod algebra
structure] \label{qu:mult-steenrod} $ $
\begin{enumerate}
\item Can the class $x$ in
Theorem~\ref{thrm:cup-products}(\ref{it:gralgiso}) always be chosen so as to
produce an isomorphism of rings, or $H^*(BG)$--algebras, when $\ell$ is odd? Is the same true for $\ell =2$,
if $\sigma = \psi^q$ with $q \equiv 1\ \mathrm{mod}\ 4$ (or if
$\sigma^*$ induces the identity on $H^*(BG;{\mathbb {Z}}/4)$)? And is there a
unique and/or canonical such class, a ``dual fundamental class''?
\item Can the class $x$ always be chosen so as to produce an
isomorphism over the subalgebra $\mathcal{A}'$ of the Steenrod algebra
of operations of even degrees (i.e., the subalgebra
leaving out the Bockstein operations for
$\ell$ odd, and $\operatorname{Sq}^{2n+1}$, $n \geq 0$, for $\ell =2$)? Or over the
whole Steenrod algebra when $\sigma = \psi^q$ with $q \equiv 1 \
\mathrm{mod}\ \ell^2$ (or when $\sigma^*$ induces the identity on
$H^*(BG;{\mathbb {Z}}/\ell^2)$)?
\item Combining the previous points, can the class $x$ be chosen so as to
yield an isomorphisms of $H^*(BG)$--algebras over $\mathcal{A}'$? (See
\cite{henn96} for more on this structure.)
\end{enumerate}
\end{Qu}
Note that for $\ell$ big enough, a candidate for a good choice
for the class $x$ exists, namely the product of the exterior
classes in cohomology; see also \cite[\S3]{solomon63} \cite[Prop.~7 and Thm.~8]{KK10}.
\subsubsection*{Outline of the paper}
The string product and the module
structure are constructed in Section~\ref{sec:products}.
In Section~\ref{sec:spectralsequences}, we set up the Serre
spectral sequences and show how they interact with the product
structures. Finally, in Sections~\ref{sec:tezuka} through
\ref{sec:funclass2} we assemble the pieces and derive
Theorems~\ref{thm:mainresult} through \ref{thm:tezukasubgrp}
stated in the introduction.
Our construction of the string product and module structures requires
a certain amount of parametrized homotopy theory which we
present in Appendix~\ref{app:parametrized-homotopy-theory}.
In the short Appendix~\ref{app:dualizability},
we briefly recall the notion of dualizability in the context
of symmetric monoidal categories. Finally, in
Appendix~\ref{appendix:pcg}, we lay out the relationship between ${\mathbb {Z}}_\ell$--root data, $\ell$--compact groups and the
$\ell$--local theory of finite groups of Lie type, and in particular prove
Theorem~\ref{thm:untwisting-intro}. Several of the results contained there
may be of independent interest.
\subsubsection*{Notation and conventions}
Throughout the paper, $\ell$ will be a prime and
$BG$ will denote a fixed connected $\ell$--compact group.
(The notion of an $\ell$--compact group
was recalled earlier in the introduction; see also \cite{grodal10}.)
We write $G$ for the based loop space $G = \Omega BG$.
Unless indicated otherwise,
homology and cohomology is with ${\mathbb {F}}_\ell$--coefficients.
\section{Construction of the products}
\label{sec:products}
Our aim in this section is to construct the
string product on $\mathbb{H}^\ast(LBG)$
and string module structure on $\mathbb{H}^\ast (BG^{h\sigma})$.
Indeed, we will construct and study a more general pairing
(Theorem~\ref{thm:pairings} below) which will yield both of these
structures as special cases. Subsection~\ref{subsec:resultsonproducts}
will introduce this pairing along with many of its properties,
and the remaining subsections are devoted to the actual construction
of the pairing along with the verification of the asserted
properties. For a quick indication of the basic idea behind the
construction, the reader should see Remark~\ref{rk:summary}.
\subsection{The pairing and its properties}
\label{subsec:resultsonproducts}
For a space $B$ and continuous maps
$f,g\colon\thinspace B \to BG$, we write $P(f,g) \to B$ for the pullback
of the evaluation fibration
\[
(\mathrm{ev}_0,\mathrm{ev}_1) \colon\thinspace BG^I \xto{\quad} BG\times BG,
\quad
\gamma \longmapsto (\gamma(0),\gamma(1))
\]
along the map
$(f,g)\colon\thinspace B \to BG$. We call $P(f,g)$
the \emph{space of paths in $BG$ from $f$ to $g$}.
Explicitly,
\[
P(f,g)
=
\big\{
(b,\gamma)\in B\times BG^I
\,|\,
\gamma(0) = f(b), \gamma(1) = g(b)
\big\},
\]
with the map $P(f,g) \to B$ given by projection onto the first coordinate.
We may thus picture a point in $P(f,g)$ as follows:
\[
\left(
b,
\begin{tikzpicture}
[
baseline=0cm,
endpoint/.style={
circle,
draw=black,
fill=black,
inner sep=0pt,
minimum size=2.6pt
},
label/.style={
font=\scriptsize
},
->-/.style={
decoration={
markings,
mark=at position .5 with {\arrow{>}}
},
postaction={decorate}
}
]
\node [endpoint] (start) at (0,0) {};
\node [endpoint] (end) at (1.5,0) {};
\node [label] at (start) [above] {$f(b)$};
\node [label] at (end) [above] {$g(b)$};
\draw [->-,thick] (start) .. controls (0.5,0.2) and (1.0,-0.26) .. (end) node [label,above,midway] {$\gamma$};
\end{tikzpicture}
\right).
\]
We will obtain the string product and the module structure
as special cases of the following result.
\begin{thrm}
\label{thm:pairings}
Let $B$ be a space.
For maps $f,g,h\colon\thinspace B \to BG$, there is a pairing
\begin{equation}
\label{eq:pairing}
\circ
\colon\thinspace
\mathbb{H}^\ast P(g,h) \otimes \mathbb{H}^\ast P(f,g)
\xto{\quad}
\mathbb{H}^\ast P(f,h).
\end{equation}
This pairing satisfies the following properties:
\begin{enumerate}[(i)]
\item (Associativity):
The equation
\[
(x \circ y) \circ z = x \circ (y \circ z)
\]
holds for all $x$, $y$, and $z$ for which
the pairings involved are defined.
\item (Existence of units):
For every $f \colon\thinspace B \to BG$,
there exists
an element $\mathbbold{1} = \mathbbold{1}_f \in \mathbb{H}^\ast P(f,f)$ such that
\[
\mathbbold{1}_f \circ x = x
\qquad\text{and}\qquad
y \circ \mathbbold{1}_f = y
\]
for all $x$ and $y$ for which the pairings are defined.
\end{enumerate}
\end{thrm}
In particular, the pairing of Theorem~\ref{thm:pairings}
makes each $\mathbb{H}^\ast P(f,f)$ into a graded ${\mathbb {F}}_\ell$--algebra.
Several special cases of the path space construction
are of particular interest:
\begin{enumerate}
\item For $B = BG$ and $f=g=\mathrm{id}_{BG}$,
the projection $(b,\gamma) \mapsto \gamma$
provides an identification
$P(\mathrm{id}_{BG},\mathrm{id}_{BG}) = LBG$,
with the projection $P(\mathrm{id}_{BG},\mathrm{id}_{BG})\to BG$
corresponding under this identification to the evaluation map
$LBG\to BG$, $\gamma \mapsto \gamma(0)=\gamma(1)$.
\item For $B = BG$ and $f=\sigma$, $g=\mathrm{id}_{BG}$,
the projection $(b,\gamma) \mapsto \gamma$
provides an identification
$P(\sigma,\mathrm{id}_{BG}) = BG^{h\sigma}$,
with the projection $P(\sigma,\mathrm{id}_{BG})\to BG$
corresponding under this identification to the evaluation fibration
$\mathrm{ev}_1\colon\thinspace BG^{h\sigma} \to BG$, $\gamma \mapsto \gamma(1)$.
\item For $B = \mathrm{pt}$ and $f=g\colon\thinspace \mathrm{pt} \to BG$
the inclusion of the basepoint,
the projection $(b,\gamma) \mapsto \gamma$
provides an identification
$P(f,f) = \Omega BG$.
\end{enumerate}
In the remainder of the paper, we will use the aforementioned
identifications without further comment.
\begin{defn}
\label{def:stringprodandmod}
The \emph{string product} on $\mathbb{H}^\ast LBG$ is the
multiplication on $\mathbb{H}^\ast LBG$ obtained by taking
$f=g=h=\mathrm{id}_{BG}$ in Theorem~\ref{thm:pairings}.
The \emph{string module structure} of
$\mathbb{H}^\ast(BG^{h\sigma})$
over
$\mathbb{H}^\ast LBG$
is the module structure
obtained by taking
$f= \sigma$ and $g=h=\mathrm{id}_{BG}$.
The string module structure
over $\mathbb{H}^\ast LBG$ on
the unshifted cohomology
$H^\ast(BG^{h\sigma})$
is obtained from the string module structure on
$\mathbb{H}^\ast(BG^{h\sigma})$
by a degree shift:
for
$x\in \mathbb{H}^\ast LBG$ and
$y\in H^\ast(BG^{h\sigma})$,
we define $x\circ y \in H^\ast(BG^{h\sigma})$ by setting
\[
x\circ y = (\sigma^{-d})^{-1} (x \circ \sigma^{-d} (y))
\]
where
$\sigma^{-d} \colon\thinspace H^\ast(BG^{h\sigma}) \to \mathbb{H}^\ast(BG^{h\sigma})$
is the degree $--d$ graded map sending each element to itself,
but now considered as an element of $\mathbb{H}^\ast(BG^{h\sigma})$.
\end{defn}
\begin{rem}
\label{rk:summary}
The reader may appreciate a quick summary of the origins
the pairing \eqref{eq:pairing}
unencumbered by the technicalities of the actual
constructions. Let $B$, $f$, $g$, and $h$ be as in
Theorem~\ref{thm:pairings}, and
write $P(f,g,h) \to B $ for the pullback of
$P(g,h) \times P(f,g) \to B\times B$ along the
diagonal map $\Delta \colon\thinspace B \to B\times B$. Explicitly,
\[
P(f,g,h)
=
\big\{
(b,\gamma_2,\gamma_1) \in B \times BG^I \times BG^I
\,|\,
\gamma_1(0) = f(b),
\gamma_1(1) = \gamma_2(0)=g(b),
\gamma_2(1) = h(b)
\big\},
\]
and we may picture a point in $P(f,g,h)$ as follows:
\[
\left(
b,
\begin{tikzpicture}
[
baseline=0cm,
endpoint/.style={
circle,
draw=black,
fill=black,
inner sep=0pt,
minimum size=2.6pt
},
label/.style={
font=\scriptsize
},
->-/.style={
decoration={
markings,
mark=at position .5 with {\arrow{>}}
},
postaction={decorate}
}
]
\node [endpoint] (start) at (0,0) {};
\node [endpoint] (mid) at (1.5,0) {};
\node [endpoint] (end) at (3,0) {};
\node [label] at (start) [above] {$f(b)$};
\node [label] at (mid) [above] {$g(b)$};
\node [label] at (end) [above] {$h(b)$};
\draw [->-,thick] (start) .. controls (0.5,-0.2) and (1.0,-0.12) .. (mid) node [label,above,midway] {$\gamma_1$};
\draw [->-,thick] (mid) .. controls (2.0,0.12) and (2.5,0.15) .. (end) node [label,above,midway] {$\gamma_2$};
\end{tikzpicture}
\right).
\]
We have the commutative diagram
\begin{equation}
\label{diag:pushpull}
\vcenter{\xymatrix@C-1em{
P(g,h) \times P(f,g)
\ar[d]
&&
P(f,g,h)
\ar[dr]
\ar[ll]_-{\mathrm{split}}
\ar[rr]^-{\mathrm{concat}}
&&
P(f,h)
\ar[dl]
\\
B\times B
&&&
B
\ar[lll]_{\Delta}
}}
\end{equation}
where the trapezoid on the left is a pullback square,
with the map labeled `split' given by
\[
\mathrm{split}
\colon\thinspace
(b,\gamma_2,\gamma_1) \longmapsto ((b,\gamma_2),(b,\gamma_1)),
\]
and where the map labeled `concat' is given by concatenation of paths:
\[
\mathrm{concat}
\colon\thinspace
(b,\gamma_2,\gamma_1) \longmapsto (b,\gamma_2 * \gamma_1).
\]
The pairing \eqref{eq:pairing} is given by
a push--pull construction in the top of row of
diagram \eqref{diag:pushpull}: it is
a degree shift of the composite
\begin{equation}
\label{eq:pairingdescr}
H^\ast P(g,h) \otimes H^\ast P(f,g)
\xrightarrow{\,\times\,}
H^\ast (P(g,h) \times P(f,g))
\xrightarrow{\, \mathrm{split}^\ast\, }
H^\ast P(f,g,h)
\xrightarrow{\, \mathrm{concat}_!\, }
H^{\ast-d} P(f,h)
\end{equation}
where $\mathrm{concat}_!$ is an ``umkehr map''
whose existence is ensured by the fact that the fibres
of $P(f,g,h) \to B$ and $P(f,h)\to B$ are dualizable and
self-dual (up to a twist) in the category of $H{\mathbb {F}}_\ell$--local spectra.
See Remark~\ref{rk:pairingrecon}.
\end{rem}
\begin{rem}
\label{rk:cmcomp}
Our product on $\mathbb{H}^\ast(LBG)$ should agree with the product
constructed by Chataur and Menichi \cite[Cor.~18]{ChataurMenichi}
(with sign corrections by Kuribayashi and Menichi \cite{KM16}),
although we will not prove the comparison here.
Indeed, Chataur and Menichi's product also arises
as a composite of an induced map and an umkehr map
as in \eqref{eq:pairingdescr},
but with the
umkehr map $\mathrm{concat}_!$
constructed in a very different way.
In addition to avoiding thorny sign issues,
our approach to the product on $\mathbb{H}^\ast(LBG)$ has the
advantage of producing a multiplication which is manifestly
unital and which, importantly for us,
descends to the Serre spectral sequence of
the evaluation fibration $LBG\to BG$.
A drawback of our approach is that it becomes difficult to
prove that the product on $\mathbb{H}^\ast(LBG)$ is commutative,
which we nevertheless expect to be the case in view of the
commutativity of the Chataur--Menichi product.
As pointed out in Remark~\ref{rk:noncomm} below,
the algebras $\mathbb{H}^\ast P(f,f)$ fail to be
commutative in general, however.
\end{rem}
As we remarked earlier,
in the case where $B=\mathrm{pt}$ and $f$ is the inclusion of
the basepoint into $BG$, the path space $P(f,f)$
recovers the based loop space $\Omega BG$.
The resulting product on $\mathbb{H}^\ast(\Omega BG)$ can be described
in more familiar terms.
\begin{thrm}
\label{thm:pontryaginproduct}
There exists an isomorphism
$\mathbb{H}^\ast (\Omega BG) \cong H_{-\ast}(\Omega BG)$
under which the product on $\mathbb{H}^\ast(\Omega BG)$
corresponds to the Pontryagin product
\[
\mathrm{concat}_\ast
\colon\thinspace
H_\ast(\Omega BG) \otimes H_\ast(\Omega BG)
\xto{\quad}
H_\ast(\Omega BG)
\]
on $H_\ast(\Omega BG)$
where $\mathrm{concat} \colon\thinspace \Omega BG\times \Omega BG \to \Omega BG$
is the concatenation map.
\end{thrm}
\begin{rem}
\label{rk:noncomm}
The Pontryagin product on $H_\ast (\Omega BG)$
is frequently noncommutative:
for $\ell$ odd,
\cite[Thm.~1.1]{Kane76}
implies that for
$BG = BK\hat{{}_\ell}$ for a compact connected Lie group $K$,
commutativity fails
precisely when $H^\ast(K;\,{\mathbb {Z}})$ has
$\ell$--torsion;
and for $\ell=2$,
an example where commutativity fails
is given by
$BG = B\mathrm{Spin}(10)\hat{{}_2}$ \cite[Thm.~16.4]{borel54}.
Thus Theorem~\ref{thm:pontryaginproduct}
implies that the graded rings $\mathbb{H}^\ast P(f,f)$ fail
to be commutative in general.
\end{rem}
The multiplication on $\mathbb{H}^\ast P(f,f)$ in fact makes it into
an augmented $H^\ast B$--algebra:
\begin{thrm}
\label{thm:algstr}
Let $B$ be a space, and let $f\colon\thinspace B\to BG$ be a map.
Equip $H^\ast B$ with the usual cup product and $\mathbb{H}^\ast P(f,f)$
with the product of Theorem~\ref{thm:pairings}.
Then there exist graded ${\mathbb {F}}_\ell$--algebra homomorphisms
\[
\iota = \iota_f \colon\thinspace H^\ast B \xto{\quad} \mathbb{H}^\ast P(f,f)
\qquad\text{and}\qquad
\rho = \rho_f \colon\thinspace \mathbb{H}^\ast P(f,f) \xto{\quad} H^\ast B
\]
such that $\rho_f \iota_f = \mathrm{id}_{H^\ast(B)}$.
\end{thrm}
\begin{rem}
\label{rk:bimodpairing}
Via the homomorphisms $\iota_g$ and $\iota_f$ and
the $\mathbb{H}^\ast P(g,g)$ and $\mathbb{H}^\ast P(f,f)$--actions
on $\mathbb{H}^\ast P(f,g)$ given by Theorem~\ref{thm:pairings},
the homology $\mathbb{H}^\ast P(f,g)$
acquires the structure of an $H^\ast B$--bimodule.
It follows formally from Theorem~\ref{thm:pairings}
that the pairing \eqref{eq:pairing} in fact gives a pairing
\[
\circ\colon\thinspace \mathbb{H}^\ast P(g,h) \otimes_{H^\ast B} \mathbb{H}^\ast P(f,g)
\xto{\quad}
\mathbb{H}^\ast P(f,h)
\]
of $H^\ast B$--bimodules.
\end{rem}
\begin{rem}
The pairings of Theorem~\ref{thm:pairings} and the maps
$\iota$ and $\rho$ of Theorem~\ref{thm:algstr}
depend on the choice of a piece of orientation data, which
we will fix (arbitrarily) once and for all.
See Remark~\ref{rk:orientation} below.
\end{rem}
In addition to the bimodule structure of
Remark~\ref{rk:bimodpairing},
$\mathbb{H}^\ast P(f,g)$ has an additional $H^\ast(B)$--module structure
induced by the projection
$P(f,g)\to B$ and the cup product on $H^\ast P(f,g)$.
\begin{defn}
\label{def:cupmodstr}
Let $\sigma^{-d} \colon\thinspace H^\ast P(f,g) \to \mathbb{H}^{\ast-d} P(f,g)$
be the degree $--d$ graded map which sends an element
$x \in H^\ast P(f,g)$ to itself, but now considered as
an element of $\mathbb{H}^\ast P(f,g)$.
Let $\pi\colon\thinspace P(f,g) \to B$ be the projection.
We define an $H^\ast(B)$--module structure
on $\mathbb{H}^\ast P(f,g)$ by setting
\begin{equation}
\label{eq:cupmodstr}
a\sigma^{-d}(x) = (-1)^{d|a|}\sigma^{-d}(\pi^\ast(a) \cupprod x)
\in \mathbb{H}^\ast P(f,g).
\end{equation}
for all $a \in H^\ast(B)$ and $\sigma^{-d}(x) \in \mathbb{H}^\ast P(f,g)$.
\end{defn}
\begin{thrm}
\label{thm:cupmodstrbilin}
Let $B$ be a space, and let $f,g,h\colon\thinspace B \to BG$ be maps.
The pairing
\[
\circ
\colon\thinspace
\mathbb{H}^\ast P(g,h) \otimes \mathbb{H}^\ast P(f,g)
\xto{\quad}
\mathbb{H}^\ast P(f,h)
\]
is $H^\ast(B)$--bilinear with respect to the
module structures of Definition~\ref{def:cupmodstr}:
for all $a\in H^{|a|}(B)$, $b\in H^{|b|} (B)$ and
$x\in \mathbb{H}^{|x|} P(g,h)$, $y\in \mathbb{H}^{|y|} P(f,g)$
we have
\[
(ax) \circ (by) = (-1)^{|b||x|}(ab)(x\circ y).
\]
\end{thrm}
The pairings for varying $B$ are compatible
in a manner we will now explain.
Let $f,g\colon\thinspace B \to BG$.
For a map of spaces $\phi\colon\thinspace A\to B$,
there is a pullback square
\begin{equation}
\label{sq:inducedmap}
\vcenter{\xymatrix{
P(f\phi,g\phi)
\ar[r]^{\bar{\phi}}
\ar[d]
&
P(f,g)
\ar[d]
\\
A
\ar[r]^{\phi}
&
B
}}
\end{equation}
where $\bar{\phi}$ is the map $\bar{\phi}(a,\gamma) = (\phi(a),\gamma)$.
Let us write
\begin{equation}
\label{eq:fphidef}
F_\phi = \bar{\phi}^\ast\colon\thinspace \mathbb{H}^\ast P(f,g) \xto{\quad} \mathbb{H}^\ast P(f\phi,g\phi)
\end{equation}
for the resulting map on cohomology groups.
We now have the following result.
\begin{thrm}
\label{thm:functors}
Given a map of spaces $\phi \colon\thinspace A \to B$,
the maps
$F_\phi \colon\thinspace \mathbb{H}^\ast P(f,g) \xto{\quad} \mathbb{H}^\ast P(f\phi,g\phi)$
for $f,g\colon\thinspace B \to BG$ have the following properties:
\begin{enumerate}[(i)]
\item $F_\phi (\mathbbold{1}_f) = \mathbbold{1}_{f\phi}$
\item $F_\phi (x\circ y) = F_\phi(x) \circ F_\phi(y)$
for all $x$ and $y$ for which the pairing is defined.
\end{enumerate}
\end{thrm}
Theorem~\ref{thm:functors} implies in particular
the following homotopy invariance property for the
pairing \eqref{eq:pairing}.
\begin{cor}
\label{cor:homotopyinvariance}
Let $f_i,g_i,h_i\colon\thinspace B \to BG$, $i=0,1$ be maps, and let
$H_f\colon\thinspace f_0\simeq f_1$,
$H_g\colon\thinspace g_0\simeq g_1$, and
$H_h\colon\thinspace h_0\simeq h_1$
be homotopies.
Then there exist induced isomorphisms
\begin{align*}
\Xi(H_f,H_g)
&\colon\thinspace
\mathbb{H}^\ast P(f_0,g_0)
\xrightarrow{\ \cong\ }
\mathbb{H}^\ast P(f_1,g_1),
\\
\Xi(H_g,H_h)
&\colon\thinspace
\mathbb{H}^\ast P(g_0,h_0)
\xrightarrow{\ \cong\ }
\mathbb{H}^\ast P(g_1,h_1),
\quad\text{and}\\
\Xi(H_f,H_h)
&\colon\thinspace
\mathbb{H}^\ast P(f_0,h_0)
\xrightarrow{\ \cong\ }
\mathbb{H}^\ast P(f_1,h_1)
\end{align*}
making the following square commutative:
\begin{equation}
\label{eq:xisquare}
\vcenter{\xymatrix{
\mathbb{H}^\ast P(g_0,h_0) \otimes \mathbb{H}^\ast P(f_0,g_0)
\ar[r]^-\circ
\ar[d]^\cong_{\Xi(H_g,H_h)\otimes \Xi(H_f,H_g)}
&
\mathbb{H}^\ast P(f_0,h_0)
\ar[d]_\cong^{\Xi(H_f,H_h)}
\\
\mathbb{H}^\ast P(g_1,h_1) \otimes \mathbb{H}^\ast P(f_1,g_1)
\ar[r]^-\circ
&
\mathbb{H}^\ast P(f_1,h_1)
}}
\end{equation}
\end{cor}
\begin{proof}
For $i=0,1$, write $j_i\colon\thinspace B \to B\times I$ for the inclusion
$b\mapsto (b,i)$. Then $P(f_i,g_i) = P(H_f\,j_i, H_g\, j_i)$.
The inclusions $P(f_i,g_i) \hookrightarrow P(H_f,H_g)$ are
homotopy equivalences, so the induced maps
\[
F_{j_i}\colon\thinspace \mathbb{H}^\ast P(H_f,H_g) \xto{\quad} \mathbb{H}^\ast P(f_i,g_i)
\]
are isomorphisms. Define
\[
\Xi(H_f,H_g) = F_{j_1} \circ F_{j_0}^{-1}
\]
and similarly for $\Xi(H_g,H_h)$ and $\Xi(H_f,H_h)$.
The commutativity of the square \eqref{eq:xisquare}
now follows from Theorem~\ref{thm:functors}.
\end{proof}
The maps $F_\phi$ satisfy the expected compatibility conditions
with the maps $\iota$ and $\rho$.
In particular, the following result implies that
the map \eqref{eq:fphidef} is $\phi^\ast$--bilinear
with respect to the bimodule structures of
Remark~\ref{rk:bimodpairing}.
\begin{thrm}
\label{thm:functorcompat}
Given a map of spaces $\phi\colon\thinspace A \to B$ and a map $f\colon\thinspace B \to BG$,
the following diagram commutes:
\begin{equation}
\label{diag:functorcompat}
\vcenter{\xymatrix@C+1em{
H^\ast B
\ar[r]^{\phi^\ast}
\ar[d]_{\iota_f}
&
H^\ast A
\ar[d]^{\iota_{f\phi}}
\\
\mathbb{H}^\ast P(f,f)
\ar[r]^{F_\phi}
\ar[d]_{\rho_f}
&
\mathbb{H}^\ast P(f\phi,f\phi)
\ar[d]^{\rho_{f\phi}}
\\
H^\ast B
\ar[r]^{\phi^\ast}
&
H^\ast A
}}
\end{equation}
\end{thrm}
We note that it is immediate from the definition
that the map \eqref{eq:fphidef} is $\phi^\ast$--linear
with respect to the module structures of
Definition~\ref{def:cupmodstr}.
The rest of the section is structured as follows.
In Subsection~\ref{subsec:strategy}, we will describe
our strategy for proving
Theorems~\ref{thm:pairings} and \ref{thm:functors}.
This strategy is then carried out in Subsections~\ref{subsec:fibfop}
through \ref{subsec:result}.
In Subsection~\ref{subsec:moreproofs}, we will prove
Theorems~\ref{thm:pontryaginproduct}, \ref{thm:algstr}, and \ref{thm:functorcompat},
and finally Subsection~\ref{subsec:pfofcupmodstrbilin}
is devoted to the proof of Theorem~\ref{thm:cupmodstrbilin}.
\subsection{The strategy for proving Theorems~\ref{thm:pairings} and \ref{thm:functors}}
\label{subsec:strategy}
Theorems~\ref{thm:pairings} and \ref{thm:functors}
should remind the reader of the definition of
a category and a functor, respectively.
We will use the language of enriched category theory
to organize their proofs.
\begin{defn}
\label{def:enrichedcat}
A \emph{category} $\mathcal{C}$
\emph{enriched in a monoidal category} $\mathcal{V}$
consists of the following data:
a collection of objects $\operatorname{Ob} \mathcal{C}$,
a hom-object $\mathcal{C}(A,B) \in \mathcal{V}$ for
every pair of objects $A,B\in \operatorname{Ob}\mathcal{C}$,
a composition law $\mathcal{C}(B,C)\otimes \mathcal{C}(A,B) \to \mathcal{C}(A,C)$
for every triple of objects $A,B,C\in\operatorname{Ob}\mathcal{C}$,
and an identity element $I \to \mathcal{C}(A,A)$ for every object $A\in\operatorname{Ob}\mathcal{C}$,
where $I$ denotes the monoidal unit in $\mathcal{V}$.
These data are supposed to satisfy
the evident analogues of the axioms of an ordinary category
\cite[\S1.2]{KellyEnriched}.
\end{defn}
\begin{defn}
\label{def:enrichedfunctor}
An \emph{enriched functor}
$F\colon\thinspace \mathcal{C} \to \mathcal{D}$ between categories enriched in $\mathcal{V}$
consists of a map $F\colon\thinspace \operatorname{Ob} \mathcal{C} \to \operatorname{Ob} \mathcal{D}$ and a
map $F=F_{A,B}\colon\thinspace\mathcal{C}(A,B) \to \mathcal{C}(FA,FB)$ for every pair of objects
$A,B\in \operatorname{Ob}\mathcal{C}$, these data being subject to
the evident analogues of the axioms
for an ordinary functor
\cite[\S1.2]{KellyEnriched}.
\end{defn}
\begin{rem}
\label{rk:enrichedcat}
In the language introduced in Definitions~\ref{def:enrichedcat}
and \ref{def:enrichedfunctor}, Theorem~\ref{thm:pairings}
can be reformulated as the assertion
that given a space $B$, there is a category enriched
in graded ${\mathbb {F}}_\ell$--vector spaces where objects are maps $B \to BG$
and the hom-object from $f\colon\thinspace B \to BG$ to $g\colon\thinspace B \to BG$
is given by $\mathbb{H}^\ast P(f,g)$; and Theorem~\ref{thm:functors}
simply asserts that the maps $F_\phi$ give an enriched
functor from the category associated to $B$ to the one
associated to $A$.
\end{rem}
\begin{terminology}[(Symmetric) monoidal functors]
\label{term:monfun}
By a (symmetric) monoidal functor $F\colon\thinspace\mathcal{C} \to \mathcal{D}$
between (symmetric) monoidal categories,
we mean a \emph{strong} (symmetric) monoidal functor
in the sense of Mac~Lane~\cite[\S XI.2]{MacLane},
meaning that the monoidality and identity constraints
\[
F_\otimes \colon\thinspace F(X) \otimes F(Y) \xto{\quad} F(X\otimes Y)
\qquad\text{and}\qquad
F_I\colon\thinspace I_\mathcal{D} \xto{\quad} F(I_\mathcal{C})
\]
are assumed to be isomorphisms. Here $I_\mathcal{C}$ and $I_\mathcal{D}$
denote the unit objects of $\mathcal{C}$ and $\mathcal{D}$, respectively.
In a \emph{lax} (symmetric) monoidal functor the requirement that the
maps are isomorphisms is dropped, and in an \emph{oplax}
(symmetric) monoidal functor the direction of the maps
is in addition reversed.
\end{terminology}
The following construction gives a basic way of
obtaining new enriched categories and functors
from existing ones.
\begin{constr}
\label{constr:newenrichedcats}
Let
$M\colon\thinspace \mathcal{V} \to \mathcal{W}$ be a lax monoidal functor.
Then from a $\mathcal{V}$--enriched
category $\mathcal{C}$ we obtain a $\mathcal{W}$--enriched category $M_\ast\mathcal{C}$
with $\operatorname{Ob} M_\ast\mathcal{C} = \operatorname{Ob} \mathcal{C}$ and hom-objects
\[
(M_\ast\mathcal{C})(A,B) = M\mathcal{C}(A,B)
\]
by taking as the composition law the composite
\[
M\mathcal{C}(B,C)\otimes M\mathcal{C}(A,B)
\xrightarrow{\ M_\otimes\ }
M(\mathcal{C}(B,C)\otimes \mathcal{C}(A,B))
\xrightarrow{\ M(\mu_{A,B,C})\ }
M\mathcal{C}(A,C)
\]
and as the identity element for an object $A$ the composite
\[
I_\mathcal{W} \xrightarrow{\ M_I\ } M(I_\mathcal{V}) \xrightarrow{\ M(\iota_A)\ } M\mathcal{C}(A,A),
\]
where $\mu_{A,B,C}$ and $\iota_A$ refer to
the composition law and the identity element in $\mathcal{C}$,
respectively, and $M_\otimes$ and $M_I$
are the monoidality and identity constraints of $M$.
Moreover, a $\mathcal{V}$--enriched functor $F\colon\thinspace \mathcal{C} \to \mathcal{D}$
induces a $\mathcal{W}$--enriched functor
$M_\ast(F) \colon\thinspace M_\ast\mathcal{C} \to M_\ast\mathcal{D}$
by letting $M_\ast(F) = F$ on objects, and by defining
\[
M_\ast(F)_{A,B} = M(F_{A,B})
\colon\thinspace
(M_\ast\mathcal{C})(A,B) \xto{\quad} (M_\ast\mathcal{D})(FA,FB)
\]
on morphisms.
\end{constr}
In view of Remark~\ref{rk:enrichedcat} and
Construction~\ref{constr:newenrichedcats},
one might try to prove Theorem~\ref{thm:pairings} by
first constructing an enriched category $\mathcal{P}_B$
where the objects are maps
$f\colon\thinspace B \to BG$,
where the hom-object from $f$ to $g$ is given by the fibration
$P(f,g) \to B$, and where the composition law is given by
diagram~\eqref{diag:pushpull}, and then applying
Construction~\ref{constr:newenrichedcats} to $\mathcal{P}_B$
with a suitable monoidal functor $M$ to obtain the category of
Theorem~\ref{thm:pairings}.
Moreover, Theorem~\ref{thm:functors}
would follow if the pullback squares \eqref{sq:inducedmap}
assembled into an enriched
functor $K_\phi\colon\thinspace \mathcal{P}_B \to \mathcal{P}_A$ which, upon
application of $M$, yielded $F_\phi$.
Modulo a few difficulties which we will point out along the way,
this strategy is indeed the one we will follow.
The functor $M$ will be the composite
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}\;}
\begin{equation}
\label{eq:functorM}
\vcenter{\xymatrix@1@!0@C=4em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]^{D_\mathrm{fw}^\mathrm{op}}
&
*!L{\;(\mathbf{hpSpectra}^\ell)^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{U_\mathrm{fw}^\mathrm{op}}
&
*!L{\;\mathbf{hpSpectra}^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{r_!^\mathrm{op}}
&
*!L{\;\ho(\mathbf{Spectra})^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{H^\ast}
&
*!L{\;\mathbf{grMod}^{{\mathbb {F}}_\ell}}
}}
\end{equation}
Here $H^\ast$ denotes cohomology with ${\mathbb {F}}_\ell$ coefficients,
$\mathbf{grMod}^{{\mathbb {F}}_\ell}$ denotes the category of graded
${\mathbb {F}}_\ell$--modules,
and
$\ho(\mathbf{Spectra})$
denotes the homotopy category of spectra.
The categories $\mathbf{hpSpectra}$ and $\mathbf{hpSpectra}^\ell$
are the categories of parametrized spectra and
parametrized $H{\mathbb {F}}_\ell$--local spectra
obtained by applying the construction
of appendix~\ref{subapp:hpcffw}
to the $\infty$--categories $\mathbf{Spectra}$ and $\mathbf{Spectra}^\ell$
of spectra and $H{\mathbb {F}}_\ell$--local spectra,
respectively, and the functor $U_\mathrm{fw}$
is the functor obtained by applying
the construction of appendix~\ref{subapp:hpcffw}
to the forgetful functor $U\colon\thinspace \mathbf{Spectra}^\ell \to \mathbf{Spectra}$.
The category $\mathbf{Fib}^\mathrm{fop}$ will be constructed in
Subsection~\ref{subsec:fibfop};
the functor $D_\mathrm{fw}$ in Subsection~\ref{subsec:dfw};
and the functor $r_!$ in Subsection~\ref{subsec:remainingfunctors},
which also contains further discussion of the functor $U_\mathrm{fw}$.
Once all the categories and functors in \eqref{eq:functorM}
have been constructed, the proof of Theorems~\ref{thm:pairings}
and \ref{thm:functors} will be completed by identifying
$H^\ast(r_!U_\mathrm{fw} D_\mathrm{fw} P(f,g))$ as $\mathbb{H}^\ast(P(f,g))$.
This will be done in Subsection~\ref{subsec:result}.
\subsection{The category \texorpdfstring{$\mathbf{Fib}^\mathrm{fop}$}{Fib\textasciicircum fop}}
\label{subsec:fibfop}
Our goal in this subsection is to construct the category
$\mathbf{Fib}^\mathrm{fop}$
appearing in \eqref{eq:functorM}.
Write $\mathcal{T}$ for the category of topological
spaces and continuous maps.
We start by constructing the category $\mathbf{Fib}$
over $\mathcal{T}$
of which $\mathbf{Fib}^\mathrm{fop}$ is the ``fibrewise opposite category.''
\begin{defn}
Call a space $X$ $H{\mathbb {F}}_\ell$--\emph{locally dualizable} if
$L \Sigma^\infty_+ X$ is a dualizable object
in the sense of appendix~\ref{app:dualizability}
in the homotopy category
$\ho(\mathbf{Spectra}^\ell)$
of $H{\mathbb {F}}_\ell$--local spectra.
Here $L \colon\thinspace \mathbf{Spectra} \to \mathbf{Spectra}^\ell$
denotes the localization functor.
\end{defn}
\begin{example}
\label{ex:loopsbgd}
By \cite[Cor.~8]{bauer04},
a sufficient condition for a space $X$ to be
$H{\mathbb {F}}_\ell$--locally dualizable is that the homology of $X$
with ${\mathbb {F}}_\ell$ coefficients is finitely generated in each
degree and vanishes outside a finite number of degrees.
In particular, the space
$\Omega BG$ is an $H{\mathbb {F}}_\ell$--locally dualizable space.
\end{example}
\begin{defn}[The category $\mathbf{Fib}$]
\label{def:Fib}
For a space $B$, write $\mathbf{Fib}_B$ for the full subcategory
of the overcategory $\mathcal{T}/B$ spanned by fibrations whose fibres are
$H{\mathbb {F}}_\ell$--locally dualizable.
For example, it follows from
Example~\ref{ex:loopsbgd} that the fibrations $P(f,g) \to B$
are objects of $\mathbf{Fib}_B$ for all $f,g \colon\thinspace B \to BG$,
as their fibres are homotopy equivalent to $\Omega BG$.
We equip $\mathbf{Fib}_B$ with the symmetric monoidal
structure given by the fibrewise direct product $\times_B$.
For all spaces $A$ and $B$, continuous maps $f\colon\thinspace A \to B$,
and objects $\pi\colon\thinspace E \to B$ of $\mathbf{Fib}_B$, choose once and for all
a pullback square
\begin{equation}
\label{sq:fibpullback}
\vcenter{\xymatrix{
f^\ast E
\ar[r]^{\bar{f}_\pi}
\ar[d]_{\pi_f}
&
E
\ar[d]^{\pi}
\\
A
\ar[r]^f
&
B
}}
\end{equation}
Now for every $f\colon\thinspace A \to B$,
the universal property of pullbacks
implies that there exists for every morphism
$\phi\colon\thinspace E \to E'$ in $\mathbf{Fib}_B$
a unique morphism $f^\ast(\phi) \colon\thinspace f^\ast E \to f^\ast E'$
in $\mathbf{Fib}_A$
with the property that the diagram
\[\xymatrix{
f^\ast E
\ar[r]^-{\bar{f}_\pi}
\ar[d]_{f^\ast(\phi)}
&
E
\ar[d]^{\phi}
\\
f^\ast E'
\ar[r]^-{\bar{f}_{\pi'}}
&
E'
}\]
commutes.
In this way,
we obtain a functor $f^\ast \colon\thinspace \mathbf{Fib}_B \to \mathbf{Fib}_A$
which is easily seen to be symmetric monoidal.
The assignments $B\mapsto \mathbf{Fib}_B$ and $f\mapsto f^\ast$
along with the natural homeomorphisms
$(g\circ f)^\ast E \cong f^\ast g^\ast E$
now define a pseudofunctor (see Definition~\ref{def:pseudofunctoretc})
from $\mathcal{T}^\mathrm{op}$ to the 2-category $\mathbf{smCat}$ of symmetric
monoidal categories, symmetric monoidal functors, and symmetric
monoidal transformations.
We define $\mathbf{Fib}$ to be the symmetric monoidal
category obtained from this pseudofunctor by
the Grothendieck construction of
appendix~\ref{subapp:grothendieckconstr}.
Notice that the symmetric monoidal product on $\mathbf{Fib}$
agrees with the cartesian product
\[
(E\xrightarrow{\ \pi\ } B) \times (E' \xrightarrow{\ \pi'\ } B')
=
(E\times E' \xrightarrow{\ \pi\times \pi'\ } B \times B').
\]
\end{defn}
\begin{rem}
\label{rk:altfibdescr}
Alternatively, the category $\mathbf{Fib}$ can be described as follows.
The objects of $\mathbf{Fib}$ are
fibrations $\pi\colon\thinspace E \to B$ with
$H{\mathbb {F}}_\ell$--locally dualizable fibres, and a map from $\pi \colon\thinspace E\to B$
to $\pi'\colon\thinspace E' \to B'$ is a pair $(\bar{f},f)$
of continuous maps making the square
\[\xymatrix{
E
\ar[r]^{\bar{f}}
\ar[d]_\pi
&
E'
\ar[d]^{\pi\smash{'}}
\\
B
\ar[r]^f
&
B'
}\]
commutative. The unique factorization of $\bar{f}$ as
a composite
\[
E
\xrightarrow{\ \phi\ }
f^\ast E
\xrightarrow{\ \bar{f}_{\pi'}\ }
E'
\]
where $\phi$ is a map over $B$ provides the link between the
two descriptions. The advantage of the admittedly more complicated
description in Definition~\ref{def:Fib}
is that it makes it simple to define the category $\mathbf{Fib}^\mathrm{fop}$.
\end{rem}
\begin{defn}[The category $\mathbf{Fib}^\mathrm{fop}$]
Writing $F$ for the pseudofunctor $\mathcal{T}^\mathrm{op} \to \mathbf{smCat}$
featuring in the construction of $\mathbf{Fib}$, we define
the category $\mathbf{Fib}^\mathrm{fop}$ to be the category obtained by
the Grothendieck construction
of appendix~\ref{subapp:grothendieckconstr}
from the pseudofunctor
\[
\mathcal{T}^\mathrm{op} \xrightarrow{\ F\ } \mathbf{smCat} \xrightarrow{\ \mathrm{op}\ } \mathbf{smCat}
\]
where $\mathrm{op}$ is the 2-functor sending a symmetric monoidal category
to its opposite.
\end{defn}
\subsection{The category \texorpdfstring{$\mathcal{P}_B$}{P\_B} enriched in \texorpdfstring{$(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$}{(Fib\textasciicircum fop)\textasciicircum op}.}
\label{subsec:calpb}
Our goal now is, roughly speaking, to construct the category $\mathcal{P}_B$
enriched in $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$ discussed at the end of
Subsection~\ref{subsec:strategy}. We start by
offering an alternative description of the category $\mathbf{Fib}^\mathrm{fop}$.
\begin{obs}
\label{obs:altfibfopdescr}
Paralleling Remark~\ref{rk:altfibdescr},
the category $\mathbf{Fib}^\mathrm{fop}$ can alternatively described
as follows:
The objects of $\mathbf{Fib}^\mathrm{fop}$ are fibrations $\pi \colon\thinspace E \to B$
whose fibres are $H{\mathbb {F}}_\ell$--locally dualizable.
For such fibrations $\pi\colon\thinspace E \to B$ and $\pi'\colon\thinspace E' \to B'$,
write $(\bar{f},f) \colon\thinspace \pi\to \pi'$ for a commutative square
\[\xymatrix{
E
\ar[r]^{\bar{f}}
\ar[d]_\pi
&
E'
\ar[d]^{\pi\smash{'}}
\\
B
\ar[r]^{f}
&
B'
}\]
and in keeping with fibred category theory, call such an
arrow $(\bar{f},f)$
\emph{cartesian} if the square is a pullback square.
A morphism from $\pi\colon\thinspace E\to B$ to $\pi'\colon\thinspace E'\to B'$ in
$\mathbf{Fib}^\mathrm{fop}$ is
then an equivalence class of zigzags
\[
\pi \xleftarrow{\ (\alpha,\mathrm{id}_B)\ } \tau \xrightarrow{\ (\bar{f},f)\ } \pi'
\]
where $(\bar{f},f)$ is cartesian.
Two such zigzags
\[
\pi \xleftarrow{\ (\alpha,\mathrm{id}_B)\ } \tau \xrightarrow{\ (\bar{f},f)\ } \pi'
\qquad\text{and}\qquad
\pi \xleftarrow{\ (\alpha',\mathrm{id}_B)\ } \tau' \xrightarrow{\ (\bar{f'},f')\ } \pi'
\]
are equivalent if $f'=f$ and there exists an arrow
$(\theta,\mathrm{id}_B)\colon\thinspace \tau \to \tau'$ such that
$\bar{f} = \bar{f}'\theta$ and $\alpha = \alpha'\theta$;
such a $\theta$ is necessarily
unique and a homeomorphism. In particular,
employing the notation of diagram \eqref{sq:fibpullback},
for a map $\phi \colon\thinspace f^\ast E' \to E$ in $\mathbf{Fib}_B$,
the zigzag
\[
\pi
\xleftarrow{\ (\phi,\mathrm{id}_B)\ }
\pi'_f
\xrightarrow{\ (\bar{f}_{\pi'},f)\ }
\pi'
\]
defines a morphism in $\mathbf{Fib}^\mathrm{fop}$ from $\pi$ to $\pi'$,
and any zigzag is equivalent to a unique zigzag of this form.
This shows the equivalence of the old and new descriptions
of morphisms in $\mathbf{Fib}^\mathrm{fop}$. In the new description,
the composite of
\[
[\pi \xleftarrow{\ (\alpha,\mathrm{id})\ } \tau \xrightarrow{\ (\bar{f},f)\ } \pi']
\qquad\text{and}\qquad
[\pi' \xleftarrow{\ (\alpha',\mathrm{id})\ } \tau \xrightarrow{\ (\bar{f}',f')\ } \pi'']
\]
is represented by the zigzag given by the composites
along the two sides of the diagram
\[\xymatrix{
&&
\sigma
\ar[dl]_{(\tilde{\alpha}',\mathrm{id})}
\ar[dr]^{(\tilde{f},f)}
\\
&
\tau
\ar[dl]_{(\alpha,\mathrm{id})}
\ar[dr]_{(\bar{f},f)}
&&
\tau'
\ar[dl]^{(\alpha',\mathrm{id})}
\ar[dr]^{(\bar{f}',f')}
\\
\pi
&&
\pi'
&&
\pi''
}\]
where $\sigma$, $\tilde{\alpha}'$ and $\tilde{f}$
are determined by the requirement that $(\tilde{f},f)$
is cartesian and $\tilde{\alpha}'$ is the unique morphism making
the diamond in the middle commutative.
The symmetric monoidal structure on $\mathbf{Fib}^\mathrm{fop}$ is given by
the direct product
\[
(E\xrightarrow{\ \pi\ } B) \times (E' \xrightarrow{\ \pi'\ } B')
=
(E\times E' \xrightarrow{\ \pi\times \pi'\ } B\times B').
\]
\end{obs}
Using the description of Observation~\ref{obs:altfibfopdescr}
for $\mathbf{Fib}^\mathrm{fop}$, we see that
diagram \eqref{diag:pushpull} defines a morphism
\begin{equation}
\label{eq:complaw}
\Big(P(f,h)\to B\Big)
\xto{\quad}
\Big(P(g,h) \to B\Big) \times \Big(P(f,g) \to B\Big)
\end{equation}
in $\mathbf{Fib}^\mathrm{fop}$.
For $f\colon\thinspace B \to BG$, we also have a morphism (in $\mathbf{Fib}^\mathrm{fop}$)
\begin{equation}
\label{eq:idmor}
\Big(P(f,f) \to B\Big) \xto{\quad} \Big(\mathrm{pt} \xrightarrow{\mathrm{id}} \mathrm{pt}\Big)
\end{equation}
into the monoidal unit
given by the diagram
\[\xymatrix@!C@C-3em{
P(f,f)
\ar[dr]
&&
B
\ar[dl]_{\mathrm{id}}
\ar[ll]_-s
\ar[rr]^-r
&&
\mathrm{pt}
\ar[d]^{\mathrm{id}}
\\
&
B
\ar[rrr]^r
&&&
\mathrm{pt}
}\]
Here the map $s$ is given by $s(b) = (b,c_{f(b)})$
where $c_{f(b)}$ denotes the constant path onto $f(b)\in BG$.
Finally, the pullback square \eqref{sq:inducedmap}
gives a map (in $\mathbf{Fib}^\mathrm{fop}$)
\begin{equation}
\label{eq:kphicomp}
(P(f\phi,g\phi)\to A) \xto{\quad} (P(f,g) \to B).
\end{equation}
The reader might now expect us to make the following
(incorrect) definition.
\begin{almostdefn}
\label{def:calpb}
The category $\mathcal{P}_B$ is the category enriched in
$(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$
whose objects are continuous maps $B\to BG$, where
the hom-object of maps from
$f\colon\thinspace B \to BG$ to $g\colon\thinspace B\to BG$
is $P(f,g) \to B$, and where the composition law
and identity elements are given by
the maps \eqref{eq:complaw} and \eqref{eq:idmor},
respectively. The functor $K_\phi \colon\thinspace \mathcal{P}_B \to \mathcal{P}_A$
is the enriched functor given by the maps \eqref{eq:kphicomp}.
\end{almostdefn}
The reason why the above definition is not quite correct is
simple: the specified data for $\mathcal{P}_B$ fail to satisfy the axioms
of an enriched category. It is readily verified that
the axioms are \emph{almost} satisfied, however,
the only problem being that
composition of paths is associative and unital
only up to homotopy, rather than strictly.
The problem could be remedied for example by
replacing $\mathbf{Fib}$ by another category
where morphisms are fibrewise homotopy classes of maps,
or by working with Moore paths instead of ordinary paths.
Instead of following either approach, we content
ourselves with noting that the object $\mathcal{P}_B$
given by Definition~\ref{def:calpb}
is close enough to being an enriched category
for the problem to disappear once we apply the functor
\[
D_\mathrm{fw}^\mathrm{op} \colon\thinspace (\mathbf{Fib}^\mathrm{fop})^\mathrm{op} \xto{\quad} (\mathbf{hpSpectra}^\ell)^\mathrm{op}
\]
constructed in the next subsection: following
Construction~\ref{constr:newenrichedcats},
we obtain a new object $(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$ which
does satisfy the axioms for a category enriched in
$(\mathbf{hpSpectra}^\ell)^\mathrm{op}$ as well as enriched functors
\[
(D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)
\colon\thinspace
(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B
\xto{\quad}
(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_A.
\]
See Remark~\ref{rk:dfwcalpb} below.
\subsection{The functor \texorpdfstring{$D_\mathrm{fw}$}{D\_fw}}
\label{subsec:dfw}
Our aim in this subsection is to construct the functor
\[
D_\mathrm{fw}\colon\thinspace \mathbf{Fib}^\mathrm{fop} \xto{\quad} \mathbf{hpSpectra}^\ell
\]
featuring in equation \eqref{eq:functorM}.
Intuitively, this functor is the fibrewise analogue
of the functor sending a space to its $H{\mathbb {F}}_\ell$--local
dual. Despite the notation,
we will not formally obtain the functor $D_\mathrm{fw}$
as a special case of Definition~\ref{def:fwfun}.
Instead, we will construct it by applying the Grothendieck construction
to the pseudo natural transformation from
the pseudofunctor $B\mapsto \mathbf{Fib}_B^\mathrm{op}$
to the pseudofunctor $B\mapsto \ho(\mathbf{Spectra}^\ell_{/B})$
given for a space $B$ by the following composite of functors:
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{\mathbf{Fib}_B^\mathrm{op}\;}
\begin{equation}
\label{eq:dfwfuns}
\vcenter{\xymatrix@1@!0@C=4em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]^{T_B^\mathrm{op}}
&
*!L{\;\ho(\mathbf{Spaces}_{/B})^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{(\Sigma^\infty_{+B})^\mathrm{op}}
&
*!L{\;\ho(\mathbf{Spectra}_{/B})^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{L_B^\mathrm{op}}
&
*!L{\;\ho(\mathbf{Spectra}^\ell_{/B})^\mathrm{op}}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{D_B}
&
*!L{\;\ho(\mathbf{Spectra}^\ell_{/B})}
}}
\end{equation}
Here the categories
$\ho(\mathbf{Spaces}_{/B})$,
$\ho(\mathbf{Spectra}_{/B})$
and
$\ho(\mathbf{Spectra}^\ell_{/B})$
are as constructed in appendix~\ref{subapp:hocoverb};
the functors $\Sigma^\infty_{+B}$ and $L_B$
are as constructed in
appendix~\ref{subapp:tb};
the functor $D_B$ is the fibrewise dual functor
\[
D_B = F_B(-,S_{B,\ell})
\colon\thinspace
\ho(\mathbf{Spectra}^\ell_{/B})^\mathrm{op}
\xto{\quad}
\ho(\mathbf{Spectra}^\ell_{/B})
\]
where $S_{B,\ell}$ denotes the unit object in
$\ho(\mathbf{Spectra}^\ell_{/B})$;
and the functor $T_B$ is the composite
\begin{equation}
\label{eq:tbfactors}
\mathbf{Fib}_B
\xrightarrow{\ i\ }
\mathcal{T}{/B}
\xrightarrow{\ \gamma\ }
\ho(\mathcal{T}/B)
\xrightarrow{\ \simeq\ }
\ho(\mathbf{Spaces}_{/B})
\end{equation}
where $i$ is the inclusion,
$\gamma$ is the projection onto the homotopy category,
and the last arrow is the equivalence of
categories alluded to in Remark~\ref{rk:spacesoverb}.
The construction of $D_\mathrm{fw}$ is now completed by
the verification of the following lemma.
\begin{lemma}
The composites of the functors \eqref{eq:dfwfuns}
define a pseudo natural transformation
between the pseudofunctors $\mathcal{T}^\mathrm{op} \to \mathbf{smCat}$
given by $B\mapsto \mathbf{Fib}_B^\mathrm{op}$
and $B \mapsto \ho(\mathbf{Spectra}^\ell_{/B})$.
\end{lemma}
\begin{proof}
The main issues are verifying that
the composite of the functors \eqref{eq:dfwfuns}
is symmetric monoidal
and that the composites
commute (up to natural isomorphism) with the
pullback functors
\[
f^\ast = (f^\ast)^\mathrm{op} \colon\thinspace \mathbf{Fib}_B^\mathrm{op} \xto{\quad} \mathbf{Fib}_A^\mathrm{op}
\]
and
\[
f^\ast
\colon\thinspace
\ho(\mathbf{Spectra}^\ell_{/B})
\xto{\quad}
\ho(\mathbf{Spectra}^\ell_{/A})
\]
for maps of spaces $f\colon\thinspace A \to B$.
As noted in
appendix~\ref{subapp:tb},
the functors $\Sigma^\infty_{+B}$ and $L_B$
are symmetric monoidal and compatible with the
pullback functors. Let us consider the functor $T_B$.
The symmetric monoidal structure on $\ho(\mathbf{Spectra}_{/B})$
is the cartesian one. Since the objects of $\mathbf{Fib}_B$
are fibrant in a model structure on $\mathcal{T}/B$
giving rise to $\ho(\mathcal{T}/B)$,
we see that $\gamma \circ i$ and hence $T_B$
preserve products. Thus $T_B$ is symmetric monoidal.
Moreover, the pullback functors
\[
f^\ast \colon\thinspace \ho(\mathcal{T}/B) \xto{\quad} \ho(\mathcal{T}/A)
\]
as constructed in \cite{MaySigurdsson}
are the total right derived functors
of the pointset-level pullback functors
\[
f^\ast \colon\thinspace \mathcal{T}/B \xto{\quad} \mathcal{T}/A.
\]
The compatibility of $\gamma\circ i$ and hence that of the
functors $T_B$ with pullback functors therefore
also follows from the fact that
the objects of $\mathbf{Fib}_B$ are fibrant in $\mathcal{T}/B$.
Let us now consider the functor $D_B$.
For $X$ and $Y$ objects of $\ho(\mathbf{Spectra}^\ell_{/B})$,
from \eqref{map:dmonconstr} we obtain a natural morphism
\begin{equation}
\label{map:dbmoncon}
D_B(X) \wedge^\ell_B D_B(Y)
\xto{\quad}
D_B(X\wedge^\ell_B Y)
\end{equation}
making $D_B$ into a lax monoidal functor (see Notation~\ref{nttn:monprod}
for the definition of $\wedge^\ell_B$).
By the criterion of Corollary~\ref{cor:fwdcrit},
all objects in the image of
the composite $L_B\Sigma^\infty_{+B} T_B$
are dualizable. It follows that the morphism
\eqref{map:dbmoncon} is an equivalence for such $X$ and $Y$.
Thus the composite
of all the functors in \eqref{eq:dfwfuns}
is symmetric monoidal, as required.
\end{proof}
\begin{rem}
\label{rk:dfwcalpb}
It is now straightforward to check that
the object $(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
obtained by following
Construction~\ref{constr:newenrichedcats}
is a category enriched in
$(\mathbf{hpSpectra}^\ell)^\mathrm{op}$,
as discussed at the end of Subsection~\ref{subsec:calpb}.
The crucial point is to observe that the
functor
\[
\gamma \colon\thinspace \mathcal{T}/B \xto{\quad} \ho(\mathcal{T}/B)
\]
in \eqref{eq:tbfactors} sends fibrewise homotopic maps
in $\mathcal{T}/B$ to the same map in $\ho(\mathcal{T}/B)$.
Moreover, for a map $\phi \colon\thinspace A \to B$, we obtain an enriched functor
\[
(D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)
\colon\thinspace
(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B
\xto{\quad}
(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_A.
\]
\end{rem}
\subsection{The functors \texorpdfstring{$U_\mathrm{fw}$}{U\_fw} and \texorpdfstring{$r_!$}{r\_!}}
\label{subsec:remainingfunctors}
In this subsection, we will discuss the functors
$U_\mathrm{fw}$ and $r_!$
featuring in \eqref{eq:functorM} and show that the
composite functor $H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op}$ is lax symmetric monoidal.
We start by promoting the functor $U_\mathrm{fw}$
(defined in appendix~\ref{subapp:hpcffw})
into a lax symmetric monoidal functor.
\begin{lemma}
The functor
\[
U_\mathrm{fw} \colon\thinspace \mathbf{hpSpectra}^\ell \xto{\quad} \mathbf{hpSpectra}
\]
admits the structure of a lax symmetric monoidal functor.
\end{lemma}
\begin{proof}
The functor $U_\mathrm{fw}$ is part of the adjunction
\eqref{eq:lfwufwadj}.
It follows formally that $U_\mathrm{fw}$, as a right adjoint
of a symmetric monoidal functor,
is a lax symmetric monoidal functor.
Explicitly, the monoidality constraint for $U_\mathrm{fw}$ is
given by the composite
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{U_\mathrm{fw} X \mathbin{\bar{\wedge}} U_\mathrm{fw} Y\;}
\begin{equation}
\label{eq:ufwmonconstr}
\vcenter{\xymatrix@!0@C=3em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]
&
*!L{\,U_\mathrm{fw} L_\mathrm{fw} (U_\mathrm{fw} X \mathbin{\bar{\wedge}} U_\mathrm{fw} Y)}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{\simeq}
&
*!L{\,U_\mathrm{fw} (L_\mathrm{fw} U_\mathrm{fw} X \mathbin{\bar{\wedge}}^\ell L_\mathrm{fw} U_\mathrm{fw} Y)}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{\simeq}
&
*!L{\,U_\mathrm{fw}(X \mathbin{\bar{\wedge}}^\ell Y)}
}}
\end{equation}
where the first map is the counit of the $(L_\mathrm{fw},U_\mathrm{fw})$
adjunction;
the second map is given by
(the inverse of) the monoidality constraint for
the symmetric monoidal functor $L_\mathrm{fw}$;
and the last map is given by the
counit of the $(L_\mathrm{fw},U_\mathrm{fw})$ adjunction.
Similarly, the identity constraint for $U_\mathrm{fw}$
is given by the composite
\begin{equation}
\label{eq:ufwidconstr}
S_\mathrm{pt} \xto{\quad} U_\mathrm{fw} L_\mathrm{fw} S_\mathrm{pt} \xrightarrow{\ \simeq\ } U_\mathrm{fw} S_{\mathrm{pt},\ell}
\end{equation}
where we have written $S_\mathrm{pt}$ and $S_{\mathrm{pt},\ell}$
for the identity objects of $\mathbf{hpSpectra}$ and $\mathbf{hpSpectra}^\ell$,
respectively, and where the first map is again
given by the counit of the
$(L_\mathrm{fw},U_\mathrm{fw})$
adjunction and the second map is induced by the
identity constraint for the symmetric monoidal functor $L_\mathrm{fw}$.
\end{proof}
\begin{notation}
For $B$ a space, we write $r_B$, $r^B$, and sometimes just $r$
for the unique map from $B$ to the one-point space.
\end{notation}
\begin{defn}
For a presentable symmetric monoidal $\infty$--category $\mathcal{C}$,
we define an oplax symmetric monoidal functor
(see Terminology~\ref{term:monfun})
\[
r_! \colon\thinspace \mathbf{hp}\mathcal{C} \xto{\quad} \ho(\mathcal{C})
\]
from the category $\mathbf{hp}\mathcal{C}$ defined in
appendix~\ref{subapp:hpcffw} as follows.
The functor $r_!$
sends an object $(B,X) \in \mathbf{hp}\mathcal{C}$
to the object $r_!^B X$
and a morphism
$(f,\phi)\colon\thinspace (B,X) \to (C,Y)$
to the composite
\[
r^B_! X
\xrightarrow{\ r_!^B(\phi)\ }
r^B_! f^\ast Y
\xrightarrow{\ \simeq\ }
r^C_! f_! f^\ast Y
\xto{\quad}
r^C_! Y
\]
where the equivalence in the middle is induced by
the natural equivalence $f^\ast (r^C)^\ast \simeq (r^B)^\ast$
between right adjoints
and the last map is induced by the counit of the
$(f_!,f^\ast)$ adjunction. (See appendix~\ref{subapp:basechangefunctors}
for the definition of the base change functors $g_!$ and $g^\ast$
associated to a map $g$ of spaces.)
The monoidality constraint of $r_!$ is given by the composite
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{r^{B\times C}_!(X \mathbin{\bar{\tensor}} Y)\;}
\begin{equation}
\label{eq:rshriekmonconstraint}
\vcenter{\xymatrix@!0@C=3em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]^{=}
&
*!L{\,
r^{B\times C}_! \big(
(\pi^{B\times C}_B)^\ast X
\otimes_{B\times C}
(\pi^{B\times C}_C)^\ast Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]
&
*!L{\,
r^{B\times C}_! \big(
(\pi^{B\times C}_B)^\ast r_B^\ast r^B_! X
\otimes_{B\times C}
(\pi^{B\times C}_C)^\ast r_C^\ast r^C_! Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B\times C}_! \big(
r_{B\times C}^\ast\, r^B_! X
\,\otimes_{B\times C} \,
r_{B\times C}^\ast\, r^C_! Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B\times C}_! r_{B\times C}^\ast \big(
r^B_! X \,\otimes \, r^C_! Y
\big)}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]
&
*!L{\,
r^B_! X \,\otimes \, r^C_! Y
}
}}
\end{equation}
Here $X$ and $Y$ are objects of $\mathbf{hp}\mathcal{C}$ over the
spaces $B$ and $C$, respectively;
$\pi^{B\times C}_B \colon\thinspace B\times C \to B$
and
$\pi^{B\times C}_C \colon\thinspace B\times C \to C$
are the projections;
the first map simply substitutes the definition
of $X \mathbin{\bar{\tensor}} Y$ (see equation~\eqref{eq:exttensordef});
the second map is given by the units of the $(r^B_!,r_B^\ast)$
and $(r^C_!,r_C^\ast)$ adjunctions;
the third map is given by the natural equivalences
$r_{B\times C}^\ast \simeq (\pi^{B\times C}_B)^\ast r^\ast_B$
and
$r_{B\times C}^\ast \simeq (\pi^{B\times C}_C)^\ast r^\ast_C$;
the fourth map is given by the monoidality constraint
for $r_{B\times C}^\ast$; and the last map is given by the
counit of the
$(r^{B\times C}_!,r_{B\times C}^\ast)$ adjunction.
Finally, the identity constraint of $r_!$ is given by the composite
\[
r^\mathrm{pt}_! S_\mathrm{pt}
\xrightarrow{\ \simeq\ }
r^\mathrm{pt}_! r_\mathrm{pt}^\ast S
\xrightarrow{\ \simeq\ }
S
\]
where $S$ and $S_\mathrm{pt}$ are the monoidal units for $\ho(\mathcal{C})$
and $\ho(\mathcal{C}_{/\mathrm{pt}})$, respectively; where the first
equivalence is given by the identity constraint for $r_\mathrm{pt}^\ast$;
and where the second equivalence is the counit of the
$(r^\mathrm{pt}_!,r_\mathrm{pt}^\ast)$ adjunction.
\end{defn}
\begin{prop}
The oplax symmetric monoidal functor
\[
r_!\colon\thinspace \mathbf{hp}\mathcal{C} \xto{\quad} \ho(\mathcal{C})
\]
is in fact a symmetric monoidal functor.
\end{prop}
\begin{proof}
Since the identity constraint of $r_!$ is an equivalence,
it is enough to show that the monoidality constraint
\eqref{eq:rshriekmonconstraint}
is.
A tedious diagram chase shows that the composite
\eqref{eq:rshriekmonconstraint}
agrees with the composite equivalence
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{r^{B\times C}_!(X \mathbin{\bar{\tensor}} Y)\;}
\[\vcenter{\xymatrix@!0@C=3em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]^{=}
&
*!L{\,
r^{B\times C}_! \big(
(\pi^{B\times C}_B)^\ast X
\otimes_{B\times C}
(\pi^{B\times C}_C)^\ast Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B}_! (\pi^{B\times C}_B)_! \big(
(\pi^{B\times C}_B)^\ast X
\otimes_{B\times C}
(\pi^{B\times C}_C)^\ast Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B}_! \big(
X
\otimes_{B}
(\pi^{B\times C}_B)_! (\pi^{B\times C}_C)^\ast Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B}_! \big(
X
\otimes_{B}
r_B^\ast r^C_! Y
\big)
}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^\simeq
&
*!L{\,
r^{B}_! X
\otimes_{B}
r^{C}_! Y
}
}}\]
where the first map substitutes the definition of $X\mathbin{\bar{\tensor}} Y$,
where the second map is given by the equivalence
$r^{B\times C}_! \simeq r^B_! (\pi^{B\times C}_C)_!$,
where the third and last equivalences follow from the
projection formula \eqref{eq:bc-projformula},
and
where
the second-last map is an instance of the commutation relation
\eqref{eq:commrelshriek}.
In performing the diagram chase, one needs to know that
the equivalence in the projection formula \eqref{eq:bc-projformula}
is given by the composite
\[
f_! (f^\ast Y \otimes_A X)
\xto{\quad}
f_! (f^\ast Y \otimes_A f^\ast f_! X)
\xrightarrow{\ \simeq\ }
f_! f^\ast (Y \otimes_B f_! X)
\xto{\quad}
Y \otimes_B f_! X
\]
where the first and last maps are
induced by the unit and the counit of the of the $(f_!,f^\ast)$
adjunction, respectively,
and where the middle map is induced by
the monoidality constraint for $f^\ast$;
and that the equivalence in the commutation relation
\eqref{eq:commrelshriek} is the composite
\[
\bar{f}_! \bar{g}^\ast
\xto{\quad}
\bar{f}_! \bar{g}^\ast f^\ast f_!
\xrightarrow{\ \simeq\ }
\bar{f}_! \bar{f}^\ast g^\ast f_!
\xto{\quad}
g^\ast f_!
\]
where the first map is induced by the unit of the $(f_!,f^\ast)$
adjunction, where the middle map is induced the equivalence
$\bar{g}^\ast f^\ast \simeq \bar{f}^\ast g^\ast$,
and where the last map is induced by the counit of the
$(\bar{f}_!, \bar{f}^\ast)$ adjunction.
\end{proof}
Our goal in the remainder of the subsection is to
show the following result.
\begin{prop}
\label{prop:hrulaxmon}
The composite functor
\begin{equation}
\label{eq:urh}
(\mathbf{hpSpectra}^\ell)^\mathrm{op}
\xrightarrow{\ U_\mathrm{fw}^\mathrm{op}\ }
\mathbf{hpSpectra}^\mathrm{op}
\xrightarrow{\ r_!^\mathrm{op}\ }
\ho(\mathbf{Spectra})^\mathrm{op}
\xrightarrow{\ H^\ast\ }
\mathbf{grMod}^{{\mathbb {F}}_\ell}
\end{equation}
appearing in \eqref{eq:functorM}
is lax symmetric monoidal.
\end{prop}
\noindent
Notice that the result is not quite obvious,
since while $r_!^\mathrm{op}$ and $H^\ast$
are lax symmetric monoidal functors,
the functor $U_\mathrm{fw}^\mathrm{op}$ is oplax rather than lax symmetric monoidal.
Write
\[
(H{\mathbb {F}}_\ell\wedge - )_\mathrm{fw} \colon\thinspace \mathbf{hpSpectra} \xto{\quad} \mathbf{hpMod}^{H{\mathbb {F}}_\ell}
\]
for the symmetric monoidal functor induced by the symmetric monoidal functor
\[
H{\mathbb {F}}_\ell\wedge - \colon\thinspace \mathbf{Spectra} \xto{\quad} \mathbf{Mod}^{H{\mathbb {F}}_\ell}
\]
given by smashing with $H{\mathbb {F}}_\ell$ (see appendix~\ref{subapp:hpcffw}).
Also write $F^{H{\mathbb {F}}_\ell}(-,-)$ for the internal hom in
$\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})$.
\begin{lemma}
\label{lm:newfactorization}
The diagram
\begin{equation}
\label{diag:newfactorization}
\vcenter{\xymatrix@C-0.4em@R+1ex{
(\mathbf{hpSpectra}^\ell)^\mathrm{op}
\ar[r]^-{U_\mathrm{fw}^\mathrm{op}}
&
\mathbf{hpSpectra}^\mathrm{op}
\ar[r]^{r_!^\mathrm{op}}
\ar[d]_{(H{\mathbb {F}}_\ell \wedge -)_\mathrm{fw}^\mathrm{op}}
&
\ho(\mathbf{Spectra})^\mathrm{op}
\ar[d]_{(H{\mathbb {F}}_\ell \wedge -)^\mathrm{op}}
\ar@/^1pc/[drrr]^{H^\ast}
\\
&
(\mathbf{hpMod}^{H{\mathbb {F}}_\ell})^\mathrm{op}
\ar[r]^{r_!^\mathrm{op}}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})^\mathrm{op}
\ar[rrr]^-{\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)}
&&&
\mathbf{grMod}^{{\mathbb {F}}_\ell}
}}
\end{equation}
commutes up to natural isomorphism.
\end{lemma}
\begin{proof}
The commutativity up to natural isomorphism
of the triangle on the right follows by the
computation
\[
\pi_{-\ast}F^{H{\mathbb {F}}_\ell}(H{\mathbb {F}}_\ell \wedge X, H{\mathbb {F}}_\ell)
\cong
\pi_{-\ast}F(X, H{\mathbb {F}}_\ell)
=
H^\ast(X).
\]
The commutativity up to natural isomorphism of the square
in the middle follows from the commutativity
up to natural equivalence of the squares
\[\xymatrix{
\ho(\mathbf{Spectra}_{/B})
\ar[r]^{r_!}
\ar[d]_{(H{\mathbb {F}}_\ell \wedge -)_B}
&
\ho(\mathbf{Spectra})^\mathrm{op}
\ar[d]^{H{\mathbb {F}}_\ell \wedge -}
\\
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})
\ar[r]^{r_!}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
}\]
for all spaces $B$, which in turn follows
from the commutativity
up to natural equivalence of the corresponding square of
right adjoints
\[\xymatrix{
\ho(\mathbf{Spectra}_{/B})
\ar@{<-}[r]^{r^\ast}
\ar@{<-}[d]_{\mathrm{forget}_B}
&
\ho(\mathbf{Spectra})^\mathrm{op}
\ar@{<-}[d]^{\ho(\mathrm{forget})}
\\
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})
\ar@{<-}[r]^{r^\ast}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
}\]
where `$\mathrm{forget}$'
stands for the forgetful functor
$\mathbf{Mod}^{H{\mathbb {F}}_\ell} \to \mathbf{Spectra}$.
\end{proof}
\begin{lemma}
\label{lm:hflsmashfw}
The functor
\[
(H{\mathbb {F}}_\ell\wedge -)_\mathrm{fw}\colon\thinspace \mathbf{hpSpectra} \xto{\quad} \mathbf{hpMod}^{H{\mathbb {F}}_\ell}
\]
sends the monoidality \eqref{eq:ufwmonconstr}
and identity \eqref{eq:ufwidconstr}
constraints
of the
lax symmetric monoidal functor
\[
U_\mathrm{fw} \colon\thinspace \mathbf{hpSpectra}^\ell \xto{\quad} \mathbf{hpSpectra}
\]
to equivalences.
\end{lemma}
\begin{proof}
As equivalences in $\mathbf{hpSpectra}$
are detected on fibres, it is enough to
show that the identity and monoidality constraints
are equivalences on fibres.
But on fibres, up to equivalence both maps
amount to unit maps of the adjunction
\[
L \colon\thinspace \ho(\mathbf{Spectra})
\mathrel{\substack{\longrightarrow \\[-.7ex] \longleftarrow}}
\ho(\mathbf{Spectra}^\ell) \colon\thinspace U.
\]
The claim follows since the unit of the $(L,U)$ adjunction is
an $H{\mathbb {F}}_\ell$--equivalence.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:hrulaxmon}]
Since the functors $r_!^\mathrm{op}$ and $H^\ast$
in \eqref{eq:urh} are lax symmetric monoidal,
it is enough to show the composite $H^\ast r_!^\mathrm{op}$
sends the identity and monoidality
constraints of $U_\mathrm{fw}^\mathrm{op}$ to isomorphisms.
The monoidality constraint of
$H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op}$
is then given by the composite
\[
H^\ast r_! U_\mathrm{fw} (X) \otimes H^\ast r_! U_\mathrm{fw} (Y)
\xto{\quad}
H^\ast r_! (U_\mathrm{fw} (X) \mathbin{\bar{\wedge}} U_\mathrm{fw} (Y))
\xrightarrow{\ \cong\ }
H^\ast r_! U_\mathrm{fw}(X \mathbin{\bar{\wedge}}^\ell Y)
\]
where the first map is the monoidality constraint
for $H^\ast r^\mathrm{op}_!$ and the second map is
the inverse of map obtained by applying
$H^\ast r_!$ to the monoidality
constraint of $U_\mathrm{fw}$,
and the identity constraint is obtained similarly.
In view of Lemma~\ref{lm:newfactorization},
the claim now follows from
Lemma~\ref{lm:hflsmashfw}.
\end{proof}
\subsection{Identifying the result}
\label{subsec:result}
We have now constructed the categories
$(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast\mathcal{P}_B$
enriched in graded ${\mathbb {F}}_\ell$--modules
we set out to construct at the end of
Subsection~\ref{subsec:strategy},
along with the enriched functors
\[
(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)
\colon\thinspace
(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast\mathcal{P}_B
\xto{\quad}
(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast\mathcal{P}_A
\]
between them.
To complete the proof of Theorems~\ref{thm:pairings}
and \ref{thm:functors}, it now suffices to prove
the following result.
\begin{thrm}
\label{thm:recognitionthm}
For all $f,g\colon\thinspace B \to BG$, there is an isomorphism
\[
H^\ast(r_!U_\mathrm{fw} D_\mathrm{fw} P(f,g)) \cong \mathbb{H}^\ast(P(f,g))
\]
natural with respect to the homomorphisms induced by diagram
\eqref{sq:inducedmap}.
\end{thrm}
The proof of Theorem~\ref{thm:recognitionthm} is based on
Proposition~\ref{prop:bgiselfdual} below,
which is essentially a fibrewise reformulation of Bauer's result on
self-duality of $\ell$--compact groups
\cite[Prop.~22]{bauer04}.
See also \cite[Prop.~3.2.3]{Rognes}.
Rather than deriving the proposition from
Bauer's (or Rognes's) result, however, we will prove it from scratch,
taking our cue from Rognes's treatment \cite{Rognes}.
In what follows,
we consider the path space $BG^I$ as a space over $BG\times BG$
via the evaluation fibration
\begin{equation}
\label{eq:bgifib}
(\mathrm{ev}_0,\mathrm{ev}_1) \colon\thinspace BG^I \xto{\quad} BG\times BG,
\quad
\alpha \longmapsto (\alpha(0),\alpha(1))
\end{equation}
and to simplify notation, we sometimes
continue to write $BG^I$ for the
corresponding object
\[
BG^I
=
L_\mathrm{fw} \Sigma^\infty_{+BG^2} BG^I
\]
in $\ho(\mathbf{Spectra}^\ell_{/BG^2})$ when this is unlikely to
cause confusion.
Given a space $B$, let us write
\[
\pi_{ij} \colon\thinspace B^3 \xto{\quad} B^2
\]
for $1\leq i < j \leq 3$
for the projection onto the $i$--th and $j$--th coordinates
and
\[
\pi_1,\pi_2 \colon\thinspace B^2 \xto{\quad} B
\]
for the projections onto the first and second coordinates.
For $B=BG$,
the following result plays the role of the ``shear equivalence''
of \cite[(4.6)]{bauer04} or \cite[Lem.~3.1.3]{Rognes}.
\begin{lemma}
\label{lm:shear}
Let $B$ be a space. Then
\begin{equation}
\label{eq:sheareqpre}
\pi_{12}^\ast X \wedge^\ell_{B^3} \pi_{23}^\ast \Delta_! S_{B,\ell}
\simeq
\pi_{13}^\ast X \wedge^\ell_{B^3} \pi_{23}^\ast \Delta_! S_{B,\ell}
\end{equation}
for any $X$ in $\ho(\mathbf{Spectra}^\ell_{/B^2})$,
where $\Delta$ denotes the diagonal map of $B$ and $S_{B,\ell}$
is the unit object in $\ho(\mathbf{Spectra}^\ell_{{\mathbf{B}} })$.
\end{lemma}
\begin{proof}
For both $\rho = \pi_{12}$ and $\rho = \pi_{13}$ we have
\[
\rho^\ast X \wedge^\ell_{B^3} \pi_{23}^\ast \Delta_! S_{B,\ell}
\simeq
\rho^\ast X \wedge^\ell_{B^3} (\mathrm{id}_B\times \Delta)_! S_{B^2,\ell}
\simeq
(\mathrm{id}_B\times \Delta)_! ((\mathrm{id}_B\times \Delta)^\ast \rho^\ast X)
\simeq
(\mathrm{id}_B\times \Delta)_! X
\]
where the first equivalence follows from the commutation relation
\eqref{eq:commrelshriek} arising from the square
\[\xymatrix@C+1em{
B^2
\ar[r]^{\mathrm{id}_B\times \Delta}
\ar[d]_{\pi_2}
&
B^3
\ar[d]^{\pi_{23}}
\\
B
\ar[r]^{\Delta}
&
B^2
}\]
where the second equivalence follows from
the projection formula \eqref{eq:bc-projformula},
and where the final equivalence follows from the
equality $\rho \circ (\mathrm{id}_B\times \Delta) = \mathrm{id}_{B^2}$.
\end{proof}
Let \[
Q
=
(\pi_2)_! D_\mathrm{fw} BG^I
\in
\ho(\mathbf{Spectra}^\ell_{/BG}).
\]
\begin{prop}
\label{prop:bgiselfdual}
There is an equivalence
\begin{equation}
\label{eq:bgiselfdualeq}
D_\mathrm{fw} BG^I
\simeq
BG^I \wedge^\ell_{BG^2} \pi_2^\ast Q
\end{equation}
in $\ho(\mathbf{Spectra}^\ell_{/BG^2})$.
\end{prop}
\begin{proof}%
The equivalence
of \eqref{eq:bgifib} and $\Delta \colon\thinspace BG \to BG\times BG$
as spaces over $BG\times BG$
implies that
$BG^I \simeq \Delta_! S_{BG,\ell}$ in $\ho(\mathbf{Spectra}^\ell_{/BG^2})$.
Making use of this equivalence and taking $B=BG$ and
$X = D_\mathrm{fw} BG^I$ in \eqref{eq:sheareqpre},
we see that
\begin{equation}
\label{eq:sheareq}
\pi_{12}^\ast D_\mathrm{fw} BG^I \wedge^\ell_{BG^3} \pi_{23}^\ast BG^I
\simeq
\pi_{13}^\ast D_\mathrm{fw} BG^I \wedge^\ell_{BG^3} \pi_{23}^\ast BG^I
\end{equation}
in $\ho(\mathbf{Spectra}^\ell_{/BG^3})$.
By Example~\ref{ex:loopsbgd} and the criterion of
Corollary~\ref{cor:fwdcrit},
$BG^I$ is dualizable as an object of
$\ho(\mathbf{Spectra}^\ell_{/BG^2})$.
Applying $(\pi_{13})_! D_\mathrm{fw}$ to the
equivalence \eqref{eq:sheareq}, we therefore obtain an equivalence
\begin{equation}
\label{eq:pi13eq}
(\pi_{13})_!\big(
\pi_{12}^\ast BG^I \wedge^\ell_{BG^3} \pi_{23}^\ast D_\mathrm{fw} BG^I
\big)
\simeq
(\pi_{13})_!\big(
\pi_{13}^\ast BG^I \wedge^\ell_{BG^3} \pi_{23}^\ast D_\mathrm{fw} BG^I
\big)
\end{equation}
in $\ho(\mathbf{Spectra}^\ell_{/BG^2})$.
Here we have used that the double dual of a dualizable objet is
the object itself, and that the dual of a smash product of
dualizable objects is the smash product of duals.
The left hand side of the equivalence is
\begin{align*}
(\pi_{13})_!\big(
\pi_{12}^\ast BG^I
\wedge^\ell_{BG^3}
\pi_{23}^\ast D_\mathrm{fw} BG^I
\big)
\simeq &\,
(\pi_{13})_!\big(
\pi_{12}^\ast \Delta_! S_{BG,\ell}
\wedge^\ell_{BG^3}
\pi_{23}^\ast D_\mathrm{fw} BG^I
\big)
\\
\simeq &\,
(\pi_{13})_!(\Delta\times \mathrm{id}_B)_! D_\mathrm{fw} BG^I
\\
\simeq &\,
D_\mathrm{fw} BG^I
\end{align*}
where the penultimate equivalance follows
by a computation similar to the one in the proof of Lemma~\ref{lm:shear}.
On the other hand, the right hand side of \eqref{eq:pi13eq} is
\begin{align*}
(\pi_{13})_!\big(
\pi_{13}^\ast BG^I \wedge^\ell_{BG^3} \pi_{23}^\ast D_\mathrm{fw} BG^I
\big)
\simeq &\,
BG^I \wedge^\ell_{BG^2} (\pi_{13})_! \pi_{23}^\ast D_\mathrm{fw} BG^I
\\
\simeq &\,
BG^I \wedge^\ell_{BG^2} \pi_2^\ast (\pi_2)_! D_\mathrm{fw} BG^I
\\
\simeq &\,
BG^I \wedge^\ell_{BG^2} \pi_2^\ast Q
\end{align*}
Here the first equivalence follows by the projection formula
\eqref{eq:bc-projformula}
and the second equivalence
follows by the commutation relation
\eqref{eq:commrelshriek}
arising from the square
\[\xymatrix{
BG^3
\ar[r]^{\pi_{23}}
\ar[d]_{\pi_{13}}
&
BG^2
\ar[d]^{\pi_2}
\\
BG^2
\ar[r]^{\pi_2}
&
BG
}\]
The last equivalence holds by the definition of $Q$.
The claim follows.
\end{proof}
The following result corresponds to \cite[Cor.~23]{bauer04}.
\begin{prop}
\label{prop:qfibres}
The fibres of $Q$ are $(-d)$--dimensional $H{\mathbb {F}}_\ell$--local spheres.
\end{prop}
\begin{proof}%
Passing to fibres over a point $(b,b)\in BG^2$ in
\eqref{eq:bgiselfdualeq}, we obtain an equivalence
\[
F(L\Sigma^\infty_+\Omega BG,S)
\simeq
L\Sigma^\infty_+ \Omega BG \wedge^\ell Q_b
\]
in $\ho(\mathbf{Spectra}^\ell)$. It follows that
\[
H^{-\ast} (\Omega BG)
\cong
H_\ast (\Omega BG) \otimes H_\ast(Q_b).
\]
We deduce that
\[
H_n (Q_b)
\cong
\begin{cases}
{\mathbb {F}}_\ell & \text{if $n = -d$}\\
0 & \text{otherwise}
\end{cases}
\]
Pick a map $f_0 \colon\thinspace Q_b \to \Sigma^{-d} H{\mathbb {F}}_\ell$
representing the
dual of a generator
of $H_{-d}(Q_b)$.
The long exact cohomology sequences
associated with the short exact sequences
\begin{equation}
\label{ses:coeffs}
0\xto{\quad} {\mathbb {Z}}/\ell \xto{\quad} {\mathbb {Z}}/{\ell^{k+1}} \xto{\quad} {\mathbb {Z}}/{\ell^k} \xto{\quad} 0
\end{equation}
of coefficients show that the map
$H^{-d}(Q_b;\,{\mathbb {Z}}/{\ell^{k+1}}) \to H^{-d}(Q_b;\,{\mathbb {Z}}/{\ell^{k}})$
is an epimorphism for all $k\geq 1$.
Thus the map $f_0$ can be lifted along the tower
\[
H{\mathbb {Z}}_\ell \simeq \operatorname{holim}_k H{\mathbb {Z}}/\ell^k
\xto{\quad}
\cdots
\xto{\quad}
H{\mathbb {Z}}/{\ell^3}
\xto{\quad}
H{\mathbb {Z}}/{\ell^2}
\xto{\quad}
H{\mathbb {Z}}/{\ell}
\]
to a map
\[
f_1
\colon\thinspace
Q_b
\xto{\quad}
\Sigma^{-d} H{\mathbb {Z}}_\ell
=
\Sigma^{-d} H\pi_{-d}(LS^{-d}).
\]
Working up the Postnikov tower of $LS^{-d}$,
we can moreover find a lift
\[
f_2 \colon\thinspace Q_b \xto{\quad} LS^{-d}
\]
of $f_1$:
an induction on $k$ using the long exact
sequences associated to the short exact sequences
\eqref{ses:coeffs}
proves that
$H^n (Q_b;\,{\mathbb {Z}}/\ell^k) = 0$ for all $n\neq -d$ and $k\geq 1$,
showing that all the obstructions for finding a lift vanish.
The map $f_2$ induces a nontrivial map on mod $\ell$ homology
since the map $f_0$ does. Thus the map $f_2$ is an equivalence.
\end{proof}
We are now ready to prove the following result.
For a space $B$, write
\[
H{\mathbb {F}}_\ell \wedge_B
\colon\thinspace
\ho(\mathbf{Spectra}_{/B})
\xto{\quad}
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})
\]
for the functor induced by the functor
$H{\mathbb {F}}_\ell \wedge \colon\thinspace\mathbf{Spectra} \to \mathbf{Mod}^{H{\mathbb {F}}_\ell}$
by the construction of appendix~\ref{subapp:tb}.
\begin{prop}
\label{prop:hfleq}
For all $f,g\colon\thinspace B \to BG$, there exists an equivalence
\[
H{\mathbb {F}}_\ell \wedge_B U_\mathrm{fw} D_\mathrm{fw} P(f,g)
\simeq
H{\mathbb {F}}_\ell\wedge_B \Sigma_B^{-d} \Sigma_{+B}^\infty P(f,g)
\]
in $\ho(\mathbf{hpMod}^{H{\mathbb {F}}_\ell})$ natural with respect to
maps induced by diagram \eqref{sq:inducedmap}.
\qedhere
\end{prop}
\begin{proof}
It suffices to construct the desired equivalence in the universal
case where
\[
f = \pi_1, g=\pi_2 \colon\thinspace BG\times BG\xto{\quad} BG
\]
in which case the space $P(\pi_1,\pi_2) \to BG\times BG$
is simply $(\mathrm{ev}_0,\mathrm{ev}_1) \colon\thinspace BG^I \to BG\times BG$.
Applying the composite functor
\begin{equation}
\label{eq:ufwhflcomp}
\mathbf{hpSpectra}^\ell
\xrightarrow{\ U_\mathrm{fw}\ }
\mathbf{hpSpectra}
\xrightarrow{\ (H{\mathbb {F}}_\ell \wedge -)_\mathrm{fw}\ }
\mathbf{hpMod}^{H{\mathbb {F}}_\ell}
\end{equation}
to the equivalence \eqref{eq:bgiselfdualeq}, we obtain
an equivalence
\[
H{\mathbb {F}}_\ell \wedge_{BG^2} U_\mathrm{fw} D_\mathrm{fw} BG^I
\simeq
(H{\mathbb {F}}_\ell \wedge_{BG^2} U_\mathrm{fw} BG^I)
\wedge^{H{\mathbb {F}}_\ell}_{BG^2}
(H{\mathbb {F}}_\ell \wedge_{BG^2} U_\mathrm{fw} \pi_2^\ast Q)
\]
in $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/BG^2})$.
To identify the right hand side, we have used the fact
that the composite \eqref{eq:ufwhflcomp} is
symmetric monoidal, as follows from Lemma~\ref{lm:hflsmashfw}.
The first factor on the right hand side is equivalent to
$H{\mathbb {F}}_\ell \wedge_{BG^2} \Sigma_{+BG^2}^\infty BG^I$,
so the claim follows from the following lemma.
\end{proof}
\begin{lemma}
\label{lm:orientation}
There exists an equivalence
\[
H{\mathbb {F}}_\ell \wedge_{BG} U_\mathrm{fw} Q
\simeq
H{\mathbb {F}}_\ell \wedge_{BG} r^\ast S^{-d}
\]
in $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/BG})$.
\end{lemma}
\begin{proof}
Since $BG$ is simply connected,
Theorem~\ref{thm:serress} below implies that there is a
strongly convergent spectral sequence
\[
E^{s,t}_2 = H^s(BG) \otimes H^t(Q_b)
\Longrightarrow
H^{s+t} (r^{BG}_! U_\mathrm{fw} Q);
\]
where $b\in BG$ is a basepoint. By Proposition~\ref{prop:qfibres},
the spectral sequence is concentrated on the $t=-d$ line,
so the spectral sequence collapses on the $E_2$ page. Let
\[
u \in H^{-d}(r^{BG}_!U_\mathrm{fw} Q)
\]
be the class corresponding to the class
$1\otimes x \in H^0(BG) \otimes H^{-d}(Q_b)$
where $x$ is a generator of $H^{-d}(Q_b) \cong {\mathbb {F}}_\ell$.
By naturality of the spectral sequence,
the class $u$ now has the property
that for every $b\in BG$, the restriction of
$u$ to $H^{-d}(r^b_! U_\mathrm{fw} Q_b) \cong H^{-d}(Q_b) \cong {\mathbb {F}}_\ell$
is a generator. The class $u$ is equivalent to the data of a map
\[
r_! U_\mathrm{fw} Q \xto{\quad} H{\mathbb {F}}_\ell \wedge S^{-d}
\]
in $\ho(\mathbf{Spectra})$, which in turn via the adjunctions
\[
r_! \colon\thinspace \ho(\mathbf{Spectra})
\mathrel{\substack{\longrightarrow \\[-.7ex] \longleftarrow}}
\ho(\mathbf{Spectra}_{/BG}) \colon\thinspace r^\ast
\]
and
\[
HF_\ell\wedge_{BG} \colon\thinspace \ho(\mathbf{Spectra}_{/BG})
\mathrel{\substack{\longrightarrow \\[-.7ex] \longleftarrow}}
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/BG}) \colon\thinspace \mathrm{forget_{BG}}
\]
is equivalent to the data of a map
\[
\tilde{u}
\colon\thinspace
H{\mathbb {F}}_\ell \wedge_{BG} U_\mathrm{fw} Q
\xto{\quad}
H{\mathbb {F}}_\ell \wedge_{BG} r^\ast S^{-d}
\]
in $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/BG})$.
Working through the adjunctions, it is easy to see that
the property that $u$ restricts to a generator for
each fibre translates precisely to the property that
$\tilde{u}$ is an equivalence on all fibres.
Since equivalences in
$\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/BG})$
are detected on fibres, the claim follows.
\end{proof}
\begin{rem}
\label{rk:orientation}
The equivalence in Lemma~\ref{lm:orientation} is not unique:
the set of all such equivalences form an ${\mathbb {F}}_\ell^\times$--torsor,
as follows by reversing the argument in the proof of
Lemma~\ref{lm:orientation}.
The choice of such an equivalence should be thought of as
an $H{\mathbb {F}}_\ell$--orientation for the ``sphere bundle'' $Q$.
For the purposes of
constructing the pairings of Theorem~\ref{thm:pairings}
and the maps $\iota$ and $\rho$ of Theorem~\ref{thm:algstr},
we (arbitrarily) fix one such an orientation. The pairings and
maps associated to different choices of orientation only differ
from each other by multiplication by an element of ${\mathbb {F}}_\ell^\times$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm:recognitionthm}]
The claim follows by applying
the functor
\begin{equation}
\label{eq:lastfuns}
(\mathbf{hpMod}^{H{\mathbb {F}}_\ell})^\mathrm{op}
\xrightarrow{\ r_!^\mathrm{op}\ }
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})^\mathrm{op}
\xrightarrow{\ \pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)\ }
\mathbf{grMod}^{{\mathbb {F}}_\ell}
\end{equation}
to the equivalence of Proposition~\ref{prop:hfleq}
and using Lemma~\ref{lm:newfactorization}
to recognize the two sides of the resulting isomorphism
as the two sides of the isomorphism of Theorem~\ref{thm:recognitionthm}.
\end{proof}
\begin{rem}
\label{rk:pairingrecon}
Let us now reconcile our construction of the pairing
of Theorem~\ref{thm:pairings}
with the description of this pairing given in
Remark~\ref{rk:summary}.
Notice that the map of $\mathbf{Fib}^\mathrm{fop}$ defined by
diagram~\eqref{diag:pushpull} splits as the composite
of the maps of $\mathbf{Fib}^\mathrm{fop}$
defined by the trapezoid and the triangle in
\eqref{diag:pushpull}.
Consider now the following diagram
in $\mathbf{hpMod}^{H{\mathbb {F}}_\ell}$:
\[\xymatrix@!0@R=8ex@C=16em{
&
\big(
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma_{+B}^\infty P(g,h)
\big)
\mathbin{\bar{\wedge}}^{H{\mathbb {F}}_\ell}
\big(
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma_{+B}^\infty P(f,g)
\big)
\ar@{<-}[dl]_\simeq
\\
\big(H{\mathbb {F}}_\ell\wedge_B U_\mathrm{fw} D_\mathrm{fw} P(g,h)\big)
\mathbin{\bar{\wedge}}^{H{\mathbb {F}}_\ell}
\big(H{\mathbb {F}}_\ell\wedge_B U_\mathrm{fw} D_\mathrm{fw} P(f,g)\big)
\\
&
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-2d} \Sigma_{+B}^\infty P(f,g,h)
\ar@{<-}[dl]_\simeq
\ar[uu]_-{H{\mathbb {F}}_\ell \wedge_\mathrm{fw} \Sigma_B^{-2d}\Sigma_{+B}^\infty (\mathrm{split})}
\\
H{\mathbb {F}}_\ell \wedge_B U_\mathrm{fw} D_\mathrm{fw} P(f,g,h)
\ar[uu]^-{H{\mathbb {F}}_\ell \wedge_\mathrm{fw} U_\mathrm{fw} D_\mathrm{fw} (\mathrm{split})}
\\
&
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma_{+B}^\infty P(f,h)
\ar@{<-}[dl]_\simeq
\ar[uu]_{\mathrm{concat}^\sharp}
\\
H{\mathbb {F}}_\ell \wedge_B U_\mathrm{fw} D_\mathrm{fw} P(f,h)
\ar[uu]^-{H{\mathbb {F}}_\ell \wedge_\mathrm{fw} U_\mathrm{fw} D_\mathrm{fw} (\mathrm{concat})}
}\]
Here the top slanted equivalence
is obtained by smashing together the equivalences of
Proposition~\ref{prop:hfleq} for $P(g,h)$ and $P(f,g)$;
the bottom slanted equivalence is the equivalence
of Proposition~\ref{prop:hfleq} for $P(f,h)$;
the middle slanted equivalence is obtained
by pulling back the equivalence on top along the
diagonal map $\Delta\colon\thinspace B \to B\times B$;
and the map $\mathrm{concat}^\sharp$ is defined by
the commutativity of the bottom
parallelogram. The diagram then commutes by construction.
Tracing through the definition of the pairing of
Theorem~\ref{thm:pairings}
and making use of
Lemma~\ref{lm:newfactorization}, one sees that
the pairing can be obtained
by applying the composite functor \eqref{eq:lastfuns}
to the above diagram and following in the resulting
diagram the path
down along the upper slanted equivalence,
down along the left-hand vertical maps,
and finally up along the bottom slanted equivalence.
The description of the pairing of Theorem~\ref{thm:pairings}
given in Remark~\ref{rk:summary} now follows
by the commutativity of the diagram:
upon applying the composite functor \eqref{eq:lastfuns},
the top right-hand vertical map yields the map $\mathrm{split}^\ast$,
while the bottom right-hand vertical map gives the
``umkehr map'' $\mathrm{concat}_!$ in \eqref{eq:pairingdescr}.
\end{rem}
\subsection{Proofs of Theorems~\ref{thm:pontryaginproduct}, \ref{thm:algstr}, and \ref{thm:functorcompat}}
\label{subsec:moreproofs}
Having concluded the proof of
Theorems~\ref{thm:pairings} and \ref{thm:functors}
in the previous subsection, we now turn to
the proof of
Theorems~\ref{thm:pontryaginproduct}, \ref{thm:algstr}, and \ref{thm:functorcompat}.
\begin{proof}[Proof of Theorem~\ref{thm:pontryaginproduct}]
The product on $\mathbb{H}^\ast \Omega BG$ agrees under the isomorphism
\[
\mathbb{H}^\ast \Omega BG \cong H^\ast D\Omega BG
\]
provided by Theorem~\ref{thm:recognitionthm}
with the composite
\defH^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;{H^\ast(D \Omega BG) \otimes H^\ast(D \Omega BG)\;}
\[\xymatrix@1@!0@C=4em{
*!R{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}
\ar[r]^{ \times }
&
*!L{\;H^\ast(D \Omega BG \wedge^\ell D \Omega BG)}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{ \cong }
&
*!L{\;H^\ast D (\Omega BG \times \Omega BG)}
\\
*!R{\phantom{H^\ast(D \loops BG) \tensor H^\ast(D \loops BG)\;}}
\ar[r]^{ (D\mathrm{concat})^\ast }
&
*!L{\;H^\ast(D \Omega BG)}.
}\]
Here $DX$
for a space $X$
denotes the dual of $L \Sigma^\infty_+ X$ in
$\ho(\mathbf{Spectra}^\ell)$,
and the middle isomorphism is induced by the equivalence
$D(\Omega BG \times \Omega BG) \simeq D\Omega BG \wedge^\ell D\Omega BG$.
The claim now follows using
the natural isomorphism $H^\ast DX \cong H_{-\ast} X$
valid for $H{\mathbb {F}}_\ell$--locally dualizable spaces $X$.
\end{proof}
\begin{defn}[The maps $\iota_f$ and $\rho_f$ of Theorem~\ref{thm:algstr}]
For a map $f\colon\thinspace B \to BG$, consider the commutative triangles
\begin{equation}
\label{triangles:algstr}
\vcenter{\xymatrix@!0@R=8ex@C=4em{
B
\ar[rr]^-s
\ar[dr]_{\mathrm{id}_B}
&&
P(f,f)
\ar[dl]^\pi
\\
&
B
}}
\qquad\text{and}\qquad
\vcenter{\xymatrix@!0@R=8ex@C=4em{
P(f,f)
\ar[rr]^-\pi
\ar[dr]_\pi
&&
B
\ar[dl]^{\mathrm{id}_B}
\\
&
B
}}
\end{equation}
where $\pi$ is the projection and $s$ is the section sending a point $b\in B$ to the pair consisting of $b$ and the constant path
at $f(b)$. The triangles define morphisms
\begin{equation}
\label{mors:iotarhopre}
(B\xrightarrow{\mathrm{id}} B) \xto{\quad} (P(f,f)\to B)
\qquad\text{and}\qquad
(P(f,f) \to B) \xto{\quad} (B\xrightarrow{\mathrm{id}} B)
\end{equation}
in the category $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$.
We define the maps
\[
\iota = \iota_f \colon\thinspace H^\ast B \xto{\quad} \mathbb{H}^\ast P(f,f)
\qquad\text{and}\qquad
\rho = \rho_f \colon\thinspace \mathbb{H}^\ast P(f,f) \xto{\quad} H^\ast B
\]
to be the morphisms obtained by first applying
the composite functor~\eqref{eq:functorM}
to the morphisms~\eqref{mors:iotarhopre}
and then using the isomorphism
\[
H^\ast(r_!U_\mathrm{fw} D_\mathrm{fw} P(f,f)) \cong \mathbb{H}^\ast(P(f,f))
\]
of Theorem~\ref{thm:recognitionthm} and the computation
\begin{multline*}
H^\ast(r_!U_\mathrm{fw} D_\mathrm{fw} B)
\cong H^\ast (r_! U_\mathrm{fw} S_{B,\ell})
\cong H^\ast (r^B_! U_B L_B S_B)
\cong H^\ast (U L r^B_! U_B L_B S_B)
\\
\cong H^\ast (U r^B_! L_B U_B L_B S_B)
\cong H^\ast (U r^B_! L_B S_B)
\cong H^\ast (U L r^B_! S_B)
\cong H^\ast (r^B_! S_B)
\cong H^\ast(B).
\end{multline*}
to recognize the source and the target.
Here we have written $S_B$ and $S_{B,\ell}$
for the unit objects in
$\ho(\mathbf{Spectra}_{/B})$ and $\ho(\mathbf{Spectra}^\ell_{/B})$,
respectively.
Here the first isomorphism in the top row
follows from the observation that $D_\mathrm{fw} B \simeq S_{B,\ell})$
(where $B$ on the left hand side is short for the object
$L_B \Sigma^\infty_{+B} B$ over $B$ in $\mathbf{hpSpectra}^\ell$);
the second isomorphism in the top row substitutes the definitions
of $r_!$ and $U_\mathrm{fw}$, and uses that $L_B$, as a symmetric monoidal functor,
preserves unit objects;
the last isomorphism in the top row and the second last one
in the bottom row are induced by the unit of the $(L,U)$ adjunction,
an $H{\mathbb {F}}_\ell$--equivalence; the first and third isomorphisms
on the bottom row are induced by the commutation relation
$L r_!^B \simeq r_!^B L_B$
implied by the commutation relation
$r_B^\ast U_B \simeq U r_B^\ast$
between right adjoints;
and the second isomorphism in the bottom row is induced by
the natural equivalence $L_B U_B \simeq \mathrm{id}$
(deriving from the natural equivalence $LU \simeq \mathrm{id}$).
\end{defn}
\begin{proof}[Proof of Theorem~\ref{thm:algstr}]
The diagrams
\[
\vcenter{\xymatrix{
B
\ar[r]^-\Delta
\ar[d]_\mathrm{id}
&
B\times B
\ar[d]^{\mathrm{id}\times\mathrm{id}}
\\
B
\ar[r]^-\Delta
&
B\times B
}}
\qquad\text{and}\qquad
\vcenter{\xymatrix{
B
\ar[r]^r
\ar[d]_\mathrm{id}
&
\mathrm{pt}
\ar[d]^\mathrm{id}
\\
B
\ar[r]^r
&
\mathrm{pt}
}}
\]
define morphisms in $\mathbf{Fib}^\mathrm{fop}$ making
the identity map $(B \xrightarrow{\mathrm{id}} B)$ into a monoid
object in the category $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$.
Ignoring the problem that some of the required identities
only hold up to fibrewise homotopy (cf.\ the discussion following
``Definition''~\ref{def:calpb}),
the morphisms \eqref{eq:idmor} and
\eqref{eq:complaw} (with $g=h=f$) make $P(f,f)\to B$
a monoid object in $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$.
It is straightforward to verify that the
morphisms~\eqref{mors:iotarhopre}
in $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$
are monoid object homomorphisms whose composite is the identity.
The claim now follows by applying the composite functor
\eqref{eq:functorM}; by recognizing the image of the monoid object
$(B\xrightarrow{\mathrm{id}} B)$ under this functor as $H^\ast(B)$ equipped
with the cup product; and by using Theorem~\ref{thm:recognitionthm}
to recognize the image of $(P(f,f)\to B)$ as the graded ring
$\mathbb{H}^\ast P(f,f)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:functorcompat}]
Consider the diagram in $(\mathbf{Fib}^\mathrm{fop})^\mathrm{op}$
\[\xymatrix{
(B\xrightarrow{\mathrm{id}} B)
\ar[r]
\ar[d]
&
(A\xrightarrow{\mathrm{id}} A)
\ar[d]
\\
(P(f,f) \to B)
\ar[r]
\ar[d]
&
(P(f\phi,f\phi) \to A)
\ar[d]
\\
(B\xrightarrow{\mathrm{id}} B)
\ar[r]
&
(A\xrightarrow{\mathrm{id}} A)
}\]
where the vertical arrows are induced by the diagrams
\eqref{triangles:algstr} and the analogous diagrams for $f\phi$,
where the top and bottom horizontal arrows are induced by the pullback
square
\[\xymatrix{
A
\ar[r]^\phi
\ar[d]_\mathrm{id}
&
B
\ar[d]^\mathrm{id}
\\
A
\ar[r]^\phi
&
B
}\]
and where the middle horizontal arrow is induced by square
\eqref{sq:inducedmap}. It is straightforward to verify
that the diagram commutes. The claim now follows by
applying the composite functor \eqref{eq:functorM}
to the diagram and recognizing the result as
diagram~\eqref{diag:functorcompat}.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:cupmodstrbilin}}
\label{subsec:pfofcupmodstrbilin}
We finish the section with the somewhat lengthy proof of
Theorem~\ref{thm:cupmodstrbilin}.
By construction, Lemma~\ref{lm:newfactorization},
and
Proposition~\ref{prop:hfleq},
the pairing in Theorem~\ref{thm:cupmodstrbilin}
arises by applying the composite functor
\[
\mathbf{hpMod}^{H{\mathbb {F}}_\ell}
\xrightarrow{\ r_!\ }
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
\xrightarrow{\ {\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)}\ }
\mathbf{grMod}^{{\mathbb {F}}_\ell}
\]
to a certain map
\begin{equation}
\label{map:modhfellcopairing}
H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,h)
\xto{\quad}
\big(H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(g,h)\big)
\mathbin{\bar{\wedge}}^{H{\mathbb {F}}_\ell}
\big(H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)\big)
\end{equation}
in $\mathbf{hpMod}^{H{\mathbb {F}}_\ell}$ covering the diagonal map
$\Delta\colon\thinspace B \to B\times B$ in $\mathcal{T}$.
The basic idea of our proof of Theorem~\ref{thm:cupmodstrbilin}
is as follows. For a general presentable
symmetric monoidal $\infty$--category $\mathcal{C}$
and space $B$, we construct
a natural comonoid structure on the object
$r^B_! S_B \in \ho(\mathcal{C})$
along with, for any object $X \in \ho(\mathcal{C}_{/B})$,
a natural $r^B_! S_B$--comodule structure
on $r^B_! X$. When $\mathcal{C} = \mathbf{Mod}^{H{\mathbb {F}}_\ell}$ and
$X = H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d}\Sigma^{\infty}_{+B} P(f,g)$,
upon application of the functor
$\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)$,
the comonoid structure recovers the cup product on $H^\ast B$
and the comodule structure recovers the module structure
of Definition~\ref{def:cupmodstr}. This reduces the proof
of Theorem~\ref{thm:cupmodstrbilin} to
proving sufficient compatibility and naturality properties
for the comonoid and comodule structures in $\ho(\mathcal{C})$.
Establishing these properties,
along with the results needed for the comparison
with the module structure of Definition~\ref{def:cupmodstr},
is done in Lemmas~\ref{lm:indmapcompat}
through \ref{lm:fcompat} below.
We begin by constructing the comonoid and comodule structures.
Let $\mathcal{C}$ be a presentable symmetric monoidal $\infty$--category
with tensor product $\otimes$.
Given $B \in \mathcal{T}$ and $X \in \ho(\mathcal{C}_{/B})$,
consider maps
\begin{equation}
\label{maps:costr}
X \xto{\quad} S_B \mathbin{\bar{\tensor}} X
\qquad
\text{and}
\qquad
S_B \xto{\quad} S_{\mathrm{pt}}
\end{equation}
in $\mathbf{hp}\mathcal{C}$ covering the diagonal map $\Delta\colon\thinspace B \to B\times B$
and the map $r_B\colon\thinspace B \to \mathrm{pt}$ in $\mathcal{T}$, respectively,
defined as follows: the first map is given by the composite
\[
X
\xrightarrow{\ \simeq\ }
S_B\otimes_B X
\xrightarrow{\ \simeq\ }
\Delta^\ast (S_B\mathbin{\bar{\tensor}} X)
\]
of the inverse of the left
unit constraint in $\ho(\mathcal{C}_{/B})$
and the isomorphism \eqref{eq:internaltensorformula},
while the second map is given by the identity constraint
\[
S_B \xrightarrow{\ \simeq\ } r_B^\ast S_{\mathrm{pt}}
\]
of the symmetric monoidal functor $r_B^\ast$.
When $X = S_B$,
it is readily verified that the maps \eqref{maps:costr}
together make $S_B$ into a comonoid object in $\mathbf{hp}\mathcal{C}$.
Moreover, for an arbitrary $X$, the first map in
\eqref{maps:costr} makes $X$ a comodule object
over the comonoid $S_B$. An application of the symmetric monoidal
functor $r_! \colon\thinspace \mathbf{hp}\mathcal{C}\to \ho(\mathcal{C})$ now produces the
desired comonoid structure on $r^B_! S_B$ and
the desired $r^B_! S_B$--comodule structure
\[
\nu_X \colon\thinspace r^B_! X \xto{\quad} r^B_! S_B \otimes r^B_! X
\]
on $r^B_! X$.
\begin{rem}
\label{rk:spacecase}
In the case $\mathcal{C} = \mathbf{Spaces}$,
for a space $B$ and a parametrized space $\pi \colon\thinspace X \to B$ over $B$,
these constructions recover the usual comonoid structure on $B$
given by the diagonal map $\Delta \colon\thinspace B \to B \times B$
and the comodule structure on $X$ given by the map
$(\pi,\mathrm{id})\colon\thinspace X \to B \times X$.
\end{rem}
The following two lemmas follow by first verifying the corresponding
assertions in $\mathbf{hp}\mathcal{C}$ and then applying the functor $r_!$.
\begin{lemma}
\label{lm:indmapcompat}
Given a map $f\colon\thinspace B \to C$ of spaces, the evident map
$\bar{f} \colon\thinspace S_B \to S_C$ in $\mathbf{hp}\mathcal{C}$ covering $f$
induces a comonoid homomorphism
\[
r_!(\bar{f}) \colon\thinspace r^B_! S_B \xto{\quad} r^C_! S_C.
\]
Moreover, given a map $\tilde{f}\colon\thinspace X \to Y$ in $\mathbf{hp}\mathcal{C}$
covering $f$, the diagram
\[\xymatrix@C+3em{
r^B_! X
\ar[r]^{r_!(\tilde{f})}
\ar[d]_{\nu_X}
&
r^C_! Y
\ar[d]^{\nu_Y}
\\
r^B_! S_B \otimes r^B_! X
\ar[r]^{r_!(\bar{f}) \otimes r_!(\tilde{f})}
&
r^C_! S_C \otimes r^C_! Y
}\]
commutes, showing that the induced map
$r_!(\tilde{f}) \colon\thinspace r^B_! X \to r^C_! Y$
is $r_!(\bar{f})$--linear. \qed
\end{lemma}
\begin{lemma}
\label{lm:exttensorcompat}
For $X \in \ho(\mathcal{C}_{/B})$, $Y\in \ho(\mathcal{C}_{/C})$,
the comodule structures $\nu_X$, $\nu_Y$ and $\nu_{X\mathbin{\bar{\tensor}} Y}$
are compatible in the sense that the following diagram commutes:
\[\xymatrix@C+4em{
r^B_! X \otimes r^C_! Y
\ar[r]^-{\nu_X \otimes \nu_Y}
\ar[ddd]_{(r_!)_\otimes}^\simeq
&
r^B_! S_B \otimes r^B_! X
\otimes
r^C_! S_C \otimes r^C_! Y
\ar[d]^{1\otimes \chi \otimes 1}_\simeq
\\
&
r^{B}_! S_{B} \otimes r^{C}_! S_{C}
\otimes
r^{B}_! X \otimes r^{C}_! Y
\ar[d]^{(r_!)_\otimes \otimes (r_!)_\otimes}_\simeq
\\
&
r^{B\times C}_! (S_{B} \mathbin{\bar{\tensor}} S_{C})
\otimes
r^{B\times C}_! (X \mathbin{\bar{\tensor}} Y)
\ar[d]_\simeq
\\
r^{B\times C}_! (X \mathbin{\bar{\tensor}} Y)
\ar[r]^-{\nu_{X\mathbin{\bar{\tensor}} Y}}
&
r^{B\times C}_! S_{B\times C}
\otimes
r^{B\times C}_! (X \mathbin{\bar{\tensor}} Y)
}\]
Here $\chi$ denotes the symmetry constraint, $(r_!)_\otimes$
denotes the monoidality constraint of $r_!$, and the bottom right
vertical map is induced by the evident equivalence
$S_B\mathbin{\bar{\tensor}} S_C \simeq S_{B\times C}$
in $\ho(\mathcal{C}_{/B\times C})$. \qed
\end{lemma}
The following lemma follows, in essence, by
specializing to the case $B = \mathrm{pt}$, $C = B$, $X=T$, and $Y=X$
in Lemma~\ref{lm:exttensorcompat}.
\begin{lemma}
\label{lm:tcompat}
Given $T \in \ho(\mathcal{C})$ and $X \in \ho(\mathcal{C}_{/B})$,
the diagram
\[\xymatrix@C+4em{
T \otimes r^B_! X
\ar[dd]_{\theta_X}^\simeq
\ar[r]^-{1 \otimes \nu_{X}}
&
T \otimes r^B_! S_B \otimes r^B_! X
\ar[d]^{\chi\otimes 1}_\simeq
\\
&
r^B_! S_B \otimes T \otimes r^B_! X
\ar[d]^{1\otimes\theta_X}_\simeq
\\
r^B_!(r_B^\ast T \otimes_B X)
\ar[r]^-{\nu_{r_B^\ast T \otimes_B X}}
&
r^B_! S_B \otimes r^B_! (r_B^\ast T \otimes_B X)
}\]
commutes, where $\theta_X$ is the composite
\[\xymatrix{
T \otimes r^B_! X
\ar[r]_-\simeq
&
r^\mathrm{pt}_! T \otimes r^B_! X
\ar[r]^-{(r_!)_\otimes}_-\simeq
&
r^{\mathrm{pt}\times B}_! (T \mathbin{\bar{\tensor}} X)
\ar[r]_-\simeq
&
r^B_! (r_B^\ast T \otimes_B X)
}\]
with the first and last arrows given by the
evident equivalences.
\qed
\end{lemma}
\begin{rem}
\label{rk:altcomdesc}
We describe an alternative construction of the comonoid structure
on $r^B_! S_B$ and the comodule structure on $r^B_! X$.
For a space $B$, the inclusion
\[
\iota_B
\colon\thinspace
(\ho(\mathcal{C}_{/B}),\otimes_B)
\xhookrightarrow{\quad}
(\mathbf{hp}\mathcal{C},\mathbin{\bar{\tensor}})
\]
is an oplax symmetric monoidal functor with monoidality
and identity constraints
\[
\iota_B(X\otimes_B Y) \xto{\quad} \iota_B(X)\mathbin{\bar{\tensor}} \iota_B(Y)
\qquad
\text{and}
\qquad
\iota_B S_B \xto{\quad} S_\mathrm{pt}
\]
given by \eqref{eq:internaltensorformula} and the equivalence
$S_B \xrightarrow{\simeq} r_B^\ast S_\mathrm{pt}$, respectively. The composite
$r^B_!\colon\thinspace \ho(\mathcal{C}_{/B}) \to \ho(\mathcal{C})$
of $\iota_B$ and the symmetric monoidal functor $r_!$
therefore obtains the structure of an oplax monoidal functor,
and hence preserves comonoid and comodule objects.
Observe now that the comonoid structure on $r^B_! S_B$
is precisely the one obtained from
the comonoid object structure $S_B$ has as the identity
object of $\ho(\mathcal{C}_{/B})$, and that the comodule structure
on $r^B_! X$ similarly agrees with the one induced by the
comodule structure
on $X$ given by the inverse
$X \to S_B\otimes X$
of the left unit constraint in
$\ho(\mathcal{C}_{/B})$.
\end{rem}
Suppose now $\mathcal{D}$ is another
presentable symmetric monoidal $\infty$--category
and $F \colon\thinspace \mathcal{C} \to \mathcal{D}$
is a symmetric monoidal $\infty$--functor
with a right adjoint $G$.
For every space $B$, the commutation
equivalence $r_B^\ast G \simeq G^{}_B r_B^\ast$
between right adjoints then induces an equivalence
\[
c\colon\thinspace r^B_! F_B \xrightarrow{\ \simeq\ } F r^B_!
\]
between left adjoints.
\begin{lemma}
\label{lm:oplaxsm}
The equivalence $c$ is oplax symmetric monoidal.
\end{lemma}
\begin{proof}
In terms of the calculus of mates \cite{KellyStreet},
the commutation equivalence
\begin{equation}
\label{diag:frshriekcommiso}
\vcenter{\xymatrix{
\ho(\mathcal{C}_{/B})
\ar[r]^{F_B}
\ar[d]_{r^B_!}
&
\ho(\mathcal{D}_{/B})
\ar[d]^{r^B_!}
\ar@{}[dl]|(0.3){}="a"|(0.7){}="b"
\ar@{=>}"a";"b" ^c_\simeq
\\
\ho(\mathcal{C})
\ar[r]^F
&
\ho(\mathcal{D})
}}
\end{equation}
can be obtained from the commutation equivalence
\begin{equation}
\label{diag:grastcommiso}
\vcenter{\xymatrix{
\ho(\mathcal{C}_{/B})
\ar@{<-}[r]^{G_B}
\ar@{<-}[d]_{r_B^\ast}
&
\ho(\mathcal{D}_{/B})
\ar@{<-}[d]^{r_B^\ast}
\ar@{}[dl]|(0.3){}="a"|(0.7){}="b"
\ar@{<=}"a";"b"_\simeq
\\
\ho(\mathcal{C})
\ar@{<-}[r]^G
&
\ho(\mathcal{D})
}}
\end{equation}
in two stages by first passing to the mate of \eqref{diag:grastcommiso}
with respect to the $(F_B,G_B)$ and $(F,G)$ adjunctions
to obtain the transformation
\begin{equation}
\label{diag:frastcomm}
\vcenter{\xymatrix{
\ho(\mathcal{C}_{/B})
\ar@{->}[r]^{F_B}
\ar@{<-}[d]_{r_B^\ast}
\ar@{}[dr]|(0.3){}="a"|(0.7){}="b"
\ar@{=>}"a";"b"%
&
\ho(\mathcal{D}_{/B})
\ar@{<-}[d]^{r_B^\ast}
\\
\ho(\mathcal{C})
\ar@{->}[r]^F
&
\ho(\mathcal{D})
}}
\end{equation}
and then passing to the mate of \eqref{diag:frastcomm}
with respect to the $(r^B_!, r_B^\ast)$ adjunctions to
obtain \eqref{diag:frshriekcommiso}.
The transformation \eqref{diag:frastcomm} is
simply the usual commutation equivalence
$F^{}_B r_B^\ast \simeq r_B^\ast F$,
which is symmetric monoidal.
Moreover, the $(r^B_!, r_B^\ast)$ adjunctions are in fact adjunctions of
oplax symmetric monoidal functors, as it is easily verified
that the monoidality and unit constraints of $r^B_!$
from Remark~\ref{rk:altcomdesc}
agree with the mates of the monoidality and unit constraints
of $r_B^\ast$, as required \cite[\S1.3]{KellyDoctrinalAdj}.
Thus the commutation equivalence \eqref{diag:frshriekcommiso}
is oplax symmetric monoidal, as claimed.
\end{proof}
In view of Remark~\ref{rk:altcomdesc},
from Lemma~\ref{lm:oplaxsm} one easily deduces the following result.
\begin{lemma}
\label{lm:fcompat}
Equip $F r^B_! S_B^\mathcal{C}$
with the comonoid structure given by the composite
\[\xymatrix@C+2em{
F r^B_! S_B^\mathcal{C}
\ar[r]%
&
F (r^B_! S_B^\mathcal{C} \otimes r^B_! S_B^\mathcal{C})
\ar[r]^{F_\otimes^{-1}}_\simeq
&
F r^B_! S_B^\mathcal{C} \otimes F r^B_! S_B^\mathcal{C}
}\]
where the first arrow is induced by the comonoid structure on
$r^B_! S_B^\mathcal{C}$.
Then the composite map
\[\xymatrix@C+2em{
r^B_! S_B^\mathcal{D}
\ar[r]^{r^B_!(F_B)_I}_\simeq
&
r^B_! F_B S_B^\mathcal{C}
\ar[r]^{c}_\simeq
&
F r^B_! S_B^\mathcal{C}
}\]
is an equivalence of comonoid objects in $\ho(\mathcal{D}_{/B})$.
Moreover, for $X \in \ho(\mathcal{C}_{/B})$, the diagram
\[\xymatrix@C+3em{
r^B_! F_B X
\ar[r]^-{\nu_{F_B X}}
\ar[ddd]_{c}^\simeq
&
r^B_! S^{\mathcal{D}}_B \otimes r^B_! F_B X
\ar[d]^{r^B_!(F_B)_I\otimes 1}_\simeq
\\
&
r^B_! F_B S^{\mathcal{C}}_B \otimes r^B_! F_B X
\ar[d]^{c\otimes c}_\simeq
\\
&
F r^B_! S^{\mathcal{C}}_B \otimes F r^B_! X
\ar[d]^{F_\otimes}_\simeq
\\
F r^B_! X
\ar[r]^-{F(\nu_X)}
&
F(r^B_! S^{\mathcal{C}}_B \otimes r^B_! X)
}\]
commutes.
\qed
\end{lemma}
We now turn to the proof of Theorem~\ref{thm:cupmodstrbilin}.
Write $(H{\mathbb {F}}_\ell)_B$ for the identity object in
$\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})$. Upon application of the functor
$\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)$, the
comonoid structure on $r^B_! (H{\mathbb {F}}_\ell)_B$ induces
an algebra structure on $H^\ast (B)$.
In view of Remark~\ref{rk:spacecase},
applying Lemma~\ref{lm:fcompat} with $F$ the functor
\[\xymatrix@C+2em{
\mathbf{Spaces}
\ar[r]^{\Sigma^\infty_+}
&
\mathbf{Spectra}
\ar[r]^{H{\mathbb {F}}_\ell \wedge -}
&
\mathbf{Mod}^{H{\mathbb {F}}_\ell},
}\]
we see that this algebra structure agrees with the usual cup product
on $H^\ast(B)$.
Moreover, upon application of the functor
$\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)$,
the comodule structure of
\[
r^B_! \big(
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)
\big)
\]
over $r^B_! (H{\mathbb {F}}_\ell)_B$
induces on $\mathbb{H}^\ast P(f,g)$ a module structure over $H^\ast(B)$.
In view of Remark~\ref{rk:spacecase},
Lemma~\ref{lm:fcompat} applied with $F$ the functor
$\Sigma^\infty_+\colon\thinspace \mathbf{Spaces} \to \mathbf{Spectra}$,
Lemma~\ref{lm:tcompat} applied with $T = S^{-d}$,
and Lemma~\ref{lm:fcompat} applied with $F$ the functor
$H{\mathbb {F}}_\ell \wedge (-) \colon\thinspace \mathbf{Spectra} \to \mathbf{Mod}^{H{\mathbb {F}}_\ell}$
imply that this module structure agrees with that
of Definition~\ref{def:cupmodstr}.
Theorem~\ref{thm:cupmodstrbilin} now follows by
applying Lemma~\ref{lm:indmapcompat} to the map
\eqref{map:modhfellcopairing}
and using Lemma~\ref{lm:exttensorcompat}.
\section{Pairings on Serre spectral sequences}
\label{sec:spectralsequences}
For a fibration $\pi\colon\thinspace X\to B$, let us write $\mathbb{E}(X)$ for the
strongly convergent spectral sequence
\[
\mathbb{E}^{s,t}_r(X) \Longrightarrow \mathbb{H}^{s+t}(X)
\]
obtained from the Serre spectral sequence $E(X)$
of $\pi$ by setting $\mathbb{E}^{s,t}_r(X) = E^{s,t+d}_r(X)$.
Our aim is to prove the following analogues of
Theorems~\ref{thm:pairings} and \ref{thm:functors}.
\begin{thrm}
\label{thm:sspairings}
Let $B$ be a space. For maps $f,g,h\colon\thinspace B \to BG$,
there is a pairing
\[
\circ
\colon\thinspace
\mathbb{E}^{s_1,t_1}_r (P(g,h)) \otimes \mathbb{E}^{s_2,t_2}_r (P(f,g))
\xto{\quad}
\mathbb{E}^{s_1+s_2,t_1+t_2}_r(P(f,h))
\]
of spectral sequences compatible
with the pairing
\[
\circ
\colon\thinspace
\mathbb{H}^\ast P(g,h) \otimes \mathbb{H}^\ast P(f,g)
\xto{\quad}
\mathbb{H}^\ast P(f,h).
\]
of Theorem~\ref{thm:pairings}
on targets. This pairing is associative, and it is unital in the
following sense: for every map $f\colon\thinspace B \to BG$,
the spectral sequence
\[
\mathbb{E}^{s,t}_r (P(f,f)) \Longrightarrow \mathbb{H}^{s+t}(P(f,f))
\]
has a permanent cycle of degree $(0,0)$ which is a unit
for the pairing on each page
of the spectral sequence
and which corresponds to the unit
$\mathbbold{1}_f \in \mathbb{H}^0(P(f,f))$ on the target.
\end{thrm}
\begin{thrm}
\label{thm:ssfunctors}
Given a map of spaces $\phi\colon\thinspace A \to B$, the maps of spectral
sequences
\begin{equation}
\label{map:ssfuncomponent}
\mathbb{E}(P(f,g)) \xto{\quad} \mathbb{E}(P(f\phi,g\phi))
\end{equation}
induced by \eqref{sq:inducedmap} preserve the
pairings and unit elements of Theorem~\ref{thm:sspairings}.
Moreover, these maps are compatible with the maps
\[
F_\phi \colon\thinspace \mathbb{H}^\ast (P(f,g)) \xto{\quad} \mathbb{H}^\ast (P(f\phi,g\phi))
\]
of \eqref{eq:fphidef} on targets.
\end{thrm}
Our strategy for proving Theorems~\ref{thm:sspairings}
and \ref{thm:ssfunctors} is similar
to the one we employed to prove Theorems~\ref{thm:pairings}
and \ref{thm:functors}.
Starting with the category $(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$ enriched in
$(\mathbf{hpSpectra}^\ell)^\mathrm{op}$ and the enriched functor
$(D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)$
constructed in
Remark~\ref{rk:dfwcalpb}, we will apply
Construction~\ref{constr:newenrichedcats}
with a lax monoidal functor
\begin{equation}
\label{eq:sscomposite}
(\mathbf{hpSpectra}^\ell)^\mathrm{op}
\xrightarrow{\ U_\mathrm{fw}^\mathrm{op}\ }
\mathbf{hpSpectra}^\mathrm{op}
\xrightarrow{\ E\ }
\mathbf{SS}
\end{equation}
to obtain a category $(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
enriched in the category $\mathbf{SS}$ of spectral sequences
along with $\mathbf{SS}$--enriched functors
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)$.
The proof is then completed by showing that the hom-objects
$E(U_\mathrm{fw} D_\mathrm{fw} P(f,g))$ in
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
are isomorphic to the shifted Serre spectral sequences
$\mathbb{E}(P(f,g))$,
and observing that under these isomorphisms, the functor
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)$ is given by the map
\eqref{map:ssfuncomponent}.
We will start by making explicit the category $\mathbf{SS}$,
after which we will turn to the construction of the
functor $E$.
\begin{defn}
A \emph{spectral sequence} $E$ consists of the following data:
a sequence $E^{\ast,\ast}_1,E^{\ast,\ast}_2,\ldots$
of bigraded ${\mathbb {F}}_\ell$--vector spaces;
a differential $d_r$ of bidegree $(r,1-r)$
on each $E^{\ast,\ast}_r$; and an isomorphism
\[
\phi_r\colon\thinspace H(E^{\ast,\ast}_r) \xrightarrow{\ \cong\ } E^{\ast,\ast}_{r+1}
\]
for each $r$. A \emph{morphism} $f\colon\thinspace E \to D$
of spectral sequences consists of a sequence
\[
f_r \colon\thinspace E^{\ast,\ast}_r \xto{\quad} D^{\ast,\ast}_r,
\quad
r= 1,2,\ldots
\]
of morphisms commuting with the differentials and having the
property that $f_{r+1}$ corresponds to $H(f_r)$ under the
isomorphisms $\phi_r$. The \emph{tensor product}
of spectral sequences $E$ and $D$ is the spectral sequence
$E \otimes D$ with
\[
(E \otimes D)^{s,t}_r
=
\bigoplus_{\substack{s_1+s_2 = s\\t_1+t_2=t}}
E^{s_1,t_1}_r \otimes D^{s_2,t_2}_r,
\]
differential
\[
d_r (x\otimes y) = d_r(x) \otimes y + (-1)^{|x|} x \otimes d_r(y)
\]
where $|x| = s+t$ for $x\in E^{s,t}_r$, and isomorphisms $\phi_r$
given by the Künneth theorem.
There results a symmetric monoidal
category of spectral sequences which we denote by $\mathbf{SS}$.
The symmetry constraint in $\mathbf{SS}$ is given by
\[
E^{s_1,t_1}_r \otimes D^{s_2,t_2}_r
\xto{\quad}
D^{s_2,t_2}_r \otimes E^{s_1,t_1}_r,
\qquad
x\otimes y \longmapsto (-1)^{(s_1+t_1)(s_2+t_2)} y\otimes x,
\]
while the monoidal unit is given by the
spectral sequence which on each page is a single copy of ${\mathbb {F}}_\ell$
concentrated in degree $(0,0)$.
\end{defn}
The functor $E\colon\thinspace \mathbf{hpSpectra} \to \mathbf{SS}$
is given by May and Sigurdsson's generalization of the
Serre spectral sequence to parametrized spectra
\cite[Thm.~20.4.1]{MaySigurdsson}, which we recall
in the relevant special case in the following theorem.
The case $X = \Sigma_{+B} Y$
recovers the ordinary Serre spectral sequence of a fibration
$Y \to B$.
\begin{thrm}
\label{thm:serress}
Let $X$ be a parametrized spectrum over a space $B$.
Then there is a spectral sequence
\begin{equation}
\label{ss:serre}
E_2^{s,t} = H^s(B;\, \mathcal{L}^t(X,H{\mathbb {F}}_\ell)) \Longrightarrow H^{s+t}(r_! X;\,{\mathbb {F}}_\ell)
\end{equation}
where $\mathcal{L}^\ast(X,H{\mathbb {F}}_\ell)$ is the local coefficient system
\cite[Def.~20.3.4]{MaySigurdsson}
with fibre over $b\in B$ given by $H^\ast(X_b;\,{\mathbb {F}}_\ell)$.
In the case where $X$ is bounded from below
in the sense that there exists a $t_0 \in {\mathbb {Z}}$ such that
$H^t(X_b;\,{\mathbb {F}}_l) = 0$ for all $t < t_0$ and $b \in B$,
the spectral sequence converges strongly to the indicated target.
\end{thrm}
\begin{proof}
Take $J= r^\ast H{\mathbb {F}}_\ell$ in \cite[Thm.~20.4.1(ii)]{MaySigurdsson}.
To identify the target, use the equivalence
$F(r_!X,H{\mathbb {F}}_\ell) \simeq r_\ast F_B(X,r^\ast H{\mathbb {F}}_\ell)$.
\end{proof}
\begin{rem}
\label{rk:ssconstruction}
Reinterpreting the construction
in \cite[Thm.~20.4.1(ii)]{MaySigurdsson},
we see that the spectral sequence \eqref{ss:serre}
can be constructed as follows: Let $\Gamma B$
be a functorial CW approximation to $B$
(such as $|\mathrm{Sing}_\bullet B|$),
and write $\Gamma_n B$ for the $n$--skeleton of $\Gamma B$.
Let $X_n$ be the restriction to $\Gamma_n B$
of the pullback of $X$ over $\Gamma B$.
Then the spectral sequence \eqref{ss:serre} is
the spectral sequence arising from the sequence
\begin{equation}
\label{eq:sequence}
r^{\Gamma_0 B}_! X_0 \xto{\quad} r^{\Gamma_1 B}_! X_1 \xto{\quad} \cdots
\end{equation}
of spectra by applying ${\mathbb {F}}_\ell$--cohomology to obtain the unrolled
exact couple
\[\xymatrix@!0@C=5.6em@R=10ex{
H^\ast(r^{\Gamma_0 B}_! X_0)
\ar@{<-}[rr]
&&
H^\ast(r^{\Gamma_1 B}_! X_1)
\ar@{<-}[rr]
&&
H^\ast(r^{\Gamma_2 B}_! X_2)
\ar@{<-}[r]
&
\cdots
\\
&
H^\ast(r^{\Gamma_1 B}_! X_1, r^{\Gamma_0 B}_! X_0)
\ar[ur]
\ar@{<-}[ul]
&&
H^\ast(r^{\Gamma_2 B}_! X_2, r^{\Gamma_1 B}_! X_1)
\ar[ur]
\ar@{<-}[ul]
&
\quad\cdots
}\]
\end{rem}
It is easily verified that the spectral sequence
of Theorem~\ref{thm:serress} is functorial.
\begin{prop}
\label{prop:ssinducedmap}
Given a map $(f,\phi) \colon\thinspace (B,X) \to (C,Y)$ in $\mathbf{hpSpectra}$,
there is an induced map
\[
E_r^{s,t} (Y) \xto{\quad} E_r^{s,t} (X)
\]
between the spectral sequences of Theorem~\ref{thm:serress}
compatible with the map
\[
H^\ast(r^C_! Y;\,{\mathbb {F}}_\ell) \xto{\quad} H^\ast(r^B_!X;\,{\mathbb {F}}_\ell)
\]
induced by $(f,\phi)$ on the targets.
\end{prop}
\begin{proof}
Using tbe cellularity of the map $\Gamma f\colon\thinspace \Gamma B \to \Gamma C$,
one sees that
the map $(f,\phi)$ induces a map from the sequence \eqref{eq:sequence}
to the corresponding sequence for $Y$.
\end{proof}
An argument analogous to
\cite[\S XIII.8]{Whitehead}
gives
\begin{thrm}
\label{thm:sspairing}
Given parametrized spectra $X$ over $B$ and $Y$ over $C$,
there is an associative pairing of spectral sequences
\[
E^{s,t}_r (X) \otimes E^{s't'}_r (Y)
\xto{\quad}
E^{s+s',t+t'}_r (X\mathbin{\bar{\wedge}} Y)
\]
which on the $E_2$--page is given by the
cross product
\[
H^s(B;\, \mathcal{L}^t(X,H{\mathbb {F}}_\ell))
\otimes
H^{s'}(C;\, \mathcal{L}^{t'}(Y,H{\mathbb {F}}_\ell))
\xrightarrow{\ \times\ }
H^{s+s'}(B\times C ;\, \mathcal{L}^{t+t'}(X\mathbin{\bar{\wedge}} Y,H{\mathbb {F}}_\ell))
\]
where the pairing on local coefficient systems is
given by the products
\[
H^t(X_b;\,{\mathbb {F}}_\ell) \otimes H^{t'}(Y_c;\,{\mathbb {F}}_\ell)
\xrightarrow{\ \times\ }
H^{t+t'}(X_b \wedge Y_c;\,{\mathbb {F}}_\ell)
\]
for $b\in B$ and $c\in C$.
If $X$ and $Y$ (and hence $X\mathbin{\bar{\wedge}} Y$) are bounded
from below in the sense of Theorem~\ref{thm:serress},
ensuring strong convergence,
the pairing on the $E_\infty$--page is the one induced by
the cross product
\[
\pushQED{\qed}
H^\ast(r^B_! X;\,{\mathbb {F}}_\ell) \otimes H^\ast(r^C_! Y;\,{\mathbb {F}}_\ell)
\xrightarrow{\ \times\ }
H^\ast(r^B_! X \wedge r^C_! Y;\,{\mathbb {F}}_\ell).
\qedhere
\popQED
\]
\end{thrm}
Summarizing and elaborating Theorem~\ref{thm:serress},
Proposition~\ref{prop:ssinducedmap}
and Theorem~\ref{thm:sspairing}, we obtain the desired
lax symmetric monoidal functor
\begin{equation}
\label{fun:ssfunctor}
E \colon\thinspace \mathbf{hpSpectra}^\mathrm{op} \xto{\quad} \mathbf{SS}.
\end{equation}
Our next goal is to show that the composite functor
\eqref{eq:sscomposite} is lax symmetric monoidal; as before,
this is not quite obvious, since the functor $U_\mathrm{fw}^\mathrm{op}$
is oplax symmetric monoidal. We need the following lemma.
\begin{lemma}
\label{lm:efactorization}
There exists a functor
$\tilde{E}\colon\thinspace (\mathbf{hpMod}^{H{\mathbb {F}}_\ell})^\mathrm{op} \to \mathbf{SS}$
such that the functor \eqref{fun:ssfunctor}
factors as the composite
\[\xymatrix@C+3em{
\mathbf{hpSpectra}^\mathrm{op}
\ar[r]^{(H{\mathbb {F}}_\ell\wedge -)_\mathrm{fw}^\mathrm{op}}
&
(\mathbf{hpMod}^{H{\mathbb {F}}_\ell})^\mathrm{op}
\ar[r]^-{\tilde{E}}
&
\mathbf{SS}.
}\]
\end{lemma}
\begin{proof}
Working as in Remark~\ref{rk:ssconstruction},
given a parametrized $H{\mathbb {F}}_\ell$--module $Y$ over $B$,
we can associate to it a sequence
\begin{equation}
\label{eq:sequence2}
r^{\Gamma_0 B}_! Y_0 \xto{\quad} r^{\Gamma_1 B}_! Y_1 \xto{\quad} \cdots
\end{equation}
of $H{\mathbb {F}}_\ell$--modules.
By applying the functor
$\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)$,
we obtain an exact couple giving rise to the spectral sequence
$\tilde{E}(Y)$.
As in the proof of Lemma~\ref{lm:newfactorization},
notice that for each space $C$, the diagram of functors
\[\xymatrix@C+2em{
\ho(\mathbf{Spectra}_{/C})
\ar[r]^{(H{\mathbb {F}}_\ell\wedge -)_C}
\ar[d]_{r_!}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/C})
\ar[d]^{r_!}
\\
\ho(\mathbf{Spectra})
\ar[r]^{H{\mathbb {F}}_\ell\wedge -}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
}\]
commutes up to natural equivalence since the corresponding
diagram of right adjoints
\[\xymatrix@C+2em{
\ho(\mathbf{Spectra}_{/C})
\ar@{<-}[r]^{\mathrm{forget}_C}
\ar@{<-}[d]_{r^\ast}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/C})
\ar@{<-}[d]^{r^\ast}
\\
\ho(\mathbf{Spectra})
\ar@{<-}[r]^{\mathrm{forget}}
&
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
}\]
does. It follows that
in the case $Y = H{\mathbb {F}}_\ell \wedge_B X$ for a parametrized
spectrum $X$ over $B$, the sequence \eqref{eq:sequence2}
is naturally equivalent to the sequence
\[
H{\mathbb {F}}_\ell \wedge r^{\Gamma_0 B}_! X_0
\xto{\quad}
H{\mathbb {F}}_\ell \wedge r^{\Gamma_1 B}_! X_1
\xto{\quad}
\cdots
\]
obtained by applying $H{\mathbb {F}}_\ell \wedge (-)$ to
\eqref{eq:sequence}.
The claim now follows from the natural isomorphism
\[
\pi_{-\ast}F^{H{\mathbb {F}}_\ell}(H{\mathbb {F}}_\ell \wedge -,H{\mathbb {F}}_\ell)
\cong
H^\ast
\]
of functors $\ho(\mathbf{Spectra}) \to \mathbf{grMod}$.
\end{proof}
\begin{cor}
The composite of the functors
\[
(\mathbf{hpSpectra}^\ell)^\mathrm{op}
\xrightarrow{\ U_\mathrm{fw}^\mathrm{op}\ }
\mathbf{hpSpectra}^\mathrm{op}
\xrightarrow{\ E\ }
\mathbf{SS}
\]
is lax symmetric monoidal.
\end{cor}
\begin{proof}
As in the proof of Proposition~\ref{prop:hrulaxmon},
it suffices to show that the functor $E$ takes
the monoidality and identity constraints of $U_\mathrm{fw}^\mathrm{op}$
to isomorphisms. That $E$ has this property follows from
Lemmas~\ref{lm:hflsmashfw} and \ref{lm:efactorization}.
\end{proof}
We can now apply Construction~\ref{constr:newenrichedcats} to
the enriched categories
$(D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
and enriched functors $(D_\mathrm{fw}^\mathrm{op})_\ast(K_\phi)$
of Remark~\ref{rk:dfwcalpb}
to obtain categories $(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
enriched in spectral sequences, along with enriched functors
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast(K_\phi)$ between them.
Explicitly, the hom-object from
$f\colon\thinspace B \to BG$ to $g\colon\thinspace B \to BG$
in $(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$ is the spectral
sequence
\[
E^{s,t}_r (U_\mathrm{fw} D_\mathrm{fw} P(f,g))
\Longrightarrow
H^{s+t}(r_! U_\mathrm{fw} D_\mathrm{fw} P(f,g))
\]
whose target is the hom-object from $f$ to $g$
in the category
$(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast\mathcal{P}_B$
constructed in
Subsections~\ref{subsec:strategy}---\ref{subsec:remainingfunctors}.
We note that the composition law in
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$
is compatible with that in
$(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast\mathcal{P}_B$,
and that the functors
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)$
and
$(H^\ast r_!^\mathrm{op} U_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast (K_\phi)$
in the two settings are also compatible.
Theorems~\ref{thm:sspairings} and \ref{thm:ssfunctors}
now follow
from the following result
identifying the hom-objects in
$(EU_\mathrm{fw}^\mathrm{op} D_\mathrm{fw}^\mathrm{op})_\ast \mathcal{P}_B$.
\begin{thrm} For all $f,g\colon\thinspace B \to BG$, there is an
isomorphism
\[
E(U_\mathrm{fw} D_\mathrm{fw} P(f,g)) \cong \mathbb{E}(P(f,g))
\]
of spectral sequences natural
with respect to the homomorphisms induced by diagram
\eqref{sq:inducedmap}.
\end{thrm}
\begin{proof}
In view of Proposition~\ref{prop:hfleq}, this follows from
Lemma~\ref{lm:efactorization}.
\end{proof}
Tracing through definitions, we obtain the following
description of the pairing between spectral sequences on the $E_2$
page.
\begin{prop}
\label{prop:e2pageprod}
Let $f,g,h\colon\thinspace B \to BG$ be maps, and pick a basepoint $b\in B$.
On the $E_2$ page, the pairing
\[
\circ
\colon\thinspace
\mathbb{E}^{s_1,t_1}_r (P(g,h)) \otimes \mathbb{E}^{s_2,t_2}_r (P(f,g))
\xto{\quad}
\mathbb{E}^{s_1+s_2,t_1+t_2}_r(P(f,h))
\]
of spectral sequences is given by the map
\[
H^\ast(B) \otimes \mathbb{H}^\ast(P(g,h)_b)
\otimes
H^\ast(B) \otimes \mathbb{H}^\ast(P(f,g)_b)
\xto{\quad}
H^\ast(B) \otimes \mathbb{H}^\ast(P(f,h)_b)
\]
induced by the cup product on $B$ and the pairing
\[
\circ
\colon\thinspace
\mathbb{H}^\ast(P(g,h)_b)
\otimes
\mathbb{H}^\ast(P(f,g)_b)
\xto{\quad}
\mathbb{H}^\ast(P(f,h)_b)
\]
arising from the identifications
$P(g,h)_b = P(gi,hi)$,
$P(f,g)_b = P(fi,gi)$, and
$P(f,h)_b = P(fi,hi)$
for $i$ is the inclusion of $b$ into $B$. \qed
\end{prop}
\section{Proofs of Theorems~\ref{thm:mainresult}--\ref{thrm:cup-products}}
\label{sec:tezuka}
In this section we put together the results of
Sections~\ref{sec:products} and \ref{sec:spectralsequences} to
to prove
Theorems~\ref{thm:mainresult}--\ref{thrm:cup-products}
stated in the introduction. We start with
Theorem~\ref{thm:mainresult}:
\begin{proof}[Proof of Theorem~\ref{thm:mainresult}]
Part~(\ref{it:mainresult-modstr}) follows from
Theorem~\ref{thm:pairings}; see
Definition~\ref{def:stringprodandmod}.
The spectral sequences of parts~(\ref{it:mainresult-ringss})
and (\ref{it:mainresult-modss})
were constructed in Theorem~\ref{thm:sspairings};
the module structure
on the Serre spectral without a degree shift
in part~(\ref{it:mainresult-modss})
is obtained from the module structure
on the degree-shifted Serre spectral sequence simply
by regrading as in Definition~\ref{def:stringprodandmod}.
In part (\ref{it:mainresult-ringss}), the assertion about
product on $\mathbb{H}^\ast(G)$ follows from Theorem~\ref{thm:pontryaginproduct}.
In part~(\ref{it:mainresult-modss}), the assertion about
the $E_2$--page follows from Proposition~\ref{prop:e2pageprod}
and Corollary~\ref{cor:homotopyinvariance} (which allows one to
compare the $\mathbb{H}^\ast(G)$--module structure
on the cohomology of the fibre of
$BG^{h\sigma} \to BG$ to that on $\mathbb{H}^\ast(G)$ itself).
\end{proof}
The following result is an elaboration of
Theorem~\ref{thm:strtoptezukacrit}.
\begin{thrm}
\label{thm:conj-red3}
Let $B$ be a path connected space,
let $f,g\colon\thinspace B \to BG$ be maps,
let $F\simeq \Omega BG$ be a fibre of the fibration $P(f,g) \to B$,
and let $i\colon\thinspace F\hookrightarrow P(f,g)$ be the inclusion.
Then the following are equivalent conditions on an element
$x \in \mathbb{H}^0 P(f,g)$:
\newcounter{savedvalue}
\begin{enumerate}
\item \label{it:freerank1}
$\mathbb{H}^\ast P(f,g)$ is free of rank 1 with basis $\{x\}$
as a graded left module over $\mathbb{H}^\ast P(g,g)$.
\item \label{it:freerank1right}
$\mathbb{H}^\ast P(f,g)$ is free of rank 1 with basis $\{x\}$
as a graded right module over $\mathbb{H}^\ast P(f,f)$.
\item \label{it:cyclic}
$\mathbb{H}^\ast P(f,g)$ is generated by $x$
as a graded left module over $\mathbb{H}^\ast P(g,g)$.
\item \label{it:cyclicright}
$\mathbb{H}^\ast P(f,g)$ is generated by $x$
as a graded right module over $\mathbb{H}^\ast P(f,f)$.
\item \label{it:nonzerores}
$i^\ast(x) \neq 0 \in H^d F$.
\item \label{it:ssiso2}
The map
\[
\mathbb{E}(P(g,g)) \xto{\quad} \mathbb{E}(P(f,g)),
\quad
z \longmapsto z \circ (1\otimes i^\ast(x))
\]
is an isomorphism from the Serre spectral
sequence of $P(g,g)\to B$ to that of $P(f,g)\to B$.
\item \label{it:ssiso2right}
The map
\[
\mathbb{E}(P(f,f)) \xto{\quad} \mathbb{E}(P(f,g)),
\quad
z \longmapsto (1\otimes i^\ast(x))\circ z
\]
is an isomorphism from the Serre spectral
sequence of $P(f,f)\to B$ to that of $P(f,g)\to B$.\setcounter{savedvalue}{\value{enumi}}
\end{enumerate}
Moreover, the following conditions are equivalent:
\begin{enumerate}
\setcounter{enumi}{\value{savedvalue}}
\item \label{it:xexists}
There exists an element $x \in \mathbb{H}^0 P(f,g)$
satisfying conditions (\ref{it:freerank1})--(\ref{it:ssiso2right}).
\item \label{it:fclass}
The map $i_\ast\colon\thinspace H_d F \to H_d P(f,g)$ is nontrivial.
\item \label{it:ssiso}
The Serre spectral sequences of $P(f,g)\to B$ and $P(g,g)\to B$
are isomorphic.
\item \label{it:ssisoright}
The Serre spectral sequences of $P(f,g)\to B$ and $P(f,f)\to B$
are isomorphic.
\item \label{it:paramiso}
The parametrized $H{\mathbb {F}}_\ell$--modules
$H{\mathbb {F}}_\ell\wedge_B \Sigma^\infty_{+B}P(f,g)$
and
$H{\mathbb {F}}_\ell\wedge_B \Sigma^\infty_{+B}P(g,g)$
are equivalent objects of $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})$.
\item \label{it:paramisoright}
The parametrized $H{\mathbb {F}}_\ell$--modules
$H{\mathbb {F}}_\ell\wedge_B \Sigma^\infty_{+B}P(f,g)$
and
$H{\mathbb {F}}_\ell\wedge_B \Sigma^\infty_{+B}P(f,f)$
are equivalent objects of $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell}_{/B})$.
\item \label{it:permanentcycle2}
The generator of
$E_2^{0,d}(P(f,g))= H^0 B \otimes H^d F \cong {\mathbb {F}}_\ell$
is a permanent cycle in the Serre spectral sequence of
$P(f,g)\to B$.
\end{enumerate}
\end{thrm}
\begin{proof}
The implications
(\ref{it:freerank1})\,$\Rightarrow$\,(\ref{it:cyclic})
and
(\ref{it:freerank1right})\,$\Rightarrow$\,(\ref{it:cyclicright})
are obvious. To show
(\ref{it:cyclic})\,$\Rightarrow$\,(\ref{it:nonzerores}),
let $j\colon\thinspace \Omega BG \hookrightarrow P(g,g)$ be the inclusion of a fibre.
By assumption, there exists some element $y\in \mathbb{H}^{-d} P(g,g)$
such that $y\circ x = 1\in \mathbb{H}^{-d}P(f,g)$.
We now have
\[
j^\ast(y) \circ i^\ast(x)
=
i^\ast(y\circ x)
=
i^\ast(1)
=
1 \in \mathbb{H}^{-d}(F),
\]
showing that $i^\ast(x) \neq 0\in \mathbb{H}^0 F$.
Here the first equality follows from
Theorem~\ref{thm:functors}.
The implication
(\ref{it:cyclicright})\,$\Rightarrow$\,(\ref{it:nonzerores})
follows similarly.
Let us now show that
(\ref{it:nonzerores})\,$\Rightarrow$\,(\ref{it:ssiso2}).
By naturality of the Serre spectral sequence, the element
$1\otimes i^\ast(x) \in \mathbb{E}_2^{0,0}(P(f,g)) = H^0(B)\otimes \mathbb{H}^0(F)$
is the image of $x$ under the composite
\[
\mathbb{H}^0 P(f,g)
\xto{\quad}
\mathbb{E}_\infty^{0,0}(P(f,g))
\xto{\quad}
\mathbb{E}_2^{0,0}(P(f,g))
\]
of the quotient and inclusion maps.
Thus $1\otimes i^\ast(x)$ is a
permanent cycle, and multiplication by it does define a
morphism of spectral sequences
\[
m_{1\otimes i^\ast(x)} \colon\thinspace \mathbb{E}(P(g,g)) \xto{\quad} \mathbb{E}(P(f,g)),
\quad
z\mapsto z\circ (1\otimes i^\ast(x)).
\]
By Proposition~\ref{prop:e2pageprod}, on the $E_2$ page,
this map is given by the map
\[
\mathrm{id} \otimes m_{i^\ast(x)}
\colon\thinspace
H^\ast(B) \otimes \mathbb{H}^\ast(\Omega BG)
\xto{\quad}
H^\ast(B) \otimes \mathbb{H}^\ast(F)
\]
where $m_{i^\ast(x)}$ is the multiplication map
$z \mapsto z \circ i^\ast(x)$
(and we interpret $i^\ast(x)$ as an element of $\mathbb{H}^0(F)$).
By Corollary~\ref{cor:homotopyinvariance},
there is an isomorphism $\mathbb{H}^\ast(F) \cong \mathbb{H}^\ast(\Omega BG)$
under which the map $m_{i^\ast(x)}$ corresponds to the map
\[
m_{y}
\colon\thinspace
\mathbb{H}^\ast(\Omega BG)\xto{\quad} \mathbb{H}^\ast(\Omega BG),
\quad
z \mapsto z \circ y
\]
for some $y \in \mathbb{H}^0(\Omega BG)$. Since $i^\ast(x)$ is nonzero,
so is $y$. Thus $y$ is a nonzero multiple of the unit
element for the product on $\mathbb{H}^\ast(\Omega BG)$.
It follows that the map $m_{y}$ and hence the map
$m_{i^\ast(x)}$ are isomorphisms. Thus the
map $m_{1\otimes i^\ast(x)}$ is an isomorphism on the $E_2$ pages,
and hence on all further pages as well, giving an isomorphism
of spectral sequences.
Again, the implication
(\ref{it:nonzerores})\,$\Rightarrow$\,(\ref{it:ssiso2right})
follows similarly.
We now prove the implication
(\ref{it:ssiso2})\,$\Rightarrow$\,(\ref{it:freerank1}).
Since $x \in \mathbb{H}^0 P(f,g)$ is a lift of the element
$1\otimes i^\ast(x) \in \mathbb{E}_\infty^{0,0}(P(f,g))$,
the multiplication map
\[
m_{x}
\colon\thinspace
\mathbb{H}^\ast P(g,g) \xto{\quad} \mathbb{H}^\ast P(f,g),
\quad
z \mapsto z\circ x
\]
induces on the associated graded modules
corresponding to the Serre spectral sequences
of $P(g,g)$ and $P(f,g)$ the isomorphism
\[
m_{1\otimes i^\ast(x)}
\colon\thinspace
\mathbb{E}^{\ast,\ast}_\infty (P(g,g))
\xrightarrow{\ \cong\ }
\mathbb{E}^{\ast,\ast}_\infty (P(f,g)).
\]
Therefore the map $m_{x}$ itself
must be an isomorphism. Thus $\mathbb{H}^\ast P(f,g)$ is free of
rank 1 over $\mathbb{H}^\ast P(g,g)$ with basis $\{x\}$.
The implication
(\ref{it:ssiso2right})\,$\Rightarrow$\,(\ref{it:freerank1right})
follows similarly.
In view of condition (\ref{it:nonzerores}), condition (\ref{it:xexists})
is equivalent to the map $i^\ast \colon\thinspace H^d P(f,g) \to H^d F$
being nontrivial, which in turn is equivalent to condition
(\ref{it:fclass}). Thus
(\ref{it:xexists})\,$\Leftrightarrow$\,(\ref{it:fclass}).
The implications
(\ref{it:xexists})\,$\Rightarrow$\,(\ref{it:ssiso})
and
(\ref{it:xexists})\,$\Rightarrow$\,(\ref{it:ssisoright})
follow from conditions (\ref{it:ssiso2}) and (\ref{it:ssiso2right}).
To show
(\ref{it:ssiso})\,$\Rightarrow$\,(\ref{it:permanentcycle2}),
it suffices to show that the generator of
$E_2^{0,d}(P(g,g)) = H^0(B) \otimes H^d (\Omega BG) \cong {\mathbb {F}}_\ell$
is a permanent cycle.
Let $j\colon\thinspace \Omega BG \hookrightarrow P(g,g)$ be the inclusion of a fibre.
As before, the element
$1\otimes j^\ast (\mathbbold{1}) \in \mathbb{E}_2^{0,0}(P(g,g))
= H^0(B) \otimes \mathbb{H}^0 (\Omega BG)$
is a permanent cycle.
Since
$j^\ast(\mathbbold{1}) = \mathbbold{1} \neq 0 \in \mathbb{H}^0 (\Omega BG)$
by Theorem~\ref{thm:functors},
it is nonzero, and hence generates
$\mathbb{E}_2^{0,0}(P(g,g)) \cong {\mathbb {F}}_\ell$.
Thus the claim follows.
Again, the implication
(\ref{it:ssisoright})\,$\Rightarrow$\,(\ref{it:permanentcycle2})
follows similarly.
Finally, to show that
(\ref{it:permanentcycle2})\,$\Rightarrow$\,(\ref{it:xexists}),
it suffices to observe that by naturality of the Serre spectral sequence,
a lift of the nontrivial permanent cycle from $E_\infty^{0,d}(P(f,g))$ to
an element of $H^d P(f,g)$ satisfies condition (\ref{it:nonzerores}).
In view of Lemma~\ref{lm:efactorization}, it is clear that
(\ref{it:paramiso})\,$\Rightarrow$\,(\ref{it:ssiso})
and
(\ref{it:paramisoright})\,$\Rightarrow$\,(\ref{it:ssisoright}).
Let us show that
(\ref{it:xexists})\,$\Rightarrow$\,(\ref{it:paramiso}).
Suppose $x\in \mathbb{H}^0 P(f,g)$ satisfies condition~(\ref{it:freerank1}).
As observed at the beginning of Subsection~\ref{subsec:pfofcupmodstrbilin},
the product
\[
\circ
\colon\thinspace
\mathbb{H}^\ast P(g,g) \otimes \mathbb{H}^\ast P(f,g)
\xto{\quad}
\mathbb{H}^\ast P(f,g)
\]
arises by applying the composite functor
\begin{equation}
\label{eq:hpmodtogrmod}
\mathbf{hpMod}^{H{\mathbb {F}}_\ell}
\xrightarrow{\ r_!\ }
\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})
\xrightarrow{\ {\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)}\ }
\mathbf{grMod}^{{\mathbb {F}}_\ell}
\end{equation}
to a certain map
\begin{equation}
\label{map:modhfellcopairing2}
H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)
\xto{\quad}
\big(H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(g,g)\big)
\mathbin{\bar{\wedge}}^{H{\mathbb {F}}_\ell}
\big(H{\mathbb {F}}_\ell \wedge_B \Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)\big)
\end{equation}
in $\mathbf{hpMod}^{H{\mathbb {F}}_\ell}$ covering the diagonal map
$\Delta\colon\thinspace B \to B\times B$ in $\mathcal{T}$.
The element $x \in \mathbb{H}^0 P(f,g)$
is represented by a map
\[
r^B_! (
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)
)
\xto{\quad}
H{\mathbb {F}}_\ell
\]
in $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})$
which by adjunction amounts to a map
\[
\tilde{x}
\colon\thinspace
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)
\xto{\quad}
H{\mathbb {F}}_\ell
\]
in $\mathbf{hpMod}^{H{\mathbb {F}}_\ell}$ covering the map $r^B \colon\thinspace B \to \mathrm{pt}$
in $\mathcal{T}$. Composing \eqref{map:modhfellcopairing2} with the
map $1\mathbin{\bar{\wedge}}^{H{\mathbb {F}}_\ell}\tilde{x}$ yields a map
\begin{equation}
\label{map:circxpre}
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma^{\infty}_{+B} P(f,g)
\xto{\quad}
H{\mathbb {F}}_\ell
\wedge_B
\Sigma_B^{-d} \Sigma^{\infty}_{+B} P(g,g)
\end{equation}
in $\mathbf{hpMod}^{H{\mathbb {F}}_\ell}$ covering the identity map of $B$;
upon application of the composite \eqref{eq:hpmodtogrmod},
this map recovers the multiplication-by-$x$ map
$\mathbb{H}^\ast P(g,g) \to \mathbb{H}^\ast P(f,g)$.
The restriction of \eqref{map:circxpre} to
fibres over a point $b\in B$ is a map
\begin{equation}
\label{map:circxbpre}
H{\mathbb {F}}_\ell \wedge \Sigma^{-d} \Sigma^\infty P(f,g)_b
\xto{\quad}
H{\mathbb {F}}_\ell \wedge \Sigma^{-d} \Sigma^\infty P(g,g)_b
\end{equation}
in $\ho(\mathbf{Mod}^{H{\mathbb {F}}_\ell})$
which, upon application of $\pi_{-\ast} F^{H{\mathbb {F}}_\ell}(-,H{\mathbb {F}}_\ell)$,
recovers the map $\mathbb{H}^\ast P(g,g)_b \to \mathbb{H}^\ast P(f,g)_b$
given by multiplication by the restriction $x_b$ of $x$ to
$\mathbb{H}^\ast P(f,g)_b$. In view of the equivalence
of conditions (\ref{it:freerank1}) and (\ref{it:nonzerores}),
the assumption on $x$ ensures that this multiplication-by-$x_b$ map
is an isomorphism for each $b\in B$.
It follows that the map \eqref{map:circxbpre}
is an equivalence for all $b$,
whence the map \eqref{map:circxpre} is an equivalence,
and the claim follows by applying $\Sigma_B^d$ to
\eqref{map:circxpre}.
The implication
(\ref{it:xexists})\,$\Rightarrow$\,(\ref{it:paramisoright})
follows similarly.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:strtoptezukacrit}]
The claim follows from the equivalence of
conditions (\ref{it:xexists}) and (\ref{it:fclass})
and the equivalence of conditions (\ref{it:freerank1})
and (\ref{it:nonzerores}) in Theorem~\ref{thm:conj-red3}
by taking $g=\mathrm{id}_{BG}$ and $f=\sigma$.
\end{proof}
We now turn to the proof of Theorem~\ref{thrm:cup-products}.
We will deduce part~(\ref{it:gralgiso}) of the theorem
from the following result.
\begin{thrm}
\label{thm:sscupstr}
Let $B$ be a path connected space,
let $f,g\colon\thinspace B \to BG$ be maps,
let $F\simeq \Omega BG$ be a fibre of the fibration $P(f,g) \to B$,
and let $i\colon\thinspace F\hookrightarrow P(f,g)$ be the inclusion.
Suppose $x\in \mathbb{H}^0 P(f,g)$ is a generator of
$\mathbb{H}^\ast P(f,g)$ as a free rank 1 module over
$\mathbb{H}^\ast P(g,g)$ satisfying
$1\circ i^\ast(x) = 1 \in H^0 F$,
where $\circ$ refers to the pairing
\[
\circ
\colon\thinspace
\mathbb{H}^\ast(\Omega BG) \otimes \mathbb{H}^\ast(F)
\xto{\quad}
\mathbb{H}^\ast(F).
\]
Then the isomorphism
\begin{equation}
\label{map:m1x}
E(P(g,g)) \xrightarrow{\ \cong\ } E(P(f,g)),
\quad
z \longmapsto z \circ (1\otimes i^\ast(x))
\end{equation}
of Theorem~\ref{thm:conj-red3}(\ref{it:ssiso2})
from the Serre spectral sequence of $P(g,g)\to B$
to that of $P(f,g) \to B$
is an isomorphism of spectral sequences of algebras,
where the spectral sequences are equipped with the
usual algebra structures induced by cup product.
In particular, the map
\[
H^\ast P(g,g) \xto{\quad} H^\ast P(f,g),
\quad
z\longmapsto z\circ x
\]
induces an algebra isomorphism
\[
\gr H^\ast P(g,g) \xrightarrow{\ \cong\ } \gr H^\ast P(f,g)
\]
on the associated graded algebras of
$H^\ast P(g,g)$ and $H^\ast P(f,g)$
corresponding to the Serre spectral sequences.
\end{thrm}
\begin{proof}
To show that the map \eqref{map:m1x} respects the algebra
structures, it is enough to show that it does so on the
$E_2$ pages. By Proposition~\ref{prop:e2pageprod},
on $E_2$ pages the map \eqref{map:m1x} is given by
\[
H^\ast(B) \otimes H^\ast(\Omega BG)
\xto{\quad}
H^\ast(B) \otimes H^\ast(F),
\quad
\alpha\otimes \beta
\longmapsto
\alpha\otimes(\beta\circ i^\ast(x)),
\]
so it is enough to show that the map
\begin{equation}
\label{map:miastx}
H^\ast(\Omega BG)
\xto{\quad}
H^\ast (F),
\quad
\beta
\longmapsto
\beta\circ i^\ast(x)
\end{equation}
is a ring homomorphism.
Picking a path connecting $f(b_0)$ and $g(b_0)$
where $b_0$ is the point in $B$ over which $F$
is a fibre of $P(f,g)\to B$ and $\Omega BG$ is a fibre of $P(g,g)\to B$,
from Corollary~\ref{cor:homotopyinvariance} we obtain
an isomorphism
\[
\Xi \colon\thinspace H^\ast(F) \xrightarrow{\ \cong\ } H^\ast(\Omega BG)
\]
of $\mathbb{H}^\ast(\Omega BG)$--modules. An inspection of the
proof of Corollary~\ref{cor:homotopyinvariance} shows
that $\Xi$ is induced by a zigzag of homotopy equivalences of spaces,
so $\Xi$ is also an algebra isomorphism with respect to
cup products. Thus it is enough to show that the
composite
\begin{equation}
\label{map:xicomp}
H^\ast(\Omega BG) \xto{\quad} H^\ast(\Omega BG),
\quad
\beta \longmapsto \beta \circ \Xi(i^\ast(x))
\end{equation}
of the map \eqref{map:miastx} and $\Xi$ is a ring homomorphism,
which we will do by showing that this map is in fact
the identity map.
We have (in $\mathbb{H}^\ast(\Omega BG)$)
\[
1 \circ \Xi(i^\ast(x))
=
\Xi(1\circ i^\ast(x))
=
\Xi(1)
=
1
\]
where the first equality follows by the
$\mathbb{H}^\ast(\Omega BG)$--linearity of $\Xi$,
the second equality holds by the assumption on $x$,
and the final equality holds since $\Xi$ is
an algebra homomorphism. Since by ${\mathbb {F}}_\ell$--linearity
the group $H^d(\Omega BG)\cong {\mathbb {F}}_\ell$
contains at most one element $y$ with the property that
$1\circ y = 1 \in H^0(\Omega BG)$, and both
$\mathbbold{1}$ and $\Xi(i^\ast(x))$ have this property,
we must have $\Xi(i^\ast(x)) = \mathbbold{1}$.
Thus the map \eqref{map:xicomp}
is the identity map as claimed.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thrm:cup-products}]
The first part of part~(\ref{it:modiso})
of the theorem is immediate from
Theorem~\ref{thm:cupmodstrbilin}.
The claimed injectivity of the induced map
$H^\ast(BG) \to H^\ast(BG^{h\sigma})$
then follows from the existence of a section
for the fibration $LBG\to BG$, which
implies that $H^\ast (LBG)$
and hence $H^\ast (BG^{h\sigma})$
are faithful as $H^\ast(BG)$--modules.
Part~(\ref{it:gralgiso}) follows from
the special case $f = \sigma$, $g=\mathrm{id}_{BG}$ of
Theorem~\ref{thm:sscupstr}
by observing that the isomorphism
\[
E_\infty^{\ast,\ast}(P(g,g))
\xrightarrow{\ \cong\ }
E_\infty^{\ast,\ast}(P(f,g)),
\quad
z \longmapsto z \circ (1\otimes i^\ast(x))
\]
agrees with the map
\[
\gr H^\ast P(g,g) \xto{\quad} \gr H^\ast P(f,g)
\]
induced by the multiplication map
\[
H^\ast P(g,g) \xto{\quad} H^\ast P(f,g),
\quad
z \longmapsto z \circ x. \qedhere
\]
\end{proof}
\section{On the existence of a fundamental class: Proof of Theorem~\ref{thm:examples}}
\label{sec:funclass1}
The aim of this section is to prove the following theorem, as well as
establish Theorem~\ref{thm:examples}.
\begin{thrm}
\label{thm:examples2} Suppose $BG$ is a connected $\ell$--compact group
that is a product of an $\ell$--compact group
with polynomial mod
$\ell$ cohomology ring %
and copies of
$B\mathrm{Spin}(n)\hat{{}_\ell}$
for various $n$. Let $\tau \in \mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$
be of finite order, with
$\ell \nmid |\tau|$ if $\ell$ odd and $\tau =1$ if $\ell =2$.
Then for every $q \in {\mathbb {Z}}_\ell^\times$, $\BtGq$ has a
$[G^{h\bbtau_e}]$--fundamental class.
\end{thrm}
It is known exactly when $BG$ is
a polynomial ring, also in the non-simply connected case, except for one
loose end at $\ell =2$; see Remark~\ref{poly-rem}.
The assumption that $\tau =1$ when
$\ell = 2$ has to do with that loose end, and can
probably be replaced by $\ell
\nmid |\tau|$. Moreover, as we mentioned earlier,
we suspect the restrictions on $BG$ in
the theorem are superfluous, and that the result in fact holds
for any connected $\ell$--compact group.
We will deal with the polynomial case in Section~\ref{subsec:poly}, the
$\mathrm{Spin}(n)$ case in Section~\ref{subsec:spin}, and put
everything together
to prove Theorem \ref{thm:examples2} and Theorem~\ref{thm:examples}
in Section~\ref{subsec:putittogether}.
\subsection{Fundamental classes for \texorpdfstring{$\ell$}{l}--compact groups with polynomial cohomology ring}\label{subsec:poly}
The aim of this section is to prove the following result.
\begin{prop}
\label{prop:poly-case}
Suppose $BG$ is an $\ell$--compact group such
that $H^*(BG;{\mathbb {F}}_\ell)$ is a polynomial ring. Then $BG(q)$ has
a $[G]$--fundamental class for every $q\in {\mathbb {Z}}_\ell^\times$
such that $q\equiv 1\ \mathrm{mod}\ \ell$.
\end{prop}
We will derive Proposition~\ref{prop:poly-case}
from the following more abstract result.
\begin{thrm}
\label{thm:polycollapse}
Suppose that $H^\ast BG$ is a polynomial ring
and $\sigma \colon\thinspace BG \to BG$ a map inducing the identity on $H^\ast
BG$. Then for the fibre sequence $G
\xrightarrow{\,i\,} BG^{h\sigma} \to BG$ of \eqref{eq:fibseq},
the map $H^*(BG^{h\sigma}) \xrightarrow{i^*} H^*(G)$ is surjective. In particular $BG^{h\sigma}$ has a
$[G]$--fundamental class.
\end{thrm}
\begin{proof}
Recall that for a pullback diagram of spaces
\[\xymatrix{
X \times_B Y
\ar[r]
\ar[d]
&
Y
\ar[d]^{g}
\\
X
\ar[r]^{f}
&
B
}\]
with at least one of $f$ and $g$ a fibration and
the space $B$ simply connected, the Eilenberg--Moore
spectral sequence is a strongly convergent second-quadrant spectral
sequence of algebras
\[
E_2^{s,t} = \operatorname{Tor}^{H^\ast(B)}_{-s,t}(H^\ast(X), H^\ast(Y))
\Longrightarrow
H^{s+t}(X\times_B Y)
\]
where $t$ is the internal degree and $--s$ the
homological degree. The entries on the $E_\infty$--page of the spectral
sequence are filtration quotients
\[
E_\infty^{s,t} = F^{s} H^{s+t}(X\times_B Y) / F^{s+1} H^{s+t}(X\times_B Y)
\]
for a certain descending filtration
\[
H^\ast(X\times_B Y)
\supset \cdots \supset
F^{-2} \supset F^{-1} \supset F^0 \supset F^1 = 0.
\]
of $H^\ast(X\times_B Y)$.
In particular, we may interpret the line
$E_\infty^{0,\ast}$ as a subring of $H^\ast (X\times_B Y)$.
By naturality of the spectral sequence, it is easy to see
that this subring is precisely the image of the
map
\[
H^\ast(X \times Y) \xto{\quad} H^\ast(X\times_B Y)
\]
induced by the inclusion of $X\times_B Y$ into $X\times Y$.
Suppose now $H^\ast(BG) = {\mathbb {F}}_\ell[x_1,\ldots,x_n]$,
and let $\sigma \colon\thinspace BG \to BG$ be a map inducing
the identity on cohomology.
Consider the Eilenberg--Moore spectral sequence for
the pullback square
\[\xymatrix{
BG^{h\sigma}
\ar[r]
\ar[d]_{\mathrm{ev}_1}
&
BG^I
\ar[d]^{(\mathrm{ev}_0,\mathrm{ev}_1)}
\\
BG
\ar[r]^-{(\sigma,1)}
&
BG\times BG
}\]
By the assumption on $\sigma$,
the $E_2$--page of the spectral sequence amounts to
\[
\operatorname{Tor}^{{\mathbb {F}}_\ell[x_1,\ldots,x_n,x'_1,\ldots,x'_n]}(
{\mathbb {F}}_\ell[x_1,\ldots,x_n],
{\mathbb {F}}_\ell[x_1,\ldots,x_n]
)
\]
where
${\mathbb {F}}_\ell[x_1,\ldots,x_n,x'_1,\ldots,x'_n]$
acts on the two copies of
${\mathbb {F}}_\ell[x_1,\ldots,x_n]$
via the map $x_i \mapsto x_i$, $x'_i \mapsto x_i$.
Interpreting
\[
{\mathbb {F}}_\ell[x_1,\ldots,x_n,x'_1,\ldots,x'_n]
=
{\mathbb {F}}_\ell[x_1,\ldots,x_n,y_1,\ldots,y_n]
\]
with $y_i = x'_i-x^{}_i$, it is now
easy to compute that the $E_2$--page is
\[
{\mathbb {F}}_\ell[x_1,\ldots,x_n] \otimes \Lambda_{{\mathbb {F}}_\ell}(z_1,\ldots,z_n)
\]
where the $x_i$'s occur on the line $s=0$ and the $z_i$'s
on the line $s=-1$. There can be no
differentials, so the spectral sequence collapses on the $E_2$--page.
From the freeness of the $E_2$--page as an $H^\ast(BG)$--module we now
deduce that the $H^\ast(BG^{h\sigma})$ is free as an
$H^\ast(BG)$ module (with the module structure
induced by $\mathrm{ev}_1\colon\thinspace BG^{h\sigma} \to BG$).
It follows that the Eilenberg--Moore spectral sequence
of the pullback diagram
\[\xymatrix{
G
\ar[r]^-{i}
\ar[d]
&
BG^{h\sigma}
\ar[d]^{\mathrm{ev}_1}
\\
\mathrm{pt}
\ar[r]
&
BG
}\]
is concentrated on the line $s=0$, which in turn implies that the
map $i^\ast \colon\thinspace H^\ast(BG^{h\sigma})\to H^\ast(G)$ is an epimorphism. In particular it is nonzero in degree $d$, so $BG^{h\sigma}$ has a
fundamental class by Definition~\ref{defn:fundclass} and the
remark that follows it.
\end{proof}
\begin{rem}
\label{rk:polysscollapse}
In the situation of Theorem~\ref{thm:polycollapse},
the Serre spectral sequence of the fibre sequence
$G \xrightarrow{i} BG^{h^\sigma} \to BG$ collapses at the $E_2$ page:
the surjectivity of $i^\ast$
guarantees that there can be no differentials
originating on the $E_r^{0,*}$ line,
whence there cannot be any nonzero
differentials at all by multiplicativity.
\end{rem}
We now establish how $\psi^q$ acts on $H^*(BG)$
when $H^*(BG)$ is a polynomial ring.
\begin{prop}
\label{prop:idmappoly}
Suppose that $H^*(BG)$ is a polynomial ring. If $\ell$ is odd then
$H^*(BG)$ is concentrated in even degrees and
$\psi^q$ acts as $q^n$ on $H^{2n}(BG)$ for any $n$; in particular it
acts as the identity if $q \equiv 1\ \mathrm{mod}\ \ell$.
If $\ell=2$ then $\psi^q$ acts as the identity on
$H^*(BG)$ for all $q\in {\mathbb {Z}}_2^\times$.
\end{prop}
\begin{proof}
If $\ell$ is odd then $H^*(BG;{\mathbb {Z}}_\ell) \xrightarrow{\cong} H^*(BT;{\mathbb {Z}}_\ell)^W$ (see
\cite[Thm.~12.1]{AGMV08}), and the result follows since $\psi^q$
induces multiplication by $q$ on $H^2(BT;{\mathbb {Z}}_\ell)$, by
definition.
Now, suppose that $\ell =2$. Here $H^*(BG) \to H^*(BT)$ need not be
injective, unless $G$ is $2$--torsion free (see
\cite[Thm.~12.1]{AGMV08}). However, by the theory of unstable modules over the
Steenrod algebra (see e.g.\ \cite[Prop.~7.2]{AG09}), there exists an elementary abelian
$2$--subgroup $V$, which, up to conjugation, contains every other elementary
abelian $2$--subgroup.
In particular, we can choose a representative which contains ${}_2T$,
the elements of order $2$ in the maximal torus. Let $W_0$ denote the stabilizer of ${}_2T$ in $W$, which by e.g.\
\cite[Lem.~11.3]{AGMV08} is an elementary abelian $2$--subgroup of
$W$. Now, recall Tits' model for $N_G(T)$ from \cite{tits66},
elaborated and extended to $2$--compact groups in \cite{DW05} and \cite{AG08auto}:
$N_G(T)$ can be constructed from the root datum by first constructing
the reflection extension $1 \to {\mathbb {Z}}[\Sigma] \to \rho(W) \to W \to
1$, where $\Sigma$ is the set of reflections in $W$, and then
constructing $N_G(T)$ as a push-forward along a $W$--map $f\colon\thinspace
{\mathbb {Z}}[\Sigma] \to T$ sending each reflection $\sigma$ to a certain element of order two
$h_\sigma$ of $T$; see \cite[\S\S2-3]{AG08auto}. Let $\rho_0$ denote
the preimage of $W_0$ in $\rho(W)$, and consider the subgroup $A$
of $N_G(T)$ generated by ${}_2T$ and the image of $\rho_0$ under
$\rho(W) \to N_G(T)$, the map to the push-forward. By construction
$A$ is an abelian subgroup of $N_G(T)$. Likewise by construction it
will contain $V$. We
hence just have to see that $\psi^q$ acts trivially on $A$. However,
this is a consequence of \cite[Thms.~B and C]{AG08auto}, which explain
exactly how $\psi^q$ acts on $N_G(T)$, namely as a quotient of a map
which multiplies by $q$ on $T$ and is the identity on $\rho(W)$ (see
Step~2 of the proof of Thm.~B in \cite{AG08auto} for the definition of
the homomorphism $s\colon\thinspace \mathrm{Out}(\mathbb{D}_G) \to \mathrm{Out}(N_G(T))$).
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:poly-case}]
The proposition is immediate from Theorem~\ref{thm:polycollapse}
and Proposition~\ref{prop:idmappoly}.
\end{proof}
\begin{rem}[Classification of $\ell$--compact groups with polynomial
cohomology ring] \label{poly-rem}
It is know which $\ell$--compact groups have polynomial cohomology ring, also in
the non-simply connected case, except for one loose
end when $\ell =2$:
\begin{itemize}
\renewcommand\labelitemi{--\ }
\item \label{polyodd} If $\ell$ is odd, $H^*(BG)$ is a polynomial ring
iff $\pi_1(G)$
is $\ell$--torsion free, and the universal cover of $G$ does not
contain a summand isomorphic to the $\ell$--completion of $F_4$,
$E_6$, $E_7$, or $E_8$ if $\ell=3$, or $E_8$ if $\ell =5$. (See \cite[Thms.~12.1 and 12.2]{AGMV08}.)
\item \label{poly2-torsionfree} If $\ell =2$ and $\pi_1(G)$ is
$2$--torsion free, then $H^*(BG)$ is a polynomial ring iff the universal
cover of $G$ does not include $2$--completions of $E_6$,
$E_7$, $E_8$, or $\mathrm{Spin}(n)$ for $n \geq 10$.
(See \cite[Thms.~1.1 and 1.4]{AG09}.)
\item If $\ell =2$ and $G$ is simple, then $H^*(BG)$ is a polynomial
ring iff $G$ is either $\mathrm{DI}(4)$
or the $2$--completion of $\mathrm{SU}(n)$, $\mathrm{Sp}(n)$, $\mathrm{Spin}(n)$ ($7 \leq n
\leq 9$),
$G_2$, $F_4$, $\mathrm{SO}(n)$ or
$\mathrm{PSp}({2n+1})$ (See \cite[Rem.~7.1]{AG09}.)
Notice that the last two examples are not simply connected.
\item In the case where $\ell =2$, $G$ is not simple, and $\pi_1(G)$
contains $2$--torsion, there are further
mixed examples of $\ell$--compact groups with polynomial cohomology.
For example, $B(\mathrm{Sp}(1) \times \cdots \mathrm{Sp}(1))/\Delta$, where $\Delta$
is a diagonally embedded copy of ${\mathbb {Z}}/2$, has polynomial cohomology.
A complete classification of such cases does not seem to be
available, however.
(Again, see \cite[Rem.~7.1]{AG09}.)
\end{itemize}
\end{rem}
\subsection{Fundamental classes for \texorpdfstring{$B\mathrm{Spin}(n)$}{BSpin(n)}}
\label{subsec:spin}
We now prove the existence of a fundamental class in the $\mathrm{Spin}(n)$
case for $\sigma = \psi^q$,
building on the work of Kameko \cite{Kameko-spin}.
\begin{prop}
\label{prop:spin-case}
$(B\mathrm{Spin}(n)\hat{{}_2})^{h\psi_q}$
has a $[\mathrm{Spin}(n)\hat{{}_2}]$--fundamental class for all
$q\in {\mathbb {Z}}_2^\times$ and $n \geq 2$.
\end{prop}
\begin{proof}
Since the mod 2 cohomology of $B\mathrm{Spin}(2)\hat{{}_2}$ is a polynomial ring,
the claim for $n=2$ follows from
Proposition~\ref{prop:poly-case}.
Let us assume that $n\geq 3$.
For brevity, let us write $BSO_n$ and $B\mathrm{Spin}_n$ for $BSO(n)\hat{{}_2}$
and $B\mathrm{Spin}(n)\hat{{}_2}$, respectively.
In view of Corollary~\ref{cor:psiqsubgrp} below,
it is enough to prove that $(B\mathrm{Spin}_n)^{h\psi^q}$
has a fundamental class when $q=3$ or $q=5$.
So let us assume that $q$ is one of these numbers;
in fact, all that matters for the argument that follows
is that $q$ is an odd prime power. By choosing suitable
models for $BSO_n$ and $B\mathrm{Spin}_n$ and for the self-maps
$\psi^q$ of $BSO_n$ and $B\mathrm{Spin}_n$,
we may assume the following:
\begin{enumerate}
\item The maps $\psi^q \colon\thinspace BSO_n \to BSO_n$ and
$\psi^q\colon\thinspace B\mathrm{Spin}_n \to B\mathrm{Spin}_n$
are basepoint-preserving.
\item The map $p\colon\thinspace B\mathrm{Spin}_n \to BSO_n$
induced by the projection $\mathrm{Spin}_n \to SO_n$
preserves basepoints and commutes with the action
of the maps $\psi^q$.
\end{enumerate}
For example, by replacing $\psi^q \colon\thinspace BSO_n \to BSO_n$
by a homotopic map if necessary, we may assume that
it preserves the basepoint, after which models for
$B\mathrm{Spin}_n$ and $\psi^q \colon\thinspace B\mathrm{Spin}_n \to B\mathrm{Spin}_n$
with the desired properties can be obtained
by passing to functorial 2-connected covers.
Let $E_{n,q}$ and $E_n$ be the spaces obtained as pullbacks
\begin{equation}
\label{eq:epbsqs}
\vcenter{\xymatrix{
E_{n,q}
\ar[r]
\ar[d]
&
BSO_n^{h\psi^q}
\ar[d]^{\mathrm{ev}_1}
\\
B\mathrm{Spin}_n
\ar[r]^{p}
&
BSO_n
}}
\qquad\text{and}\qquad
\vcenter{\xymatrix{
E_{n}
\ar[r]
\ar[d]
&
LBSO_n^{\vphantom{h\psi^q}}
\ar[d]^{\mathrm{ev}_1}
\\
B\mathrm{Spin}_n
\ar[r]^{p}
&
BSO_n
}}
\end{equation}
Let
\[
\pi_{n,q} \colon\thinspace B\mathrm{Spin}_n^{h\psi^q} \xto{\quad} E_{n,q}
\qquad\text{and}\qquad
\pi_{n} \colon\thinspace LB\mathrm{Spin}_n \xto{\quad} E_n
\]
be the maps over $B\mathrm{Spin}_n$ induced by the evaluation maps
\[
\mathrm{ev}_1 \colon\thinspace B\mathrm{Spin}_n^{h\psi^q} \xto{\quad} B\mathrm{Spin}_n
\qquad\text{and}\qquad
\mathrm{ev}_1 \colon\thinspace LB\mathrm{Spin}_n \xto{\quad} B\mathrm{Spin}_n
\]
and the maps
\[
B\mathrm{Spin}_n^{h\psi^q} \xto{\quad} BSO_n^{h\psi^q}
\qquad\text{and}\qquad
LB\mathrm{Spin}_n \xto{\quad} LBSO_n
\]
induced by $p$. Then, up to homotopy, $\pi_{n,q}$ and $\pi_n$
are two-fold covering spaces, and there are maps
of fibre sequences
\[
\vcenter{\xymatrix{
{\mathbb {Z}}/2
\ar[r]
\ar@{=}[d]
&
\Omega B\mathrm{Spin}_n
\ar[r]^-{\Omega p}
\ar[d]^{i_{n,q}}
&
\Omega BSO_n
\ar[d]^{j_{n,q}}
\\
{\mathbb {Z}}/2
\ar[r]
&
B\mathrm{Spin}_n^{h\psi^q}
\ar[r]^-{\pi_{n,q}}
&
E_{n,q}
}}
\qquad\text{and}\qquad
\vcenter{\xymatrix{
{\mathbb {Z}}/2
\ar[r]
\ar@{=}[d]
&
\Omega B\mathrm{Spin}_n
\ar[r]^-{\Omega p}
\ar[d]^{i_n}
&
\Omega BSO_n
\ar[d]^{j_n}
\\
{\mathbb {Z}}/2
\ar[r]
&
LB\mathrm{Spin}_n^{\phantom{h}}
\ar[r]^-{\pi_{n}}
&
E_{n}
}}
\]
where the vertical arrows are inclusions of fibres of
the various spaces over $B\mathrm{Spin}_n$ over the basepoint.
We obtain the following commutative
diagram whose rows are long exact Gysin sequences:
\[\xymatrix{
&
\mathllap{\cdots\xto{\quad}}\,
H^d(E_{n,q})
\ar[r]^-{\pi_{n,q}^\ast}
\ar[d]_{j_{n,q}^\ast}
&
H^d (B\mathrm{Spin}_n^{h\psi^q})
\ar[r]^-{(\pi_{n,q})_!}
\ar[d]_{i_{n,q}^\ast}%
&
H^d(E_{n,q})
\ar[r]^-{\cup e_{n,q}}
\ar[d]_{j_{n,q}^\ast}
&
H^{d+1}(E_{n,q})
\ar[d]_{j_{n,q}^\ast}
\,\mathrlap{\xto{\quad}\cdots}
&
\\
&
\mathllap{\cdots\xto{\quad}}\,
H^d(\Omega BSO_n)
\ar[r]^{(\Omega p)^\ast}%
&
H^d(\Omega B\mathrm{Spin}_n)
\ar[r]^-{(\Omega p)_!}
&
H^d(\Omega BSO_n)
\ar[r]^-{\cup e}
&
H^{d+1}(\Omega BSO_n)
\,\mathrlap{\xto{\quad}\cdots}
&
\\
&
\mathllap{\cdots\xto{\quad}}\,
H^d(E_{n})
\ar[r]^-{\pi_{n}^\ast}
\ar[u]^{j_n^\ast}
&
H^d (LB\mathrm{Spin}_n)
\ar[r]^-{(\pi_{n})_!}
\ar[u]^{i_n^\ast}
&
H^d(E_{n})
\ar[r]^-{\cup e_{n}}
\ar[u]^{j_n^\ast}
&
H^{d+1}(E_{n})
\ar[u]^{j_n^\ast}
\,\mathrlap{\xto{\quad}\cdots}
&
}\]
Here $d= \dim SO_n = \dim \mathrm{Spin}_n$
and $e$, $e_n$ and $e_{n,q}$ are
the Euler classes for the respective double covers.
Our task is to show that the map $i_{n,q}^\ast$ in the above
diagram is nonzero. To do this,
it suffices to show that the composite map
\[
(\Omega p)_! \circ i_{n.q}^\ast
=
j_{n,q}^\ast \circ (\pi_{n,q})_!
\colon\thinspace
H^d(B\mathrm{Spin}_n^{h\psi^q})
\xto{\quad}
H^d(\Omega BSO_n)
\]
is nonzero. Thus it is enough to show that there
exists an element $x\in H^d(E_{n,q})$ such that $j_{n,q}^\ast(x) \neq 0$
and $x \cup e_{n,q} = 0$.
To show the existence of such an element $x$,
we will compare the upper part of the diagram
with the lower part.
As in \cite[Props.~4.1 and 4.3]{Kameko-spin},
the Eilenberg--Moore spectral sequences for the
pullback squares \eqref{eq:epbsqs} yield
ring isomorphisms
\[
H^\ast(E_{n,q})
\cong
H^\ast(B\mathrm{Spin}_n) \otimes_{H^\ast (BSO_n)} H^\ast(BSO_n^{h\psi^q})
\]
and
\[
H^\ast(E_{n})
\cong
H^\ast(B\mathrm{Spin}_n) \otimes_{H^\ast (BSO_n)} H^\ast(LBSO_n).
\]
Let
\[
k_{n,q} \colon\thinspace \Omega BSO_n \xto{\quad} BSO_n^{h\psi^q}
\qquad\text{and}\qquad
k_{n} \colon\thinspace \Omega BSO_n \xto{\quad} LBSO_n
\]
be inclusions of fibres of the evaluation fibrations
$BSO_n^{h\psi^q}\to BSO_n$ and $LBSO_n \to BSO_n$.
Under the above isomorphisms, the maps
$j_{n,q}^\ast$ and $j_n^\ast$
then correspond to the maps induced by
\[
k_{n,q}^\ast\colon\thinspace H^\ast(BSO_n^{h\psi^q}) \xto{\quad} H^\ast(\Omega BSO_n)
\qquad\text{and}\qquad
k_{n}^\ast\colon\thinspace H^\ast(LBSO_n) \xto{\quad} H^\ast (\Omega BSO_n)
\]
(along with the augmentation $H^\ast(B\mathrm{Spin}_n) \to {\mathbb {F}}_2)$,
respectively.
By \cite[Thm.~1.7]{Kameko-spin},
there exists a $H^\ast(BSO_n)$--algebra isomorphism
\[
H^\ast(BSO_n^{h\psi^q}) \cong H^\ast(LBSO_n).
\]
Under this isomorphism, the maps $k_{n,q}^\ast$ and $k_n^\ast$ agree;
to see this, observe that both halves of the diagram on p.\ 525
of Kameko's paper \cite{Kameko-spin} are pullbacks, so
$k_{n,q}$ and $k_n$ factor (up to homotopy) through the same
map from $\Omega BSO_n$ to Kameko's space $\tilde{B} A_{n-1}$.
We conclude that there exists a ring isomorphism
\begin{equation}
\label{iso:enqen}
H^\ast(E_{n,q}) \cong H^\ast(E_n)
\end{equation}
under which the maps $j_{n,q}^\ast$ and $j_n^\ast$ agree.
Using for example the Serre spectral sequence, it is easy to see
that $H^1 E_{n,q} \cong H^1 E_n \cong {\mathbb {F}}_2$.
The Euler classes $e_{n,q} \in H^1 E_{n,q}$ and $e_n \in H^1 E_n$
both pull back to the class $e\in H^1 \Omega BSO_n$, which in
turn pulls back to the (nontrivial) Euler class of the
double cover $\mathrm{Spin}_2 \to SO_2$. Thus the classes $e_{n,q}$
and $e_n$ must be the unique nontrivial degree 1 classes
in their respective cohomology groups, and hence they must correspond
under the isomorphism \eqref{iso:enqen}.
We have reduced the task of constructing the desired class
$x\in H^d E_{n,q}$ to the task of finding a class
$y\in H^d E_n$ such that $j_n^\ast(y) \neq 0$ and $y\cup e_n = 0$.
By Theorem~\ref{thm:functors}, the map $i_n^\ast$
sends the class $\mathbbold{1}\in H^d(LB\mathrm{Spin}_n)$ to the nontrivial class
$\mathbbold{1}\in H^d(\Omega B\mathrm{Spin}_n)$.
Moreover, since $H^{d+1}(\Omega BSO_n) = 0$, the map $(\Omega p)_!$ is an
epimorphism and hence an isomorphism, as its source and
target are both one-dimensional.
Thus $(\Omega p)_! i_n^\ast(\mathbbold{1}) \neq 0$. Now the class
$y = (\pi_n)_!(\mathbbold{1}) \in H^d(E_n)$ is as desired.
\end{proof}
\subsection{Proofs of Theorem~\ref{thm:examples2} and Theorem~\ref{thm:examples}}
\label{subsec:putittogether}
\begin{lemma}\label{lem:productsfund}
Suppose $BG$ and $BH$ are connected $\ell$--compact
groups, and let $\sigma \colon\thinspace BG \to BG$ and $\tau \colon\thinspace BH \to BH$
be maps such that $BG^{h\sigma}$ and $BH^{h\tau}$ have
$[G]$-- and $[H]$--fundamental classes, respectively.
Then $(BG \times BH)^{h(\sigma\times \tau)}$
has a $[G \times H]$--fundamental class.
\end{lemma}
\begin{proof}
The claim follows from the observation that the fibre
sequence
\[
G\times H
\xto{\quad}
(BG \times BH)^{h(\sigma\times \tau)}
\xto{\quad}
BG\times BH
\]
of equation~\eqref{eq:fibseq} for $G\times H$
is the product of the analogous fibre sequences for $G$ and $H$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:examples2}]
Let $q\in {\mathbb {Z}}^\times_\ell$.
Consider first the case $\ell=2$.
In this case $e$, the multiplicative order of $q$ mod $\ell$,
is $1$.
Moreover, by assumption, $\tau=1$, so we are reduced to showing that
$BG(q)=BG^{h\psi^q}$ has a $[G]$--fundamental class.
In view of Lemma~\ref{lem:productsfund},
the claim now follows from
Propositions~\ref{prop:poly-case} and~\ref{prop:spin-case}
and the observation that the unstable Adams operation $\psi^q$
for a product of $\ell$--compact groups is the product of the
corresponding operations $\psi^q$ for the factors.
Suppose now that $\ell$ is odd.
Then the cohomology of $B\mathrm{Spin}(n)\hat{{}_\ell}$
is a polynomial cohomology ring
(see~Remark~\ref{poly-rem}), so
the theorem is reduced to the case where $BG$ has polynomial
cohomology ring. By Proposition~\ref{prop:ghksplit},
the cohomology of the fixed points $BG^{h\bbtau_e}$ is also
a polynomial ring, so the claim follows from
Proposition~\ref{prop:poly-case}.
\end{proof}
Before proving Theorem~\ref{thm:examples}, we need a small lemma which
is an analog of a well-known result for finite groups of Lie type.
\begin{lemma}\label{lem:permute}
Let $BH$ be a simple $\ell$--compact group, and let
$BG = BH^n$.
Suppose $\sigma \in \mathrm{Out}(BG)$ is an element
(of any order, infinite or finite) such that
the permutation of $\{1,\ldots,n\}$ induced by $\sigma$
\cite[Prop.~8.14]{AG09} is transitive.
Then $\sigma^n = \operatorname{diag}(\rho,\ldots,\rho)$ for some $\rho \in
\mathrm{Out}(BH)$ and
\[
BG^{h\langle\sigma\rangle} \simeq BH^{h\langle \rho\rangle}.
\]
\end{lemma}
\begin{proof}
By \cite[Thm.~1.2 and Prop.~8.14]{AG09} $\mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)
\cong \mathrm{Out}(\mathbb{D}_H)\wr \mathfrak{S}_n$.
Hence $\sigma^n = \operatorname{diag}(\rho,\ldots, \rho)$
for some $\rho \in \mathrm{Out}(\mathbb{D}_H)$.
Note that by Proposition~\ref{autBTtoBG}, $\sigma$ and $\rho$
determine well-defined homotopy actions on $BG$ and $BH$.
Furthermore,
$$(BG)^{h\langle \sigma\rangle} \simeq
((BG)^{h\langle\sigma^n\rangle})^{hC_n} \simeq
((BH^{h\langle\rho\rangle})^n)^{hC_n} \simeq BH^{h\langle \rho\rangle}$$
where the first homotopy equivalence follows by transitivity of homotopy
fixed points, see\ e.g.\ \cite[Lem.~10.5]{DW94}, and the last one by
``Shapiro's lemma,'' see e.g.\ \cite[Lem.~10.8]{DW94}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:examples}]
Suppose we are away from the 8 listed exceptional cases.
Consider first the case $\ell$ odd. In this case,
by \cite[Thms.~12.1 and 12.2]{AGMV08},
our list of exclusions ensures that $BG$ has
polynomial cohomology, and the claim follows
from Theorem~\ref{thm:examples2}.
Consider now the case $\ell=2$.
Using the classification of $2$--compact groups \cite[Thm.~1.1]{AG09},
our list of exclusions ensures that
$BG$ is a product of a $2$--compact group with
polynomial cohomology and $B\mathrm{Spin}(n)\hat{{}_2}$'s for
various $n\geq 10$; see \cite[Thm.~1.4]{AG09}.
When $\tau=1$, the claim therefore follows from
Theorem~\ref{thm:examples2}. It remains to consider
the case $\tau \neq 1$.
We have $BG \simeq \prod_{i=1}^s (BK_i)^{m_i}$ for some
non-isomorphic simple simply-connected $2$--compact groups $BK_i$.
Again by \cite[Thm.~1.2 and Prop.~8.14]{AG09}, $\mathrm{Out}(BG) \cong
\prod_{i=1}^s\mathrm{Out}({BK_i})\wr \mathfrak{S}_{m_i}$. Hence we can write $BG
\simeq \prod_{i=1}^t (BH_i)^{n_i}$ for (potentially isomorphic)
simple and simply-connected
$BH_i$ such that $\tau_e$
permutes the factors of
each product $(BH_i)^{n_i}$ transitively, and
\[
BG^{h\bbtau_e} \simeq \prod_{i=1}^t (BH_i)^{h\langle \rho_i\rangle}
\]
for some elements $\rho_i \in \mathrm{Out}(BH_i)$ by Lemma~\ref{lem:permute}.
Notice that the assumption that $\tau$ is of finite order prime
to $\ell$ implies that each of the $\rho_i$'s also is.
In view of Lemma~\ref{lem:productsfund},
we are therefore reduced to considering the case
where $BG$ is simple.
The possible twistings $\tau$ are now
tabulated in \cite[Thm.~13.1]{AGMV08}.
A glance at that list reveals that there is in fact only one possibility
for a nontrivial $\tau$ of odd order,
namely the triality graph automorphism when
$BG \simeq B\mathrm{Spin}(8)\hat{{}_2}$.
The resulting $2$--compact group
$(B\mathrm{Spin}(8)\hat{{}_2})^{h\langle \tau \rangle}$ certainly does not have
an $E_i$--summand for rank reasons (in fact it is easily seen to be
$(BG_2)\hat{{}_2}$, as one would expect). Thus the
claim follows from the case $\tau=1$ already proven.
\end{proof}
\begin{rem}[The assumption $\ell \nmid |\tau|$]
\label{rk:ellnmidtau}
Theorem~\ref{thm:examples}
does have an assumption that the order of $\tau$ is
prime to $\ell$. When $G$ is simple this excludes
$\mathrm{Spin}_{4n}^-(q)$, $\mathrm{SO}^-_{4n}(q)$, $(P\mathrm{Spin})_{4n}^-(q)$, and ${}^2G_2(3^{f})$ for $\ell =2$ and the case
${}^3D_4(q)$ for $\ell = 3$, where the picture is a bit
different. It would be interesting to completely work out directly
what
happens in these case.
\end{rem}
\section{On the existence of a fundamental class for general \texorpdfstring{$\sigma$}{sigma}: Proof of Theorem~\ref{thm:tezukasubgrp}}
\label{sec:funclass2}
In this section, we prove Theorem~\ref{thm:tezukaconjsubgrp2} and
Corollary~\ref{cor:psiqsubgrp} below, which imply and elaborate
Theorem~\ref{thm:tezukasubgrp}. For this we define
\[
D = \{
[\sigma] \in \mathrm{Out}(BG)
\,|\,
BG^{h\sigma} \text{ has a $[G]$--fundamental class}
\}
\]
and
equip $\mathrm{Out}(BG)$ with the
$\ell$--adic topology induced by the action on $H^*(BG;{\mathbb {Z}}/\ell^k)$,
$k\geq1$, as explained e.g.\ in \cite[p.~7]{BrotoMoellerOliver}.
Under the isomorphism $\mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$,
this topology agrees with the natural topology on $\mathrm{Out}(\mathbb{D}_G)$;
see Proposition~\ref{prop:topcomparison} below.
\begin{thrm}
\label{thm:tezukaconjsubgrp2}
The set $D$ has the following properties:
\begin{enumerate}[(i)]
\item \label{it:inker}
Suppose $\sigma \in D$.
Then $\sigma^\ast \colon\thinspace H^\ast BG \to H^\ast BG$
is the identity map.
\item \label{it:subgrp}
$D$ is a subgroup of $\mathrm{Out}(BG)$.
\item \label{it:closureprop}
$D$ has the following closure property:
Suppose $x,y\in \mathrm{Out}(BG)$ generate the same closed subgroup of
$\mathrm{Out}(BG)$. Then $x\in D$ if and only if $y\in
D$.
\item \label{it:ssclosed}
When $BG$ is semisimple, $D$ is a closed subgroup of $\mathrm{Out}(BG)$.
\item %
\label{it:nontrivialq}
$\psi^q \in D$ for some $q \in {\mathbb {Z}}_\ell^\times$ of
infinite order.
\end{enumerate}
\end{thrm}
\begin{cor}
\label{cor:psiqsubgrp}
If $\ell$ is odd, the set of $q \in {\mathbb {Z}}_\ell^\times$
for which $BG^{h\psi^q}$ has a $[G]$--fundamental class
is a subgroup of ${\mathbb {Z}}_\ell^\times$ splitting as an internal direct product
$\mu_e H_n$ for some $e\mid(\ell-1)$ and $1 \leq n < \infty$ where
$\mu_e \leq {\mathbb {Z}}_\ell^\times$ is the subgroup of $e$--th roots of unity and
\[
H_n = \{ q \in {\mathbb {Z}}_\ell^\times \,|\, q\equiv 1\ \mathrm{mod}\ \ell \text{ and } v_\ell(q-1) \geq n\}.
\]
If $\ell =2$,
it is a subgroup of ${\mathbb {Z}}_2^\times$ of one of the following forms:
\[
\begin{gathered}
H'_n \text{ or } \pm H'_n\,\,\,\, \text{ for $2\leq n <
\infty$,} \qquad or \qquad
H'_n \cup (-1+2^{n-1}) H'_n\,\,\,\, \text{ for $3\leq n < \infty$,}
\end{gathered}
\]
where
\[
H'_n = \{
q \in {\mathbb {Z}}_2^\times
\,|\,
q \equiv 1\ \mathrm{mod}\ 4 \text{ and } v_2(q-1) \geq n
\}.
\]
\end{cor}
\begin{rem}
As remarked in the introduction, we do not know an example of an
$\ell$--compact group $BG$ for which $D$
is a proper subgroup of the kernel of the
homomorphism $\mathrm{Out}(BG)\to \mathrm{Aut}(H^\ast(BG))$ in
Theorem~\ref{thm:tezukaconjsubgrp2}(\ref{it:subgrp}).
If $D$ is the whole kernel, it of course is a closed subgroup of
$\mathrm{Out}(BG)$.
\end{rem}
We will prove Theorem~\ref{thm:tezukaconjsubgrp2} in
a series of auxiliary results.
\begin{lemma}
\label{lm:tezukaimpliesid}
Suppose $\sigma \in D$. Then $\sigma^\ast \colon\thinspace H^\ast BG \to H^\ast BG$
is the identity map.
\end{lemma}
\begin{proof}
The existence of
a section for the evaluation fibration
$LGB\to BG$ implies that the Serre spectral
sequence for $LGB\to BG$ has no differentials hitting the
bottom row. By Theorem~\ref{thm:conj-red3},
the same is therefore true for the Serre spectral sequence
of the evaluation fibration $\mathrm{ev}_1\colon\thinspace BG^{h\sigma} \to BG$.
It follows that the map
\[
\mathrm{ev}_1^\ast \colon\thinspace H^\ast BG \xto{\quad} H^\ast BG^{h\sigma}
\]
is injective.
Since the maps $\mathrm{ev}_0,\mathrm{ev}_1 \colon\thinspace BG^{h\sigma} \to BG$
are homotopic, we have $\mathrm{ev}_0^\ast = \mathrm{ev}_1^\ast$.
The claim now follows from the equation
\[
\mathrm{ev}_1^\ast \sigma^\ast
=
(\sigma\circ \mathrm{ev}_1)^\ast
=
\mathrm{ev}_0^\ast
=
\mathrm{ev}_1^\ast.
\qedhere
\]
\end{proof}
\begin{lemma}
\label{lm:dsubgrp}
$D$ is a subgroup of $\mathrm{Out}(BG)$.
\end{lemma}
\begin{proof}
The cohomology of $BG^{h\mathrm{id}_{BG}} = LBG$ is certainly
free of rank 1 over itself, so $[\mathrm{id}_{BG}] \in D$
by Theorem~\ref{thm:strtoptezukacrit}.
Suppose now $[\sigma]\in D$, and let $\sigma^{-1}$ be a homotopy
inverse of $\sigma$. We then have the following commutative
diagram:
\[\xymatrix@C=0em{
\mathllap{BG^{h\sigma^{-1}}=\,} P(\sigma^{-1},\mathrm{id}_{BG})
\ar[rr]^-{\simeq}
\ar[dr]
&&
P(\sigma^{-1},\sigma\sigma^{-1})
\ar[rrr]^-\simeq
\ar[dl]
&&&
P(\mathrm{id}_{BG},\sigma)
\ar[rr]^-\approx
\ar[dr]
&&
P(\sigma,\mathrm{id}_{BG})
\mathrlap{\,= BG^{h\sigma}}
\ar[dl]
\\
&
BG
\ar[rrrrr]^{\sigma^{-1}}_\simeq
&&&&&
BG
}\]
Here the top left hand map is a fibre homotopy equivalence
induced by a homotopy between $\mathrm{id}_{BG}$ and $\sigma\sigma^{-1}$;
the trapezoid in the middle is a pullback square;
and the top right hand map is the homeomorphism
given by path reversal. The middle map in the top row is
a homotopy equivalence because the map $\sigma^{-1}$ is.
It follows that $[\sigma^{-1}] \in D$.
Finally, suppose $[\sigma],[\sigma'] \in D$.
Our aim is to show that $[\sigma\sigma'] \in D$.
In view of Theorem~\ref{thm:strtoptezukacrit} and the
equivalence between (\ref{it:xexists})
and (\ref{it:permanentcycle2}) in Theorem~\ref{thm:conj-red3},
it suffices to show that the generator of
\[
\mathbb{E}^{0,0}_2 (P(\sigma\sigma',\mathrm{id}_{BG}))\cong {\mathbb {F}}_\ell
\]
in the shifted Serre spectral sequence of
$P(\sigma\sigma',\mathrm{id}_{BG}) \to BG$
is a permanent cycle. Under our pairings between spectral
sequences, this generator factors as the product of the
generators of $\mathbb{E}^{0,0}_2 (P(\sigma',\mathrm{id}_{BG}))$
and $\mathbb{E}^{0,0}_2 (P(\sigma\sigma',\sigma'))$,
so it is enough to show that these two factors
are permanent cycles.
That the first factor is a permanent cycle
follows from the assumption that $[\sigma'] \in D$
by Theorem~\ref{thm:strtoptezukacrit} and
the equivalence between
(\ref{it:xexists})
and (\ref{it:permanentcycle2})
in Theorem~\ref{thm:conj-red3}.
That the second factor is also
a permanent cycle follows similarly from the
assumption that $[\sigma] \in D$
by making use of the pullback diagram
\[\xymatrix{
P(\sigma\sigma',\sigma')
\ar[r]^{\simeq}
\ar[d]
&
P(\sigma,\mathrm{id}_{BG})
\ar[d]
\\
BG
\ar[r]^{\sigma'}_\simeq
&
BG
}\]
where the top horizontal map is a homotopy equivalence
because $\sigma'$ is.
\end{proof}
\begin{prop}
\label{prop:closedsubgrp}
Suppose $[\sigma],[\sigma'] \in \mathrm{Out}(BG)$
generate the same
closed subgroup of $\mathrm{Out}(BG)$.
Then $BG^{h\sigma}$ and $BG^{h\sigma'}$ are homotopy equivalent over
$BG$, i.e., we can choose a homotopy equivalence so that the following
diagram commutes
\[\xymatrix{
BG^{h\sigma}
\ar[dr]
\ar[rr]^\simeq
&&
BG^{h\sigma'}
\ar[dl]
\\
&
BG
}\]
In particular $[\sigma] \in D$
if and only if $[\sigma'] \in D$.
\end{prop}
\begin{proof}
We have a homotopy equivalence
$BG^{h\sigma} \xrightarrow{\simeq} BG^{h\sigma'}$,
by
\cite[Thm.~2.4]{BrotoMoellerOliver}, generalizing
\cite[Prop.~6.5]{BM07}, and an inspection of the proof shows that this homotopy equivalence
can be chosen so that the indicated diagram commutes.
The hypotheses of the theorem are satisfied
in our situation: $H^*(BG)$
is Noetherian by \cite[Thm.~2.4]{DW94}, and $\mathrm{Out}(BG)$ is detected on
$\hat H^*(BG;{\mathbb {Z}}_\ell) = H^*(BG;{\mathbb {Z}}_\ell)$ since $H^*(BG;{\mathbb {Z}}_\ell)
\otimes {\mathbb {Q}} \cong (H^*(BT;{\mathbb {Z}}_\ell)\otimes {\mathbb {Q}})^{W_G}$ by
\cite[Thm.~9.7(iii)]{DW94} and $\mathrm{Out}(BG)$ is detected on $BT$, as part
of the classification of connected $\ell$--compact groups
\cite[Thm.~1.2]{AG09} (in fact $\mathrm{Out}(BG) \cong \mathrm{Out}(\mathbb{D}_G)$).
The statement about $D$ is a consequence of the first claim.
\end{proof}
The following result can be read off from
\cite[Thm.~13.1]{AGMV08}.
\begin{lemma}
\label{lm:irredfinind}
Suppose $\mathbb{D}$ is an irreducible ${\mathbb {Z}}_\ell$--root datum.
Then the image of the homomorphism
${\mathbb {Z}}_\ell^\times \to \mathrm{Out}(\mathbb{D})$ sending
$q\in{\mathbb {Z}}_\ell^\times$ to the multiplication-by-$q$ map
has finite index in $\mathrm{Out}(\mathbb{D})$.
\qed
\end{lemma}
\begin{lemma}
\label{lm:homzlmodtooutbg}
Any group homomorphism from a finitely generated free ${\mathbb {Z}}_\ell$--module
into $\mathrm{Out}(BG)$ is continuous.
\end{lemma}
\begin{proof}
The topology on $\mathrm{Out}(BG)$ is easily seen to be
induced from the discrete topologies on
the groups $\mathrm{Aut}(H^\ast(BG;\,{\mathbb {Z}}/\ell^k))$, $k\geq 1$ via
the evident maps $\mathrm{Out}(BG) \to \mathrm{Aut}(H^\ast(BG;\,{\mathbb {Z}}/\ell^k))$,
so that a homomorphism into $\mathrm{Out}(BG)$ is continuous if and only if its
composite with all these maps is.
By \cite[Thm.~4.2]{ACFJS}, the cohomology ring
$H^\ast(BG;\,{\mathbb {Z}}/\ell^k)$ is Noetherian and hence
\cite[Thm.~13.1]{Matsumura}
a finitely generated ${\mathbb {Z}}/\ell^k$--algebra.
It follows that $\mathrm{Aut}(H^\ast(BG;\,{\mathbb {Z}}/\ell^k))$
is finite for every $k$.
The claim now follows if we can show that
all group homomorphisms from finitely generated
free ${\mathbb {Z}}_\ell$--modules into finite groups are continuous,
or, what is the same, that every finite-index subgroup
of a finitely generated free ${\mathbb {Z}}_\ell$--module is
open. But the latter claim holds as a special case of
a theorem of Serre,
which asserts that
finite-index subgroups of topologically finitely generated
pro-$\ell$--groups
are open (see e.g.\ \cite[\S4.2]{SerreGaloisCohomology}
for an indication of the proof).
\end{proof}
\begin{lemma}
\label{lm:ssclosed}
When $BG$ is semisimple, $D$ is a closed subgroup of $\mathrm{Out}(BG)$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lm:dsubgrp}, $D$ is a subgroup of $\mathrm{Out}(BG)$.
It remains to show that it is closed.
In view of Lemma~\ref{lm:irredfinind},
the isomorphisms
\[
{\mathbb {Z}}_\ell^\times \cong {\mathbb {Z}}_\ell \times ({\mathbb {Z}}/\ell)^\times
\quad\text{(for $\ell$ odd)}
\qquad\text{and}\qquad
{\mathbb {Z}}_2^\times \cong {\mathbb {Z}}_2 \times ({\mathbb {Z}}/4)^\times
\quad\text{(for $\ell=2$)}
\]
(see e.g.\ \cite[Cor.~4.5.10]{Gouvea}),
and the description of the group $\mathrm{Out}(BG)\cong \mathrm{Out}(\mathbb{D}_G)$
given in
\cite[Props.~8.14 and 8.15]{AG09},
the assumption that $BG$ is semisimple
implies that we can find
a group homomorphism $\phi\colon\thinspace E \to \mathrm{Out}(BG)$
where $E$ is a finitely generated free ${\mathbb {Z}}_\ell$--module
and the image $\phi E$ has finite index in $\mathrm{Out}(BG)$.
The preimage $\phi^{-1}(D)$ is, of course, a subgroup of $E$.
We wish to show that it in fact is a ${\mathbb {Z}}_\ell$--submodule.
Suppose $x\in \phi^{-1}(D)$.
By Lemma~\ref{lm:homzlmodtooutbg},
the homomorphism
$\psi\colon\thinspace {\mathbb {Z}}_\ell \to \mathrm{Out}(BG)$, $\psi(b) = \phi(bx)$
is continuous.
For $a\in {\mathbb {Z}}_\ell$, we therefore have
$
\psi \overline{\langle a\rangle}
\subset
\overline{\psi \langle a \rangle}
\subset
\psi \overline{\langle a\rangle}
$
where the first inclusion follows from the continuity of $\psi$
and the second one from the observation that
$\psi \overline{\langle a\rangle}$
is closed,
being the image of a compact set in a continuous map
to a Hausdorff space. Thus
$
\overline{\langle\phi(ax)\rangle}
=
\overline{\langle\psi(a)\rangle}
=
\overline{\psi\langle a\rangle}
=
\psi \overline{\langle a\rangle}
$,
so that the closed subgroup of $\mathrm{Out}(BG)$
generated by $\phi (ax)$
only depends on the closed subgroup of ${\mathbb {Z}}_\ell$
generated by $a$.
For any $b\in {\mathbb {Z}}_\ell$, we can find an integer $n$
generating the same closed subgroup of ${\mathbb {Z}}_\ell$ as $b$ does.
Since $nx \in \phi^{-1}(D)$, Proposition~\ref{prop:closedsubgrp}
implies that $bx \in \phi^{-1}(D)$. Thus
$\phi^{-1}(D)$ is a ${\mathbb {Z}}_\ell$--submodule of $E$
as claimed.
Since ${\mathbb {Z}}_\ell$ is a principal ideal domain,
as a submodule of the
finitely generated free ${\mathbb {Z}}_\ell$--module $E$,
the preimage $\phi^{-1}(D)$
is also such.
In particular, $\phi^{-1}(D)$ is compact.
Since by Lemma~\ref{lm:homzlmodtooutbg}
the homomorphism $\phi| \colon\thinspace \phi^{-1}(D) \to \mathrm{Out}(BG)$
is continuous,
it follows that the image
$\phi \phi^{-1}(D)$ is closed in $\mathrm{Out}(BG)$.
Moreover, the index of
$\phi \phi^{-1}(D)$ in $D$
is finite because the index of
$\phi E$ in $\mathrm{Out}(BG)$ is. Thus $D$ is a finite union of
closed sets and therefore closed, as claimed.
\end{proof}
\begin{lemma}
\label{lm:nontrivialq}
$[\psi^q]\in D$ for some $q \in {\mathbb {Z}}_\ell^\times$ of infinite order.
\end{lemma}
\begin{proof}
Suppose first that $BG$ is the $\ell$--completion of the classifying
space of a compact connected Lie group.
In the proof of \cite[Thm.~1.5]{Kameko-pb},
it is shown that the Serre spectral sequences of the fibrations
$LBG \to BG$ and $BG^{h\psi^q} \to BG$ are isomorphic
when $q$ is a suitable prime power congruent to $1$ modulo $\ell$. The claim now follows from
the equivalence of (\ref{it:ssiso}) and (\ref{it:xexists})
in Theorem~\ref{thm:conj-red3}.
Now for the general case we know by the classification of
$\ell$--compact groups (specifically \cite[Thm.~1.1]{AG09}
\cite[Thm.~1.2]{AGMV08}) that $BG$ splits a product $BH \times BK$,
where $BH$ is the
$\ell$--completion of the
classifying space of a compact connected Lie group, and $BK$ is an exotic
$\ell$--compact group with polynomial cohomology ring. Hence by
the already-proven polynomial case of
Theorem~\ref{thm:examples},
if $\psi^q \in D$ for $BH$, we will also
have $\psi^q \in D$ for the product.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:tezukaconjsubgrp2}]
The various parts of the theorem were established
in Proposition~\ref{prop:closedsubgrp} and
Lemmas~\ref{lm:tezukaimpliesid}, \ref{lm:dsubgrp},
\ref{lm:ssclosed} and \ref{lm:nontrivialq}
above.
\end{proof}
We now proceed to derive Corollary~\ref{cor:psiqsubgrp}
from Theorem~\ref{thm:tezukaconjsubgrp2}.
\begin{lemma}
\label{lm:thetaiscont}
The homomorphism
\[
\theta \colon\thinspace {\mathbb {Z}}_\ell^\times \xto{\quad} \mathrm{Out}(BG),
\quad
q \longmapsto [\psi^q]
\]
is continuous.
\end{lemma}
\begin{proof}
The group ${\mathbb {Z}}_\ell^\times$ has an open subgroup
isomorphic to ${\mathbb {Z}}_\ell$.
See e.g.\ \cite[Cor.~4.5.10]{Gouvea}.
By Lemma~\ref{lm:homzlmodtooutbg},
the restriction of $\theta$ to this subgroup is continuous,
whence $\theta$ also is.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:psiqsubgrp}]
Let
\[
K = \{
q\in{\mathbb {Z}}_\ell^\times
\,|\,
BG^{h\psi^q} \text{ has a fundamental class}
\}
\subset
{\mathbb {Z}}_\ell^\times.
\]
As an inverse image of the subgroup $D \leq \mathrm{Out}(BG)$
under the homomorphism $\theta$ of
Lemma~\ref{lm:thetaiscont}, $K$ is a subgroup of ${\mathbb {Z}}_\ell^\times$.
Moreover,
Theorem~\ref{thm:tezukaconjsubgrp2}(\ref{it:closureprop})
and
Lemma~\ref{lm:thetaiscont}
imply that $K$ has the following closure property:
if $q,q'\in {\mathbb {Z}}_\ell^\times$ generate the same closed subgroup of
${\mathbb {Z}}_\ell^\times$, then $q\in K$ if and only if $q'\in K$.
Consider now the case $\ell$ odd. Recall that in this case
there exists a canonically split short exact sequence
\begin{equation}
\label{ses:ladicunits}
\xymatrix@C+0.8em{
0
\xto{\quad}
{\mathbb {Z}}_\ell
\ar[r]^-{\exp \circ \ell}
&
{\mathbb {Z}}_\ell^\times
\ar[r]
&
({\mathbb {Z}}/\ell)^\times
\xto{\quad}
0
}
\end{equation}
where the $\ell$ in $\exp \circ \ell$ refers to multiplication by
$\ell$, the map out of ${\mathbb {Z}}_\ell^\times$
is given by reduction mod $\ell$,
and the image of the splitting is precisely
the set of roots of unity in ${\mathbb {Z}}_\ell^\times$.
See e.g.\ \cite[Cor.~4.5.10]{Gouvea}.
Let $K'\leq {\mathbb {Z}}_\ell$ be the inverse image of $K \leq {\mathbb {Z}}_\ell^\times$.
By the aforementioned closure property of $K$ and
continuity of $\exp\circ\ell$,
the question of whether an element $x\in {\mathbb {Z}}_\ell$
belongs to $K'$ only depends on the closed subgroup generated by $x$.
Since this subgroup consists
of all elements of ${\mathbb {Z}}_\ell$ with $\ell$--adic valuation
at least equal to that of $x$, we see that
\[
K' = \{x\in {\mathbb {Z}}_\ell \,|\, v_\ell(x) \geq k \}
\]
where $0\leq k \leq \infty$ is the minimum of
the $\ell$--adic valuations of elements of $K'$.
Moreover,
Theorem~\ref{thm:tezukaconjsubgrp2}(\ref{it:nontrivialq})
implies that $k < \infty$.
It follows that the image $H$ of $K'$ under $\exp\circ\ell$ is
\[
H = \{
q \in {\mathbb {Z}}_\ell^\times
\,|\,
q \equiv 1\ \mathrm{mod}\ \ell \text{ and } v_\ell(q-1) \geq k+1
\} = H_n
\]
for $n = k+1$. Now let $e$ be the order of the image of $K$ in
$({\mathbb {Z}}/\ell)^\times \cong {\mathbb {Z}}/(\ell-1)$. To prove the claim, it
suffices to show that $K$ contains a primitive $e$--th root of unity.
Let $x$ be any element of $K$ mapping to an element of order $e$
in $({\mathbb {Z}}/\ell)^\times$. By the split exact sequence \eqref{ses:ladicunits},
we can write $x$ as $x = uy$, where $u\in{\mathbb {Z}}_\ell^\times$ is
a primitive $e$--th root of unity and $y \equiv 1$ mod $\ell$.
Then $y^e = x^e \in H$. Since $\ell \nmid e$, we have
$v_\ell(y^e-1) = v_\ell(y-1)$, and therefore also $y\in H$.
Thus $u = xy^{-1} \in K$, as desired.
Consider now the case $\ell = 2$.
In this case, there exists a (canonically split) short exact sequence
\[\xymatrix@C+0.8em{
0
\xto{\quad}
{\mathbb {Z}}_2
\ar[r]^-{\exp \circ 4}
&
{\mathbb {Z}}_2^\times
\ar[r]
&
({\mathbb {Z}}/4)^\times
\xto{\quad}
0
}\]
where the map out of ${\mathbb {Z}}_2^\times$ is given by
reduction mod $4$. Let $K'$ be the preimage of $K$
under the embedding $\exp \circ 4$. Again,
we have
\[
K' = \{x \in {\mathbb {Z}}_2 \,|\, v_2(x) \geq k\}
\]
for some $0\leq k < \infty$. The image $H$ of $K'$
under $\exp\circ 4$ is then
\[
H = \{
q \in {\mathbb {Z}}_2^\times
\,|\,
q \equiv 1\ \mathrm{mod}\ 4 \text{ and } v_2(q-1) \geq k+2
\} = H'_n
\]
where $n=k+2$.
The quotient $K/H$ embeds into $({\mathbb {Z}}/4)^\times \cong {\mathbb {Z}}/2$, so the
index of $H$ in $K$ is 1 or 2. In the former case $K = H$.
Assume that the latter case holds.
Then $K$ contains an element
$x$ such that $x\equiv -1\ \mathrm{mod}\ 4$, and $K = H \cup xH$.
If in fact $--1 \in K$, we thus have $K = \pm H$.
Let us assume $--1 \not\in K$.
Then $x = -1 + 2^e x'$ for some $2\leq e < \infty$
and $x'\in {\mathbb {Z}}_2^\times$, and we may assume that $x$ has been
chosen so that $e$ is minimal. Since $x^2 \in H$, we must have
$e \geq n-1$. On the other hand, if we had $e > n$,
consideration of the element
$(1+2^{n})x$ would lead to a contradiction with
the minimality of $e$. Thus $e = n$ or $e = n-1$.
In the former case, we would have $1+2^e x'x^{-1} \in H$, and
therefore
\[
-1 = x(1+2^e x'x^{-1}) \in K,
\]
contradicting the assumption that $--1 \not\in K$. Thus $e=n-1$.
Now $1+2^e (1-x')x^{-1} \in H$, so
\[
xH = x(1+2^e (1-x')x^{-1}) H = (-1+2^e) H.
\]
Thus the claim follows.
\end{proof}
For a ${\mathbb {Z}}_\ell$--root datum $\mathbb{D}$,
the outer automorphism group $\mathrm{Out}(\mathbb{D})$
is by definition a quotient of a closed subgroup of $\mathrm{GL}_{{\mathbb {Z}}_\ell}(L)$
by a finite group, where $L$ denotes
the underlying finitely generated free ${\mathbb {Z}}_\ell$--module
of $\mathbb{D}$. See \cite[p.~388]{AG09}.
Thus the natural topology on $\mathrm{GL}_{{\mathbb {Z}}_\ell}(L)$ induces
a topology on $\mathrm{Out}(\mathbb{D})$ which makes
$\mathrm{Out}(\mathbb{D})$ into a profinite group.
\begin{prop}
\label{prop:topcomparison}
Under the isomorphism $\mathrm{Out}(BG) \cong\mathrm{Out}(\mathbb{D}_G)$,
the topologies on $\mathrm{Out}(BG)$ and $\mathrm{Out}(\mathbb{D}_G)$
coincide.
\end{prop}
\begin{proof}
Since $\mathrm{Out}(\mathbb{D}_G)$ is compact and $\mathrm{Out}(BG)$ is Hausdorff,
it suffices to show that the isomorphism
$\mathrm{Out}(\mathbb{D}_G) \xrightarrow{\ \cong\ } \mathrm{Out}(BG)$
is continuous.
As in the proof of Lemma~\ref{lm:homzlmodtooutbg},
the continuity of this homomorphism follows
if we can show that finite-index subgroups of $\mathrm{Out}(BG)$
are open.
It is a result of Nikolov and Segal \cite{NikolovSegal}
(generelizing the theorem of Serre
used in the proof of Lemma~\ref{lm:homzlmodtooutbg})
that finite-index subgroups of \emph{all}
topologically finitely generated profinite groups are open.
Thus it is enough to show that
the group $\mathrm{Out}(\mathbb{D})$ is topologically finitely generated
for every ${\mathbb {Z}}_\ell$--root datum $\mathbb{D}$.
When $\mathbb{D}$ is irreducible, $\mathrm{Out}(\mathbb{D})$
is topologically finitely generated by Lemma~\ref{lm:irredfinind}.
That $\mathrm{Out}(\mathbb{D})$ is topologically finitely
generated in general now follows from the
description of $\mathrm{Out}(\mathbb{D})$ given in
\cite[Thm.~8.13 and Props.~8.14 and 8.15]{AG09}
and the observation that
open subgroups of topologically finitely
generated profinite groups are topologically finitely generated.
\end{proof}
\begin{rem}
The Nikolov--Segal theorem used in the above proof
is a very deep result
ultimately relying on the classification of finite simple groups.
It would be preferable to find a more direct proof for
Proposition~\ref{prop:topcomparison}.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm:tezukasubgrp}]
The theorem is immediate from
Theorem~\ref{thm:tezukaconjsubgrp2}(\ref{it:inker}), (\ref{it:subgrp}),
and (\ref{it:ssclosed});
Proposition~\ref{prop:topcomparison};
Corollary~\ref{cor:psiqsubgrp},
which implies that the set of $q\in{\mathbb {Z}}_\ell^\times$ for
which $BG^{h\psi^q}$ has a $[G]$--fundamental class is uncountable;
and the fact, evident from the defining property of $\psi^q$,
that the kernel of the
homomorphism ${\mathbb {Z}}_\ell^\times \to \mathrm{Out}(BG)$, $q \mapsto [\psi^q]$
is finite.
\end{proof}
\begin{rem}
\label{rk:generalcoefficients}
Most of our results generalize from cohomology with
${\mathbb {F}}_\ell$--coefficients
to cohomology
with coefficients in an arbitrary commutative ring $R$
such that all maps inducing an isomorphism on homology
with ${\mathbb {F}}_\ell$--coefficients do so on homology
with $R$--coefficients;
examples of such rings $R$
include $R={\mathbb {Z}}/\ell^k$ for all $k\geq1$.
All of the results in
sections \ref{sec:products} and \ref{sec:spectralsequences},
and hence Theorem~\ref{thm:mainresult},
hold for cohomology with $R$--coefficients
with little change required in the proofs.
(In the description of the $E_2$--pages of the spectral
sequences in Theorem~\ref{thm:mainresult}, the tensor
product $H^\ast(BG) \otimes \mathbb{H}^\ast(G)$ should, of course,
be replaced with $H^\ast(BG;\, \mathbb{H}^\ast(G;\,R))$,
and similarly elsewhere.)
Moreover, the resulting products are natural with respect to
the maps induced by ring homomorphisms $R \to R'$ as long
as long as the orientations over $R$ and $R'$ (see Remark~\ref{rk:orientation}) are chosen
in a compatible way.
Theorem~\ref{thm:conj-red3} and its proof also generalize, with the
following modifications: Condition~(\ref{it:fclass}) should be removed, in
condition~(\ref{it:nonzerores}) replace $i^\ast(x) \neq 0$ with the
statement that $i^\ast(x) \in H^d(F;\,R)$ is an $R$--module generator;
and in conditions~(\ref{it:ssiso2})
and (\ref{it:ssiso2right}) replace $1\otimes i^\ast(x)$ with the element
$\bar{x} \in \mathbb{E}_2^{0,0} = H^0(B;\,\mathbb{H}^0(F;\,R))$
corresponding to
$i^\ast(x)\in \mathbb{H}^0(F;\,R)$
under the isomorphism $H^0(B;\,\mathbb{H}^0(F;\,R)) \cong \mathbb{H}^0(F;\,R)$.
Theorem~\ref{thm:sscupstr} also generalizes in a straightforward
fashion, as does Theorem~\ref{thm:tezukaconjsubgrp2}
(with the exception of part (\ref{it:nontrivialq}))
when we interpret $D$ as
\[
D = \{
[\sigma] \in \mathrm{Out}(BG)
\mid
\text{$H^\ast(BG^{h\sigma};\,R)$ is free of rank 1 over
$\mathbb{H}^\ast(LBG;\,R)$}
\}.
\]
Even though ${\mathbb {Z}}_\ell$ does not satisfy the aforementioned condition on $R$,
one also obtains a multiplication on
$\mathbb{H}^\ast(LBG;\,{\mathbb {Z}}_\ell) \cong \lim_{k} \mathbb{H}^\ast(LBG;\,{\mathbb {Z}}/\ell^k)$
and a module structure on
$H^\ast(BG^{h\sigma};\,{\mathbb {Z}}_\ell) \cong \lim_{k} \mathbb{H}^\ast(BG^{h\sigma};\,{\mathbb {Z}}/\ell^k)$
by passing to limits over ${\mathbb {Z}}/\ell^k$--coefficients. The resulting
module structure is never free of rank 1, however,
when $BG^{h\sigma}$ recovers $\ell$--completion
of the classifying space of a finite group of Lie type,
as in this case
$H^\ast(BG^{h\sigma};\,{\mathbb {Z}}_\ell)$
is torsion in positive degrees, unlike $H^\ast(LBG;\,{\mathbb {Z}}_\ell)$.
This illustrates the difficulty for
the module structure to be free of rank 1 over ${\mathbb {Z}}/\ell^k$
for $k$ large. We will not describe the behavior
over ${\mathbb {Z}}/\ell^k$ or ${\mathbb {Z}}_\ell$--coefficients further in this paper,
but note that
a key question seems to be understanding the
(non-)divisibility and order of the ``integral fundamental class'',
namely the image of the fundamental class of $G$ under the map
$H^{{\mathbb {Z}}_\ell}_d(G) \to H^{{\mathbb {Z}}_\ell}_d(BG^{h\sigma})$,
where we have written $H^{{\mathbb {Z}}_\ell}_\ast(X)$
for $\lim_k H_\ast(X;\,{\mathbb {Z}}/\ell^k)$.
\end{rem}
| {
"timestamp": "2020-03-18T01:15:28",
"yymm": "2003",
"arxiv_id": "2003.07852",
"language": "en",
"url": "https://arxiv.org/abs/2003.07852",
"abstract": "We show that the mod $\\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\\ell$ admits the structure of a module over the mod $\\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration.We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\\ell$-compact fixed point group depending on the order of $q$ mod $\\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.",
"subjects": "Algebraic Topology (math.AT); Group Theory (math.GR); Representation Theory (math.RT)",
"title": "String topology of finite groups of Lie type",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846640860382,
"lm_q2_score": 0.72487026428967,
"lm_q1q2_score": 0.7092019500330066
} |
https://arxiv.org/abs/1810.03772 | The existence of perfect codes in Doob graphs | We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers. | \section{Introduction}
{T}{he} codes in Doob graphs are special cases of codes over Eisenstein--Jacobi integers, see, e.g., \cite{Huber94,MSBG:2008}, which can be used for the information transmission in the channels with two-dimensional or complex-valued modulation. The vertices of a Doob graph can be considered as words in the mixed alphabet consisting of the elements of the quotient (modulo $4$ and modulo $2$) rings of the ring of Eisenstein--Jacobi integers, see, e.g., \cite{Kro:perfect-doob}. In contrast to the cases considered in \cite{Huber94,MSBG:2008}, $4$ is not a prime number, and the quotient ring is not a field. This fact is not a problem from the point of view of the modern coding theory, which has a reach set of algebraic and combinatorial tools to deal with rings, see, e.g., \cite{SAS:codes&rings}; moreover, studying codes in the Doob graphs is additionally motivated by the application of association schemes in coding theory \cite{Delsarte:1973}:
the algebraic parameters of the schemes associated with these graphs are the same as for the quaternary Hamming scheme (this fact can be also treated from the point of view of the corresponding distance-regular graphs).
In this paper, we completely solve the problem of existence of perfect codes in the class of Doob graphs.
Namely,
we show the existence of $1$-perfect codes
in the Doob graph $D(m,n)$ for all $m$ and $n$ that satisfy the obvious necessary condition: the size $6m+3n+1$ of a ball of radius $1$ divides the number $4^{2m+n}$ of vertices. In the previous papers \cite{KoolMun:2000,Kro:perfect-doob,SHK:additive}, the problem was solved only for the cases when the parameters satisfy additional conditions admitting the existence of linear or additive perfect codes, or for small values of $m$.
The class of Doob graphs is a class of distance-regular graphs of unbounded diameter, and the problem considered can be viewed in the general context of the problem of existence of perfect codes in distance-regular graphs. We mention some known results in this area,
mainly concentrating on the distance-regular graphs important for coding theory.
A connected graph is called distance-regular if there are constants $s_{ij}$ such that for every $i$, $j$ and for every vertex $x$, every vertex $y$ at distance $i$ from $x$ has exactly $s_{ij}$ neighbors at distance $j$ from $x$.
In the Hamming graphs $H(n,q)$, the problem of complete characterization of parameters of perfect codes is solved only for the case when $q$ is a prime power \cite{ZL:1973,Tiet:1973}: there are no nontrivial perfect codes except the $e$-perfect repetition codes in $H(2e+1,2)$, the $3$- and $2$-perfect Golay codes \cite{Golay:49} in $H(23,2)$ and $H(11,3)$, respectively, and the $1$-perfect codes in $H((q^k-1)/(q-1),q)$. In the case of a non-prime-power $q$, no nontrivial perfect codes are known, and the parameters for which
the nonexistence is not proven are restricted by $1$- and $2$-perfect codes (the last case is solved for some values of $q$), see \cite{Heden:2010:non-prime} for a survey of the known results in this
area. The problem of the (non)existence of perfect codes in the Johnson graphs $J(n,w)$ is known as Delsarte's conjecture, see \cite{Etzion:2007:Johnson} and \cite{Gordon:2006} for the known nonexistence results; in general, the problem remains open.
An interesting open problem is connected with the problem of existence of perfect codes in the doubled Johnson graph
$J(2w+1,w,w+1)$ (the subgraph of $H(2w+1,2)$ induced by the words of weight $w$ and $w+1$): the existence of such codes is equivalent to the existence of Steiner systems $S(w,w+1,2w+2)$; in particular, the Steiner quadruple system $S(3,4,8)$ and the small Witt design $S(5,6,12)$
\cite{Carmichael:31,Witt:37} correspond to nontrivial perfect codes in $J(7,3,4)$ and $J(11,5,6)$ (in general, the problem remains open).
In the Grassmann graphs $J_q(n,w)$ and the bilinear forms graphs $B_q(m,n)$, nontrivial perfect codes do not exist \cite{Chihara87}, see also~\cite{MarZhu:1995}.
The Doob graph $D(m, n)$ is the Cartesian product
of $m$ copies of the Shrik\-hande graph
and $n$ copies of the complete graph of order $4$
(detailed definitions are given in the next
section).
It is a distance regular graph of diameter $2m + n$
with the same parameters (intersection array) as the
Hamming graph $H(2m + n, 4)$.
On the other hand,
the vertices of the Doob graph can be naturally associated
with the elements of the module
$\mathrm{GR}(4^2)^m \times {\mathbb F}_4^n$ over the Galois ring $\mathrm{GR}(4^2)$
or with the elements of the module
${\mathbb Z}_4^{2m} \times {\mathbb Z}_2^{2n'} \times {\mathbb Z}_4^{n''}$ over ${\mathbb Z}_4$, where $n'+n''=n$.
In this way,
the Doob graph is a Cayley graph on the corresponding module.
The submodules of the first module are called the linear codes in $D(m,n)$;
the submodules of ${\mathbb Z}_4^{2m} \times {\mathbb Z}_2^{2n'} \times {\mathbb Z}_4^{n''}$
are called the additive codes in $D(m,n)$.
The history of studying perfect codes in Doob graphs started from the paper
\cite{KoolMun:2000}, where it was shown that nontrivial $e$-perfect codes
in $D(m, n)$ can only exist when $e = 1$ and $2m + n = (4^k- 1)/3$
for some integer $k$ and two $1$-perfect codes, in $D(2,1)$ and $D(1,3)$,
were constructed.
In~\cite{Kro:perfect-doob}, infinite series of perfect codes in Doob graphs were obtained. In particular, it was shown that the necessary condition $2m + n = (4^k-1)/3$ is sufficient if $m < n-o(2m+n)$;
the class of linear perfect codes was completely characterized; a class of additive perfect codes was constructed and necessary conditions on $m$, $n'$, $n''$ for the existence of additive perfect codes in $D(m,n'+n'')$ were obtained (in a recent work \cite{SHK:additive}, it was shown that those conditions are also sufficient).
\section{Definitions}
The Shrikhande graph $\mathrm{Sh}$ can be naturally defined on the pairs of elements from ${\mathbb Z}_4$. Two such pairs $(x_1,x_2)$ and $(y_1,y_2)$ are adjacent if their difference
$(x_1-y_1,x_2-y_2)$ is one of $(0,1)$, $(0,3)$, $(1,0)$, $(3,0)$, $(1,1)$, $(3,3)$ (so, $\mathrm{Sh}$ is a Cayley graph on ${\mathbb Z}_4^2$).
We will use two representations of the complete graph $K_4$.
In the first one, $K_4({\mathbb Z}_4)$, its vertices are the elements $0$, $1$, $2$, $3$ of ${\mathbb Z}_4$; in the second, $K_4({\mathbb F}_4)$, the elements $0$, $1$, $\xi$, $\xi^2$ of the finite field ${\mathbb F}_4$ of order $4$.
If $m$ is even, then $D(m,n)$ will be considered as the Cartesian product of $m$ copies of $\mathrm{Sh}$ and $n$ copies of $K_4({\mathbb F}_4)$ (in particular, $D(0,n)$ is the Hamming graph $H(n,4)$).
If $m$ is odd, then $D(m,n)$ will be considered as the Cartesian product of $m$ copies of $\mathrm{Sh}$, two copies of $K_4({\mathbb Z}_4)$ and $n-2$ copies of $K_4({\mathbb F}_4)$.
So, the vertex set is the set of words of length $2m+n$ from $({\mathbb Z}_4^{2})^m\times {\mathbb F}_4^{n}$ or
$({\mathbb Z}_4^{2})^m\times {\mathbb Z}_4^2\times {\mathbb F}_4^{n-2}$,
and two vertices are adjacent if their coordinatewise difference has exactly one non-zero position $i$, $i>2m$, or exactly one non-zero position $i$, $i\le 2m$, with value $1$ or $3$, or exactly two nonzero positions $2i-1$, $2i$, $i\in \{1,\ldots,m\}$ with values $1,1$ or $3,3$.
The distance between
two vertices $\bar x$ and $\bar y$ of $D(m,n)$ (as well as in any other connected graph) is defined as the number of edges in the shortest path connecting $\bar x$ and $\bar y$.
Equivalently, the distance is equal to the sum of distances between the corresponding components of $\bar x$ and $\bar y$: $m$ Shrikhande components and $n$ $K_4$-components.
In any graph, an $e$-perfect code is defined as a set of vertices such that every ball of radius $e$ contains exactly one code vertex. We define a $1$-perfect Hamming code $\mathcal H$ in $H(n,4)$, $n=(4^k-1)/3$, by the check matrix consisting of all columns of height $k$ whose first nonzero element is $1$. To be explicit, we require the columns to be inverse-lexicographically ordered, for example ($k=3$),
$$\!\!\!\left[\!\!
\begin{array}{c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c}
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0&0 \\
\xi^2&\xi^2&\xi^2&\xi^2&
\xi&\xi&\xi&\xi&
1&1&1&1&0&0&0&0 &1&1&1&1&0 \\
\xi^2&\xi&1&0&
\xi^2&\xi&1&0&
\xi^2&\xi&1&0&
\xi^2&\xi&1&0&
\xi^2&\xi&1&0&1
\end{array}\!\!\right]
$$
\section{Construction}
The approach of the construction for $1$-perfect codes in $D(m,n)$ is partially similar to that of \cite{KoolMun:2000} for tight $2$-designs (the codes formally dual to $1$-perfect). We start with the Hamming code $\mathcal H$ over ${\mathbb F}_4$ in $H(2m+n,4)$ and replace subwords of length $4$ corresponding to the positions
$4i-3$, $4i-2$, $4i-1$, $4i$ of the codewords by
subwords of length $4$ over ${\mathbb Z}_4$, treated
as elements of $D(2,0)$ if $i\le [m/2]$
or $D(1,2)$ if $i=(m+1)/2$.
In details, there are some differences with the construction in \cite{KoolMun:2000}.
For the code dual to $\mathcal H$,
there are only $16$ possibilities for
subwords in the considered quadruples of coordinates, and the substitution function used in \cite{KoolMun:2000} is an isometry from the corresponding subcode in $H(4,4)$ into
$D(2,0)$ ($D(1,2)$).
In our case, all $256$ possible length-$4$ words occur as subwords, and there is no such isometry (indeed, the graphs $H(4,4)$,
$D(1,2)$, $D(2,0)$ are not isomorphic). However, for the resulting code being $1$-perfect, we need not control the distance between any two codewords;
it is sufficient only to ensure that this distance cannot be $1$ or $2$.
To do this, we construct the substitution bijection between $H(4,4)$
and $D(2,0)$ ($D(1,2)$) using the principles of the generalized concatenated construction \cite{Zin1976:GCC}.
It occurs that the resulting construction is close to a variant of the generalized concatenated construction for $1$-perfect codes in $H(n,q)$ presented in
\cite{Romanov:concat}.
\subsection{Codes in $D(1,2)$, $D(2,0)$ and $H(4,4)$.}
To construct a substitution function with the desired properties, in each of graphs $H(4,4)$, $D(1,2)$, $D(2,0)$, we need two additive codes, of distance $3$ and $2$ and cardinality $16$ and $64$, respectively.
\begin{lemma}\label{lemma:d4}
Denote
$$\bar x = (0,1,2,3),\
\bar y = (1,0,1,2) \in {\mathbb Z}_4^4;$$
$$\bar z = (0,0,1,1) \in {\mathbb Z}_4^4;$$
$$\bar u = (0,0,0,2),\ \bar v = (0,0,2,0) \in {\mathbb Z}_4^4;$$
$$\bar x'=(1,1,1,1),\ \bar y'=(0,1,\xi,\xi^2) \in {\mathbb F}_4^4;$$
$$\bar z'= (0,0,1,1)\in {\mathbb F}_4^4.$$
Define
$$C''=
\langle \bar x, \bar y \rangle,\quad C'=\langle \bar x, \bar y, \bar z \rangle;$$
$$
D''=
\langle \bar x, \bar y \rangle,
\quad D'=\langle \bar x, \bar y, \bar u, \bar v \rangle;$$
$$E''=
\langle \bar x', \bar y' \rangle,\quad E'=
\langle \bar x', \bar y', \bar z' \rangle.$$
Then
\begin{itemize}
\item[(a)] $C''\subset C'$,
$D''\subset D'$,
$E''\subset E'$;
\item[(b)] $C'$, $D'$, $E'$
are distance-$2$ codes
of cardinality $64$ in
$D(1,2)$, $D(2,0)$, $H(4,4)$,
respectively;
\item[(c)] $C''$, $D''$, $E''$
are distance-$3$ codes
of cardinality $16$ in
$D(1,2)$, $D(2,0)$, $H(4,4)$,
respectively.
\end{itemize}
\end{lemma}
\begin{proof}
(a) is trivial.
(b). Every codeword of $C'$ is orthogonal to $(1,1,1,3)$. It is easy to see that such a word cannot have weight $1$ in $D(1,2)$. The cardinality of $C'$ is $4\cdot 4\cdot 4$, as $\bar x$, $\bar y$, $\bar z$ are linearly independent.
The proof for $D'$ and $E'$ is similar. $D'$ is orthogonal to $(0,2,0,2)$ and $(2,0,2,0)$; and $E'$ is orthogonal to $(1,1,1,1)$.
(c). The cardinalities of the codes $C''$, $D''$, $E''$ are easy to check. Next, it is readable that a nontrivial linear combination of $\bar x'$ and $\bar y'$ cannot have less than $3$ nonzeros; so, $E''$ is distance-$3$.
The minimum weight of $C''$ and $D''$ is easy to see from the complete list of codewords:
$$
\begin{array}{r@{\ }c@{\ }c@{\ }l}
C''=D''=
\{(0,0,0,0),&(0,1,2,3),&(0,2,0,2),& (0,3,2,1),\\
(1,0,1,2),& (1,1,3,1),& (1,2,1,0),& (1,3,3,3),\\
(2,0,2,0),& (2,1,0,3),& (2,2,2,2),& (2,3,0,1),\\
(3,0,3,2),& (3,1,1,1),& (3,2,3,0),& (3,3,1,3)\}.
\end{array}
$$
\end{proof}
\begin{lemma}\label{l:e}
Let $\bar c=(c_1,\ldots,c_n)$ be a codeword of the Hamming code $\mathcal H$, and let $\bar e = (e_1,e_2,e_3,e_4)$ be a codeword of the code $E''$ defined in Lemma~\ref{lemma:d4}. Then for every $j$, $0\le j< (n-1)/4$, the word
$\bar b=(b_1,\ldots,b_n)$ whose components are
$$ b_i=
\left\{
\begin{array}{ll}
c_i+e_{i-4j}
& \mbox{if}\ i\in\{4j+1,4j+2,4j+3,4j+4\}, \\
c_i & \mbox{otherwise}
\end{array}
\right.
$$
is also a codeword of $\mathcal H$.
\end{lemma}
\begin{proof}
It is sufficient to prove the statement for the case when $\bar c$ is the all-zero word.
For the all-zero $\bar c$,
the word $\bar b$ has the form
$(0,\ldots,0,e_1,e_2,e_3,e_4,0,\ldots,0)$,
and its syndrome $P \bar b$ coincides
with $P_{(4j+1,4j+2,4j+3,4j+4)} \bar e$, where the matrix $P_{(4j+1,4j+2,4j+3,4j+4)}$ is composed from the four corresponding columns of $P$. By the construction of $P$ (recall, it consists of all different columns whose first nonzero element is $1$ placed in the inverse lexicographical order), the considered submatrix has the last row $(\xi^2,\xi,1,0)$, while the other rows are multiples of $(1,1,1,1)$.
From the definition of the code $E''$ in Lemma~\ref{lemma:d4}, we see that its
codewords are orthogonal to both $(\xi^2,\xi,1,0)$ as $(1,1,1,1)$ (indeed, this is true for the base codewords $\bar x'$ and $\bar y'$). It follows that
$P_{(4j+1,4j+2,4j+3,4j+4)} \bar e = \bar 0$ and, hence, $P \bar b=\bar 0$. That is, $\bar b$ belongs to $\mathcal H$.
\end{proof}
\begin{lemma}\label{l:cde}
For every two cosets $C''_1$, $C''_2$
of $C''$ that are not subsets of the same coset of $C'$, for every $\bar x$ from $C''_1$, there is $\bar y$ from $C''_2$ at distance $1$ from $\bar x$. The same holds for the cosets of $D''$
that are not in one coset of $D'$, and for the cosets of $E''$ that are not in one coset of $E'$.
\end{lemma}
\begin{proof}
The statement is proven by the following counting argument. The word $\bar x$ has exactly $12$ neighbors. Two neighbors cannot belong to the same coset of $C''$, because $C''$ is distance-$3$. No one of these $12$ neighbors belongs to the same coset of $C'$ as $\bar x$, because $C'$ is distance-$2$. Since there are $16$ cosets of $C''$ and $4$ of them are subsets of the same coset of $C'$ containing $\bar x$, each of the remaining $12$ cosets contains exactly one neighbor of $\bar x$.
\end{proof}
\subsection{Main theorem}
\begin{theorem}\label{th:main}
Let $\mathcal H$ be the Hamming code in $H((4^k-1)/3,4)$ whose check matrix consists
of all columns with first nonzero element $1$, in the inverse lexicographical order.
Let the codes $E''$, $E'$
in $H(4,4)$,
the codes $C''$, $C'$
in $D(1,2)$,
the codes $D''$, $D'$
in $D(2,0)$ be defined as in Lemma~\ref{lemma:d4}.
Let $\phi$ be a bijection between
the vertices of $H(4,4)$ and $D(2,0)$
such that
\begin{enumerate}
\item[\rm (a)] $\bar a$ and $\bar b$
belong to the same coset of $E''$ if and only if $\phi(\bar a)$ and $\phi(\bar b)$ belong to the same coset of~$D''$;
\item[\rm (b)] $\bar a$ and $\bar b$ belong to the same coset of $E'$ if and only if $\phi(\bar a)$ and $\phi(\bar b)$
belong to the same coset of~$D'$.
\end{enumerate}
Similarly, let $\psi$ be a bijection between
the vertices of $H(4,4)$ and $D(1,2)$
such that
\begin{enumerate}
\item[\rm (c)]$\bar a$ and $\bar b$
belong to the same coset of~$E''$
if and only if
$\psi(\bar a)$ and $\psi(\bar b)$
belong to the same coset~of~$C''$;
\item[\rm (d)] $\bar a$ and $\bar b$
belong to the same coset of~$E'$
if and only if
$\psi(\bar a)$ and $\psi(\bar b)$
belong to the same coset of~$C'$.
\end{enumerate}
Let $m$ and $n$ be positive integers
such that $2m+n=(4^k-1)/3$.
If $m$ is even, then
\begin{eqnarray*}
\mathcal C = \left\{ \left(\vphantom{|^2}\phi(x_1,x_2,x_3,x_4), \ldots,
\phi(x_{2m-3},x_{2m-2},x_{2m-1},x_{2m}),
x_{2m+1}, \ldots,x_{2m+n} \right):
\vphantom{|^2}
(x_{1}, \ldots,x_{2m+n})
\in \mathcal{H}\right\}
\end{eqnarray*}
is a $1$-perfect code in $D(m,n)$.
If $m$ is odd, then
\begin{eqnarray*}
\mathcal C = \left\{ (\vphantom{|^2}\phi(x_1,x_2,x_3,x_4), ...,
\phi(x_{2m{-}5},x_{2m{-}4},x_{2m{-}3},x_{2m{-}2}),
\psi(x_{2m{-}1},x_{2m},x_{2m{+}1},x_{2m{+}2}),\left.
x_{2m+1}, ...,x_{2m+n} \right):\right. \\
\phantom{aaaaaaaaaaaaaaaa}
\left.\vphantom{|^2}
(x_{1}, \ldots,x_{2m+n})
\in \mathcal{H}\right\}
\end{eqnarray*}
is a $1$-perfect code in $D(m,n)$.
\end{theorem}
\begin{proof}
We will consider the case when $m$ is even; the odd case is similar.
Assume the receiver get a word
$\bar y =(y_1,\ldots,y_{2m+n}) \in {\mathbb Z}_4^{2m} \times {\mathbb F}_4^n $, associated
with a vertex of $D(m,n)$.
To decode the message under the assumption that
an error of weight at most $1$ occurred,
one should find a codeword $\bar c$
at distance at most $1$ from $\bar y$.
Consider
\begin{eqnarray*}
&&\bar x = (\phi^{-1} (y_1,y_2), \ldots, \phi^{-1}
(y_{2m-1},y_{2m}), y_{2m+1}, \ldots, y_{2m+n}) \in {{\mathbb F}_4}^{2m+n}.
\end{eqnarray*}
If $\bar x$ is a codeword of $\mathcal H$, then, by the definition of $\mathcal C$, we have $\bar c=\bar y\in \mathcal C$.
Assume that $\bar x\not\in\mathcal H$. Since $\mathcal H$ is a $1$-perfect code, there is $\bar b=(b_1,\ldots,b_{2m+n})\in\mathcal H$ at distance $1$ from $\bar x$.
We consider the codeword $\bar z\in \mathcal C$ defined as
\begin{eqnarray*}
&&\bar z=(z_1,\ldots,z_{2m+n}) = (\phi(b_1,b_2,b_3,b_4), \ldots,
\phi(b_{2m-3},b_{2m-2},b_{2m-1},b_{2m}), b_{2m+1},\ldots,b_{2m+n}).
\end{eqnarray*}
Note that $\bar z$ is not necessarily the required $\bar c$.
However, we can state the following.
\begin{itemize}
\item[(i)] \emph{If $\bar b$ differs from $ \bar x$ in one of the last $n$ coordinates, then $ \bar z$ and $ \bar y$ differ in exactly one, the same as $\bar b$ and $\bar x$, coordinate; so, $\bar c= \bar z$ in this case.}
Indeed, $\bar z$ and $\bar y$ trivially coincide in the other coordinates.
\item[(ii)] \emph{If $\bar b$ differs from $\bar x$ in one of the first $2m$ coordinates,
say,
$(b_{4i-3},b_{4i-2},b_{4i-1},b_{4i})\ne (x_{4i-3},x_{4i-2},x_{4i-1},x_{4i})$, then there is $(c_{4i-3},c_{4i-2},c_{4i-1},c_{4i}) \in {\mathbb Z}_4^4$ in the same coset of $D''$ as $(z_{4i-3},z_{4i-2},z_{4i-1},z_{4i})$ such that
$$ \bar c=(z_1,\ldots,z_{4i-4}, c_{4i-3},c_{4i-2},c_{4i-1},c_{4i}, z_{4i+1},\ldots,z_{2m+n})$$
at distance $1$ from $\bar y$.
Moreover, $ \bar c$ is a codeword of $\mathcal C$.
} Indeed, the first part of the claim is straightforward from Lemma~\ref{l:cde} and the definition of the map $\phi$. From Lemma~\ref{l:e} and the construction of $\mathcal C$, we have $ \bar c \in \mathcal C$.
\end{itemize}
In any case, there is a codeword
$\bar c \in \mathcal C$
at distance at most $1$ from $\bar y$. From standard counting arguments (the size of the space equals the size of the code multiplied by the size of a radius-$1$ ball), we see that such codeword is unique. Therefore, the code is $1$-perfect.
\end{proof}
So, if there is a $1$-perfect code in a $4$-ary Hamming graph, then there is a $1$-perfect code in every Doob graph of the same diameter.
\begin{corollary}
The Doob graph $D(m, n)$ has a non-trivial $e$-perfect code if and only if $e = 1$ and
there is a positive integer $k$ such that
$2m + n = (4^k-1)/3$.
\end{corollary}
\begin{proof}
For the ``only if'' part of the statement, see \cite[Theorem~3]{KoolMun:2000}. Theorem~\ref{th:main} provides the ``if'' part.
\end{proof}
\section{Conclusion}
For every Doob graph $D(m,n)$ that satisfies
the obvious ball-packing necessary condition
on the existence of $1$-perfect codes,
we can construct such a code
by Theorem~\ref{th:main}.
In general, the code constructed
is not linear or even additive
(closed with respect to addition).
Moreover, as was shown in
\cite[Theorem~1]{Kro:perfect-doob},
existence of additive $1$-perfect codes
implies additional conditions
on the parameters $m$ and $n$.
Namely,
$2m+n=(2^{\Gamma+2\Delta}-1)/3$,
$3n=2^{\Gamma+\Delta}-1-2n''$,
$1\ne n''\le 2^\Delta -1$
for some nonnegative integer
$\Gamma$, $\Delta$, $n''$.
Examples of Doob graphs
for which additive $1$-perfect codes do not exist,
while unrestricted $1$-perfect codes
can be constructed by Theorem~\ref{th:main}, are
$D(6,9)$, $D(9,3)$, $D(10,1)$.
As can be seen from the proof of the theorem,
we do not need additivity
to have a good decoding algorithm.
Indeed, decoding the constructed code
in the Doob graph
is not more complicate
than decoding the original
$4$-ary Hamming code of length $2m+n$;
all additional operations
(mainly, applying $\phi$ and $\phi^{-1}$)
take $o(2m+n)$ time.
\providecommand\href[2]{#2} \providecommand\url[1]{\href{#1}{#1}}
\def\DOI#1{{\small {DOI}:
\href{http://dx.doi.org/#1}{#1}}}\def\DOIURL#1#2{{\small{DOI}:
\href{http://dx.doi.org/#2}{#1}}}
| {
"timestamp": "2018-10-10T02:04:13",
"yymm": "1810",
"arxiv_id": "1810.03772",
"language": "en",
"url": "https://arxiv.org/abs/1810.03772",
"abstract": "We solve the problem of existence of perfect codes in the Doob graph. It is shown that 1-perfect codes in the Doob graph D(m,n) exist if and only if 6m+3n+1 is a power of 2; that is, if the size of a 1-ball divides the number of vertices. Keywords: perfect codes, distance-regular graphs, Doob graphs, Eisenstein-Jacobi integers.",
"subjects": "Information Theory (cs.IT); Discrete Mathematics (cs.DM); Combinatorics (math.CO)",
"title": "The existence of perfect codes in Doob graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9783846634557751,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7092019495761477
} |
https://arxiv.org/abs/1210.1252 | On Permutation Binomials over Finite Fields | Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (d -1)d$, where $d = {gcd}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\mathbb{F}_{q}$ from a permutation binomial over $\mathbb{F}_{q}$. | \section{Introduction}
Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements. A polynomial $f(x) \in \mathbb{F}_{q}$ is called a permutation polynomial of $\mathbb{F}_{q}$ if the induced map $f: \mathbb{F}_{q} \rightarrow \mathbb{F}_{q}$ is one to one. The study of permutation polynomials goes back to Hermite \cite{Her} for $\mathbb{F}_{p}$ and Dickson \cite{Dic} for $\mathbb{F}_{q}$. The interest on permutation polynomials increased in part because of their application in cryptography and coding theory. Despite the interest of numerous people on the subject, characterizing permutation polynomials and finding new families of permutation polynomials remain open questions. Carlitz conjectured that given an even positive integer $n$ there exists a constant $C(n)$ such that for $q > C(n)$, then there are no permutation polynomials of degree $n$ over $\mathbb{F}_{q}$. Fried, Guralnick and Saxl \cite{Fri} proved Carlitz's conjecture. Permutation monomials are completely understood, however permutation binomials are not well understood. Niederreiter and Robinsom \cite{Nie} proved the following theorem.
\begin{thm}
Given a positive integer $n$, there is a constant $C(n)$ such that for $q > C(n)$, no polynomial of the form $ax^{n} + bx^{m} + c \in \mathbb{F}_{q}[x]$, with $n > m> 1$, $\gcd(n, m) = 1$, and $ab \neq 0$, permutes $\mathbb{F}_{q}$.
\label{thm:Neiderreiter}
\end{thm}
The constant $C(n)$ in Theorem 1 is not explicit. Turnwald \cite{Tur} improved Theorem 1 and proved the following.
\begin{thm}
If $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{q}$, where $n>m>0$ and $a \in {\mathbb{F}_{q}}^{*}$, then either $q \leq (n-2)^{4} + 4n -4$ or $n = mp^{i}$.
\end{thm}
Turnwald's proof uses Weil's lower bound \cite{Wei} for the number of the points on the curve $(f(x)-f(y)/(x-y)$ over $\mathbb{F}_{q}$. For $q$ a prime number, Turnwald \cite{Tur} proved the following.
\begin{thm}
If $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p < n{\mbox{max}}(m,n-m)$.
\end{thm}
For $m =1$, Wan \cite{Wan2} proved the following.
\begin{thm}
If $f(x) = ax^{n} + x$ permutes $\mathbb{F}_{p}$, where $n>1$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (n-1) \cdot{\mbox{gcd}}(n-1,p-1)$.
\end{thm}
The bounds in Theorem 3 and Theorem 4 are of different nature. The bound in Theorem 3 is given in term of ${\mbox{max}}(m,n-m)$, whereas the bound in Theorem 4 is given in term of ${\mbox{gcd}}(n-1,p-1)$.
Theorem 3 and Theorem 4 have been improved (\cite{Mas1}) as follow.
\begin{thm}
If $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, and $d = {\mbox{gcd}}(n-m,p-1)$, then $p -1 \leq (n -1) \cdot {\mbox{max}}(m,d)$.
\end{thm}
The bounds in the theorems above are not given in term of $d$ only, and one can ask whether the prime $p$ can be bounded in term of $d$ only. The answer was given by Masuda and Zieve \cite{Mas1} who proved the following.
\begin{thm}
If $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$. Then $p -1 \leq (d +1)d$.
\end{thm}
Clearly, Theorem 6 improves Theorem 5 whenever $d-1 \leq n-1$, which is always the case except when $m =1$ and $(n-1) \mid (p-1)$. In section 2, we prove the following.
\begin{thm}
If $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$. Then $p -1 \leq (d -1)d$.
\end{thm}
Clearly, Theorem 7 implies Theorem 6 and Theorem 5 in all cases. When $m =1$ and $(n-1) \mid (p-1)$, we will see in corollary 5 that $ p-1 \leq (n-1)(n-3)$, which improve Theorem 5.
It would be interesting to have a bound for $p$ in term of $d = {\mbox{gcd}}(n-m,p-1)$ when $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{q}$ and $q$ a power of the prime $p$. In Theorem 9, we will show how in certain cases, one can obtain from a permutation binomial $f(x) \in \mathbb{F}_{q}[x]$ a new permutation binomial $g(x) \in \mathbb{F}_{p}[x]$, and deduce in Corollary 6, a bound of $p$ in term of $d = {\mbox{gcd}}(n-m,p-1)$. Some consequences of this theorem are stated in sections $2$ and $3$
We fix some notation which will be used through this paper. The letter $p$ always denotes a prime number, and $\mathbb{F}_{q}$ the finite field containing $q = p^{r}$ elements. For any polynomial $g(x) \in \mathbb{F}_{q}[x]$, we denote by $\overline{g(x)}$ the unique polynomial of degree at most $q-1$, with coefficients in $\mathbb{F}_{q}$ such that $g(x) \equiv \overline{g(x)} \pmod{(x^{q}-x)}$. When we refer to a binomial $f(x)$ over $\mathbb{F}_{q}$, we always mean a polynomial $f(x) \in \mathbb{F}_{q}[x]$ of the form $f(x) = ax^{n} + x^{m}$ with the nonrestrictive condition $\gcd(m, n) = 1$ (see \cite[Ex. 2. 1]{Sma1}), $n>m$, and $a \neq 0$. The integer $d = \gcd(n-m, q-1)$ will play an important role. It is well known that if $-a\in(\mathbb{F}_{q}^{\star})^d$ , then the equation $f(x)=0$ has $d+1$ distinct solutions in $\mathbb{F}_{q}$, hence $f(x)$ is not a permutation of $\mathbb{F}_{q}$\cite{Sma2}. In particular this claim is true if $d=1$.
\section{Non existence of Permutation Binomials of Certain Shapes}
An old and stricking result in the theory of permutation polynomials, is the following theorem proved by Hermite for the prime fields and Dickson in the general case.
\begin{thm}
\label{thm:Hermite/Dickson}
Let $p$ be a prime number, $q = p^{r}$, and $g(x) \in \mathbb{F}_{q}[x]$. Then $g(x)$ is a permutation polynomial if and only if
\begin{itemize}
\item [(i)] $g(x) = 0$ has a unique solution in $\mathbb{F}_{q}$.
\item[(ii)] For every $l \in \lbrace 1, \ldots, q-2 \rbrace, {\mbox{deg}}\,\overline{g^{l}(x)} \leq q-2$.
\end{itemize}
\end{thm}
For binomials, we deduce from Theorem \ref{thm:Hermite/Dickson} the following corollary.
\begin{cor}
Let $f(x) = ax^{n} + x^{m} \in \mathbb{F}_{q}[x]$, such that $a\neq 0$ and $(m, n) = 1$. Let $d = \gcd(n-m, q-1)$. Suppose $d \geq 2$. Then $f(x)$ is a permutation polynomial of $\mathbb{F}_{q}$ if and only if
\begin{itemize}
\item [(i)] $f(x) = 0$ has a unique solution in $\mathbb{F}_{q}$.
\item[(ii)] For every $l \in \lbrace 1, \ldots, q-2\rbrace$ such that $d \mid l$, we have deg $\overline{f^{l}(x)} \leq q - 2$.
\end{itemize}
\end{cor}
\begin{proof}
From Theorem \ref{thm:Hermite/Dickson}, we have only to prove that if $l \in \lbrace 1,\ldots, q-2 \rbrace$ and $d \nmid l$, then deg $\overline{f^{l}(x)} \leq q-2$. Let $k$ be an integer and let $\overline{k}$ be the integer in $\lbrace1,\ldots,q-1 \rbrace$ such that $k \equiv \overline{k} \pmod{q-1}$. Then
\begin{equation*}
x^{k} \equiv \left\lbrace \begin{array}{ll}
1 & \mbox{if } k = 0\\
x^{\overline{k}} & \mbox{if } k \neq 0 \end{array}\right.
\end{equation*}
It follows that if $k > 0$, then $x^{k} \equiv x ^{q-1} \pmod{x^{q}-x}$ if and only if $k \equiv 0 \pmod{q-1}$. Suppose that there exists $l \in \lbrace 1,\ldots, q-2 \rbrace$ with $d \nmid l$ such that deg $\overline{f^{l}(x)} = q-1$. We deduce from
\begin{align}
(ax^{n}+x^{m})^l & = \sum^{l}_{j=0}\binom{l}{j}a^{j}x^{nj+m(l-j)}\nonumber\\
& = \sum^{l}_{j=0}\binom{l}{j}a^{j}x^{(n-m)j+lm}
\label{eq: 1}
\end{align}
that there exists an integer $j \in \lbrace 0, \ldots, l \rbrace$ such that
\begin{equation*}
x^{(n-m)j + lm} \equiv x^{q-1} \pmod{x^{q}-x}
\end{equation*}
Hence, $(n-m)j + lm > 0$ and $(n-m)j+lm \equiv 0 \pmod{q-1}$. Since $d = \gcd(n-m, q-1)$, then $d \mid (n-m)$ and $d \mid q-1$. But $gcd(n, m) = 1$ implies that $\gcd(d, m) = 1$. Then $d \mid l$ which is a contradiction.
\end{proof}
Corollary 1 reduces enormously the calculations when applying Theorem 8 to check whether a given a binomial permutes $\mathbb{F}_{q}$ or not. One needs to check the degress of only $ \[\frac{q-2}{d}\]$ polynomials instead of $q-2$ polynomials as given by Theorem 8 (ii).
For the proof of Theorem 7, we need the following lemma.
\begin{lem}
Let $f(x)$ be a binomial such that $d > 1$. Let $l \in \lbrace 1, \ldots, q-2 \rbrace$ such that $d \mid l$. Then the following assertions are equivalent
\begin{itemize}
\item[(i)] \begin{equation} {\mbox{deg}}f^{l}(x) \leq q-2 \label{eq:2} \end{equation}
\item[(ii)] \begin{equation} \sum_{\substack{j=0\\(n-m)j + lm\equiv 0\pmod{q-1}}}^{l}\binom{l}{j}a^{j} = 0 \label{eq:3} \end{equation}
\item[(iii)] \begin{equation} \sum_{\lambda = 0}^{\gamma_{l}} \binom{l}{j_{0}+\lambda (q-1)/d} \left( a^{(q-1)/d} \right)^{\lambda} = 0 \label{eq:4} \end{equation}
where $j_{0}$ is the smallest integer $\geq 0$ satisfying
\begin{equation*} j_{0} \equiv \frac{- lm}{(n-m)} \pmod{\frac{q-1}{d}} \equiv \frac{-lm}{d} / \frac{n-m}{d} \pmod{\frac{q-1}{d}} \end{equation*}
and $\gamma_{l}$ is the largest integer $\lambda$ such that \begin{equation*} j_{0} + \lambda(q-1)/d \leq l \end{equation*}
\end{itemize}
\end{lem}
\begin{proof}
From equation 1, deg $\overline{f^{l}(x)} \leq q-2$ if and only if
\begin{equation}
\sum_{\substack{j=0\\(n-m)j + lm\equiv 0\pmod{q-1}}}^{l}\binom{l}{j}a^{j} = 0
\label{eq:5}
\end{equation}
The condition $(n-m) j + lm \equiv 0 \pmod{q-1}$ is equivalent to $\frac{(n-m)}{d}j + \frac{l}{d}m \equiv 0 \pmod{\frac{q-1}{d}}$, which is equivalent to
\begin{equation}
j \equiv \frac{-lm}{(n-m)} \pmod{\frac{q-1}{d}}
\label{eq: 6}
\end{equation}
Let $j_{0}$ be the smallest integer satisfying (6). Then $j \equiv j_{0} \pmod{(q-1)/d}$. Hence, equation (5) is equivalent to
\begin{equation*}
\sum_{\lambda = 0}^{\alpha_{l}} \binom{l}{j_{0} + \lambda (\frac{a-1}{d})} \left( \left( a \right)^{\frac{q-1}{d}} \right)^{\lambda} = 0
\end{equation*}
where $\gamma_{l}$ is the largest integer $\lambda$ such that $j_{0} + \lambda \left( \frac{q-1}{d}\right) \leq l$.
\end{proof}
\noindent {\bf Proof of Theorem 7.} In the following proof, we repetedly use Equation (4). The integer $j_{0}$ appearing in this equation depends on $l$. So, it will be denoted by $j_{0}(l)$. Suppose that there exists a permutation binomial $f(x)$ over $\mathbb{F}_{p}$ such that $p - 1 > d(d-1)$, then $(p-1)/d > d-1$, i.e., $(p-1)/d \geq d$ . From Theorem 10, $(p-1)/d \neq d$, then $(p-1)/d > d$. Set $(p-1)/d = \alpha d - z$ where $\alpha > 1$ and $z$ are integers such that $z \in \lbrace 0, \ldots, d-1 \rbrace$. By Theorem 10, we may suppose that $z \in \lbrace 1, \ldots, d-1 \rbrace$. Let $j_{0}(d)$ be the unique integer determined by (4) for $l = d$. Set $j_{0}(d) = \beta d + \delta$ with $\delta \in \lbrace 0, \ldots, d-1 \rbrace$, then
\begin{equation}
j_{0}(d) < (p-1)/d < \alpha d.
\end{equation}
\begin{enumerate}
\item[(Case 1):] $j_{0}(d) \leq d$.
In this case, because $j_{0}(d) + \lambda(p-1)/d > d$ for $\lambda \geq 1$, Equation (4) reduces to $\binom{d}{j_{0}(d)} = 0$. Since $j_{0}(d) \leq d < p$, then $\binom{d}{j_{0}(d)} \neq 0$, which is a contradiction, and we can exclude this case.
\item[(Case 2):] $j_{0}(d) > d$.
Clearly, $\beta \geq 1$, and From (7), we deduce that $\beta < \alpha$. Consider Equation (4) for $l = \alpha d$. We have
\begin{align*}
\alpha j_{0}(d)& = (\frac{p-1}{d^{2}} + \frac{z}{d})j_{0}(d)\\
& = \frac{p-1}{d}\beta + z\beta + \alpha\delta \\
&\equiv z\beta + \alpha\delta \pmod{(p-1)/d}.
\end{align*}
\item[(Case 2.1):] $z\beta + \alpha\delta < (p-1)/d$.
In this case, we have
\begin{equation}
d < j_{0}(\alpha d) = z \beta + \alpha\delta < (p-1)/d < \alpha d = l.
\end{equation}
Let $\lambda$ be a positive integer, then
\begin{align*}
j_{0}(\alpha d) + \lambda(p-1)/d & \geq j_{0}(\alpha d) + (p-1)/d \\
&= z \beta + \alpha\delta + (p-1)/d \\
&> d + (p-1)/d \\
&> z + (p-1)/d \\
&= \alpha d
= l.
\end{align*}
Hence there is only one term in the left hand side of equation (4) corresponding to $l = \alpha d$, namely $\binom{\alpha d}{j_{0}(\alpha d)} = \binom{\alpha d}{z\beta + \alpha \delta}$. Since $(p-1)/d \geq d$, we have $ \alpha d < \frac{p-1}{d} + d < p $. Hence, from (8), we obtain that $j_{0}(\alpha d) = z \beta + \alpha\delta < \alpha d < p$. Then $\binom{\alpha d}{j_{0}(\alpha d)} = \binom{\alpha d}{z\beta + \alpha \delta} \neq 0$, and we reject this case.
\item[(Case 2.2):] $z\beta + \alpha\delta \geq (p-1)/d$.
Suppose that $\delta = 0$, then $z\beta + \alpha\delta = z\beta$ and since $\beta \leq \alpha - 1$, we deduce that $z\beta + \alpha\delta \leq (\alpha - 1)z \leq(\alpha - 1)(d-1)$, hence
\begin{align*}
z\beta + \alpha\delta &\leq\alpha d - \alpha - d + 1\\
& < \alpha d - z = (p-1)/d,
\end{align*}
which is a contradiction. We many suppose that $\delta$ is positive. Consider Equation (4) for $l = (\alpha - 1)d$. We have
\begin{align*}(\alpha - 1)j_{0}(d) &= (\frac{p-1}{d^{2}} + \frac{z-d}{d})j_{0}(d)\\
& = (\frac{p-1}{d^{2}} + \frac{z-d}{d})(\beta d + \delta) \\
&\equiv (z-d)\beta + (\alpha - 1)\delta \pmod{(p-1)/d}.
\end{align*}
In order to prove that $(z-d)\beta + (\alpha-1)\delta =j_{0}\left( (\alpha - 1)d \right)$, we have to show that $0 \leq (z-d)\beta + (\alpha-1)\delta < (p-1)/d$. Since $z < d$, then $(z-d)\beta < 0$, hence
\begin{align*}
(z-d)\beta + (\alpha - 1)\delta &< (\alpha - 1)\delta \leq(\alpha - 1)(d-1) \\
&= \alpha d - d - \alpha + 1 \\
&< \alpha d - z = (p-1)/d.
\end{align*}
We now look at the sign of $(z-d)\beta + (\alpha - 1)\delta$.
On the one hand side, we have $\alpha > \beta \geq 1$, hence $\alpha \geq 2$. Furthermore since $z - d < 0$ and $\beta \leq \alpha - 1$, then $(z-d)\beta \geq (z-d)(\alpha - 1)$, hence
\begin{align*}
(z-d)\beta + (\alpha - 1)\delta &\geq (z-d)(\alpha - 1) + (\alpha - 1)\delta \\
&= (\alpha - 1)(z-d+\delta) \\
&\geq z- d + \delta.
\end {align*}
On the other hand side, since $z\beta + \alpha\delta \geq (p-1)/d = \alpha d - z$, then
\begin{align*}
(z-d)\beta + (\alpha - 1)\delta &= z\beta + \alpha\delta - d\beta - \delta \\
&\geq \alpha d - z - d\beta - \delta \\
&= (\alpha -\beta)d - z - \delta \\
&\geq d - z - \delta.
\end {align*}
We have shown that $(z-d)\beta + (\alpha - 1)\delta \geq |A|$, where $A = z - d - \delta$. Hence $(z - d)\beta + (\alpha - 1)\delta \geq 0$ and then $j_{0}\left( (\alpha - 1)d \right) = (z-d)\beta + (\alpha - 1)\delta$. As in the preceding cases we prove that in the left hand side of equation (4), for $l = (\alpha - 1)d$, there is only one term. For any integer $\lambda \geq 1$, we have $(z-d)\beta + (\alpha - 1)\delta + \lambda(p-1)/d \geq (p-1)/d > l$. Equation (4) reads $\binom{(\alpha - 1)d}{(z-d)\beta + (\alpha - 1)\delta)} = 0$. But, since $(z-d)\beta < 0$, then $(z-d)\beta + (\alpha - 1)\delta < (\alpha - 1)d$. Hence $\binom{(\alpha - 1)d}{(z-d)\beta + (\alpha - 1)\delta)} \neq 0$, and the proof of Theorem 7 is complete.
\end{enumerate}
\begin{cor}
Let $f(x)$ be a permutation binomial over $\mathbb{F}_{p}$. Then $p-1 \leq d ( d -2)$ except possibly in the case $d \equiv 0 \pmod{3}$, $p = d^{2} - d + 1$ and one of the two possibilities: $n \equiv 0 \pmod{(p-1)/d}$ or $m \equiv 0 \pmod{(p-1)/d}$.
\end{cor}
\begin{proof}
Since there is no permutation binomial over $\mathbb{F}_2$ and over $\mathbb{F}_3$, we may suppose that $p\geq 5$. From Theorem 7, we have $(p-1)/d \leq d-1$. It remains to consider the case $(p-1)/d = d-1$, i.e. $p = d^{2}-d+1$. Suppose that there exists a permutation binomial over $\mathbb{F}_{p}$, $f(x) = ax^{n}+x^{m}$ such that $p = d^2 - d + 1$. Consider Equation (4) for $l = d$ and let $j_{0}$ be the integer appearinig in this equation. Since $j_{0} \in \lbrace 0, \ldots, (p-1)/d \rbrace$, then $j_{0} < d$. For any positive integer $\lambda$, we have $j_{0} + \lambda(p-1)/d \geq j_{0} + (p-1)/d > d$ except if $j_{0} = 0$ or $j_{0} = 1$. Beyond these exceptions, Equation (4) reads $\binom{d}{j_{0}} = 0$. Since $j_{0} < d < p$ this equation is impossible and we get a contradiction.
\noindent (Case $j_{0} = 0$)
Equation (4) reads $$\binom{d}{0} + \binom{d}{\frac{p-1}{d}} (a)^{\frac{p-1}{d}} = 0,$$ hence $1 + d(a)^{(d-1)} \equiv 0 \pmod p$. We deduce that $d^d\equiv(-1)^d\pmod{p}$, hence $(-d)^d\equiv 1\pmod{p}$. We have $(-d)^2\equiv d-1\pmod{p}$ and $(-d)^3\equiv 1\pmod{p}$, hence the order of $-d$ in $\mathbb{F}_p$ which is a divisor of $d$ is equal to $1$ or $3$. Since $d(d-1)=p-1$, the first possibility is excluded, hence $d\equiv 0\pmod 3$. On the other hand side, the condition $(n-m) j_{0} + dm \equiv 0 \pmod{p-1}$ implies $dm \equiv 0 \pmod{p-1}$, i.e., $m \equiv 0 \pmod {\frac{p-1}{d}}$.
\noindent (Case $j_{0} = 1$)
Equation (4) reads $$\binom{d}{1} + \binom{d}{1 + \frac{p-1}{d}} (a)^{\frac{p-1}{d}} = 0,$$ hence $d + (a)^{d} \equiv 0 \pmod p$. As in the preceding case we find $d^d\equiv(-1)^d\pmod{p}$, hence $d\equiv 0\pmod 3$. On the other hand side, the condition $(n-m) j_{0} + dm \equiv 0 \pmod{p-1}$ implies $(n-m) + dm \equiv 0 \pmod{p-1}$, i.e., $n \equiv 0 \pmod{\frac{p-1}{d}}$.
\end{proof}
The condition $d \equiv 0 \pmod{3}$, $p = d^{2} - d + 1$ in Corollary 4 occurs, for instance, for $p = 7$ and $d =3$ or $p = 31$ and $d =6$ (see \cite[Cor. 2. 5]{Mas1}). This shows that the bound of $p$ in term of $d$ in Theorem 7 is sharp.
If $f(x) = ax^{n} + x$ permutes $\mathbb{F}_{p}$, where $n>1$, $a \in {\mathbb{F}_{p}}^{*}$ and $(n-1) \mid (p-1)$, then Theorem 6 does not generalize Theorem 5 which implies that $p-1 \leq (n-1)^{2}$. Theorem 7 proved above generalizes Theorem 5 even in this case as shown by the following corollary.
\begin{cor}
If $f(x) = ax^{n} + x$ permutes $\mathbb{F}_{p}$, where $n>1$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p-1 \leq (n-1)(n-3)$.
\end{cor}
\begin{proof}
From Corollary 4, we have $p-1 \leq d(d-2)$, which implies that $p-1 \leq (n-1)(n-3)$, except if $d \equiv 0 \pmod {3}$, $\frac{p-1}{d} = d-1$, and $n \equiv 0 \pmod{ \frac{p-1}{d}}$ (because $m=1$). So we have only to consider the exceptionnel case. In this case, we have $n \equiv 0 \pmod{d-1}$ and $n-1 \equiv 0 \pmod d$. Clearly $n \neq d-1$. It follows that $n \geq 2(d-1)$. We conclude that $3 \leq d \leq \frac{n}{2} +1$. It is now easy to deduce the inequality $p-1 \leq (n-1)(n-3)$.
\end{proof}
The following result is similar to Corollary 2.4 of \cite{Mas1} except that the four polynomials arizing for $d=3$ and $p=7$ were forgoten.
\begin{cor}
If $f(x) = x^{n} + ax^{m}$ permutes $\mathbb{F}_{p}$, where $1\leq m<n<p$ and $a \in {\mathbb{F}_{p}}^{*}$, then $\gcd(n-m,p-1) > 4$ except if $d=3$, $p=7$ and $f(x)$ is one of the followings.
\noindent (i) $f(x) = x^{4} + 3 x$.
\noindent (ii) $f(x) = x^{4} - 3 x$.
\noindent (iii) $f(x) = x^{5} + 2x^{2}$.
\noindent (iv) $f(x) = x^{5} - 2x^{2}$.
\end{cor}
\begin{proof}
We conclude from Corollary 4, that if $d = 4$, then $p-1 \leq 8$, i.e., $p \leq 7$. We see from table 7.1 of \cite{Lid2} that there are no permutation binomials in this case. When $d =2$, we conclude from Corollary 4, that there are no permutation binomials in this case. When $d =3$, Corollary 4 implies that $p =7$. We see from table 7.1 of \cite{Lid2} that the only possible cases are the one listed above.
\end{proof}
\section{Permutation binomials over a subfield of $\mathbb{F}_{q}$ arising from permutation binomials over $\mathbb{F}_{q}$}
Before stating a result about the possibilty to deduce, in some cases, a permutation binomial of a subfield of $\mathbb{F}_q$ from a given permutation binomial of $\mathbb{F}_{q}$, we make the following definition.
\begin{definition}Fix the integers $m$ and $n$ such that $1\leq m<n\leq q-1$ and let $d=gcd(n-m,q-1)$. We say that the polynomials $f(x)=ax^n+x^m$ and $g(x)=bx^n+x^m$, with coefficients in $\mathbb{F}_{q}$, are $d$-equivalent and we write $f\stackrel{d}{\sim}g$ if and only if there exists $\epsilon\in(\mathbb{F}_{q})^d$ such that $b=\epsilon a$.
\end{definition}
Obviously the above relation in the set of binomials over $\mathbb{F}_{q}$, of degree at most $q-1$, where the couple $(m,n)$ is fixed, is an equivalence relation and that each equivalence class contains $(q-1)/d$ elements.
\begin{lemma}Suppose that the polynomials $f(x)=ax^n+x^m$ and $g(x)=bx^n+x^m\in\mathbb{F}_{q}[x]$, are $d$-equivalent and that $f(x)$ permutes $\mathbb{F}_{q}$, then so does $g(x)$.
\end{lemma}
\begin{proof} Since $gcd(n-m,q-1)=d$, there exit two integers $u$ and $v$ such that
\begin{equation}
u(n-m)+v(q-1)=d.
\label{eq:9}\end{equation}
The binomials $f(x)$ and $g(x)$ being $d$-equivalent, there exists $\eta\in\mathbb{F}_{q}$ such that $b=\eta^da$. Using $(9)$, we obtain $b=\eta^{u(n-m)}a$. We deduce that
\begin{align*}
g(x)&=\eta^{u(n-m)}ax^n+x^m=\eta^{-um)}[\eta^{un}ax^n+\eta^{um}x^m]\\
&=\eta^{-um}[a(\eta^{u}x)^{n}+(\eta^{u}x)^m]=\eta^{-um}f(\eta^{u}x)),
\end{align*}
and this proves our lemma.
\end{proof}
\begin{thm}
\label{theorem 3}
Let $f(x) = ax^{n} + x^{m}$ be a permutation binomial of $\mathbb{F}_{q}$ with $q=p^r$ and $s$ be a positive divisor of $r$. Let $d = {\mbox{gcd}}(n-m,q-1)$.
\begin{itemize}
\item[( 1)] There exists a binomial $g(x) =b x^{n} + x^{m}\in\mathbb{F}_{p^s}[x]$ $d$-equivalent to $f(x)$ if and only if the order of $a$ in $(\mathbb{F}_{q})^{\star }$ divides $lcm(p^s-1, (q-1)/d).$
\item [(2)]If these equivalent conditions hold, then the number of $g(x) =b x^{n} + x^{m}\in\mathbb{F}_{p^s}[x]$, $d$-equivalent to $f(x)$ is equal to $gcd(p^s-1,(q-1)/d)$ and they are all distinct as permutations of $\mathbb{F}_{p^s}$. Moreover, we have $g(x) \equiv bx^{n_{1}} + x^{m_{1}} \pmod{x^{p^s} - x}$ if
$p^s - 1 \nmid d$ and $g(x) \equiv (b+1)x^{k} \pmod {x^{p^s}-x}$ if $p^s-1 \mid d$ where $k, m_{1}, n_{1}$ are positive integers less than $p^s-1$, $m_{1} \neq n_{1}$, $\gcd(p^s-1, k) = 1$.
\item[(3)]Let $t$ be a posive integer (not necesseraly dividing $r$). There exists a binomial $g(x) =b x^{n} + x^{m}\in\big(\mathbb{F}_{p^t}\cap\mathbb{F}_{q}\big)[x]$, $d$-equivalent to $f(x)$ if and only if the order of $a$ in $(\mathbb{F}_{q})^{\star }$ divides $lcm(p^t-1, (q-1)/d).$
\end {itemize}
\end{thm}
\begin{proof}
\noindent (1) Suppose first that the order of $a$ in $(\mathbb{F}_{q})^{\star }$ divides $lcm(p^s-1, (q-1)/d).$ We will use the following claim for which the proof is omited
\textbf{Claim 1} Let $\delta, u, v$ be positive integers. Then $\delta \mid lcm(u, v)$
if and only if there exist positive integers $\delta_{1}, \delta_{2}$ such that
$\delta_{1} \mid u$, $\delta_{2} \mid v$ and $\delta = lcm(\delta_{1}, \delta_{2})$.
Let $\delta$ be the order of $a$ in $\mathbb{F}_{q}^{\star}$, then $\delta = lcm(\delta_{1}, \delta_{2})$, where
$\delta_{1}$ and $\delta_{2}$ are positive integers such that $\delta_{1} \mid p^s-1$
and $\delta_{2} \mid (q-1)/d$. Let $\xi$ be a generator of $\mathbb{F}_{q}^{\star}$, then $a
= (\xi^{(q-1)/\delta_{1}})^{i}(\xi^{(q-1)/\delta_{2}})^{j}$ for some nonnegative
integers $i$ and $j$. Let $\epsilon = \xi^{-j(q-1)/\delta_{2}}$, $b = \epsilon
a$ and $g(x)=b x^{n} + x^{m}$. Then $\epsilon^{(q-1)/d} = (\xi^{-j(q-1)/d\delta_{2}})^{q-1} = 1$ and $b^{p^s-1} = (\xi^{i(p^s-1)/ \delta_{1}})^{q-1} = 1$, hence $\epsilon\in\mathbb{F}_{q})^d$, $b\in\mathbb{F}_{p^s}$ an $g(x)\stackrel{d}{\sim}f(x)$.
Conversely, suppose that there exist $g(x)=b x^{n} + x^{m}\in\mathbb{F}_{p^s}$, $d$-equivalent to $f(x)$, then we may find a $(q-1)/d$-th root of unity $\epsilon$ such that $a=\epsilon b$. Hence $a^{ lcm(p^{s}-1, (q-1)/d)}=(\epsilon b) ^{\frac{(p^s-1)(q-1)/d}{\delta_s}}=1$, where $\delta_s=gcd(p^s-1,(q-1)/d)$. It follows that the order of $a$ in $(\mathbb{F}_{q})^{\star }$ divides $lcm(p^s-1, (q-1)/d)$.
\noindent (2) Let $\delta_s=gcd(p^s-1, (q-1)/d)$. By (1), there exists at least one permutation binomial of $\mathbb{F}_{q}$, $g(x)=c_s x^{n} + x^{m}$ with $c_s\in\mathbb{F}_{p^s}$, $d$-equivalent to $f(x)$. Let $h(x)=b_s x^n+x^m$ be any permutation binomial of $\mathbb{F}_{q}$with $b_s\in\mathbb{F}_{p^s}$, $d$-equivalent to $f(x)$, then $g(x)\stackrel{d}{\sim}h(x)$, hence there exists $\epsilon\in\mathbb{F}_{q})^d$ such that $b_s=\epsilon c_s$. We deduce that $\epsilon=b_s/c_s\in\mathbb{F}_{p^s}$. It follows that $\epsilon^{p^s-1}=1=\epsilon^{(q-1)/d}$ and then $\epsilon^{\delta_s}=1$. We conclude that $h(x)$ has the form $h(x)=\epsilon c_s x^n+x^m$ with $\epsilon$ satsfying the condition $\epsilon^{\delta_s}=1$. On the other hand any polynomial $h(x)$ of this form is $d$-equivalent to $g(x)$ and then to $f(x)$. Clearly all these $h(x)$, as permutations of $\mathbb{F}_{q}$, are distinct. Because all of them take different values at the argument $x=1$, they are distinct as permutations of $\mathbb{F}_{p^s}$. We conclude that the number of these $h$'s is equal to $\delta_s$.
To prove the last part of the theorem we reduce $g(x)$ modulo $x^{p^s} - x$. Denote by $\overline{g(x)}$
the unique polynomial over $\mathbb{F}_{p^s}$ of degree at most $p^s-1$ such that $g(x)\equiv\overline{g(x)}\pmod{x^{p^s} - x}$. Set $n =
(p^s-1)\lambda + n_{1}$ and $m = (p^s-1)\mu + m_{1}$ with $0 \leq m_{1}, n_{1}\leq p^s-2$.
If $m_{1} = 0$ or $n_{1} = 0$, then the degree of $\overline{g(x)}$ is equal to
$p^s-1$, which is excluded by the fact that $g(x)$ is a permutation polynomial of
$\mathbb{F}_{p^s}$. If $m_{1} = n_{1}$, then clearly $p^s - 1 \mid d$ and
$\overline{g(x)} = (b+1)x^{k}$, where $k = n_{1} = m_{1}$ and $\gcd(k, p^s-1) = 1$.
Suppose now that $m_{1} \neq 0$, $n_{1} \neq 0$ and $m_{1} \neq n_{1}$, then $p^s-1
\nmid d$ and $\overline{g(x)} = bx^{n_{1}} + x^{m_{1}}$. Let $k = \gcd(m_{1},
n_{1})$, then the polynomial $g_{1}(x) = bx^{n_{1}/k} + x^{m_{1}/k}$ is a permutation
binomial of $\mathbb{F}_{p^s}$.
\noindent (3) We will use the following which is certainly well known.
\textbf{Claim 2} Let $a$, $b$, $c$ be nonzero integers, then $$gcd\big(lcm(a,b),lcm(a,c)\big)=lcm\big(a,gcd(b,c)\big)$$.
Suppose that there exists a binomial $g(x) =b x^{n} + x^{m}\in\big(\mathbb{F}_{p^t}\cap\mathbb{F}_{q}\big)[x]$, $d$-equivalent to $f(x)$. Let $s=gcd(r,t)$, then $\mathbb{F}_{q}\cap\mathbb{F}_{p^t}=\mathbb{F}_{p^s}$ and by $(1)$, the order of $a$ in $\mathbb{F}_{q}$ divides $lcm(p^s-1, (q-1)/d)$. Therefore this order divides $lcm(p^t-1, (q-1)/d)$. Suppose now that the order of $a$ in $\mathbb{F}_{q}$ divides $lcm(p^t-1, (q-1)/d)$, then applying the above Claim with $a=(q-1)/d$, $b=p^t-1$ and $c=p^r-1$, we conclude that this order divides $lcm(p^s-1, (q-1)/d)$ and then by $(1)$ there exists a binomial $g(x) =b x^{n} + x^{m}\in\big(\mathbb{F}_{p^t}\cap\mathbb{F}_{q}\big)[x]$, $d$-equivalent to $f(x)$.
\end{proof}
\begin{rem}
Suppose that $p$ is odd, then under the hypothesis $(1)$ of the above theorem, we have
$\gcd(d, p^s-1) \neq 1$. Indeed if this gcd is equal to 1, then $lcm(p^s-1, (q-1)/d) =
(q-1)/d$. But it is knownn that if $(-1/a)^{(q-1)/d} = 1$, then the corresponding
binomial is not a permutation binomial of $\mathbb{F}_{q}$(see \cite{Sma2}).
\end{rem}
\begin{corollary}
Let $f(x) = ax^{n} + x^{m}$ be a permutation binomial of $\mathbb{F}_{q}$. Suppose
that the order of $a$ in $\mathbb{F}_{q}^{*}$ divides $lcm(p-1, (q-1)/d).$ Then $p-1 \leq d(d-1)$.
\end{corollary}
\begin{proof} If $p-1\mid d$, the corollary is clear. If not, the proof is a direct consequence of Theorem 7 and Theorem 9.
\end{proof}
\begin{corollary} Suppose that there exists a permutation binomial $f(x)=ax^n+x^m$ of $\mathbb{F}_q$ with q= $p^r$ such that for any prime number, $l\mid d$, we have $\mbox{gcd}(l(l-1),r)=1$. Then $d=p-1$ or there exist a permutation binomial of $\mathbb{F}_p$, $g_1(x)=cx^{n_1}+x^{m_1}$ such that $n\equiv kn_1\pmod{p-1}$, $m\equiv km_1\pmod{p-1}$, $0<km_1<kn_1<p-1$, where $k$ is a positive integer coprime with $p-1$, and $p-1 \leq d(d-1)$. Moreover the two possibilites exclude each other.
\end{corollary}
\begin{proof} Let $l$ be any prime factor of $d$. We have $p^r\equiv 1\pmod {l}$, by asymption and $p^{l-1}\equiv 1\pmod {l}$, by Fermat's little theorem. Since $r$ and $l-1$ are coprime, then $p\equiv 1\pmod l$. It is easy to see that $p\equiv 1\pmod d$ and $\mbox{lcm}(p-1,(q-1)/d)=q-1$ so that Theorem 9 may be applied to any permutation binomial of $\mathbb{F}_q$. Let $g(x)$ be the permutation binomial of $\mathbb{F}_q$ with coefficients in $\mathbb{F}_p$ deduced from $f(x)$, using Theorem 9. Let $g_1(x)$ be the reduced polynomial of $g(x)$ modulo $x^p-x$. Then $g_1(x)$ is a monomial or $g_1(x)$ is a sum of $2$ monomials of degree $n'$ and $m'$ respectively satisfying $0<m'<n'<p-1$. Moreover the first case holds if and only if $p-1\mid d$. Since $p\equiv 1\pmod d$ then the first case holds if and only if $d=p-1$. To complete the proof let $k=\mbox{gcd}(m',n')$, $m_1=m'/k$ and $n_1=n'/k$, and by applying Corollary 6, we have $p-1 \leq d(d-1)$.
\end{proof}
The following result is a generalization of \cite[Corallary 2. 4, Corollary 2. 5]{Mas1}.
\begin{corollary} There does not exist a permutation binomial of $\mathbb{F}_q$ with $q=p^r$ if one of the following conditions holds.
\begin{itemize}
\item[(i)] $r$ odd, $d=2$, $p\neq3$.
\item[(ii)] $r$ odd, $d=4$, $p\neq5$.
\item[(iii)] $\mbox{gcd}(r,6)=1$, $d=3$, $p\neq 7$.
\item[(iv)] $\mbox{gcd}(r,10)=1$, $d=5$ $p\neq 11$.
\item[(v)] $\mbox{gcd}(r,6)=1$, $d=6$, $p\neq 7, 13, 19, 31$.
\item[(vi)] $\mbox{gcd}(r,42)=1$, $d=7$, $p\neq 29$.
\item[(vii)] $r$ odd, $d=8$, $p\neq 17$.
\end{itemize}
\end{corollary}
\begin{proof} We prove the case $(v)$ using corollaries 6 and 8 and \cite[Corollary 2. 5]{Mas1}. The proof of the other statements will be omitted. Suppose that there exist a permutation binomial of $\mathbb{F}_q$ with $d=6$ and $\mbox{gcd}(r,6)=1$, then the hypotheses of the above corollary holds. We deduce that $p=d+1=7$ or there exists some permutation binomial $g_1(x)=cx^{n_1}+x^{m_1}$ of $\mathbb{F}_p$. It is evident that $\mbox{gcd}(n_1-m_1,p-1)$ divides $d=6$ and is not trivial. The possible values of this gcd are $2$ or $3$ or $6$. According to \cite[ Corollary 2. 5]{Mas1}, the possible values of $p$ are $p=7, 13, 19, 31$.
\end{proof}
\begin{rem} It is of interest to improve the conditions on $r$ and $p$ in the above corollary. We use the results of \cite[Table 7.1] {Lid2} to
make some observations in this direction. Since $ax^3+x$ is a permutation polynomial of $\mathbb{F}_q$ for $q\equiv 0\pmod{3}$ and $-a$ not a square, then the condition $p\neq3$ is necessary for $d=2$. The polynomial $ax^5+x$ is a permutation of $\mathbb{F}_q$ for $q\equiv 0\pmod{5}$ and $-a$ is not fourth power, hence the condition $p\neq5$ is necessary for $d=4$. Let $a\in\mathbb{F}_9$ such that $a^2=-1$, then $ax^5+x$ permutes $\mathbb{F}_9$, hence the condition $r$ odd is necessary for $d=4$.
\end{rem}
\begin{proposition} Le $q=p^r$ and $f(x) =ax^{n} + x^{m}\in\mathbb{F}_q[x]$ be a permutation binomial. Let $\mathbb{F}_{p^{s_1}},\ldots,\mathbb{F}_{p^{s_u}}$ be subfields of $\mathbb{F}_q$ such that for each $i$, $\mathbb{F}_{p^{s_i}}$ contains the coefficients of some binomial $g_i(x)$, $d$-equivalent to $f(x)$, then $\cap_{i=1}^{u}\mathbb{F}_{p^{s_i}}$ contains the coefficients of some binomial $g(x)$, $d$-equivalent to $f(x)$.
\end{proposition}
\begin {proof} We prove the result for $u=2$. The proposition may be completed easily by induction. By Theorem 9, $(1)$, the order of $a $ divides both $lcm(p^{s_1}-1,(q-1)/d)$ and $lcm(p^{s_2}-1,(q-1)/d)$, hence by Claim 2, this order divides $lcm\big((q-1)/d,gcd(p^{s_1}-1,p^{s_2}-1)\big)$. It is well known that $gcd(p^{s_1}-1,p^{s_2}-1)=p^{gcd(s_1,s_2)}-1$ and that $\mathbb{F}_{p^{s_1}}\cap\mathbb{F}_{p^{s_2}}=\mathbb{F}_{p^{gcd(s_1,s_2)}}$. By Theorem 9 again this last field contains the coefficients of some binomial $g(x)$, $d$-equivalent to $f(x)$.
\end{proof}
If we consider all the subfields $\mathbb{F}_{p^{s_i}}$ of $\mathbb{F}_q$ satisfying the given property in the preceding proposition we may conclude that the field $F_0=\cap_{i}\mathbb{F}_{p^{s_i}}$ contains the coefficients of some binomial $g(x)$, $d$-equivalent to $f(x)$. We call this field {\itshape the smallest field containing the coefficients of some $d$-equivalent to $f(x)$.}
The next proposition shows that binomials that are conjugate over $\mathbb{F}_q$ or in the same $d$-class have the same smallest field.
\begin{proposition} Let $f(x) =ax^{n} + x^{m}\in\mathbb{F}_q[x]$ be a permutation binomial of $\mathbb{F}_q$ and let $F_0$ be the smallest field containing the coefficients of some $d$-equivalent to $f(x)$.
\begin{itemize}
\item[(1)] Let $g(x)\in\mathbb{F}_q[x].$ If $f\stackrel{d}{\sim}g$, then $F_0$ is the smallest field containing the coefficients of some $d$-equivalent to $g(x)$.
\item[(2)] Let $\tilde{f}(x) =a^{p^e}x^{n} + x^{m}$, then $F_0$ is the smallest field containing the coefficients of some $d$-equivalent to $\tilde{f}(x)$.
\end{itemize}
\end{proposition}
\begin{proof} $(1)$ Let $F_1$ be the smallest field corresponding to $g(x)$. For the proof of $(1)$ and by symmetry it is sufficient to prove that $F_0\subset F_1$. Let $g_1(x)$ be a $d$-equivalent of $g(x)$ with coefficients in $F_1$, the $g_1\stackrel{d}{\sim}g\stackrel{d}{\sim}f$, hence $F_1$ contains the coefficients of some $d$-equivalent to $f(x)$. Therefore $F_0\subset F_1$.
$(2)$ The result follows from Theorem 9, $(1)$ and the observation that $a$ and $a^{p^e}$ have the same order.
\end{proof}
| {
"timestamp": "2012-10-05T02:01:10",
"yymm": "1210",
"arxiv_id": "1210.1252",
"language": "en",
"url": "https://arxiv.org/abs/1210.1252",
"abstract": "Let $\\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{n} + x^{m}$ permutes $\\mathbb{F}_{p}$, where $n>m>0$ and $a \\in {\\mathbb{F}_{p}}^{*}$, then $p -1 \\leq (d -1)d$, where $d = {gcd}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\\mathbb{F}_{q}$ from a permutation binomial over $\\mathbb{F}_{q}$.",
"subjects": "Number Theory (math.NT)",
"title": "On Permutation Binomials over Finite Fields",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9912886147702209,
"lm_q2_score": 0.7154240018510025,
"lm_q1q2_score": 0.7091916677682483
} |
https://arxiv.org/abs/2203.05418 | On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds | Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon \left|\nabla m\right|^2 + \frac{1}{\epsilon}\left(1-\|m\|^2\right)^2\right) \, dx,\] for $\Omega\subset\mathbb R^2$ and $m\colon\Omega\to\mathbb R^2$ satisfying $\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, we obtain the following. First, we prove compactness in $L^p$ of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in $BV$, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case $\|\cdot\|=|\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\|\cdot\|$. | \section{Introduction}
The Aviles-Giga functional
\begin{align*}
AG_\varepsilon(u)=\int_\Omega \left( \varepsilon |\nabla^2 u|^2 +\frac 1\varepsilon (1-| \nabla u|^2)^2\right)\, dx,\quad\Omega\subset{\mathbb R}^2,\quad u\colon\Omega\to{\mathbb R},
\end{align*}
is a second order functional that (subject to appropriate boundary conditions) models phenomena from thin film blistering to smectic liquid crystals, and is also a natural higher order generalization of the Cahn-Hilliard functional.
The conjecture on the $\Gamma$-limit of the Aviles-Giga energy,
which roughly states that the energy concentrates on a one-dimensional jump set as $\varepsilon\to 0$,
has attracted a great deal of attention, yet remains open; see for example \cite{avilesgig,avgig1,ADM,mul2,ottodel1, contidel, ark}.
The second term in the Aviles-Giga functional penalizes values of the divergence-free vector field $m=\nabla^\perp u $ that are far from the euclidean unit circle $\mathbb S^1\subset{\mathbb R}^2$.
In the present work we continue the study, initiated in \cite{boch}, of a generalized Aviles-Giga functional where $\mathbb S^1$ is replaced by the unit circle of a more general norm on ${\mathbb R}^2$. Specifically, we let $\|\cdot\|$ be a strictly convex norm on $\mathbb R^2$ that is $C^1$ in ${\mathbb R}^2\setminus\{0\}$ (strictly convex $C^1$ norm for simplicity), and consider the generalized Aviles-Giga functional
\begin{align}\label{eq:Ieps}
&I_\varepsilon(m)= I_\varepsilon(m;\Omega)=\int_\Omega \left(\varepsilon |\nabla m|^2 +\frac{1}{\varepsilon} (1-\|m\|^2)^2\right)\, dx,\\
& \Omega\subset{\mathbb R}^2,\quad m\colon\Omega\to{\mathbb R}^2, \quad \nabla\cdot m=0\text{ in }\mathcal D'(\Omega).
\nonumber
\end{align}
Here the constraint $\nabla\cdot m=0$ is equivalent to $m=\nabla^\perp u$ if the domain $\Omega$ is simply connected, so $I_\varepsilon$ can effectively be seen as a second order functional generalizing the Aviles-Giga functional. In \cite{boch} Bochard and Pegon
obtain some preliminary results on the characterization of zero-energy states of $I_\varepsilon$
(limits of sequences of asymptotically vanishing energy). In this work we carry out a rather comprehensive analysis of this generalized Aviles-Giga functional. Our goal is to investigate to which extent the results and methods that have been developed for the classical Aviles-Giga functional can be extended to this more general setting.
In doing so, we hope to shed some light on what parts of the theory are contingent on specific algebraic properties of $\mathbb S^1$, and what parts are more flexible.
Similar generalized Aviles-Giga functionals have also been studied in \cite{ignat-monteil-20},
with a focus on symmetry properties of entire critical points.
Here we concentrate on four aspects:
\begin{itemize}
\item compactness in $L^p$ and energy lower bounds for sequences of bounded energy;
\item characterization of zero-energy states;
\item optimal regularity estimates for limits of sequences of bounded energy;
\item comparison of upper and lower bounds for sequences converging to a map of bounded variation ($BV$).
\end{itemize}
For the first two aspects we obtain complete generalizations of the analogous results in the classical case.
For the last two aspects, our results demonstrate the effects induced by possible anisotropy and degenerate convexity of $\|\cdot\|$.
A central tool, introduced in \cite{mul2} for the classical Aviles-Giga functional, is the notion of entropies, imported from scalar conservation laws. Formally (and this is justified by the compactness result), limits of sequences of bounded energy should satisfy the generalized Eikonal equation
\begin{align}\label{eq:geneikon}
\|m\|=1\text{ a.e.},\qquad\nabla\cdot m=0\text{ in }\mathcal D'(\Omega).
\end{align}
Writing locally the unit circle $\partial\mathsf{B}=\lbrace z\in{\mathbb R}^2\colon \|z\|=1\rbrace$ as the graph of a convex function $f$, this equation can formally be rewritten as the scalar conservation law
\begin{align}\label{eq:scl}
\partial_t u +\partial_x f(u) =0.
\end{align}
In direct analogy with the entropy-entropy flux pairs for this scalar conservation law, entropies for the generalized Eikonal equation \eqref{eq:geneikon} are $C^1$ maps $\Phi\colon\partial \mathsf{B}\to{\mathbb R}^2$ with the property that $\nabla\cdot\Phi(m)=0$ for any smooth solution $m$ of \eqref{eq:geneikon}. For weak solutions, the distributions $\nabla\cdot\Phi(m)$, called \textit{entropy productions}, encode the presence of singularities and can therefore be used to understand compactness and regularity properties. The key property used in \cite{mul2} is that, in the classical case $\|\cdot\|=|\cdot|$, entropy productions are controlled by the energy. This provides compactness \cite{ADM,mul2}, and an energy lower bound. Thanks to the strict convexity of $\|\cdot\|$, this analysis can be adapted to our generalized setting; see Theorem~\ref{t:comp} and Proposition~\ref{p:lowerbound}.
A further consequence of the energy lower bound is that zero-energy states, that is, limits of sequences $\{m_n\}$ such that $I_{\varepsilon_n}(m_n)\to 0$, have vanishing entropy productions. This is exploited in \cite{otto} for the classical Aviles-Giga functional to obtain a kinetic equation which roughly speaking ensures that zero-energy states are, in a weak way, constant along characteristics. As a consequence, zero-energy states in the classical case $\|\cdot\|=|\cdot|$ are shown in \cite{otto} to be locally Lipschitz outside a locally finite set of singular points, and around each singular point they must coincide with a vortex $m(x)=\pm ix/|x|$. In \cite{boch} this rigidity result is generalized (with appropriate modifications) to $I_\varepsilon$ associated with any $C^1$ norm $\|\cdot\|$ of power type $p$ for some $p\in [2,\infty)$ (a quantitative form of strict convexity, see Remark~\ref{r:zeroenergytypep}). Here we extend this further to $I_\varepsilon$ associated with any strictly convex $C^1$ norm (see Theorem~\ref{T:zeroenergy}) using an elementary argument that reduces it to the classical case $\|\cdot\|=|\cdot|$.
Finite-energy states, that is, limits of sequences of bounded energy, can have a much more complicated structure.
The energy lower bound ensures that entropy productions are finite Radon measures, and a central question to solve the $\Gamma$-convergence conjecture for the classical Aviles-Giga functional is whether these measures are concentrated on a one-dimensional rectifiable set. Substantial progress on that question has been made in \cite{ottodel1,marconi21ellipse} but it remains open.
For scalar conservation laws \eqref{eq:scl} with $f$ uniformly convex (Burgers' equation), this rectifiability property has recently been proved in \cite{marconi20burgers}.
The results of \cite{ottodel1} and \cite[Proposition~1.7]{marconi21ellipse} can likely be generalized to the class of energy functionals \eqref{eq:Ieps} associated with any strictly convex $C^1$ norm $\|\cdot\|$ (using the kinetic formulation obtained in Lemma~\ref{l:kin}),
but here we don't address that question and concentrate instead on optimal regularity estimates for solutions of the generalized Eikonal equation \eqref{eq:geneikon} whose entropy productions are locally finite Radon measures.
In the classical case $\|\cdot\|=|\cdot|$, it is proved in \cite{GL} (adapting an argument of \cite{golseperthame13} for scalar conservation laws) that such solutions must locally have the Besov regularity $B^{\frac 13}_{3,\infty}$, i.e. $\sup_h |h|^{-\frac 13}\|m-m^h\|_{L^3_{loc}}<\infty$ where $m^h=m(\cdot+h)$.
Moreover this estimate is \emph{strongly optimal} in the sense that it is \emph{equivalent} to entropy productions being locally finite Radon measures.
In the general case, the coercivity provided by the strict convexity of the norm $\|\cdot\|$ depends on the direction $z$ on its unit circle $\partial\mathsf{B}$, and optimal estimates must take that into account. We prove therefore a regularity estimate of the form
$\sup_h |h|^{-1}\| \Pi(m,m^h)\|_{L^1_{loc}}<\infty$ for some function $\Pi\colon\partial\mathsf{B}\times\partial\mathsf{B}\to [0,\infty)$ that is sensitive to the anisotropy of $\|\cdot\|$,
and show that it is strongly optimal (equivalent to entropy productions being locally finite Radon measures) at least when the norm $\|\cdot\|$ is analytic; see Theorems~\ref{thmbes} and \ref{thmbesrev}.
(For a norm $\|\cdot\|$ of power type $p$ convexity this estimate implies in particular Besov regularity agreeing with the results of \cite{golseperthame13} for scalar conservation laws \eqref{eq:scl} when the flux $f$ has degenerate convexity; see Remark~\ref{R6}.)
Furthermore, if $\|\cdot\|$ is merely $C^1$ then the quantity
$\sup_h |h|^{-1}\| \Pi(m,m^h)\|_{L^1_{loc}}$ is comparable to the total entropy production when $m$ is $BV$, hinting that the regularity estimate could be strongly optimal for all strictly convex $C^1$ norms $\|\cdot\|$.
The $\Gamma$-convergence of the classical Aviles-Giga functional in the $BV$ setting is well understood \cite{ADM, contidel, ark}. For a solution $m$ of the generalized Eikonal equation \eqref{eq:geneikon} which is $BV$, an upper bound can be obtained for the minimal energy of approximating sequences $m_n\to m$ by pasting optimal one-dimensional transitions along the jump set $J_m$ \cite{contidel,ark,ark2}.
In the classical case $\|\cdot\|=|\cdot|$, this upper bound happens to coincide with the lower bound provided by a particular class of entropy productions \cite{ADM}, thus characterizing the $\Gamma$-limit at $BV$ maps $m$.
This perfect agreement of entropy lower bound and 1D upper bound is very likely linked to specific algebraic properties of the euclidean norm $|\cdot|$ (as are the symmetry results of \cite{ignat-monteil-20}).
In fact it is known \cite[\S~4]{kohn} that for general $\|\cdot\|$ optimal transition profiles may not be one-dimensional, and in that case the 1D upper bound is strictly larger than any lower bound (see \cite{poliakovsky13,poliakovsky15} for more results related to such issues).
It is however interesting to find out whether these two bounds (the entropy lower bound and the 1D upper bound) are of the same order of magnitude, or can instead be very far apart.
Like optimal regularity estimates, this question is sensitive to the possibly anisotropic behavior of $\|\cdot\|$. We prove that these upper and lower bounds do agree up to a multiplicative constant; see Theorem~\ref{t:lowerupper}.
In the rest of this introduction we present the precise statements of our results.
In Section~\ref{s:entropyprod} we derive some useful properties of the entropies in our generalized setting. In Section~\ref{s:compactness} we prove the compactness result. In Section~\ref{s:zeroenergy} we prove the rigidity of zero-energy states. In Section~\ref{s:reg} we prove regularity estimates for finite-energy states and their optimality. And in Section~\ref{s:boundsBV} we compare upper and lower bounds for $BV$ limits.
\subsection{Notations and assumptions}
Let $\Omega\subset{\mathbb R}^2$ be a bounded open set and $\|\cdot\|$ be a strictly convex $C^1$ norm on ${\mathbb R}^2$ unless otherwise specified. We denote by $\mathsf{B}=\left\lbrace z\in{\mathbb R}^2\colon \|z\|<1\right\rbrace$ the open unit disk for the norm $\|\cdot\|$. The properties of $\|\cdot\|$ are equivalent to strict convexity of $\mathsf{B}$ and $\partial \mathsf{B}$ being a $C^1$ manifold.
Without loss of generality, we assume that $\partial\mathsf{B}$ has length $2\pi$, and let
$\gamma:{\mathbb R}/2\pi{\mathbb Z}\rightarrow \partial \mathsf{B}$ be the counterclockwise arc-length parametrization of $\partial\mathsf{B}$ (unique up to translation of the variable). By assumption, $\gamma\in C^1({\mathbb R}/2\pi\mathbb{Z};{\mathbb R}^2)$. In many places we identify $\mathbb{R}^2$ with $\mathbb{C}$ and in particular we let $i$ denote the counterclockwise rotation by $\frac{\pi}{2}$.
We will use the symbols $\lesssim$ and $\gtrsim$ to denote inequality up to a multiplicative constant that depends only on $\mathsf{B}$.
\subsection{Compactness and lower bound}
Our first result generalizes the compactness result obtained independently in \cite[Theorem~3.3]{ADM} and \cite[Proposition~1]{mul2} for the Aviles-Giga functional.
\begin{thm}
\label{t:comp}
Suppose the sequence $\{m_{n}\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$ satisfies $\nabla\cdot m_n=0$ and
\begin{equation*}
\sup_{n} I_{\epsilon_n}(m_{n})<\infty.
\end{equation*}
Then $\{m_{n}\}$ is precompact in $L^2(\Omega)$.
\end{thm}
As explained above, this compactness result relies heavily on the notion of entropies for the generalized Eikonal equation
\begin{align}\tag{\ref{eq:geneikon}}
\|m\|=1\text{ a.e.,}\quad \nabla\cdot m=0\text{ in }\mathcal D'(\Omega).
\end{align}
Equivalently, the first constraint $\|m\|=1$ means that $m$ takes values into $\partial\mathsf{B}$.
Entropies for this equation are $C^1$ maps $\Phi\colon\partial \mathsf{B}\to{\mathbb R}^2$ such that, if $m$ is a $C^1$ solution of \eqref{eq:geneikon}, then $\Phi(m)$ is also divergence-free $\nabla\cdot\Phi(m)=0$. It is a lengthy but straightforward exercise to see that this is equivalent to requiring that, for all $\theta\in{\mathbb R}$,
\begin{align*}
\frac{d}{d\theta}\Phi(\gamma(\theta)) \text{ is tangent to }\partial\mathsf{B}\text{ at }\gamma(\theta).
\end{align*}
For a weak solution $m$ of \eqref{eq:geneikon}, the entropy production $\nabla\cdot\Phi(m)$ is in general not zero, and encodes the presence of singularities. The proof of
Theorem~\ref{t:comp} relies on the control of entropy productions provided by the energy. This control is possible for regular enough entropies: we define
\begin{align}\label{eq:ENT}
\mathrm{ENT}=\Big\lbrace \Phi\in C^1(\partial\mathsf{B};{\mathbb R}^2)\colon &\frac{d}{d\theta}\Phi(\gamma(\theta))=\lambda_\Phi(\theta)\gamma'(\theta)
\nonumber\\
&\text{for some function }\lambda_\Phi\in C^1({\mathbb R}/2\pi{\mathbb Z})\Big\rbrace.
\end{align}
The control of entropy productions used to establish compactness also provides a lower bound for the energy. From this point on all entropies for equation \eqref{eq:geneikon} in statements and proofs will be taken to be the ones from $\mathrm{ENT}$.
\begin{prop}\label{p:lowerbound}
Let $m\colon\Omega\to{\mathbb R}^2$ be such that $m=\lim_{n\to\infty} m_n$ in $L^2(\Omega)$ for some sequence $\{m_{n}\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$ with $\nabla\cdot m_n=0$ and $\sup_n I_{\varepsilon_n}(m_n) <\infty$. Then $m$ satisfies the generalized Eikonal equation \eqref{eq:geneikon}, its entropy productions satisfy $\nabla\cdot\Phi(m)\in\mathcal{M}(\Omega)$ for all $\Phi\in \mathrm{ENT}$, and they provide the lower bound
\begin{align}\label{eq:lowerbound}
\left(\bigvee_{\|\lambda_\Phi'\|_{\infty}\leq 1}|\nabla\cdot\Phi(m)|\right)(U) \leq C_0 \,\liminf_{n\to\infty} I_{\varepsilon_n}(m_n;U),
\end{align}
for any open subset $U\subset\Omega$ and some constant $C_0>0$ depending only on $\mathsf{B}$. Here $\bigvee$ denotes the lowest upper bound measure \cite[Definition~1.68]{ambrosio} of a family of measures.
\end{prop}
\begin{rem} The hypothesis that $\|\cdot \|$ is strictly convex is necessary for Theorem~\ref{t:comp}: Suppose that $\partial \mathsf{B}$ contains a line segment $\left[\zeta_0,\zeta_1\right]$ then without loss of generality we can assume $\zeta_0=e_1+\delta e_2$ and $\zeta_1=e_1-\delta e_2$.
Setting
$m_\varepsilon(x)=e_1 + \delta \sin \left(x_1/\sqrt{\varepsilon}\right) e_2$, then $\nabla \cdot m_\varepsilon=0$ and $\|m_\varepsilon\|=1$ everywhere in $\Omega$. Thus $\sup_{\varepsilon>0} I_{\varepsilon}\left(m_\varepsilon\right)<\infty$, but $m_\varepsilon$ converges weakly to $\tilde{m}\equiv e_1$ in $L^p$ as $\varepsilon\to 0$ and $\|m_\varepsilon-\tilde{m}\|_{L^p}\gtrsim \delta$ for all $\varepsilon>0$ and all $p\geq 1$.
\end{rem}
\subsection{Zero-energy states}
As stated previously, Jabin, Otto and Perthame showed in \cite[Theorem~1.1]{otto} that zero-energy states of the Aviles-Giga functional are rigid.
This result
has several interesting implications \cite{ottodel1,DeI,LP,llp}.
It is proved in two steps: first, zero-energy states have vanishing entropy productions and satisfy as a consequence the kinetic equation $e^{it}\cdot\nabla_x\mathbf 1_{m(x)\cdot e^{it}>0}=0$, which expresses in a weak way the fact that $m$ is constant along characteristics of the classical Eikonal equation; second, solutions of this kinetic equation are shown to be rigid.
In \cite{boch}, Bochard and Pegon generalize the second step to solutions of the kinetic equation $\gamma'(t)\cdot\mathbf 1_{m(x)\cdot i\gamma(t)>0}=0$ naturally associated with the generalized Eikonal equation \eqref{eq:geneikon}, under the assumption that the $C^1$ norm $\|\cdot\|$ is of power type $p$ (see Remark~\ref{r:zeroenergytypep}). They do not however prove the first step, namely that zero-energy states of $I_\varepsilon$ satisfy this kinetic equation.
Here we do establish that missing step, and generalize their rigidity result to any strictly convex $C^1$ norm $\|\cdot\|$, with a somewhat more direct proof.
\begin{thm}
\label{T:zeroenergy}
Let $m:\Omega\to {\mathbb R}^2$ be such that $m=\lim_{n\to\infty} m_n$ in $L^2(\Omega)$, where the sequence $\{m_n\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$ satisfies $\nabla\cdot m_n=0$ and
\begin{equation*}
\lim_{n\to\infty}I_{\epsilon_n}(m_n) = 0.
\end{equation*}
Then $m$ is continuous outside a locally finite set of singular points. For every singular point $x_0$, there exists $\beta\in\{-1,1\}$ such that in any convex neighborhood $\mathcal{U}$ of $x_0$, we have $m(x)=\beta V_{\mathsf{B}}(i(x-x_0))$, where $V_{\mathsf{B}}(\cdot):=\nabla \|\cdot \|_{*}$ is the vortex associated to $\|\cdot\|$
and $\|\cdot \|_{*}$ is the dual norm of $\|\cdot \|$.
\end{thm}
\begin{rem}\label{r:zeroenergytypep}
Our proof also recovers the result, obtained in \cite{boch}, that if the $C^1$
norm $\|\cdot\|$ is of power type $p$ for some $p\geq 2$, that is,
\begin{align*}
1-\left\|\frac{x+y}{2}\right\| \geq K \|x-y\|^p\qquad\forall x,y\in\partial \mathsf{B},
\end{align*}
for some constant $K>0$,
then $m$ is locally $\frac{1}{p-1}$-H\"{o}lder outside a locally finite set of singular points (see the end of Section \ref{s:zeroenergy}).
\end{rem}
\subsection{Optimal regularity estimates}
Proposition \ref{p:lowerbound} motivates the study of \emph{finite-entropy solutions} of the generalized Eikonal equation, i.e. solutions of \eqref{eq:geneikon} satisfying $\nabla\cdot\Phi(m)\in\mathcal{M}_{loc}(\Omega)$ for all $\Phi\in\mathrm{ENT}$.
We present here regularity estimates for these solutions, that are strongly optimal in the sense that a converse estimate is valid: regularity implies locally finite entropy productions.
In the context of scalar conservation laws, this type of optimality is related to ``Onsager conjecture-type'' statements: see e.g. \cite{bardos19} where the authors investigate minimal regularity requirements that are sufficient to ensure that entropy productions vanish.
In the classical case $\|\cdot\|=|\cdot|$ it was shown in \cite{GL} that finite-entropy solutions coincide with solutions of \eqref{eq:geneikon} that live in the Besov space $B^{1/3}_{3,\infty,loc}(\Omega)$: such Besov estimates are strongly optimal.
This was obtained by adapting methods of \cite{golseperthame13} for scalar conservation laws \eqref{eq:scl} with convex flux $f$.
The $1/3$ order of regularity is valid for uniformly convex fluxes $f$ and
an example in \cite{delelliswestdickenberg03} had also demonstrated its optimality, in a different sense than the one we wish to study here: there exist finite-entropy solutions which don't have a better order of regularity.
For fluxes with degenerate convexity, quantified by the inequality
\begin{align*}
f'(v)-f'(w)\gtrsim \left|v-w\right|^{p-1}\qquad \forall v>w,
\end{align*}
for some $p\geq 2$, the regularity obtained in \cite{golseperthame13} is $B^{\frac 1{p+1}}_{p+1,\infty,loc}$. This applies for instance to $f(w)=|w|^p$, and is shown to be optimal in \cite[Proposition~3.2]{delelliswestdickenberg03}, again in the sense that there exist finite-entropy solutions which don't have a better order of regularity.
However it is clear (considering solutions whose values stay away from the point $w=0$ at which convexity degenerates) that this Besov regularity does not provide a converse estimate: it is not strongly optimal.
Here, following \cite{GL} we adapt the methods of \cite{golseperthame13} to the generalized Eikonal equation \eqref{eq:geneikon} in order to obtain regularity estimates that take into account the anisotropy of $\|\cdot\|$, and in particular the fact that the convexity of $\|\cdot\|$ may degenerate differently in different directions. For a precise statement, we introduce the (unique up to an additive constant integer multiple of $2\pi$) continuous function $\alpha\colon{\mathbb R}\to{\mathbb R}$ such that
\begin{align*}
\gamma'(\theta)=e^{i\alpha(\theta)}\qquad\forall \theta\in{\mathbb R}.
\end{align*}
The strict convexity of $\mathsf{B}$ ensures that this function $\alpha$ is increasing, and the symmetry of $\mathsf{B}$ implies $\alpha(t+\pi)=\alpha(t)+\pi$ for all $t\in{\mathbb R}$. We define a function $\Pi\colon\partial \mathsf{B}\times \partial \mathsf{B}\to [0,\infty)$ by
\begin{align}
\label{eq:Lambda}
\Pi\left(\gamma(\theta_1),\gamma(\theta_2)\right) =\int_{\theta_1}^{\theta_2} \int_{\theta_1}^{\theta_2} |\alpha(t)-\alpha(s)|\, dt ds \qquad\text{ for }|\theta_1-\theta_2|\leq \pi.
\end{align}
Using this function $\Pi$ as a ``metric'' for the increments, we have the following regularity estimate for finite-entropy solutions of \eqref{eq:geneikon}.
\begin{thm}
\label{thmbes} Let $m$ satisfy the generalized Eikonal equation \eqref{eq:geneikon}. Suppose
\begin{align}
\label{eqbes1}
\nabla \cdot \Phi(m)\in \mathcal{M}_{loc}(\Omega)\quad\text{ for all }\Phi\in \mathrm{ENT},
\end{align}
then
\begin{align}
\label{eqbes2}
\sup_{\left|h\right|<\mathrm{dist}(\Omega',\partial \Omega)} \frac{1}{\left|h\right|}\int_{\Omega'} \Pi\left(m(x+h),m(x)\right)\, dx<\infty\quad \text{ for any }\Omega'\subset \subset \Omega.
\end{align}
\end{thm}
\begin{rem}
\label{R6}
If $\|\cdot\|$ is of power type $p$ for some $p\geq 2$ (see Remark~\ref{r:zeroenergytypep}), the estimate \eqref{eqbes2} directly implies that $m\in B^{\frac{1}{p+1}}_{p+1, \infty, loc}(\Omega)$, as explained in Remark~\ref{remppower}. This corollary is analogous to the regularity results obtained in \cite[Theorem~4.1]{golseperthame13} for convex scalar conservation laws.
\end{rem}
The main interest of this regularity estimate is that the ``metric'' $\Pi$ is sensitive enough to the local convexity of $\partial\mathsf{B}$ to ensure the validity of a converse estimate, at least when $\|\cdot\|$ is analytic in ${\mathbb R}^2\setminus \lbrace 0\rbrace$ (or equivalently $\partial\mathsf{B}$ is analytic):
\begin{thm}
\label{thmbesrev}
Let $m$ satisfy \eqref{eq:geneikon}. Assume that the strictly convex norm $\|\cdot\|$ is analytic in ${\mathbb R}^2\setminus\lbrace 0\rbrace$, then \eqref{eqbes2} implies \eqref{eqbes1}.
\end{thm}
\begin{rem}
Note that Theorem~\ref{thmbesrev} applies in particular to $\|\cdot\|=\|\cdot\|_{\ell^p}$ for any $1<p<\infty$ (see Remark \ref{rem:lp-sphere}).
\end{rem}
We don't know whether the analyticity assumption on $\|\cdot\|$ is necessary for the validity of the converse estimate \eqref{eqbes2} implying \eqref{eqbes1}. An indication that it might not be needed is given by the following.
\begin{thm}\label{t:regconvBV}
Let $\|\cdot\|$ be a strictly convex $C^1$ norm on ${\mathbb R}^2$ and let $m\in BV(\Omega; {\mathbb R}^2)$ satisfy \eqref{eq:geneikon}. Then for any open subset $\Omega'\subset\subset\Omega$ we have
\begin{align*}
\left(\bigvee_{\|\lambda_\Phi'\|_\infty\leq 1} |\nabla\cdot\Phi(m)|\right)(\Omega')
\leq C_0 \,
\sup_{|h|<\dist(\Omega',\partial\Omega)}\frac{1}{|h|}\int_{\Omega'}\Pi(m(x+h),m(x))\, dx,
\end{align*}
for some absolute constant $C_0>0$.
\end{thm}
Note that for a $BV$ solution of \eqref{eq:geneikon} as in Theorem~\ref{t:regconvBV}, both the entropy productions and the quantity appearing in the regularity estimate \eqref{eqbes2} are finite. Here the point is that the latter controls the former, without any further regularity assumption on the norm $\|\cdot\|$.
\subsection{Comparison of upper and lower bounds}
For general maps $m$, finding an upper bound that matches (at least up to a multiplicative constant) the lower bound of Proposition~\ref{p:lowerbound} is a famously hard problem even in the classical case. However, when the limiting solution $m$ of \eqref{eq:geneikon} additionally belongs to $BV(\Omega;{\mathbb R}^2)$, then it is known \cite{ark2} that an upper bound (in the sense of $\Gamma$-convergence) is obtained by pasting optimal one-dimensional transitions at scale $\varepsilon$ along the jump set $J_m$. Specifically, for any solution $m\in BV(\Omega;{\mathbb R}^2)$ of the generalized Eikonal equation \eqref{eq:geneikon} and any smooth simply connected open subset $U\subset\Omega$, there exists a sequence $m_\varepsilon\to m$ in $L^p(U;{\mathbb R}^2)$ for $1\leq p <\infty$, such that
\begin{align}\label{eq:upperboundBV}
\limsup_{\varepsilon\to 0}I_{\varepsilon}(m_\varepsilon;U) \leq \int_{U\cap J_m} \mathrm{c}^{\mathrm{1D}}(m^+,m^-)\, d\mathcal H^1,
\end{align}
where $m^\pm$ are the traces of $m$ along $J_m$, and $\mathrm{c}^{\mathrm{1D}}\colon\partial\mathsf{B}\times\partial\mathsf{B}\to [0,\infty)$ is given by
\begin{align}\label{eq:c1D}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)
&
=2\left|\int_{z^{-}\cdot i\nu}^{z^{+}\cdot i\nu} \left(1-\|a \nu + s i \nu \|^2\right) ds\right|,\\
\nu&=i\frac{z^+-z^-}{|z^+-z^-|},\qquad a=z^+\cdot\nu =z^-\cdot \nu.\nonumber
\end{align}
Here the unit vector $\nu$ represents a normal vector to the jump set $J_m$ at a jump between $z^+$ and $z^-$. The divergence-free constraint $\nabla\cdot m=0$ forces $\nu$ to satisfy $(z^+-z^-)\cdot\nu=0$, and this characterizes $\nu$ up to a sign. Note that it is known that the upper bound provided by one-dimensional profiles will in general not be optimal \cite[\S~4]{kohn} (see \cite{poliakovsky13,poliakovsky15} for a discussion of optimal upper bounds), but here we are only interested in optimality up to a multiplicative constant.
We wish to compare this 1D upper bound to the lower bound provided by the entropy productions in Proposition~\ref{p:lowerbound}. For
a solution $m$ of \eqref{eq:geneikon} which additionally belongs to $BV(\Omega;{\mathbb R}^2)$, the $BV$ chain rule implies that the entropy productions are absolutely continuous with respect to $\mathcal H^1_{\lfloor J_m}$. Thanks to \cite[Remark~1.69]{ambrosio}, the resulting lowest upper bound measure is also absolutely continuous with respect to $\mathcal H^1_{\lfloor J_m}$, and (see Lemma \ref{l:cENT}) we have
\begin{align*}
\bigvee_{\|\lambda'_\Phi\|_{\infty}\leq 1}|\nabla\cdot\Phi(m)| & =\mathrm{c}^{\mathrm{ENT}}(m^+,m^-)\,\mathcal H^1_{\lfloor J_m},
\end{align*}
where $m^\pm$ are the traces of $m$ along $J_m$, and the jump cost $\mathrm{c}^{\mathrm{ENT}}\colon \partial\mathsf{B}\times\partial\mathsf{B}\to\ [0,\infty)$ is given by
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)=\sup_{\|\lambda'_\Phi\|_{\infty}\leq 1}
\int_{\theta^-}^{\theta^+}\lambda_\Phi(s)\,\gamma'(s)\cdot\nu\, ds\qquad\text{for }z^\pm=\gamma(\theta^\pm),
\end{align*}
and $\nu$ as in \eqref{eq:c1D}.
The value of the last integral does not depend on the choices of $\theta^\pm$ modulo $2\pi$, because the definition of $\lambda_\Phi$ in \eqref{eq:ENT} implies $\int_{{\mathbb R}/2\pi{\mathbb Z}}\lambda_\Phi(s)\gamma'(s)\, ds=0$ for any entropy $\Phi\in \mathrm{ENT}$.
In other words, for a $BV$ map $m$, the lower bound \eqref{eq:lowerbound} becomes
\begin{align}\label{eq:lowerboundBV}
\liminf_{n\to\infty} I_{\varepsilon_n}(m_n;U)
\gtrsim
\int_{J_m\cap U} \mathrm{c}^{\mathrm{ENT}}(m^+,m^-)\, d\mathcal H^1.
\end{align}
We show that these lower and upper bounds \eqref{eq:lowerboundBV} and \eqref{eq:upperboundBV} for $BV$ maps $m$ are comparable:
\begin{thm}\label{t:lowerupper}
There exists a constant $C_0>0$ depending only on $\mathsf{B}$ such that
\begin{align*}
C_0^{-1}\,\mathrm{c}^{\mathrm{1D}}(z^+,z^-)\leq \mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
\leq C_0\, \mathrm{c}^{\mathrm{1D}}(z^+,z^-)
\end{align*}
for all $z^\pm\in\partial\mathsf{B}$.
\end{thm}
\section{Entropy productions}\label{s:entropyprod}
In this section we compute entropy productions of divergence-free $m\in W^{1,2}$, and as a direct consequence we prove Proposition \ref{p:lowerbound}. Since $\mathsf{B}$ is convex and centered, for any $z\in {\mathbb R}^2\setminus\lbrace 0\rbrace$ there is a unique $(r,\theta)\in (0,\infty)\times{\mathbb R}/2\pi{\mathbb Z}$ such that $z=r\gamma(\theta)$. In order to make use of classical polar coordinates, we introduce the bijection $X\colon{\mathbb R}^2\to{\mathbb R}^2$ given by
\begin{align}\label{eq:X}
X(r e^{i\theta})= r \gamma(\theta)\qquad\forall r\geq 0,\;\theta\in{\mathbb R}.
\end{align}
The map $X$ is $C^1$ in ${\mathbb R}^2\setminus\lbrace 0\rbrace$, and its jacobian determinant is
\begin{align*}
\det (\nabla X(re^{i\theta})) =i\gamma(\theta)\cdot\gamma'(\theta) \geq\alpha_0>0,
\end{align*}
where $\alpha_0$ is the radius of the largest euclidean ball contained in $\overline\mathsf{B}$. This last inequality follows from the convexity of $\mathsf{B}$: for any $z\in \overline\mathsf{B}$ we have $(z-\gamma(\theta))\cdot i\gamma'(\theta) \geq 0$, and applying this to $z=-i\alpha_0\gamma'(\theta)$ gives $i\gamma(\theta)\cdot\gamma'(\theta) \geq\alpha_0$.
As a consequence, $X^{-1}$ is $C^1$ in ${\mathbb R}^{2}\setminus\lbrace 0\rbrace$. Moreover $X$ is a bi-Lipschitz homeomorphism.
In the following, we take $\eta(r)\in C^1([0,\infty))$ so that $0\leq\eta\leq 1$, $\eta\equiv 0$ in $[0,\frac 12]\cup[2,\infty)$ and $\eta(1)=1$. For $\Phi\in \mathrm{ENT}$, define $\widehat\Phi\in C^1({\mathbb R}^2;{\mathbb R}^2)$ by
\begin{equation}\label{eq:extPhi}
\widehat\Phi\left(r\gamma(\theta)\right) = \eta(r)\Phi\left(\gamma(\theta)\right).
\end{equation}
\begin{lem}
\label{l:extPhi}
Let $m\in W^{1,2}(\Omega;{\mathbb R}^2)$ satisfy $\nabla\cdot m=0$. Then for any $\Phi\in \mathrm{ENT}$,
%
we have
\begin{align} \label{eq:dvextPhim}
\nabla\cdot\widehat\Phi(m)=
\frac 12\Psi(m)\cdot\nabla\left(1-\|m\|^2\right),
\end{align}
where
\begin{equation*}
\Psi\left(r\gamma(\theta)\right)=\frac{\eta\left(r\right)\lambda_{\Phi}\left(\theta\right)}{r^2}\gamma\left(\theta\right) - \frac{\eta'\left(r\right)}{r}\Phi\left(\gamma(\theta)\right)
\end{equation*}
and
\begin{equation*}
\lambda_{\Phi}(\theta)=\frac{d}{d\theta}\left(\Phi\left(\gamma(\theta)\right)\right)\cdot \gamma'(\theta).
\end{equation*}
\end{lem}
\begin{proof}
It suffices to prove \eqref{eq:dvextPhim} for a smooth map $m\colon\Omega\to{\mathbb R}^2$, because we can then approximate a $W^{1,2}$ map $m$ with smooth maps $m_n\to m$ in $W^{1,2}$ and a.e., satisfying in addition $\nabla\cdot m_n=0$, so that \eqref{eq:dvextPhim} passes to the limit in $\mathcal D'(\Omega)$.
It is convenient to change variable in order to use classical polar coordinates: we set
$\widetilde\Phi =\widehat\Phi\circ X$, $\tilde m = X^{-1}(m)$,
so that
\begin{align}
\label{eqfcc1}
\widetilde\Phi(r e^{i\theta})=\eta(r)\Phi(\gamma(\theta)),
\qquad\widetilde\Phi(\tilde m)=\widehat\Phi(m), \qquad\text{and }\nabla\cdot X(\tilde m)=0.
\end{align}
And note that $\|X(v)\|=\left|v\right|$ for all $v\in \mathbb{R}^2$, and so
\begin{align}
\label{eqfcc2}
\left|\tilde{m}\right|=\|m\|.
\end{align}
In the following we perform calculations in the open set $\{m\ne 0\}$. Using that $\Phi \in\mathrm{ENT}$, we have
\begin{align*}
\frac{d}{d\theta}\Phi(\gamma(\theta))=\lambda_{\Phi}(\theta)\gamma'(\theta),
\end{align*}
so computing $D\widetilde\Phi$ using polar coordinates we find
\begin{align*}
D\widetilde\Phi(re^{i\theta})&=\eta'(r)\Phi(\gamma(\theta))\otimes e^{i\theta}
+\frac{\eta(r)}{r}\lambda_{\Phi}(\theta) \,\gamma'(\theta)\otimes ie^{i\theta}.
\end{align*}
Note that for any $v,w\in \mathbb{R}^2$ we have the identity $\mathrm{tr}\left(v\otimes w Dm\right)=\left(\left(v\cdot \nabla\right)m \right)\cdot w$. So, writing $\tilde m=re^{i\theta}$, we obtain
\begin{align*}
\nabla\cdot\widetilde\Phi(\tilde m)&=\mathrm{tr}(D\widetilde \Phi(\tilde m)D\tilde m)\\
&=\eta'(|\tilde m|)\left[\left(\Phi(\gamma(\theta))\cdot\nabla\right)\tilde m\right] \cdot \frac{\tilde m}{|\tilde m|}
+\frac{\eta(|\tilde m|)}{|\tilde m|}\lambda_{\Phi}(\theta) \left[\left(\gamma'(\theta)\cdot\nabla\right)\tilde m\right] \cdot i
\frac{\tilde m}{|\tilde m|}.
\end{align*}
Applying this to $\eta(r)=r$ (the above calculations only require $\eta$ to be $C^1$) and $\Phi(z)=z$ gives in particular
\begin{align*}
\nabla\cdot X(\tilde m)=
\left[\left(\gamma(\theta)\cdot\nabla\right)\tilde m\right] \cdot \frac{\tilde m}{|\tilde m|}
+ \left[\left(\gamma'(\theta)\cdot\nabla\right)\tilde m\right] \cdot i
\frac{\tilde m}{|\tilde m|},
\end{align*}
so the previous expression for $\nabla\cdot\widetilde\Phi(\tilde m)$ can be rewritten as
\begin{align*}
\nabla\cdot\widetilde\Phi(\tilde m)&=
\frac{\eta(|\tilde m|)}{|\tilde m|}\lambda_{\Phi}(\theta) \,\nabla\cdot X(\tilde m) \\
&\quad
+
\left[\left(\left(\eta'(|\tilde m|)\Phi(\gamma(\theta))
-\frac{\eta(|\tilde m|)}{|\tilde m|}\lambda_{\Phi}(\theta) \gamma(\theta)\right)\cdot\nabla\right)\tilde m \right]
\cdot \frac{\tilde m}{|\tilde m|}.
\end{align*}
Using $\nabla\cdot X(\tilde m)=0$ and $\partial_j\tilde m\cdot\tilde m = \partial_j |\tilde m|^2/2$, this becomes
\begin{align*}
\nabla\cdot\widetilde\Phi(\tilde m)&=
\frac 12 \widetilde\Psi(\tilde m)\cdot\nabla (1- |\tilde m|^2),\\
\widetilde\Psi(\tilde m)&=
\frac{\eta(|\tilde m|)}{|\tilde m|^2}\lambda_{\Phi}(\theta) \gamma(\theta)
-
\frac{\eta'(|\tilde m|)}{|\tilde m|}\Phi(\gamma(\theta)).
\end{align*}
The above calculations are valid in $\lbrace m\neq 0\rbrace$, but since $\eta(r)=\eta'(r)=0$ for $0\leq r <1/2$, this last expression makes sense everywhere.
Recalling from \eqref{eqfcc1}-\eqref{eqfcc2} that $\widetilde\Phi(\tilde m)=\widehat\Phi(m)$, and $|\tilde m|=\|m\|$, setting $\Psi=\widetilde{\Psi}\circ X^{-1}$ this is exactly the claimed expression \eqref{eq:dvextPhim} for $\nabla\cdot\widehat\Phi(m)$.
\end{proof}
Proposition~\ref{p:lowerbound} is a rather direct consequence of the identity obtained in Lemma~\ref{l:extPhi}.
\begin{proof}[Proof of Proposition~\ref{p:lowerbound}]
Let $m\colon\Omega\to {\mathbb R}^2$ be such that $m=\lim_{n\to\infty} m_n$ in $L^2(\Omega)$
for some $\{m_n\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$
with $\nabla\cdot m_n=0$ and $\sup_n I_{\varepsilon_n}(m_n)<\infty$.
The fact that $\nabla\cdot m=0$ in $\mathcal{D}'(\Omega)$ follows from $\nabla\cdot m_n=0$ and $L^2$ convergence.
The assumption $\sup_n I_{\varepsilon_n}(m_n)<\infty$ implies that $\|m_n\|\to 1$ in $L^2(\Omega)$.
This together with $m_n\to m$ in $L^2(\Omega)$ gives $\|m\|=1$ a.e., and thus $m$ satisfies the generalized Eikonal equation \eqref{eq:geneikon}.
Let $\Phi\in \mathrm{ENT}$ and its extension $\widehat\Phi$ defined in \eqref{eq:extPhi}.
First note that, in order to estimate $\nabla\cdot\Phi(m)$, we may assume without loss of generality that
\begin{align}\label{eq:Phizeroavg}
\int_{{\mathbb R}/2\pi{\mathbb Z}}\lambda_\Phi =0\quad\text{and}\quad \int_{\partial\mathsf{B}}\Phi =0.
\end{align}
This is due to the fact that, for any $a\in{\mathbb R}$ and $b\in{\mathbb R}^2$ the entropy given by $\Phi^{a,b}(z)=\Phi(z)+az+b$ for $z\in\partial\mathsf{B}$ satisfies $\nabla\cdot\Phi^{a,b}(m)=\nabla\cdot\Phi(m)$ since $\nabla\cdot m=0$, and $\lambda_{\Phi^{a,b}}=\lambda_\Phi + a$. Hence we may choose $a$ such that $\lambda_{\Phi^{a,b}}$ has zero average for any $b\in{\mathbb R}^2$, and $b$ such that $\Phi^{a,b}$ has zero average.
Thanks to Lemma~\ref{l:extPhi}, for any test function $\zeta\in C_c^\infty(\Omega)$ with support inside an open subset $V\subset\Omega$, we have
\begin{align*}
\langle \nabla\cdot\widehat\Phi(m_n),\zeta\rangle
& = - \frac 12\int_\Omega \zeta\, (1-\|m_n\|^2) \nabla\cdot\Psi(m_n) \, dx\nonumber\\
& \quad
-\frac 12 \int_\Omega \Psi(m_n)\cdot\nabla\zeta\, (1-\|m_n\|^2)\, dx \nonumber\\
& \lesssim \|\nabla\Psi\|_\infty \|\zeta\|_\infty I_{\varepsilon_n}(m_n;V) +
\|\Psi\|_\infty \|\nabla\zeta\|_\infty |V|^{\frac 12} \varepsilon_n^{\frac 12} I_{\varepsilon_n}(m_n)^{\frac 12},
\end{align*}
so taking the limit $n\to\infty$ we deduce
\begin{align}\label{eq:dvPhimzeta}
\langle\nabla\cdot\Phi(m),\zeta\rangle \lesssim \|\nabla\Psi\|_\infty \|\zeta\|_\infty \liminf_{n\to\infty}I_{\varepsilon_n}(m_n;V).
\end{align}
This implies in particular that $\nabla\cdot\Phi(m)$ is a finite Radon measure. From the proof of Lemma \ref{l:extPhi}, we have $\Psi=\widetilde\Psi\circ X^{-1}$ with
\begin{align*}
\widetilde\Psi(re^{i\theta})=\frac{\eta\left(r\right)\lambda_{\Phi}\left(\theta\right)}{r^2}\gamma\left(\theta\right) - \frac{\eta'\left(r\right)}{r}\Phi\left(\gamma(\theta)\right).
\end{align*}
Recalling that $X^{-1}$ is Lipschitz we deduce
\begin{align*}
\|\nabla\Psi\|_\infty \lesssim \|\Phi\|_{C^1} + \|\lambda_{\Phi}\|_{C^1}.
\end{align*}
Recall that $\lambda_\Phi$ and $\Phi$ have zero average thanks to \eqref{eq:Phizeroavg} and thus $\|\lambda_{\Phi}\|_{C^1}$ is controlled by $\|\lambda_{\Phi}'\|_{\infty}$.
Further, as $(d/d\theta)\Phi(\gamma(\theta))=\lambda_\Phi(\theta)\gamma'(\theta)$, we also have that $\|\Phi\|_{C^1}$ is controlled by $\|\lambda_{\Phi}\|_{\infty}$ and hence controlled by $\|\lambda_{\Phi}'\|_{\infty}$. So we have
\begin{align*}
\|\nabla\Psi\|_\infty \leq C_0 \|\lambda_{\Phi}'\|_{\infty}.
\end{align*}
Plugging this into \eqref{eq:dvPhimzeta} and taking the supremum over all test functions $\zeta\in C^{\infty}_c(V)$ with $\|\zeta\|_{\infty}\leq 1$ we deduce
\begin{align*}
|\nabla\cdot\Phi(m)|(V)\leq C_0 \,\liminf_{n\to\infty} I_{\varepsilon_n}(m_n;V)\quad\forall \Phi\in\mathrm{ENT}\text{ with }\|\lambda'_\Phi\|_{\infty}\leq 1.
\end{align*}
Hence for any open subset $U\subset\Omega$ and any disjoint open subsets $V_1,\ldots, V_k\subset U$ and entropies $\Phi_1,\ldots,\Phi_k\in \mathrm{ENT}$ with $\|\lambda'_{\Phi_j}\|_{\infty}\leq 1$ we have
\begin{align*}
\sum_j |\nabla\cdot\Phi_j(m)|(V_j) &\leq C_0 \sum_j \liminf_{n\to\infty} I_{\varepsilon_n}(m_n;V_j)\\
&\leq C_0 \liminf_{n\to\infty} I_{\varepsilon_n}(m_n;U).
\end{align*}
Given any disjoint compact sets $A_1,\ldots,A_k\subset U$, we can find disjoints open sets containing them, and so for entropies $\Phi_1,\ldots,\Phi_k\in \mathrm{ENT}$ with $\|\lambda'_{\Phi_j}\|_{\infty}\leq 1$ we have
\begin{align*}
\sum_j |\nabla\cdot\Phi_j(m)|(A_j)
&\leq C_0 \liminf_{n\to\infty} I_{\varepsilon_n}(m_n;U).
\end{align*}
By inner regularity of the Radon measures $\nabla\cdot\Phi(m)$ this is in fact valid for any disjoint measurable sets $A_1,\ldots,A_k$, and then for any countable disjoint family of measurable sets $\lbrace A_j\rbrace$.
Recalling the definition
\cite[Definition~1.68]{ambrosio} of the lowest upper bound measure, this implies the lower bound \eqref{eq:lowerbound}.
\end{proof}
\section{Compactness}\label{s:compactness}
In this section we prove Theorem~\ref{t:comp}: let $\{m_n\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$ satisfy $\nabla\cdot m_n=0$ and $\sup_n I_{\varepsilon_n}(m_n)<\infty$, then $\{m_n\}$ is precompact in $L^2(\Omega)$. The proof follows very closely the arguments in \cite[Proposition~1.2]{mul2}. We only briefly sketch the main ideas and highlight the steps that require adaptation. We refer to \cite{mul2} for the details that stay unchanged.
The proof consists in showing that any Young measure $\lbrace\mu_x\rbrace_{x\in\Omega}$ generated by a subsequence of $\{m_n\}$ must be a family of Dirac measures. As in \cite[(3.17)]{mul2}, the energy bound $\sup_n I_{\varepsilon_n}(m_n)<\infty$ implies that $\mu_x$ is concentrated on $\partial\mathsf{B}$ for a.e. $x\in\Omega$. The first main step is to prove that, for any entropy $\Phi\in\mathrm{ENT}$ the sequence $\nabla\cdot\widehat\Phi(m_n)$ is precompact in $H^{-1}(\Omega)$. This follows from the identity obtained in Lemma~\ref{l:extPhi}, exactly as in \cite[(3.1)]{mul2}.
The div-curl lemma therefore implies that for any entropies $\Phi_1,\Phi_2\in\mathrm{ENT}$, the weak* limit of the product $\widehat\Phi_1(m_n)\cdot i\widehat\Phi_2(m_n)$ in measures is the product of the weak limits of $\widehat\Phi_1(m_n)$ and $i\widehat\Phi_2(m_n)$ in $L^2(\Omega)$. Hence, for a.e. $x\in\Omega$, $\mu=\mu_x$ is a probability measure concentrated on $\partial\mathsf{B}$ and satisfying
\begin{align}\label{eq:Phi1Phi2mu}
\int \Phi_1\cdot i \Phi_2\, d\mu = \int \Phi_1\, d\mu \cdot i \int \Phi_2\, d\mu.
\end{align}
The conclusion of Theorem~\ref{t:comp} then follows from the next Lemma, which is the counterpart of \cite[Lemma~2.6]{mul2}.
\begin{lem}
\label{l:muDirac}
Let $\mu$ be a probability measure on ${\mathbb R}^2$ that is supported on $\partial\mathsf{B}$ and satisfies \eqref{eq:Phi1Phi2mu}
for all $\Phi_1, \Phi_2\in\mathrm{ENT}$. Then $\mu$ is a Dirac measure.
\end{lem}
\begin{rem}
In the euclidean case $\|\cdot\|=|\cdot|$, building on earlier work by Aviles and Giga \cite{avilesgig}, Jin and Kohn \cite{kohn} introduced two fundamental entropies $\Sigma_1, \Sigma_2:\mathbb{S}^1\to{\mathbb R}^2$ given by
\begin{align*}
\Sigma_1\left(e^{i\theta}\right)
= \frac i 2\left(\frac{e^{i 3\theta}}{3}+e^{-i\theta}\right),
\quad \Sigma_2\left(e^{i\theta}\right)=\frac 12\left(\frac{e^{i3\theta}}{3}-e^{-i\theta}\right).
\end{align*}
The proof of compactness given by \cite{ADM} uses only $\Sigma_1,\Sigma_2$ but is somewhat intricate, and the one in \cite{mul2} uses an infinite family of entropies.
We indicate here a somewhat shorter proof using only $\Sigma_1,\Sigma_2$ and \v{S}ver\'ak's theorem \cite{sverak}.
To see this, rewrite \eqref{eq:Phi1Phi2mu} applied to $\Sigma_1,\Sigma_2$ as
\begin{align*}
\det\left(\int_{{\mathbb R}^{2\times 2}} X \,d\nu(X)\right) = \int_{{\mathbb R}^{2\times 2}} \det (X)\, d\nu(X),
\end{align*}
where $\nu=\mathcal{P}_{\sharp}\mu$ is the pushforward of $\mu$ by the matrix-valued map $\mathcal{P}\colon\mathbb{S}^1\rightarrow {\mathbb R}^{2\times 2}$ whose rows are $\Sigma_1,\Sigma_2$.
Hence $\nu$ is a Null Lagrangian measure (in the sense of
\cite{LPnull}) supported on $K=\mathcal{P}(\mathbb S^1)$ (which is the same set as in \cite[(42)]{LP}).
By \cite[Lemma~7]{LP}, the set $K$ has no Rank-$1$ connections, so \cite[Lemma~3]{sverak} ensures that $\nu$ is a Dirac measure.
\end{rem}
The proof of Lemma~\ref{l:muDirac} follows closely \cite[Lemma~2.6]{mul2}. Nevertheless we give some details, because this is where the crucial assumption that $\mathsf{B}$ is strictly convex is used. In the proof we will need the following construction.
\begin{lem}
\label{l:genent}
Given $\xi=\gamma(\theta_0)\in \partial \mathsf{B}$, define $\Phi^{\xi}:\partial\mathsf{B}\to{\mathbb R}^2$ by
\begin{align*}
\Phi^{\xi}(z)=\mathbf 1_{z\cdot i\gamma(\theta_0)>0}\,\gamma'(\theta_0)=\mathbf 1_{z\cdot i\xi>0}\,in_\mathsf{B}(\xi),
\end{align*}
where $n_{ \mathsf{B}}(\xi)$ denotes the outer unit normal to $\partial \mathsf{B}$ at $\xi$.
%
Then $\Phi^{\xi}$ is a generalized entropy for the equation \eqref{eq:geneikon} in the sense that there exists a sequence $\left\{\Phi^{\xi}_{\delta}\right\}_{\delta>0}\subset\mathrm{ENT}$ that is uniformly bounded and satisfies
\begin{equation}
\label{eqb2}
\Phi^{\xi}_{\delta}(z)\rightarrow\Phi^{\xi}(z)\text{ for all }z\in \partial\mathsf{B}.
\end{equation}
\end{lem}
\begin{proof}
For any $\lambda\in C^1({\mathbb R}/2\pi{\mathbb Z})$ such that $\int_{{\mathbb R}/2\pi{\mathbb Z}}\lambda(\theta)\gamma'(\theta)\, d\theta =0$, the map $\Phi\colon\partial \mathsf{B}\to{\mathbb R}^2$ given by
\begin{align*}
\Phi(\gamma(\theta))=\int_{\theta_0}^\theta \lambda(t)\gamma'(t)\, dt,
\end{align*}
is well defined and belongs to $\mathrm{ENT}$. We define a sequence of functions $\lambda_\delta$ such that the corresponding $\Phi=\Phi^\xi_\delta$ has the desired properties. We fix a smooth nonnegative kernel $\rho\in C_c^\infty({\mathbb R})$ with support $\supp\rho\subset (0,1)$ and unit integral $\int\rho=1$, denote $\rho_\delta(t)=\delta^{-1}\rho(t/\delta)$, and define a function $\hat\lambda_\delta\in C^\infty({\mathbb R}/2\pi{\mathbb Z})$ by setting
\begin{align*}
\hat\lambda_\delta(\theta)= \rho_\delta(\theta-\theta_0)
+ \rho_\delta(\pi+\theta_0-\theta)
\qquad\text{for }\theta_0 < \theta \leq \theta_0 +2\pi,
\end{align*}
and $\hat\lambda_\delta$ extended as a $2\pi$-periodic function. Note that $\hat\lambda_\delta$ is supported in
\begin{align*}
(\theta_0,\theta_0+\delta) \cup (\theta_0+\pi-\delta,\theta_0 +\pi) +2\pi{\mathbb Z}.
\end{align*}
Moreover, the map $\Psi_\delta\colon (\theta_0,\theta_0+2\pi]\to{\mathbb R}^2$ defined by
\begin{align*}
\Psi_\delta(\theta)=\int_{\theta_0}^\theta \hat\lambda_\delta(t)\gamma'(t)\, dt\qquad\text{for }\theta_0 < \theta \leq \theta_0 +2\pi,
\end{align*}
satisfies $|\Psi_\delta|\leq 2$ since $|\gamma'|=1$, and
\begin{align*}
\Psi_\delta(\theta)
&
=
\begin{cases}
\int_{\mathbb R} \rho_\delta(t-\theta_0)\gamma'(t)\, dt &\text{ if }\theta_0+\delta <\theta <\theta_0+\pi-\delta,\\
\int_{\mathbb R} \rho_\delta(t-\theta_0)\gamma'(t)\, dt
+\int_{\mathbb R} \rho_\delta(\theta_0+\pi-t)\gamma'(t)\, dt
&\text{ if }\theta_0+\pi \leq \theta \leq \theta_0+2\pi.\\
\end{cases}
\end{align*}
Using that $\gamma'$ is continuous and $\gamma'(\theta_0+\pi)=-\gamma'(\theta_0)$ we obtain, for any $\theta\in (\theta_0,\theta_0+2\pi]$, the limit
\begin{align*}
\lim_{\delta\to 0}\Psi_\delta(\theta)
&=
\begin{cases}
\gamma'(\theta_0) &\quad\text{ if }\theta_0 <\theta <\theta_0+\pi ,\\
0
&\quad \text{ if }\theta_0+\pi \leq \theta \leq \theta_0+2\pi.\\
\end{cases}
\end{align*}
This corresponds exactly to $\Phi^\xi(\gamma(\theta))$. The only issue is that $\hat\lambda_\delta$ does not satisfy the constraint
$\int_{{\mathbb R}/2\pi{\mathbb Z}}\hat\lambda_\delta(\theta)\gamma'(\theta)\, d\theta =0$, hence $\Psi_\delta$ cannot be extended to a $2\pi$-periodic function and does not define an entropy. So we need to modify $\hat\lambda_\delta$.\footnote{Note in order to get the pointwise convergence in \eqref{eqb2} for every $z\in \partial \mathsf{B}$ we can not define $\hat\lambda_{\delta}$ via the standard symmetric (across zero) kernel centered on $\theta_0$, $\theta_0+\pi$. This is why we do the two step procedure of defining $\hat\lambda_\delta$ then modifying it.} We claim that there exists $\mu_\delta\in C^1({\mathbb R}/2\pi{\mathbb Z})$ such that
\begin{align}\label{eq:mudelta}
\begin{aligned}
&\int_{{\mathbb R}/2\pi{\mathbb Z}}\mu_\delta(\theta)\gamma'(\theta)\, d\theta
=
\int_{{\mathbb R}/2\pi{\mathbb Z}}\hat\lambda_\delta(\theta)\gamma'(\theta)\, d\theta,\\
\text{and }&\int_{{\mathbb R}/2\pi{\mathbb Z}}|\mu_\delta(\theta)|\,d\theta \longrightarrow 0\qquad\text{as }\delta\to 0.
\end{aligned}
\end{align}
Granted \eqref{eq:mudelta}, we define $\lambda_\delta =\hat\lambda_\delta(\theta)-\mu_\delta$. This function does satisfy $\int_{{\mathbb R}/2\pi{\mathbb Z}}\lambda_\delta(\theta)\gamma'(\theta)\, d\theta =0$ thanks to \eqref{eq:mudelta}, so the formula
\begin{align*}
\Phi_\delta^\xi(\gamma(\theta))=\int_{\theta_0}^\theta \lambda(t)\gamma'(t)\, dt,
\end{align*}
defines an entropy $\Phi_\delta^\xi\in\mathrm{ENT}$. Moreover for any $\theta\in (\theta_0,\theta_0+2\pi]$ we have
\begin{align*}
|\Psi_\delta(\theta)-\Phi_\delta^\xi(\gamma(\theta))|\leq
\int_{{\mathbb R}/2\pi{\mathbb Z}}|\mu_\delta(\theta)|\,d\theta \longrightarrow 0\qquad\text{as }\delta\to 0.
\end{align*}
Thanks to the convergence of $\Psi_\delta$ established above, this implies $\Phi_\delta^\xi(z)\to \Phi^\xi(z)$ for all $z\in\partial\mathsf{B}$, and uniform boundedness of $\Phi_\delta^\xi$ follows from the uniform boundedness of $\Psi_\delta$. Therefore the proof of Lemma~\ref{l:genent} will be complete once we prove the existence of $\mu_\delta\in C^1({\mathbb R}/2\pi{\mathbb Z})$ satisfying \eqref{eq:mudelta}.
This construction is possible thanks to the fact that
\begin{align*}
v^\delta:=\int_{{\mathbb R}/2\pi{\mathbb Z}}\hat\lambda_\delta(\theta)\gamma'(\theta)\, d\theta\longrightarrow \gamma'(\theta_0)+\gamma'(\theta_0+\pi)=0,
\end{align*}
as $\delta\to 0$.
To explicitly construct $\mu_\delta$, we introduce a (small) parameter $\eta>0$, to be fixed later, and two functions $f_1^\eta,f_2^\eta\in C^1({\mathbb R}/2\pi{\mathbb Z})$ such that
\begin{align*}
\|f_j^\eta-\gamma_j'\|_{L^2({\mathbb R}/2\pi{\mathbb Z})}\leq\eta\qquad\text{for }j=1,2,
\end{align*}
and look for $\mu_\delta$ in the form
\begin{align*}
\mu_\delta (\theta)=\alpha_\delta f_1^\eta(\theta) +\beta_\delta f_2^\eta(\theta),\qquad \alpha_\delta,\beta_\delta\in{\mathbb R}.
\end{align*}
With these notations, the constraint
$\int \mu_\delta \gamma'
=
\int \hat\lambda_\delta \gamma' $ in \eqref{eq:mudelta} turns into
\begin{align*}
v^\delta=A_\eta \left(\begin{array}{c} \alpha_\delta\\ \beta_\delta\end{array}\right),
\qquad
A_\eta=\left(\begin{array}{cc}
\int f_1^\eta\gamma'_1 & \int f_2^\eta\gamma'_1\\
\int f_1^\eta\gamma'_2 & \int f_2^\eta\gamma'_2
\end{array}
\right).
\end{align*}
Next we show that we may fix $\eta>0$ such that $A_\eta$ is invertible. As a consequence, defining $(\alpha_\delta,\beta_\delta)^{T}=A_\eta^{-1}v^\delta$ ensures that $\mu_\delta$ satisfies the constraint
$\int \mu_\delta \gamma'
=
\int \hat\lambda_\delta \gamma' $
in \eqref{eq:mudelta}, and the convergence $\mu_\delta\to 0$ in $L^1$ follows from $v^\delta\to 0$. This concludes the proof of \eqref{eq:mudelta} and of Lemma~\ref{l:genent}.
It remains to prove that $A_\eta$ is invertible for small enough $\eta>0$. To that end we remark that
thanks to the convergence $f_j^\eta\to \gamma_j'$ in $L^2$, we have
\begin{align*}
A_\eta \longrightarrow A_0 =
\left(\begin{array}{cc}
\int (\gamma'_1)^2 & \int \gamma'_2\gamma'_1\\
\int \gamma'_1\gamma'_2 & \int (\gamma'_2)^2
\end{array}
\right)\qquad\text{ as }\eta\to 0.
\end{align*}
The Cauchy-Schwarz inequality $(\int \gamma'_1\gamma'_2)^2\leq\int (\gamma'_1)^2\int (\gamma'_2)^2$ ensures that $\det(A_0)\geq 0$, and in fact $\det(A_0)>0$ because equality cannot occur in the Cauchy-Schwarz inequality: otherwise $\gamma'_1,\gamma'_2$ would be colinear in $L^2$, implying that $\gamma'$ takes values in a fixed line, which is incompatible with it being the unit tangent of $\partial\mathsf{B}$. So the matrix $A_0$ is invertible, and we may fix $\eta>0$ such that $A_\eta$ is invertible.
\end{proof}
With the construction of Lemma~\ref{l:genent} at hand, we turn to the proof of Lemma~\ref{l:muDirac}.
\begin{proof}[Proof of Lemma~\ref{l:muDirac}]
Let $\xi_1,\xi_2\in \partial \mathsf{B}$.
By Lemma \ref{l:genent}, for $j=1,2$, we can find $\left\{\Phi_{\delta}^{\xi_j}\right\}_{\delta}\subset \mathrm{ENT}$ such that
\begin{equation*}
\Phi^{\xi_j}_{\delta}(z)\overset{\delta\rightarrow 0}{\rightarrow} \Phi^{\xi_j}(z)\quad\text{ for all }z\in \partial \mathsf{B}.
\end{equation*}
Also recall that $\left\{\Phi^{\xi_j}_{\delta}\right\}$ is uniformly bounded. Hence, applying \eqref{eq:Phi1Phi2mu} to $\Phi_j=\Phi^{\xi_j}_\delta$ and passing to the limit $\delta\to 0$ we obtain, by dominated convergence,
\begin{equation*}
\int_{\partial \mathsf{B}} \Phi^{\xi_1}(z)\cdot i \Phi^{\xi_2}(z) d\mu(z) = \left(\int_{\partial \mathsf{B}} \Phi^{\xi_1}(z) d\mu(z)\right)\cdot i\left(\int_{\partial \mathsf{B}} \Phi^{\xi_2}(z) d\mu(z)\right).
\end{equation*}
In other words, recalling the definition of $\Phi^\xi$ in Lemma~\ref{l:genent}, for any $\xi_1,\xi_2\in \partial \mathsf{B}$ we have
\begin{align*}
& in_\mathsf{B}(\xi_1)\cdot n_\mathsf{B}(\xi_2) \,\mu( \lbrace z\cdot i\xi_1>0\rbrace\cap \lbrace z\cdot i\xi_2>0\rbrace)
\nonumber\\
&
= in_\mathsf{B}(\xi_1)\cdot n_\mathsf{B}(\xi_2) \,\mu( \lbrace z\cdot i\xi_1>0\rbrace)\,\mu( \lbrace z\cdot i\xi_2>0\rbrace).
\end{align*}
(Here and in the rest of the proof, we use the shortened notation $\lbrace z\cdot i\xi>0\rbrace$ to denote the subset of points $z\in\partial \mathsf{B}$ satisfying this inequality.)
By strict convexity of $\partial\mathsf{B}$, for $\xi_1\ne \pm \xi_2$ we have $in_{\mathsf{B}}(\xi_1)\cdot n_{\mathsf{B}}(\xi_2)\ne 0$, and the last equation becomes
\begin{align*}
\mu( \lbrace z\cdot i\xi_1>0\rbrace\cap \lbrace z\cdot i\xi_2>0\rbrace)
= \mu( \lbrace z\cdot i\xi_1>0\rbrace)\,\mu( \lbrace z\cdot i\xi_2>0\rbrace).
\end{align*}
From that point on the proof follows exactly \cite[Lemma~2.6]{mul2}, for the reader's convenience we recall here the short argument. Letting $\xi_2\to\xi_1$ with $\xi_2\neq \pm\xi_1$
we obtain
\begin{equation*}
\mu\left( \lbrace z\cdot i\xi_1>0\rbrace\right)\leq\mu\left( \lbrace z\cdot i\xi_1>0\rbrace\right)\mu\left( \lbrace z\cdot i\xi_1\geq0\rbrace\right),
\end{equation*}
which implies
\begin{equation*}
\mu\left( \lbrace z\cdot i\xi>0\rbrace\right)=0\text{ or }\mu\left( \lbrace z\cdot i\xi\geq0\rbrace\right)=1\text{ for all }\xi\in \partial \mathsf{B}.
\end{equation*}
This is equivalent to
\begin{equation}
\label{eqb26}
\mu\left( \lbrace z\cdot i\xi>0\rbrace\right)=0\text{ or }\mu\left( \lbrace z\cdot i\xi<0\rbrace\right)=0\text{ for all }\xi\in \partial \mathsf{B}.
\end{equation}
This implies that $\mu$ is a Dirac measure: otherwise we can find $\xi\in \partial \mathsf{B}$ such that
\begin{equation*}
\mu\left(\lbrace z\cdot i\xi>0\rbrace\right)>0 \text{ and }\mu\left(\lbrace z\cdot i\xi<0\rbrace\right)>0,
\end{equation*}
which contradicts \eqref{eqb26}.
\end{proof}
\section{Zero-energy states}\label{s:zeroenergy}
In this section we prove Theorem~\ref{T:zeroenergy} on zero-energy states:
let $m\colon\Omega\to {\mathbb R}^2$ satisfy $m=\lim_{n\to\infty} m_n$ in $L^2(\Omega)$ for some $\{m_n\}\subset W^{1,2}(\Omega;{\mathbb R}^2)$
with $\nabla\cdot m_n=0$ and $\lim_{n\to \infty} I_{\varepsilon_n}(m_n)=0$ for some $\varepsilon_n\to 0$,
then $m$ must be continuous outside a locally finite set of vortices associated to the norm $\|\cdot\|$.
The first step, similar to \cite[Proposition~1.1]{otto} is to obtain the kinetic formulation
\begin{align}\label{eq:kinzero}
\gamma'(t)\cdot \nabla_x\mathbf 1_{m(x)\cdot i\gamma(t)>0} = 0\quad\text{ in }\mathcal{D}'(\Omega)\text{ for all }t\in{\mathbb R}.
\end{align}
This follows from the fact that entropy productions vanish: thanks to Proposition~\ref{p:lowerbound}, $\nabla\cdot\Phi(m)=0$ in $\mathcal{D}'(\Omega)$ for all $\Phi\in\mathrm{ENT}$. For any $t\in{\mathbb R}$, we may apply this to the entropies $\Phi_\delta^{\gamma(t)}$ provided by Lemma~\ref{l:genent}, hence
\begin{align*}
\int_\Omega \Phi_\delta^{\gamma(t)}(m(x))\cdot \nabla\zeta(x)\, dx=0\qquad\text{for all }\zeta\in C_c^\infty(\Omega).
\end{align*}
Thanks to the pointwise convergence $\Phi_\delta^{\gamma(t)}(z)\to \Phi^{\gamma(t)}(z)=
\mathbf 1_{z\cdot i\gamma(t)>0}\gamma'(t)$ and the uniform boundedness of $\Phi_\delta^{\gamma(t)}$, we can pass to the limit $\delta\to 0$ by dominated convergence, and deduce
\begin{align*}
\int_\Omega \mathbf 1_{m(x)\cdot i\gamma(t)>0}\,\gamma'(t)\cdot \nabla\zeta(x)\, dx=0\qquad\text{for all }\zeta\in C_c^\infty(\Omega),
\end{align*}
which is exactly \eqref{eq:kinzero}.
Next we define $\tilde m\colon\Omega\to\mathbb S^1$ by setting
\begin{align*}
\tilde m =n_\mathsf{B}(m),\quad\text{where } n_\mathsf{B}\colon\partial\mathsf{B}\to\mathbb S^1\text{ is the outer unit normal to }\partial\mathsf{B}.
\end{align*}
The symmetries of $\mathsf{B}$ ensure that, for
any $z\in\partial \mathsf{B}$,
\begin{align*}
z\cdot i\gamma(t) >0 \qquad\Longleftrightarrow\qquad n_\mathsf{B}(z)\cdot \gamma'(t)>0.
\end{align*}
Therefore, for a fixed $t\in {\mathbb R}$, and $\theta_t=\alpha(t)$ where $\alpha\colon{\mathbb R}\to{\mathbb R}$ is the (unique up to an additive constant) continuous function such that $\gamma'(t)=e^{i\alpha(t)}$, we have
\begin{align*}
\mathbf 1_{m(x)\cdot i\gamma(t)>0}\,\gamma'(t)
= \mathbf 1_{\tilde m(x)\cdot e^{i\theta_t}>0}\, e^{i\theta_t}.
\end{align*}
As $t\mapsto\theta_t$ is a bijection from ${\mathbb R}$ into itself we deduce from \eqref{eq:kinzero} that $\tilde m$ solves the kinetic equation
\begin{align*}
e^{i\theta}\cdot\nabla_x\mathbf 1_{\tilde m(x)\cdot e^{i\theta}>0} =0
\quad\text{ in }\mathcal{D}'(\Omega)\text{ for all } \theta\in{\mathbb R}.
\end{align*}
This is the kinetic formulation that characterizes zero-energy states for the classical Aviles-Giga functional: it follows from \cite[Theorem~1.3]{otto} that $\tilde m$ is locally Lipschitz outside a locally finite set. Moreover, in any convex neighborhood of a singularity $x_0$ we have $\tilde m(x)=\beta i(x-x_0)/|x-x_0|$ for some $\beta\in\lbrace \pm 1\rbrace$.
Note that, since $\partial \mathsf{B}$ is $C^1$ and strictly convex, the map
$n_\mathsf{B}\colon\partial\mathsf{B}\to\mathbb S^1$ is a homeomorphism. We deduce that $m=n_\mathsf{B}^{-1}(\tilde m)$ is continuous outside a locally finite set. Moreover, from \cite[Proposition~2.2]{boch} we know that $n_\mathsf{B}^{-1}(x/|x|)=V_\mathsf{B}(x)$ for any $x\in{\mathbb R}^2$, where $V_\mathsf{B}=\nabla\|\cdot\|_*$ is the vortex associated to $\|\cdot\|$ and $\|\cdot \|_{*}$ is the dual norm of $\|\cdot \|$, and $V_\mathsf{B}(-x)=-V_\mathsf{B}(x)$, so we deduce $m(x)=\beta V_\mathsf{B}(i(x-x_0))$ in any convex neighborhood of a singularity $x_0$. This concludes the proof of Theorem~\ref{T:zeroenergy}.
To prove the assertion of Remark~\ref{r:zeroenergytypep}, simply note that $n_\mathsf{B}^{-1}$ is $1/(p-1)$-H\"older whenever $\|\cdot\|$ is of power type $p$ for $p\geq 2$ \cite[Theorem~2.6]{boch}.
\section{Regularity estimates}\label{s:reg}
In this section we give the proofs of Theorems \ref{thmbes} and \ref{thmbesrev} in Subsections \ref{ss:reg} and \ref{ss:revreg}, respectively. The proof of Theorem \ref{t:regconvBV} relies on explicit calculations in the $BV$ setting and is postponed to Subsection \ref{besbv}.
\subsection{Finite entropy production implies regularity estimates}
\label{ss:reg}
In this subsection we find it more convenient to work with maps $m\colon\Omega\to\mathbb{R}^2$ solving
\begin{align}\label{eq:mX}
|m|=1\text{ a.e.},\quad \nabla\cdot X(m)=0\text{ in }\mathcal D'(\Omega).
\end{align}
Solutions of \eqref{eq:mX} and of the generalized Eikonal equation \eqref{eq:geneikon} are in correspondence via the Lipschitz homeomorphism $X$ defined in \eqref{eq:X}. Specifically, a map $m$ solves \eqref{eq:mX} if and only if $\overline m = X(m)$ solves \eqref{eq:geneikon}.
This transformation also induces a correspondence between entropies.
A $C^1$ map $\Phi\colon \mathbb S^1\to{\mathbb R}^2$ is an entropy for equation \eqref{eq:mX} if and only if
$\nabla\cdot\Phi(m)=0$ for any smooth solution of \eqref{eq:mX}.
It is an exercise to see that this is equivalent to $(d/d\theta)\Phi(e^{i\theta})$ being colinear to $\gamma'(\theta)$.
As in the definition of $\mathrm{ENT}$ in \eqref{eq:ENT}, we consider the subclass of entropies where we require a $C^1$ colinearity coefficient:
\begin{align*}
\frac{d}{d\theta}\Phi(e^{i\theta})=\lambda(\theta)\gamma'(\theta)\quad\text{for some }\lambda\in C^1({\mathbb R}/2\pi{\mathbb Z}).
\end{align*}
Therefore, $\Phi$ is an entropy for equation \eqref{eq:mX} in this subclass
if and only if $\overline\Phi=\Phi\circ X^{-1}\colon\partial\mathsf{B}\to{\mathbb R}^2$ belongs to $\mathrm{ENT}$, as follows directly from the definition \eqref{eq:ENT} of the class $\mathrm{ENT}$. Moreover we have $\overline\Phi(\overline m)=\Phi(m)$. In particular, the entropy productions $\nabla\cdot\Phi(m)$ of a solution $m$ of \eqref{eq:mX} are measures if and only if the entropy productions $\nabla\cdot\overline\Phi(\overline m)$ of the solution $\overline m=X(m)$ of \eqref{eq:geneikon} are measures. Thanks to the above discussion, all results we prove in this section about solutions of \eqref{eq:mX} directly translate into corresponding results about solutions of \eqref{eq:geneikon}.
We use the family of entropies $\Phi_\psi$ for equation \eqref{eq:mX} given by
\begin{align}\label{eq:Phipsi}
\Phi_\psi(e^{i\theta})=\int_{\mathbb{R}/ 2\pi \mathbb{Z}} \mathbf 1_{e^{i\theta}\cdot e^{is}>0}\,\psi(s)\gamma'(s-\pi/2)\, ds,\qquad\psi\in C^1({\mathbb R}/2\pi\mathbb Z).
\end{align}
Note that since $\gamma'(t+\pi)=-\gamma'(t)$ we have that $(d/d\theta)\Phi_{\psi}\left(e^{i\theta}\right)=\lambda(\theta)\gamma'(\theta)$ with
$\lambda(\theta)=\psi(\theta+\pi/2)+\psi(\theta-\pi/2)$.
Therefore \eqref{eq:Phipsi} does define an entropy for equation \eqref{eq:mX}.
Recall that $\alpha\in C^0({\mathbb R})$ is such that
\begin{align}\label{eq:alpha}
e^{i\alpha(\theta)}=\gamma'(\theta)\qquad \forall \theta\in{\mathbb R}.
\end{align}
The continuous function $\alpha$ is uniquely determined up to a constant, is strictly increasing, and satisfies $\alpha(\theta+\pi)=\alpha(\theta)+\pi$ for all $\theta\in{\mathbb R}$.
\begin{prop}\label{p:besov}
If $m$ satisfies \eqref{eq:mX} and
\begin{align*}
\nabla\cdot\Phi_\psi(m)\in\mathcal M_{loc}(\Omega)\qquad\forall\psi\in C^1(\mathbb R/2\pi\mathbb Z),
\end{align*}
then we have
\begin{align}
\label{eqbes22}
\sup_{0<|h|<\dist(\Omega',\partial\Omega)} \frac{1}{|h|}\int_{\Omega'} \Lambda(m(x),m(x+h))\, dx
<\infty\qquad\forall\Omega'\subset\subset\Omega,
\end{align}
where $\Lambda\colon\mathbb S^1\times \mathbb S^1\to [0,+\infty)$ is given by
\begin{align}\label{eq:Lambda1}
\Lambda(e^{i\theta_1},e^{i\theta_2}) =\int_{\theta_1}^{\theta_2} \int_{\theta_1}^{\theta_2} |\alpha(t)-\alpha(s)|\, dt ds \quad\text{ for }|\theta_1-\theta_2|\leq \pi.
\end{align}
Moreover, for $m_1, m_2\in\mathbb{S}^1$, we have
\begin{align}
\label{eqbes21}
\Lambda(m_1,m_2)\gtrsim \delta^2\omega^{-1}(\delta/2),\qquad\delta=|m_1-m_2|,
\end{align}
where $\omega(\delta)=\sup \lbrace |\alpha^{-1}(t)-\alpha^{-1}(s)|\colon |t-s|<\delta \rbrace$ is the minimal modulus of continuity of $\alpha^{-1}$.
\end{prop}
Theorem \ref{thmbes} is a direct consequence of Proposition \ref{p:besov} and the correspondence between the generalized Eikonal equation \eqref{eq:geneikon} and equation \eqref{eq:mX}. In particular, the regularity estimate \eqref{eqbes2} of Theorem \ref{thmbes} is equivalent to \eqref{eqbes22} by noting that the function $\Pi$ defined in \eqref{eq:Lambda} satisfies $\Pi(\gamma(\theta_1),\gamma(\theta_2)) = \Lambda(e^{i\theta_1},e^{i\theta_2})$.
\begin{rem}
\label{remppower}
If $\|\cdot\|$ is in addition of power type $p$ for some $p\geq 2$ then $\alpha^{-1}$ is $1/(p-1)$-H\"older \cite[Theorem 2.6]{boch}, so $\omega^{-1}(\delta)\gtrsim \delta^{p-1}$ and therefore \eqref{eqbes22} and \eqref{eqbes21} imply
\begin{align*}
\sup_{0<|h|<\dist(\Omega',\partial\Omega)}\frac{1}{|h|}\int_{\Omega'} |m(x+h)-m(x)|^{p+1}\, dx <\infty\qquad\forall\Omega'\subset\subset\Omega,
\end{align*}
that is, $m$ has the local Besov regularity $B^{\frac{1}{p+1}}_{p+1,\infty}$.
\end{rem}
\begin{rem}
\label{remlam}
The function $\alpha$ is increasing and therefore $D\alpha$ is a positive measure. Hence for $0\leq \theta_2-\theta_1\leq\pi$,
one can rewrite the quantity $\Lambda$ defined in \eqref{eq:Lambda1} as
\begin{align*}
\Lambda(e^{i\theta_1},e^{i\theta_2})&= \iiint_{[\theta_1,\theta_2]^3}\left(\mathbf 1_{s<\tau<t} +\mathbf 1_{s>\tau>t}\right)\, dtds\, D\alpha(d\tau)
\nonumber\\
&=2\int_{\theta_1}^{\theta_2} (\tau-\theta_1)(\theta_2-\tau)\, D\alpha(d\tau)
\nonumber\\
&\geq \frac {2}{9} (\theta_2-\theta_1)^2 D\alpha([\theta_1+(\theta_2-\theta_1)/3,\theta_1+2(\theta_2-\theta_1)/3]).
\end{align*}
(We used the identity $\iint_{[\theta_1,\theta_2]^2}\left(\mathbf 1_{s<\tau<t} +\mathbf 1_{s>\tau>t}\right)\, dtds=2(\tau-\theta_1)(\theta_2-\tau)$ to obtain the second equality.)
\end{rem}
\begin{proof}[Proof of Proposition~\ref{p:besov}]
The proof is a direct combination of Lemmas~\ref{l:kin} to \ref{l:lowerLambda} below.
\end{proof}
\begin{lem}\label{l:kin}
If $m$ satisfies \eqref{eq:mX} and
\begin{align*}
\nabla\cdot\Phi_\psi(m)\in\mathcal M(\Omega)\qquad\forall\psi\in C^1\left(\mathbb{R}/2\pi\mathbb{Z}\right),
\end{align*}
then $m$ satisfies the kinetic equation
\begin{align}\label{eq:kin}
\gamma'(s-\pi/2)\cdot \nabla_x \mathbf 1_{m(x)\cdot e^{is}>0} =\partial_s\sigma(s,x)\qquad \text{in }\mathcal{D}'(\mathbb{R}/2\pi\mathbb{Z}\times\Omega),
\end{align}
for some $\sigma\in\mathcal M(\mathbb{R}/2\pi\mathbb{Z}\times\Omega)$.
\end{lem}
\begin{proof}
For any fixed $\zeta\in C^{\infty}_c(\Omega)$,
the operator $T_{\zeta}:\psi \mapsto \langle \nabla \cdot \Phi_{\psi}(m), \zeta\rangle $ is a linear operator from $C^1({\mathbb R}/2\pi\mathbb{Z})$ to ${\mathbb R}$.
Further the estimate $\left|\langle \nabla \cdot \Phi_{\psi}(m), \zeta\rangle \right|\lesssim \|\psi\|_{C^1\left( \mathbb{R}/2\pi\mathbb{Z}\right)} \|\nabla \zeta\|_{L^\infty(\Omega)}$
implies that $T_{\zeta}$ is a bounded linear operator.
Thus the same Banach-Steinhaus argument as in \cite[Lemma~3.4]{GL} provides the bound
\begin{align*}
|\langle \nabla\cdot\Phi_\psi(m),\zeta\rangle | \lesssim \|\psi\|_{C^1(\mathbb{R}/2\pi\mathbb{Z})}\|\zeta\|_{L^\infty(\Omega)},
\end{align*}
for all $\zeta\in C^0_c(\Omega)$ and $\psi\in C^1(\mathbb{R}/2\pi\mathbb{Z})$. Moreover, when $\psi$ is a constant $\psi\equiv c$, we have $\nabla\cdot\Phi_\psi(m)=2c\nabla\cdot X(m)=0$, so in the above we can consider the quotient space $C^1(\mathbb{R}/2\pi\mathbb{Z})/{\mathbb R}\approx C^0({\mathbb R}/2\pi{\mathbb Z})$. Explicitly, for any $f\in C^0(\mathbb{R}/2\pi\mathbb{Z})$ consider the function $\psi[f]\in C^1(\mathbb{R}/2\pi\mathbb{Z})$ given by
\begin{align*}
\psi[f](t)=\int_0^t\left(f-\Xint{-}_{\mathbb{R}/2\pi\mathbb{Z}}f\right),
\end{align*}
then we have
\begin{align*}
\langle \nabla\cdot\Phi_{\psi[f]}(m),\zeta\rangle \lesssim \|f\|_{L^\infty(\mathbb{R}/2\pi\mathbb{Z})}\|\zeta\|_{L^\infty(\Omega)},
\end{align*}
for all $\zeta\in C^0_c(\Omega)$ and $f\in C^0(\mathbb{R}/2\pi\mathbb{Z})$. As a consequence (see \cite[Appendix~B]{LPfacto} for a detailed proof) there exists a measure $\sigma\in \mathcal M(\mathbb{R}/2\pi\mathbb{Z}\times\Omega)$ such that, for all $\zeta\in C^{\infty}_c(\Omega)$ and $f\in C^0({\mathbb R}/2\pi\mathbb{Z})$ with $\Xint{-}_{\mathbb{R}/2\pi\mathbb{Z}}f=0$, we have
\begin{align*}
\langle \nabla\cdot\Phi_{\psi[f]}(m),\zeta\rangle=-\langle \sigma,f\otimes\zeta\rangle =\langle\partial_s\sigma,\psi[f]\otimes \zeta\rangle.
\end{align*}
From the the definition of $\Phi_\psi$ \eqref{eq:Phipsi} we see that this is equivalent to
\begin{align}
\label{eqb11}
\langle \gamma'(s-\pi/2) \cdot\nabla_x \mathbf 1_{m(x)\cdot e^{is}>0} -\partial_s\sigma,\psi(s)\zeta(x)\rangle =0,
\end{align}
for $\psi=\psi[f]$, that is, for all $\psi\in C^1(\mathbb{R}/2\pi\mathbb{Z})$ such that $\psi(0)=0$ and $\zeta\in C^{\infty}_c(\Omega)$. For constant $\psi =c$, equation \eqref{eqb11} amounts to $2 c\,\nabla\cdot X(m)=0$, so it is in fact valid for any $\psi\in C^1(\mathbb{R}/2\pi\mathbb{Z})$. This proves the kinetic equation \eqref{eq:kin}.
\end{proof}
\begin{rem}
Note that the kinetic equation \eqref{eq:kin} only uniquely determines $\partial_s\sigma$. We may choose the unique $\sigma$ satisfying in addition $\langle\sigma(s,x), \zeta(x)\rangle=0$ for all $\zeta\in C^0_c(\Omega)$.
\end{rem}
\begin{lem
If $m$ satisfies $|m|=1$ a.e. and the kinetic equation \eqref{eq:kin}, then
for any $\varphi\in BV({\mathbb R}/2\pi{\mathbb Z})$ which is odd, i.e. $\varphi(-\theta)=-\varphi(\theta)$, the quantity
\begin{align}\label{eq:Deltavarphi}
\Delta_\varphi(e^{i\theta_1},e^{i\theta_2})&=
\iint_{{\mathbb R}/2\pi{\mathbb Z}\times{\mathbb R}/2\pi{\mathbb Z}}\varphi(t-s)\,\gamma'(s-\pi/2)\wedge\gamma'(t-\pi/2)
\\
&\qquad\qquad
\left(\mathbf 1_{e^{is}\cdot e^{i\theta_2} >0}-\mathbf 1_{e^{is}\cdot e^{i\theta_1} >0}\right)\left(\mathbf 1_{e^{it}\cdot e^{i\theta_2} >0}-\mathbf 1_{e^{it}\cdot e^{i\theta_1} >0}\right)\, dtds, \nonumber
\end{align}
satisfies
\begin{align*}
\frac{1}{|h|}\int_{\Omega'}\Delta_\varphi(m(x),m(x+h))\, dx \lesssim \frac{\|\varphi\|_{L^1({\mathbb R}/2\pi\mathbb{Z})}}{\dist(\Omega',\partial\Omega)}+|D\varphi|({\mathbb R}/2\pi\mathbb{Z})
|\sigma|({\mathbb R}/2\pi\mathbb{Z} \times\Omega),
\end{align*}
for all $\Omega'\subset\subset \Omega$ and $h\in{\mathbb R}^2$ such that $|h| <\dist(\Omega',\partial\Omega)$.
\end{lem}
\begin{proof}
This essentially follows \cite[Lemma~3.9]{GL} in a slightly modified setting; we provide some details here for the reader's convenience. We set
\begin{align*}
\chi(t,x)=\mathbf 1_{e^{it}\cdot m(x)>0},
\end{align*}
and for a small parameter $\varepsilon>0$ we consider regularized (with respect to $x$) maps
\begin{align*}
\chi_\varepsilon=\chi *_x \rho_\varepsilon,\quad \sigma_\varepsilon =\sigma *_{x} \rho_\varepsilon,
\end{align*}
where $\rho_\varepsilon$ is a regularizing kernel. We have the regularized kinetic equation
\begin{align*}
\gamma'(s-\pi/2)\cdot\nabla_{x}\chi_\varepsilon=\partial_s\sigma_\varepsilon.
\end{align*}
Let $\Omega'\subset\subset\Omega$ be fixed and $h=u e$ for some $e\in\mathbb{S}^1$ and $u\in{\mathbb R}$ such that $|u|=|h|<\dist(\Omega',\partial\Omega)$. Without loss of generality, assume $e=e_1$. We denote by $\chi^u(t, x)=\chi(t, x+ue_1)$ and $D^u\chi(t,x)=\chi^u(t,x)-\chi(t,x)$.
Define the
quantity
\begin{align*}
\Delta^{\varepsilon}_\varphi(x,u)&=\iint_{{\mathbb R}/2\pi{\mathbb Z}\times{\mathbb R}/2\pi{\mathbb Z}}\varphi(t-s)\,\gamma'(s-\pi/2)\wedge\gamma'(t-\pi/2) \\
&\hspace{12em}
D^{u}\chi_\varepsilon(s,x) D^{u}\chi_\varepsilon(t,x)\, dtds,
\end{align*}
for $x\in\Omega$ and $|u|+\epsilon < \dist(x,\partial\Omega)$.
Note that, as $\varepsilon\to 0$, we have the pointwise limit
\begin{align*}
\Delta^{\varepsilon}_\varphi(x,u)\longrightarrow \Delta_\varphi(m(x),m(x+ ue_1))\quad\text{ for a.e. } x\in\Omega.
\end{align*}
A long but direct calculation (detailed in \cite[Lemma~3.9]{GL} in the case $\gamma(t)=e^{it}$) provides, for any smooth odd $\varphi$, the identity
\begin{align*}
\frac{\partial}{\partial u}\Delta^{\varepsilon}_\varphi & = I^{\epsilon}
+\nabla \cdot A^{\varepsilon},\\
I^{\epsilon}& = 2\iint \varphi(t-s)\gamma'_2(t-\pi/2)\left[
\chi_\varepsilon^{u}(t,x)\partial_s \sigma_\varepsilon(s,x)
-\chi_\varepsilon(t,x)\partial_s\sigma_\varepsilon^{u}(s,x)
\right] \, dtds,\\
A_1^\varepsilon&=2\iint \varphi(t-s)\gamma'_2(t-\pi/2)\gamma'_1(s-\pi/2)\chi_\varepsilon^{u}(t,x)D^{u}\chi_\varepsilon(s,x)\, dtds, \\
A_2^\varepsilon&= 2\iint \varphi(t-s)\gamma'_2(t-\pi/2)\gamma'_2(s-\pi/2)\chi_\varepsilon(t,x)\chi^u_\varepsilon(s,x)\, dtds.
\end{align*}
Note that $|A^{\varepsilon}|\lesssim \|\varphi\|_{L^1({\mathbb R}/2\pi\mathbb{Z})}$. Integrating with respect to $u$ and against a smooth cut-off function in $x$ we deduce
\begin{align*}
\frac{1}{|h|}\int_{\Omega'} \Delta^{\varepsilon}_\varphi(x,u)\, dx
&
\lesssim
\frac{\|\varphi\|_{L^1({\mathbb R}/2\pi\mathbb{Z})}}{\dist(\Omega',\partial\Omega)} +
|\sigma|({\mathbb R}/2\pi\mathbb{Z}\times\Omega)\int_{{\mathbb R}/2\pi\mathbb{Z}} |\varphi'(t)|\, dt .
\end{align*}
Letting $\varepsilon\to 0$ we infer
\begin{align*}
\frac{1}{|h|}\int_{\Omega'}\Delta_\varphi(m(x),m(x+u e_1))\, dx \lesssim \frac{\|\varphi\|_{L^1({\mathbb R}/2\pi\mathbb{Z})}}{\dist(\Omega',\partial\Omega)}+
|D\varphi|({\mathbb R}/2\pi\mathbb{Z})|\sigma|({\mathbb R}/2\pi\mathbb{Z}\times\Omega),
\end{align*}
for all smooth odd $\varphi$, and by approximation for any odd $\varphi\in BV({\mathbb R}/2\pi\mathbb{Z})$.
\end{proof}
\begin{lem}\label{l:lowerDelta}
There exists an odd function $\varphi\in BV({\mathbb R}/2\pi{\mathbb Z})$ such that the quantity $\Delta_\varphi$ defined in \eqref{eq:Deltavarphi} satisfies
$\Delta_\varphi\gtrsim \Lambda$, where $\Lambda$ is defined in \eqref{eq:Lambda1}.
\end{lem}
\begin{proof}
We define an odd function $\varphi\in BV({\mathbb R}/2\pi{\mathbb Z})$ by setting
\begin{align*}
\varphi(\theta)=\begin{cases}
1 &\text{ for }0< \theta<\delta,\\
-1 &\text{ for }-\delta < \theta < 0,\\
0 &\text{ for } \delta < |\theta | <\pi,
\end{cases}
\end{align*}
where $\delta\in (0,\pi/2)$ is a parameter to be chosen later.
Recalling the definitions of $\Delta_\varphi$ \eqref{eq:Deltavarphi} and $\alpha$ \eqref{eq:alpha}, we have, for $m_1, m_2\in\mathbb{S}^1$,
\begin{align}\label{eq:LambdaXi}
\Delta_\varphi(m_1,m_2)&=\iint_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta} |\sin(\alpha(t-\pi/2)-\alpha(s-\pi/2))|\, \Xi(t)\,\Xi(s)\, dtds,\\
\Xi(t)&=\Xi(t,m_1,m_2)=\mathbf 1_{e^{it}\cdot m_2>0}-\mathbf 1_{e^{it}\cdot m_1>0}.\nonumber
\end{align}
Here and in what follows we let $\dist_{\mathbb{S}^1}$ denote the geodesic distance in $\mathbb S^1$. The function $\Xi(\cdot,m_1,m_2)$ is supported in two opposite arcs of length $\dist_{\mathbb{S}^1}(m_1,m_2)$:
\begin{align*}
\Xi(t,m_1,m_2)&=\mathbf 1_{t\in A} -\mathbf 1_{t\in -A}\qquad \text{for a.e. }t\in {\mathbb R}/2\pi\mathbb{Z}, \\
A = A(m_1,m_2)&=\left\lbrace t\in {\mathbb R}/2\pi\mathbb{Z}\colon e^{it}\cdot m_2>0\text{ and }e^{it}\cdot m_1 < 0\right\rbrace.
\end{align*}
If $\dist_{\mathbb{S}^1}(m_1,m_2)\leq \pi-\delta$, then the distance between these arcs is at least $\delta$, so we have
\begin{align*}
\Xi(t)\Xi(s)&=\mathbf 1_{s,t\in A} + \mathbf 1_{s,t\in -A}\qquad\text{ for }\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta,
\end{align*}
and therefore
\begin{align*}
\Delta_\varphi(m_1,m_2)&=\iint_{A\times A} \mathbf 1_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta} |\sin(\alpha(t-\pi/2)-\alpha(s-\pi/2))|\, dtds\\
& \quad + \iint_{-A\times -A} \mathbf 1_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta}
|\sin(\alpha(t-\pi/2)-\alpha(s-\pi/2))|\, dtds\\
&=2 \iint_{A\times A} \mathbf 1_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta}
|\sin(\alpha(t-\pi/2)-\alpha(s-\pi/2))|\, dtds.
\end{align*}
The last equality follows from $\alpha(t+\pi)=\alpha(t)+\pi$ for all $t\in {\mathbb R}/2\pi\mathbb{Z}$. Because $\alpha$ is uniformly continuous we may choose $\delta_0>0$ small enough to ensure that $|\alpha(t-\pi/2)-\alpha(s-\pi/2)|\leq\pi/2$ for $\dist_{\mathbb{S}^1}(e^{it},e^{is})<\delta$ provided $\delta\leq \delta_0$, and then we have
\begin{align*}
\Delta_\varphi(m_1,m_2)&\geq \frac 4\pi \iint_{ A\times A} \mathbf 1_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta}
|\alpha(t-\pi/2)-\alpha(s-\pi/2)|\, dtds.
\end{align*}
Letting
$m_1=e^{i\theta_1}$,
$m_2=e^{i\theta_2}$ with $|\theta_1-\theta_2|=\dist_{\mathbb{S}^1}(m_1,m_2)$, this turns into
\begin{align}\label{eq:lowDelta1}
\Delta_\varphi(m_1,m_2)&\geq \frac 4\pi \int_{\theta_1+\frac\pi 2}^{\theta_2 +\frac\pi 2}
\int_{\theta_1+\frac\pi 2}^{\theta_2 +\frac\pi 2} \mathbf 1_{|t-s|<\delta} |\alpha(t-\pi/2)-\alpha(s-\pi/2)|\, dtds \nonumber\\
&= \frac 4\pi \int_{\theta_1}^{\theta_2}
\int_{\theta_1}^{\theta_2} \mathbf 1_{|t-s|<\delta} |\alpha(t)-\alpha(s)|\, dtds.
\end{align}
Recalling the definition \eqref{eq:Lambda1} of $\Lambda(m_1,m_2)$, we deduce
\begin{align*
\Delta_\varphi(m_1,m_2)\geq
\frac 4\pi \Lambda(m_1,m_2) \qquad \text{ if }\dist_{\mathbb{S}^1}(m_1,m_2)\leq \delta.
\end{align*}
For $\delta<\dist_{\mathbb{S}^1}(m_1,m_2)\leq\pi-\delta$, from \eqref{eq:lowDelta1} we have
\begin{align*}
\Delta_\varphi(m_1,m_2)& \geq C_1(\delta):=\frac 4\pi \inf_{\theta\in{\mathbb R}}\int_\theta^{\theta+\delta}\int_{\theta}^{\theta+\delta}|\alpha(t)-\alpha(s)|\, dtds >0
\end{align*}
(where this infimum is indeed positive because $\alpha$ is strictly increasing thanks to the strict convexity of $\mathsf{B}$),
so
\begin{align*
\Delta_{\varphi}(m_1,m_2)&\geq \frac{C_1(\delta)}{\sup_{\mathbb S^1\times\mathbb S^1}\Lambda} \Lambda(m_1,m_2)\qquad\text{ if }\delta<\dist_{\mathbb{S}^1}(m_1,m_2)\leq\pi-\delta.
\end{align*}
Finally we turn to the case $\pi- \delta < \dist_{\mathbb{S}^1}(m_1,m_2) \leq\pi$, where the product $\Xi(s)\Xi(t)$ can take negative values. We have
\begin{align*}
\Xi(t)\Xi(s)&=\mathbf 1_{s,t\in A} + \mathbf 1_{s,t\in -A} -\mathbf 1_{t\in A,s\in -A} -\mathbf 1_{t\in -A,s\in A}\\
&\geq \mathbf 1_{s,t\in A} + \mathbf 1_{s,t\in -A} -\mathbf 1_{s,t\in \hat A_\delta}-\mathbf 1_{s,t\in -\hat A_\delta},
\end{align*}
where $\hat A_\delta$ is the arc of length $2\delta$ given by
\begin{align*}
\hat A_\delta=\left\lbrace t\in{\mathbb R}/2\pi\mathbb{Z}\colon \dist_{\mathbb{S}^1}(e^{it},ie^{i\theta_0})<
\delta\right\rbrace,\qquad\theta_0=\frac{\theta_1+\theta_2}{2}-\frac\pi 2.
\end{align*}
Plugging this into the expression of $\Delta_\varphi$ in \eqref{eq:LambdaXi} and using again $\alpha(\theta+\pi)=\alpha(\theta)+\pi$ and $|\alpha(t-\pi/2)-\alpha(s-\pi/2)|\leq\pi/2$ for $\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta$, we find
\begin{align}\label{eq:lowDeltalarge1}
\Delta_\varphi(m_1,m_2)&\geq \frac{4}{\pi}\iint_{A\times A} \mathbf 1_{\dist_{\mathbb{S}^1}(e^{is},e^{it})<\delta} |\alpha(t-\pi/2)-\alpha(s-\pi/2)|\, dtds
\nonumber \\
&\quad - 2\iint_{\hat A_\delta\times\hat A_\delta}|\alpha(t-\pi/2)-\alpha(s-\pi/2)|\, dtds
\nonumber \\
& \geq \frac{4}{\pi}\int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2}\int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2} \mathbf 1_{|t-s|<\delta} |\alpha(t)-\alpha(s)|\, dtds \nonumber \\
&\quad - 2\int_{\theta_0 - \delta}^{\theta_0 +\delta}
\int_{\theta_0 -\delta}^{\theta_0 +\delta}
|\alpha(t)-\alpha(s)|\, dtds.
\end{align}
Since $\alpha$ is increasing, its derivative $D\alpha$ is a nonnegative Radon measure. We use this to calculate
\begin{align*}
& \int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2}\int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2} \mathbf 1_{|t-s|<\delta} |\alpha(t)-\alpha(s)|\, dtds\\
& =\int_{\theta_0+\frac\delta 2}^{\theta_0+\pi-\frac\delta 2} W(\tau) \, D\alpha(d\tau),
\\
W(\tau)&=\iint \mathbf 1_{|t-s|<\delta} \mathbf 1_{\theta_0+\delta/2 < t< \tau}
\mathbf 1_{\tau < s< \theta_0+\pi-\delta/2}\,dtds\\
&\quad + \iint \mathbf 1_{|t-s|<\delta}\mathbf 1_{\theta_0+\delta/2 < s< \tau}
\mathbf 1_{\tau < t< \theta_0+\pi-\delta/2}
\, dtds.
\end{align*}
For $\tau\in [\theta_0 +3\delta/2,\theta_0+\pi-3\delta/2]$ the quantity $W(\tau)$ is the sum of the areas of two isosceles right-angled triangles of height $\delta$, so $W(\tau)=\delta^2$; see Figure \ref{tri}.
\begin{figure}[t]
\centering
\includegraphics[width=.55\linewidth]{pic4}
\caption{The two isosceles triangles in $W(\tau)$.}
\label{tri}
\end{figure}
Thus we deduce
\begin{align}\label{eq:lowDeltalarge2}
& \int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2}\int_{\theta_0+\frac\delta 2}^{\theta_0+\pi -\frac\delta 2} \mathbf 1_{|t-s|<\delta} |\alpha(t)-\alpha(s)|\, dtds
\nonumber
\\
&\geq \delta^2 D\alpha([\theta_0+ 3\delta/2,\theta_0+\pi-3\delta/2]).
\end{align}
Similarly we write
\begin{align}\label{eq:lowDeltalarge3}
& \int_{\theta_0 -\delta}^{\theta_0 +\delta}
\int_{\theta_0 -\delta}^{\theta_0 +\delta}
|\alpha(t)-\alpha(s)|\, dtds \nonumber\\
&=\int_{\theta_0 -\delta}^{\theta_0 +\delta}
\left( \iint_{[\theta_0-\delta,\theta_0+\delta]^2} (\mathbf 1_{t<\tau<s} +\mathbf 1_{s<\tau<t})\, dtds \right)\, D\alpha(d\tau), \nonumber\\
&\leq
4\delta^2 D\alpha([\theta_0-\delta,\theta_0+\delta]).
\end{align}
Plugging \eqref{eq:lowDeltalarge2}-\eqref{eq:lowDeltalarge3} into \eqref{eq:lowDeltalarge1} we obtain
\begin{align*}
\Delta_\varphi(m_1,m_2)&\geq 4\delta^2\left(\frac{1}{\pi} D\alpha([\theta_0+3\delta/2,\theta_0+\pi-3\delta/2]) - 2 D\alpha([\theta_0-\delta,\theta_0+\delta])\right),
\end{align*}
if $\pi-\delta <\dist_{\mathbb{S}^1}(m_1,m_2)\leq\pi$.
Since $\alpha$ is continuous, the measure $D\alpha$ has no atoms, and we deduce the convergence
\begin{align*}
&\frac{1}{\pi} D\alpha([\theta_0+3\delta/2,\theta_0+\pi-3\delta/2]) - 2 D\alpha([\theta_0-\delta,\theta_0+\delta])\\
&\longrightarrow \frac{1}{\pi}D\alpha([\theta_0,\theta_0+\pi])=1,
\end{align*}
as $\delta\to 0$, uniformly with respect to $\theta_0\in{\mathbb R}$. In particular we may choose $\delta\in(0,\delta_0)$ sufficiently small such that
\begin{align*}
\Delta_\varphi(m_1,m_2)&\geq \delta^2 \geq \frac{\delta^2}{\sup_{\mathbb S^1\times S^1}\Lambda}\Lambda(m_1,m_2),
\end{align*}
for $\pi-\delta <\dist_{\mathbb{S}^1}(m_1,m_2)\leq\pi$, and this concludes the proof of Lemma~\ref{l:lowerDelta}.
\end{proof}
\begin{lem}\label{l:lowerLambda}
For all $m_1,m_2\in\mathbb S^1$, the function $\Lambda$ defined in \eqref{eq:Lambda1} satisfies
\begin{align*}
\Lambda(m_1,m_2)\gtrsim \delta^2\omega^{-1}\left(\delta/2\right),\qquad\delta=|m_1-m_2|,
\end{align*}
where $\omega$ is the minimal modulus of continuity of $\alpha^{-1}$.
\end{lem}
\begin{proof}
By definition of the modulus of continuity $\omega$ we have
\begin{align*}
|\alpha(t)-\alpha(s)|\geq \omega^{-1}(|t-s|)\qquad \forall s,t\in{\mathbb R}.
\end{align*}
Using this and the fact that $\omega^{-1}$ is increasing in the definition \eqref{eq:Lambda1} of $\Lambda$ we obtain, for $|\theta_2-\theta_1|\leq\pi$,
\begin{align*}
\Lambda(e^{i\theta_1},e^{i\theta_2})& \geq \int_{\theta_1}^{\theta_2}\int_{\theta_1}^{\theta_2} \omega^{-1}(|t-s|)\, dtds\\
& \geq \int_{\theta_1}^{\theta_2}\int_{\theta_1}^{\theta_2}\mathbf 1_{\frac 12 |\theta_2-\theta_1| \leq |t-s|\leq |\theta_1-\theta_2|} \, dtds\, \omega^{-1}\left(\frac{|\theta_2-\theta_1|}{2}\right)\\
&\gtrsim |\theta_2-\theta_1|^2\, \omega^{-1}\left( \frac{|\theta_2-\theta_1|}{2}\right).
\end{align*}
Since $\delta=|e^{i\theta_1}-e^{i\theta_2}|\leq|\theta_1-\theta_2|$ and $\omega^{-1}$ is increasing we deduce
\begin{align*}
\Lambda(m_1,m_2)&\gtrsim \delta^2 \,\omega^{-1}\left(\delta/2 \right),\qquad \delta=|m_1-m_2|,
\end{align*}
for all $m_1,m_2\in\mathbb S^1$.
\end{proof}
\subsection{Regularity implies finite entropy production for analytic norms}
\label{ss:revreg}
Recall the definition of $\alpha$ in \eqref{eq:alpha}, i.e. $\alpha\in C^0({\mathbb R})$ satisfies
\begin{equation*}
e^{i\alpha(\theta)} = \gamma'(\theta)\quad\qd\forall \theta\in {\mathbb R},
\end{equation*}
and that $\alpha$ is increasing and therefore $D\alpha$ is a nonnegative measure on ${\mathbb R}$.
Here we prove Theorem \ref{thmbesrev}, which follows from the following Lemmas \ref{LBV1} and \ref{LD1}.
\begin{lem}
\label{LBV1}
Let $m$ satisfy \eqref{eq:geneikon}. Assume $D\alpha$ forms a doubling measure and for any $\Omega'\subset \subset \Omega$ we have
\begin{align}
\label{eqbe2}
\sup_{\left|h\right|<\mathrm{dist}(\Omega', \partial\Omega)}
\frac{1}{\left|h\right|}\int_{\Omega'} \Pi\left(m(x),m(x+h)\right)\, dx <\infty,
\end{align}
where $\Pi$ is defined by \eqref{eq:Lambda}. Then the entropy productions of $m$ satisfy $\nabla\cdot\Phi(m)\in\mathcal{M}_{loc}(\Omega)$ for all $\Phi\in\mathrm{ENT}$, and their lowest upper bound measure satisfies the estimate
\begin{align*}
\left(\bigvee_{\|\lambda_\Phi'\|_\infty\leq 1}
\left| \nabla\cdot\Phi(m) \right|\right)(\Omega')
& \leq C\,
\sup_{\left|h\right|<\mathrm{dist}(\Omega', \partial\Omega)}
\frac{1}{\left|h\right|}\int_{\Omega'} \Pi\left(m(x),m(x+h)\right)\, dx,
\end{align*}
for some constant $C>0$ depending on $\mathsf{B}$ and the doubling constant of $D\alpha$.
\end{lem}
\begin{proof}
Let $m_{\varepsilon} = m\ast \rho_{\varepsilon}$ for a regularizing kernel $\rho_{\varepsilon}$. For any $\Phi\in\mathrm{ENT}$ and any test function $\zeta\in C_c^{\infty}(\Omega)$ with support inside an open subset $V\subset\Omega$, using estimates similar to those leading to \eqref{eq:dvPhimzeta}, we deduce that
\begin{align}\label{eq:dvPhimzeta2}
\left|\langle\nabla\cdot\Phi(m),\zeta\rangle\right| \lesssim \|\nabla\Psi\|_\infty \|\zeta\|_\infty \liminf_{\epsilon\to\infty}\int_V \left|D m_{\epsilon}\right| \left(1-\|m_{\epsilon}\|^2\right)\,dx.
\end{align}
Given $x\in V$, note that
\begin{align}
\label{apxeq7}
\left|D m_{\epsilon}(x)\right|&\leq \int_{B_{\epsilon}(x)} \left|m\left(z\right)-m\left(x\right)\right|\left|\nabla \rho_{\epsilon}\left(x-z\right)\right| dz\nonumber\\
&\lesssim \epsilon^{-1} \Xint{-}_{B_{\epsilon}(x)} \left|m\left(z\right)-m\left(x\right)\right| dz.
\end{align}
Define the function $F:{\mathbb R}^2\to{\mathbb R}$ by $F(z)=\|z\|^2$ for any $z\in \mathbb{R}^2$. By convexity of $F$, we have
\begin{align*}
1-\|m_{\epsilon}(x)\|^2&=
F\left(m(x)\right)-F\left(m_{\epsilon}(x)\right) \\
&\leq
\nabla F\left(m(x)\right)\cdot\left(m(x)-m_{\epsilon}(x)\right)
\\
&
= \nabla F\left(m(x)\right)\cdot \int_{B_{\epsilon}(x)} \left(m(x)-m(z)\right)\rho_{\epsilon}(x-z) dz .
\end{align*}
As the level sets $\lbrace F=\lambda^2\rbrace$ are the curves $\lbrace \lambda\gamma(\theta)\rbrace_{\theta\in{\mathbb R}}$, the gradient of $F$ at $m=\gamma(\theta)$ is in the direction of $-i\gamma'(\theta)$. Since moreover $F$ is locally Lipschitz we have
\begin{align*}
\nabla F(\gamma(\theta))=-c(\theta)i\gamma'(\theta),\qquad 0 < c(\theta) \leq C,
\end{align*}
for some constant $C>0$ depending on $\mathsf{B}$. (Explicitly, $c(\theta)=2/(i\gamma(\theta)\cdot\gamma'(\theta))$.)
We write
$m(x)=\gamma(\theta(x))$ for some $\theta(x)\in{\mathbb R}$ and $m(z)=\gamma(\theta_x(z))$ for some $\theta_x(z)\in{\mathbb R}$ such that $\dist_{\partial \mathsf{B}}(m(x),m(z))=|\theta(x)-\theta_x(z)|$, where $\dist_{\partial \mathsf{B}}$ denotes the geodesic distance in $\partial\mathsf{B}$, and plug the above expression for $\nabla F$ into the previous inequality. This yields
\begin{align*}
1-\|m_{\epsilon}(x)\|^2&\leq C
(-i\gamma'(\theta(x)))\cdot
\int_{B_{\epsilon}(x)} \int_{\theta_x(z)}^{\theta(x)} \gamma'(s)\, ds\,\rho_{\epsilon}(x-z) dz\\
&= C \,
\int_{B_{\epsilon}(x)}
\int_{\theta_x(z)}^{\theta(x)}
\sin(\alpha(\theta(x))-\alpha(s))
\,ds
\,\rho_{\epsilon}(x-z) dz.
\end{align*}
For the last equality we used the definition of the continuous increasing function $\alpha$ characterized by $\gamma'=e^{i\alpha}$. Letting $g(x,z)=\dist_{\partial \mathsf{B}}(m(x),m(z))$, we infer
\begin{align}\label{apxeq4.5}
1-\|m_{\epsilon}(x)\|^2&\leq
C \,
\int_{B_{\epsilon}(x)}
\int_{\theta(x)-g(x,z)}^{\theta(x)+g(x,z)}
|\alpha(\theta(x))-\alpha(s)|
\,ds
\,\rho_{\epsilon}(x-z) dz.
\end{align}
For all $\theta, r\in{\mathbb R}$, define
\begin{align}
\label{apxeq80.3}
\mathcal{G}_{\theta}(r)=\int_{\theta-r}^{\theta+r}\int_{\theta-r}^{\theta+r} \left|\alpha(t)-\alpha(s)\right| dt ds.
\end{align}
It follows that
\begin{align}
\label{apxeq81.3}
\mathcal{G}'_{\theta}(r)=2\int_{\theta-r}^{\theta+r}\left(\left|\alpha\left(\theta+r\right)-\alpha(s)\right|+ \left|\alpha\left(\theta-r\right)-\alpha(s)\right|\right) ds.
\end{align}
Using that $\alpha$ is strictly increasing, we find that $\mathcal{G}'_\theta$ is strictly increasing, and thus $\mathcal{G}_\theta$ is strictly convex.
Moreover, for $\theta-r<s<\theta$ we have $|\alpha(\theta)-\alpha(s)|<|\alpha(\theta+r)-\alpha(s)|$, and for $\theta <s<\theta+r$ we have
$|\alpha(\theta)-\alpha(s)|<|\alpha(\theta-r)-\alpha(s)|$. So we deduce from the estimate \eqref{apxeq4.5} and the expression \eqref{apxeq81.3} of $\mathcal{G}_\theta'$ that
\begin{align*}
1-\|m_{\epsilon}(x)\|^2 &\leq \frac{C}{2}\int_{B_{\epsilon}(x)}\mathcal{G}_{\theta(x)}'\left(g(x,z)\right)\rho_{\epsilon}(x-z) dz.
\end{align*}
Putting this together with \eqref{apxeq7}, we obtain
\begin{equation*}
\left|Dm_{\epsilon}(x)\right|\left(1-\|m_{\epsilon}(x)\|^2\right)\lesssim \frac{C}{\varepsilon}\,\Xint{-}_{B_{\epsilon}(x)} \Xint{-}_{B_{\epsilon}(x)} \mathcal{G}_{\theta(x)}'\left(g(x,z)\right) g(x,y) \, dz\, dy.
\end{equation*}
Let $\mathcal{H}_{\theta}$ denote the Legendre transform of $\mathcal{G}_{\theta}$, i.e. $\mathcal{H}_{\theta}(p) = \sup_{r\in{\mathbb R}}\{pr-\mathcal{G}_{\theta}(r)\}$ for all $p\in{\mathbb R}$. It follows that
\begin{align}
\label{eq:product}
\frac{\varepsilon}{C}\, \left|Dm_{\epsilon}(x)\right|\left(1-\|m_{\epsilon}(x)\|\right)&\lesssim \Xint{-}_{B_{\epsilon}(x)} \mathcal{H}_{\theta(x)}\left(\mathcal{G}_{\theta(x)}'\left(g(x,z)\right)\right)dz\nonumber\\
&\quad + \Xint{-}_{B_{\epsilon}(x)} \mathcal{G}_{\theta(x)}\left(g(x,y)\right)dy.
\end{align}
Note that $\mathcal{G}_{\theta}(r)\geq cr^2$ for all $r$ sufficiently large and for some universal constant $c>0$. Therefore, for all $p\in{\mathbb R}$, we have $\mathcal{H}_{\theta}(p) = pr^*-\mathcal{G}_{\theta}(r^*)$ for the unique $r^*\in{\mathbb R}$ characterized by $\mathcal{G}_\theta'(r^*)=p$. Thus for all $\theta,r\in{\mathbb R}$ we have
\begin{align*}
\mathcal{H}_{\theta}(\mathcal{G}_{\theta}'(r))&=\mathcal{G}_{\theta}'(r)r-\mathcal{G}_{\theta}(r)\\
&\leq \mathcal{G}_{\theta}'(r)r =2r \int_{\theta-r}^{\theta+r}\left(\left|\alpha\left(\theta+r\right)-\alpha(s)\right|+ \left|\alpha\left(\theta-r\right)-\alpha(s)\right|\right) ds \\
&\leq 8\, r^2\, |\alpha(\theta+r)-\alpha(\theta-r)|
=8\, r^2\, D\alpha\left([\theta-r, \theta+r]\right).
\end{align*}
For the last inequality we used again the fact that $\alpha$ is increasing.
On the other hand, it is clear from \eqref{apxeq80.3} that
\begin{equation*}
\mathcal{G}_{\theta}\left(r\right)\leq 4\, r^2\, D\alpha\left([\theta-r, \theta+r]\right).
\end{equation*}
Plugging these two inequalities for $\mathcal{H}_\theta(\mathcal{G}_\theta')$ and $\mathcal{G}_\theta$ into \eqref{eq:product} and changing $y$ to $z$ in the second integral on the right-hand side we obtain
\begin{align}\label{eq:product2}
&\left|Dm_{\epsilon}(x)\right|\left(1-\|m_{\epsilon}(x)\|\right)
\nonumber
\\
&
\lesssim \frac{C}{\varepsilon}\,\Xint{-}_{B_{\epsilon}(x)}g(x,z)^2D\alpha\left([\theta(x)-g(x,z), \theta(x)+g(x,z)]\right)dz.
\end{align}
Recall from Remark~\ref{remlam} that, for any $\theta_1,\theta_2\in{\mathbb R}$ with $r=|\theta_1-\theta_2|\leq\pi$, we have
\begin{align*}
\Pi(\gamma(\theta_1),\gamma(\theta_2))&=
\Lambda(e^{i\theta_1},e^{i\theta_2})
\gtrsim r^2 D\alpha\left(\left[\frac{\theta_1+\theta_2}{2}-\frac r6,
\frac{\theta_1+\theta_2}{2}+\frac r6 \right]\right).
\end{align*}
Using the fact that $D\alpha$ is a doubling measure, we deduce
\begin{align*}
\Pi(\gamma(\theta_1),\gamma(\theta_2))&\gtrsim C_0\, r^2 D\alpha\left(\left[\frac{\theta_1+\theta_2}{2}-\frac{3 r}{2},
\frac{\theta_1+\theta_2}{2}+\frac{3 r}{2} \right]\right)\\
&\geq C_0 \, r^2 D\alpha\left([\theta_1-r, \theta_1+r]\right),
\end{align*}
for some constant $C_0>0$ depending on the doubling constant of $D\alpha$. Applying this to $\theta_1=\theta(x)$, $\theta_2=\theta_x(z)$ and $r=g(x,z)$ and plugging the resulting inequality into \eqref{eq:product2} we find
\begin{align*}
\left|Dm_{\epsilon}(x)\right|\left(1-\|m_{\epsilon}(x)\|\right)
\leq \frac{C_1}{\varepsilon}\,\Xint{-}_{B_\varepsilon(x)}\Pi(m(x),m(z)) \, dz,
\end{align*}
for a constant $C_1>0$ depending on the doubling constant of $D\alpha$ and $\mathsf{B}$.
Integrating this estimate with respect to $x\in V$ and recalling \eqref{eq:dvPhimzeta2} we deduce
\begin{align*}
\left|\langle\nabla\cdot\Phi(m),\zeta\rangle\right|
\lesssim C_1 \|\nabla\Psi\|_\infty \|\zeta\|_\infty
\liminf_{\varepsilon\to 0}
\frac{1}{\varepsilon}\Xint{-}_{B_\varepsilon(0)}
\int_V \Pi(m(x),m(x+h))\, dx\, dh,
\end{align*}
for any $\zeta\in C_c^\infty(V)$. Thanks to \eqref{eqbe2}, the limit in the right-hand side is finite. This implies in particular that $\nabla\cdot\Phi(m)$ is a locally finite Radon measure, such that
\begin{align*}
\left|\nabla\cdot\Phi(m) \right|(V)\lesssim C_1
\|\nabla\Psi\|_\infty
\liminf_{\varepsilon\to 0}
\frac{1}{\varepsilon}\Xint{-}_{B_\varepsilon(0)}
\int_V \Pi(m(x),m(x+h))\, dx\, dh.
\end{align*}
Moreover, from this estimate we infer (arguing as in the proof of Proposition~\ref{p:lowerbound}) that
\begin{align*}
\left(\bigvee_{\|\lambda_\Phi'\|_\infty\leq 1}
\left| \nabla\cdot\Phi(m) \right|\right)(\Omega')
& \lesssim C_1\, \liminf_{\varepsilon\to 0}
\frac{1}{\varepsilon}\Xint{-}_{B_\varepsilon(0)}
\int_{\Omega'} \Pi(m(x),m(x+h))\, dx\, dh,
\end{align*}
for any open subset $\Omega'\subset\subset\Omega$. This implies the conclusion of Lemma~\ref{LBV1}.
\end{proof}
\begin{lem}
\label{LD1}
Suppose $\partial \mathsf{B}$ is analytic. Then $\alpha$ is analytic and $\alpha'(t)\,dt$ forms a doubling measure.
\end{lem}
\begin{proof} The lemma comes down to the fact that an absolutely continuous measure whose density is a nonnegative analytic function is doubling. This is presumably a well known fact, but we found no direct reference for it beyond a (more general) theorem in \cite{gar} for the square of an analytic function. Since $\alpha'\geq 0$, the function $\beta(t):=\sqrt{\alpha'(t)}$ is well defined. As the square root function is analytic in $(0,\infty)$, it follows that $\beta(t)$ is analytic at all $t$ such that $\alpha'(t)>0$ (see \cite[Proposition 1.4.2]{krantz}). Given $t_0$ with $\alpha'(t_0)=0$, again by the fact that $\alpha'\geq 0$, we can write, for $t$ in a sufficiently small neighborhood $I$ of $t_0$, $\alpha'(t)=(t-t_0)^{2p} h(t)$ for some integer $p\geq 1$ and some analytic function $h$ with $h(t)\ne 0$ in $I$. Thus $\beta(t)=(t-t_0)^p\sqrt{h(t)}$ is analytic in $I$. This shows that $\beta(t)$ is analytic in ${\mathbb R}$, and thus \cite[Theorem 1]{gar} can be applied to $\beta$ to conclude that $\alpha'(t)\,dt=\beta^2(t)\,dt$ is a doubling measure.
\end{proof}
\begin{rem}
\label{rem:lp-sphere}
The unit sphere of the $\ell^p$ norm in ${\mathbb R}^2$ defined by $|x|^p+|y|^p=1$ is analytic for $1<p<\infty$. This follows from the analyticity of the function $\left(1-|x|^p\right)^{1/p}$ in $(-1,1)$, which in turn is a consequence of the analyticity of the functions $1-|x|^p$ and $x^{1/p}$ in the intervals $(-1,1)$ and $(0,\infty)$, respectively.
\end{rem}
\section{Comparison of upper and lower bounds when $m$ is $BV$}\label{s:boundsBV}
In this section we consider $m\in BV(\Omega;{\mathbb R}^2)$ that satisfies \eqref{eq:geneikon} and assume that $\Omega\subset{\mathbb R}^2$ is a bounded simply connected smooth domain. Therefore the constraint $\nabla\cdot m=0$ is equivalent to the existence of a function $u$ such that $m=i\nabla u$. Using that correspondence, it is somewhat lengthy but straightforward to see that the upper bound construction in \cite{ark2} directly applies (taking $F(A,w)=\left|A\right|^2+\left(1-\|iw\|^2\right)^{2}$ for $A\in{\mathbb R}^{2\times 2}$ and $w\in{\mathbb R}^2$ in \cite[Theorem~1.2]{ark2}) to provide the existence of a $C^1$ sequence $m_\varepsilon\to m$ in $L^p(\Omega;{\mathbb R}^2)$ for $1\leq p <\infty$, such that $\nabla\cdot m_{\varepsilon}=0$ and
\begin{align*}
\limsup_{\varepsilon\to 0} I_\varepsilon(m_\varepsilon) \leq \int_{J_m} \mathrm{c}^{\mathrm{1D}}(m^+,m^-)\, d\mathcal H^1.
\end{align*}
Here $m^\pm$ are the traces of $m$ along its jump set $J_m$, and $\mathrm{c}^{\mathrm{1D}}(z^+,z^-)$ is the optimal energy of a one-dimensional transition between two states $z^\pm\in\partial\mathsf{B}$. In other words
\begin{align}\label{eq:c1Dinf}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)&=\inf_{\zeta\in Y} \left\lbrace \int_{-\infty}^{+\infty} \left( (\zeta'(x))^2 + (1-\|a\,\nu +\zeta(x) \, i\nu\|^2)^2\right)\, dx\right\rbrace,
\\
\text{where }
Y&=\left\lbrace \zeta\in C^1({\mathbb R})\colon \lim_{x\to\pm\infty}\zeta =z^\pm\cdot i\nu\right\rbrace,
\nonumber
\end{align}
and
\begin{align}\label{eq:nuzpm}
\nu &=\nu_{z^+,z^-}=i\frac{z^+-z^-}{|z^+-z^-|},\qquad a=a_{z^+,z^-}=z^+\cdot\nu=z^-\cdot\nu.
\end{align}
Classically (see e.g. \cite{sternberg91} for details), the infimum in \eqref{eq:c1Dinf} can be explicitly calculated. Indeed, assuming without loss of generality that $z^-\cdot i\nu \leq z^+\cdot i\nu$, for any admissible function $\zeta\in Y$ we have
\begin{align*}
&\int_{-\infty}^{+\infty} \left( (\zeta'(x))^2 + (1-\|a\,\nu +\zeta(x) \, i\nu\|^2)^2\right)\, dx \\
&
\geq 2\int_{-\infty}^{+\infty} (1-\|a\,\nu +\zeta(x)\,i\nu\|^2) \, \zeta'(x) \, dx \\
&= 2 \int_{-\infty}^{+\infty} \frac{d}{dx}\left[\int_{-\infty}^{\zeta(x)} (1-\|a\,\nu + s \, i\nu\|^2)\, ds\right]\, dx\\
&=2\int_{z^-\cdot i\nu}^{z^+\cdot i\nu} (1-\|a\,\nu + s \, i\nu\|^2)\, ds,
\end{align*}
and conversely, one can check that any solution of $\zeta'=1-\|a\,\nu +\zeta\, i\nu\|^2$ with initial condition $\zeta(0)\in (z^-\cdot i\nu,z^+\cdot i\nu)$ is admissible, i.e. belongs to the class $Y$, and achieves equality. So we have
\begin{align*}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)=
2\left|\int_{z^{-}\cdot i\nu}^{z^{+}\cdot i\nu} \left(1-\|a \nu + s\, i \nu \|^2\right) ds\right|,
\end{align*}
which corresponds to the expression \eqref{eq:c1D} given in the introduction.
Our goal in this section is to prove Theorem \ref{t:lowerupper} by comparing this upper bound with the lower bound \eqref{eq:lowerbound} provided by the entropy productions:
\begin{align*}
\left(\bigvee_{\|\lambda_\Phi'\|_{\infty}\leq 1}|\nabla\cdot\Phi(m)|\right)(\Omega) \leq C_0 \,\liminf_{\varepsilon\to 0} I_{\varepsilon}(m_{\varepsilon}).
\end{align*}
This follows from the estimate \eqref{eq:cENTleqc1D} and Lemma \ref{l:c1DleqcENT} below.
\begin{lem}\label{l:cENT}
For $m\in BV(\Omega;{\mathbb R}^2)$ satisfying \eqref{eq:geneikon} we have
\begin{align*}
\bigvee_{\|\lambda_\Phi'\|_{\infty}\leq 1}|\nabla\cdot\Phi(m)| = \mathrm{c}^{\mathrm{ENT}}(m^+,m^-)\, \mathcal H^1_{\lfloor J_m},
\end{align*}
where
\begin{align}\label{eq:cENT}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)=\sup_{\lambda\in\Lambda_*} \left\lbrace \int_{\theta^-}^{\theta^+}\lambda(t)\,\gamma'(t)\cdot\nu_{z^+,z^-}\, dt\right\rbrace \qquad\text{for }z^\pm=\gamma(\theta^\pm),
\end{align}
and
\begin{align*
\Lambda_*=\left\lbrace\lambda\in C^1({\mathbb R}/2\pi{\mathbb Z})\colon \int_{{\mathbb R}/2\pi {\mathbb Z}}\lambda(t)\gamma'(t)\, dt = 0\text{ and }\|\lambda'\|_\infty\leq 1\right\rbrace.
\end{align*}
\end{lem}
\begin{proof}
For any $\Phi\in \mathrm{ENT}$, the function $\lambda_\Phi$ defined by $\frac{d}{d\theta}\Phi(\gamma(\theta))=\lambda_\Phi(\theta)\gamma'(\theta)$ belongs to $\Lambda_*$ if it satisfies $\|\lambda_\Phi'\|_\infty\leq 1$, and reciprocally, to any $\lambda\in\Lambda_*$
one can associate an entropy
$\Phi_\lambda\in \mathrm{ENT}$ by setting
\begin{align*}
\Phi_\lambda(\gamma(\theta))=\int_0^\theta \lambda(t)\, \gamma'(t)\, dt.
\end{align*}
With these notations we therefore have
\begin{align*}
\bigvee_{\|\lambda_\Phi'\|_{\infty}\leq 1}|\nabla\cdot\Phi(m)|
=\bigvee_{\lambda\in\Lambda_*}|\nabla\cdot\Phi_{\lambda}(m)|.
\end{align*}
For a $BV$ map $m$, the $BV$ chain rule implies that the entropy productions are absolutely continuous with respect to $\mathcal H^1_{\lfloor J_m}$, and
\begin{align*}
|\nabla\cdot\Phi_\lambda(m)|&= |(\Phi_{\lambda}(m^+)-\Phi_{\lambda}(m^-))\cdot\nu | \, d\mathcal H^1_{\lfloor J_m}\\
&=|\mathrm{c}_{\lambda}(m^+,m^-)|\, \, d\mathcal H^1_{\lfloor J_m},\\
\mathrm{c}_{\lambda}(z^+,z^-)&=\int_{\theta^-}^{\theta^+}\lambda(t)\,\gamma'(t)\cdot \nu_{z^+,z^-}\, dt\quad \text{for }z^\pm=\gamma(\theta^\pm).
\end{align*}
Therefore, restricting the supremum to a countable dense subset of $\Lambda_*$ and applying \cite[Remark~1.69]{ambrosio}, we see that the lowest upper bound measure is also absolutely continuous with respect to $\mathcal H^1_{\lfloor J_m}$ and given by
\begin{align*}
\bigvee_{\lambda\in\Lambda_*}|\nabla\cdot\Phi_\lambda(m)|
&=\left(\sup_{\lambda\in\Lambda_*} |\mathrm{c}_\lambda(m^+,m^-)|\right)\, \mathcal H^1_{\lfloor J_m}.
\end{align*}
Since $\lambda\mapsto\mathrm{c}_\lambda$ is linear we can remove the absolute value in the right-hand side, concluding the proof of Lemma~\ref{l:cENT}.
\end{proof}
Combining the lower and upper bounds, we see that for any $m\in BV(\Omega;{\mathbb R}^2)$ satisfying \eqref{eq:geneikon} we have
\begin{align*}
\int_{J_m}\mathrm{c^{\mathrm{ENT}}}(m^+,m^-)\,d\mathcal H^1 \leq C_0\int_{J_m}\mathrm{c}^{\mathrm{1D}}(m^+,m^-)\, d \mathcal H^1.
\end{align*}
For any fixed $z^\pm\in\partial\mathsf{B}$ we may apply this to a divergence-free map taking only the two values $z^\pm$, and we immediately deduce the inequality
\begin{align}\label{eq:cENTleqc1D}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)\leq C_0\, \mathrm{c}^{\mathrm{1D}}(z^+,z^-)\qquad\forall z^\pm\in\partial\mathsf{B}.
\end{align}
Next we prove the reverse inequality. To that end we start by obtaining a more explicit expression of $\mathrm{c}^{\mathrm{ENT}}$ for small jumps.
\begin{lem}
\label{LL11.3}
For all $z^{\pm}=\gamma(\theta^\pm) \in \partial\mathsf{B}$ with $|\theta^+-\theta^-|=\mathrm{dist}_{\partial\mathsf{B}}(z^+,z^-)<\pi/2$, we have
\begin{equation}
\label{eqaqq82}
\mathrm{c}^{\mathrm{ENT}}\left(z^{+},z^{-}\right) =\left|\int_{\theta^-}^{\theta^+}\left(\theta-\tilde\theta_{z^+,z^-}\right)\left(\gamma'(\theta)\cdot \nu_{z^+,z^-}\right)\,d\theta\right|,
\end{equation}
where
$\tilde\theta_{z^+,z^-}\in{\mathbb R}$ is the unique point between $\theta^-$ and $\theta^+$ satisfying
\begin{equation*}
\gamma'\left(\tilde\theta_{z^+,z^-}\right)\cdot \nu_{z^+,z^-}=0.
\end{equation*}
Moreover we also have
\begin{equation}
\label{eqkk11.22}
\inf_{\mathrm{dist}_{\partial\mathsf{B}}(z^+,z^-)\geq\frac\pi 2}\mathrm{c}^{\mathrm{ENT}}\left(z^{+},z^{-}\right) >0.
\end{equation}
\end{lem}
\begin{proof}
Let $z^{\pm}=\gamma(\theta^\pm) \in \partial \mathsf{B}$ with $|\theta^+-\theta^-|<\pi/2$. Assume without loss of generality that $\theta^- < \theta^+ < \theta^-+\pi/2$.
To simplify notations, in this proof we drop the indices $(z^+,z^-)$ and simply write $\nu=\nu_{z^+,z^-}$ and $\tilde\theta=\tilde\theta_{z^+,z^-}$.
Recall that $\mathrm{c}^{\mathrm{ENT}}$ is given by
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)&=\sup_{\lambda\in\Lambda_*} \mathrm{c}_\lambda(z^+,z^-),\\
\text{where }\mathrm{c}_\lambda(z^+,z^-)&=\int_{\theta^-}^{\theta^+}\lambda(\theta)\gamma'(\theta)\cdot \nu \,d\theta\\
&=\int_{\theta^-}^{\theta^+}\left(\lambda(\theta)-\lambda(\tilde\theta )\right)\gamma'(\theta)\cdot \nu \,d\theta.
\end{align*}
The last equality is valid because $\int_{\theta^-}^{\theta^+}\gamma'(\theta)\cdot\nu\, d\theta=(z^+-z^-)\cdot\nu=0$ by definition of $\nu$. Since for
all $\lambda\in\Lambda_*$ we have $\|\lambda'\|_{\infty}\leq 1$, and therefore
$|\lambda (\theta)-\lambda (\tilde\theta)|\leq |\theta-\tilde\theta|$. This implies
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)&\leq \int_{\theta^-}^{\theta^+}
|\theta-\tilde\theta |\,\left|\gamma'(\theta)\cdot \nu \right|\,d\theta.
\end{align*}
The definition of $\tilde\theta$ and convexity of $\mathsf{B}$ ensure that $(\theta-\tilde\theta)(\gamma'(\theta)\cdot\nu)\geq 0$ for $\theta\in (\theta^-,\theta^+)$, so the above becomes
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)&\leq \left|\int_{\theta^-}^{\theta^+}
(\theta-\tilde\theta )\,(\gamma'(\theta)\cdot \nu) \,d\theta\right|.
\end{align*}
Conversely, since $|\theta^+-\theta^-|<\pi/2$ we may choose a $\pi$-periodic $\lambda_0\in C^1({\mathbb R}/\pi{\mathbb Z})$ with $\|\lambda_0'\|_{\infty}\leq 1$ and such that $\lambda_0(\theta)-\lambda_0(\tilde\theta)=\theta-\tilde\theta$ for $\theta^-<\theta<\theta^+$. Note that the $\pi$-periodicity of $\lambda_0$ ensures $\int_{{\mathbb R}/2\pi{\mathbb Z}}\lambda_0\gamma'=0$ since $\gamma'(t+\pi)=-\gamma'(t)$, so $\lambda_0\in\Lambda_*$. Therefore we have $\mathrm{c}^{\mathrm{ENT}}\geq\mathrm{c}_{\lambda_0}$ and
we deduce that $\mathrm{c}^{\mathrm{ENT}}$ is given by \eqref{eqaqq82}.
To prove \eqref{eqkk11.22}, note that $\mathrm{c}^{\mathrm{ENT}}$ is defined in \eqref{eq:cENT} as a supremum of continuous functions, and is therefore lower semicontinuous on $\partial\mathsf{B}\times\partial\mathsf{B}$. Hence the infimum in \eqref{eqkk11.22} is attained at some $z^\pm\in \partial \mathsf{B}$, $\mathrm{dist}_{\partial\mathsf{B}}\left(z^{+},z^{-}\right)\geq \frac{\pi}{2}$.
As $\mathsf{B}$ is strictly convex, the function $\theta\mapsto \gamma'(\theta)\cdot\nu$ cannot be identically zero on any open interval, which prevents $\mathrm{c}^{\mathrm{ENT}}$ from vanishing unless $z^+=z^-$. So the infimum in \eqref{eqkk11.22} is positive.
\end{proof}
\begin{lem}\label{l:c1DleqcENT}
We have
\begin{align*}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)\leq C\, \mathrm{c}^{\mathrm{ENT}}(z^+,z^-)\qquad\forall z^\pm\in\partial \mathsf{B},
\end{align*}
for some constant $C>0$ depending only on $\mathsf{B}$.
\end{lem}
\begin{proof}
Let $z^\pm=\gamma(\theta^\pm)\in\partial\mathsf{B}$ be two distinct points, with $|\theta^+-\theta^-|=\dist_{\partial \mathsf{B}}(z^+,z^-)$, and $\theta^-<\theta^+\leq \theta^- +\pi$. Dropping the indices $(z^+,z^-)$ we denote
\begin{align*}
\nu=i\frac{z^+-z^-}{|z^+-z^-|},\qquad a=z^+\cdot\nu=z^-\cdot\nu\leq 0,
\end{align*}
and recall that $\mathrm{c}^{\mathrm{1D}}$ is given by
\begin{align*}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)=2\int_{z^+\cdot i\nu}^{z^-\cdot i\nu}\left(1-\|a\,\nu + s\,i\nu\|^2\right)\, ds.
\end{align*}
Since $\mathsf{B}$ is strictly convex, for any $\theta\in (\theta^-,\theta^+)$ there is a unique $s(\theta)\in (z^+\cdot i\nu, z^-\cdot i\nu)$ such that
\begin{align*}
a\,\nu + s(\theta)\,i\nu = \beta(\theta) \gamma(\theta)\text{ for some }0<\beta(\theta)<1.
\end{align*}
The function $s\colon (\theta^-,\theta^+)\to (z^+\cdot i\nu, z^-\cdot i\nu)$ is a decreasing bijection. Taking the scalar product of the above with $i\gamma(\theta)$ and with $\nu$ we have
\begin{align*}
s(\theta)=a\frac{\gamma(\theta)\cdot i\nu}{\gamma(\theta)\cdot\nu},\qquad \beta(\theta)=\frac{a}{\gamma(\theta)\cdot\nu}.
\end{align*}
The change of variable $s=s(\theta)$ therefore gives
\begin{align}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)&=-2\int_{\theta^- }^{\theta^+ }\left(1-\beta(\theta)^2\right)\, s'(\theta)\, d\theta
\nonumber\\
&=-2\int_{\theta^- }^{\theta^+ }\left(1-\frac{a^2}{(\gamma(\theta)\cdot\nu)^2}\right)\, s'(\theta)\, d\theta
\nonumber \\
&=-2\int_{\theta^- }^{\theta^+ }(\gamma(\theta)\cdot\nu -a) \frac{\gamma(\theta)\cdot\nu +a}{(\gamma(\theta)\cdot\nu)^2} \, s'(\theta)\, d\theta.
\label{eq:Achangevar}
\end{align}
Since $|\gamma'|=1$, from the explicit expression of $s(\theta)$ we have $|s'(\theta)|\leq 2 |a| \,|\gamma(\theta)|/(\gamma(\theta)\cdot\nu)^2$. For $|\theta^+-\theta^-| < \pi$, using moreover the inequality $\gamma(\theta)\cdot\nu \leq a< 0$, which follows from the convexity of $\mathsf{B}$ and implies in particular $(\gamma(\theta)\cdot\nu)^2\geq a^2$, we deduce
\begin{align}\label{eq:Aleqinta}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)&
\leq \frac{4}{|a|^3}\int_{\theta^- }^{\theta^+ } (a - \gamma(\theta)\cdot\nu)\,
|\gamma(\theta)|\, |\gamma(\theta)\cdot\nu +a|\, d\theta \nonumber\\
&\lesssim \frac{1}{|a|^3}\int_{\theta^- }^{\theta^+ } (a - \gamma(\theta)\cdot\nu)\, d\theta.
\end{align}
The last inequality follows from $\mathrm{diam}(\mathsf{B})\lesssim 1$.
Recalling from Lemma \ref{LL11.3} the definition of $\tilde\theta$ as the unique $\tilde\theta\in (\theta^-,\theta^+)$ such that $\gamma'(\tilde\theta)\cdot\nu=0$, and that $a=z^+\cdot\nu=z^-\cdot\nu$, we write
\begin{align*}
a-\gamma(\theta)\cdot\nu
&=
\mathbf 1_{\tilde \theta <\theta<\theta^+} \int_{\theta}^{\theta^+}\gamma'(t)\cdot\nu\, dt
-
\mathbf 1_{\theta^-<\theta<\tilde\theta} \int_{\theta^-}^\theta \gamma'(t)\cdot\nu\, dt,
\end{align*}
and
\begin{align}
\int_{\theta^- }^{\theta^+ } (a - \gamma(\theta)\cdot\nu)\, d\theta
&=\int_{\tilde\theta}^{\theta^+} \int_{\theta}^{\theta^+}\gamma'(t)\cdot\nu\, dt\,d\theta
-\int_{\theta^-}^{\tilde\theta} \int_{\theta^-}^{\theta}\gamma'(t)\cdot\nu\, dt\,d\theta
\nonumber
\\
&=\int_{\tilde\theta}^{\theta^+} \int_{\tilde\theta}^{t}\, d\theta\, \gamma'(t)\cdot\nu\, dt
-\int_{\theta^-}^{\tilde\theta} \int_{t}^{\tilde \theta}\, d\theta\, \gamma'(t)\cdot\nu\, dt
\nonumber\\
&=\int_{\theta^-}^{\theta^+}(t-\tilde\theta)(\gamma'(t)\cdot\nu)\,dt.
\label{eq:intequalsB}
\end{align}
For $|\theta^+-\theta^-| < \pi/2$, this last integral is exactly the expression of $\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)$ given by Lemma~\ref{LL11.3}. Therefore, combining this with \eqref{eq:Aleqinta} we deduce
\begin{align*}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)&
\lesssim \frac{1}{|a_{z^+,z^-}|^3} \mathrm{c}^{\mathrm{ENT}}(z^+,z^-)\quad\text{for }\dist_{\partial\mathsf{B}}(z^+,z^-) < \frac\pi 2.
\end{align*}
The function
\begin{align*}
(z^+,z^-)\mapsto |a_{z^+,z^-}|,
\end{align*}
is continuous on $\partial \mathsf{B}\times\partial\mathsf{B} \setminus \lbrace z^+=z^-\rbrace$, vanishes exactly when $\mathrm{dist}_{\partial\mathsf{B}}(z^+,z^-)=\pi$, and for $z^+$ close to $z^-$ it satisfies
\begin{align*}
|a_{z^+,z^-}|\longrightarrow | i\gamma'(\theta)\cdot \gamma(\theta)|,
\quad\text{ as }(z^+,z^-)\to (\gamma(\theta),\gamma(\theta)).
\end{align*}
By convexity of $\mathsf{B}$ we have $| i\gamma'(\theta)\cdot \gamma(\theta)|\geq\alpha_0>0$, where $\alpha_0$ is the largest radius of a euclidean ball contained in $\mathsf{B}$. From these properties we deduce that
\begin{align*}
\inf_{\dist_{\partial\mathsf{B}}(z^+,z^-) \leq \frac\pi 2 } |a_{z^+,z^-}| >0,
\end{align*}
and the above bound on $\mathrm{c}^{\mathrm{1D}}$ implies
\begin{align*}
\mathrm{c}^{\mathrm{1D}}(z^+,z^-)&
\leq C \mathrm{c}^{\mathrm{ENT}}(z^+,z^-)\quad\text{for }\dist_{\partial\mathsf{B}}(z^+,z^-) < \frac\pi 2,
\end{align*}
for some constant $C>0$ depending only on $\mathsf{B}$. Since $\mathrm{c}^{\mathrm{1D}}\leq 2 \pi$ and $\mathrm{c}^{\mathrm{ENT}}$ is bounded away from zero for $\dist_{\partial \mathsf{B}}(z^+,z^-)\geq\pi/2$ thanks to Lemma~\ref{LL11.3}, this inequality is true for all $z^\pm\in\partial \mathsf{B}$.
\end{proof}
\begin{rem}
Combining the expressions \eqref{eq:Achangevar} and \eqref{eq:intequalsB} obtained in the proof of Lemma~\ref{l:c1DleqcENT} and passing to the limit as $\theta^+-\theta^-\to 0$ one obtains
\begin{align*}
\frac{\mathrm{c}^{\mathrm{1D}}(z^+,z^-)}{\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)}\longrightarrow \frac{2}{|i\gamma'(\theta)\cdot\gamma(\theta)|}\qquad\text{as }(z^+,z^-)\to (\gamma(\theta),\gamma(\theta)).
\end{align*}
Hence for infinitesimally small jumps, the costs $\mathrm{c}^{\mathrm{1D}}$ and $\mathrm{c}^{\mathrm{ENT}}$ differ by the above multiplicative factor, which depends on the direction of the jump.
\end{rem}
\subsection{Regularity controls entropy productions when $m$ is $BV$}
\label{besbv}
In this subsection we prove Theorem~\ref{t:regconvBV}. To that end we first compare the jump cost $c^{\mathrm{ENT}}$ to the regularity cost $\Pi$ defined in \eqref{eq:Lambda}.
\begin{lem}\label{l:cENTleqLambda}
We have
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)\leq \, \Pi(z^+,z^-),
\end{align*}
for all $z^\pm\in \partial\mathsf{B}$.
\end{lem}
\begin{proof}
We let $z^{\pm}=\gamma(\theta^\pm)$ for some $\theta^\pm\in{\mathbb R}$ such that $|\theta^+-\theta^-|=\dist_{\partial\mathsf{B}}(z^+,z^-)\leq \pi$, and assume without loss of generality that $\theta^-<\theta^+$.
From the proof of Lemma~\ref{LL11.3} we have
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
&\leq \int_{\theta^-}^{\theta^+}(\theta-\tilde\theta)\gamma'(\theta)\cdot\nu\, d\theta,
\end{align*}
where $\tilde\theta\in (\theta^-,\theta^+)$ is such that $\nu=i\gamma'(\tilde\theta)$ and $\nu=\nu_{z^+,z^-}$ is defined in \eqref{eq:nuzpm}.
Recalling that the continuous increasing function $\alpha$ is defined by $\gamma'=e^{i\alpha}$, we rewrite this as
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
&\leq \int_{\theta^-}^{\theta^+}(\theta-\tilde\theta)\,e^{i\alpha(\theta)}\cdot ie^{i\alpha(\tilde\theta)}\, d\theta\\
&=\int_{\theta^-}^{\tilde\theta}(\tilde\theta-\theta)\,\sin(\alpha(\tilde\theta)-\alpha(\theta))\, d\theta\\
&\quad
+
\int_{\tilde\theta}^{\theta^+}(\theta-\tilde\theta)\,
\sin(\alpha(\theta)-\alpha(\tilde\theta))\, d\theta \\
&\leq (\tilde\theta-\theta^-)A^- +(\theta^+-\tilde\theta) A^+,\\
\text{where }
A^-&=\int_{\theta^-}^{\tilde\theta} \sin(\alpha(\tilde\theta)-\alpha(\theta))\, d\theta,
\quad
A^+=\int_{\tilde\theta}^{\theta^+}
\sin(\alpha(\theta)-\alpha(\tilde\theta))\, d\theta.
\end{align*}
Next recall that $(z^+-z^-)\cdot\nu=0$ by definition of $\nu$, and rewrite this as
\begin{align*}
0&=\int_{\theta^-}^{\theta^+}\gamma'(\theta)\cdot\nu\, d\theta
=\int_{\theta^-}^{\theta^+}\sin(\alpha(\theta)-\alpha(\tilde\theta))\, d\theta =A^+-A^-,
\end{align*}
so we have in fact $A^+=A^-$ and the above estimate for $\mathrm{c}^{\mathrm{ENT}}$ becomes
\begin{align}\label{eq:cENTA}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)&\leq (\theta^+-\theta^-) A,
\nonumber\\
\text{where } A&=\int_{\theta^-}^{\tilde\theta} \sin(\alpha(\tilde\theta)-\alpha(\theta))\, d\theta=\int_{\tilde\theta}^{\theta^+}
\sin(\alpha(\theta)-\alpha(\tilde\theta))\, d\theta.
\end{align}
Note that, using the fact that $\alpha$ is absolutely continuous, we have
\begin{align*}
A&\leq \int_{\theta^-}^{\tilde\theta}(\alpha(\tilde\theta)-\alpha(\theta))\, d\theta
=
\int_{\theta^-}^{\tilde\theta}\int_{\theta}^{\tilde\theta} D\alpha(d\tau)\, d\theta \
=\int_{\theta^-}^{\tilde \theta } (\tau-\theta^-)\, D\alpha(d\tau),
\end{align*}
and similarly
\begin{align*}
A&\leq \int_{\tilde\theta}^{\theta^+}(\theta^+-\tau)\, D\alpha(d\tau).
\end{align*}
So from \eqref{eq:cENTA} we infer
\begin{align}\label{eq:cENTleqmin}
&\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
\nonumber\\
&\leq (\theta^+-\theta^-) \,
\min\left \lbrace
\int_{\theta^-}^{\tilde \theta } (\tau-\theta^-)\, D\alpha(d\tau),
\int_{\tilde\theta}^{\theta^+}(\theta^+-\tau)\, D\alpha(d\tau)
\right\rbrace.
\end{align}
Next we consider two cases, depending on whether $\tilde\theta\in (\theta^-,\theta^+)$ is closer to $\theta^-$ or to $\theta^+$. If $\tilde\theta$ is closer to $\theta^-$ we have
$\theta^+-\theta^- \leq 2 (\theta^+-\tilde\theta)$,
so from \eqref{eq:cENTleqmin} (recalling that $\alpha$ is increasing and thus $D\alpha$ is a nonnegative measure) we deduce
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)&\leq 2 (\theta^+-\tilde\theta)
\int_{\theta^-}^{\tilde \theta } (\tau-\theta^-)\, D\alpha(d\tau) \\
&\leq 2 \int_{\theta^-}^{\tilde \theta } (\theta^+-\tau) (\tau-\theta^-)\, D\alpha(d\tau).
\end{align*}
Otherwise, $\tilde\theta$ is closer to $\theta^{+}$ so we have $\theta^+-\theta^- \leq 2 (\tilde\theta-\theta^-)$ and we find
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
&\leq 2
\int_{\tilde \theta }^{\theta^+} (\theta^+-\tau) (\tau-\theta^-)\, D\alpha(d\tau).
\end{align*}
In both cases, we have
\begin{align*}
\mathrm{c}^{\mathrm{ENT}}(z^+,z^-)
&\leq 2
\int_{\theta^- }^{\theta^+} (\theta^+-\tau) (\tau-\theta^-)\, D\alpha(d\tau),
\end{align*}
and thanks to Remark~\ref{remlam} and the symmetry of $\Pi$, this last expression is exactly $\Lambda(e^{i\theta^-}, e^{i\theta^+})=\Pi(\gamma(\theta^-), \gamma(\theta^+))=\Pi(z^+,z^-)$.
\end{proof}
Next we deduce from Lemma~\ref{l:cENTleqLambda} and properties of $BV$ maps
that the regularity estimate provided by $\Pi$ controls the entropy productions, proving Theorem~\ref{t:regconvBV}.
Recall for an entropy $\Phi\in \mathrm{ENT}$ the $C^1$ function $\lambda_{\Phi}$ is defined by $\frac{d}{d\theta}\Phi(\gamma(\theta))=\lambda_\Phi(\theta)\gamma'(\theta)$.
\begin{lem}
\label{misbv}
Let $m\in BV\left(\Omega;{\mathbb R}^2\right)$ satisfy \eqref{eq:geneikon}. Then for any open set $\Omega'\subset \subset \Omega$
we have
\begin{align*}
&\left(\bigvee_{\|\lambda_{\Phi}'\|_{\infty}\leq 1} \left|\nabla \cdot \Phi(m)\right|\right)\left(\Omega'\right)
\leq C_0 \,\limsup_{\left|h\right|\to 0} \frac{1}{|h|}\int_{\Omega'}
\Pi\left(m(x+h),m(x)\right) \, dx,
\end{align*}
where $C_0>0$ is an absolute constant.
\end{lem}
\begin{proof}
We know from Lemma \ref{l:cENT} that if $m$ is $BV$ then
\begin{align*}
\left(\bigvee_{\|\lambda_{\Phi}'\|_{\infty}\leq 1} \left|\nabla \cdot \Phi(m)\right|\right)
\left(\Omega'\right)=\int_{J_m\cap \Omega'} c^{\mathrm{ENT}}\left(m^{+},m^{-}\right) \,d\mathcal{H}^1,
\end{align*}
where $c^{\mathrm{ENT}}$ is defined by \eqref{eq:cENT}. Thanks to the inequality $\mathrm{c}^{\mathrm{ENT}}\leq\Pi$ provided by Lemma~\ref{l:cENTleqLambda}, we deduce
\begin{align*}
\left(\bigvee_{\|\lambda_{\Phi}'\|_{\infty}\leq 1} \left|\nabla \cdot \Phi(m)\right|\right)
\left(\Omega'\right)\leq
\int_{J_m\cap \Omega'} \Pi\left(m^{+},m^{-}\right) \,d\mathcal{H}^1.
\end{align*}
Hence the proof of Lemma~\ref{misbv} follows from the inequality
\begin{align}
\label{eq:intJmPi}
\int_{J_m\cap \Omega'} \Pi\left(m^{+},m^{-}\right) \,d\mathcal{H}^1 \leq C_0 \,
\limsup_{\left|h\right|\to 0} \left|h\right|^{-1}\int_{\Omega'}
\Pi\left(m(x+h),m(x)\right) \, dx.
\end{align}
This inequality is valid for any Lipschitz function $\Pi$ and $BV$ map $m$, as a consequence of the rectifiability of $J_m$ and the trace properties of $m$ (see e.g. \cite{ambrosio}), via a Besicovitch covering argument which we detail next.
Let $\delta\in (0,1)$. There exists $\epsilon_0>0$ and a subset $\widetilde{J}_m\subset J_m$ with $\mathcal{H}^1\left(J_m\cap \Omega' \setminus \widetilde{J}_m\right)<\delta$ and $\widetilde J_m +B_{\varepsilon_0}(0)\subset \Omega'$, such that for any $x_0\in \widetilde{J}_m$ and $0<r<\varepsilon_0$,
\begin{align}
\label{eqmisbv11}
\Xint{-}_{B_r(x_0)\cap J_m} \left|m^{\pm}\left(x\right)-m^{\pm}(x_0)\right| d\mathcal{H}^1 (x)&<\delta,
\nonumber\\
\left|\mathcal{H}^1\left(B_r(x_0)\cap J_m\right)-2r \right|&<\delta \, r,\nonumber\\
\text{ and } \Xint{-}_{B^{\pm}_r(x_0)} \left|m\left(x\right)-m^{\pm}(x_0)\right| dx &<\delta\quad\text{ for all }0<r<\epsilon_0,
\end{align}
where $B^{\pm}_r(x)$ denote the two half balls obtained by splitting $B_r(x)$ along the tangent line to $J_m$ at $x$. Let $\epsilon\in (0,\epsilon_0)$. By Besicovitch's covering theorem \cite[Theorem~2.18]{ambrosio} there
exists an absolute constant $Q\in \mathbb{N}$ and families $\mathcal{B}_1, \mathcal{B}_2,\dots, \mathcal{B}_Q$ of pairwise disjoint balls in the set $\left\{B_{\epsilon}(x):x\in \widetilde{J}_m\right\}$ such that
\begin{equation*}
\widetilde{J}_m\subset \bigcup_{k=1}^Q \bigcup_{B\in \mathcal{B}_k} B.
\end{equation*}
In particular for some $k_0\in \left\{1,2,\dots, Q\right\}$ we have
\begin{equation}\label{eq:intPileqQsumPij}
\int_{\widetilde{J}_m} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1 \leq Q \sum_{B\in \mathcal{B}_{k_0}} \int_{\widetilde{J}_m\cap B} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1.
\end{equation}
%
We have $\mathcal{B}_{k_0}=\lbrace B_\varepsilon(x_j)\rbrace_{j=1,\ldots,p}$ for some $x_1,\ldots ,x_p\in\widetilde J_m$.
Using that $\Pi$ is
Lipschitz thanks to its definition \eqref{eq:Lambda}, and the properties \eqref{eqmisbv11} of $\widetilde J_m$, we find
\begin{align*}
\int_{\widetilde{J}_m\cap B_{\epsilon}(x_j)} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1
& \leq
2\epsilon \Pi\left(m^{+}(x_j),m^{-}(x_j)\right) + 2L\,\delta \,\mathcal{H}^1\left(J_m\cap B_{\epsilon}(x_j)\right)\\
&\leq
2\varepsilon
\Xint{-}_{B^{+ }_{\epsilon}(x_j)}\Xint{-}_{B^{- }_{\epsilon}(x_j)}
\Pi\left(m(x), m\left(\tilde{x}\right)\right) \,d\tilde{x}\, dx
+5 L\delta\varepsilon,
\end{align*}
for some $L>0$ depending only on $\Pi$. Summing over $j=1,\ldots,p$ and taking \eqref{eq:intPileqQsumPij} into account, we deduce
\begin{align*}
\int_{\widetilde{J}_m} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1
&\leq 2 Q\varepsilon
\sum_{j=1}^p \Xint{-}_{B^{+ }_{\epsilon}(x_j)}\Xint{-}_{B^{- }_{\epsilon}(x_j)}
\Pi\left(m(x), m\left(\tilde{x}\right)\right) \,d\tilde{x}\, dx +5 L\delta\, p\varepsilon.
\end{align*}
Noting from the properties \eqref{eqmisbv11} of $\widetilde J_m$ that
\begin{align*}
\mathcal{H}^1\left(J_m\cap \Omega'\right)\geq \sum_{k=1}^p \mathcal{H}^1\left(B_{\epsilon}(x_k)\cap J_m\right)\geq p \epsilon,
\end{align*}
this implies
\begin{align*}
\int_{\widetilde{J}_m} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1
&\leq 2 Q\varepsilon
\sum_{j=1}^p \Xint{-}_{B^{+ }_{\epsilon}(x_j)}\Xint{-}_{B^{- }_{\epsilon}(x_j)}
\Pi\left(m(x), m\left(\tilde{x}\right)\right) \,d\tilde{x}\, dx\\
&\quad +5 L\delta\, \mathcal{H}^1\left(J_m\cap \Omega'\right).
\end{align*}
Moreover we have
\begin{align*}
&
\varepsilon
\sum_{j=1}^p \Xint{-}_{B^{+ }_{\epsilon}(x_j)}\Xint{-}_{B^{- }_{\epsilon}(x_j)}
\Pi\left(m(x), m\left(\tilde{x}\right)\right) \,d\tilde{x}\, dx \\
&\leq
\frac{16}{\pi\varepsilon}
\sum_{j=1}^p \int_{B^{+ }_{\epsilon}(x_j)}\left(\Xint{-}_{B_{2\epsilon}(0)}
\Pi\left(m(x), m\left(x+h\right)\right)\,dh\right)\, dx \\
&\leq
\frac{16}{\pi\varepsilon} \int_{\Omega'}\Xint{-}_{B_{2\epsilon}(0)}
\Pi\left(m(x), m\left(x+h\right)\right)\,dh \, dx\\
&\leq \frac{32}{\pi}
\sup_{|h|<2\varepsilon}
\frac{1}{|h|}
\int_{\Omega'}
\Pi\left(m(x), m\left(x+h\right)\right) \, dx,
\end{align*}
provided $\epsilon<1/2\dist(\Omega',\partial\Omega)$, so plugging this into the previous inequality we deduce
\begin{align*}
\int_{\widetilde{J}_m} \Pi\left(m^{+},m^{-}\right) d\mathcal{H}^1
&
\leq \frac{64 Q}{\pi}
\sup_{|h|<2\varepsilon}
\frac{1}{|h|}
\int_{\Omega'}
\Pi\left(m(x), m\left(x+h\right)\right) \, dx
+ 5 L \delta \, \mathcal{H}^1\left(J_m\right).
\end{align*}
Taking the limits $\varepsilon\to0$ and then $\delta\to 0$, we obtain \eqref{eq:intJmPi}, which concludes the proof of Lemma~\ref{misbv}.
\end{proof}
\bibliographystyle{acm}
| {
"timestamp": "2022-03-11T02:23:22",
"yymm": "2203",
"arxiv_id": "2203.05418",
"language": "en",
"url": "https://arxiv.org/abs/2203.05418",
"abstract": "Given any strictly convex norm $\\|\\cdot\\|$ on $\\mathbb{R}^2$ that is $C^1$ in $\\mathbb{R}^2\\setminus\\{0\\}$, we study the generalized Aviles-Giga functional \\[I_{\\epsilon}(m):=\\int_{\\Omega} \\left(\\epsilon \\left|\\nabla m\\right|^2 + \\frac{1}{\\epsilon}\\left(1-\\|m\\|^2\\right)^2\\right) \\, dx,\\] for $\\Omega\\subset\\mathbb R^2$ and $m\\colon\\Omega\\to\\mathbb R^2$ satisfying $\\nabla\\cdot m=0$. Using, as in the euclidean case $\\|\\cdot\\|=|\\cdot|$, the concept of entropies for the limit equation $\\|m\\|=1$, $\\nabla\\cdot m=0$, we obtain the following. First, we prove compactness in $L^p$ of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in $BV$, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case $\\|\\cdot\\|=|\\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\\|\\cdot\\|$.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9912886142555052,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7091916613853356
} |
https://arxiv.org/abs/1306.2423 | Numerical Radii for Tensor Products of Matrices | For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\otimes B)\le\|A\|w(B)$ holds, where $w(\cdot)$ and $\|\cdot\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\|A\|=\|A^k\|=1$ for some $k$, $1\le k<\infty$, then $w(A)\ge\cos(\pi/(k+2))$, and, moreover, the equality holds if and only if $A$ is unitarily similar to the direct sum of the $(k+1)$-by-$(k+1)$ Jordan block $J_{k+1}$ and a matrix $B$ with $w(B)\le\cos(\pi/(k+2))$, and (3) if $B$ is a nonnegative matrix with its real part (permutationally) irreducible, then $w(A\otimes B)=\|A\|w(B)$ if and only if either $p_A=\infty$ or $n_B\le p_A<\infty$ and $B$ is permutationally similar to a block-shift matrix \[[{array}{cccc}0 & B_1 & && 0 & \ddots && & \ddots & B_k& & & 0{array}]\] with $k=n_B$, where $p_A=\sup\{\ell\ge 1: \|A^{\ell}\|=\|A\|^{\ell}\}$ and $n_B=\sup\{\ell\ge 1 : B^{\ell}\neq 0\}$. | \section{Introduction and Preliminaries}
For any $n$-by-$n$ complex matrix $A$, its \emph{numerical range} $W(A)$ is, by definition, the subset $\{\langle Ax, x\rangle : x\in \mathbb{C}^n, \|x\|=1\}$ of the complex plane $\mathbb{C}$, where $\langle\cdot, \cdot\rangle$ and $\|\cdot\|$ denote the standard inner product and its associated norm in $\mathbb{C}^n$, respectively. The \emph{numerical radius} $w(A)$ of $A$ is $\max\{|z| : z\in W(A)\}$. It is known that $W(A)$ is a nonempty compact convex subset of $\mathbb{C}$, and $w(A)$ satisfies $\|A\|/2\le w(A)\le \|A\|$, where $\|A\|$ denotes the usual operator norm of $A$. For other properties of the numerical range and numerical radius, the reader may consult \cite{7}, \cite[Chapter 22]{9} or \cite[Chapter 1]{12}.
\vspace{4mm}
The \emph{tensor product} (or \emph{Kronecker product}) $A\otimes B$ of an $n$-by-$n$ matrix $A=[a_{ij}]_{i,j=1}^n$ and an $m$-by-$m$ matrix $B$ is the $(mn)$-by-$(mn)$ matrix
\[\left[
\begin{array}{ccc}
a_{11}B & \cdots & a_{1n}B \\
\vdots & & \vdots \\
a_{n1}B & \cdots & a_{nn}B \\
\end{array}
\right].\]
It is known that $A\otimes B$ and $B\otimes A$ are unitarily similar and $\|A\otimes B\|=\|A\|\cdot\|B\|$. Other properties of the tensor product can be found in \cite[Chapter 4]{12}.
\vspace{4mm}
The main concern of this paper is the relations between the numerical radius of $A\otimes B$ and those of $A$ and $B$. For one direction, we have $w(A\otimes B)\le\min\{\|A\|w(B), \|B\|w(A)\}$. This can be proven by using the unitary dilation of contractions, as to be done below. On the other hand, we also have $w(A\otimes B)\ge w(A)w(B)$. We are interested in when these become equalities. In the present paper, we obtain various conditions, necessary or sufficient, for $w(A\otimes B)=\|A\|w(B)$ to hold. The discussions on the equality $w(A\otimes B)=w(A)w(B)$ will be the subject of a subsequent paper of ours.
\vspace{4mm}
For the ease of exposition, we introduce two indices for an $n$-by-$n$ matrix $A$: the \emph{power norm index} $p_A$ and \emph{nilpotency index} $n_A$ of $A$. They are defined, respectively, by
\[p_A=\sup\{k\ge 1 : \|A^k\|=\|A\|^k\}\]
and
\[n_A=\left\{\begin{array}{ll} \sup\{k\ge 1 : A^k\neq 0_n\} & \ \ \ \mbox{if } \ A\neq 0_n,\\ 0 & \ \ \ \mbox{if } \ A=0_n,\end{array}\right.\]
where $0_n$ denotes the $n$-by-$n$ zero matrix.
\vspace{4mm}
We start in Section 2 by proving that if $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and $W(B)$ is a circular disc centered at the origin (Theorem \ref{2.2}). The proof depends on the dilation of $A$ to a direct sum of $S_{\ell}$-matrices with $\ell\le n$, the Poncelet property of the numerical ranges of matrices of the latter class, and Anderson's theorem on the circular disc numerical range. As a by-product, we obtain a lower bound for $w(A)$ when $A$ satisfies $\|A\|=\|A^k\|=1$ for some $k$, $1\le k< n$: $w(A)\ge\cos(\pi/(k+2))$, and determine exactly when this bound is attained: this is the case if and only if $A$ is unitarily similar to $J_{k+1}\oplus B$, where $J_{k+1}$ is the $(k+1)$-by-$(k+1)$ \emph{Jordan block}
\[\left[
\begin{array}{cccc}
0 & 1 & & \\
& 0 & \ddots & \\
& & \ddots & 1 \\
& & & 0 \\
\end{array}
\right]\]
and $B$ is a finite matrix with $w(B)\le\cos(\pi/(k+2))$ (Theorem \ref{2.10}). This generalizes the classical result of Willams and Crimmins \cite{17} for $k=1$. We conclude Section 2 with a result on nilpotent contractions, namely, we prove that if $A$ is an $n$-by-$n$ matrix with $\|A\|=1$, then a necessary and sufficient condition for $p_A=n_A<\infty$ to hold is that $A$ be unitarily similar to a direct sum $J_{k+1}\oplus B$, where $k=p_A$ and $B^{k+1}=0$ (Theorem \ref{2.13}).
\vspace{4mm}
Finally, in Section 3, we consider $B$ to be a nonnegative matrix with $\re B$ ($=(B+B^*)/2$) (permutationally) irreducible. We obtain in Theorem \ref{3.1} a complete characterization for $w(A\otimes B)=\|A\|w(B)$, namely, this is the case if and only if either $p_A=\infty$ or $n_B\le p_A<\infty$ and $B$ is permutationally similar to a block-shift matrix of the form
\[\left[
\begin{array}{cccc}
0 & B_1 & & \\
& 0 & \ddots & \\
& & \ddots & B_k \\
& & & 0 \\
\end{array}
\right]\]
with $k=n_B$.
\vspace{4mm}
As was mentioned before, the inequality $w(A\otimes B)\le\|A\|w(B)$ for $n$-by-$n$ and $m$-by-$m$ matrices $A$ and $B$ is known. It is a consequence of \cite[Theorem 3.4]{10} because $A\otimes B$ is the product of $A\otimes I_m$ and $I_n\otimes B$, and the latter two matrices \emph{doubly commute}, that is, $A\otimes I_m$ commutes with both $I_n\otimes B$ and its adjoint $I_n\otimes B^*$. Here we give a simple proof based on the unitary dilation of contractions.
\vspace{3mm}
\begin{proposition}\label{1.1}
If $A$ and $B$ are $n$-by-$n$ and $m$-by-$m$ matrices, respectively, then $w(A\otimes B)\le\min\{\|A\|w(B), \|B\|w(A)\}$.
\end{proposition}
\vspace{1pt}
\begin{proof}
We need only prove that $w(A\otimes B)\le \|A\|w(B)$, and may assume that $\|A\|=1$. Then the $(2n)$-by-$(2n)$ matrix
\[U=\left[
\begin{array}{cc}
A & (I_n-AA^*)^{1/2} \\
(I_n-A^*A)^{1/2} & -A^* \\
\end{array}
\right]\]
is unitary. Let $U$ be unitarily similar to the diagonal matrix $\dia(u_1, \ldots, u_{2n})$, where $|u_j|=1$ for all $j$. Then
\[w(A\otimes B)\le w(U\otimes B) = w(\sum_{j=1}^{2n}\oplus u_jB)=\max_j w(u_jB)=w(B)=\|A\|w(B).\qedhere\]
\end{proof}
\vspace{1mm}
We conclude this section with some basic properties of the indices $p_A$ and $n_A$ of a matrix $A$.
\vspace{3mm}
\begin{proposition}\label{1.2}
Let $A$ be an $n$-by-$n$ matrix. Then
\begin{enumerate}
\item[\rm (a)] $1\le p_A\le n-1$ or $p_A=\infty$,
\item[\rm (b)] $p_A=n-1$ if and only if $A$ is a nonzero multiple of a $S_n$-matrix, and
\item[\rm (c)] the following conditions are equivalent:
\begin{enumerate}
\item[\rm (1)] $p_A=\infty$,
\item[\rm (2)] $\|A\|=\rho(A)$,
\item[\rm (3)] $\|A\|=w(A)$,
\end{enumerate}
and if $\|A\|=1$, then the above are also equivalent to
\begin{enumerate}
\item[\rm (4)] $A$ has a unitary part.
\end{enumerate}
\end{enumerate}
\end{proposition}
\vspace{3mm}
Here $\rho(A)$ denotes the \emph{spectral radius} of $A$, that is, $\rho(A)$ is the maximum modulus of eigenvalues of $A$.
\vspace{3mm}
Recall that an $n$-by-$n$ matrix $A$ is of \emph{class} $S_n$ if it is a contraction ($\|A\|\le 1$), its eigenvalues are all in $\mathbb{D}\equiv\{z\in\mathbb{C} : |z|<1\}$, and $\rank(I_n-A^*A)=1$. Any contraction $A$ is unitarily similar to the direct sum of a unitary matrix $U$, called the \emph{unitary part} of $A$, and a completely nonunitary contraction $A'$, called the \emph{c.n.u. part} of $A$. The latter means that $A'$ is not unitarily similar to any direct sum with a unitary summand.
\vspace{3mm}
\begin{proof}[Proof of Proposition $\ref{1.2}$]
(a) was obtained by Pt\'{a}k in 1960 (cf. \cite[Theorem 2.1]{15}) and (b) was proven in \cite[Theorem 3.1]{4}. As for (c), the implication (1) $\Rightarrow$ (2) is by \cite[Problem 88]{9}, (2) $\Rightarrow$ (3) by the known inequalities $\rho(A)\le w(A)\le\|A\|$, (3) $\Rightarrow$ (2) by \cite[Problem 218 (b)]{9}, and (2) $\Rightarrow$ (1) by the inequalities $\rho(A)\le\|A^k\|^{1/k}\le\|A\|$ for all $k \ge 1$. If $\|A\|=\rho(A)=1$, then, letting $\lambda$ be an eigenvalue of $A$ with $|\lambda|=1$, we have the unitary similarity of $A$ and a matrix of the form
{\footnotesize$\left[ \begin{array}{cc} \lambda & B \\ 0 & C \\ \end{array} \right]$}.
Since $\|A\|=|\lambda|=1$ implies that $B=0$, $A$ is unitarily similar to $[\lambda]\oplus C$ and thus has a unitary part. This proves (2) $\Rightarrow$ (4). That (4) $\Rightarrow$ (2) is trivial.
\end{proof}
\vspace{1mm}
\begin{proposition}\label{1.3}
Let $A$ be an $n$-by-$n$ matrix. Then
\begin{enumerate}
\item[\rm (a)] $0\le n_A\le n-1$ or $n_A=\infty$,
\item[\rm (b)] $n_A=n-1$ if and only if $A$ is similar to the $n$-by-$n$ Jordan block $J_n$,
\item[\rm (c)] $n_A=\infty$ if and only if $A$ is not nilpotent, and
\item[\rm (d)] $p_A\le n_A$ for $A\neq 0_n$.
\end{enumerate}
\end{proposition}
\vspace{3mm}
We omit its easy proofs.
\vspace{3mm}
In the following, we use $\sigma(A)$ to denote the \emph{spectrum} of $A$, that is, $\sigma(A)$ is the set of eigenvalues of $A$. An $n$-by-$n$ matrix $A$ is a \emph{dilation} of an $m$-by-$m$ matrix $B$ (or $B$ is a \emph{compression} of $A$) if there is an $n$-by-$m$ matrix $V$ such that $B=V^*AV$ and $V^*V=I_m$. This is equivalent to $A$ being unitarily similar to a matrix of the form {\footnotesize$\left[ \begin{array}{cc} B & * \\ * & * \\ \end{array} \right]$}.
\vspace{5mm}
\section{Contractions}
We start with a simple condition which yields the equality $w(A\otimes B)=\|A\|w(B)$.
\vspace{1mm}
\begin{lemma}\label{2.1}
If $A$ is an $n$-by-$n$ matrix with $p_A=\infty$, then $w(A\otimes B)=\|A\|w(B)$ for any $m$-by-$m$ matrix $B$. In particular, this is the case for $A$ a contraction with a unitary part.
\end{lemma}
\vspace{1pt}
\begin{proof}
Since $p_A=\infty$ implies, by Proposition \ref{1.2} (c), that $\|A\|=w(A)$. If $\lambda$ is a number in $W(A)$ with $|\lambda|=w(A)$, then $|\lambda|=\|A\|$. Since $A$ is unitarily similar to a matrix of the form {\footnotesize$\left[\begin{array}{cc} \lambda & * \\ * & * \\ \end{array} \right]$}, we have the unitary similarity of $A\otimes B$ and {\footnotesize$\left[\begin{array}{cc} \lambda B & * \\ * & * \\ \end{array} \right]$}. It follows that $\|A\|w(B)=w(\lambda B)\le w(A\otimes B)$. On the other hand, we also have $w(A\otimes B)\le\|A\|w(B)$ by Proposition \ref{1.1}. Thus $w(A\otimes B)=\|A\|w(B)$ holds.
\end{proof}
\vspace{1mm}
The next theorem is one of the main results of this section. It gives a necessary condition for the equality $w(A\otimes B)=\|A\|w(B)$.
\vspace{1mm}
\begin{theorem}\label{2.2}
Let $A$ and $B$ be $n$-by-$n$ and $m$-by-$m$ matrices, respectively. If $\|A\|=1$ and $w(A\otimes B)=w(B)$, then either $A$ has a unitary part or A is c.n.u. and $W(B)$ is a circular disc centered at the origin.
\end{theorem}
\vspace{1mm}
We first prove this for the case when $A$ is an $S_n$-matrix. The numerical ranges of such matrices are known to have the \emph{Poncelet property}, namely, if $A$ is of class $S_n$, then, for any point $\lambda$ on the unit circle $\partial \mathbb{D}$, there is a unique (up to unitary similarity) $(n+1)$-by-$(n+1)$ unitary dilation $U$ of $A$ such that $\lambda$ is an eigenvalue of $U$ and each edge of the $(n+1)$-gon $\partial W(U)$ intersects $W(A)$ at exactly one point (cf. \cite[Theorem 2.1 and Lemma 2.2]{2}).
\vspace{1mm}
\begin{lemma}\label{2.3}
Let $A$ be an $S_n$-matrix and $B$ an $m$-by-$m$ matrix. If $w(A\otimes B)=w(B)$, then $W(B)$ is a circular disc centered at the origin.
\end{lemma}
\vspace{1pt}
\begin{proof}
Let $U_1, \ldots, U_{m+1}$ be $(n+1)$-by-$(n+1)$ unitary dilations of $A$ with $\sigma(U_i)\cap\sigma(U_j)=\emptyset$ for all $i$ and $j$, $1\le i\neq j\le m+1$. We may assume that $U_j=\dia(\lambda_{1 j}, \ldots, \lambda_{n+1, j})$ for each $j$, where $|\lambda_{i j}|=1$ for all $i$ and $j$. Let $V_j$ be an $(n+1)$-by-$n$ matrix such that $A=V_j^*U_jV_j$ and $V_j^*V_j=I_n$ for each $j$. Since $\|A\|=1$ and
\[ w(A\otimes \lambda B)=w(A\otimes B)=w(B)=w(\lambda B)\]
for any $\lambda$, $|\lambda|=1$, we may further assume that $w(B)$ is in $W(A\otimes B)$. Let $x$ be a unit vector in $\mathbb{C}^n\otimes \mathbb{C}^m$ such that $\langle (A\otimes B)x, x\rangle=w(B)$. We decompose $(V_j\otimes I_m)x$ as $y_{1j}\oplus\cdots\oplus y_{n+1, j}$ with $y_{ij}$, $1\le i\le n+1$, in $\mathbb{C}^m$ for each $j$. Then
\begin{align*}
w(B) &= \langle (A\otimes B)x, x\rangle\\
&= \langle (U_j\otimes B)(V_j\otimes I_m)x, (V_j\otimes I_m)x\rangle\\
&= \langle (\lambda_{1 j}B\oplus\cdots\oplus\lambda_{n+1, j}B)(y_{1 j}\oplus\cdots\oplus y_{n+1, j}), y_{1 j}\oplus\cdots\oplus y_{n+1, j}\rangle\\
&= \sum_{i=1}^{n+1}\langle \lambda_{i j}By_{i j}, y_{i j}\rangle\\
&\le \sum_{i=1}^{n+1}|\langle By_{i j}, y_{i j}\rangle|.
\end{align*}
Letting $\eta_{ij}=\langle B(y_{i j}/\|y_{i j}\|), y_{i j}/\|y_{i j}\|\rangle$ for each $y_{ij}\neq 0$, we obtain
\[w(B)=\sum_{y_{ij}\neq 0}\lambda_{ij}\|y_{ij}\|^2\eta_{ij}\le\sum_{y_{ij}\neq 0}\|y_{ij}\|^2|\eta_{ij}|\le\sum_{y_{ij}\neq 0}\|y_{ij}\|^2 w(B)=w(B)\]
since
\[\sum_{i}\|y_{ij}\|^2=\|(V_j\otimes I_m)x\|^2=\|x\|^2=1.\]
Thus we have equalities throughout the above sequence, which yields that $w(B)=\lambda_{ij}\eta_{ij}$ for $y_{ij}\neq 0$. Since $\sum_i\|y_{ij}\|^2=1$, this must hold for at least one $i$, say, $i_j$. Hence $\overline{\lambda}_{i_j j}w(B)=\eta_{i_j j}$ is in $\partial W(B)$ for each $j$. Note that such $\overline{\lambda}_{i_j j}w(B)$'s, $1\le j\le m+1$, are distinct from each other by our assumption on the disjointness of the spectra of the $U_j$'s. This shows that the boundary of $W(B)$ and the circle $|z|=w(B)$ intersect at at least $m+1$ points. Since $W(B)$ is contained in $\{z\in\mathbb{C} : |z|\le w(B)\}$, we apply Anderson's theorem (cf. \cite[Theorem]{3} or \cite{20}) to infer that $W(B)=\{z\in\mathbb{C}: |z|\le w(B)\}$.
\end{proof}
\vspace{2mm}
\begin{proof}[Proof of Theorem $\ref{2.2}$]
We assume that $A$ is c.n.u. Then $A$ can be dilated to the direct sum $A'\oplus\cdots\oplus A'$ of $\rank(I_n-A^*A)$ many copies of some $S_{\ell}$-matrix $A'$ with $\ell\le n$ (cf. \cite[Theorem 1.4]{18} or \cite[Lemma 3 (a)]{21}). Hence $A\otimes B$ dilates to $(A'\oplus\cdots\oplus A')\otimes B=(A'\otimes B)\oplus\cdots\oplus (A'\otimes B)$. We have
\[ w(B)=w(A\otimes B)\le w((A'\otimes B)\oplus\cdots\oplus (A'\otimes B))= w(A'\otimes B)\le\|A'\|w(B)=w(B).\]
Thus $w(A'\otimes B)=w(B)$. It follows from Lemma \ref{2.3} that $W(B)$ is a circular disc centered at the origin.
\end{proof}
\vspace{3mm}
An easy consequence of Theorem \ref{2.2} is that the converse of Lemma \ref{2.1} is also true.
\vspace{3mm}
\begin{corollary}\label{2.4}
For an $n$-by-$n$ matrix $A$, the equality $w(A\otimes B)=\|A\|w(B)$ holds for all matrices $B$ if and only if $p_A=\infty$.
\end{corollary}
\vspace{1pt}
\begin{proof}
For the necessity, assume that $\|A\|=1$ and let $B$ be any matrix with its numerical range not a circular disc centered at the origin. Theorem \ref{2.2} yields that $A$ has a unitary part. Then $p_A=\infty$ follows immediately.
\end{proof}
\vspace{3mm}
In Theorem \ref{2.2}, if $B$ is the Jordan block $J_m$, then we have the following characterizations for $w(A\otimes B)=\|A\|w(B)$.
\vspace{3mm}
\begin{theorem}\label{2.5}
Let $A$ be an $n$-by-$n$ matrix with $\|A\|=1$. Then the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $W(A\otimes J_m)=W(J_m)$,
\item[\rm (b)] $w(A\otimes J_m)=w(J_m)$,
\item[\rm (c)] $A\otimes J_m$ is unitarily similar to $J_m\oplus B$ for some matrix $B$ with $w(B)\le w(J_m)$, and
\item[\rm (d)] $\|A^{m-1}\|=1$.
\end{enumerate}
If, in addition, $n=m$, then the above conditions are also equivalent to
\begin{enumerate}
\item[\rm (e)] either $A$ has a unitary part or $A$ is of class $S_n$, and
\item[\rm (f)] $p_A=\infty$ or $n-1$.
\end{enumerate}
\end{theorem}
\vspace{3mm}
Note that $W(J_m)=\{z\in\mathbb{C}: |z|\le\cos(\pi/(m+1))\}$ (cf. \cite[Proposition 1]{8}).
\vspace{3mm}
\begin{proof}[Proof of Theorem $\ref{2.5}$]
The implication (a) $\Rightarrow$ (b) is trivial. To prove (b) $\Rightarrow$ (c), note that $(A\otimes J_m)^m=A^m\otimes J_m^m=0_{nm}$ and $\|A\otimes J_m\|=\|A\|\|J_m\|=1$. If $x$ is a unit vector in $\mathbb{C}^n\otimes \mathbb{C}^m$ such that $|\langle (A\otimes J_m)x, x\rangle|=w(A\otimes J_m)$, then $w(A\otimes J_m)=w(J_m)=\cos(\pi/(m+1))$ implies that the subspace $K$ of $\mathbb{C}^n\otimes \mathbb{C}^m$ generated by the vectors $x, (A\otimes J_m)x, \ldots, (A\otimes J_m)^{m-1}x$ is reducing for $A\otimes J_m$, and the restriction of $A\otimes J_m$ to $K$ is unitarily similar to $J_m$ (cf. \cite[Theorem 1 (2)]{8}). Hence $A\otimes J_m$ is unitarily similar to $J_m\oplus B$, where $B$ is the restriction of $A\otimes J_m$ to $K^{\perp}$. We obviously have $w(B)\le w(A\otimes J_m)=w(J_m)$.
\vspace{3mm}
For (c) $\Rightarrow$ (d), note that $A^{m-1}\otimes J_m^{m-1}$ is unitarily similar to $J_m^{m-1}\oplus B^{m-1}$ under (c). Hence
\[\|A^{m-1}\|=\|A^{m-1}\otimes J_m^{m-1}\|=\|J_m^{m-1}\oplus B^{m-1}\|=\max\{\|J_m^{m-1}\|, \|B^{m-1}\|\}=1.\]
\vspace{3mm}
To prove (d) $\Rightarrow$ (c), let $x$ be a unit vector in $\mathbb{C}^n$ such that $\|A^{m-1}x\|=1$. Then $\|A^{m-j}x\|=1$ for all $j$, $1\le j\le m$. Let $\{e_1, \ldots, e_m\}$ be the standard basis for $\mathbb{C}^m$, let $x_j=A^{m-j}x\otimes e_j$, $1\le j\le m$, and let $K$ be the subspace of $\mathbb{C}^n\otimes \mathbb{C}^m$ generated by $x_1, \ldots, x_m$. Then $(A\otimes J_m)x_1=0$ and $(A\otimes J_m)x_j=x_{j-1}$ for $2\le j\le m$. Since $\{x_1, \ldots, x_m\}$ is an orthonormal basis of $K$, this shows that $(A\otimes J_m)K\subseteq K$ and the restriction of $A\otimes J_m$ to $K$ is unitarily similar to $J_m$. On the other hand, it follows from $\|A\otimes J_m\|=\|A\|\|J_m\|=1$ and
\[ (A\otimes J_m)^*x_m=(A^*\otimes J_m^*)(x\otimes e_m)=(A^*x)\otimes(J_m^*e_m)=(A^*x)\otimes 0=0\]
that $K$ is reducing for $A\otimes J_m$, and hence $A\otimes J_m$ is unitarily similar to $J_m\oplus B$, where $B$ is the restriction of $A\otimes J_m$ to $K^{\perp}$. Obviously, we have
\[ w(B)\le w(A\otimes J_m)\le\|A\|w(J_m)=w(J_m).\]
\vspace{3mm}
To prove (c) $\Rightarrow$ (a), note that the unitary similarity of $J_m$ and $e^{i\theta}J_m$ for all real $\theta$ implies the same for $A\otimes J_m$ and $e^{i\theta}(A\otimes J_m)$. Thus $W(A\otimes J_m)$ is a circular disc centered at the origin. (c) implies that $w(A\otimes J_m)=w(J_m)$, which means that the radii of the two circular discs $W(A\otimes J_m)$ and $W(J_m)$ are equal. Therefore, $W(A\otimes J_m)=W(J_m)$ holds.
\vspace{3mm}
Now assume that $n=m$ and that $\|A^{n-1}\|=1$. If $\|A^n\|=1$, then $p_A=\infty$ and hence $A$ has a unitary part by Proposition \ref{1.2} (a) and (c). On the other hand, if $\|A^n\|<1$, then $A$ is of class $S_n$ by \cite[Theorem 3.1]{4}. This shows that (d) $\Rightarrow$ (e). Next, if (e) is true, then $p_A=\infty$ or $n-1$ depending on whether $A$ has a unitary part or $A$ is of class $S_n$ (cf. \cite[Theorem 3.1]{4} for the latter). This proves (f). Finally, if $p_A=\infty$, then $\|A^k\|=1$ for all $k\ge 1$, and, in particular, $\|A^{n-1}\|=1$. On the other hand, if $p_A=n-1$, then $\|A^{n-1}\|=\|A\|^{n-1}=1$. This proves (f) $\Rightarrow$ (d).
\end{proof}
\vspace{3mm}
The next proposition gives a characterization of $w(A\otimes B)=\|A\|w(B)$ when $B$ is of class $S_m$.
\vspace{3mm}
\begin{proposition}\label{2.6}
Let $A$ be an $n$-by-$n$ matrix with $\|A\|=1$, and $B$ be an $S_m$-matrix. Then $w(A\otimes B)=w(B)$ if and only if either $A$ has a unitary part or $A$ is c.n.u., $\|A^{m-1}\|=1$ and $B$ is unitarily similar to $J_m$.
\end{proposition}
\vspace{3mm}
Its proof depends on a special property of $S_n$-matrices. The following lemma is from \cite[Lemma 5]{19}. Here we give a shorter geometric proof.
\vspace{3mm}
\begin{lemma}\label{2.7}
Let $A$ be an $S_n$-matrix. Then $W(A)$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to $J_n$.
\end{lemma}
\vspace{1pt}
\begin{proof}
If $W(A)$ is as asserted, then the Poncelet property of $W(A)$ says that it is circumscribed by $(n+1)$-gons with vertices on the unit circle. As the circular disc $\{z\in\mathbb{C}: |z|\le\cos(\pi/(n+1))\} (=W(J_n))$ is circumscribed by any regular $(n+1)$-gon on the unit circle, if the radius of $W(A)$ is not equal to $\cos(\pi/(n+1))$, then we infer from a geometrical consideration that $W(A)$ cannot have the Poncelet property. Thus $W(A)$ must equal $W(J_n)$. The unitary similarity of $A$ and $J_n$ then follows from \cite[Theorem 3.2]{2}. The converse is trivial.
\end{proof}
\vspace{1mm}
\begin{proof}[Proof of Proposition $\ref{2.6}$]
If $w(A\otimes B)=w(B)$, then, by Theorem \ref{2.2}, either $A$ has a unitary part or $A$ is c.n.u. and $W(B)$ is a circular disc centered at the origin. In the latter case, Lemma \ref{2.7} yields the unitary similarity of $B$ and $J_m$, and then Theorem \ref{2.5} gives $\|A^{m-1}\|=1$. The converse also follows from Theorem \ref{2.5}.
\end{proof}
\vspace{3mm}
Note that, under the conditions of Proposition \ref{2.6}, if $A$ is c.n.u., then we automatically have $m\le n$. This is because if, otherwise, $m>n$, then $\|A^{m-1}\|=1$ yields, by Proposition \ref{1.2} (a) and (c), that $A$ has a unitary part.
\vspace{3mm}
A specific example of the results obtained so far is in the next proposition.
\vspace{3mm}
\begin{proposition}\label{2.8}
Let $n$ and $m$ be positive integers. Then $W(J_n\otimes J_m)=W(J_{\ell})$, where $\ell=\min\{n, m\}$, and thus $w(J_n\otimes J_m)=\min\{w(J_n), w(J_m)\}$.
\end{proposition}
\vspace{1pt}
\begin{proof}
Assume that $m\le n$. Since the principal submatrix of $J_n\otimes J_m$ formed by its rows and columns numbered $1, m+2, 2m+3, \ldots$, and $(m-1)m+m$ is $J_m$, we have that $J_n\otimes J_m$ is a dilation of $J_m$. Thus $w(J_m)\le w(J_n\otimes J_m)$. The reversed inequality $w(J_n\otimes J_m)\le\|J_n\|w(J_m)=w(J_m)$ is by Proposition \ref{1.1}. Therefore, $w(J_n\otimes J_m)=w(J_m)$ holds. As was seen in the proof of (c) $\Rightarrow$ (a) in Theorem \ref{2.5}, $W(J_n\otimes J_m)$ is a circular disc centered at the origin. Thus the equality of $w(J_n\otimes J_m)$ and $w(J_m)$ implies that of $W(J_n\otimes J_m)$ and $W(J_m)$.
\end{proof}
\vspace{3mm}
Besides $S_n$-matrices, another generalization of the Jordan blocks is the companion matrices. Recall that a companion matrix is one of the form
\[\left[ \begin{array}{ccccccc}
0 & 1 & & & & & \\
& 0 & 1& & & & \\
& & \cdot & \cdot & & & \\
& & & \cdot & \cdot & & \\
& & & & \cdot & \cdot & \\
& & & & & 0 & 1 \\
-a_n & -a_{n-1} & \cdot & \cdot & \cdot & -a_2 & -a_1
\end{array} \right],\]
whose characteristic and minimal polynomials are both equal to $z^n+\sum_{j=1}^na_jz^{n-j}$. The numerical ranges of such matrices have been studied in \cite{5,6,1}.
\vspace{3mm}
\begin{proposition}\label{2.9}
Let $A$ be an $n$-by-$n$ $(n\ge 2)$ companion matrix. Then the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $w(A\otimes A)=\|A\|w(A)$,
\item[\rm (b)] $A$ is unitary, $A=J_n$, or $A$ is unitarily similar to a direct sum $[a\omega_n^j]\oplus B$, where $|a|>1$, $\omega_n=e^{i(2\pi/n)}$, $0\le j\le n-1$, and $B$ is an $S_{n-1}$-matrix with eigenvalues $(1/\overline{a})\omega_n^k$, $0\le k\le n-1$ and $k\neq j$, and
\item[\rm (c)] $p_A=n_A=\infty$ or $n-1$.
\end{enumerate}
\end{proposition}
\vspace{1pt}
\begin{proof}
To prove (a) $\Rightarrow$ (b), let $A'=A/\|A\|$. Then (a) gives $w(A'\otimes A')=w(A')$. By Theorem \ref{2.2}, either $A'$ has a unitary part or it is c.n.u. with numerical range a circular disc centered at the origin. In the former case, either $A$ is normal or is unitarily similar to a matrix of the form $[a\omega_n^j]\oplus B$, where $|a|=\|A\|\ge 1$ and $B$ is of size $n-1$ with eigenvalues $(1/\overline{a})\omega_n^k$, $0\le k\le n-1$ and $k\neq j$ (cf. \cite[Theorem 1.1 and Corollary 1.3]{5}). If $A$ is normal or $|a|=1$, then $A$ is unitary by \cite[Corollary 1.2]{5}. Hence we may assume that $|a|>1$. Thus the eigenvalues of $B$ are all contained in $\mathbb{D}$. Moreover, by \cite[Theorem 2.1]{1}, we have $\rank(I_{n-1}-B^*B)=1$. These two together imply, by way of the singular value decomposition of $B$, that $\|B\|=1$. Hence $B$ is of class $S_{n-1}$. On the other hand, if it is the latter case, then $W(A)$ is also a circular disc centered at the origin. Therefore, $A=J_n$ by \cite[Theorem 2.9]{5}. This proves (b).
\vspace{3mm}
For (b) $\Rightarrow$ (c), if $A$ is unitary (resp., $A=J_n$), then, obviously, $p_A=n_A=\infty$ (resp., $p_A=n_A=n-1$). On the other hand, if $A$ is unitarily similar to the asserted $[a\omega_n^j]\oplus B$, then $\|A\|=\max\{|a|, \|B\|\}=|a|=\rho(A)$. Thus $p_A=n_A=\infty$ by Proposition \ref{1.2} (c) and \ref{1.3}.
\vspace{3mm}
Finally, for (c) $\Rightarrow$ (a), if $p_A=n_A=\infty$, then (a) is a consequence of Lemma \ref{2.1}. On the other hand, if $p_A=n_A=n-1$, then $A^n=0_n$. This implies that $A=J_n$ and thus (a) holds by Proposition \ref{2.8}.
\end{proof}
\vspace{3mm}
The next theorem is a consequence of Theorem \ref{2.5}. It gives a lower bound, in terms of $p_A$, for $w(A)$ when $A$ is an $n$-by-$n$ matrix with $\|A\|=1$.
\vspace{3mm}
\begin{theorem}\label{2.10}
If $A$ is an $n$-by-$n$ matrix with $\|A\|=\|A^k\|=1$ for some $k\ge 1$, then $w(A)\ge\cos(\pi/(k+2))$. Moreover, in this case, the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $w(A)=\cos(\pi/(k+2))$,
\item[\rm (b)] $A$ is unitarily similar to $J_{k+1}\oplus B$, where $B$ is a finite matrix with $w(B)\le\cos(\pi/(k+2))$, and
\item[\rm (c)] $W(A)=\{z\in\mathbb{C}: |z|\le\cos(\pi/(k+2))\}$.
\end{enumerate}
\end{theorem}
\vspace{3mm}
For the proof of (a) $\Rightarrow$ (b), we need the following lemma.
\vspace{3mm}
\begin{lemma}\label{2.11}
Let
\[ A=\left[
\begin{array}{ccccc}
0 \ & a_1 & & & \\
& 0 & \ \ddots & & \\
& & \ddots \ & a_{n-2} & \\
& & & 0 & a_{n-1} \\
& & & & a \\
\end{array}
\right] \ \ \ and \ \ \ B=\left[
\begin{array}{cccc}
0 \ & a_1 & & \\
& 0 & \ \ddots & \\
& & \ddots \ & a_{n-2} \\
& & & 0 \\
\end{array}
\right] \]
be $n$-by-$n$ and $(n-1)$-by-$(n-1)$ matrices, respectively, where $n\ge 2$ and $a_j$ is nonzero for all $j$. Then $w(A)>w(B)$.
\end{lemma}
\vspace{1pt}
\begin{proof}
We prove this by induction on $n$. If $n=2$, then $A=${\footnotesize$\left[ \begin{array}{cc} 0 & a_1 \\ 0 & a \\ \end{array} \right]$} and $B=[0]$, in which case we obviously have $w(A)>0=w(B)$. Assume now that the assertion is true for the matrix $A$ of size at most $n-1$ ($n\ge 3$), and let $A$ and $B$ be of the above form. By considering $e^{i\theta}A$ for a suitable real $\theta$ instead of $A$, we may assume that $w(A)$ equals the largest eigenvalue of $\re A$. Let
\[ C=\left[ \begin{array}{cccc}
0 \ & a_1 & & \\
& 0 & \ \ddots & \\
& & \ddots \ & a_{n-3} \\
& & & 0 \\
\end{array}
\right], \]
and let $p(z)$, $q(z)$ and $r(z)$ be the characteristic polynomials of $\re A$, $\re B$ and $\re C$, respectively. We expand the determinant of
\[ \left[
\begin{array}{ccccc}
z & -a_1/2 & & & \\
-\overline{a}_1/2 & z & \ddots & & \\
& \ddots & \ddots & \ddots & \\
& & \ddots & z & -a_{n-1}/2 \\
& & & -\overline{a}_{n-1}/2 & z-\re a \\
\end{array}
\right]\]
by minors on its last row to obtain $p(z)=(z-\re a)q(z)-(|a_{n-1}|^2/4)r(z)$. Let $\alpha$, $\beta$ and $\gamma$ be the largest eigenvalues of $\re A$, $\re B$ and $\re C$, respectively. Then $\alpha=w(A)$, $\beta=w(B)$ and $\gamma=w(C)$. Since $\re B$ (resp., $\re C$) is a principal submatrix of $\re A$ (resp., $\re B$), we have $\beta\le \alpha$ (resp., $\gamma\le\beta$). Assume that $\alpha=\beta$. Then the above equation yields
\[ 0=p(\alpha)=(\alpha-\re a)q(\beta)-\frac{1}{4}|a_{n-1}|^2\gamma(\beta)=-\frac{1}{4}|a_{n-1}|^2\gamma(\beta).\]
Since $a_{n-1}\neq 0$ and $\beta$ is larger than or equal to all eigenvalues of $\re C$, we infer from $\gamma(\beta)=0$ that $\beta=\gamma$ or $w(B)=w(C)$. This contradicts our induction hypothesis for $B$ and $C$. Hence we must have $\alpha>\beta$ or $w(A)>w(B)$.
\end{proof}
\vspace{2mm}
\begin{proof}[Proof of Theorem $\ref{2.10}$]
By Theorem \ref{2.5}, the assumption $\|A\|=\|A^k\|=1$ implies that $w(A\otimes J_{k+1})=w(J_{k+1})$. Hence
\[ w(A)=\|J_{k+1}\|w(A)\ge w(A\otimes J_{k+1})=w(J_{k+1})=\cos\frac{\pi}{k+2} \]
as asserted.
\vspace{3mm}
We now prove the equivalence of (a), (b) and (c). The implications (b) $\Rightarrow$ (c) and (c) $\Rightarrow$ (a) are trivial. To prove (a) $\Rightarrow$ (b), let $x$ be a unit vector in $\mathbb{C}^n$ such that $\|A^kx\|=1$. Then $\|A^jx\|=1$ for all $j$, $0\le j\le k$. We now check that $A^{k+1}x=0$. Assuming otherwise that $\|A^{k+1}x\|>0$, let $u_t=[u_{t 1} \ \ldots \ u_{t, k+2}]^T$ in $\mathbb{C}^{k+2}\otimes \mathbb{C}^n$, where
\[ u_{t j}=\left\{ \begin{array}{ll} \frac{\textstyle\sqrt{1-t^2}}{\textstyle\|A^{k+1}x\|} A^{k+1}x & \ \ \ \mbox{if } \ j=1, \vspace{3mm} \\
t\sqrt{\frac{\textstyle 2}{\textstyle k+2}} \sin\frac{\textstyle (j-1)\pi}{\textstyle k+2} A^{k-j+2}x & \ \ \ \mbox{if } \ j=2, \ldots, k+2
\end{array}\right.\]
for any $t$, $0<t<1$. Note that
\[v\equiv \sqrt{\frac{2}{k+2}}\left[\sin\frac{\pi}{k+2} \ \ \sin\frac{2\pi}{k+2} \ \ \ldots \ \ \sin\frac{(k+1)\pi}{k+2}\right]^T \]
is a unit vector in $\mathbb{C}^{k+1}$ with $\langle J_{k+1}v, v\rangle=\cos(\pi/(k+2))$ (cf. \cite[Proposition 1 (3)]{8}). Hence $\|u_t\|=((1-t^2)+t^2\|v\|^2)^{1/2}=1$, and
\begin{eqnarray*}
\langle (J_{k+2}\otimes A)u_t, u_t\rangle &= & t\sqrt{1-t^2}\sqrt{\frac{2}{k+2}}\sin\frac{\pi}{k+2}\|A^{k+1}x\|\\
&& +t^2\frac{2}{k+2}\sum_{j=1}^k\sin\frac{j\pi}{k+2}\sin\frac{(j+1)\pi}{k+2}\|A^{k-j+1}x\|^2\\
&= & t\sqrt{1-t^2}\sqrt{\frac{2}{k+2}}\sin\frac{\pi}{k+2}\|A^{k+1}x\| + t^2\langle J_{k+1}v, v\rangle\\
&= & t\sqrt{1-t^2}\sqrt{\frac{2}{k+2}}\sin\frac{\pi}{k+2}\|A^{k+1}x\| + t^2\cos\frac{\pi}{k+2}.
\end{eqnarray*}
To reach a contradiction, we need to find some $t_0$, $0<t_0<1$, such that $\langle (J_{k+2}\otimes A)u_{t_0}, u_{t_0}\rangle > \cos(\pi/(k+2))$. This is the same as
\[ t_0\sqrt{1-t_0^2}\sqrt{\frac{2}{k+2}}\sin\frac{\pi}{k+2}\|A^{k+1}x\| > (1-t_0^2)\cos\frac{\pi}{k+2} \]
or
\[ \frac{t_0}{\sqrt{1-t_0^2}} > \sqrt{\frac{k+2}{2}}\frac{\cot\frac{\pi}{k+2}}{\|A^{k+1}x\|}.\]
Since $\lim_{t\rightarrow 1^-}t/\sqrt{1-t^2}=\infty$, the existence of such a $t_0$ is guaranteed. On the other hand, we also have
\[ \langle (J_{k+2}\otimes A)u_{t_0}, u_{t_0}\rangle \le w(J_{k+2}\otimes A)\le\|J_{k+2}\|w(A)=w(A)=\cos\frac{\pi}{k+2},\]
hence a contradiction. Thus we must have $A^{k+1}x=0$. Let $K$ be the subspace of $\mathbb{C}^n$ generated by $x, Ax, \ldots, A^kx$. Then $AK\subseteq K$. If $A'$ is the restriction of $A$ to $K$, then ${A'}^{k+1}=0$ and $\|{A'}^jx\|=\|A^jx\|=1$ for all $j$, $0\le j\le k$. Hence $\|{A'}^j\|=1$ for all such $j$'s. Together with ${A'}^{k+1}=0$, this says that $p_{A'}=k$ and thus $\dim K=k+1$ by Proposition \ref{1.2} (a). Therefore, $A'$ is unitarily similar to a matrix of the form $[a_{ij}]_{i,j=1}^{k+1}$ with $a_{ij}=0$ for all $i\ge j$. Since $1=\|{A'}^k\|=|a_{12}\cdots a_{k, k+1}|$, we infer that $|a_{12}|=\cdots= |a_{k, k+1}|=1$, and thus all the other $a_{ij}$'s are zero. Therefore, $[a_{ij}]_{i,j=1}^{k+1}$, and hence $A'$, is unitarily similar to $J_{k+1}$. Then $A$ is unitarily similar to a matrix of the form
\[\left[\begin{array}{c|c} \ \ J_{k+1} \ \ & \begin{array}{c} 0 \\ \, b_1 \ \cdots \ b_{n-k-1}\end{array} \\ \hline 0 & \begin{array}{ccc} c_1 & & *\\ & \ddots & \\ * & & c_{n-k-1}\end{array}\end{array}\right].\]
To show that all the $b_j$'s are zero, we appeal to Lemma \ref{2.11}. Indeed, for each $j$, $1\le j\le n-k-1$, consider the $(k+2)$-by-$(k+2)$ matrix
\[ A_j=\left[\begin{array}{c|c} \ \ J_{k+1} \ \ & \begin{array}{c} 0 \\ \vdots \\ 0 \\ b_j \end{array} \\ \hline 0 & c_j \end{array}\right].\]
If $b_j\neq 0$, then $w(A_j)>w(J_{k+1})=\cos(\pi/(k+2))$ by Lemma \ref{2.11}, which contradicts $w(A_j)\le w(A)=\cos(\pi/(k+2))$. This proves (a) $\Rightarrow$ (b).
\end{proof}
\vspace{3mm}
Theorem \ref{2.10} generalizes the classical result of Williams and Crimmins \cite{17} for $k=1$. The following corollary is for $k=n-1$. Part of it has been proven in \cite{19}: the equivalence of (b) and (c) is in \cite[Theorem 1]{19} and that of (b) and (d) in \cite[p. 352]{19}.
\vspace{3mm}
\begin{corollary}\label{2.12}
The following conditions are equivalent for an $n$-by-$n$ matrix $A$ with $\|A\|=1$:
\begin{enumerate}
\item[\rm (a)] $\|A^{n-1}\|=1$ and $w(A)=\cos(\pi/(n+1))$,
\item[\rm (b)] $A$ is unitarily similar to $J_{n}$,
\item[\rm (c)] $W(A)=\{z\in\mathbb{C}: |z|\le\cos(\pi/(n+1))\}$,
\item[\rm (d)] $\|A^{n-1}\|=1$ and $A^n=0_n$, and
\item[\rm (e)] $p_A=n_A=n-1$.
\end{enumerate}
\end{corollary}
\vspace{1pt}
\begin{proof}
The equivalence of (a) and (b) is by Theorem \ref{2.10}. The other implications are either in \cite{19} or trivial.
\end{proof}
\vspace{3mm}
Note that, in the preceding corollary, the conditions that $\|A\|=1$ and $w(A)=\cos(\pi/(n+1))$ for an $n$-by-$n$ matrix $A$ are not sufficient to guarantee that $A$ be unitarily similar to $J_n$. One example is $A=J_{n-1}\oplus[\cos(\pi/(n+1))]$.
\vspace{3mm}
We end this section with a characterization of matrices $A$ satisfying $p_A=n_A$. This is related to the previous results.
\vspace{3mm}
\begin{theorem}\label{2.13}
Let $A$ be an $n$-by-$n$ matrix with $\|A\|=1$. Then
\begin{enumerate}
\item[\rm (a)] $A$ satisfies $p_A=n_A\ (\le\infty)$ if and only if either it has a unitary part or is unitarily similar to a direct sum $J_{k+1}\oplus B$, where $k=p_A<\infty$ and $B^{k+1}=0_{n-k-1}$, and
\item[\rm (b)] if $p_A=n_A\ (\le \infty)$, then $w(A\otimes A)=w(A)$ holds, but not conversely.
\end{enumerate}
\end{theorem}
\vspace{1pt}
\begin{proof}
(a) For the necessity, we may assume, in view of Proposition \ref{1.2} (c), that $k\equiv p_A=n_A<\infty$ and prove that $A$ is unitarily similar to the asserted direct sum. Since $A^{k+1}=0_n$, $A$ is unitarily similar to a block matrix $A'$ of the form $[A_{ij}]_{i,j=1}^{k+1}$ with $A_{ij}=0$ for $1\le j\le i\le k+1$. Hence
\[ {A'}^k=\left[
\begin{array}{cccc}
0 & \cdots & 0 & \prod_{i=1}^kA_{i, i+1} \\
& 0 & & 0 \\
& & \ddots & \vdots \\
& & & 0 \\
\end{array}
\right]. \]
Since $\|{A'}^k\|=\|A^k\|=\|A\|^k=1$, we have $\|\prod_{i=1}^kA_{i, i+1}\|=1$. Let $x$ be a unit vector such that $\|(\prod_{i=1}^kA_{i, i+1})x\|=1$. Then $\|(\prod_{i=j}^kA_{i, i+1})x\|=1$ for all $j$, $1\le j\le k$. Let $\{e_1, \ldots, e_{k+1}\}$ be the standard basis for $\mathbb{C}^{k+1}$, and let $x_j=e_j\otimes(\prod_{i=j}^kA_{i, i+1})x$ if $1\le j\le k$, and $x_{k+1}=e_{k+1}\otimes x$. Then $x_1, \ldots, x_{k+1}$ are orthonormal vectors in $\mathbb{C}^n$, and $A'x_1=0$ and $A'x_j=x_{j-1}$ for $2\le j\le k+1$. Thus if $K$ is the subspace generated by $x_1, \ldots, x_{k+1}$, then $\dim K=k+1$, $A'K\subseteq K$, and the restriction of $A'$ to $K$ is unitarily similar to $J_{k+1}$. We infer from $\|A'\|=1$ and ${A'}^*x_{k+1}=0$ that $K$ reduces $A'$, and thus $A'$ is unitarily similar to $J_{k+1}\oplus B$ with $B^{k+1}=0$.
\vspace{3mm}
For the converse, if $A$ has a unitary part, then $p_A=n_A=\infty$ by Proposition \ref{1.2} (c). On the other hand, if $A$ is unitarily similar to $J_{k+1}\oplus B$ with the asserted properties, then $A^{k+1}=0$ implies that $p_A\le n_A\le k$. But
\[ \|A^k\|=\|J_{k+1}^k\oplus B^k\|=\max\{\|J_{k+1}^k\|, \|B^k\|\}=1=\|A\|^k \]
and $\|A^{k+1}\|=0<1=\|A\|^{k+1}$ together yield $p_A=n_A=k$.
\vspace{3mm}
(b) If $A$ has a unitary part, then $w(A\otimes A)=w(A)$ by Proposition \ref{2.1}. On the other hand, if $A$ is unitarily similar to $J_{k+1}\oplus B$ as in (a), then $A\otimes A$ is unitarily similar to $(J_{k+1}\otimes J_{k+1})\oplus (J_{k+1}\otimes B)\oplus(B\otimes J_{k+1})\oplus(B\otimes B)$. Note that $w(J_{k+1}\otimes J_{k+1})=w(J_{k+1})$ by Proposition \ref{2.8}, and
\begin{equation}\label{eq1}
w(J_{k+1}\otimes B)=w(B\otimes J_{k+1})\le\|J_{k+1}\|w(B)=w(B)
\end{equation}
by Proposition \ref{1.1}. Since $B^{k+1}=0$ and $\|B\|\le 1$, \cite[Lemma 3 (a)]{21} implies that $B$ can be dilated to the direct sum of $\rank(I-B^*B)$ copies of $J_m$ for some $m\le k+1$. Thus $w(B)\le w(J_m)\le w(J_{k+1})$. Combined with \eqref{eq1}, this yields $w(J_{k+1}\otimes B)\le w(J_{k+1})$. Also,
\[w(B\otimes B)\le\|B\|w(B)\le w(B)\le w(J_{k+1}).\]
Therefore,
\begin{align*}
w(A\otimes B) &= \max\{w(J_{k+1}\otimes J_{k+1}), w(J_{k+1}\otimes B), w(B\otimes B)\}\\
&= w(J_{k+1})\\
&= \max\{w(J_{k+1}), w(B)\}\\
&= w(A).
\end{align*}
\vspace{1mm}
That $w(A\otimes A)=w(A)$ does not imply $p_A=n_A$ is seen by $A=J_2\oplus[a]$, where $0<|a|\le 1/2$, in which case, $\|A\|=1$ and $w(A\otimes A)=w(A)=1/2$, but $p_A=1$ and $n_A=\infty$.
\end{proof}
\vspace{3mm}
The final result of this section is conditions for a matrix $A$ with $p_A=n_A$ so that it be unitarily similar to a block-shift matrix
\begin{equation}\label{eq2}
A'=\left[
\begin{array}{cccc}
0 & A_1 & & \\
& 0 & \ddots & \\
& & \ddots & A_k \\
& & & 0 \\
\end{array}
\right]
\end{equation}
with $\|A_1\cdots A_k\|=\|A\|$.
\vspace{3mm}
\begin{proposition}\label{2.14}
Let $A$ be an $n$-by-$n$ matrix with $p_A=n_A\equiv k<\infty$. If either {\rm (a)} $k=1$, $n-2$ or $n-1$, or {\rm (b)} $n=2, 3, 4$ or $5$, then $A$ is unitarily similar to the block-shift matrix $A'$ in \eqref{eq2} with $\|A_1\cdots A_k\|=\|A\|$.
\end{proposition}
\vspace{1pt}
\begin{proof}
We may assume that $\|A\|=1$.
\vspace{3mm}
(a) If $k=n_A=1$, then $A^2=0_n$. Hence $A$ is unitarily similar to a block-shift matrix of the form {\footnotesize $\left[
\begin{array}{cc}
0 & A_1 \\
0 & 0
\end{array}
\right]$} with $\|A_1\|=\|A\|$.
\vspace{3mm}
If $k=p_A=n_A=n-1$ (resp., $n-2$), then Theorem \ref{2.13} (a) implies that $A$ is unitarily similar to $J_n$ (resp., $J_{n-1}\oplus[0]$). The latter matrix plays the role of $A'$ with $k=n-1$ (resp., $n-2$) and $A_1=\cdots=A_{n-1}=[1]$ (resp., $A_1=\cdots=A_{n-3}=[1]$ and $A_{n-2}=[1 \ \ 0]$).
\vspace{3mm}
(b) In light of (a), we need only prove for $n=5$ and $k=2$. Invoking Theorem \ref{2.13} to obtain the unitary similarity of $A$ and $J_3\oplus${\footnotesize $\left[
\begin{array}{cc}
0 & b \\
0 & 0
\end{array}
\right]$}, where $|b|\le 1$. The latter matrix is permutationally similar to a block-shift matrix $A'$ with $k=2$, $A_1=${\footnotesize $\left[
\begin{array}{cc}
1 & 0 \\
0 & b
\end{array}
\right]$} and $A_2=${\footnotesize $\left[
\begin{array}{c}
1 \\
0
\end{array}
\right]$}. We obviously have $\|A_1A_2\|=\|${\footnotesize $\left[
\begin{array}{c}
1 \\
0
\end{array}
\right]$}$\|=1=\|A\|$.
\end{proof}
\vspace{3mm}
We remark that the preceding proposition fails for $n=6$ and $k=2$. Here is an example. Let $A=J_3\oplus B$, where
\[B=b \left[
\begin{array}{ccc}
0 & 1 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{array}
\right]\]
with $b=\sqrt{2/(3+\sqrt{5})}$. Then $\|A^2\|=1=\|A\|^2$ and $A^3=0_6$. This shows that $p_A=n_A=2$. Since $w(B)=2b>\sqrt{2}/2=w(J_3)$ and $w(B)$ is not a circular disc centered at the origin (cf. \cite[Theorem 4.1 (2)]{13}), we infer that nor is $W(A)$ ($=$ the convex hull of $W(J_3)\cup W(B)$). This implies that $A$ cannot be unitarily similar to a block-shift matrix.
\vspace{10mm}
\section{Nonnegative Matrices}
Recall that a matrix $A=[a_{ij}]_{i,j=1}^n$ is \emph{nonnegative} (resp., \emph{positive}), denoted by $A\succcurlyeq 0$ (resp., $A\succ 0$), if $a_{ij}\ge 0$ (resp., $a_{ij}>0$) for all $i$ and $j$. Two $n$-by-$n$ matrices $A$ and $B$ are \emph{permutationally similar} if there is an $n$-by-$n$ \emph{permutation matrix} $P$ (one with each row and column has exactly one 1 and all other entries 0) such that $P^TAP=B$. $A$ is said to be (\emph{permutationally}) \emph{reducible} if either $A$ is the 1-by-1 zero matrix or $n\ge 2$ and it is permutationally similar to a matrix of the form {\footnotesize$\left[ \begin{array}{cc} B & C \\ 0 & D \\ \end{array} \right]$}, where $B$ and $D$ are square matrices; otherwise, it is (\emph{permutationally}) \emph{irreducible}. It is known that if $A$ is nonnegative with $\re A$ irreducible, then it is permutationally similar to a block-shift matrix if and only if its numerical range is a circular disc centered at the origin (cf. \cite[Theorem 1 (a)$\Leftrightarrow$(r)]{16}). Other properties of nonnegative matrices can be found in \cite[Section 6.2 and Chapter 8]{11}.
\vspace{3mm}
The main result of this section is the following theorem, which essentially generalizes Theorem \ref{2.5}.
\vspace{3mm}
\begin{theorem}\label{3.1}
Let $A$ be an $n$-by-$n$ matrix and $B$ an $m$-by-$m$ nonnegative matrix with $\re B$ irreducible. Then the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $w(A\otimes B)=\|A\|w(B)$,
\item[\rm (b)] either $p_A=\infty$ or $n_B\le p_A<\infty$ and $W(B)$ is a circular disc centered at the origin, and
\item[\rm (c)] either $p_A=\infty$ or $n_B\le p_A<\infty$ and $B$ is permutationally similar to a block-shift matrix
\[\left[
\begin{array}{cccc}
0 & B_1 & & \\
& 0 & \ddots & \\
& & \ddots & B_k \\
& & & 0 \\
\end{array}
\right]\]
with $k=n_B$.
\end{enumerate}
\end{theorem}
\vspace{3mm}
For its proof, we need the following two lemmas.
\vspace{3mm}
\begin{lemma}\label{3.2}
Let $A=[a_{ij}]_{i,j=1}^n$ be a nonnegative matrix. Then the following hold:
\begin{enumerate}
\item[\rm (a)] The index $n_A$ is finite if and only if there is no sequence of indices $i_0, i_1, \ldots, i_{k-1}, i_k\ (k\ge 1)$ with $i_0=i_k$ such that $a_{i_0 i_1}, \ldots, a_{i_{k-1} i_k}$ are all nonzero. In particular, we have $n_A=\sup\{k\ge 1 : \mbox{there are distinct } i_j, 0\le j\le k, \mbox{such that } a_{i_j i_{j+1}}\neq 0 \mbox{ for all } j\}$.
\item[\rm (b)] $n_A=\infty$ if and only if there is a $k\ge 1$ such that some diagonal entry of $A^k$ is nonzero.
\item[\rm (c)] If $a_{ii}\neq 0$ for some $i$, $1\le i\le n$, then $n_A=\infty$.
\item[\rm (d)] If $A$ is irreducible, then $n_A=\infty$.
\item[\rm (e)] If $A$ is the block-shift matrix
\[\left[
\begin{array}{cccc}
0_{n_1} & A_1 & & \\
& 0_{n_2} & \ddots & \\
& & \ddots & A_k \\
& & & 0_{n_{k+1}} \\
\end{array}
\right] \ \ \mbox{on} \ \ \mathbb{C}^n=\mathbb{C}^{n_1}\oplus\cdots\oplus \mathbb{C}^{n_{k+1}}\]
and $\re A$ is irreducible, then $k=n_A$.
\end{enumerate}
\end{lemma}
\vspace{1mm}
\begin{proof}
(a) Assume first that the indices $i_0, i_1, \ldots, i_{k-1}, i_k=i_0$ ($k\ge 1$) are such that $a_{i_0 i_1},\ldots, a_{i_{k-1} i_k}\neq 0$. \cite[Theorem 6.2.16]{11} says that this is the case if and only if $(A^k)_{i_0 i_0}$, the $(i_0, i_0)$-entry of $A^k$, is nonzero. Hence $A^k\neq 0_n$. Similarly, considering the sequence $i_0, \ldots, i_k, i_1, \ldots, i_k, \ldots, i_1, \ldots, i_k$ of $\ell k+1$ indices for any $\ell\ge 1$, we also obtain $A^{\ell k}\neq 0_n$. It follows that $n_A=\infty$. Conversely, assume that $n_A=\infty$. Then $A^k\neq 0_n$ for some $k\ge n$. \cite[Theorem 6.2.16]{11} yields that, for some $i$ and $j$, there are indices $i_0=i, i_1, \ldots, i_{k-1}, i_k=j$ such that $a_{i_0 i_1},\ldots, a_{i_{k-1} i_k}$ are all nonzero. By the pigeonhole principle, we infer that $i_s=i_t$ for some $s$ and $t$, $0\le s< t\le k$. Then $i_s, \ldots, i_t$ are such that $i_s=i_t$ and $a_{i_s i_{s+1}},\ldots, a_{i_{t-1} i_t}\neq 0$. This proves the converse. The expression for $n_A$ is an easy consequence of \cite[Theorem 6.2.16]{11} and the above arguments. So are (b) and (c).
\vspace{3mm}
(d) Note that the irreducibility of $A$ is equivalent to the existence, for every distinct pair $i$ and $j$, of indices $i_0=i, i_1, \ldots, i_{k-1}, i_k=j$ ($k\ge 1$) such that $a_{i_0 i_1},\ldots, a_{i_{k-1} i_k}$ are all nonzero. Combining such indices from $i$ to $j$ with those from $j$ to $i$ yields one from $i$ to $i$ with the corresponding entries nonzero. Thus $n_A=\infty$ by \cite[Theorem 6.2.16]{11} and (b).
\vspace{3mm}
(e) Since $A^{k+1}=0_n$, we have $n_A\le k$. If $n_A<k$, then $A^k=0_n$, which implies that $A_1\cdots A_k=0$. If there are any nonzero $a_{i_0 i_1}, a_{i_1 i_2}, \ldots, a_{i_{k-1} i_k}$, where $(\sum_{j=1}^{\ell}n_j)+1\le i_{\ell}\le\sum_{j=1}^{\ell+1}n_j$ for $0\le\ell\le k$, then the $(i_0, n_{k+1} - (n-i_{k}))$-entry of $A_1\cdots A_k$, being larger than or equal to $\prod_{j=0}^{k-1}a_{i_j i_{j+1}}$, is nonzero, which contradicts the zeroness of the product $A_1\cdots A_k$. Thus no such nonzero sequence exists. This results in the reducibility of $\re A$, a contradiction. Hence we must have $n_A=k$.
\end{proof}
\vspace{3mm}
We remark that the conditions in the preceding lemma can all be expressed equivalently in terms of the directed graph associated with the matrix $A$ (cf. \cite[Section 6.2]{11}).
\vspace{3mm}
\begin{lemma}\label{3.3}
Let $A$ and $B$ be $n$-by-$n$ and $m$-by-$m$ matrices, respectively. If $B$ is unitarily similar to a block-shift matrix
\begin{equation}\label{eq3}
\left[
\begin{array}{cccc}
0_{m_1} & B_1 & & \\
& 0_{m_2} & \ddots & \\
& & \ddots & B_k \\
& & & 0_{m_{k+1}} \\
\end{array}
\right] \ \ \mbox{on} \ \ \mathbb{C}^m=\mathbb{C}^{m_1}\oplus\cdots\oplus \mathbb{C}^{m_{k+1}}
\end{equation}
with $k\le p_A\le\infty$, then $w(A\otimes B)=\|A\|w(B)$.
\end{lemma}
\vspace{1mm}
\begin{proof}
We may assume that $\|A\|=1$ and $B$ is equal to the block-shift matrix \eqref{eq3}. Since $k\le p_A\le\infty$, we have $\|A^k\|=\|A\|^k=1$. Let $x$ be a unit vector in $\mathbb{C}^n$ such that $\|A^kx\|=1$, and let $y=[y_1 \ \ldots \ y_{k+1}]^T$, where $y_j$ is in $\mathbb{C}^{m_j}$, $1\le j\le k+1$, be a unit vector in $\mathbb{C}^m$ such that $|\langle By, y\rangle|=w(B)$. Let $u=[y_1\otimes A^kx \ \ y_2\otimes A^{k-1}x \ \ \ldots \ \ y_{k+1}\otimes x]^T$. Then $u$ is a vector in $\mathbb{C}^m\otimes \mathbb{C}^n$ with
\begin{align*}
\|u\| &= (\sum_{j=1}^{k+1}\|y_j\otimes A^{k-j+1}x\|^2)^{1/2} = (\sum_{j=1}^{k+1}\|y_j\|^2 \|A^{k-j+1}x\|^2)^{1/2} \\
&= (\sum_{j=1}^{k+1}\|y_j\|^2)^{1/2} = \|y\| = 1.
\end{align*}
Moreover, we have
\begin{eqnarray*}
&& |\langle (B\otimes A)u, u\rangle|\\
&=& \left| \left\langle \left[
\begin{array}{cccc}
0_{m_1n} & B_1\otimes A & & \\
& 0_{m_2n} & \ddots & \\
& & \ddots & B_k\otimes A \\
& & & 0_{m_{k+1}n} \\
\end{array}
\right] \left[
\begin{array}{c}
y_1\otimes A^kx \\
y_2\otimes A^{k-1}x \\
\vdots \\
y_{k+1}\otimes x
\end{array}
\right], \left[
\begin{array}{c}
y_1\otimes A^kx \\
y_2\otimes A^{k-1}x \\
\vdots \\
y_{k+1}\otimes x
\end{array}
\right]\right\rangle \right| \\
&=& | \sum_{j=1}^{k}\langle (B_jy_{j+1})\otimes(A^{k-j+1}x), y_j\otimes(A^{k-j+1}x)\rangle | \\
&=& | \sum_{j=1}^{k}\langle B_jy_{j+1}, y_j\rangle\|A^{k-j+1}x\|^2 | \\
&=& | \sum_{j=1}^{k}\langle B_jy_{j+1}, y_j\rangle | \\
&=& |\langle By, y\rangle| = w(B).
\end{eqnarray*}
This shows that $w(B)\le w(B\otimes A) = w(A\otimes B)$. But $w(A\otimes B)\le \|A\|w(B)=w(B)$ always holds by Proposition \ref{1.1}. Hence $w(A\otimes B)=w(B)$ as asserted.
\end{proof}
\vspace{3mm}
We are now ready to prove Theorem \ref{3.1}.
\vspace{3mm}
\begin{proof}[Proof of Theorem $\ref{3.1}$]
For (a) $\Rightarrow$ (b), We assume that $\|A\|=1$ and $A$ is c.n.u. In view of Theorem \ref{2.2} and Proposition \ref{1.2} (c), we need only check that $w(A\otimes B)=w(B)$ implies $n_B\le p_A\ (<\infty)$. Let $B=[b_{ij}]_{i,j=1}^m$, and let $x$ be a unit vector in $\mathbb{C}^m\otimes \mathbb{C}^n$ such that $w(B\otimes A)=|\langle (B\otimes A)x, x\rangle|$. If $x=[x_1 \ \ldots \ x_m]^T$, where $x_j$ is in $\mathbb{C}^n$ for $1\le j\le m$, then
\begin{align}
w(B) &= w(B\otimes A) = |\langle [b_{ij}A]x, x\rangle| \nonumber \\
&\le \sum_{i,j}b_{ij}|\langle Ax_j, x_i\rangle| \nonumber \\
&\le \sum_{i,j} b_{ij}\|Ax_j\|\|x_i\| \label{eq4}\\
&\le \|A\|\sum_{i,j} b_{ij}\|x_j\|\|x_i\| \label{eq5} \\
&\le \langle Bx', x'\rangle \nonumber \\
&\le w(B), \label{eq6}
\end{align}
where $x'=[\|x_1\| \ \ldots \ \|x_{m}\|]^T$ is a unit vector in $\mathbb{C}^m$. This shows that the above inequalities are equalities throughout. Since $B\succcurlyeq 0$ and $\re B$ is irreducible, there is a unique unit vector $y$ in $\mathbb{C}^m$ with $y\succ 0$ such that $\langle By, y\rangle=w(B)$ (cf. \cite[Proposition 3.3]{14}). The equality in \eqref{eq6} yields that $x'=y$ and thus $x_j\neq 0$ for all $j$. Also, the equalities in \eqref{eq4} and \eqref{eq5} imply that $|\langle Ax_{j}, x_i\rangle|=\|Ax_j\|\|x_i\|=\|x_j\|\|x_i\|$ for all those $b_{ij}$'s with $b_{ij}>0$. Thus $Ax_j=\lambda_{ij}x_i$ for some $\lambda_{ij}$ satisfying $|\lambda_{ij}|=\|x_j\|/\|x_i\|$. Assume first that $k\equiv n_B<\infty$. Thus $B^k\neq 0_m$. By Lemma \ref{3.2} (a), there are distinct indices $i_0, \ldots, i_k$ such that $b_{i_0 i_1}, \ldots, b_{i_{k-1} i_k} >0$. It thus follows from above that $Ax_{i_j}=\lambda_{i_{j-1} i_j}x_{i_{j-1}}$ for $1\le j\le k$. Hence $A^kx_{i_k}=(\prod_{j=1}^k\lambda_{i_{j-1} i_j})x_{i_0}$. Since
\[ \|A^kx_{i_k}\| = (\prod_{j=1}^k\frac{\|x_{i_j}\|}{\|x_{i_{j-1}}\|})\|x_{i_0}\|=\|x_{i_k}\|,\]
we obtain $\|A^k\|=1$ or $p_A\ge k=n_B$. On the other hand, if $n_B=\infty$, then the same arguments as above with $k$ arbitrarily large yield that $p_A=\infty$, which contradicts our assumption that $A$ is c.n.u. This proves (a) $\Rightarrow$ (b).
\vspace{3mm}
That (b) $\Leftrightarrow$ (c) is a consequence of \cite[Theorem 1 (a)$\Leftrightarrow$(r)]{16}, and (c) $\Rightarrow$ (a) is by Lemma \ref{3.2} (e) and Lemma \ref{3.3}.
\end{proof}
\vspace{3mm}
Note that, in Theorem \ref{3.1}, the implication (a) $\Rightarrow$ (b) or (a) $\Rightarrow$ (c) is no longer true if $B$ is nonnegative but without the irreducibility of $\re B$. One example is $A=B=J_2\oplus[a]$, where $0<a\le 1/2$ (cf. the end of the proof of Theorem \ref{2.13} (b)). The next example shows that the same can be said if $B$ is not nonnegative but $\re B$ is irreducible.
\vspace{3mm}
\begin{example}\label{3.4}
Let $A=J_3$ and
\[B = \left[
\begin{array}{ccc}
0 & -\sqrt{2} & 1 \\
0 & 0 & 1 \\
0 & 0 & \sqrt{2}/2
\end{array}
\right].\]
Then $\re B$ is easily seen to be irreducible. We now show that $W(B)=\overline{\mathbb{D}}$. This is seen via \cite[Corollary 2.5]{13} by letting $u=0$ and $\lambda=\sqrt{2}/2$ therein and checking that
\[ \tr(B^*B^2)=\tr\left[
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 0 & \sqrt{2}/4
\end{array}
\right]=\frac{\sqrt{2}}{4}=\lambda|\lambda|^2\]
and $\tr(B^*B)=9/2\ge 5|\lambda|^2$, where $\tr(\cdot)$ denotes the trace of a matrix. We next prove that 1 is an eigenvalue of $\re(A\otimes B)$. Indeed, since
\[ \re(A\otimes B)=\frac{1}{2}\left[
\begin{array}{ccc}
0_3 & B & 0_3 \\
B^* & 0_3 & B \\
0_3 & B^* & 0_3
\end{array}
\right],\]
we need to check that
\[ \det\left[
\begin{array}{ccc}
2I_3 & -B & 0_3 \\
-B^* & 2I_3 & -B \\
0_3 & -B^* & 2I_3
\end{array}
\right]=0.\]
By a repeated use of the Schur decomposition, the above determinant is seen to be equal to
\begin{eqnarray*}
&& \det\,(2I_3) \, \det\left( \left[
\begin{array}{cc}
2I_3 & -B \\
-B^* & 2I_3
\end{array}
\right] - \left[\begin{array}{c} -B^* \\ 0_3 \end{array} \right](\frac{1}{2}I_3)\left[-B \ \ 0_3\right]\right)\\
&=& 8 \det \left[
\begin{array}{cc}
2I_3-(1/2)B^*B & -B \\
-B^* & 2I_3
\end{array}
\right]\\
&=& 8 \det \, (4I_3-B^*B-BB^*)\\
&=& 8 \det \left[
\begin{array}{ccc}
1 & -1 & -\sqrt{2}/2 \\
-1 & 1 & \sqrt{2}/2 \\
-\sqrt{2}/2 & \sqrt{2}/2 & 1
\end{array}
\right]\\
&=& 0
\end{eqnarray*}
as required. Since $W(A\otimes B)$ is a circular disc centered at the origin (by the unitary similarity of $A\otimes B$ and $e^{i\theta}(A\otimes B)$ for all real $\theta$) and $w(A\otimes B)\le\|A\|w(B)=1$, we infer from $1\in\sigma(\re(A\otimes B))$ that $W(A\otimes B)=\overline{\mathbb{D}}$. Hence $w(A\otimes B)=1=\|A\|w(B)$. But, obviously, we have $n_B=\infty$ and $p_A=2$. \hfill $\square$
\end{example}
\vspace{3mm}
The next corollary gives a more concrete equivalent condition, in terms of block-shift matrices, for $w(A\otimes B)=\|A\|w(B)$ when $A=B\succcurlyeq 0$ and $\re B$ is irreducible.
\vspace{3mm}
\begin{corollary}\label{3.5}
Let $A$ be an $n$-by-$n$ nonnegative matrix with $\re A$ irreducible. Then the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $w(A\otimes A)=\|A\|w(A)$,
\item[\rm (b)] $p_A=n_A\ (\le\infty)$, and
\item[\rm (c)] either $A$ is unitarily similar to $[a]\oplus A'$ with $|a|\ge\|A'\|$, or $A$ is permutationally similar to a block-shift matrix
\[A''=\left[
\begin{array}{cccc}
0 & A_1 & & \\
& 0 & \ddots & \\
& & \ddots & A_k \\
& & & 0 \\
\end{array}
\right] \]
with $\|A_1\cdots A_k\|=\|A\|$.
\end{enumerate}
\end{corollary}
\vspace{1mm}
\begin{proof}
We may assume that $\|A\|=1$. The implication (a) $\Rightarrow$ (b) is by Theorem \ref{3.1} and Proposition \ref{1.3} (d). For (b) $\Rightarrow$ (c), if $p_A=n_A=\infty$, then $A$ has a unitary part by Proposition \ref{1.2} (c), and hence $A$ is unitarily similar to $[a]\oplus A'$ with $|a|=1\ge\|A'\|$ as asserted. On the other hand, if $p_A=n_A<\infty$, then $w(A\otimes A)=w(A)$ by Theorem \ref{2.13} (b). Hence Theorem \ref{2.2} implies that $W(A)$ is a circular disc centered at the origin. For a nonnegative $A$ with $\re A$ irreducible, this is equivalent to $A$ being permutationally similar to the block-shift matrix $A''$ (cf. \cite[Theorem 1 (a)$\Leftrightarrow$(r)]{16}). As $n_{A''}=k$ by Lemma \ref{3.2} (e), we also have $p_A=k$. Thus $\|A^k\|=\|A\|^k=1$, which yields that $\|A_1\cdots A_k\|=1=\|A\|$ as required. Finally, for (c) $\Rightarrow$ (a), if $A$ is unitarily similar to $[a]\oplus A'$ with $|a|\ge\|A'\|$, then $w(A\otimes A)=w(A)$ by Lemma \ref{2.1}. On the other hand, if $A$ is permutationally similar to the block-shift matrix $A''$ with $\|A_1\cdots A_k\|=1$, then
\[ \|A^k\|=\|{A''}^k\|=\|A_1\cdots A_k\|=1=\|A\|^k.\]
Thus $p_A\ge k=n_A$. The equality $w(A\otimes A)=w(A)$ then follows from Theorem \ref{3.1}.
\end{proof}
\vspace{3mm}
\begin{corollary}\label{3.6}
Let $A=[a_{ij}]_{i,j=1}^n$, where $a_{ij}\ge 0$ for all $i$ and $j$, $a_{ij}=0$ for $i\ge j$, and $a_{i, i+1}>0$ for all $i$. Then the following conditions are equivalent:
\begin{enumerate}
\item[\rm (a)] $w(A\otimes A)=\|A\|w(A)$,
\item[\rm (b)] $p_A=n_A=n-1$, and
\item[\rm (c)] $a_{1 2}= \cdots =a_{n-1, n}$ and $a_{ij}=0$ for all other pairs of $i$ and $j$.
\end{enumerate}
\end{corollary}
\vspace{1mm}
\begin{proof}
In this case, $A$ is nonnegative, $\re A$ is irreducible and $n_A=n-1$. Consequently, Corollary \ref{3.5} yields the equivalence of (a), (b) and the condition (c') that $A$ is permutationally similar to a block-shift matrix $A''$ as in Corollary \ref{3.5} (c). Since $k=n_{A''}=n_A$ by Lemma \ref{3.2} (e), $A''$ is necessarily equal to $A$ with $|a_{1 2}\cdots a_{n-1, n}|=\|A\|$ and $a_{ij}=0$ for all other pairs of $i$ and $j$. The norm condition above yields that $a_{12}=\cdots=a_{n-1, n}=\|A\|$. Thus (c') is the same as (c), and we have the equivalence of (a), (b) and (c).
\end{proof}
\vspace{5mm}
\noindent
{\large Acknowledgements}
This research was partially supported by the National Science Council of the Republic of China under projects NSC-101-2115-M-008-006, NSC-101-2115-M-009-001 and NSC-101-2115-M-009-004 of the respective authors. P. Y. Wu was also supported by the MOE-ATU. This paper was presented by him at the 4th International Conference on Matrix Analysis and Applications in Konya, Turkey. He thanks the organizers for their works with the conference.
\newpage
| {
"timestamp": "2013-06-12T02:01:19",
"yymm": "1306",
"arxiv_id": "1306.2423",
"language": "en",
"url": "https://arxiv.org/abs/1306.2423",
"abstract": "For $n$-by-$n$ and $m$-by-$m$ complex matrices $A$ and $B$, it is known that the inequality $w(A\\otimes B)\\le\\|A\\|w(B)$ holds, where $w(\\cdot)$ and $\\|\\cdot\\|$ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if $\\|A\\|=1$ and $w(A\\otimes B)=w(B)$, then either $A$ has a unitary part or $A$ is completely nonunitary and the numerical range $W(B)$ of $B$ is a circular disc centered at the origin, (2) if $\\|A\\|=\\|A^k\\|=1$ for some $k$, $1\\le k<\\infty$, then $w(A)\\ge\\cos(\\pi/(k+2))$, and, moreover, the equality holds if and only if $A$ is unitarily similar to the direct sum of the $(k+1)$-by-$(k+1)$ Jordan block $J_{k+1}$ and a matrix $B$ with $w(B)\\le\\cos(\\pi/(k+2))$, and (3) if $B$ is a nonnegative matrix with its real part (permutationally) irreducible, then $w(A\\otimes B)=\\|A\\|w(B)$ if and only if either $p_A=\\infty$ or $n_B\\le p_A<\\infty$ and $B$ is permutationally similar to a block-shift matrix \\[[{array}{cccc}0 & B_1 & && 0 & \\ddots && & \\ddots & B_k& & & 0{array}]\\] with $k=n_B$, where $p_A=\\sup\\{\\ell\\ge 1: \\|A^{\\ell}\\|=\\|A\\|^{\\ell}\\}$ and $n_B=\\sup\\{\\ell\\ge 1 : B^{\\ell}\\neq 0\\}$.",
"subjects": "Functional Analysis (math.FA)",
"title": "Numerical Radii for Tensor Products of Matrices",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9912886137407894,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7091916610170955
} |
https://arxiv.org/abs/0910.3442 | Counting the spanning trees of a directed line graph | The line graph LG of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix-Tree Theorem to prove a formula for the number of spanning trees of LG, and he asked for a bijective proof. In this paper, we give a bijective proof of a generating function identity due to Levine which generalizes Knuth's formula. As a result of this proof we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2^{n-1}. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine. | \section{Introduction} \label{introduction}
In a directed graph $G=(V,E)$, each edge $e\in E$ is directed from its source $s(e)$ to its target $t(e)$. The directed line graph $\mathcal{L} G$ of $G$ with vertex set $E$, and with an edge $(e,f)$ for every pair of edges in $G$ such that $t(e)=s(f)$. A spanning tree of $G$ rooted at a vertex $r$ is an edge-induced subgraph of $G$ in which there is a unique path from $v$ to $r$, for all $v\in V$.
We denote the indegree and outdegree of a vertex $v$ by $\text{indeg}(v)$ and $\text{outdeg}(v)$, respectively, and we denote the number of spanning trees of $G$ by $\kappa(G)$. Knuth proved that if every vertex of $G$ has indegree greater than 0, then
\[\kappa(\mathcal{L} G)=\kappa(G)\prod_{v\in V} \text{outdeg}(v)^{\text{indeg}(v)-1}\]
Knuth's proof relied on the Matrix-Tree Theorem. In his paper, he noted that the simple form of this result suggested that a bijective proof was possible, but that it was not at all obvious how to find such a bijection \cite{Kn}.
In fact, there are even stronger relations between $\kappa(\mathcal{L} G)$ and $\kappa(G)$. Let $\{x_v|v\in V\}$ and $\{x_e|e\in E\}$ be variables indexed by the vertices and edges of $G$. The vertex and edge generating functions of $G$ are defined as follows, where the sums are taken over all rooted spanning trees $T$ of $G$.
\[\kappa^{edge}(G)=\sum_T \prod_{e\in T}x_e,\ \ \kappa^{vertex}(G)=\sum_T \prod_{e\in T}x_{t(e)}\]
Levine used linear algebraic methods to prove the following generalization of Knuth's result. Our first result in this paper is a bijective proof of Levine's theorem, which yields a bijective proof on Knuth's theorem as a special case.
\\ \\
\noindent \textbf{Theorem~\ref{thm1.1}.} \textit{Let $G=(V,E)$ be a directed graph in which every vertex has indegree greater than 0. Then}
\[\kappa^{vertex}(\mathcal{L}G)=\kappa^{edge}(G)\prod_{v\in V} \left(\sum_{s(e)=v} x_e\right)^{\text{indeg}(v)-1} \]
\\
Using this bijection, we are able to answer the following open question posed by Stanley.
\\ \\
\textbf{Exercise 5.73 from \cite{St}.} Let $\mathcal{B}(n)$ be the set of binary de Bruijn sequences of degree $n$, and let $\mathcal{S}_n$ be the set of all binary sequences of length $2^n$. Find an explicit bijection $\mathcal{B}(n)\times \mathcal{B}(n)\rightarrow \mathcal{S}(n)$.
\\ \\
The critical group $K(G)$ of a graph $G$ is a finite abelian group whose order is the number of spanning trees of $G$. Critical groups have applications in statistical physics \cite{Ho}, algebraic combinatorics \cite{Le}, and arithmetic geometry \cite{Be}. We review the definition of this group in section \ref{definitions}.
The Kautz graphs $\text{Kautz}_n(m)$ and the de Bruijn graphs $DB_n(m)$ are families of iterated line graphs. $\text{Kautz}_1(m)$ is the complete directed graph on $m+1$ vertices, without self-loops, and $DB_1(m)$ is the complete graph on $m$ vertices, with self-loops. These families are defined for $n>1$ as follows.
\[\text{Kautz}_n(m)=\mathcal{L}^{n-1}\text{Kautz}_1(m),\ \ DB_n(m)=\mathcal{L}^{n-1}DB_1(m)\]
Levine recently determined $K(DB_n(2))$ and $K(\text{Kautz}_n(m))$, where $m$ is prime \cite{Le}. We generalize these results, proving the following characterizations of the critical groups of all the Kautz and de Bruijn graphs.
\\ \\
\noindent \textbf{Theorem~\ref{db}.} \textit{The critical group of $DB_n(m)$ is}
\[K\left(DB_n(m)\right)=\left(\mathbb{Z}_{m^n}\right)^{m-2}\oplus\bigoplus_{i=1}^{n-1} \left(\mathbb{Z}_{m^i}\right)^{m^{n-1-i}(m-1)^2}\]
\\
\noindent \textbf{Theorem~\ref{kautz}.} \textit{The critical group of $\text{Kautz}_n(m)$ is}
\[K\left(\text{Kautz}_n(m)\right)=\left(\mathbb{Z}_{m+1}\right)^{m-1}\oplus \left(\mathbb{Z}_{m^{n-1}}\right)^{m^2-2}\oplus\bigoplus_{i=1}^{n-2} \left(\mathbb{Z}_{m^i}\right)^{m^{n-2-i}(m-1)^2(m+1)}\]
\\
The rest of this paper is organized as follows. In Section~\ref{definitions} we provide background and definitions. In Section~\ref{bijection}, we introduce a bijection which proves Theorem~\ref{thm1.1}. We apply this bijection in Section~\ref{dbsequence} to construct a bijection betweeen binary de Bruijn sequences of order $n$ and binary sequences of length $2^{n-1}$. Finally, in Section~\ref{kautzsection}, we prove Theorem~\ref{db} and ~\ref{kautz}, giving a complete description of the critical groups of the Kautz and de Bruijn graphs.
\section{Background and definitions} \label{definitions}
In a directed graph $G=(V,E)$, each edge $e\in E$ is directed from its \textit{source} $s(e)$ to its \textit{target} $t(e)$.
\begin{definition}[Directed line graph] Let $G=(V,E)$ be a directed graph. The \textit{directed line graph} $\mathcal{L}G$ is a directed graph with vertex set $E$, and with an edge $(e,f)$ for every pair of edges $e$ and $f$ of $G$ with $t(e)=s(f)$. \end{definition}
\begin{center}\includegraphics[height=2.4cm]{dlg}
\\ \textit{Figure 2.1. A directed graph and its line graph.}
\end{center}
At times we may speak of a subset $F$ of $E$ as a subgraphs of $G$ - in this case we mean the subgraph $(V,F)$. If $H$ is a subgraph of $G$ and $v$ is in $H$, we denote the indegree of $v$ in $H$ by $\text{indeg}_H(v)$, and the outdegree by $\text{outdeg}_H(v)$.
\begin{definition}[Oriented spanning tree] Let $G=(V,E)$ be a directed graph. An \textit{oriented spanning tree} of $G$ is an acyclic subgraph of $G$ with a distinguished node, the \textit{root}, in which there is a unique path from every vertex $v\in V$ to the root. We refer to these trees as \textit{spanning trees}. \end{definition}
Let $T$ be a spanning tree of $G$. Every vertex of $G$ has outdegree 1 in $T$, except the root, which has outdegree 0. We denote the number of spanning trees of $G$ by $\kappa(G)$, and the number of spanning trees rooted at $r$ by $\kappa(G,r)$.
Let $G=(V,E)$ be a strongly-connected directed graph, and let $\mathbb{Z}^V$ be the free abelian group generated by vertices of $G$ -- the group of of formal linear combinations of vertices of $G$. We define $\Delta_v\in \mathbb{Z}^V$, for all $v\in V$, as follows.
\[\Delta_v = \sum_{e\in E\ s.t\ s(e)=v} (t(e)-v) \]
The \textit{sandpile group} $K(G,r)$ with \textit{sink} $r$ is the quotient group
\[K(G,r)=\mathbb{Z}^V/(r,\Delta_v|v\in V\backslash r)\]
It is well-known that the order of $K(G,r)$ is $\kappa(G,r)$.
A directed graph $G$ is \textit{Eulerian} if $\text{indeg}(v)=\text{outdeg}(v)$ for all vertices $v$ in $V$. According to Lemma 4.12 of \cite{Ho}, if $G$ is Eulerian, the sandpile groups $K(G,r_1)$ and $K(G,r_2)$ are isomorphic for any two $r_1,r_2$ in $V$. In this case, we call the group the \textit{critical group} $K(G)$.
\begin{definition}[The Laplacian] Let $G=(V,E)$ be a finite directed graph with vertices $v_1,v_2,\ldots v_{|V|}$. The \textit{adjacency matrix} $A(G)$ of $G$ is the $|V|\times |V|$ matrix in which $A(G)_{ij}$ is the multiplicity of the edge $(v_i,v_j)$ in $E$. The \textit{degree matrix} $D(G)$ is the $|V|\times |V|$ diagonal matrix in which $D_{ii}=\text{outdeg}(v_i)$. The \textit{Laplacian} $L(G)$ of $G$ is defined as $A(G)-D(G)$.
\end{definition}
Note that the row vectors of $L(G)$ are the elements $\Delta_v$. We consider $L(G)^T$ as a $\mathbb{Z}$-linear operator on $\mathbb{Z}^V$ -- its image is the subgroup generated by the $\Delta_v$. For a strongly-connected Eulerian graph $G$, the Laplacian has exactly one eigenvalue 0, so for such a graph $G$, we have
\[\mathbb{Z}^V/\text{im}\ L(G)\cong \mathbb{Z}^V/\text{im}\ L(G)^T\cong \mathbb{Z}\oplus K(G)\]
The following elementary row and column operations on matrices with entries in a ring $R$ are invertible over $R$.
\begin{enumerate}[-]
\item Permuting two rows (columns)
\item Adding a multiple of a row (column) by an element of $R$ to another row (column)
\item Multiplying the entries of a row (column) by a unit
\end{enumerate}
If $L'$ is obtained from $L(G)$ by invertible row and column operations over $\mathbb{Z}$, then $\mathbb{Z}^V/\text{Im}\ L' \cong \mathbb{Z}^V/\text{Im}\ L(G)$.
Suppose that $R$ is a principal ideal domain. Under these operations, any matrix with entries in $R$ is equivalent to a matrix in \textit{Smith normal form}. A matrix in this form is diagonal, and its diagonal entries $x_{11},x_{22},\ldots x_{nn}$ are elements of $R$ such that $x_{(i+1)(i+1)}$ is a multiple of $x_{ii}$ for all $i<n$. These entries are called the \textit{invariant factors} of the original integer matrix, and they are unique up to multiplication by units. If the invariant factors of $L(G)$ over $\mathbb{Z}$ are $x_{11},x_{22},\ldots x_{nn}$ then
\[\mathbb{Z}^V/\text{Im}\ L(G) = \bigoplus_{i=1}^n \mathbb{Z}_{x_{ii}}\]
Thus, row-reducing the Laplacian yields information about the critical group.
\section{Counting spanning trees} \label{bijection}
Let $G=(V,E)$ be a directed graph, and let $\{x_v\}_{v\in V}$ and $\{x_e\}_{e\in E}$ be variables indexed by the vertices and edges of $G$. The \textit{edge and vertex generating functions}, which enumerate the spanning trees of $G$, are defined as follows
\[\kappa^{edge}(G)=\sum_T \prod_{e\in T}x_e\]
\[\kappa^{vertex}(G)=\sum_T \prod_{e\in T}x_{t(e)}\]
\noindent where $T$ ranges over all spanning trees of $G$. In this section, we give a bijective proof of the following identity, solving a problem posed by Levine in \cite{Le}
\begin{theorem} \label{thm1.1} Let G=(V,E) be a directed graph in which every vertex has indegree greater than 0. Then:
\begin{equation} \label{kthm} \kappa^{vertex}(\mathcal{L}G)=\kappa^{edge}(G)\prod_{v\in V} \left(\sum_{s(e)=v} x_e\right)^{\text{indeg}(v)-1} \end{equation} \end{theorem}
In order to find a bijection, we adopt the following strategy. We put an arbitrary total order on the edges in $E$.
\begin{enumerate}[-]
\item We provide a bijection between monomial terms on the right-hand side of Eq. (\ref{kthm}) and \textit{tree arrays}, which are arrays of lists, one list for each vertex $v\in V$.
\item Then we present a map $\sigma$ that take a tree array to a spanning tree of $\mathcal{L} G$ which contributes the same term to the left-hand side of Eq. (\ref{kthm}).
\item Finally, we show that $\sigma$ is bijective by constructing an inverse map $\pi$ which takes a spanning tree of $\mathcal{L} G$ to a tree array.
\end{enumerate}
We define a \textit{list} to be an ordered tuple of edges. We \textit{append} an element $x$ to a list $l$ by adding $x$ to the end of $l$. We \textit{pop} list $l$ by removing the first element of $l$. We denote the number of times an element $e$ appears in a list $l$ by $N(l,e)$ .
Let $v$ be a vertex of $G$ and let $l'_v$ be a list with $\text{indeg}(v)-1$ elements, all of which are edges with source $v$. We map $l'_v$ to a monomial term of $(\sum_{s(e)=v} x_e)^{\text{indeg}(v)-1}$, as follows.
\[l'_v=(e_1,e_2,\ldots e_{\text{indeg}(v)-1})\rightarrow x_{e_1}x_{e_2}\ldots x_{e_{\text{indeg}(v)-1}}\]
This map provides a bijection between lists $l'_v$ and terms of $(\sum_{s(e)=v} x_e)^{\text{indeg}(v)-1}$. Therefore, a term on the right-hand side of Eq. (\ref{kthm}) corresponds to a choice of spanning tree $T$ of $G$ and a choice of one such list $l'_v$ for each vertex $v$.
Suppose a monomial term on the right-hand side of Eq. (\ref{kthm}) corresponds to a spanning tree $T$ rooted at $r$ and an array of lists $\langle l'_v\rangle$. For each vertex $v\in V\backslash r$, we obtain $l_v$ by appending the unique edge $e$ in $T$ with source $v$ to the list $l'_v$. We obtain $l_r$ by appending a new variable $\Omega$ to $l'_r$.
Each list $l_v$ has length $\text{indeg}(v)$, for $v\in V$. We call an array of lists $\langle l_v \rangle_{v\in V}$ obtained in this way a \textit{tree array}. By construction, terms on the right-hand side of Eq. (\ref{kthm}) are in bijection with tree arrays.
We now define the bijective map $\sigma$, which takes a tree array of $G$ to a spanning tree of $\mathcal{L} G$. \vspace{.1in}
\noindent \textbf{The bijection $\sigma$:} We start with a tree array $\langle l_v \rangle$ and an empty subgraph $T'$ of $\mathcal{L} G$. Then we run the following algorithm.
\begin{enumerate}[Step 1.]
\item Let $R$ be the subset of edges $e$ of $G$ for which $N(l_{s(e)},e)=0$ and $\text{outdeg}_{T'}(e)=0$. Let $f$ be the smallest edge in $R$ under the order on $E$.
\item Pop the first element $g$ from the list $l_{t(f)}$. If $g$ is $\Omega$, then $\sigma(\langle l_v\rangle)=T'$.
\item Otherwise, $g\in E$ and $s(g)=t(f)$. Add the edge $(f,g)$ to $T'$, and then return to step 1.
\end{enumerate}
We also define a map $\pi$ which takes a spanning tree of $\mathcal{L} G$ to a tree array of $G$. \vspace{.1in}
\noindent \textbf{The inverse map $\pi$:} We start with a spanning tree $T'$ of $\mathcal{L} G$, and an empty list $l_v$ at each vertex $v\in V$. This map is given by another algorithm.
\begin{enumerate}[Step 1.]
\item Let $S$ be the set of leaves of $T'$. Let $f$ be the smallest edge in $S$ under the order on $E$.
\item If $f$ is not the root of $T'$, remove $f$ and its outedge $(f,g)$ from $T'$, and append $g$ to $l_{t(f)}$. Go back to step 2.
\item If $f$ is the root of $T'$, append $\Omega$ to $l_{t(f)}$, and return the array of lists.
\end{enumerate}
As an example, we apply $\sigma$ to a tree array in a small directed graph $G$. We order the edges of $G$ by the lexigraphic order.
\begin{center}\includegraphics[height=2.4cm]{bij1}
\\ \textit{Figure 3.1. The graph $G$, with a spanning tree $T$ highlighted in red. Below the graph is a monomial term of $\kappa^{vertex}(G)$, where $x_{ij}$ is the variable for edge $(i,j)$. The tree array corresponding to this term is shown to the right. In the term and the tree array, red elements correspond to edges of the tree.}
\end{center} \vspace{.025in}
\begin{center}\includegraphics[height=2.1cm]{bij2}\vspace{.025in}
\\ \textit{Figure 3.2. The first two edges added to $T'$ by the algorithm for $\sigma$. Initially, the edges $(1,2)$ and $(1,3)$ do not appear in the lists. We pop $(2,5)$ from $l_2$ and add the edge $((1,2),(2,5))$ to $T'$. Then the edges $(1,3)$ and $(5,4)$ have outdegree 0 in $T'$ and do not appear in the lists. We pop $(3,4)$ from $l_3$ and add $((1,3),(3,4))$ to $T'$. }
\end{center}
\begin{center}\includegraphics[height=5.1cm]{bij3}\vspace{.025in}
\\ \textit{Figure 3.3. The last three edges added to $T'$, and the final tree. The last element left in the lists of the tree array is $\Omega$.}
\end{center}
In order to prove Theorem~\ref{thm1.1}, we first prove three lemmas. In the definition of the algorithm for the map $\sigma$, we assumed that the set $R$ is always non-empty in step 1 and that the the list $l_{t(f)}$ is always non-empty in step 2. In Lemma~\ref{welldef}, we show that both assumptions are valid.
\begin{lemma} \label{welldef} The algorithm used to define map $\sigma$ is well-defined: at step 1, the set $R$ is non-empty, and at step 2, the list $l_{t(f)}$ is non-empty. \end{lemma}
\begin{proof} After $k$ edges have been added to $T'$, there are $|E|-k$ elements left in all the lists $l_v$, where one of the elements is $\Omega$. There are $|E|-k-1$ edges left in the lists, but there are $|E|-k$ edges of $G$ which do not have an outedge in $T'$, so $R$ must be non-empty in step 1.
Every time we pop $l_v$, we add an edge $(f,g)$ to $T'$, where $t(f)=v$. When we are at step 2, $\text{outdeg}_{T'}(f)=0$, so at most $\text{indeg}(t(f))-1$ of the elements of $l_{t(f)}$ have been popped. Therefore, the list $l_{t(f)}$ is always nonempty at step 2. The algorithm is well-defined. \end{proof} \vspace{.05in}
The following lemma shows that $\sigma$ takes a tree array corresponding to a term on the right-hand side of Eq. (\ref{kthm}) to a spanning tree which contributes the same term to the left-hand side.
\begin{lemma} \label{correctterm} Suppose that $\langle l_v \rangle$ is a tree array and that $\sigma(\langle l_v \rangle)=T'$. Then $T'$ is a spanning tree of $\mathcal{L} G$, and $\text{indeg}_{T'}(e)=N(l_{s(e)},e)$, for all $e\in E$. \end{lemma}
\begin{proof} Let $I(e)$ be the initial value of $N(l_{s(e)},e)$. By the definition of a tree array, the edges which are the last elements of the lists $l_v$ form a spanning tree $T$ of $G$.
We claim that $T'$ is acylic, because the last edge of a cycle is never included in $T'$. While the algorithm is running, suppose that $(e_n,e_1)$ is not an edge of $T'$, and that it completes a cycle $(e_1,e_2),(e_2,e_3),\ldots (e_{n-1},e_n)$ of edges in $T'$. Since $(e_1,e_2)$ was already added to $T'$, $N(l_{s(e_1)},e_1)$ must be 0. Therefore, $(e_n,e_1)$ will never be added to $T'$.
We say a vertex $v\in V$ is \textit{cleared} if all the elements of its list are popped. Suppose that $e=(v,w)$ is an edge in $T$. The list $l_w$ is cleared when all the edges of $G$ with target $w$ have an outedge in $T'$. Then $w$ can only be cleared after an outedge $(e,f)$ of $e$ is added to $T$. The edge $(e,f)$ can only be added to $T'$ when $N(l_v,e)=0$. Because $e$ is an edge of $T$, it is the last element of $l_v$, so $v$ must be cleared before $w$ can be cleared.
The algorithm terminates when $\Omega$ is popped from $l_r$, which occurs when $r$ is cleared. There is a path $(v,v_1,v_2,\ldots v_k, r)$ in $T$ from any vertex $v$ to $r$. Therefore $r$ can only be cleared after all the vertices on this path are cleared. Thus, all the vertices of $G$ are cleared when the algorithm finishes, so there are $|E|-1$ edges in the subgraph $T'$.
All the vertices of $\mathcal{L} G$ has an outedge in $T'$, except one. Since $T'$ is acyclic, it is a spanning tree of $\mathcal{L} G$. Because $\text{indeg}_{T'}(e)+N(l_{s(e)},e)$ is constant, when the algorithm returns $T'$, $\text{indeg}_{T'}(e)=I(e)$ for all $e\in E$. \end{proof} \vspace{.05in}
In our final lemma, we show that $\pi$ will take a spanning tree $T'$ of $\mathcal{L} G$ and reconstruct a tree array $\langle l_v\rangle$.
\begin{lemma} \label{Ttree} Suppose $T'$ is a spanning tree of $\mathcal{L} G$ with root $r'$, and that $\pi$ takes $T'$ to the array of lists $\langle l_v \rangle$. Then $\langle l_v \rangle$ is a tree array, which means that
\begin{enumerate}[(a)]
\item The length of $l_v$ is $\text{indeg}(v)$, for all $v\in V$.
\item Every element of $l_v$ is an edge with source $v$, for all vertices $v$ except $t(r')$. The last element of $l_{t(r')}$ is $\Omega$, and every other element of $l_{t(r')}$ is an edge with source $t(r')$.
\item The set $T$ of edges which are the last elements of the lists $\{l_v | v\in V\backslash t(r')\}$ is a spanning tree of $G$.
\end{enumerate}
\end{lemma}
\begin{proof} We first show parts (a) and (b). Each time an edge $e\in E$ is removed from $T'$, an element is appended to the list $l_{t(e)}$. Since $r'$ can only be removed after all the other vertices of $T'$, this algorithm adds $\text{indeg}(v)$ elements to $l_v$ for all $v\in V$, so part (a) holds. Every element of the list $l_v$ is an edge with source $v$, with the exception of $\Omega$, which is the last element of $l_r$, so part (b) holds.
While the algorithm $\pi$ is running, say a vertex $v\in V$ is \textit{filled} if $l_v$ has $\text{indeg}(v)$ elements. Every vertex is eventually filled, so the order in which vertices are filled is a total order on $V$.
We claim that this order is a topological sort of the subgraph $T$. Suppose that $f=(v,w)$ is the last element of $l_v$ for some vertex $v$ other than $r$. Vertex $v$ was filled at step 2, right after some leaf $e$ and some edge $(e,f)$ were removed from $T'$ in step 1. However, $w$ cannot be filled until $f$ is removed from $T'$, which happens after $e$ and $(e,f)$ are removed from $T'$. Therefore $v$ is filled before $w$, so the filling ordering on $V$ is a topological sort of $T$, and $T$ is acyclic.
After $\pi$ terminates, $t(r')$ has no outedge in $T$ and every other vertex of $G$ has one outedge. Then $T$ is a spanning tree of $G$, and part (c) holds. \end{proof} \vspace{.05in}
We now prove the main result.
\begin{proof}[\textbf{Proof of Theorem~\ref{thm1.1}}] Let $\langle l_v \rangle$ be a tree array and let $T'=\sigma(\langle l_v \rangle)$. We show the following claim by induction on $n$: after $n$ edges have been added to $T'$ by the algorithm for $\sigma(\langle l_v \rangle)$, and $n$ edges have been removed from $T'$ by the algorithm for $\pi(T')$, we have
\begin{enumerate}[(a)]
\item The set of edges added to $T'$ by $\sigma$ is the set of edges removed by $\pi$.
\item The elements popped from $l_v$ by $\sigma$ are exactly the elements added to $l_v$ by $\pi$, in the same order.
\end{enumerate}
In the base case $n=0$, both claims hold trivially. Suppose both results hold for $n=k$. The edge $e$ is a leaf of $T'$ in $\pi$ if and only if it satisfies $N(l_{s(e)},e)=0$ and $\text{outdeg}_{T'}(e)=0$ in $\sigma$.
Therefore, the $(k+1)$st edge $(f,g)$ added to $T'$ by $\sigma$ is also the $(k+1)$st edge removed from $T'$ by $\pi$, and the element $g$ popped from $l_{t(f)}$ in $\sigma$ is also the element appended to $l_{t(f)}$ by $\pi$. Both claims hold for $n=k+1$. By induction, they hold for all $n\le |E|-1$.
When $n=|E|-1$, condition (b) implies that $\pi(T')=\langle l_v \rangle$. Then $\pi$ is a left inverse of $\sigma$, and $\sigma$ is injective.
By similar reasoning, $\pi$ is a right inverse of $\sigma$, and $\sigma$ is surjective. So $\sigma$ is a bijection between tree arrays in $G$ and spanning trees of $\mathcal{L} G$. The bijection $\sigma$ induces between equal terms in Eq. (\ref{kthm}) proves Theorem~\ref{thm1.1}.
\end{proof}
\section{The de Bruijn bijection} \label{dbsequence}
A \textit{binary de Bruijn sequence of degree $n$} is a cyclic binary sequence $B$ such that every binary sequence of length $n$ appears as a subsequence of consecutive elements of $B$ exactly once. For example, 0011 is a binary de Bruijn sequence of degree 2, since its cyclic subsequences of length 2 are 00, 01, 11, and 10.
It is well-known that there are $2^{2^{n-1}}$ binary de Bruijn sequences of degree $n$. Stanley posed the following open problem in \cite{St}. \vspace{.1in}
\noindent \textbf{Exercise 5.73 of \cite{St}.} Let $\mathcal{B}(n)$ be the set of binary de Bruijn sequences of degree $n$, and let $\mathcal{S}_n$ be the set of all binary sequences of length $2^n$. Find an explicit bijection $\mathcal{B}(n)\times \mathcal{B}(n) \rightarrow \mathcal{S}_n$. \vspace{.1in}
Our solution to this problem involves the de Bruijn graphs, which are closely related to de Bruijn sequences.
\begin{definition}[de Bruijn graph] The \textit{de Bruijn graph} $DB_n(m)$ has $m^n$ vertices, which are identified with the strings of length $n$ on $m$ symbols. The edges of the graph are labeled with the strings of length $n+1$ on $m$ symbols. The edge $s_0s_1\ldots s_n$ has source $s_0s_1\ldots s_{n-1}$ and target $s_1s_2\ldots s_n$. \end{definition}
An edge of $DB_n(m)$ can be identified with the vertex of $DB_{n+1}(m)$ that is labeled with the same string of length $n+1$. With this identification, we have
\[DB_n(m)=\LiDB_{n-1}(m)\]
Each vertex $v=s_0s_1\ldots s_{n-1}$ of $DB_n(2)$ has two outedges, $s_1s_2,\ldots s_{n-1}0$ and $s_1s_2\ldots s_{n-1}1$. We call these edges the \textit{zero edge} of $v$ and the \textit{one edge} of $v$, respectively.
It is well-known that binary de Bruijn sequences of degree $n$ are in bijection with Hamiltonian paths in $DB_n(2)$. Let $B=b_0b_1\ldots b_{2^n-1}$ be a binary de Bruijn sequence of degree $n$. Let $v_i=b_ib_{i+1}\ldots b_{i+n-1}$, for $0\le i\le 2^n-1$, where indices are taken mod $2^n$. The path $(v_0,v_1,\ldots v_{2^n-1})$ is the corresponding Hamiltonian path in $DB_n(2)$.
\begin{theorem} There is an explicit bijection between $\mathcal{B}(n)$ and the set of binary sequences of length $2^{n-1}$, for $n>1$. \end{theorem}
\begin{proof} We describe a bijection between Hamiltonian paths in $DB_n(2)$ and binary sequences of length $2^{n-1}$. By composing this bijection with the map between de Bruijn sequences and Hamiltonian paths, we construct the desired bijection.
We order the vertices in $DB_k(2)$ by the lexicographic order on their associated binary strings, for $1\le k\le n$. Let $(v_1,\ldots v_{2^n})$ be a Hamiltonian path in $DB_n(2)$. This path is an oriented spanning tree of $DB_n(2)$, so we can apply the inverse map $\pi$ defined in Section \ref{bijection} to it.
Let $A_{n-1}$ be the tree array $A_{n-1}=\pi(v_1,\ldots v_{2^n})$. We recursively define a sequence of tree arrays $A_k$, for $1\le k\le n-1$. Suppose we have a tree array $A_{k+1}$ in $DB_{k+1}(2)$. Let $T_{k+1}$ be the spanning tree consisting of the edges which are the last elements of the lists in $A_{k+1}$. We define $A_k$ to be $\pi(T_{k+1})$.
We construct a binary sequence $s_1s_2\ldots s_{2^{n-1}}$ from these tree arrays. We denote vertex $w$'s list in the tree array $A_k$ by $(A_k)_w$. Let $s_{2^{n-1}}$ be 0 if the first element of $(A_{n-1})_{s(v_{2^n})}$ is the zero edge of $s(v_{2^n})$, and 1 otherwise.
We define $s_{2^k}$ through $s_{2^{k+1}-1}$, for $1\le k\le n-2$, as follows. Let $w_1,w_2,\ldots w_{2^k}$ be the vertices of $DB_k(2)$, in lexicographic order. Let $s_{2^k+i-1}$ be 0 if the first element of $(A_k)_{w_i}$ is the zero edge of $w_i$, and 1 otherwise. Let $s_1$ be 0 if $T_1$ is rooted at vertex 0, and 1 otherwise.
The string $s_1s_2\ldots s_{2^{n-1}}$ is the binary sequence that corresponds to the Hamiltonian path we began with.
Now we construct the inverse map, from binary sequences to Hamiltonian paths. Given any binary sequence $S$ of length $2^{n-1}$, we use the first $2^{n-1}-1$ characters of the sequence to invert the previous procedure and construct a sequence of spanning trees $T_1,T_2,\ldots T_{n-1}$. The tree $T_k$ will be a spanning tree of $DB_k(2)$.
We determine $T_k$ recursively. The tree $T_1$ in $DB_1(2)$ is rooted at 0 if $s_1$ is 0, and rooted at 1 otherwise. Assume that the first $2^k-1$ characters of $S$ determine a spanning tree $T_k$ of $DB_k(2)$, where $k\le n-2$. We choose a tree array $A_k$ of $DB_k(2)$ using this tree and the next $2^k$ characters of $S$, as follows.
Let the vertices of $DB_k(2)$ be $w_1,w_2,\ldots w_{2^k}$, in lexicographic order. The first element of $(A_k)_{w_i}$ is the zero edge of $w_i$ if $s_{2^k+i-1}$ is 0, and the one edge of $w_i$ otherwise. The second element of $(A_k)_{w_i}$ comes from $T_k$. We define $T_{k+1}$ to be $\sigma(A_k)$, using the map defined in Section \ref{bijection}.
We use $T_{n-1}$ to construct a tree array $A_{n-1}$ such that $\sigma(A_{n-1})$ is a Hamiltonian path in $DB_n(2)$. Let $r$ be the root of $T_{n-1}$, and let $v$ be another arbitrary vertex. The list $l_v$ in the array $A_{n-1}$ must contain two distinct edges, if $\sigma(A_{n-1})$ is a Hamiltonian path. The second edge in $l_v$ must be the unique edge in $T_{n-1}$ with source $v$, so $l_v$ is determined. Our only remaining choice is which of the two edges of $DB_{n-1}(2)$ with source $r$ to include in $l_r$, which we determine by $s_{2^{n-1}}$.
Clearly, this map from binary sequences to Hamiltonian paths inverts the map from Hamiltonian paths to binary sequences. Therefore, our first map is the bijection we need. \end{proof} \vspace{.05in}
This bijection can easily be generalized to count the $k$-ary de Bruijn sequences, in which the 2-symbol alphabet $\{0,1\}$ is replaced with the $k$-symbol alphabet $\{0,1,\ldots k-1\}$.
\section{The Kautz and de Bruijn graphs} \label{kautzsection}
In this section, we determine the critical groups of all the Kautz graphs and the de Bruijn graphs. The critical groups of these graphs have been found in some special cases by Levine \cite{Le}.
The Kautz graphs are similar to the de Bruijn graphs, except that the vertices are indexed by \textit{Kautz strings}. A Kautz string is a string in which no two adjacent characters are the same.
\begin{definition}[Kautz graph] The \textit{Kautz graph} $\text{Kautz}_n(m)$ has $(m+1)m^{n-1}$ vertices, identified with the Kautz strings of length $n$ on $m+1$ symbols. The edges of the graph are labeled with the Kautz strings of length $n+1$ on $m+1$ symbols, such that the edge $s_0s_1\ldots s_n$ has source $s_0s_1\ldots s_{n-1}$ and target $s_1s_2\ldots s_n$. \end{definition}
We also consider the Kautz and de Bruijn graphs as families of iterated line graphs. $\text{Kautz}_1(m)$ is the complete directed graph on $m+1$ vertices, without self-loops, and $DB_1(m)$ is the complete directed graph on $m$ vertices, with self-loops. Then for $n>1$, we have
\[\text{Kautz}_{n+1}(m)=\mathcal{L}\text{Kautz}_n(m)=\mathcal{L}^n\text{Kautz}_1(m)\]
\[DB_{n+1}(m)=\LiDB_n(m)=\mathcal{L}^nDB_1(m)\]
We say a directed graph $G=(V,E)$ is \textit{balanced k-regular} if $\text{indeg}(v)=\text{outdeg}(v)=k$ for all $v\in V$. Both $\text{Kautz}_n(m)$ and $DB_n(m)$ are balanced $m$-regular, for all $n\in \mathbb{N}$, which implies that they are Eulerian. Since these graphs are also strongly-connected, their critical groups are defined.
Levine found the critical groups of the de Bruijn graphs $DB_n(2)$ and the Kautz graphs $K_n(p)$, where $p$ is prime \cite{Le}. In this section we characterize the critical groups of all the Kautz and de Bruijn graphs. We prove the following theorems.
\begin{theorem} \label{db} The critical group of $DB_n(m)$ is
\[K\left(DB_n(m)\right)=\left(\mathbb{Z}_{m^n}\right)^{m-2}\oplus\bigoplus_{i=1}^{n-1} \left(\mathbb{Z}_{m^i}\right)^{m^{n-1-i}(m-1)^2}\]
\end{theorem}
\begin{theorem} \label{kautz} The critical group of $\text{Kautz}_n(m)$ is
\[K\left(\text{Kautz}_n(m)\right)=\left(\mathbb{Z}_{m+1}\right)^{m-1}\oplus \left(\mathbb{Z}_{m^{n-1}}\right)^{m^2-2}\oplus\bigoplus_{i=1}^{n-2} \left(\mathbb{Z}_{m^i}\right)^{m^{n-2-i}(m-1)^2(m+1)}\]
\end{theorem}
In order to prove these theorems, we first prove two lemmas about row-reducing the Laplacians $L(\text{Kautz}_n(m))$ and $L(DB_n(m))$. We refer to the row and column of a vertex $v$ in the Laplacian by $R(v)$ and $C(v)$, respectively. We also use $L(v,w)$ to denote the entry in the row of $v$ and the column of $w$.
We say two strings of length $n$ are \textit{similar} if their last $n-1$ characters are equal. Similarity is an equivalence relation. We partition the vertices of $\text{Kautz}_n(m)$ and $DB_n(m)$ into equivalence classes, by grouping vertices labeled with similar strings in the same class. There are $m$ vertices in each class.
\begin{lemma} \label{cycle} Let $G=(V,E)$ be a Kautz graph $\text{Kautz}_{n+1}(m)$ or a de Bruijn graph $DB_{n+1}(m)$, where $n\in \mathbb{N}$. Then $G$ contains a cycle $(v_1,v_2,\ldots v_c)$ of length $c=|V|/m$ which contains one vertex from each class. \end{lemma}
\begin{proof} Let $G'$, the predecessor of $G$, be $\text{Kautz}_n(m)$ if $G$ is $\text{Kautz}_{n+1}(m)$, and $DB_n(m)$ if $G$ is $DB_{n+1}(m)$.
First we show that there is a Hamiltonian cycle in $G'$. Such a cycle exists in the complete graphs $K_m$ and $K_{m+1}$, so the case $n=1$ is done. There is an Eulerian tour of $\text{Kautz}_{n-1}(m)$ and of $DB_{n-1}(m)$ for $n>1$, since graphs in both families are Eulerian. Because $G'$ is either $\mathcal{L} \text{Kautz}_{n-1}(m)$ or $\mathcal{L} DB_{n-1}(m)$, one of these Eulerian tours induces a Hamiltonian cycle in $G'$, for $n>1$.
The Hamiltonian cycle in $G'$ can be represented as a string $S=s_1s_2\ldots s_{n+c-1}$, where the $i$th vertex of the cycle is labeled with $s_is_{i+1}\ldots s_{i+n-1}$.
We use string $S$ to find a cycle in $G$. Let $v_i=s_is_{i+1}\ldots s_{i+n}$ for $i<c$, and let $v_c=s_cs_{c+1}\ldots s_{n+c-1}s_1$. By the construction of $S$, $(v_1,v_2,\ldots v_c)$ is a cycle which contains one vertex from each class. \end{proof} \vspace{.05cm}
In the next lemma we show that every invariant factor of $L(\text{Kautz}_{n+1}(m))$ and $L(DB_n(m))$ is either a multiple of $m$ or relatively prime to $m$. We prove this lemma by row-reducing the Laplacian in an order derived from the cycle in Lemma~\ref{cycle}.
\begin{lemma} \label{divbym} Let $G=(V,E)$ be a Kautz graph $\text{Kautz}_{n+1}(m)$ or a de Bruijn graph $DB_{n+1}(m)$, where $n\in \mathbb{N}$. The first $c=|V|/m$ invariant factors of $L(G)$ are relatively prime to $m$, and all of the rest are divisible by $m$. \end{lemma}
\begin{proof} We reduce the Laplacian $L(G)$ over the principal ideal domain $\mathbb{Z}_m$. Let the invariant factors of $L(G)$ over $\mathbb{Z}$ be $x_1,x_2,\ldots x_{|V|}$. Any invertible row or column operation over $\mathbb{Z}$ descends to an invertible operation over $\mathbb{Z}_m$, so the invariant factors of $L(G)$ over $\mathbb{Z}_m$ are the $x_i$ mod $m$.
Let $(v_1,v_2,\ldots v_c)$ be the cycle in $G$ from Lemma~\ref{cycle}, and let $[v_i]$ be the set of vertices in the class of $v_i$. We take indices mod $c$, so $v_{c+1}$ is $v_1$.
Note that if $u$ and $v$ are vertices in the same class, then $(u,w)$ is an edge if and only if $(v,w)$ is, for all $w$. Therefore, the rows of $u$ and $v$ in the adjacency matrix $A(G)$ are the same.
Because every vertex of $G$ has outdegree $m$, $L(G)\equiv A(G)$ mod $m$. Therefore rows $R(u)$ and $R(v)$ are congruent mod $m$ if $u$ and $v$ are in the same class. We reduce the Laplacian in $c$ stages. In the $i$th stage, we subtract row $R(v_i)$ from $R(v)$ for all $v\in [v_i]\backslash v_i$. After this operation, every entry of $R(v)$ is divisible by $m$.
{\small \[\mbox{\bordermatrix
& 01 & 02 & 10 & 12 & 20 & 21 \cr
01 & -2 & 0 & 1 & 1 & 0 & 0 \cr
02 & 0 & -2 & 0 & 0 & 1 & 1 \cr
10 & 1 & 1 & -2 & 0 & 0 & 0 \cr
12 & 0 & 0 & 0 & -2 & 1 & 1 \cr
20 & 1 & 1 & 0 & 0 & -2 & 0 \cr
21 & 0 & 0 & 1 & 1 & 0 & -2 \cr}} \rightarrow \mbox{\bordermatrix
& 01 & 02 & 10 & 12 & 20 & 21 \cr
01 & -2 & 0 & 1 & 1 & 0 & 0 \cr
02 & 0 & -2 & 0 & 2 & 0 & 0 \cr
10 & 0 & 0 & -2 & 0 & 2 & 0 \cr
12 & 0 & 0 & 0 & -2 & 1 & 1 \cr
20 & 1 & 1 & 0 & 0 & -2 & 0 \cr
21 & 2 & 0 & 0 & 0 & 0 & -2 \cr}} \] }
\begin{center} \textit{Figure 5.1. Reducing $L(\text{Kautz}_2(2))$ using the cycle $(01,12,20)$. The original Laplacian is on the left. We obtain the reduced Laplacian on the right by subtracting $R(01)$ from $R(21)$, $R(12)$ from $R(02)$, and $R(20)$ from $R(10)$. Every entry of rows $R(02)$, $R(10)$, and $R(21)$ is divisible by 2.}\end{center}
The entry $L(v_i,v_{i+1})$ is 1 before and after these row operations, for $1\le i\le c$. We claim that in the reduced Laplacian, every entry of $C(v_{i+1})$ is divisible by $m$ except $L(v_i,v_{i+1})$. There are $m$ edges with target $v_{i+1}$ in $G$. The sources of these edges are the $m$ vertices in $[v_i]$. After the row operations, every entry of $R(v)$ is divisible by $m$ for $v\in [v_i]\backslash v_i$, so $L(v_i,v_{i+1})$ is the only entry in $C(v_{i+1})$ which is non-zero mod $m$, for $1\le i\le c$.
By permuting rows and columns, we move $L(v_i,v_{i+1})$ to the $i$th diagonal entry $L_{ii}$ of the Laplacian. The reduced Laplacian is now in the form
\[\begin{pmatrix}I_c & A \\ 0 & 0\end{pmatrix}\ \text{mod } m\]
where $I_c$ is the $c\times c$ identity matrix. Using column operations, we can make all the entries in A divisible by $m$, without changing the rest of the matrix mod $m$. After this column operations, the Laplacian is in Smith normal form.
The first $c$ invariant factors of $L(G)$ over $\mathbb{Z}_m$ are 1, so the first $c$ invariant factors of $L(G)$ over $\mathbb{Z}$ are relatively prime to $m$. The last $|V|-c$ invariant factors of $L(G)$ over $\mathbb{Z}_m$ are 0, so the last $|V|-c$ invariant factors of $L(G)$ over $\mathbb{Z}$ are divisible by $m$. The lemma holds.
\end{proof}
We use Lemmas~\ref{cycle} and \ref{divbym} to characterize the critical group of the Kautz and de Bruijn graphs. The first step is finding the orders of these groups. We apply Theorem~\ref{thm1.1} to $DB_n(m)$, and we let all the variables $x_e$ equal 1, to find that
\[\kappa\left(DB_{n+1}(m)\right)=\kappa\left(DB_n(m) \right)\left( m^{(m-1)m^n}\right)\]
The number of spanning trees of the complete graph $DB_1(m)$ is $m^{m-1}$ \cite{Ho}. By a simple induction, we have
\[\kappa\left(DB_n(m)\right)=m^{m^n-1}\]
In an Eulerian graph, the sandpile groups $K(G,v)$ are isomorphic for all vertices $v$, so $|K(G)|=\kappa(G)/|V|$. Therefore, we have
\begin{equation}\label{dborder} |K\left(DB_n(m)\right)|=m^{m^n-n-1}\end{equation}
Similarly, we have
\[\kappa\left(\text{Kautz}_n(m)\right)=(m+1)^mm^{\left(m^n-1\right)(m+1)}\]
\begin{equation}\label{order}|K\left(\text{Kautz}_n(m)\right)|=(m+1)^{m-1}m^{\left(m^n+m^{n-1}-m-n\right)}\end{equation}
We are ready to prove theorems~\ref{db} and ~\ref{kautz}.
\begin{proof}[\textbf{Proof of Theorem~\ref{db}}] We proceed by induction on $n$. The critical group of the complete graph on $m$ vertices is $(\mathbb{Z}_m)^{m-2}$, so the base case holds.
Assume that Theorem~\ref{db} holds for $n-1$, where $n>1$. We prove it for $n$. As shown by Levine \cite{Le}, if $G$ is a balanced $k$-regular graph, then
\begin{equation} \label{homo} k K(\mathcal{L} G) \cong K(G)\end{equation}
We will use this fact to determine $Syl_p(K(DB_n(m)))$, the Sylow-$p$ subgroup of $K(DB_n(m))$, for any prime $p$. We break into two cases: either $p$ does not divide $m$, or $p$ divides $m$.
If $p$ does not divide $m$, then by Eq. (\ref{homo}), we have
\[\text{Syl}_p\left(K\left(DB_n(m)\right)\right)\cong \text{Syl}_p\left(K\left(DB_{n-1}(m)\right)\right)\]
By the inductive hypothesis, $\text{Sylow}_p(K(DB_n(m)))$ is the trivial group.
Now let $p$ be a prime that divides $m$, and suppose $p^k$ is the largest power of $p$ that divides $m$. Let the Sylow-$p$ subgroup of $DB_n(m)$ be
\[\text{Syl}_p\left(K\left(DB_n(m)\right)\right) = \mathbb{Z}_p^{a_1}\oplus \mathbb{Z}_{p^2}^{a_2} \oplus \ldots \oplus \mathbb{Z}_{p^l}^{a_l}\]
By Lemma~\ref{divbym}, $K(DB_n(m))$ can be written as a direct sum of cyclic groups, such that the order of each group is either non-zero mod $p$ or divisible by $p^k$. Thus, $a_i=0$ for $i<k$. Further, we can derive the order of $\text{Syl}_p(K(DB_n(m)))$ from Eq. (\ref{dborder}). We find that
\begin{equation}\label{dbporder} \sum_{i=n}^l ia_i = k\left(m^n-n-1\right) \end{equation}
because the expression on the right-hand side equals the number of factors of $p$ in $m^{m^n-n-1}$. By Eq. (\ref{homo}) and the inductive hypothesis, we know that
\[p^k\text{Syl}_p\left(K\left(DB_n(m)\right)\right)=\mathbb{Z}_p^{a_{k+1}}\oplus \mathbb{Z}_{p^2}^{a_{k+2}} \oplus \ldots \oplus \mathbb{Z}_{p^{l-k}}^{a_{l}} \cong \text{Syl}_p\left(K\left(DB_{n-1}(m)\right)\right)\]
\begin{equation}\label{dbindstep} \mathbb{Z}_p^{a_{k+1}}\oplus \mathbb{Z}_{p^2}^{a_{k+2}} \oplus \ldots \oplus \mathbb{Z}_{p^{l-k}}^{a_{l}} \cong \left(\mathbb{Z}_{p^{nk}}\right)^{m-2}\oplus \bigoplus_{i=1}^{n-2}\left(\mathbb{Z}_{p^{ik}}\right)^{m^{n-2-i}(m-1)^2} \end{equation}
Eq. (\ref{dbindstep}) implies that $a_{nk}=m-2$, that $a_{(i+1)k}=m^{n-2-i}(m-1)^2$ for $1\le i \le n-2$, and that $a_i=0$ for $i>nk$ or $k\nmid i$. The only $a_i$ which we have not yet determined is $a_k$. We solve Eq. (\ref{dbporder}) for $a_k$, by moving all the other $a_{ik}$ to the right-hand side and dividing by $k$.
\[a_k = \left(m^n-n-1\right)-n(m-2)-\sum_{i=2}^{n-1} i\left(m^{n-1-i}(m-1)^2\right)\]
By evaluating the geometric series, we find that $a_k=m^{n-2}(m-1)^2$. With these values, we may write
\[\text{Syl}_p\left(K\left(DB_n(m)\right)\right)=\left(\mathbb{Z}_{p^{nk}}\right)^{m-2}\oplus \bigoplus_{i=1}^{n-1}\left(\mathbb{Z}_{p^{in}}\right)^{m^{n-1-i}(m-1)^2}\]
The Sylow-$p$ subgroups of $K(DB_n(m))$ are trivial for $p\nmid m$. Taking the direct sum of the Sylow-$p$ subgroups over primes $p$ which divide $m$, we find
\[ K\left(DB_n(m)\right)\cong \bigoplus_{p\mid m}\text{Sylow}_p\left(K\left(DB_n(m)\right)\right)\cong\left(\mathbb{Z}_{m^n}\right)^{m-2}\oplus\bigoplus_{i=1}^{n-1} \left(\mathbb{Z}_{m^i}\right)^{m^{n-1-i}(m-1)^2} \]
With this equation we complete the inductive step, as desired.
\end{proof} \vspace{.05in}
\begin{proof}[\textbf{Proof of Theorem~\ref{kautz}}] This proof is similar to the proof of Theorem~\ref{db}. Again, we induct on $n$. Because the critical group of the complete graph on $m+1$ vertices is $(\mathbb{Z}_{m+1})^{m-1}$, the base case holds.
Assume that Theorem~\ref{kautz} holds for $n-1$, where $n>1$. Using Eq. (\ref{homo}), we calculate the direct sum of the Sylow-$p$ subgroups of $K(\text{Kautz}_n(m))$ over primes $p$ which do not divide $m$, as follows
\begin{equation}\label{nodivide} \bigoplus_{p\nmid m} \text{Syl}_p\left( K\left(\text{Kautz}_n(m)\right) \right)= \bigoplus_{p\nmid m} \text{Syl}_p \left(K\left(\text{Kautz}_1(m)\right)\right) = \left(\mathbb{Z}_{m+1}\right)^{m-1} \end{equation}
Now let $p$ be a prime that divides $m$. Suppose that $p^k$ is the largest power of $p$ that divides $m$, and that the Sylow-$p$ subgroup of $K(\text{Kautz}_n(m))$ is
\[\text{Syl}_p\left(K\left(\text{Kautz}_n(m)\right)\right) = \mathbb{Z}_p^{a_1}\oplus \mathbb{Z}_{p^2}^{a_2} \oplus \ldots \oplus \mathbb{Z}_{p^l}^{a_l}\]
Lemma~\ref{divbym} implies that $a_i=0$ for $i<k$. Furthermore, we know the order of the Sylow-$p$ subgroup of $K(\text{Kautz}_n(m))$ from Eq. (\ref{order}), which implies that
\begin{equation}\label{porder} \sum_{i=k}^l ia_i = k\left(m^n+m^{n-1}-m-n\right) \end{equation}
because the expression on the right-hand side equals the number of factors of $p$ in $m^{(m^n+m^{n-1}-m-n)}$. By Eq. (\ref{homo}), we have
\[p^k\text{Syl}_p\left(K\left(\text{Kautz}_n(m)\right)\right)=\mathbb{Z}_p^{a_{k+1}}\oplus \mathbb{Z}_{p^2}^{a_{k+2}} \oplus \ldots \oplus \mathbb{Z}_{p^{l-k}}^{a_{l}} \cong \text{Syl}_p\left(K\left(\text{Kautz}_{n-1}(m)\right)\right)\]
By the inductive hypothesis, we find that $a_{(i+1)k}={m^{n-3-i}(m-1)^2(m+1)}$ for $1\le i \le n-3$, that $a_{(n-1)k}=m^2-2$, and that $a_i=0$ for $i>(n-1)k$ or $k\nmid i$.
We solve Eq. (\ref{porder}) for $a_k$. We find that $a_k=m^2-2$ if $n=2$ and that $a_k=m^n-m^{n-1}-m^{n-2}+1$ if $n>2$. Then we have
\begin{equation} \label{divide} \text{Syl}_p\left(K\left(\text{Kautz}_n(m)\right)\right)=\left(\mathbb{Z}_{p^{(n-1)k}}\right)^{m^2-2}\oplus \bigoplus_{i=1}^{n-2}\left(\mathbb{Z}_{p^{ik}}\right)^{m^{n-2-i}(m-1)^2(m+1)} \end{equation}
By taking the direct sum of Eq. (\ref{nodivide}) and Eq. (\ref{divide}) over all primes which divide $m$, we complete the inductive step and prove Theorem~\ref{kautz}, as desired.
\end{proof}\vspace{.1in}
| {
"timestamp": "2009-12-04T20:37:51",
"yymm": "0910",
"arxiv_id": "0910.3442",
"language": "en",
"url": "https://arxiv.org/abs/0910.3442",
"abstract": "The line graph LG of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix-Tree Theorem to prove a formula for the number of spanning trees of LG, and he asked for a bijective proof. In this paper, we give a bijective proof of a generating function identity due to Levine which generalizes Knuth's formula. As a result of this proof we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2^{n-1}. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine.",
"subjects": "Combinatorics (math.CO)",
"title": "Counting the spanning trees of a directed line graph",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9912886173437991,
"lm_q2_score": 0.7154239836484144,
"lm_q1q2_score": 0.7091916515654295
} |
https://arxiv.org/abs/1610.04122 | Modules and Structures of Planar Upper Triangular Rook Monoids | In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule of V. In particular, we provide a recursive formula for calculating the dimension of the cyclic module generated by v_S, and show that some dimensions are Catalan numbers, giving rise to new combinatorial identities. | \section{Introduction}
A matrix is a rook matrix if each entry is $0$ or $1$ and each row and column have at most one $1$. A rook matrix $A$ is {\it planar} or {\it order preserving} if the matrix obtained from $A$ by deleting all the zero rows and all the zero columns is an identity matrix.
The structure and representation theory of the rook monoid, consisting of all rook matrices, are intensively studied \cite{M2, S}. Herbig gives a structure and representation theory of a planar rook monoid \cite{H}. The planar upper triangular rook monoid $B_n$ consists of planar upper triangular rook matrices of size $n$.
It is natural to ask: What are the representation and structure properties of the planar upper triangular rook monoid? More specifically, how do we construct interesting modules over $B_n$, and what do irreducible $B_n$-modules look like? What is the order of $B_n$ and what are the dimensions of the modules of interest? How are the order and the dimensions related to combinatorics? What are the generators and defining relations of $B_n$? These questions are closely related to the theory of linear algebraic monoids, since it was made clear in \cite{LLC14, R1} that we are here dealing with the most familiar interesting case of planar upper triangular Renner monoids of reductive monoids. For more information on Renner monoids, see \cite{LR03, LLC14, P1, R2, S}.
In this paper we answer the questions above, and our discussion goes a little deeper, showing that the $B_n$-module properties of $V$ are dramatically different from those of $V$ as a module over the planar rook monoid.
In Section 2 after gathering basic definitions and concepts related to planar upper triangular rook monoids $B_n$, we give a new interpretation of $B_n$ using generalized reduced echelon matrices. We then calculate in Section 3 the order of $B_n$ in two different ways and show that it is a Catalan number.
Section 4 is devoted to the investigation of $B_n$-modules over a field $F$ of characteristic $0$. Let $V_k$ be a vector space over $F$ generated by a set of elements $v_S$ indexed by the $k$-subsets of $\n=\{1, ..., n\}$. Then $V_k$ is a $B_n$-module under the action (\ref{moddef}). We are particularly interested in $B_n$-submodules of $V_k$ and of $V=\bigoplus_{k=0}^n V_k$. We find that
every nonzero submodule of $V$ is completely decomposable, and that a submodule of $V$ is indecomposable if any only if it lies in some $V_k$. Furthermore, we show that every submodule of $V$ is cyclic, and that each irreducible submodule of $V$ is $1$-dimensional and is contained in all nonzero submodules of some $V_k$. We also show that any two different submodules of $V$ are not isomorphic. Moreover, we give a formula for calculating the dimension of every submodule of $V$ using the inclusion-exclusion principle. In particular, we provide a recursive formula for calculating the dimensions of the modules generated by a single basis vector, and find that some of these dimensions are Catalan numbers again, connecting to combinatorics. Viewed as $B_t$-modules with $t<n$, we are able to decompose some $B_n$-submodules of $V_k$ into indecomposable $B_t$-submodules.
Section 5 describes the generators and defining relations of $B_n$.
{\bf Acknowledgement} {We would like to thank Dr. M. Can for useful email communications and Dr. R. Koo for valuable comments.}
\section{Preliminaries}
\begin{definition}
An {\em injective partial map} $f$ of
$\mathbf{n}$ is a one-to-one map of a subset $D(f)$
of $\mathbf{n}$ onto a subset $R(f)$ of $\mathbf{n}$ where $D(f)$ is
the domain of $f$ and $R(f)$ is the range of $f$.
\end{definition}
We agree that there is a map with empty domain and range and call it 0 map. We can write an injective partial map $f$ of $\mathbf{n}$ in 2-line notation by writing the numbers $s_1,\dots, s_k$ in the top line if $D(f)=\{s_1,\dots, s_k\}$, and then below each number we write its image. Equivalently, we can represent such a map by an $n\times n$ rook matrix, where the entry in the
$i$th row and the $j$th column is 1 if the map takes $j$ to $i$, and is 0 otherwise.
For example, the map $\sigma$ given below is an injective partial map of $\mathbf{5}$,
{
\begin{eqnarray*}
\sigma&=& \left(
\begin{array}{cccc}
1 & 2 & 3 & 5 \\
1 & 2 & 4 & 5 \\
\end{array}
\right)
\\
&=& \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)~.
\end{eqnarray*}
}
\begin{definition} The {\em rook monoid} $R_n$ is the monoid of
injective partial maps from $\mathbf{n}$ to $\mathbf{n}$, whose operation is the composition of partial maps and the identity element is the identity map of $\n$.
\end{definition}
Since elements of $R_n$ are not necessarily invertible, $R_n$ is not a group. The map with empty domain and empty range behaves as
a zero element. In matrix form, the composition of $R_n$ is consistent with the usual matrix multiplication. Here is an example: for $g=\left(
\begin{array}{ccc}
2 & 3 & 4 \\
1 & 5 & 2 \\
\end{array}
\right), ~
f=\left(
\begin{array}{cccc}
1 & 3 & 4 & 5 \\
1 & 2 & 3 & 4 \\
\end{array}
\right)\in R_5,
$ we have
$$
gf=
\left(
\begin{array}{ccc}
2 & 3 & 4 \\
1 & 5 & 2 \\
\end{array}
\right)\circ
\left(
\begin{array}{cccc}
1 & 3 & 4 & 5 \\
1 & 2 & 3 & 4 \\
\end{array}
\right)
=\left(
\begin{array}{ccc}
3 &4& 5 \\
1 &5& 2 \\
\end{array}
\right)~.
$$
The corresponding matrix form of the operation reads as
$$
gf= \left(
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
\end{array}
\right) \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
\end{array}
\right)= \left(
\begin{array}{ccccc}
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
\end{array}
\right)~. $$
An injective partial map from $\mathbf{n}$ to $\mathbf{n}$ is {\em order preserving} if whenever $a<b$ in the domain of the map, then $f(a)<f(b)$. An injective partial map $f$ is order preserving if and only if the matrix obtained from the matrix form of $f$ by deleting all the zero rows and all the zero columns is an identity matrix; equivalently the graph obtained from the 2-line notation of $f$ by joining all defined $f(a)$ in the range of the map to $a$ is a planar graph, which justifies the name in the following definition.
\begin{definition}
The {\em planar rook monoid}, denoted by $PR_n$, is the monoid of {\em order preserving} injective partial maps from $\mathbf{n}$
to $\mathbf{n}$.
\end{definition}
\noindent Obviously, $PR_n$ is a submonoid of $R_n$. The structure and representation of the planar rook monoid is studied in Herbig \cite{H}. In particular, $V_k$ is an irreducible $PR_n$-module.
The next definition will give a different interpretation of an order preserving injective partial map.
\begin{definition}
A rectangular matrix is a {\em generalized (row and column) reduced echelon matrix if}
\vspace{-3mm}
\begin{enumerate}[{\rm(1)}]
\item Each leading entry of a row is $1$ and is in a column to the right of the leading entry of the row above it.
\vspace{-3mm}
\item Each leading entry of a column is $1$ and is in a row below the leading entry of the column to the left of it.
\vspace{-3mm}
\item Each leading 1 is the only nonzero entry in its column and its row.
\end{enumerate}
\end{definition}
\noindent
This definition does not require that all nonzero rows are above any zero rows nor all nonzero columns are to the left of any zero columns. Since the row and column reduced echelon form of a matrix is equivalent to the normal form of the matrix, we can consider a generalized reduced echelon matrix to be a generalization of the normal form of a matrix.
An injective partial map is order preserving if and only if its matrix form is a generalized reduced echelon matrix. Thus, the set of all the generalized reduced echelon matrices of size $n$ is a monoid with respect to the multiplication of matrices, and the order of this monoid is $\binom{2n}{n}$, since the order of $PR_n$ is
\[
|PR_n| = \binom{2n}{n}~.
\]
An injective partial map is called {\em order decreasing} if for all $a$ in the domain of the map, we have $f(a)\leq a$. Equivalently, an injective partial map is order decreasing if and only if its matrix form is an upper triangular rook matrix, which motivates the name in the following definition.
\begin{definition}
The {\em planar upper triangular rook monoid}, denoted by $B_n$, is the monoid of order preserving, order decreasing injective partial maps from $\mathbf{n}$ to $\mathbf{n}$.
\end{definition}
An injective partial map is in $B_n$ if and only if its matrix form is an upper triangular generalized reduced echelon matrix. In the previous example, we have $f\in B_5$, but $g\notin B_5$.
\section{Order of $B_n$}
We first show that the order of $B_n$ is a Catalan number, which is defined by
$c_0=c_1=1$ and $c_n=\sum_{i=0}^{n-1}c_ic_{n-1-i}$ for
$n>1$ (see \cite{St}).
\begin{proposition}\label{bnCatalan}
Let $n\ge 0$. Then the order of the planar upper triangular rook monoid $B_n$
is the Catalan number $c_{n+1}$, that is, $b_n=c_{n+1}$.
\end{proposition}
\begin{proof}
To prove the proposition, we set up a one-to-one correspondence
between the set $B_n$ and the set $C_{n+1}$ of all sequences
$a_1,a_2,\dots,a_{2n+2}$ of $n+1$ copies of 1's and $n+1$ copies of $-1$'s, such that
$a_1+a_2+\dots+a_l\geq 0$ for all $1\leq l\leq 2n+2$.
Let $f$ be an element of $B_n$
with domain $S=\{s_1<s_2<\dots<s_k\}$ and range
$T=\{t_1<t_2<\dots<t_k\}$. Define $s_1^\p=s_1, \,s_i^\p=s_i-s_{i-1}$
for $2\leq i\leq k$, and $s_{k+1}^\p=n+1-s_k$. Also define
$t_1^\p=t_1, \,t_i^\p=t_i-t_{i-1}$ for $2\leq i\leq k$, and
$t_{k+1}^\p=n+1-t_k$. Let $f^\p$ be the sequence
\begin{equation}\label{sequence}
\underbrace{1,\,\dots,\,1}_{s_1^\p},\, \underbrace{-1,\,\dots,\,-1}_{t_1^\p},\,
\underbrace{1,\,\dots,\,1}_{s_2^\p}, \underbrace{-1,\dots,-1}_{t_2^\p}, \,
\dots, \,
\underbrace{1,\,\dots,\,1}_{s_{k+1}^\p},\, \underbrace{-1,\,\dots,\,-1}_{t_{k+1}^\p} ~.
\end{equation}
Now we prove $f^\p\in C_{n+1}$. By definition of $B_n$, we have
$t_i\leq s_i$ for all $1\leq i\leq n$. Thus
$s_1^\p-t_1^\p=s_1-t_1\geq 0, \,(s_1^\p+s_2^\p+\dots+s_i^\p)
-(t_1^\p+t_2^\p+\dots+t_i^\p)=s_i-t_i\geq 0$ for $2\leq i\leq k$ and
$(s_1^\p+s_2^\p+\dots+s_{k+1}^\p)
-(t_1^\p+t_2^\p+\dots+t_{k+1}^\p)=(n+1)-(n+1)\geq 0$. Denote the
partial sum of the first $l$ items in (\ref{sequence}) by $a_l$.
Then our previous argument shows that $a_l\geq 0$ for
$l=s_1+t_1+s_2+t_2+\dots+s_h+t_h$, where $1\leq h\leq k+1$, and this
implies $a_l\geq 0$ for $1\leq l\leq 2+2n$ by the format of
(\ref{sequence}). Hence, $f^\p\in C_{n+1}$.
We next show that the mapping from $B_n$ to $C_{n+1}$ defined by
\begin{eqnarray*}
\sigma:B_n&\rightarrow& C_{n+1} \\
f &\mapsto& f^\p
\end{eqnarray*}
is bijective. From the definition of the map $\sigma$, it is straightforward to
see that $\sigma$ is injective. Now we prove the onto property of
the map $\sigma$. For arbitrary element
$f^\p=\{a_1,a_2,\dots,a_{2n+2}\}$ of $C_{n+1}$, the condition
$a_1+a_2+\dots+a_l\geq 0$ for all $1\leq l\leq 2n+2$ implies $a_1=1$
and $a_{2n+2}=-1$. As in (\ref{sequence}), let $s_1^\p$ be the number
of the first consecutive 1's in $f^\p$, let $t_1^\p$ be the number of
the consecutive $-1$'s that follow, and define similarly $s_i^\p$ and
$t_i^\p$ as before for $2\leq i\leq k+1$. Let
$s_1=s_1^\p,\, t_1=t_1^\p,\, \,s_i=s_1^\p+s_2^\p+\dots+s_i^\p$, and $t_i=t_1^\p+t_2^\p+\dots+t_i^\p$ for $2\leq i\leq k$. Then the condition $a_1+a_2+\dots+a_l\geq 0$ for all
$1\leq l\leq 2n+2$ implies $t_i\leq s_i$ for $1\leq i\leq k$.
Define $f$ to be the mapping $f(s_i) = t_i$ with domain $s_1,s_2,\dots,s_k$
and range $t_1,t_2,\dots,t_k$. Then
$f\in B_n$ and $\sigma(f)=f^\p$. Hence $\sigma$
is a one-to-one correspondence between the set $B_n$ and the set
$C_{n+1}$. It is well known that the order of $C_{n+1}$ is the
Catalan number $c_{n+1}$, so the proposition is proved.
\end{proof}
\begin{remark}
{\rm
Clearly $B_n$ is a submonoid of the monoid $\mathscr B_n$ consisting of all order decreasing (not necessarily planar) injective partial maps. See \cite{BRR} for the order of $\mathscr B_n$.
}
\end{remark}
The following corollary is immediate.
\begin{corollary}
The number of upper triangular generalized reduced echelon matrices of size $n$ is the Catalan number $c_{n+1}$.
\end{corollary}
Our next proposition provides a recursive formula for calculating the order $b_n$ of
\begin{equation*}\label{bn}
B_n=\{f\in R_n \mid f(j)\leq j \;\mathrm{for}\; j\in D(f);\, f(i)<f(j)
\;\mathrm{for}\; i,j\in D(f) \;\mathrm{and}\; i<j\}~.
\end{equation*}
For $0\leq p,\,q\leq n-1$, let $b_{p,\,q}=|B_{p,\,q}|$ where
\[
B_{p,\,q}=\{f\in B_n \mid D(f)\subseteq \{n-q,\, \dots,\,n\} \text{ and }R(f)\subseteq \{n-p,\, \dots,\, n\}\}~.
\]
\begin{proposition}\label{bnre} Let $n\ge 1$. Then
\vspace{-2mm}
\begin{enumerate}[{\rm(1)}]
\item $b_0=1$,\, $b_1=2$~.
\vspace{-2mm}
\item $b_{p,\,0}=p+2$\hspace{3.5cm} for $0\le p\le n-1$~.
\vspace{-2mm}
\item $b_{p,\,p}=b_{p+1}$\quad\quad\quad\quad\quad\quad\quad\quad\quad\, for $0\le p\le n-1$~.
\vspace{-2mm}
\item $b_n = 2b_{n-1} + 1 + \sum_{q=0}^{n-3}b_{n-2,\,q}$\quad\, for $n\geq 2$~.
\vspace{-2mm}
\item $b_{p,\,q}=1+\sum_{r=0}^q b_{p-1,\,r}$\quad\quad\quad\quad\, for $1\leq q<p\leq n-1$~.
\end{enumerate}
\end{proposition}
\begin{proof}
Parts (1), (2), and (3) are clear. To prove (4), divide the elements of $B_n$ into two groups: the elements whose ranges contain $1$, and those whose ranges do not contain $1$. Part (4) follows from the following three identities:
\vspace{-2mm}
\begin{align*}
b_{n-1}&=|\{f\in B_n \mid f(1)=1\}| = |\{f\in B_n \mid 1\notin R(f)\}|~. \\
b_{n-2,\,q}&=|\{f\in B_n\mid f(n-1-q)=1\}|\quad \text{for}\quad 0\leq q\leq n-3 ~. \\
1 &= |\{f\in B_n\mid f(n)=1\}|~.
\end{align*}
Similarly, for part (5), we divide the elements of $B_{p, q}$ into two groups: the elements whose ranges contain $n-p$, and those whose ranges do not contain $n-p$. Part (5) follows from the following three identities:
\begin{align*}
b_{p-1,\,q}&=|\{f\in B_{p,q}\mid n-p\notin R(f)\}|~.\\
b_{p-1,\,r}&=|\{f\in B_{p,q}\mid f(n-1-r)=n-p\}|\quad\text{for}\quad 0\leq r\leq q-1~.\\
1 &=|\{f\in B_{p,q}\mid f(n)=n-p\}|~.
\end{align*}
\end{proof}
\section{Modules for $B_n$}
A vector space $V$ over a field $F$ of characteristic $0$ is called a $B_n$-module if $B_n$ acts on $V$ satisfying, for all $f, f_1, f_2\in B_n$, $u, v\in V$, and $\lambda\in F$,
\begin{eqnarray*}
f\cdot (u+v) &= f\cdot u + f\cdot v, \quad\quad\quad f_1\cdot (f_2\cdot u) &= (f_1f_2)\cdot u, \\
f\cdot (\lambda u) &= \lambda (f\cdot u),~\quad\quad\quad\quad\quad\quad\quad 1\cdot u &= u.
\end{eqnarray*}
From now on, $V$ denotes a vector space with a basis
$
\mathcal{B} = \{v_S\mid S\subseteq\n\}
$
indexed by all the subsets of $\n$. Then $V=\bigoplus_{S\subseteq\n} Fv_S$ as subspaces is a $B_n$-module with respect to the following action: for $f\in B_n$ and $S\subseteq\n$,
\begin{equation}\label{moddef}
f\cdot v_S=
\left\{
\begin{array}{ll}
v_{S^\p}, & \hbox{if\; $S\subseteq D(f)$} \\
0, & \hbox{otherwise,}
\end{array}
\right.
\end{equation}
where $S^\p=\{f(s_1),\dots,f(s_k)\}$ if $S=\{s_1,\dots,s_k\}$. For $0\le k \le n$, let
$$
V_k=\mathrm{span}\{v_S\in\mathcal{B}\mid k=|S|\}.
$$
Then $V=\bigoplus^n_{k=0}V_k$ is a direct sum of $B_n$-submodules.
Every module under consideration is a $B_n$-module over $F$, unless otherwise stated.
Our intent below is to describe the $B_n$-module structure of $V_k$ and $V$. To this end, we
define a partial order on the power set of $\n$. For any $k$-subsets
$S=\{s_1<\dots<s_k\}$ and $T=\{t_1<\dots<t_k\}$ of $\n$, define
\[
T\leq S \quad \Leftrightarrow \quad t_i\leq s_i \quad\text{for all}\quad i\in \mathbf{k}~,
\]
and a $k$-subset is not comparable to any $l$-subset if $k\ne l$.
For $v\in V$ we use $B_nv$ to denote the cyclic submodule of $V$ generated by $v$. If $S$ is a $k$-subset of $\n$, then $B_nv_S$ is a submodule of $V_k$. Indeed, for any $f\in B_n$ if $S\subseteq D(f)$ then $f(S)$ is a $k$-subset, so $f\cdot v_S = v_{f(S)}\in V_k$; if $S$ is not a subset of $D(f)$ then $f\cdot v_S = 0\in V_k$. Some further properties of the module $B_nv_S$ are described in the next result.
\begin{lemma}\label{mod1} Let $S,T$ be $k$-subsets of $\n$.
{\rm(1)} $B_nv_S = \bigoplus_{S'\subseteq\n,\, S'\leq S}Fv_{S'}$ as vector spaces. In particular, $V_k=B_nv_{\{n-k+1,\, \ldots,\, n\}}$.
{\rm(2)} $B_nv_T\subseteq B_nv_S$ if and only if $T\leq S$.
{\rm(3)} $B_nv_S\cap B_nv_T = B_nv_{S\wedge T}$, where $S\wedge T$ is the greatest lower bound of $S$ and $T$.
\end{lemma}
\begin{proof}
To prove (1) notice that two subsets $S'\leq S$ if and only if $S=D(f)$ and $S'=R(f)$ for a unique $f\in B_n$. Let $S'\le S$. Then $v_{S'} = f\cdot v_S\in B_nv_S$. Hence $\bigoplus_{S'\subseteq\n,\, S'\leq S}Fv_{S'}$ is included in $B_nv_S$. Conversely, let $x=g\cdot v_S \ne 0$ for some $g\in B_n$. We have $S\subseteq D(g)$, $g(S)\le S$, and hence $x=v_{g(S)}\in \bigoplus_{S'\subseteq\n,\, S'\leq S}Fv_{S'}$. The second part of (i) is now clear.
The proof of (2) follows from (1) since $\{T'\mid T'\subseteq\n,\, T'\leq T\}\subseteq\{S'\mid S'\subseteq\n,\, S'\leq S\}$ if and only if $T\le S$.
To prove (3) let $g\cdot v_S = h\cdot v_T\ne 0$ for some $g, h\in B_n$. Then $g(S)=h(T)$. Suppose
\[
S = \{s_1< \ldots < s_k\}\quad\text{and}\quad T = \{t_1< \ldots < t_k\}~.
\]
Then
$
S \wedge T =\{\min(s_1, t_1),\, \ldots,\, \min(s_k, t_k)\},
$
and $g(s_i)=h(t_i)$. We define $f\in B_n$ with $D(f)=S \wedge T $ and $R(f)=g(S)$ by
$
f(\min(s_i,\, t_i)) = g(s_i),
$
where $1\le i\le k$. Then $g\cdot v_S = f\cdot v_{S\wedge T} \in B_n v_{S\wedge T}$, and hence $B_nv_S\cap B_nv_T \subseteq B_nv_{S\wedge T}$.
Conversely, for any given $0\ne f\cdot v_{S\wedge T} \in B_n v_{S\wedge T}$ define $g(s_i)=h(t_i)= f(\min(s_i,\, t_i))$ for $1\le i\le k$. Then $f\cdot v_{S\wedge T} = g\cdot v_S=h\cdot v_T\in B_nv_S\cap B_nv_T$. The proof of (3) is complete.
\end{proof}
Let $v=\sum_{S\subseteq \n}\lambda_Sv_S, \lambda_S\in F$ be a vector of $V$. The {\em support} of $v$ is defined to be
\[
{\rm supp}(v) = \{S\subseteq\n\mid \lambda_S \ne 0\}~.
\]
\begin{definition}\label{redGen}
A vector of the form $w=\sum_{S\in {\rm supp}(w)}v_S\in V$ is called a {\em reduced generator} of a submodule $W$ of $V$ if $W=B_nw$ and $W$ cannot be generated by any other vector whose support contains fewer elements than {\rm supp(}$w${\rm )}. We agree that $0$ is the reduced generator of the zero submodule.
\end{definition}
The next proposition gives some properties of submodules of $V$.
\begin{proposition}\label{cyclic} Let $v=\sum_{S\in\,{\rm supp(}v{\rm )}}\lambda_Sv_S\in V$.
{\rm (1)} If $S$ is in {\rm supp}$(v)$, then $v_S\in B_nv$~.
{\rm (2)} $B_nv = \bigoplus_{T\in \mathcal{P}(v)} Fv_T$ as subspaces, where $\mathcal{P}(v) = \bigcup_{S\in {\rm supp}(v)}\{T\subseteq\n\mid T\le S\}$.
{\rm (3)} Every submodule of $V$ is cyclic and contains a unique reduced generator.
\end{proposition}
\begin{proof} To prove (1) let $\min \big\{\,|S| \,\big|\, S \in {\rm supp}(v)\big\}=r$. Then there exists an $r$-subset $T=\{t_1<\cdots < t_r\}\subseteq\n$ such that $T\in {\rm supp}(v)$; if $r=0$, then $T=\emptyset$. Let $f\in B_n$ such that $D(f)=R(f)=T$. By the choice of $r$, for every $S\in{\rm supp}(v)$ with $S\neq T$, there is at least one $s\in S$ such that $s\notin T$, so $f\cdot v_S=0$. Hence
$$
f\cdot v=f\cdot \sum_{S\in\,{\rm supp(}v{\rm )}}\lambda_Sv_S=\sum_{S\in\,{\rm supp(}v{\rm )}}\lambda_S(f\cdot v_S)=\lambda_Tv_T~.
$$
Thus $v_T\in B_nv$ since $\lambda_T\neq 0$. It is easily seen that
$$
\sum_{S\in\,{\rm supp(}v{\rm )}\atop |S|>r}\lambda_Sv_S=v-\sum_{S\in\,{\rm supp(}v{\rm )}\atop |S|=r}\lambda_Sv_S\in B_nv~.
$$
Applying the above procedure to
$
\sum_{S\in\,{\rm supp(}v{\rm )},\,|S|>r}\lambda_Sv_S
$
and iteratively using this procedure, if needed, we get $v_S\subseteq B_nv$ for all $S\in \text{supp}(v)$. The proof of (1) is complete.
From (1) and Lemma \ref{mod1} (1), we have
\begin{eqnarray*}
B_nv&=& \sum_{S \in\text{\rm supp}(v)}\lambda_SB_nv_S \\
&=&\sum_{S \in\text{\rm supp}(v)}\mathrm{span}\,\{v_T\in\mathcal{B}\mid T\leq S\}\\
&=& \bigoplus_{S\in \mathcal{P}(v)} Fv_S, \quad\text{as subspaces}.
\end{eqnarray*}
This completes the proof of (2).
We now prove (3). It is trivial for $W=\{0\}$. Let W be a nonzero submodule of $V$. We claim that $W$ has a basis $\{v_S\in\mathcal{B}\mid S\in \mathcal{P}\}$ for some subset $\mathcal{P}$ of the power set of $\n$. Indeed, suppose $\mathcal{B}_1$ is a basis of $W$ and write every element of $\mathcal B_1$ as a linear combination of basis vectors in $\mathcal{B}=\{v_S\mid S\subseteq \n\}$. Let $\mathcal P$ be the set of all the different subsets $S$ where $S$ runs through the support of every element of $\mathcal B_1$. By (1) the set $\{v_S\in\mathcal{B}\mid S\in \mathcal{P}\}$ is a subset of $W$, and hence a basis of $W$ since it is linearly independent and spans $W$. Let
$
w=\sum_{S\in\mathcal{P}} v_S.
$
By (1) again, $W$ is generated by $w$, and hence $W$ is cyclic.
We now show how to deduce a reduced generator of $W$ from $w$. Indeed, if $w$ contains two vectors $v_S, \,v_T$ with $T\le S$ and $T\ne S$ in supp($w$), then we can remove the term $v_T$ from $w$, and by Lemma \ref{mod1} (i) the sum of the remaining terms is still a generator. Repeat this process until we obtain the set
\[
{\rm Red}(w) = \{S\mid S \;\text{is maximal in supp}(w)\},
\]
and then we define the corresponding generator $w_{\rm red}$ of $W$ by
\[
w_{\rm red}=\sum_{S\in{\rm Red}(w)}v_S~.
\]
We claim that $w_{\rm red}$ is a reduced generator of $W$. Let $v = \sum_{S\in\text{supp}(v)}\lambda_Sv_S$ be another generator of $W$. From Definition \ref{redGen} it suffices to show that $|{\rm supp(}v{\rm )}| \ge |{\rm Red(}w{\rm )}|$.
From (2) we find $W = \bigoplus_{T\in \mathcal{P}(v)} Fv_T = \bigoplus_{T\in \mathcal{P}(w)} Fv_T$ where $\mathcal{P}(v)$ and $\mathcal{P}(w)$ are as in (2),
and hence
$
\mathcal{P}(v) = \mathcal{P}(w).
$
Define
\begin{equation}\label{redv}
{\rm Red}(v) = \{S\mid S \text{ is maximal in supp}(v)\}.
\end{equation}
Thus, ${\rm Red}(v) = \{S\mid S \text{ is maximal in }\mathcal{P}(v)\}$ and
${\rm Red}(w) = \{S\mid S \text{ is maximal in }\mathcal{P}(w)\}$.
So, Red($v$) = Red($w$) and $|{\rm supp(}v{\rm )}| \ge |{\rm Red(}v{\rm )}| = |{\rm Red(}w{\rm )}|$, showing that $w_{\rm red}$ is reduced.
Suppose that $v = \sum_{S\in{\rm supp}(v)}v_S$ is another reduced generator of $W$. By the definition of reduced generators we know $|{\rm supp(}v{\rm )}| = |{\rm Red(}w{\rm )}|$. Hence $|{\rm supp(}v{\rm )}| = |{\rm Red(}v{\rm )}|$ since Red($v$) = Red($w$). It follows that ${\rm supp(}v{\rm )} = {\rm Red(}v{\rm )}$. Let $v_{\rm red}=\sum_{S\in{\rm Red}(v)}v_S$. Then $v=v_{\rm red}=w_{\rm red}$. Therefore $w_{\rm red}$ is the unique reduced generator of $W$.
\end{proof}
\begin{definition} The set ${\rm Red}(v)$ in {\rm (\ref{redv})} is called the {\em reduced support} of $v$, and the element $v_{\rm red}=\sum_{S\in{\rm Red}(v)}v_S$ is termed the {\em reduced form} of $v$. The reduced support of $0$ is empty, and the reduced form of $0$ is itself.
\end{definition}
For example, if $n=7$ and $v = v_\emptyset - 2v_{\{1\}} + v_{\{3\}} + 5 v_{\{1, \,2\}} + 3v_{\{4,\, 7\}} - 2v_{\{5, \,6\}} + v_{\{1, \,2, \,3\}}$, then Red($v$) = $\{\emptyset,\,\{3\},\,\{5,\, 6\},\,\{4,\, 7\}, \{1, \,2, \,3\}\}$ is the reduced support of $v$, and its reduced form is $v_{\rm red} = v_\emptyset + v_{\{3\}} + v_{\{4,\, 7\}} + v_{\{5, \,6\}} + v_{\{1, \,2, \,3\}}$.
It is sometimes convenient to call the reduced support of $v$ the {\em reduced support} of the module $B_nv$.
A direct calculation yields that the reduced generator of $V_k$ is $v_{\{n-k+1,\,\ldots,\,n\}}$ for $1\le k\le n$, and the reduced support of $V_k$ is the set $\{n-k+1,\,\ldots,\,n\}$. The module $V_0$ has the element $v_\emptyset$ as its reduced generator, and its reduced support is the set $\{\emptyset\}$.
The next result is a consequence of Lemma \ref{mod1} (i) and Proposition \ref{cyclic} (3).
\begin{corollary}\label{eq}
If $v, w\in V$, then $B_n v = B_n w$ if and only if they have the same reduced support {\rm Red(}v{\rm)} = {\rm Red(}w{\rm)} if and only if they have the same reduced generator $v_{\rm red} = w_{\rm red}$.
\end{corollary}
We can now describe the irreducible submodules of $V_k$ for $0\le k\le n$. Write $\bold k = \{1,\,\dots,\,k\}$. If $k=0$, we agree that $\bold k=\emptyset$ and $v_\bold k = v_\emptyset$.
\begin{proposition}\label{irreVk}
For each $0\le k\le n$, the 1-dimensional submodule $B_nv_{\bold k}$ is the only irreducible submodule of $V_k$, and every nonzero submodule of $V_k$ contains $B_nv_{\bold k}$.
\end{proposition}
\begin{proof} Since ${\bold k}$ is the smallest element of the set of all $k$-subsets and the elements of $B_n$ are order decreasing as well as order preserving injective maps, from action (\ref{moddef}) we find $B_nv_{{\bold k}}=Fv_{\bold k}$ is an irreducible submodule of $V_k$, and it is $1$-dimensional.
If $W$ is another nonzero irreducible submodule of $V_k$, by Proposition \ref{cyclic} (3) there exists a generator $w\in V_k$ such that $W = B_nw$. The irreducibility of $W$ forces that supp($w$) contains only the $k$-subset ${\bold k}$, since if supp($w$) contains another $k$-subset $S$ different from ${\bold k}$, then by Lemma \ref{mod1} (1), $v_S\in W\setminus B_nv_{\bold k}$ and hence $B_nv_{\bold k}$ would be a nonzero proper submodule of $W$. We conclude $W=B_nv_{{\bold k}}$.
Now let $W$ be any nonzero submodule of $V_k$. From Proposition \ref{cyclic} (3) we know that $W$ is generated by a nonzero element $v\in V_k$. Pick any $S\in$ supp($v$). Then $v_S\in W$ by Proposition \ref{cyclic} (1). Since there exists a unique map $f\in B_n$ such that $D(f)=S$ and $R(f)={\bold k}$, we find $v_{\bold k}=f\cdot v_S\in B_nv_S\subseteq W$. Therefore $B_nv_{\bold k}\subseteq W$.
\end{proof}
The next result describes irreducible submodules of $V$ in terms of those of $V_k$.
\begin{proposition}\label{irreV}
If $W$ is an irreducible submodule of $V$, then $W=B_nv_{{\bold k}}$ for some $0\le k\le n$ and $\dim W =1.$ Moreover, $\{B_nv_{{\bold k}} \mid k = 0,\,\ldots,\, n\}$ is a complete set of irreducible submodules of $V$.
\end{proposition}
\begin{proof}
From Proposition \ref{cyclic} (3) we find $W = B_nw$ for a reduced generator $w\in V$. Since $W$ is nonzero, supp($w$) is not empty. Assume that $S,\,T\in$ supp($w$) and $S\ne T$. Since $S,\,T$ are different maximal elements in supp($w$), from Lemma \ref{mod1} (i) we find $v_T\in W\setminus B_nv_S$, and hence $B_nv_S$ is a nonzero proper submodule of $W$, which contradicts the irreducibility of $W$. Therefore, supp($w$) contains only one subset of $\n$, showing that $W$ is a submodule of $V_k$ for some $0\le k\le n$. It follows from Proposition \ref{irreVk} that $W=B_nv_{{\bold k}}$ and $\dim W =1$. The second part of the proposition is now straightforward.
\end{proof}
Recall that a $B_n$-module is {\em indecomposable} if it is nonzero and cannot be written as a direct sum of two nonzero submodules, and that a $B_n$-module is called {\em completely decomposable} if it is nonzero and is a direct sum of indecomposable submodules.
\begin{proposition}\label{indDecom}
Let $W$ be a nonzero submodule of $V$. Then $W$ is indecomposable if and only if $W$ is a submodule of some $V_k$ where $0\le k\le n$.
\end{proposition}
\begin{proof}
If $W$ is a nonzero submodule of $V_k$ where $0\le k\le n$, then by Proposition \ref{irreVk} any two nonzero submodules of $W$ both contain $B_nv_{{\bold k}}$, so their sum cannot be direct, and hence $W$ is indecomposable.
Conversely, if $W$ is a nonzero indecomposable submodule of $V$, from Proposition \ref{cyclic} (3) it follows that $W=B_nv$ for a unique reduced generator $v = \sum_{S\in {\rm Red}(v)}v_S\in V$. Let $\mathcal{P}(i)$ be the set of all $i$-subsets of $\n$ where $0\le i\le n$. For each $i$ let Red$_i(v) = {\rm Red(}v{\rm)}\cap \mathcal{P}(i)$. Forgetting all the possible empty Red$_i(v)$, we obtain a partition of
\[
{\rm Red}(v) = {\rm Red}_{i_1}(v) \sqcup \cdots \sqcup {\rm Red}_{i_s}(v), \quad\text{for some } 1\le s\le n+1,
\]
where $0\le i_1 < \cdots < i_s \le n.$ Let $v_{i_j} = \sum_{S\in {\rm Red}_{i_j}(v)} v_S$ be the reduced vector with support Red$_{i_j}(v)$ where $1\le j\le s$. Then
\begin{equation}\label{comDom}
W = \bigoplus_{j=1}^{s} B_nv_{i_j}~, \quad\text{direst sum of submodules}.
\end{equation}
Since $W$ is indecomposable and each $B_nv_{i_j}$ is a nonzero proper submodule of $W$, there exists some $k=i_j$ such that $W = B_nv_k$, which is a submodule of $V_k$.
\end{proof}
\begin{corollary}
Every nonzero submodule of $V$ is completely decomposable. In particular, $V$ is completely decomposable and $V=\bigoplus_{k=0}^{n}V_k$ is a direct sum of indecomposable submodules.
\end{corollary}
\begin{proof}
Let $W$ be a nonzero submodule of $V$. Then from Proposition \ref{cyclic} (3) we have $W=B_nv$ for a unique reduced generator $v = \sum_{S\in {\rm Red}(v)}v_S\in V$. A similar argument to that of Proposition \ref{indDecom} shows that $W$ has the decomposition (\ref{comDom}). From Proposition \ref{indDecom} each $B_nv_{i_j}$ in (\ref{comDom}) is indecomposable. The first part of the desired result follows. The second part follows immediately.
\end{proof}
\begin{proposition}
No two different submodules of $V$ are isomorphic.
\end{proposition}
\begin{proof}
Let $W$ and $U$ be two submodules of $V$ and $\sigma:W\rightarrow U$ be a module isomorphism. Let $x=\sum_{S\in\, \mathcal I} v_S$ be a reduced generator of $W$, where $\mathcal I$ is an index set. Now for $v_S,S\in \mathcal I$, suppose $\sigma(v_S)=\sum_{T\in\, \mathcal J} \lambda_T v_T$ for some index set $\mathcal J$, where $\lambda_T\in F$. Take $f_S\in B_n$ with $D(f)=R(f)=S$. Then
\[
\sigma(v_S)=\sigma(f_S\cdot v_S)=f_S\cdot\sigma(v_S)=f_S\cdot\sum_{T\in \mathcal J} \lambda_T
v_T=\sum_{T\in \mathcal J,\,T\subseteq S} \lambda_T (f_S\cdot v_T)=\sum_{T\in
\mathcal J,\,T\subseteq S} \lambda_T v_T.
\]
We show that $\sigma(v_S)=\lambda_Sv_S$ with $\lambda_S\neq 0$. If in the sum on the right there is some $T^\p\in \mathcal J,\,T^\p\subsetneq S$ with $\lambda_{T^\p}\neq 0$, let $f_{T^\p}\in B_n$ with $D(f_{T^\p})=R(f_{T^\p})=T^\p$. We have
\[
f_{T^\p}\cdot\sigma(v_S) = f_{T^\p}\cdot\sum_{T\in \mathcal J,\,T\subseteq S}\lambda_T v_T=\sum_{T\in \mathcal J,\,T\subseteq S}\lambda_T (f_{T^\p}\cdot v_T)=\sum_{T\in \mathcal J,\,T\subseteq T^\p}\lambda_T v_T\ne 0,
\]
but $f_{T^\p}\cdot\sigma(v_S) = \sigma(f_{T^\p}\cdot v_S) =\sigma(0)=0$, a contradiction. Thus we get $\sigma(v_S)=\lambda_Sv_S$ and $\lambda_S\neq 0$, so $B_nv_S=B_n\sigma(v_S)$. By Proposition 4.3 (1), we find
\[
W=B_nx=\sum_{S\in \mathcal I} B_nv_S=\sum_{S\in \mathcal I}B_n\sigma(v_S)=\sigma\Big(\sum_{S\in \mathcal I}B_nv_S\Big)=\sigma\Big(B_n\big(\sum_{S\in \mathcal I}v_S\big)\Big)=\sigma(B_nx)=U.
\] \end{proof}
We now describe the dimension of any nonzero submodule of $V$. Proposition \ref{cyclic} (3) assures that the submodule is equal to the module $B_nv$ generated by some $v\in V$.
\begin{proposition}
Let $v\in V$ and {\rm Red(}v{\rm )}= $\{S_1,\,\ldots,\,S_m\}$. For any $J\subseteq$ {\rm Red(}v{\rm )} denote by $S_J$ the greatest lower bound of $\{S_j\mid j\in J\}$. Then the dimension of $B_nv$ is given by
\[
\dim B_nv = \sum_{\emptyset\,\ne J\,\subseteq {\bold m}} (-1)^{|J|-1} \dim B_nv_{S_J}.
\]
\end{proposition}
\begin{proof}
From Proposition \ref{cyclic} (2) and (3) the dimension of $B_nv$ is equal to the cardinality of the set
$\mathcal{P}(v) = \bigcup_{S\in {\rm Red(}v{\rm )}}\{T\subseteq\n\mid T\le S\}$. Let $A_j=\{T\subseteq\n\mid T\le S_j\}$, $j\in{\bold m}$. Then $\mathcal{P}(v) = \bigcup_{j\in{{\bold m}}}A_j$, and $\dim B_nv_{S_j} = |A_j|$ by Proposition \ref{cyclic} (2). With Lemma \ref{mod1} (3) in mind and applying the inclusion-exclusion principle to count the cardinality of $\mathcal{P}(v)$, we obtain the desired formula for $\dim B_nv$.
\end{proof}
We further describe the dimension of $B_nv_S$ for any $S\subseteq\n$. In what follows we agree that if $x>y$, then $\binom{y}{x}=0$ and the empty sum $\sum_{i=x}^y\square_i = 0$.
\begin{theorem}\label{dim}
If $S=\{s_1<\dots<s_k\}$ is a $k$-subset of $\n$, let $d_k$ be the dimension of the module $B_nv_S$. We have $d_1 = s_1$, and for $k\ge 2$,
\begin{align}
d_k &= \sum^{k-1}_{i=1}{s_{k-i+1} \choose k+1-i}\gamma_i -\sum^{k-1}_{i=1}{s_{k-i+1}-s_1 \choose k+1-i}\gamma_i-\sum^{k-2}_{i=1}s_1 {s_{k-i+1}-s_2 \choose k-i}\gamma_i ~, \label{dk}
\end{align}
where $\gamma_1=1$ and for $2\leq j\leq k-1$,
\begin{equation*}\label{}
\gamma_j=-\sum^{j-2}_{i=1}
{s_{k+1-i}-s_{k+2-j} \choose j-i}\gamma_i ~.
\end{equation*}
\end{theorem}
\begin{proof}
By Lemma \ref{mod1} (i) we know that $d_k$ is equal to the number of $k$-subsets $T$ of $\n$ such that $T\leq S$. Let
$
\la_i=s_{k-i+1}-(k-i+1)\text{ for } 1\le i \le k.
$
Then $\lambda_i\ge\lambda_{i+1}$ since $s_{k-i+1}>s_{k-i}$.
Because the smallest $k$-subset is $\{1,\dots,k\}$, we have
\begin{equation}\label{lambdaSequence}
\la_1\geq \dots\geq\la_k\geq 0,
\end{equation}
and the number of $k$-subsets $T$ of $\n$ with $T\leq S$ is equal to the number of all the sequences
\begin{equation}\label{partition}
\mu_1\geq \dots\geq \mu_k\geq 0\quad\text{with}\quad \mu_i\leq \lambda_i\quad\text{for}\quad i=1,\dots,k.
\end{equation}
To find $d_k$ it suffices to compute the number of the sequences in (\ref{partition}) for the given sequence (\ref{lambdaSequence}).
If $k=1$, then $d_1 = \lambda_1 + 1 = s_1$.
If $k\ge 2$, let $2\le j\le k$. For each fixed nonnegative integer $\mu\le\la_j$ we calculate iteratively on $j$ the number $\alpha_j(\mu)$ of the sequences
\begin{equation}\label{aSj}
\mu_1\geq \dots\geq \mu_{j-1}\ge\mu\quad\text{with}\quad \mu_i\leq \lambda_i\quad\text{for}\quad i=1,\dots,j-1,
\end{equation}
and the required dimension $d_k = \sum_{\mu = 0}^{\lambda_k}\alpha_k(\mu)$.
Let $\xi_j=\la_j-\mu$. Then $0\leq \xi_j\leq\la_j$. Our aim now is to prove
\begin{equation}\label{alphaj}
\alpha_j(\mu) = \beta_j + \gamma_j,
\end{equation}
where
$\beta_j=\sum^{j-1}_{i=1}{\la_i-\la_j+\xi_j+j-i \choose j-i}\gamma_i$ and $\gamma_j = -\sum^{j-2}_{i=1}{\la_i-\la_{j-1}+j-i-1 \choose j-i}\gamma_i$ with $\gamma_1=1$.
Notice that $\alpha_j(\mu)$ is a sum of two numbers $\beta_j$ and $\gamma_j$, of which $\gamma_j$ depends on $\la_1,\dots,\la_{j-1}$, whereas $\beta_j$ depends on $\la_1,\dots,\la_j$ and $\xi_j$.
We use induction on $j$ to prove (\ref{alphaj}) for $2 \le j\le k$. If $j=2$, for each fixed nonnegative integer $\mu\le\lambda_2$
we have $\xi_2=\la_2-\mu$ and $0\leq \xi_2\leq\la_2$. Let $\xi_1=\la_1-\mu_1$. To ensure that (\ref{aSj}) holds for this case, namely $\mu_1\geq\mu$ and $\mu_1\leq\lambda_1$, we must have $0\leq \xi_1\leq\la_1-\la_2+\xi_2$, and conversely. So
\begin{equation*}
\alpha_2(\mu)= \la_1-\la_2+\xi_2+1 = \beta_2+\gamma_2
\end{equation*}
where $\beta_2=\la_1-\la_2+\xi_2+1$ and $\gamma_2=0$, and this is (\ref{alphaj}) for $j=2$.
Suppose (\ref{alphaj}) holds for $j=l$ with $2\leq l\leq k-1$, that is, for each fixed nonnegative integer $\mu\le\lambda_l$
we have $\xi_l=\la_l-\mu$ with $0\leq \xi_l\leq\la_l$, and the number of sequences $\mu_1\geq\dots\geq\mu_{l-1}\geq\mu$ with $\mu_i\leq\lambda_i$ for $i=1,\ldots,l-1$ is
\begin{equation}\label{hypothesis}
\alpha_l(\mu) = \beta_l + \gamma_l,
\end{equation}
where
$\beta_l=\sum^{l-1}_{i=1}{\la_i-\la_l+\xi_l+l-i \choose l-i}\gamma_i$ and $\gamma_l = -\sum^{l-2}_{i=1}{\la_i-\la_{l-1}+l-i-1 \choose l-i}\gamma_i$.
We now prove (\ref{alphaj}) for $j=l+1$. For a fixed nonnegative integer $\nu\le\lambda_{l+1}$ we have $\xi_{l+1} = \la_{l+1}-\nu$ with $0\leq \xi_{l+1}\leq\la_{l+1}$. Let $\mu=\la_l-\xi_l$. To ensure that the condition (\ref{aSj})
\[
\mu_1\geq\dots\geq\mu_{l-1}\geq\mu\ge\nu\quad\text{ with}\quad\mu_i\le\lambda_i,\, i=1,\ldots,{l-1}\text{ and } \mu\le\lambda_l
\]
holds here, we must have $0\leq \xi_l\leq\rho_l$ where $\rho_l=\la_l-\la_{l+1}+\xi_{l+1}$, and conversely. Adding all $\alpha_l(\mu)$ up for $\nu\le\mu\le\lambda_l$ and using the induction hypothesis (\ref{hypothesis}), we obtain
\begin{eqnarray
\nonumber\alpha_{l+1}(\nu)&=&\sum_{\mu = \nu}^{\lambda_{l}} \alpha_l(\mu) = \sum_{\mu = \nu}^{\lambda_{l}} (\beta_l + \gamma_l)\\
&=&\sum^{\rho_l}_{\xi_l=0}\sum^{l-1}_{i=1}{\la_i-\la_l+\xi_l+l-i \choose l-i}\gamma_i+\sum^{\rho_l}_{\xi_l=0}\gamma_l \label{second}\\
\nonumber &=&\sum^{l-1}_{i=1}\left\{{\la_i-\la_{l+1}+\xi_{l+1}+(l+1)-i \choose l+1-i}\gamma_i
-{\la_i-\la_l+l-i \choose l+1-i}\gamma_i\right\}\\
&&\qquad\qquad\qquad\qquad\qquad+{\la_l-\la_{l+1}+\xi_{l+1}+1\choose 1}\gamma_l \label{alp1}\\
\nonumber &=&\sum^l_{i=1}{\la_i-\la_{l+1}+\xi_{l+1}+(l+1)-i \choose l+1-i}\gamma_i-\sum^{l-1}_{i=1}{\la_i-\la_l+l-i \choose l+1-i}\gamma_i\\
\nonumber &=&\beta_{l+1} + \gamma_{l+1},
\end{eqnarray}
where
\begin{eqnarray*}\label{}
\beta_{l+1}&=&\sum^l_{i=1}
{\la_i-\la_{l+1}+\xi_{l+1}+(l+1)-i \choose l+1-i}\gamma_i ~,\\
\gamma_{l+1}&=&-\sum^{l-1}_{i=1}{\la_i-\la_l+l-i \choose
l+1-i}\gamma_i ~.
\end{eqnarray*}
Here we have made use of the identity $\sum_{z=a}^{a+b-1}\binom{z}{p} = \binom{a+b}{p+1} - \binom{a}{p+1}$ in which $a, b, p$ are natural numbers to obtain (\ref{alp1}) from (\ref{second}) by assigning $a=\lambda_i - \lambda_l + l - i,\, b=\lambda_l - \lambda_{l+1} + \xi_{l+1} + 1$ and $p=l-i\ge 1$.
Therefore, (\ref{alphaj}) is valid for $j=l+1$, and we complete the proof of (\ref{alphaj}) by induction.
We are now able to calculate the dimension $d_k$ of the module $B_nv_S$ for $k\ge 2$ by summing all $\alpha_k(\mu)$ in (\ref{alphaj}) up where $\mu$ runs from $0$ to $\la_k$, yielding
\begin{eqnarray*}\label{dimlambda3}
\nonumber d_k &=& \sum^{\la_k}_{\mu=0}\alpha_k(\mu) \\
\nonumber &=&\sum^{\la_k}_{\xi_k=0}\sum^{k-1}_{i=1}
{\la_i-\la_k+\xi_k+k-i \choose k-i}\gamma_i-\sum^{\la_k}_{\xi_k=0}\sum^{k-2}_{i=1}
{\la_i-\la_{k-1}+k-i-1 \choose k-i}\gamma_i\\
\nonumber &=&\sum^{k-1}_{i=1}{\la_i+k-i+1 \choose k+1-i}\gamma_i -\sum^{k-1}_{i=1}{\la_i-\la_k+k-i \choose k+1-i}\gamma_i \\
&&\qquad\qquad -\sum^{k-2}_{i=1}(\la_k+1) {\la_i-\la_{k-1}+k-i-1 \choose k-i}\gamma_i~.
\end{eqnarray*}
From $\la_i=s_{k-i+1}-(k-i+1)$, we conclude that $d_k$ is given by (\ref{dk}).
\end{proof}
The following four corollaries are consequences of Theorem \ref{dim}. Considering the sequence $\la_1\geq \dots\geq\la_k\geq 0$ given in (\ref{lambdaSequence}) a partition of $d=\sum_{i=1}^k\la_i$ into at most $k$-parts, we obtain the next result.
\begin{corollary}
Let $\la_1\geq\dots\geq\la_k\geq 0$ be a given partition
of some nonnegative integer. Then the number of distinct Young diagrams obtained from the Young diagram of $\la$ by removing zero or more boxes from the rows is $d_k$, which is given in Theorem {\rm \ref{dim}}.
\end{corollary}
\begin{proof}
It is easily seen that the number of distinct Young diagrams obtained from the Young diagram of $\la$ by removing zero or more boxes from the rows is equal to the number of partitions $\mu_1\geq\dots\geq\mu_k\geq 0$ such that $\mu_i\leq \la_i$ for all $i=1,\dots,k$. The desired result follows from the proof of Theorem \ref{dim}.
\end{proof}
\begin{corollary}\label{dimspeci1}
If $S=\{2,4,\ldots,2k\} \subseteq \mathbf{n}$, then the dimension of the submodule $B_nv_S$ is the Catalan number
$c_{k+1}$.
\end{corollary}
\begin{proof}
The sequence $\la_1\geq \dots\geq\la_k\geq 0$ in (\ref{lambdaSequence}) associated to $S$ is now $k \geq k-1\geq\cdots\geq 1$. In this case, it is well known that the number of sequences $\mu_1\ge\mu_2\ge\ldots\ge\mu_k\ge 0$ such that $\mu_i\leq\la_i$ for $1\leq i\leq k$ is the Catalan number $c_{k+1}$. The desired result follows from Theorem \ref{dim}.
\end{proof}
We find a combinatorial identity below for the Catalan number. To our knowledge, the identity is new.
\begin{corollary}\label{comId}
If $k\geq 2$, then
\begin{equation}\label{catalannewform1}
c_{k+1}=\sum^{k-1}_{i=1}{2(k-i+1) \choose k+1-i}\gamma_i
-\sum^{k-1}_{i=1}{2(k-i) \choose k+1-i}\gamma_i-\sum^{k-2}_{i=1}2
{2(k-i-1) \choose k-i}\gamma_i~,
\end{equation}
where $\gamma_1=1$ and for $2\leq i\leq k$,
\begin{equation}\label{catalannewform2}
\gamma_i =-\sum^{i-2}_{j=1}
{2(i-j-1) \choose i-j}\gamma_j~.
\end{equation}
\end{corollary}
\begin{proof}
The combinatorial identity is obtained from Theorem \ref{dim} and Corollary \ref{dimspeci1}.
\end{proof}
\begin{corollary}\label{dimspeci2}
For the $k$-subset $\{m+1,\ldots,m+k\}$ of $\mathbf{n}$, the
dimension of the submodule $B_nv_S$ is ${m+k \choose k}$.
\end{corollary}
\begin{proof}
The sequence in (\ref{lambdaSequence}) corresponding to $S$ is the $k$-subset $\{m, \ldots,m\}$. A direct calculation of $d_k$ for $k\geq 1$ using the formulas given in Theorem \ref{dim} yields
\begin{equation}\label{}
d_k={m+k \choose k}~,
\end{equation}
which is the desired result.
\end{proof}
We now compute the dimension $d_{k,\, m}$ of the $B_n$-module $B_nv_{S_{k,\,m}}$, where $k\ge m$ and
$$
S_{k,\,m}=\{2,\,4,\,\ldots,\,2m,\,2m+1,\,2m+2,\,\ldots,\,2m+(k-m)\}
$$
is a subset of $\mathbf{n}$, which is a mixture of the two types of subsets in Corollaries \ref{dimspeci1} and \ref{dimspeci2}. The sequence (\ref{lambdaSequence}) associated to $S_{k,\,m}$ is $\{m,\, m,\,\ldots,m,\,m-1,\,m-2,\,\ldots,\,1\}$ of length $k$.
Recall that if $x>y$, then $\binom{y}{x}=0$ and the empty sum $\sum_{i=x}^y\square_i = 0$. Without showing the details, from Theorem \ref{dim} we obtain, for $m\ge 2$,
\begin{eqnarray}\label{catagene2}
\nonumber d_{k,\,m}&=&{m+k \choose k}-{m+k-2 \choose k}-2
{m+k-4 \choose k-1} +\sum^{k-1}_{i=k-m+3}{2(k-i+1) \choose k+1-i}\gamma_i \\
&&{}-\sum^{k-1}_{i=k-m+3}{2(k-i) \choose k+1-i}\gamma_i-\sum^{k-2}_{i=k-m+3}2
{2(k-i-1) \choose k-i}\gamma_i
\end{eqnarray}
where $\gamma_1=1,\gamma_2=\gamma_3=\dots=\gamma_{k-m+2}=0,
\gamma_{k-m+3}=-1$, and for $i\geq k-m+4$
\begin{equation}\label{catagene3}
\gamma_i =-{m-k+2i-4\choose i-1}-\sum^{i-2}_{j=k-m+3}
{2(i-j-1) \choose i-j}\gamma_j~.
\end{equation}Notice that $d_{k,\,k}$ is just the Catalan number $c_{k+1}$ by Corollary (\ref{dimspeci1}). Formula (\ref{catagene2}) will be used in Corollary \ref{ccd}.
Identifying an element of $B_t$ $(t< n)$ with an element of $B_n$ that fixes $t+1, \ldots,n$, we can regard $B_t$ as
a submonoid of $B_n$, so we are allowed to view any $B_n$-submodules of $V$, for example $V_k$ and its submodules, as $B_t$-modules.
Our aim below is to investigate decompositions of the $B_n$-module $W^m_k =B_nv_{S^m_k}$ into indecomposable submodules for $S^m_k=\{m+1,\ldots,m+k\}$ where $k$ and $m$ are positive integers and $k+m\le n$. Using the notation above, we obtain the following result.
\begin{proposition}\label{decomposition}
Viewed as a $B_{m+l}\,(1\leq l<k)$ module, $W^m_k =B_nv_{S^m_k}$ is decomposed into a direct sum of indecomposable submodules
\vspace {-2mm}
\begin{equation}\label{moddecom1}
W^m_k \downarrow_{B_{m+l}}^{B_{m+k}} \,\cong\, \bigoplus_{a=0}^{k-l} {k-l \choose a}W_{k-a}^{m+l-k+a} ~,
\end{equation}
\vspace {-2mm}
where ${k-l \choose a}$ is the multiplicity of the indecomposable
submodule $W_{k-a}^{m+l-k+a}$.
\end{proposition}
\begin{proof}
By Lemma \ref{mod1} (1) we have
\[
W^m_k = B_{m+k}v_{S^m_k} = \bigoplus_{k\text{-subsets } T\subseteq \{1,\, ...,\, m+k\}} Fv_T~.
\]
To obtain the desired indecomposable $B_{m+l}$-submodules on the right of (\ref{moddecom1}), we group the $1$-dimensional subspaces $Fv_T$ with $k$-subsets $T\subseteq\{1,\, ...,\, m+k\}$ into categories according to the intersection $\{t'_1, \ldots, t'_a\} = T\cap \{m+l+1,\,\ldots,\, m+k\}$ where $t'_1 < \cdots < t'_a$,\, $0\leq a\leq k-l$, and $1\leq l<k$.
Let $\mathcal{T}$ denote the set of all these $k$-subsets $T$. For any $T\in \mathcal{T}$, we have
\[
T=\{t_1,\,\ldots,\,t_{k-a}, \,t'_1, \ldots, t'_a\}~,
\]
for some subset $\{t_1,\,\ldots,\,t_{k-a}\}\subseteq\{1, ..., m+l\}$ with $t_1 < \cdots < t_{k-a}$, so $m+l\ge k-a.$
Write $p = m+l-(k-a)$ and let $T_a = \{p+1,\,\ldots,\, m+l, \,t'_1, \ldots, t'_a\}.$ Then $T_a$ is a $k$-subset of $\{1,\ldots,m+k\}$, and
$T\le T_a$. Define $f\in B_{m+k}$ by
\[
f(p+1)=t_1,\, \ldots,\, f(m+l) = t_{k-a},\, f(m+l+1) = m+l+1,\, \ldots,\, f(m+k) = m+k.
\]
Then $T=f(T_a)$. Identifying $f$ with an element of $B_{m+l}$, we have $v_{T}=f\cdot v_{T_a}\in B_{m+l}v_{T_a}$. From Lemma \ref{mod1} (i) we have
\[
B_{m+l}v_{T_a} = \bigoplus_{T\in\, \mathcal{T}}Fv_T ~.
\]
We next show that
$
B_{m+l}v_{T_a} \cong B_{m+l}v_{S_{k-a}^{p}}
$
as $B_{m+l}$-modules. Notice that
\[
S_{k-a}^{p}=\{p+1,\,\ldots,\, p+(k-a)\} = \{p+1,\,\ldots,\, m+l\} \subseteq \{1, ..., m+l\}~.
\]
Let
$
\mathcal{U} = \{U\subseteq \{1, ..., m+l\}\mid U\le S_{k-a}^{p}\}.
$
By Lemma \ref{mod1} (i) we have
\[
B_{m+l}v_{S_{k-a}^{p}} = \bigoplus_{U\in\, \mathcal{U}}Fv_U~.
\]
Since
$
T_a = S_{k-a}^{p}\cup \{t'_1,\, \ldots, t'_a\},
$
the map of $\mathcal{T}$ to $\mathcal{U}$ defined by
\[
T \mapsto T\cap\{1,\,\cdots,\,m+l\} = \{t_1,\,\ldots,\,t_{k-a}\}
\]
is one-to-one and onto. Since $B_{m+l}$ fixes $\{t'_1,\, \ldots, t'_a\}$ pointwise, the map defined by
$
v_T \mapsto v_{\{t_1,\, \ldots,\, t_{k-a}\}}
$
leads to a $B_{m+l}$-module isomorphism of $B_{m+l}v_{T_a}$ onto $B_{m+l}v_{S_{k-a}^{p}}$, which is indecomposable by Proposition \ref{indDecom}.
Since there are ${k-l \choose a}$ ways to choose $\{t'_1,\,\ldots,\, t'_a\}$
from $\{m+l+1,\ldots,m+k\}$, there are the same number of corresponding
modules $B_{m+l}v_{T_a}$ in $W^m_k$. Thus
\[
W^m_k \downarrow_{B_{m+l}}^{B_{m+k}} \,\cong\, \bigoplus_{a=0}^{k-l} {k-l \choose a}B_{m+l}v_{T_a}~,
\]
and the proof is complete.
\end{proof}
\begin{corollary} Let $k, l, m$ be positive integers with $l<k$.
\begin{equation}\label{moddecomform}
{m+k \choose k}=\sum_{a=0}^{k-l} {k-l \choose a}{m+l \choose k-a}~.
\end{equation}
\end{corollary}
\begin{proof}
Corollary \ref{dimspeci2} shows that $\dim W^m_k={m+k \choose k}$ and $\dim W_{k-a}^{m+l-k+a} = {m+l \choose k-a}$. Inserting them into (\ref{moddecom1}), we complete the proof.
\end{proof}
We apply the same procedure in the proof of Proposition \ref{decomposition} to deal with, without showing all the details, the decomposition of the $B_{2k}$-module $W_k=B_nv_{S_k}$ into indecomposable $B_{2(k-1)}$-modules for $S_k=\{2,4,\ldots,2k\}$ with $2k\le n$. We regard $W_k$ as a $B_{2(k-1)}$-module.
Let $S$ be a $k$-subset of $\mathbf{n}$ such that $S\leq S_k$. If $S$ contains $2k$, then $v_S=fv_{S_k}$ for some $f\in B_{2(k-1)}$ since $B_{2(k-1)}$ fixes $2k$. Thus the module $B_{2(k-1)}v_{S_k}$ contains basis vectors $v_S$ where $S$ runs through all $k$-subsets containing $2k$, so $B_{2(k-1)}v_{S_k}$ is isomorphic to the submodule
$W_{k-1} = B_{2(k-1)}v_{S_{k-1}}$.
If $S$ does not contain $2k$ but contains $2k-1$, then $S\leq T=\{2, \,4,\,\ldots,\,2(k-1),\,2k-1\}$, and since $B_{2(k-1)}$
fixes $2k-1$, we have $v_S=fv_{T}$ for some $f\in B_{2(k-1)}$. Thus the module $B_{2(k-1)}v_{T}$ contains basis vectors $v_S$ where $S$ runs through all $k$-subsets containing $2k-1$, so it is isomorphic to the submodule $W_{k-1} = B_{2(k-1)}v_{S_{k-1}}$.
If $S$ contains neither $2k$ nor $2k-1$, then $S\leq S_{k,\, k -2}=\{2,4,\ldots,2(k-2),2k-3,2k-2\}$, and we have
$v_S=fv_{S_{k,\, k -2}}$ for some $f\in B_{2(k-1)}$. The module $B_{2(k-1)}v_{k,\, k -2}$ then contains basis vectors $v_S$ where $S$ runs through all $k$-subsets containing neither $2k$ nor $2k-1$. Therefore, we have shown
\begin{proposition}
For the $k$-subset $S_k=\{2,4,\ldots,2k\}$ of $\mathbf{n}$, let
$W_k=B_nv_{S_k}$. Then viewed as a $B_{2(k-1)}$ module, we have the
decomposition of $W_k$ into a direct sum of indecomposable
submodules
\begin{equation}\label{moddecom2}
W_k\downarrow_{B_{2(k-1)}}^{B_{2k}}\cong 2W_{k-1} \oplus
B_{2(k-1)}v_{S_{k,\,k-2}}~.
\end{equation}
\end{proposition}
\begin{corollary}\label{ccd} Let $k\ge 2$. Then the $(k+1)${\rm st} Catalan number
\begin{equation}\label{ccd} c_{k+1}=2c_k + d_{k,\,k-2}~,
\end{equation}
where $c_k$ is the $k$th Catalan number and $d_{k,\,k-2}$ is the dimension of $B_{2(k-1)}v_{S_{k,\,k-2}}$.
\end{corollary}
\begin{proof}
By Corollary \ref{dimspeci1}, the dimension of $W_k$ is $c_{k+1}$ and that of $W_{k-1}$ is $c_{k}$.
The dimension $d_{k,\,k-2}$ of $B_{2(k-1)}v_{S_{k,\,k-2}}$ is given in (\ref{catagene2}). Putting them into (\ref{moddecom2}), we obtain the desired combinatorial identity (\ref{ccd}).
\end{proof}
\section{Presentation on generators and relations}
We use the method of Section 3 of Herbig \cite{H} to describe generators and relations of $B_n$.
Some preparations are needed. We define a new monoid $\hat{B}_n$ generated by symbols $\l_i,\e_i, \hat 1$ subject to the relations:
\vspace{-2mm}
\begin{enumerate}[{\rm(i)}]
\item $\e_i^2=\e_i$
\vspace{-2mm}
\item $\l_i\l_{i+1}\l_i=\l_i\l_{i+1}=\l_{i+1}\l_i\l_{i+1}$
\vspace{-2mm}
\item $\l_i\e_i=\l_i=\e_{i+1}\l_i$
\vspace{-2mm}
\item $\l_i\e_{i+1}=\e_i\e_{i+1}=\e_i\l_i=\l_i^3=\l_i^2$
\vspace{-2mm}
\item $\e_i\l_j=\l_j\e_i$ for $i\neq j,j+1$
\vspace{-2mm}
\item $\l_i\l_j=\l_j\l_i$ for $|i-j|\geq 2$
\vspace{-2mm}
\item $\e_i\e_j=\e_j\e_i$ for all $i,j$~.
\end{enumerate}
\vspace{-2mm}
For $a, b\in \mathbf{n}$ with $a>b$, let $\L^{a,\,a}=\hat{1}$ and
$
\L^{a,\,b}=\l_b\l_{b+1}\cdots \l_{a-2}\l_{a-1}.
$
For any subsets $S=\{s_1<\dots<s_k\}$ and $T=\{t_1<\dots<t_k\}$ of $\mathbf{n}$ satisfying $s_j\geq t_j$ for all $1\leq j\leq k$,
let
\begin{equation}\label{uv}
U=\mathbf{n}-S=\{u_1<\cdots<u_{n-k}\}\quad\text{and}\quad V=\mathbf{n}-T=\{v_1<\cdots<v_{n-k}\}~.
\end{equation}
Define
$
\E_S = \e_{u_1}\cdots \e_{u_{n-k}},\,\L^{S,\,T} =\L^{s_k,\,t_k}\cdots \L^{s_1,\,t_1},\text{ and } \E_T=\e_{v_1}\cdots \e_{v_{n-k}}~,
$
where we agree that $\E_\mathbf{n} = \hat{1}$. The word $W^S_T=\E_T\L^{S,\,T}\E_S$ is called a {\em standard word} of $\hat{B}_n$. The following proposition shows that every element of $\hat{B}_n$ is equivalent, under the relations (i) -- (vii), to one of the standard words. First, we note that the generators $\l_i,\e_i$ and $\hat{1}$ are themselves standard words.
\begin{proposition}\label{ml}
\begin{enumerate}[{\rm(1)}]
\item $W^S_T\l_i=W^{S^\prime}_{T^\prime},$ where
$$(S^\prime,\,T^\prime)=
\left\{
\begin{array}{ll}
(S,\,T) & \text{\rm if }\, i,\,i+1\notin S \\
(S\backslash\{i\},\,T\backslash\{t_{c+1}\}) & \text{\rm if }\, i,\,i+1\in S \\
((S\backslash\{i\})\cup\{i+1\},\,T) & \text{\rm if }\, i\in S,\,i+1\notin S \\
(S\backslash\{i+1\},\,T\backslash\{t_{c+1}\}) & \text{\rm if }\, i\notin S,\,i+1\in S \\
\end{array}
\right.
$$where $i+1$ is mapped to $t_{c+1}$ under $W^S_T$.
\item $W^S_T\e_i=W^{S^\prime}_{T^\prime},$ where
$$(S^\prime,\,T^\prime)=
\left\{
\begin{array}{ll}
(S,\,T) & \text{\rm if }\, i\notin S \\
(S\backslash\{i\},\,T\backslash\{t_c\})& \text{\rm if }\, i\in S \\
\end{array}
\right.
$$where $i$ is mapped to $t_c$ under $W^S_T$.
\item $\l_iW^S_T=W^{S^\prime}_{T^\prime},$ where
$$(S^\prime,\,T^\prime)=
\left\{
\begin{array}{ll}
(S,\,T) & \text{\rm if }\, i,\,i+1\notin T \\
(S\backslash\{s_c\},\,T\backslash\{i+1\}) & \text{\rm if }\, i,\,i+1\in T \\
(S\backslash\{s_c\},\,T\backslash\{i\}) & \text{\rm if }\, i\in T,\,i+1\notin T \\
(S,\,(T\backslash\{i+1\})\cup\{i\}) & \text{\rm if }\, i\notin T,\,i+1\in T \\
\end{array}
\right.
$$where $s_c$ is mapped to $i$ under $W^S_T$.
\item $\e_iW^S_T=W^{S^\prime}_{T^\prime},$ where
$$(S^\prime,\,T^\prime)=
\left\{
\begin{array}{ll}
(S,\,T) & \text{\rm if }\, i\notin T \\
(S\backslash\{s_c\},\,T\backslash\{i\}) & \text{\rm if }\, i\in T \\
\end{array}
\right.
$$where $s_c$ is mapped to $i$ under $W^S_T$.
\end{enumerate}
\end{proposition}
\begin{proof}
We use relations (i) to (vii) repeatedly, but sometimes we do not mention them.
We divide the proof of part (1) into four cases.
Case 1.1: neither $i$ nor $i+1$ is in $S$. Then $i,i+1\in U$ and $\hat{e_i},\e_{i+1}$ appear in $\E_S$. Since the items in $\E_S$ commute with $\hat{l_i}$ except $\hat{e_i},\e_{i+1}$, we have
\begin{eqnarray*}
W^S_T\l_i &=& \E_T\L^{S,\,T}\e_{u_1}c\dots \e_i\e_{i+1}\cdots
\e_{u_{n-k}}\l_i \\
&=& \E_T\L^{S,\,T}\e_{u_1}\cdots\e_i\e_{i+1}\l_i\cdots
\e_{u_{n-k}} \\
&=& \E_T\L^{S,\,T}\e_{u_1}\cdots\e_i\l_i\cdots \e_{u_{n-k}} \quad\quad\text{(by \rm (iii))}\\
&=& \E_T\L^{S,\,T}\e_{u_1}\cdots\e_i\e_{i+1}\cdots \e_{u_{n-k}} \quad\text{(by \rm (iv))} \\
&=& W^S_T~.
\end{eqnarray*}
Case 1.2: both $i$ and $i+1$ are in $S$. We have $i,i+1\notin U$, and every term in $\E_S$ commutes with $\l_i$ by (v).
From (iii) we get
$$
\E_S\l_i=\E_S\l_i\e_i =\l_i\E_S\e_i=\l_i\E_{S^\prime}~,
$$
where $S^\prime=S\backslash\{i\}.$
Let $s_c=i$. Then $s_{c+1}=i+1$ since $i+1\in S$. For $j<c$, by (vi) the terms in $\L^{s_j,t_j}$ commute with $\l_i$, as
their indices are less than or equal to $i-2$. Thus
\begin{eqnarray}\label{wstl}
\nonumber W^S_T\l_i &=& \E_T\L^{S,\,T}\E_S\l_i \\
\nonumber &=& \E_T\L^{S,\,T}\l_i\E_{S^\p} \\
&=& \E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\L^{i+1,\,t_{c+1}}\L^{i,\,t_{c}}\l_i)\L^{s_{c-1},\,t_{c-1}}\cdots\L^{s_1,\,t_1}\E_{S^\p}~.
\end{eqnarray}
We now show
\begin{equation}\label{lll}
\L^{i+1,\,t_{c+1}}\L^{i,\,t_c}\l_i = L^{i+1,\, t_{c}}~.
\end{equation}
Indeed, by (vi) the term $\l_i$ in $\L^{i+1,\,t_{c+1}}=\l_{t_{c+1}}\cdots\l_{i-1}\l_i$ commutes with all of the terms of $\L^{i,\,t_{c}}=\l_{t_c}\cdots\l_{i-2}\l_{i-1}$ until $\l_{i-1}$, where we use (ii): $\l_i\l_{i-1}\l_i=\l_{i-1}\l_i$ to simplify.
Repeating the same procedure for each of the remaining terms of $\L^{i+1,\,t_{c+1}}$, we conclude
\begin{eqnarray*}
\L^{i+1,\,t_{c+1}}\L^{i,\,t_c}\l_i &=& (\l_{t_{c+1}}\cdots\l_{i-1}\l_i)(\l_{t_c}\cdots\l_{i-1})\l_i \\
&=& (\l_{t_{c+1}}\cdots\l_{i-1})(\l_{t_c}\cdots\l_i\l_{i-1}\l_i) \quad\text{\rm by (vi)} \\
&=& (\l_{t_{c+1}}\cdots\l_{i-1})(\l_{t_c}\cdots\l_{i-1}\l_i) \quad\text{\rm by (ii)}\\
&\vdots& \\
&=& \l_{t_c}\cdots\l_{i-1}\l_i \\
&=& L^{i+1,\, t_{c}}~.
\end{eqnarray*}
Putting (\ref{lll}) into (\ref{wstl}), we obtain
\begin{eqnarray}\label{wstl1}
\nonumber W^S_T\l_i &=& \E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}\L^{i+1,\,t_{c}}\L^{s_{c-1},\,t_{c-1}}\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=& \E_T\L^{S^\prime,\,T^\p}\E_{S^\prime}
\end{eqnarray}
where $S'=S\setminus\{i\}$ and $T'=T\setminus\{t_{c+1}\}$.
The right hand side of (\ref{wstl1}) is not a standard word yet because $\e_{t_{c+1}}$ is still missing in $\E_T$. Since
$t_c\leq t_{c+1}-1\leq i$, the term $\l_{(t_{c+1}-1)}$ appears in $\L^{i+1,\,t_{c}}=\l_{t_c}\cdots\l_i$. Notice that
$\l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)}=\l_{(t_{c+1}-1)}\l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)}$ and $\e_{t_{c+1}}\l_{(t_{c+1}-1)}=\l_{(t_{c+1}-1)}$. We have
\begin{eqnarray*}
L^{i+1,\, t_{c}} &=& \l_{t_c}\cdots\l_{(t_{c+1}-3)}(\l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)}) \cdots\l_i \\
&=& \l_{t_c}\cdots\l_{(t_{c+1}-3)}(\l_{(t_{c+1}-1)}\l_{(t_{c+1}-2)} \l_{(t_{c+1}-1)}) \cdots\l_i \\
&=& \l_{t_c}\cdots\l_{(t_{c+1}-3)}(\e_{t_{c+1}}\l_{(t_{c+1}-1)})\l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)} \cdots\l_i \\
&=& \l_{t_c}\cdots\l_{(t_{c+1}-3)}\e_{t_{c+1}} \l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)} \cdots\l_i\\
&=& \e_{t_{c+1}}(\l_{t_c}\cdots \l_{(t_{c+1}-3)}\l_{(t_{c+1}-2)}\l_{(t_{c+1}-1)} \cdots\l_i)\\
&=& \e_{t_{c+1}}L^{i+1,\, t_{c}}~,
\end{eqnarray*}
where we have used that $\e_{t_{c+1}}$ commutes with all terms on its left.
Inserting the above result into (\ref{wstl1}) and knowing that $\e_{t_{c+1}}$ commutes with $\L^{s_k,\,t_k}\cdots\L^{s_{c+2},t_{c+2}}$, we deduce
\begin{eqnarray*}
W^S_T\l_i &=& \E_T (\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}})(\e_{t_{c+1}}L^{i+1,\, t_{c}}) \L^{s_{c-1},\,t_{c-1}}\cdots\L^{s_1,\,t_1}\E_{S^\prime} \\
&=& \E_T\e_{t_{c+1}}(\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}} L^{i+1,\, t_{c}} \L^{s_{c-1},\,t_{c-1}}\cdots\L^{s_1,\,t_1})\E_{S^\prime} \\ &=&\E_{T^\p}\L^{S^\p,\,T^\p}\E_{S^\p} \\
&=&W^{S^\p}_{T^\p}~.
\end{eqnarray*}
Case 1.3: $i$ is in $S$, but $i+1$ is not. It follows immediately that $i\notin U,\,i+1\in
U$, so $\e_{i+1}$ appears in $\E_{S}$, but $\e_i$ does not. Using (iii): $\e_{i+1}\l_i=\l_i\e_i$ and (v): $\l_i\e_j=\e_j\l_i$ for $j\ne i, i+1$, we have
\begin{eqnarray*}
\E_S\l_i&=&\e_{u_1}\cdots\e_{i+1}\cdots\e_{u_{n-k}}\l_i\\
&=& \e_{u_1}\cdots\e_{i+1}\l_i\cdots\e_{u_{n-k}} \\
&=& \e_{u_1}\cdots\l_i\e_i\cdots\e_{u_{n-k}} \\
&=& \l_i\E_{S^\p}~,
\end{eqnarray*}
where $S^\p=(S\backslash\{i\})\cup\{i+1\}$. Let $s_c=i$. Then $s_{c+1}>i+1$ since $i+1\notin S$. For
$j<c$, all the terms in $\L^{s_j,t_j}$ commute with $\l_i$ since their indices are at most $i-2$, giving
\begin{eqnarray*}
W^S_T\l_i &=& \E_T\L^{S,\,T}\E_S\l_i \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{i,t_c}
\cdots\L^{s_1,\,t_1}\l_i\E_{S^\p}\\
&=&\E_T\L^{s_k,\,t_k}\cdots(\l_{t_c}\cdots\l_{i-2}\l_{i-1})\l_i
\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_{T^\p}\L^{S^\p,\,T^\p}\E_{S^\p}\\
&=&W^{S^\p}_{T^\p}~,
\end{eqnarray*}
where $T'=T$.
Case 1.4: $i+1$ is in $S$, but $i$ is not. We have $i\in U,i+1\notin U$. The item $\e_i$ is in $\E_S$, but $\e_{i+1}$ is not. By $\e_i\l_i=\l_i\e_{i+1}$, $\l_i=\l_i\e_i$, and $\l_i\e_j=\e_j\l_i$ for $j\ne i, i+1$, we find
\begin{eqnarray*}
\E_S\l_i&=&\e_{u_1}\cdots\e_i\cdots\e_{u_{n-k}}\l_i\\
&=& \e_{u_1}\cdots\e_i\l_i\cdots\e_{u_{n-k}} \\
&=& \e_{u_1}\cdots\l_i\e_{i+1}\cdots\e_{u_{n-k}} \\
&=& \e_{u_1}\cdots\l_i\e_i\e_{i+1}\cdots\e_{u_{n-k}} \\
&=& \l_i\e_{u_1}\cdots\e_i\e_{i+1}\cdots\e_{u_{n-k}} \\
&=& \l_i\E_{S^\p}~,
\end{eqnarray*}
where $S^\p=S\backslash\{i+1\}$.
Let $s_{c+1}=i+1$. Then for $j\le c$, the indices of the terms in $\L^{s_j,\,t_j}$ are at most $i-2$, so they commute with
$\l_i$, leading to
\begin{eqnarray*}
W^S_T\l_i &=& \E_T\L^{S,\,T}\E_S\l_i \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}\L^{i+1,\,t_{c+1}}\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\l_i\E_{S^\p} \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\l_{t_{c+1}}\cdots \l_{i-1}\l_i)\l_i\L^{s_{c},\,t_{c}} \cdots\L^{s_1,\,t_1}\E_{S^\p}~.
\end{eqnarray*}
Since $\l_i^2=\e_i\e_{i+1}$ and $\l_{j-1}\e_j=\e_{j-1}\e_j$ for all
$t_{c+1}+1\leq j\leq i$ (use them repeatedly below), we obtain
\begin{eqnarray*}
W^S_T\l_i&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\l_{t_{c+1}}\cdots
\l_{i-1}\l_i^2)\L^{s_{c},\,t_{c}} \cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\l_{t_{c+1}}\cdots
\l_{i-1}\e_i\e_{i+1})\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\l_{t_{c+1}}\cdots
\l_{i-2}\e_{i-1}\e_i\e_{i+1})\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&\vdots& \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}(\e_{t_{c+1}}\cdots
\e_{i-1}\e_i\e_{i+1})\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\E_{S^\p}~.
\end{eqnarray*}
As the indices of all the terms $\l$ on the left of $\e_{t_{c+1}}$ are at least $t_{c+1}+1$, it follows from (v) that $\e_{t_{c+1}}$ commutes with $\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}.$ We obtain
\begin{eqnarray*}
W^S_T\l_i &=&\E_T\e_{t_{c+1}}\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}
(\e_{t_{c+1}+1}\cdots\e_i\e_{i+1})
\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_{T^\p}\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}
(\e_{t_{c+1}+1}\cdots\e_i\e_{i+1})
\L^{s_{c},\,t_{c}}\cdots\L^{s_1,\,t_1}\E_{S^\p}~,
\end{eqnarray*}
where $T^\p=T\backslash\{t_{c+1}\}$.
Our aim now is to switch $\e_j$ for $t_{c+1}+1\leq j\leq i+1$ with the terms $\l$ one by one on the right of $\e_j$ until it encounters either $\l_j$ or $\l_{j-1}$, or until it commutes past all of the $\l$. If $t_{c+1}+1\leq j\leq s_c$, then $\e_j$ will run into some $\l_{j-1}$ in $\L^{s_c,\,t_c}=\l_{t_c}\l_{t_c+1}\cdots\l_{s_c-1}$, leading to $\e_j\l_{j-1} = \l_{j-1}$ by (iii).
If $s_c+1\le j\leq i+1$, then $\e_j$ does not run into any $\l_j$ or $\l_{j-1}$ in $\L^{s_c,\,t_c}$ since the maximal index of the $\l$ there is $s_c-1$, and $\e_j$ does not run into any $\l_j$ or $\l_{j-1}$ to the right of $\L^{s_c,\,t_c}$, either, because all the indices of the $\l$ are at most $s_c-2$. But for $s_{c+1}\le j\leq i+1$, we have $j\in \mathbf{n}-S^\p$, so $\e_j$ will run into $\e_j$ in $\E_{S^\p}$. By $\e_j\l_{j-1}=\l_{j-1}$ and
$\e^2_j=\e_j$, we get
\begin{eqnarray*}
W^S_T\l_i&=&\E_{T^\p}\L^{s_k,\,t_k}\cdots\L^{s_{c+2},\,t_{c+2}}
\L^{s_c,\,t_c}
\cdots\L^{s_1,\,t_1}\E_{S^\p} \\
&=&\E_{T^\p}\L^{S^\p,\,T^\p}\E_{S^\p} \\
&=&W^{S^\p}_{T^\p}~.
\end{eqnarray*}
This completes the proof of part (1).
We next show part (2). If $i\notin S$, then $i\in U$ and $\e_i$ is in $\E_S$. By $\e^2_i=\e_i$, we have
\begin{eqnarray*}
W^S_T\e_i &=&\E_T\L^{S,\,T}\e_{u_1}
\cdots\e_i\cdots\e_{u_{n-k}}\e_i \\
&=&\E_T\L^{S,\,T}\e_{u_1}\cdots\e_i\e_i\cdots\e_{u_{n-k}} \\
&=&\E_T\L^{S,\,T}\e_{u_1}\cdots\e_i\cdots\e_{u_{n-k}}\\
&=& \E_T\L^{S,\,T}\E_S \\
&=& W^S_T~.
\end{eqnarray*}
If $i\in S$, let $s_c=i$, and then for $j<c$, all of the
indices of the terms in $\L^{s_j,\,t_j}$ are at most $i-2$, so they
commute with $\e_i$. From $\e_i^2=\e_i$, we obtain
\begin{eqnarray*}
W^S_T\e_i&=&\E_T\L^{s_k,\,t_k}\cdots\L^{i,\,t_c}
\cdots\L^{s_1,\,t_1}\E_S\e_i \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{i,\,t_c}
\cdots\L^{s_1,\,t_1}\E_S\e_i\e_i \\
&=&\E_T\L^{s_k,\,t_k}\cdots\L^{i,\,t_c}\e_i
\cdots\L^{s_1,\,t_1}\E_{S^\p}~.
\end{eqnarray*}
where $S^\p=S\backslash\{i\}$.
For $t_c+1\leq j\leq i$, using (iv):
$\l_{j-1}\e_j=\e_{j-1}\e_j$ repeatedly, we get
\begin{eqnarray*}
W^S_T\e_i&=&\E_T\L^{s_k,\,t_k}\cdots(\l_{t_c}\l_{t_c+1}\cdots
\l_{i-1})\e_i\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_T\L^{s_k,\,t_k}\cdots(\l_{t_c}\l_{t_c+1}\cdots\l_{i-2}
\e_{i-1}\e_i)\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&\vdots& \\
&=&\E_T\L^{s_k,\,t_k}\cdots(\e_{t_c}\e_{t_c+1}\cdots
\e_{i-1}\e_i)\cdots\L^{s_1,\,t_1}\E_{S^\p}~.
\end{eqnarray*}
Now the situation is very similar to Case 1.3. We commute $\e_{t_c}$ past all $\l$ on the left and get
$\E_T\e_{t_c}=\E_{T^\p}$, where $T^\p=T\backslash\{t_c\}$.
For
$t_c+1\leq j\leq s_c-1$, commute $\e_j$ past all $\l$ on the right
until they reach an $\l_{j-1}$, then use $\e_j\l_{j-1}=\l_{j-1}$. For $s_{c}-2\le j\leq i$, commute $\e_j$ past all $\l$ on the right and until it meets an $\e_j$ in $\E_{S^\p}$, then use $\e_j^2=\e_j$. We find
\begin{eqnarray*}
W^S_T\e_i&=&\E_T\L^{s_k,\,t_k}\cdots(\e_{t_c}\cdots
\e_{i-1}\e_i)\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_T\e_{t_c}\L^{s_k,\,t_k}\cdots(\e_{t_c+1}\cdots
\e_{i-1}\e_i)\cdots\L^{s_1,\,t_1}\E_{S^\p}\\
&=&\E_{T^\p}L^{S^\p,\,T^\p}E_{S^\p}\\
&=&W^{S^\p}_{T^\p}~.
\end{eqnarray*}
Parts (3) and (4) are similar.
\end{proof}
\begin{remark}
{\rm
One can prove that the standard words can also be chosen as $\E_{S,\,T}\L^{S,\,T}$ where $\E_{S,\,T}=\e_{w_1}\e_{w_2}\dots\e_{w_h}$ if $\mathbf{n}-S-T=\{w_1,w_2,\dots,w_h\}$, and $L^{S,\,T}$ is the same as Proposition \ref{ml}.
}
\end{remark}
Let $G$ be the subset of $B_n$ consisting of the elements $1$, $l_i$, and $e_j$, where
\begin{align*}
l_i&=\left(
\begin{array}{ccccccc}
1 & \dots & i-1 & i+1 & i+2 & \dots & n \\
1 & \dots & i-1 & i & i+2 & \dots & n \\
\end{array}
\right)\quad \text{ and} \\
e_j&=\left(
\begin{array}{cccccc}
1 & \dots & j-1 & j+1 & \dots & n \\
1 & \dots & j-1 & j+1 & \dots & n \\
\end{array}
\right)~,
\end{align*}
where $1\le i\leq n-1$ and $1\leq j\leq n$.
Our intention here is to show that $G$ generates $B_n$. For any arbitrary $a, b\in \mathbf{n}$ with $a>b$, we define $L^{a,\,b}=l_b\cdots l_{a-2}l_{a-1}$
and $L^{a,\,a}=1$.
For $f\in B_n$, let
\[
S=\{s_1<\cdots<s_k\} \quad\text{and}\quad T=\{t_1<\dots<t_k\}
\]
be its respective domain and range. Let $U, V$ be as in (\ref{uv}) and let
\[
E_S=e_{u_1}\cdots e_{u_{n-k}},\quad L^{S,\,T}=L^{s_k,\,t_k}\cdots L^{s_1,\,t_1},\quad E_T=e_{v_1}\cdots e_{v_{n-k}}~,
\]
where we agree that $E_\mathbf{n}=1$. Then $x=E_TL^{S,\,T}E_S$, and we have shown
\begin{theorem}\label{gene}
Every element of $B_n$ is a product of elements of $G$.
\end{theorem}
The elements of $G$ satisfy the following relations (we omit the details, which are straightforward), where the indices $i$ and $j$ are such that all expressions in the relations are meaningful.
We use $R$ to denote the set of these relations:
\vspace{-2mm}
\begin{enumerate}[{\rm(1)}]
\item $e_i^2=e_i$~.
\vspace{-2mm}
\item $l_il_{i+1}l_i=l_il_{i+1}=l_{i+1}l_il_{i+1}$~.
\vspace{-2mm}
\item $l_ie_i=l_i=e_{i+1}l_i$~.
\vspace{-2mm}
\item $l_ie_{i+1}=e_ie_{i+1}=e_il_i=l_i^3=l_i^2$~.
\vspace{-2mm}
\item $e_il_j=l_je_i$ \quad for $i\neq j,j+1$~.
\vspace{-2mm}
\item $l_il_j=l_jl_i$ \quad for $|i-j|\geq 2$~.
\vspace{-2mm}
\item $e_ie_j=e_je_i$ \quad for all $i,j$~.
\end{enumerate}
\begin{theorem}\label{main}
The monoid $B_n$ has presentation $\langle\, G \mid R\,\rangle $.
\end{theorem}
\begin{proof}
The mapping $\phi:\hat{B}_n\rightarrow B_n$ defined by $\l_i\mapsto l_i$ and $\e_i\mapsto e_i$ and $\hat 1\mapsto 1$ induces a monoid homomorphism of $\hat{B}_n$ onto $B_n$. It follows from Proposition \ref{ml} that the mapping is injective.
\end{proof}
| {
"timestamp": "2016-10-14T02:06:09",
"yymm": "1610",
"arxiv_id": "1610.04122",
"language": "en",
"url": "https://arxiv.org/abs/1610.04122",
"abstract": "In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule of V. In particular, we provide a recursive formula for calculating the dimension of the cyclic module generated by v_S, and show that some dimensions are Catalan numbers, giving rise to new combinatorial identities.",
"subjects": "Representation Theory (math.RT)",
"title": "Modules and Structures of Planar Upper Triangular Rook Monoids",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9912886168290836,
"lm_q2_score": 0.7154239836484143,
"lm_q1q2_score": 0.7091916511971895
} |
https://arxiv.org/abs/1212.5934 | Some Bounds on the Rainbow Connection Number of 3-, 4- and 5-connected Graphs | The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show that for $\kappa =3$ or $\kappa = 4$, every $\kappa$-connected graph $G$ on $n$ vertices with diameter $\frac{n}{\kappa}-c$ satisfies $rc(G) \leq \frac{n}{\kappa} + 15c + 18$. We also show that for every maximal planar graph $G$, $rc(G) \leq \frac{n}{\kappa} + 36$. This proves a conjecture of Li et al. for graphs with large diameter and maximal planar graphs. | \section{Introduction}
Let $G$ be a simple, undirected, connected graph on $n$ vertices, such that its edges are colored by some edge coloring $c$. We say that a path $P$ in $G$ is a \textit{rainbow path} if no two edges of $P$ are the same color. We say that the edge-colored graph $(G, c)$ is \textit{rainbow-connected} if every pair of vertices is connected by a rainbow path, and that the coloring $c$ is a \textit{rainbow coloring} of the graph $G$. The \textit{rainbow connection number} of a graph $G$, $rc(G)$, is the minimum number of colors required to rainbow color $G$. For example, the rainbow connection number of a complete graph is $1$, that of a path, or in general, any tree, is $n-1$, and that of a cycle is $\lceil \frac{n}{2} \rceil$. A basic introduction to the subject can be found in Chapter 11 of \cite{surveybook}.
The concept of rainbow coloring was introduced by Chartrand, Johns, McKeon and Zhang in 2008 \cite{chartrand}. Computing the rainbow connection number of a graph was later shown to be NP-hard by Chakraborty, Fischer, Matsligh and Yuster \cite{chakraborty}. However, the rainbow connection number is still of interest as a `quantifiable' extension of the concept of connectivity in a graph \cite{caro, chakraborty}. In particular, there has been much interest in finding tight upper bounds for the rainbow connection number in terms of other measures of connectivity.
Recent results presented in the literature are as follows. Basavaraju et al. proved an upper bound in terms of radius, $rc(G) \leq r(r+2)$, and showed that it is tight \cite{bas}. Chandran et al. proved an upper bound in terms of minimum degree, $rc(G) \leq 3n/(\delta+1)+3$, and showed that it is tight up to additive factors \cite{chandran}. Upper bounds in terms of vertex and edge connectivity, $rc(G) \leq 3n/(\kappa+1)+3$ and $rc(G) \leq 3n/(\lambda+1)+3$, follow trivially, and Li et al. showed that the bound in terms of edge connectivity is tight up to additive factors for infinitely many values of $n$ and $\lambda$ \cite{li12}. Improving the bound in terms of vertex connectivity, Ekstein et al. and Li et al. independently proved that $rc(G) \leq \left\lceil\frac{n}{2}\right\rceil$ for 2-connected graphs, and that the bound is tight \cite{ekstein, li12}. In addition, Li et al. proved that $rc(G) \leq (2 + \varepsilon) \frac{n}{\kappa} + \frac{23}{\varepsilon^2}$ for any $\varepsilon > 0 $, so that $rc(G) \leq \frac{n}{\kappa} + C_0$ for graphs of high girth \cite{li12}. This led them to make the following conjecture.
\begin{conj}[Li et al. \cite{li12}]\label{mainconj}
Let $G$ be a $\kappa$-connected graph on $n$ vertices. Then there exists a constant $C$ such that
\begin{eqnarray*}
rc(G) < \frac{n}{\kappa} + C.
\end{eqnarray*}
\end{conj}
\noindent We prove Conjecture \ref{mainconj} for graphs with large diameter and maximal planar graphs.
\section{Preliminary Definitions and Results}
\begin{defn}
\end{defn} \vspace{-23pt}$~~~~~~~~~~~~~~$ Let $G$ be a simple, connected, undirected graph. $G$ is \textit{$k$-connected}, or \textit{$k$-vertex-connected}, if the removal of less than $k$ vertices does not disconnect the graph. Similarly, $G$ is \textit{$k$-edge-connected} if the remove of less than $k$ edges does not disconnect the graph. The \textit{vertex connectivity} of a graph, $\kappa(G)$, is the maximum value of $k$ such that $G$ is $k$-connected, and the \textit{edge connectivity}, $\lambda(G)$, is the maximum value of $k$ such that $G$ is $k$-edge-connected. Given a set $X \subset V(G)$, the graph $G|_X$ is the subgraph of $G$ induced by the set $X$, with vertex set $X$ and edge set given by the subset of edges of $G$ with both ends in $X$. The \textit{degree} of a vertex $deg(v)$ is the number of other vertices adjacent to it, and the \textit{minimum degree of $G$} is $\delta(G): = \min_{v \in V(G)} deg(v)$. A \textit{leaf} of a graph is a vertex with degree $1$.
\begin{defn}
\end{defn} \vspace{-23pt}$~~~~~~~~~~~~~~$ The \textit{length} of a path or cycle is the number of edges in the path or cycle. The \textit{girth} of a graph is the length of shortest cycle in the graph. The \textit{distance} between two vertices $u$ and $v$ of $G$, $d(u,v)$, is the length of the shortest path between them in $G$. The \textit{eccentricity} of a vertex $v$ is $ecc(v) := \max_{u \in V(G)} d(u,v)$. The \textit{diameter} of $G$ is $diam(G) := \max_{u,v \in V(G)} d(u,v)$, and the \textit{radius} of $G$ is $rad(G) := \max_{v\in V(G)} ecc(v)$. The distance from a vertex $v$ to a set $X\subset V(G)$ is $d(v,X) = \min{x \in X} d(v,x)$. The \textit{$l$-step open neighborhood} of a set $X \subset V(G)$ is $N^l(X) = \{v \in V(G) | d(v, X) = l\}$, and the \textit{$l$-step neighborhood} of $X$ is $N^l[X] = \{v \in V(G) | d(v, X) \leq l\}$. A set $D^l \subset V(G)$ is called an \textit{$l$-step dominating set of $G$} if every vertex in $G$ is in the $l$-step neighborhood of $D^l$, and if $G|_D$ is a connected subgraph of $G$, then $D^l$ is called a \textit{connected $l$-step dominating set of $G$}. The \textit{connected $l$-step domination number} of a graph $G$, $\gamma_c^l(G)$, is the minimum possible size for a connected $l$-step dominating set of $G$.
\begin{defn}
\end{defn} \vspace{-23pt} $~~~~~~~~~~~~~~~~~$ A \textit{planar graph} is a graph $G$ that can be embedded in the plane so as to preserve incidences and prevent crossing edges. Such an embedding of a graph in the plane is known as a \textit{planar embedding} of the graph. A \textit{face} of a planar graph is a minimal region bounded by edges. A \textit{maximal planar graph} $G$ is a planar graph where all faces are triangles.\\
\noindent Li et al. proved that Conjecture \ref{mainconj} holds for high girth graphs.
\begin{thm}[Li et al. \cite{li12}]
Let $G$ be a $\kappa$-connected graph.
\begin{itemize}
\item[(i)] If $\kappa \geq 3$ and $girth(G) \geq 7$, then $rc(G) \leq \frac{n}{\kappa} + 41$.
\item[(ii)] If $\kappa \geq 5$ and $girth(G) \geq 5$, then $rc(G) \leq \frac{n}{\kappa} +19$.
\end{itemize}
\end{thm}
\noindent Hence it remains to show that Conjecture \ref{mainconj} holds for low girth graphs.
\section{Bounds on $rc(G)$ for Graphs with Large Diameter}
Let $G$ be a graph on $n$ vertices with vertex connectivity $\kappa$. Then $rc(G) \geq diam(G)$, and as $G$ is $\kappa$-connected, the diameter of $G$ is at most $\frac{n}{\kappa} +1$. Hence for graphs with large diameter, the lower bound is close to the conjectured upper bound. We show that, in the cases $\kappa=3$ and $\kappa=4$, $rc(G)$ is close to $\frac{n}{\kappa}$ for graphs with diameter close to $\frac{n}{\kappa}$.
\begin{comment}
\begin{lem}
Let $G$ be a $3-connected$ graph on $n$ vertices. Suppose that there exist a pair of vertices $u_1, u_2 \in V(G)$ such that there are $k$ internally vertex-disjoint induced $u_1 \rightarrow u_2$ paths $P_1, \ldots, P_k$, and $t$ vertices not contained in any of these paths. Then
\begin{eqnarray*}
rc(G) \leq 3t + \max_i |P_i| +1.
\end{eqnarray*}
\end{lem}
\begin{proof}
Let $X_i = V(P_i)-\{u_1,u_2\}$ be the interior vertices of each path. Let $\{Y_1, \ldots Y_s\}$ be the vertex sets of the connected components of $G - \cup_i P_i$. Let $G'$ be the graph obtained from $G$ by contracting each component $G|_{Y_j}$ to a single vertex $y_j$. It is clear that $G'$ is still 3-connected.
We construct a rainbow coloring of $G'$ using $\max_i |P_i| + 3s + 1$ colors.
We first partition the $P_i$ into two non-empty sets $(A,B)$. For every path $P_i \in A$, we color its edges $c_1, c_2, \ldots, c_{|P_i|}$ in order from $u_1$ to $u_2$. For every path $P_j \in B$, we color its edges $c_1, c_2, \ldots, c_{|P_j|}$ in order from $u_2$ to $u_2$.
\end{proof}
\end{comment}
\begin{thm}
Let $G$ be a $\kappa$-connected graph on $n$ vertices and let $diam(G) = \frac{n}{\kappa}-c$, where $c \geq 0$.
\begin{itemize}
\item[(i)] If $\kappa = 3$, then $rc(G) \leq \frac{n}{3}+11c +6$.
\item[(ii)] If $\kappa = 4$, then $rc(G) \leq \frac{n}{4}+15c + 18$.
\end{itemize}
\end{thm}
\begin{proof}[Proof of (i)]$~$\\
Let $u_1, u_2 \in V(G)$ be a pair of vertices of $G$ such that $d(u_1,u_2) = diam(G)$. As $G$ is $3$-connected, there exist three internally vertex disjoint $u_1-u_2$ paths, $P_1$, $P_2$ and $P_3$. Note that we may choose the $P_i$ to be induced subgraphs, and we may assume without loss of generality that $|P_1| \leq |P_2| \leq |P_3|$. Let $X_i = V(P_i)-\{u_1,u_2\}$ be the interior vertices of each path, and let $X = X_1 \cup X_3 \cup \{u_1, u_2\}$. Let $\{Y_j\}$ be the vertex sets of the connected components of $G - \cup_i P_i$.
Let $G'$ be the graph obtained from $G$ by contracting each component $G|_{Y_j}$ to a single vertex $y_j$, and let $Y = \{y_j\}$ be the set of vertices obtained by this contraction. It is clear that $G'$ is still 3-connected. This means, in particular, that any vertex $v \in X_i$ is incident with some vertex $w \not\in X_i$, as we have chosen the $P_i$ to be induced subgraphs of $G$.
Color the cycle $C=u_1 \rightarrow P_1 \rightarrow u_2 \rightarrow P_3 \rightarrow u_1$ using $\left\lceil\frac{1}{2}(|P_1|+|P_3|-2)\right\rceil$ colors, in cyclic order $c_1, c_2, \ldots, c_{m}, c_1, \ldots$, so that $G$ restricted to $C$ is rainbow colored. Color the edges of the path $P_2$ using $|P_2|-1$ colors, one for each edge, reusing as many of the colors $\{c_1, \ldots, c_{m}\}$ as possible. Following this, all edges in the paths $P_i$ are now colored using $m := \max\left\{\left\lceil\frac{1}{2}(|P_1|+|P_3|-2)\right\rceil, |P_2|-1\right\}$ colors.
We now color the edges in $E(G') - \cup_i E(P_i)$. For each vertex of $G'$ not in $X$, we let $l(v)$ be the minimum length of a path from $v$ to $X$, with the constraint that the path does not contain adjacent edges in $X_2$ and uses the fewest edges of $P_2$ possible for any $v\rightarrow X$ path satisfying this constraint. As $G'$ is 3-connected, $l(v)$ is well-defined for every vertex $v \in X_2 \cup Y$.
Let $p(v)$ be a fixed vertex adjacent to $v$ such that $l(p(v)) < l(v)$. If $v \in X$, we take $l(v) = 0$. Hence for each $v$, the vertices $v, p(v), p(p(v)), \ldots, p^{(l(v))}(v)$ form a minimal $v \rightarrow X$ path that uses as few edges of $P_2$ as possible. Let $E_p$ be the set of edges of the form $\{v, p(v)\}$ for some $v \in V(G)-X$.
Now color each edge $e \not \in \cup_i P_i$ and recolor some of the edges $e \in P_2$ by the color $c_e$, where $c_e$ is given by
\vspace{-5pt}
\begin{eqnarray*}
c_e = \left\{
\begin{array}{cc}
c_{m + l(v)} & \text{ if } e = \{v, p(v)\} \text{ and } v \in X_2 \\
c_{v} & \text{ if } e = \{v, p(v)\} \text{ and } v \in Y \\
d & \text{ if } e = \{u,v\}\not\in E_p \text{ and } u \in Y, v \in X \cup \{u_1, u_2\} \\
e & \text{ if } e = \{u,v\}\not\in E_p \text{ and } u \not\in X_2, v \in X_2.
\end{array}
\right.
\end{eqnarray*}
Then each of the edges of $G'$ are colored using one of $m + \max_{v} l(v) + |Y| + 2$ colors.
We may bound $\max_{v} l(v)$ as follows. Each $v \in Y$ is incident only to vertices in $X \cup X_2$. Moreover, if $v \in X_2$ and $l(v) > 1$ then $v$ is incident to a vertex in $Y$. Finally, there is no pair of adjacent edges in any path $v \rightarrow p(v) \rightarrow p(p(v)) \cdots p^{(l(v))}(v)$ that are both in $P_2$. Hence, if $D_l$ is the set of vertices with domination distance $l$, then every three sets $D_l, D_{l+1}, D_{l+2}$ contains at least one member of $Y$. Hence $\max_v l(v) \leq 3|Y|$.
Finally, we show that such a coloring, applied after first coloring the edges of $P_1 \cup P_3$, is a rainbow coloring of $G'$. We first note the following useful facts.
\begin{itemize}
\item \underline{If $u, v \in X$ then there is a rainbow $u \rightarrow v$ path using only colors in $\{c_1, \ldots, c_{m}\}$.}\\
It is easy to verify that the cycle $G'|_{P_1 \cup P_3}$ is rainbow colored. Hence there is a rainbow $u\rightarrow v$ path contained in $P_1 \cup P_3$.
\item \underline{If $u, v \in X_2$ then there is a rainbow $u \rightarrow v$ path using only colors in $\{c_1, \ldots, c_{3|Y|}\}$.}\\
It is easy to verify that $P_2$ is a rainbow path, and hence any subpath is also a rainbow path. Moreover, as $\max_v l(v) \leq 3|Y|$, all the colors used are in the set $\{c_1, \ldots, c_{3|Y|}\}$.
\item \underline{If $u \not\in X$ then there is a rainbow $u \rightarrow X$ path using only colors in $\{c_v\}$.}\\
This is also clear, as the $u \rightarrow X$ path $u, p(u), \ldots, p^{(l(u))}(u)$ is such a path.
\item \underline{If $u \not\in X$ then there is a rainbow $u \rightarrow X_2 \cup X$ path using only colors in $\{d, e\}$.}\\
This follows from 3-connectivity of $G'$. For if $u$ is a vertex of $G$, either $u \in X_2$ and the $u \rightarrow X_2$ path exists trivially, or $u \in Y$ and is adjacent only to vertices in $X$ and $X_2$. Since $G'$ is 3-connected, $u$ is incident with at least one edge not in $E_p$, which forms the required $u \rightarrow X_2 \cup X$ path.
\end{itemize}
Let $u,v$ be a pair of vertices in $G'$. We show that there is a rainbow $u\rightarrow v$ path.
\begin{itemize}
\item \underline{Case 1: Both $u$ and $v$ are in $X$, or both $u$ and $v$ are in $X_2$.} These cases have been shown above.
\item \underline{Case 2: Exactly one of $u$ and $v$ is in $X$.}\\
We may assume without loss of generality that $v \in X$, $u \not\in X$. Then there exists some $w \in X$ such that there is a $u \rightarrow w$ path using only colors in $\{c_v\}$, and as $w \in X$ there is a $w \rightarrow v$ path using colors in $\{c_1, \ldots, c_{m}\}$, so concatenating these gives the required rainbow $u\rightarrow v$ path.
\item \underline{Case 3: Both $u$ and $v$ are not in $X$.}\\
If either $u$ or $v$ is adjacent, via an edge not in $E_p$, to a vertex in $X$, then we may choose the following rainbow $u\rightarrow v$ path. Without loss of generality let $v$ be adjacent to $v' \neq p(v)$ in $X$, so that there is a $v'\rightarrow v$ path using only the colors in $\{d,e\}$. As $u \not\in X$, there is also a $u \rightarrow X$ path using only colors in $\{c_v\}$ ending at some vertex $u'$. Finally, there is a $u' \rightarrow v'$ path using only colors in $\{c_1, \ldots, c_{m}\}$. Concatenating these three paths, we obtain a rainbow $u \rightarrow u' \rightarrow v' \rightarrow v$ path.
Otherwise the 1-step neighborhoods of $u$ and $v$ contain only vertices in $X_2 \cup Y$. Let $u' = p(u)$ and $v'$ be the end of a rainbow $v \rightarrow X \cup X_2$ path using only colors in $\{d,e\}$, where we may choose both $u'$ and $v'$ to be in $X_2$. Then we may concatenate the edge $u \rightarrow u'$, colored by the color $c_u$, with a $u' \rightarrow v'$ path using only colors in $\{c_1, \ldots, c_{3|Y|}\}$ and the $v'\rightarrow v$ path using only colors in $\{d,e\}$ to obtain a $u\rightarrow v$ rainbow path.
\end{itemize}
Hence $rc(G') \leq m + 4|Y| + 2$.
If we then take a spanning forest of $G - \cup_i P_i$ and color each of its edges a different color, we obtain a rainbow coloring of $G$ using at most $m+ 4|Y| + 2 + \sum_i(|Y_i|-1) \leq m+ 3|Y|+\sum_i |Y_i|+2$ colors. A bit of arithmetic yields
\begin{eqnarray*}
\left\lceil\frac{1}{2}(|P_1|+|P_3|-2)\right\rceil + \left\lfloor\frac{1}{2}(|P_1|+|P_3|-2)\right\rfloor + (|P_2|-1) - 1&=& n - \sum_i |Y_i|
\end{eqnarray*}
\vspace{-15pt}
\begin{eqnarray*}
m + 11diam(G) - 4 &\leq& 4n - 4\sum_i |Y_i| \\
m + 4\sum_i |Y_i| &\leq& \frac{n}{3} +4 + 11c \\
rc(G) ~\leq~ m+ 3|Y| +\sum_i |Y_i|+2 &\leq& \frac{n}{3} + 11c + 6.
\end{eqnarray*}
Hence $rc(G) \leq \frac{n}{3} + 11c + 6$, as required.
\end{proof}
\begin{proof}[Proof of (ii)]
As before, take $u_1, u_2 \in V(G)$ a pair of vertices with distance $d(u_1, u_2) = diam(G)$, and four internally vertex disjoint $u_1\rightarrow u_2$ paths $P_1, P_2, P_3, P_4$. Similarly define $X_i$, define $\{Y_i\}$ as the vertex sets of the components of $G - \cup_i P_i$, let the graph $G'$ be obtained from $G$ by contracting each set $Y_i$ to a single point, and define $Y$ as before.
Color the paths $P_1$ and $P_4$ in order from $u_1$ to $u_2$ using the colors $\{c_1, \ldots, c_{|P_i|-1}\}$ and color the paths $P_2$, $P_3$ in reverse order, from $u_2$ to $u_1$.
Let $Z_i$ be the set of vertices in $Y$ and only adjacent to members of $X_i$, and for each $v \in X_i \cup Z_I$ let $l_i(v)$ to be the minimum length of a path from $v$ to $X_1$, using as few of the edges in $P_i$ as possible. Let $E_{p_i}$ be the set of all edges of the form $\{v, l_i(v)\}$.
Now color each edge $e \not\in \cup_i P_i$ and recolor some of the edges $e \in \cup_i P_i$ by the color $c_e$, where $c_e$ is given by
\vspace{-5pt}
\begin{eqnarray*}
c_e = \left\{
\begin{array}{cc}
c_{i,l_i(v) } & \text{ if } e = \{v, l_i(v)\} \text{ and } v \in X_i \\
c_{i, v} & \text{ if } e = \{v, l_i(v)\} \text{ and } v \in Z_i \\
d_i & \text{ if } e = \{u,v\}\not\in E_{p_i} \text{ and } u \in Y, v \in X_i \\
e_{i,j} & \text{ if } e = \{u,v\}\not\in E_{p_i} \text{ and } u \in X_i, v \in X_j.
\end{array}
\right.
\end{eqnarray*}
As in the proof of (i), for any pair of vertices in $u \in X_j, v \in X_k$, unless $j + k = 5$, there is clearly a $u\rightarrow v$ rainbow path along the $P_i$. For any $u \in Z_i$ there is a rainbow $u \rightarrow \cup_{j \neq i} X_j$ path using only colors in $\{c_{i,v}\}$ and a rainbow $u \rightarrow X_i$ edge using the color $\{d_i\}$. Following a similar case analysis as in (i), concatenating rainbow paths within $X_i \cup Z_i$ with rainbow paths outside of $X_i \cup Z_i$, it can be shown that the assigned colors form a rainbow coloring of $G'$.
Again as in the proof of (i), it can also be seen that this rainbow coloring uses at most $\max_i (|P_i|-1) + \sum_i \max_v l_i(v) + \sum_i |Z_i| + 4 + {4 \choose 2}$ colors, where $\max_v l_i(v) \leq 3|Z_i|$ for each $i$. Hence
\begin{eqnarray*}
rc(G') &\leq& \max_i (|P_i|-1) + 4\sum_i |Z_i| + 10~~~~~
\end{eqnarray*}
\vspace{-17pt}
\begin{eqnarray*}
rc(G) &\leq& \left(\max_i (|P_i|-1) + 4\sum_i |Y_i| \right)+ 10\\
&\leq& \left(4n - 15(diam(G)) +8 \right) + 10 \\
&=& 4n - 15\left(\frac{n}{4}-c\right) +18 \\
&=& \frac{n}{4} + 15c + 18.
\end{eqnarray*}
This completes the proof.
\end{proof}
\section{Bounds on $rc(G)$ for Maximal Planar Graphs}
In this section, we prove that if $G$ is a maximal planar graph, then $rc(G) \leq \frac{n}{\kappa} + 36$. We remark that the proof can be quite simply extended to show that $rc(G) \leq \frac{n}{\kappa} + C_f$ for any planar graph graph with maximum face size $f$ and constants $C_f$ which grow at least linearly in $f$.
\begin{thm}\label{thm:max}
Let $G$ be a $\kappa$-connected maximal planar graph on $n$ vertices.
\begin{itemize}
\item[(i)] If $\kappa = 3$, then $rc(G) \leq \frac{n}{3} + 16$.
\item[(ii)] If $\kappa = 4$, then $rc(G) \leq \frac{n}{3} + 25$.
\item[(iii)] If $\kappa = 5$, then $rc(G) \leq \frac{n}{3} + 36$.
\end{itemize}
\end{thm}
We note that any planar graph has a vertex of degree at most $5$, and therefore has vertex connectivity at most $5$. Thus Theorem \ref{thm:max} proves Conjecture \ref{mainconj} for $C=36$ and all maximal planar graphs.
In proving the theorem, we make use of the following lemmas.
\begin{lem}[Basavaraju et al. \cite{bas}]\label{lem:bas}
Let $G$ be a bridgeless graph and let $D^l$ be a connected $l$-step dominating set of $G$. Then
\begin{eqnarray*}
rc(G) \leq rc(D^l)+l^2+2l.
\end{eqnarray*}
\end{lem}
\begin{lem}\label{lem:2conn_neighbs}
Let $G$ be a planar embedding of a maximal planar graph. Then for any cycle $C$, $N^1(C)$, the one-step neighborhood of $C$, is comprised of two 2-connected components, one inside $C$ and one outside $C$.
\end{lem}
\begin{proof}
By symmetry, it suffices to show that the component outside of $C$ is 2-connected.
Let $v_i$ be a vertex of $C$ and let $v_{i,1}, v_{i,2}, \ldots, v_{i,m_i}$ be the vertices outside of $C$ adjacent to $v$, in counterclockwise order. Then as $G$ is maximal planar, $v_{i,j}$ and $v_{i,j+1}$ are adjacent for all $i, 1 \leq j < m_i$.
Let $v_i, v_{i+1}$ be two adjacent vertices in $C$, in counterclockwise order. Then as $G$ is maximal planar, $v_{i,m_i} = v_{i+1,1}$.
Hence the subgraph induced by those vertices of $N^1(C)$ outside of $C$ is Hamiltonian, and hence 2-connected.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:max}]$~$\\
\noindent Let $F$ be the vertex set of an arbitrary face of $G$, and let $N_k = N^k(F)$ be the $k$-step open neighborhood of $F$. Let $t$ be maximal such that $N_t$ is non-empty.
\begin{comment}
We define a series of vertex sets $N_k$ as follows. Let $N_0 F \subset V(G)$ be the vertex set of an arbitrary face of $G$. Given $N_k$, let $N'_{k+1}$ be the 1-step neighborhood of $N_k$ on the outside of the boundary of $N_k$. If $|N'_{k+1}|$ is even, let $N_{k+1}=N'_{k+1}$, and if $|N'_{k+1}|$ is odd, choose some vertex $v$ outside $N'_{k+1}$ that is incident with two adjacent vertices of $N'_{k+1}$, and let $N_{k+1} = N'_{k+1} \cup \{v\}$. Such a vertex exists as $G$ is a maximal planar graph.
We claim first that the $N_k$ may be constructed so that, for each $0 \leq s < t$, there exists a path from $F$ to the outer boundary of $N_k$ of length $s$ containing exactly one vertex from each $N_k$. It will become clear later why this is important.
The case $s=0$ is trivial, so suppose $N_k$ has been constructed for $0 \leq k \leq s$. If $|N^1(N_s)| = |N'_{s+1}|$ is even, then we may take $N_{s+1} = N'_{s+1}$. Otherwise, we must add a vertex.
Suppose the graph induced by $N^2(N_s)$ is not a single cycle. Then, as $G$ is maximal planar, the area between the inner boundary and the outer boundary of $N^2(N_k)}$ is triangulated. Hence there exists a
Suppose that the process stops at $N_t \neq \emptyset$. By induction, each $N'_k$ is 2-connected and has a cycle for its boundary, and so by construction $N_k$ is also 2-connected and has a cycle for its boundary. Moreover, as $G$ is $\kappa$-connected, $|N_k| \geq \kappa$ is even for $1 \leq l < t$, and $|N_0| = |F| = 3$.
\end{comment}
We construct a connected $\kappa$-step dominating set $D$ of $G$ and show that $rc(D) \leq \frac{n}{\kappa} + 1$. Then by Lemma \ref{lem:bas}, $rc(G) \leq rc(D) + \kappa^2+2\kappa = \frac{n}{\kappa} + 1 + \kappa^2 + 2\kappa$ and the theorem is proved.
Let $A_1 = \{k : |N_k| \leq 2\kappa-1\}$, $A_2 =\{k: |N_k| = 2\kappa\}$ and $A_3 =\{k: |N_k| \geq 2\kappa+1\}$, with cardinalities $n_i=|A_i|$. We label the elements of $A_2 \cup A_3$ in ascending order, $A_2 \cup A_3 = \{k_1, \ldots, k_{|A_2 \cup A_3|}$. We choose a path $P$ of length $t$ containing exactly one vertex from each of the $N_k$. We also choose $A = \{k_i: i \equiv a ~(mod~ \kappa)\}$ to be the congruence class, modulo $\kappa$, such that $\sum_{k \in A} \left\lceil \frac{|N_k|}{2} \right\rceil$ is minimized. Let $D = F \cup \{N_k : k \in A\} \cup V(P)$.
We show first that $D$ is a connected $l$-step dominating set of $G$. Connectivity is clear, as $P$ is connected, $N_k$ is connected for each $k$ and $P \cap N_k$ is non-empty for each $k$. To show that $D$ is an $l$-step dominating set of $G$, consider some $v \in N_k$. Let $k' \leq k$ be maximal such that either $|N_{k'}| \leq 2 l_1+1$, or $k' \in A$. Let $v' \in N_{k'}$ be chosen such that $d(v,v') = k-k'$. Then $d(v, D) \leq d(v,v') + d(v',D)$. Note that $k-k' < l_2$.
If $k'$ is such that $|N_{k'}| \leq 2 l_1+1$, then there is a vertex $v_{k'} \in N_{k'} \cap D$, so $d(v', D) \leq d(v', v_{k'}) \leq l_1$, as $N_{k'}$ is 2-connected. Hence $d(v, D) \leq l_2-1+l_1 = l$. If $k'$ is such that $k'=k_i$ for some $i$, then $v' \in D$ so $d(v, D) = d(v,v') < l$. Hence $D$ is a connected $l$-step dominating set of $G$.
We now show that $rc(D) \leq \frac{n}{\kappa}$ by constructing a rainbow coloring of $D$. Rainbow color $P$ using $|P|$ colors, and as each $N_k$ is 2-connected, rainbow color each $N_k$ using at most $\left\lceil \frac{|N_k|}{2} \right\rceil$ colors. As new colors are used for each $N_k$, this clearly gives a rainbow coloring of $D$.
Moreover, the rainbow coloring uses at most $|P| + \sum_{k \in A} \left\lceil \frac{|N_k|}{2} \right\rceil$ colors. Hence
\vspace{-5pt}
\begin{eqnarray*}
rc(D) &\leq& (n_1+n_2+n_3-1) + \frac{1}{\kappa}\sum_{k \in A_2 \cup A_3} \left\lceil \frac{|N_k|}{2} \right\rceil ~~~~~~~~~~~~~\text{ (by choice of $A$)}\\
&\leq& (n_1+n_2+n_3-1) + \frac{1}{2\kappa}\sum_{k \in A_2} |N_k| + \frac{1}{2\kappa}\sum_{k \in A_3} \left(|N_k| + 1\right)
\end{eqnarray*}
\begin{equation}\label{eqn:rc}
rc(D) ~\leq~ n_1+\frac{2\kappa n_2+ (2\kappa + 1)n_3}{2\kappa}-1 + \frac{1}{2\kappa}\sum_{k \in A_2 \cup A_3} |N_k|. ~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation}
To simplify this, we note that if $k \in A_2$ then $|N_k| = 2\kappa$, and if $k \in A_3$ then $|N_k| \geq 2\kappa +1$. Hence
\begin{equation}\label{eqn:A2A3}
\sum_{k \in A_2 \cup A_3}|N_k| ~\geq~ 2\kappa n_2 + (2\kappa +1)n_3.
\end{equation}
Moreover, if $k\neq 0, t \in A_1$ then $|N_k| \geq \kappa$, we also have $|N_0| = 3$ and $|N_t| \geq 1$. Hence
\begin{equation}\label{eqn:ns}
\sum_{k \in A_2 \cup A_3} |N_k| ~=~ n - \sum_{k \in A_1} |N_k| ~\leq~ n - \kappa n_1 + 2\kappa - 4.
\end{equation}
Putting together equations (\ref{eqn:rc}), (\ref{eqn:A2A3}) and (\ref{eqn:ns}), we obtain
\begin{eqnarray*}
rc(D) ~\leq~ n_1 + \frac{1}{\kappa}\sum_{k \in A_2 \cup A_2} |N_k| -1 ~\leq~ n_1 + \frac{1}{\kappa} (n - \kappa n_1 + 2\kappa -4) - 1 ~\leq~ \frac{n}{\kappa} + 1.
\end{eqnarray*}
This gives the required bound on $rc(D)$ and completes the proof of the theorem.
\end{proof}
We note that Theorem \ref{thm:max} can be extended to planar graphs with maximum face size $f$, by defining the $N_k$ recursively, taking $N_{k+1}$ to be a 2-connected subset of $N^{\left\lceil\frac{f-1}{2}\right\rceil}(N_k)$-step open neighborhood of $N_k$. However, the constant $C$ grows at least linearly with $f$, so this would not lead to a proof of the general case.
\section{Acknowledgements}
This research was performed at the University of Minnesota Duluth REU run by Joe Gallian. The REU was supported by the National Science Foundation, the Department of Defense (grant number DMS 1062709), the National Security Agency (grant number H98230-11-1-0224) and Princeton University.
I would like to thank Joe Gallian for supervising this research. I am also grateful to Eric Riedl, David Rolnick, Samuel Elder and the other participants at the 2012 Duluth REU for their insightful comments and support.
\bibliographystyle{plain}
| {
"timestamp": "2012-12-27T02:03:17",
"yymm": "1212",
"arxiv_id": "1212.5934",
"language": "en",
"url": "https://arxiv.org/abs/1212.5934",
"abstract": "The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show that for $\\kappa =3$ or $\\kappa = 4$, every $\\kappa$-connected graph $G$ on $n$ vertices with diameter $\\frac{n}{\\kappa}-c$ satisfies $rc(G) \\leq \\frac{n}{\\kappa} + 15c + 18$. We also show that for every maximal planar graph $G$, $rc(G) \\leq \\frac{n}{\\kappa} + 36$. This proves a conjecture of Li et al. for graphs with large diameter and maximal planar graphs.",
"subjects": "Combinatorics (math.CO)",
"title": "Some Bounds on the Rainbow Connection Number of 3-, 4- and 5-connected Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575178175919,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542250430822
} |
https://arxiv.org/abs/2302.07595 | Macaulay's theorem for vector-spread algebras | Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\ldots,t_{d-1})\in{\mathbb{Z}}_{\ge 0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a ${\bf t}$-spread ideal $I$ in $S$, we associate a unique $f_{\bf t}$-vector and we prove that if $I$ is ${\bf t}$-spread strongly stable, then there exists a unique ${\bf t}$-spread lex ideal which shares the same $f_{\bf t}$-vector of $I$ via the combinatorics of the ${\bf t}$-spread shadows of special sets of monomials of $S$. Moreover, we characterize the possible $f_{\bf t}$-vectors of ${\bf t}$-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal-Katona. Finally, we prove that among all ${\bf t}$-spread strongly stable ideals with the same $f_{\bf t}$-vector, the ${\bf t}$-spread lex ideals have the largest Betti numbers. | \section*{Introduction}
One of the main well--studied and important numerical invariant of a graded ideal in a standard graded polynomial ring is its Hilbert function which gives the sizes of the graded components of the ideal. There is an extensive literature on this topic (see, for instance, \cite{JT} and the references therein). Usually, Hilbert functions are described using the well--known Macaulay's expansion with binomials. This fact often implies the use of combinatorics tools and furthermore the arguments consist of very clever computations with binomials. The crucial idea of Macaulay is that there exist special monomial ideals, the so called \emph{lex ideals}, that attain all possible Hilbert functions. The pivotal property is that a lex ideal grows as slowly as possible. The ``squarefree'' analogue of Macaulay’s theorem is known as the Kruskal–Katona theorem. Indeed, if Macaulay’s theorem describes the possible Hilbert functions of the graded ideals in polynomial rings, the possible $f$-vectors of a simplicial complex are characterized in the theorem of Kruskal–Katona \cite{JT, GK, JK}. In fact, the Hilbert function of the Stanley–Reisner ring of a simplicial complex $\Delta$ is determined by the $f$-vector of $\Delta$, and vice versa. Kruskal–Katona's theorem is a fundamental result in topological combinatorics and discrete geometry which quickly have aroused much interest in face enumeration questions for various classes of simplicial complexes, polytopes, and manifolds. Furthermore, it may be also interpreted as a theorem on Hilbert functions of quotients of exterior algebras \cite{AHH}.
The lex ideals as well as squarefree lex ideals play a key role in the study of the minimal free resolutions of monomial ideals. Indeed, if one consider the stable and squarefree stable ideals and the formulas for computing their graded Betti numbers \cite{JT} one can deduce the Bigatti-Hulett theorem \cite{BAM, HH} which says that lex ideals have the largest graded Betti numbers among all graded ideals with the same Hilbert function (see also \cite{HH2, KP}).
Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\dots,t_{d-1})\in{\NZQ Z}_{\ge0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. Recently in \cite{F1}, the class of ${\bf t}$\textit{-spread strongly stable ideals} has been introduced. It is a special class of monomial ideals which generalizes the class of $t$-spread strongly stable ideals in \cite{EHQ}, $t$ non negative integer. More in detail, a monomial $u=x_{j_1}x_{j_2}\cdots x_{j_\ell}$ ($1\le j_1\le j_2\le\cdots\le j_\ell\le n$) of degree $\ell\le d$ of $S$ is called a \textit{vector-spread monomial of type ${\bf t}$} or simply a \textit{${\bf t}$-spread monomial} if $j_{i+1}-j_{i}\ge t_{i}$, for $i=1,\dots,\ell-1$ and a \textit{${\bf t}$-spread monomial ideal} is a monomial ideal of $S$ generated by ${\bf t}$-spread monomials. A \textit{${\bf t}$-spread strongly stable ideal} is a \textit{${\bf t}$-spread monomial ideal} with an additional combinatorial property (Definition \ref{def:stronglylex}). For ${\bf t}\in \{(0, \ldots, 0), (1, \ldots, 1)\}$, one obtains the classical notions of strongly stable ideal and squarefree strongly stable ideal, respectively \cite{JT}.
The aim of this article is to generalize Macaulay's theorem for the class of ${\bf t}$-spread strongly stable ideals. The crucial role is played by the class of \textit{${\bf t}$-spread lex ideal} (Definition \ref{def:stronglylex}). Since, ${\bf t} \in {\NZQ Z}_{\ge0}^{d-1}$, \emph{i.e.}, the entries of ${\bf t}$ can be also zero, in order to unify the theory about the classification of the Hilbert functions of graded ideals of $S$, we put our attention on the classification of the possible $f_{{\bf t}}$-vector of ${\bf t}$-spread strongly stable ideals. More in detail, we answer to the following question: \emph{under which conditions a given sequence of positive integers $f = (f_{-1}, f_{0}, \ldots, f_{d-1})$
is the $f_{{\bf t}}$-vector of a ${\bf t}$-spread strongly stable ideal?}
The plain of the paper is as follows. Section \ref{sec1} contains some preliminaries and notions that will be used in the article. We introduce the notion of \textit{${\bf t}$-spread shadow} of a set of monomials of $S$ and the notions of ${\bf t}$\textit{-spread strongly stable set (ideal)} and ${\bf t}$\textit{-spread lex set (ideal)} (Definitions \ref{def:stronglylexset}, \ref{def:stronglylex}). The combinatorics of such sets is deeply analyzed in Section \ref{sec2}. The key result in the section is Theorem \ref{Thm:BayerVectSpread} which allows us to prove that to every ${\bf t}$-spread strongly stable ideal $I$ one can associate a unique ${\bf t}$-spread lex ideal which shares the same $f_{\bf t}$-vector of $I$ (Corollary \ref{Cor:SubstituteTLex}). Moreover, we point out why this is not possible for an arbitrary ${\bf t}$-spread monomial ideal. Section \ref{sec3} contains the main result in the article which gives the classification of all possible $f_{\bf t}$-vectors of a ${\bf t}$-spread strongly stable ideal (Theorem \ref{thm:main}). The classification is obtained by introducing a new operator (Definition \ref{Def:vector-spreadOp}) which, for suitable values of ${\bf t}$, is analog either to the operator $a\longrightarrow a^{\langle d\rangle}$ or to the operator $a\longrightarrow a^{(d)}$ which are involved in the Macaulay theorem and in the Kruskal--Katona theorem, respectively \cite{JT}. Finally, in Section \ref{sec4}, as an application of the results in the previous sections, we state an upper bound for the graded Betti numbers of the class of all ${\bf t}$-spread strongly stable ideals with a given $f_{{\bf t}}$-vector (Theorem \ref{thm:upperbound}). We prove that the ${\bf t}$-spread lex ideals give the maximal Betti numbers among all ${\bf t}$-spread strongly stable ideals with a given $f_{{\bf t}}$-vector. Such a statement generalizes the well--known result proved independently by Bigatti \cite{BAM} and Hulett \cite{HH} for graded ideals in a polynomial ring with coefficient in a field of characteristic zero and afterwards generalized by Pardue \cite{KP} in any characteristic. The article contains some examples illustrating the main results developed using \emph{Macaulay2} \cite{GDS}.
\section{Preliminaries and notations}\label{sec1}
Let $S=K[x_1,\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\bf t}=(t_1,\dots,t_{d-1})\in{\NZQ Z}_{\ge0}^{d-1}$, $d\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. A monomial $u=x_{j_1}x_{j_2}\cdots x_{j_\ell}$ ($1\le j_1\le j_2\le\cdots\le j_\ell\le n$) of degree $\ell\le d$ of $S$ is called a \textit{vector-spread monomial of type ${\bf t}$} or simply a \textit{${\bf t}$-spread monomial} if $j_{i+1}-j_{i}\ge t_{i}$, for $i=1,\dots,\ell-1$.
If $I$ is a graded ideal of $S$ we denote by $I_j$ the $j$-graded component of $I$ and by $\indeg(I)$ the initial degree of $I$, \emph{i.e.}, the smallest $j$ such that $I_j\neq 0$. Moreover, for a monomial ideal $I\subset S$, we denote by $G(I)$ the unique minimal set of monomial generators. Furthermore, if $j\ge0$ is an integer, we set $G(I)_j=\{u\in G(I):\deg(u)=j\}$.
A \textit{${\bf t}$-spread monomial ideal} is a monomial ideal of $S$ generated by ${\bf t}$-spread monomials.
For instance, $I=(x_1x_4^2x_5,x_1x_4^2x_6,x_1x_5x_7)$ is a $(3,0,1)$-spread monomial ideal of the polynomial ring $S=K[x_1, \ldots, x_7]$, but it is not $(3,0,2)$-spread as $x_1x_4^2x_5\in G(I)$ is not a $(3,0,2)$-spread monomial.
Note that any monomial (ideal) is ${\bf 0}$-spread, where ${\bf 0}=(0,0,\dots,0)$. If $t_i\ge1$, for all $i$, a ${\bf t}$-spread monomial (ideal) is a \textit{squarefree} monomial (ideal).
We denote by $M_{n,\ell,{\bf t}}$ the set of all ${\bf t}$-spread monomials of degree $\ell$ in $S$. If $\ell \leq d$, by \cite[Corollary 2.4]{F1},
\begin{equation}\label{Formula:|Mn,l,t|}
\lvert M_{n,\ell,{\bf t}} \rvert = \binom{n+(\ell-1)-\sum_{j=1}^{\ell-1}t_j}{\ell}.
\end{equation}
For a set $L$ of monomials of $S$, one defines the \textit{vector-spread shadow} or simply the \textit{${\bf t}$-spread shadow} of $L$ to be the set
\begin{align*}
\Shad_{\bf t}(L)\ &=\ \big\{x_iw:w\in L, x_iw\ \textup{is}\ {\bf t}\textup{-spread}, \, i=1,\dots,n \big\}.
\end{align*}
Note that $\Shad_{\bf t}(M_{n,\ell,{\bf t}})=M_{n,\ell+1,{\bf t}}$ for all $\ell\ge 0$ and that $\Shad_{\bf t}(L)=\emptyset$ whenever all monomials in $L$ have degrees $\ge d$. Moreover, one can quickly observe that if $L$ is a set of monomials of $S$, then the definition of $\Shad_{\bf 0}(L)$ coincides with the classical notion of shadow of $L$ \cite[Chapter 6]{JT}.
If $I$ is a monomial ideal of $S$, we denote by $[I_j]_{\bf t}$ the set of all ${\bf t}$-spread monomials in $I_j$. Furthermore, we set
\[f_{{\bf t},\ell-1}(I) = \vert M_{n,\ell,{\bf t}}\vert - \vert [I_\ell]_{\bf t}\vert, \qquad 0\le\ell\le d.\]
and define the vector
\[f_{\bf t} (I)= (f_{{\bf t},-1}(I), f_{{\bf t},0}(I), \ldots, f_{{\bf t}, d-1}(I)).\]
Such a vector is called the \textit{$f_{\bf t}$-vector} of $I$. Note that $f_{{\bf t},-1}(I)=1$.
Let $i,j$ integers, we set $[i,j]=\{k\in{\NZQ Z}: i\leq k\leq j\}$. Note that $[i,j]\ne\emptyset$ if and only if $i\le j$. Moreover, if $i=1$, we denote the set $[1,j]$ simply by $[j]$.
One can observe that for ${\bf t} =(t_1, \ldots, t_{d-1})$ with $t_i\ge 1$, for all $i$, then $I$ is the Stanley--Reisner ideal $I_\Delta$ of a uniquely determined simplicial complex $\Delta$ on vertex set $[n]$ with $f_{\bf 1}(I)$ as $f$-vector, where ${\bf 1} =(1, \ldots, 1)$.
\begin{Definition} \label{def:stronglylexset}
\rm Let $L\subseteq M_{n,\ell,{\bf t}}$. $L$ is called a ${\bf t}$\textit{-spread strongly stable set} if for all $u\in L$, $j<i$ such that $x_i$ divides $u$ and $x_j(u/x_{i})$ is ${\bf t}$-spread, then $x_j(u/x_{i})\in L$. $L$ is called a ${\bf t}$-\textit{spread lex set}, if for all $u\in L$, $v\in M_{n,\ell,{\bf t}}$ such that $v\ge_{\lex}u$, then $v\in L$.
\end{Definition}
Here $\ge_{\lex}$ stands for the lex order induced by $x_1>\dots>x_n$ \cite{JT}.
For our convenience, throughout the article, we assume the empty set to be both a ${\bf t}$-spread strongly stable set and a lex set.
\begin{Definition} \label{def:stronglylex}
\rm Let $I$ be a ${\bf t}$\textit{-spread ideal}. $I$ is said to be a ${\bf t}$\textit{-spread strongly stable ideal} if $[I_{\ell}]_{\bf t}$ is a ${\bf t}$\textit{-spread strongly stable set}, for all $\ell$.
$I$ is said to be a ${\bf t}$\textit{-spread lex ideal} if $[I_{\ell}]_{\bf t}$ is a ${\bf t}$\textit{-spread lex set}, for all $\ell$.
\end{Definition}
One can observe that any {\bf t}-spread lex set (ideal) is a {\bf t}-spread strongly stable set (ideal).
Moreover, for ${\bf t}={\bf 0}$
(${\bf t}={\bf 1}$)
one obtains the classical notions of (squarefree) strongly stable ideal and (squarefree) lex ideal \cite{JT}.
\section{Combinatorics on vector-spread shadows}\label{sec2}
In this section, if ${\bf t} =(t_1, \ldots, t_{d-1})\in{\NZQ Z}_{\ge0}^{d-1}$, $d\ge 2$, we deal with the combinatorics of the {\bf t}-spread shadows of {\bf t}-spread strongly stable sets and {\bf t}-spread lex sets. As a consequence, given a {\bf t}-spread strongly stable ideal $I$ of the polynomial ring $S$, we prove the existence of a unique {\bf t}-spread lex ideal of $S$ with the same $f_{\bf t}$-vector of $I$.\\
Let $u$ be a monomial. We set $\max(u)=\max\{i:x_i\ \textup{divides}\ u\}$.
\begin{Lemma}\label{Lemma:ShadVectSS}
Let $L\subseteq M_{n,\ell,{\bf t}}$ be a ${\bf t}$-spread strongly stable set. Then
$$
\Shad_{\bf t}(L)=\big\{wx_j:w\in L,\ j\ge\max(w),\ wx_j\ \textup{is}\ {\bf t}\textup{-spread}\big\}.
$$
\end{Lemma}
\begin{proof}
Let $u\in\Shad_{\bf t}(L)$. Then, $u=w x_j$ for some $w\in L$. If $\max (w)\leq j$ there is nothing to prove. Suppose $j<\max(w)$. We can write $u=w'x_{\max (u)}$, with $w' = x_j(u/x_{\max (u)})$ and $\max(w')\le\max(u)$. The proof is complete if we show that $w'\in L$. Let $u=x_{j_1}\cdots x_{j_{\ell+1}}$ with $j_1\le\dots\le j_{\ell+1}$. Then, $j=j_p$ for some $p<\ell+1$ and $w'=x_{j_1}\cdots x_{j_\ell}=x_{j_p}(u/x_{\max(u)})$ is a ${\bf t}$-spread monomial
because $w'$ is ${\bf t}$-spread. Moreover, $w'\in L$ since $j_p<\max(u)=j_{\ell+1}$ and $L$ is a ${\bf t}$-spread strongly stable set.
\end{proof}
\begin{Proposition}\label{Prop:Shadt(L)}
Let $L\subseteq M_{n,\ell,{\bf t}}$ be a ${\bf t}$-spread set.
\begin{enumerate}
\item[\textup{(a)}] If $L$ is a ${\bf t}$-spread strongly stable set, then $\Shad_{\bf t}(L)\subseteq M_{n,\ell+1,{\bf t}}$ is a ${\bf t}$-spread strongly stable set.
\item[\textup{(b)}] If $L$ is a ${\bf t}$-spread lex set, then $\Shad_{\bf t}(L)\subseteq M_{n,\ell+1,{\bf t}}$ is a ${\bf t}$-spread lex set.
\end{enumerate}
\end{Proposition}
\begin{proof}
Let $u=wx_j\in\Shad_{\bf t}(L)$. For the proofs of both (a) and (b), by Lemma \ref{Lemma:ShadVectSS}, we can assume that $\max(w)\le j$.\medskip\\
(a) Let $i<k$ such that $x_k$ divides $u$ and $u'=x_i(u/x_k)$ is ${\bf t}$-spread. We prove that $u'\in\Shad_{\bf t}(L)$. If $k=j$, then $u'=x_iw\in\Shad_{\bf t}(L)$ by definition. Suppose $k\ne j$. Since $j=\max(u)$, then $i<k<j$ and consequently $x_k$ divides $w$. Therefore, $u'=x_i(w/x_k)x_j\in\Shad_{\bf t}(L)$ because $x_i(w/x_k)\in L$ as $x_i(w/x_k)$ is a ${\bf t}$-spread monomial, $i<k$ and $L$ is a ${\bf t}$-spread strongly stable set.\smallskip\\
(b) Let $v\in M_{n,\ell+1,{\bf t}}$ with $v>_{\lex}u$. We prove that $v\in\Shad_{\bf t}(L)$. By definition of the lex order, it follows that $v/x_{\max(v)}\ge_{\lex} u/x_{\max(u)}$. The hypothesis on $L$ guarantees that $v/x_{\max(v)}\in L$. Hence, $v=(v/x_{\max(v)})x_{\max(v)}\in\Shad_{\bf t}(L)$.
\end{proof}
Let $L\subseteq M_{n,\ell,{\bf t}}$ be a set of monomials, where $\ell\leq d$. For every $i\in\{1,\ldots,n\}$ we denote by $m_i (L)$ the number of monomials $u\in L$ such that $\max(u)=i$ and then we set $m_{\leq j}(L) = \sum_{i=1}^{j} m_i(L)$. Note that $m_i(L)=0$ if $i\leq \sum_{j=1}^{\ell-1} t_j$.
\begin{Lemma}\label{Lemma:m_i}
Let $L\subseteq M_{n,\ell,{\bf t}}$ be a $\bf t$-spread strongly stable set with $\ell<d$. Then
\begin{enumerate}
\item[\textup{(a)}] $m_i(\Shad_{\bf t}(L))=m_{\leq i-t_\ell}(L)$ for all $i$;
\item[\textup{(b)}] $\left\lvert \Shad_{\bf t}(L) \right\rvert = \sum_{k=1+\sum_{j=1}^{\ell-1} t_j}^{n-t_\ell} m_{\leq k}(L)$.
\end{enumerate}
\end{Lemma}
\begin{proof} (a) If $i\leq\sum_{j=1}^{\ell} t_j$ the proof is trivial. Let $i\geq 1+\sum_{j=1}^{\ell} t_j$. Consider the map
\[\varphi : \left\{ u\in\Shad_{\bf t}(L)\, :\, \max(u)=i \right\} \longrightarrow \left\{ w \in L \, :\, \max(w)\leq i-t_\ell \right\},\]
defined as follows. Let $u\in\Shad_{\bf t}(L)$ with $\max(u)=i$. By Lemma \ref{Lemma:ShadVectSS}, $u=w x_i$ where $w\in L$ is the unique monomial such that $\max(u)=i$. Thus we set $\varphi(u)=w$. The map $\varphi$ is well defined by the uniqueness of $w$. To prove (a), it is enough to show that $\varphi$ is a bijection. $\varphi$ is clearly injective. To prove that $\varphi$ is surjective, let $w\in L$ with $\max(w)\leq i-t_\ell$. Then, $u=w x_i$ is ${\bf t}$-spread because $\max(w)\leq i - t_\ell$. Since $\max(u)=i$, then $u$ belongs to the domain of $\varphi$ and $\varphi(u)=w$, as desired.\medskip
\noindent(b) Since
$$
\Shad_{\bf t}(L)=\bigcup_{i=1+\sum_{j=1}^{\ell}t_j}^{n}\big\{u\in\Shad_{\bf t}(L):\max(u)=i\big\},
$$
where the union is disjoint, by (a), $|\{u\in\Shad_{\bf t}(L):\max(u)=i\}|=m_i(\Shad_{\bf t}(L))=m_{\le i-t_\ell}(L)$, and so
\begin{align*}
\left\lvert \Shad_{\bf t}(L) \right\rvert & = \sum_{i=1+\sum_{j=1}^{\ell}t_j}^{n} m_i (\Shad_{\bf t}(L)) \\
& = \sum_{i=1+\sum_{j=1}^{\ell}t_j}^{n} m_{\leq i-t_\ell} (L) \\
& = \sum_{k=1+\sum_{j=1}^{\ell-1}t_j}^{n-t_\ell} m_{\leq k} (L).
\end{align*}
\end{proof}
The following result is a vector-spread analogue of a well known theorem due to Bayer, see \cite[Theorem 6.3.3]{JT}. The proof is very similar to \cite[Theorem 2]{CAC}, but we include it in all the details for the convenience of the reader stressing the needed changes.
\begin{Theorem}\label{Thm:BayerVectSpread}
Let $L\subset M_{n,\ell,{\bf t}}$ be a {\bf t}-spread lex set and let $N\subset M_{n,\ell,{\bf t}}$ be a {\bf t}-spread strongly stable set. Suppose $\left\lvert L \right\rvert \leq \left\lvert N \right\rvert$. Then $m_{\leq i}(L)\leq m_{\leq i}(N)$.
\end{Theorem}
\begin{proof}
We first observe that $N=N_0 \union N_1 x_n$, where $N_0$ and $N_1$ are the unique {\bf t}-spread strongly stable sets such that
\begin{align*}
N_0\ &=\ \{u\in N:\max(u)<n\},&
N_1\ &=\ \{u/x_n:u\in N,\max(u)=n\}.
\end{align*}
Similarly, we can write $L=L_0\cup L_1x_n$, where $L_0$ and $L_1$ are ${\bf t}$-spread lex sets defined as above.\smallskip
We proceed by induction on $n\ge1$, with the base case being trivial. Let $n>1$. Firstly, observe that $m_{\leq n}(L)=\left\lvert L \right\rvert$ and $\left\lvert N \right\rvert = m_{\leq n}(N)$. Hence the assertion holds for $i=n$. Note that $m_{\leq n-1}(L)=|L_0|$ and $m_{\leq n-1}(N)=|N_0|$. Thus, to say that $m_{\leq n-1}(L)\leq m_{\leq n-1}(N)$ is equivalent to prove that
\begin{equation}\label{eq:N0L0Crucial}
|L_0|\leq|N_0|.
\end{equation}
Assume for a moment that inequality \eqref{eq:N0L0Crucial} holds. Then, applying our inductive hypothesis to the sets $L_0,N_0\subset M_{n-1,\ell,{\bf t}}$ we obtain
$$
m_{\le i}(L)=m_{\le i}(L_0)\le m_{\le i}(N_0)=m_{\le i}(N)\ \ \ \text{for}\ \ \ i=1,\dots,n-1,
$$
as desired. Thus it remains to prove the inequality \eqref{eq:N0L0Crucial}.\smallskip
Let $N_0^* \subset M_{n-1,\ell,{\bf t}}$ be a {\bf t}-spread lex set with $\left\lvert N_0^* \right\rvert = \left\lvert N_0 \right\rvert$ and $N_1^* \subset M_{n-t_{\ell-1},\ell-1,{\bf t}}$ be a {\bf t}-spread lex set with $\left\lvert N_1^* \right\rvert = \left\lvert N_1 \right\rvert$. Let $N^* = N_0^*\cup N_1^* x_n$. We claim that $N^*$ is a {\bf t}-spread strongly stable set. Let $u\in N^*$. We shall prove that for every $j<i$ such that $x_i$ divides $u$ and $x_j(u/x_{i})$ is ${\bf t}$-spread, then $x_j(u/x_{i}) \in N^*$. If $u\in N_0^*$ there is nothing to prove since $N_0^*$ is a ${\bf t}$-spread lex set. Suppose $u\in N_1^* x_n$, then we can write $u=wx_n$, where $w\in N_1^*$. If $i<n$, then $w'= x_j(w/x_{i})$ belongs to $N_1^*$ and $x_j(u/x_i)=w'x_n\in N_1^* x_n$. If $i=n$, then $x_j(u/x_i)=x_jw$. Now, if $x_n$ divides $x_jw$, then again $x_jw\in N_1^*x_n$. Otherwise, if $x_n$ does not divide $x_jw$, then $x_jw\in N^*$ if and only if $x_jw\in N_0^*$. Thus, we must show that $\Shad_{\bf t}(N_1^*)\subset N_0^*$. For this aim, it is sufficient to prove that $|\Shad_{\bf t}(N_1^*)|\leq |N_0^*|$, as both sets are ${\bf t}$-spread lex sets (Proposition \ref{Prop:Shadt(L)}(b)). By Lemma \ref{Lemma:m_i} and the induction hypothesis we obtain
\begin{align*}
|\Shad_{\bf t}(N_1^*)| & = \sum_{i=1+\sum_{j=1}^{\ell-2}t_j}^{n-t_{\ell-1}} m_{\leq i}(N_1^*)\leq \sum_{i=1+\sum_{j=1}^{\ell-2}t_j}^{n-t_{\ell-1}} m_{\leq i}(N_1) \\
& = |\Shad_{\bf t}(N_1)|\leq |N_0|= |N_0^*|.
\end{align*}
Finally,
$N^*$ is a {\bf t}-spread strongly stable set.\smallskip
Since $|N|=|N^*|$, we may replace $N$ by $N^*$ and assume that $N_0$ is a {\bf t}-spread lex set. We suppose $n>1 + \sum_{j=1}^{\ell-1} t_j$, otherwise $M_{n,\ell,{\bf t}}=\{x_1x_{1+t_1}\cdots x_{1+\sum_{j=1}^{\ell-1} t_j}\}$ and the assertion is trivial.\smallskip
Let $m=x_{j_1}\cdots x_{j_{\ell}}$ be a {\bf t}-spread monomial and $\alpha : M_{n,\ell,{\bf t}} \rightarrow M_{n,\ell,{\bf t}}$ be the map defined as follows:
\begin{enumerate}
\item[{\normalfont(a)}] if $j_{\ell} \neq n$, then $\alpha(m)=m$;
\item[{\normalfont(b)}] if $j_{\ell} = n$ and $m\neq \min_{>_{\lex}}M_{n,\ell,{\bf t}}=x_{n-\sum_{j=1}^{\ell -1}t_j} x_{n-\sum_{j=2}^{\ell -1}t_j}\cdots x_{n-t_{\ell -1}}x_n$, then there exists $r\in [2,\ell]$ such that $j_r > j_{r-1} + t_{r-1}$. Hence, if $r$ is the largest integer with this property, we define
$$\alpha(m) = x_{j_1}\cdots x_{j_{r-1}} x_{j_r - 1}\cdots x_{j_{\ell -1}-1}x_{n-1};$$
\item[{\normalfont(c)}] if $j_{\ell} = n$ and $m=\min_{>_{\lex}}M_{n,\ell,{\bf t}}=x_{n-\sum_{j=1}^{\ell -1}t_j} x_{n-\sum_{j=2}^{\ell -1}t_j}\cdots x_{n-t_{\ell -1}}x_n$, then
$$\alpha(m)=x_{n-1-\sum_{j=1}^{\ell -1}t_j} x_{n-1-\sum_{j=2}^{\ell -1}t_j}\cdots x_{n-1-t_{\ell -1}}x_{n-1}.$$
\end{enumerate}
Such map $\alpha$ is well defined and is easily seen to be a lexicographic order preserving map, \textit{i.e.}, if $m_1,m_2\in M_{n,\ell,{\bf t}}$ and $m_1 <_{\lex} m_2$, then $\alpha(m_1)<_{\lex}\alpha(m_2)$, too.\smallskip
To prove \eqref{eq:N0L0Crucial}, since both $L_0$ and $N_0$ are {\bf t}-spread lex sets, it is enough to show that $\min_{>_{\lex}} L_0 \geq_{\lex} \min_{>_{\lex}} N_0$.
Let $u=\min_{>_{\lex}} L=x_{i_1}\cdots x_{i_{\ell}}$ and $v=\min_{>_{\lex}} N=x_{j_1}\cdots x_{j_{\ell}}$. We claim that $\alpha(u)=\min_{>_{\lex}}L_0$ and $\alpha(v)=\min_{>_{\lex}}N_0$. Indeed,
\begin{enumerate}
\item[{\normalfont(a)}] if $v\in N_0$, then $\alpha(v)=v\in N_0$
\item[{\normalfont(b)}] if $v\in N_1 x_n$ and $\alpha(v)=x_{j_1}\cdots x_{j_{r-1}} x_{j_r - 1}\cdots x_{j_{\ell -1}-1}x_{n-1}$, where $r\in [2,\ell]$ is the largest integer such that $j_r > j_{r-1} + t_{r-1}$, then
\begin{enumerate}
\item[{\normalfont(i)}] if $r=\ell$, then $\alpha(v)=x_{j_1}\cdots x_{j_{\ell-1}} x_{n-1}=(v/x_n)x_{n-1}\in N_0$, because $N$ is a {\bf t}-spread strongly stable set.
\item[{\normalfont(ii)}] if $r<\ell$, since $N$ is a {\bf t}-spread strongly stable set, we have
\begin{align*}
v_1&= x_{j_k -1}(v/x_{j_k})\in N,\\
v_2&= x_{j_{k+1} -1}(v_1/x_{j_{k+1}})\in N,\\
&\phantom{..}\vdots\\
v_{\ell-r}&= x_{j_{\ell-1} -1}(v_{\ell -k-1}/x_{j_{\ell-1}})\in N,
\end{align*}
then $\alpha(v)=x_{n-1}(v_{\ell -k}/x_n)\in N_0$.
\end{enumerate}
\item[{\normalfont(c)}] if $v\in N_1 x_n$ and $\alpha(v)=x_{j_1 -1}\cdots x_{j_{\ell -1}-1}x_{n-1}$, we have
\begin{align*}
v_1&=x_{j_1 -1}(v/x_{j_1})\in N,\\
v_2&=x_{j_2 -1}(v_1/x_{j_2})\in N,\\
&\phantom{..}\vdots\\
v_{\ell-1}&=x_{j_{\ell} -1}(v_{\ell-2}/x_{j_{\ell}-1})\in N,
\end{align*}
and $\alpha(v)=x_{n-1}(v_{\ell -1}/x_n)\in N_0$, because $N$ is a {\bf t}-spread strongly stable set.
\end{enumerate}
Hence, in all possible cases $\alpha(v)\in N_0$. Thus, we have $\min_{>_{\lex}}N_0 \leq_{\lex} \alpha(v)$, and $\min_{>_{\lex}}N_0 \geq_{\lex} v =\min_{>_{\lex}}N$. Since $\max(\min_{>_{\lex}}N_0)<n$, we have
$$
\min_{>_{\lex}}N_0 = \alpha(\min_{>_{\lex}} N_0) \geq_{\lex} \alpha(v) \geq_{\lex} \min_{>_{\lex}} N_0,
$$
and so $\min_{>_{\lex}} N_0 = \alpha(v)$. Similarly one can prove that $\min_{>_{\lex}} L_0 = \alpha(u)$.
Finally, since $L$ is a {\bf t}-spread lex set and $|L|\leq |N|$, we have $u\geq_{\lex}v$. Consequently, $\min_{>_{\lex}}L_0 = \alpha(u) \geq_{\lex} \alpha(v)= \min_{>_{\lex}}N_0$ and the proof is completed.
\end{proof}
As a striking consequence of the previous result, we prove that to every ${\bf t}$-spread strongly stable ideal $I$ one can associate a unique ${\bf t}$-spread lex ideal which shares the same $f_{\bf t}$-vector of $I$. It is necessary to highlight that if $I$ is an arbitrary ${\bf t}$-spread ideal of $S$, then such a ${\bf t}$-spread lex ideal does not always exist
\cite[Remark 2]{CAC}. Nevertheless, there can exist a ${\bf t}$-spread ideal $I$ of $S$ which is not ${\bf t}$-spread strongly stable but for which there exists a ${\bf t}$-spread lex ideal with the same $f_{\bf t}$-vector of $I$ (see, for instance, \cite[Remark 4.10]{ACF} and \cite[Remark 2]{CAC}).
\begin{Corollary}\label{Cor:SubstituteTLex}
Let $I\subset S$ be a ${\bf t}$-spread strongly stable ideal. Then there exists a unique ${\bf t}$-spread lex ideal $I^{{\bf t},\lex}\subset S$ such that $f_{\bf t}(I)=f_{\bf t}(I^{{\bf t},\lex})$.
\end{Corollary}
\begin{proof}
We construct a ${\bf t}$-spread lex ideal $J$ verifying $f_{\bf t}(J)=f_{\bf t}(I)$ as follows. For all $0\leq \ell\leq d$, let $L_\ell$ be the unique {\bf t}-spread lex set of $M_{n,\ell,{\bf t}}$ with $|L_\ell|=|[I_\ell]_{\bf t}|$. Whereas, for $\ell>d$ we set $L_{\ell}=\emptyset$. For all $\ell\ge0$, we denote by $J_\ell$ the $K$-vector space spanned by the monomials in the set
$$
L_\ell\cup\Shad_{\bf 0}(B_{\ell-1}),
$$
where $B_{-1}=\emptyset$ and for $\ell\ge1$, $B_{\ell-1}$ is the set of monomials in $J_{\ell-1}$.
Then, we set
$$
J=\bigoplus_{\ell\ge0}J_{\ell}.
$$
By abuse of notation, we denote by $[J_{\ell}]$ the set of monomials spanning $J_{\ell}$. We claim that $J$ satisfies our statement. Firstly, we must show that $J$ is a ${\bf t}$-spread lex ideal. For this purpose, it is enough to show that
$$
\Shad_{\bf 0}([J_{\ell-1}])\subseteq[J_\ell],
$$
for all $\ell\ge1$. But $B_{\ell-1}=[J_{\ell-1}]$, and so
$$
\Shad_{\bf 0}([J_{\ell-1}])=\Shad_{\bf 0}(B_{\ell-1})\subseteq L_{\ell}\cup\Shad_{\bf 0}(B_{\ell-1})=[J_\ell].
$$
It remains to prove that $f_{\bf t}(J)=f_{\bf t}(I)$, \emph{i.e.}, $|[J_\ell]_{\bf t}|=|[I_{\ell}]_{\bf t}|$ for $0\leq \ell\leq d$. Let $\delta=\indeg(I)= \indeg(J)$. Then $\delta\le d$ and $|[J_\ell]_{\bf t}|=|[I_{\ell}]_{\bf t}|=0$ for all $0\le\ell<\delta$. Now, let $\ell\ge\delta$. Since $|L_{\ell}|=|[I_\ell]_{\bf t}|$, then
$$
|[J_\ell]_{\bf t}|=|L_\ell\cup\Shad_{\bf t}(B_{\ell-1})|=|[I_\ell]_{\bf t}|
$$
if and only if $\Shad_{\bf t}(B_{\ell-1})\subseteq L_\ell$. We proceed by finite induction on $\delta\le\ell\le d$. For the base case $\ell=\delta$, just note that $B_{\delta-1}=\emptyset$. Now, let $\ell>\delta$, then $\Shad_{\bf t}(B_{\ell-2})\subseteq L_{\ell-1}$ by the inductive hypothesis. Hence,
\begin{align*}
\Shad_{\bf t}(B_{\ell-1})\ &=\ \Shad_{\bf t}(L_{\ell-1}\cup\Shad_{\bf 0}(B_{\ell-2}))\\
&=\ \Shad_{\bf t}(L_{\ell-1}\cup\Shad_{\bf t}(B_{\ell-2}))\\
&=\ \Shad_{\bf t}(L_{\ell-1}).
\end{align*}
Indeed,
$$
\Shad_{\bf t}(\Shad_{\bf 0}(B_{\ell-2}))=\Shad_{\bf t}(\Shad_{\bf t}(B_{\ell-2})).
$$
It is clear that the second set is included in the first one. For the other inclusion, let $u\in\Shad_{\bf t}(\Shad_{\bf 0}(B_{\ell-2}))$. Then $u=vx_ix_j$ with $\max(v)\le i\le j$ and $\deg(v)=\ell-2$. Since $u$ is ${\bf t}$-spread and clearly not a generator of $J$, there exists $w\in G(J)$ that properly divides $u$. Note that $\deg(w)\le\ell-2$. By the ${\bf t}$-spread lex property, there exists also a $w'\in G(J)$ that divides $v$. Hence, $v\in[J_{\ell-2}]$ and $u\in\Shad_{\bf t}(\Shad_{\bf t}(B_{\ell-2}))$.
Thus, it remains to prove that $\Shad_{\bf t}(L_{\ell-1})\subseteq L_{\ell}$. Both sets are {\bf t}-spread lex sets. Therefore, the previous inclusion holds if and only if $|\Shad_{\bf t}(L_{\ell-1})|\le|L_{\ell}|$. By Lemma \ref{Lemma:m_i}(b) and Theorem \ref{Thm:BayerVectSpread} applied to the sets $L_{\ell-1}$ and $[I_{\ell-1}]_{\bf t}$ satisfying $|L_{\ell-1}|=|[I_{\ell-1}]_{\bf t}|$, we have,
\begin{align*}
\left\lvert \Shad_{\bf t}(L_{\ell-1}) \right\rvert\ &= \sum_{k=1+\sum_{j=1}^{\ell-2} t_j}^{n-t_{\ell-1}} m_{\leq k}(L_{\ell-1})\le\sum_{k=1+\sum_{j=1}^{\ell-2} t_j}^{n-t_{\ell-1}} m_{\leq k}([I_{\ell-1}]_{\bf t})\\
&=\ |\Shad_{\bf t}([I_{\ell-1}]_{\bf t})|\le|[I_{\ell}]_{\bf t}|=|L_\ell|.
\end{align*}
The inductive proof is complete.
We denote $J$ by $I^{{\bf t},\lex}$. It is clear that $I^{{\bf t},\lex}$ is the unique ideal meeting the requirements of the statement.
\end{proof}
\begin{Example}\label{Ex:Itlex}
\rm Let ${\bf t}=(1,0,2)$ and $n=6$. Consider the following ${\bf t}$-spread strongly stable ideal of $S=K[x_1,\dots,x_6]$:
$$
I=(x_1x_2,x_1x_3,x_1x_4,x_2x_3,x_2x_4^2,x_3x_4^2x_6).
$$
Then,
\begin{align*}
[I_\ell]_{\bf t}\ &=\ \emptyset,\ \text{for}\ \ell=0,1,\\[0.3em]
[I_2]_{\bf t}\ &=\ \{x_1x_2,x_1x_3,x_1x_4,x_2x_3\},\\[0.3em]
[I_3]_{\bf t}\ &=\ \{x_1x_2^2,x_1x_2x_3,x_1x_2x_4,x_1x_2x_5,x_1x_2x_6,x_1x_3^2,x_1x_3x_4,x_1x_3x_5,x_1x_3x_6,\\
&\phantom{=\ \{.}x_1x_4^2,x_1x_4x_5,x_1x_4x_6,x_2x_3^2,x_2x_3x_4,x_2x_3x_5,x_2x_3x_6,x_2x_4^2\},\\[0.3em]
[I_4]_{\bf t}\ &=\ \{x_1x_2^2x_4,x_1x_2^2x_5,x_1x_2^2x_6,x_1x_2x_3x_5,x_1x_2x_3x_6,x_1x_2x_4x_6,x_1x_3^2x_5,x_1x_3^2x_6,\\
&\phantom{=\ \{.}x_1x_3x_4x_6,x_1x_4^2x_6,x_2x_3^2x_5,x_2x_3^2x_6,x_2x_3x_4x_6,x_2x_4^2x_6,x_3x_4^2x_6\},\\[0.3em]
[I_\ell]_{\bf t}\ &=\ \emptyset,\ \text{for all}\ \ell\ge5.
\end{align*}
Therefore,
\begin{align*}
f_{\bf t}(I)\ &=\ (f_{{\bf t},-1}(I),f_{{\bf t},0}(I),f_{{\bf t},1}(I),f_{{\bf t},2}(I),f_{{\bf t},3}(I))\\
&=\ (1,6,11,18,0).
\end{align*}
Note that the value of $f_{{\bf t},3}(I)$ depends on the fact that $[I_4]_{\bf t}=M_{6,4,{\bf t}}$.
Moreover, $L_\ell=\emptyset$ for $\ell=0,1$ and for $\ell\ge5$.
Whereas, for $\ell=2,3,4$, we have
\begin{align*}
L_2\ &=\ \{x_1x_2,x_1x_3,x_1x_4,x_1x_5\},\\[0.3em]
L_3\ &=\ \{x_1x_2^2,x_1x_2x_3,x_1x_2x_4,x_1x_2x_5,x_1x_2x_6,x_1x_3^2,x_1x_3x_4,x_1x_3x_5,x_1x_3x_6,\\
&\phantom{=\ \{.}x_1x_4^2,x_1x_4x_5,x_1x_4x_6,x_1x_5^2,x_1x_5x_6,x_1x_6^2,x_2x_3^2,x_2x_3x_4\},\\[0.3em]
L_4\ &=\ \{x_1x_2^2x_4,x_1x_2^2x_5,x_1x_2^2x_6,x_1x_2x_3x_5,x_1x_2x_3x_6,x_1x_2x_4x_6,x_1x_3^2x_5,x_1x_3^2x_6,\\
&\phantom{=\ \{.}x_1x_3x_4x_6,x_1x_4^2x_6,x_2x_3^2x_5,x_2x_3^2x_6,x_2x_3x_4x_6,x_2x_4^2x_6,x_3x_4^2x_6\}.
\end{align*}
Hence,
$$
I^{{\bf t},\lex}=(x_1x_2,x_1x_3,x_1x_4,x_1x_5,x_1x_6^2,x_2x_3^2,x_2x_3x_4,x_2x_4^2x_6,x_3x_4^2x_6).
$$
\end{Example}
\section{The vector-spread Macaulay theorem}\label{sec3}
The purpose of this section is to give a classification of all possible $f_{\bf t}$-vectors of a ${\bf t}$-spread strongly stable ideal. We follow the steps of the classical Macaulay theorem, see \cite[Theorem 6.3.8]{JT}.
We quote the following result from \cite[Lemma 6.3.4]{JT}.
\begin{Lemma}\label{Lemma:BinExp}
Let $\ell$ be a positive integer. Then each positive integer $a$ has a unique expansion
$$
a=\sum_{j=p}^{\ell}\binom{a_j}{j},
$$
with $a_\ell>a_{\ell-1}>\dots>a_{p}\ge p\ge1$.
\end{Lemma}
The previous expansion is called the \textit{binomial or Macaulay expansion of $a$ with respect to $\ell$}.\smallskip
\begin{Definition}\label{Def:vector-spreadOp}
\rm Let $n,\ell$ be positive integers, ${\bf t}=(t_1,\dots,t_{d-1})\in{\NZQ Z}_{\ge0}^d$, $d\ge2$ such that $n>\sum_{j=1}^{d-1}t_j$ and $\ell<d$. For all $1\le\ell<d$, we define a \textit{{\bf t}-spread operator} as follows: for any positive integer $a\le|M_{n,\ell,{\bf t}}|$, let $a=\sum_{j=p}^{\ell}\binom{a_j}{j}$ be the binomial expansion of $a$ with respect to $\ell$. We define
$$
a^{(n,\ell,{\bf t})}=\sum_{j=p+1}^{\ell+1}\binom{a_{j-1}+1-t_\ell}{j}.
$$
\end{Definition}
Let $u\in M_{n,\ell,{\bf t}}$. We define the \textit{initial ${\bf t}$-spread lexsegment set determined by $u$}
to be the se
\begin{align*}
\mathcal{L}_{\bf t}^i(u)\ &=\ \{v\in M_{n,\ell,{\bf t}}:v\ge_{\lex}u\}
\end{align*}
Note that any ${\bf t}$-spread lex set $L\subset M_{n,\ell,{\bf t}}$ is an initial ${\bf t}$-spread lexsegment set.\smallskip
Definition \ref{Def:vector-spreadOp} is justified by the next result.
\begin{Theorem}\label{Thm:M{n,l,t}VectorOp}
Let $u\in M_{n,\ell,{\bf t}}$ with $\ell<d$ and $a=|M_{n,\ell,{\bf t}}\setminus\mathcal{L}_{\bf t}^i(u)|$. Then,
$$
\big|M_{n,\ell+1,{\bf t}}\setminus\Shad_{\bf t}(\mathcal{L}_{\bf t}^i(u))\big|\ =\ a^{(n,\ell,{\bf t})}.
$$
\end{Theorem}
In order to prove the theorem, we need some preliminary lemmata. Hereafter, suppose we can write a positive integer $a$ as
\begin{equation}\label{eq:fakeBinExp}
a=\binom{a_\ell}{\ell}+\dots+\binom{a_p}{p}+\dots+\binom{a_1}{1},
\end{equation}
where $a_\ell>a_{\ell-1}>\dots>a_p\ge p$ and $a_j<j$ for $j=1,\dots,p-1$. Then, by Lemma \ref{Lemma:BinExp}, $a=\sum_{j=p}^{\ell}\binom{a_j}{j}$ is the (unique) binomial expansion of $a$ with respect to $\ell$. However, for our convenience, we refer to \eqref{eq:fakeBinExp} also as a binomial expansion of $a$.\smallskip
Given $\emptyset\neq A \subseteq [n]$, we set
\[
M_{A,\ell,{\bf t}} = M_{n,\ell,{\bf t}} \cap K[x_a : a\in A].
\]
Moreover, if ${\bf t}=(t_1,\dots,t_{d-1})\in{\NZQ Z}_{\ge0}^d$, we set ${\bf t}_{\ge k}=(t_k,\dots,t_{d-1})$.
\begin{Lemma}\label{Lemma:utilda}
Let $u=x_{i_1}\cdots x_{i_{\ell}}\in M_{n,\ell,{\bf t}}$. Then
\begin{equation}\label{Lemma:M-Lu}
M_{n,\ell,{\bf t}} \setminus \mathcal{L}_{\bf t}^i(u) = \Union_{k=1}^{\ell} x_{i_1}\cdots x_{i_{k-1}} M_{[i_{k}+1,n],\ell-(k-1),\, {\bf t}_{\ge k}}.
\end{equation}
This union is disjoint, and the binomial expansion of $\left\lvert M_{n,\ell,{\bf t}} \setminus \mathcal{L}_{\bf t}^i(u)\right\rvert$ is
\begin{equation}\label{Lemma:|M-Lu|}
\left\lvert M_{n,\ell,{\bf t}} \setminus \mathcal{L}_{\bf t}^i(u)\right\rvert = \sum_{j=1}^{\ell} \binom{a_j}{j},
\end{equation}
where $a_j=n-i_{\ell-(j-1)}+j-1-\sum_{h=\ell-(j-1)}^{\ell-1}t_h$, for all $j\in[\ell]$.
\end{Lemma}
\begin{proof} Since $\geq_{\lex}$ is a total order, we have $M_{n,\ell,{\bf t}} \setminus \mathcal{L}_{\bf t}^i(u) = \left\{v\in M_{n,\ell,{\bf t}}: v<_{\lex}u\right\}$.\linebreak Let $v=x_{j_1}\cdots x_{j_\ell} \in M_{n,\ell,{\bf t}}$, with $v<_{\lex} u$. Then $i_1=j_1,\ldots,i_{k-1}=j_{k-1}$ and $i_k<j_k$, for some $k\in [\ell]$. Hence $v=x_{i_1}\cdots x_{i_{k-1}} w$, where $w\in M_{[i_{k}+1,n],\ell-(k-1), {\bf t}_{\ge k}}$ and \eqref{Lemma:M-Lu} follows. To prove \eqref{Lemma:|M-Lu|} one can apply \eqref{Formula:|Mn,l,t|}, observing that the union in \eqref{Lemma:M-Lu} is disjoint and that $\lvert M_{[i_{k}+1,n],\ell-(k-1), {\bf t}_{\ge k}} \rvert = \lvert M_{n-i_k,\ell-(k-1), {\bf t}_{\ge k}} \rvert$.
It remains to prove that \eqref{Lemma:|M-Lu|} is the binomial expansion of $\left\lvert M_{n,\ell,{\bf t}} \setminus \mathcal{L}_{\bf t}^i(u)\right\rvert$.
Let $p=\min\{j:a_j\ge j\}$. By Lemma \ref{Lemma:BinExp}, it is enough to show the following facts:
\begin{enumerate}
\item[(i)] $a_\ell>a_{\ell-1}>\dots>a_p\ge p$, and
\item[(ii)] $a_j<j$, for $j=1,\dots,p-1$.
\end{enumerate}
Statement (ii) follows from the definition of $p$. For the proof of (i), let $\ell>j\ge p$. Then, we have
$$
a_{j+1}-a_{j}=i_{\ell-(j-1)}-i_{\ell-j}+1-t_{\ell-j}\ge t_{\ell-j}+1-t_{\ell - j}=1,
$$
since $i_{\ell-(j-1)}-i_{\ell-j}\ge t_{\ell-j}$. Thus,
$$
a_{j+1}\ge a_j+1,
$$
and so $a_\ell>a_{\ell-1}>\dots>a_p\ge p$, as desired.
\end{proof}
Let $L\subseteq M_{n,\ell,{\bf t}}$ be a ${\bf t}$-spread lex set, with $\ell<d$. By Proposition \ref{Prop:Shadt(L)}(b), $\Shad_{\bf t}(L)\subseteq M_{n,\ell+1,{\bf t}}$ is again a ${\bf t}$-spread lex set. Let
$$
u=\min_{>_{\lex}}L=x_{i_1}x_{i_2}\cdots x_{i_\ell}.
$$
Then $L=\mathcal{L}_{\bf t}^i(u)$. Hence, if we set $\widetilde{L}=\Shad_{\bf t}(L)$ and $\widetilde{u}=\min_{>_{\lex}}\Shad_{\bf t}(L)$, then $\widetilde{L}= \mathcal{L}_{\bf t}^i(\widetilde{u})$. Therefore, to determine the ${\bf t}$-spread shadow $\widetilde{L}$ of $L$ it is enough to determine the monomial $\widetilde{u}$. This is accomplished in the next lemma.
\begin{Lemma}\label{Lemma:utildavect}
With the notation and assumptions as above, we have
\begin{equation}\label{eq:utilda}
\widetilde{u}=\big(\prod_{h=1}^{\ell-r}x_{i_h}\big)\big(\prod_{h=1}^{r+1}x_{n-\sum_{p=\ell-(r-h)}^{\ell}t_p}\big),
\end{equation}
where we set $i_0=t_0=0$ and
\begin{equation}\label{eq:rwidetilde(u)}
r=\min\Big\{s\in [0,\ell] : n-i_\ell+\sum_{h=1}^{s}(i_{\ell-(h-1)}-i_{\ell-h}-t_{\ell-h})\ge t_\ell \Big\}.
\end{equation}
\end{Lemma}
\begin{proof}
Let us prove that $\widetilde{u}$ belongs to $\widetilde{L}$. For this aim, it is enough to show that $v=\widetilde{u}/x_n\in L$. Note that
$$
n-i_\ell+\sum_{h=1}^{s}(i_{\ell-(h-1)}-i_{\ell-h}-t_{\ell-h})=n-i_{\ell-s}-\sum_{h=1}^s t_{\ell-h},
$$
for all $s\in[0,\ell]$. Thus, $r=\min\big\{s\in [0,\ell] : n-i_{\ell-s}-\sum_{h=1}^s t_{\ell-h}\ge t_\ell \big\}$. Hence,
$$
\big(n-\sum_{h=1}^r t_{\ell-h}\big)-i_{\ell-r}\ge t_\ell.
$$
By definition of $>_{\lex}$ we have $v\ge_{\lex}u$. Since $v$ is ${\bf t}$-spread, it follows that $v\in L$.
To prove that $\widetilde{u}=\min_{>_{\lex}}\widetilde{L}$, suppose by contradiction that there exists $w\in\widetilde{L}$ such that $w<_{\lex}\widetilde{u}$. Write $w=x_{j_1}\cdots x_{j_{\ell+1}}$, $\widetilde{u}=x_{k_1}\cdots x_{k_{\ell+1}}$. Then, $j_1=k_1,\dots,j_{q-1}=k_{q-1}$ and $j_q>k_q$, for some $q\in[\ell+1]$.
If $q\ge \ell-r+1$, then $j_q>k_q=n-\sum_{p=q}^{\ell}t_p$. This is absurd, because all monomials $x_{s_1}\cdots x_{s_{\ell+1}}\in M_{n,\ell+1,{\bf t}}$ satisfy the inequalities $s_{q}\le n-\sum_{h=q}^{\ell}t_h$, $q\in[\ell+1]$.
If $1\le q\le \ell-r$, then $j_q>k_q=i_q$. By Lemma \ref{Lemma:ShadVectSS}, $w/x_{j_{\ell+1}}=w'\in L$. Hence $\min_{>_{\lex}} L=u>_{\lex}w'$, a contradiction. Finally, $\widetilde{u}=\min_{>_{\lex}}\widetilde{L}$.
\end{proof}
The next example illustrates the previous lemma.
\begin{Example}
\rm Let ${\bf t}=(2,1,2)$, $S=K[x_1,\dots,x_8]$, $L=\mathcal{L}^i(u)$ for some $u\in M_{8,3,{\bf t}}$. Set $\Shad_{\bf t}(L)=\widetilde{L}$ and $\widetilde{u}=\min_{>_{\lex}}\widetilde{L}$. Let $r$ the integer defined in (\ref{eq:rwidetilde(u)}).\smallskip
Let $u=x_2x_4x_6$. Since $n-\max(u)=2=t_3$, then $r=0$ and $\widetilde{u}=ux_n=x_2x_4x_6x_8$.\smallskip
Let $u=x_2x_6x_7$. Then, $r=2$ and $\widetilde{u}=x_2x_5x_6x_8$.\smallskip
Let $u=x_4x_6x_7$. Then, $r=3$ and $\widetilde{u}=x_3x_5x_6x_8$. In such a case $\Shad_{\bf t}(L)=M_{8,4,{\bf t}}$.
\end{Example}
\begin{proof}[Proof of Theorem \ref{Thm:M{n,l,t}VectorOp}]
As before, let $L=\mathcal{L}_{\bf t}^i(u)$, $\widetilde{L}=\Shad_{\bf t}(L)$ and $\widetilde{u}=\min_{>_{\lex}}\widetilde{L}$. Write $\widetilde{u}=x_{k_1}x_{k_2}\cdots x_{k_{\ell+1}}$, where the indices $k_j$ are determined in (\ref{eq:utilda}) and
$$
r=\min\Big\{s\in [0,\ell] : n-i_\ell+\sum_{h=1}^{s}(i_{\ell-(h-1)}-i_{\ell-h}-t_{\ell-h})\ge t_\ell \Big\}.
$$
Then, by Lemma \ref{Lemma:utilda}, we have the binomial expansions
\begin{align*}
|M_{n,\ell,{\bf t}}\setminus L|\ =\ \sum_{j=1}^{\ell}\binom{a_j}{j},\ \ \ \ \ \ \ \
|M_{n,\ell+1,{\bf t}}\setminus\widetilde{L}|\ =\ \sum_{j=1}^{\ell+1}\binom{\widetilde{a}_j}{j},
\end{align*}
where
\begin{enumerate}
\item[(a)] $a_j=n-i_{\ell-(j-1)}+j-1-\sum_{h=\ell-(j-1)}^{\ell-1}t_h$, for all $j\in[\ell]$, and
\item[(b)] $\widetilde{a}_j=n-k_{\ell+1-(j-1)}+j-1-\sum_{h=\ell+1-(j-1)}^{\ell}t_h$, for all $j\in[\ell+1]$.
\end{enumerate}
It remains to prove that $|M_{n,\ell+1,{\bf t}}\setminus\widetilde{L}|=a^{(n,\ell,{\bf t})}$. Firstly, we establish how the coefficients $a_j$ and $\widetilde{a}_j$ are related. Note that, for $j\in[r+1]$, we have
\begin{align*}
\widetilde{a}_j&=n-k_{\ell+1-(j-1)}+j-1-\sum_{h=\ell+1-(j-1)}^{\ell}t_h\\
&=n-\Big(n-\sum_{p=\ell+1-(j-1)}^{\ell}t_p\Big)+j-1-\sum_{h=\ell+1-(j-1)}^{\ell}t_h\\
&=j-1.
\end{align*}
Since $\binom{j-1}{j}=0$, we may write as well
$$
|M_{n,\ell+1,{\bf t}}\setminus\widetilde{L}|=\sum_{j=r+2}^{\ell+1}\binom{\widetilde{a}_j}{j}.
$$
Instead, since $k_{\ell+1-(j-1)}=i_{\ell-(j-1)}$, for $j\in[r+2,\ell+1]$, we have
$$
\widetilde{a}_j=a_{j-1}+1-t_\ell.
$$
Therefore,
\begin{align*}
\big| M_{n,\ell+1,{\bf t}} \setminus\widetilde{L}\big| &= \sum_{j=r+2}^{\ell+1} \binom{a_{j-1}+1-t_{\ell}}{j}.
\end{align*}
Let $p=\min\{j:a_j\ge j\}$. The theorem is proved if we show that
$$
\big| M_{n,\ell+1,{\bf t}} \setminus\widetilde{L}\big|=\big| M_{n,\ell,{\bf t}} \setminus L\big|^{(n,\ell,{\bf t})}=\sum_{j=p+1}^{\ell+1}\binom{a_{j-1}+1-t_{\ell}}{j}.
$$
If $p+1=r+2$ this is clear. Suppose $p+1>r+2$. Then, it is enough to show that
$$
\binom{a_{j-1}+1-t_{\ell}}{j}=0, \ \ \ \text{for all}\ r+2\le j\le p.
$$
If $j\le p$, then $a_{j-1}<j-1$. Hence $a_{j-1}+1-t_\ell\le a_{j-1}+1<j$ and $\binom{a_{j-1}+1-t_{\ell}}{j}=0$, as desired. Now let
$r+2>p+1$. We must prove that
$$
\binom{a_{j-1}+1-t_{\ell}}{j}=0,
$$
for all $p+1\le j\le r+1$. Set $a_{\ell+1}=n-\sum_{j=1}^{\ell-1}t_j$. Then
$$
r=\min\{s\in[0,\ell]:a_{s+1}-s\ge t_\ell\}.
$$
If $j\le r+1$, then $j-2\le r-1$. Hence $a_{(j-2)+1}-(j-2)=a_{j-1}-(j-2)<t_\ell$. It follows that $a_{j-1}+1-t_\ell<a_{j-1}+2-t_\ell<j$ and $\binom{a_{j-1}+1-t_{\ell}}{j}=0$, as desired.
\end{proof}
\begin{Example}
\rm Let $n=31$, ${\bf t}=(0,1,3,1)$, $a=2023$ and $\ell=3$. Then
$$
a=\sum_{j=1}^{\ell}\binom{a_j}{j}=\binom{23}{3}+\binom{22}{2}+\binom{21}{1}
$$
is the binomial expansion of $a$ with respect to $\ell$. Therefore, since $r=0$, we have
\begin{align*}
a^{(n,\ell,{\bf t})}=2023^{(31,3,(0,1,3,1))}\ &=\ \sum_{j=r+2}^{\ell+1}\binom{a_{j-1}+1-t_\ell}{j}\\
&=\ \binom{19}{2}+\binom{20}{3}+\binom{21}{4}=7296.
\end{align*}
\end{Example}
Now, we can state and prove the main result in the article.
\begin{Theorem}\label{thm:main}
Let $f=(f_{-1},f_0,\dots,f_{d-1})$ be a sequence of non-negative integers. The following conditions are equivalent:
\begin{enumerate}
\item[\textup{(i)}] there exists a ${\bf t}$-spread strongly stable ideal $I\subset S=K[x_1,\dots,x_n]$ such that $$f_{\bf t}(I)=f;$$
\item[\textup{(ii)}] $f_{-1}=1$ and $f_{\ell+1}\le f_{\ell}^{(n,\ell+1,{\bf t})}$, for all $\ell=-1,\dots,d-2$.
\end{enumerate}
\end{Theorem}
\begin{proof}
(i) $\implies$ (ii). Assume that $f_{\bf t}(I)=f$. By Corollary \ref{Cor:SubstituteTLex}, we may replace $I$ by $I^{{\bf t},\lex}$ without changing the $f_{\bf t}$-vector. Thus, we may assume as well that $I$ is a ${\bf t}$-spread lex ideal. Then $f_{-1}=f_{{\bf t},-1}(I)=1$ and for all $-1\leq\ell\leq d-2$ we have $\Shad_{\bf t}([I_{\ell+1}]_{\bf t})\subseteq[I_{\ell+2}]_{\bf t}$. Hence,
\begin{align*}
f_{\ell+1}=f_{{\bf t},\ell+1}(I)=|M_{n,\ell+2,{\bf t}}|-|[I_{\ell+2}]_{\bf t}|\ &\le\ |M_{n,\ell+2,{\bf t}}|-|\Shad_{\bf t}([I_{\ell+1}]_{\bf t})|\\&=\ |M_{n,\ell+2,{\bf t}}\setminus\Shad_{\bf t}([I_{\ell+1}]_{\bf t})|\\&=\ f_{{\bf t},\ell}(I)^{(n,\ell+1,{\bf t})}=f_\ell^{(n,\ell+1,{\bf t})},
\end{align*}
where the last equality follows from Theorem \ref{Thm:M{n,l,t}VectorOp}. Statement (ii) is proved.\smallskip
\noindent(ii) $\implies$ (i). First we prove that
$$
f_{\ell}\ \le\ |M_{n,\ell+1,{\bf t}}|,\ \ \ \text{for all}\ \ \ \ell=-1,\dots,d-1.
$$
For $\ell=-1$, $f_{-1}=1=|M_{n,0,{\bf t}}|$ because there is only one ${\bf t}$-spread monomial of degree 0, namely $u=1$. Now we proceed by induction. Let $\ell\ge0$. By the hypothesis (ii), we have $f_{\ell}\le f_{\ell-1}^{(n,\ell,{\bf t})}$ and, by induction, $f_{\ell-1}\le|M_{n,\ell,{\bf t}}|$. Thus, there exists a unique monomial $u\in M_{n,\ell,{\bf t}}$ such that $|M_{n,\ell,{\bf t}}\setminus\mathcal{L}_{\bf t}^i(u)|=f_{\ell-1}$. By Theorem \ref{Thm:M{n,l,t}VectorOp}, we have $f_{\ell-1}^{(n,\ell,{\bf t})}=|M_{n,\ell+1,{\bf t}}\setminus\Shad_{\bf t}(\mathcal{L}_{\bf t}^i(u))|$. This shows that $f_{\ell-1}^{(n,\ell,{\bf t})}\le|M_{n,\ell+1,{\bf t}}|$ and consequently we have $f_{\ell}\le|M_{n,\ell+1,{\bf t}}|$, as desired.
For all $\ell\in [0,d]$,
let $L_\ell$ be the unique ${\bf t}$-spread lex set of $M_{n,\ell,{\bf t}}$ such that $|L_{\ell}|=|M_{n,\ell,{\bf t}}|-f_{\ell-1}$. For $\ell>d$ we set $L_{\ell}=\emptyset$. As in Corollary \ref{Cor:SubstituteTLex}, we construct the ideal $I=\bigoplus_{\ell\ge0}I_\ell$ where $I_\ell$ is the $K$-vector space spanned by the set
$$
L_\ell\cup\Shad_{\bf 0}(B_{\ell-1}),
$$
where $B_{-1}=\emptyset$ and for $\ell \geq 1$, $B_{\ell-1}$ is the set of monomials generating $I_{\ell-1}$.
As in Corollary \ref{Cor:SubstituteTLex}, one shows that $I$ is a {\bf t}-spread lex ideal. Hence, it remains to prove that $f_{\bf t}(I)=f$. As in the proof of Corollary \ref{Cor:SubstituteTLex}, this boils down to proving that $\Shad_{\bf t}(L_\ell)\subseteq L_{\ell+1}$, for all $\ell\in [0,d-1]$.
Since $f_{\ell}\le f_{\ell-1}^{(n,\ell,{\bf t})}$ we have
$$
|M_{n,\ell+1,{\bf t}}\setminus L_{\ell+1}|\le|M_{n,\ell,{\bf t}}\setminus L_\ell|^{(n,\ell,{\bf t})}=|M_{n,\ell+1,{\bf t}}\setminus\Shad_{\bf t}(L_{\ell})|,
$$
where the last equality follows from Theorem \ref{Thm:M{n,l,t}VectorOp}. Thus $|\Shad_{\bf t}(L_\ell)|\le|L_{\ell+1}|$. Hence $\Shad_{\bf t}(L_\ell)\subseteq L_{\ell+1}$, because both are ${\bf t}$-spread lex sets. The proof is complete.
\end{proof}
\begin{Example}
\rm Let ${\bf t}=(1,0,2)$, $d=4$ and $n=6$. Consider the following vector
\begin{align*}
f\ &=\ (f_{-1},f_{0},f_{1},f_{2},f_{3})\ =\ (1,6,11,18,0).
\end{align*}
Then $f_{-1}=1$ and $f_{\ell+1}\le f_{\ell}^{(6,\ell+1,{\bf t})}$, for all $\ell=-1,\dots,2$. Therefore, from Theorem \ref{thm:main} there exists a ${\bf t}$-spread strongly stable ideal of $S=K[x_1,\dots,x_6]$ that has $f$ as a $f_{\bf t}$-vector. The ideal $I$ of Example \ref{Ex:Itlex} is such an ideal.
\end{Example}
\section{An application} \label{sec4}
In this final section, as an application we recover the vector-spread version of the well--known result proved by Bigatti \cite{BAM} and Hulett \cite{HH}, independently (see, also, \cite{HH2, KP}). More precisely, we prove that in the class of all ${\bf t}$-spread strongly stable ideals with a given $f_{\bf t}$-vector, the ${\bf t}$-spread lex ideals have the largest
graded Betti numbers.
\begin{Theorem}\label{thm:upperbound}
Let $I\subset S=K[x_1,\dots,x_n]$ be a ${\bf t}$-spread strongly stable ideal. Then,
$$
\beta_{i,j}(I)\ \le\ \beta_{i,j}(I^{{\bf t},\lex}), \ \ \ \text{for all}\ i\ \text{and}\ j.
$$
\end{Theorem}
\begin{proof}
By \cite[Corollary 5.2]{F1},
we have
\begin{equation}\label{eneherzogqureshiformulabetti}
\beta_{i,i+j}(I)\ =\ \sum_{u\in G(I)_j}\binom{\max(u)-1-\sum_{h=1}^{j-1}t_h}{i}.
\end{equation}
We are going to write (\ref{eneherzogqureshiformulabetti}) in a more suitable way. We observe that $I$ is a {\bf t}-spread ideal and thus
$$
G(I)_j\ =\ [I_j]_{\bf t}\setminus \Shad_{\bf t}([I_{j-1}]_{\bf t}).
$$
Hence, we can write the Betti number in (\ref{eneherzogqureshiformulabetti}) as a difference $A-B$, where
\begin{align*}
A\ =&\ \sum_{u\in G([I_j]_{\bf t})} \binom{\max(u)-1-\sum_{h=1}^{j-1}t_h}{i}=\sum_{k=1}^n m_k([I_j]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}\\
=&\ \sum_{k=1}^n\Big(m_{\le k}([I_j]_{\bf t})-m_{\le k-1}([I_j]_{\bf t})\Big)\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}\\
=&\ \sum_{k=1}^nm_{\le k}([I_j]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}-\sum_{k=1}^{n-1}m_{\le k}([I_j]_{\bf t})\binom{k-\sum_{h=1}^{j-1}t_h}{i}
\end{align*}
and
\begin{align*}
B\ =&\ \sum_{u\in \Shad_{\bf t}([I_{j-1}]_{\bf t})}\binom{\max(u)-1-\sum_{h=1}^{j-1}t_h}{i}\\=&\ \sum_{k=1+\sum_{h=1}^{j-1}t_h}^n\!\!\!\!m_k(\Shad_{\bf t}([I_{j-1}]_{\bf t}))\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}\\=&\ \sum_{k=1+\sum_{h=1}^{j-1}t_h}^n m_{\le k-t_{j-1}}([I_{j-1}]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i},
\end{align*}
where the last equality follows from Lemma \ref{Lemma:m_i}(a).
Furthermore, we can write $A=A_1-A_2$ with
\begin{align*}
A_1\ =&\ m_{\le n}([I_j]_{\bf t})\binom{n-1-\sum_{h=1}^{j-1}t_h}{i},\\[0.8em]
A_2\ =&\ \sum_{k=1}^{n-1}m_{\le k}([I_j]_{\bf t})\bigg[\binom{k-\sum_{h=1}^{j-1}t_h}{i}-\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}\bigg]\\
=&\ \sum_{k=1+\sum_{h=1}^{j-1}t_h}^{n-1}m_{\le k}([I_j]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i-1}.
\end{align*}
Therefore, we obtain
\begin{equation}\label{presentationbettinumbers}
\begin{aligned}
\beta_{i,i+j}(J)\ &=\ m_{\le n}([I_j]_{\bf t})\binom{n-1-\sum_{h=1}^{j-1}t_h}{i}\\
&-\sum_{k=1+\sum_{h=1}^{j-1}t_h}^{n-1}m_{\le k}([I_j]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i-1}\\
&-\sum_{k=1+\sum_{h=1}^{j-1}t_h}^n m_{\le k-t_{j-1}}([I_{j-1}]_{\bf t})\binom{k-1-\sum_{h=1}^{j-1}t_h}{i}.
\end{aligned}
\end{equation}
Now, we compute the graded Betti numbers $\beta_{i,i+j}(I^{{\bf t},\lex})$.
Recall that $I$ and $I^{{\bf t},\lex}$ share the same $f_{\bf t}$-vector. Therefore, $|[I^{{\bf t},\lex}_j]_{\bf t}|=|[I_{j}]_{\bf t}|$, for all $j$.
Applying Theorem \ref{Thm:BayerVectSpread}, we have $m_{\le k}([I^{{\bf t},\lex}_j]_{\bf t})\le m_{\le k}([I_j]_{\bf t})$ for all $k\in[n]$. Moreover,
$$
m_{\le n}([I_j]_{\bf t})=|[I_j]_{\bf t}|=|[I^{{\bf t},\lex}_j]_{\bf t}|=m_{\le n}([I^{{\bf t},\lex}_j]_{\bf t}).
$$
Therefore, replacing in (\ref{presentationbettinumbers}), for all $k$ and $j$, every occurrence of $m_{\le k}([I_j]_{\bf t})$ with $m_{\le k}([I^{{\bf t},\lex}_j]_{\bf t})$, we get the Betti number $\beta_{i,i+j}(I^{{\bf t},\lex})$. Finally, $\beta_{i,i+j}(I)\le\beta_{i,i+j}(I^{{\bf t},\lex})$, for all $i,j\ge0$.
\end{proof}
\begin{Remark}\em Note that in the previous result, we allow $K$ to be an arbitrary field.
\end{Remark}
\begin{Example}
\rm Consider again the ${\bf t}$-spread strongly stable ideal $I$ of Example \ref{Ex:Itlex}. Then, the Betti tables of $I$ and $I^{{\bf t},\lex}$ are, respectively,
$$
\begin{matrix}
& 0 & 1 & 2\\
\hline
2:& 4 & 4 & 1 \\
3:& 1 & 2 & 1 \\
4:& 1 & 2 & 1
\end{matrix}\qquad\qquad\qquad
\begin{matrix}
& 0 & 1 & 2 & 3 & 4\\
\hline
2: & 4 & 6 & 4 & 1 & .\\
3: & 3 & 7 & 7 & 4 & 1\\
4: & 2 & 4 & 2 & . & .
\end{matrix}
$$
From these tables we see that $\beta_{i,i+j}(I)\le\beta_{i,i+j}(I^{{\bf t},\lex})$ for all $i$ and $j$.
\end{Example}
| {
"timestamp": "2023-02-21T02:30:57",
"yymm": "2302",
"arxiv_id": "2302.07595",
"language": "en",
"url": "https://arxiv.org/abs/2302.07595",
"abstract": "Let $S=K[x_1,\\dots,x_n]$ be the standard graded polynomial ring, with $K$ a field, and let ${\\bf t}=(t_1,\\ldots,t_{d-1})\\in{\\mathbb{Z}}_{\\ge 0}^{d-1}$, $d\\ge 2$, be a $(d-1)$-tuple whose entries are non negative integers. To a ${\\bf t}$-spread ideal $I$ in $S$, we associate a unique $f_{\\bf t}$-vector and we prove that if $I$ is ${\\bf t}$-spread strongly stable, then there exists a unique ${\\bf t}$-spread lex ideal which shares the same $f_{\\bf t}$-vector of $I$ via the combinatorics of the ${\\bf t}$-spread shadows of special sets of monomials of $S$. Moreover, we characterize the possible $f_{\\bf t}$-vectors of ${\\bf t}$-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal-Katona. Finally, we prove that among all ${\\bf t}$-spread strongly stable ideals with the same $f_{\\bf t}$-vector, the ${\\bf t}$-spread lex ideals have the largest Betti numbers.",
"subjects": "Commutative Algebra (math.AC)",
"title": "Macaulay's theorem for vector-spread algebras",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.982557516796073,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542243058078
} |
https://arxiv.org/abs/2209.06304 | A note on conjectures generalizing the road colouring theorem | The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence for the conjectures. | \section{Introduction}
The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree. The first of these conjectures is the $O(G)$ conjecture, which was first proposed by Ashley-Marcus-Tuncel in \cite{AMT} as part of a characterization of isomorphism of one-sided markov chains. It was also observed in \cite{AMT} that a proof of the $O(G)$ conjecture would yield a proof of the (at the time unproven) road colouring theorem. The second conjecture, called the bunchy factor conjecture, strengthens the $O(G)$ conjecture (in some sense), and gives a more direct generalization of the road colouring theorem. In this paper we give reasons to believe that both conjectures are true. Our main results focus on two classes of graphs, proving the bunchy factor conjecture for both classes and the $O(G)$ conjecture for one. We also present computer simulations that give some empirical evidence for the conjectures.
\section{Background}
Throughout this paper, graphs are finite, directed, and allow loops and multiple edges. Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a path $w$ in $G$, we write $i(w)$ and $t(w)$ for the initial and terminal vertices of $w$. For a vertex $I\in V(G)$, we write $E_I(G)=\{e\in E(G): i(e)=I\}$ for the set of outgoing edges from $I$, $E^I(G)=\{e\in E(G): t(e)=I\}$ for the set of incoming edges to $I$, and $E_I^J(G)=E_I(G)\cap E^J(G)$ for the set of edges from $I$ to a vertex $J\in V(G)$. The follower states (terminal vertices of outgoing edges) of $I\in V(G)$ are denoted by $F(I)=t(E_I(G))$. We use $L(G)$ to denote the set of (finite) paths on $G$, and write $L_I(G)=\{w\in L(G): i(w)=I\}$ and $L^I(G)=\{w\in L(G): t(w)=I\}$ for the sets of paths that start and end at a vertex $I\in V(G)$. We say that $G$ is strongly connected if there is a directed path from $I$ to $J$ for all ordered pairs $(I,J)\in V(G)\times V(G)$. The period of $G$, denoted $\text{per}(G)$, is defined as the $\gcd$ of the lengths of the cycles in $G$, and we say that $G$ is aperiodic if $\text{per}(G)=1$.
The ``road colourings" of Trahtman's road colouring theorem are edge colourings of a graph where each colour is assigned to exactly one outgoing edge from each vertex (ie. a colouring that is bijective when restricted to the outgoing edges of any vertex). A road colouring is said to be synchronizing if there is a sequence of colours such that every path coloured by that sequence ends at the same vertex. We can now state the road colouring theorem.
\begin{theorem}[Trahtman, \cite{Trahtman}]
Every strongly connected aperiodic graph with constant out-degree admits a synchronizing road colouring.
\end{theorem}
Note that the constant out-degree assumption is required for $G$ to be road-colourable (ie. for $G$ to admit any road colouring at all). To generalize road colourings to graphs with non-constant out degree, we introduce the notion of right resolving graph homomorphisms. We also introduce left and bi-resolving homomorphisms, which will be important in later sections.
\begin{definition}[Left resolvers, right resolvers, and bi-resolvers]
Let $\Phi:G\to H$ be a surjective graph homomorphism with edge map $\Phi:E(G)\to E(H)$ and vertex map $\partial\Phi: V(G)\to V(H)$. We say $\Phi$ is right resolving if the restriction $\Phi|_{E_I(G)}:E_I(G)\to E_{\partial\Phi(I)}(H)$ is a bijection for all $I\in V(G)$. We say $\Phi$ is left resolving if the restriction $\Phi|_{E^I(G)}:E^I(G)\to E^{\partial\Phi(I)}(H)$ is a bijection for all $I\in V(G)$. We say $\Phi$ is bi-resolving if it is both left and right resolving.
\end{definition}
If there is a right resolver from graphs $G$ to $H$, we say that $H$ is a right resolving factor of $G$ and write $H\leq_R G$. The set of right resolvers is closed under composition, and if $H\leq_R G$ and $G\leq_R H$ then $H\cong G$ (a right resolver between graphs of the same size is an isomorphism), so the relation $\leq_R$ is a partial ordering on isomorphism classes of directed graphs. Ashley-Marcus-Tuncel proved the following uniqueness property under this order.
\begin{theorem}[Ashley-Marcus-Tuncel, \cite{AMT}]
\label{M(G)}
For any graph $G$, there is a unique $\leq_R$-minimal graph $M(G)\leq_R G$. Moreover, there is a unique vertex map $\Sigma_G:G\to M(G)$ such that $\partial\Phi=\Sigma_G$ for all right resolvers $\Phi:G\to M(G)$.
\end{theorem}
Now consider again the case of a graph $G$ with some constant out-degree $k$. A road colouring of $G$ corresponds to a right resolving homomorphism from $G$ to the graph with one vertex and $k$ self loops, which we call $M_k$. Such a homomorphism ``colours" $G$ with the self loops of $M_k$. Since $M_k$ is clearly $\leq_R$-minimal, we have $M(G)=M_k$ in this case.
In the same way that a sequence of colours can be followed from each vertex in graph under a road colouring, a transition map on the vertices of a graph can be defined by ``lifting" paths through a right resolver from the range graph to the domain graph.
\begin{definition}
Let $\Phi:G\to H$ be a right resolver. For $I\in V(G)$ and $w\in L_{\partial\Phi(I)}(H)$, we write $I\cdot_\Phi w$ to denote the terminal vertex of the unique path $\pi\in L_I(G)$ with $\Phi(\pi)=w$. For $u\in L^{\partial\Phi(I)}(H)$, we write $u\cdot_\Phi I=\{J\in \partial\Phi^{-1}(i(u)):J\cdot_\Phi u=I\}$ for the set of vertices that lift to $I$ under $u$.
\end{definition}
Since a path $w$ in the range graph of a right resolver $\Phi: G\to H$ can only be lifted to paths that start in $\partial\Phi^{-1}(i(w))$, we define the synchronization $\Phi$ in terms of an equivalence relation that refines the fibers of $\partial\Phi$.
\begin{definition}[Stability relation]
Let $G$ and $H$ be graphs and let $\Phi:G\to H$ be a right resolver. The stability relation of $\Phi$, denoted by $\sim_\Phi$, is defined as follows: for $I\in V(H)$ and $I_1',I_2'\in \partial\Phi^{-1}(I)$, we say $I_1'\sim_\Phi I_2'$ if and only if, for all $u\in L_I(H)$ there is a $v\in L_{t(u)}(H)$ such that $I_1'\cdot_\Phi uv=I_2'\cdot_\Phi uv$.
\end{definition}
\begin{definition}[Synchronizing right resolver]
A right resolver $\Phi:G \to H$ is synchronizing if the equivalence classes of $\sim_\Phi$ are the entire fibers of $\partial\Phi$.
\end{definition}
We could have equivalently defined a synchronizing right resolver as a right right resolver $\Phi:G \to H$ such that, for all $I\in V(H)$, there is a $u\in L_I(H)$ such that $|\partial\Phi^{-1}(I)\cdot_\Phi u|=1$. Since the only fiber of a right resolver $\Phi: G\to M_k$ is all of $V(G)$, this clearly generalizes the notion of a synchronizing road colouring. The stability relation, however, will be crucially important in the inductive strategy used in later sections.
It will also be useful to extend the stability relation of a right resolver $\Phi:G\to H$ to the edges of $G$: for $e,f\in E(G)$, say that $e\sim_\Phi f$ if and only if $i(e)\sim_\Phi i(f)$ and $\Phi(e)=\Phi(f)$. Note also that if $I\sim_\Phi J$ for some right resolver $\Phi:G\to H$, then $I\cdot_\Phi e\sim_\Phi J\cdot_\Phi e$ for any $e\in E_{\partial\Phi(I)}(H)$. This congruence property allows us to define quotient graphs over stability relations.
\begin{definition}[Stability quotient]
Let $G$ and $H$ be graphs, and let $\Phi:G\to H$ be a right resolver. Define a quotient graph $G/\sim_\Phi$ by setting $V(G/\sim_\Phi)=V(G)/\sim_\Phi$ and $E(G/\sim_\Phi)=E(G)/\sim_\Phi$, where $i([e]_{\sim_\Phi})=[i(e)]_{\sim_\Phi}$ and $t([e]_{\sim_\Phi})=[t(e)]_{\sim_\Phi}$.
\end{definition}
We will use the following results from \cite{Mac}, which describe the behaviour of synchronizers and stability quotients.
\begin{theorem}[MacDonald, \cite{Mac}]
Let $G$ and $H$ be graphs such that there is a right resolver $\Phi:G\to H$. Let $\Psi:G\to G/\sim_\Phi$ be the quotient map. Then $\Psi$ is a synchronizing right resolver, and there is a right resolver $\Delta:G/\sim_\Phi\to H$ such that $\Phi = \Delta\circ\Psi$. For any such $\Delta$, $\sim_\Delta$ is trivial.
\end{theorem}
\begin{theorem}[Macdonald \cite{Mac}, proposition 9.1.16 in \cite{LM} for strongly connected graphs]
\label{degMult}
Let $G$, $K$, and $H$ be graphs such that there are right resolvers $\Phi:G\to K$ and $\Psi:K\to H$. Then $\Psi\circ\Phi$ is synchronizing if and only if both $\Phi$ and $\Psi$ are synchronizing.
\end{theorem}
\section{The $O(G)$ and bunchy factor conjectures}
Similar to the partial ordering $\leq_R$, we can define a relation $\leq_S$ on graphs by saying $H\leq_S G$ if there is a synchronizing right resolver from $G$ to $H$. The relation $\leq_S$ is reflexive and antisymmetric on graphs up to isomorphism. Theorem \ref{degMult} shows that $\leq_S$ is also transitive, so $\leq_S$ is a partial ordering. The $O(G)$ conjecture can be formulated as a statement that parallels theorem \ref{M(G)}.
\begin{conjecture}[$O(G)$ conjecture]
For all strongly connected graphs $G$, there is a unique $\leq_S$-minimal graph $O(G)\leq_S G$.
\end{conjecture}
The implication from the $O(G)$ conjecture to the road colouring theorem follows by a result from \cite{AGW}, which shows that, for all strongly connected aperiodic graphs $G$ with constant out degree $k$, $G$ and $M_k$ have a common strongly connected synchronizing extension (ie. a strongly connected graph $K$ such that $G\leq_S K$ and $M_k\leq_S K$). Assuming the $O(G)$ conjecture, it follows that $O(G)=O(M_k)$. But $O(M_k)=M_k$, so $O(G)=M_k$, and in particular $M_k\leq_S G$. If we consider a synchronizing right resolver from $G$ to $M_k$ as a road colouring, we then get a synchronizing road colouring of $G$ on $k$ colours.
The related bunchy factor conjecture requires a definition of the class of bunchy graphs. We also define the classes of almost bunchy and weakly almost bunchy graphs, which will be relevant in later sections.
\begin{definition}[Bunchy, almost bunchy, and weakly almost bunchy graphs]
Let $G$ be a graph. We say $G$ is bunchy if $\Sigma_G|_{F(I')}:F(I')\to F(\Sigma_G(I'))$ is a bijection for all $I'\in V(G)$. We say $G$ is almost bunchy if, for all pairs of states $(I,J)\in V(M(G))\times V(M(G))$, there is at most one state $I'\in \Sigma^{-1}(I)$ such that $|\Sigma^{-1}(J)\cap F(I')|\geq 2$. We say $G$ is weakly almost bunchy if, for every pair of states $(I,J)\in V(M(G))\times V(M(G))$ such that $|E_I^J(M(G))|\geq 2$, there is at most one state $I'\in \Sigma^{-1}(I)$ such that $|\Sigma^{-1}(J)\cap F(I')|=|E_I^J(M(G))|$.
\end{definition}
Note that bunchy graphs are almost bunchy and almost bunchy graphs are weakly almost bunchy. MacDonald proved in \cite{Mac} that the $O(G)$ conjecture holds for almost bunchy (and therefore bunchy) graphs. We will also use three other properties of bunchy graphs proved in \cite{Mac}.
\begin{theorem}[MacDonald, \cite{Mac}]
\label{bunchy}
Let $G$ be a graph.
\begin{enumerate}
\item The set of right resolving bunchy factors of $G$ has a unique $\leq_R$-maximal element $B(G)$.
\item For any bunchy $H\leq_R G$ and right resolver $\Phi: G \to H$, there are right resolvers $\Psi:G\to B(G)$ and $\Delta: B(G)\to H$ such that $\Phi = \Delta\circ\Psi$.
\label{5.10.2}
\item $G$ has at most one $\leq_S$-minimal synchronizing bunchy factor.
\label{5.9}
\end{enumerate}
\end{theorem}
The results of theoreom \ref{bunchy} lead us to the bunchy factor conjecture. Suppose it can be shown that every strongly connected graph has a bunchy synchronizing factor. Then every minimal synchronizing factor must be bunchy, and theorem \ref{bunchy} (\ref{5.9}) would show that every strongly connected graph has a unique synchronizing factor. We are thereby lead to the following conjecture.
\begin{conjecture}[Bunchy factor conjecture, version 1]
\label{BFC0}
Every strongly connected graph has a bunchy synchronizing factor.
\end{conjecture}
Note that, while the bunchy factor conjecture implies the $O(G)$ conjecture, if an individual graph $G$ has a bunchy synchronizing factor, it does not follow by theorem \ref{bunchy} (\ref{5.9}) that $G$ has a unique minimal synchronizing factor.
By theorem \ref{bunchy} (\ref{5.10.2}) and theorem \ref{degMult}, a graph $G$ has a bunchy synchronizing factor if and only if $B(G)\leq_S G$. The bunchy factor conjecture can therefore be equivalently stated as follows.
\begin{conjecture}[Bunchy factor conjecture, version 2]
\label{BFC}
For all strongly connected graphs $G$, $B(G)\leq_S G$.
\end{conjecture}
The bunchy factor conjecture was first posed by MacDonald in \cite{Mac}, where two more equivalent versions are also given.
It is not difficult to see that the only bunchy right resolving factor of a strongly connected aperiodic graph with constant out-degree $k$ is $M_k$. Thus $B(G)=M_k$, and so the statement of conjecture \ref{BFC} reduces to Trahtman's road colouring theorem when restricted to these graphs. The bunchy factor conjecture might also be provable using an inductive strategy similar to the one employed by Trahtman: by assuming that a strongly connected graph $G$ is not bunchy, try to find a right resolver $\Phi$ on $G$ with non-trivial stability congruence. We would then have $G/\sim_\Phi\leq_S G$ with $|V(G/\sim_\Phi)|<V(G)$. By induction on number of states, we get a bunchy graph $B$ such that $B\leq_S G/\sim_\Phi$ and therefore $B\leq_S G$. This would prove the first version of the bunchy factor conjecture. The next two sections prove the bunchy factor conjecture for two classes of graphs using a variation of this strategy.
\section{Proof of both conjectures for weakly almost bunchy graphs}
We now prove the $O(G)$ and bunchy factor conjectures for the class of weakly almost bunchy graphs. The proof mirrors the proof of the almost bunchy case given in \cite{Mac}. As in \cite{Mac}, we use the notion of a minimal image to find a non-trivial stability relation.
\begin{definition}[Minimal image]
Let $G$ and $H$ be strongly connected graphs with a right resolver $\Phi: G\to H$. A minimal image is a set $U=\partial\Phi^{-1}(I)\cdot_\Phi u$ for some $I\in V(G)$ and $u\in L_I(H)$, such that for all $v\in L_{t(u)}(H)$, $|U\cdot_\Phi v|=|U|$.
\end{definition}
The following properties of minimal images are contained in \cite{Mac}, but for completeness we give a proof.
\begin{proposition}
\label{minImg}
Let $G$ and $H$ be strongly connected graphs with a right resolver $\Phi:G\to H$. Every minimal image of $\Phi$ is of the same size, and every vertex in $G$ is contained in some minimal image.
\end{proposition}
\begin{proof}
For the first assertion, let $U_1=\partial\Phi^{-1}(I_1)\cdot_\Phi u_1$ and $U_2= \partial\Phi^{-1}(I_2)\cdot_\Phi u_2$ be minimal images, where $I_i\in V(H)$ and $u_i\in L_{I_i}(H)$. By strong connectedness, there is a path $w$ from $t(u_1)$ to $I_2$ in $H$. Then $U_1\cdot_\Phi wu_2\subset U_2$, and by minimality of $U_1$, we have $|U_1\cdot_\Phi wu_2|=|U_1|$. Hence $|U_1|\leq|U_2|$. Similarly, $|U_2|\leq|U_1|$, so $|U_1|=|U_2|$.
For the second assertion, let $I'\in V(G)$ be arbitrary, let $U$ be a minimal image, and let $J'\in U$. By strong connectedness, there is a path $w$ from $J'$ to $I'$ in $G$. Then $U\cdot_\Phi \Phi(w)$ is a minimal image with $I'=J'\cdot_\Phi \Phi(w)\in U\cdot_\Phi \Phi(w)$.
\end{proof}
We will use the following sufficient condition for stability to find a right resolver with a non-trivial stability relation. This result is contained in \cite{Mac}, but was motivated by a similar result in the road colouring context due to Trahtman.
\begin{proposition}
\label{minImgStab}
Let $G$ and $H$ be strongly connected graphs with a right resolver $\Phi:G\to H$. If there are minimal images $U_1, U_2\subset\partial\Phi^{-1}(I)$ for some $I\in H$ such that $U_1\Delta U_2=\{J_1,J_2\}$, then $J_1\sim_\Phi J_2$.
\end{proposition}
We now show that we can find a non-trivial stability congruence for weakly almost bunchy graphs that are not bunchy, and then proceed by induction.
\begin{proposition}
\label{wabStab}
Let $G$ be a strongly connected weakly almost bunchy graph. If $G$ is not bunchy, then there is a right resolver $\Phi:G\to M(G)$ such that $\sim_\Phi$ is non-trivial.
\end{proposition}
\begin{proof}
Since $G$ is not bunchy, there are states $I, J\in V(M(G))$ such that there is a vertex $I_0'\in \Sigma_G^{-1}(I)$ with distinct follower states $J_1',J_2'\in\Sigma_G^{-1}(J)$ and edges $e_i\in E_{I_0'}^{J_i'}(G)$. If there is a state $I_1'\in\Sigma_G^{-1}(I)$ such that $|\Sigma_G^{-1}(J)\cap F(I_1')|=|E_I^J(M(G))|$, choose $I_0'=I_1'$. By weakly almost bunchiness, for each $I'\in \Sigma_G^{-1}(I)\setminus\{I_0'\}$, there is a pair of distinct edges $f_{I',1},f_{I',2}\in E_{I'}(G)$ with $t(f_{I',1})=t(f_{I',2})\in\Sigma_G^{-1}(J)$. Define a right-resolver $\Phi:G\to M(G)$ by setting $\Phi(f_{I',i})=\Phi(e_i)$ for all $I'\in \Sigma_G^{-1}(I)\setminus\{I_0'\}$, and making $\Phi$ otherwise arbitrary. By proposition \ref{minImg}, there is a minimal image $U\subset \Sigma_G^{-1}(I)$ that contains $I_0'$. Let $U_0=U\setminus\{I_0'\}$. Then $U_0\cdot_\Phi\Phi(e_1)=U_0\cdot_\Phi\Phi(e_2)$ since $I'\cdot_\Phi \Phi(e_1)=I'\cdot_\Phi \Phi(e_2)$ for all $I'\in \Sigma_G^{-1}(I)\setminus \{I_0\}$. By minimality of $U$, $J_i'\notin U_0\cdot_\Phi\Phi(e_i)$, so $(U\cdot_\Phi \Phi(e_i))\Delta( U\cdot_\Phi\Phi(e_1))=\{J_1',J_2'\}$. Therefore $J_1'\sim_\Phi J_2'$ by proposition \ref{minImgStab}.
\end{proof}
The following closure property of weakly almost bunchy graphs is used in the induction step. It is also used to prove the $O(G)$ conjecture for weakly almost bunchy graphs as a corollary of the bunchy factor conjecture.
\begin{proposition}
\label{wabRRClose}
The class of weakly almost bunchy graphs is closed under right resolvers.
\end{proposition}
\begin{proof}
Let $G$ be weakly almost bunchy and let $\Phi:G\to H$ be a right-resolver. Let $I,J\in V(M)$ be such that $E_I^J(M(G))\geq 2$. Let $I'\in \Sigma_H^{-1}(I)$. By theorem \ref{M(G)} and the surjectivity of $\partial\Phi$, there is an $I''\in\Sigma^{-1}_G(I)$ such that $\partial\Phi(I'')=I'$. Let $J'\in \Sigma_H^{-1}(J)\cap F(I')$ and $e\in E_{I'}^{J'}$. Since $\Phi$ is right resolving, there is an $a\in E_{I''}(G)$ such that $\Phi(a)=e$. Then $t(a)\in \Sigma_G^{-1}(J)\cap F(I'')$ such that $\partial\Phi(t(a))=J'$. Hence $|\Sigma^{-1}_H(J)\cap F(I')|\leq|\Sigma^{-1}_G(J)\cap F(I'')|$. Since $G$ is weakly almost bunchy, it follows that $H$ is also weakly almost bunchy.
\end{proof}
The following two propositions show that the bunchy factor conjecture and the $O(G)$ conjecture are true for weakly almost bunchy graphs.
\begin{proposition}
\label{wabBF}
For all strongly connected weakly almost bunchy graphs $G$, $B(G)\leq_S G$.
\end{proposition}
\begin{proof}
If $|V(G)|=1$, the claim clearly holds. Suppose $|V(G)|=n>1$ and that the claim holds for graphs with fewer than $n$ vertices. If $G$ is bunchy, then $B(G)=G$ and so $B(G)\leq_S G$. Suppose $G$ is non-bunchy. By proposition \ref{wabStab}, there is a right resolver $\Phi:G\to M(G)$ such that $|V(G/\sim_\Phi)|<|V(G)|$. By proposition \ref{wabRRClose}, $G/\sim_\Phi$ is weakly almost bunchy. By the induction hypothesis, we then get $B(G/\sim_\Phi)\leq_S G/\sim_\Phi\leq_S G$. Let $\Psi: G\to B(G/\sim_\Phi)$ be a synchronizing right resolver. By theorem \ref{bunchy} (\ref{5.10.2}), there are right resolvers $\Delta: G\to B(G)$ and $\Theta: B(G)\to B(G/\sim_\Phi)$ such that $\Psi=\Theta\circ\Delta$. By theorem \ref{degMult}, $\Delta$ is synchronizing. Hence $B(G)\leq_S G$.
\end{proof}
\begin{proposition}
Every strongly connected weakly almost bunchy graph has a unique $\leq_S$-minimal synchronizing factor.
\end{proposition}
\begin{proof}
Let $G$ be strongly connected and weakly almost bunchy. Let $H\leq_S G$. By proposition \ref{wabRRClose}, $H$ is weakly almost bunchy. By proposition \ref{wabStab}, if $H$ is not bunchy then $H$ is not $\leq_S$-minimal. Hence every $\leq_S$-minimal sychronizing factor of $G$ is bunchy. Theorem \ref{bunchy} (\ref{5.9}) then shows that $G$ has a unique $\leq_S$-minimal synchronizing factor.
\end{proof}
\section{The bunchy factor conjecture for bi-resolving graphs}
J. Kari \cite{Kari} solved the road problem for Eulerian graphs a few years before Trahtman gave a proof of the general case. We now adapt Kari's proof to prove the bunchy factor conjecture for a class of graphs that generalizes Eulerian graphs. The key property used in Kari's proof is that Eulerian graphs with constant out-degree admit a road colouring for which no pair of states is synchronizable. A right resolver on a strongly connected graph has no synchronizable pair of states if and only if it is bi-resolving. It is therefore natural to generalize Kari's approach to prove the bunchy factor conjecture for graphs $G$ that admit a bi-resolving homomorphism onto $B(G)$. We will refer to these graphs themselves as bi-resolving.
\begin{definition}
A graph $G$ is bi-resolving if there is a bi-resolving homomorphism $\Phi:G\to B(G)$.
\end{definition}
Note that this is weaker than the condition that $G$ admit a bi-resolver $\Phi:G \to M(G)$. If there is such a bi-resolver, by theorem \ref{bunchy} (\ref{5.10.2}) there are right resolvers $\Psi:G\to B(G)$ and $\Delta:B(G)\to M(G)$ such that $\Phi=\Delta\circ\Psi$. The factors $\Psi$ and $\Delta$ are then also bi-resolving.
The following will be used in proposition \ref{BRStab}. The converse holds similarly, but is not needed.
\begin{proposition}
\label{injIfSurj}
Let $G$ be a strongly connected graph, and let $\Phi:G\to H$ be right resolving. If $\Phi|_{E^I(G)}:E^I(G)\to E^{\partial\Phi(I)}(H)$ is surjective for all $I\in V(G)$, then it is injective for all $I\in V(G)$. Hence $\Phi$ is bi-resolving in this case.
\end{proposition}
\begin{proof}
Let $e_0\in E(H)$ be arbitrary. By strong connectedness, there is a path $e_1,\dots, e_n$ in $H$ from $t(e_0)$ to $i(e_0)$. Since $\Phi$ is right resolving and $\Phi|_{E^I(G)}$ is surjective for all $I\in\partial\Phi^{-1}(t(e_k))$, the lifting of each edge $e_k$ defines a surjection from $\partial\Phi^{-1}(i(e_k))$ to $\partial\Phi^{-1}(t(e_k))$. The composition of these surjections is then a surjection from $\partial\Phi^{-1}(i(e_0))$ to itself, and is therefore a bijection. The lifting of each edge $e_k$ must then define a bijection, so no two preimages of $e_0$ can have the same terminal vertex. Hence $\Phi|_{E^I(G)}$ is injective for all $I\in V(G)$.
\end{proof}
We now aim to find, for any strongly connected bi-resolving graph $G$, a right resolver $\Phi:G\to B(G)$ with a non-trivial stability relation. As in Kari's proof, we use the notion of a maximal synchronized set.
\begin{definition}[Synchronized and maximal synchronized sets]
Let $\Phi:G\to H$ be right-resolving and let $G$ be strongly connected. A synchronized set is a set of the form $S=u\cdot_\Phi I'$ for some $I'\in V(G)$ and $u\in L^{\partial\Phi(I')}(H)$. A synchronized set $S\subset \partial\Phi^{-1}(J)$ for some $J\in V(H)$ is a maximal synchronized set (MSS) if $|v\cdot_\Phi S|\leq |S|$ for all $v\in L^{J}(H)$.
\end{definition}
If $S\subset \partial\Phi^{-1}(I)$ is a synchronized set under a right-resolver $\Phi:G\to H$, then so is $u\cdot_\Phi S$ for any $u\in L^I(H)$. The size of $u\cdot_\Phi S$ is also bounded above by $|V(G)|$, so there must be some $w\in L^I(H)$ such that, for all $v\in L^{i(v)}(H)$, $|vw\cdot_\Phi S|\leq |w\cdot_\Phi S|$. Therefore every right-resolver has an MSS contained in some fiber.
\begin{proposition}
\label{BRMSS}
Let $G$ be an strongly connected bi-resolving graph. Let $\Phi:G\to B(G)$ be a bi-resolver, and let $\Phi':G\to B(G)$ be a right resolver such that $\partial\Phi'=\partial\Phi$. Let $S\subset \partial\Phi'^{-1}(I)$ be a maximal synchronized set under $\Phi'$. Then $u\cdot_{\Phi'} S$ is an MSS under $\Phi'$ for any $u\in L^I(B(G))$. Consequently, every fiber of $\partial\Phi'$ contains an MSS under $\Phi'$.
\end{proposition}
\begin{proof}
Let $I'\in S\subset\partial\Phi'^{-1}(I)$. Since $\Phi$ is bi-resolving, $\Phi|_{E^{I'}(G)}:E^{I'}(G)\to E^I(B(G))$ is a bijection, and so $|E^{I'}(G)|=|E^I(B(G))|$. Since $\Phi'$ is right resolving, we then have
$$
\sum_{e\in E^{I}(B(G))} |e\cdot_{\Phi'} I'|= |E^{I'}(G)|=|E^I(B(G))|
$$
and using the right resolving property to sum over $S$ gives
\begin{align}
\sum_{e\in E^{I}(B(G))} |e\cdot_{\Phi'} S| = |E^I(B(G))||S|\label{eq1}
\end{align}
Since $S$ is an MSS, $|e\cdot_{\Phi'} S|\leq |S|$ for all $e\in E^I(B(G))$. By (\ref{eq1}), it must then be that $|e\cdot_{\Phi'} S| = |S|$ for all $e\in E^I(B(G))$. Therefore $e\cdot_{\Phi'} S$ is an MSS for all $e\in E^I(B(G))$ and so, by induction, $u\cdot_{\Phi'} S$ is an MSS for all $u\in L^I(B(G))$.
By the strong connectedness of $B(G)$, for any $J\in V(B(G))$, there is a $v$ from $J$ to $I$. The set $v\cdot_{\Phi'} S$ is then an MSS contained in $\partial\Phi'^{-1}(J)$. Therefore every fiber of $\partial\Phi'$ contains an MSS.
\end{proof}
Suppose $\Phi: G \to H$ is a right resolver. For any $I\in V(H)$ and $w\in L_I(H)$, the equivalence classes of the relation on $\partial\Phi^{-1}(I)$ defined by synchronization by $w$ partitions $\partial\Phi^{-1}(I)$ into synchronized sets. The next proposition shows that, under some assumptions, $w$ can be chosen so that these synchronized sets are maximal.
\begin{proposition}
Let $G$ be a strongly connected bi-resolving graph. Let $\Phi:G\to B(G)$ be a right resolver for which there is a bi-resolver onto $B(G)$ with the same vertex map. For any $I\in V(B(G))$, there is a path $w\in L_I(B(G))$ that partitions $\partial\Phi^{-1}(I)$ into MSS's under $\Phi$.
\end{proposition}
\begin{proof}
Let $S_1,\dots, S_n\subset \partial\Phi^{-1}(I)$ be MSS's synchronized by a path $w$ to vertices $J_1',\dots, J_n'\in\partial\Phi^{-1}(t(w))$ respectively. By proposition \ref{BRMSS}, we can choose $w$ so that $n\geq 1$. Suppose there is an $I'\in\partial\Phi^{-1}(I)\setminus\bigcup_{i=1}^nS_i$. By strong connectedness of $G$, there is a path $u\in L_{t(w)}(B(G))$ such that $S_1\cdot_\Phi wu=I'$. Consider the MSS's synchronized by $wuw$. These include $wu\cdot_\Phi S_1,\dots,wu\cdot_\Phi S_n$, which are synchronized to $J_1',\dots, J_n'$ respectively, and $S_1$, which is synchronized to $I'\cdot_\Phi w$. Because these sets are synchronized to distinct vertices by the same path, they are disjoint. Repeating this process gives a path that partitions $\partial\Phi^{-1}(I)$ into maximal synchronized sets.
\end{proof}
The following is an illustration of the right resolvers $\Phi$ and $\Phi'$ in the proof of proposition \ref{BRStab}. For each of $\Phi$ and $\Phi'$, the left (resp. right) states in $G$ map to the left (resp. right) states in $B(G)$. The solid and dashed edges in $G$ map to the corresponding edges in $B(G)$.
\begin{center}
\begin{tikzpicture}
\tikzstyle{vtx} = [circle, minimum width = 2mm, fill, inner sep = 0pt]
\node (G) at (-1.5,2) {$G:$};
\node (M_G) at (-1.5,-1) {$B(G):$};
\node (P) at (.75, 4.5) {$\Phi:$};
\node[vtx] (I) at (0,2){};
\node[vtx] (J_1) at (1.5,2.75){};
\node[vtx] (J_2) at (1.5,1.25){};
\node[vtx] (N_1) at (0,3.5){};
\node[vtx] (N_2) at (0,.5){};
\node (I_1) [right = 0cm of J_1]{$I_1'$};
\node (I_2) [right = 0cm of J_2]{$I_2'$};
\draw[- Stealth, thick, dashed] (N_1) to node[above]{$f_1$} (J_1);
\draw[- Stealth, thick] (N_2) to node[below]{$f_2$} (J_2);
\draw[- Stealth, thick] (I) to node[above]{$e_1$} (J_1);
\draw[- Stealth, thick, dashed] (I) to node[below]{$e_2$} (J_2);
\node[vtx] (M_1) at (0,-1){};
\node[vtx] (M_2) at (1.5,-1){};
\node (I) [right = 0cm of M_2] {$I$};
\draw[- Stealth, thick] (M_1) to [bend left = -20] (M_2);
\draw[- Stealth, thick, dashed] (M_1) to [bend left = 20] (M_2);
\node (P') at (3.75, 4.5) {$\Phi':$};
\node[vtx] (I') at (3,2){};
\node[vtx] (J_1') at (4.5,2.75){};
\node[vtx] (J_2') at (4.5,1.25){};
\node[vtx] (N_1') at (3,3.5){};
\node[vtx] (N_2') at (3,.5){};
\node (I_1') [right = 0cm of J_1']{$I_1'$};
\node (I_2') [right = 0cm of J_2']{$I_2'$};
\draw[- Stealth, thick, dashed] (N_1') to node[above]{$f_1$} (J_1');
\draw[- Stealth, thick] (N_2') to node[below]{$f_2$} (J_2');
\draw[- Stealth, thick, dashed] (I') to node[above]{$e_1$} (J_1');
\draw[- Stealth, thick] (I') to node[below]{$e_2$} (J_2');
\node[vtx] (M_1') at (3,-1){};
\node[vtx] (M_2') at (4.5,-1){};
\node (I) [right = 0cm of M_2'] {$I$};
\draw[- Stealth, thick] (M_1') to [bend left = -20] (M_2');
\draw[- Stealth, thick, dashed] (M_1') to [bend left = 20] (M_2');
\end{tikzpicture}
\end{center}
\begin{proposition}
\label{BRStab}
Let $G$ be a non-bunchy, strongly connected graph such that there is a bi-resolving homomorphism $\Phi:G\to B(G)$. Then there is a right-resolving homomorphism $\Phi':G\to B(G)$ that has a non-trivial stability congruence $\sim_{\Phi'}$. Moreover, $\Phi'$ induces a bi-resolving homomorphism from $G/\sim_{\Phi'}$ onto $B(G)$.
\end{proposition}
\begin{proof}
Since $G$ is non-bunchy, there are distinct vertices $I_1',I_2'\in V(G)$ and edges $e_i\in E^{I_i'}(G)$ such that $i(e_1)=i(e_2)$, and $\Sigma_G(I_1')=\Sigma_G(I_2')$. Since $B(G)$ is bunchy, we must then have $\partial\Phi(I_1')=\partial\Phi(I_2')=I$ for some $I\in V(B(G))$. Since $\Phi$ is bi-resolving, there are edges $f_i\in E^{I_i'}(G)$ such that $\Phi(f_1)=\Phi(e_2)$ and $\Phi(f_2)=\Phi(e_1)$. Define $\Phi':G\to B(G)$ by setting $\partial\Phi=\partial\Phi'$, $\Phi'(e_1)=\Phi(e_2)$, $\Phi'(e_2)=\Phi(e_1)$, and $\Phi'(e)=\Phi(e)$ for all $e\in E(G)\setminus\{e_1,e_2\}$.
Let $S\subset \partial\Phi^{-1}(I)$ be such that $I_1'\in S$ and $I_2'\notin S$. Since $\Phi'$ is bi-resolving on every vertex but $I_1'$ and $I_2'$, $|\Phi'(e_1)\cdot_{\Phi'} (S\setminus\{I_1'\})|=|S|-1$. Also, $\Phi'(e_1)\cdot_{\Phi'} I_1'=\{i(e_1),i(f_1)\}$. Since $\Phi'$ is still right-resolving, we then have $|\Phi'(e_1)\cdot_{\Phi'} S|=|S|+1$. Therefore $S$ is not an MSS. Similarly, if $S\subset \partial\Phi^{-1}(I)$ such that $I_2'\in S$ but $I_1'\notin S$, then $S$ is not an MSS. Thus, in any partition of $\partial\Phi^{-1}(I)$ into MSS's, $I_1'$ and $I_2'$ must belong to the same partition element.
To show that $I_1'$ and $I_2'$ are stable under $\Phi'$, let $v\in L_I(B(G))$, and extend $v$ by strong connectedness to a cycle $v'$ from $I$ to $I$. Let $w\in F(I)$ be a path that partitions $\partial\Phi^{-1}(I)$ into some MSS's $S_1,\dots, S_n$. Then $v'w$ partitions $\partial\Phi^{-1}(I)$ into the MSS's $v'\cdot_{\Phi'} S_1, \dots, v'\cdot_{\Phi'} S_n$, and $\{I_1',I_2'\}\subset v'\cdot_{\Phi'} S_i$ for some $i\in\{1,\dots, n\}$, so $I_1'\cdot_{\Phi'} v'w=I_2' \cdot_{\Phi'} v'w$. Hence $I_1'\sim_{\Phi'} I_2'$.
We now show that the right-resolver from $G/\sim_{\Phi'}$ to $B(G)$ induced by $\Phi'$ is bi-resolving. Let $\Psi: G/\sim_{\Phi'}\to B(G)$ be the right-resolver given by $\partial\Psi([I']_{\sim_{\Phi'}}) = \partial\Phi(I')$ for $I'\in V(G)$ and $\Psi([e]_{\sim_{\Phi'}})= \Phi'(e)$ for $e\in E(G)$. By proposition \ref{injIfSurj}, it suffices to show that $\Psi|_{E^{[I']_{\sim_{\Phi'}}}}:E^{[I']_{\sim_{\Phi'}}}\to E^{\partial\Phi(I')}$ is surjective for all $[I']_{\sim_{\Phi'}}\in V(G/\sim_{\Phi'})$. Let $[I']_{\sim_{\Phi'}}\in V(G/\sim_{\Phi'})$, and let $I=\partial\Phi(I')$. If there is a $I_3'\in[I']_{\sim_{\Phi'}}\setminus\{I_1',I_2'\}$ then $\Phi'(E^{I_3'})=E^I$ so $\Psi(E^{[I']_{\sim_{\Phi'}}}) = E^I$. Otherwise, $[I']_{\sim_{\Phi'}}\subset\{I_1',I_2'\}$, and since $I_1'\sim_{\Phi'}I_2'$ we have $[I']_{\sim_{\Phi'}}=\{I_1',I_2'\}$. Since $\Phi'(E^{I_1'}\cup E^{I_2'})=E^I$, we still have $\Psi(E^{[I']_{\sim_{\Phi'}}})=E^I$. Hence $\Psi|_{E^{[I']_{\sim_{\Phi'}}}}$ is surjective.
\end{proof}
The following proposition shows that the bunchy factor conjecture holds for bi-resolving graphs.
\begin{proposition}
Let $G$ be a strongly connected bi-resolving graph. Then $B(G)\leq_S G$.
\end{proposition}
\begin{proof}
If $|V(G)|=1$, the claim clearly holds. Suppose $|V(G)|=n>1$ and that the claim holds for graphs with fewer than $n$ vertices. If $G$ is bunchy, then $B(G)=G$ and so $B(G)\leq_S G$. If $G$ is non-bunchy, by proposition \ref{BRStab}, there is a right-resolver $\Phi':G\to M(G)$ such that $|V(G/\sim_{\Phi'})|<|V(G)|$, and there is a bi-resolver from $G/\sim_{\Phi'}$ to $B(G)=B(G/\sim_{\Phi'})$. By the induction hypothesis, $B(G/\sim_{\Phi'})\leq_S G/\sim_{\Phi'}\leq_S G$, so $B(G)\leq_S G$.
\end{proof}
We have now shown that the bunchy factor conjecture holds for bi-resolving graphs, but we have not shown that the $O(G)$ conjecture holds similarly. In the case of weakly almost bunchy graphs, we used the fact that that class of graphs is closed under right resolvers to show that the bunchy factor conjecture implies the $O(G)$ conjecture. The class of bi-resolving graphs, however, is not closed under right resolvers, so a different or modified strategy would be needed.
\section{Empirical evidence for the bunchy factor conjecture}
We now present computer generated data that supports the bunchy factor conjecture, and by extension the $O(G)$ conjecture. In \cite{AMT}, a polynomial time algorithm is given for constructing $M(G)$, and deciding isomorphism between $M(G)$ and $M(H)$ for two graphs $G$ and $H$. In \cite{Mac}, polynomial time algorithms are given for constructing $B(G)$ and the stability relation of a right resolver. By implementing these algorithms we can estimate the probability that a given right resolver $\Phi: G\to B(G)$ is synchronizing. The bunchy factor conjecture states that this probability is nonzero for all strongly connected graphs. Not only did all of the graphs tested have a positive associated probability, but our results suggest that most right resolvers from a graph $G$ to its $B(G)$ are synchronizing.
For an irreducible graph $G$, the probability that a right resolver $\Phi:G\to B(G)$ is synchronizing was estimated by generating random right resolvers until 100 synchronizing right resolvers from $G$ to $B(G)$ were found (the fact that the testing procedure never failed to find a synchronizer shows that the bunchy factor conjecture is true for all graphs tested). The probability is then given by $p=100/\textit{\# of right resolvers generated}$. This estimator is derived from the fact that the total number of right resolvers generated is modeled by the sum of 100 geometric random variables, which has expectation $100/p$ where $p$ is the probability of success, in this case the probability of synchronization.
The following table records the average of the associated probability of graphs according to their minimal right resolving factor (given by an adjacency matrix) and number of states. Each entry in the table is an average over 10,000 graphs with the given $M(G)$ and $|V(G)|$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$M(G)$ / $|V(G)|$ & 4 & 5 & 6 & 7 & 8\\
\hline
$\begin{bmatrix}
2&1\\
1&0
\end{bmatrix}$ & .983705 & .997026 & .996073 & .999273 & .999419\\
\hline
$\begin{bmatrix}
1&2\\
1&0
\end{bmatrix}$ & .981192 & .982191 & .988474 & .993667 & .995971\\
\hline
$\begin{bmatrix}
0&3\\
1&0
\end{bmatrix}$ & .945870 & .926934 & .938034 & .957786 & .975323\\
\hline
\end{tabular}
\end{center}
The generally high probability (greater than $.9$) that a right resolver $\Phi:G \to B(G)$ is synchronizing is consistent with the constant out-degree case, where it is known that most road colourings are synchronizing. We might also expect that right resolvers are synchronizing with high probability (in a rigorous sense) since this was shown by Berlinkov in \cite{Berlinkov} to be true for road colourings. The small size of the graphs tested in our results, however, limits the insight we can provide on this point.
For most graphs, every right resolver from $G$ to $B(G)$ tested was synchronizing. The few graphs with an estimated associated probability of less than one often had a bimodal or multi-modal distribution. The following histograms illustrate these results. For each $M(G)$ and $|V(G)|$, the left histogram includes all graphs tested, while the right histogram includes only those with probability less than one.
\begin{center}
\begin{tikzpicture}
\node (M) at (-1,0) {$M(G):$};
\node[vtx] (g1) at (0,0){};
\node[vtx] (g2) at (1,0){};
\draw[-Stealth, thick] (g1) to [bend left] (g2);
\draw[-Stealth, thick] (g2) to [bend left] (g1);
\draw[Stealth-, thick] (g2)+(.7mm,-.9mm) arc (-160:160:2.5mm);
\draw[Stealth-, thick] (g2)+(-.3mm,-1mm) arc (-165:165:4mm);
\node at (-4,-3) {\includegraphics[scale=.33]{histograms/V_G=6.png}};
\node at (4,-3) {\includegraphics[scale=.33]{histograms/V_G=6,_P_1.png}};
\node at (-4,-8) {\includegraphics[scale=.33]{histograms/V_G=7.png}};
\node at (4,-8) {\includegraphics[scale=.33]{histograms/V_G=7,_P_1.png}};
\node at (-4,-13) {\includegraphics[scale=.33]{histograms/V_G=8.png}};
\node at (4,-13) {\includegraphics[scale=.33]{histograms/V_G=8,_P_1.png}};
\end{tikzpicture}
\begin{tikzpicture}
\node (M) at (-1,0) {$M(G):$};
\node[vtx] (g1) at (0,0){};
\node[vtx] (g2) at (1,0){};
\draw[-Stealth, thick] (g1) to [bend left=35] (g2);
\draw[-Stealth, thick] (g2) to [bend left = 0] (g1);
\draw[-Stealth, thick] (g2) to [bend left = 35] (g1);
\draw[Stealth-, thick] (g2)+(.4mm,-.8mm) arc (-163:163:3mm);
\node at (-4,-3) {\includegraphics[scale=.33]{histograms/2_V_G=6.png}};
\node at (4,-3) {\includegraphics[scale=.33]{histograms/2_V_G=6,_P_1.png}};
\node at (-4,-8) {\includegraphics[scale=.33]{histograms/2_V_G=7.png}};
\node at (4,-8) {\includegraphics[scale=.33]{histograms/2_V_G=7,_P_1.png}};
\node at (-4,-13) {\includegraphics[scale=.33]{histograms/2_V_G=8.png}};
\node at (4,-13) {\includegraphics[scale=.33]{histograms/2_V_G=8,_P_1.png}};
\end{tikzpicture}
\end{center}
\newpage
\section{Acknowledgments}
Many thanks to Sophie MacDonald and Brian Marcus for their generous feedback and support. The author also acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC).
\bibliographystyle{plain}
| {
"timestamp": "2022-09-15T02:03:20",
"yymm": "2209",
"arxiv_id": "2209.06304",
"language": "en",
"url": "https://arxiv.org/abs/2209.06304",
"abstract": "The road colouring theorem characterizes the class of strongly connected directed graphs with constant out-degree that admit a synchronizing road colouring. The subject of this paper is a pair of related conjectures that generalize the road colouring theorem to graphs with non-constant out-degree; we give reasons to believe that both of these conjectures are true. Our main results focus on two classes of graphs, proving both conjectures for one class of graphs and one of the conjectures for an additional class of graphs. We also present computer simulations that give some empirical evidence for the conjectures.",
"subjects": "Dynamical Systems (math.DS); Combinatorics (math.CO)",
"title": "A note on conjectures generalizing the road colouring theorem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575137315161,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542220939847
} |
https://arxiv.org/abs/2203.11550 | Running Time Analysis of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) using Binary or Stochastic Tournament Selection | Evolutionary algorithms (EAs) have been widely used to solve multi-objective optimization problems, and have become the most popular tool. However, the theoretical foundation of multi-objective EAs (MOEAs), especially the essential theoretical aspect, i.e., running time analysis, has been still largely underdeveloped. The few existing theoretical works mainly considered simple MOEAs, while the non-dominated sorting genetic algorithm II (NSGA-II), probably the most influential MOEA, has not been analyzed except for a very recent work considering a simplified variant without crossover. In this paper, we present a running time analysis of the standard NSGA-II for solving LOTZ, OneMinMax and COCZ, the three commonly used bi-objective optimization problems. Specifically, we prove that the expected running time (i.e., number of fitness evaluations) is $O(n^3)$ for LOTZ, and $O(n^2\log n)$ for OneMinMax and COCZ, which is surprisingly as same as that of the previously analyzed simple MOEAs, GSEMO and SEMO. Next, we introduce a new parent selection strategy, stochastic tournament selection (i.e., $k$ tournament selection where $k$ is uniformly sampled at random), to replace the binary tournament selection strategy of NSGA-II, decreasing the required expected running time to $O(n^2)$ for all the three problems. Experiments are also conducted, suggesting that the derived running time upper bounds are tight for LOTZ, and almost tight for OneMinMax and COCZ. | \section{Introduction}
Multi-objective optimization~\cite{Steuer86}, which requires to optimize several objective functions simultaneously, arises in many areas. Since the objectives are usually conflicting, there doesn't exist a single solution which can perform well on all these objective functions. Thus,
the goal of multi-objective optimization is to find a set of Pareto optimal solutions (or the Pareto front), representing different optimal trade-offs between these objectives.
Evolutionary algorithms (EAs)~\cite{back:96} are a kind of randomized heuristic optimization algorithms, inspired by natural evolution. They maintain a set of solutions (i.e., a population), and iteratively improve the population by reproducing new solutions and selecting better ones. Due to their population-based nature, EAs are very popular for solving multi-objective optimization problems, and have been widely used in many real-world applications~\cite{coello2004applications}.
Compared with practical applications, the theoretical foundation of EAs is still underdeveloped, which is mainly because the sophisticated behaviors of EAs make theoretical analysis quite difficult. Though much effort has been devoted to the essential theoretical aspect, i.e., running time analysis, leading to a lot of progresses~\cite{neumann2010bioinspired,auger2011theory,doerr-telo21-survey} in the past 25 years, most of them focused on single-objective optimization, while only a few considered the more complicated scenario of multi-objective optimization. Next, we briefly review the results of running time analyses on multi-objective EAs (MOEAs).
The running time analysis of MOEAs started from GSEMO, a simple MOEA which employs the bit-wise mutation operator to generate an offspring solution in each iteration and keeps the non-dominated solutions generated-so-far in the population.
For GSEMO solving the bi-objective optimization problems LOTZ and COCZ, the expected running time has been proved to be $O(n^3)$~\cite{Giel03} and $O(n^2 \log n)$~\cite{Qian13,bian2018tools}, respectively, where $n$ is the problem size.
SEMO is a counterpart of GSEMO, which employs the local mutation operator, one-bit mutation, instead of the global bit-wise mutation operator. \citet{LaumannsTEC04} proved that the expected running time of SEMO solving LOTZ and COCZ are $\Theta(n^3)$ and $O(n^2\log n)$, respectively.
\citet{Giel10} considered another bi-objective problem OneMinMax,
and proved that both GSEMO and SEMO can solve it in $O(n^2\log n)$ expected running time.
\citet{doerr2013lower} also proved a lower bound $\Omega(n^2/p)$ for GSEMO solving LOTZ, where $p<n^{-7/4}$ is the mutation rate, i.e., the probability of flipping each bit when performing bit-wise mutation.
Later, the analyses of GSEMO were conducted on multi-objective combinatorial optimization problems.
For bi-objective minimum spanning trees (MST), GSEMO was proved to be able to find a 2-approximation of the Pareto front in expected pseudo-polynomial time~\cite{Neumann07}.
For multi-objective shortest paths, a variant of GSEMO can achieve an $(1+\epsilon)$-approximation in expected pseudo-polynomial time~\cite{Horoba09,Neumann10}, where $\epsilon>0$.
\citet{laumanns-nc04-knapsack} considered the GSEMO and its variant for solving a special case of the multi-objective knapsack problem, and proved that the expected running time of the two algorithms for finding all the Pareto optimal solutions are $O(n^6)$ and $O(n^5)$, respectively.
There are also studies that analyze the GSEMO for solving single-objective constrained optimization problems. By optimizing a reformulated bi-objective optimization problem that optimizes the original objective and a constraint-related objective simultaneously, the GSEMO can reduce the expected running time significantly for achieving a desired approximation ratio. For example, by reformulating the MST problem into a bi-objective problem, \citet{neumann-nc06-mst-easier} proved that GSEMO and SEMO can solve MST in $O(mn(n+\log w_{\max}))$ expected running time, which is better than the $O(m^2(\log n+\log w_{\max}))$ expected running time achieved by (1+1)-EA and RLS~\cite{neumann-tcs07-mst}, i.e., single-objective counterparts to GSEMO and SEMO, for $m=\Theta(n^2)$, where $m,n$ and $w_{\max}$ denote the number of edges, the number of nodes and the largest weight of the graph, respectively.
More evidences have been proved on the problems of minimum cuts~\cite{neumann-algo11-cut}, set cover~\cite{friedrich-ecj10-cover} and submodular optimization~\cite{friedrich2015maximizing}. Note that we concern multi-objective optimization problems in this paper.
Based on the GSEMO and SEMO, the effectiveness of some strategies for multi-objective evolutionary optimization have been analyzed. For example,
\citet{LaumannsTEC04} showed the effectiveness of greedy selection by proving that using this strategy can reduce the expected running time of SEMO from $O(n^2\log n)$ to $\Theta(n^2)$ for solving the COCZ problem.
\citet{Qian13} showed that crossover can accelerate filling the Pareto front by comparing the expected running time of GSEMO with and without crossover for solving the artificial problems COCZ and weighted LPTNO (a generalization of LOTZ), as well as the combinatorial problem multi-objective MST.
The effectiveness of some other mechanisms, e.g., heuristic selection~\cite{qian-ppsn16-hyper}, diversity~\cite{plateaus10}, fairness~\cite{LaumannsTEC04,friedrich2011illustration}, and diversity-based parent selection~\cite{osuna2020diversity} have also been examined.
Though the GSEMO and SEMO share the general structure of MOEAs, they have been much simplified. To characterize the behaviors of practical MOEAs, some efforts have been devoted to analyzing MOEA/D, which is a popular MOEA based on decomposition~\cite{zhang2007moea}.
\citet{li2015primary} analyzed a simplified variant of MOEA/D without crossover for solving COCZ and weighted LPTNO, and proved that the expected running time is $\Theta(n\log n)$ and $\Theta(n^2)$, respectively.
\citet{huang2021runtime} also considered a simplified MOEA/D, and examined the effectiveness of different decomposition approaches by comparing the running time for solving two many-objective problems $m$LOTZ and $m$COCZ, where $m$ denotes the number of objectives.
Surprisingly, the running time analysis of the non-dominated sorting genetic algorithm II (NSGA-II)~\cite{deb-tec02-nsgaii}, the probably most influential MOEA, has been rarely touched. The NSGA-II enables to find well-spread Pareto-optimal solutions by incorporating two substantial features, i.e., non-dominated sorting and crowding distance, and has become the most popular MOEA for solving multi-objective optimization problems~\cite{deb2011multi}.
To the best of our knowledge, the only attempt is a very recent work, which, however, considered a simplified version of NSGA-II without crossover, and proved that the expected running time is $O(n^2 \log n)$ for OneMinMax and $O(n^3)$ for LOTZ~\cite{zheng2021first}.
In this paper, we present a running time analysis for the standard NSGA-II. We prove that for NSGA-II solving LOTZ, the expected running time is $O(n^3)$; while for OneMinMax and COCZ, the expected running time is $O(n^2\log n)$. Note that these running time upper bounds are as same as that of GSEMO and SEMO~\cite{LaumannsTEC04,Giel03,Qian13,Giel10}, implying that the NSGA-II does not have advantage over simplified MOEAs on these problems if the derived upper bounds are tight.
Next, we introduce a new parent selection strategy, i.e., stochastic tournament selection, which samples a number $k$ uniformly at random and then performs $k$ tournament selection. By replacing the original binary tournament selection of NSGA-II with stochastic tournament selection, we prove that the expected running time of NSGA-II can be improved to $O(n^2)$ for LOTZ, OneMinMax and COCZ.
We also conduct experiments to show that the derived upper bounds are almost tight.
The goal of this work is to take a step towards analyzing the running time of practical MOEAs, and meanwhile, the introduced stochastic tournament selection strategy may be helpful in practical applications.
\section{Preliminaries}
In this section, we first introduce multi-objective optimization and the procedure of NSGA-II, and then present a new tournament selection strategy, i.e., stochastic tournament selection.
\subsection{Multi-objective Optimization}
Multi-objective optimization requires to simultaneously optimize two or more objective functions, as shown in Definition~\ref{def_MO}. We consider maximization here, while minimization can be defined similarly. The objectives are usually conflicting, and thus there is no canonical complete order in the solution space $\mathcal{X}$.
The comparison between solutions relies on the \emph{domination} relationship, as presented in Definition~\ref{def_Domination}. A solution is \emph{Pareto optimal} if there is no other solution in $\mathcal{X}$ that dominates it. The set of objective vectors of all the Pareto optimal solutions constitutes the \emph{Pareto front}. The goal of multi-objective optimization is to find the Pareto front, that is, to find at least one corresponding solution for each objective vector in the Pareto front.
\begin{definition}[Multi-objective Optimization]\label{def_MO}
Given a feasible solution space $\mathcal{X}$ and objective functions $f_1,f_2,\ldots, f_m$, multi-objective optimization can be formulated as\vspace{-0.3em}
\begin{align}
\max\nolimits_{\bm{x} \in
\mathcal{X}}\; \big(f_1(\bm{x}),f_2(\bm{x}),...,f_m(\bm{x})\big).
\end{align}
\end{definition}
\begin{definition}[Domination]\label{def_Domination}
Let $\bm f = (f_1,f_2,\ldots, f_m):\mathcal{X} \rightarrow \mathbb{R}^m$ be the objective vector. For two solutions $\bm{x}$ and $\bm{y}\in \mathcal{X}$:\vspace{-0.1em}
\begin{itemize}
\item $\bm{x}$ \emph{weakly dominates} $\bm{y}$ (denoted as $\bm{x} \succeq \bm{y}$) if $\forall 1 \leq i \leq m, f_i(\bm{x}) \geq f_i(\bm{y})$;
\item $\bm{x}$ \emph{dominates} $\bm{y}$ (denoted as $\bm{x}\succ \bm{y}$) if $\bm{x} \succeq \bm{y}$ and $f_i(\bm{x}) > f_i(\bm{y})$ for some $i$;
\item $\bm{x}$ and $\bm{y}$ are \emph{incomparable} if neither $\bm{x}\succeq \bm{y}$ nor $\bm{y}\succeq \bm{x}$.
\end{itemize}
\end{definition}
\subsection{NSGA-II}\label{sec:nsgaii}
The NSGA-II algorithm~\cite{deb-tec02-nsgaii} as presented in Algorithm~\ref{alg:nsgaii} is a popular MOEA, which incorporates two substantial features, i.e., non-dominated sorting in Algorithm~\ref{alg:fastsort}, and crowding distance in Algorithm~\ref{alg:crowdist}.
NSGA-II starts from an initial population of $N$ random solutions (line~1). In each generation, it employs binary tournament selection $N$ times to generate a parent population $P'$ (line~4), and then applies one-point crossover and bit-wise mutation on the $N/2$ pairs of parent solutions to generate $N$ offspring solutions (lines~5--9).
Note that the two adjacent selected solutions form a pair, and thus the $N$ selected solutions form $N/2$ pairs.
The one-point crossover operator first selects a crossover point $i\in \{1,2,\ldots,n\}$ uniformly at random, where $n$ is the problem size, and then exchanges the first $i$ bits of two solutions. The bit-wise mutation operator flips each bit of a solution independently with probability $1/n$.
The binary tournament selection presented in Definition~\ref{def:bin-tour} picks two solutions randomly from the population $P$ with or without replacement, and then selects a better one (ties broken uniformly). Note that we consider the strategy with replacement in this paper.
\begin{algorithm}[t]
\caption{NSGA-II Algorithm~\cite{deb-tec02-nsgaii}}
\label{alg:nsgaii}
\begin{flushleft}
\textbf{Input}: objective functions $f_1,f_2\ldots,f_m$, population size $N$\\
\textbf{Output}: $N$ solutions from $\{0,1\}^n$\\
\textbf{Process}:
\end{flushleft}
\begin{algorithmic}[1]
\STATE $P\!\leftarrow\!\! N$ solutions uniformly and randomly selected from $\{0,\! 1\}^{\!n}$;
\WHILE{criterion is not met}
\STATE $Q=\emptyset$;
\STATE apply binary tournament selection $N$ times to generate a parent population $P'$ of size $N$;
\FOR{each pair of the parent solutions $\bm{x}$ and $\bm{y}$ in $P'$}
\STATE apply one-point crossover on $\bm{x}$ and $\bm{y}$ to generate two solutions $\bm{x}'$ and $\bm{y}'$, with probability 0.9;
\STATE apply bit-wise mutation on $\bm{x}'$ and $\bm{y}'$ to generate $\bm{x}''$ and $\bm{y}''$, respectively;
\STATE add $\bm{x}''$ and $\bm{y}''$ into $Q$
\ENDFOR
\STATE apply Algorithm~\ref{alg:fastsort} to partition $P\cup Q$ into non-dominated sets $F_1,F_2,\ldots$;
\STATE let $P=\emptyset$, $i=1$;
\WHILE{$|P\cup F_i|<N$}
\STATE $P=P\cup F_i$, $i=i+1$
\ENDWHILE
\STATE apply Algorithm~\ref{alg:crowdist} to assign each solution in $F_i$ with a crowding distance;
\STATE sort the solutions in $F_i$ by crowding distance in descending order, and add the first $N-|P |$ solutions into $P$
\ENDWHILE
\RETURN $P$
\end{algorithmic}
\end{algorithm}
\begin{definition}[Binary Tournament Selection]\label{def:bin-tour}
The binary tournament selection strategy first picks two solutions from the population $P$ uniformly at random, and then selects a better one with ties broken uniformly.
\end{definition}
After generating $N$ offspring solutions, the best $N$ solutions in the current population $P$ and the offspring population $Q$ are selected as the population in the next generation (lines~10--16).
In particular, the solutions in the current and offspring populations are partitioned into non-dominated sets $F_1,F_2,\ldots$ (line~10), where $F_1$ contains all the non-dominated solutions in $P \cup Q$, and $F_i$ ($i\ge 2$) contains all the non-dominated solutions in $(P \cup Q) \setminus \cup_{j=1}^{i-1} F_j$. The fast implementation of not-dominated sorting is presented in Algorithm~\ref{alg:fastsort}. Not that we use the notion $\rank{\bm{x}}=i$ to denote that $\bm{x}$ belongs to $F_i$.
Then, the solutions in $F_1,F_2,\ldots$ are added into the next population (lines~12--14), until the population size exceeds $N$. For the critical set $F_i$, i.e., the inclusion of which can make the population size larger than $N$, Algorithm~\ref{alg:crowdist} is used to compute the crowding distance for each of the solutions in it (line~15). Finally, the solutions in $F_i$ with large crowding distance are selected to fill the remaining population slots (line~16).
\begin{algorithm}[t]
\caption{Fast Non-dominated Sorting~\cite{deb-tec02-nsgaii}}
\label{alg:fastsort}
\begin{flushleft}
\textbf{Input}: a population $P$\\
\textbf{Output}: non-dominated sets $F_1,F_2,\ldots$\\
\textbf{Process}:
\end{flushleft}
\begin{algorithmic}[1]
\STATE $F_1=\emptyset$;
\FOR{ each $\bm{x} \in P$}
\STATE $S_{\bm{x}}=\emptyset ; n_{\bm{x}}=0$;
\FOR{ each $y \in P$}
\IF{$\bm{x}\succ \bm{y}$}
\STATE $S_{\bm{x}}=S_{\bm{x}} \cup\{\bm{y}\}$
\ELSIF {$\bm{x} \prec \bm{y}$}
\STATE $n_{\bm{x}}=n_{\bm{x}}+1$
\ENDIF
\ENDFOR
\IF{$n_{\bm{x}}=0$}
\STATE $\rank{\bm{x}}=1 ; F_{1}=F_{1}\cup \{\bm{x}\}$
\ENDIF
\ENDFOR
\STATE $i=1$;
\WHILE{ $F_{i} \neq \emptyset$}
\STATE $Q=\emptyset$;
\FOR{ each $\bm{x} \in F_{i}$ }
\FOR{ each $\bm{y} \in S_{\bm{x}}$ }
\STATE $n_{\bm{y}}=n_{\bm{y}}-1$;
\IF{ $n_{\bm{y}}=0$}
\STATE $\rank{\bm{y}}=i+1$; $Q=Q \cup\{\bm{y}\}$
\ENDIF
\ENDFOR
\ENDFOR
\STATE $i=i+1$; $F_i=Q$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[t]
\caption{Crowding Distance Assignment~\cite{deb-tec02-nsgaii}}
\label{alg:crowdist}
\begin{flushleft}
\textbf{Input}: $Q=\{\bm{x}^1,\bm{x}^2,\ldots,\bm{x}^l\}$ with the same rank\\
\textbf{Output}: the crowding distance $\dist{\cdot}$ for each solution in $Q$\\
\textbf{Process}:
\end{flushleft}
\begin{algorithmic}[1]
\STATE let $\dist{\bm{x}^j}=0$ for any $j\in\{1,2,\ldots,l\}$;
\FOR{$i=1$ to $m$}
\STATE sort the solutions in $Q$ w.r.t. $f_i$ in ascending order;
\STATE $\dist{Q[1]}=\dist{Q[l]}=\infty$;
\FOR{$j=2$ to $l-1$}
\STATE $\dist{Q[j]}=\dist{Q[j]}+\frac{f_i(Q[j+1])-f_i(Q[j-1])}{f_i(Q[l])-f_i(Q[1])}$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
When using binary tournament selection (line~4), the selection criterion is based on the crowded-comparison, that is, a solution $\bm{x}$ is superior to $\bm{y}$ (denoted as $\bm{x}\succ_{\mathrm{c}}\bm{y}$) if
\begin{align}\label{eq:crowd-comp}
&\rank{\bm{x}}<\rank{\bm{y}} \text{ or }\\ &\rank{\bm{x}}=\rank{\bm{y}}\wedge \dist{\bm{x}}>\dist{\bm{y}}.
\end{align}
Intuitively, the crowding distance of a solution means the distance between its closest neighbour solutions, and a solution with larger crowding distance is preferred so that the diversity of the population can be preserved as much as possible.
Note that in Algorithm~\ref{alg:crowdist}, we assume that the relative positions of the solutions with the same objective vector are unchanged or totally reversed when the solutions are sorted w.r.t. some objective function (line~3).
In line~6 of Algorithm~\ref{alg:nsgaii}, the probability of using crossover has been set to 0.9, which is the same as the original setting and also commonly used~\cite{deb-tec02-nsgaii}. However, the theoretical results derived in this paper can be directly generalized to the scenario where the probability of using crossover belongs to $[\Omega(1),1-\Omega(1)]$.
\subsection{Stochastic Tournament Selection}
As the crowded-comparison $\succ_{\mathrm{c}}$ in Eq.~\eqref{eq:crowd-comp} actually gives a total order of the solutions in the population $P$,
binary tournament selection can be naturally extended to $k$ tournament selection~\cite{eiben-book}, as presented in Definition~\ref{def:k-tour}, where $k$ is a parameter such that $1\le k\le N$.
That is, $k$ solutions are first picked from $P$ uniformly at random, and then the solution with the smallest rank is selected. If several solutions have the same smallest rank, the one with the largest crowding distance is selected, with ties broken uniformly.
\begin{definition}[$k$ Tournament Selection]\label{def:k-tour}
The $k$ tournament selection strategy first picks $k$ solutions from the population $P$ uniformly at random, and then selects the best one with ties broken uniformly.
\end{definition}
Note that a larger $k$ implies a larger selection pressure, i.e., a larger probability of selecting a good solution, and thus the value of $k$ can be used to control the selection pressure of EAs~\cite{eiben-book}. However, this also brings about a new issue, i.e., how to set $k$ properly.
In order to reduce the risk of setting improper values of $k$ as well as the overhead of tuning $k$, we introduce a natural strategy, i.e., stochastic tournament selection in Definition~\ref{def:sto-tour}, which first selects a number $k$ randomly, and then performs the $k$ tournament selection. In this paper, we consider that the tournament candidates are picked with replacement from the population.
\begin{definition}[Stochastic Tournament Selection]\label{def:sto-tour}
The stochastic tournament selection strategy first selects a number $k$ from $\{1,2,\ldots,N\}$ uniformly at random, where $N$ is the size of the population $P$, and then employs the $k$ tournament selection to select a solution from the population $P$.
\end{definition}
In each generation of NSGA-II, we need to select $N$ parent solutions independently, and each selection may involve the comparison of several solutions, which may lead to a large number of comparisons. To improve the efficiency of stochastic tournament selection, we can first sort the solutions in the population $P$, and then perform the parent selection procedure. Specifically, each solution $\bm{x}_i$ ($1\le i\le N$) in $P$ is assigned a number $\pi(i)$, where $\pi: \{1,2,\ldots,N\}\rightarrow \{1,2,\ldots,N\}$ is a bijection such that
\begin{equation}\label{eq:pi}
\forall 1\le i,j\le N, i\neq j: \bm{x}_i\succ_{\mathrm{c}} \bm{x}_j\Rightarrow \pi(i)<\pi(j).
\end{equation}
That is, a solution with a smaller number is better.
Note that the number $\pi(\cdot)$ is assigned randomly if several solutions have the same rank and crowding distance.
Then, we sample a number $k$ randomly from $\{1,2,\ldots,N\}$ and pick $k$ solutions from $P$ at random, where the solution with the lowest $\pi(\cdot)$ value is finally selected.
Lemma~\ref{lem:stotour-prob} presents the property of stochastic tournament selection, which will be used in the following theoretical analysis.
It shows that any solution (even the worst solution) in $P$ can be selected with probability at least $1/N^2$, and any solution belonging to the best $O(1)$ solutions in $P$ (with respect to $\succ_{\mathrm{c}}$) can be selected with probability at least $\Omega(1)$.
Note that for binary tournament selection, the probability of selecting the worst solution (denoted as $\bm{x}^{\mathrm{w}}$) is $1/N^2$, because $\bm{x}^{\mathrm{w}}$ is selected if and only if the two solutions picked for competition are both $\bm{x}^{\mathrm{w}}$; the probability of selecting the best solution (denoted as $\bm{x}^{\mathrm{b}}$) is $1-(1-1/N)^2=2/N-1/N^2$, because $\bm{x}^{\mathrm{b}}$ is selected if and only if $\bm{x}^{\mathrm{b}}$ is picked at least once.
Thus, compared with binary tournament selection, stochastic tournament selection can increase the probability of selecting the top solutions, and meanwhile maintaining the probability of selecting the bottom solutions.
\begin{lemma}\label{lem:stotour-prob}
If using stochastic tournament selection, any solution in $P$ can be selected with prob. at least $1/N^2$.
Furthermore, a solution $\bm{x}_i\in P$ with $\pi(i)\!=\!O(1)$ can be selected with prob. $\Omega(1)$, where $\pi: \{1,2,\ldots,N\}\rightarrow \{1,2,\ldots,N\}$ is a bijection satisfying Eq.~\eqref{eq:pi}.
\end{lemma}
\begin{proof}
For any solution $\bm{x} \in P$, it can be selected if $k=1$ and the solution picked for competition is exactly $\bm{x}$. The probabilities of the two events are both $1/N$, implying a lower bound $1/N^2$ on the probability of selecting $\bm{x}$ as a parent solution.
For the furthermore clause, we consider the case that $k\ge N/2$. Suppose that $\bm{x}_i$ is a solution with $\pi(i)=O(1)$. Then, it can be selected if $\bm{x}_i$ is picked for competition, while any solution $\bm{x}_j$ with $\pi(j)<\pi(i)$ is not picked. The probability of not picking any $\bm{x}_j$ with $\pi(j)<\pi(i)$ is $(1-(\pi(i)-1)/N)^k$, and conditional on this event, the probability of picking $\bm{x}_i$ is $1-(1-1/(N-\pi(i)+1))^k$.
Thus, the probability of selecting $\bm{x}_i$ given $k\ge N/2$ is
\begin{align}
&\bigg(1\!-\!\frac{\pi(i)-1}{N}\bigg)^k \cdot \bigg(1-\bigg(1-\frac{1}{N-\pi(i)+1}\bigg)^k\bigg)\\
&=\bigg(1-\frac{\pi(i)-1}{N}\bigg)^k - \bigg(\frac{N-\pi(i)}{N}\bigg)^k \\
&\ge \binom{k}{0}\bigg(1-\frac{\pi(i)}{N}\bigg)^k\bigg(\frac{1}{N}\bigg)^0 \!+\binom{k}{1}\bigg(1-\frac{\pi(i)}{N}\bigg)^{k-1}\bigg(\frac{1}{N}\bigg) \!-\!\bigg(1\!-\!\frac{\pi(i)}{N}\bigg)^k\\
&= \frac{k}{N}\bigg(1-\frac{1}{N/\pi(i)}\bigg)^{(N/\pi(i)-1)\cdot (k-1)/(N/\pi(i)-1)}\\
& \ge \frac{k}{N}\cdot \bigg(\frac{1}{e}\bigg)^{(k-1)/(N/\pi(i)-1)}=\Omega(1),
\end{align}
where the last equality is by $N/2\le k\le N$ and $\pi(i)=O(1)$. Note that the probability of selecting a $k$ such that $k\ge N/2$ is 1/2, and thus the lemma holds.
\end{proof}
\section{Running Time Analysis of NSGA-II}
In this section, we analyze the expected running time of the standard NSGA-II in Algorithm~\ref{alg:nsgaii}, i.e., NSGA-II using binary tournament selection, solving three bi-objective pseudo-Boolean problems LOTZ, OneMinMax and COCZ, which are widely used in MOEAs’ theoretical analyses~\cite{Giel03,LaumannsTEC04,Giel10,doerr2013lower,Qian13}.
The LOTZ problem presented in Definition~\ref{def:LOTZ} aims to maximize the number of leading 1-bits and the number of trailing 0-bits of a binary bit string.
The Pareto front of LOTZ is $\mathcal{F}=\{(0,n),(1,n-1),\ldots,(n,0)\}$, and the corresponding Pareto optimal solutions are $0^n,10^{n-1},\ldots,1^{n}$.
\begin{definition}[LOTZ~\cite{LaumannsTEC04}]\label{def:LOTZ}
The LOTZ problem of size $n$ is to find $n$ bits binary strings which maximize
\begin{align}
{\bm{f}}(\bm{x})= \left(\sum\nolimits^n_{i=1} \prod\nolimits^{i}_{j=1}x_j, \sum\nolimits^{n}_{i=1} \prod\nolimits^{n}_{j=i}(1-x_j)\right),
\end{align}
where $x_j$ denotes the $j$-th bit of $\bm{x} \in \{0,1\}^n$.
\end{definition}
We prove in Theorem~\ref{thm:bintour-lotz} that the NSGA-II can find the Pareto front in $O(n^2)$ expected number of generations, i.e., $O(n^3)$ expected number of fitness evaluations, because the generated $N$ offspring solutions need to be evaluated in each iteration.
Note that the running time of an EA is usually measured by the number of fitness evaluations, because evaluating the fitness of a solution is often the most time-consuming step in practice.
The main proof idea can be summarized as follows.
The NSGA-II first employs the mutation operator to find the two solutions with the largest number of leading 1-bits and the largest number of trailing 0-bits, i.e., $1^n$ and $0^n$, respectively; then employs the recommendation operator to find the whole Pareto front.
\begin{theorem}\label{thm:bintour-lotz}
For the NSGA-II solving LOTZ, if using binary tournament selection and a population size $N$ such that $2n+2\le N=O(n)$, then the expected number of generations for finding the Pareto front is $O(n^2)$.
\end{theorem}
\begin{proof}
We divide the running process of NSGA-II into two phases. The first phase starts after initialization and finishes until $1^n$ and $0^n$ are both found; the second phase starts
after the first phase and finishes when the Pareto front is found. We will show that the expected number of generations of the two phases are both $O(n^2)$, and thus prove the theorem.
In the following proof, we will use $\mathrm{LO}(\cdot)$ to denote the first objective value, i.e., the number of leading 1-bits of a solution, and $\mathrm{TZ}(\cdot)$ to denote the second objective value, i.e., the number of trailing 0-bits of a solution.
\vspace{0.6em}
\textbf{Analysis of the first phase.} \vspace{0.1em}
For the first phase,
we will prove that the expected number of generations for finding $1^n$ is $O(n^2)$, and then the same bound also holds for $0^n$ analogously.
Let $\mathrm{LO}_{\max}$ denote the maximum number of leading 1-bits of a solution in the current population $P$, i.e.,
$\mathrm{LO}_{\max}=\max\{\mathrm{LO}(\bm{x})\mid \bm{x}\in P \}. $
We first show that $\mathrm{LO}_{\max}$ will not decrease during the optimization procedure of NSGA-II. Let $A=\{\bm{x}\in P\cup Q \mid \mathrm{LO}(\bm{x})=\max_{\bm{x}\in P\cup Q} \mathrm{LO}(\bm{x})\}$ denote the set of solutions in $P\cup Q$ with the maximum leading 1-bits, and $A^*=\{\bm{x}\in A\mid \mathrm{TZ}(\bm{x})=\max_{\bm{x}\in A}\mathrm{TZ}(\bm{x})\}$ denote the set of solutions in $A$ with the maximum trailing 0-bits,
where
$Q$ denotes the set of offspring solutions generated from $P$ in lines~5--9 of Algorithm~\ref{alg:nsgaii}.
Then, the rank of any solution $\bm{x}\in A^*$ is 1 (i.e., $\bm{x}$ cannot be dominated by any other solution in $P\cup Q$), because $\bm{x}$ has the largest $\mathrm{LO}$ value and any solution with the same $\mathrm{LO}$ value cannot have larger $\mathrm{TZ}$ value.
Next, we consider two cases for $|F_1|$, where $F_1$ denotes the set of solutions in $P\cup Q$ with rank 1, and $|\cdot|$ denotes the size of a set. \\
(1) $|F_1|\le N$. Then, all the solutions in $A^*$ will be included in the next population. \\
(2) $|F_1|>N$. Then, we need to compute the crowding distances of the solutions in $F_1$ using Algorithm~\ref{alg:crowdist}, and preserve $N$ solutions with the largest crowding distance. Note that $A$ consists of the solutions with the largest $\mathrm{LO}$ value, and the solutions in $A\setminus A^*$ must have rank larger than 1, thus $A^*$ actually consists of all the solutions in $F_1$ with the largest $\mathrm{LO}$ value. Therefore, one of the solutions in $A^*$ will be put in the last slot when sorting the solutions in $F_1$ according to the $\mathrm{LO}$ value, which implies such solution will have an crowding distance of $\infty$. For the bi-objective problem LOTZ, we need to sort the solutions in $F_1$ twice, i.e., according to the $\mathrm{LO}$ value and the $\mathrm{TZ}$ value, respectively,
implying that there are at most four solutions whose crowding distance can be assigned to $\infty$. Consequently, at least one solution in $A^*$ belongs to the four best solutions in $F_1$ (w.r.t. $\succ_{\mathrm{c}}$), and thus can be included in the next population. \\
Combining the two cases, we have shown that there exists one solution in the next population whose number of leading 1-bits is $\max_{\bm{x}\in P\cup Q}\mathrm{LO}(\bm{x})$, which is obviously not smaller than $\mathrm{LO}_{\max}$.
Next, we show that $\mathrm{LO}_{\max}$ ($<n$) can increase by at least 1 with probability at least $\Omega(1/n)$ in each generation.
Similar to the analysis in the above paragraph, there exists one solution $\bm{x}^*\in \{\bm{x}\in P\mid \mathrm{LO}(\bm{x})=\mathrm{LO}_{\max}\} $ such that $\rank{\bm{x}^*}=1$ and $\dist{\bm{x}^*}=\infty$.
Recall that when using the tournament selection to select a parent solution, the competition between the two randomly selected solutions is based on their ranks and crowding distances (in case of equal ranks).
Thus, once $\bm{x}^*$ is selected for competition (whose probability is $1/N$), it will always win, if the other solution selected for competition has larger rank or finite crowding distance; or win with probability $1/2$, if the other solution has the same rank and crowding distance as $\bm{x}^*$, resulting in a tie which is broken uniformly at random.\\
Suppose $\bm{x}^*$ becomes a parent solution, then it will generate two offspring solutions together with another parent solution by crossover and mutation.
In the reproduction procedure, $\bm{x}^*$ can keep unchanged after crossover with probability at least 0.1 by line~6 of Algorithm~\ref{alg:nsgaii}, and flip only its $(\mathrm{LO}(\bm{x}^*)+1)$-th bit (which is a 0-bit) with probability $(1/n)\cdot (1-1/n)^{n-1}\ge 1/(en)$ by bit-wise mutation. Thus, the probability of generating an offspring solution $\bm{y}$ with $\mathrm{LO}(\bm{y})\ge \mathrm{LO}(\bm{x}^*)+1= \mathrm{LO}_{\max}+1$ is at least $0.1\cdot 1/(en)$. \\
In each generation, $N$ (i.e., $N/2$ pairs of) parent solutions will be selected and produce $N/2$ pairs of offspring solutions, thus the probability of generating a solution with more than $\mathrm{LO}_{\max}$ leading 1-bits is at least
\begin{equation}\label{eq:LOmax-onestep}
\begin{aligned}
&1-\Big(1-\frac{1}{2N}\cdot 0.1\cdot \frac{1}{en}\Big)^{N/2}
\ge 1-e^{-(1/(20enN))\cdot (N/2)}\\
& = 1-\frac{1}{e^{1/(40en)}}\ge 1-\frac{1}{1+1/(40en)}= \frac{1}{40en+1}=\Omega\Big(\frac{1}{n}\Big),
\end{aligned}
\end{equation}
where the inequalities hold by $1+a\le e^a$ for any $a\in \mathbb{R}$.
By the analysis in the previous paragraph, there must exist a solution $\bm{y}^*\in P\cup Q$ with the largest number of leading 1-bits such that $\rank{\bm{y}^*}=1$ and $\dist{\bm{y}^*}=\infty$, and thus will be maintained in the next population.
Hence, once an offspring solution with the number of leading 1-bits larger than $\mathrm{LO}_{\max}$ is generated, $\mathrm{LO}_{\max}$ will increase by at least 1 in the next population.
Note that the initial value of $\mathrm{LO}_{\max}$ is at least 0, thus the expected number of generations for increasing $\mathrm{LO}_{\max}$ to $n$, i.e., finding $1^n$, is at most $O(n^2)$. Analogously, we can derive that the expected number of generations for finding $0^n$ is also $O(n^2)$.
\vspace{0.6em}
\textbf{Analysis of the second phase.} \vspace{0.1em}
Now we consider the second phase, i.e., finding the whole Pareto front. We first show that once an objective vector $\bm{f}^*$ in the Pareto front has been found, it will always be maintained in the population. To this end, we first show that for any $i\in \{0,1,\ldots,n\}$, there exist at most two solutions in $P\cup Q$ with $i$ leading 1-bits, such that their ranks are equal to 1 and crowding distances are larger than 0.
Given any $i\in \{0,1,\ldots,n\}$, we simply assume that there exists at least one solution in $P\cup Q$ with $i$ leading 1-bits, because otherwise, the claim already holds. Let
\begin{equation}\label{eq:bintour-lotz-Bi}
\begin{aligned}
B^i= \{\bm{x}\in P\cup Q\mid \mathrm{LO}(\bm{x})=i \wedge
\mathrm{TZ}(\bm{x})=\max_{\bm{x}'\in P\cup Q, \mathrm{LO}(\bm{x}')=i}\mathrm{TZ}(\bm{x}')\}
\end{aligned}
\end{equation}
denote the set of solutions which have $i$ leading 1-bits and meanwhile have the maximum number of trailing 0-bits. Then, for any $\bm{x}\in (P\cup Q)\setminus B^i$ satisfying $\mathrm{LO}(\bm{x})=i$, it must hold $\rank{\bm{x}}>1$, because $\bm{x}$ can be dominated by any solution in $B^i$. Thus, we only need to consider the solutions in $B^i$.
If $|B^i|\le 2$, where $|\cdot|$ denotes the cardinality of a set, then the claim trivially holds.
If $|B^i|\ge 3$, then at most two solutions in $B^i$ can have crowding distances larger than 0, because all the solutions in $B^i$ have the same objective vector, and one solution can be assigned a crowding distance larger than 0 only if it is put in the first or the last position among the solutions in $B^i$ when the solutions are sorted according to some objective function.
Note that here we use the assumption in Section~\ref{sec:nsgaii}, i.e., when the solutions with the same objective vector are sorted according to some objective function $f_j$, theirs positions are unchanged or totally reversed.\\
Now we show that there exists at least one solution $\bm{x}$ corresponding to $\bm{f}^*$ such that $\rank{\bm{x}}=1$ and $\dist{\bm{x}}>0$, and then conclude the statement, i.e., $\bm{f}^*$ will always be maintained in the population. Let $C$ denote the set of solutions in $P\cup Q$ whose objective vectors are identical to $\bm{f}^*$. Then, any solution in $C$ has rank 1, because it cannot be dominated by any other solution. When the solutions in $C$ are sorted according to some objective function, one solution (denoted as $\hat{\bm{x}}$) will be put in the first or the last position, thus having a crowding distance larger than 0 by line~6 of Algorithm~\ref{alg:crowdist}. Thus, $\hat{\bm{x}}$ will not be inferior to $(2n+2)$ solutions in $P\cup Q$, because otherwise there must exist other $(2n+2)$ solutions with rank 1 and crowding distance larger than 0, which leads to a contradiction. Note that the population size $N\ge 2n+2$, we can derive that $\hat{\bm{x}}$ will be kept in the next population, which implies $\bm{f}^*$ will also be maintained.
Next, we consider the expansion of the Pareto fount.
We first analyze the probability of selecting $1^n$ or $0^n$ as a parent solution.
Note that at least one $1^n$ in $P$ has rank 1 and crowding distance $\infty$, thus once it is selected for competition, whose probability is $1/N$, it will win the competition with probability at least $1/2$, where $1/2$ is the probability of breaking a tie. Hence, the probability of selecting $1^n$ as a parent solution is least $1/(2N)$, and the same bound also holds for $0^n$ by a similar argument.
We now analyze the probability of generating a new Pareto optimal solution.
Let
\begin{equation}\label{eq:bintoru-lotz-D}
D=\{(j,n-j)\mid\bm{x} \in P \wedge \bm{x}=1^j0^{n-j}, j \in\{1,2,\ldots,n-1\} \}
\end{equation}
denote the set of objective vectors of the Pareto optimal solutions in $P$ (except $1^n$ and $0^n$).
Suppose currently the size of $D$ is equal to $i$. If $i=0$, then selecting $1^n$ and $0^n$ as a pair of parent solution, and exchanging the first $j$ bits ($1\le j\le n-1$) can generate a new Pareto optimal solution $1^j0^{n-j}$. The probability of such event is at least
\begin{equation}\label{eq:second-path-i0}
\begin{aligned}
\frac{2}{(2N)^2}\cdot 0.9\cdot \frac{n-1}{n} \cdot \big(1-\frac{1}{n}\big)^n
\ge \frac{0.9}{2eN^2}\cdot \big(1-\frac{1}{n}\big)^2\ge \frac{1}{4eN^2},
\end{aligned}
\end{equation}
where the term $2/(2N)^2$ denotes the probability of selecting $1^n$ and $0^n$, or $0^n$ and $1^n$ as a pair of solutions, the term 0.9 denotes the probability of performing the crossover operator, the term $(n-1)/n$ denotes the probability of selecting one of the $(n-1)$ crossover points, the term $(1-1/n)^n$ denotes the probability of not flipping any bits by mutation, the first inequality holds by $(1-1/n)^{n-1}\ge 1/e$, and the last inequality holds for $n\ge 4$. \\
Then, we consider the case $i>0$.
In one binary tournament selection procedure, the probability of selecting two solutions with objective vectors in $D$ is at least $i^2/N^2$, and we denote the winning solution, i.e., the parent solution, as $\tilde{\bm{x}}$.
Suppose the number of leading 1-bits of $\tilde{\bm{x}}$ is $k$,
and let $D_1=\{j\mid (j,n-j)\in D\wedge j<k\}$,
$D_2=\{j\mid (j,n-j)\in D\wedge j>k\}$.
If the other parent solution is $0^n$, then exchanging the first $k_1$ ($1\le k_1\le k-1,k_1\notin D_1$) bits of $\tilde{\bm{x}}$ and $0^n$ can generate a new Pareto optimal solution $1^{k_1}0^{n-k_1}$, whose probability is $(k-1-|D_1|)/n$; if the other parent solution is $1^n$, then exchanging the first $k_2$ ($k+1\le k_2\le n-1,k_2\notin D_2$) bits of $\tilde{\bm{x}}$ and $1^n$ can generate a new Pareto optimal solution $1^{k_2}0^{n-k_2}$, whose probability is $(n-1-k-|D_2|)/n$. Note that the probability of selecting $1^n$ (or $0^n$) as a parent solution is lower bounded by $1/(2N)$, thus we can derive that the probability of generating a new Pareto optimal solution in $P$ is at least
\begin{equation}
\begin{aligned}\label{eq:second-path-i>0}
&\frac{i^2}{N^2}\cdot \frac{1}{2N}\cdot 0.9\cdot \Big(\frac{k-1-|D_1|}{n}+\frac{n-1-k-|D_2|}{n}\Big)\cdot \big(1-\frac{1}{n}\big)^n \\
&\ge \frac{i^2}{N^2}\cdot\bigg( \frac{1}{2N}\cdot \frac{n-2-|D_1|-|D_2|}{n}\cdot \frac{1}{2e}\bigg)
=\frac{i^2(n-1-i)}{4enN^3},
\end{aligned}
\end{equation}
where the term $0.9$ denotes the probability of using the crossover operator,
the term $(1-1/n)^n$ denotes the probability of not flipping any bits by mutation, and the inequality holds by $(1-1/n)^{n-1}\ge 1/e$ and $0.9\cdot (1-1/n)>1/2$ for $n\ge 3$.\\
Note that NSGA-II performs $N/2$ reproduction procedures, i.e., selection, crossover and mutation, in each generation. Thus, by Eqs.~\eqref{eq:second-path-i0} and~\eqref{eq:second-path-i>0}, the probability of generating a new objective vector in Pareto front is at least
\begin{equation}
\begin{aligned}
1-\Big(1-\frac{1}{4eN^2}\Big)^{N/2}\ge 1-e^{-1/(8eN)}
\ge 1-\frac{1}{1+1/(8eN)}=\frac{1}{8eN+1}
\end{aligned}
\end{equation}
for $i=0$, and
\begin{equation}
\begin{aligned}
&1-\Big(1-\frac{i^2(n-1-i)}{4enN^3}\Big)^{N/2}
\ge 1-e^{-i^2(n-1-i)/(8enN^2)}\\
\ge 1-\frac{1}{1+i^2(n-1-i)/(8enN^2)}
\ge \frac{i^2(n-1-i)}{8enN^2+i^2(n-1-i)}
\end{aligned}
\end{equation}
for $i>0$, where the inequalities hold by $1+a\le e^a$ for any $a\in \mathbb{R}$.
Now, we can derive that the expected number of generations for finding the Pareto front is at most
\begin{equation}\label{eq:second-path-sum}
\begin{aligned}
&8eN\!+\!1\!+\sum_{i=1}^{n-2}\Big(\frac{8enN^2}{i^2(n-1-i)}+1\Big)
=O(n)+8enN^2\sum_{i=1}^{n-2}\frac{1}{i^2(n-1-i)},
\end{aligned}
\end{equation}
where the equality is by $N=O(n)$. Note that
\begin{align}
\frac{1}{i^2(n-1-i)}
=\frac{1}{n-1}\cdot \frac{1}{i^2}+\frac{1}{(n-1)^2}\cdot \Big(\frac{1}{i}+\frac{1}{n-1-i}\Big),
\end{align}
thus Eq.~\eqref{eq:second-path-sum} continues with
\begin{align}
&=O(n)+8enN^2\sum_{i=1}^{n-2}\bigg(\frac{1}{n-1}\cdot \frac{1}{i^2}+\frac{1}{(n-1)^2}\cdot \Big(\frac{1}{i}+\frac{1}{n-1-i}\Big)\bigg)\\
&=O(n)+\frac{8enN^2}{n-1}\bigg(1+\sum_{i=2}^{n-2}\frac{1}{i^2}+\frac{1}{n-1}\sum_{i=1}^{n-2}\Big(\frac{1}{i}+\frac{1}{n-1-i}\Big)\bigg)\\
&\le O(n)+\frac{8enN^2}{n-1}\bigg(1+\sum_{i=2}^{n-2}\Big(\frac{1}{i-1}-\frac{1}{i}\Big)+\frac{2(1+\ln (n-2))}{n-1}\bigg)\\
&\le O(n)+\frac{8enN^2}{n-1}\bigg(1+1-\frac{1}{n-2}+o(1)\bigg)
=O(n^2),
\end{align}
where the first inequality is by $\sum_{i=1}^{j}1/i\le 1+\ln j$ for any $j\ge 1$. Hence, we have shown an upper bound $O(n^2)$ for the expected number of generations of the second phase.
\end{proof}
The OneMinMax problem presented in Definition~\ref{def:OMM} aims to maximize the number of 0-bits and the number of 1-bits of a binary bit string.
The Pareto front of OneMinMax is $\mathcal{F}=\{(0,n),(1,n-1),\ldots,(n,0)\}$, and
any solution $\bm{x}\in\{0,1\}^n$ is Pareto optimal, corresponding to the objective vector $(n-|\bm{x}|_1,|\bm{x}|_1)$ in the Pareto front, where $|\cdot|_1$ denotes the number of 1-bits of a solution.
\begin{definition}[OneMinMax~\cite{Giel10}]\label{def:OMM}
The OneMinMax problem of size $n$ is to find $n$ bits binary strings which maximize
\begin{align}
{\bm f}(\bm{x})=\left(n-\sum\nolimits^n_{i=1}x_i, \sum\nolimits^{n}_{i=1} x_i\right),
\end{align}
where $x_i$ denotes the $i$-th bit of $\bm{x} \in \{0,1\}^n$.
\end{definition}
We prove in Theorem~\ref{thm:bintour-omm} that the NSGA-II can find the Pareto front in $O(n\log n)$ expected number of generations, i.e., $O(n^2\log n)$ expected running time.
\begin{theorem}\label{thm:bintour-omm}
For the NSGA-II solving OneMinMax, if using binary tournament selection and a population size $N$ such that $2n+2\le N= O(n)$, then the expected number of generations for finding the Pareto front is $O(n\log n)$.
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem~\ref{thm:bintour-lotz}, i.e., we divide the optimization procedure into two phases, where the first phase starts after initialization and finishes until $1^n$ and $0^n$ are both found; the second phase starts after the first phase and finishes when the Pareto front is found. However, for OneMinMax, we will show that the expected numbers of generations of the two phases are both $O(n\log n)$, instead of $O(n^2)$.
\textbf{For the first phase}, we need to consider the increment of a quantity $\mathrm{O}_{\max}$, which is defined as $\mathrm{O}_{\max}=\max\{|\bm{x}|_1\mid \bm{x}\in P\}$, where $|\cdot |_1$ denotes the number of 1-bits of a solution. Then, similar to the analysis of $\mathrm{LO}_{\max}$, $\mathrm{O}_{\max}$ will not decrease, and we need to analyze the probability that $\mathrm{O}_{\max}$ increases by at least 1 in each generation. Let $\bm{x}^*$ be a solution in $\{\bm{x}\in P\mid |\bm{x}|_1=\mathrm{O}_{\max}\} $ such that $\rank{\bm{x}^*}=1$ and $\dist{\bm{x}^*}=\infty$, then $\bm{x}^*$ will be selected for competition in the binary tournament selection with probability at least $1/N$, and can win with probability at least 1/2. In the reproduction procedure, $\bm{x}^*$ can generate an offspring solution $\bm{y}$ such that $|\bm{y}|_1\ge |\bm{x}^*|_1+1$ with probability at least $0.1\cdot ((n-|\bm{x}^*|_1)/n)\cdot (1-1/n)^{n-1}\ge (n-|\bm{x}^*|_1)/(10en)$, where the term $0.1$ denotes the probability of not using recommendation operator, and the term $((n-|\bm{x}^*|_1)/n)\cdot (1-1/n)^{n-1}$ denotes the probability of flipping one 0-bit of $\bm{x}^*$. Then, similar to Eq.~\eqref{eq:LOmax-onestep}, the probability of generating a solution with more than $|\bm{x}^*|_1$ 1-bits in each generation is at least
\begin{equation}\label{eq:Omax-onestep}
\begin{aligned}
&1-\Big(1-\frac{1}{2N}\cdot \frac{n-|\bm{x}^*|_1}{10en}\Big)^{N/2}
\ge 1-\frac{1}{e^{(n-|\bm{x}^*|_1)/(40en)}}\\
&\ge 1-\frac{1}{1+(n-|\bm{x}^*|_1)/(40en)}= \frac{n-|\bm{x}^*|_1}{40en+(n-|\bm{x}^*|_1)}\\
&=\Omega\Big(\frac{n-|\bm{x}^*|_1}{n}\Big).
\end{aligned}
\end{equation}
Thus, the expected number of generations for finding $1^n$ is at most
\begin{equation}\label{eq:Omax-time}
\sum_{i=0}^{n-1}O\Big(\frac{n}{n-i}\Big)=O(n\log n),
\end{equation}
and the bound also holds for $0^n$ by a similar analysis procedure.
\textbf{Now we consider the second phase}. Similar to Eq.~\eqref{eq:bintour-lotz-Bi}, we define a set $B^i=\{\bm{x}\in P\cup Q \mid |\bm{x}|_1=i\}$ ($0\le i\le i$), which denotes the set of solutions which have $i$ 1-bits. Note that we do not add any restriction to the other objective value, i.e., the number of 0-bits, because the number of 0-bits of a solution can be decided by the number of 1-bits. Then, following the analysis in the proof of Theorem~\ref{thm:bintour-lotz}, we can show that there exist at most two solutions in $P\cup Q$ with $i$ 1-bits such that their ranks are equal to 1 and crowding distances are larger than 0, and thus prove that an objective vector $\bm{f}^*$ in the Pareto front will always be maintained in the population once it has been found. \\
Now we examine the probability of generating a new Pareto optimal solution in each generation. Similar to Eq.~\eqref{eq:bintoru-lotz-D}, let $D=\{(n-|\bm{x}|_1,|\bm{x}|_1)\mid\bm{x} \in P \wedge 1\le |\bm{x}|_1\le n-1 \}$ denote the set of objective vectors of the solutions in $P$ (except $1^n$ and $0^n$), and suppose currently the size of $D$ is equal to $i$.
Note that for OneMinMax, any solution $\bm{x}\in\{0,1\}^n$ is Pareto optimal, thus the following analysis is a little easier. First, we consider the case that the number of $1^n$ or the number of $0^n$ is larger than $N/4$. Without loss of generality, we assume that the number of $0^n$ in $P$ is at least $N/4$. Then, the probability of selecting $0^n$ as a parent solution is at least $(1/4)^2=1/16$, because we only need to select $0^n$ twice in binary tournament selection. By the analysis in Theorem~\ref{thm:bintour-lotz}, the probability of selecting $1^n$ as the other parent solution is at least $1/(2N)$. Then, exchanging the first $j$ ($1\le j\le n-1 \wedge (n-j,j)\notin D$) bits of $0^n$ and $1^n$ can generate a new Pareto optimal solution $1^j0^{n-j}$, whose probability is at least $(n-1-i)/n$. Thus, combining the above-mentioned probabilities, we can derive that the probability of generating a new Pareto
optimal solution not in $P$ is at least $\Omega((n-1-i)/(nN))$.\\
Then, we consider the case that the number of $1^n$ and the number of $0^n$ in $P$ are both smaller than or equal to $N/4$. In one binary tournament selection procedure, the probability of selecting two solutions with objective vectors in $D$ is at least $(N-N/4-N/4)^2/N^2=1/4$, because it is sufficient to not select $0^n$ or $1^n$. Let $\tilde{\bm{x}}$ denote the winning solution, i.e., the parent solution, and suppose $|\tilde{\bm{x}}|_1=k$.
If the other parent solution is $0^n$, then for any $1\le k_1\le k-1,(n-k_1,k_1)\notin D$, there must exist a crossover point $k_1'$ such that exchanging the first $k_1'$ bits of $\bm{x}$ and $0^n$ can generate a Pareto optimal solution with $k_1$ 1-bits.
If the other parent solution is $1^n$, then for any $k+1\le k_2\le n-1,(n-k_2,k_2)\notin D$, there must exist a crossover point $k_2'$ such that exchanging the first $k_2'$ bits of $\bm{x}$ and $1^n$ can generate a Pareto optimal solution with $k_2$ 1-bits.
Note that the probability of selecting $1^n$ (or $0^n$) as a parent solution is at least $1/(2N)$, thus similar to Eq.~\eqref{eq:second-path-i>0}, we can derive that the probability of generating a new Pareto optimal solution not in $P$ is at least
\begin{equation}
\begin{aligned}
\frac{1}{4}\cdot \frac{1}{2N}\cdot 0.9\cdot \frac{n-1-i}{n}\cdot \big(1-\frac{1}{n}\big)^n \ge \frac{n-1-i}{16enN}=\Omega\Big(\frac{n-1-i}{nN}\Big).
\end{aligned}
\end{equation}
Thus, in each generation, the probability of generating a new objective vector in Pareto front is at least
\begin{equation}\label{eq:bintour-omm-phase2-prob}
\begin{aligned}
&1-\Big(1-\Omega\Big(\frac{n-1-i}{nN}\Big)\Big)^{N/2}\ge 1-e^{-\Omega((n-1-i)/(2n))}\\
&\ge 1-\frac{1}{1+\Omega((n-1-i)/(2n))}
\end{aligned}
\end{equation}
Then, we can derive that the expected number of generations for finding the whole Pareto front is at most
\begin{equation}\label{eq:bintour-omm-phase2-time}
\begin{aligned}
&\sum_{i=0}^{n-2}\Big(1-\frac{1}{1+\Omega((n-1-i)/(2n))}\Big)^{-1}\\
&= \sum_{i=0}^{n-2}\Big(1+\frac{1}{\Omega((n-1-i)/(2n))}\Big)\\
&=n-1+\sum_{i=0}^{n-2}O\Big(\frac{2n}{n-1-i}\Big)=
O(n\log n),
\end{aligned}
\end{equation}
where the last equality is by $\sum_{i=1}^{j}1/i\le 1+\ln j$ for any $j\ge 1$. Thus, combining the analyses for the two phases, the Theorem holds.
\end{proof}
The COCZ problem as presented in Definition~\ref{def:COCZ} is similar to OneMinMax, but is a little complicated. Its first objective is to maximize the number of 1-bits of a solution, and the other objective is to maximize the number of 1-bits in the first half of the solution plus the number of 0-bits in the second half. That is, the two objectives are consistent in maximizing the number of 1-bits in the first half of the solution, but conflict in the second half.
The Pareto front of COCZ is $\mathcal{F}=\{(n/2,n),(n/2+1,n-1),\ldots,(n,n/2)\}$, and any solution $\bm{x}$ satisfying $\sum_{i=1}^{n/2}x_i=n/2$
is Pareto optimal, corresponding to the objective vector $(n/2+\sum_{i=n/2+1}^{n}x_i,n/2+\sum_{i=n/2+1}^{n}(1-x_i))$ in the Pareto front.
\begin{definition}[COCZ~\cite{LaumannsTEC04}]\label{def:COCZ}
The COCZ problem of size $n$ is to find $n$ bits binary strings which maximize
\begin{align}
{\bm f}(\bm{x})=\left(\sum\nolimits^n_{i=1} x_i, \sum\nolimits^{n/2}_{i=1} x_i +\sum\nolimits^{n}_{i=n/2+1} (1-x_i)\right),
\end{align}
where $n$ is even and $x_i$ denotes the $i$-th bit of $\bm{x} \in \{0,1\}^n$.
\end{definition}
We prove in Theorem~\ref{thm:bintour-cocz} that the NSGA-II can find the Pareto front in $O(n\log n)$ expected number of generations, i.e., $O(n^2\log n)$ expected running time.
\begin{theorem}\label{thm:bintour-cocz}
For the NSGA-II solving COCZ, if using binary tournament selection and a population size $N$ such that $n+2\le N= O(n)$, then the expected number of generations for finding the Pareto front is $O(n\log n)$.
\end{theorem}
\begin{proof}
The proof is similar to that of Theorems~\ref{thm:bintour-lotz} and~\ref{thm:bintour-omm}, i.e., we will divide the optimization procedure into two phases. However, the target of the first phase is a little different, i.e., we need to find $1^n$ and $1^{n/2}0^{n/2}$, instead of $1^n$ and $0^n$. We will show that the expected numbers of generations of the two phases are both $O(n\log n)$, i.e., the same as that of OneMinMax.
In the following discussion, we will use $\mathrm{CZ}(\bm{x})$ to denote the second objective value of a solution $\bm{x}$, i.e., $\mathrm{CZ}(\bm{x})=\sum^{n/2}_{i=1} x_i +\sum^{n}_{i=n/2+1} (1-x_i)$.
\textbf{For the first phase}, the analysis for $1^n$ is almost the same as that of Theorem~\ref{thm:bintour-omm}, and we mainly examine the expected number of generations for finding $1^{n/2}0^{n/2}$. Let $\mathrm{CZ}_{\max}=\max\{\mathrm{CZ}(\bm{x}) \mid \bm{x}\in P\}$ denote the maximum second objective value of a solution in $P$, then similar to the analysis in the previous theorems, $\mathrm{CZ}_{\max}$ will not decrease, and we need to consider the increase of $\mathrm{CZ}_{\max}$. Let $\bm{x}^*$ be a solution in $\{\bm{x}\in P\mid \mathrm{CZ}(\bm{x})=\mathrm{CZ}_{\max}\} $ such that $\rank{\bm{x}^*}=1$ and $\dist{\bm{x}^*}=\infty$, then following the analysis in the Theorem~\ref{thm:bintour-omm}, $\bm{x}^*$ can be selected as a parent solution with probability at least $1/(2N)$. Suppose the number of 1-bits in the first half of $\bm{x}^*$ is $k_1$, i.e., $\sum^{n/2}_{i=1} x_i^*=k_1$, and the number of 0-bits in the second half of $\bm{x}^*$ is $k_2$, i.e., $\sum^{n/2}_{i=1}(1- x_i^*)=k_2$. Then, flipping one of the $(n/2-k_1)$ 0-bits in the first half of $\bm{x}^*$, or flipping one of the $(n/2-k_2)$ 1-bits in the second half of $\bm{x}^*$ can generate an offspring solution $\bm{y}$ such that $\mathrm{CZ}(\bm{y})\ge \mathrm{CZ}(\bm{x}^*)+1=\mathrm{CZ}_{\max}+1$.
Then, similar to Eq.~\eqref{eq:Omax-onestep}, the probability of generating a solution with the $\mathrm{CZ}$ value larger than $\mathrm{CZ}_{\max}$ in each generation is at least
\begin{equation}
\begin{aligned}
&1-\Big(1-\frac{1}{2N}\cdot \frac{n/2-k_1+n/2-k_2}{10en}\Big)^{N/2}
\\
&=1-\Big(1-\frac{1}{2N}\cdot \frac{n-\mathrm{CZ}_{\max}}{10en}\Big)^{N/2}
=\Omega\Big(\frac{n-\mathrm{CZ}_{\max}}{n}\Big),
\end{aligned}
\end{equation}
where the first equality is by $k_1+k_2=\mathrm{CZ}(\bm{x}^*)=\mathrm{CZ}_{\max}$, and the last equality is by the same derivation in Eq.~\eqref{eq:Omax-onestep}.
Thus, similar to Eq.~\eqref{eq:Omax-time}, the expected number of generations for increasing $\mathrm{CZ}_{\max}$ to $n$, i.e., finding $1^{n/2}0^{n/2}$, is at most $O(n\log n)$.
\textbf{For the second phase}, we first need to show that once an objective vector $\bm{f}^*$ in the Pareto front is found, it will always be maintained. Similar to Eq.~\eqref{eq:bintour-lotz-Bi}, let
\begin{align}
B^i=\Big\{\bm{x}\in P\cup Q\mid &\sum\nolimits^n_{i=n/2+1} x_i = i \wedge\\ &\sum\nolimits^{n/2}_{i=1} x_i=\max_{\bm{x}'\in P\cup Q, \sum\nolimits^n_{i=n/2+1} x_i'=i} \sum\nolimits^{n/2}_{i=1} x_i'\Big\}
\end{align}
denote the set of solutions which have $i$ 1-bits in the second half and meanwhile have the maximum number of 1-bits in the first half. Then, following the analysis after Eq.~\eqref{eq:bintour-lotz-Bi}, we can prove that for any $i\in \{0,1,\ldots,n/2\}$, there exist at most two solutions in $P\cup Q$ with the numbers of the 1-bits in the second half equal to $i$, such that their ranks are equal to 1 and crowding distances are larger than 0. Then, similar to the statement in Theorem~\ref{thm:bintour-lotz}, we can prove the claim, i.e., an objective vector $\bm{f}^*$ in the Pareto front will always be maintained once it has been found.
Now we consider the expansion of the Pareto front. Let $D=\{(|\bm{x}|_1,\mathrm{CZ}(\bm{x}))\mid\bm{x} \in P \wedge n/2<|\bm{x}|_1<n \wedge \sum\nolimits^{n/2}_{i=1} x_i = n/2\} $ denote the set of objective vectors of the solutions in $P$ (except $1^n$ and $1^{n/2}0^{n/2}$), and suppose currently the size of $D$ is equal to $i$.
Note that in one reproduction procedure, we need to select two parent solutions. Suppose we are given a parent solution $\tilde{\bm{x}}$ such that $\sum\nolimits^{n}_{i=n/2+1} \tilde{x}_i =k$, we will show that the probability of generating a new Pareto optimal solution not in $P$ is at least $\Omega((n/2-1-i)/(nN))$.
If $k=0$, then selecting $1^n$ as the other parent solution and exchanging the first $k'$ (where $n/2<k'<n,(k',n/2+n-k')\notin D$) bits of two parent solutions can generate a new Pareto optimal solution $1^{k'}0^{n-k'}$; if $k=n/2$, then selecting $1^{n/2}0^{n/2}$ as the other parent solution and exchanging the first $k'$ (where $n/2<k'<n,(n/2+n-k',n/2+k'-n/2)\notin D$) bits of two parent solutions can generate a new Pareto optimal solution $1^{n/2}0^{k'-n/2}1^{n-k'}$. Note that the probability of selecting $1^n$ (or $0^n$) as a parent solution is at least $1/(2N)$, thus in both cases, the probability of generating a new Pareto optimal solution not in $P$ is at least $\Omega((n/2-1-i)/(nN))$. If $0<k<n/2$, then similar to the analysis of the second phase in Theorems~~\ref{thm:bintour-lotz} and~\ref{thm:bintour-omm}, we can also derive that the probability of generating a new Pareto optimal solution not in $P$ is at least $\Omega((n/2-1-i)/(nN))$.\\
Thus, similar to Eqs.~\eqref{eq:bintour-omm-phase2-prob} and~\eqref{eq:bintour-omm-phase2-time}, the expected number of generations for finding the whole Pareto front is at most $O(n\log n)$.
\end{proof}
\section{Analysis of NSGA-II Using Stochastic Tournament Selection}
In the previous section, we have proved that the expected running time of the standard NSGA-II is $O(n^3)$ for LOTZ, and $O(n^2\log n)$ for OneMinMax and COCZ, which is as same as that of the previously analyzed simple MOEAs, GSEMO and SEMO~\cite{LaumannsTEC04,Giel03,Qian13,Giel10}.
Next, we will show that by employing stochastic tournament selection in Definition~\ref{def:sto-tour} instead of binary tournament selection, the NSGA-II needs much less time to find the whole Pareto front.
In particular, we prove that the expected number of generations of the NSGA-II using stochastic tournament selection is $O(n)$ (implying $O(n^2)$ expected running time) for solving all the three problems, in Theorems~\ref{thm:stotour-lotz}--\ref{thm:stotour-cocz}.
The working principle of the NSGA-II observed in the proofs of these theorems is similar to that observed in the previous section. That is, the NSGA-II first employs the mutation operator to find the solutions that maximize each objective function, and then employs the crossover operator to quickly find the remaining objective vectors in the Pareto front.
However, the utilization of stochastic tournament selection can make the NSGA-II select prominent solutions, i.e., solutions maximizing each objective function, with larger probability,
making the crossover operator easier fill in the remaining Pareto front and thus reducing the total running time.
\begin{theorem}\label{thm:stotour-lotz}
For the NSGA-II solving LOTZ, if using stochastic tournament selection and a population size $N$ such that $2n+2\le N= O(n)$, then the expected number of generations for finding the Pareto front is $O(n)$.
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem~\ref{thm:bintour-lotz}. The main difference is the probability of selecting a specific solution from $P$, which will affect the running time complexity of both the first phase and the second phase.
\textbf{For the first phase}, the probability of selecting the solution $\bm{x}^*$ is $q=\Omega(1)$ by Lemma~\ref{lem:stotour-prob}, instead of $\Omega(1/N)$. Then, similar to Eq.~\eqref{eq:LOmax-onestep}, the probability of generating a solution with more than $\mathrm{LO}_{\max}$ leading 1-bits in each generation is at least
\begin{equation}\label{eq:LOmax-onestep-stotour}
\begin{aligned}
&1-\Big(1-\frac{q}{2}\cdot \frac{1}{en}\Big)^{N/2}
\ge 1-\frac{1}{e^{qN/(4en)}}\\
&\ge 1-\frac{1}{1+qN/(4en)} \ge 1-\frac{1}{1+q/(2e)}= \frac{q}{2e+q}=\Omega(1),
\end{aligned}
\end{equation}
where the last inequality is by $N\ge 2n+2$. Thus, we can derive an upper bound $O(n)$ on
the expected number of generations for finding $1^n$, and the bound also holds for $0^n$ similarly.
\textbf{For the second phase}, we consider the case that the parent solutions are exactly $0^n$ and $1^n$, instead of the case in the proof of Theorem~\ref{thm:bintour-lotz}, i.e., one parent solution is selected from the set of Pareto optimal solutions in $P$, and the other parent solution is selected from $0^n$ or $1^n$.
By Lemma~\ref{lem:stotour-prob}, the probability of selecting $0^n$ (or $1^n$) as a parent solution is $\Omega(1)$, thus the probability that $0^n$ and $1^n$ are selected as a pair of parent solutions is also $\Omega(1)$. Suppose currently $i$ objective vectors in the Pareto front (except $(n,0)$ and $(0,n)$) have been found. Note that exchanging the first $j$-th ($1\le j\le n-1$) bits of $1^n$ and $0^n$ can generate a Pareto optimal solution $1^j0^{n-j}$, thus the probability of generating a new objective vector in the Pareto front is at least
\begin{equation}\label{eq:second-path-stotour}
\begin{aligned}
\Omega(1)\cdot 0.9\cdot \frac{n-1-i}{n}\cdot \Big(1-\frac{1}{n}\Big)^n = \Omega\Big(\frac{n-1-i}{n}\Big),
\end{aligned}
\end{equation}
where the term $0.9$ denotes the probability of using the crossover operator,
the term $(n-1-i)/n$ denotes the probabilities of selecting one crossover point, and the term $(1-1/n)^n$ denotes the probability of not flipping any bits by mutation.
In each generation, NSGA-II produces $N/2$ pairs of offspring solutions, thus, the probability of generating a new objective vector in Pareto front is at least
\begin{equation}
\begin{aligned}
&1-\Big(1-\Omega\Big(\frac{n-1-i}{n}\Big)\Big)^{N/2}\ge 1-e^{-\Omega(N(n-1-i)/(2n))}\\
&\ge 1-e^{-\Omega(n-1-i)}=1-\frac{1}{e^{\Omega(n-1-i)}}\ge 1-\frac{1}{1+\Omega(n-1-i)}
\end{aligned}
\end{equation}
where the first and third inequalities hold by $1+a\le e^a$ for any $a\in \mathbb{R}$, and the second inequality holds by $N\ge 2n+2$. Then, we can derive that the expected number of generations for finding the whole Pareto front is at most
\begin{equation}
\begin{aligned}
&\sum_{i=0}^{n-2}\frac{1}{1-1/(1+\Omega(n-1-i))}= \sum_{i=0}^{n-2}\Big(1+\frac{1}{\Omega(n-1-i)}\Big)\\
&=n-1+O(1+\ln (n-1))=O(n),
\end{aligned}
\end{equation}
where the first inequality is by $\sum_{i=1}^{j}1/i\le 1+\ln j$ for any $j\ge 1$. Thus, combining the analyses for the two phases leads to the theorem.
\end{proof}
The proofs of Theorems~\ref{thm:stotour-omm} and~\ref{thm:stotour-cocz} are omitted, because they are almost as same as that of Theorem~\ref{thm:stotour-lotz}. We only need to incorporate the properties of OneMinMax and COCZ revealed in the proofs of Theorems~\ref{thm:bintour-omm} and~\ref{thm:bintour-cocz}.
That is, an objective vector in the Pareto front will always be maintained once it has been found, if the population size $N$ is at least $2n+2$ for OneMinMax and at least $n+2$ for COCZ.
\begin{theorem}\label{thm:stotour-omm}
For the NSGA-II solving OneMinMax, if using stochastic tournament selection and a population size $N$ such that $2n+2\le N= O(n)$, then the expected number of generations for finding the Pareto front is $O(n)$.
\end{theorem}
\begin{theorem}\label{thm:stotour-cocz}
For the NSGA-II solving COCZ, if using stochastic tournament selection and a population size $N$ such that $n+2\le N= O(n)$, then the expected number of generations for finding the Pareto front is $O(n)$.
\end{theorem}
\section{Experiments}
\begin{figure}[t!]\centering
\hspace{0.1em}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_LOTZ_BinTour_rep-5_size-10-10-100_50filecombined}
\end{minipage}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_LOTZ_StoTour_rep-10_size-10-10-100_50filecombined}
\end{minipage}\\\vspace{0.3em}
\begin{minipage}[c]{1\linewidth}\centering
\small(a) \text{LOTZ}
\end{minipage}\\\vspace{1em}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_OMM_BinTour_rep-100_size-10-10-100_10filecombined}
\end{minipage}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_OMM_StoTour_rep-100_size-10-10-100_10filecombined}
\end{minipage}\\\vspace{0.3em}
\begin{minipage}[c]{1\linewidth}\centering
\small(b) \text{OneMinMax}
\end{minipage}\\\vspace{1em}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_COCZ_BinTour_rep-100_size-10-10-100_popsize-n_10filecombined}
\end{minipage}
\begin{minipage}[c]{0.48\linewidth}\centering
\includegraphics[width=1\linewidth]{result_nsgaii_COCZ_StoTour_rep-100_size-10-10-100_popsize-n_10filecombined}
\end{minipage}\\\vspace{0.3em}
\begin{minipage}[c]{1\linewidth}\centering
\small(c) \text{COCZ}
\end{minipage}\\\vspace{1em}
\caption{Average \#generations of the NSGA-II
using binary tournament selection or stochastic tournament selection
for solving the LOTZ, OneMinMax and COCZ problems.
Left subfigure: average \#generations of the NSGA-II using binary tournament selection vs. problem size $n$;
right subfigure: average \#generations of the NSGA-II using stochastic tournament selection vs. problem size $n$.}\label{fig:nsgaii-two-methods}
\end{figure}
In the previous sections, we have proved that when binary tournament selection is used in the NSGA-II, the expected number of generations is $O(n^2)$ for LOTZ, and $O(n\log n)$ for OneMinMax and COCZ; when stochastic tournament selection is used, the expected number of generations can be improved to $O(n)$ for all the three problems.
But as the lower bounds on the running time have not been derived, the comparison may be not strict. Thus, we conduct experiments to examine the tightness of these upper bounds.
For each problem, we examine the performance of NSGA-II when the problem size $n$ changes from 10 to 100, with a step of 10. On each problem size $n$, we run the NSGA-II 1000 times independently, and record the number of generations until the Pareto front is found. Then, the average number of generations and the standard deviation of the 1000 runs are reported in Figure~\ref{fig:nsgaii-two-methods}.
From the left subfigure of Figure~\ref{fig:nsgaii-two-methods}(a), we can observe that the average number of generations increases by a factor of nearly four when the problem size $n$ doubles. Thus, the average number of generations is approximately $\Theta(n^2)$, implying that the upper bound $O(n^2)$ derived in Theorem~\ref{thm:bintour-lotz} is tight. By the right subfigure of Figure~\ref{fig:nsgaii-two-methods}(a), the average number of generations is clearly a linear function of $n$, which implies that the upper bound $O(n)$ derived in Theorem~\ref{thm:stotour-lotz} is also tight. As the problem size $n$ increases, the average number of generations in the left subfigures of Figure~\ref{fig:nsgaii-two-methods}(b) and Figure~\ref{fig:nsgaii-two-methods}(c) both increases at a faster pace, implying that the expected number of generations is both $\omega(n)$, and thus the upper bound $O(n\log n)$ derived in Theorems~\ref{thm:bintour-omm} and~\ref{thm:bintour-cocz} is almost tight.
From the right subfigures of Figure~\ref{fig:nsgaii-two-methods}(b) and Figure~\ref{fig:nsgaii-two-methods}(c), we can observe that the average number of generations increases by about 40\% when the problem size $n$ doubles, suggesting that the expected number of generations is approximately $\Omega(n^{1/2})$.
\section{Conclusion}
In this paper, we theoretically study the running time of the NSGA-II solving three bi-objective problems, LOTZ, OneMinMax and COCZ, and derive upper bounds that are as same as that of the previously analyzed simple MOEAs, GSEMO and SEMO. Then, we propose a new parent selection strategy, stochastic tournament selection, to replace the binary tournament selection strategy of the NSGA-II, and prove that the NSGA-II using the new strategy can find the Pareto front of the three problems with much less time. Experiments are also conducted to examine the tightness of the upper bonds.
In the future, we will analyze the lower bounds on the running time
to make the comparison strict, and also conduct a more comprehensive experiment (e.g., increase the problem size $n$ to 500) to better reflect the tendency of the running time.
It is also interesting and expected to study the running time of the NSGA-II on multi-objective combinatorial optimization problems.
\bibliographystyle{plainnat}
| {
"timestamp": "2022-03-23T01:20:53",
"yymm": "2203",
"arxiv_id": "2203.11550",
"language": "en",
"url": "https://arxiv.org/abs/2203.11550",
"abstract": "Evolutionary algorithms (EAs) have been widely used to solve multi-objective optimization problems, and have become the most popular tool. However, the theoretical foundation of multi-objective EAs (MOEAs), especially the essential theoretical aspect, i.e., running time analysis, has been still largely underdeveloped. The few existing theoretical works mainly considered simple MOEAs, while the non-dominated sorting genetic algorithm II (NSGA-II), probably the most influential MOEA, has not been analyzed except for a very recent work considering a simplified variant without crossover. In this paper, we present a running time analysis of the standard NSGA-II for solving LOTZ, OneMinMax and COCZ, the three commonly used bi-objective optimization problems. Specifically, we prove that the expected running time (i.e., number of fitness evaluations) is $O(n^3)$ for LOTZ, and $O(n^2\\log n)$ for OneMinMax and COCZ, which is surprisingly as same as that of the previously analyzed simple MOEAs, GSEMO and SEMO. Next, we introduce a new parent selection strategy, stochastic tournament selection (i.e., $k$ tournament selection where $k$ is uniformly sampled at random), to replace the binary tournament selection strategy of NSGA-II, decreasing the required expected running time to $O(n^2)$ for all the three problems. Experiments are also conducted, suggesting that the derived running time upper bounds are tight for LOTZ, and almost tight for OneMinMax and COCZ.",
"subjects": "Neural and Evolutionary Computing (cs.NE)",
"title": "Running Time Analysis of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) using Binary or Stochastic Tournament Selection",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575116884779,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542206194358
} |
https://arxiv.org/abs/1909.06221 | A proximal average for prox-bounded functions | In this work, we construct a proximal average for two prox-bounded functions, which recovers the classical proximal average for two convex functions. The new proximal average transforms continuously in epi-topology from one proximal hull to the other. When one of the functions is differentiable, the new proximal average is differentiable. We give characterizations for Lipschitz and single-valued proximal mappings and we show that the convex combination of convexified proximal mappings is always a proximal mapping. Subdifferentiability and behaviors of infimal values and minimizers are also studied. | \section{Introduction}
The proximal average provides a novel technique for averaging convex functions, see \cite{convmono,proxbas}.
The proximal average has been used widely in applications such as machine learning \cite{reidwill,Yu13a}, optimization \cite{resaverage,wolenski,boyd14,planwang2016,zaslav}, matrix analysis \cite{kimlaws, lim18} and
modern monotone operator theory
\cite{simons}. The proximal mapping of the proximal average is precisely
the average of proximal mappings of the convex
functions involved. Averages of proximal mappings are
important in convex and nonconvex optimization algorithms;
see, e.g., \cite{convmono, aveproj}.
A proximal average for
possible nonconvex functions has long been sought.
In this work, we have proposed a proximal average for prox-bounded functions, which enjoy rich
theory in variational analysis and optimization.
Our proximal average
significantly extends the works of \cite{proxbas} from convex functions to
possibly nonconvex functions. The new average function provides an epicontinuous transformation
between proximal hulls of functions, and reverts to the convex proximal average
definition in the case of convex functions.
When studying the proximal average of possibly nonconvex functions, two fundamental issues arise. The first is when
the proximal mapping is convex-valued; the second is when the function can
be recovered from its proximal mapping. It turns out that
resolving both difficulties requires the `proximal'
condition in variational analysis.
\subsection{Outline}
The plan of the paper is as follows. In the following three subsections, we give basic concepts from variational analysis,
review related work in the literature and state the blanket assumptions of the paper.
In Section \ref{s:prel}, we prove some interesting and new
properties of proximal functions, proximal mappings and envelopes. Section \ref{s:conv}
gives
an explicit relationship between the convexified proximal mapping and the Clarke
subdifferential of the Moreau envelope. Section \ref{s:char} provides
characterizations of Lipschitz and single-valued proximal mappings.
In Section \ref{s:main}, we
define the proximal average for prox-bounded functions and give a systematic study
of its properties. Relationships to arithmetic average and epi-average and
full epi-continuity of the proximal average are studied
in Section \ref{s:rela}. Section \ref{s:opti} is devoted to optimal
value and minimizers and convergence in minimization of the proximal average.
In Section \ref{s:subd}, we investigate the subdifferentiability and
differentiability of the proximal average.
As an example, the
proximal average for quadratic functions is given in Section \ref{s:quad}.
Finally, Section \ref{s:theg} illustrates the difficulty when the proximal mapping
is not convex-valued.\par
Two distinguished features
of our proximal average
deserve to be singled out: whenever one of the function is differentiable,
the new proximal average is differentiable and the convex combinations of
convexified proximal mappings is always a proximal mapping.
While epi-convergence
\cite{attouch1984, beertopologies} plays a dominant role in our analysis of
convergence
in minimization, the class of proximal functions, which
is significantly broader than the class of convex functions, is indispensable
for studying the proximal average.
In carrying out the proofs later, we often cite results from
the standard reference Rockafellar--Wets \cite{rockwets}.
\subsection{Constructs from variational analysis}
In order to define the proximal average of possibly nonconvex functions, we utilize the Moreau envelope
and proximal hull. In what follows, $\operatorname{\mathbb{R}}^n$ is the $n$-dimensional Euclidean space
with Euclidean norm $\|x\|=\sqrt{\scal{x}{x}}$ and inner product
$\scal{x}{y}=\sum_{i=1}^{n}x_{i}y_{i}$ for $x,y\in\operatorname{\mathbb{R}}^n$.
\begin{df}
For a proper function $f:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ and parameters $0<\mu<\lambda$, the
{Moreau envelope}
function $e_{\lambda}f$ and {proximal mapping} are defined, respectively, by
$$e_{\lambda}f(x)=\inf_{w}\left\{f(w)+\frac{1}{2\lambda}\|w-x\|^2\right\},
\quad \operatorname{Prox}_{\lambda}f(x)=\operatornamewithlimits{argmin}_{w}\left\{f(w)+\frac{1}{2\lambda}\|w-x\|^2\right\};$$
the {proximal hull} function $h_{\lambda}f$ is defined by
$$h_{\lambda}f(x)=\inf_{w}\left\{e_{\lambda}f(w)-\frac{1}{2\lambda}\|x-w\|^2\right\};$$
the {Lasry--Lions envelope}
$e_{\lambda,\mu}f$ is defined by
$$e_{\lambda,\mu}f(x)=\sup_{w}\left\{e_{\lambda}f(w)-\frac{1}{2\mu}\|x-w\|^2\right\}.$$
\end{df}
\begin{df} The function $f:\operatorname{\mathbb{R}}^n\rightarrow \ensuremath{\,\left]-\infty,+\infty\right]}$ is prox-bounded if
there exist $\lambda>0$ and $x\in \operatorname{\mathbb{R}}^n$ such that
$e_{\lambda}f(x)>-\infty.$
The supremum of the set of all such $\lambda$ is the threshold $\lambda_{f}$
of prox-boundedness for $f$.
\end{df}
Any function $f:\operatorname{\mathbb{R}}^{n}\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ that is bounded
below by an affine function has threshold of prox-boundedness $\lambda_{f}=\infty$; cf.
\cite[Example 3.28]{rockwets}. A differentiable function $f$ with a Lipschitz continuous
gradient has $\lambda_{f}>0$.
Our notation is standard. For every nonempty set $S\subset\operatorname{\mathbb{R}}^n$, $\operatornamewithlimits{conv} S$, $\operatorname{cl} S$ and $\iota_{S}$ denote the \emph{convex hull}, \emph{closure} and
\emph{indicator function} of set $S$, respectively.
For a proper, lower semicontinuous (lsc) function $f:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$, $\operatornamewithlimits{conv} f$ is its convex hull and $f^*$ is its \emph{Fenchel conjugate}.
We let $\inf f$ and $\operatornamewithlimits{argmin} f$ denote the infimum and
the set of minimizers of $f$ on $\operatorname{\mathbb{R}}^n$, respectively. We call
$f$ \emph{level-coercive} if
$$\liminf_{\|x\|\rightarrow\infty}\frac{f(x)}{\|x\|}>0,$$
and \emph{coercive} if
$$\liminf_{\|x\|\rightarrow\infty}\frac{f(x)}{\|x\|}=\infty.$$
We use $\partial f$, $\hat{\partial }f, \partial_{L}f, \partial_{C}f$ for the Fenchel subdifferential, Fr\'echet subdifferential, limiting subdifferential
and Clarke subdifferential of $f$, respectively. More precisely,
at a point $x\in\operatorname{dom} f$, the \emph{Fenchel subdifferential}
of $f$ at $x$ is the set
$$\partial f(x)=\{s\in\operatorname{\mathbb{R}}^n:\ f(y) \geq f(x)+\scal{s}{y-x} \text{ for all $y\in\operatorname{\mathbb{R}}^n$}\};$$
the \emph{Fr\'echet subdifferential} of $f$ at $x$ is the set
$$\hat{\partial} f(x)=\{s\in\operatorname{\mathbb{R}}^n:\ f(y) \geq f(x)+\scal{s}{y-x}+o(\|y-x\|)\};$$
the \emph{limiting subdifferential} of $f$ at $x$ is
$$\partial_{L}f(x)=\{v\in\operatorname{\mathbb{R}}^n:\ \exists \text{ sequences } x_{k} \stackrel{\operatorname{f}}{\rightarrow} x \text{ and }
s_{k}\in\hat{\partial}f(x_{k}) \text{ with } s_{k}\rightarrow v\},$$
where $x_{k} \stackrel{\operatorname{f}}{\rightarrow} x$ means $x_{k}\rightarrow x$ and $f(x_{k})\rightarrow f(x)$.
We let $\operatorname{Id}:\operatorname{\mathbb{R}}^n\rightarrow\operatorname{\mathbb{R}}: x\mapsto x$ be the identity mapping and $\ensuremath{\,\mathfrak{q}}=\frac{1}{2}\|\cdot\|^2$.
The mapping $J_{\mu\partial_{L}f}=(\operatorname{Id}+\mu\partial_{L}f)^{-1}$ is
called the \emph{resolvent} of $\mu\partial_{L}f$;
cf. \cite[page 539]{rockwets}.
When $f$ is locally Lipschitz at $x$, the \emph{Clarke subdifferential}
$\partial_{C}f$ at $x$ is $\partial_{C} f(x)=
\operatornamewithlimits{conv} \partial_{L}f(x)$.
For further details on subdifferentials, see \cite{optanal,mordukhovich2006variational,rockwets}.
For $f_{1},f_{2}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$, the \emph{infimal convolution}
(or epi-sum)
of $f_{1}, f_{2}$ is defined by
$$(\forall x\in\operatorname{\mathbb{R}}^n)\ f_{1}\Box f_{2}(x)=\inf_{w}\{f_{1}(x-w)+f_{2}(w)\},$$
and it is exact at $x$ if $\exists~w\in\operatorname{\mathbb{R}}^n$ such that $f_{1}\Box f_{2}(x)=f_{1}(x-w)+f_{2}(w)$;
$f_{1}\Box f_{2}$ is exact if it is exact at every point of its domain.
\subsection{Related work}
A comparison to known work in the literature is in order.
In \cite{zhang2,zhang1}, Zhang et. al. defined a lower compensated convex
transform for $0<\mu<+\infty$ by
$$C_{\mu}^{l}(f)=\operatornamewithlimits{conv}(2\mu\ensuremath{\,\mathfrak{q}}+f)-2\mu\ensuremath{\,\mathfrak{q}}.$$
The lower compensated convex transform is the proximal
hull.
In \cite{zhang2}, Zhang, Crooks and Orlando gave a comprehensive
study on the average compensated
convex approximation,
which is an arithmetic average of the proximal hull and the upper proximal hull.
While the proximal hull is a common ingredient, our work and theirs
are completely different.
By nature, the proximal mapping of the
proximal average for convex functions
is exactly the convex combination
of proximal mappings of individual convex functions \cite{proxbas}.
In \cite{proxave}, Hare proposed a proximal average by
$$\operatorname{\mathcal{PA}}_{1/\mu}=-e_{1/(\mu+\alpha(1-\alpha))}(-\alpha e_{1/\mu}f-(1-\alpha)e_{1/\mu}g).$$
For this average,
$x\mapsto\operatorname{\mathcal{PA}}_{1/\mu}(x)$ is $\mathcal{C}^{1+}$ for every $\alpha\in]0,1[$, and enjoys other nice stabilities
with respect to $\alpha$, see, e.g., \cite[Theorem 4.6]{proxave}. However,
this average definition has two disadvantages.\\\noindent (i) Even when both $f,g$ are convex, it does not
recover the proximal average for convex functions: $$-e_{1/\mu}(-\alpha e_{1/\mu}f-
(1-\alpha)e_{1/\mu}g).$$ (ii) Neither the proximal mapping
$\operatorname{Prox}_{1/(\mu+\alpha(1-\alpha))}\operatorname{\mathcal{PA}}_{1/\mu}$ nor
$\operatorname{Prox}_{1/\mu}\operatorname{\mathcal{PA}}_{1/\mu}$
is the average of the proximal mappings
$\operatorname{Prox}_{1/\mu}f$ and $\operatorname{Prox}_{1/\mu}g$.\par
In \cite{goebel2010proximal}, Goebel introduced a proximal average for saddle functions
by using extremal convolutions:
$$\mathcal{P}_{\mu,\eta}^{\cup\cap}
=\big(\lambda_{1}\timess (f_{1}+\mu\timess\ensuremath{\,\mathfrak{q}}_{x}-\eta\timess \ensuremath{\,\mathfrak{q}}_{y})\big)\pluss \big(\lambda_{2}\timess (f_{2}
+\mu\timess\ensuremath{\,\mathfrak{q}}_{x}-\eta\timess\ensuremath{\,\mathfrak{q}}_{y})\big)-
\mu\timess\ensuremath{\,\mathfrak{q}}_{x}+\eta\timess\ensuremath{\,\mathfrak{q}}_{y},$$
in which $f_{1}, f_{2}: \operatorname{\mathbb{R}}^m\times \operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left[-\infty,+\infty\right]} $ are
saddle functions, $\ensuremath{\,\mathfrak{q}}_{x}(x,y)=\ensuremath{\,\mathfrak{q}}(x), \ensuremath{\,\mathfrak{q}}_{y}(x,y)
=\ensuremath{\,\mathfrak{q}}(y)$, $\mu, \eta>0$, $\lambda_{1}+\lambda_{2}=1$ with $\lambda_{i}>0$, and
$\pluss$ is the extremal convolution.
Some nice results about self-duality with respect to saddle function conjugacy and partial conjugacy are put forth and proved by Goebel \cite{goebel2010proximal}.
Goebel's average is the proximal average for convex functions when each $f_{i}$
is convex. However, the proximal mapping of $\operatorname{Prox}_{\lambda}\mathcal{P}_{\mu,\eta}^{\cup\cap}$ is not
the convex combination of $\operatorname{Prox}_{\lambda}f_1$ and $\operatorname{Prox}_{\lambda} f_2$.
\subsection{Blanket assumptions}\label{s:assump}
Throughout the paper,
the functions $f,g:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ are proper, lsc
and prox-bounded with thresholds $\lambda_{f}, \lambda_{g}>0$ respectively,
$\operatorname{\bar{\lambda}}=\min\{\lambda_{f},\lambda_{g}\}$, $\lambda>0$,
$\mu>0$ and $\alpha\in [0,1]$.
\section{Preliminaries}\label{s:prel}
In this section, we
collect several facts and present some auxiliary results
on proximal mappings of proximal functions, Moreau envelopes
and proximal hulls, which will be used in the sequel.
\subsection{Relationship among three regularizations: $e_{\lambda}f$, $h_{\lambda}f$, and $e_{\lambda,\mu}f$}
Some key properties about these regularizations come as follows.
\begin{Fact}\emph{(\cite[Example 11.26]{rockwets})}\label{l:dcform}
Let $0<\lambda<\lambda_{f}$.
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:mor:l}
The Moreau envelope
$$e_{\lambda}f=
-\left(f+\frac{1}{2\lambda}\|\cdot\|^2\right)^*\bigg(\frac{\cdot}{\lambda}\bigg)+
\frac{1}{2\lambda}\|\cdot\|^2$$
is locally Lipschitz.
\item The proximal hull satisfies
$$h_{\lambda}f+\frac{1}{2\lambda}\|\cdot\|^2=\bigg(f+\frac{1}{2\lambda}\|\cdot\|^2\bigg)^{**}.$$
\end{enumerate}
\end{Fact}
\begin{Fact}\emph{(\cite[Examples 1.44, 1.46, Exercise 1.29]{rockwets})}\label{f:m-p-l}
Let $0<\mu<\lambda<\lambda_{f}$. One has
\begin{enumerate}[label=\rm(\alph*)]
\item $h_{\lambda}f=-e_{\lambda}(-e_{\lambda}f)$,
\item \label{i:p:hull}
$e_{\lambda} f=e_{\lambda}(h_{\lambda}f)$,
\item $h_{\lambda}(h_{\lambda}f)=h_{\lambda}f$,
\item\label{i:d:env}
$e_{\lambda,\mu}f=-e_{\mu}(-e_{\lambda}f)=h_{\mu}(e_{\lambda-\mu}f)=e_{\lambda-\mu}(h_{\lambda}f)$,
\item $e_{\lambda_{1}}(e_{\lambda_{2}}f)=e_{\lambda_{1}+\lambda_{2}}f$ for
$\lambda_{1}, \lambda_{2}>0$.
\end{enumerate}
\end{Fact}
For more details about these regularizations, we refer the reader to
\cite{attouch1990approximation,infconv,proxhilbert,diffprop} and \cite[Chapter 1]{rockwets}.
\subsection{Proximal functions}
The concept of $\lambda$-proximal functions will play an important role. This
subsection is dedicated to properties of $\lambda$-proximal functions.
\begin{df} We say that $f$ is \emph{$\lambda$-proximal}
if $f+\frac{1}{2\lambda}\|\cdot\|^2$ is convex.
\end{df}
\begin{lem}\label{l:env:neg}
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:e:1} The negative Moreau envelope $-e_{\lambda}f$ is always $\lambda$-proximal.
\item\label{i:e:2}
If $e_{\lambda}f$ is $\mathcal{C}^{1}$, then $f+\frac{1}{2\lambda}\|\cdot\|^2$ is convex,
i.e., $f$ is $\lambda$-proximal.
\end{enumerate}
\end{lem}
\begin{proof} By Fact~\ref{l:dcform},
\begin{equation}\label{e:moreau}
(\forall x\in\operatorname{\mathbb{R}}^n)\ \frac{1}{2\lambda}\|x\|^2-e_{\lambda}f(x)=
\bigg(f+\frac{1}{2\lambda}\|\cdot\|^2\bigg)^{*}\bigg(\frac{x}{\lambda}\bigg).
\end{equation}
\ref{i:e:1}: This is clear from \eqref{e:moreau}.
\noindent\ref{i:e:2}: By \eqref{e:moreau}, the assumption ensures that
$\big(f+\frac{1}{2\lambda}\|\cdot\|^2\big)^{*}\big(\frac{x}{\lambda}\big)$ is differentiable.
It follows from Soloviov's theorem \cite{soloviov}
that $f+\frac{1}{2\lambda}\|\cdot\|^2$ is convex.
\end{proof}
While for convex functions, proximal mappings and resolvents are the same, they
differ for nonconvex functions in general.
\begin{Fact}\emph{(\cite[Example 10.2]{rockwets})}
For any proper, lsc function $f:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ and
any $\mu>0$, one has
$$(\forall x\in\operatorname{\mathbb{R}}^n)\ P_{\mu}f(x)\subseteq J_{\mu\partial_{L}f}(x).$$
When $f$ is convex, the inclusion holds as an equation.
\end{Fact}
\noindent However, proximal functions have surprising properties.
\begin{prop}\label{p:resolventf}
Let $0<\mu<\lambda_{f}$. Then
the following are equivalent:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:resolvent1}
$\operatorname{Prox}_{\mu}f=J_{\mu\partial_{L}f}$,
\item\label{i:resolvent2} $f$ is $\mu$-proximal,
\item\label{i:resolvent3} $\operatorname{Prox}_{\mu}f$ is maximally monotone,
\item\label{i:resolvent4} $\operatorname{Prox}_{\mu}f$ is convex-valued.
\end{enumerate}
\end{prop}
\begin{proof}
\ref{i:resolvent2}$\Rightarrow$\ref{i:resolvent1}: See \cite[Proposition 12.19]{rockwets}
\& \cite[Example 11.26]{rockwets}.
\noindent\ref{i:resolvent1}$\Rightarrow$\ref{i:resolvent2}: As $\operatorname{Prox}_{\mu}f$ is always monotone,
$(\operatorname{Prox}_{\mu}f)^{-1}=(\operatorname{Id}+\mu\partial_{L}f)$ is monotone and it suffices to apply
\cite[Proposition 12.19(c)$\Rightarrow$(b)]{rockwets}.
\noindent\ref{i:resolvent2}$\Leftrightarrow$\ref{i:resolvent3}:
See \cite[Proposition 12.19]{rockwets}.
\noindent\ref{i:resolvent3}$\Rightarrow$\ref{i:resolvent4}: This is clear.
\noindent\ref{i:resolvent4}$\Rightarrow$\ref{i:resolvent3}: By \cite[Example 1.25]{rockwets},
$\operatorname{Prox}_{\mu}f$ is nonempty, compact-valued and monotone with full domain. As $\operatorname{Prox}_{\mu}f$ is convex-valued, it suffices to apply \cite{lohne08}.
\end{proof}
\begin{lem}\label{l:prox:map}
Let $f$ be $\lambda$-proximal and $0<\mu<\lambda$. Then
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:p:convex} $\operatorname{Prox}_{\lambda}f$ is convex-valued,
\item\label{i:p:single} $\operatorname{Prox}_{\mu}f$ is single-valued.
\end{enumerate}
Consequently, $\operatorname{Prox}_{\mu}f$ is maximally monotone if $0<\mu\leq\lambda$.
\end{lem}
\begin{proof}
\ref{i:p:convex}:
Observe that
$$e_{\lambda}f(x)=\inf_{y}\left\{f(y)+\frac{1}{2\lambda}\|y\|^2-\langle\frac{x}{\lambda}, y
\rangle\right\}+\frac{1}{2\lambda}\|x\|^2.$$
Since
$f+\frac{1}{2\lambda}\|\cdot\|^2-\langle{\frac{x}{\lambda}},\cdot\rangle$ is convex, $\operatorname{Prox}_{\lambda}f(x)$ is convex.
\noindent\ref{i:p:single}: This follow from the fact that
$f+\frac{1}{2\mu}\|\cdot\|^2-\langle\frac{x}{\mu},\cdot\rangle$ is strictly convex
and coercive.
When $0<\mu<\lambda$, $\operatorname{Prox}_{\mu}f$ is continuous and monotone, so maximally monotone by
\cite[Example 12.7]{rockwets}. For the maximal monotonicity of $\operatorname{Prox}_{\lambda}f$, apply
\ref{i:p:convex} and
\cite{lohne08} or Lemma~\ref{l:prox:grad}.
\end{proof}
The set of proximal functions is a convex cone. In particular, one has the following.
\begin{prop}\label{p:p:cone}
Let $f_1$ be $\lambda_1$-proximal and $f_2$ be $\lambda_2$-proximal. Then for any $\alpha,\beta>0$, the function $\alpha f_1+\beta f_2$ is $\frac{\lambda_1\lambda_2}{\beta\lambda_1+\alpha\lambda_2}$-proximal.
\end{prop}
\begin{proof}
Since $f_1+\frac{1}{2\lambda_1}\|\cdot\|^2$ and $f_2+\frac{1}{2\lambda_2}\|\cdot\|^2$ are convex, so are $\alpha\left(f_1+\frac{1}{2\lambda_1}\|\cdot\|^2\right)$, $\beta\left(f_2+\frac{1}{2\lambda_2}\|\cdot\|^2\right)$ and their sum:
$$\alpha f_1+\beta f_2+\left(\frac{\alpha}{2\lambda_1}+\frac{\beta}{2\lambda_2}\right)\|\cdot\|^2=\alpha f_1+\beta f_2+\frac{\beta\lambda_1+\alpha\lambda_2}{2\lambda_1\lambda_2}\|\cdot\|^2.$$
Therefore, $\alpha f_1+\beta f_2$ is $\frac{\lambda_1\lambda_2}{\beta\lambda_1+\alpha\lambda_2}$-proximal.
\end{proof}
\subsection{The proximal mapping of the proximal hull}
\begin{lem}\label{l:e:h}Let $0<\lambda<\lambda_{f}$.
One has
\begin{equation}\label{i:hull:prox}
\operatorname{Prox}_{\lambda}(h_{\lambda}f)=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f.
\end{equation}
\end{lem}
\begin{proof}
Applying \cite[Example 10.32]{rockwets} to
$-e_{\lambda} f=-e_{\lambda}(h_{\lambda}f)$ yields
$$\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}(h_{\lambda}f)=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f.$$
Since $h_{\lambda}$ is $\lambda$-proximal, by Lemma~\ref{l:prox:map} we have
$\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}(h_{\lambda}f)=\operatorname{Prox}_{\lambda}(h_{\lambda}f).$
Hence \eqref{i:hull:prox} follows.
\end{proof}
\begin{lem} Let $0<\lambda<\lambda_{f}$. The following are equivalent:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{p:hull} $\operatorname{Prox}_{\lambda}(h_{\lambda}f)=\operatorname{Prox}_{\lambda}f$,
\item\label{p:function} $f$ is $\lambda$-proximal.
\end{enumerate}
\end{lem}
\begin{proof}
\ref{p:hull}$\Rightarrow$\ref{p:function}: Since $\operatorname{Prox}_{\lambda}(h_{\lambda}f)=
\operatornamewithlimits{conv} \operatorname{Prox}_{\lambda}(h_{\lambda}f)$, $\operatorname{Prox}_{\lambda} f$ is upper
semicontinuous, convex and compact valued, and monotone with full domain, so
maximally monotone in view of \cite{lohne08} or Lemma~\ref{l:prox:grad}.
By \cite[Proposition 12.19]{rockwets},
$f+\frac{1}{2\lambda}\|\cdot\|^2$ is convex, equivalently,
$f$ is $\lambda$-proximal by \cite[Example 11.26]{rockwets}.
\noindent\ref{p:function}$\Rightarrow$\ref{p:hull}: As $f$ is $\lambda$-proximal,
$\operatorname{Prox}_{\lambda}f$ is convex-valued by Lemma~\ref{l:prox:map}. Then Lemma~\ref{l:e:h} gives
$\operatorname{Prox}_{\lambda}(h_{\lambda}f)=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f=\operatorname{Prox}_{\lambda}f.$
\end{proof}
\begin{cor} If $f\neq h_{\lambda}f$, then $\operatorname{Prox}_{\lambda}(h_{\lambda}f)
\neq\operatorname{Prox}_{\lambda}f.$
\end{cor}
\subsection{Proximal mappings and envelopes}
\begin{lem}\label{l:env:prox}
Let $0<\mu<\lambda<\operatorname{\bar{\lambda}}$.
The following are equivalent:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:env:fg}$e_{\lambda}f=e_{\lambda}g$,
\item\label{i:phull:fg}
$h_{\lambda}f=h_{\lambda}g$,
\item \label{i:conv:fg} $\operatornamewithlimits{conv}\left(f+\frac{1}{2\lambda}\|\cdot\|^2\right)=\operatornamewithlimits{conv}\left(g+\frac{1}{2\lambda}\|\cdot\|^2\right)$,
\item\label{i:double:fg}
$e_{\lambda,\mu}f=e_{\lambda,\mu}g$,
\item\label{i:prox:initial}
$\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}g$, and for some $x_{0}\in\operatorname{\mathbb{R}}^n$ one has
$e_{\lambda}f(x_{0})=e_{\lambda}g(x_{0})$.
\end{enumerate}
Under any one of the conditions \ref{i:env:fg}--\ref{i:prox:initial}, one has
\begin{equation}\label{e:convexhull}
\overline{\operatornamewithlimits{conv}} f=\overline{\operatornamewithlimits{conv}} g.
\end{equation}
\end{lem}
\begin{proof}
\ref{i:env:fg}$\Rightarrow$\ref{i:phull:fg}:
We have
$-e_{\lambda}f=-e_{\lambda}g$ implies $-e_{\lambda}(-e_{\lambda}f)=-e_{\lambda}(-e_{\lambda}g)$,
which is \ref{i:phull:fg}.
\noindent\ref{i:phull:fg}$\Rightarrow$\ref{i:env:fg}:
This follows from $e_{\lambda}f=e_{\lambda}(h_{\lambda}f)=e_{\lambda}(h_{\lambda}g)=e_{\lambda}g.$
\noindent\ref{i:phull:fg}$\Leftrightarrow$\ref{i:conv:fg}: Since $\lambda<\operatorname{\bar{\lambda}}$, we have that
$f+\frac{1}{2\lambda}\|\cdot\|^2$ and $g+\frac{1}{2\lambda}\|\cdot\|^2$ are coercive, so
$\operatornamewithlimits{conv}\left(f+\frac{1}{2\lambda}\|\cdot\|^2\right)$ and $\operatornamewithlimits{conv}\left(f+\frac{1}{2\lambda}\|\cdot\|^2\right)$
are lsc. Fact~\ref{l:dcform} gives
$$h_{\lambda}f=\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\lambda}\|\cdot\|^2\bigg)-\frac{1}{2\lambda}\|\cdot\|^2,$$
$$h_{\lambda}g=\operatornamewithlimits{conv}\bigg(g+\frac{1}{2\lambda}\|\cdot\|^2\bigg)-\frac{1}{2\lambda}\|\cdot\|^2.$$
\noindent\ref{i:double:fg}$\Leftrightarrow$\ref{i:env:fg}: Invoking Fact~\ref{f:m-p-l}, we have
\begin{align*}
e_{\lambda,\mu}f =e_{\lambda,\mu}g &\Leftrightarrow h_{\mu}(e_{\lambda-\mu}f)=h_{\mu}(e_{\lambda-\mu}g)\\
& \Leftrightarrow e_{\mu}(h_{\mu}(e_{\lambda-\mu}f))=e_{\mu}(h_{\mu}(e_{\lambda-\mu}g))\\
& \Leftrightarrow e_{\mu}(e_{\lambda-\mu}f)=e_{\mu}(e_{\lambda-\mu}g)\\
& \Leftrightarrow e_{\lambda}f=e_{\lambda}g. \
\end{align*}
\noindent\ref{i:env:fg}$\Rightarrow$\ref{i:prox:initial}: The Moreau envelope $e_{\lambda}f(x)=e_{\lambda}g(x)$
for every $x\in\operatorname{\mathbb{R}}^n$. Apply \cite[Example 10.32]{rockwets}
to $-e_{\lambda}f = -e_{\lambda}g$ to get
$$(\forall x\in\operatorname{\mathbb{R}}^n)\ \frac{\operatornamewithlimits{conv} \operatorname{Prox}_{\lambda}f(x)-x}{\lambda}=\frac{\operatornamewithlimits{conv} \operatorname{Prox}_{\lambda}g(x)-x}{\lambda},
$$
which gives \ref{i:prox:initial} after simplifications.
\noindent\ref{i:prox:initial}$\Rightarrow$\ref{i:env:fg}: Since
both $e_{\lambda}f$ and $e_{\lambda}g$ are locally Lipschitz,
$\operatornamewithlimits{conv} \operatorname{Prox}_{\lambda}f=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}g$
implies
$-e_{\lambda}f=-e_{\lambda}g+\text{constant}$ by
\cite[Example 10.32]{rockwets}. The $\text{constant}$ has to be zero by
$e_{\lambda}f(x_{0})=e_{\lambda}g(x_{0})$. Thus, \ref{i:env:fg} holds.
Equation~\eqref{e:convexhull} follows from the equivalence of \ref{i:env:fg}--\ref{i:double:fg}
and taking the Fenchel conjugate to $e_{\lambda}f=e_{\lambda}g$, followed
by cancelation of terms and taking the Fenchel conjugate again.
\end{proof}
The notion of `proximal' is instrumental.
\begin{cor}\label{c:needed1} Let $0<\mu\leq \lambda <\operatorname{\bar{\lambda}}$, and
let $f,g$ be $\lambda$-proximal.
Then $e_{\mu}f=e_{\mu}g$ if and only if $f=g$
\end{cor}
\begin{proof} Since $\mu\leq\lambda$, both $f,g$ are also $\mu$-proximal, so
$f=h_{\mu}f, g=h_{\mu}g$. Lemma~\ref{l:env:prox}\ref{i:env:fg}$
\Leftrightarrow$\ref{i:phull:fg} applies.
\end{proof}
\begin{prop}\label{p:needed1}
Let $0<\mu <\operatorname{\bar{\lambda}}$, and let
$\operatorname{Prox}_{\mu}f=\operatorname{Prox}_{\mu}g$. If $f, g$ are $\mu$-proximal, then $f-g\equiv
\text{constant}$.
\end{prop}
\begin{proof}
As $\operatorname{Prox}_{\mu}f=\operatorname{Prox}_{\mu}g$,
by \cite[Example 10.32]{rockwets},
$\partial(-e_{\mu}f)=\partial(-e_{\mu}g)$. Since
both $-e_\mu f, -e_{\mu}g$ are locally Lipschitz and Clarke regular,
we obtain that there exists $-c\in\operatorname{\mathbb{R}}$ such that
$-e_\mu f=-e_{\mu}g-c$. Because $f, g$ are $\mu$-proximal, we have
$$f=-e_{\mu}(-e_{\mu}f)=-e_{\mu}(-e_{\mu}g-c)=-e_{\mu}(-e_{\mu}g)+c=g+c,$$
as required.
\end{proof}
\subsection{An example}
The following example shows that one cannot remove the assumption of
$f, g$ being $\mu$-proximal in Proposition~\ref{p:resolventf}, Corollary~\ref{c:needed1}
and Proposition~\ref{p:needed1}.
\begin{ex}\label{e:proximal:fk}
Consider the function
$$f_{k}(x)=\max\{0,(1+\varepsilon_{k})(1-x^2)\},$$
where $\varepsilon_{k}>0$.
It is easy to check that $f_{k}$ is $1/(2(1+\varepsilon_{k}))$-proximal, but not $1/2$-proximal.
\end{ex}
\noindent {\sl Claim 1: The functions $f_{k}$ have the same proximal mappings
and Moreau envelopes for all $k\in\operatorname{\mathbb{N}}$. However, whenever $\varepsilon_{k_{1}}\neq\varepsilon_{k_{2}}$, $f_{k_{1}}-f_{k_{2}}=(\varepsilon_{k_{1}}-\varepsilon_{k_{2}})f\neq \text{constant}$.}
Indeed, simple calculus gives that for every $\varepsilon_{k}>0$ one has
\begin{equation*}\label{e:proximal2}
\operatorname{Prox}_{1/2}f_{k}(x)=\begin{cases}
x &\text{ if $x\geq 1$,}\\
1 &\text{ if $0<x<1$,}\\
\{-1,1\} &\text{ if $x=0$,}\\
-1 &\text{ if $-1<x<0$,}\\
x &\text{ if $x\leq -1$,}
\end{cases}
\end{equation*}
and
$$e_{1/2}f_{k}(x)=\begin{cases}
0 &\text{ if $x\geq 1$,}\\
(x-1)^2 &\text{ if $0\leq x<1$,}\\
(x+1)^2 &\text{ if $-1<x<0$,}\\
0 &\text{ if $x\leq -1$.}
\end{cases}
$$
\noindent{\sl Claim 2: $\operatorname{Prox}_{1/2}f_{k}\neq J_{1/2\partial_{L}f_{k}},$ i.e., the
proximal mapping
differs from the resolvent.}
Since $J_{1/2\partial_{L}f_{k}}=(\operatorname{Id}+1/2\partial_{L}f_{k})^{-1}$ and
$$\partial_{L}f_{k}(x)
=\begin{cases}
0 & \text{ if $x<-1$,}\\
[0,2(1+\varepsilon_{k})] &\text{ if $x=-1$,}\\
-2(1+\varepsilon_{k})x &\text{ if $-1<x<1$,}\\
[-2(1+\varepsilon_{k}),0] &\text{ if $x=1$,}\\
0 & \text{ if $x>1$},
\end{cases}
$$
we obtain
$$J_{1/2\partial_{L}f_{k}}(x)=
\begin{cases}
x & \text{ if $x<-1$,}\\
-1 &\text{ if $-1\leq x\leq \varepsilon_{k}$,}\\
-\frac{x}{\varepsilon_{k}} &\text{ if $-\varepsilon_{k}<x<\varepsilon_{k}$,}\\
1 &\text{ if $-\varepsilon_{k}\leq x\leq 1$,}\\
x & \text{ if $x>1$},
\end{cases}
$$
equivalently,
$$J_{1/2\partial_{L}f_{k}}(x)=
\begin{cases}
x & \text{ if $x<-1$,}\\
-1 &\text{ if $-1\leq x< -\varepsilon_{k}$,}\\
\left\{-1,-\frac{x}{\varepsilon_{k}},1\right\} &\text{ if $-\varepsilon_{k}\leq x\leq \varepsilon_{k}$,}\\
1 &\text{ if $\varepsilon_{k}< x\leq 1$,}\\
x & \text{ if $x>1$},
\end{cases}
$$
which does not equal \eqref{e:proximal2}.
\section{The convexified proximal mapping and Clarke subdifferential of the Moreau envelope}\label{s:conv}
The following result gives the relationship between the Clarke subdifferential
of the Moreau envelope and the convexified proximal mapping.
\begin{lem}\label{l:prox:grad}
For $0<\mu<\lambda_{f}$, the following hold.
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:fplus:p}The convex hull
\begin{equation*}\label{e:convp}
\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f=\partial \bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*.
\end{equation*}
In particular, $\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f$ is maximally monotone.
\item\label{i:fplus:n}The limiting subdifferential
\begin{equation*}
-\partial_{L} \bigg(-\bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*\bigg)
\subseteq \operatorname{Prox}_{\mu}f.
\end{equation*}
\item \label{e:env:clarke} The Clarke subdifferential
\begin{equation}\label{e:clarkesub}
\partial_{C}(e_{\mu}f)=-\partial_{L}(-e_{\mu}f)=\frac{\operatorname{Id}-\operatornamewithlimits{conv} \operatorname{Prox}_{\mu}f}{\mu}.
\end{equation}
If, in addition, $f$ is $\mu$-proximal, then
\begin{equation}\label{e:clarke:prox}
\partial_{C}(e_{\mu}f)=\frac{\operatorname{Id}-\operatorname{Prox}_{\mu}f}{\mu}.
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof} \ref{i:fplus:p}: By Fact~\ref{l:dcform},
\begin{equation}\label{e:moreau:d}
-e_{\mu}f(x)=-\frac{1}{2\mu}\|x\|^2+\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right).
\end{equation}
Using \cite[Example 10.32]{rockwets} and the subdifferential sum rule \cite[Corollary 10.9]{rockwets}, we get
$$
\frac{\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f(x)-x}{\mu} =\partial_{L} (-e_{\mu}f)(x)=-\frac{x}{\mu}
+\partial \bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right).
$$
Simplification gives
\begin{align*}
\operatornamewithlimits{conv} \operatorname{Prox}_{\mu}f(x) &
=\partial \mu \bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right)\\
& =\partial \bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*(x).
\end{align*}
Since $\mu f+\frac{1}{2}\|\cdot\|^2$ is coercive, we conclude that
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ is a continuous convex function, so
$\operatornamewithlimits{conv} \operatorname{Prox}_{\mu}f$ is maximally monotone \cite[Theorem 12.17]{rockwets}.
\noindent\ref{i:fplus:n}: By \eqref{e:moreau:d},
\begin{align*}
-\bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*(x) & =-\mu\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right)\\
&=\mu e_{\mu}f(x)-\frac{1}{2}\|x\|^2.
\end{align*}
From \cite[Example 10.32]{rockwets} we obtain
\begin{align*}
\partial_{L} \bigg(-\bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*\bigg)
(x) &= \partial_{L} (\mu e_{\mu}f)(x)-x\\
&\subseteq \mu \frac{x-\operatorname{Prox}_{\mu}f(x)}{\mu}-x=-\operatorname{Prox}_{\mu}f(x).
\end{align*}
Therefore, $-\partial_{L} \bigg(-\bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*\bigg)(x)\subseteq \operatorname{Prox}_{\mu}f(x)$.
\noindent\ref{e:env:clarke}: As $-e_{\mu}f$ is Clarke regular, using \cite[Example 10.32]{rockwets}
we obtain
$$\partial_{C}e_{\mu}f(x)=-\partial_{C}(-e_{\mu}f)(x)=-\partial_{L} (-e_{\mu}f)(x)=
\frac{x-\operatornamewithlimits{conv} \operatorname{Prox}_{\mu}f(x)}{\mu}.$$
If $f$ is $\mu$-proximal, then $\operatorname{Prox}_{\mu}f(x)$ is convex for every $x$, so
\eqref{e:clarke:prox} follows from \eqref{e:clarkesub}.
\end{proof}
\begin{rem} {\rm Lemma~\ref{l:prox:grad}\ref{i:fplus:p}} \& {\rm\ref{e:env:clarke}}
extend {\rm\cite[Exercise 11.27]{rockwets}} and {\rm\cite[Theorem 2.26]{rockwets}},
respectively, from convex functions to possibly nonconvex functions.
\end{rem}
It is tempting to ask whether
\begin{equation*}\label{e:boris}
\partial_L (e_{\mu}f)=\frac{\operatorname{Id}-\operatorname{Prox}_{\mu}f}{\mu}
\end{equation*}
holds. This is answered negatively
below.
\begin{prop} Let
$0<\lambda<\lambda_{f}$
and $\psi=h_{\lambda}f$. Suppose
that there exists $x_{0}\in\operatorname{\mathbb{R}}^n$ such that $\operatorname{Prox}_{\lambda}f(x_{0})$ is not convex.
Then
\begin{equation}\label{e:sub:point}
\partial_{L} e_{\lambda}\psi(x_{0})\neq \frac{x_{0}-
\operatorname{Prox}_{\lambda}\psi(x_{0})}{\lambda};
\end{equation}
consequently,
$$\partial_{L} e_{\lambda}\psi\neq \frac{\operatorname{Id}-\operatorname{Prox}_{\lambda}\psi}{\lambda}.$$
\end{prop}
\begin{proof} We prove by contrapositive. Suppose \eqref{e:sub:point} fails,
i.e.,
\begin{equation}\label{e:prox:f}
\partial_{L} e_{\lambda}\psi(x_{0})=\frac{x_{0}-\operatorname{Prox}_{\lambda}\psi(x_{0})}{\lambda}.
\end{equation}
In view of $e_{\lambda}\psi=e_{\lambda}f$ and \cite[Example 10.32]{rockwets},
we have
\begin{equation}\label{e:prox:s}
\partial_{L} e_{\lambda}\psi(x_{0})= \partial_L e_{\lambda}f(x_{0})
\subseteq \frac{x_{0}-\operatorname{Prox}_{\lambda}f(x_{0})}{\lambda}.
\end{equation}
Since $\operatorname{Prox}_{\lambda}\psi=\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f$ by Lemma~\ref{l:e:h}, \eqref{e:prox:f} and \eqref{e:prox:s}
give
$$\frac{x_{0}-\operatornamewithlimits{conv}\operatorname{Prox}_{\lambda}f(x_{0})}{\lambda}\subseteq \frac{x_{0}-\operatorname{Prox}_{\lambda}f(x_{0})}{\lambda},$$
which implies that $\operatorname{Prox}_{\lambda}f(x_{0})$ is a convex set.
This is a contradiction.
\end{proof}
\section{Characterizations of Lipschitz and single-valued proximal mappings}\label{s:char}
Simple examples show that proximal mappings can be wild, although always monotone.
\begin{ex} The function $f(x)=-\frac{1}{2}\|\cdot\|^2$ is prox-bounded with threshold
$\lambda_{f}=1$. We have $\operatorname{Prox}_{1}f=N_{\{0\}}$ the normal cone map at $0$,
i.e.,
$$N_{\{0\}}(x)=\begin{cases}
\operatorname{\mathbb{R}}^n & \text{ if $x=0$,}\\
\varnothing & \text{ otherwise.}
\end{cases}
$$
When $0<\mu<1$, $$\operatorname{Prox}_{\mu}f=\frac{\operatorname{Id}}{1-\mu},$$
which is Lipschitz continuous
with constant $1/(1-\mu)$.
\end{ex}
\begin{Fact}\emph{(\cite[Example 7.44]{rockwets})}
Let $f:\operatorname{\mathbb{R}}^n\rightarrow \ensuremath{\,\left]-\infty,+\infty\right]}$ be proper, lsc and prox-bounded with threshold $\lambda_{f}$, and $0<\mu<\lambda_{f}$. Then
$\operatorname{Prox}_{\mu}f$ is always upper semicontinuous and locally bounded.
\end{Fact}
The following characterizations of the proximal mapping are of independent interest.
\begin{prop}[Lipschitz proximal mapping]
Let
$0<\mu<\lambda_{f}$.
Then the following are equivalent.
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:map} The proximal mapping $\operatorname{Prox}_{\mu}f$ is Lipschitz continuous with constant $\kappa>0$.
\item\label{i:function}
The function
$$f+\frac{\kappa-1}{2\mu\kappa}\|\cdot\|^2$$
is convex.
\end{enumerate}
\end{prop}
\begin{proof}
\ref{i:map}$\Rightarrow$\ref{i:function}: By Lemma~\ref{l:prox:grad}\ref{i:fplus:p},
$\bigg(\mu f+\frac{1}{2}\|\cdot\|^2\bigg)^*$ is differentiable and its
gradient is Lipschitz continuous with constant $\kappa$. By Soloviov's
theorem \cite{soloviov},
$\mu f+\frac{1}{2}\|\cdot\|^2$ is convex. Then the convex function
$\mu f+\frac{1}{2}\|\cdot\|^2$
has differentiable Fenchel conjugate $\big(\mu f+\frac{1}{2}\|\cdot\|^2\big)^*$ and
$\triangledown \big(\mu f+\frac{1}{2}\|\cdot\|^2\big)^*$ is Lipschitz continuous
with constant $\kappa$. It follows from \cite[Proposition 12.60]{rockwets} that
$\mu f+\frac{1}{2}\|\cdot\|^2$ is $\frac{1}{\kappa}$-strongly convex, i.e.,
$$\mu f+\frac{1}{2}\|\cdot\|^2-\frac{1}{\kappa}\frac{1}{2}\|\cdot\|^2$$
is convex. Equivalently,
$$f+\frac{\kappa-1}{2\mu\kappa}\|\cdot\|^2$$
is convex.
\noindent\ref{i:function}$\Rightarrow$\ref{i:map}: We have
$$\mu f+\frac{1}{2}\|\cdot\|^2-\frac{1}{\kappa}\frac{1}{2}\|\cdot\|^2$$
is convex, i.e.,
$\mu f+\frac{1}{2}\|\cdot\|^2$ is strongly convex with constant $\frac{1}{\kappa}$.
Then \cite[Proposition 12.60]{rockwets} implies that
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ is differentiable and its
gradient is Lipschitz continuous with constant $\kappa$.
In view of Lemma~\ref{l:prox:grad}\ref{i:fplus:p},
$\operatorname{Prox}_{\mu}f$ is Lipschitz continuous with constant $\kappa$.
\end{proof}
\begin{cor}\label{c:lip}
Let
$0<\mu<\lambda_{f}$.
Then the following are equivalent.
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:map1}The proximal mapping $\operatorname{Prox}_{\mu}f$ is Lipschitz continuous with constant $1$, i.e., nonexpansive.
\item\label{i:function1}
The function
$f$
is convex.
\end{enumerate}
\end{cor}
\begin{df}\emph{(See \cite[Section 26]{rockconv} or \cite[page 483]{rockwets})}
A proper, lsc, convex function $f:\operatorname{\mathbb{R}}^n\rightarrow (-\infty, +\infty]$
is
\begin{enumerate}[label=\rm(\alph*)]
\item essentially strictly convex if $f$ is strictly convex on every convex subset
of $\operatorname{dom} \partial f$;
\item essentially differentiable if $\partial f(x)$ is a singleton whenever
$\partial f(x)\neq\varnothing$.
\end{enumerate}
\end{df}
\begin{prop}[single-valued proximal mapping]\label{p:single}
Let
$0<\mu<\lambda_{f}$.
Then the following are equivalent.
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:maps}The proximal mapping $\operatorname{Prox}_{\mu}f$ is single-valued, i.e., $\operatorname{Prox}_{\mu}f(x)$
is a singleton for every $x\in\operatorname{\mathbb{R}}^n$.
\item\label{i:functionc}
The function
$$f+\frac{1}{2\mu}\|\cdot\|^2$$
is essentially strictly convex and coercive.
\end{enumerate}
\end{prop}
\begin{proof}
\ref{i:maps}$\Rightarrow$\ref{i:functionc}: By Lemma~\ref{l:prox:grad}\ref{i:fplus:p},
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ is differentiable. By Soloviov's
theorem \cite{soloviov},
$\mu f+\frac{1}{2}\|\cdot\|^2$ is convex. The convex function
$\mu f+\frac{1}{2}\|\cdot\|^2$
has differentiable Fenchel conjugate $\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$. It follows from \cite[Proposition 11.13]{rockwets} that
$\mu f+\frac{1}{2}\|\cdot\|^2$ is essentially strictly convex.
Since $\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ has full domain and
$\mu f+\frac{1}{2}\|\cdot\|^2$ is convex, the function
$\mu f+\frac{1}{2}\|\cdot\|^2$ is coercive by \cite[Theorem 11.8]{rockwets}.
\noindent\ref{i:functionc}$\Rightarrow$\ref{i:maps}:
Since
$\mu f+\frac{1}{2}\|\cdot\|^2$ is essentially strictly convex,
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ is essentially differentiable by \cite[Theorem 11.13]{rockwets}.
Because $\mu f+\frac{1}{2}\|\cdot\|^2$ is coercive,
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$
has full domain. Then
$\left(\mu f+\frac{1}{2}\|\cdot\|^2\right)^*$ is differentiable on $\operatorname{\mathbb{R}}^n$.
In view of Lemma~\ref{l:prox:grad}\ref{i:fplus:p},
$\operatorname{Prox}_{\mu}f(x)$ is single-valued for every $x\in\operatorname{\mathbb{R}}^n$.
\end{proof}
Recall that for a nonempty, closed set $S\subseteq\operatorname{\mathbb{R}}^n$ and every $x\in\operatorname{\mathbb{R}}^n$,
the projection $P_{S}(x)$
consists of the points in $S$ nearest to $x$, so
$P_{S}=\operatorname{Prox}_{1}\iota_{S}$.
Combining Corollary~\ref{c:lip} and Proposition~\ref{p:single},
we can derive the following result
due to Rockafellar and Wets, \cite[Corollary 12.20]{rockwets}.
\begin{cor} Let $S$ be a nonempty, closed set in $\operatorname{\mathbb{R}}^n$. Then the following are
equivalent:
\begin{enumerate}[label=\rm(\alph*)]
\item $P_{S}$ is single-valued,
\item $P_{S}$ is nonexpansive,
\item $S$ is convex.
\end{enumerate}
\end{cor}
\section{The proximal average for prox-bounded functions}\label{s:main}
The goal of this section is to establish a proximal average function that works for any two prox-bounded functions. Our framework
will generalize the convex proximal average of
\cite{proxpoint} to include nonconvex functions, in a manner
that recovers the original definition in the convex case.
Remembering the standing assumptions in Subsection \ref{s:assump},
we define the \emph{proximal average} of $f, g$ associated with parameters $\mu, \alpha$ by
\begin{equation}\label{e:prox:def}
\ensuremath{\varphi^{\alpha}_{\mu}}=-e_{\mu}(-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g),
\end{equation}
which essentially relies on the Moreau envelopes.
\begin{thm}[basic properties of the proximal average]\label{t:prox}
Let
$0<\mu<\operatorname{\bar{\lambda}}$,
and let $\ensuremath{\varphi^{\alpha}_{\mu}}$ be defined as in \eqref{e:prox:def}.
Then the following hold.
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:env:conhull}
The Moreau envelope $e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})=\alpha e_{\mu}f+(1-\alpha)e_{\mu} g.$
\item\label{i:lowers}The proximal average $\ensuremath{\varphi^{\alpha}_{\mu}}$ is proper, lsc and prox-bounded with threshold
$\lambda_{\ensuremath{\varphi^{\alpha}_{\mu}}}\geq\operatorname{\bar{\lambda}}$.
\item\label{i:epi:sum}
The proximal average $\ensuremath{\varphi^{\alpha}_{\mu}}(x)=$
\begin{equation}\label{e:func}
\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)
-\frac{1}{2\mu}\|x\|^2,
\end{equation}
where the inf-convolution $\Box$ is exact;
consequently, $\operatorname{epi}(\ensuremath{\varphi^{\alpha}_{\mu}}+1/2\mu\|\cdot\|^2)=$
\begin{equation}\label{e:epig}
\alpha \operatorname{epi}\operatornamewithlimits{conv}(f+1/2\mu\|\cdot\|^2)+(1-\alpha)\operatorname{epi}\operatornamewithlimits{conv}(g+1/2\mu\|\cdot\|^2).
\end{equation}
\item\label{i:dom:convhull}
The domain $\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha \operatornamewithlimits{conv}\operatorname{dom} f+(1-\alpha)\operatornamewithlimits{conv}\operatorname{dom} g$.
In particular, $\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatorname{\mathbb{R}}^n$ if either one of $\operatornamewithlimits{conv}\operatorname{dom} f$ and $\operatornamewithlimits{conv}\operatorname{dom} g$ is
$\operatorname{\mathbb{R}}^n$.
\item\label{i:hull:ave} The proximal average of $f$ and $g$ is the same
as the proximal average of proximal hulls $h_{\mu}f$ and $h_{\mu}g$, respectively.
\item\label{i:alpha}
When $\alpha=0$, $\varphi_{\mu}^{0}=h_{\mu}g$; when $\alpha=1$, $\varphi_{\mu}^{1}=h_{\mu}g$.
\item\label{i:phi:mu}
Each $\ensuremath{\varphi^{\alpha}_{\mu}}$ is $\mu$-proximal, or equivalently, $\mu$-hypoconvex.
\item\label{i:f=g}
When $f=g$, $\ensuremath{\varphi^{\alpha}_{\mu}}=h_{\mu}f$; consequently, $\ensuremath{\varphi^{\alpha}_{\mu}}=f$ when $f=g$ is $\mu$-proximal.
\item\label{i:g=c}
When $g\equiv c\in\operatorname{\mathbb{R}}$, $\ensuremath{\varphi^{\alpha}_{\mu}}=e_{\mu/\alpha,\mu}(\alpha f+(1-\alpha)c)$,
the Lasry-Lions envelope of $\alpha f+(1-\alpha)c$.
\end{enumerate}
\end{thm}
\begin{proof}
\ref{i:env:conhull}:
Since $-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g$ is $\mu$-proximal by
Lemma~\ref{l:env:neg}\ref{i:e:1} and Proposition~\ref{p:p:cone},
we have
\begin{align*}
-e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})& =-e_{\mu}(-e_{\mu}(-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g))\\
&=h_{\mu}(-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g)\\
&=-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g.
\end{align*}
\noindent\ref{i:lowers}: Because
$0<\mu<\operatorname{\bar{\lambda}}$,
both $e_{\mu}f$ and $e_{\mu}g$ are continuous, see, e.g., \cite[Theorem 1.25]{rockwets}.
By \ref{i:env:conhull}, $e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})$ is real-valued and continuous. If
$\ensuremath{\varphi^{\alpha}_{\mu}}$ is not proper, then $e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})\equiv-\infty$ or
$e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})\equiv\infty$, which is a contradiction. Hence,
$\ensuremath{\varphi^{\alpha}_{\mu}}$ must be proper.
Lower semicontinuity follows from
the definition of the Moreau envelope.
To show that $\lambda_{\ensuremath{\varphi^{\alpha}_{\mu}}}\geq \operatorname{\bar{\lambda}}$, take any $\delta\in ]0,\operatorname{\bar{\lambda}}-\mu[$. By \cite[Exercise 1.29(c)]{rockwets} and \ref{i:env:conhull}, we have
\begin{align*}
e_{\delta+\mu}(\ensuremath{\varphi^{\alpha}_{\mu}}) &=e_{\delta}(e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}}))\\
& =e_{\delta}(\alpha e_{\mu}f+(1-\alpha)e_{\mu}g)\\
&\geq \alpha e_{\delta}(e_{\mu}f)+(1-\alpha)e_{\delta}(e_{\mu}g)\\
&=\alpha e_{\delta+\mu}f+(1-\alpha)e_{\delta+\mu}g>-\infty.
\end{align*}
Since $\delta\in ]0,\operatorname{\bar{\lambda}}-\mu[$ was arbitrary, $\ensuremath{\varphi^{\alpha}_{\mu}}$ has prox-bound
$\lambda_{\ensuremath{\varphi^{\alpha}_{\mu}}}\geq \operatorname{\bar{\lambda}}$.
\noindent\ref{i:epi:sum}: Since $\mu<\operatorname{\bar{\lambda}}$, both
$e_{\mu}f$ and $e_{\mu}g$ are locally Lipschitz with full domain by
Fact~\ref{l:dcform}\ref{i:mor:l},
so
$$\operatorname{dom} \bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*=
\operatorname{dom} \bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*
=\operatorname{\mathbb{R}}^n.$$ It follows from \cite[Theorem 11.23(a)]{rockwets} that
\begin{align*}
& \left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\right]^*\\
&=
\bigg(\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\bigg)^{*}\Box
\bigg((1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\bigg)^{*}
\end{align*}
where the $\Box$ is exact; see, e.g., \cite[Theorem 16.4]{rockconv}.
By Fact~\ref{l:dcform},
\begin{align*}
& -\alpha e_{\mu}f-(1-\alpha)e_{\mu}g \\
&=\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right)
+(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\left(\frac{x}{\mu}\right)-\frac{1}{2\mu}\|\cdot\|^2.
\end{align*}
Substitute this into the definition of $\ensuremath{\varphi^{\alpha}_{\mu}}$ and use Fact~\ref{l:dcform} again
to obtain $\ensuremath{\varphi^{\alpha}_{\mu}}(x)=$
\begin{align}
&\left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\big(\frac{\cdot}{\mu}\big)+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\big(\frac{\cdot}{\mu}\big)
\right]^*\big(\frac{x}{\mu}\big)
-\frac{1}{2\mu}\|x\|^2\nonumber\\
=&\left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\right]^*\big(\mu\frac{x}{\mu}\big)
-\frac{1}{2\mu}\|x\|^2\nonumber\\
= & \left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^{**}\big(\frac{\cdot}{\alpha}\big)\Box
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^{**}
\big(\frac{\cdot}{1-\alpha}\big)
\right](x)
-\frac{1}{2\mu}\|x\|^2\nonumber\\
=& \left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\big(\frac{\cdot}{\alpha}\big)\Box
(1-\alpha)\operatornamewithlimits{conv}\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)(\frac{\cdot}{1-\alpha}\big)
\right](x)
-\frac{1}{2\mu}\|x\|^2,\label{e:box:exact}
\end{align}
in which
$$\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^{**}=\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)$$
$$\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^{**}=\operatornamewithlimits{conv}\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)$$
because $f+\frac{1}{2\mu}\|\cdot\|^2$ and $g+\frac{1}{2\mu}\|\cdot\|^2$
are coercive; see, e.g., \cite[Example 11.26(c)]{rockwets}.
Also, in \eqref{e:box:exact}, the infimal convolution is exact because
$\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)^*$ and $\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)^*$
have full domain and \cite[Theorem 16.4]{rockconv}
or \cite[Theorem 11.23(a)]{rockwets}.
\eqref{e:epig} follows from \eqref{e:func} and
\cite[Proposition 12.8(ii)]{convmono} or \cite[Exercise 1.28]{rockwets}.
\noindent\ref{i:dom:convhull}: This is immediate from \ref{i:epi:sum} and
\cite[Proposition 12.6(ii)]{convmono}.
\noindent\ref{i:hull:ave}: Use
\eqref{e:prox:def}, and the fact that
$e_{\mu}(h_{u}f)=e_{\mu}f$ and $e_{\mu}(h_{u}g)=e_{\mu}g$.
\noindent\ref{i:alpha}: When $\alpha=0$, this follows from $\varphi_{\mu}^{0}=-e_{\mu}(-e_{\mu}g)=h_{\mu}g$;
the proof for $\alpha=1$ case is similar.
\noindent\ref{i:phi:mu}: This follows from Fact~\ref{l:dcform}\ref{i:mor:l}.
\noindent\ref{i:f=g}: When $f=g$, we have $e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=e_{\mu}f$ so that $-e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=-e_{\mu}f$. Since
$\ensuremath{\varphi^{\alpha}_{\mu}}$ is $\mu$-proximal by \ref{i:phi:mu}, it follows that
$\ensuremath{\varphi^{\alpha}_{\mu}}=-e_{\mu}(-e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}})=-e_{\mu}(-e_{\mu}f)=h_{\mu}f$.
\noindent\ref{i:g=c}: This follows from
\begin{align*}
\ensuremath{\varphi^{\alpha}_{\mu}} &=-e_{\mu}(-\alpha e_{\mu}f-(1-\alpha)c)=-e_{\mu}(-e_{\mu/\alpha}(\alpha f)-(1-\alpha)c)\\
&=-e_{\mu}[-e_{\mu/\alpha}(\alpha f+(1-\alpha)c)],
\end{align*}
and Fact~\ref{f:m-p-l}\ref{i:d:env}.
\end{proof}
\begin{prop}
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:regular} The proximal average $\ensuremath{\varphi^{\alpha}_{\mu}}$ is always Clarke regular, prox-regular and strongly
amenable on $\operatorname{\mathbb{R}}^n$.
\item\label{i:full:d}
If one of the sets $\operatornamewithlimits{conv}\operatorname{dom} f$ or $\operatornamewithlimits{conv}\operatorname{dom} g$ is $\operatorname{\mathbb{R}}^n$,
then $\ensuremath{\varphi^{\alpha}_{\mu}}$ is locally Lipschitz on $\operatorname{\mathbb{R}}^n$.
\item\label{i:u:proximable} When $f, g$ are both $\mu$-proximal, $\ensuremath{\varphi^{\alpha}_{\mu}}$ is the proximal
average for convex functions.
\end{enumerate}
\end{prop}
\begin{proof}
One always has
$$\ensuremath{\varphi^{\alpha}_{\mu}}=\bigg(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\bigg)-\frac{1}{2\mu}\|\cdot\|^2$$
where $\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2$ is convex
by Theorem~\ref{t:prox}\ref{i:phi:mu}.
\noindent\ref{i:regular}:
Use \cite[Example 11.30]{rockwets} and \cite[Exercise 13.35]{rockwets}
to conclude that $\ensuremath{\varphi^{\alpha}_{\mu}}$ is prox-regular. \cite[Example 10.24(g)]{rockwets}
shows that $\ensuremath{\varphi^{\alpha}_{\mu}}$ is strongly amenable.
Also, being a sum of a convex function and a $\mathcal{C}^2$ function, $\ensuremath{\varphi^{\alpha}_{\mu}}$ is Clarke regular.
\noindent\ref{i:full:d}:
By Theorem~\ref{t:prox}\ref{i:dom:convhull},
$\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatorname{\mathbb{R}}^n$, then $(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2)$ is
a finite-valued convex function on $\operatorname{\mathbb{R}}^n$, so it is
locally Lipschitz, hence $\ensuremath{\varphi^{\alpha}_{\mu}}$.
\noindent\ref{i:u:proximable}: Since both $f+\frac{1}{2\mu}\|\cdot\|^2$ and
$g+\frac{1}{2\mu}\|\cdot\|^2$
are convex, the result follows from
Theorem~\ref{t:prox}\ref{i:epi:sum}
and
\cite[Definition 4.1]{proxbas}.
\end{proof}
\begin{cor}
Let
$0<\mu<\operatorname{\bar{\lambda}}$
and let $\ensuremath{\varphi^{\alpha}_{\mu}}$ be defined as in \eqref{e:prox:def}.
Then
$$-\partial_L\left[-\bigg(\mu\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2}\|\cdot\|^2\bigg)^*\right]\subseteq
\alpha \operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g.$$
\end{cor}
\begin{proof}
By Theorem~\ref{t:prox}\ref{i:env:conhull},
$e_{\mu}(\ensuremath{\varphi^{\alpha}_{\mu}})=\alpha e_{\mu}f+(1-\alpha)e_{\mu} g.$ Since both $e_{\mu}f, e_{\mu}g$ are locally
Lipschitz, the sum rule for $\partial_{L}$ \cite[Corollary 10.9]{rockwets} gives
\begin{align*}
\partial_{L} e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}(x) & \subseteq \alpha\partial_{L} e_{\mu}f(x)
+(1-\alpha)\partial_{L} e_{\mu}g(x)\\
&\subseteq \alpha \frac{x-\operatorname{Prox}_{\mu}f(x)}{\mu}+(1-\alpha)\frac{x-\operatorname{Prox}_{\mu}g(x)}{\mu}\\
&=\frac{x}{\mu}-\frac{\alpha \operatorname{Prox}_{\mu}f(x)+(1-\alpha)\operatorname{Prox}_{\mu}g(x)}{\mu},
\end{align*}
from which
$$\partial_{L}\left(e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}-\frac{1}{2\mu}\|x\|^2\right)\subseteq -\frac{\alpha \operatorname{Prox}_{\mu}f(x)+(1-\alpha)\operatorname{Prox}_{\mu}g(x)}{\mu}.$$
As
$$e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}(x)-\frac{1}{2}\|x\|^2=-\left(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\right)^{*}\left(\frac{x}{\mu}\right)
=-\frac{\left(\mu\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2}\|\cdot\|^2\right)^*(x)}{\mu},$$
we have
$$-\partial_L\left (-\left(\mu\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2}\|\cdot\|^2\right)^{*}\right)(x)\subseteq\alpha \operatorname{Prox}_{\mu}f(x)+(1-\alpha)\operatorname{Prox}_{\mu}g(x).$$
\end{proof}
A natural question to ask is whether $\alpha \operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g$ is still a
proximal mapping. Although this is not clear in general, we have the following.
\begin{thm}[the proximal mapping of the proximal average]\label{prop:convcomb}
Let $0<\mu<\operatorname{\bar{\lambda}}$ and let $\ensuremath{\varphi^{\alpha}_{\mu}}$ be defined as in \eqref{e:prox:def}.
Then
\begin{equation}\label{e:prox:conv}
\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f+(1-\alpha)\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}g.
\end{equation}
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:u:prox} When both $f$ and $g$ are $\mu$-proximal, one has
$$\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g.$$
\item\label{i:r:prox} Suppose that on an open subset $U\subset\operatorname{\mathbb{R}}^n$
both $\operatorname{Prox}_{\mu}f, \operatorname{Prox}_{\mu}g$ are
single-valued (e.g., when
$e_{\mu}f$ and $e_{\mu}g$ are
continuously differentiable).
Then $\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}$ is single-valued, and
$$\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g \text{ on $U$.}$$
\item\label{i:r:prox2} Suppose that on an open subset $U\subset\operatorname{\mathbb{R}}^n$
both $\operatorname{Prox}_{\mu}f, \operatorname{Prox}_{\mu}g$ are
single-valued and Lipschitz continuous (e.g., when $f$ and $g$ are prox-regular).
Then $\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}$ is single-valued and Lipschitz continuous, and
$$\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g \text{ on $U$.}$$
\end{enumerate}
\end{thm}
\begin{proof}
By Theorem~\ref{t:prox},
$$-e_{\mu}(\varphi^\alpha_{\mu})=-\alpha e_{\mu}f-(1-\alpha)e_{\mu}g.$$
Since both $-e_{\mu}f, -e_{\mu}g$ are Clarke regular, the sum rule \cite[Corollary 10.9]{rockwets}
gives
$$\partial_L(-e_{\mu}(\varphi^\alpha_{\mu}))=\alpha\partial_L (- e_{\mu}f)
+(1-\alpha)\partial_L(-e_{\mu}g).$$
Apply \cite[Example 10.32]{rockwets} to get
$$\frac{\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}(x)-x}{\mu}=\alpha \frac{\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f(x)-x}{\mu}+
(1-\alpha)\frac{\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}g(x)-x}{\mu}$$
from which
$$
\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}f+(1-\alpha)\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}g.
$$
Since $\ensuremath{\varphi^{\alpha}_{\mu}}$ is $\mu$-proximal, $\operatornamewithlimits{conv}\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}$,
therefore, \eqref{e:prox:conv} follows.
\noindent\ref{i:u:prox}: Since $f,g$ are $\mu$-proximal,
$\operatorname{Prox}_{\mu}f$ and $\operatorname{Prox}_{\mu}g$ are convex-valued
by Proposition~\ref{p:resolventf}.
\noindent\ref{i:r:prox}: When $e_{\mu}f$ and $e_{\mu}g$ are continuously differentiable,
both $\operatorname{Prox}_{\mu}f, \operatorname{Prox}_{\mu}g$ are
single-valued on $U$ by \cite[Proposition 5.1]{proxhilbert}.
\noindent\ref{i:r:prox2}: When $f$ and $g$ are prox-regular on $U$,
both $\operatorname{Prox}_{\mu}f, \operatorname{Prox}_{\mu}g$ are
single-valued and Lipschitz continuous on $U$ by \cite[Proposition 5.3]{proxhilbert}
or \cite[Proposition 13.37]{rockwets}.
\end{proof}
\begin{cor}
Let $0<\mu<\operatorname{\bar{\lambda}}$ and
let $\ensuremath{\varphi^{\alpha}_{\mu}}$ be defined as in \eqref{e:prox:def}.
Then
$$
\operatorname{Prox}_{\mu}\varphi^\alpha_{\mu}
=\alpha\operatorname{Prox}_{\mu}(h_{\mu}f)+(1-\alpha)\operatorname{Prox}_{\mu}(h_{\mu}g).
$$
\end{cor}
\begin{proof}
Combine Theorem~\ref{prop:convcomb}
and Lemma~\ref{l:e:h}.
\end{proof}
\begin{cor} Let $\mu>0$. The following set of proximal mappings
$$\{\operatorname{Prox}_{\mu}f|\ \text{$f$ is $\mu$-proximal and $\mu<\lambda_{f}$}\}$$
is a convex set. Moreover, for every $\mu$-proximal function,
$\operatorname{Prox}_{\mu}f=(\operatorname{Id}+\mu\partial_{L} f)^{-1}$.
\end{cor}
\begin{proof}
Apply Theorem~\ref{prop:convcomb}\ref{i:u:prox},
Theorem~\ref{t:prox}\ref{i:lowers}\&\ref{i:phi:mu} and Proposition~\ref{p:resolventf}.
\end{proof}
\section{Relationships to the arithmetic average and epi-average}\label{s:rela}
\begin{df}[epi-convergence and epi-topology]
\emph{(See \cite[Chapter~6]{rockwets}.)}
Let $f$ and $(f_k)_\ensuremath{{k \in \N}}$ be functions from $\operatorname{\mathbb{R}}^n$ to $\ensuremath{\,\left]-\infty,+\infty\right]}$. Then
$(f_k)_{k\in\operatorname{\mathbb{N}}}$ \emph{epi-converges} to $f$, in symbols $f_k\stackrel{\operatorname{e}}{\rightarrow} f$,
if for every $x\in \operatorname{\mathbb{R}}^n$ the following hold: \hfill
\begin{enumerate}[label=\rm(\alph*)]
\item $\big(\forall\,(x_k)_{\ensuremath{{k \in \N}}}\big)$ $x_k\to x \Rightarrow
f(x)\leq\liminf f_k(x_k)$;
\item $\big(\exists (y_k)_\ensuremath{{k \in \N}}\big)$ $y_k\to x$
and $\limsup f_k(y_k) \leq f(x)$.
\end{enumerate}
We write $\operatornamewithlimits{e-lim}_{k\rightarrow\infty}f_{k}=f$ to say that $f_k$ epi-converges to $f$.
The \emph{epi-topology} is the topology induced by epi-convergence.
\end{df}
\begin{rem} The threshold $\operatorname{\bar{\lambda}}=+\infty$ whenever both $f,g$ are bounded
from below by an affine function.\end{rem}
\begin{thm}\label{t:go:infinity} Let $0<\mu<\operatorname{\bar{\lambda}}$.
One has the following.
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:mono:phi} For every fixed $x\in \operatorname{\mathbb{R}}^n$,
the function $\mu\mapsto\ensuremath{\varphi^{\alpha}_{\mu}}(x)$ is monotonically decreasing and
left-continuous on $]0,\operatorname{\bar{\lambda}}]$.
\item\label{i:inf:phi1}
The pointwise limit
$\lim_{\mu\uparrow \operatorname{\bar{\lambda}}}\ensuremath{\varphi^{\alpha}_{\mu}}=\inf_{\operatorname{\bar{\lambda}}>\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}=$
$$
\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\operatorname{\bar{\lambda}}}\|\cdot\|^2\bigg)\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\operatorname{\bar{\lambda}}}\|\cdot\|^2\bigg)\bigg(\frac{\cdot}{1-\alpha}\bigg)\right](x)
-\frac{1}{2\operatorname{\bar{\lambda}}}\|x\|^2.
$$
\item\label{i:inf:phi2} When $\operatorname{\bar{\lambda}}=\infty$,
the pointwise limit
\begin{equation}\label{e:pointwise}
\lim_{\mu\uparrow \infty}\ensuremath{\varphi^{\alpha}_{\mu}}=\inf_{\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}=
\alpha\operatornamewithlimits{conv} f\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
g\bigg(\frac{\cdot}{1-\alpha}\bigg), \text{ and }
\end{equation}
the epigraphical limit
\begin{equation}\label{i:epi:limit}
\operatornamewithlimits{e-lim}_{\mu\uparrow\infty}\ensuremath{\varphi^{\alpha}_{\mu}}=
\operatorname{cl}\left[\alpha\operatornamewithlimits{conv} f\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
g\bigg(\frac{\cdot}{1-\alpha}\bigg)\right].
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
\ref{i:mono:phi}:
We have $\ensuremath{\varphi^{\alpha}_{\mu}}(x)=$
\footnotesize\begin{align}
&
\inf_{u+v=x}\left(\alpha\inf_{\sum_{i}\alpha_{i}x_{i}=u/\alpha\atop{\sum_{i}\alpha_{i}=1,
\alpha_{i}\geq 0}}\left(\sum_{i}\alpha_{i}f(x_{i})
+\alpha_{i}\frac{1}{2\mu}\|x_{i}\|^2\right)+(1-\alpha)\inf_{\sum_{j}\beta_{j}y_{j}=v/(1-\alpha)
\atop{\sum_{j}\beta_{j}=1,
\beta_{j}\geq 0}}\left(\sum_{j}\beta_{j}g(y_{j})
+\beta_{j}\frac{1}{2\mu}\|y_{j}\|^2\right)\right)\nonumber\\
&\quad -\frac{1}{2\mu}\|x\|^2\nonumber\\
&=
\inf_{{\alpha\sum_i\alpha_{i}x_{i}+
(1-\alpha)\sum_{j}\beta_{j}y_{j}=x}\atop{\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1},\alpha_{i}\geq 0, \beta_{j}\geq 0}\bigg(\alpha\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\sum_{j}\beta_{j}g(y_{j})+\nonumber\\
&\quad \frac{1}{2\mu}\underbrace{\left(\alpha\sum_{i}\alpha_{i}\|x_{i}\|^2
+(1-\alpha)\sum_{j}\beta_{j}\|y_{j}\|^2-\bigg\|\alpha\sum_{i}\alpha_{i}x_{i}+(1-\alpha)\sum_{j}
\beta_{j}y_{j}\bigg\|^2\right)}\bigg).\nonumber
\end{align}\normalsize
The underbraced part is nonnegative because $\|\cdot\|^2$ is convex, $\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1$. It follows that
$\mu\mapsto\ensuremath{\varphi^{\alpha}_{\mu}}$ is a monotonically decreasing function on $]0,+\infty[$.
Let $\operatorname{\bar{\mu}}\in ]0,\operatorname{\bar{\lambda}}]$. Then $\lim_{\mu\uparrow\operatorname{\bar{\mu}}}\ensuremath{\varphi^{\alpha}_{\mu}}
=\inf_{\operatorname{\bar{\mu}}>\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}=$
\begin{align}
&\inf_{\operatorname{\bar{\mu}}>\mu>0}
\inf_{{\alpha\sum_i\alpha_{i}x_{i}+
(1-\alpha)\sum_{j}\beta_{j}y_{j}=x}\atop{\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1},\alpha_{i}\geq 0, \beta_{j}\geq 0}\bigg(\alpha\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\sum_{j}\beta_{j}g(y_{j})+\nonumber\\
&\quad \frac{1}{2\mu}\left(\alpha\sum_{i}\alpha_{i}\|x_{i}\|^2
+(1-\alpha)\sum_{j}\beta_{j}\|y_{j}\|^2-\bigg\|\alpha\sum_{i}\alpha_{i}x_{i}+(1-\alpha)\sum_{j}
\beta_{j}y_{j}\bigg\|^2\right)\bigg)\label{e:k1}\\
&=
\inf_{{\alpha\sum_i\alpha_{i}x_{i}+
(1-\alpha)\sum_{j}\beta_{j}y_{j}=x}\atop{\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1},\alpha_{i}\geq 0, \beta_{j}\geq 0}\inf_{\operatorname{\bar{\mu}}>\mu>0}\bigg(\alpha\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\sum_{j}\beta_{j}g(y_{j})+\nonumber\\
&\quad \frac{1}{2\mu}\left(\alpha\sum_{i}\alpha_{i}\|x_{i}\|^2
+(1-\alpha)\sum_{j}\beta_{j}\|y_{j}\|^2-\bigg\|\alpha\sum_{i}\alpha_{i}x_{i}+(1-\alpha)\sum_{j}
\beta_{j}y_{j}\bigg\|^2\right)\bigg)\\
&=
\inf_{{\alpha\sum_i\alpha_{i}x_{i}+
(1-\alpha)\sum_{j}\beta_{j}y_{j}=x}\atop{\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1},\alpha_{i}\geq 0, \beta_{j}\geq 0}
\bigg(\alpha\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\sum_{j}\beta_{j}g(y_{j})+\nonumber\\
&\quad \frac{1}{2\operatorname{\bar{\mu}}}\left(\alpha\sum_{i}\alpha_{i}\|x_{i}\|^2
+(1-\alpha)\sum_{j}\beta_{j}\|y_{j}\|^2-\bigg\|\alpha\sum_{i}\alpha_{i}x_{i}+(1-\alpha)\sum_{j}
\beta_{j}y_{j}\bigg\|^2\right)\bigg)\label{e:k2}\\
&=
\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\operatorname{\bar{\mu}}}
\|\cdot\|^2\bigg)\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\operatorname{\bar{\mu}}}\|\cdot\|^2\bigg)\bigg(\frac{\cdot}{1-\alpha}\bigg)\right](x)
-\frac{1}{2\operatorname{\bar{\mu}}}\|x\|^2\nonumber.
\end{align}
\noindent\ref{i:inf:phi1}: This follows from \ref{i:mono:phi}.
\noindent\ref{i:inf:phi2}:
By \ref{i:mono:phi}, we have $\lim_{\mu\rightarrow\infty}\ensuremath{\varphi^{\alpha}_{\mu}}
=\inf_{\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}$. Using similar arguments as \eqref{e:k1}--\eqref{e:k2},
we obtain $\inf_{\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}=$
\begin{align*}
&\inf_{{\alpha\sum_i\alpha_{i}x_{i}+
(1-\alpha)\sum_{j}\beta_{j}y_{j}=x}\atop{\sum_{i}\alpha_{i}=1,
\sum_{j}\beta_{j}=1},\alpha_{i}\geq 0, \beta_{j}\geq 0}\bigg(\alpha\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\sum_{j}\beta_{j}g(y_{j})\bigg)\\
&=\inf_{{u+v=x}}\bigg(\alpha\inf_{\sum_i\alpha_{i}x_{i}=u/\alpha
\atop{\sum_{i}\alpha_{i}=1,\alpha_{i}\geq 0}}\sum_{i}\alpha_{i}f(x_{i})
+(1-\alpha)\inf_{\sum_{j}\beta_{j}y_{j}=v/(1-\alpha)\atop{\sum_{j}\beta_{j}=1}, \beta_{j}\geq 0}\sum_{j}\beta_{j}g(y_{j})\bigg)\\
&=\inf_{{u+v=x}}\bigg(\alpha(\operatornamewithlimits{conv} f)(u/\alpha)
+(1-\alpha)(\operatornamewithlimits{conv} g)(v/(1-\alpha))\bigg),
\end{align*}
as required. To get \eqref{i:epi:limit}, we combine \eqref{e:pointwise} and
\cite[Proposition 7.4(c)]{rockwets}.
\end{proof}
In order to study the limit behavior when $\mu\downarrow 0$, a lemma helps.
We omit its simple proof.
\begin{lem}\label{l:concave:e}
The Moreau envelope function respects the inequality$$e_{\mu}(\alpha f_{1}+(1-\alpha)f_{2})\geq \alpha e_{\mu}f_{1}+(1-\alpha)e_{\mu}f_{2}.$$
\end{lem}
\begin{thm}\label{t:u:0}
Let $0<\mu<\operatorname{\bar{\lambda}}$. One has
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:three:c}
\begin{equation}\label{e:three:c}
\alpha e_{\mu}f+(1-\alpha) e_{\mu}g \leq\ensuremath{\varphi^{\alpha}_{\mu}}\leq \alpha h_{\mu}f+(1-\alpha)h_{\mu}g
\leq\alpha f+(1-\alpha) g \text{ and }
\end{equation}
\item when $\mu\downarrow 0$, the pointwise limit and epi-graphical limit agree with
\begin{equation}\label{e:lim:0}
\lim_{\mu\downarrow 0}\ensuremath{\varphi^{\alpha}_{\mu}}=\sup_{\mu>0}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha f+(1-\alpha)g.
\end{equation}
Furthermore, the convergence in \eqref{e:lim:0}
is uniform on compact subsets of $\operatorname{\mathbb{R}}^n$ when $f,g$ are continuous.
\end{enumerate}
\end{thm}
\begin{proof}
Apply Lemma~\ref{l:concave:e} with $f_{1}=-e_{\mu}f, f_{2}=-e_{\mu}g$ to obtain
$e_{\mu}(\alpha(-e_{\mu}f)+(1-\alpha)(-e_{\mu}g))\geq \alpha e_{\mu}(-e_{\mu}f)+(1-\alpha)
e_{\mu}(-e_{\mu}g).$ Then
\begin{equation}\label{e:1}
\ensuremath{\varphi^{\alpha}_{\mu}}\leq \alpha (-e_{\mu}(-e_{\mu}f))+(1-\alpha)(-e_{\mu}(-e_{\mu}g))=
\alpha h_{\mu}f+(1-\alpha)h_{\mu}g.
\end{equation}
On the other hand,
$e_{\mu}(\alpha(-e_{\mu}f)+(1-\alpha)(-e_{\mu}g))\leq \alpha(-e_{\mu}f)+(1-\alpha)(-e_{\mu}g)$
so
\begin{equation}\label{e:2}
\ensuremath{\varphi^{\alpha}_{\mu}}\geq \alpha e_{\mu}f +(1-\alpha) e_{\mu}g.
\end{equation}
Combining \eqref{e:1} and \eqref{e:2} gives
$$\alpha e_{\mu}f +(1-\alpha) e_{\mu}g \leq\ensuremath{\varphi^{\alpha}_{\mu}}\leq \alpha h_{\mu}f+(1-\alpha)h_{\mu}g
\leq \alpha f+(1-\alpha) g,$$
which is \eqref{e:three:c}. Equation \eqref{e:lim:0} follows from \eqref{e:three:c} by sending $\mu\downarrow 0$.
The pointwise and epigraphical limits agree because of
\cite[Proposition 7.4(d)]{rockwets}.
Now assume that $f,g$ are continuous.
Since both $e_{\mu}f$ and $f$ are continuous, and $e_{\mu}f\uparrow f$.
Dini's theorem says that $e_{\mu}f\uparrow f$ uniformly on compact subsets
of $\operatorname{\mathbb{R}}^n$. The same can be said about
$e_{\mu}g\uparrow g$.
Hence, the convergence in \eqref{e:lim:0} is uniform
on compact subsets of $\operatorname{\mathbb{R}}^n$ by \eqref{e:three:c}.
\end{proof}
To study the epi-continuity of proximal average, we recall the following two standard notions.
\begin{df} A sequence of functions $(f_{k})_{k\in\operatorname{\mathbb{N}}}$ is eventually prox-bounded
if there exists $\lambda>0$ such that $\liminf_{k\rightarrow\infty}e_{\lambda}f_{k}(x)>-\infty$
for some $x$. The supremum of all such $\lambda$ is then
the threshold of eventual prox-boundedness of the sequence.
\end{df}
\begin{df}
A sequence of functions $(f_{k})_{k\in \operatorname{\mathbb{N}}}$ converges continuously to $f$ if
$f_{k}(x_{k})\rightarrow f(x)$ whenever $x_{k}\rightarrow x$.
\end{df}
The following key result is implicit in the proof of \cite[Theorem 7.37]{rockwets}.
We provide its proof for completeness.
Define $\operatorname{\mathcal{N}_{\infty}}=\{N\subset\operatorname{\mathbb{N}}|\ \operatorname{\mathbb{N}}\setminus N \text{ is finite}\}.$
\begin{lem}\label{l:env:cont}
Let $(f_{k})_{k\in\operatorname{\mathbb{N}}}$ and $f$ be proper, lsc functions on $\operatorname{\mathbb{R}}^n$.
Suppose that $(f_{k})_{k\in\operatorname{\mathbb{N}}}$ is eventually prox-bounded,
$\bar{\lambda}$ is the threshhold of eventual prox-boundedness,
and $f_{k}\stackrel{\operatorname{e}}{\rightarrow} f$. Suppose also that
$\mu_{k}, \mu\in ]0,\bar{\lambda}[$, and $\mu_{k}\rightarrow\mu$.
Then $f$ is prox-bounded with threshold $\lambda_{f}\geq \bar{\lambda}$, and
$e_{\mu_{k}}f_{k}$ converges continuously to $e_{\mu}f$.
In particular,
$e_{\mu_{k}}f_{k}\stackrel{\operatorname{e}}{\rightarrow} e_{\mu}f$, and $e_{\mu_{k}}f_{k}\stackrel{\operatorname{p}}{\rightarrow} e_{\mu}f$.
\end{lem}
\begin{proof} Let $\varepsilon \in ]0,\operatorname{\bar{\lambda}}[$. The eventual prox-boundness of $(f_{k})_{k\in\operatorname{\mathbb{N}}}$
means that
there exist $b\in\operatorname{\mathbb{R}}^n$,
$\beta\in\operatorname{\mathbb{R}}$ and $N\in\operatorname{\mathcal{N}_{\infty}}$ such that
$$(\forall k\in N)(\forall w\in \operatorname{\mathbb{R}}^n)\ f_{k}(w)
\geq \beta-\frac{1}{2\varepsilon}\|b-w\|^2.$$
Let $\mu\in ]0,\varepsilon[$.
Consider any $x\in\operatorname{\mathbb{R}}^n$ and any sequence $x_{k}\rightarrow x$
in $\operatorname{\mathbb{R}}^n$, any sequence $\mu_{k}\rightarrow\mu$ in $(0,\operatorname{\bar{\lambda}})$.
Since $f_{k}\stackrel{\operatorname{e}}{\rightarrow} f$, the functions
$f_{k}+(1/2\mu_{k})\|\cdot-x_{k}\|^2$ epi-converge to $f+(1/2\mu)\|\cdot-x\|^2$.
Take $\delta\in ]\mu,\varepsilon[$. Because
$\mu_{k}\rightarrow\mu$, there exists $N'\subseteq N$, $N'\in\operatorname{\mathcal{N}_{\infty}}$
such that $\mu_{k}\in (0,\delta)$ when $k\in N'$.
Then $\forall k\in N'$,
\begin{align*}
f_{k}(w)+\frac{1}{2\mu_{k}}\|x_{k}-w\|^2 &\geq \beta-\frac{1}{2\varepsilon}\|b-w\|^2+ \frac{1}{2\delta}\|x_{k}-w\|^2\\
&=\beta-\frac{1}{2\varepsilon}\|b-w\|^2+ \frac{1}{2\delta}\|(x_{k}-b)+(b-w)\|^2\\
&\geq \beta+\bigg(\frac{1}{2\delta}-\frac{1}{2\varepsilon}\bigg)
\|b-w\|^2-\frac{1}{\delta}\|x_{k}-b\|\|b-w\|.
\end{align*}
In view of $x_{k}\rightarrow x$, the sequence $(\|x_{k}-b\|)_{k\in\operatorname{\mathbb{N}}}$ is bounded,
say by $\rho>0$. We have
$$
(\forall k\in N')\ f_{k}(w)+\frac{1}{2\mu_{k}}\|x_{k}-w\|^2 \geq h(w):=
\beta+\bigg(\frac{1}{2\delta}-\frac{1}{2\varepsilon}\bigg)\|b-w\|^2-\frac{\rho}{\delta}
\|b-w\|.
$$
The function $h$ is level-bounded because $\delta<\varepsilon$. Hence, by
\cite[Theorem 7.33]{rockwets},
$$\lim_{k\rightarrow\infty}\inf_{w}\bigg(f_{k}(w)+\frac{1}{2\mu_{k}}
\|x_{k}-w\|^2\bigg)
=\inf_{w}\bigg(f(w)+\frac{1}{2\mu}\|x-w\|^2\bigg),$$
i.e., $e_{\mu_{k}}f_{k}(x_{k})\rightarrow e_{\mu}f(x)$.
Also, $e_{\mu}f(x)$ is finite, so $\lambda_{f}\geq \mu$.
Since $\varepsilon\in ]0,\operatorname{\bar{\lambda}}[$ and $\mu\in ]0,\varepsilon[$ were
arbitrary, the result holds whenever $\mu\in ]0,\operatorname{\bar{\lambda}}[$. This in turn implies
$\lambda_{f}\geq\operatorname{\bar{\lambda}}$.
\end{proof}
For the convenience of analyzing the full epi-continuity,
below we write the proximal average $\ensuremath{\varphi^{\alpha}_{\mu}}$ explicitly in the form
$\ensuremath{\varphi_{f,g,\alpha,\mu}}$.
\begin{thm}[full epi-continuity of proximal average]
Let the sequences of functions $(f_{k})_{k\in \operatorname{\mathbb{N}}}$, $(g_{k})_{k\in\operatorname{\mathbb{N}}}$ on $\operatorname{\mathbb{R}}^n$
be eventually prox-bounded
with threshold of eventual prox-boundedness $\bar{\lambda}>0$.
Let $(\mu_{k})_{k\in\operatorname{\mathbb{N}}}$ be a sequence and $\mu$ in
$]0,\bar{\lambda}[$ and let $(\alpha_{k})_{k\in\operatorname{\mathbb{N}}}$ be a sequence and $\alpha$
in
$[0,1]$. Suppose that $f_{k}\stackrel{\operatorname{e}}{\rightarrow} f$, $g_{k}\stackrel{\operatorname{e}}{\rightarrow} g$,
$\mu_{k}\rightarrow\mu$, and $\alpha_{k}\rightarrow\alpha$.
Then $\ensuremath{\varphi_{f_{k},g_{k},\alpha_{k},\mu_{k}}}\stackrel{\operatorname{e}}{\rightarrow}\ensuremath{\varphi_{f,g,\alpha,\mu}}$.
\end{thm}
\begin{proof} Consider any $x\in\operatorname{\mathbb{R}}^n$ and any sequence $x_{k}\rightarrow x$.
By \cite[Example 11.26]{rockwets},
$$e_{\mu_{k}}f_{k}(\mu_{k}x_{k})=\frac{\mu_{k}\|x_{k}\|^2}{2}-
\bigg(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*(x_{k}).$$
Lemma~\ref{l:env:cont} shows that
\begin{align*}
\lim_{k\rightarrow\infty}
\bigg(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*(x_{k})
& = \lim_{k\rightarrow\infty} \frac{\mu_{k}\|x_{k}\|^2}{2}-e_{\mu_{k}}f_{k}(\mu_{k}x_{k})
=\frac{\mu\|x\|^2}{2}-e_{\mu}f(\mu x)\\
&=\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*(x).
\end{align*}
Therefore, the functions $\left(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\right)^*$ converge
continuously to $\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)^*$.
It follows that
$$\alpha_{k}\bigg(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*+
(1-\alpha_{k})\bigg(g_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*$$
converges continuously
to
$$\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*,$$
so epi-converges.
Then by Wijsman's theorem \cite[Theorem 11.34]{rockwets},
$$\bigg[\alpha_{k}\bigg(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*+
(1-\alpha_{k})\bigg(g_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*\bigg]^*$$
epi-converges to
$$\bigg[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\bigg]^*.$$
Since $(\mu, x)\mapsto \frac{1}{2\mu}\|x\|^2$ is continuous
on $]0,+\infty[\times\operatorname{\mathbb{R}}^n$, we have that
$$\ensuremath{\varphi_{f_{k},g_{k},\alpha_{k},\mu_{k}}}
=\bigg[\alpha_{k}\bigg(f_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*+
(1-\alpha_{k})\bigg(g_{k}+\frac{1}{2\mu_{k}}\|\cdot\|^2\bigg)^*\bigg]^*
-\frac{1}{2\mu_{k}}\|\cdot\|^2$$
epi-converges to
$$\ensuremath{\varphi_{f,g,\alpha,\mu}}
=\bigg[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\bigg]^*
-\frac{1}{2\mu}\|\cdot\|^2.$$
\end{proof}
\begin{cor}[epi-continuity of the proximal average]\label{t:alpha:epi}
Let $0<\mu<\operatorname{\bar{\lambda}}$.
Then the function
$\alpha\mapsto \ensuremath{\varphi^{\alpha}_{\mu}}$ is continuous with respect to the epi-topology. That is,
$\forall (\alpha_{k})_{k\in\operatorname{\mathbb{N}}}$ and $\alpha$ in $[0,1]$,
$$\alpha_{k}\rightarrow\alpha \quad \Rightarrow \quad \ensuremath{\varphi^{\alpha_{k}}_{\mu}}\stackrel{\operatorname{e}}{\rightarrow}\ensuremath{\varphi^{\alpha}_{\mu}}.$$
In particular, $\ensuremath{\varphi^{\alpha}_{\mu}}\stackrel{\operatorname{e}}{\rightarrow} h_{\mu}g$ when $\alpha\downarrow 0$, and
$\ensuremath{\varphi^{\alpha}_{\mu}}\stackrel{\operatorname{e}}{\rightarrow} h_{\mu}f$ when $\alpha\uparrow 1$.
\end{cor}
\section{Optimal value and minimizers of the proximal average}\label{s:opti}
\subsection{Relationship of infimum and minimizers among
$\ensuremath{\varphi^{\alpha}_{\mu}}$, $f$ and $g$.}
\begin{prop}\label{p:minimizer} Let $0<\mu<\operatorname{\bar{\lambda}}$. One has
\begin{enumerate}[label=\rm(\alph*)]
\item \label{e:env:conv}
\begin{align*}
\inf\ensuremath{\varphi^{\alpha}_{\mu}} &= \inf[\alpha e_{\mu}f+(1-\alpha)e_{\mu}g], \text{ \emph{and} }\\
\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}} &=\operatornamewithlimits{argmin} [\alpha e_{\mu}f+(1-\alpha)e_{\mu}g];
\end{align*}
\item \label{e:hull:arith}
\begin{align*}
\alpha\inf f+(1-\alpha)\inf g& \leq \inf\ensuremath{\varphi^{\alpha}_{\mu}}
\leq\inf[\alpha h_{\mu}f+(1-\alpha)h_{\mu}g]
\leq \inf[\alpha f+(1-\alpha) g].
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof}
For \ref{e:env:conv}, apply Theorem~\ref{t:prox}\ref{i:env:conhull}
and $\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatornamewithlimits{argmin} e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}$.
For \ref{e:hull:arith}, apply Theorem~\ref{t:u:0}\ref{i:three:c}
and $\inf e_{\mu}f=\inf f$, and $\inf e_{\mu}g=\inf g$.
\end{proof}
\begin{thm} Suppose that $\operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g\neq\varnothing$ and
$\alpha\in ]0,1[$.
Then the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:arithmin}
\begin{equation}\label{e:arith}
\min(\alpha f+(1-\alpha)g)=\alpha\min f+(1-\alpha)\min g, \text{ \emph{and} }
\end{equation}
\begin{equation}\label{e:arith:minimizer}
\operatornamewithlimits{argmin}(\alpha f+(1-\alpha)g)=\operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g;
\end{equation}
\item \label{i:proxmin}
\begin{equation}\label{e:minvalue}
\min\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha\min f+(1-\alpha)\min g, \text{ \emph{and} }
\end{equation}
\begin{equation*}
\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g.
\end{equation*}
\end{enumerate}
\end{thm}
\begin{proof} Pick $x\in \operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g$. We have
\begin{equation}\label{e:minvalue:two}
\inf[\alpha f+(1-\alpha)g]=\alpha f(x)+(1-\alpha) g(x)=\alpha\min f+(1-\alpha)\min g.
\end{equation}
\ref{i:arithmin}: Equation \eqref{e:minvalue:two} gives \eqref{e:arith} and
\begin{equation*}\label{e:setinclus}
(\operatornamewithlimits{argmin} f\cap\operatornamewithlimits{argmin} g)\subseteq \operatornamewithlimits{argmin} (\alpha f+(1-\alpha)g).
\end{equation*}
To see the converse inclusion of \eqref{e:minvalue},
let $x\in\operatornamewithlimits{argmin}(\alpha f+(1-\alpha)g)$. Then \eqref{e:arith} gives
$$\alpha \min f+(1-\alpha)\min g=\min(\alpha f+(1-\alpha)g)=\alpha f(x)+(1-\alpha)g(x),$$
from which
$$\alpha (\min f-f(x))+(1-\alpha)(\min g-g(x))=0.$$
Since $\min f\leq f(x), \min g\leq g(x)$, we obtain
$\min f= f(x), \min g= g(x)$, so $x\in\operatornamewithlimits{argmin} f\cap\operatornamewithlimits{argmin} g$. Thus,
$\operatornamewithlimits{argmin} (\alpha f+(1-\alpha)g)\subseteq (\operatornamewithlimits{argmin} f\cap\operatornamewithlimits{argmin} g)$. Hence,
\eqref{e:arith:minimizer} holds.
\noindent\ref{i:proxmin}: Equation \eqref{e:minvalue} follows from Proposition~\ref{p:minimizer} and Theorem~\ref{t:u:0}\ref{i:three:c}. This also gives
$$(\operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g)\subseteq \operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}.$$
To show $(\operatornamewithlimits{argmin} f\cap \operatornamewithlimits{argmin} g)\supseteq \operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}$, take any $x\in \operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}$.
By \eqref{e:minvalue} and Theorem~\ref{t:u:0}\ref{i:three:c}, we have
$$\alpha\min f+(1-\alpha)\min g=\ensuremath{\varphi^{\alpha}_{\mu}}(x)\geq \alpha e_{\mu}f(x)+(1-\alpha)e_{\mu}g(x),$$
from which
$$\alpha(e_{\mu}f(x)-\min f)+(1-\alpha)(e_{\mu}g(x)-\min g)\leq 0.$$
Since $\min f=\min e_{\mu}f$ and $\min g=\min e_{\mu}g$, it follows that
$e_{\mu}f(x)=\min e_{\mu}f$ and $e_{\mu}g(x)=\min e_{\mu}g$, so
$x\in(\operatornamewithlimits{argmin} e_{\mu}f\cap\operatornamewithlimits{argmin} e_{\mu}g)=(\operatornamewithlimits{argmin} f\cap\operatornamewithlimits{argmin} g)$
because of $\operatornamewithlimits{argmin} e_{\mu}f=\operatornamewithlimits{argmin} f$ and $\operatornamewithlimits{argmin} e_{\mu}g=\operatornamewithlimits{argmin} g$.
\end{proof}
To explore further optimization properties of $\ensuremath{\varphi^{\alpha}_{\mu}}$, we need the following three
auxiliary results.
\begin{lem}\label{l:box} Suppose that $f_{1}, f_{2}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ are
proper and lsc, and that
$f_{1}\Box f_{2}$ is exact. Then
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:box:inf}
\begin{equation}\label{e:epis:inf}
\inf (\ensuremath{f_{1}\Box f_{2}})=\inf f_{1}+\inf f_{2}, \text{ and }
\end{equation}
\item\label{i:box:min}
\begin{equation}\label{e:epis:min}
\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})=\operatornamewithlimits{argmin} f_{1}+\operatornamewithlimits{argmin} f_{2}.
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof} Equation \eqref{e:epis:inf} follows from
\begin{align*}
\inf\ensuremath{f_{1}\Box f_{2}} &=\inf_{x}\inf_{x=y+z}[f_{1}(y)+f_{2}(z)]\\
&=\inf_{y,z}[f_{1}(y)+f_{2}(z)]=\inf f_{1}+\inf f_{2}.
\end{align*}
To see \eqref{e:epis:min}, we first show
\begin{equation}\label{e:setsum}
\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})\subseteq\operatornamewithlimits{argmin} f_{1}+\operatornamewithlimits{argmin} f_{2}.
\end{equation}
If $\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})=\varnothing$, the inclusion holds trivially. Let us
assume that $\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})\neq \varnothing$ and
let $x\in\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})$. Since $\ensuremath{f_{1}\Box f_{2}}$ is exact, we have $x=y+z$ for some $y,z$ and
$\ensuremath{f_{1}\Box f_{2}}(x)=f_{1}(y)+f_{2}(z)$. In view of \eqref{e:epis:inf},
\begin{align*}
f_{1}(y)+f_{2}(z) &=\ensuremath{f_{1}\Box f_{2}}(x)=\min \ensuremath{f_{1}\Box f_{2}}=\inf f_{1}+\inf f_{2},
\end{align*}
from which
$$(f_{1}(y)-\inf f_{1})+(f_{2}(z)-\inf f_{2})=0.$$
Then $f_{1}(y)=\inf f_{1}, f_{2}(z)=\inf f_{2}$, which gives
$y\in\operatornamewithlimits{argmin} f_{1}, z\in\operatornamewithlimits{argmin} f_{2}$. Therefore,
$x\in \operatornamewithlimits{argmin} f_{1}+\operatornamewithlimits{argmin} f_{2}$. Next, we show
\begin{equation}\label{e:setsum2}
\operatornamewithlimits{argmin} (\ensuremath{f_{1}\Box f_{2}})\supseteq\operatornamewithlimits{argmin} f_{1}+\operatornamewithlimits{argmin} f_{2}.
\end{equation}
If one of $\operatornamewithlimits{argmin} f_{1}, \operatornamewithlimits{argmin} f_{2}$ is empty, the inclusion holds
trivially. Assume that
$\operatornamewithlimits{argmin} f_{1}\neq\varnothing$ and $\operatornamewithlimits{argmin} f_{2}\neq\varnothing$.
Take
$y\in\operatornamewithlimits{argmin} f_{1}, z\in\operatornamewithlimits{argmin} f_{2}$, and put $x=y+z$. The definition of
$\Box$ and \eqref{e:epis:inf} give
$$\ensuremath{f_{1}\Box f_{2}}(x)\leq f_{1}(y)+f_{2}(z)=\min f_{1}+\min f_{2}=
\inf \ensuremath{f_{1}\Box f_{2}},$$
which implies $x\in\operatornamewithlimits{argmin}(\ensuremath{f_{1}\Box f_{2}})$. Since $y\in\operatornamewithlimits{argmin} f_{1}$, $z\in\operatornamewithlimits{argmin} f_{2}$
were arbitrary,
\eqref{e:setsum2} follows. Combining \eqref{e:setsum} and \eqref{e:setsum2}
gives \eqref{e:epis:min}.
\end{proof}
\begin{lem}\label{l:epimulti}
Let $f_{1}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ be proper and lsc, and let $\beta>0$. Then
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:epimul:inf}
$$\inf\left[\beta f_{1}\left(\frac{\cdot}{\beta}\right)\right]=\beta\inf f_{1},\text{ and }
$$
\item \label{i:epimul:min}
$$\operatornamewithlimits{argmin} \left[\beta f_{1}
\left(\frac{\cdot}{\beta}\right)\right]=\beta\operatornamewithlimits{argmin} f_{1}.$$
\end{enumerate}
\end{lem}
\begin{lem}\label{l:convexhull:f} Let $f_{1}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ be proper and lsc.
Then the following
hold:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:convexh:inf}
$$\inf (\operatornamewithlimits{conv} f_1) =\inf f_{1};$$
\item\label{i:convexh:min}
if, in addition, $f_{1}$ is coercive, then
$$\operatornamewithlimits{argmin} (\operatornamewithlimits{conv} f_{1})
=\operatornamewithlimits{conv}(\operatornamewithlimits{argmin} f_{1}),$$
and $\operatornamewithlimits{argmin} (\operatornamewithlimits{conv} f_{1})\neq\varnothing$.
\end{enumerate}
\end{lem}
\begin{proof} Combine \cite[Comment 3.7(4)]{benoist} and
\cite[Corollary 3.47]{rockwets}.
\end{proof}
We are now ready for the main result of this section.
\begin{thm}\label{t:shifted}
Let $0<\mu<\operatorname{\bar{\lambda}}$, and
let $\ensuremath{\varphi^{\alpha}_{\mu}}$ be defined as in \eqref{e:prox:def}.
Then the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:shifted:inf}
\begin{align*}
&\inf\left(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\right)\\
&=\alpha\inf\left(
f+\frac{1}{2\mu}\|\cdot\|^2\right)+(1-\alpha)
\inf \left(
g+\frac{1}{2\mu}\|\cdot\|^2\right);
\end{align*}
\item\label{i:shifted:min}
\begin{align*}
&\operatornamewithlimits{argmin}\left(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\right)\\
& =\alpha\operatornamewithlimits{conv} \left[\operatornamewithlimits{argmin}\left(
f+\frac{1}{2\mu}\|\cdot\|^2\right)\right]+(1-\alpha)
\operatornamewithlimits{conv} \left[\operatornamewithlimits{argmin}\left(
g+\frac{1}{2\mu}\|\cdot\|^2\right)\right]\neq\varnothing.
\end{align*}
\end{enumerate}
\end{thm}
\begin{proof} Theorem~\ref{t:prox}\ref{i:epi:sum} gives
\begin{align*}
& \ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\\
&=\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\right]
\Box \left[(1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right],
\end{align*}
in which the inf-convolution $\Box$ is exact.
\noindent\ref{i:shifted:inf}: Using Lemma \ref{l:box}\ref{i:box:inf} and Lemma \ref{l:convexhull:f}\ref{i:convexh:inf},
we deduce
\begin{align*}
&\inf\left(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\right)\\
&=\inf \left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\right]
+\inf \left[(1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right]\\
&=\alpha\inf \left[\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right]
+(1-\alpha)\inf \left[\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right]\\
&=\alpha\inf\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)
+(1-\alpha)\inf
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg).
\end{align*}
\noindent\ref{i:shifted:min}: Note that
$f+\frac{1}{2\mu}\|\cdot\|^2$ and
$g+\frac{1}{2\mu}\|\cdot\|^2$ are coercive
because of $0<\mu<\operatorname{\bar{\lambda}}$.
Using Lemma~\ref{l:box}\ref{i:box:min}-Lemma~\ref{l:convexhull:f}\ref{i:convexh:min},
we deduce
\begin{align*}
&\operatornamewithlimits{argmin}\left(\ensuremath{\varphi^{\alpha}_{\mu}}+\frac{1}{2\mu}\|\cdot\|^2\right)\\
&=\operatornamewithlimits{argmin} \left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\right]
+\operatornamewithlimits{argmin} \left[(1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right]\\
&=\alpha\operatornamewithlimits{argmin} \left[\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right]
+(1-\alpha)\operatornamewithlimits{argmin} \left[\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right]\\
&=\alpha\operatornamewithlimits{conv}\left[\operatornamewithlimits{argmin} \bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right]
+(1-\alpha)\operatornamewithlimits{conv}\left[\operatornamewithlimits{argmin}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\right].
\end{align*}
Finally, these three sets of minimizers are nonempty by
Lemma~\ref{l:convexhull:f}\ref{i:convexh:min}.
\end{proof}
\begin{rem} \emph{Theorem~\ref{t:shifted}\ref{i:shifted:min}} is just a rewritten form of
$$\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}(0)=\alpha\operatornamewithlimits{conv}[\operatorname{Prox}_{\mu}f(0)]+(1-\alpha)\operatornamewithlimits{conv}[\operatorname{Prox}_{\mu}g(0)].$$
\end{rem}
In view of
Theorem~\ref{t:go:infinity}\ref{i:inf:phi2},
when $\operatorname{\bar{\lambda}}=\infty$, as $\mu\rightarrow\infty$ the pointwise limit is
$$\ensuremath{\varphi^{\alpha}_{\mu}}\stackrel{\operatorname{p}}{\rightarrow} \left[
\alpha\operatornamewithlimits{conv} f\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
g\bigg(\frac{\cdot}{1-\alpha}\bigg)\right],$$
and the epi-limit is
$$\ensuremath{\varphi^{\alpha}_{\mu}}\stackrel{\operatorname{e}}{\rightarrow} \operatorname{cl}\left[
\alpha\operatornamewithlimits{conv} f\bigg(\frac{\cdot}{\alpha}\bigg)\Box (1-\alpha)\operatornamewithlimits{conv}
g\bigg(\frac{\cdot}{1-\alpha}\bigg)\right].$$
We conclude this section with a result on minimization of this limit.
\begin{prop}\label{p:coercive}
Suppose that both $f$ and $g$ are coercive. Then the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:coercive:1}
$\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
g\left(\frac{\cdot}{1-\alpha}\right)$ is proper, lsc and convex;
\item\label{i:coercive:2}
$$\min \left[\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
g\left(\frac{\cdot}{1-\alpha}\right)\right]=
\alpha\min f+(1-\alpha)\min g;$$
\item\label{i:coercive:3}
\begin{align*}
& \operatornamewithlimits{argmin}\left[\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
g\left(\frac{\cdot}{1-\alpha}\right)\right]\\
& =
\alpha\operatornamewithlimits{conv}\operatornamewithlimits{argmin} f+(1-\alpha)\operatornamewithlimits{conv}\operatornamewithlimits{argmin} g\neq\varnothing.
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof} Since both $f$ and $g$ are coercive, by \cite[Corollary 3.47]{rockwets},
$\operatornamewithlimits{conv} f$ and $\operatornamewithlimits{conv} g$ are lsc, convex and coercive.
As
$$(\alpha f^*+(1-\alpha) g^*)^*=\operatorname{cl}\left[\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
g\left(\frac{\cdot}{1-\alpha}\right)\right]$$
and $\operatorname{dom} f^*=\operatorname{\mathbb{R}}^n=\operatorname{dom} g^*$, the closure operation on the right-hand side is superfluous.
This establishes \ref{i:coercive:1}.
Moreover, the
infimal convolution
\begin{equation}\label{e:coercive0}
\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
g\left(\frac{\cdot}{1-\alpha}\right)
\end{equation}
is exact.
For
\ref{i:coercive:2}, \ref{i:coercive:3}, it suffices to apply Lemma~\ref{l:box} to
\eqref{e:coercive0} for functions $\alpha\operatornamewithlimits{conv} f\left(\frac{\cdot}{\alpha}\right)$ and
$\alpha\operatornamewithlimits{conv} g\left(\frac{\cdot}{\alpha}\right)$, followed by invoking
Lemma~\ref{l:epimulti} and Lemma~\ref{l:convexhull:f}.
\end{proof}
\subsection{Convergence in minimization}
We need the following result on coercivity.
\begin{lem}\label{l:psi}
Let $0<\mu<\operatorname{\bar{\lambda}}$, and let $\psi:\operatorname{\mathbb{R}}^n\rightarrow\operatorname{\mathbb{R}}$ be a convex function. If
$f\geq \psi, g\geq\psi$, then $\ensuremath{\varphi^{\alpha}_{\mu}}\geq \psi$.
\end{lem}
\begin{proof}
Recall
$\ensuremath{\varphi^{\alpha}_{\mu}}(x)=$
$$\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)
-\frac{1}{2\mu}\|x\|^2.
$$
As $f+\frac{1}{2\mu}\|\cdot\|^2\geq \psi+\frac{1}{2\mu}\|\cdot\|^2$ and the latter is convex,
we have
$$\operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)\geq \psi+\frac{1}{2\mu}\|\cdot\|^2;$$
similarly,
$$\operatornamewithlimits{conv}\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\geq \psi+\frac{1}{2\mu}\|\cdot\|^2.$$
Then
\begin{align*}
&\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\\
&\geq \alpha\bigg(\psi+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)
\Box
(1-\alpha)\bigg(\psi+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\\
&=\psi+\frac{1}{2\mu}\|\cdot\|^2,
\end{align*}
in which we have used the convexity of $\psi+\frac{1}{2\mu}\|\cdot\|^2$.
The result follows.
\end{proof}
\begin{thm} Let $0<\mu<\operatorname{\bar{\lambda}}$. One has the following.
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:coercive0}If $f, g$ are bounded from below,
then $\ensuremath{\varphi^{\alpha}_{\mu}}$ is bounded from below.
\item\label{i:coercive1}
If $f, g$ are level-coercive, then $\ensuremath{\varphi^{\alpha}_{\mu}}$ is level-coercive.
\item\label{i:coercive2}
If $f,g$ are coercive, then $\ensuremath{\varphi^{\alpha}_{\mu}}$ is coercive.
\end{enumerate}
\end{thm}
\begin{proof}
\ref{i:coercive0}: Put $\psi=\min\{\inf f, \inf g\}$ and apply Lemma~\ref{l:psi}.
\noindent\ref{i:coercive1}:
By \cite[Theorem 3.26(a)]{rockwets}, there exist $\gamma\in (0,\infty)$, and $\beta\in\operatorname{\mathbb{R}}$
such that $f\geq \psi, g\geq \psi$ with $\psi=\gamma\|\cdot\|+\beta$.
Apply Lemma~\ref{l:psi}.
\noindent\ref{i:coercive2}: By \cite[Theorem 3.26(b)]{rockwets}, for
every $\gamma\in (0,\infty)$, there exists $\beta\in\operatorname{\mathbb{R}}$
such that $f\geq \psi, g\geq \psi$ with $\psi=\gamma\|\cdot\|+\beta$.
Apply Lemma~\ref{l:psi}.
\end{proof}
\begin{thm} Suppose that the proper, lsc functions $f, g$ are level-coercive. Then
for every $\operatorname{\bar{\alpha}}\in [0,1]$, we have
\begin{align*}
\lim_{\alpha\rightarrow\operatorname{\bar{\alpha}}}\inf\ensuremath{\varphi^{\alpha}_{\mu}} & =\inf \ensuremath{\varphi^{\balpha}_{\mu}} \text{ (finite)}, \text{ and }
\\
\limsup_{\alpha\rightarrow\operatorname{\bar{\alpha}}}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}} & \subseteq\operatornamewithlimits{argmin} \ensuremath{\varphi^{\balpha}_{\mu}}.
\end{align*}
Moreover, $(\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}})_{\alpha\in[0,1]}$ lies in a bounded set.
Consequently,
$$\lim_{\alpha\downarrow 0}\inf\ensuremath{\varphi^{\alpha}_{\mu}}=\inf g, \text{ and } \limsup_{\alpha\downarrow 0}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}\subseteq\operatornamewithlimits{argmin} g;$$
$$\lim_{\alpha\uparrow 1}\inf\ensuremath{\varphi^{\alpha}_{\mu}}=\inf f, \text{ and }
\limsup_{\alpha\uparrow 1}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}}\subseteq\operatornamewithlimits{argmin} f.$$
\end{thm}
\begin{proof} By assumption, there exist $\gamma>0$ and $\beta\in\operatorname{\mathbb{R}}$ such that
$f\geq\gamma\|\cdot\|+\beta, g\geq\gamma\|\cdot\|+\beta$. Lemma~\ref{l:psi}
shows that $\ensuremath{\varphi^{\alpha}_{\mu}}\geq \gamma\|\cdot\|+\beta$ for every $\alpha\in [0,1]$.
Since $\gamma\|\cdot\|+\beta$ is level-bounded,
$(\ensuremath{\varphi^{\alpha}_{\mu}})_{\alpha\in [0,1]}$ is uniformly level-bounded (so
eventually level-bounded). Corollary~\ref{t:alpha:epi} says that
$\alpha\mapsto \ensuremath{\varphi^{\alpha}_{\mu}}$ is epi-continuous on $[0,1]$.
As $\lambda_{f}=\lambda_{g}=\infty$,
$\ensuremath{\varphi^{\alpha}_{\mu}}$ and $\ensuremath{\varphi^{\balpha}_{\mu}}$ are proper and lsc for every $\mu>0$. Hence
\cite[Theorem 7.33]{rockwets} applies.
\end{proof}
\begin{thm} Suppose that the proper, lsc functions $f, g$ are level-coercive and
$\operatorname{dom} f\cap\operatorname{dom} g\neq\varnothing$. Then
\begin{align*}
\lim_{\mu\downarrow 0}\inf\ensuremath{\varphi^{\alpha}_{\mu}} & =\inf (\alpha f+(1-\alpha)g), \text{ and }
\\
\limsup_{\mu\downarrow 0}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}} & \subseteq\operatornamewithlimits{argmin} (\alpha f+(1-\alpha)g).
\end{align*}
Moreover, $(\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}})_{\mu>0}$ lies in a bounded set.
\end{thm}
\begin{proof}
Note that each $\ensuremath{\varphi^{\alpha}_{\mu}}$ is proper and lsc, and $f+g$ is proper and lsc.
By Theorem~\ref{t:u:0}, when $\mu\downarrow 0$, $\ensuremath{\varphi^{\alpha}_{\mu}}$ epi-converges to
$f+g$.
By assumption, there exist $\gamma>0$ and $\beta\in\operatorname{\mathbb{R}}$ such that
$f\geq\gamma\|\cdot\|+\beta, g\geq\gamma\|\cdot\|+\beta$. Lemma~\ref{l:psi}
shows that $\ensuremath{\varphi^{\alpha}_{\mu}}\geq \gamma\|\cdot\|+\beta$ for every $\mu\in ]0,\infty[$.
Since $\gamma\|\cdot\|+\beta$ is level-bounded,
$(\ensuremath{\varphi^{\alpha}_{\mu}})_{\mu\in ]0,\infty[}$ is uniformly level-bounded (so
eventually level-bounded).
It remains to
apply \cite[Theorem 7.33]{rockwets}.
\end{proof}
\begin{thm} Suppose that the proper and lsc functions $f, g$ are coercive.
Then
for every $\operatorname{\bar{\mu}}\in ]0,\infty]$, we have
\begin{align}\label{e:bcoercive}
\lim_{\mu\uparrow\operatorname{\bar{\mu}}}\inf\ensuremath{\varphi^{\alpha}_{\mu}} & =\inf \ensuremath{\varphi_{f,g,\alpha,\bmu}} \text{ (finite)}, \text{ and }
\nonumber\\
\limsup_{\mu\uparrow \operatorname{\bar{\mu}}}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}} & \subseteq\operatornamewithlimits{argmin} \ensuremath{\varphi_{f,g,\alpha,\bmu}}.
\end{align}
Moreover, $(\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}})_{\mu>0}$ lies in a bounded set.
Consequently,
\begin{align}\label{e:coercive:inf}
\lim_{\mu\uparrow \infty}\inf\ensuremath{\varphi^{\alpha}_{\mu}} & =\alpha\min f+(1-\alpha)\min g, \text{ and }
\nonumber \\
\limsup_{\mu\uparrow \infty}\operatornamewithlimits{argmin}\ensuremath{\varphi^{\alpha}_{\mu}} & \subseteq
(\alpha\operatornamewithlimits{conv}\operatornamewithlimits{argmin} f+(1-\alpha)\operatornamewithlimits{conv}\operatornamewithlimits{argmin} g).
\end{align}
\end{thm}
\begin{proof} Note that each $\ensuremath{\varphi^{\alpha}_{\mu}}$ is proper and lsc for $\mu\in]0,\infty[$.
When $\mu=\infty$, Proposition~\ref{p:coercive} gives that
the epi-limit is proper, lsc and convex. By Theorem~\ref{t:go:infinity}\ref{i:mono:phi},
when $\mu\uparrow\operatorname{\bar{\mu}}$, $\ensuremath{\varphi^{\alpha}_{\mu}}$ monotonically decrease to $\ensuremath{\varphi_{f,g,\alpha,\bmu}}$.
Since $\ensuremath{\varphi_{f,g,\alpha,\bmu}}$ is lsc, so $\ensuremath{\varphi^{\alpha}_{\mu}}$ epi-converges to $\ensuremath{\varphi_{f,g,\alpha,\bmu}}$.
By assumption, for every $\gamma>0$ there exists $\beta\in\operatorname{\mathbb{R}}$ such that
$f\geq\gamma\|\cdot\|+\beta, g\geq\gamma\|\cdot\|+\beta$. Lemma~\ref{l:psi}
shows that $\ensuremath{\varphi^{\alpha}_{\mu}}\geq \gamma\|\cdot\|+\beta$ for every $\mu\in ]0,\infty[$.
Since $\gamma\|\cdot\|+\beta$ is level-bounded,
$(\ensuremath{\varphi^{\alpha}_{\mu}})_{\mu\in ]0,\infty[}$ is uniformly level-bounded (so
eventually level-bounded). Hence \eqref{e:bcoercive} follows from
\cite[Theorem 7.33]{rockwets}.
Combining \eqref{e:bcoercive}, Theorem~\ref{t:go:infinity}
and Proposition~\ref{p:coercive} yields \eqref{e:coercive:inf}.
\end{proof}
\section{Subdifferentiability of the proximal average}\label{s:subd}
In this section, we focus on the subdifferentiability and differentiability of proximal average.
Following Benoist and Hiriart-Urruty \cite{benoist},
we say that a family of points $\{x_{1},\ldots, x_{m}\}$ in $\operatorname{dom} f$
is
called by $x\in\operatorname{dom}\operatornamewithlimits{conv} f$ if
$$x=\sum_{i=1}^{m}\alpha_{i}x_{i}, \text{ and } \operatornamewithlimits{conv} f(x)=\sum_{i=1}^{m}\alpha_{i}f(x_{i}),$$
where $\sum_{i=1}^{m}\alpha_{i}=1$ and $(\forall i)\ \alpha_{i}>0$.
The following result is the central one of this section.
\begin{thm}[subdifferentiability of the proximal average]\label{t:vphi:sub}
Let
$0<\mu<\operatorname{\bar{\lambda}}$,
let $x\in\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}$ and $x=y+z$.
Suppose the following conditions hold:
\begin{enumerate}[label=\rm(\alph*)]
\item \label{i:function1}
\begin{align*}
&\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)\\
&=\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{y}{\alpha}\right)+(1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{z}{1-\alpha}\right),
\end{align*}
\item \label{i:function2}$\{y_{1},\ldots,y_{l}\}$ are called by
$y/\alpha$ in $\operatornamewithlimits{conv}(f+1/2\mu\|\cdot\|^2)$, and
\item \label{i:function3}
$\{z_{1},\ldots,z_{m}\}$ are called by
$z/(1-\alpha)$ in $\operatornamewithlimits{conv}(g+1/2\mu\|\cdot\|^2)$.
\end{enumerate}
Then
\begin{align*}
\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x) &=
\partial_{L}\ensuremath{\varphi^{\alpha}_{\mu}}(x)
=\partial_{C}\ensuremath{\varphi^{\alpha}_{\mu}}(x)
\\
&=\left[\cap_{i=1}^{l}\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\left(y_{i}\right)\right]\cap
\left[\cap_{j=1}^{m}\partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\left(z_{j}\right)\right]-
\frac{x}{\mu}.
\end{align*}
\end{thm}
\begin{proof} By Theorem~\ref{t:prox}\ref{i:epi:sum}, the Clarke regularity of
$\ensuremath{\varphi^{\alpha}_{\mu}}$ and sum rule of limiting subdifferentials, we have
$\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)=\partial_{C}\ensuremath{\varphi^{\alpha}_{\mu}}(x)=\partial_{L}\ensuremath{\varphi^{\alpha}_{\mu}}(x)=$
\begin{align}\label{e:diff:conv}
&\partial_{L}\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)
-\frac{x}{\mu}\\
&=\partial \left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)
-\frac{x}{\mu}.\nonumber
\end{align}
Using the subdifferential formula for infimal convolution \cite[Proposition 16.61]{convmono} or \cite[Corollary 2.4.7]{zalinescu2002convex}, we obtain
\begin{align*}\label{e:infsub}
& \partial\left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{\alpha}\right)\Box (1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{\cdot}{1-\alpha}\right)\right](x)\\
&=
\partial \left[\alpha\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{y}{\alpha}\right)\right]\cap
\partial \left[(1-\alpha)\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\left(\frac{z}{1-\alpha}\right)\right]\\
&=
\partial\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)(\bar{y})\cap
\partial\operatornamewithlimits{conv}
\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)(\bar{z})
\end{align*}
where $\bar{y}=\frac{y}{\alpha}$, $\bar{z}=\frac{z}{1-\alpha}$.
The subdifferential formula for the convex hull of a coercive function
\cite[Corollary 4.9]{benoist} or \cite[Theorem 3.2]{rabier} gives
\begin{equation*}\label{e:convf}
\partial\operatornamewithlimits{conv}\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\big(\bar{y}\big)=
\cap_{i=1}^{l}\partial\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)\big(y_{i}),
\end{equation*}
\begin{equation}\label{e:convg}
\partial\operatornamewithlimits{conv}\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\big(\bar{z}\big)=
\cap_{j=1}^{m}\partial\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)\big(z_{j}).
\end{equation}
Therefore, the result follows by combining \eqref{e:diff:conv} and \eqref{e:convg}.
\end{proof}
\begin{cor}
Let $0<\mu<\operatorname{\bar{\lambda}}$,
let $\alpha_{i}>0, \beta_{j}>0$ with
$\sum_{i=1}^{l}\alpha_{i}=1, \sum_{j=1}^{m}\beta=1$ and let
$\alpha\in ]0,1[$.
Suppose that
\begin{equation*}\label{e:x=yz}
x=\alpha\sum_{i=1}^{l}\alpha_{i}y_{i}+(1-\alpha)\sum_{j=1}^{m}\beta_{j}z_{j},
\end{equation*}
and
\begin{equation}\label{e:commonsub}
\left[\cap_{i=1}^{l}\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\left(y_{i}\right)\right]\cap
\left[\cap_{j=1}^{m}\partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\left(z_{j}\right)\right]
\neq \varnothing.
\end{equation}
Then
\begin{align*}
\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x) &=
\partial_{L}\ensuremath{\varphi^{\alpha}_{\mu}}(x)
=\partial_{C}\ensuremath{\varphi^{\alpha}_{\mu}}(x)\nonumber\\
&=\left[\cap_{i=1}^{l}\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\left(y_{i}\right)\right]\cap
\left[\cap_{j=1}^{m}\partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\left(z_{j}\right)\right]-
\frac{x}{\mu}.
\end{align*}
\end{cor}
\begin{proof} We will show that
\begin{equation}\label{e:called:y0}
\operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)\sum_{i=1}^{l}\alpha_{i}y_{i}
=\sum_{i}^{l}\alpha_{i}
\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i}).
\end{equation}
By \eqref{e:commonsub}, there exists
$$y^*\in \left[\cap_{i=1}^{l}\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\left(y_{i}\right)\right]\cap
\left[\cap_{j=1}^{m}\partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\left(z_{j}\right)\right].$$
For every $y_{i}$, we have
$$(\forall u\in\operatorname{\mathbb{R}}^n)\ \left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(u)\geq \left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})
+\scal{y^*}{u-y_{i}}.$$ Multiplying each inequality by $\alpha_i$, followed by summing
them up, gives
$$(\forall u\in\operatorname{\mathbb{R}}^n)\ \left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(u)\geq \sum_{i=1}^{l}\alpha_{i}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})
+\scal{y^*}{u-\sum_{i=1}^{l}\alpha_{i}y_{i}}.$$
Then
$$(\forall u\in\operatorname{\mathbb{R}}^n)\ \operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(u)\geq \sum_{i=1}^{l}\alpha_{i}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})
+\scal{y^*}{u-\sum_{i=1}^{l}\alpha_{i}y_{i}},$$
from which
\begin{equation}\label{e:called:y1}
\operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)\sum_{i=1}^{l}\alpha_{i}y_{i}
\geq \sum_{i=1}^{l}\alpha_{i}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i}).
\end{equation}
Since $\operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(\sum_{i=1}^{l}\alpha_{i}y_{i})
\leq \sum_{i=1}^{l}\alpha_{i}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})$
always holds, \eqref{e:called:y0} is established.
Moreover, \eqref{e:called:y0}
and \eqref{e:called:y1} implies
\begin{equation}\label{e:called:y2}
y^*\in \partial \operatornamewithlimits{conv}\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\sum_{i=1}^{l}\alpha_{i}y_{i}.
\end{equation}
Similar arguments give
\begin{equation}\label{e:called:z0}
\operatornamewithlimits{conv}\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)(\sum_{j=1}^{m}\beta_{j}z_{j})
=\sum_{j}^{m}\beta_{j}
\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)(z_{j}),
\end{equation}
and
\begin{equation}\label{e:called:z1}
y^*\in \partial \operatornamewithlimits{conv}\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)\sum_{j=1}^{m}\beta_{j}y_{j}.
\end{equation}
Put
$x=y+z$ with $y=\alpha\sum_{i=1}^{l}\alpha_{i}y_{i}$ and $z=(1-\alpha)
\sum_{j=1}^{m}\beta_{j}z_{j}.$
Equations \eqref{e:called:y2} and \eqref{e:called:z1} guarantee
the assumption \ref{i:function1} of Theorem~\ref{t:vphi:sub}; \eqref{e:called:y0} and \eqref{e:called:z0} guarantee the assumptions \ref{i:function2} and \ref{i:function3}
of Theorem~\ref{t:vphi:sub} respectively.
Hence, Theorem~\ref{t:vphi:sub} applies.
\end{proof}
\begin{cor}
Let $0<\mu<\operatorname{\bar{\lambda}}$.
Suppose that
$$\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(x)\cap \partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)(x)\neq\varnothing.$$
Then
\begin{align*}
\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x) &=
\partial_{L}\ensuremath{\varphi^{\alpha}_{\mu}}(x)
=\partial_{C}\ensuremath{\varphi^{\alpha}_{\mu}}(x)\\
&=\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
(x)\cap
\partial\left(g+\frac{1}{2\mu}\|\cdot\|^2\right)(x)-
\frac{x}{\mu}.
\end{align*}
\end{cor}
Armed with Theorem~\ref{t:vphi:sub}, we now turn to the differentiability of $\ensuremath{\varphi^{\alpha}_{\mu}}$.
\begin{df}
A function $f_{1}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ is almost differentiable
if $\hat{\partial} f_{1}(x)$ is a singleton
for every $x\in \operatorname{int}(\operatorname{dom} f_{1})$, and $\hat{\partial} f_{1}(x)=\varnothing$
for every $x\in\operatorname{dom} f_{1}\setminus\operatorname{int}(\operatorname{dom} f_{1})$, if any.
\end{df}
\begin{lem}\label{l:sumrule}
Let $f_{1},f_{2}:\operatorname{\mathbb{R}}^n\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ be proper, lsc functions
and let $x\in\operatorname{dom} f_{1}\cap\operatorname{dom} f_{2}$. If $f_{2}$ is
continuously differentiable at $x$, then
$$\partial (f_{1}+f_{2})(x)
\subset\hat{\partial}(f_{1}+f_{2})(x)=
\hat{\partial}f_{1}(x)+\triangledown f_{2}(x).$$
\end{lem}
\begin{lem}\label{l:diff:hypo}
Let $f_{1}:\operatorname{\mathbb{R}}^n\rightarrow \ensuremath{\,\left]-\infty,+\infty\right]}$ be proper, lsc and $\mu$-proximal, and
let $x\in\operatorname{int}\operatorname{dom} f_{1}$. If $\hat{\partial}f_{1}(x)$ is a singleton, then
$f_{1}$ is differentiable at $x$.
\end{lem}
\begin{proof} Observe that
$f_{2}=f_{1}+\frac{1}{2\mu}\|\cdot\|^2$ is convex, and
$$\partial f_{2}(x)=\hat{\partial}f_{2}(x)
=\hat{\partial}f_{1}(x)+\frac{x}{\mu}.$$
When $\hat{\partial}f_{1}(x)$ is a singleton, $\partial f_{2}(x)$
is a singleton. This implies that $f_{2}$ is differentiable at $x$
because $f_{2}$ is convex and $x\in\operatorname{int}\operatorname{dom} f_{2}$.
Hence, $f_{1}$ is differentiable at $x$.
\end{proof}
\begin{cor}[differentiability of the proximal average] \label{c:interior:diff}
Let $0<\mu<\operatorname{\bar{\lambda}}$.
Suppose that either $f$ or $g$ is almost differentiable (in particular, if $f$ or $g$ is differentiable
at every point of its domain). Then
$\ensuremath{\varphi^{\alpha}_{\mu}}$ is almost differentiable. In particular, $\ensuremath{\varphi^{\alpha}_{\mu}}$ is differentiable on the interior of
its domain
$\operatorname{int}\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}$.
\end{cor}
\begin{proof} Without loss of generality, assume that
$f$ is almost differentiable.
By Lemma~\ref{l:sumrule},
\begin{equation}\label{e:frechet}
\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)
\left(y_{i}\right)\subset \hat{\partial} f(y_{i})+\frac{y_{i}}{\mu}.
\end{equation}
It follows that
$\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})$ is
at most single-valued whenever
$\hat{\partial}f(y_{i})$ is single-valued.
With the same notation as in Theorem~\ref{t:vphi:sub}, we consider two cases.
{\sl Case 1:} $x\in\operatorname{bdry}\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}$. As $x=\alpha(y/\alpha)+(1-\alpha)(z/(1-\alpha))$,
we must have
$y/\alpha\in(\operatorname{bdry}\operatornamewithlimits{conv} \operatorname{dom} f)$ and $z/(1-\alpha)\in\operatorname{bdry}(\operatornamewithlimits{conv} \operatorname{dom} g)$;
otherwise $x\in\operatorname{int}(\alpha\operatornamewithlimits{conv}\operatorname{dom} f+(1-\alpha)\operatornamewithlimits{conv}\operatorname{dom} g)
=\operatorname{int}\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}$,
which is a contradiction. Then the family of $\{y_{1},\ldots, y_{m}\}$ called
by $y/\alpha$ must be from $\operatorname{bdry}\operatorname{dom} f$. As $f$ is almost differentiable,
$\hat{\partial}f(y_{i})=\varnothing$, then $\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)=\varnothing$ by
Theorem~\ref{t:vphi:sub} and \eqref{e:frechet}.
{\sl Case 2:} $x\in\operatorname{int}(\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}})$. As $\ensuremath{\varphi^{\alpha}_{\mu}}$ is $\mu$-proximal,
$\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)\neq\varnothing$. We claim that
the family of $\{y_{1},\ldots, y_{m}\}$ called by
$y/\alpha$ in Theorem~\ref{t:vphi:sub} are necessarily from $\operatorname{int}\operatorname{dom} f$.
If not, then
$\partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})=\varnothing$
because of \eqref{e:frechet} and
$\hat{\partial}f(y_{i})=\varnothing$ for $y_{i}\in\operatorname{bdry}(\operatorname{dom} f)$.
Then Theorem~\ref{t:vphi:sub} implies
$\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)=\varnothing$, which is a contradiction.
Now $\{y_{1},\ldots, y_{m}\}$ are from $\operatorname{int}\operatorname{dom} f$ and $f$ is almost
differentiable, so $(\forall i)\ \hat{\partial}f(y_{i})$
is a singleton. Using \eqref{e:frechet} again and
$\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)\neq\varnothing$,
we see that $(\forall i)\ \partial\left(f+\frac{1}{2\mu}\|\cdot\|^2\right)(y_{i})$
is a singleton. Hence, $\hat{\partial}\ensuremath{\varphi^{\alpha}_{\mu}}(x)$ is a singleton by
Theorem~\ref{t:vphi:sub}.
Case 1 and Case 2 together show that $\ensuremath{\varphi^{\alpha}_{\mu}}$ is almost differentiable.
Finally, $\ensuremath{\varphi^{\alpha}_{\mu}}$ is differentiable on
$\operatorname{int}\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}$ by Lemma~\ref{l:diff:hypo}.
\end{proof}
\begin{cor} Let $0<\mu<\operatorname{\bar{\lambda}}$.
Suppose that either $f$ or $g$ is almost differentiable and that either $\operatornamewithlimits{conv}\operatorname{dom} f=\operatorname{\mathbb{R}}^n$
or $\operatornamewithlimits{conv}\operatorname{dom} g=\operatorname{\mathbb{R}}^n$. Then
$\ensuremath{\varphi^{\alpha}_{\mu}}$ is differentiable on $\operatorname{\mathbb{R}}^n$.
\end{cor}
\begin{proof}
By Theorem~\ref{t:prox}\ref{i:dom:convhull}, $\operatorname{dom}\ensuremath{\varphi^{\alpha}_{\mu}}=\operatorname{\mathbb{R}}^n$.
It suffices to apply Corollary~\ref{c:interior:diff}.
\end{proof}
We end this section with a result on Lipschitz continuity of the gradient of $\ensuremath{\varphi^{\alpha}_{\mu}}$.
\begin{prop} Suppose that $f$ (or $g$) is differentiable with
a Lipschtiz continuous gradient and $\mu$-proximal.
Then, for every $\alpha\in ]0,1[$, the function $\ensuremath{\varphi^{\alpha}_{\mu}}$ is differentiable
with a Lipschitz continuous gradient.
\end{prop}
\begin{proof}
As $f$ is $\mu$-proximal and differentiable with a Lipschtiz continuous
gradient,
the function $f+\frac{1}{2\mu}\|\cdot\|^2$ is convex and differentiable with a Lipschitz
continuous gradient. By \cite[Proposition 12.60]{rockwets},
$\big(f+\frac{1}{2\mu}\|\cdot\|^2\big)^*$ is strongly convex,
so $$\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*$$ is strongly convex.
By \cite[Proposition 12.60]{rockwets} again,
$$\left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\right]^*$$
is convex and differentiable with a Lipschitz continuous gradient.
Since
$\ensuremath{\varphi^{\alpha}_{\mu}}=$
$$\left[\alpha\bigg(f+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*+
(1-\alpha)\bigg(g+\frac{1}{2\mu}\|\cdot\|^2\bigg)^*\right]^*
-\frac{1}{2\mu}\|\cdot\|^2,$$
we see that $\ensuremath{\varphi^{\alpha}_{\mu}}$ is differentiable with a Lipschitz continuous gradient.
\end{proof}
\section{The proximal average for quadratic functions}\label{s:quad}
In this section, we illustrate the above results for quadratic functions.
For an $n\times n $ symmetric matrix $A$, define the quadratic function
$\ensuremath{\,\mathfrak{q}}_{A}:\operatorname{\mathbb{R}}^n\rightarrow\operatorname{\mathbb{R}}$ by $x\mapsto \frac{1}{2}\scal{x}{Ax}.$
We use $\ensuremath{\, \lambda_{\min}} A$ to denote the smallest eigenvalue of $A$.
\begin{lem}\label{l:quad} For an $n\times n$ symmetric matrix $A$, one has
\begin{enumerate}[label=\rm(\alph*)]
\item\label{i:quad1}
$\ensuremath{\,\mathfrak{q}}_{A}$ is prox-bounded with threshold
\begin{equation}\label{e:boundp}
\lambda_{\ensuremath{\,\mathfrak{q}}_{A}}=\frac{1}{\max\{0,-\ensuremath{\, \lambda_{\min}} A\}}>0
\end{equation}
and $\mu$-proximal for every $0<\mu\leq \lambda_{\ensuremath{\,\mathfrak{q}}_{A}}$;
\item \label{e:boundp1}
the prox-bound $\lambda_{\ensuremath{\,\mathfrak{q}}_{A}}=+\infty$ if and only if $A$ is positive semidefinite;
\item\label{i:quad2} if $0<\mu<\lambda_{\ensuremath{\,\mathfrak{q}}_{A}}$, then
\begin{equation}\label{e:quadenv}
e_{\mu}\ensuremath{\,\mathfrak{q}}_{A}=\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\operatorname{Id}-(\mu A+\operatorname{Id})^{-1}]};\text { and }
\end{equation}
\begin{equation*}\label{e:quadprox}
\operatorname{Prox}_{\mu}\ensuremath{\,\mathfrak{q}}_{A}=(\mu A+\operatorname{Id})^{-1}.
\end{equation*}
\end{enumerate}
\end{lem}
\begin{proof}
\ref{i:quad1}: As $A$ can be diagonalized,
$\ensuremath{\,\mathfrak{q}}_{A}\geq \ensuremath{\, \lambda_{\min}} A \ensuremath{\,\mathfrak{q}}_{\operatorname{Id}}$. Apply \cite[Exercise 1.24]{rockwets} to
obtain \eqref{e:boundp}. When $0<\mu\leq \lambda_{\ensuremath{\,\mathfrak{q}}_{A}}$,
$A+\frac{1}{\mu}\operatorname{Id}$ has nonnegative eigenvalues,
so $\ensuremath{\,\mathfrak{q}}_{A}+\frac{1}{\mu}\ensuremath{\,\mathfrak{q}}_{\operatorname{Id}}$ is convex.
\noindent\ref{e:boundp1}: This follows from \ref{i:quad1}.
\noindent\ref{i:quad2}: When $0<\mu<\lambda_{\ensuremath{\,\mathfrak{q}}_{A}}$, the function
$\ensuremath{\,\mathfrak{q}}_{A}+\frac{1}{\mu}\ensuremath{\,\mathfrak{q}}_{\operatorname{Id}}$ is strictly convex. To find
\begin{equation}\label{e:quade}
e_{\mu}\ensuremath{\,\mathfrak{q}}_{A}(x)=\inf_{w}\left(\ensuremath{\,\mathfrak{q}}_{A}(w)+\frac{1}{\mu}\ensuremath{\,\mathfrak{q}}_{\operatorname{Id}}(x-w)\right),
\end{equation}
one directly takes derivative with repect to $w$ to find
\begin{equation}\label{e:quadp}
\operatorname{Prox}_{\mu}\ensuremath{\,\mathfrak{q}}_{A}(x)=(\mu A+\operatorname{Id})^{-1}(x).
\end{equation}
Substitute \eqref{e:quadp} into \eqref{e:quade} to get \eqref{e:quadenv}.
\end{proof}
\begin{ex} Let $A_{1}, A_{2}$ be two $n\times n$ symmetric matrices and
let $0<\mu<\operatorname{\bar{\lambda}}=\min\{\lambda_{\ensuremath{\,\mathfrak{q}}_{A_{1}}},\lambda_{\ensuremath{\,\mathfrak{q}}_{A_{2}}}\}$.
Then the following hold:
\begin{enumerate}[label=\rm(\alph*)]
\item\label{e:proxq1}
$\ensuremath{\varphi^{\alpha}_{\mu}}=\ensuremath{\,\mathfrak{q}}_{\mu^{-1}A_{3}}$ with
$$A_{3}=[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]^{-1}-\operatorname{Id},$$
and
$$\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1};$$
\item \label{e:proxq2}
$\lim_{\alpha\downarrow 0}\ensuremath{\varphi^{\alpha}_{\mu}}=\ensuremath{\,\mathfrak{q}}_{A_{2}}$ and $\lim_{\alpha\uparrow 1}\ensuremath{\varphi^{\alpha}_{\mu}}=\ensuremath{\,\mathfrak{q}}_{A_{1}};$
\item\label{e:proxq3}
$\lim_{\mu\downarrow 0}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha\ensuremath{\,\mathfrak{q}}_{A_{1}}+(1-\alpha)\ensuremath{\,\mathfrak{q}}_{A_{2}};$
\item \label{e:proxq3.5} when $\operatorname{\bar{\lambda}}<\infty$,
$$\lim_{\mu\uparrow \operatorname{\bar{\lambda}}}\ensuremath{\varphi^{\alpha}_{\mu}}=
\ensuremath{\,\mathfrak{q}}_{\alpha^{-1}(A_{1}+\operatorname{\bar{\lambda}}^{-1}\operatorname{Id})}\Box \ensuremath{\,\mathfrak{q}}_{(1-\alpha)^{-1}(A_{2}+\operatorname{\bar{\lambda}}^{-1}\operatorname{Id})}-
\ensuremath{\,\mathfrak{q}}_{\operatorname{\bar{\lambda}}^{-1}\operatorname{Id}};$$
\item \label{e:proxq4}
when both $A_{1}, A_{2}$ are positive definite, $\operatorname{\bar{\lambda}}=+\infty$,
$$\lim_{\mu\uparrow \infty}\ensuremath{\varphi^{\alpha}_{\mu}}=\ensuremath{\,\mathfrak{q}}_{(\alpha A_{1}^{-1}+(1-\alpha)A_{2}^{-1})^{-1}}.$$
\end{enumerate}
\end{ex}
\begin{proof}
\ref{e:proxq1}: By Lemma~\ref{l:quad},
\begin{align*}
& -\alpha e_{\mu}\ensuremath{\,\mathfrak{q}}_{A_{1}}-(1-\alpha)e_{\mu}\ensuremath{\,\mathfrak{q}}_{A_{2}}\\
&=-\alpha \ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\operatorname{Id}-(\mu A_{1}+\operatorname{Id})^{-1}]}-(1-\alpha)\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\operatorname{Id}-(\mu A_{2}+\operatorname{Id})^{-1}]}\\
&=\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})-\operatorname{Id}]}.
\end{align*}
Thus, applying Lemma~\ref{l:quad} again,
\begin{align*}
\ensuremath{\varphi^{\alpha}_{\mu}} &=-e_{\mu}(\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}-\operatorname{Id}]})
\\
&=-\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\operatorname{Id}-(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})^{-1}]}\\
&=\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})^{-1}-\operatorname{Id}]}.
\end{align*}
Again, using Lemma~\ref{l:quad},
\begin{align*}
e_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}} &= e_{\mu}\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})^{-1}-\operatorname{Id}]}\\
&=\ensuremath{\,\mathfrak{q}}_{\mu^{-1}[\operatorname{Id}-(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})]},
\end{align*}
so
$$\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}.$$
\noindent\ref{e:proxq2}: Note that the matrix function $A\mapsto A^{-1}$ is continuous whenever
$A$ is invertible. Then \ref{e:proxq2} is immediate because
$$\lim_{\alpha\downarrow 0}(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})^{-1}
=((\mu A_{2}+\operatorname{Id})^{-1})^{-1}=\mu A_{2}+\operatorname{Id}, \text{ and }
$$
$$\lim_{\alpha\uparrow 1}(\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1})^{-1}
=((\mu A_{1}+\operatorname{Id})^{-1})^{-1}=\mu A_{1}+\operatorname{Id}.
$$
\noindent\ref{e:proxq3}: It suffices to show
$$\lim_{\mu\downarrow 0}\frac{[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]^{-1}-\operatorname{Id}}{\mu}=\alpha A_{1}+(1-\alpha)A_{2},$$
equivalently,
\begin{equation}\label{e:matrix}
\small\lim_{\mu\downarrow 0}\frac{[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]^{-1}-[\alpha(\mu A_{1}+\operatorname{Id})+(1-\alpha)(\mu A_{2}+\operatorname{Id})]}{\mu}=0.
\end{equation}
Since $\lim_{\mu\downarrow 0}[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]
=\operatorname{Id}$, \eqref{e:matrix} follows from the following calculation:
\begin{align*}
&[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]\cdot
\nonumber\\
&\frac{[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]^{-1}-[\alpha(\mu A_{1}+\operatorname{Id})+(1-\alpha)(\mu A_{2}+\operatorname{Id})]}{\mu}\\
&= \frac{\operatorname{Id}-[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]
[\alpha(\mu A_{1}+\operatorname{Id})+(1-\alpha)(\mu A_{2}+\operatorname{Id})]}{\mu}\\
&=-\alpha(1-\alpha)[(\mu A_{1}+\operatorname{Id})^{-1}(A_{2}-A_{1})
+(\mu A_{2}+\operatorname{Id})^{-1}(A_{1}-A_{2})]\\
& \rightarrow -\alpha(1-\alpha)[(A_{2}-A_{1})
+(A_{1}-A_{2})]= 0.
\end{align*}
\noindent\ref{e:proxq3.5}: The matrices $A_{1}+\operatorname{\bar{\lambda}}^{-1}\operatorname{Id}$ and $A_{2}+\operatorname{\bar{\lambda}}^{-1}\operatorname{Id}$
are positive semidefinite, so the convex hulls are superfluous.
\noindent\ref{e:proxq4}: As $\mu\rightarrow\infty$, we have
\begin{align*}
&\frac{[\alpha(\mu A_{1}+\operatorname{Id})^{-1}+(1-\alpha)(\mu A_{2}+\operatorname{Id})^{-1}]^{-1}-\operatorname{Id}}{\mu}\\
&=\left[\alpha\left(A_{1}+\frac{\operatorname{Id}}{\mu}\right)^{-1}
+(1-\alpha)\left(A_{2}+\frac{\operatorname{Id}}{\mu}\right)^{-1}\right]-\frac{\operatorname{Id}}{\mu}\\
&\rightarrow (\alpha A_{1}^{-1}+(1-\alpha)A_{2}^{-1})^{-1}.
\end{align*}
\end{proof}
\begin{rem} When both $A_{1}, A_{2}$ are positive semidefinite matrices,
we refer the reader to \emph{\cite{respos}}.
\end{rem}
\section{The general question is still unanswered}\label{s:theg}
According to Theorem~\ref{prop:convcomb}, suppose that $0<\mu<\operatorname{\bar{\lambda}}$,
$0<\alpha<1$
and $\operatorname{Prox}_{\mu}f$ and $\operatorname{Prox}_{\mu}g$ are convex-valued. Then
there exists a proper, lsc function
$\ensuremath{\varphi^{\alpha}_{\mu}}$ such that
$\operatorname{Prox}_{\mu}\ensuremath{\varphi^{\alpha}_{\mu}}=\alpha\operatorname{Prox}_{\mu}f+(1-\alpha)\operatorname{Prox}_{\mu}g$.
When the proximal mapping is not convex-valued, the situation is subtle.
We illustrate this by revisiting Example~\ref{e:proximal:fk}.
Recall that for $\varepsilon_{k}>0$, the function
$$f_{k}(x)=\max\{0,(1+\varepsilon_{k})(1-x^2)\}$$
has
$$\operatorname{Prox}_{1/2}f_{k}(x)=\begin{cases}
x &\text{ if $x\geq 1$,}\\
1 &\text{ if $0<x<1$,}\\
\{-1,1\} &\text{ if $x=0$,}\\
-1 &\text{ if $-1<x<0$,}\\
x &\text{ if $x\leq -1$.}
\end{cases}
$$
With $\alpha=1/2$, we have
\begin{equation}\label{e:prox:half}
(\alpha \operatorname{Prox}_{1/2}f_{1}+(1-\alpha)\operatorname{Prox}_{1/2}f_{2})(x)
=\begin{cases}
x &\text{ if $x\geq 1$,}\\
1 &\text{ if $0<x<1$,}\\
\{-1,0,1\} &\text{ if $x=0$,}\\
-1 &\text{ if $-1<x<0$,}\\
x &\text{ if $x\leq -1$.}
\end{cases}
\end{equation}
Because $\operatorname{Prox}_{1/2}f_{i}(0)$ is not convex-valued, $(\alpha \operatorname{Prox}_{1/2}f_{1}+(1-\alpha)\operatorname{Prox}_{1/2}f_{2})(0)$ is neither
$\operatorname{Prox}_{1/2}f_{1}(0)$ nor $\operatorname{Prox}_{1/2}f_{2}(0)$, although
$\operatorname{Prox}_{1/2}f_{1}(0)=\operatorname{Prox}_{1/2}f_{2}(0)$.
One can verify that \eqref{e:prox:half} is indeed
$\operatorname{Prox}_{1/2}g(x)$ where
$$g(x)=\begin{cases}
0 &\text{ if $x>1$,}\\
-x(x-1)-x^2+1 &\text{ if $0<x\leq 1$,}\\
-x(x+1)-x^2+1 &\text{ if $-1<x\leq 0$,}\\
0 &\text{ if $x\leq -1$.}
\end{cases}
$$
Regretfully, we do not have a systematic way to find $g$
when $\operatorname{Prox}_{\mu}g$ is not convex-valued.
The challenging question is still open:
\emph{Is a convex combination of proximal mappings of possibly nonconvex functions
always a proximal mapping?}
\section*{Acknowledgment}
Xianfu Wang was partially supported by the Natural Sciences and
Engineering Research Council of Canada.
\bibliographystyle{plain}
| {
"timestamp": "2019-09-16T02:11:59",
"yymm": "1909",
"arxiv_id": "1909.06221",
"language": "en",
"url": "https://arxiv.org/abs/1909.06221",
"abstract": "In this work, we construct a proximal average for two prox-bounded functions, which recovers the classical proximal average for two convex functions. The new proximal average transforms continuously in epi-topology from one proximal hull to the other. When one of the functions is differentiable, the new proximal average is differentiable. We give characterizations for Lipschitz and single-valued proximal mappings and we show that the convex combination of convexified proximal mappings is always a proximal mapping. Subdifferentiability and behaviors of infimal values and minimizers are also studied.",
"subjects": "Functional Analysis (math.FA)",
"title": "A proximal average for prox-bounded functions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575116884778,
"lm_q2_score": 0.7217432182679957,
"lm_q1q2_score": 0.7091542206194358
} |
https://arxiv.org/abs/1402.4911 | Finite element eigenvalue enclosures for the Maxwell operator | We propose employing the extension of the Lehmann-Maehly-Goerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollution-free finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives complementary bounds for the eigenfrequencies which are adjacent to a given real parameter. We present a concrete numerical scheme which provides certified enclosures in a suitable asymptotic regime. We illustrate the applicability of this scheme by means of some numerical experiments on benchmark data using Lagrange elements and unstructured meshes. | \section{Introduction}
The framework developed by Zimmermann and Mertins \cite{ZM95} which generalizes the Lehmann-Maehly-Goerisch method \cite{1985Goerisch2,1980Goerisch,1949Lehmann,1950Lehmann,1952Maehly} (also \cite[Chapter~4.11]{1974Weinberger}), is a reliable tool for the numerical computation of bounds for the eigenvalues of linear operators in the spectral pollution regime \cite{Behnke:2009p3097,BouStr:2011man,theoretical}. In its most basic formulation \cite{davies-plum,theoretical,BoHo2013}, this framework relies on fixing a parameter $t\in \mathbb{R}$ and then characterizing the spectrum which is adjacent to $t$ by means of a combination of the Variational Principle with the Spectral Mapping Theorem. In the present paper we show that
this formulation can be effectively implemented for computing sharp estimates for the angular frequencies and electromagnetic field phasors of the resonant cavity problem by means of the finite element method.
Let $\Omega\subset \RR^3$ be a polyhedron. Denote by $\bomega$ the boundary of this region and by $\nn$ its outer normal vector. Consider the anisotropic Maxwell eigenvalue problem: find $\omega\in \R$ and $({\boldsymbol E},{\boldsymbol H}) \not=0$ such that
\begin{equation} \label{maxwell}
\left\{
\begin{aligned}
&
\begin{aligned} &\curl {\boldsymbol E} = i\omega\mu {\boldsymbol H} \\
& \curl {\boldsymbol H} = -i\omega\epsilon {\boldsymbol E}
\end{aligned} & \text{in }\Omega \\
& {\boldsymbol E}\times\nn =0& \text{on } \bomega.
\end{aligned} \right.
\end{equation}
The physical phenomenon of electromagnetic oscillations in a resonator is described by \eqref{maxwell}, assuming that the field phasor satisfies Gauss's law
\begin{equation} \label{ansatz_div}
\div (\epsilon {\boldsymbol E})=0=\div (\mu {\boldsymbol H}) \quad \text{in }\Omega.
\end{equation}
Here $\epsilon$ and $\mu$, respectively, are the given electric permittivity and magnetic permeability at each point of the resonator.
The orthogonal complement in a suitable inner product \cite{1990Birman} of the solenoidal space \eqref{ansatz_div} is the gradient space. This gradient space has infinite dimension and is part of the kernel of the densely defined linear self-adjoint operator
\[
\mathcal{M}:\operatorname{D} (\mathcal{M})\longrightarrow L^2(\Omega)^6
\]
associated to \eqref{maxwell}. In turns, this means that \eqref{maxwell}-\eqref{ansatz_div} and the unrestricted problem \eqref{maxwell}, have exactly the same non-zero spectrum and exactly the same eigenvectors orthogonal to the kernel. For general data, the numerical computation of $\omega$ by means of the finite element method is extremely challenging, due to a combination of variational collapse ($\mathcal{M}$ is strongly indefinite) and the fact that finite element bases seldom satisfy the ansatz \eqref{ansatz_div}.
Several ingenious methods for the finite element treatment of the eigenproblem
\eqref{maxwell}-\eqref{ansatz_div} have been developed in the recent past.
Perhaps the most effective among these methods \cite{BFGP1999, Boffi-Act-Num}
consists in re-writing the spectral problem associated to $\mathcal{M}^2$ in a mixed
form and employing edge elements. This turns out to be linked to deep
mathematical ideas on the rigorous treatment of finite elements
\cite{Arnold:2010p3067} and it is at the core of an elegant geometrical
framework. Other approaches include, \cite{Bramble05} combining nodal elements
with a least squares formulation of \eqref{maxwell}-\eqref{ansatz_div}
re-written in weak form, \cite{BCJ09} employing continuous finite element spaces
of Taylor-Hood-type by coupling \eqref{maxwell} with \eqref{ansatz_div} via a
Lagrange multiplier, and \cite{BG11} enhancing the divergence of the electric
field in a fractional order negative Sobolev norm.
In spite of the fact that some of these techniques are convergent, unfortunately, none of them provides \textit{a priori} guaranteed one-sided bounds for the exact eigenfrequencies. In turns, detecting the presence of a spectral cluster (or even detecting multiplicities) is extremely difficult. Below we argue that the most basic formulation of the pollution-free technique described in \cite{ZM95} can be successfully implemented for determining certified upper and lower bounds for the eigenfrequencies and
corresponding approximated field phasors of \eqref{maxwell}. Remarkably the classical family of nodal finite elements renders sharp numerical approximations.
In Section~\ref{settings} we fix the rigorous setting of the self-adjoint
operator $\mathcal{M}$ and set
our concrete assumptions on the data of the problem. For these concrete
assumptions we consider both a region $\Omega$ with and without cylindrical
symmetries, generally non-convex and not even Lipschitz. In
Section~\ref{feceb} we describe the finite element realization of the
computation of complementary eigenvalue bounds. Based on this realization, in
Section~\ref{numstratnut} an algorithm providing certified
eigenvalue enclosures in a given interval is presented and analyzed. This
algorithm is then implemented and its results are reported in
Sections~\ref{convex-domains}-\ref{transmission}.
\section{Abstract setting of the Maxwell eigenvalue problem}
\label{settings}
\subsection{Concrete assumptions on the data}
\label{Assumptions}
The concrete assumptions on the data of equation \eqref{maxwell} made below are as follows.
The polyhedron $\Omega\subset \mathbb{R}^3$ will always be open, bounded and simply connected.
The permittivities will always be such that
\begin{equation} \label{bdd_away_from_0}
\epsilon,\,\frac 1\epsilon,\,\mu,\,\frac 1\mu \in L^\infty(\Omega).
\end{equation}
Without further mention, the non-zero spectrum of $\mathcal{M}$ will be assumed to be purely discrete and it does not accumulate at $\omega=0$. This hypothesis is verified, for example, whenever $\Omega$ is a polyhedron with a Lipschitz boundary, \cite[Corollary 3.49]{Monk2003} and \cite[Lemma
1.3]{1990Birman}. A more systematic analysis of the spectral properties
of $\mathcal{M}$ on more general regions $\Omega$ is being carried out elsewhere
\cite{BBBP}.
\subsection{The self-adjoint Maxwell operator}
\label{3dmaxwell}
We follow closely \cite{1990Birman}. Let
\[
\begin{aligned}
\mathcal{H}(\curl;\Omega)&=\left\{\boldsymbol u \in L^2(\Omega)^3 : \curl \boldsymbol u \in
L^2(\Omega)^3 \right\} \\
\mathcal{H}_0(\curl;\Omega)
& = \{ \boldsymbol u \in \mathcal{H}(\curl;\Omega) :\int_\Omega \curl \boldsymbol u \cdot \boldsymbol v =
\int_\Omega \boldsymbol u \cdot \curl \boldsymbol v
\quad \forall \boldsymbol v \in \mathcal{H}(\curl;\Omega)\}.
\end{aligned}
\]
The linear space $\mathcal{H}(\curl;\Omega)$ becomes a Hilbert space for the norm
\begin{equation*}
\| \boldsymbol u \|_{\curl,\Omega}=\left(\|\boldsymbol u\|_{0,\Omega}^2+\|\curl\boldsymbol u\|_{0,\Omega}^2\right)^{1/2},
\end{equation*}
where
\[
\|\boldsymbol v\|_{0,\Omega}=\left( \int_\Omega | \boldsymbol v|^2 \right)^{1/2}
\]
is the corresponding norm of $L^2(\Omega)^3$.
By virtue of Green's identity for the rotational \cite[Theorem~I.2.11]{1986Giraultetal}, if $\Omega$ is a Lipschitz
domain \cite[Notation~2.1]{ABDG98}, then $\boldsymbol u\in
\mathcal{H}_0(\curl;\Omega)$ if and only if $\boldsymbol u\in \mathcal{H}(\curl;\Omega)$ and
$\boldsymbol u\times\nn ={\mathbf 0} \;\mathrm{on}\;\bomega$.
Moreover
\begin{equation} \label{closure}
\mathcal{H}_0(\curl;\Omega)^3=\overline{C^\infty_0(\Omega)^3},
\end{equation}
where the closure is in the norm $\|\cdot\|_{\curl,\Omega}$.
A domain of self-adjointness of the operator associated to \eqref{maxwell} for
$\epsilon=\mu=1$ is \[\mathcal{D}_1=\mathcal{H}_0(\curl;\Omega)\times \mathcal{H}(\curl;\Omega)\subset L^2(\Omega)^6\]
and its action is given by
\[
\mathcal{M}_1 =\begin{bmatrix} 0 & i \curl \\ -i \curl
& 0\end{bmatrix} : \mathcal{D}_1 \longrightarrow L^2(\Omega)^6.
\]
Let
\[
\mathcal{P}=\begin{bmatrix}\epsilon^{1/2} I_{3\times 3}& 0\\
0 & \mu^{1/2} I_{3\times 3} \end{bmatrix}.
\]
Condition \eqref{bdd_away_from_0} ensures that $\mathcal{P}:L^2(\Omega)^6\longrightarrow L^2(\Omega)^6$ is bounded and invertible. Moreover, \[\left(\omega,\begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H}\end{bmatrix} \right)\in \mathbb{R}\times \mathcal{D}_1\] is a solution of \eqref{maxwell}, if and only if
\[\begin{bmatrix}\tilde{{\boldsymbol E}} \\ \tilde{{\boldsymbol H}}\end{bmatrix}=\mathcal{P}\begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H}\end{bmatrix}\] is a solution of
\[
\mathcal{P}^{-1}\mathcal{M}_1 \mathcal{P}^{-1}\begin{bmatrix} \tilde{\boldsymbol E} \\ \tilde{\boldsymbol H} \end{bmatrix} =\omega \begin{bmatrix} \tilde{\boldsymbol E} \\ \tilde{\boldsymbol H} \end{bmatrix}.
\]
Therefore $\mathcal{M}=\mathcal{P}^{-1}\mathcal{M}_1 \mathcal{P}^{-1}$ on $\operatorname{D}(\mathcal{M})=\mathcal{P}\mathcal{D}_1$ is the self-adjoint operator associated to \eqref{maxwell}.
As $\mathcal{M}$ anticommutes with complex conjugation, the spectrum
is symmetric with respect to $0$. Moreover, $\ker(\mathcal{M})$ is infinite dimensional, because it always contains the gradient space, see~\cite{1990Birman}.
\subsection{Isotropic cylindrical symmetries} \label{cylindrical}
If $\Omega=\tilde{\Omega}\times (0,\pi)$ for $\tilde{\Omega}\subset \mathbb{R}^2$ an open simply connected polygon, then \eqref{maxwell} decouples by separating the variables for $\epsilon=\mu=1$. In turns, a non-zero $\omega$ is an eigenvalue of $\mathcal{M}_1$, if and only if either $\omega^2=\lambda^2$ where $\lambda^2$ is a Dirichlet eigenvalue of the Laplacian in $\tilde{\Omega}$, or $\omega^2=\nu^2+\rho^2$ where $\nu^2$ is a non-zero Neumann eigenvalue of the Laplacian in $\tilde{\Omega}$ and $\rho\in \mathbb{N}$.
The Neumann problem can be re-written as ($\nu=\omega$)
\begin{equation} \label{maxwell_2d}
\left\{
\begin{aligned}
& \begin{aligned}&\curl {\boldsymbol E}= i\omega H \\ & \curl H=
-i\omega{\boldsymbol E} \end{aligned} & \text{in }\tilde{\Omega} \\
& {\boldsymbol E} \cdot \mathbf{t} = \mathbf{0}& \text{on } \partial \tilde{\Omega}\,,
\end{aligned}\right.
\end{equation}
for \[\left(\omega,\begin{bmatrix}{\boldsymbol E} \\ H\end{bmatrix}\right)\in \R\times (\tilde{\mathcal{D}}_1\setminus\{0\}).\]
Here
\[
{\boldsymbol E}=\begin{bmatrix}E_1 \\ E_2 \end{bmatrix}, \qquad \curl {\boldsymbol E}=\partial_x E_2-\partial_y E_1, \qquad
\curl H= \begin{bmatrix}\partial_y H \\ -\partial_x H\end{bmatrix},
\]
$\mathbf{t}$ is the unit tangent to $\partial \tilde{\Omega}$ and
\[
\tilde{\mathcal{D}}_1\!=\!
\left\{\!\boldsymbol u \in L^2(\Omega)^2 : \curl \boldsymbol u \in
L^2(\Omega)\text{ and }\boldsymbol u \cdot \mathbf{t} = \mathbf{0}\! \right\} \times
\left\{\!u \in L^2(\Omega) : \curl u \in
L^2(\Omega)^2\!\right\}.
\]
This two-dimensional Maxwell problem exhibits all the complications concerning spectral pollution as its three-dimensional counterpart.
We denote by $\tilde{\mathcal{M}}:\tilde{\mathcal{D}}\longrightarrow L^2(\tilde{\Omega})^3$ the self-adjoint operator associated to \eqref{maxwell_2d}. This operator has often been employed for tests which can then be validated against numerical calculations for the original Neumann Laplacian via the Galerkin method, \cite{2004Dauge}. Note that the latter is a semi-definite operator with a compact resolvent, so it does not exhibit spectral pollution.
\section{Finite element computation of the eigenvalue bounds} \label{feceb}
The basic setting of the general method proposed in \cite{ZM95} is achieved by deriving eigenvalue bounds directly from \cite[Theorem~1.1]{ZM95}, as described in \cite[Section~6]{davies-plum} and \cite{theoretical}. We will see next that, from this setting, a general finite element scheme for computing guaranteed bounds for the eigenvalues of $\mathcal{M}$ which are in the vicinity of a given non-zero $t\in \mathbb{R}$ can be established.
\subsection{Formulation of the weak problem and eigenvalue bounds} \label{weak_prob}
Let $\{\mathcal{T}_h\}_{h>0}$ be a family of shape-regular \cite{EG04} triangulations of $\overline{\Omega}$, where the elements
$K\in {\mathcal{T}}_h$ are simplexes with diameter $h_K$ and $h=\max_{K\in{\mathcal{T}}_h}h_K$.
For $r\ge 1$, let
\begin{align*}
\mathbf{V}_h^r &=\{\boldsymbol v_h\in C^0(\overline{\Omega})^3: \boldsymbol v_h|_K \in
\mathbb{P}_r(K)^3 \
\forall K\in \mathcal{T}_h \} \\
\mathbf{V}_{h,0}^r &=\{\boldsymbol v_h\in \mathbf{V}_h^r: \boldsymbol v_h\x\nn ={\mathbf 0}
\;\textrm{on}\;\bomega \}.
\end{align*}
Then
\begin{equation}\label{fe-space}
\L\equiv \L_{h}=\mathbf{V}_{h,0}^r\x \mathbf{V}_h^r \subset \mathcal{D}_1.
\end{equation}
For $t\in \mathbb{R}$, let $\mathfrak{m}^p_t:\mathcal{D}_1\times \mathcal{D}_1 \longrightarrow \mathbb{C}$ be given by
\[
\begin{aligned}
\mathfrak{m}^1_t\left( \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix}, \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix} \right)&=
\int_\Omega \left((\mathcal{M}_1-t\mathcal{P}^2) \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix} \right) \cdot \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix}
\\
\mathfrak{m}^2_t\left( \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix}, \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix} \right)&=
\int_\Omega \left((\mathcal{P}^{-1}\mathcal{M}_1-t\mathcal{P}) \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix}\right) \cdot
\left((\mathcal{P}^{-1}\mathcal{M}_1-t\mathcal{P}) \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix} \right)
\end{aligned}
\]
The following weak eigenvalue problem \cite{ZM95,davies-plum,theoretical} plays a central role below:
\begin{equation} \label{weak}
\begin{aligned}
&\text{find }
\left(\tau,\begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H}\end{bmatrix} \right)\in \mathbb{R}\times (\L\setminus \{0\}) \text{ such that } \\
&\mathfrak{m}^1_t\left( \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix}, \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix} \right) =
\tau \mathfrak{m}^2_t\left( \begin{bmatrix}{\boldsymbol E} \\ {\boldsymbol H} \end{bmatrix}, \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix} \right)
\qquad \forall \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix}\in \L .
\end{aligned}
\end{equation}
Let $m^{\pm}(t)\equiv m^\pm(t,h)$ be the number of negative and positive eigenvalues of \eqref{weak}, respectively. Let
$\tau^{\pm}_j(t)\equiv \tau^{\pm}_j(t,h)$,
\[\tau^-_1(t) \leq \ldots \leq \tau^-(t)_{m^-(t)}\] be the negative eigenvalues of \eqref{weak} and
\[\tau^+_{m^+(t)}(t)\leq \ldots \leq \tau^+_{1}(t)\] be the positive eigenvalues of \eqref{weak}, if they exist at all.
Let
\[
\rho_j^\pm(t,h)=t+\frac{1}{\tau_j^\pm(t)}.
\]
As we will see next, the latter quantities provide bounds for the spectrum of $\mathcal{M}$ in the vicinity of $t$.
By counting multiplicities, let
\[
\ldots \leq \nu_2^-(t) \leq \nu_1^-(t) < t < \nu_1^+(t) \leq \nu_2^+(t) \leq \ldots
\]
be the eigenvalues of $\mathcal{M}$ which are adjacent to $t$. That is
$\nu_j^-(t)$ is the $j$-th eigenvalue strictly to the left of $t$ and $\nu_j^+(t)$ is the $j$-th eigenvalue strictly
to the right of $t$. The following crucial statement is a direct consequence of \cite[Theorem~2.4]{ZM95} or \cite[Corollary~7]{theoretical} (see also
\cite[Theorem~11]{davies-plum}).
\begin{theorem} \label{bounds}
Let $t\in\R$. Then
\[
\rho_j^-(t,h) \leq \nu_j^-(t) \quad \forall j=1,\ldots,m^-(t) \quad \text{and} \quad \nu_j^+(t) \leq \rho_j^+(t,h)
\quad \forall j=1,\ldots,m^+(t).
\]
\end{theorem}
\begin{remark} \label{2d}
In the case of the lower-dimensional Maxwell operator $\tilde{\mathcal{M}_1}$, the finite element spaces on a
corresponding triangulation $\mathcal{T}_h$ of $\tilde{\Omega}$ are chosen as
\[
\L_h=\left\{\begin{bmatrix}\boldsymbol u_h \\ v_h \end{bmatrix} \in C^0\left(\overline{\tilde{\Omega}}\right)^{2+1}: \left.\begin{bmatrix}\boldsymbol u_h \\ v_h \end{bmatrix} \right|_K \in
\mathbb{P}_r(K)^{2+1} \ \forall K\in \mathcal{T}_h \text{ and } \boldsymbol u_h\cdot \mathbf{t} =0
\text{ on } \partial\tilde{\Omega} \right\}.
\]
The weak problem analogous to \eqref{weak} and a corresponding version of Theorem~\ref{bounds} (and further statements below) are formulated by substituting $\mathfrak{m}^p_t$ with the corresponding lower-dimensional forms.
\end{remark}
\subsection{Convergence of the eigenvalue bounds} \label{conv_evb}
According to \cite[Theorem~12]{theoretical}, if $\L$ captures an eigenspace of $\mathcal{M}$
within a certain order of precision $\mathcal{O}(\varepsilon)$ for small $\varepsilon$, then
the eigenvalue bounds found in Theorem~\ref{bounds} are within $\mathcal{O}(\varepsilon^2)$.
We now show a consequence of this statement in the present setting.
Consider an open bounded segment $J\subset\R$, such that $0\not\in J$. Denote by $\mathcal{E}_J$ the eigenspace associated to this segment and assume that $t\in J$. Here and elsewhere the relevant set where the indices $j$ move is
\[
\mathcal{F}_J^{\pm}(t)=\{j\in \mathbb{N}:\nu_j^\pm(t)\in J\}.
\]
\begin{theorem} \label{order_maxwell}
Let $r\in \mathbb{N}$ be fixed. Then
\[
\lim_{h\to 0} \left| \rho_j^\pm (t,h)-\nu_j^\pm(t)\right|= 0 \qquad \forall j\in \mathcal{F}_J^{\pm}(t).
\]
If in addition $\mathcal{P}^{-1}\mathcal{E}_J \subseteq \mathcal{H}^{r+1}(\Omega)^6$, then there exist $C_t^\pm\equiv C_t^\pm(r) >0$ such that
\begin{equation} \label{order_conv_evalue}
\left| \rho_j^\pm (t,h)-\nu_j^\pm(t) \right|\leq
C_t^\pm h^{2r} \qquad \forall j\in \mathcal{F}_J^{\pm}(t)
\end{equation}
for $h$ sufficiently small.
\end{theorem}
\begin{proof}
By combining \cite[Theorem~3.26]{Monk2003} with \eqref{closure} and standard interpolation estimates (cf. \cite{EG04}), it follows that \[\text{ for any } \begin{bmatrix}\boldsymbol F \\ \boldsymbol G \end{bmatrix}\in \mathcal{D}_1 \text{ there exists } \begin{bmatrix}\boldsymbol F_h \\ \boldsymbol G_h\end{bmatrix} \in \L_h\]
such that
\begin{equation} \label{approx_max_fe_1}
\lim_{h\to 0}\Big( \|\boldsymbol F-\boldsymbol F_h\|_{\curl,\Omega} +
\|\boldsymbol G-\boldsymbol G_h\|_{\curl,\Omega}\Big) = 0\,.
\end{equation}
Since $\mathcal{P}$ is a bounded operator, then for any
\[\begin{bmatrix}\tilde{\boldsymbol F} \\ \tilde{\boldsymbol G}\end{bmatrix}={\mathcal P}\begin{bmatrix}\boldsymbol F \\ \boldsymbol G\end{bmatrix}\in \operatorname{D}(\mathcal{M}),\] we have \begin{equation} \label{approx_max_fe_11}
\lim_{h\to 0}\left( \left\|\mathcal{M}\begin{bmatrix}
\tilde{\boldsymbol F}-\tilde{\boldsymbol F}_h\\
\tilde{\boldsymbol G}-\tilde{\boldsymbol G}_h\end{bmatrix}\right\|_{0,\Omega}
+ \left\|\begin{bmatrix} \tilde{\boldsymbol F}-\tilde{\boldsymbol F}_h\\
\tilde{\boldsymbol G}-\tilde{\boldsymbol G}_h\end{bmatrix}\right\|_{0,\Omega} \right) = 0
\end{equation}
where
\[\begin{bmatrix} \tilde{\boldsymbol F}_h \\ \tilde{\boldsymbol G}_h \end{bmatrix}= \mathcal{P} \begin{bmatrix} \boldsymbol F_h \\ \boldsymbol G_h \end{bmatrix} \in \operatorname{D}(\mathcal{M}).\]
In turns, this is exactly the hypothesis required in \cite[Theorem~12]{theoretical} which ensures the claimed statement.
Let $\mathcal{I}_{r,h}$ denote the interpolation operator associated to the finite element spaces $\mathbf{V}^r_h$ (cf. \cite{EG04}) and let
\[
\begin{bmatrix} {\boldsymbol F}_h \\ {\boldsymbol G}_h \end{bmatrix}=\mathcal{I}_{r,h} \begin{bmatrix} {\boldsymbol F} \\ {\boldsymbol G} \end{bmatrix}.
\]
If
\[
\begin{bmatrix} \boldsymbol F \\ \boldsymbol G \end{bmatrix}\in \mathcal{E}_J \subset \mathcal{H}^{r+1}(\Omega)^6,
\]
then
\begin{equation} \label{approx_max_fe_2}
\left\|\begin{bmatrix} {\boldsymbol F}-{\boldsymbol F}_h\\
{\boldsymbol G}-{\boldsymbol G}_h\end{bmatrix}\right\|_{\curl,\Omega}\leq c(r)
h^{r}\left\|\begin{bmatrix}\boldsymbol F \\ \boldsymbol G\end{bmatrix} \right\|_{r+1,\Omega} ,
\end{equation}
so that
\begin{equation} \label{approx_max_fe_22}
\left\| \mathcal{M} \begin{bmatrix} \tilde{\boldsymbol F}-\tilde{\boldsymbol F}_h\\
\tilde{\boldsymbol G}-\tilde{\boldsymbol G}_h\end{bmatrix}\right\|_{0,\Omega}
+ \left\|\begin{bmatrix} \tilde{\boldsymbol F}-\tilde{\boldsymbol F}_h\\
\tilde{\boldsymbol G}-\tilde{\boldsymbol G}_h\end{bmatrix}\right\|_{0,\Omega}\leq C h^{r}
\end{equation}
where $C>0$ is a constant independent of $h$.
This is precisely the condition \cite[(35)]{theoretical}.
Thus, \cite[Theorem~12]{theoretical} ensures the claimed statement.
\end{proof}
\subsection{Eigenfunctions}
The statement established in \cite[Corollary~13]{theoretical}, provides an insight on how the eigenspace $\mathcal{E}_J$ in the framework of Theorem~\ref{order_maxwell} is also captured by the trial subspaces $\L$ as $h\to 0$. Let
\[
\dist_{1}[({\boldsymbol F},{\boldsymbol G}),\mathcal{E}] =\inf_{\begin{bmatrix}{\boldsymbol X} \\ {\boldsymbol Y}\end{bmatrix} \in \mathcal{E}}
\left\| \begin{bmatrix} {\boldsymbol F}- {\boldsymbol X}\\ {\boldsymbol G}- {\boldsymbol Y} \end{bmatrix} \right\|_{\curl,\Omega}
\]
be the Hausdorff distance between a given vector
\[
\begin{bmatrix}\boldsymbol F \\ \boldsymbol G\end{bmatrix}\in \mathcal{D}_1 \qquad \text{ and } \qquad \mathcal{E}\subseteq \mathcal{D}_1.
\]
Denote by
\[
\begin{bmatrix}\boldsymbol F^\pm_{j}(t,h) \\ {\boldsymbol G}^\pm_{j}(t,h)\end{bmatrix}\in {\L}_h
\]
the eigenvectors of \eqref{weak} associated to $\tau_j^\pm(t)$ respectively and
assume that
\[
\left\|\begin{bmatrix}\boldsymbol F^\pm_{j}(t,h) \\ {\boldsymbol G}^\pm_{j}(t,h)\end{bmatrix}\right\|_{0,\Omega}=1.
\]
Then, the following result concerning approximation of eigenspaces can be stated.
\begin{theorem} \label{order_maxwell_eigenfunction}
Let $r\in \mathbb{N}$ be fixed. Then,
\[
\lim_{h\to 0} \dist_{1} [({\boldsymbol F}^\pm_{j}(t,h) ,
{\boldsymbol G}^\pm_{j}(t,h)),\mathcal{E}_{J}] = 0.
\]
If in addition $\mathcal{P}^{-1}\mathcal{E}_J \subseteq \mathcal{H}^{r+1}(\Omega)^6$, then there exist
$C_t^\pm(r) >0$ such that
\[
\dist_{1} [({\boldsymbol F}^\pm_{j}(t,h) ,
{\boldsymbol G}^\pm_{j}(t,h)),\mathcal{E}_{J}] \leq C^{\pm}_t(r) h^r
\]
for $h$ sufficiently small.
\end{theorem}
\begin{proof}
Proceed as in the proof of Theorem~\ref{order_maxwell} in order to verify the hypotheses of \cite[Corollary~13]{theoretical}.
\end{proof}
Convergence of the eigenvalue bounds in Theorem~\ref{bounds} are therefore ensured, in spite of the fact that $\L$ are spaces of nodal finite elements with no particular mesh structure.
Note that this is guaranteed, even in the case where $\epsilon$ and $\mu$ are rough, however, since $\mathcal{P}^{-1}\mathcal{E}_J \not\subseteq \mathcal{H}^{r+1}(\Omega)^6$ unless these coefficients are smooth themselves, an estimate on the convergence rate in this situation is beyond the scope of Theorem~\ref{order_maxwell_eigenfunction}. For a highly heterogeneous medium, a deterioration of the convergence speed is to be expected.
We remark that the above analysis relies on the regularity of the eigenspaces associated to the interval $J$ only. Thus, for non-convex $\Omega$,
this allows the possibility of approximating eigenvalues associated to regular eigenfunctions with high
accuracy, if some a priori information about their location is at hand.
\section{A certified numerical strategy} \label{numstratnut}
Let us now describe a procedure which, in an asymptotic regime,
renders small intervals which are guaranteed to contain spectral points. Convergence will be derived from Theorem~\ref{order_maxwell}.
Denote by $0<t_{\mathrm{up}}<t_{\mathrm{low}}$ the corresponding parameters $t$
in the weak problem \eqref{weak}, which are set for
computing $\rho^-_j(t_{\mathrm{low}},h)$ (lower bounds) and $\rho^+_j(t_{\mathrm{up}},h)$ (upper bounds) in the segment $(t_{\mathrm{up}},t_{\mathrm{low}})$. The scheme described next aims at finding intervals of enclosure for the eigenvalues of $\mathcal{M}$ which lie in this segment, for a prescribed tolerance set by the parameter $\delta>0$. According to Lemma~\ref{lem_certified} below, these intervals will be certified in the regime $\delta\to 0$.
\begin{algorithm} \label{alg3}
\
\begin{itemize}
\item[] \underline{Input}.
\begin{itemize}
\item Initial $t_{\mathrm{up}}>0$.
\item Initial $t_{\mathrm{low}}>t_{\mathrm{up}}$ such that $t_{\mathrm{low}}-t_{\mathrm{up}}$ is fairly large.
\item A sub-family $\mathcal{F}$ of finite element spaces ${\mathcal{L}}_h$ as in
\eqref{fe-space}, dense as $h\to 0$.
\item A tolerance $\delta>0$ fairly small compared with $t_{\mathrm{low}}-t_{\mathrm{up}}$.
\end{itemize}
\item[] \underline{Output}.
\begin{itemize}
\item A prediction $\tilde{m}(\delta)\in \mathbb{N}$ of $\tr \1_{(t_{\mathrm{up}},t_{\mathrm{low}})}(\mathcal{M})$.
\item Predictions $\omega_{j,\delta}^\pm$ of the endpoints of enclosures for the eigenvalues in $\spec(\mathcal{M})\cap (t_{\mathrm{up}},t_{\mathrm{low}})$, such that
$0<\omega_{j,\delta}^+- \omega_{j,\delta}^-< \delta$ for $j=1,\ldots,\tilde{m}(\delta)$.
\end{itemize}
\item[] \underline{Steps}.
\begin{enumerate}
\item Set initial ${\mathcal{L}}_h\in \mathcal{F}$.
\item \label{alg3a} While
\[
\rho^+_{j,h} -\rho^-_{j,h} \geq \delta \text{ or } \rho_{j,h}^->\rho_{j,h}^+ \text{ for some } j=1,\ldots,\tilde{m},
\]
do \ref{alg3b} - \ref{alg3d}.
\item \label{alg3b} Compute
\[
\rho_{j,h}^+=\rho_j^+(t_{\mathrm{up}},h) \qquad \text{for} \qquad
j=1,\ldots,\tilde{m}_{\mathrm{up}}
\]
where $\tilde{m}_{\mathrm{up}}$ is such that $\rho_{\tilde{m}_{\mathrm{up}},h}^+<t_{\mathrm{low}}$ and
\[\rho_{\tilde{m}_{\mathrm{up}}+1}^+(t_{\mathrm{up}},h)\geq t_{\mathrm{low}}.\]
\item \label{alg3c} Compute
\[
\rho_{\tilde{m}_{\mathrm{low}}-k+1,h}^-=\rho_{k}^-(t_{\mathrm{low}},h) \qquad \text{for}
\qquad k=1,\ldots,\tilde{m}_{\mathrm{low}}
\]
where $\tilde{m}_{\mathrm{low}}$ is such that $\rho_{\tilde{m}_{\mathrm{low}},h}^->t_{\mathrm{up}}$ and
\[
\rho_{\tilde{m}_{\mathrm{low}}+1}^-(t_{\mathrm{low}},h)\leq t_{\mathrm{up}}.
\]
\item \label{alg3d} If $\tilde{m}_{\mathrm{low}}\not=\tilde{m}_{\mathrm{up}}$, decrease $h$, set new $\mathcal{L}_h\in \mathcal{F}$ and
go back to \ref{alg3b}. Otherwise set $\tilde{m}=\tilde{m}_{\mathrm{low}}=\tilde{m}_{\mathrm{up}}$, decrease $h$, set new $\mathcal{L}_h\in \mathcal{F}$ and continue from \ref{alg3a}.
\item Exit with $\tilde{m}(\delta)=\tilde{m}$ and $\omega_{j,\delta}^\pm=\rho_{j,h}^\pm$
for $j=1,\ldots,\tilde{m}$.
\end{enumerate}
\end{itemize}
\end{algorithm}
Assume that
\[
(t_{\mathrm{up}},t_{\mathrm{low}})\cap \spec(\mathcal{M})=\{\omega_{k+1},\ldots,\omega_{k+m}\}
\]
where
\[
m=\tr \1_{(t_{\mathrm{up}},t_{\mathrm{low}})}(\mathcal{M})>0 \qquad \text{ and } \qquad k\geq 0.
\]
\textit{A priori}, an interval $(\omega_{j,\delta}^-,\omega_{j,\delta}^+)$ obtained as the output of
Procedure~\ref{alg3} is not guaranteed to have a non-empty intersection with the spectrum of $\mathcal{M}$ or in fact include precisely the eigenvalue $\omega_{k+j}$. However, as it is established by the following lemma, the latter is certainly true for $\delta$ small enough.
\begin{lemma} \label{lem_certified}
There exist $t^0>0$ and $\delta_0>0$,
ensuring
all the next items for all $t_{\mathrm{low}}\geq t^0$ and $\delta<\delta_0$.
\begin{enumerate}
\item The conditional loop in Procedure~\ref{alg3} always exits in the regime $h\to 0$.
\item $m(\delta)=m$.
\item $\omega^{-}_{j,\delta}\leq \omega_{k+j} \leq \omega_{j,\delta}^+$ for all $j=1,\ldots, m$.
\end{enumerate}
\end{lemma}
\begin{proof}
Set $t^0>\omega_1^+(t_{\mathrm{up}})$ sufficiently large, ensuring
$
m \neq 0
$
for any $t_{\mathrm{low}}\geq t^0$. Since $\nu_j^+(t_{\mathrm{up}})=\omega_{k+j}=\nu^-_{m-j+1}(t_{\mathrm{low}})$ for all
$j=1,\ldots,m$, Theorem~\ref{order_maxwell} alongside with the assumption on
$\mathcal{F}$, ensures the existence of $\rho^{\pm}_{j,h}$ in
Procedure~\ref{alg3}-\ref{alg3b} and \ref{alg3c}, for all $j=1,\ldots, m$
whenever $h$ is small enough. Moreover
\[
\rho_{j,h}^+\downarrow \omega_{k+j} \qquad \text{and} \qquad \rho^-_{j,h}\uparrow \omega_{k+j}
\qquad \text{as } h\to 0
\]
as needed.
\end{proof}
If the eigenfunctions of $\mathcal{M}$ lie in $\mathcal{H}^{r+1}(\Omega)^6$, then
\[
\rho_{j,h}^+ - \rho_{j,h}^- =O(h^{2r}).
\]
This means that the exit rate of the conditional loop in Procedure~\ref{alg3} is
also $O(h^{2r})$ as $h\to 0$.
Observe that in the above procedure, a good choice of $t_{\mathrm{up}}$ and $t_{\mathrm{low}}$ has a noticeable impact in performance. See Section~\ref{non-lip}. The results of the recent manuscript \cite{BoHo2013}, suggest\footnote{An upper bound is provided in \cite[Corollary~11]{theoretical}
and the value $-1$ seems to be the right exponent.} that the constants involved in the
estimates of Theorem~\ref{order_maxwell} are
of order $|t-\nu^\pm_1(t)|^{-1}$. Table~\ref{tab_slit_square} strongly
suggest that the accuracy improves significantly, as $t_{\mathrm{up}}\downarrow
\nu_1^-(t_{\mathrm{up}})$ and $t_{\mathrm{low}}\uparrow \nu_1^+(t_{\mathrm{low}})$.
In the subsequent sections we proceed to illustrate the practical applicability
of the ideas discussed above by means of several examples.
Two canonical references for benchmarks on the Maxwell
eigenvalue problem are \cite{2004Dauge} and \cite{BFGP1999}. We validate some of our
numerical bounds against these benchmarks. Everywhere below we will write $\omega_j^\pm\equiv \omega_{j,\delta}^\pm$ (see Procedure \ref{alg3}) where $\delta$ might or might not be specified. In the latter case, we have taken its value small enough to ensure the reported accuracy.
We consider constant $\epsilon=\mu=1$ in sections~\ref{convex-domains} and \ref{non-con}, and
$\epsilon\not=1$ with jumps in Section~\ref{transmission}.
\section{Convex domains}\label{convex-domains}
The eigenfunctions of \eqref{maxwell} and \eqref{maxwell_2d} are regular in the
interior of a convex domain, see~\cite{Monk2003,ABDG98}. In this, the best possible case scenario, the method of sections~\ref{feceb} and \ref{numstratnut} achieves an optimal order of convergence for finite elements.
Without further mention, the following convention will be in place here and everywhere below. The index $k$ for eigenvalues and eigenvalue bounds will be used, whenever multiplicities are not counted. Otherwise the index $j$ (as in previous sections) will be used.
\subsection{The square}
\label{accuracy}
\begin{figure}[t]
\centerline{
\includegraphics[height=8cm, angle=0]{\dir many_eigenvalues_square_bw.pdf}}
\caption{Semi-log graph associated to $\tilde{\Omega}_\mathrm{sqr}$. Vertical axis:
$\omega^+_k -\omega^-_k$. Horizontal axis: eigenvalue index $k$ (not counting
multiplicity). Here we consider elements of order $r=1,3,5$ on unstructured uniform
meshes rendering roughly the same number of degrees of freedom. For each $r$,
we have used exactly the same trial subspace $\mathcal{L}_h$ for all the eigenvalues.
\label{many_eigenvalues}
}
\end{figure}
Let
$
\tilde{\Omega}\equiv \tilde{\Omega}_\mathrm{sqr}=(0,\pi)^2\subset \RR^2.
$
The eigenvalues
of $\tilde{\mathcal{M}}$ are $\omega=\pm \sqrt{l^2+m^2}$ for $l,m\in \N\cup\{0\}$. Pick
\[
t_{\mathrm{up}}=\frac{1}{4}\omega_{k-1} + \frac{3}{4} \omega_k \qquad \text{and}
\qquad t_{\mathrm{low}}=\frac{3}{4}\omega_{k} + \frac{1}{4} \omega_{k+1}
\]
to machine precision. In our first experiment we have computed enclosure widths
$\omega_k^+-\omega_k^-$ for $k=1,\ldots,100$
and $r=1,3,5$. We have chosen $h=h(r)$ such that the corresponding trial subspaces have
roughly the same dimension $\approx61$K. We have then found all the eigenvalue bounds for a fixed $r$, from exactly the same
trial subspace. Figure~\ref{many_eigenvalues} shows the outcomes of this
experiment. In the graph, we have excluded enclosures with size above $10^{-1}$.
As it is natural to expect, for a fixed $\L_h$, the accuracy
deteriorates as the eigenvalue counting number increases: high energy
eigenfunctions have more oscillations, so their approximation requires a higher number of degrees of freedom. The accuracy increases with the polynomial order.
The first 100 eigenvalues are approximated fairly accurately (note
that $\omega_{(k=100)}=\sqrt{261}$ with polynomial order $r=5$).
\subsection{The slashed cube}
Let
$
\Omega\equiv \Omega_{\mathrm{sla}}=(0,\pi)^3 \setminus T \subset \mathbb{R}^3,
$
where $T$ is the closed tetrahedron with vertices $(0,0,0),(\pi/2,0,0),(0,\pi/2,0)$ and $(0,0,\pi/2)$. This domain does not have
symmetries allowing a reduction into two-dimensions.
\begin{figure}[t]
\begin{minipage}{5.5cm}
\includegraphics[height=5.5cm, angle=0]{\dir mesh_cut}
\end{minipage} \hspace{1cm}
\begin{tabular}{c|c|c|c}
$j$ & $\omega_j\ _-^+$ & $t_{\mathrm{up}}$ ($l$) & $t_{\mathrm{low}}$ ($l$) \\
\hline
$1$ & $1.412^{236}_{000}$ & $0.5$ ($1$) & $1.6$ ($3$) \\
$2$ & $1.430^{672}_{560}$ & $0.5$ ($2$) & $1.6$ ($2$) \\
$3$ & $1.430^{673}_{577}$ & $0.5$ ($3$) & $1.6$ ($1$) \\
$4$ & $1.755^{308}_{043}$ & $1.5$ ($1$) & $2.1$ ($2$) \\
$5$ & $1.755^{329}_{063}$ & $1.5$ ($2$) & $2.1$ ($1$) \\
$6$ & $2.22^{200}_{053}$ & $1.8$ ($1$)& $2.6$ ($5$) \\
$7$ & $2.237^{667}_{434}$ & $1.8$ ($2$) & $2.6$ ($4$) \\
$8$ & $2.237^{684}_{459}$ & $1.8$ ($3$) & $2.6$ ($3$) \\
$9$ & $2.239^{533}_{387}$ & $1.8$ ($4$) & $2.6$ ($2$) \\
$10$ & $2.270^{778}_{558}$ & $1.8$ ($5$) & $2.6$ ($1$) \\
\hline
\end{tabular} \hspace{1cm}
\caption{Benchmark spectral approximation for $\Omega_{\mathrm{sla}}$. In the table we
compute interval of enclosure for the first 10 eigenvalues, by means of an implementation of Procedure~\ref{alg3}. The
trial spaces are made of Lagrange elements of order $r=3$. The final mesh is the
one shown on the right side. Total number of DOF=117102. \label{cut_cube_table}}
\end{figure}
In our first experiment on this region, we determine benchmark eigenvalue
enclosures for \eqref{maxwell}. The table in Figure~\ref{cut_cube_table} shows the outcomes of implementing a numerical scheme
based on Procedure~\ref{alg3}. We have iterated our algorithm for three fixed
choices of $t_{\mathrm{up}}$ and $t_{\mathrm{low}}$ (third and fourth columns), with
$\delta=10^{-2}$. We have picked the family of meshes so that no more than five
iterations were required to achieve the needed accuracy. We have chosen trial spaces made out of Lagrange elements of
order $r=3$. All the final eigenvalue enclosures have a length of at most
$2\times 10^{-3}$. The mesh used in the last iteration is depicted on the
left of Figure~\ref{cut_cube_table}. The parameter $l$ in
this table counts the number of eigenvalues to the right of $t_\mathrm{up}$ or to the
left of $t_\mathrm{low}$, respectively.
From the table, it is natural to conjecture that there is a cluster of eigenvalues at the
bottom of the positive spectrum near $\sqrt{2}$. The latter is the first
positive eigenvalue for $\Omega\equiv \Omega_{\mathrm{cbe}}=(0,\pi)^3$, which is of multiplicity 3 for that region. See \cite[Section~5.1]{theoretical}. As we deform $\Omega_{\mathrm{cbe}}$ into $\Omega_{\mathrm{sla}}$, it appears that
this eigenvalue splits into a single eigenvalue at the bottom of the spectrum
and a seemingly double eigenvalue slightly above it.
Another cluster occurs at $\omega_4$ and $\omega_5$ with strong indication that
this is a double eigenvalue. This pair is near $\sqrt{3}$, the second eigenvalue
for $\Omega_{\mathrm{cbe}}$, which is indeed double. The next eigenvalues for $\Omega_{\mathrm{cbe}}$ are
$2$ and $\sqrt{5}$ with total multiplicity 5. We conjecture that $\omega_j$ for $j=6,\ldots,10$ are indeed perturbations of these eigenvalues.
For our second experiment on the region $\Omega_{\mathrm{sla}}$, we have estimated numerically the
electromagnetic fields corresponding to index up to $j=6$.
The purpose of the experiment is to set benchmarks
for the eigenfunctions on $\Omega_{\mathrm{sla}}$ and simultaneously illustrate
Theorem~\ref{order_maxwell_eigenfunction}.
In Figure~\ref{cornerless_1} we depict the density of electric and magnetic
fields, $|{\boldsymbol E}|$ and $|{\boldsymbol H}|$ both re-scaled to having maximum equal to 1. We also
show arrows pointing towards the direction of these fields on $\partial
\Omega_{\mathrm{sla}}$. The mesh employed for these calculations is the one shown in
Figure~\ref{cut_cube_table}.
It is remarkable that for both experiments on $\Omega_\mathrm{sla}$,
a reasonable accuracy has been achieved even for the fairly coarse mesh depicted.
\section{Non-convex domains} \label{non-con}
The numerical approximation of the eigenfrequencies and electromagnetic fields
in the resonant cavity is known to be extremely challenging when the domain is not convex.
The main reason for this is the fact that the
electromagnetic field might have a singularity and a low degree of regularity at
re-entrant corners. See for example the discussion after
\cite[Lemma~3.56]{Monk2003} and references therein.
In some of the examples of this section we consider a mesh adapted to the geometry of the region.
However, we do not pursue any specialized mesh refinement strategy.
We show below that, even in the case where there is poor approximation due to
low regularity of the eigenspace, the scheme in Procedure~\ref{alg3} provides
a stable approximation.
\begin{figure}
\centerline{
\begin{tabular}{c|c|c|cc}
$j$ &$\omega_j$ from \cite{BFGP1999} & $\omega_j\ _-^+$ & $t_{\mathrm{up}}$ ($l$) &
$t_{\mathrm{low}}$ ($l$) \\
& (from \cite{2004Dauge}) & & & \\
\hline
$1$ &$0.768192684$ & $0.773334_{694}^{991}$ & $0.1$ ($1$) & $2.1$ ($4$) \\
& ($0.773334985176$) &&& \\
$2$ &$1.196779010$ & $1.1967827557_{026}^{761}$ & $0.1$ ($2$) & $2.1$ ($3$) \\
& ($1.19678275574$) &&& \\
$3$ &$1.999784988$ & $_{1.99999999933}^{2.00000000064}$ & $1.5$ ($1$) & $2.5$
($4$) \\
& ($2.00000000000$) &&& \\
$4$ &$1.999784988$ & $_{1.99999999936}^{2.00000000067}$ & $1.5$ ($2$) & $2.5$
($3$) \\
& ($2.00000000000$) &&& \\
$5$ &$2.148306309$ & $2.14848368_{199}^{365}$ & $1.5$ ($3$) & $3.1$ ($5$) \\
& ($2.14848368266$) &&& \\
$6$ &$2.252760528$ & $2.25729_{776}^{896}$ & $1.5$ ($4$) & $3.1$ ($4$) \\
$7$ &$2.828075317$ & $2.8284271_{186}^{354}$ & $1.5$ ($5$) & $3.7$ ($4$) \\
$8$ &$2.938491109$ & $2.94671_{112}^{343}$ & $1.5$ ($6$) & $3.7$ ($3$) \\
$9$ &$3.075901493$ & $3.0758929_{571}^{738}$ & $1.5$ ($7$) & $3.7$ ($2$) \\
$10$ &$3.390427701$ & $3.3980_{676}^{724}$ & $1.5$ ($8$) & $3.7$ ($1$) \\
\hline
\end{tabular}
\begin{minipage}{7.5cm}
\includegraphics[height=7cm, angle=0]{\dir mesh_lshape_refined}
\end{minipage}
}
\caption{Enclosures for the first 10 positive eigenvalues of $\tilde{\mathcal{M}}$ on
$\tilde{\Omega}_{\mathrm{L}}$. The next eigenvalue is above 3.7. Here
Procedure~\ref{alg3} has been implemented on Lagrange elements of order 3. The
final mesh shown on the right has a number of DOF=56055. The mesh has a maximum
element size $h=0.1$ and has been refined at $(\pi/2,\pi/2)\in \partial \tilde{\Omega}_{\mathrm{L}}$.
For comparison
on the second column we include the eigenvalue estimations
found in \cite{BFGP1999} and \cite{2004Dauge}.\label{table_eigenvalues_lshape}}
\end{figure}
\subsection{A re-entrant corner in two dimensions} \label{Lshape}
The region \[\tilde{\Omega}\equiv \tilde{\Omega}_{\mathrm{L}}=(0,\pi)^2\setminus [0,\pi/2]^2\subset \R^2\]
is a classical benchmark domain both for the Maxwell and the Helmholtz problems,
and it has been extensively examined in the past. Numerical computations for the
eigenvalues of $\tilde{\mathcal{M}}$, via an
implementation based on a mixed formulation of \eqref{maxwell_2d} and edge finite elements, were reported in \cite[Table~5]{BFGP1999}. See also \cite{2004Dauge}. We now show estimation of sharp enclosures for these eigenvalues by means of
the method described in sections~\ref{convex-domains} and \ref{non-con}.
\begin{figure}[t]
\centerline{\includegraphics[height=8cm, angle=0]{\dir order_conv_lshape_bw}}
\caption{
Compared order of approximation for different eigenvalues in the region
$\tilde \Omega_{\mathrm{L}}$. The log-log plot shows residual versus maximum element size $h$
for the calculation of enclosures for $\omega_j$ where $j=1,2,3$ and
$\L$ is generated by Lagrange elements of order $r=3$ and $r=5$. Note
that $({\boldsymbol E},H)\not \in \mathcal{H}^s(\Omega_{\mathrm{L}})^3$ for $j=1$ and $s=1$, and
for $j=2$ and $s=1.5$. On the other hand, for $j=3$ we have $({\boldsymbol E},H)$
smooth, as the eigenfunction is also solution of \eqref{maxwell} on a square of
side $\pi/2$.
\label{lshape_orders}
}
\end{figure}
For the next set of experiments we consider unstructured triangulations of the
domain, refined around the re-entrant corner $(\pi/2,\pi/2)\in \partial \tilde\Omega_{\mathrm{L}}$. The polynomial order is set to
$r=3$. Figures~\ref{table_eigenvalues_lshape}, \ref{lshape_orders} and \ref{Lshape_eigenfunctions}
summarize our findings.
We produced the table in Figure~\ref{table_eigenvalues_lshape} by implementing
Procedure~\ref{alg3} in the same fashion as
for the case of $\Omega_\mathrm{sla}$ described previously.
For comparison, in the second column of this table we have included the benchmark
eigenvalue estimations
found in \cite{BFGP1999} and \cite{2004Dauge}. Note that some of the
approximations made by means of the mixed formulation are lower bounds of the true
eigenvalues, and some (see the row for $j=9$ in the table) are upper bounds. This confirms that the latter approach is in general un-hierarchical
as previously suggested in the literature.
From the third column of the table, it is clear that the accuracy depends on the
regularity of the corresponding eigenspaces.
The eigenfunctions associated to $\omega=2$ and $\omega=\sqrt{8}$
are found by the translation and gluing in an appropriate fashion, of eigenfunctions in the sub-region $\tilde{\Omega}=(0,\pi/2)^2\subset \tilde{\Omega}_{\mathrm{L}}$.
These eigenfunctions are smooth in the interior of $\tilde \Omega_\mathrm{L}$ and they achieve a maximum order of convergence.
The eigenfunctions associated to $\omega_1$ and
$\omega_2$, on the other hand, are singular at the re-entrant corner. Moreover, the electric field component for index $j=1$ is known to be outside $\mathcal{H}^1(\Omega_{\mathrm{L}})^2$ while that for index $j=2$ is in
$\mathcal{H}^1(\tilde \Omega_{\mathrm{L}})^2$. This explains the significant gain in accuracy in the calculation of $\omega_2$ with respect to the one for $\omega_1$. Here the computation of the eigenvalues with smooth eigenspace ($j=3,4$ or $7$) is less accurate than that for the index $j=2$, because of the mesh chosen.
Figure~\ref{lshape_orders} depicts
in log-log scale residuals versus maximum element size. We have considered here
Lagrange elements of order $r=3$ and $r=5$. The hierarchy of meshes (not shown)
was chosen unstructured, but with an uniform distribution of nodes. Since the
eigenfunctions associated to $\omega_1$ and $\omega_2$ have a limited regularity,
then there is no noticeable improvement on the convergence order as $r$ changes from $3$ to $5$.
Since the third eigenfunction is smooth, it does obey the estimate \eqref{order_conv_evalue}.
Benchmark approximated eigenfunctions are depicted in
Figure~\ref{Lshape_eigenfunctions}. The mesh employed to produce
these graphs is the one shown on the right of Figure~\ref{table_eigenvalues_lshape}. As some of the electric fields have a
singularity at $(\pi/2,\pi/2)\in\partial \tilde{\Omega}_{\mathrm{L}}$ we have re-scaled each individual plot to a
range in the interval $[0,1]$.
\begin{figure}
\begin{minipage}{7cm}
\includegraphics[height=7cm, angle=0]{\dir mesh_fichera_2}
\end{minipage} \hspace{1cm}
\begin{tabular}{c|c}
$j$ & $\omega_j\ _-^+$ \\
\hline
$1$ & $1.1^{441}_{298}$ \\ \hline
$2$ & $1.54^{391}_{228}$ \\
$3$ & $1.54^{392}_{233}$ \\ \hline
$4$ & $2.0^{815}_{690}$\\
$5$ & $2.0^{820}_{796}$ \\
$6$ & $2.0^{820}_{796}$\\ \hline
$7$ & $2.2^{348}_{180}$ \\
$8$ & $2.2^{348}_{187}$ \\ \hline
$9$ & $2.32^{679}_{547}$\\ \hline
$10$ & $2.33^{250}_{044}$ \\
$11$ & $2.33^{255}_{066}$\\ \hline
$12$ & $2.^{400}_{375}$\\ \hline
$13$ & $2.60^{366}_{023}$\\
$14$ & $2.605_{62}^{98}$\\
$15$ & $2.605_{65}^{97}$\\
\hline
\end{tabular} \hspace{1cm}
\caption{Spectral enclosures for the Fichera domain $\Omega_{\mathrm{F}}$. Here we have fixed $t_{\mathrm{up}}=0.1$ and
$t_{\mathrm{low}}=2.8$. The final mesh in the iteration is shown on the left side. Its number of DOF=347460. \label{fichera_table}}
\end{figure}
\subsection{The Fichera domain}\label{Section:Fishera}
In this next experiment we consider the region
\[\Omega\equiv \Omega_{\mathrm{F}}=(0,\pi)^3 \setminus [0,\pi/2]^3 \subset \R^3.\]
See also \cite[\S5.2]{theoretical} for related results.
The table on right of Figure~\ref{fichera_table} shows numerical
estimation of the first 15 positive eigenvalues. Here we have fixed
$t_{\mathrm{up}}=0.1$ and $t_{\mathrm{low}}=2.8$. We have considered meshes refined along the re-entrant edges. The final mesh is shown on
the left side of Figure~\ref{fichera_table}. We have
stopped the algorithm when the tolerance $\delta=0.03$ has been achieved.
However, note that the accuracy is much higher for the indices
$j=2,3,9,10,11,13,14,15$.
Figure~\ref{fichera} includes the corresponding approximated eigenfunctions. The
mesh employed for this calculation is the same as that of
Figure~\ref{fichera_table}.
\subsection{A non-Lipschitz domain} \label{non-lip}
As mentioned earlier, for a single trial space
$\mathcal{L}$, the accuracy of the eigenvalue bounds established in Theorem~\ref{bounds}
depends on the position of $t$ relative to adjacent components of the spectrum. In this
experiment we demonstrate that this dependence might vary significantly
with $t$. The numerical evidence below suggests that a good choice of $t_{\mathrm{up}}$ and
$t_{\mathrm{low}}$ plays a major role in the design of efficient algorithms for eigenvalue
calculation based on this method.
\begin{figure}
\begin{tabular}{c|c||c|c||c|c}
RF & DOF & $t_{\mathrm{low}}=1.95$ & $t_{\mathrm{low}}=2.05$ &
$t_{\mathrm{up}}=1.05$ & $t_{\mathrm{up}}=0.7$ \\
& & ($l=1$ $\omega_3^-)$& ($l=3$ $\omega_3^-)$ &
($l=1$ $\omega_3^+)$ & ($l=3$ $\omega_3^+)$ \\
\hline
1 & 4143 & 1.24764 & 1.26640 & 1.50395 & 1.3436 \\
0.1 & 9648 & 1.25029 & 1.26830 & 1.49282 & 1.3336 \\
0.01 & 74226 & 1.25063 & 1.26846 & 1.48899 & 1.3274 \\ \hline
\end{tabular}
\caption{Dependence of the accuracy of the bounds from Theorem~\ref{bounds} on the choice of $t$ for the region $\tilde{\Omega}_{\mathrm{cut}}$. It is preferable to
pick $t_{\mathrm{up}}$ and $t_{\mathrm{low}}$ as far as possible from $\omega$, than to increase
the dimension of the trial subspace. \label{tab_slit_square}}
\end{figure}
Let $\tilde{\Omega}\equiv \tilde{\Omega}_{\mathrm{cut}}= (0,\pi)^2\setminus S$ for $S=[\pi/2,\pi]\times \{\pi/2\}$.
Benchmarks \cite{2004Dauge} on the eigenvalues of \eqref{maxwell_2d} are found by means of
solving numerically the corresponding Neumann Laplacian problem.
The first seven positive eigenvalues are
\begin{gather*}
\omega_1\approx 0.647375015,\, \omega_2=1,\, \omega_3\approx 1.280686161, \\
\omega_4=\omega_5=2,\, \omega_6\approx 2.096486081 \quad\text{and} \quad
\omega_7\approx 2.229523505.
\end{gather*}
The eigenfunctions associated to $\omega_2$, $\omega_4$ and $\omega_5$ are
smooth, as they are also eigenfunctions on $\tilde \Omega_{\mathrm{sqr}}$. On the other hand,
$\omega_1$ and $\omega_3$ correspond to singular eigenfunctions.
Standard nodal elements are completely unsuitable for the computation of these
eigenvalues, even with a significant refinement of the mesh on $S$.
The table in Figure~\ref{tab_slit_square} shows computation of $\omega_3^\pm$ on a mesh that
is increasingly refined at $S$ with a factor RF for two pairs of choices of $t_{\mathrm{up}}$ and $t_{\mathrm{low}}$. Here $h=0.1$ and we consider Lagrange elements of order $r=1$. The choice of $t_{\mathrm{up}}$ and $t_{\mathrm{low}}$ further from $\omega_3$, even with the very coarse mesh, provides a sharper estimate of
$\omega_3^\pm$ than the other choices even with a finer mesh.
\section{The transmission problem}\label{transmission}
In this final example, we consider a non-constant electric permittivity. Let
\begin{gather*}
\Omega_{\mathrm{sqr},1} =\left(0,\frac\pi 2\right)\times \left(0,\frac\pi 2\right) \qquad \Omega_{\rm sqr,2}=\left(\frac\pi 2,\pi\right)\times \left(\frac\pi 2,\pi\right)
\\ \Omega_{\rm sqr,3}=\left(\frac \pi 2,\pi\right)\times\left(0,\frac \pi 2\right) \qquad
\text{and} \qquad \Omega_{\rm sqr,4}=\left(0,\frac\pi 2\right)\times\left(\frac\pi 2,\pi\right).
\end{gather*}
so that \[\overline{\tilde \Omega_{\mathrm{sqr}}}=\overline{\bigcup_{l=1}^4\tilde\Omega_{\mathrm{sqr},l}}.\]
Set $\mu=1$ and
\begin{equation*}
\epsilon(x)= \left\{\begin{array}{ll}
1 & x\in \Omega_{\mathrm{sqr},1} \cup \Omega_{\mathrm{sqr},2} \\
\frac12 & x\in \Omega_{\mathrm{sqr},3} \cup \Omega_{\mathrm{sqr},4}.
\end{array} \right.
\end{equation*}
Numerical estimations of the eigenvalues of $\tilde\mathcal{M}$ on $\tilde{\Omega}\equiv \tilde{\Omega}_\mathrm{sqr}$ for this data were found in \cite{2004Dauge}.
\begin{figure}\label{table_eigenvalues_transmission}
\centerline{
\begin{tabular}{c|c|c|lcc}
$j$ &$\omega_j$ from \cite{2004Dauge} & $\omega_j\ _-^+$ & $l$ & up &
low \\
\hline
$1$ & $1.15954813181$ & $ 1.159^{555}_{456}$& & $1$ & $85$ \\
$2$ & $1.16804100636$ & $1.16^{807}_{770}$ && $2$ & $84$ \\
$3$ & $1.5834295853$ & $1.5834^{453}_{229}$ && $3$ & $83$ \\
$4$ & $2.3757369919$ & $2.375^{788}_{452}$ & &$4$ & $82$ \\
$5$ & $2.4724291674$ & $2.472^{479}_{212}$ & &$5$ & $81$ \\
$6$ & $ 2.5288205712$ & $2.528^{884}_{634}$ & &$6$ & $80$ \\
$7$ & $2.7487894882$ & $2.748^{868}_{693}$ && $7$ & $79$ \\
$8$ & $3.2334726763$ & $3.23^{362}_{280}$ & &$8$ & $78$ \\
$9$ & $3.47832176265$ & $3.47^{8478}_{775}$ && $9$ & $77$\\
$10$ & $ 3.51802898831$ & $3.51^{822}_{718}$ && $10$ & $76$ \\
\hline
\end{tabular}
}
\caption{Enclosures for the first 10 positive eigenvalues of
$\tilde\mathcal{M}$ for the transmission problem (Section~\ref{transmission}).
For comparison, on the second column we include the upper bounds
found in \cite{2004Dauge}. Here the trial subspace is made out of Lagrange elements of order 1,
$t_{\mathrm{up}}=10^{-9}$ and $t_{\mathrm{low}}=11.74$.
The mesh employed was constructed in an unstructured fashion in the
four sub-domains $\tilde\Omega_{\mathrm{sqr},l}$. The maximum element size is set to
$h=.01$ and the total number of DOF=399720. }
\end{figure}
We have set the experiment reported in Figure~\ref{table_eigenvalues_transmission}, on a family of meshes (not shown), which is unstructured but of equal maximum element sizes in each one of the subdomains $\tilde\Omega_{\mathrm{sqr},l}$. We implemented Procedure~\ref{alg3} as discussed previously, with fix $t_{\mathrm{up}}=10^{-9}$ and $t_{\mathrm{low}}=11.74$. For comparison, in the second column of the table we have included the benchmark upper bounds from \cite{2004Dauge}.
As we pointed out in sections~\ref{convex-domains} and \ref{non-con}, accuracy depends on the regularity of the corresponding eigenspace. Moreover, finding conclusive lower bounds for the ninth and tenth eigenvalues turns out to be extremely expensive, if $t_{\mathrm{low}}\approx 3.5$. Observe that, from the reproduced values in the second column of the table, these
two eigenvalues form a cluster of multiplicity 2. It seems that in fact
they are part of a larger cluster. The resulting narrow gap from this cluster seems to be the cause of
the dramatic deterioration in accuracy. Recall the observations made in Section~\ref{non-lip}.
The data has a natural symmetry with respect to the diagonals of $\tilde{\Omega}_{\mathrm{sqr}}$.
Four types of eigenvectors arise from these symmetries, and the analytical problem reduces
to four different eigen-problems which give rise to degenerate eigenspaces. As we are not considering a mesh
that completely respects these symmetries, the multiplicities arising from them are not shown completely in the numerics.
In order to find reasonable bounds for $\omega_9$ and $\omega_{10}$, we had to resource to exploiting
the fact that $\rho^{-}_{j}(t,h)$ is locally non-increasing in $t$, and it respects ordering in $j$. An analytical proof of this property is achieved by extending to the indefinite case the results of \cite[\S3]{BoHo2013}, but in the present context we have examined them only from a numerical perspective. Note that, when $t_\mathrm{low}$ is near to cross an eigenvalue, $\rho^{-}_{j}(t_{\mathrm{low}},h)$ jumps. These jumps appear to be small (respecting the order of the $j$) as long as the subspace captures well the eigenvectors. This effect will disappear eventually as we increase $t_\mathrm{low}$ further, due to the fact that $\L$ is finite-dimensional. In our experiments, we have determined that $t=t_{\mathrm{low}}\approx 11.74$ is near to optimal for the trial subspaces employed. Note that $t_{\mathrm{low}}=11.74$ gives $85$ eigenvalues in the segment $(10^{-9},11.74)$ for these trial subspaces.
\section*{Acknowledgements}
We kindly thank Universit\'e de Franche-Comt{\'e}, University College London and the Isaac Newton Institute for Mathematical Sciences, for their hospitality. Funding was provided by the British-French project PHC Alliance (22817YA), the British Engineering and Physical Sciences Research Council (EP/I00761X/1 and \linebreak EP/G036136/1) and the French Ministry of Research (ANR-10-BLAN-0101).
\bibliographystyle{siam}
\def$'${$'$}
| {
"timestamp": "2014-03-03T19:57:26",
"yymm": "1402",
"arxiv_id": "1402.4911",
"language": "en",
"url": "https://arxiv.org/abs/1402.4911",
"abstract": "We propose employing the extension of the Lehmann-Maehly-Goerisch method developed by Zimmermann and Mertins, as a highly effective tool for the pollution-free finite element computation of the eigenfrequencies of the resonant cavity problem on a bounded region. This method gives complementary bounds for the eigenfrequencies which are adjacent to a given real parameter. We present a concrete numerical scheme which provides certified enclosures in a suitable asymptotic regime. We illustrate the applicability of this scheme by means of some numerical experiments on benchmark data using Lagrange elements and unstructured meshes.",
"subjects": "Analysis of PDEs (math.AP); Numerical Analysis (math.NA)",
"title": "Finite element eigenvalue enclosures for the Maxwell operator",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575116884778,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542206194357
} |
https://arxiv.org/abs/1808.06525 | First Occurring Singularities of Functions in Symplectic Semi-Space | We explain how a classical theorem by Arnol'd and Melrose on non-singular functions on a symplectic manifold with boundary can be proved in few lines, and we use the same method to obtain a new result, which is a normal form with functional invariants for the first occurring singularities | \section{Introduction and main results}
\label{sec-intr-results}
All objects in the paper are either $C^{\infty}$ or analytic germs at $0$.
By a symplectic semi-space we mean the symplectic space $(\mathbb{R}^{2n},\omega )$ where $\omega $ is a symplectic form, endowed with a smooth hypersurface $\mathcal H$, which can be interpreted as the boundary.
The Darboux-Givental' theorem (see \cite{A3}) implies that all symplectic semi-spaces are locally equivalent.
We deal with the problem of classification of functions $f$ on a symplectic semi-space $(\mathbb R^{2n},\omega, \mathcal H)$. Two function germs are called equivalent if they can be brought one to the other by a local symplectomorphism of $(\mathbb R^{2n}, \omega )$ which preserves the hypersurface $\mathcal H$. An equivalent problem
is the classification of triples $(\omega,\mathcal H,f)$ with respect to the whole group of local diffeomorphisms.
\medskip
{\sc Notation}. In what follows $f$ and $h$ are function germs such that
\begin{equation}
\label{h}
\begin{tabular}{l}
$f(0)=0$, \ and \\
$h(0)=0, \ dh(0)\ne 0, \ \mathcal H =\{h=0\}$.
\end{tabular}
\end{equation}
\noindent {\bf Definition}. {\it
The triple $(\omega , \mathcal H, f)$ is non-singular if $\{f,h\}(0)\ne 0$.}
\medskip
Here and in what follows $\{., .\}$ is the Poisson bracket with respect to
$\omega $.
\begin{thm}[Melrose \cite{M}, Arnol'd \cite{A1}]
\label{thm-Ar-Me}
Any non-singular triple $(\omega , \mathcal H, f)$ on $\mathbb R^{2n}$ is equivalent to
\begin{equation}
\label{nf-Ar-Me}
\begin{tabular}{l}
$\omega = \sum _{i=1}^ndx_i\wedge dy_i, \ \mathcal H=\{x_1=0\}, \ f=y_1$.
\end{tabular}
\end{equation}
\end{thm}
In \cite{M} and \cite{A1} the proofs are given without details, but in fact
Theorem \ref{thm-Ar-Me} is a simple corollary of the Darboux
theorem on odd-symplectic (or quasi-symplectic) forms, and we find worth to give a very short proof in section \ref{sec-proofs}; the proofs of our much more difficult theorems on singular $(\omega , \mathcal H, f)$ are based on the same approach.
\medskip
The paper is devoted to first occurring singularities of $(\omega, \mathcal H, f)$ on $\mathbb R^{2n}$.
\medskip
\noindent {\bf Definition}.
{\it By $S_1$ we denote the singularity class consisting of $(\omega , \mathcal H, f)$ such that}
\begin{equation*}
\begin{tabular}{l}
for $n=1$: \ $\{f,h\}(0)=0$, \ $\{f,\{f,h\}\}(0)\ne 0$;
\\
for $n\ge 2$: \ $\{f,h\}(0)=0$, \ $\{f,\{f,h\}\}(0)\ne 0$, \ $\{h,\{f,h\}\}(0)\ne 0$, \ $df\wedge dh(0)\ne 0$.
\end{tabular}
\end{equation*}
It is easy to check that the choice of $h$ in (\ref{h}) in the given definitions is irrelevant: if they hold for $h$ then they also hold for $Qh$ where $Q$ is any non-vanishing function.
The difference in the definition of $S_1$ for $n=1$ and $n\ge 2$ is explained as follows: it is easy to see that for $n=1$ the assumptions
$\{f,h\}(0)=0$, \ $\{f,\{f,h\}\}(0)\ne 0$ imply that $\{h,\{f,h\}\}(0)\ne 0$ and $(df\wedge dh)(0)=0$.
\medskip
The problem of classification of triples $(\omega , \mathcal H, f)\in S_1$
was raised by R. B. Melrose in \cite{M} where
he studied a tied problem of classification of
triples $(\omega , \mathcal H, \mathcal F)$, $\mathcal H=\{h=0\}, \ \mathcal F=\{f=0\}$
where $\mathcal H$ and $\mathcal F$ are two
smooth hypersurfaces and $\omega $ is a symplectic form.
Melrose studied the first occurring singularities of such triples,
called glancing hypersurfaces in a symplectic space,
which are distinguished by the same conditions as in the definition of $S_1$.
The main theorem in \cite{M} states that
in the $C^\infty $ category the glancing hypersurfaces in the symplectic space $\mathbb{R}^{2n+2}$
can be described by the normal form
\begin{equation}
\label{Melrose-glancing-nf}
\begin{tabular}{l}
$\omega = dx\wedge dy + \sum _{i=1}^{n} dp_i\wedge dq_i$, \
$\mathcal H = \{y+x^2+p_1=0\}$, \ $\mathcal F=\{y=0\}$.
\end{tabular}
\end{equation}
(without $dp_i\wedge dq_i$ in $\omega $ and without $p_1$ in $\mathcal H$ for $n=0)$. In \cite{M}, p. 176 he noticed that replacing the hypersurface $\mathcal F$ by a function $f$
makes the problem substantially different - starting from the first occurring singularities moduli occur. Melrose claimed, and left to the reader to check, that in the family of triples
\begin{equation}
\label{Melrose-example}
\begin{tabular}{l}
$\omega = dx\wedge dy + \sum _{i=1}^{n}dp_i\wedge dq_i$, \
$\mathcal H = \{y+x^2+p_1=0\}$, \ $f = sy$
\end{tabular}
\end{equation}
the parameter $s$ is an invariant.
\medskip
The problem of classifying singularities of $(\omega, \mathcal H, \mathcal F)$ is
tied with a number of other classification problems, many of them
were solved ( a good part - by V. I. Arnol'd and his school, c.f. \cite{A1}, \cite{A3}).
Nevertheless, the problem of classification of singularities of
functions in a symplectic semi-space remains, as we know, open.
The purpose of this paper is to fill in this gap.
\begin{thm}
\label{thm-n-1}
Any triple $(\omega , \mathcal H, f)\in S_1$ on $\mathbb R^2$ can be brought to
the normal form
\begin{equation}
\label{nf-n-1}
\begin{tabular}{l}
$(\omega , \mathcal H, f)_{g(y)}$:\\
$\omega = dx\wedge dg(y), \ \mathcal H=\{y+x^2=0\}, \ f = g(y)$,
\ \
$g^\prime (0)\ne 0.$
\end{tabular}
\end{equation}
The triple (\ref{nf-n-1}) is equivalent to $(\omega , \mathcal H, f)_{\widetilde g(y)}$ if and only if \ $\widetilde g(y)=g(y)$.
\end{thm}
We have a functional invariant - a function of one variable $g(\cdot)$.
In \cite{K} the functional invariant was constructed in a canonical (coordinate-free) way using powerful tools from Gauss-Manin theory. In the same paper
another normal form
\begin{equation}
\label{nf-K}
\begin{tabular}{l}
$\omega = \phi(y+x^2)dx\wedge dy, \ \mathcal H=\{y=0\}, \ f=y+x^2$, \
$\phi(0)\ne 0$
\end{tabular}
\end{equation}
with the functional invariant $\phi(\cdot )$ was obtained. Note that (\ref{nf-K})
can be easily obtained from (\ref{nf-n-1}) and the proof of
(\ref{nf-n-1}) in section \ref{sec-proofs} is elementary and takes just few lines.
\begin{thm}
\label{thm-n-from-2}
In the space of $(2n)$-jets of triples $(\omega , \mathcal H, f)\in S_1$
on $\mathbb R^{2n+2}$, $n\ge 1$, there exists an open set $U$ such that any triple $(\omega , \mathcal H, f)\in S_1$ with $j^{2n}(\omega , \mathcal H, f)\in U$
can be brought to the normal form
\begin{equation}
\label{nf-n-from-2}
\begin{tabular}{l}
$(\omega , \mathcal H, f)_{\mu , g(y), \phi(y,p,q)}$: \\
$\omega = dx\wedge dF(y,p,q) + \mu , \ \mathcal H=\{y+x^2=0\}, \ f = F(y,p,q)$
\\
$F(y,p,q)= g(y)+\sum_{i=1}^n(p_iy^{2i-2}+q_iy^{2i-1})+y^{2n}\phi(y,p,q)$, \ $g^\prime (0)\ne 0$,
\end{tabular}
\end{equation}
where $\mu $ is a symplectic 2-form on $\mathbb R^{2n}(p,q)$. The triple (\ref{nf-n-from-2})
is equivalent to
$(\omega , \mathcal H, f)_{\widetilde \mu , \widetilde g(y), \widetilde \phi(y,p,q)}$
if and only if $\widetilde \mu =\mu$, \ $\widetilde g(y)=g(y)$,\ $\widetilde \phi(y,p,q) = \phi(y,p,q)$.
\end{thm}
In both normal forms (\ref{nf-n-1}) and (\ref{nf-n-from-2}) the numerical invariant $g^\prime (0)$ has a simple canonical meaning
by the following lemma.
\begin{prop}
\label{lem-first-invariant}
In the classification of triples $(\omega , \mathcal H, f)\in S_1$
the following number $\kappa $ is an invariant:
\begin{equation*}
\begin{tabular}{l}
$\kappa =\frac{\{h,\{f,h\}\}(0)}{(\{f,\{f,h\}\}(0))^2}$
\end{tabular}
\end{equation*}
In terms of normal forms (\ref{nf-n-1}) and (\ref{nf-n-from-2}) one has $\kappa = \frac{1}{2g^\prime (0)}$.
\end{prop}
\begin{proof}
Using that $h(0)=0$ and $\{f,h\}(0)=0$ it is easy to see that when multiplying $h$ by a non-vanishing function $Q$ one has
$\{f,\{f,h\}\}(0)\to Q(0)\{f,\{f,h\}\}(0)$ and
$\{h,\{f,h\}\}(0)\to Q^2(0)\{h,\{f,h\}\}(0)$. It follows that $\kappa $
does not depend of the choice of $h$. The formula $\kappa = \frac{1}{2g^\prime (0)}$
can be easily computed.
\end{proof}
\medskip
{\sc Remark}. Due to Lemma \ref{lem-first-invariant} we can do Melrose's homework:
the invariant $\kappa $ in his example (\ref{Melrose-example}) is equal to
$\frac{1}{2s}$, where $s=g'(0)$.
\medskip
Theorems \ref{thm-n-1} and \ref{thm-n-from-2} are proved, along with
much simpler and known Theorem \ref{thm-Ar-Me}, in the same way in section \ref{sec-proofs}. The proofs are very short provided that one uses the two lemmas in section \ref{sec-lemmas}, the proof of each one of them occupying only a few lines.
Probably the reader would ask why we do not use
the normal form $\omega = \sum dx_i\wedge dy_i, \ \mathcal H=\{x_1=0\}$ for
the pair $(\omega , \mathcal H)$ which seems a natural way to obtain normal forms for $(\omega , \mathcal H, f)$ by normalizing the function $f$ with respect
to local diffeomorphism preserving this normal form.
One can check that this way leads to very involved computations, and even if the computational obstacles can be resolved the proof would be very long.
\section{Auxiliary lemmas}
\label{sec-lemmas}
{\bf Definition}. {\it
A local odd-symplectic form on $\mathbb R^{2n+1}$ is the germ $\widehat w$ at $0$ of closed 2-form $\widehat \omega $ such that $\widehat \omega ^n(0)\ne 0$.}
\medskip
In the following lemma and its proof
\begin{equation}
\label{notations}
\begin{tabular}{l}
$p=(p_1,..., p_n), q = (q_1,..., q_n)$, $dp\wedge dq = \sum _{i=1}^ndp_i\wedge dq_i$.
\end{tabular}
\end{equation}
\begin{lem}
\label{lem-cor-Darboux-odd-1}
Any odd-symplectic form $\widehat \omega $ on $\mathbb R^{2n+1}(y,p,q)$ such that $(\widehat \omega ^n\wedge dy)(0)\ne 0$ can be brought to the form $dp\wedge dq$ by a local diffeomorphism preserving the coordinate $y$.
If $\Psi $ is a local diffeomorphism of $\mathbb R^{2n+1}(y,p,q)$
which preserves $dp\wedge dq$ and the coordinate $y$ then $\Psi :(y,p,q)\to (y, A(p,q),B(p,q))$ where $(p,q)\to (A,B)$ is a local symplectomorphism of
$(\mathbb R^{2n}, dp\wedge dq)$.
\end{lem}
\begin{proof}
By the classical Darboux theorem for odd-symplectic forms (the proof can be found, for example, in \cite{Zh}) $\widehat \omega $ can be brought to
$dp\wedge dq$ by some local diffeomorphism $\Phi $. This diffeomorphism
brings $y$ to a function $Y(y,p,q)$; the assumption $(\widehat \omega ^n\wedge dy)(0)\ne 0$ implies $\frac{\partial Y}{\partial y}(0)\ne 0$.
The local diffeomorphism $(y,p,q)\to (Y,p,q)$ brings $Y(y,p,q)$ to $y$ and preserves $dp\wedge dq$. It proves the first statement. To prove the second statement it suffices to note that any diffeomorphism which preserves
both $y$ and the line field $\mathcal L$ generated by
$V=\frac{\partial }{\partial y}$ has the form given in the lemma, and that
$\mathcal L$ is invariantly related to $\widehat \omega = dp\wedge dq$:
$V$ is defined, up to multiplication by a function, by $V\rfloor \Omega = \widehat \omega ^n$ where $\Omega $ is a volume form.
\end{proof}
\begin{lem}
\label{lem-Z-H-1}
Let $Z$ be a non-singular vector field on $\mathbb R^{2n+2}$ which has the simple tangency with
a smooth hypersurface $\mathcal H$. In suitable local coordinates $Z = \frac{\partial}{\partial x}, \ \mathcal H=\{y+x^2=0\}$. Any local diffeomorphism of $\mathbb R^{2n+2}$ which preserves this normal form preserves the coordinates $x$ and $y$.
\end{lem}
\begin{proof}
The given normal form is well-known, see for example
\cite{A0}. Let us prove the second statement. Any local diffeomorphism
preserving the vector field $\frac{\partial}{\partial x}$ changes $x$ to
$x+A(y,p,q)$ and $y$ to $B(y,p,q)$. It preserves the hypersurface
$y+x^2=0$ if and only if the function $(x+A)^2+B$ vanishes on the hypersurface $y=-x^2$. It means $x^2+B(-x^2,p,q)+A^2(-x^2,p,q)+ 2xA(-x^2,p,q)\equiv 0$.
Taking the even and the odd part with respect to $x$ we obtain $A=0, B=y$.
\end{proof}
\section{Proof of Theorems \ref{thm-Ar-Me}, \ref{thm-n-1}, \ref{thm-n-from-2}}
\label{sec-proofs}
In what follows we denote by $Z_f$ the Hamiltonian vector field
defined by $f$:
\begin{equation}
\label{Ham-vf}
Z_f \ \rfloor \ \omega = df.
\end{equation}
We use the coordinates $x,y$ in the 2-dim case. For higher dimensions
we work on $\mathbb R^{2n+2}(x,y,p,q)$ and we use notations (\ref{notations}).
\begin{lem}
\label{lem-omega-hat}
Let $Z_f = \frac{\partial }{\partial x}$. Then in the same coordinates
\begin{equation}
\label{eq-with-omega-hat}
\begin{tabular}{l}
\text{\rm in the 2-dim case} $f = g(y)$, \ $\omega = dx\wedge dg(y)$
\\
\text{\rm for higher dimensions}:
\
$f = F(y,p,q)$, \ $\omega = dx\wedge dF(y,p,q)+\widehat \omega $
\\
\text{\rm where} $\widehat \omega $ \text{\rm is an odd-symplectic form on} $\mathbb R^{2n+1}(y,p,q)$.
\end{tabular}
\end{equation}
\end{lem}
\begin{proof}
The given form of $f$ follows
from the equation $Z_f(f)=0$ which is a corollary of (\ref{Ham-vf}) with
$Z_f = \frac{\partial }{\partial x}$. This equation also implies
$\omega = dx\wedge dg(y)$ for $n=0$ and
$\omega = dx\wedge dF(y,p,q)+\widehat \omega $, \ $\frac{\partial }{\partial x}\rfloor \omega = 0$ for $n\ge 1$. Since $d\omega =0$ and
$\omega ^{n+1}(0)\ne 0$
it follows that $\widehat \omega $ is an odd-symplectic form.
\end{proof}
\smallskip
{\bf Proof of Theorem \ref{thm-Ar-Me}}. The condition $\{f,h\}(0)\ne 0$
means that $Z_f$ is transversal to $\mathcal H$.
Take coordinates in which $Z_f=\frac{\partial}{\partial x}, \ \mathcal H = \{x=0\}$. By Lemma \ref{lem-omega-hat}, $\omega $ and $f$ have form
(\ref{eq-with-omega-hat}). Changing the coordinate $y$ and using Lemma \ref{lem-cor-Darboux-odd-1} we obtain (\ref{nf-Ar-Me}), up to notations of the coordinates.
\medskip
{\bf Proof of Theorem \ref{thm-n-1}}.
The conditions $\{f,h\}(0)=0$, $\{f,\{f,h\}\}(0)\ne 0$ mean that $Z_f$ is a non-singular vector field which has the simple tangency with $\mathcal H$.
By Lemma \ref{lem-Z-H-1} the pair $(Z_f,\mathcal H)$ can be brought to the normal form $Z_f=\frac{\partial}{\partial x}, \ \mathcal H = \{y+x^2=0\}$.
By Lemma \ref{lem-omega-hat}, $\omega $ and $f$ have form
$\omega = dx\wedge dg(y), \ f = g(y)$ so that we have normal form (\ref{nf-n-1}) for the triple $(\omega , \mathcal H, f)$. The fact that $g(y)$ is a functional invariant is a direct corollary of the second statement of Lemma \ref{lem-Z-H-1}.
\medskip
{\bf Proof of Theorem \ref{thm-n-from-2}}.
As in the proof of Theorem \ref{thm-n-1}, we take coordinates in which
$Z_f=\frac{\partial}{\partial x}, \ \mathcal H = \{y+x^2=0\}$, and by
Lemma \ref{lem-omega-hat}, $\omega $ and $f$ have form
(\ref{eq-with-omega-hat}).
Now we will show that
the condition $\{h,\{f,h\}\}(0)\ne 0$ in the definition of the singularity class $S_1$ is equivalent to $(\widehat \omega ^n\wedge dy)(0)\ne 0$.
Let $h = y+x^2$. Let $Z_h$ be the Hamiltonian vector field defined by $h$. The condition $\{h,\{f,h\}\}(0)\ne 0$ means $(Z_h(x))(0)\ne 0$. Indeed, since $Z_f=\frac{\partial }{\partial x}$ we have
$\{f,h\}=2x$ and $\{h, \{f,h\}\} = 2Z_h(x)$.
To find $Z_h(x)$ we have by definition
$Z_h \ \rfloor \ (dx\wedge dF(y,p,q) + \widehat \omega ) = dh = dy + 2xdx$.
Express this equation in the form
\begin{equation}
\label{eq-t}
\begin{tabular}{l}
$Z_h(x)\cdot dF(y,p,q) = dy - Z_h\rfloor \widehat \omega + (2x+(Z_h\ \rfloor\ dF(y,p,q))dx$
\end{tabular}
\end{equation}
Since $\widehat \omega ^{n+1}=0$ we have $(Z_h\rfloor \widehat \omega )\wedge \widehat \omega ^n=0$. Therefore taking the external product of
(\ref{eq-t}) with $\widehat \omega ^n\wedge dx$ we obtain
$Z_h(x)\cdot dF(y,p,q)\wedge \widehat \omega ^n\wedge dx = dy\wedge \widehat \omega ^n\wedge dx$ and it follows
$dy\wedge \widehat \omega ^n =
Z_h(x)\cdot (dF(y,p,q)\wedge \widehat \omega ^n)$. Since $\omega ^{n+1}\ne 0$ we have $dF(y,p,q)\wedge \widehat \omega ^n\ne 0$. Therefore
the condition $(dy\wedge \widehat \omega ^n)(0)\ne 0$ is the same as
$(Z_h(x))(0)\ne 0$ which, as is shown above, is the same as $\{h,\{f,h\}\}(0)\ne 0$.
\smallskip
Now we can use Lemma \ref{lem-cor-Darboux-odd-1} which allows to bring $\widehat \omega $ to
$dp\wedge dq$ preserving $Z_f = \frac{\partial }{\partial x}$ and
$\mathcal H=\{y+x^2=0\}$. We obtain the normal form
\begin{equation}
\label{nf-pre-final}
\begin{tabular}{l}
$(\omega , \mathcal H, f)_{F(y,p,q)}: $
\\
$\omega = dx\wedge dF(y,p,q) + dp\wedge dq, \ \mathcal H=\{y+x^2=0\}, \ f = F(y,p,q)$.
\end{tabular}
\end{equation}
The next step is using the second statements of Lemmas \ref{lem-cor-Darboux-odd-1} and \ref{lem-Z-H-1}. They imply that $(\omega , \mathcal H, f)_{F(y,p,q)}$ is equivalent to $(\omega , \mathcal H, f)_{\widetilde F(y,p,q)}$ if and only if
the functions $F(y,p,q)$ and $\widetilde F(y,p,q)$ can be brought one to the other by a symplectomorphisms of $\mathbb R^{2n}(p,q)$.
Therefore reducing (\ref{nf-pre-final}) to exact normal form is the same problem as classification of functions $F(y,p,q)$ with respect to symplectomorphisms of $\mathbb R^{2n}(p,q)$.
An equivalent problem is as follows:
\medskip
\noindent (*) classification of pairs $(\mu, F(y,p,q))$ where $\mu $ is any symplectic form on $\mathbb R^{2n}(p,q)$ with respect to all diffeomorphisms of the form $(y,p,q)\to (y,\Phi (p,q), \Psi (p,q))$.
\medskip
Let $F(y,p,q)=g(y) +F_0(p,q) + F_1(p,q)y + \cdots $ where $F_i(0,0)=0$. Assuming that the functions $F_0,..., F_{2n-1}$ are differentially independent at $0$, which defines the open set $U$ in Theorem \ref{thm-n-from-2}, we obtain, in the problem (*), the exact
normal form with
$F(y,p,q)=g(y)+p_1+q_1y + p_2y^2+q_2y^3\cdots + q_ny^{2n-1}+ \phi(y,p,q)y^{2n}$ where $g(y)$ and $\phi(y,p,q)$ are functional invariants, and the symplectic form $\mu $ in the pair $(\mu, F(y,p,q))$ is a functional invariant too.
This exact normal form leads to the exact normal form (\ref{nf-n-from-2}).
| {
"timestamp": "2018-08-21T02:18:16",
"yymm": "1808",
"arxiv_id": "1808.06525",
"language": "en",
"url": "https://arxiv.org/abs/1808.06525",
"abstract": "We explain how a classical theorem by Arnol'd and Melrose on non-singular functions on a symplectic manifold with boundary can be proved in few lines, and we use the same method to obtain a new result, which is a normal form with functional invariants for the first occurring singularities",
"subjects": "Symplectic Geometry (math.SG)",
"title": "First Occurring Singularities of Functions in Symplectic Semi-Space",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575116884778,
"lm_q2_score": 0.7217432182679956,
"lm_q1q2_score": 0.7091542206194357
} |
https://arxiv.org/abs/1905.10355 | Taylor expansions of groups and filtered-formality | Let $G$ be a finitely generated group, and let $\Bbbk{G}$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\Bbbk{G}$ which generalizes the Magnus expansion of a free group. The group $G$ is said to be filtered-formal if its Malcev Lie algebra is isomorphic to the degree completion of its associated graded Lie algebra. We show that $G$ is filtered-formal if and only if it admits a Taylor expansion, and derive some consequences. | \section{Introduction}
\label{sect:intro}
\subsection{Expansions of groups}
\label{intro:exp}
Group expansions were first introduced by Magnus in \cite{Magnus35},
in order to show that finitely generated free groups are residually nilpotent.
This technique has been generalized and used in many ways.
For instance, the exponential expansion of a free group was used
to give a presentation for the Malcev Lie algebra
of a finitely presented group by Papadima \cite{Papadima95}
and Massuyeau \cite{Massuyeau12}.
Expansions of pure braid groups and their applications in knot theory
have been studied since the 1980s by several authors,
see for instance Kohno's papers \cite{Kohno85, Kohno88, Kohno16}.
X.-S. Lin studied in \cite{Linxiaosong97} expansions of fundamental
groups of smooth manifolds, using K.T. Chen's theory \cite{Chen77} of formal power
series connections and their induced monodromy representations.
More generally, Bar-Natan has explored in \cite{Bar-Natan16}
the Taylor expansions of an arbitrary ring.
Let $G$ be a finitely generated group, and fix a coefficient
field $\k$ of characteristic zero. We let ${\gr}(\k{G})$ be the associated
graded algebra of $\k{G}$ with respect to the filtration by powers of the
augmentation ideal, and we let
$\widehat{\gr}(\k{G})$ be the degree completion of this algebra.
Developing an idea from \cite{Bar-Natan16}, we say that a map
$E\colon G\to \widehat{\gr}(\k{G})$ is a multiplicative expansion
of $G$ if the induced algebra morphism,
$\bar{E}\colon \k{G}\to \widehat{\gr}(\k{G})$,
is filtration-preserving and induces the identity at the associated graded level.
Such a map $E$ is called a \emph{Taylor expansion}\/ if it sends
each element of $G$ to a group-like element of the Hopf algebra
$\widehat{\gr}(\k{G})$.
\subsection{Expansions and filtered-formality}
\label{intro:exp formal}
Once again, let $G$ be a finitely generated group.
The concept of filtered-formality relates an object from rational homotopy
theory to a group-theoretic object. The first object is the Malcev Lie
algebra $\mathfrak{m}(G,\k)$, defined by Quillen \cite{Quillen69}
as the set of primitive elements of the
$I$-adic completion of the group algebra of $G$, where $I$ is the augmentation
ideal of $\k{G}$. This Lie algebra comes endowed with a (complete) filtration
induced from the natural filtration on $\widehat{\k{G}}$,
and is isomorphic to the dual of Sullivan's $1$-minimal model of
a $K(G,1)$ space.
The second object is the graded Lie algebra $\gr(G,\k)$, defined by taking
the direct sum of the successive quotients of the lower central series of $G$,
tensored with $\k$. As shown by Quillen in \cite{Quillen68}, the associated
graded algebra ${\gr}(\k{G})$ is isomorphic to the universal enveloping
algebra of $\gr(G,\k)$.
The group $G$ is called {\em filtered-formal}\/ if its
Malcev Lie algebra, $\mathfrak{m}(G,\k)$, is isomorphic
to $\widehat{\gr}(G;\k)$, the degree completion of its associated
graded Lie algebra, as filtered Lie algebras.
If, in addition, the graded Lie algebra $\gr(G;\k)$ is
quadratic, the group $G$ is said to be {\em $1$-formal}.
For more details on these notions we refer to
\cite{PS04-imrn, Papadima-Suciu09, SW-formal}
and references therein.
The following result, which elucidates the relationship
between Taylor expansions and formality properties,
is a combination of Theorem \ref{thm:expansionFiltered} and
Corollary \ref{cor:expansionFormal}.
\begin{theorem}
\label{thm:intro expansion}
Let $G$ be a finitely generated group. Then:
\begin{enumerate}
\item \label{tt1}
$G$ is filtered-formal if and only if
$G$ has a Taylor expansion $G\to \widehat{\gr}(\k{G})$.
\item \label{tt2} $G$ is $1$-formal if and only if
$G$ has a Taylor expansion and $\gr(\k{G})$ is a quadratic algebra.
\end{enumerate}
\end{theorem}
Combining this theorem with our results on filtered-formality from \cite{SW-formal},
we conclude that the following propagation property of Taylor expansions holds.
This is a combination of Theorems \ref{thm:Taylorpropagation} and \ref{thm:nilp-taylor}.
\begin{prop}
\label{prop:intro propagation}
The existence of a Taylor expansion is preserved under
field extensions, and taking finite products and coproducts,
split injections, nilpotent quotients or solvable quotients of groups.
\end{prop}
In particular, if a finitely generated group $G$
has a Taylor expansion over $\mathbb{C}$, then it also has a Taylor expansion
over $\mathbb{Q}$.
\subsection{Residual properties and Taylor expansions}
\label{intro:rtfn-intro}
A group $G$ is said to be \emph{residually torsion-free nilpotent}\/
if any non-trivial element of $G$ can be detected in a
torsion-free nilpotent quotient. If $G$ is finitely generated,
this condition is equivalent to the injectivity of the canonical
map to the Malcev group completion, $\kappa\colon G\to \mathfrak{M}(G,\k)$.
An expansion $\bar{E}\colon\k G\to \widehat{\gr}(\k G)$ is said to be
{\em faithful}\/ if the map $E\colon G\to \widehat{\gr}(\k{G})$ is injective.
The next proposition relates the property
of being residually torsion-free nilpotent to the existence of
a faithful Taylor expansion.
\begin{prop}
\label{prop:fftaylor-intro}
A finitely generated group $G$ has a faithful Taylor expansion
if and only if $G$ is residually torsion-free nilpotent and filtered-formal.
\end{prop}
The work of Magnus \cite{Magnus35,Magnus-K-S} shows that all the
free groups $F_n$ are residually torsion-free nilpotent (RTFN).
Furthermore, as shown by Hain \cite{Hain97} and Berceanu--Papadima
\cite{Berceanu-Papadima09}, the Torelli groups
$\IA_n=\ker(\Aut(F_n)\to \Aut((F_n)_{\ab}))$
are also RTFN.
Consequently, all the subgroups of $\IA_n$, for instance,
the pure braid group $P_n$, the McCool group $wP_n$, and the
upper McCool group $wP_n^+$, inherit this property.
Let $\Pi_n$ be the direct product of the free groups $F_{1}, \dots, F_{n-1}$.
The graded Lie algebras $\gr(P_n,\k)$, $\gr(\Pi_n,\k)$ and $\gr(wP^+_n,\k)$
are isomorphic as vector spaces. Hence, their universal enveloping algebras,
which are domains for the Taylor expansions of $P_n$, $\Pi_n$, and $wP^n$,
are also isomorphic as vector spaces. The next proposition shows that
they are not isomorphic as algebras.
\begin{prop}
\label{prop:theta-intro}
For each $n\geq 4$, the graded Lie algebras $\gr(P_n,\k)$, $\gr(\Pi_n,\k)$, and
$\gr(wP^+_n,\k)$ are pairwise non-isomorphic.
\end{prop}
\subsection{Braid-like groups and further directions}
\label{subsec:motivation}
Explicit Taylor expansions have been constructed for
several classes of filtered-formal groups, including
finitely generated free groups, free abelian groups, surface groups,
the pure braid groups, and the McCool groups.
When $G$ is the fundamental group of a smooth manifold $M$,
an important construction for a Taylor expansion arises from Chen's theory of
formal power series connections and their induced monodromy representations.
Using this technique, Kohno \cite{Kohno88, Kohno16} gave explicit
Taylor expansions for the pure braid groups $P_n$. Using a completely
different approach, Papadima constructed in \cite{Papadima02}
{\em integral}\/ Taylor expansions for the braid groups $B_n$.
In another direction,
Hain studied expansions for link groups \cite{Hain85}, fundamental groups
of algebraic varieties \cite{Hain86}, and the Torelli groups \cite{Hain97},
while Lin \cite{Linxiaosong97} further investigated the relationship between
expansions and link invariants, including Vassiliev invariants, Milnor's link
variants and the Kontsevich integral.
There is also a strong interplay between Taylor expansions of
the pure braid groups and the finite-type (or Vassiliev)
invariants in knot theory. In this context, the relevant formal power series
connection is a version of the Knizhnik--Zamolodchikov connection.
The Taylor expansions of the groups constructed from Chen's theory of
formal power series connections yield finite-type invariants for pure braids,
and provide a prototype for the Kontsevich integral for knots.
For more on all of this, we refer the reader to \cite{Habegger-Massbaum,
Mostovoy-Willerton, Papadima02,Linxiaosong97, Bar-Natan95, Bar-Natan16}.
\section{Hopf algebras and expansions of groups}
\label{sec:expansion}
\subsection{Group algebras, completions, and associated graded algebras}
\label{subsec:grkg}
Let $G$ be a finitely generated group, and let $\k{G}$
be its group algebra over a field $\k$.
Let $\varepsilon\colon \k{G}\to \k$ be the augmentation
homomorphism, defined by $\varepsilon(g)=1$ for all $g\in G$.
The powers of the augmentation ideal, $I=\ker(\varepsilon)$,
define the $I$-adic filtration on the group algebra, $\{I^k\}_{k\ge 0}$.
This filtration is multiplicative, in the sense that
$I^k\cdot I^{\ell}\subset I^{k+\ell}$.
The corresponding completion,
\begin{equation}
\label{eq:hat-kg}
\widehat{\k{G}}=\varprojlim_k \k{G}/I^k,
\end{equation}
comes equipped with the inverse limit filtration,
$\{\widehat{I^k}\}_{k\ge 0}$. The multiplication in $\k{G}$
extends to a multiplication in $\widehat{\k{G}}$, compatible
with this filtration.
On the other hand, the associated graded group,
\begin{equation}
\label{eq:grkG}
{\gr}(\k{G})=\bigoplus_{k\geq 0}I^k/I^{k+1},
\end{equation}
is a graded algebra, with multiplication inherited from the
product in $\k{G}$.
This algebra comes endowed
with the degree filtration,
$\mathcal{F}_k(\gr(\k{G}))=\bigoplus_{j\geq k}I^j/I^{j+1}$.
The completion of $\gr(\k{G})$ with respect to this filtration,
\begin{equation}
\label{eq:completegr}
\widehat{\gr}(\k{G})=\prod_{k\geq 0}I^k/I^{k+1},
\end{equation}
comes endowed with the inverse limit filtration,
\begin{equation}
\label{eq:completefil}
\widehat{\mathcal{F}}_k(\widehat{\gr}(\k{G}))=\prod_{j\geq k}I^j/I^{j+1}.
\end{equation}
The associated graded algebra of $\widehat{\gr}(\k{G})$
is canonically identified with $\gr(\k{G})$.
For example, if $G=F_n$ is a free group of rank $n\ge 1$,
then $\gr(\k{G})$ is the tensor $\k$-algebra on $n$ generators
$t_i$ while the completion $\widehat{\gr}(\k{G})$ is the power series
ring in $n$ non-commuting variables $x_i=t_i-1$.
\subsection{Hopf algebras}
\label{subsec:hopf}
A {\em Hopf algebra}\/ is an associative and coassociative bialgebra
over a field $\k$, with multiplication $\nabla\colon A\otimes A\to A$,
comultiplication $\Delta\colon A\to A\otimes A$, unit
$\eta\colon \k\to A$, and counit $\varepsilon \colon A\to \k$,
endowed with a $\k$-linear map $T\colon A\to A$
(called the antipode), such that the following diagram commutes:
\[
\xymatrixrowsep{24pt}
\xymatrixcolsep{14pt}
\xymatrix{
& A\otimes A \ar^{T\otimes \id}[rr] && A\otimes A \ar^{\nabla}[dr]\\
A \ar^{\varepsilon}[rr] \ar^{\Delta}[ur] \ar_{\Delta}[dr]&& \k \ar^{\eta}[rr] && A\\
& A\otimes A \ar^{\id\otimes T}[rr] && A\otimes A \ar_{\nabla}[ur]
}
\]
An element $x\in A$ is called {\em group-like}\/ if
$\Delta(x)=x\otimes x$, and it is called {\em primitive}\/ if
$\Delta x=x {\otimes} 1+ 1{\otimes} x$.
The set of group-like elements of $A$
form a group, with multiplication inherited from $A$ and
inverse given by the antipode,
while the set of primitive elements of $A$ form a Lie algebra,
with Lie bracket $[x,y]=\nabla(x,y)-\nabla(y,x)$.
For instance, if $\mathfrak{g}$ is a Lie algebra, then its universal enveloping algebra,
$U(\mathfrak{g})$, is a Hopf algebra, with $\Delta x=x {\otimes} 1+ 1{\otimes} x$,
$\varepsilon(x)=0$, and $T(x)=-x$ for all $x\in \mathfrak{g}$.
By construction, the set of primitive elements in $U(\mathfrak{g})$
coincides with $\mathfrak{g}$. Suppose now that $\mathfrak{g}\cong \k^n$,
with Lie bracket equal to $0$. We may then identify $U(\mathfrak{g})$
with the polynomial ring $\k[x_1,\dots , x_n]$. Likewise,
if $\widehat{U}(\mathfrak{g})$ denotes the completion of $U(\mathfrak{g})$
with respect to the filtration by powers of the augmentation ideal
$J=\ker(\varepsilon)$, we may then identify $\widehat{U}(\mathfrak{g})$
with the power series ring $\k\com{x_1,\dots , x_n}$.
From now on, we will assume that $\k$ is a field of characteristic $0$.
As is well-known, the group algebra $\k{G}$ of a group $G$ is a Hopf algebra,
with comultiplication $\Delta\colon \k{G}\to \k{G}\otimes \k{G}$
given by $\Delta(g)=g\otimes g$ for $g\in G$, counit $\varepsilon \colon \k{G}\to \k$
the augmentation map, and antipode $T\colon \k{G}\to \k{G}$
given by $T(g)=g^{-1}$.
In \cite{Quillen69}, Quillen showed that the $I$-adic completion
of the group algebra,
$\widehat{\k{G}}$, is a complete Hopf algebra, with comultiplication map
\begin{equation}
\label{eq:hatdelta}
\xymatrixcolsep{16pt}
\xymatrix{\widehat{\Delta} \colon \widehat{\k{G}}\ar[r]& \widehat{\k{G}}\,
\hat{\otimes}\, \widehat{\k{G}}
}.
\end{equation}
where $\hat{\otimes}$ denotes the completed tensor product, defined
in this case as $\widehat{\k{G}}\, \hat{\otimes}\, \widehat{\k{G}}=
\varprojlim_k \k{G}/I^k \otimes \k{G}/I^k$.
Identifying the associated graded algebra $\gr\big(\k{G}\otimes \k{G}\big)$
with $\gr(\k{G})\otimes \gr(\k{G})$,
we see that the degree completion $\widehat{\gr}(\k{G})$ is also
a complete Hopf algebra, with comultiplication map
\begin{equation}
\label{eq:bardelta}
\xymatrixcolsep{18pt}
\xymatrix{\bar{\Delta}:=\widehat{\gr}(\Delta) \colon\widehat{\gr}(\k{G})\ar[r]
&\widehat{\gr}(\k{G})\, \hat{\otimes}\, \widehat{\gr}(\k{G})}.
\end{equation}
\subsection{Multiplicative expansions and Taylor expansions}
\label{subsec:exp}
Given a map $f\colon G\to R$, where $R$ is a ring, we will
denote by $\bar{f} \colon \k{G}\to R$ its linear extension to
the group algebra.
\begin{definition}
\label{def:exp}
A \emph{\textup{(}multiplicative\textup{)} expansion}\/
of a group $G$ is a map
\begin{equation}
\label{eq:expansion}
\xymatrixcolsep{16pt}
\xymatrix{
E\colon G \ar[r]& \widehat{\gr}(\k G)
}
\end{equation}
such that the linear extension $\bar{E}\colon\k G\to \widehat{\gr}(\k G)$
is a filtration-preserving algebra morphism with the property that $\gr(\bar{E})=\id$.
Furthermore, we say that the expansion $E$ is {\em faithful}\/ if $E$ is injective.
\end{definition}
Alternatively, an expansion of $G$ is a (multiplicative) monoid
map $E\colon G \to \widehat{\gr}(\k{G})$ such that the following
property holds: If $f\in I^k\setminus I^{k+1}$, then $\bar{E}(f)$
starts with $[f]\in I^k/I^{k+1}$, that is, $\bar{E}(f)=(0,\dots,0,[f],*,*,\dots)$.
Following Bar-Natan \cite{Bar-Natan16}, we make the following definition.
\begin{definition}
\label{def:taylor}
An expansion $E\colon G \to \widehat{\gr}(\k G)$ is
called a \emph{Taylor expansion}\/
(or, a \emph{group-like}\/ expansion) if it sends all elements of
$G$ to group-like elements of $\widehat{\gr}(\k G)$,
that is,
\begin{equation}
\label{eq:DeltaBar}
\xymatrix{\bar\Delta (E(g))=E(g) \hat{\otimes} E(g)
}
\end{equation}
for all $g\in G$.
\end{definition}
Equivalently, an expansion $E$ is a Taylor expansion if it is \emph{co-multiplicative},
i.e., the following diagram commutes:
\begin{equation}
\label{eq:grouplike}
\begin{gathered}
\xymatrix{
\k{G} \ar[d]^{{\bar{E}}} \ar[r]^(.45){{\Delta}} &\k{G}\otimes \k{G} \ar[d]^{{\bar{E}} {\otimes}{\bar{E}}}\\
\widehat{\gr}(\k G) \ar[r]^(.4){\bar\Delta} &\widehat{\gr}(\k G) \hat{\otimes}\widehat{\gr}(\k G) .
}
\end{gathered}
\end{equation}
\begin{prop}
\label{prop:Taylor}
A Taylor expansion $E\colon G\to \widehat{\gr}(\k{G})$ induces a
filtration-preserving isomorphism of complete Hopf algebras,
$\widehat{E}\colon \widehat{\k G}\to \widehat{\gr}(\k G)$, such that
$\gr(\widehat{E})$ is the identity on $\gr(\k{G})$.
\end{prop}
\begin{proof}
As in the above definition, the expansion $E$ induces a filtration-preserving algebra morphism,
$\bar{E}\colon \k G\to \widehat{\gr}(\k G)$. Applying the $I$-adic completion functor,
we obtain an algebra morphism, $\widehat{E} \colon \widehat{\k G}\to \widehat{\gr}(\k G)$.
By the above discussion, the expansion $E$ is group-like if and only if
$\widehat{E}$ is co-multiplicative. Applying the completion functor to
diagram \eqref{eq:grouplike} yields another commuting diagram,
\begin{equation}
\begin{gathered}
\xymatrix{
\widehat{\k G} \ar[d]^{\widehat{E}} \ar[r]^(.45){\widehat{\Delta}}
&\widehat{\k G}\hat{\otimes}\widehat{\k G}
\ar[d]^{\widehat{E}\hat{\otimes}\widehat{E}}\\
\widehat{\gr}(\k G) \ar[r]^(.4){\bar{\Delta}}
&\widehat{\gr}(\k G) \hat{\otimes}\widehat{\gr}(\k G).
}
\end{gathered}
\end{equation}
Since $\bar{E}$ is filtration-preserving and $\gr(\bar{E})=\id$, this implies
that the Hopf algebra morphism $\widehat{E}$
preserves filtrations and that $\gr(\widehat{E})=\id$.
By induction on $k$, all induced maps
$\widehat{\k G}/\widehat{I^k} \to\widehat{\gr}(\k G)/\widehat{\mathcal{F}}_k$
are isomorphisms,
where $\widehat{\mathcal{F}}_k$ is the filtration from display \eqref{eq:completefil}.
It follows from the next lemma that $\widehat{E}$ is an isomorphism.
\end{proof}
\begin{lemma}
\label{lem:completefiltration}
Let $f\colon A\to B$ be a morphism of
filtered, complete, and separated algebras.
If $\gr(f)\colon \gr^{\mathcal{F}}(A)\to \gr^{\mathcal{G}}(B)$
is an isomorphism, then $f$ is also an isomorphism.
\end{lemma}
\begin{proof}
By assumption, the homomorphisms
$\gr_k(f)\colon \mathcal{F}_{k}A/\mathcal{F}_{k+1}A \to \mathcal{G}_k B/\mathcal{G}_{k+1} B$
are isomorphisms, for all $k\ge 1$. An easy induction on $k$
shows that all maps $f_k\colon A/\mathcal{F}_{k+1}A \to B/\mathcal{G}_{k+1}B $
are isomorphisms. Therefore, the map
$\hat{f} \colon \widehat{A} \to \widehat{A}$ is
an isomorphism. On the other hand, both $A$ and $B$ are
complete and separated, and so $A=\widehat{A}$
and $B=\widehat{B}$. Hence $f=\hat{f}$, and we are done.
\end{proof}
\subsection{On the existence of Taylor expansions}
\label{subsec:alt-taylor}
As we shall see, not all finitely generated groups admit a Taylor
expansion. We conclude this section with an if-and-only-if criterion
for the existence of a such expansion.
\begin{prop}
\label{prop:TaylorExp}
A filtration-preserving isomorphism of complete Hopf algebras,
$\phi\colon \widehat{\k G}\to \widehat{\gr}(\k G)$,
induces a Taylor expansion $E\colon G\to \widehat{\gr}(\k{G})$.
\end{prop}
\begin{proof}
The isomorphism $\phi$ induces
a filtration-preserving isomorphism of complete Hopf algebras,
$\tilde\phi:=(\widehat{\gr}(\phi))^{-1}\circ \phi$, from $\widehat{\k{G}}$ to
$\widehat{\gr}(\k G)$, such that $\gr(\tilde\phi)=\id$.
Let $E\colon G\to \widehat{\gr}(\k G)$ be the composite
\begin{equation}
\label{eq:gkg}
\xymatrixcolsep{20pt}
\xymatrix{
G\ar@{^{(}->}[r] & \k{G}\ar^{\jmath}[r] & \widehat{\k{G}}\ar[r]^(.4){\tilde\phi}
& \widehat{\gr}(\k G).
}
\end{equation}
Since both $\tilde\phi$ and $\jmath$ are morphisms of Hopf algebras, and
since the inclusion $G\hookrightarrow \k{G}$ is a monoid map sending $G$ to
the group-like elements of $\k{G}$, the composite
$E$ is also a monoid map.
It is clear that $\widehat{E}=\tilde\phi$ and $\bar{E}=\tilde\phi\circ \jmath$.
Since both $\tilde\phi$ and $\jmath$ are filtration-preserving,
and $\gr(\jmath)=\gr(\widehat{E})=\id$, we infer that $\bar{E}$ is filtration-preserving
and $\gr(\bar{E})=\id$.
Finally, by construction, $E$ is a group-like expansion.
\end{proof}
Propositions \ref{prop:Taylor} and \ref{prop:TaylorExp} can be summarized as follows.
\begin{theorem}
\label{thm:TaylorHopf}
The assignment $E\leadsto \widehat{E}$ establishes a
one-to-one correspondence between Taylor expansions
$G\to \widehat{\gr}(\k{G})$ and filtration-preserving isomorphisms
of complete Hopf algebras $\widehat{\k G}\to \widehat{\gr}(\k G)$ for which
the associated graded morphism is the identity on $\gr(\k{G})$.
\end{theorem}
This theorem generalizes a result of Massuyeau (\cite[Proposition~2.10]{Massuyeau12}),
from finitely generated free groups to arbitrary finitely generated groups.
Proposition \ref{prop:TaylorExp} and Theorem \ref{thm:TaylorHopf} have
as an immediate corollary the aforementioned
criterion for the existence of a Taylor expansion.
\begin{corollary}
\label{cor:te}
A finitely generated group $G$ has a Taylor expansion if and only if
there is an isomorphism of filtered Hopf algebras,
$\widehat{\k G}\cong\widehat{\gr}(\k G)$.
\end{corollary}
\section{Chen iterated integrals and Taylor expansions}
\label{sec:chenTaylor}
\subsection{Chen iterated integrals}
\label{subsec:chen}
In \cite{Chen73,Chen77}, Chen developed a theory of formal power series
connections and iterated integrals on smooth manifolds.
His original motivation was to describe the homology
of the loop space of a smooth manifold $M$
in terms of the differential graded algebra formed by tensoring the de
Rham algebra $\Omega_{\textrm{DR}}(M)$ with the tensor algebra
on the vector space $H_{>0}(M,\mathbb{R})$, completed with respect
to the powers of the augmentation ideal. As summarized below,
Chen's theory leads to monodromy representations of the fundamental
group of $M$ (see also Lin \cite{Linxiaosong97} and Kohno \cite{Kohno16}
for further details).
For simplicity, we will assume the manifold $M$ has the homotopy type
of a connected, finite-type CW-complex.
Upon choosing a basis $\mathbf{X}=\{X_i\}_i$ for $\widetilde{H}_*(M,\k)$,
we may identify the algebra
$\Omega_{\textrm{DR}}(M) \otimes_{\k} \widehat{T}(\widetilde{H}_*(M,\k))$
with $\Omega_{\textrm{DR}}(M)\ncom{\mathbf{X}}$. (Here, $\k=\mathbb{R}$ or $\mathbb{C}$.)
A {\em formal power series connection}\/
on $M$ is an element $\omega\in \Omega_{\textrm{DR}}(M)\ncom{\mathbf{X}}$.
We may write such an element (which may also be viewed as a usual connection
on the trivial bundle $M\times \k\ncom{\mathbf{X}}$) as
\begin{equation}
\label{eq:connection}
\omega=
\sum w_iX_i + \cdots + \sum w_{i_1\ldots i_r}X_{i_1}\cdots X_{i_r}+\cdots,
\end{equation}
where the coefficients are smooth forms of positive degree on $M$.
A connection $\omega$ as above is said to be {\em flat}\/ if it satisfies
the Maurer--Cartan equation, $d\omega-\omega\wedge \omega=0$.
For a homology class $X\in \widetilde{H}_{p}(M,\k)$
we set $\deg X= p-1$; more generally, we set
$\deg (X_{i_1}\cdots X_{i_r}):=\deg X_{i_1}+\cdots +\deg X_{i_r}$.
We denote by $\omega_0\in \Omega^1_{\textrm{DR}}(M)\otimes_{\k}
\widehat{T}(H_1(M,\k))$ the degree $0$ part of $\omega$.
Now let $G=\pi_1(M,x_0)$ and suppose $\widehat{\gr}(\k{G})$
admits a presentation of the form $\widehat{T}(H_1(M,\k))/ I$,
for some closed Hopf ideal $I$ in the completed tensor algebra on $H_1(M,\k)$.
If the connection $\omega_0$ is flat modulo the relations in $I$,
the corresponding holonomy homomorphism,
$J\colon G\to \widehat{T}(H_1(M,\k))/I$,
may be defined by means of iterated integrals, as follows:
\begin{align}
\label{eq:chenint}
J(g)&=
1+\sum_{k=1}^{\infty} \int_{0\leq t_1\leq \cdots\leq t_k\leq 1}
\omega_0(\dot{\gamma}(t_k))\wedge \cdots\wedge \omega_0(\dot{\gamma}(t_1))\\ \notag
&=1+\sum_{k=1}^{\infty} \int_{0}^{1}
\omega_0(\dot{\gamma}(t_k)) \cdots \int_{0}^{t_3}\omega_0(\dot{\gamma}(t_2))
\int_{0}^{t_2}\omega_0(\dot{\gamma}(t_1))\, ,
\end{align}
where $g\in G$ is represented by a piecewise smooth loop $\gamma\colon [0,1] \to M$
at $x_0$. As shown in \cite{Chen57} (see also \cite{Linxiaosong97,Kohno16}),
the holonomy homomorphism $J\colon G\to \widehat{\gr}(\k{G})$
is multiplicative and maps $G$ to group-like elements in
$\widehat{\gr}(\k{G})$; thus, $J$ is a Taylor expansion for $G$.
\subsection{Expansions of free groups}
\label{sec:magnus}
Let $F_n$ be a finitely-generated free group on generators $x_1,\dots,x_n$.
The complete Hopf algebra $\widehat{\gr}(\k F_n)$ can be identified with
$\k\ncom{\mathbf{X}}=\k\ncom{X_1,\dots,X_n}$,
the power series ring over $\k$ in $n$ non-commuting variables.
There are three well-known expansions of this group.
\begin{enumerate}
\item The first one is the Magnus expansion,
$M\colon F_n\to \k\ncom{\mathbf{X}}$,
given by $M(x_i)=1+X_i$, see \cite{Magnus-K-S}.
This expansion is multiplicative but not co-multiplicative
if $n>1$; thus, it is not a Taylor expansion.
\item The second one is the power series expansion,
$L\colon F_n\to \k\ncom{\mathbf{X}}$, given by
$L(x_i)= \exp(X_i)$. As shown by Lin in \cite{Linxiaosong97},
this is a Taylor expansion.
\item The third type of expansion arises from the construction outlined in
\S \ref{subsec:chen}, with $\k=\mathbb{C}$. Let $C_n=\mathbb{C}\setminus \{1,\ldots,n\}$
be the complex plane $\mathbb{C}$ with $n$ punctures, so that $F_n=\pi_1(C_n, 0)$. Let
$w_i=\dfrac{1}{2\pi \sqrt{-1}}\cdot\dfrac{dz}{z-i}$
be closed $1$-forms on $C_n$ dual to the cycles $x_i$.
Then $\omega=\sum_{i=1}^n w_i X_i$ is a degree~$0$ flat connection on the
trivial bundle $C_n\times \mathbb{C}\ncom{\mathbf{X}}\to C_n$.
The corresponding monodromy representation,
$J\colon F_n\to \mathbb{C}\ncom{\mathbf{X}}$,
is given by
\begin{equation}
\label{eq:jeff}
J(f)=1+ \sum_{k=1}^{\infty}\sum\limits_{0\leq i_1, \dots, i_k\leq n}
\left(\dfrac{1}{2\pi \sqrt{-1}} \right)^k
\int\limits_{0\leq t_1\leq \cdots \leq t_k\leq 1} \bigwedge_{r=1}^k
\dfrac{d\gamma(t_r)}{\gamma(t_r)-i_r}\, X_{i_1}\cdots X_{i_k}\, ,
\end{equation}
where $f\in F_n$ is represented by a piecewise smooth loop $\gamma\colon [0,1] \to C_n$ at $0$.
This gives another Taylor expansion over $\mathbb{C}$ for the free group $F_n$.
\end{enumerate}
\subsection{Expansions of free abelian groups}
\label{sec:magnus-abel}
Let $\mathbb{Z}^n$ be the free abelian group of rank $n>0$.
This group admits a presentation of the form
$\mathbb{Z}^n=F_n/N$, where $N$ is the normal subgroup of $F_n$ generated
by the commutators $[x_i,x_j]:=x_ix_jx_i^{-1}x_j^{-1}$ for $1\leq i<j\leq n$.
The complete Hopf algebra $\widehat{\gr}(\k \mathbb{Z}^n)$ may be identified with
$\k\com{\mathbf{X}}=\k\com{X_1,\dots,X_n}$,
the power series ring over $\k$ in $n$ commuting variables.
The power series expansion of the free group $F_n$ induces a Taylor expansion
of the free abelian group $\mathbb{Z}^n$; this expansion, $L\colon \mathbb{Z}^n\to \k\com{\mathbf{X}}$,
is given by $L(x_i)= \exp(X_i)$.
\subsection{Taylor expansions for surface groups }
\label{subsec:Surfacegroups}
Let $G=\pi_{1}(S_g)=F_{2g}/\langle r\rangle$ be the fundamental
group of a compact, connected, orientable surface of genus $g\ge 1$.
Such a group has a presentation with generators
$x_i, y_i$ for $i=1, \dots, g$ and a single relator
$r=\sum_{i=1}^g[x_i,y_i]$. It is well-known that $G$ is $1$-formal.
In particular, there is a Taylor expansion $G\to \widehat{\gr}(\k{G})$,
for any field $\k$ of characteristic $0$.
Here, the complete Hopf algebra $\widehat{\gr}(\k G)$ is generated by
$X_i$, $Y_i$ for $i=1, \dots, g$, and subjects to a relation
$\sum_{i=1}^g[X_i,Y_i]=0$.
However, actually constructing such an expansion is not an easy task.
Using Chen's theory of iterated integrals, Lin constructed in \cite{Linxiaosong97}
an explicit Taylor expansion over $\k=\mathbb{C}$ for the group $G=\pi_{1}(S_g)$.
Let $\alpha_i, \beta_i$ be closed $1$-forms dual to $x_i, y_i$, respectively.
Set $\omega=\sum_{r=1}^{\infty}\omega^{(r)}$, where
$\omega^{(1)}=\sum_{i=1}^g \alpha_iX_i+ \sum_{i=1}^g \beta_iY_i$,
and $\omega^{(r)}$ is the homogeneous polynomial of degree $r$
defined inductively by solving the equation $d \omega -\omega\wedge \omega=0$.
Then $\omega$ is a flat formal power series connection
on $S_g$. The corresponding expansion, $J\colon G\to \widehat{\gr}(\k G)$,
is defined by means of the iterated integral
\eqref{eq:chenint}.
By Theorem \ref{thm:Taylorpropagation}\eqref{item:fieldextensionT},
there exists a rational Taylor expansion for $G$.
Recently, Massuyeau \cite{Massuyeau12} constructed
rational Taylor expansions for the surface groups $G=\pi_{1}(S_g)$
by suitably deforming the power series expansion of the free groups $F_{2g}$.
\section{Lower central series and holonomy Lie algebras}
\label{sec:malcev}
\subsection{Associated graded Lie algebras}
\label{subsec:Associatedgraded}
Let $G$ be a group.
The \emph{lower central series}\/ of $G$ is the sequence
of subgroups $\{\Gamma_k G\}_{k\geq 1}$ \/ defined inductively
by $\Gamma_1G=G$ and
\begin{equation}
\label{eq:lcs}
\Gamma_{k+1} G=[\Gamma_k G,G]
\end{equation}
for $k\geq 1$.
Here, for any subgroups $H$ and $K$ of $G$, we denote $[H,K]$
the subgroup of $G$ generated by all group commutators
$[h,g]:=hgh^{-1}g^{-1}$ with $h\in H$ and $g\in K$. In particular,
$\Gamma_{2}G$ equals $G'$, the commutator subgroup of $G$.
Clearly, each term in the LCS series is a normal subgroup
(in fact, a characteristic subgroup) of $G$. Moreover,
$\Gamma_{k+1} G$ contains the commutator subgroup of $\Gamma_k G$,
and so the quotient group, $\gr_k(G):=\Gamma_k G/\Gamma_{k+1} G$,
is abelian.
Let us fix a coefficient field $\k$ of characteristic $0$. The
associated graded Lie algebra of $G$ over $\k$ is defined by
\begin{equation}
\label{eq:gradedLiealgebra}
\gr(G,\k):=\bigoplus\limits_{k\geq 1} \gr_k(G) \otimes \k,
\end{equation}
with the Lie bracket induced by the group commutator.
This construction is functorial: if $\varphi\colon G\to H$
is a group homomorphism, then $\varphi$ preserves the respective
lower central series, and so it induces a morphism of graded Lie algebras,
$\gr(\varphi, \k)\colon \gr(G,\k) \to \gr(H,\k)$.
Assume now that $G$ is a finitely generated group. Then each
LCS quotient $\gr_k(G)$ is a finitely generated abelian group.
Furthermore, $\gr(G,\k)$ is a finitely generated graded Lie algebra,
that can be presented as $\gr(G,\k)=\mathfrak{lie}(V)/\mathfrak{r}$, where $\mathfrak{lie}(V)$
is the free Lie algebra on a finite-dimensional $\k$-vector space $V$
(with non-zero elements in degree $1$), and $\mathfrak{r}$ is a homogeneous
Lie ideal. We let $\phi_k(G):=\dim \gr_k(G,\k)$ be the LCS ranks of $G$.
\subsection{Chen Lie algebras}
\label{subsec:ChenLie}
Another descending series associated to a group $G$ is the
{\em derived series}, starting at
$G^{(0)}=G$, $G^{(1)}=G'$, and $G^{(2)}=G''$,
and defined inductively by $G^{(i+1)}=[G^{(i)},G^{(i)}]$. Note
that any homomorphism $G\to H$ takes $G^{(i)}$ to $H^{(i)}$.
The quotient groups, $G/G^{(i)}$, are solvable; in particular, $G/G'=G_{\ab}$,
while $G/G''$ is the maximal metabelian quotient of $G$.
Assume now that $G$ is finitely generated.
For each $i\ge 2$, the \textit{$i$-th Chen Lie algebra}\/ of $G$ is defined to
be the associated graded Lie algebra of the corresponding solvable quotient,
\begin{equation}
\label{eq:Chen Lie}
\gr(G/G^{(i)},\k).
\end{equation}
Clearly, this construction is functorial.
The quotient map, $p_i\colon G\twoheadrightarrow G/G^{(i)}$, induces a surjective
morphism $\gr(p_i)$ between associated graded Lie algebras
$\gr_k(G,\k)$ and $\gr_k(G/G^{(i)},\k)$. Plainly,
this morphism is the canonical identification in degree $1$.
In fact, the map $\gr(p_i)$ is an isomorphism for each
$k\leq 2^i-1$, see \cite{SW-holo}.
We now specialize to the case when $i=2$,
originally studied by K.-T. Chen in \cite{Chen51}.
The {\em Chen ranks}\/ of $G$ are defined as
$\theta_k(G):=\dim_{\k} (\gr_k(G/G^{''},\k))$.
By the above remarks, $\phi_k(G)\ge \theta_k(G)$, with
equality for $k\le 3$.
\subsection{Holonomy Lie algebras}
\label{subsec:holoLie}
Once again, let $G$ be a finitely generated group. Write $V=H_1(G,\k)$
and let $\mu_G^{\vee}\colon H_2(G,\k)\to V\wedge V$ be the dual of
the cup product map $\mu_G\colon H^1(G,\k)\wedge H^1(G,\k) \to H^2(G,\k)$.
The \emph{holonomy Lie algebra}\/ of $G$ is the
quadratic Lie algebra defined as
\begin{equation}
\label{eq:HolonomyLieAlgebra}
\mathfrak{h}(G,\k)=\mathfrak{lie}(V)/\langle \im \mu_G^{\vee} \rangle\, .
\end{equation}
Clearly, this construction is functorial. Furthermore,
there is a natural surjective morphism of graded Lie algebras,
\begin{equation}
\label{eq:ComparisonMap}
\xymatrixcolsep{16pt}
\xymatrix{
\psi_G\colon \mathfrak{h}(G,\k) \ar@{->>}[r]& \gr(G,\k)\, ,
}
\end{equation}
inducing isomorphisms in degree $1$ and $2$. (See \cite[Lemma 6.1]{SW-holo}
and references therein.) If the map $\psi_G$ is an isomorphism, then we say that
the group $G$ is \emph{graded-formal} (over $\k$).
\subsection{Free groups and surface groups}
\label{subsec:free surf}
We conclude this section with some simple examples illustrating
the above concepts.
\begin{example}
\label{ex:gr-free}
Let $F_n=\langle x_1,\dots, x_n\rangle$ be the free group
of rank $n$. Then $\gr(F_n,\k)=\mathfrak{lie}(x_1,\dots,x_n)$, the free
Lie algebra on $n$ generators, and the map $\psi\colon
\mathfrak{h}(F_n, \k)\to \gr(F_n,\k)$ is an isomorphism. Moreover, as
shown by Witt \cite{Witt37} and Magnus \cite{Magnus37}, the LCS ranks
are given by
\begin{equation}
\label{eq:lcs-free}
\prod\nolimits_{k\geq 1}(1-t^k)^{\phi_k(F_n)}=1-n t ,
\end{equation}
or, equivalently, $\phi_k(F_n)=\tfrac{1}{k}\sum_{d\mid k} \mu(d) n^{k/d}$,
where $\mu$ denotes the M\"{o}bius function. Finally, as shown in
\cite{Chen51}, the Chen ranks of the free groups are given by $\theta_1(F_n)=n$
and
\begin{equation}
\label{eq:chen-free}
\theta_k(F_n)=(k-1)\binom{n+k-2}{k}
\quad
\text{for $k\ge 2$}.
\end{equation}
\end{example}
\begin{example}
\label{ex:gr-surface}
Let $S_g$ be a closed, orientable surface of genus $g\ge 1$.
Its fundamental group, $\Pi_g=\pi_1(S_g)$, has a presentation
with generators $ x_1,y_1,\dots, x_g,y_g$ and a single relator,
$[x_1,y_1]\cdots [x_g,y_g]$. As shown by Labute \cite{Labute70},
$\gr(\Pi_g,\k)= \mathfrak{lie}(x_1,y_1,\dots,x_g,y_g )/\langle
\sum_{i=1}^{g} [x_i,y_i]\rangle$. Again, it is readily seen that
$\mathfrak{h}(\Pi_g), \k)\cong \gr(\Pi_g,\k)$. Furthermore, the LCS ranks
of $\Pi_g$ are given by
\begin{equation}
\label{eq:lcs-surf}
\prod\nolimits_{k\geq 1}(1-t^k)^{\phi_k(\Pi_g)}=1-2gt+t^2,
\end{equation}
while the Chen ranks are given by $\theta_1(\Pi_g)=2g$,
$\theta_2(\Pi_g)=2g^2-g-1$, and
\begin{equation}
\label{eq:chen-surf}
\theta_k(\Pi_g)=(k-1)\binom{2g+k-2}{k}-\binom{2g+k-3}{k-2}
\quad\text{for $k\ge 3$}.
\end{equation}
\end{example}
\section{Malcev Lie algebras and formality properties}
\label{sect:malcev-formal}
\subsection{Malcev Lie algebras}
\label{subsec:malcev}
As before, let $G$ be a finitely generated group, let $\k$ be a field
of characteristic $0$, and let $\widehat{\k{G}}$
be the $I$-adic completion of the group algebra of $G$, where $I$ is the
augmentation ideal of $\k{G}$.
Following Quillen \cite{Quillen69}, we define the \emph{Malcev Lie algebra}\/
of $G$ as the set $\mathfrak{m}(G,\k)$ of all primitive elements in $\widehat{\k{G}}$,
with bracket $[x,y]=xy-yx$. By construction, $\mathfrak{m}(G,\k)$ is a complete, filtered
Lie algebra. Moreover, if we complete
the universal enveloping algebra $U(\mathfrak{m}(G,\k))$
with respect to the powers of its augmentation ideal, then
$\widehat{U}(\mathfrak{m}(G,\k))\cong \widehat{\k{G}}$, as
complete Hopf algebras.
The set of all primitive elements in $\gr(\k{G})$ forms a graded Lie algebra,
which is isomorphic to $\gr(G,\k)$.
An important connection between the Malcev Lie algebra $\mathfrak{m}(G,\k)$
and the associated graded Lie algebra $\gr(G;\k)$ was discovered
by Quillen, who showed in \cite{Quillen68} that there is an isomorphism
of graded Lie algebras,
\begin{equation}
\gr(\mathfrak{m}(G,\k))\cong \gr(G,\k).
\end{equation}
The set of all group-like elements in
$\widehat{\k{G}}$ forms a group, denoted $\mathfrak{M}(G;\k)$. This group comes
endowed with a complete, separated filtration, whose $k$-th term
is $\mathfrak{M}(G;\k) \cap (1+\widehat{I^k})$.
As explained for instance in \cite{Massuyeau12},
there is a one-to-one, filtration-preserving
correspondence between primitive elements
and group-like elements of $\widehat{\k{G}}$ via the exponential
and logarithmic maps
\begin{equation}
\label{eq:explog}
\xymatrix{
\mathfrak{M}(G;\k)\subset 1+\widehat{I} \ar@/_0.9pc/[rr]|{~\log~} &
& \widehat{I} \supset\mathfrak{m}(G;\k)\ar@/_1.2pc/[ll]|{~\exp~} }.
\end{equation}
Let $G$ be a group which admits a finite presentation of the form $G=F/R$.
Using a Taylor expansion for the finitely generated free group $F$, we may find
a presentation for the Malcev Lie algebra
$\mathfrak{m}(G;\k)$, using the approach of Papadima \cite{Papadima95} and
Massuyeau \cite{Massuyeau12}, which may be summarized in
the following theorem.
\begin{theorem}[\cite{Massuyeau12, Papadima95}]
\label{thm:Massuyeau}
Let $G$ be a group with generators $x_1,\dots,x_n$ and relators $r_1,\dots, r_m$.
Let $E$ be a Taylor expansion of the free group $F=\langle x_1,\dots,x_n\rangle$.
There exists then a unique filtered Lie algebra isomorphism
\[
\mathfrak{m}(G;\k) \cong\widehat{\mathfrak{lie}}(\k^n)/
\langle\!\langle W\rangle\!\rangle,
\]
where $\langle\!\langle W \rangle\!\rangle$ denotes
the closed ideal of the completed free Lie algebra $\widehat{\mathfrak{lie}}(\k^n)$ generated
by the subset $\{\log(E(r_1)),\dots,\log(E(r_m))\}$.
\end{theorem}
\subsection{Formality and filtered-formality}
\label{subsec:ff}
The notion of formality first appeared in the study of rational
homotopy types of topological spaces initiated by Sullivan \cite{Sullivan, DGMS}.
Since then, it has been broadly used in investigating a variety
of differential graded objects. We recall now a formality notion
introduced in \cite{SW-formal}.
\begin{definition}
\label{def:MalcevLie}
A finitely generated group $G$ is called \emph{filtered-formal} (over $\k$),
if there is a filtered Lie algebra isomorphism from the Malcev Lie algebra $\mathfrak{m}(G;\k)$
to the degree completion $\widehat{\gr}(G;\k)$ inducing the identity on associated
graded Lie algebras.
\end{definition}
As shown in \cite[Lemma 2.4]{SW-formal}, the following holds:
if $\mathfrak{m}(G;\k)$ is isomorphic (as a filtered Lie algebras) to the
degree completion of a graded Lie algebra $\mathfrak{g}$,
then the group $G$ is filtered-formal (over $\k$).
The notion of filtered-formality satisfies the following
propagation properties.
\begin{theorem}[\cite{SW-formal}]
\label{thm:propagation}
Let $G$ be a finitely generated group.
\begin{enumerate}
\item \label{item:splitinjection} Suppose there is a split
monomorphism $\iota\colon K\rightarrow G$.
If $G$ is filtered-formal, then $K$ is also filtered-formal.
\item \label{item:fieldextension} The group $G$ is filtered-formal over
a field $\k$ of characteristic $0$ if and only if $G$ is filtered-formal over $\mathbb{Q}$.
\item \label{item:products}
$G_1$ and $G_2$ are filtered-formal if and only if
$G_1* G_2$ is filtered-formal if and only if
$G_1\times G_2$ is filtered-formal.
\end{enumerate}
\end{theorem}
\begin{proof}
This theorem is a combination of the following results
from \cite{SW-formal}:
Theorem 5.11 for \eqref{item:splitinjection};
Theorem 6.6 for \eqref{item:fieldextension};
Theorem 7.17 for \eqref{item:products}.
\end{proof}
In particular, if a finitely generated group $G$ if filtered-formal
over $\mathbb{C}$, then it also filtered-formal over $\mathbb{Q}$.
A finitely generated group group $G$ is said to be \emph{$1$-formal}\/ (over $\k$)
if $\mathfrak{m}(G,\k)\cong\widehat{\mathfrak{h}}(G,\k)$ as filtered Lie algebras. It is readily seen
that $G$ is $1$-formal if and only if it is
graded-formal and filtered-formal.
\subsection{Chen Lie algebras and formality}
\label{subsec:chen formal}
The next theorem is the Lie algebra version of Theorem 3.5 from \cite{PS04-imrn},
which describes the relationship between the Malcev Lie algebras of the derived quotients
of a group $G$ and the corresponding quotients of the Malcev Lie algebra of $G$.
\begin{theorem}[\cite{PS04-imrn}]
\label{thm:DerivedQuotientMalcev}
Let $G$ be a finitely generated group.
There is an isomorphism of complete, separated filtered Lie algebras,
\begin{equation*}
\mathfrak{m}(G/G^{(i)};\k)\cong \mathfrak{m}(G;\k)/\overline{\mathfrak{m}(G;\k)^{(i)}},
\end{equation*}
for each $i\ge 2$, where $\overline{\mathfrak{m}(G;\k)^{(i)}}$ is the closure of $\mathfrak{m}(G;\k)^{(i)}$
with respect to the filtration topology on $\mathfrak{m}(G;\k)$.
\end{theorem}
One important application of Theorem \ref{thm:DerivedQuotientMalcev} is the
next theorem, which delineates the relationship between associated graded Lie algebras of
derived quotients and derived quotients of associated graded Lie algebras.
This theorem also shows that filtered-formality is preserved under the operation
of taking derived quotients.
\begin{theorem}[\cite{SW-formal}]
\label{thm:DerivedQuotientMalcevISO}
The quotient map $p_i\colon G\twoheadrightarrow G/G^{(i)}$ induces a
natural epimorphism of graded $\k$-Lie algebras,
\begin{equation*}
\xymatrix{\Psi_G^{(i)}\colon \gr(G;\k)/\gr(G;\k)^{(i)} \ar@{->>}[r]
& \gr(G/G^{(i)};\k)},
\end{equation*}
for each $i\ge 2$.
Moreover, if the group $G$ is filtered-formal, then $\Psi_G^{(i)}$ is
an isomorphism and the derived quotient $G/G^{(i)}$ is filtered-formal.
\end{theorem}
\subsection{filtered-formality and Chen Lie algebras}
\label{subsec:ff-chen}
As mentioned previously, any homomorphism $G_1\to G_2$ induces morphisms
of graded Lie algebras,
$\gr(G_1;\k)\to \gr(G_2;\k)$ and $\gr(G_1/G_1^{(i)};\k)\to \gr(G_2^{(i)};\k)$.
On the other hand, it is not {\it a priori}\/ clear that a morphism
$\gr(G_1;\k)\to \gr(G_2;\k)$ should induce
morphisms between the corresponding Chen Lie algebras.
Nevertheless, as the next theorem shows, this happens
for filtered-formal groups.
\begin{theorem}
\label{thm:Chenobstruction}
Let $G_1$ and $G_2$ be two $\k$-filtered-formal groups.
Then every morphism of graded Lie algebras,
$\alpha \colon \gr(G_1;\k)\to \gr(G_2,\k)$, induces
morphisms $\alpha_i\colon \gr(G_1/G_1^{(i)};\k)\to \gr(G_2/G_2^{(i)};\k)$
for all $i\ge 1$. Consequently, if $\gr(G_1;\k)\cong \gr(G_2;\k)$, then
$\gr(G_1/G_1^{(i)};\k)\cong \gr(G_2/G_2^{(i)};\k)$, for all $i$.
\end{theorem}
\begin{proof}
Fix an index $i\ge 1$, and consider the following diagram of
graded Lie algebras:
\begin{equation}
\begin{gathered}
\xymatrix{
\gr(G_1;\k)\ar[d]^{\alpha} \ar@{->>}[r] &\gr(G_1;\k)/\gr(G_1;\k)^{(i)}
\ar[r]^(.58){\Psi_{G_1}^{(i)}}\ar@{.>}[d]^{\beta_i} &\gr(G_1/G_1^{(i)};\k)\ar@{.>}[d]^{\alpha_i}\\
\gr(G_2;\k) \ar@{->>}[r] &\gr(G_2;\k)/\gr(G_2;\k)^{(i)}
\ar[r]^(.58){\Psi_{G_2}^{(i)}} &\gr(G_2/G_2^{(i)};\k)
}
\end{gathered}
\end{equation}
The morphism $\alpha$ induces a morphism $\beta_i$ between the
respective solvable quotients. By Theorem \ref{thm:DerivedQuotientMalcevISO},
the maps $\Psi_{G_1}^{(i)}$ and $\Psi_{G_2}^{(i)}$
are isomorphisms. We define the desired morphism $\alpha_i$
to be the composition $\Psi_{G_2}^{(i)}\circ\beta_i\circ \big(\Psi_{G_1}^{(i)}\big)^{-1}$.
The last claim follows at once.
\end{proof}
Taking $i=2$ in the above theorem, we obtain the following corollary.
\begin{corollary}
\label{cor:noniosLie}
Suppose $G_1$ and $G_2$ are two $\k$-filtered-formal groups.
If $\theta_k(G_1)\neq \theta_k(G_2)$ for some $k\geq 1$,
then $\gr(G_1,\k)\not\cong \gr(G_2,\k)$, as graded Lie algebras.
\end{corollary}
\section{Taylor expansions and formality properties}
\label{sec:expformal}
In this section we relate the notions of Taylor expansion and filtered-formality
for a finitely generated group $G$.
\subsection{Taylor expansions and isomorphisms of filtered Lie algebras}
\label{subsec:tff}
As the next theorem shows, Taylor expansions of $G$
are intimately related to isomorphisms between the Malcev Lie
algebra $\mathfrak{m}(G;\k)$ and the LCS completion of the associated
graded Lie algebra $\gr(G;\k)$.
\begin{theorem}
\label{thm:expansionFiltered}
There is a one-to-one correspondence between Taylor expansions
$G\to \widehat{\gr}(\k G)$ and filtration-preserving Lie algebra isomorphisms
$\mathfrak{m}(G;\k)\to \widehat{\gr}(G;\k)$ inducing the identity on $\gr(G,\k)$.
\end{theorem}
\begin{proof}
First suppose $E\colon G\to \widehat{\gr}(\k G)$ is a Taylor expansion.
Then, by Proposition \ref{prop:Taylor},
there is a filtration-preserving Hopf algebra isomorphism
$\widehat{E}\colon \widehat{\k{G}} \to \widehat{\gr}(\k G)$, inducing the identity
on $\gr(\k{G})$.
Recall that $\widehat{\k{G}}\cong U(\mathfrak{m}(G;\k))$ and $\widehat{\gr}(\k G)\cong
U(\widehat{\gr}(G;\k))$, as filtered Hopf algebras. Taking primitives,
we obtain a filtration-preserving isomorphism of complete Lie algebras,
$\Prim(\widehat{E})\colon \mathfrak{m}(G;\k)\to \widehat{\gr}(G;\k)$, inducing the identity
on $\gr(G;\k)$.
Now suppose there is an isomorphism of filtered, complete Lie algebras,
$\alpha\colon \mathfrak{m}(G;\k)\to \widehat{\gr}(G;\k)$, such that $\gr(\alpha)=\id$.
Taking universal enveloping algebras, we obtain an
isomorphism of filtered, complete Hopf algebras,
$U(\alpha)\colon\widehat{\k G}\isom\widehat{\gr}(\k G)$,
such that $\gr(\phi)=\id$.
By Proposition \ref{prop:TaylorExp}, the map $U(\alpha)$ induces
a Taylor expansion $E\colon G\to \widehat{\gr}(\k G)$.
\end{proof}
Using this theorem, we obtain in Corollaries \ref{cor:TaylorFilteredFormal}
and \ref{cor:expansionFormal} alternate interpretations of filtered-formality and
$1$-formality.
\begin{corollary}
\label{cor:TaylorFilteredFormal}
A finitely generated group $G$ has a Taylor expansion if and only if $G$ is filtered-formal.
\end{corollary}
\begin{proof}
Follows at once from Theorem \ref{thm:expansionFiltered} and Definition \ref{def:MalcevLie}.
\end{proof}
\begin{theorem}
\label{thm:Taylorpropagation}
Let $G$ be a finitely generated group.
\begin{enumerate}
\item \label{item:splitinjectionT}
Suppose there is a split monomorphism $\iota\colon K\rightarrow G$.
If $G$ has a Taylor expansion, then $K$ also has a Taylor expansion.
\item \label{item:fieldextensionT} The group $G$ has a Taylor expansion
over a field $\k$ of characteristic $0$ if and only if $G$ has a Taylor expansion
over $\mathbb{Q}$.
\item \label{item:productsT}
$G_1$ and $G_2$ have a Taylor expansion if and only if
$G_1* G_2$ has a Taylor expansion if and only if
$G_1\times G_2$ has a Taylor expansion.
\item \label{item:solvablequo} If $G$ has a Taylor expansion, then
all the solvable quotients $G/G^{(i)}$ have a Taylor expansion.
\end{enumerate}
\end{theorem}
\begin{proof}
The first three claims follow from Corollary \ref{cor:TaylorFilteredFormal}
and Theorem \ref{thm:propagation}.
Claim \eqref{item:solvablequo} follows from Corollary \ref{cor:TaylorFilteredFormal}
and Theorem \ref{thm:DerivedQuotientMalcevISO}.
\end{proof}
\begin{corollary}
\label{cor:expansionFormal}
A finitely generated group $G$ is $1$-formal if and only if there is
a Taylor expansion $G \to \widehat{\gr}(\k{G})$
and $\gr(\k G)$ is a quadratic algebra.
\end{corollary}
\begin{proof}
We know that $G$ is $1$-formal if
and only if $G$ is filtered-formal and graded-formal. By
Corollary \ref{cor:TaylorFilteredFormal},
$G$ is filtered-formal if and only if it has a Taylor expansion.
On the other hand, $G$ is graded-formal if and only if $\gr(G;\k)$ admits a
quadratic presentation.
As shown in \cite[\S 2.2.3]{Lee}, this latter condition is equivalent to
the quadraticity of $\gr(\k{G})$. This completes the proof.
\end{proof}
\begin{example}
\label{ex:ReducedFree}
The reduced free group $RF_n$, introduced by J.~Milnor
in his study of link homotopy \cite{Milnor54} is the quotient of the free group
$F_n=\langle x_1,\dots, x_n\rangle$ by the normal subgroup
generated by all elements of the form $[x_i, gx_ig^{-1}]$ with $g\in F_n$.
The relations in $RF_n$ can be reduced to multiple group commutators
in $x_1, \dots, x_n$ with some $x_i$ appears at least twice.
In \cite{Linxiaosong97}, Lin showed that $RF_n$ has
Taylor expansions induced from certain expansions of the
free group $F_n$ (the power series expansion and
the expansion arising from formal power series connections,
as described in \S\ref{sec:magnus}).
It follows from Corollary \ref{cor:TaylorFilteredFormal} that the
group $RF_n$ is filtered-formal.
\end{example}
\subsection{Taylor expansions of nilpotent groups }
\label{sec: TaylorNilpotent}
As before, let $G$ be a finitely generated group.
The next result shows that the Taylor expansions of $G$ are
inherited by the nilpotent quotients $G/\Gamma_{i}G$.
\begin{theorem}
\label{thm:nilp-taylor}
Suppose $G$ admits a
Taylor expansion $E\colon G\to \widehat{\gr}(\k G)$.
Then each nilpotent quotient $G/\Gamma_{i}G$ admits
an induced Taylor expansion,
$E_i\colon G/\Gamma_{i}G\to \widehat{\gr}(\k [G/\Gamma_iG])$.
\end{theorem}
\begin{proof}
By Theorem \ref{thm:expansionFiltered}, the Taylor expansion
$E\colon G\to \widehat{\gr}(\k G)$ determines a filtered
Lie algebra isomorphism, $\alpha\colon \mathfrak{m}(G;\k)\to \widehat{\gr}(G;\k)$.
From the proof of \cite[Theorem 7.13]{SW-formal}, we deduce that
$\alpha$ induces filtered Lie algebra isomorphisms,
$\alpha_i\colon \mathfrak{m}(G/\Gamma_iG;\k)\to \widehat{\gr}(G/\Gamma_iG;\k)$.
Using Theorem \ref{thm:expansionFiltered} again, we obtain the desired
Taylor expansions, $E_i\colon G/\Gamma_{i}G \to \widehat{\gr}(G/\Gamma_{i}G;\k)$.
\end{proof}
\begin{example}
\label{ex:nstep}
As noted in \S \ref{sec:magnus}, the finitely generated free group $F$ admits
Taylor expansions. By Theorem \ref{thm:nilp-taylor},
the $k$-step, free nilpotent group $F/\Gamma_{k+1}F$ admits Taylor expansions
for each $k\geq 1$.
\end{example}
\begin{example}
\label{ex:step2nilpotent}
Let $G$ be a finitely generated, torsion-free, $2$-step nilpotent group,
and suppose $G_{\ab}$ is also torsion-free. As shown in \cite{SW-formal},
the group $G$ is filtered-formal. Thus, by Corollary \ref{cor:TaylorFilteredFormal},
$G$ admits a Taylor expansion.
\end{example}
\begin{example}
\label{ex:Cornulier}
Let $\mathfrak{m}$ be the $5$-dimensional, nilpotent Lie algebra with non-zero
Lie brackets given by $[e_1,e_3]=e_4$ and $[e_1,e_4]=[e_2,e_3]=e_5$.
This Lie algebra may be realized as the Malcev Lie algebra of
a finitely generated, torsion-free nilpotent group $G$.
As noted in \cite{Cornulier14, SW-formal}, this group is not filtered-formal.
Thus, the group $G$ admits no Taylor expansion.
\end{example}
\section{Automorphisms of free groups and almost-direct products}
\label{sec:exp-rtfn}
\subsection{Braid groups}
\label{subsec:BraidGroups}
An automorphism of the free group $F_n=\langle x_1,\dots, x_n\rangle$
is a permutation-conju\-gacy automorphism if it sends each generator
$x_i$ to a conjugate of some other generator $x_j$.
The Artin braid group $B_n$ is the subgroup of $\Aut(F_n)$
consisting of all permutation-conjugacy automorphisms
which fix the product $x_1\cdots x_n$. As shown for instance in \cite{Birman74},
the group $B_n$ is generated by the elementary braids
$\sigma_1,\dots ,\sigma_{n-1}$ (where $\sigma_i$ sends $x_i$ to $x_{i+1}$
and $x_{i+1}$ to $x_{i+1}^{-1}x_ix_{i+1}$ while fixing the other $x_k$'s),
subject to the relations
\begin{equation}
\label{eq:braidrelations}
\begin{cases}
\sigma_{i} \sigma_{j}=
\sigma_{j} \sigma_{i}, &\abs{i-j}\geq 2,\\
\sigma_{i} \sigma_{i+1} \sigma_{i} =
\sigma_{i+1} \sigma_{i} \sigma_{i+1},
& 1\le i\le n-2.
\end{cases}
\end{equation}
The pure braid group $P_n$ is the kernel of the canonical projection
$B_n\to S_n$ that sends a generator $\sigma_i$ to the
transposition $(i,i+1)$. This group is generated by the braids
\begin{equation}
\label{eq:GeneratorsPureBraid}
A_{ij}:=(\sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+1}) \sigma_i^2
(\sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+1})^{-1},
\textrm{ for } 1\leq i<j\leq n.
\end{equation}
The pure braid group $P_n$ decomposes as a semidirect product,
$P_n=F_{n-1}\rtimes P_{n-1}$, where $P_{n-1}$ acts on $F_{n-1}$
by restriction of the Artin representation $B_{n-1}\subset \Aut(F_{n-1})$.
The group $P_n$ is $1$-formal. The associated graded Lie algebra $\gr(P_n,\k)$
is generated by $\{t_{ij} \mid 1\leq i < j\leq n\}$ subject to the relations
$[t_{ij}, t_{kl}]=0$ and $[t_{ij}, t_{ik}+t_{jk}]=0$ whenever $i,j,k,l$ are distinct.
\subsection{Taylor expansions for the pure braid groups}
\label{subsec:TaylorPn}
Explicit Taylor expansions for the pure braid groups $P_n$ over
$\k=\mathbb{C}$ can be constructed using Chen's method of iterated integrals,
see e.g. \cite{Linxiaosong97, Bar-Natan16, Kohno16}.
Let $\textrm{Conf}_n(\mathbb{C})=\mathbb{C}^n\setminus \bigcup_{1\leq i< j\leq n} \{ z_i= z_j \}$
be the configuration space of $n$ ordered points in $\mathbb{C}$,
so that $P_n=\pi_1(\textrm{Conf}_n(\mathbb{C}), 0)$. Consider
the logarithmic $1$-forms on $\textrm{Conf}_n(\mathbb{C})$ given by
\begin{equation}
\label{eq:log-forms}
w_{ij}=w_{ji}=\dfrac{1}{2\pi \sqrt{-1}}\cdot d \log(z_i-z_j)\, .
\end{equation}
Clearly, these $1$-forms are closed. Furthermore, as shown by Arnold
\cite{Arnold69}, these $1$-forms satisfy the relations
$w_{ij}\wedge w_{jl}+w_{jl}\wedge w_{li}+w_{li}\wedge w_{ij}=0$.
As shown by Kohno \cite{Kohno85},
the complete Hopf algebra $\widehat{\gr}(\k{P_n})$ admits a presentation
with generators $\{X_{ij}=X_{ji}; 1\leq i<j\leq n\}$,
subject to the infinitesimal pure braid relations
\begin{equation}
\label{eq:purebraidalgebrarelations}
\begin{cases}
[X_{ij}, X_{kl}]=0 \\
[X_{ij}, X_{il}+X_{lj}]=0\, .
\end{cases}
\end{equation}
The formal power series connection $\omega=\sum_{1\leq i<j\leq n} w_{ij} X_{ij}$
on $\textrm{Conf}_n(\mathbb{C})$ is flat. The corresponding monodromy representation
yields a (faithful) Taylor expansion for the pure braid group,
$J\colon P_n\to \widehat{\gr}(\k P_n)$, given by \eqref{eq:chenint}, more explicitly,
as stated in \cite{Bar-Natan16}, (first appeared in \cite{Kohno88} )
\begin{equation}
\label{eq:chenint-pn}
J(g)=1+ \sum_{k=1}^{\infty}\sum\limits_{1\leq i_1<j_1, \dots, i_k<j_k\leq n} \left(\dfrac{1}{2\pi \sqrt{-1}} \right)^k
\int\limits_{0\leq t_1\leq \cdots \leq t_k\leq 1} \bigwedge_{s=1}^k
d \log(z_{i_s}-z_{j_s} )\, X_{i_1,j_1}\cdots X_{i_k,j_k}\, ,
\end{equation}
where $g\in P_n$ is represented by a piecewise smooth loop
$\gamma\colon [0,1] \to \textrm{Conf}_n(\mathbb{C})$ at $0$,
and $z_i$ is the $i$-th coordinate of the loop $\gamma$.
The Taylor expansion $J$ is called the monodromy of the flat connection in \cite{Kohno85},
and the holonomy homomorphism in \cite{Kohno16}. This expansion is a finite type invariant
for the pure braid groups, and a prototype for the Kontsevich integral in knot theory.
\subsection{Welded braid groups}
\label{subsec:WeldedBraidGroups}
The welded braid group (or, the braid-permutation group)
$wB_n$ is the subgroup of $\Aut(F_n)$
consisting of all permutation-conjugacy automorphisms of $F_n$.
The welded pure braid group (also known as the group of
basis-conjugating automorphisms, or McCool group) $wP_n$
is the kernel of the canonical projection $wP_n\to S_n$.
In \cite{McCool86}, McCool gave a finite presentation for
$wP_n$; the generators are the automorphisms $\alpha_{ij}$
($1\leq i\neq j\leq n$) sending $x_i$ to $x_jx_ix_j^{-1}$.
The subgroup of $wP_n$ generated by the elements $\alpha_{ij}$ with
$i>j$ is called the {\em upper welded pure braid group} (or, upper
triangular McCool group), and is denoted by $wP_n^+$.
As shown in \cite{Cohen-P-V-Wu08}, the upper welded
pure braid group $wP_n^+$ also decomposes as a
semidirect product, $wP_n^+=F_{n-1}\rtimes wP_{n-1}^+$.
Work of Berceanu and Papadima from \cite{Berceanu-Papadima09}
establishes the $1$-formality of the groups $wP_n$ and $wP_n^+$.
Bar-Natan and Dancso, in \cite{Bar-Natan14}, investigate expansions
of welded braid groups. The Chen ranks of the
groups $P_n$, $wP_n$, and $wP_n^+$ were computed
in \cite{Cohen-Suciu95}, \cite{Cohen-Schenck15}, and \cite{SW-mccool},
respectively. We summarize those results, as follows.
\begin{theorem}[\cite{Cohen-Suciu95,Cohen-Schenck15,SW-mccool}]
\label{thm:ChenRanksLCSRanks}
The Chen ranks of $P_n$, $wP_n$, and $wP_n^+$ are given by
\begin{enumerate}
\item
$\theta_1(P_n)=\binom{n}{2}$, $\theta_2(P_n)=\binom{n}{3}$, and
$\theta_k(P_n)= (k-1)\binom{n+1}{4}$ for $k\geq 3$.
\item
$\theta_k(wP_n) = (k-1)\binom{n}{2} + (k^2-1) \binom{n}{3}$ for $k\gg 0$.
\item
$\theta_1(wP_n^+)=\binom{n}{2}$, $\theta_2(wP_n^+)=\binom{n}{3}$,
and $\theta_k(wP_n^+)= \binom{n+1}{4} + \sum_{i=3}^k\binom{n+i-2}{i+1}$
for $k\geq 3$.
\end{enumerate}
\end{theorem}
\subsection{Distinguishing some related Lie algebras}
\label{subsec:discuss}
Both the pure braid groups $P_n$ and the upper McCool groups $wP_n^+$
are iterated semidirect products of the form $F_{n-1}\rtimes \dots \rtimes F_2\rtimes F_1$.
Clearly, $P_1=wP_1^+ =\{1\}$ and $P_2=wP_2^+ =\mathbb{Z}$; it is also known that
$P_3\cong wP_3^+\cong F_2\times F_1$. Furthermore,
both $P_n$ and $wP_n^+$ share the same LCS ranks and the same Betti numbers
as the corresponding direct product of free groups, $\Pi_n= \prod_{i=1}^{n-1}F_{i}$, see
\cite{Arnold69, Cohen-P-V-Wu08, FalkRandell, Kohno85}.
\begin{prop}
\label{prop:theta4}
For each $n\geq 4$, the graded Lie algebras $\gr(P_n,\k)$, $\gr(wP^+_n,\k)$,
and $\gr(\Pi_n,\k)$ are pairwise non-isomorphic.
\end{prop}
\begin{proof}
Using the computations recorded in Theorem \ref{thm:ChenRanksLCSRanks},
we find that $\theta_4(P_n)=3\binom{n+1}{4}$ and
$\theta_{4}(wP_n^+)= 2\binom{n+1}{4}+\binom{n+2}{5}$.
Furthermore, the computation of K.-T.~Chen recorded in Example \ref{ex:gr-free}
implies that $\theta_4(\Pi_n)=3\binom{n+2}{5}$, cf. \cite{Cohen-Suciu95}.
Comparing these ranks and using Corollary \ref{cor:noniosLie}
shows that the graded Lie algebras $\gr(P_n,\k)$, $\gr(\Pi_n,\k)$, and
$\gr(wP^+_n,\k)$ are pairwise non-isomorphic, as claimed
\end{proof}
This proposition recovers (in stronger form) the following
result from \cite{SW-mccool}: For each $n\geq 4$,
the groups $P_n$, $wP_n^+$, and $\Pi_n$ are
pairwise non-isomorphic.
\subsection{Almost-direct products}
\label{subsec:Almostdirect}
A semi-direct product of groups, $H \rtimes Q$, is called
an \emph{almost-direct product} of $H$ and $Q$,
if the action of $Q$ on $H$ induces a trivial action on
the abelianization $H_{\ab}$, that is,
$qhq^{-1} \equiv h$ modulo $[H,H]$
for any $q\in Q$ and $h\in H$.
\begin{theorem}
Let $G=H \rtimes Q$ be a almost-direct product. Then,
\begin{enumerate}
\item[(1)] $\gr(G;\k)\cong \gr(H;\k) \rtimes \gr(Q;\k)$ as graded Lie algebras.
\item[(2)] $\widehat{\gr}(\k G)\cong \widehat{\gr}(\k H)\hat{\otimes}
\widehat{\gr}(\k Q)$ as graded vector spaces.
\end{enumerate}
\end{theorem}
\begin{proof}
The first claim follows from \cite[Theorem (3.1)]{FalkRandell}, while the second claim
follows from \cite[Theorem 3.1]{Papadima02}.
\end{proof}
In general, an almost-direct product of $1$-formal groups need not
be $1$-formal, or even filtered-formal.
\begin{example}
\label{ex:semiproduct}
Let $L$ be the link of $5$ great circles in $S^3$ corresponding to the
arrangement of transverse planes through the origin of $\mathbb{R}^4$
denoted as $\mathcal{A}(31425)$ in Matei--Suciu \cite{MateiSuciu00}.
The link group $G=\pi_1(S^3\setminus L)$ is isomorphic to the almost-direct
product $F_4\rtimes_{\alpha} F_1$, where $\alpha=A_{1,3}A_{2,3}A_{2,4}\in P_4$.
From \cite{SW-holo}, based on the work of Berceanu and Papadima
\cite{Berceanu-Papadima94}, the group $G$ is graded-formal. On the
other hand, as noted by Dimca, Papadima, and Suciu in
\cite[Example 8.2]{Dimca-Papadima-Suciu}, the Tangent Cone theorem
does not hold for this group, and thus $G$ is not $1$-formal.
Consequently, $G$ is not filtered-formal.
\end{example}
\section{Faithful Taylor expansions and the RTFN property}
\label{sect:RTFN}
\subsection{Residually torsion free-nilpotent groups}
\label{subsec:rtfn}
A group $G$ is said to be \emph{residually torsion-free nilpotent}\/
(for short RTFN) if for any $g\in G$, $g\neq 1$,
there exists a torsion-free nilpotent group $Q$, and an epimorphism
$\psi\colon G\to Q$ such that $\psi(g)\neq 1$.
Equivalently, $G$ is residually torsion-free nilpotent if and only if
$\bigcap_{k\geq 1} \tau_{k}G=\{1\}$, where
\begin{equation}
\label{eq:tau-filtration}
\tau_{k}G=\{g\in G\mid \text{$g^n \in \Gamma_{k}G$, for some $n\in \mathbb{N}$} \}.
\end{equation}
For a group $G$, the property of being residually torsion-free nilpotent
is inherited by all subgroups, and is preserved under direct products and free products.
By \cite[Ch.~VI, Thm.~2.26]{Passi79}, a group $G$ is residually
torsion-free nilpotent if and only if the group-algebra
$\k{G}$ is residually nilpotent, that is, $\bigcap_{k\geq 1}I^k=\{0\}$,
where $I$ is the augmentation ideal.
Therefore, if $G$ is finitely generated, the RTFN condition is
equivalent to the injectivity of the canonical map to the prounipotent
completion, $\kappa\colon G\to \mathfrak{M}(G,\k)$, where
recall $\mathfrak{M}(G,\k)$ is the set of group-like elements in $\widehat{\k G}$.
If $G$ is residually nilpotent and $\gr_{k} (G)$ is torsion-free for
$k\ge 1$, then $G$ is residually torsion-free nilpotent.
Residually torsion-free nilpotent implies residually nilpotent, which
in turn implies residually finite.
Examples of residually torsion-free nilpotent groups include
torsion-free nilpotent groups, free groups and surface groups;
more examples will be discussed below.
\subsection{Torelli groups}
\label{subsec:torelli}
Let $G$ be a finitely generated group, and let $\Aut(G)$ be its
group of automorphisms. The {\em Torelli group}\/ of $G$ is
the subgroup of $\Aut(G)$ consisting of all automorphisms
inducing the identity on abelianization; that is,
\begin{equation}
\label{eq:iag}
\IA(G)= \ker (\Aut(G) \to \Aut(G/[G,G]).
\end{equation}
\begin{example}
Let $F_n$ be the free group of rank $n$,
and let $\mathbb{Z}^n$ be its abelianization. Identify the automorphism
group $\Aut(\mathbb{Z}^n)$ with the general linear group $\GL_n(\mathbb{Z})$.
As is well-known, the map $\Aut(F_n)\to \GL_n (\mathbb{Z})$
which sends an automorphism to the induced map on the
abelianization is surjective.
The Torelli group $\IA(F_n)=\ker (\Aut(F_n) \twoheadrightarrow \GL_n(\mathbb{Z}))$ is
classically denoted by $\IA_n$. Magnus showed that this group is
finitely generated.
Clearly, $\IA_1=\{1\}$, while, as noted by Magnus,
$\IA_2=\Inn(F_2)\cong F_2$.
On the other hand, Krsti\'{c} and McCool showed
that $\IA_3$ admits no finite presentation. It is still unknown
whether $\IA_n$ admits a finite presentation for $n\ge 4$.
\end{example}
\begin{example}
Let $\Sigma_g$ be a Riemann surface of genus $g$,
and let $\mathcal{I}_g=\IA(\pi_1(\Sigma_g))$ be the associated
Torelli group. For $g \le 1$, the group $\mathcal{I}_g$ is trivial,
while for $g=2$, it is not finitely generated. On the other hand,
it is known that $\mathcal{I}_g$ is finitely generated for $n\ge 3$.
\end{example}
As noted by Hain \cite{Hain97} in the case of the Torelli group of
a Riemann surface and proved by Berceanu and Papadima
\cite{Berceanu-Papadima09} in full generality, a stronger
assumption on $G$ leads to a stronger conclusion on $\IA(G)$.
\begin{theorem}[\cite{Hain97, Berceanu-Papadima09}]
\label{thm:hbp}
Let $G$ be a finitely generated, residually nilpotent group, and suppose
$\gr_k(G)$ is torsion-free for all $k\ge 1$. Then the Torelli group
$\IA(G)$ is residually torsion-free nilpotent.
\end{theorem}
As shown by Magnus, all free group $F_n$ are residually torsion-free nilpotent.
Hence, the Torelli groups $\IA(F_n)$ are residually torsion-free nilpotent.
Furthermore, all its subgroups, such as the pure braid group
$P_n$, the McCool group $wP_n$, and the
upper McCool group $wP_n^+$ are also residually torsion-free nilpotent.
We refer to \cite{BB09, Marin12, SW-braids} for more details and references
on this subject.
\subsection{The RTFN property and Taylor expansions}
\label{subsec:rtfn-taylor}
The next result relates the RTFN property of a filtered-formal group
to the injectivity of the corresponding Taylor expansion.
\begin{prop}
\label{prop:fftaylor}
A finitely generated group $G$ has a faithful Taylor expansion
if and only if $G$ is residually torsion-free nilpotent and filtered-formal.
\end{prop}
\begin{proof}
By Corollary \ref{cor:TaylorFilteredFormal}, the group
$G$ is filtered-formal if and only if there is
a Taylor expansion $E\colon G\to \widehat{\gr}(\k G)$.
In this case, by Propositions \ref{prop:Taylor} and \ref{prop:TaylorExp},
the map $\widehat{E}\colon \widehat{\k G}\to \widehat{\gr}(\k G)$ is an
isomorphism of filtered Hopf algebras, which fits into the commuting diagram
\begin{equation}
\begin{gathered}
\xymatrix{
G \ar[r]^{\kappa}\ar[rd]_{E} & \widehat{\k G} \ar[d]^{\widehat{E}} \\
& \widehat{\gr}(\k G)\, .
}
\end{gathered}
\end{equation}
Hence, $E$ is injective if and only if $\kappa$ is injective.
That is to say, the expansion $E$ is faithful if and only if
the group $G$ is RTFN.
\end{proof}
\begin{example}
\label{ex:braid-rtfn}
Consider the braid group $B_n$, with $n\ge 3$. Let us identify the complete Hopf algebra
$\widehat{\gr}(\k B_n)$ with $\k\com{X}$, the power series ring over $\k$ in one variable.
The homomorphism $L\colon B_n\to \k\com{X}$ given by $L(\sigma_i)= \exp(X)$
is a Taylor expansion of the braid group, since
$\log(\exp(\sigma_i)\exp(\sigma_j))=2X$
is a group-like element in $\k\com{X}$.
It is clear that this expansion is not faithful, since
$L([\sigma_1,\sigma_2])=0$ but $[\sigma_1,\sigma_2]\neq 1\in B_n$.
In fact, it is known that the braid groups $B_n$ ($n\ge 3$) are
not RTFN, see \cite{RolfsenZhu-98}.
\end{example}
\begin{ack}
We wish to thank Dror Bar-Natan for an inspiring conversation that led us to
work on this project.
\end{ack}
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| {
"timestamp": "2019-11-20T02:18:30",
"yymm": "1905",
"arxiv_id": "1905.10355",
"language": "en",
"url": "https://arxiv.org/abs/1905.10355",
"abstract": "Let $G$ be a finitely generated group, and let $\\Bbbk{G}$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\\Bbbk{G}$ which generalizes the Magnus expansion of a free group. The group $G$ is said to be filtered-formal if its Malcev Lie algebra is isomorphic to the degree completion of its associated graded Lie algebra. We show that $G$ is filtered-formal if and only if it admits a Taylor expansion, and derive some consequences.",
"subjects": "Group Theory (math.GR); Rings and Algebras (math.RA)",
"title": "Taylor expansions of groups and filtered-formality",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575111777183,
"lm_q2_score": 0.7217432182679957,
"lm_q1q2_score": 0.7091542202507986
} |
https://arxiv.org/abs/1905.03456 | Expanding polynomials on sets with few products | In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \in \mathbb{R}[x,y]$ we have that $|f(A,A)| = \Omega_{K,\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form $f(x,y) = g(M(x,y))$ for some monomial $M$ and some univariate polynomial $g$. | \section{Introduction}
\label{intro}
Given polynomials $f\in \ensuremath{\mathbb R}[x]$ and $g\in \ensuremath{\mathbb R}[x,y]$, and sets $A,B\subset \ensuremath{\mathbb R}$, we write
\[ f(A) = \{f(a) :\ a\in A \} \ \text{ and } \ g(A,B) = \{g(a,b) :\ a\in A,\ b\in B\}. \]
That is, $g(A,B)$ is the set of distinct values that can be obtained by applying $g$ on the cartesian product $A\times B$. When $g(x,y)=x+y$ or $g(x,y)=xy$, the more convenient notation $g(A,B)=A+B$ or $g(A,B)=AB$ is generally preferred. This paper will be concerned with understanding the growth of sets such as $g(A,B)$ with respect to $|A|$ and $|B|$. We will only focus on polynomials over the reals, so our story begins with the result of Elekes and R\'onyai, who in \cite{ER00} uncovered that $|f(A,B)|$ must be asymptotically larger than $|A|$ or $|B|$, unless the polynomial $f \in \ensuremath{\mathbb R}[x,y]$ has one of the special forms $f = h(g_1(x)+g_2(y))$ and $f = h(g_1(x)\cdot g_2(y))$, for some $h,g_1,g_2\in \ensuremath{\mathbb R}[x]$. The current best bound for this problem is the following one by Raz, Sharir, and Solymosi \cite{RSS16}.
\bigskip
\begin{theorem} \label{th:ElekesRonyai}
Let $A, B \subset \ensuremath{\mathbb R}$ be finite sets, and let $f\in \ensuremath{\mathbb R}[x,y]$ be of a constant degree.
Then, unless $f = h(g_1(x)+g_2(y))$ or $f = h(g_1(x)\cdot g_2(y))$ for some $h,g_1,g_2\in \ensuremath{\mathbb R}[x]$, we have
\[ f(A,B) = \Omega\left(\min\left\{|A|^{2/3}|B|^{2/3},|A|^2,|B|^2\right\}\right).\]
\end{theorem}
\bigskip
Theorem \ref{th:ElekesRonyai} generalizes many problems from discrete geometry and additive combinatorics, and so has many applications (for example, see \cite{RSS16} or \cite{RSZ16}). While we will not aim to give a complete background, it is important to also mention that the analogue problem has been considered over different fields instead of $\mathbb{R}$, where many interesting results are also available. See for instance \cite{Tao} for a more complete account. In particular, over finite fields it is worth pointing out the result of Vu from \cite{Vu}, who classified the two variable polynomials $f(x,y) \in \mathbb{F}_{q}[x,y]$ such that $|f(A,A)|$ is large whenever $|A+A|$ is small.
\bigskip
\begin{theorem} \label{th:Vu}
Let $A$ be a subset of $\mathbb{F}_{q}$ and let $f(x,y) \in \mathbb{F}_{q}[x,y]$ be a polynomial which cannot be written as $g(L(x,y))$ for some linear polynomial $L$ and some univariate polynomial $g$. Then,
$$\max \left\{ |A+A|, f(A,A) \right\} = \Omega\left(\min \left\{ |A|^{2/3} q^{1/3}, |A|^{3/2} q^{-1/4}\right\}\right).$$
\end{theorem}
\bigskip
Motivated by Theorem \ref{th:Vu}, Shen then considered the analogue question over the reals and proved in \cite{Shen12} the following very interesting result, which in part preceeded Theorem \ref{th:ElekesRonyai}.
\bigskip
\begin{theorem} \label{th:Shen}
Let $f\in \ensuremath{\mathbb R}[x,y]$ be a polynomial of a constant degree that is not of the form $g(L(x,y))$ for some linear polynomial $L$ and some univariate polynomial $g$. If $A$ is a finite set of real numbers, then
\[ |A + A| |f(A, A)| = \Omega\left(|A|^{5/2}\right). \]
\end{theorem}
\bigskip
The proof of Theorem \ref{th:Shen} is in some sense a generalization of Elekes's argument from \cite{Elekes} (for the particular case when $f(x,y) = xy$), which manages to replace the spectral graph theory from the finite field case \cite{Vu} with tools from incidence geometry over the reals. In particular, Theorem \ref{th:Shen} implies that when $A \subset \mathbb{R}$ satisfies $|A+A| = O(|A|)$, we have that $|f(A,A)| = \Omega(|A|^{3/2})$ for every polynomial that is not of the form $g(L(x,y))$ for some linear polynomial $L$ and some univariate polynomial $g$. Like Theorem \ref{th:ElekesRonyai}, however, this is not optimal and it is widely believed that the exponent $3/2$ could probably be replaced with $2-\epsilon$ for every $\epsilon > 0$ in general (as it is the case for $f(x,y)=xy$; see for example \cite{Soly09} for more details).
In this paper, we address the ``dual'' problem of classifying the two variable polynomials $f(x,y) \in \mathbb{R}[x,y]$ such that $|f(A,A)|$ is large whenever $|AA|$ is small. As above, it is easy to check that there are some polynomials for which this fails. For instance, consider $A = \left\{2,2^{2},\ldots,2^{N}\right\}$ for which $|AA| = 2N-1$. If we let $f$ be a single monomial such as $f(x,y)=x^{2}y^{3}$, then it is easy to check that $f(A,A) = 5N-4$. More generally, if we choose $g$ to be a polynomial in one variable and $M(x,y)$ to be a single monomial, the $f(A,A)$ will also be usually small for $f(x,y)=g(M(x,y))$. Indeed, consider say $f(x,y) = xy+x^2y^2$; then we also have that $f(A,A) = 2N-1$. Our main result shows that $g(M(x,y))$ is the {\it{only}} real enemy, in the following strong sense.
\bigskip
\begin{theorem} \label{th:main}
Let $f\in \ensuremath{\mathbb R}[x,y]$ be a polynomial of a constant degree that is not of the form $g(M(x,y))$ for some single monomial $M$ and some univariate polynomial $g$. If $A$ is a finite set of real numbers such that $|AA| = K|A|$, then
$$|f(A,A)| = \Omega_{K}(|A|^2).$$
\end{theorem}
\bigskip
This is also optimal up to the dependence on $K$. We prove Theorem \ref{th:main} in Section 3, after introducing the required ingredients in the upcoming Section 2.
\bigskip
\section{Preliminaries}
\bigskip
The proof of Theorem \ref{th:main} is in some sense in spirit with the proofs of Theorem \ref{th:ElekesRonyai} and Theorem \ref{th:Shen}, but it does not rely on any incidence geometry. The main new ingredient is a quantitative version of the celebrated Schmidt subspace theorem \cite{Sch} due to Amoroso and Viada \cite{AV}.
\smallskip
\begin{theorem} \label{subspace}
Let $a_{1},\ldots,a_{n} \in K$ be nonzero elements of an algebraically closed field $K$, and let $\Gamma$ be a subgroup of $K$ of finite rank $r$. Then, the number of solutions of the equation
$$a_{1}z_{1} + \ldots + a_{n}z_{n} = 1$$
with $z_{i} \in \Gamma$ and no subsum on the left hand side vanishing is at most
$$C(n,r) := (8n)^{4n^{4}(n+nr+1)}.$$
\end{theorem}
\smallskip
Schmidt's subspace theorem (together with its different variants) represents a powerful result in number theory, particularly famous for its many applications in diophantine approximation and complexity of algebraic numbers. We will not remind them here, since many excellent surveys have been written about it, so we refer the reader for instance to \cite{Bilu} and \cite{SS}. In fact, Theorem \ref{subspace} has already manifested itself in additive combinatorics as well in \cite{Chang}, where Chang noticed that one can use it to prove that $|AA| = O(|A|)$ implies $|A+A| = \Omega(|A|^{2})$. Theorem \ref{th:main} can therefore also be seen as a generalization of this phenomenon.
The next ingredient is a multiplicative version of a somewhat more unusual version of Freiman's theorem from additive combinatorics, which is essentially a combination of \cite{Chang2} and Freiman's Lemma \cite{Freiman}. See \cite{GT06} for more details.
\smallskip
\begin{theorem} \label{FreimanM}
Let $t \geq 1$ be an integer, let $\epsilon > 0$, and let $A$ be a finite set of real numbers with $|AA| = K|A|$ and $|A| \geq CK^{2}/\epsilon$ for some absolute constant $C > 0$. Then, $A$ is a subset of a set $G$, which is of the form\footnote{If $n$ is a positive integer, $[n]$ denotes the set $\left\{0,1,\ldots,n-1\right\}$.}
$$G:= g_{1}^{[H_{1}]} \cdot \ldots \cdot g_{r}^{[H_{r}]} =\left\{ \prod_{i=1}^{r}g_{i}^{\mu_{i}}:\ \mu_{i} \in \mathbb{Z}, \mu_{i} \in [H_{i}]\right\},$$
where $r \leq \lfloor K - 1 + \epsilon\rfloor$, all the products in
$$G^{(t)} := \left\{\prod_{i=1}^{r} g_{i}^{\mu_i}:\ \mu_{i} \in \mathbb{Z}, \mu_{i} \in [tH_{i}]\right\}$$
are pairwise distinct, and
$$|G| = H_{1} \cdot \ldots \cdot H_{r} \leq t^{K} \exp(CK^{2} \log^{3}{K})|B|.$$
\end{theorem}
\bigskip
The last two ingredients are more algebraic in nature. First, recall that a polynomial $f\in \ensuremath{\mathbb R}[x,y]$ is said to be \emph{reducible} if there exist polynomials $f_1,f_2\in \ensuremath{\mathbb R}[x,y]$ of positive degrees such that $f(x,y) =f_1(x,y) \cdot f_2(x,y)$. A polynomial that is not reducible is said to be \emph{irreducible}. Furthermore, we say that a polynomial $p\in \ensuremath{\mathbb R}[x,y]$ is \emph{decomposable} if there exists a univariate polynomial $p_1$ of degree at least two and $p_2\in \ensuremath{\mathbb R}[x,y]$ such that $p(x, y) = p_1(p_2(x, y))$. Similarly, a polynomial that is not decomposable is said to be \emph{indecomposable}.
We will need a consequence of a theorem of Stein \cite{Stein89}, which follows from the main result of \cite{Ayad02}. See \cite{RSS16} for more details.
\smallskip
\begin{theorem} \label{Stein}
If $f\in \ensuremath{\mathbb R}[x,y]$ is indecomposable, then the polynomial $f(x,y) - \lambda$ is reducible for at most $\mathrm{deg} f$ values of $\lambda\in \ensuremath{\mathbb R}$.
\end{theorem}
\smallskip
Last but not least, we will also need the classical B\'ezout theorem \cite{Cox}, which again we only state for real polynomials, as these are the main objects of our paper.
\smallskip
\begin{theorem} \label{Bezout}
Let $f$ and $g$ be two polynomials in $\mathbb{R}[x,y]$. If $f$ and $g$ vanish simultaneously on more than $(\mathrm{deg} f)(\mathrm{deg} g)$ points of $\mathbb{R}^{2}$, then $f$ and $g$ have a common non-trivial factor.
\end{theorem}
\bigskip
\section{Proof of Theorem \ref{th:main}}
\bigskip
Let $f\in \ensuremath{\mathbb R}[x,y]$ be a polynomial that is not of the form $g(M(x,y))$ for some single monomial $M$ and some univariate polynomial $g$, and let $d$ be the degree of $f$. We will prove that
$$|f(A,A)| = \Omega_{d,K}(|A|^{2})$$
whenever $A \subset \mathbb{R}$ satisfies $|AA|=K|A|$. The dependence on $d$ and $K$ is going to be explicit, but since it is not a priority from time to time we will reserve the right to hide certain expressions under the asymptotic notation whenever it is more convenient.
First, recall that if $f$ is decomposable, then there exist a univariate $f_1$ of degree at least two and $f_2\in \ensuremath{\mathbb R}[x,y]$ such that $f(x,y)=f_1(f_2(x,y))$. Let $(f_1,f_2)$ be a pair of such polynomials that minimizes the degree of $f_2$. In particular, this implies that $f_2$ is indecomposable. Since $f$ is of degree at most $d$, so are $f_1$ and $f_2$. Since $f_1$ is univariate, for every $a\in \ensuremath{\mathbb R}$ there exist at most $d$ numbers $b\in \ensuremath{\mathbb R}$ such that $f_1(b)=a$. Thus, if $|f_2(A,A)|\geq T$ holds for some positive quantity $T$, then $|f(A,A)|\ge T/d$. It then remains to derive the lower bound for the indecomposable $f_2$, which we also know it is not a single monomial $M(x,y)$ from the hypothesis. Abusing of notation, we will refer to $f_2$ as $f$ from now on, and therefore assume without loss of generality that $f$ is indecomposable and not a single monomial as well.
Next, we naturally define the following polynomial energy of $A$ by
\[ E_{f}(A) := \left|\left\{(x,y,x',y') \in A^4:\ f(x,y) = f(x',y')\right\}\right|.\]
For each $\alpha \in f(A,A)$, we also let $m_{A}(\alpha)$ denote the number of pairs $(x,y) \in A \times A$ such that $f(x,y) = \alpha$. In particular,
$$m_{A}(\alpha)^{2} = \left|\left\{(x,y,x',y') \in A^4:\ f(x,y) = f(x',y') = \alpha \right\}\right|,$$
so by Cauchy-Schwarz,
$$E_{f}(A) = \sum_{\alpha \in f(A,A)} m_{A}(\alpha)^{2} \geq \frac{1}{|f(A,A)|} \cdot \left(\sum_{\alpha \in f(A,A)} m_{A}(\alpha)\right)^{2} = \frac{|A|^{4}}{|f(A,A)|}.$$
In order to prove that $|f(A,A)| = \Omega_{d,K}(|A|^{2})$, it thus suffices to show that $E_{f}(A) = O_{d,K}(|A|^{2})$ instead. To achieve this, we will show that for most values of $\alpha \in f(A,A)$, the number of solutions in $A \times A$ to the equation $f(x,y) = \alpha$ is at most a constant which depends solely on $d$ and $K$. More precisely, we claim that
\begin{equation} \label{key}
\left| \left\{\alpha \in f(A,A):\ m_{A}(\alpha) > C\left({d +2 \choose 2},K\right) + d^{2}2^{{d+2\choose 2}} \right\} \right| = O_{d}(1),
\end{equation}
where $C\left({d+2 \choose 2},K\right)$ is the explicit constant from Theorem \ref{subspace}.
Let us first check that this claim implies that $E_{f}(A) = O_{d,K}(|A|^{2})$. For convenience, let
$$\Upsilon(A) := \left\{\alpha \in f(A,A):\ m_{A}(\alpha) > C\left({d +2 \choose 2},K\right) + d^{2}2^{{d+2\choose 2}} \right\},$$
and write
\begin{equation} \label{energy}
E_{f}(A) = \sum_{\alpha \in \Upsilon(A)} m_{A}(\alpha)^{2} + \sum_{\alpha \in f(A,A) \backslash \Upsilon(A)} m_{A}(\alpha)^{2}.
\end{equation}
For every $\alpha \in f(A,A)$, it is easy to see that $m_{A}(\alpha)^{2} = O_{d}(|A|^{2})$. Indeed, recall that this quantity is the number of solutions in $A^{4}$ to $f(x,y) = f(x',y') = \alpha$, so once $x$ and $x'$ are chosen in $A$, there are at most $d$ value for each $y$ and $y'$ that can satisfy the equality. In particular, if $|\Upsilon(A)| = O_{d}(1)$, this implies that the first term in \eqref{energy} satisfies
$$\sum_{\alpha \in \Upsilon(A)} m_{A}(\alpha)^{2} = O_{d}(|A|^{2}).$$
For the second term, note that if $\alpha \not \in \Upsilon(A)$ then
$$M:= \max_{\alpha \in f(A,A) \backslash \Upsilon(A)} m_{A}(\alpha) \leq C\left({d +2 \choose 2},K\right) + d^{2}2^{{d+2\choose 2}}= O_{d,K}(1),$$
therefore
$$\sum_{\alpha \in f(A,A) \backslash \Upsilon(A)} m_{A}(\alpha)^{2} \leq M \sum_{\alpha \in f(A,A) \backslash \Upsilon(A)} m_{A}(\alpha) \leq M|A|^{2} = O_{d,K}(|A|^{2}).$$
Putting these two estimates together, we indeed get that $E_{f}(A) = O_{d,K}(|A|^{2})$. We are now left to prove \eqref{key}, which will require the tools from Section 2.
Recall that $A$ satisfies $|AA| = K|A|$. If the size of $A$ is upper bounded by a constant in terms of $K$, then there is nothing to prove since $|f(A,A)| = \Omega_{d,K}(|A|^{2})$ is trivially true, so we can safely apply Theorem \ref{FreimanM} with $\epsilon = 1$ and $t=d$. This implies that $A$ is a subset of a set $G$, which is of the form
$$G:= g_{1}^{[H_{1}]} \cdot \ldots \cdot g_{r}^{[H_{r}]} =\left\{ \prod_{i=1}^{r}g_{i}^{\mu_{i}}:\ \mu_{i} \in \mathbb{Z}, \mu_{i} \in [H_{i}]\right\},$$
where $r \leq \lfloor K \rfloor \leq K$ and all the products all the products in
$$G^{(d)} := \left\{\prod_{i=1}^{r} g_{i}^{\mu_i}:\ \mu_{i} \in \mathbb{Z}, \mu_{i} \in [dH_{i}]\right\}$$
are pairwise distinct. We also have a quantitative estimate for $|G|$, but it is not required.
For each $\alpha \in f(A,A)$, we now analyze the number of solutions in $A \times A$ to $f(x,y)=\alpha$. Write $f$ explicitly as
$$f(x,y) := \sum_{(i,j) \in S} a_{i,j} x^{i} y^{j},$$
where $S$ is some subset of the set of pairs $\left\{(i,j):\ i,j \geq 0, i+j \leq d\right\}$ and $a_{i,j}$ is a real coefficient for each $(i,j) \in S$.
We begin with a first key lemma.
\bigskip
\begin{lemma} \label{lem}
For every $\alpha \in f(A,A)$, the number of solutions in $A \times A$ to
$$\sum_{(i,j) \in S} a_{i,j} x^{i} y^{j} = \alpha$$
with no subsum on the left hand side vanishing is at most $C\left({d + 2 \choose 2},K\right)$.
\end{lemma}
\smallskip
\begin{proof} Let $\Gamma$ be the multiplicative subgroup of $\mathbb{C}^{*}$ generated by $g_{1},\ldots,g_{r}$, which has rank $r \leq K$ and contains $G$ (and thus also $A$). The number of solutions to
\begin{equation} \label{sub}
\sum_{(i,j) \in S} a_{i,j} z_{i,j} = \alpha
\end{equation}
with $z_{i,j} \in G^{(d)}$ for each $(i,j) \in S$ and no subsum on the left hand side vanishing is at most the number of solutions to \eqref{sub} with the $z_{i,j}$ in $\Gamma$ and no subsum on the left hand side vanishing, so by Theorem \ref{subspace} it is at most $C\left({d + 2 \choose 2},K\right)$. If we also can argue that for each such solution $(z_{i,j})_{(i,j) \in S}$ to \eqref{sub}, there is at most one solution $(x,y) \in G \times G$ (and thus in $A \times A$) to the system of equations
\begin{equation} \label{system}
x^{i} y^{j} = z_{i,j}\ \ \ \ \text{for each}\ (i,j) \in S,
\end{equation}
then the claim follows.
For $x$ and $y$ in $G$, write
$$x = g_{1}^{x_{1}} \cdot \ldots \cdot g_{r}^{x_{r}}\ \ \text{and}\ \ y = g_{1}^{y_{1}} \cdot \ldots \cdot g_{r}^{y_{r}},$$
where $x_{k},y_{k} \in [H_{k}]$ for each $k \in \left\{1,\ldots,r\right\}$. Similarly, for $z_{i,j} \in G^{(d)}$, let
$$z_{i,j} =g_{1}^{z_{i,j,1}} \cdot \ldots \cdot g_{r}^{z_{i,j,r}},$$
where $z_{i,j,k} \in [dH_{k}]$ for each $k \in \left\{1,\ldots,r\right\}$. Plugging these expressions into \eqref{system}, we get
$$g_{1}^{ix_{1}+jy_{1}} \cdot \ldots \cdot g_{r}^{ix_{r}+jy_{r}} = g_{1}^{z_{i,j,1}} \cdot \ldots \cdot g_{r}^{z_{i,j,r}} \ \ \ \ \text{for each}\ (i,j) \in S.$$
Furthermore, since $i+j \leq d$, we also have that $ix_{k}+jy_{k} \in [dH_{k}]$ for each $k$, so by the fact that $G^{(d)}$ has all its products pairwise distinct, it follows that \eqref{system} translates into the following system of equalities, call it $\mathcal{S}_{i,j}$, satisfied by the exponents above for each $(i,j) \in S$:
$$ix_{k} + jy_{k} = z_{i,j,k}\ \ \ \text{for all}\ k \in \left\{1,\ldots,r\right\}.$$
At this point, recall that $f$ is indecomposable by our assumption and is also not a single monomial, so it must contain at least two monomials, say $x^{i}y^{j}$ and $x^{i'}y^{j'}$, for which the two-dimensional vectors $(i,j)$ and $(i',j')$ are not a (rational) scalar multiple of each other. In particular, if a pair $(x,y) \in A \times A \subset G \times G$ exists to satisfy both $\mathcal{S}_{i,j}$ and $\mathcal{S}_{i',j'}$, then each pair $(x_{k},y_{k})$ is uniquely determined in terms of $i,j,i',j'$ and $z_{i,j,k}$, $z_{i',j',k}$ for each $k \in \left\{1,\ldots,r\right\}$, which implies that $(x,y)$ is then uniquely determined. This proves the claim.
\end{proof}
\smallskip
We now analyze what happens if there are vanishing subsums on the left hand side of $f(x,y) = \alpha$. In this sense, we prove the following second key lemma.
\bigskip
\begin{lemma} \label{steinbez}
For all but possibly at most $d+1$ values of $\alpha \in f(A,A)$, the number of pairs $(x,y) \in A \times A$ satisfying $f(x,y) = \alpha$ with some vanishing subsum on the left hand side is at most $d^{2}2^{{d+2\choose 2}}$.
\end{lemma}
\smallskip
\begin{proof}
Recall $f(x,y) := \sum_{(i,j) \in S} a_{i,j} x^{i} y^{j}$ with $|S| \geq 2$, and now suppose that
$$\sum_{(i’,j’) \in S’} a_{i’,j’} x^{i’} y^{j’} = 0$$
for some nontrivial subset $S' \subset S$. Let $N_{S'}(\alpha)$ be number of common solutions in $A \times A$ to
\begin{equation} \label{bez}
f(x,y)-\alpha = 0\ \ \text{and}\ \ g_{S'}(x,y) = 0,
\end{equation}
where $g_{S'} \in \mathbb{R}[x,y]$ is the polynomial defined by
$$g_{S'}(x,y):= \sum_{(i’,j’) \in S'} a_{i’,j’} x^{i’} y^{j’}$$
for a nontrivial subset $S'$ of $S$. By a union bound, it suffices to prove that
$$\sum_{S' \subset S} N_{S'}(\alpha) \leq d^{2} 2^{{d+2 \choose 2}}$$
holds for all but possibly at most $d+1$ values of $\alpha \in f(A,A)$.
For each $S' \subset S$, note that $g_{S'}$ has degree at most $d$, since $\mathrm{deg} f = d$. By Theorem \ref{Stein} there are at most $d$ values of $\alpha$ for which $f(x,y) - \alpha$ is reducible, and at most one value for which $f(x,y)-\alpha$ may be identical to $g_{S'}(x,y)$, for some $S' \subset S$ ($\alpha$ may be equal to the free term in $f$). For each of the other $\alpha \in f(A,A)$, we have that $N_{S'}(\alpha) \leq d^{2}$ for every proper $S' \subset S$. Indeed, if $\alpha$ is such that the polynomial $f(x,y) - \alpha$ is irreducible in $\mathbb{R}[x,y]$ and does not coincide with $g_{S'}(x,y)$, then this simply follows from Theorem \ref{Bezout}, since \eqref{bez} must have at most $d^{2}$ solutions if there is no common factor. Therefore,
$$\sum_{S' \subset S} N_{S'}(\alpha) \leq d^{2} 2^{|S|} \leq d^{2} 2^{{d + 2 \choose 2}}$$
is indeed satisfies by all $\alpha \in f(A,A)$, except for perhaps at most $d+1$ values. This completes the proof of Lemma \ref{steinbez}.
\end{proof}
\bigskip
Claim \eqref{key} now follows by combining Lemma \ref{lem} and Lemma \ref{steinbez}. Indeed, together these two imply that for all but possibly at most $d+1$ values of $\alpha \in f(A,A)$, the number of pairs $(x,y) \in A \times A$ with $f(x,y) = \alpha$ is at most $C\left({d + 2 \choose 2},K\right) + d^{2}2^{{d+2\choose 2}}$. In other words, $\left|\Upsilon(A)\right| \leq d+1$, where
$$\Upsilon(A) := \left\{\alpha \in f(A,A):\ m_{A}(\alpha) > C\left({d +2 \choose 2},K\right) + d^{2}2^{{d+2\choose 2}} \right\}.$$
This completes the proof of Theorem \ref{th:main}.
\bigskip
\bigskip
{\bf{Acknowledgements}}. I would like to thank Vlad Matei, Adam Sheffer and Dmitrii Zhelezov for useful discussions.
\bigskip
| {
"timestamp": "2019-05-10T02:09:34",
"yymm": "1905",
"arxiv_id": "1905.03456",
"language": "en",
"url": "https://arxiv.org/abs/1905.03456",
"abstract": "In this note, we prove that if $A$ is a finite set of real numbers such that $|AA| = K|A|$, then for every polynomial $f \\in \\mathbb{R}[x,y]$ we have that $|f(A,A)| = \\Omega_{K,\\operatorname{deg} f}(|A|^2)$, unless $f$ is of the form $f(x,y) = g(M(x,y))$ for some monomial $M$ and some univariate polynomial $g$.",
"subjects": "Combinatorics (math.CO)",
"title": "Expanding polynomials on sets with few products",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575188391107,
"lm_q2_score": 0.7217432122827969,
"lm_q1q2_score": 0.7091542198995545
} |
https://arxiv.org/abs/1801.00082 | An HDG Method for Distributed Control of Convection Diffusion PDEs | We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a distributed optimal control problem governed by an elliptic convection diffusion PDE. We derive optimal a priori error estimates for the state, adjoint state, their fluxes, and the optimal control. We present 2D and 3D numerical experiments to illustrate our theoretical results. | \section{Introduction}
\label{intro}
We consider the following distributed control problem: Minimize the functional
\begin{align}
\min J(u)=\frac{1}{2}\| y- y_{d}\|^2_{L^{2}(\Omega)}+\frac{\gamma}{2}\|u\|^2_{L^{2}(\Omega)}, \quad \gamma>0, \label{cost1}
\end{align}
subject to
\begin{equation}\label{Ori_problem}
\begin{split}
-\Delta y+\bm \beta\cdot\nabla y&=f+u \quad\text{in}~\Omega,\\
y&=g\qquad\quad\text{on}~\partial\Omega,
\end{split}
\end{equation}
where $\Omega\subset \mathbb{R}^{d} $ $ (d\geq 2)$ is a Lipschitz polyhedral domain with boundary $\Gamma = \partial \Omega$, $ f \in L^2(\Omega) $, and the vector field $\bm{\beta}$ satisfies
\begin{align}\label{beta_con}
\nabla\cdot\bm{\beta} = 0.
\end{align}
It is well known that the optimal control problem \eqref{cost1}-\eqref{Ori_problem} is equivalent to the optimality system
\begin{subequations}\label{eq_adeq}
\begin{align}
-\Delta y+\bm \beta\cdot\nabla y &=f+u\quad~\text{in}~\Omega,\label{eq_adeq_a}\\
y&=g\qquad~~~~\text{on}~\partial\Omega,\label{eq_adeq_b}\\
-\Delta z-\nabla\cdot(\bm{\beta} z) &=y_d-y\quad~\text{in}~\Omega,\label{eq_adeq_c}\\
z&=0\qquad\quad~~\text{on}~\partial\Omega,\label{eq_adeq_d}\\
z-\gamma u&=0\qquad\quad~~\text{in}~\Omega.\label{eq_adeq_e}
\end{align}
\end{subequations}
Optimal control problems for convection diffusion equations arise in applications \cite{MR1766429} and are also an important step towards optimal control problems for fluid flows. Therefore, researchers have developed many different numerical methods for this type of problem including approaches based on finite differences \cite{MR3451511}, standard finite element discretizations \cite{MR2851444,MR2719819,MR2475653}, stabilized finite elements \cite{MR2302057,MR3144702}, the symmetric stabilization method \cite{MR2486088}, the SUPG method \cite{MR2595051,MR3451479}, the edge-stabilization method \cite{MR2463111,MR2068903}, mixed finite elements \cite{MR2550371,MR2851444,MR2971662}, and discontinuous Galerkin (DG) methods \cite{MR2595051,MR2587414,MR2773301,MR3416418,MR3149415,MR3022208,MR2644299}.
DG methods are well suited for problems with convection, but they often have a higher computational cost compared to other methods. Hybridizable discontinuous Galerkin (HDG) methods keep the advantages of DG methods, but have a lower number of globally coupled unknowns. HDG methods were introduced in \cite{MR2485455}, and now have been applied to many different problems \cite{MR2772094,MR2513831,MR2558780,MR2796169,MR3626531,MR3522968,MR3463051,MR3452794,MR3343926}.
HDG methods have recently been successfully applied to two PDE optimal control problems. Zhu and Celiker \cite{MR3508834} obtained optimal convergence rates for an HDG method for a distributed optimal control problem governed by the Poisson equation. The authors have also studied an HDG method for a difficult Dirichlet optimal boundary control problem for the Poisson equation in \cite{HuShenSinglerZhangZheng_HDG_Dirichlet_control1}. We proved an optimal superlinear convergence rate for the control in polygonal domains. Despite the large amount of work on this problem, a superlinear convergence result of this type had only been previously obtained for one other numerical method on a special class of meshes \cite{ApelMateosPfeffererRosch17}.
Due to these recent results and the favorable properties of HDG methods, we continue to investigate HDG for optimal control problems for PDEs in this work. Specifically, we consider the above distributed control problem for the elliptic convection diffusion equation, and apply an HDG method with polynomials of degree $k$ to approximate all the variables of the optimality system \eqref{eq_adeq}, i.e., the state $y$, dual state $z$, the numerical traces, and the fluxes $\bm q = -\nabla y $ and $ \bm p = -\nabla z$. We describe the HDG method and its implementation in \Cref{sec:HDG}. In \Cref{sec:analysis}, we obtain the error estimates
\begin{align*}
&\norm{y-{y}_h}_{0,\Omega}=O( h^{k+1}),\quad \quad \;\norm{z-{z}_h}_{0,\Omega}=O( h^{k+1}),\\
&\norm{\bm{q}-\bm{q}_h}_{0,\Omega} = O( h^{k+1}),\quad \quad\;\; \norm{\bm{p}-\bm{p}_h}_{0,\Omega} = O( h^{k+1}),
\end{align*}
and
\begin{align*}
&\norm{u-{u}_h}_{0,\Omega} = O( h^{k+1}).
\end{align*}
We present 2D and 3D numerical results in \Cref{sec:numerics} and then briefly discuss future work.
\section{HDG scheme for the optimal control problem}
\label{sec:HDG}
We begin by setting notation.
Throughout the paper we adopt the standard notation $W^{m,p}(\Omega)$ for Sobolev spaces on $\Omega$ with norm $\|\cdot\|_{m,p,\Omega}$ and seminorm $|\cdot|_{m,p,\Omega}$ . We denote $W^{m,2}(\Omega)$ by $H^{m}(\Omega)$ with norm $\|\cdot\|_{m,\Omega}$ and seminorm $|\cdot|_{m,\Omega}$. Specifically, $H_0^1(\Omega)=\{v\in H^1(\Omega):v=0 \;\mbox{on}\; \partial \Omega\}$. We denote the $L^2$-inner products on $L^2(\Omega)$ and $L^2(\Gamma)$ by
\begin{align*}
(v,w) &= \int_{\Omega} vw \quad \forall v,w\in L^2(\Omega),\\
\left\langle v,w\right\rangle &= \int_{\Gamma} vw \quad\forall v,w\in L^2(\Gamma).
\end{align*}
Define the space $H(\text{div},\Omega)$ as
\begin{align*}
H(\text{div},\Omega) = \{\bm{v}\in [L^2(\Omega)]^d, \nabla\cdot \bm{v}\in L^2(\Omega)\}.
\end{align*}
Let $\mathcal{T}_h$ be a collection of disjoint elements that partition $\Omega$. We denote by $\partial \mathcal{T}_h$ the set $\{\partial K: K\in \mathcal{T}_h\}$. For an element $K$ of the collection $\mathcal{T}_h$, let $e = \partial K \cap \Gamma$ denote the boundary face of $ K $ if the $d-1$ Lebesgue measure of $e$ is non-zero. For two elements $K^+$ and $K^-$ of the collection $\mathcal{T}_h$, let $e = \partial K^+ \cap \partial K^-$ denote the interior face between $K^+$ and $K^-$ if the $d-1$ Lebesgue measure of $e$ is non-zero. Let $\varepsilon_h^o$ and $\varepsilon_h^{\partial}$ denote the set of interior and boundary faces, respectively. We denote by $\varepsilon_h$ the union of $\varepsilon_h^o$ and $\varepsilon_h^{\partial}$. We finally introduce
\begin{align*}
(w,v)_{\mathcal{T}_h} = \sum_{K\in\mathcal{T}_h} (w,v)_K, \quad\quad\quad\quad\left\langle \zeta,\rho\right\rangle_{\partial\mathcal{T}_h} = \sum_{K\in\mathcal{T}_h} \left\langle \zeta,\rho\right\rangle_{\partial K}.
\end{align*}
Let $\mathcal{P}^k(D)$ denote the set of polynomials of degree at most $k$ on a domain $D$. We introduce the discontinuous finite element spaces
\begin{align}
\bm{V}_h &:= \{\bm{v}\in [L^2(\Omega)]^d: \bm{v}|_{K}\in [\mathcal{P}^k(K)]^d, \forall K\in \mathcal{T}_h\},\\
{W}_h &:= \{{w}\in L^2(\Omega): {w}|_{K}\in \mathcal{P}^{k}(K), \forall K\in \mathcal{T}_h\},\\
{M}_h &:= \{{\mu}\in L^2(\mathcal{\varepsilon}_h): {\mu}|_{e}\in \mathcal{P}^k(e), \forall e\in \varepsilon_h\}.
\end{align}
Let $M_h(o)$ and $M_h(\partial)$ denote the subspaces of $M_h$ containing each $e\in \varepsilon_h^o$ and $e\in \varepsilon_h^{\partial}$, respectively. Note that $M_h$ consists of functions which are continuous inside the faces (or edges) $e\in \varepsilon_h$ and discontinuous at their borders. In addition, for any function $w\in W_h$ we use $\nabla w$ to denote the piecewise gradient on each element $K\in \mathcal T_h$. A similar convention applies to the divergence $\nabla\cdot\bm r$ for all $\bm r\in \bm V_h$.
\subsection{The HDG Formulation}
The mixed weak form of the optimality system \eqref{eq_adeq_a}-\eqref{eq_adeq_e} is given by
\begin{subequations}\label{mixed}
\begin{align}
(\bm q,\bm r_1)-( y,\nabla\cdot \bm r_1)+\langle y,\bm r_1\cdot \bm n\rangle&=0,\label{mixed_a}\\
(\nabla\cdot(\bm q+\bm \beta y), w_1)&= ( f+ u, w_1), \label{mixed_b}\\
(\bm p,\bm r_2)-(z,\nabla \cdot\bm r_2)+\langle z,\bm r_2\cdot\bm n\rangle&=0,\label{mixed_c}\\
(\nabla\cdot(\bm p-\bm \beta z), w_2)&= (y_d- y, w_2), \label{mixed_d}\\
( z-\gamma u,v)&=0,\label{mixed_e}
\end{align}
\end{subequations}
for all $(\bm r_1, w_1,\bm r_2, w_2,v)\in H(\text{div},\Omega)\times L^2(\Omega)\times H(\text{div},\Omega)\times L^2(\Omega)\times L^2(\Omega)$. Recall we assume $ \bm \beta $ is divergence free; this allows us to rewrite the convection term $ \bm \beta \cdot \nabla y $ in \eqref{eq_adeq_a} as $ \nabla \cdot( \bm \beta y ) $ in \eqref{mixed_b}.
To approximate the solution of this system, the HDG method seeks approximate fluxes ${\bm{q}}_h,{\bm{p}}_h \in \bm{V}_h $, states $ y_h, z_h \in W_h $, interior element boundary traces $ \widehat{y}_h^o,\widehat{z}_h^o \in M_h(o) $, and control $ u_h \in W_h$ satisfying
\begin{subequations}\label{HDG_discrete2}
\begin{align}
(\bm q_h,\bm r_1)_{\mathcal T_h}-( y_h,\nabla\cdot\bm r_1)_{\mathcal T_h}+\langle \widehat y_h^o,\bm r_1\cdot\bm n\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}&=-\langle g,\bm r_1\cdot\bm n\rangle_{\varepsilon_h^\partial}, \label{HDG_discrete2_a}\\
-(\bm q_h+\bm \beta y_h, \nabla w_1)_{\mathcal T_h} +\langle\widehat {\bm q}_h\cdot\bm n,w_1\rangle_{\partial\mathcal T_h} \quad \nonumber \\
+\langle \bm \beta\cdot\bm n\widehat y_h^o,w_1\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial} - ( u_h, w_1)_{\mathcal T_h} &= - \langle \bm \beta\cdot\bm n g,w_1\rangle_{\varepsilon_h^\partial} + ( f, w_1)_{\mathcal T_h} \label{HDG_discrete2_b
\end{align}
for all $(\bm{r}_1, w_1)\in \bm{V}_h\times W_h$.
\begin{align}
(\bm p_h,\bm r_2)_{\mathcal T_h}-(z_h,\nabla\cdot\bm r_2)_{\mathcal T_h}+\langle \widehat z_h^o,\bm r_2\cdot\bm n\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}&=0,\label{HDG_discrete2_c}\\
-(\bm p_h-\bm \beta z_h, \nabla w_2)_{\mathcal T_h}+\langle\widehat{\bm p}_h\cdot\bm n,w_2\rangle_{\partial\mathcal T_h} \quad \nonumber\\
-\langle\bm \beta\cdot\bm n\widehat z_h^o,w_2\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial} + ( y_h, w_2)_{\mathcal T_h}&= (y_d, w_2)_{\mathcal T_h}, \label{HDG_discrete2_d}
\end{align}
for all $(\bm{r}_2, w_2)\in \bm{V}_h\times W_h$.
\begin{align}
\langle\widehat {\bm q}_h\cdot\bm n+\bm \beta\cdot\bm n\widehat y_h^o,\mu_1\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}&=0\label{HDG_discrete2_e},\\
\langle\widehat{\bm p}_h\cdot\bm n-\bm \beta\cdot\bm n\widehat z_h^o,\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}&=0,\label{HDG_discrete2_f}
\end{align}
for all $\mu_1,\mu_2\in M_h(o)$, and the optimality condition
\begin{align}
(z_h-\gamma u_h, w_3)_{\mathcal T_h} &= 0\label{HDG_discrete2_g},
\end{align}
for all $ w_3\in W_h$. The numerical traces on $\partial\mathcal{T}_h$ are defined as
\begin{align}
\widehat{\bm{q}}_h\cdot \bm n &=\bm q_h\cdot\bm n+\tau_1 (y_h-\widehat y_h^o) \quad \mbox{on} \; \partial \mathcal{T}_h\backslash\varepsilon_h^\partial, \label{HDG_discrete2_h}\\
\widehat{\bm{q}}_h\cdot \bm n &=\bm q_h\cdot\bm n+\tau_1 (y_h-g) \quad \ \mbox{on}\; \varepsilon_h^\partial, \label{HDG_discrete2_i}\\
\widehat{\bm{p}}_h\cdot \bm n &=\bm p_h\cdot\bm n+\tau_2(z_h-\widehat z_h^o)\quad \mbox{on} \; \partial \mathcal{T}_h\backslash\varepsilon_h^\partial,\label{HDG_discrete2_j}\\
\widehat{\bm{p}}_h\cdot \bm n &=\bm p_h\cdot\bm n+\tau_2 z_h\quad\quad\quad\quad\mbox{on}\; \varepsilon_h^\partial,\label{HDG_discrete2_k}
\end{align}
\end{subequations}
where $\tau_1$ and $\tau_2$ are positive stabilization functions defined on $\partial \mathcal T_h$. We specify these functions in the next section.
\subsection{Implementation}
For the numerical implementation, we follow a similar procedure to our earlier work \cite{HuShenSinglerZhangZheng_HDG_Dirichlet_control1}. First, we perform some basic manipulations to the above system \eqref{HDG_discrete2_a}-\eqref{HDG_discrete2_k} to find that
\[ ({\bm{q}}_h,{\bm{p}}_h, y_h,z_h,{\widehat{y}}_h^o,{\widehat{z}}_h^o)\in \bm{V}_h\times\bm{V}_h\times W_h \times W_h\times M_h(o)\times M_h(o) \]
is the solution of the following weak formulation:
\begin{subequations}\label{imple}
\begin{align}
(\bm{q}_h, \bm{r_1})_{{\mathcal{T}_h}}- (y_h, \nabla\cdot \bm{r_1})_{{\mathcal{T}_h}}+\langle \widehat{y}_h^o, \bm{r_1}\cdot \bm{n} \rangle_{\partial{{\mathcal{T}_h}}\backslash \varepsilon_h^{\partial}} &= - \langle g, \bm{r_1}\cdot \bm{n} \rangle_{\varepsilon_h^{\partial}} , \label{imple_a}\\
(\bm{p}_h, \bm{r_2})_{{\mathcal{T}_h}}- (z_h, \nabla\cdot \bm{r_2})_{{\mathcal{T}_h}}+\langle \widehat{z}_h^o, \bm{r_2}\cdot \bm{n} \rangle_{\partial{{\mathcal{T}_h}}\backslash \varepsilon_h^{\partial}} &=0, \label{imple_b}\\
(\nabla\cdot\bm{q}_h, w_1)_{{\mathcal{T}_h}} - (\bm{\beta} y_h, \nabla w_1)_{\mathcal T_h} + \langle \tau_1 y_h, w_1\rangle_{\partial\mathcal T_h} \quad \nonumber\\
- (\gamma^{-1} z_h,w_1)_{\mathcal T_h} +\langle (\bm{\beta}\cdot\bm n - \tau_1)\widehat y_h^o, w_1 \rangle_{\partial{{\mathcal{T}_h}}\backslash\varepsilon_h^\partial} &= -\langle (\bm{\beta}\cdot\bm n - \tau_1) g, w_1 \rangle_{\varepsilon_h^\partial} \nonumber\\
&\quad + (f, w_1)_{{\mathcal{T}_h}}, \label{imple_c}\\
(\nabla\cdot\bm{p}_h, w_2)_{{\mathcal{T}_h}} + (y_h, w_2)_{\mathcal T_h} + (\bm{\beta} z_h, \nabla w_2)_{\mathcal T_h} \quad & \nonumber\\
+ \langle \tau_2 z_h, w_2\rangle_{\partial \mathcal T_h} -\langle (\tau_2 + \bm{\beta}\cdot \bm n) \widehat{z}_h^o, w_2 \rangle_{\partial{{\mathcal{T}_h}}\backslash \varepsilon_h^{\partial}} &=(y_d, w_2)_{{\mathcal{T}_h}}, \label{imple_d}\\
\langle{\bm{q}_h}\cdot \bm{n}, \mu_1 \rangle_{\partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}}+\langle \tau_1 y_h ,\mu_1 \rangle_{\partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}} \quad \nonumber \\
+\langle (\bm{\beta}\cdot\bm n - \tau_1)\widehat{y}_h^{o} ,\mu_1 \rangle_{\partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}}&=0, \label{imple_e}\\
\langle {\bm{p}_h}\cdot \bm{n}, \mu_2\rangle_{ \partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}}+\langle \tau_2 z_h, \mu_2\rangle_{ \partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}} \quad \nonumber\\
-\langle (\bm{\beta}\cdot\bm n+\tau_2) \widehat{z}_h^o, \mu_2\rangle_{ \partial\mathcal{T}_{h}\backslash {\varepsilon_h^{\partial}}} &=0,
\label{imple_f}
\end{align}
\end{subequations}
for all $({\bm{r}_1},{\bm{r}_2},w_1,w_2,\mu_1,\mu_2)\in \bm{V}_h\times\bm{V}_h\times W_h \times W_h\times M_h(o)\times M_h(o)$.
Note that we have used the optimality condition \eqref{HDG_discrete2_g} to eliminate $ u_h $ from the discrete equations. Once the above system \eqref{imple} is solved numerically, $ u_h $ can be easily found using the optimality condition: $ u_h = \gamma^{-1} z_h $.
\subsection{Matrix equations}
Assume $\bm{V}_h = \mbox{span}\{\bm\varphi_i\}_{i=1}^{N_1}$, $W_h=\mbox{span}\{\phi_i\}_{i=1}^{N_2}$, $M_h^{o}=\mbox{span}\{\psi_i\}_{i=1}^{N_3} $. Then
\begin{equation}\label{expre}
\begin{split}
&\bm q_{h}= \sum_{j=1}^{N_1}q_{j}\bm\varphi_j, \quad y_h = \sum_{j=1}^{N_2}y_{j}\phi_j, \quad \widehat{y}_h^o = \sum_{j=1}^{N_3}\alpha_{j}\psi_{j},\\
&\bm p_{h} = \sum_{j=1}^{N_1}p_{j}\bm\varphi_j, \quad z_h = \sum_{j=1}^{N_2} z_{j}\phi_j, \quad \widehat{z}_h^o = \sum_{j=1}^{N_3}\gamma_{j}\psi_{j}.
\end{split}
\end{equation}
Substitute \eqref{expre} into \eqref{imple_a}-\eqref{imple_f} and use the corresponding test functions to test \eqref{imple_a}-\eqref{imple_f}, respectively, to obtain the matrix equation
\begin{align}\label{system_equation}
\begin{bmatrix}
A_1 &0 &-A_2&0 & A_{15}&0 \\
0 & A_1 &0 &-A_2& 0 &A_{15} \\
A_2^T &0&A_{12}&-\gamma^{-1}A_4&A_{16} &0\\
0 &A_2^T & A_4 &A_{13} &0&A_{17} \\
A_{14}^T & 0 &A_{18}&0 &A_{20}&0 \\
0& A_{15}^T &0&A_{19} &0&A_{21}
\end{bmatrix}
\left[ {\begin{array}{*{20}{c}}
\mathfrak{q}\\
\mathfrak{p}\\
\mathfrak{y}\\
\mathfrak{z}\\
\mathfrak{\widehat y}\\
\mathfrak{\widehat z}
\end{array}} \right]
=\left[ {\begin{array}{*{20}{c}}
-b_1\\
0\\
-b_5\\
b_4\\
0\\
0\\
\end{array}} \right],
\end{align}
where $\mathfrak{q},\mathfrak{p},\mathfrak{y},\mathfrak{z},\mathfrak{\widehat y},\mathfrak{\widehat z}$ are the coefficient vectors for $\bm q_h,\bm p_h,y_h,z_h,\widehat y_h^o, \widehat z_h^o$, respectively, and
\begin{align*}
A_1 &= [(\bm\varphi_j,\bm\varphi_i )_{\mathcal{T}_h}], &
A_2 &= [(\phi_j,\nabla\cdot\bm{\varphi_i})_{\mathcal{T}_h}], &
A_3 &= [(\psi_j,\bm{\varphi}_i\cdot \bm n)_{\mathcal{T}_h}],\\
A_4 &= [(\phi_j,\phi_i)_{\mathcal{T}_h}], &
A_5 &= [(\bm\beta\phi_j,\nabla\phi_i)_{\mathcal{T}_h}], &
A_6 &= [\langle \tau_1\phi_j, \phi_i \rangle_{\partial{{\mathcal{T}_h}}}],\\
A_7 &= [\langle \bm{\beta}\cdot\bm n\phi_j, \phi_i \rangle_{\partial{{\mathcal{T}_h}}}], &
A_8 &= [\left\langle \tau_1\psi_j,{\varphi_i}\right\rangle_{\partial\mathcal{T}_h}], &
A_9 &= [\left\langle \bm{\beta}\cdot\bm n\psi_j,{\varphi_i}\right\rangle_{\partial\mathcal{T}_h}],\\
A_{10} &= [\left\langle \tau_1\psi_j,\psi_i\right\rangle_{\partial\mathcal{T}_h}], &
A_{11} &= [\left\langle \bm{\beta}\cdot\bm n \psi_j,\psi_i\right\rangle_{\partial\mathcal{T}_h}], &
A_{12} &= A_6-A_5\\
A_{13} &= A_5+ A_6-A_7, &
b_1 &= [\langle g, \bm\varphi_i \cdot \bm{n} \rangle_{\varepsilon_h^{\partial}}],&
b_2 &= [\langle (\bm{\beta}\cdot\bm n - \tau_1) g, \phi \rangle_{\varepsilon_h^\partial}], \\
b_3 &= [(f,\phi_i )_{\mathcal{T}_h}], &
b_4 &= [(y_d,\phi_i )_{\mathcal{T}_h}], &
b_5 &= b_3-b_2.
\end{align*}
The remaining matrices are constructed by extracting the corresponding rows and columns from linear combinations of $A_3 $, $A_8$, $A_{9}$, $A_{10}$, and $A_{11}$.
\subsection{Local solver}
Next, we use the discontinuous nature of the approximation spaces $\bm{V_h}$ and ${W_h}$ to eliminate all unknowns except the coefficient vectors of the numerical traces.
The matrix equation \eqref{system_equation} can be rewritten as
\begin{align}\label{system_equation2}
\begin{bmatrix}
B_1 & B_2&B_3\\
-B_2^T & B_4&B_5\\
B_6&B_7&B_8\\
\end{bmatrix}
\left[ {\begin{array}{*{20}{c}}
\bm{\alpha}\\
\bm{\beta}\\
\bm{\gamma}
\end{array}} \right]
=\left[ {\begin{array}{*{20}{c}}
\bm b_1\\
\bm b_2\\
0
\end{array}} \right],
\end{align}
where $\bm{\alpha}=[\mathfrak{q};\mathfrak{p}]$, $\bm{\beta}=[\mathfrak{y};\mathfrak{z}]$, $\bm{\gamma}=[\mathfrak{\widehat y};\mathfrak{\widehat z}]$, $ \bm b_1 = [- b_1; 0 ] $, and $ \bm b_2 = [ -b_5; b_4 ] $, and also $\{B_i\}_{i=1}^8$ are the corresponding blocks of the coefficient matrix of \eqref{system_equation}.
In the appendix, we show how the first two equations of \eqref{system_equation2} can be used to eliminate both $\bm{\alpha}$ and $\bm{\beta}$ in an element-by-element fashion. We obtain
\begin{align}\label{local_solver}
\left[ {\begin{array}{*{20}{c}}
\bm{\alpha}\\
\bm{\beta}
\end{array}} \right]= \begin{bmatrix}
G_1 & H_1\\
G_2 & H_2
\end{bmatrix}
\left[ {\begin{array}{*{20}{c}}
\bm \gamma\\
b
\end{array}} \right]
\end{align}
and
\begin{align}
B_6\bm{\alpha}+B_7\bm{\beta}+B_8\bm{\gamma} = 0,
\end{align}
where $G_1,G_2,H_1,H_2$ are sparse. This gives a globally coupled equation for $\bm{\gamma}$ only:
\begin{align}\label{global_eq}
\mathbb{K}\bm{\gamma} = \mathbb{F},
\end{align}
where
\begin{align*}
\mathbb{K}= B_6G_1+B_7G_2+B_8\quad\text{and}\quad\mathbb{F} = B_6H_1+B_7H_2.
\end{align*}
Once $\bm{\gamma}$ is computed, $\bm{\alpha}$ and $\bm{\beta}$ can be quickly and easily computed using \eqref{local_solver}.
\section{Error Analysis}
\label{sec:analysis}
Next, we provide a convergence analysis of the above HDG method for the optimal control problem. Throughout this section, we assume $ \bm \beta \in [W^{1,\infty}(\Omega)]^d $, $ \Omega $ is a bounded convex polyhedral domain, $ h \leq 1 $, and the solution of the optimality system \eqref{eq_adeq} is smooth enough.
\subsection{Main result}
For our theoretical results, we require the stabilization functions $\tau_1$ and $\tau_2$ are chosen to satisfy
\begin{description}
\item[\textbf{(A1)}] $\tau_1$ is piecewise constant on $\partial \mathcal T_h$.
\item[\textbf{(A2)}] $\tau_1 = \tau_2 + \bm{\beta}\cdot \bm n$.
\item[\textbf{(A3)}] For any $K\in\mathcal T_h$, $\min{(\tau_1-\frac 1 2 \bm \beta \cdot \bm n)}|_{\partial K} >0$.
\end{description}
We note that \textbf{(A2)} and \textbf{(A3)} imply
\begin{equation}\label{eqn:tau1_condition}
\min{(\tau_2 + \frac 1 2 \bm \beta \cdot \bm n)}|_{\partial K} >0 \quad \mbox{for any $K\in\mathcal T_h$.}
\end{equation}
\begin{theorem}\label{main_res}
We have
\begin{align*}
\|\bm q-\bm q_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|\bm p-\bm p_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|y-y_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|z-z_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|u-u_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align*}
\end{theorem}
\subsection{Preliminary material}
\label{sec:Projectionoperator}
Next, we introduce the projection operators $\bm{\Pi}_V$ and $\Pi_W$ defined in \cite{MR2991828} that we use frequently in our proof. The value of the projection on each element $K\in \mathcal{T}_h$ is determined by requiring that the components satisfy the equations
\begin{subequations}\label{projection_operator}
\begin{align}
(\bm\Pi_V\bm q+\bm\beta\Pi_Wy,\bm r)_K&=(\bm q+\bm\beta y,\bm r)_K, \label{projection_operator_a}\\
(\Pi_Wy,w)_K&=(y,w)_K, \label{projection_operator_b}\\
\langle\bm\Pi_V\bm q\cdot\bm n+\bm\beta\cdot\bm nP_My+\tau_1\Pi_Wy,\mu\rangle_{e}&=\langle\bm q\cdot\bm n+\bm\beta\cdot\bm ny+\tau_1 y,\mu\rangle_{e}, \label{projection_operator_c}
\end{align}
for all $(\bm{r} , w, \mu)\in \bm{\mathcal P}_{k-1}(K)\times {\mathcal P}_{k-1}(K)\times {\mathcal P}_{k}(e)$ and for all faces $e$ of the simplex $K$.
\end{subequations}
Here, $P_M$ denotes the $L^2$-orthogonal projection from $L^2(\varepsilon_h)$ into $M_h$ satisfying
\begin{align}\label{P_M}
\left\langle P_M y-y, \mu\right\rangle_e = 0, \quad \forall e\in \varepsilon_h, \;\forall\mu\in M_h.
\end{align}
The following lemma from \cite{MR2991828} provides the approximation properties of the projection operator \eqref{projection_operator}.
\begin{lemma}\label{pro_error}
Suppose $k\ge0$, and $ \tau_1 $ satisfies \textbf{(A3)}. Then the system \eqref{projection_operator} is uniquely solvable for $\bm{\Pi}_V\bm{q}$ and $\Pi_W y$. Moreover, we have the following approximation properties
\begin{subequations}
\begin{align}
\|\bm\Pi_V\bm q-\bm q\|_K &\le Ch^{k+1}|\bm q|_{k+1,K}+Ch^{k+1}|y|_{k+1,K},\\
\|\Pi_Wy-y\|_K&\le Ch^{k+1}|\bm q|_{k+1,K}+Ch^{k+1}|y|_{k+1,K},
\end{align}
\end{subequations}
where $C$ is a constant depending on the polynomial degree and the shape-regularity parameters of the elements.
\end{lemma}
For the convection diffusion optimal control problem, we introduce another projection operator associated to the dual problem. The projection $\widetilde{\bm \Pi}_{V}$ and $\widetilde{\Pi}_W$ is determined by the following equations
\begin{subequations}\label{projection_operator1}
\begin{align}
(\widetilde{\bm\Pi}_V\bm p-\bm\beta\widetilde{\Pi}_Wz,\bm r)_K&=(\bm p-\bm\beta z,\bm r)_K, \label{projection_operator1_a}\\
(\widetilde {\Pi}_Wz,w)_K&=(z,w)_K,\label{projection_operator1_b}\\
\langle\widetilde{\bm\Pi}_V\bm p\cdot\bm n-\bm\beta\cdot\bm nP_Mz+\tau_2\widetilde{\Pi}_Wz,\mu\rangle_{e}&=\langle\bm p\cdot\bm n-\bm\beta\cdot\bm nz+\tau_2 z,\mu\rangle_{e},\label{projection_operator1_c}
\end{align}
for all $(\bm{r} , w, \mu)\in \bm{\mathcal P}_{k-1}(K)\times {\mathcal P}_{k-1}(K)\times {\mathcal P}_{k}(e)$ and for all faces $e$ of the simplex $K$.
\end{subequations}
Again, results from \cite{MR2991828} give the following estimates.
\begin{lemma}\label{pro_error1}
Suppose $k\ge0$, and $ \tau_2 $ satisfies \eqref{eqn:tau1_condition}. Then the system \eqref{projection_operator1} is uniquely solvable for $\widetilde{\bm{\Pi}}_V\bm{p}$ and $\widetilde{\Pi}_W z$, an
\begin{subequations}
\begin{align}
\|\widetilde{\bm\Pi}_V\bm p-\bm p\|_{K}&\le Ch^{k+1}|\bm p|_{k+1,K}+Ch^{k+1}|z|_{k+1,K},\\
\|\widetilde{\Pi}_Wz-z\|_{K}&\le Ch^{k+1}|\bm p|_{k+1,K}+Ch^{k+1}|z|_{k+1,K},
\end{align}
\end{subequations}
where $C$ is a constant depending on the polynomial degree and the shape-regularity parameters of the elements.
\end{lemma}
Next, we present a basic approximation of the function $\bm \beta$. Let $ \bm P_0 $ be the vectorial piecewise-constant $ L^2 $ projection. We have the following estimate:
\begin{align*}
\norm {\bm\beta - \bm P_0 \bm\beta}_{0,\infty,\Omega} \le C h \|\bm \beta\|_{1,\infty,\Omega}.
\end{align*}
\begin{lemma}\label{tau_lem}
For any $e\in \partial K$, define $\widetilde \tau_2|_e = \tau_1|_e - \bm P_0 \bm\beta|_K\cdot \bm n_{e}$, we have
\begin{align*}
\norm{\tau_2 - \widetilde \tau_2}_{0,\infty,\partial \mathcal T_h} \le C_{\bm \beta}h\|\bm \beta\|_{1,\infty,\Omega}.
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
\norm{\tau_2 - \widetilde \tau_2}_{0,\infty,\partial\mathcal T_h} &= \sum_{K\in\mathcal T_h}\norm{\tau_2 - \widetilde \tau_2}_{0,\infty,\partial K} \\
& = \sum_{K\in\mathcal T_h} \norm{\tau_1 - \bm\beta\cdot\bm n - \tau_1 + \bm P_0\bm \beta\cdot \bm n}_{0,\infty,\partial K}\\
& = \sum_{K\in\mathcal T_h} \norm{\bm\beta\cdot\bm n - \bm P_0\bm \beta\cdot \bm n}_{0,\infty, K}\\
& \le \norm {\bm\beta - \bm P_0 \bm\beta}_{0,\infty,\Omega}\\
& \le Ch\|\bm \beta\|_{1,\infty,\Omega}.
\end{align*}
\end{proof}
We define the following HDG operators $ \mathscr B_1$ and $ \mathscr B_2 $.
\begin{equation}\label{def_B1}
\begin{split}
\hspace{1em}&\hspace{-1em} \mathscr B_1( \bm q_h,y_h,\widehat y_h^o;\bm r_1,w_1,\mu_1) \\
&=(\bm q_h,\bm r_1)_{\mathcal T_h}-( y_h,\nabla\cdot\bm r_1)_{\mathcal T_h}+\langle \widehat y_h^o,\bm r_1\cdot\bm n\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial} \\
&\quad-(\bm q_h+\bm \beta y_h, \nabla w_1)_{\mathcal T_h} +\langle {\bm q}_h\cdot\bm n +\tau_1y_h,w_1\rangle_{\partial\mathcal T_h}+\langle (\bm\beta\cdot\bm n -\tau_1) \widehat y_h^o,w_1\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
&\quad -\langle {\bm q}_h\cdot\bm n+\bm \beta\cdot\bm n\widehat y_h^o +\tau_1(y_h-\widehat y_h^o),\mu_1\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h},\\
\hspace{1em}&\hspace{-1em} \mathscr B_2(\bm p_h,z_h,\widehat z_h^o;\bm r_2, w_2,\mu_2)\\
&=(\bm p_h,\bm r_2)_{\mathcal T_h}-( z_h,\nabla\cdot\bm r_2)_{\mathcal T_h}+\langle \widehat z_h^o,\bm r_2\cdot\bm n\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}\\
&\quad-(\bm p_h-\bm \beta z_h, \nabla w_2)_{\mathcal T_h}
+\langle {\bm p}_h\cdot\bm n +\tau_2 z_h ,w_2\rangle_{\partial\mathcal T_h} -\langle (\bm \beta\cdot\bm n + \tau_2)\widehat z_h^o ,w_2\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}
\\
&\quad-\langle {\bm p}_h\cdot\bm n-\bm \beta\cdot\bm n\widehat z_h^o +\tau_2(z_h-\widehat z_h^o),\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}.
\end{split}
\end{equation}
By the definition in \eqref{def_B1}, we can rewrite the HDG formulation of the optimality system \eqref{HDG_discrete2} as follows: find $({\bm{q}}_h,{\bm{p}}_h,y_h,z_h,u_h,\widehat y_h^o,\widehat z_h^o)\in \bm{V}_h\times\bm{V}_h\times W_h \times W_h\times W_h\times M_h(o)\times M_h(o)$ such that
\begin{subequations}\label{HDG_full_discrete}
\begin{align}
\mathscr B_1(\bm q_h,y_h,\widehat y_h^o;\bm r_1,w_1,\mu_1)&=( f+ u_h, w_1)_{\mathcal T_h}\ \nonumber\\
&\quad-\langle g, (\bm\beta\cdot\bm n-\tau_1)w_1+\bm r_1\cdot\bm n \rangle_{\varepsilon_h^\partial},\label{HDG_full_discrete_a}\\
\mathscr B_2(\bm p_h,z_h,\widehat z_h^o;\bm r_2,w_2,\mu_2)&=(y_d-y_h,w_2)_{\mathcal T_h},\label{HDG_full_discrete_b}\\
(z_h-\gamma u_h,w_3)_{\mathcal T_h}&= 0,\label{HDG_full_discrete_e}
\end{align}
\end{subequations}
for all $\left(\bm{r}_1, \bm{r}_2,w_1,w_2,w_3,\mu_1,\mu_2\right)\in \bm{V}_h\times\bm{V}_h\times W_h \times W_h\times W_h\times M_h(o)\times M_h(o)$.
Next, we present a basic property of the operators $\mathscr B_1$ and $\mathscr B_2$, and show the HDG equations \eqref{HDG_full_discrete} have a unique solution.
\begin{lemma}\label{property_B}
For any $ ( \bm v_h, w_h, \mu_h ) \in \bm V_h \times W_h \times M_h(o) $, we have
\begin{align*}
\hspace{2em}&\hspace{-2em} \mathscr B_1(\bm v_h,w_h,\mu_h;\bm v_h,w_h,\mu_h)\\
&=(\bm v_h,\bm v_h)_{\mathcal T_h}+ \langle (\tau_1 - \frac 12 \bm \beta\cdot\bm n)(w_h-\mu_h),w_h-\mu_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
&\quad+\langle (\tau_1-\frac12\bm \beta\cdot\bm n) w_h,w_h\rangle_{\varepsilon_h^\partial},\\
\hspace{2em}&\hspace{-2em}\mathscr B_2(\bm v_h,w_h,\mu_h;\bm v_h,w_h,\mu_h)\\
&=(\bm v_h,\bm v_h)_{\mathcal T_h}+ \langle (\tau_2 + \frac 12 \bm \beta\cdot\bm n)(w_h-\mu_h),w_h-\mu_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
&\quad+\langle (\tau_2+\frac12\bm \beta\cdot\bm n) w_h,w_h\rangle_{\varepsilon_h^\partial}.
\end{align*}
\end{lemma}
\begin{proof}
We only prove the first identity; the second can be obtained by the same argument.
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_1(\bm v_h,w_h,\mu_h;\bm v_h,w_h,\mu_h)\\
&=(\bm v_h,\bm v_h)_{\mathcal T_h}-( w_h,\nabla\cdot\bm v_h)_{\mathcal T_h}+\langle \mu_h,\bm v_h\cdot\bm n\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
& \quad -(\bm v_h+\bm \beta w_h, \nabla w_h)_{\mathcal T_h} +\langle {\bm v}_h\cdot\bm n +\tau_1 w_h,w_h\rangle_{\partial\mathcal T_h} \\
&\quad+\langle (\bm\beta\cdot\bm n -\tau_1) \mu_h,w_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
& \quad-\langle {\bm v}_h\cdot\bm n+\bm \beta\cdot\bm n\mu_h +\tau_1(w_h - \mu_h),\mu_h \rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}\\
&=(\bm v_h,\bm v_h)_{\mathcal T_h}-(\bm \beta w_h, \nabla w_h)_{\mathcal T_h}+\langle \tau_1 w_h,w_h\rangle_{\partial\mathcal T_h} \\
&\quad+\langle (\bm\beta\cdot\bm n -\tau_1) \mu_h,w_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial} -\langle \bm \beta\cdot\bm n \mu_h +\tau_1(w_h - \mu_h ),\mu_h\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}.
\end{align*}
Moreover,
\begin{align*}
(\bm \beta w_h,\nabla w_h)_{\mathcal T_h}=(\nabla\cdot(\bm \beta w_h),w_h)_{\mathcal T_h} =\langle\bm \beta\cdot\bm n w_h,w_h\rangle_{\partial\mathcal T_h}-(\bm \beta w_h,\nabla w_h)_{\mathcal T_h},
\end{align*}
which implies
\begin{align*}
(\bm \beta w_h,\nabla w_h)_{\mathcal T_h}=\frac12\langle\bm \beta\cdot\bm n w_h,w_h\rangle_{\partial\mathcal T_h}.
\end{align*}
Then we obtain
\begin{align*}
\hspace{1em}&\hspace{-1em} \mathscr B_1 (\bm v_h,w_h,\mu_h;\bm v_h,w_h,\mu_h)\\
&=(\bm v_h,\bm v_h)_{\mathcal T_h}+ \langle ( \tau_1 - \frac 12 \bm \beta\cdot\bm n)(w_h-\mu_h),w_h-\mu_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}\\
&\quad + \langle (\tau_1-\frac12\bm \beta\cdot\bm n) w_h,w_h\rangle_{\varepsilon_h^\partial}-\frac12\langle\bm \beta\cdot\bm n \mu_h,\mu_h\rangle_{\partial\mathcal T_h\backslash \varepsilon_h^\partial}.
\end{align*}
Since $ \mu_h$ is single-valued across the interfaces, we have
\begin{align*}
-\frac12\langle\bm \beta\cdot\bm n\mu_h,\mu_h\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}=0.
\end{align*}
This completes the proof.
\end{proof}
Next, we give a property of the HDG operators $\mathscr B_1$ and $\mathscr B_2$ that is critical to our error analysis of the method.
\begin{lemma}\label{identical_equa}
If \textbf{(A2)} holds, then
$$\mathscr B_1 (\bm q_h,y_h,\widehat y_h^o;\bm p_h,-z_h,-\widehat z_h^o) + \mathscr B_2 (\bm p_h,z_h,\widehat z_h^o;-\bm q_h,y_h,\widehat y_h^o) = 0.$$
\end{lemma}
\begin{proof}
By the definition of $ \mathscr B_1 $ and $ \mathscr B_2 $,
\begin{align*}
%
\hspace{1em}&\hspace{-1em} \mathscr B_1 (\bm q_h,y_h,\widehat y_h^o;\bm p_h,-z_h,-\widehat z_h^o) + \mathscr B_2 (\bm p_h,z_h,\widehat z_h^o;-\bm q_h,y_h,\widehat y_h^o)\\
%
&=(\bm{q}_h, \bm p_h)_{{\mathcal{T}_h}}- (y_h, \nabla\cdot \bm p_h)_{{\mathcal{T}_h}}+\langle \widehat{y}_h^o, \bm p_h\cdot \bm{n} \rangle_{\partial{{\mathcal{T}_h}}\backslash {\varepsilon_h^{\partial}}} \\
%
&\quad + (\bm{q}_h + \bm{\beta} y_h, \nabla z_h)_{{\mathcal{T}_h}} - \langle\bm q_h\cdot\bm n +\tau_1 y_h , z_h \rangle_{\partial{{\mathcal{T}_h}}} - \langle(\bm\beta\cdot \bm n-\tau_1)\widehat y_h^o, z_h \rangle_{\partial{{\mathcal{T}_h}}\backslash \varepsilon_h^{\partial}} \\
%
&\quad+ \langle\bm q_h\cdot\bm n + \bm{\beta}\cdot\bm n \widehat y_h^o +\tau_1 (y_h-\widehat y_h^o), \widehat z_h^o \rangle_{\partial{{\mathcal{T}_h}}\backslash\varepsilon_h^{\partial}}\\
%
&\quad-(\bm{p}_h, \bm q_h)_{{\mathcal{T}_h}}+ (z_h, \nabla\cdot \bm q_h)_{{\mathcal{T}_h}} -\langle \widehat{z}_h^o, \bm q_h \cdot \bm{n} \rangle_{\partial{{\mathcal{T}_h}}\backslash {\varepsilon_h^{\partial}}}\\
%
&\quad - (\bm{p}_h - \bm{\beta} z_h, \nabla y_h)_{{\mathcal{T}_h}} +\langle\bm p_h\cdot\bm n +\tau_2 z_h , y_h \rangle_{\partial{{\mathcal{T}_h}}} - \langle (\bm{\beta}\cdot \bm n+\tau_2 ) \widehat z_h^o, y_h \rangle_{\partial{{\mathcal{T}_h}}\backslash \varepsilon_h^{\partial}}\\
%
&\quad- \langle\bm p_h\cdot\bm n -\bm{\beta} \cdot\bm n\widehat z_h^o + \tau_2 (z_h-\widehat z_h^o), \widehat y_h^o \rangle_{\partial{{\mathcal{T}_h}}\backslash\varepsilon_h^{\partial}}.
\end{align*}
Integration by parts gives
\begin{align*}
%
\mathscr B_1 &(\bm q_h,y_h,\widehat y_h^o;\bm p_h,-z_h,-\widehat z_h^o) + \mathscr B_2 (\bm p_h,z_h,\widehat z_h^o;-\bm q_h,y_h,\widehat y_h^o)\\
%
&=\langle (\tau_2 + \bm{\beta}\cdot\bm n-\tau_1) y_h, z_h \rangle_{\partial\mathcal T_h} + \langle (\tau_2 + \bm{\beta}\cdot\bm n-\tau_1) \widehat y_h^o, \widehat z_h^o \rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}.
\end{align*}
The proof is complete by assumption \textbf{(A2)}.
\end{proof}
\begin{proposition}\label{ex_uni}
There exists a unique solution of the HDG equations \eqref{HDG_full_discrete}.
\end{proposition}
\begin{proof}
Since the system \eqref{HDG_full_discrete} is finite dimensional, we only need to prove the uniqueness. Therefore, we assume $y_d = f =g= 0$ and we show the system \eqref{HDG_full_discrete} only has the trivial solution.
Take $(\bm r_1,w_1,\mu_1) = (\bm p_h,-z_h,-\widehat z_h^o)$, $(\bm r_2,w_2,\mu_2) = (-\bm q_h,y_h,\widehat y_h^o)$, and $w_3 = z_h -\gamma u_h $ in the HDG equations \eqref{HDG_full_discrete_a}, \eqref{HDG_full_discrete_b}, and \eqref{HDG_full_discrete_e}, respectively, and sum to obtain
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_1 (\bm q_h,y_h,\widehat y_h^o;\bm p_h,-z_h,-\widehat z_h^o) + \mathscr B_2 (\bm p_h,z_h,\widehat z_h^o;-\bm q_h,y_h,\widehat y_h^o) \\
& = \gamma (y_h,y_h)_{\mathcal T_h} + (z_h,z_h)_{\mathcal T_h
\end{align*}
Since $\gamma>0$, Lemma \ref{identical_equa} implies $y_h = u_h = z_h= 0$.
Next, take $(\bm r_1,w_1,\mu_1) = (\bm q_h,y_h,\widehat y_h^o)$ and $(\bm r_2,w_2,\mu_2) = (\bm p_h,z_h,\widehat z_h^o)$ in the HDG equations \eqref{HDG_full_discrete_a}-\eqref{HDG_full_discrete_b}. Lemma \eqref{property_B} and \textbf{(A2)} and \textbf{(A3)} give $\bm q_h= \bm p_h= \bm 0 $ and $ \widehat y_h^o = \widehat z_h^o=0$.
\end{proof}
\subsection{Proof of Main Result}
To prove the main result, we follow the strategy of our earlier work \cite{HuShenSinglerZhangZheng_HDG_Dirichlet_control1} and split the proof into
five steps. We consider the following auxiliary problem: find $$({\bm{q}}_h(u),{\bm{p}}_h(u), y_h(u), z_h(u), {\widehat{y}}_h^o(u), {\widehat{z}}_h^o(u))\in \bm{V}_h\times\bm{V}_h\times W_h \times W_h\times M_h(o)\times M_h(o)$$ such that
\begin{subequations}\label{HDG_inter_u}
\begin{align}
\mathscr B_1(\bm q_h(u),y_h(u),\widehat{y}_h(u);\bm r_1, w_1,\mu_1)&=( f+ u,w_1)_{\mathcal T_h} \ \nonumber\\
& \quad- \langle g, (\bm\beta\cdot\bm n-\tau_1)w_1+\bm r_1\cdot\bm n \rangle_{\varepsilon_h^\partial},\label{HDG_u_a} \\
\mathscr B_2(\bm p_h(u),z_h(u),\widehat{z}_h(u);\bm r_2, w_2,\mu_2)&=(y_d-y_h(u), w_2)_{\mathcal T_h},\label{HDG_u_b}
\end{align}
\end{subequations}
for all $\left(\bm{r}_1, \bm{r}_2,w_1,w_2,\mu_1,\mu_2\right)\in \bm{V}_h\times\bm{V}_h \times W_h\times W_h\times M_h(o)\times M_h(o)$. We begin by bounding the error between the solutions of the auxiliary problem and the mixed form \eqref{mixed_a}-\eqref{mixed_d} of the optimality system.
\subsubsection{Step 1: The error estimates for $\norm{ q- q_h(u)}_{\mathcal T_h}$ and $\norm {y-y_h(u)}_{\mathcal T_h}$.} \label{subsec:proof_step1}
The auxiliary HDG equation \eqref{HDG_u_a} is precisely the standard HDG discretization of the convection diffusion PDE \eqref{eq_adeq_a}-\eqref{eq_adeq_b} for $ y $ since the exact optimal control $ u $ is fixed in \eqref{HDG_u_a}. The HDG error estimates for this problem have already been obtained in \cite{MR2991828}:
\begin{lemma}[\cite{MR2991828}]\label{le}
If conditions {\bf{(A1)}} and {\bf{(A2)}} hold, we have
\begin{align}\label{err_y_yhu}
\|y-y_h(u)\|_{\mathcal T_h}+\|\bm q-\bm q_h(u)\|_{\mathcal T_h}\le Ch^{k+1}(|\bm q|_{k+1}+|y|_{k+1}).
\end{align}
\end{lemma}
\subsubsection{Step 2: The error equation for part 2 of the auxiliary problem \eqref{HDG_u_b}.} \label{subsec:proof_step2}
Next, we bound the error between the solution of the dual convection diffusion equation \eqref{eq_adeq_c}-\eqref{eq_adeq_d} for $ z $ and the auxiliary HDG equation \eqref{HDG_u_b}. We split the errors in the variables using the HDG projections. Define
\begin{equation}\label{notation_1}
\begin{split}
\delta^{\bm p} &=\bm p- \widetilde{\bm\Pi}_V\bm p, \qquad\qquad\qquad \qquad\qquad\qquad\;\;\;\;\varepsilon^{\bm p}_h=\widetilde{\bm\Pi}_V \bm p-\bm p_h(u),\\
\delta^z&=z- \widetilde{\Pi}_W z, \qquad\qquad\qquad \qquad\qquad\qquad\;\;\;\; \;\varepsilon^{z}_h=\widetilde{\Pi}_W z-z_h(u),\\
\delta^{\widehat z} &= z-P_Mz, \qquad\qquad\qquad\qquad\qquad\qquad\quad\;\; \varepsilon^{\widehat z}_h=P_M z-\widehat{z}_h(u),\\
\widehat {\bm\delta}_2 &= \delta^{\bm p}\cdot\bm n+\tau \delta^z, \qquad\qquad\qquad\qquad\quad\quad\quad\quad
\widehat {\bm \varepsilon }_2= \varepsilon_h^{\bm p}\cdot\bm n+\tau (\varepsilon^z_h-\varepsilon_h^{\widehat z}).
\end{split}
\end{equation}
where $\widehat z_h(u) = \widehat z_h^o(u)$ on $\varepsilon_h^o$ and $\widehat z_h(u) = 0$ on $\varepsilon_h^{\partial}$. This gives $\varepsilon_h^{\widehat z} = 0$ on $\varepsilon_h^{\partial}$.
\begin{lemma}\label{lemma:step1_first_lemma}
We have
\begin{align}\label{error_z}
\hspace{3em}&\hspace{-3em} \mathscr B_2(\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat{z}}_h;\bm r_2, w_2,\mu_2) \ \nonumber\\
&=(\delta^{\bm p},\bm r_2)_{\mathcal T_h}+(y_h(u)- y, w_2)_{\mathcal T_h}+ \langle (\tau_2 - \widetilde \tau_2)\delta^{\widehat z},w_2 - \mu_2\rangle_{\partial\mathcal T_h}.
\end{align}
\end{lemma}
\begin{proof}
By the definition of operator $\mathscr B_2$ \eqref{def_B1}, we have
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_2 (\widetilde{\bm \Pi}_V\bm p,\widetilde{\Pi}_W z,P_Mz;\bm r_2,w_2,\mu_2) \\
& =(\widetilde{\bm \Pi}_V\bm p,\bm r_2)_{\mathcal T_h}-(\widetilde{\Pi}_W z,\nabla\cdot\bm r_2)_{\mathcal T_h}+\langle P_M z,\bm r_2\cdot\bm n\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}\\
&\quad-(\widetilde{\bm \Pi}_V\bm p -\bm \beta \widetilde{\Pi}_W z, \nabla w_2)_{\mathcal T_h} +\langle \widetilde{\bm \Pi}_V\bm p \cdot\bm n +\tau_2 \widetilde{\Pi}_W z ,w_2\rangle_{\partial\mathcal T_h}\nonumber\\
&\quad-\langle (\bm \beta\cdot\bm n + \tau_2)P_Mz ,w_2\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}\\
&\quad-\langle \widetilde{\bm \Pi}_V\bm p \cdot\bm n-\bm \beta\cdot\bm nP_Mz+\tau_2( \widetilde{\Pi}_W z -P_Mz ),\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}.
\end{align*}
By properties of the HDG projections $\widetilde {\bm \Pi}_V$ and $\widetilde \Pi_W$ in \eqref{projection_operator1_c} and the $L^2 $ projection $P_M$ in \eqref{P_M}, we have
\begin{align*}
\langle\widetilde{\bm\Pi}_V\bm p\cdot\bm n+\tau_2\widetilde{\Pi}_Wz,w_2\rangle_{\partial\mathcal T_h}=\langle\bm p\cdot\bm n +\bm\beta\cdot\bm nP_Mz -\bm\beta\cdot\bm nz+\tau_2 z,w_2\rangle_{\partial\mathcal T_h},\\
\langle\widetilde{\bm\Pi}_V\bm p\cdot\bm n-\bm\beta\cdot\bm nP_Mz+\tau_2\widetilde{\Pi}_Wz,\mu\rangle_{\partial \mathcal T_h\backslash \varepsilon_h^\partial}=\langle\bm p\cdot\bm n-\bm\beta\cdot\bm nz+\tau_2 z,\mu\rangle_{\partial \mathcal T_h\backslash \varepsilon_h^\partial}.
\end{align*}
By \eqref{projection_operator1_a}-\eqref{projection_operator1_b}, we have
\begin{align*}
\hspace{1em}&\hspace{-1em} \mathscr B_2 (\widetilde{\bm \Pi}_V\bm p,\widetilde{\Pi}_W z,P_Mz;\bm r_2,w_2,\mu_2)\\
& = (\bm p,\bm r_2)_{\mathcal T_h} - (\delta^{\bm p},\bm r_2)_{\mathcal T_h}-(z,\nabla\cdot\bm r_2)_{\mathcal T_h}+\langle z,\bm r_2\cdot\bm n\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}\\
&\quad-(\bm p -\bm \beta z, \nabla w_2)_{\mathcal T_h} +\langle \bm p\cdot\bm n -\bm\beta\cdot\bm nz ,w_2\rangle_{\partial\mathcal T_h} + \langle \bm\beta\cdot\bm nP_Mz + \tau_2 z ,w_2\rangle_{\partial\mathcal T_h}\\
&\quad-\langle (\bm \beta\cdot\bm n + \tau_2)P_Mz ,w_2\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial}
-\langle \bm p\cdot\bm n-\bm\beta\cdot\bm nz ,\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}\\
& \quad- \langle \tau_2 z -\tau_2 P_Mz ,\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}.
\end{align*}
Note that the exact solution $\bm p$ and $z$ satisfies
\begin{align*}
(\bm p,\bm r_2)_{\mathcal T_h}-(z,\nabla\cdot\bm r_2)_{\mathcal T_h}+\langle z,\bm r_2\cdot\bm n\rangle_{\partial\mathcal T_h}&=0,\\
-(\bm p-\bm \beta z, \nabla w_2)_{\mathcal T_h}+\langle {\bm p}\cdot\bm n-\bm \beta\cdot\bm n z,w_2\rangle_{\partial\mathcal T_h}&= (y_d- y, w_2)_{\mathcal T_h},\\
\langle {\bm p}\cdot\bm n-\bm \beta\cdot\bm n z,\mu_2\rangle_{\partial\mathcal T_h\backslash\varepsilon^{\partial}_h}&=0,
\end{align*}
for all $\left(\bm{r}_2, w_2,\mu_2\right)\in \bm V_h\times W_h\times M_h(o)$. Since $z=0$ on $\varepsilon_h^\partial$, we have
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_2 (\widetilde{\bm \Pi}_V\bm p,\widetilde{\Pi}_W z,P_Mz;\bm r_2,w_2,\mu_2) \\
& = - (\delta^{\bm p},\bm r_2)_{\mathcal T_h} + (y_d - y,w_2)_{\mathcal T_h} + \langle \tau_2\delta^{\widehat z} ,w_2-\mu_2\rangle_{\partial\mathcal T_h}.
\end{align*}
By the definition of $P_M$ in \eqref{P_M} and since $\widetilde \tau_2$ from Lemma \ref{tau_lem} is piecewise constant on $\partial \mathcal T_h$, we have
\begin{align*}
\langle \tau_2 \delta^{\widehat z}, w_2-\mu_2\rangle_{\partial \mathcal T_h} &= \langle (\tau_2 - \widetilde \tau_2)\delta^{\widehat z}, w_2-\mu_2\rangle_{\partial \mathcal T_h}.
\end{align*}
This gives
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_2 (\widetilde{\bm \Pi}_V\bm p,\widetilde{\Pi}_W z,P_Mz;\bm r_2,w_2,\mu_2) \\
& = - (\delta^{\bm p},\bm r_2)_{\mathcal T_h} + (y_d - y,w_2)_{\mathcal T_h} + \langle (\tau_2 - \widetilde \tau_2) \delta^{\widehat z} ,w_2 - \mu_2\rangle_{\partial\mathcal T_h}.
\end{align*}
Subtract part 2 of the auxiliary problem \eqref{HDG_u_b} from the above equality to obtain the result.
\end{proof}
\subsubsection{Step 3: Estimates for $\varepsilon_h^p$ and $\varepsilon_h^z$ by an energy and duality argument.} \label{subsec:proof_step3}
\begin{lemma}\label{e_sec}
We have
\begin{align}
\|\varepsilon_h^{\bm p}\|_{\mathcal T_h}&+\|\varepsilon_h^z-\varepsilon_h^{\widehat z}\|_{\partial\mathcal T_h} \le \mathbb E + \kappa \| \varepsilon^z_h \|_{\mathcal T_h},\label{error_es_p}
\end{align}
where
\begin{align*}
\mathbb E = C\| \delta^{\bm p} \|_{\mathcal T_h} + \frac C {\kappa} \| y_h(u) - y \|_{\mathcal T_h} +C\| \tau_2 - \widetilde \tau_2\|_{0, \infty, \partial\mathcal T_h} \| \delta^{\widehat z}\|_{\partial \mathcal T_h}
\end{align*}
and $ \kappa $ is any positive constant and $ C $ does not depend on $ \kappa $.
\end{lemma}
\begin{proof}
Taking $(\bm r_2,w_2,\mu_2) = (\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat z}_h)$ in \eqref{error_z} in Lemma \ref{lemma:step1_first_lemma} gives
\begin{align*}
\hspace{1em}&\hspace{-1em} \mathscr B_2 ( \varepsilon^{ \bm p}_h, \varepsilon^z_h, \varepsilon^{\widehat z}_h;\varepsilon^{\bm p}_h, \varepsilon^z_h, \varepsilon^{\widehat z}_h )\\
& = (\delta^{\bm p},\varepsilon^{\bm p}_h)_{\mathcal T_h}+(y_h(u)- y, \varepsilon^z_h)_{\mathcal T_h}+ \langle (\tau_2 - \widetilde \tau_2)\delta^{\widehat z},\varepsilon_h^z-\varepsilon_h^{\widehat z}\rangle_{\partial\mathcal T_h}\\
&\le \| \delta^{\bm p} \|_{\mathcal T_h} \| \varepsilon_ h^ {\bm p} \|_{\mathcal T_h} + \| y_h(u) - y \|_{\mathcal T_h} \| \varepsilon^z_h \|_{\mathcal T_h} \\
&\quad+\| \tau_2 - \widetilde \tau_2\|_{0, \infty, \partial\mathcal T_h} \| \delta^{\widehat z}\|_{\partial \mathcal T_h} \|\varepsilon_h^z-\varepsilon_h^{\widehat z}\|_{\partial\mathcal T_h}.
\end{align*}
Lemma \eqref{property_B} gives
\begin{align*}
\|\varepsilon_h^{\bm p}\|_{\mathcal T_h}&+\|\varepsilon_h^z-\varepsilon_h^{\widehat z}\|_{\partial\mathcal T_h}\nonumber\\
&\le C\| \delta^{\bm p} \|_{\mathcal T_h} + \frac C {\kappa} \| y_h(u) - y \|_{\mathcal T_h} +C\| \tau_2 - \widetilde \tau_2\|_{0, \infty, \partial\mathcal T_h} \| \delta^{\widehat z}\|_{\partial \mathcal T_h} + \kappa \| \varepsilon^z_h \|_{\mathcal T_h},
\end{align*}
where $\kappa$ is any positive constant.
\end{proof}
Next, we introduce the dual problem for any given $\Theta$ in $L^2(\Omega):$
\begin{equation}\label{Dual_PDE}
\begin{split}
\bm\Phi-\nabla\Psi&=0\qquad~\text{in}\ \Omega,\\
-\nabla\cdot\bm{\Phi}+\bm\beta\cdot\nabla\Psi&=\Theta \qquad\text{in}\ \Omega,\\
\Psi&=0\qquad~\text{on}\ \partial\Omega.
\end{split}
\end{equation}
Since the domain $\Omega$ is convex, we have the following regularity estimate
\begin{align}\label{dual_esti}
\norm{\bm \Phi}_{1,\Omega} + \norm{\Psi}_{2,\Omega} \le C_{\text{reg}} \norm{\Theta}_\Omega,
\end{align}
Before we estimate $\varepsilon_h^{\bm p}$ and $\varepsilon_h^z$, we introduce the following notation, which is similar to the earlier notation in \eqref{notation_1}:
\begin{align}
\delta^{\bm \Phi} &=\bm \Phi-\widetilde{\bm\Pi}_V\bm \Phi, \quad \delta^\Psi=\Psi- \widetilde{\Pi}_W \Psi, \quad
\delta^{\widehat \Psi} = \Psi-P_M\Psi.
\end{align}
\begin{lemma}
We have
\begin{subequations}
\begin{align}
\norm{\varepsilon_h^{\bm p}}_{\mathcal T_h} &\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\label{var_p}\\
\|\varepsilon^z_h\|_{\mathcal T_h} &\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).\label{var_z}
\end{align}
\end{subequations}
\end{lemma}
\begin{proof}
Consider the dual problem \eqref{Dual_PDE} and let $\Theta = \varepsilon_h^z$. Take $(\bm r_2,w_2,\mu_2) = (\widetilde {\bm\Pi}_V\bm{\Phi},\widetilde{\Pi}_W\Psi,P_M\Psi)$ in \eqref{error_z} in Lemma \ref{lemma:step1_first_lemma}. Since $\Psi=0$ on $\varepsilon_h^{\partial}$ we have
\begin{align*}
\hspace{1em}&\hspace{-1em} \mathscr B_2 (\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat z}_h;\widetilde{\bm\Pi}_V\bm{\Phi},\widetilde{\Pi}_W\Psi,P_M\Psi)\\
&= (\varepsilon^{\bm p}_h,\widetilde{\bm\Pi}_V\bm{\Phi})_{\mathcal T_h}-( \varepsilon^z_h,\nabla\cdot\widetilde{\bm\Pi}_V\bm{\Phi})_{\mathcal T_h}+\langle \varepsilon^{\widehat z}_h,\widetilde{\bm\Pi}_V\bm{\Phi} \cdot\bm n\rangle_{\partial\mathcal T_h}\\
&\quad-(\varepsilon^{\bm p}_h-\bm \beta \varepsilon^z_h, \nabla \widetilde{\Pi}_W\Psi)_{\mathcal T_h}
+\langle \varepsilon^{\bm p}_h\cdot\bm n-\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h +\tau_2(\varepsilon^z_h-\varepsilon^{\widehat z}_h ), \widetilde{\Pi}_W\Psi -P_M \Psi\rangle_{\partial\mathcal T_h}\\
&=(\varepsilon^{\bm p}_h,\bm{\Phi})_{\mathcal{T}_h}-(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h}-(\varepsilon^z_h,\nabla\cdot\bm{\Phi})_{\mathcal{T}_h}+(\varepsilon^z_h,\nabla\cdot \delta ^{\bm{\Phi}})_{\mathcal{T}_h}
\\
&\quad-\langle \varepsilon^{\widehat z}_h, \delta^{\bm \Phi}\cdot \bm{n}\rangle_{\partial\mathcal{T}_h\backslash\varepsilon_h^\partial} -(\varepsilon^{\bm p}_h-\bm \beta \varepsilon^z_h, \nabla \Psi)_{\mathcal{T}_h}+(\varepsilon^{\bm p}_h-\bm \beta \varepsilon^z_h, \nabla \delta^{\Psi})_{\mathcal{T}_h}\\
&\quad - \langle \varepsilon^{\bm p}_h\cdot\bm n-\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h +\tau_2(\varepsilon^z_h-\varepsilon^{\widehat z}_h ),\delta^{\Psi} - \delta^{\widehat \Psi}\rangle_{\partial\mathcal T_h}\\
&= -(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h} + \| \varepsilon_h^z\|_{\mathcal T_h}^2 + (\varepsilon^z_h,\nabla\cdot \delta ^{\bm{\Phi}})_{\mathcal{T}_h} -\langle \varepsilon^{\widehat z}_h, \delta^{\bm \Phi}\cdot \bm{n}\rangle_{\partial\mathcal{T}_h}\\
& \quad+(\varepsilon^{\bm p}_h-\bm \beta \varepsilon^z_h, \nabla \delta^{\Psi})_{\mathcal{T}_h} -\langle \varepsilon^{\bm p}_h\cdot\bm n-\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h +\tau_2(\varepsilon^z_h-\varepsilon^{\widehat z}_h ),\delta^{\Psi} - \delta^{\widehat \Psi}\rangle_{\partial\mathcal T_h}.
\end{align*}
Here, we used $\langle\varepsilon^{\widehat z}_h,\bm \Phi\cdot\bm n\rangle_{\partial\mathcal T_h}=0$, which holds since $\varepsilon^{\widehat z}_h$ is single-valued function on interior edges and $\varepsilon^{\widehat z}_h=0$ on $\varepsilon^{\partial}_h$.
Next, integration by parts gives
\begin{align*}
(\varepsilon^z_h,\nabla\cdot\delta^{\bm \Phi})_{\mathcal{T}_h}
&=\langle \varepsilon^z_h,\delta^{\bm \Phi} \cdot\bm n\rangle_{\partial\mathcal T_h}-(\nabla\varepsilon^z_h,\delta^{\bm \Phi})_{\mathcal{T}_h} = \langle \varepsilon^z_h,\delta^{\bm \Phi}\cdot\bm n\rangle_{\partial\mathcal T_h}-(\nabla\varepsilon^z_h,\bm{\beta}\delta^{\Psi})_{\mathcal T_h},\\
(\varepsilon^{\bm p}_h, \nabla \delta^{ \Psi})_{\mathcal{T}_h}&=\langle \varepsilon^{\bm p}_h \cdot\bm n, \delta^{ \Psi}\rangle_{\partial\mathcal T_h}-(\nabla\cdot \varepsilon^{\bm p}_h , \delta^{ \Psi})_{\mathcal T_h} = \langle \varepsilon^{\bm p}_h \cdot\bm n, \delta^{ \Psi}\rangle_{\partial\mathcal T_h},\\
(\bm\beta \varepsilon_h^z, \nabla \delta^{ \Psi})_{\mathcal{T}_h}&=\langle \bm{\beta} \cdot\bm n \varepsilon_h^z, \delta^{ \Psi}\rangle_{\partial\mathcal T_h} - (\bm{\beta} \nabla\varepsilon_h^z, \delta^{ \Psi})_{\mathcal T_h}.
\end{align*}
We have
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_2(\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat z}_h;\widetilde{\bm\Pi}_V\bm{\Phi},\widetilde{\Pi}_W\Psi,P_M\Psi)\\
& = -(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h} + \| \varepsilon_h^z\|_{\mathcal T_h}^2 + \langle \varepsilon^z_h,\delta^{\bm \Phi}\cdot\bm n\rangle_{\partial\mathcal T_h} -\langle \varepsilon^{\widehat z}_h, \delta^{\bm \Phi}\cdot \bm{n}\rangle_{\partial\mathcal{T}_h\backslash\varepsilon_h^\partial}
\\
&\quad-\langle \bm{\beta} \cdot\bm n \varepsilon_h^z, \delta^{ \Psi}\rangle_{\partial\mathcal T_h} -\langle -\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h +\tau_2(\varepsilon^z_h-\varepsilon^{\widehat z}_h ),\delta^{\Psi} - \delta^{\widehat \Psi}\rangle_{\partial\mathcal T_h}.
\end{align*}
Remembering that $\varepsilon^{\widehat z}_h$ is single-valued function on interior edges and $\varepsilon^{\widehat z}_h=0$ on $\varepsilon^{\partial}_h$ gives
\begin{align*}
\langle\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h,P_M\Psi\rangle_{\partial\mathcal T_h}=0=\langle\bm \beta\cdot\bm n\varepsilon^{\widehat z}_h,\Psi\rangle_{\partial\mathcal T_h}.
\end{align*}
This implies
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_2 (\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat z}_h;\widetilde{\bm\Pi}_V\bm{\Phi},\widetilde{\Pi}_W\Psi,P_M\Psi)\\
&=-(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h} + \| \varepsilon_h^z\|_{\mathcal T_h}^2 + \langle \varepsilon^z_h - \varepsilon^{\widehat z}_h,\delta^{\bm \Phi}\cdot\bm n\rangle_{\partial\mathcal T_h}
\\
&\quad-\langle \bm{\beta} \cdot\bm n (\varepsilon_h^z - \varepsilon_h^{\widehat z}), \delta^{ \Psi}\rangle_{\partial\mathcal T_h\backslash\varepsilon_h^\partial} -\langle \tau_2(\varepsilon^z_h-\varepsilon^{\widehat z}_h ),\delta^{\Psi} - \delta^{\widehat \Psi}\rangle_{\partial\mathcal T_h}\\
&=-(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h} + \| \varepsilon_h^z\|_{\mathcal T_h}^2 + \langle \varepsilon^z_h - \varepsilon^{\widehat z}_h,\delta^{\bm \Phi}\cdot\bm n - \bm{\beta}\cdot\bm n\delta^{\Psi}-\tau_2(\delta^{\Psi} - \delta^{\widehat \Psi})\rangle_{\partial\mathcal T_h}.
\end{align*}
On the other hand,
\begin{align*}
\mathscr B_2 &(\varepsilon^{\bm p}_h,\varepsilon^z_h,\varepsilon^{\widehat z}_h;\widetilde{\bm\Pi}_V\bm{\Phi},\widetilde{\Pi}_W\Psi,P_M\Psi)\\
& = (\delta^{\bm p},\widetilde{\bm\Pi}_V\bm{\Phi} )_{\mathcal T_h}+(y_h(u)- y, \widetilde{\Pi}_W\Psi)_{\mathcal T_h}+ \langle (\tau_2 - \widetilde \tau_2)\delta^{\widehat z},\widetilde{\Pi}_W\Psi - P_M\Psi\rangle_{\partial\mathcal T_h}.
\end{align*}
Comparing the above two equalities gives
\begin{align*}
\| \varepsilon_h^z\|_{\mathcal T_h}^2 & = (\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h} -\langle \varepsilon^z_h - \varepsilon^{\widehat z}_h,\delta^{\bm \Phi}\cdot\bm n - \bm{\beta}\cdot\bm n\delta^{\Psi}-\tau_2(\delta^{\Psi} - \delta^{\widehat \Psi})\rangle_{\partial\mathcal T_h}\\
&\quad+(\delta^{\bm p},\widetilde{\bm\Pi}_V\bm{\Phi} )_{\mathcal T_h}+(y_h(u)- y, \widetilde{\Pi}_W\Psi)_{\mathcal T_h}+ \langle (\tau_2 - \widetilde \tau_2) \delta^{\widehat z},\widetilde{\Pi}_W\Psi - P_M\Psi\rangle_{\partial\mathcal T_h}\\
&= \sum_{i=1}^7 R_i.
\end{align*}
Let $ C_0 = \max\{C, 1\} $, where $ C $ is the constant defined in Lemma \ref{pro_error1}. For the terms $R_1$ and $R_2$, Lemma \ref{e_sec} gives
\begin{align*}
R_1&=-(\varepsilon^{\bm p}_h,\delta^{\bm \Phi})_{\mathcal{T}_h}\le \|\varepsilon^{\bm p}_h\|_{\mathcal{T}_h} \|\delta^{\bm \Phi}\|_{\mathcal{T}_h}
\le \left( \mathbb E+\kappa\|\varepsilon^z_h\|_{\mathcal T_h}\right) C_0(\|\bm \Phi\|_1 + \| \Psi\|_1)\\
&\le C_0C_{\text{reg}}\left( \mathbb E+\kappa\|\varepsilon^z_h\|_{\mathcal T_h}\right) \|\varepsilon^z_h\|_{\mathcal T_h},\\
R_2 &= -\langle \varepsilon^z_h - \varepsilon^{\widehat z}_h,\delta^{\bm \Phi}\cdot\bm n - \bm{\beta}\cdot\bm n\delta^{\Psi}-\tau_2(\delta^{\Psi} - \delta^{\widehat \Psi})\rangle_{\partial\mathcal T_h}\\
&\le \| {\varepsilon^z_h - \varepsilon^{\widehat z}_h}\|_{\partial \mathcal T_h}(\|{\delta^{\bm \Phi}}\|_{\partial \mathcal T_h}+\| {\tau_1}\|_{0,\infty,\partial\mathcal T_h}\|{\delta^{\Psi}}\|_{\partial \mathcal T_h} +\| {\tau_2}\|_{0,\infty,\partial\mathcal T_h}\|{\delta^{\widehat\Psi}}\|_{\partial \mathcal T_h})\\
&\le 3\left( \mathbb E+\kappa\|\varepsilon^z_h\|_{\mathcal T_h}\right)(1+\norm {\tau_1}_{0,\infty,\partial\mathcal T_h} + \norm {\tau_2}_{0,\infty,\partial\mathcal T_h})C_0(\|\bm \Phi\|_1 + \| \Psi\|_1)\\
&\le 3C_0C_{\text{reg}}\left( \mathbb E+\kappa\|\varepsilon^z_h\|_{\mathcal T_h}\right)(1+\norm {\tau_1}_{0,\infty,\partial\mathcal T_h} + \norm {\tau_2}_{0,\infty,\partial\mathcal T_h})\|\varepsilon^z_h\|_{\mathcal T_h}.
\end{align*}
For the terms $R_3$, $R_4$ and $R_5$, we use the triangle inequality, the regularity estimate \eqref{dual_esti}, and the assumption $ h \leq 1 $ to give
\begin{align*}
R_3&=(\delta^{\bm p},\widetilde{\bm\Pi}_V\bm \Phi)_{\mathcal T_h}\le \|\delta^{\bm p}\|_{\mathcal T_h}(\|\widetilde{\bm\Pi}_V\bm \Phi- \bm \Phi\|_{\mathcal T_h} + \|\bm \Phi\|_{\mathcal T_h})\\
&\le C_0\|\delta^{\bm p}\|_{\mathcal T_h}(\norm {\bm{\Phi}}_{1,\Omega} + \norm {{\Psi} }_{1,\Omega} +\|\bm \Phi\|_{\mathcal T_h})\\
&\le 2C_0C_{\text{reg}}\|\delta^{\bm p}\|_{\mathcal T_h}\|\varepsilon^z_h\|_{\mathcal T_h},\\
R_4&=(y-y_h(u),\widetilde\Pi_W\Psi)_{\mathcal T_h}\le \|y-y_h(u)\|_{\mathcal T_h} \|\widetilde\Pi_W\Psi\|_{\mathcal T_h}\\
&\le \|y-y_h(u)\|_{\mathcal T_h}(\|\widetilde\Pi_W\Psi-\Psi\|_{\mathcal T_h}+\|\Psi\|_{\mathcal T_h})\\
&\le C_0\|y-y_h(u)\|_{\mathcal T_h}(\|\Psi\|_{1,\Omega}+\|\bm\Phi\|_{1,\Omega}+\|\Psi\|_{\mathcal T_h})\\
&\le 2C_0C_{\text{reg}}\|y-y_h(u)\|_{\mathcal T_h}\|\varepsilon^z_h\|_{\mathcal T_h},\\
R_5 &= \langle (\tau_2 - \widetilde \tau_2) \delta^{\widehat z}, \widetilde{\Pi}_W\Psi - P_M\Psi\rangle_{\partial\mathcal T_h} \\
&\le \| {\tau_2 - \widetilde \tau_2}\|_{0,\infty,\partial\mathcal T_h} \|{\delta^{\widehat z}}\|_{\partial\mathcal T_h} \| {\delta^{\Psi} - \delta^{\widehat \Psi}}\|_{\partial\mathcal T_h}\\
&\le C_{\bm \beta}\|\bm \beta\|_{1,\infty,\Omega} h^{1/2}\|{\delta^{\widehat z}}\|_{\partial\mathcal T_h}C_0(\|\Psi\|_{1,\Omega}+\|\bm\Phi\|_{1,\Omega}+\|\Psi\|_{1,\Omega})\\
&\le 2C_0C_{\text{reg}} C_{\bm \beta}\|\bm \beta\|_{1,\infty,\Omega} h^{1/2}\|{\delta^{\widehat z}}\|_{\partial\mathcal T_h} \|\varepsilon^z_h\|_{\mathcal T_h}.
\end{align*}
Summing $R_1$ to $R_5$ gives
\begin{align*}
\|\varepsilon^z_h\|_{\mathcal T_h} &\le\mathbb C(\mathbb E +\kappa\|\varepsilon^z_h\|_{\mathcal T_h} ) + C ( \norm {\delta^{\bm p}}_{\mathcal T_h} +\norm{y - y_h(u)}_{\mathcal T_h} + h^{1/2}\|{\delta^{\widehat z}}\|_{\partial\mathcal T_h}),
\end{align*}
where
\begin{align*}
\mathbb C =4 C_0C_{\text{reg}} (1+\norm {\tau_1}_{0,\infty,\partial\mathcal T_h} + \norm {\tau_2}_{0,\infty,\partial\mathcal T_h}).
\end{align*}
Choose $\kappa=\frac{1}{2\mathbb C}$ gives
\begin{align*}
\|\varepsilon^z_h\|_{\mathcal T_h}\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align*}
Finally, \eqref{error_es_p} and \eqref{var_z} imply \eqref{var_p}.
\end{proof}
As a consequence, a simple application of the triangle inequality gives optimal convergence rates for $\|\bm p -\bm p_h(u)\|_{\mathcal T_h}$ and $\|z -z_h(u)\|_{\mathcal T_h}$:
\begin{lemma}\label{lemma:step3_conv_rates}
\begin{subequations}
\begin{align}
\|\bm p -\bm p_h(u)\|_{\mathcal T_h} &\le \|\delta^{\bm p}\|_{\mathcal T_h} + \|\varepsilon_h^{\bm p}\|_{\mathcal T_h} \ \nonumber \\
&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|z -z_h(u)\|_{\mathcal T_h} &\le \|\delta^{z}\|_{\mathcal T_h} + \|\varepsilon_h^{z}\|_{\mathcal T_h} \ \nonumber \\
& \lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align}
\end{subequations}
\end{lemma}
\subsubsection{Step 4: Estimate for $\|u-u_h\|_{\mathcal T_h}$, $\norm {y-y_h}_{\mathcal T_h}$ and $\norm {z-z_h}_{\mathcal T_h}$.}
Next, we bound the error between the solutions of the auxiliary problem and the HDG discretization of the optimality system \eqref{HDG_full_discrete}. We use these error bounds and the error bounds in Lemmas \ref{le} and \ref{lemma:step3_conv_rates} to obtain the main result.
For the remaining steps, we denote
\begin{equation*}
\begin{split}
\zeta_{\bm q} &=\bm q_h(u)-\bm q_h,\quad\zeta_{y} = y_h(u)-y_h,\quad\zeta_{\widehat y} = \widehat y_h(u)-\widehat y_h,\\
\zeta_{\bm p} &=\bm p_h(u)-\bm p_h,\quad\zeta_{z} = z_h(u)-z_h,\quad\zeta_{\widehat z} = \widehat z_h(u)-\widehat z_h.
\end{split}
\end{equation*}
Subtracting the auxiliary problem and the HDG problem gives the following error equations
\begin{subequations}\label{eq_yh}
\begin{align}
\mathscr B_1(\zeta_{\bm q},\zeta_y,\zeta_{\widehat y};\bm r_1, w_1,\mu_1)&=(u-u_h,w_1)_{\mathcal T_h}\label{eq_yh_yhu}\\
\mathscr B_2(\zeta_{\bm p},\zeta_z,\zeta_{\widehat z};\bm r_2, w_2,\mu_2)&=-(\zeta_y, w_2)_{\mathcal T_h}\label{eq_zh_zhu}.
\end{align}
\end{subequations}
\begin{lemma}
We have
\begin{align}\label{eq_uuh_yhuyh}
\hspace{3em}&\hspace{-3em} \gamma\|u-u_h\|^2_{\mathcal T_h}+\|y_h(u)-y_h\|^2_{\mathcal T_h}\nonumber\\
&=( z_h-\gamma u_h,u-u_h)_{\mathcal T_h}-(z_h(u)-\gamma u,u-u_h)_{\mathcal T_h}.
\end{align}
\end{lemma}
\begin{proof}
First, we have
\begin{align*}
\hspace{3em}&\hspace{-3em} ( z_h-\gamma u_h,u-u_h)_{\mathcal T_h}-( z_h(u)-\gamma u,u-u_h)_{\mathcal T_h}\\
&=-(\zeta_{ z},u-u_h)_{\mathcal T_h}+\gamma\|u-u_h\|^2_{\mathcal T_h}.
\end{align*}
Next, Lemma \ref{identical_equa} gives
\begin{align*}
\mathscr B_1 &(\zeta_{\bm q},\zeta_y,\zeta_{\widehat{y}};\zeta_{\bm p},-\zeta_{z},-\zeta_{\widehat z}) + \mathscr B_2(\zeta_{\bm p},\zeta_z,\zeta_{\widehat z};-\zeta_{\bm q},\zeta_y,\zeta_{\widehat{y}}) = 0.
\end{align*}
On the other hand, using the definition of $ \mathscr B_1 $ and $ \mathscr B_2 $ gives
\begin{align*}
\hspace{3em}&\hspace{-3em} \mathscr B_1 (\zeta_{\bm q},\zeta_y,\zeta_{\widehat{y}};\zeta_{\bm p},-\zeta_{z},-\zeta_{\widehat z}) + \mathscr B_2(\zeta_{\bm p},\zeta_z,\zeta_{\widehat z};-\zeta_{\bm q},\zeta_y,\zeta_{\widehat{y}})\\
&= - ( u- u_h,\zeta_{ z})_{\mathcal{T}_h}-\|\zeta_{ y}\|^2_{\mathcal{T}_h}.
\end{align*}
Comparing the above two equalities gives
\begin{align*}
-(u-u_h,\zeta_{ z})_{\mathcal{T}_h}=\|\zeta_{ y}\|^2_{\mathcal{T}_h}.
\end{align*}
This completes the proof.
\end{proof}
\begin{theorem}\label{thm:estimates_u_y_z}
We have
\begin{subequations}
\begin{align}\label{err_yhu_yh}
\|u-u_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|y-y_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\\
\|z-z_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}
Recall the continuous and discretized optimality conditions \eqref{eq_adeq_e} and \eqref{HDG_full_discrete_e} gives $ \gamma u = z $ and $ \gamma u_h = z_h $. These equations and the previous lemma give
\begin{align*}
\hspace{3em}&\hspace{-3em} \gamma\|u-u_h\|^2_{\mathcal T_h}+\|\zeta_{ y}\|^2_{\mathcal T_h}\\
&=( z_h-\gamma u_h,u-u_h)_{\mathcal T_h}-( z_h(u)-\gamma u,u-u_h)_{\mathcal T_h}\\
&=-( z_h(u)- z,u-u_h)_{\mathcal T_h}\\
&\le \| z_h(u)- z\|_{\mathcal T_h} \|u-u_h\|_{\mathcal T_h}\\
&\le\frac{1}{2\gamma}\| z_h(u)- z\|^2_{\mathcal T_h}+\frac{\gamma}{2}\|u-u_h\|^2_{\mathcal T_h}.
\end{align*}
By Lemma \ref{lemma:step3_conv_rates}, we have
\begin{align}\label{eqn:estimate_u_zeta_y}
\|u-u_h\|_{\mathcal T_h}+\|\zeta_{ y}\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align}
Then, by the triangle inequality and Lemma \ref{le} we obtain
\begin{align*}
\|y-y_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align*}
Finally, since $z = \gamma u $ and $z_h = \gamma u_h$ we have
\begin{align*}
\|z-z_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).
\end{align*}
\end{proof}
\subsubsection{Step 5: Estimate for $\|q-q_h\|_{\mathcal T_h}$ and $\|p-p_h\|_{\mathcal T_h}$.}
\begin{lemma}
We have
\begin{subequations}
\begin{align}
\|\zeta_{\bm q}\|_{\mathcal T_h} &\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\label{err_Lhu_qh}\\
\|\zeta_{\bm p}\|_{\mathcal T_h} &\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}).\label{err_Lhu_ph}
\end{align}
\end{subequations}
\end{lemma}
\begin{proof}
By Lemma \ref{property_B}, the error equation \eqref{eq_yh_yhu}, and the estimate \eqref{eqn:estimate_u_zeta_y} we have
\begin{align*}
\|\zeta_{\bm q}\|^2_{\mathcal T_h} &\lesssim \mathscr B_1(\zeta_{\bm q},\zeta_y,\zeta_{\widehat y};\zeta_{\bm q},\zeta_y,\zeta_{\widehat y})\\
&=( u- u_h,\zeta_{ y})_{\mathcal T_h}\\
&\le\| u- u_h\|_{\mathcal T_h}\|\zeta_{ y}\|_{\mathcal T_h}\\
&\lesssim h^{2k+2}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1})^2.
\end{align*}
Similarly, by Lemma \ref{property_B}, the error equation \eqref{eq_zh_zhu}, Lemma \ref{lemma:step3_conv_rates}, and Theorem \ref{thm:estimates_u_y_z} we have
\begin{align*}
\|\zeta_{\bm p}\|^2_{\mathcal T_h} &\lesssim \mathscr B_2(\zeta_{\bm p},\zeta_z,\zeta_{\widehat z};\zeta_{\bm p},\zeta_z,\zeta_{\widehat z})\\
&=-(\zeta_{ y},\zeta_{ z})_{\mathcal T_h}\\
&\le\|\zeta_{y}\|_{\mathcal T_h}\|\zeta_{ z}\|_{\mathcal T_h}\\
&\le\|\zeta_{y}\|_{\mathcal T_h} ( \| z_h(u) - z \|_{\mathcal T_h} + \| z - z_h \|_{\mathcal T_h} )\\
&\lesssim h^{2k+2}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1})^2.
\end{align*}
\end{proof}
The above lemma along with the triangle inequality, Lemma \ref{le}, and Lemma \ref{lemma:step3_conv_rates} complete the proof of the main result:
\begin{theorem}
We have
\begin{subequations}
\begin{align}
\|\bm q-\bm q_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1}),\label{err_q}\\
\|\bm p-\bm p_h\|_{\mathcal T_h}&\lesssim h^{k+1}(|\bm q|_{k+1}+|y|_{k+1}+|\bm p|_{k+1}+|z|_{k+1})\label{err_p}.
\end{align}
\end{subequations}
\end{theorem}
\section{Numerical Experiments}
\label{sec:numerics}
In this section, we present three numerical examples to confirm our theoretical results. We consider two 2D problems on a square domain $\Omega = [0,1]\times [0,1] \subset \mathbb{R}^2$, and a 3D problem on a cubic domain $\Omega = [0,1]\times [0,1]\times [0,1] \subset \mathbb{R}^3$. For the three examples, we take $\gamma = 1$ and specify the exact state, dual state, and function $\bm \beta$. The data $f$, $ g $, and $y_d$ is generated from the optimality system \eqref{eq_adeq}. Also, we chose $ \tau_1 = 1 $ and set $ \tau_2 $ using \textbf{(A2)}. For all three examples, conditions \textbf{(A1)}-\textbf{(A3)} are satisfied.
Numerical results for $ k = 0 $ and $ k = 1 $ for the three examples are shown in Table \ref{table_1}--Table \ref{table_6}. The observed convergence rates exactly match the theoretical results.
\begin{example}\label{example1}
We take $\bm{\beta} = [1,1]$, state $ y(x_1,x_2) = \sin(\pi x_1) $, and dual state $ z(x_1,x_2) = \sin(\pi x_1)\sin(\pi x_2)$.
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/8$& $1/16$&$1/32$ &$1/64$ & $1/128$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$&1.7818e-01 &8.6412e-02 &4.2357e-02 &2.0948e-02 &1.0415e-02 \\
\hline
order&-& 1.04& 1.03 &1.02& 1.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$& 4.2057e-01 &2.1839e-01 &1.1116e-01 &5.6062e-02 &2.8151e-02 \\
\hline
order&-& 0.94&0.97 &0.99 & 1.00 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&1.6300e-01 &8.4087e-02 &4.2612e-02 &2.1437e-02 &1.0750e-02\\
\hline
order&-& 0.95&0.98&0.99 & 1.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 2.1310e-01 &1.0803e-01 &5.4219e-02 &2.7138e-02 &1.3573e-02 \\
\hline
order&-& 0.98&0.99&1.00& 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example1}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=0$.}\label{table_1}
\end{table}
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/8$& $1/16$&$1/32$ &$1/64$ & $1/128$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$&1.3708e-02 &3.5192e-03 &8.8851e-04 &2.2301e-04 &5.5850e-05 \\
\hline
order&-& 2.00& 2.00 &2.00& 2.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$& 3.4995e-02 &8.9472e-03 &2.2581e-03 &5.6694e-04 &1.4202e-04 \\
\hline
order&-& 2.00&2.00 &2.00 & 2.00 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&1.1705e-02 &2.9528e-03 &7.4012e-04 &1.8519e-04 &4.6315e-05\\
\hline
order&-& 2.00&2.00&2.00 & 2.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 2.3361e-02 &5.9059e-03 &1.4810e-03 &3.7059e-04 &9.2676e-05 \\
\hline
order&-& 2.00&2.00&2.00& 2.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example1}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=1$.}\label{table_2}
\end{table}
\end{example}
\begin{example}\label{example2}
We take $\bm{\beta} = [x_2,x_1]$, state $ y(x_1,x_2) = \sin(\pi x_1) $, and dual state $ z(x_1,x_2) = \sin(\pi x_1)\sin(\pi x_2)$.
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/8$& $1/16$&$1/32$ &$1/64$ & $1/128$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$ &1.7838e-01 &8.6461e-02 &4.2375e-02 &2.0957e-02 &1.0419e-02 \\
\hline
order&-& 1.04& 1.03 &1.02& 1.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$& 4.2050e-01 &2.1848e-01 &1.1123e-01 &5.6101e-02 &2.8171e-02 \\
\hline
order&-& 0.95&0.97 &0.99 & 0.99 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&1.6285e-01 &8.4032e-02 &4.2588e-02 &2.1426e-02 &1.0744e-02\\
\hline
order&-& 0.95&0.98&0.99 & 1.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 2.1223e-01 &1.0773e-01 &5.4094e-02 &2.7081e-02 &1.3546e-02 \\
\hline
order&-& 0.98&0.99&1.00& 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example2}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=0$.}\label{table_3}
\end{table}
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/8$& $1/16$&$1/32$ &$1/64$ & $1/128$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$&1.3713e-02 &3.5195e-03 &8.8853e-04 &2.2301e-04 &5.5850e-05 \\
\hline
order&-& 2.00& 2.00 &2.00& 2.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$ & 3.5010e-02 &8.9481e-03 &2.2581e-03 &5.6694e-04 &1.4202e-04 \\
\hline
order&-& 2.00&2.00 &2.00 & 2.00 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&1.1712e-02 &2.9532e-03 &7.4015e-04 &1.8520e-04 &4.6315e-05\\
\hline
order&-& 2.00&2.00&2.00 & 2.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 2.3368e-02 &5.9064e-03 &1.4810e-03 &3.7059e-04 &9.2676e-05 \\
\hline
order&-& 2.00&2.00&2.00& 2.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example2}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=1$.}\label{table_4}
\end{table}
\end{example}
\begin{example}\label{example3}
We take $\bm{\beta} = [1,1,1]$, state $ y(x_1,x_2,x_3) = \sin(\pi x_1) $, and dual state $ z(x_1,x_2,x_3) = \sin(\pi x_1)\sin(\pi x_2)\sin(\pi x_3)$.
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/2$& $1/4$&$1/8$ &$1/16$ & $1/32$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$ &6.3167e-01 &3.4472e-01 &1.7715e-01 &8.9373e-02 &4.4778e-02 \\
\hline
order&-& 0.87& 0.96 &0.99& 1.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$& 4.9907e-01 &2.9505e-01 &1.5339e-01 &7.7393e-02 &3.8724e-02 \\
\hline
order&-& 0.76&0.94 &0.99 & 1.00 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&1.7959e-01 &1.0026e-01 &5.3061e-02 &2.7275e-02 &1.3646e-02\\
\hline
order&-& 0.84&0.92&0.96 & 1.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 2.3121e-01 &1.3646e-01 &7.2318e-02 &3.7004e-02 &1.8587e-02 \\
\hline
order&-& 0.76&0.92&0.97& 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example3}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=0$.}\label{table_5}
\end{table}
\begin{table
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$h/\sqrt 2$ &$1/2$& $1/4$&$1/8$ &$1/16$ & $1/32$ \\
\hline
$\norm{\bm{q}-\bm{q}_h}_{0,\Omega}$&9.2498e-02 &2.7594e-02 &7.4959e-03 &1.9486e-03 &4.8720e-04\\
\hline
order&-& 1.75& 1.90 &1.94& 2.00\\
\hline
$\norm{\bm{p}-\bm{p}_h}_{0,\Omega}$ & 1.8360e-01 &5.3637e-02 &1.3921e-02 &3.5138e-03 &8.7857e-04 \\
\hline
order&-& 1.80&1.95 &1.99 & 2.00 \\
\hline
$\norm{{y}-{y}_h}_{0,\Omega}$&4.4822e-02 &1.1780e-02 &2.9545e-03 &7.3644e-04 &1.8423e-04\\
\hline
order&-& 1.93&2.00&2.00 & 2.00 \\
\hline
$\norm{{z}-{z}_h}_{0,\Omega}$& 9.1413e-02 &2.7583e-02 &7.3069e-03 &1.8623e-03 &4.6575e-04 \\
\hline
order&-& 1.73&1.92&1.97& 2.00 \\
\hline
\end{tabular}
\end{center}
\caption{Example \ref{example3}: Errors for the state $y$, adjoint state $z$, and the fluxes $\bm q$ and $\bm p$ when $k=1$.}\label{table_6}
\end{table}
\end{example}
\section{Conclusions}
We proposed an HDG method to approximate the solution of an optimal distributed control problems for an elliptic convection diffusion equation. We obtained optimal a priori error estimates for the control, state, dual state, and their fluxes. The next step is to study optimal control problems governed by more complicated PDEs governing fluids. It would also be of interest to investigate if postprocessing gives superconvergence for this optimal control problem.
\section*{Appendix}
\label{sec:local_solver_details}
Before we investigate the local elimination, we give the following proposition.
\begin{proposition}
The matrices $A_{12}$ and $A_{13}$ in \eqref{system_equation} are positive definite.
\end{proposition}
\begin{proof}
We only prove $A_{12}$ is positive definite; a similar argument applies to $A_{13}$. The matrix $A_{12}$ is positive definite if and only if $\bm x^TA_{12}\bm x>0$ for any $\bm x=[x_1,x_2,\cdots,x_{N_2}]\in\mathbb R^{N_2} $. For $ x = \sum_{j=1}^{N_2} x_j \phi_j$, we have
\begin{align*}
\bm x^T A_{12}\bm x = \langle \tau_1x, x\rangle_{\partial \mathcal T_h} - (\bm{\beta}x, \nabla x)_{\mathcal T_h}.
\end{align*}
Moreover
\begin{align*}
(\bm \beta x,\nabla x)_{\mathcal T_h}=\langle\bm \beta\cdot\bm n x,x\rangle_{\partial\mathcal T_h}-(\bm \beta x,\nabla x)_{\mathcal T_h},
\end{align*}
this implies
\begin{align*}
(\bm \beta x,\nabla x)_{\mathcal T_h}&=\frac12\langle\bm \beta\cdot\bm n x,x\rangle_{\partial\mathcal T_h}.
\end{align*}
Then,
\begin{align*}
\bm x^T A_{12}\bm x = \langle (\tau_1-\frac 1 2 \bm{\beta}\cdot\bm n) x, x\rangle_{\partial\mathcal T_h}>0,
\end{align*}
by the assumption concerning $ \tau_1 $.
\end{proof}
By simple algebraic operations in equation \eqref{system_equation2}, we obtain the following formulas for the matrices $ G_1 $, $ G_2 $, $ H_1 $, and $ H_2 $ in \eqref{local_solver}:
\begin{align*}
G_1 &= B_1^{-1}B_2(B_4+B_2^TB_1^{-1}B_2)^{-1}(B_5+B_2^TB_1^{-1}B_3)-B_1^{-1}B_3,\\
G_2 &= -(B_4+B_2^TB_1^{-1}B_2)^{-1}(B_5+B_2^TB_1^{-1}B_3),\\
H_1 &= -B_1^{-1}B_2(B_4+B_2^TB_1^{-1}B_2)^{-1},\\
H_2 &= (B_4+B_2^TB_1^{-1}B_2)^{-1}.
\end{align*}
We briefly describe how these matrices can be easily computed using the HDG method described in this work.
Since the spaces $ \bm{V}_h $ and $ W_h $ consist of discontinuous polynomials, some of the system matrices are block diagonal and each block is small and symmetric positive definite. The matrix $ B_1 $ is this type, and therefore $ B_1^{-1} $ is easily computed and is also a matrix of the same type. Therefore, the the matrices $ G_1 $, $ G_2 $, $ H_1 $, and $ H_2 $ are easily computed if $ B_4 + B_2^T B_1^{-1} B_2 $ is also easily inverted
It can be checked that $B_2^T B_1^{-1} B_2$ is block diagonal with small nonnegative definite blocks. Next, $B_4 = \begin{bmatrix}
A_{12} & -\gamma^{-1}A_4\\
A_4 & A_{13}
\end{bmatrix}$, where $A_4$ is symmetric positive block diagonal, $A_{12}$ and $A_{13}$ are positive block diagonal. Due to the structure of $ B_1 $ and $ B_2 $, the matrix $B_2^TB_1^{-1}B_2 + B_4$ has the form
$\begin{bmatrix}
C_1 & -\gamma^{-1}A_4\\
A_4 & C_2
\end{bmatrix},
$
where $ C_1 $ and $ C_2 $ are symmetric positive block diagonal. The inverse can be easily computed using the formula
$$\begin{bmatrix}
C_1 & -\gamma^{-1}A_{4}\\
A_4 & C_2
\end{bmatrix}^{-1} = \\
\begin{bmatrix}
C_1^{-1}-\gamma^{-1}C_1^{-1}A_4 D^{-1}A_4C_1^{-1} & \gamma^{-1}C_1^{-1}A_4 D^{-1}\\
-D^{-1}A_4C_1^{-1} &D^{-1}
\end{bmatrix},
$$
where $D = C_2 +\gamma^{-1}A_4C_1^{-1}A_4$.
Furthermore, $ C_1^{-1} $ and $D^{-1}$ are both symmetric positive block diagonal.
\bibliographystyle{plain}
| {
"timestamp": "2018-01-03T02:02:32",
"yymm": "1801",
"arxiv_id": "1801.00082",
"language": "en",
"url": "https://arxiv.org/abs/1801.00082",
"abstract": "We propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a distributed optimal control problem governed by an elliptic convection diffusion PDE. We derive optimal a priori error estimates for the state, adjoint state, their fluxes, and the optimal control. We present 2D and 3D numerical experiments to illustrate our theoretical results.",
"subjects": "Numerical Analysis (math.NA)",
"title": "An HDG Method for Distributed Control of Convection Diffusion PDEs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575178175919,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7091542191622802
} |
https://arxiv.org/abs/math/0608746 | A parabolic free boundary problem with Bernoulli type condition on the free boundary | Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular perturbation problem $$\Delta u_\epsilon - \partial_t u_\epsilon = \beta_\epsilon(u_\epsilon) \textrm{as} \epsilon\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity.In particular, we show that the topological free boundary $\partial\{u>0\}$ can be decomposed into an {\em open} regular set (relative to $\partial\{u>0\}$) which is locally a surface with Hölder-continuous space normal, and a closed singular set.Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems. | \section{Introduction}
The parabolic free boundary problem
\begin{equation}\label{bernoulli}\Delta u - \partial_t u = 0 \textrm{ in } \{ u>0\}\> , \>
|\nabla u|=1 \textrm{ on } \partial\{ u>0\}\end{equation}
has originally been derived as singular limit
from a model for the propagation of equidiffusional premixed
flames with high activation energy (\cite{buckmaster});
here $u=\lambda(T_c-T)\> ,$ $T_c$ is the flame
temperature, which is assumed to be constant, $T$ is
the temperature outside the flame and $\lambda$ is a
normalization factor.
\\
Let us shortly summarize the mathematical results
directly relevant in this context, beginning with
the limit problem (\ref{bernoulli}):
in the brilliant paper \cite{AC}, H.W. Alt and L.A. Caffarelli
proved via minimization of the energy $\int (\vert\nabla u\vert^2
\> + \> \chi_{\{u>0\}})$ --
here $\chi_{\{u>0\}}$ denotes the characteristic function
of the set ${\{u>0\}}$ --
existence of a stationary solution
of (\ref{bernoulli}) in the sense of distributions.
They also derived regularity of the free boundary
$\partial\{u>0\}$ up to a set of vanishing ${n-1}$-dimensional
Hausdorff measure.
By \cite{cpde} existence of singular minimizers implies the existence of
singular minimizing cones.
L.A. Caffarelli-D. Jerison-C. Kenig showed that singular minimizing cones
do not exist in dimension $3$ (\cite{david1}).
Moreover it is known that singular minimizing cones exist
for $n\ge 7$ (\cite{david2}).
{\em Non-minimizing} singular cones appear already
for $n= 3$ (see \cite[example 2.7]{AC}). Moreover it is known,
that solutions of the Dirichlet problem in two space dimensions
are not unique (see
\cite[example 2.6]{AC}).
\\
For the time-dependent (\ref{bernoulli}), both ``trivial non-uniqueness''
(the positive solution of
the heat equation is always another solution of (\ref{bernoulli}))
and ``non-trivial uniqueness'' (see \cite{nonunique}) occur.
Even for flawless initial data, classical solutions of
(\ref{bernoulli}) develop singularities after a finite time
span; consider e.g. the example of two colliding traveling waves
\begin{equation}
\begin{array}{ll}\label{ex1}
u(t,x) = & \chi_{\{x+t>1\}} (\exp(x+t-1)-1)
\\
& \quad + \> \chi_{\{-x+t>1\}} (\exp(-x+t-1)-1) \textrm{ for } t \in [0,1)
\end{array}
\end{equation}
(see Figure \ref{travel}).
\begin{figure}[hb]
\begin{center}
\input{parberfig1.pstex_t}
\end{center}
\caption{Colliding traveling waves}\label{travel}
\end{figure}
\\[1cm]
There are several approaches concerning the construction
of a solution of the time-dependent problem, all of which
are based in some form on the convergence of
the solution $u_\varepsilon$ of the reaction-diffusion equation
\begin{equation}\label{perturb}
\Delta u_\varepsilon - \partial_t u_\varepsilon = \beta_\varepsilon(u_\varepsilon)
\end{equation}
to (\ref{bernoulli}) as $\varepsilon\to 0$;
here $\beta_\varepsilon(z)
= {1\over \varepsilon} \beta({z\over \varepsilon})\> , \>
\beta\in C^1_0([0,1])\> , \> \beta > 0$ in $(0,1)$ and
$\int \beta = {1\over 2}\> .$\\
L.A. Caffarelli and J.L. Vazquez
proved in \cite{cava}
uniform estimates for (\ref{perturb})
and a convergence result: for initial data $u^0$
that are strictly mean concave in the interior of their
support, a sequence of $\varepsilon$-solutions converges
to a solution of (\ref{bernoulli}) in the sense of
distributions.\\
Let us also mention several results on the corresponding
two-phase problem, which are relevant as solutions of
the one-phase problem are automatically solutions of the
corresponding two-phase problem. In \cite{leder1} and \cite{leder2},
L.A. Caffarelli, C. Lederman and N.
Wolanski prove convergence to a barrier solution
in the case that the limit function satisfies $\{u=0\}^\circ=\emptyset\> .$
\\
Then, there is the convergence to a solution in the sense of domain
variations \cite{calc} which seems to contain more information
than the barrier solutions in \cite{leder1} and \cite{leder2}.
For more general two-phase problems see \cite{ejde}.
Domain variation solutions play an important rule in this paper and
will be discussed in more detail in Section \ref{notion}.
\\[.3cm]
Here let it suffice to say that domain variation solutions
are pairs $(u,\chi)$ where the order parameter $\chi$
shares many properties with the characteristic function
$\chi_{\{ u>0\}}$ but does not necessarily coincide with
it. By \cite{calc}, {\em all limits} of the singular
perturbation problem (\ref{perturb}) are domain variation solutions,
so all results in the present paper hold for all limits of (\ref{perturb}).\\
Our main result Theorem \ref{theo:main} states
-- leaving out inessential assumptions --
that if
$(0,\rho^2)$ is a point on the topological free boundary
and if the set $\{ \chi >0\}$ is flat enough, i.e.
\begin{displaymath}
\chi(x,t)=0 \textrm{ when } (x,t)\in Q_\rho \textrm{ and }x_n\ge \sigma \rho,
\end{displaymath}
for some $\sigma\le \sigma_0$ (see Figure \ref{onesided}), then
the free boundary
$Q_{\rho/4} \cap \partial\{ u>0\}$ is
a surface with H\"older-continuous
space normal.\\
\begin{figure}[b]
\begin{center}
\input{parberfig2.pstex_t}
\end{center}
\caption{One-sided flatness in the case $\rho=1$}\label{onesided}
\end{figure}
As a consequence we obtain that the regular set is open relative
to $\partial\{ u>0\}$ (Corollary \ref{regular}, cf. Figure \ref{example}).\begin{figure}[t]
\begin{center}
\input{parberfig3.pstex_t}
\end{center}
\caption{Example of the set of regular free boundary points (stationary)}\label{example}
\end{figure}
\\
Note that even in the stationary case our result
extends the result in \cite{AC}
as our assumptions {\em do not exclude}
degenerate points or cusps close to the origin
(excluded by the definition of weak solutions \cite[5.1]{AC}),
{\em our result does that}.
\\
In the proof of our result we use ingenious tools developed in
\cite{AC}:
We prove that flatness on the side of $\{ \chi=0\}$
implies flatness on the side of $\{ \chi>0\}$ which in turn yields
uniform convergence of an inhomogeneously
scaled sequence of free boundaries.
\\
However we replace
the core in the
method of H.W. Alt-L.A. Caffarelli, relying on non-positive
mean curvature of $\partial\{ u>0\}$ at singularities,
by a method based on {\em scaling discrepancies}
(Proposition \ref{discr}). This original component
gives hope that the method may now be applicable
to more general free boundary or free discontinuity
problems, in particular two-phase free boundary problems.
\section{Notation}
Throughout this article ${\bf R}^n$ will be equipped with the Euclidean
inner product $x\cdot y$ and the induced norm $\vert x \vert\> ,\>
B_r(x_0)$ will denote the open $n$-dimensional ball of center
$x_0$, radius $r$ and volume $r^n\> \omega_n\> ,
\> B'_r(0)$ the open $n-1$-dimensional ball of center $0$ and radius
$r$,
and $e_i$ the $i$-th unit vector in ${\bf R}^n$.
We define $Q_r(x_0,t_0) := B_r(x_0)\times (t_0-r^2 , t_0+ r^2)$ to be
the cylinder
of radius $r$ and height $2r^2$,
$Q^-_r(x_0,t_0) := B_r(x_0)\times (t_0-r^2 , t_0)$ its ``negative part''
and
$T_r^-(t_0) := {\bf R}^n \times (t_0 - 4r^2 , t_0 - r^2)$
the horizontal layer from $t_0 - 4r^2$ to $t_0-r^2$.
Let us also introduce the parabolic distance
$\textrm{\rm pardist}((t,x),A) := \inf_{(s,y)\in A}
\sqrt{\vert x-y\vert^2 + \vert t-s\vert}$.
Considering
a function $\phi \in H^{1,2}_{\rm loc}({\bf R}^n;{\bf R}^n)$
we denote by $\textrm{\rm div }\phi := \sum_{i=1}^n \partial_i \phi_i$
the space
divergence and
by \[ D\phi := \left(\begin{array}{ccc}
\partial_1 \phi_1 & \dots & \partial_n \phi_1\\
& \dots & \\
\partial_1 \phi_n & \dots & \partial_n \phi_n\end{array}\right)\]
the matrix of the spatial partial derivatives.\\
Given a set $A\subset {\bf R}^n\> ,$ we denote its interior by $A^\circ$
and its characteristic function by $\chi_A\> .$
In the text we use the $n$-dimensional Lebesgue-measure
${\mathcal L}^n$ and
the $m$-dimensional Hausdorff measure
${\mathcal H}^m$.
When considering a given set $A\subset {\bf R}^n$,
let \[\partial_M A := \{x\in {\bf R}^n\> : \>
\limsup_{r\to 0} {{\mathcal L}^n(B_r(x)\cap A)\over
{{\mathcal L}^n(B_r)}} > 0 \textrm{ and }
\limsup_{r\to 0} {{\mathcal L}^n(B_r(x) - A)\over
{{\mathcal L}^n(B_r)}} > 0 \}\] be
the measure-theoretic boundary of $A$, let
$\partial^* A := \{x\in {\bf R}^n\> : \>
\textrm{ there is } \nu(x) \in \partial B_1(0)
\textrm{ such that }
r^{-n} \int_{B_r(x)} \vert \chi_A-\chi_{\{y:(y-x)\cdot \nu(x)<0\}}
\vert \to 0 \textrm{ as } r\to 0\}$
(by \cite[Corollary 5.6.8]{ziemer} $\partial^* A$ coincides
${\mathcal H}^{n-1}$-a.e. with the reduced boundary of a set
of finite perimeter defined in \cite[Definition 5.5.1]{ziemer}),
and let
$\nu:\partial^* A\to \partial B_1(0)$ denote this measure theoretic
outward normal to $\partial A$.
We shall often use
abbreviations for inverse images like $\{u>0\} :=
\{x\in \Omega\> : \> u(x)>0\}\> , \> \{x_n>0\} :=
\{x \in {\bf R}^n \> : \> x_n > 0\}\> , \>
\{s=t\} := \{(s,y)\in {\bf R}^{n+1}\> : \> s=t\}$ etc.
as well as $A(t) := A \cap \{s=t\}$ for a set $A\subset {\bf R}^{n+1}$,
and occasionally
we employ the decomposition $x=(x',x_n)$ of a vector $x\in {\bf R}^n$
as well as the corresponding decompositions of the gradient
and the Laplace operator,
$$ \nabla u = (\nabla' u,\partial_n u) \textrm{ and } \Delta u = \Delta' u\> + \> \partial_{nn} u\; .$$
\\
Finally, ${\bf C}^{\beta,\mu}:={\bf H}^{\mu,\beta}$
denotes the parabolic
H\"older-space defined in
\cite{lady}.
\section{Notion of solution and Preliminaries}\label{notion}
In this section we gather some results from \cite{calc}. As degenerate points are
unavoidable in the parabolic problem (see the introduction of \cite{calc}
for examples), an extension of the {\em weak solutions} in \cite{AC}
does not seem to be the right choice. Instead we use the solutions of
\cite[Definition 6.1]{calc}, which are, roughly speaking, solutions in the
sense of domain variations. The advantage is that the class of
solutions defined in \cite[Definition 6.1]{calc} is closed under the blow-up
process. Moreover, {\em all} limits of the singular perturbation problem
discussed in \cite{cava}
{\em are} domain variation solutions and
satisfy \cite[Definition 6.1]{calc} (see \cite[Section 6]{calc}).
Let us recall the definition of solutions
and the monotonicity formula used therein:
\begin{theorem}[Monotonicity Formula, \textrm{cf. \cite[Theorem 5.2]{calc}}]
\label{mon}
Let $(x_0,t_0)\in {\bf R}^n\times (0,\infty)\> , \>
T_r^-(t_0)$ $= {\bf R}^n \times (t_0 - 4r^2 , t_0 - r^2)\> ,$
$0<\rho<\sigma<{\sqrt{t_0}\over 2}$ and
\[ G_{(x_0,t_0)}(x,t) =
4\pi (t_0-t)\> {\vert 4\pi (t_0-t)\vert}^{-{n \over 2}-1}
\; \exp\left(-{\vert x-x_0\vert^2 \over
{4 (t_0-t)}}\right)\; \; .\]
Then
\[ \begin{array}{l}
\Psi_{(x_0,t_0)}(r)\;
=\;
r^{-2} \int_{T_r^-(t_0)} \left(
{\vert \nabla u \vert}^2 \> +\>
\chi
\right)\> G_{(x_0,t_0)}
- \; {1 \over 2} \> r^{-{2}}
\int_{T_r^-(t_0)}
{1 \over {t_0-t}} \> u^2 \> G_{(x_0,t_0)}
\end{array}\]
satisfies the monotonicity formula
\[ \Psi_{(x_0,t_0)}(\sigma)\> - \> \Psi_{(x_0,t_0)}(\rho)
\]\[ \ge \;
\int_\rho^\sigma r^{-1-{2}} \>
\int_{T_r^-(t_0)} {1 \over {t_0-t}}\> \Bigg(\nabla u
\cdot (x-x_0) \> - \>
2 (t_0-t) \partial_t u \> - \>
u \Bigg)^2 \> G_{(x_0,t_0)}\> dr\; \ge \; 0\; \; .
\]
\end{theorem}
\begin{definition}[\textrm{cf. \cite[Definition 6.1]{calc}}]\label{solution}
We call $(u,\chi)$ a solution in $\Omega_0 := {\bf R}^n\times (0,\infty)$
(in which case we set $\tau := 0$) or $\Omega_1 := {\bf R}^n\times (-\infty,\infty)$
(in which case we set $\tau := 1$), if:\\
1) $u \in {\bf C}^{1,{1\over 2}}_{\rm loc}(\Omega_\tau)\cap
C^2(\Omega_\tau\cap \{u>0\})\cap H^{1,2}_{\rm loc}(\Omega_\tau)$ and
$\chi\in L^1((-\tau R,R);BV(B_R(0)))$ for each $R\in (0,\infty)\> .$
For each $R\in (0,\infty)$ and $\delta\in (0,1)$ there exists
$C_1<\infty$ such that for $Q_r(x_0,t_0)\subset \Omega_\tau\cap Q_R(0)$
\[ \int_{Q_r(x_0,t_0)} \vert \nabla\chi\vert \> \le \> C_1 \> r^{n+1},\]
\[ \int_{Q_r(x_0,t_0)} \vert \partial_t u\vert^2 \> \le \> C_1 \> r^{n}, \textrm{ and}\]
\[ \int_{B_r(x_0)\times (t_0+S_1r^2,t_0+S_2r^2)} \vert\partial_t (\vert\nabla u\vert^2
\> + \> \chi)*\phi_{r\delta} \vert \; \le \; C_1 \sqrt{
S_2-S_1}\> r^n\]
for $0< S_1 < S_2 < \infty$;
here the mollifier $(\phi_\delta)_{\delta\in (0,1)}$ should be non-negative
and satisfy
$\phi_\delta(\cdot) = {1\over {\delta^n}}\phi({\cdot\over {\delta}}),
\phi\in C^{0,1}_0({\bf R}^n)\> ,$ $\int \phi = 1$ and
$\textrm{\rm\small supp } \phi \subset B_1(0)\> .$
\\
Moreover, $\chi\in \{0,1\}$ a.e. in $\Omega_\tau$ and
$\chi_{\{u>0\}} \le \chi$ a.e. in $\Omega_\tau\> .$\\
2) The solution $u$ satisfies the monotonicity formula Theorem \ref{mon}
(in the case of $\tau=1$ for $(x_0,t_0)\in {\bf R}^{n+1}$
and $\sigma\in (0,\infty)$).
\\
\[ \textrm{3) } 0 \; = \;
\int_{-\infty}^{\infty} \int_{{\scriptsize\bf R}^n} [-2\partial_t u \> \nabla u\cdot \xi
\> + \> (\vert\nabla u\vert^2 \> + \> \chi)\> \textrm{\rm div } \xi
\> - \> 2 \nabla u D\xi \nabla u]\]
for every $\xi\in C^{0,1}_0(\Omega_\tau;{\bf R}^n)\> .$
\\
4) The solution $u$ is non-negative.\\
5) The solution $u$ attains the initial data $u^0\in C^{0,1}_0({\bf R}^n)$
in $L^2_{\rm loc}({\bf R}^n)$ in the case that $\tau=0\> .$
\\
6) For each $\kappa>0$ there is $\delta>0$ such
that $Q_r(x_0,t_0)\subset \Omega_\tau$ and
$\Vert{u(x_0+rx,t_0+r^2 t)\over r}-\theta \vert x_n \vert\Vert_{
C^0(Q_1(0))} < \delta$ imply
$\theta < 1 + \kappa\> .$
\\
7) For $\delta\in (0,1)\> , \>
\psi_\delta \in C^{0,1}_0(\{ \vert y \vert^2 + s^2 < \delta^2
\})\> , \> u_r(y,s) := {u(t_0+r^2 s,x_0+ry)\over r}$ and
$\chi_r(y,s) := \chi(x_0+ry,t_0+r^2 s)$ the following holds:
\[ \textrm{a) }
\; \int_{Q_\rho(x_1,t_1)} \vert (\nabla\chi_r
\cdot x \> + \> 2t\partial_t \chi_r)*\psi_\delta\vert
\]\[ \le \; C(\delta,Z,T,S,\rho)
\left(\Psi_{(x_0,t_0)}(r
\sqrt{{-t_1+\delta+\rho^2\over 2}})
\> - \> \Psi_{(x_0,t_0)}(r
\sqrt{{-t_1-\delta-\rho^2\over 2}})\right)\]
for $-S\le t_1 \le -T<0\> , \> \delta+\rho^2\le {T\over 2}\> , \>
\vert x_1 \vert \le Z$
and, in the case of $\tau = 0\> ,\> t_0-2r^2(-t_1+\rho^2+\delta)>0\> .$
\[ \textrm{b) }
\; \int_{Q_\rho(t_1,x_1)} \vert (\nabla \chi_r
\cdot \xi) * \psi_\delta\vert
\; \le \; C(\delta) \int_{Q_{\sqrt{\delta}+\rho}(t_1,x_1)}
\vert \nabla u_{r}\cdot \xi\vert \]
for $\xi \in \partial B_1(0)\> , \> t_1<0$ and, in the case of $\tau = 0
\> , \> t_0-r^2(-t_1+(\rho+\sqrt{\delta})^2)>0\> .$
\[ \textrm{c) }
\; \int_{t_1}^{t_2} \partial_t((\vert \nabla u_r\vert^2
+\chi_r)
*\phi_\delta)(t,x_0)
\; \le \; \int_{t_1}^{t_2} \int_{{\scriptsize\bf R}} 2\partial_t u_{r}(t,z)
\nabla u_{r}(t,z)\cdot\nabla \phi_\delta(x_0-z)\> dz\]
for $-\infty<t_1<t_2<\infty$
and, in the case of $\tau = 0\> , \> t_0+r^2t_1>0\> .$
\end{definition}
\begin{remark}
As the function $\chi$ is defined only almost everywhere, all
pointwise equalities/inequalities involving $\chi$ should
be understood as equalities/inequalities that hold almost everywhere
with respect to the Lebesgue measure.
\end{remark}
The reader may wonder whether a solution in the sense of distributions
(possibly defined by the identity in \cite[Lemma 11.3]{calc})
would not be good enough for the purposes of this paper.
It turns however out that the information yielded by the order parameter $\chi$ in Definition \ref{solution}
carries information that is essential in what follows.
Incidentally, $\chi$ may be different from $\chi_{\{ u>0\}}$ (see \cite[Remark 4.1]{calc}).
\begin{lemma}\label{prem}
Let $(u,\chi)$ be a solution in the sense of Definition \ref{solution} and suppose that
for some $(x_0,t_0)$ in the set of definition and for
some sequence $r_m\to 0,m\to \infty$
\[ u_{r_m}(y,s) := {u(x_0+{r_m}y,t_0+{r_m}^2 s)\over {r_m}} \to 0
\textrm{ locally in } \{ y_n<0\}\times (-\infty,0) \textrm{ as } m\to\infty\]
and
\[ \chi_{r_m}(y,s) := \chi(x_0+{r_m}y,t_0+{r_m}^2 s)
\to 0 \textrm{ a.e. in } \{ y_n >0\}\times(-\infty,0)\textrm{ as } m\to\infty\; .\]
Then for some $\delta>0$, $u$ is caloric in
$Q_\delta(x_0,t_0)$ and satisfies
\[ u=0 \textrm{ in } Q^-_\delta(x_0,t_0)\; .\]
\end{lemma}
\proof
The assumptions imply
by Definition \ref{solution} 1) that
\[ u_{r_m}
\to 0 \textrm{ a.e. in } \{ x_n >0\}\times(-\infty,0)\textrm{ as } m\to\infty\; .\]
Moreover, they imply by \cite[Proposition 10.1 2)]{calc} that the density
\[ \Psi_{(x_0,t_0)}(0+) \in \{ 0 \} \cup \{ H_n \}\; ,\]
where $H_n$ is the energy of the half-plane solution
defined in \cite[Section 10]{calc}.
In the case
\[ \Psi_{(x_0,t_0)}(0+)=0\]
we obtain from \cite[Proposition 10.1 2)]{calc} immediately
the statement of the lemma.\\
In the case \[ \Psi_{(x_0,t_0)}(0+)=H_n\]
it follows from \cite[Proposition 10.1 1)]{calc} that
the limit of $u_{r_m}(y,s)$ as $m\to \infty$ must after rotation be the half-plane
solution $\max(-x_n,0)$, a contradiction to the limit of $u_{r_m}$ being $0$.
\qed
\section{Flatness Classes}
\begin{definition}\label{flatness}
Let $0<\sigma_+, \sigma_- < 1$ and $\tau\ge 0$. We say that
\begin{displaymath}
u\in F(\sigma_+,\sigma_-,\tau) \quad \textrm{in } Q_{\rho}\quad \textrm{in direction }e_n
\end{displaymath}
if
\begin{enumerate}
\item $(u,\chi)$ is a solution in the sense of Definition \ref{solution} in a domain containing $Q_{\rho}$.
\item
\begin{displaymath}
(0,\rho^2) \in \partial \{u>0\},
\end{displaymath}
\begin{displaymath}
u(x,t)=\chi(x,t)=0 \textrm{ when } (x,t)\in Q_\rho \textrm{ and }x_n\ge \sigma_+ \rho,
\end{displaymath}
\begin{displaymath}
\chi(x,t)=1 \textrm{ and }
u(x,t)\ge -(x_n+\sigma_- \rho) \textrm{ when } (x,t)\in Q_\rho \textrm{ and }x_n\le -\sigma_- \rho\; .
\end{displaymath}
\item
\begin{displaymath}
|\nabla u|\le 1+\tau \textrm{ in } Q_{\rho}.
\end{displaymath}
\end{enumerate}
When the origin is replaced by $(x_0,t_0)$ and the flatness direction
$e_n$ is replaced by $\nu$ then we define $u$ to belong to the flatness
class $F(\sigma_+,\sigma_-,\tau) $ in $Q_{\rho}(x_0,t_0)$ in direction $\nu$.
\end{definition}
\section{Flatness on the side of $\{ \chi=0\}$ implies
flatness on the side of $\{ \chi>0\}$}
The aim of this and the following sections is to draw information from
properties of an inhomogeneous blow-up limit. One of the central problems when
using blow-up arguments is ``{\em not-strong convergence}'' or
``{\em energy loss}'' in the limit. Here we avoid
those problems by working with {\em uniform convergence} (not
some Sobolev norm). The approach is based on a powerful idea by H.W. Alt-L.A. Caffarelli
who used ``flatness on the side of $\{ u=0\}$ implies
flatness on the side of $\{ u>0\}$'' to prove uniform convergence
to an inhomogeneous blow-up limit (cf \cite[Section 7]{AC}). In this section we extend their
result to a
weaker class of solutions and to
the parabolic case, using results in \cite{calc}.\\
The following Lemma is the parabolic version of \cite[Lemma 4.10]{AC}.
\begin{lemma}\label{lem:ball}
Let $(u,\chi)$ be a solution in the sense of Definition \ref{solution} in a domain containing
the closure of a non-empty open ball $B = \{ (y,s): |(y,s)-(y_0,s_0)| < c\}$
such that
$B\subset \{ \chi=0\}$ and
$B$ touches the set $\{ u>0\}$ at the origin.\\
Then
\begin{displaymath}
\limsup_{\{ u>0\}\ni (x,t)\rightarrow 0}\frac{u(x,t)}{\textrm{\rm pardist}((x,t),B)} = 1.
\end{displaymath}
\end{lemma}
\proof
Let $Y_k=(y_k,s_k)\in {\bf R}^{n+1}$ be a sequence such that
\begin{displaymath}
\ell=\limsup_{\{ u>0\}\ni (x,t)\rightarrow 0}\frac{u(x,t)}{\textrm{\rm pardist}((x,t),B)}=
\lim_{k\rightarrow \infty}\frac{u(Y_k)}{\textrm{\rm pardist}(Y_k,B)}.
\end{displaymath}
Set $d_k :=\textrm{\rm pardist}(Y_k,B)$ and let $(x_k,t_k)=X_k\in \partial B$ be such that
$\textrm{\rm pardist}(Y_k,X_k)=d_k$.
\\
We consider the blow-up sequence
\begin{displaymath}
u_k(x,t)=\frac{u(d_kx+x_k,d_k^2t+t_k)}{d_k}, \chi_k(x,t)=\chi(d_kx+x_k,d_k^2t+t_k).
\end{displaymath}
We know that, passing to a subsequence if necessary,
$u_k\rightarrow u_0$ locally uniformly in ${\bf R}^{n+1}$
and $\chi_k\rightharpoonup \chi_0$ weakly-* in $L^\infty_{\rm loc}({\bf R}^{n+1})$ as $k\to\infty$.
Also, after
a rotation and translation, the scaled
$B$ converges to $\{ x_n > 0 \}$ and $\left((y_k-x_k)/d_k, (s_k-t_k)/d_k^2\right)\rightarrow (\xi,\tau)
\in \partial Q_1(0)$ as $k\to \infty$.
The limit function $u_0$ satisfies
\begin{displaymath}
\begin{array}{ll}
\Delta u_0 -\partial_t u_0= 0 & \textrm{in } {\bf R}^{n+1}\cap
\{ u_0>0 \} \; ,\\
u_0(\xi,\tau)=\ell & \textrm{ and}\\
u_0(x,t)=0 & \textrm{in } \{x_n > 0\}.
\end{array}
\end{displaymath}
By the definition of the limit superior we know also that
\begin{displaymath}
u_0(x,t)\le -\ell x_n \quad \textrm{ in } \{x_n<0\}.
\end{displaymath}
The strong maximum principle (applied to $u_0(x,t)+\ell x_n$) tells us therefore
that $u_0(x,t)=\ell \max(-x_n,0)$ for $t<\tau$. We have to
show that $\ell = 1$.
\\
In the case $\ell>0$ we obtain from
the fact that $(u,\chi)$ is a solution in the sense of
Definition \ref{solution}, that $\chi_0 =1$ in
$\{ x_n<0\}\cap \{ t<\tau\}$. Furthermore, we infer from the assumption that
$\chi_0 =0$ in $\{ x_n>0\}$. But then
$(u_0(\cdot,t+\tau),\chi_0(\cdot,t+\tau))$ is in $\{ t<0\}$ a solution in the sense of
Definition \ref{solution} whose
energy
$M(u_0(\cdot,t+\tau),\chi_0(\cdot,t+\tau))=H_n$ (cf. \cite[Section 10]{calc}), whence \cite[Proposition 10.1]{calc}
implies that $\ell=1$.
\\
In the case $\ell=0$ we apply Lemma \ref{prem} to obtain
for some $\delta>0$ that $u$
is caloric in $ Q_\delta$ and satisfies
\[ u=0 \textrm{ in } Q^-_\delta\; .\]
As $\{ u= 0\}$ contains $B$, $u$ being caloric in $Q_\delta$
and therefore analytic with respect to the space variables implies
\[ u=0 \textrm{ in } Q_{\delta_1}\]
for some $\delta_1>0$.
This is a contradiction in view of the origin being a free boundary point.
\qed
\\
The following theorem extends \cite[Lemma 7.2]{AC}.
\begin{theorem} \label{lem:flatabove}
There exists a constant $C\in(0,+\infty)$
depending only on the space dimension $n$
such that if
$u\in F(\sigma,1,\sigma)$ in $Q_{\rho}$ then $u\in F(C\sigma,C\sigma,\sigma)$
in $Q_{\rho/2}(0,y_n,0))$ for some $|y_n|\le C\sigma$.
\end{theorem}
\proof
The idea is to touch the boundary $\partial \{ \chi=0\}$ with the graph of a $C^2$-function,
to apply Lemma \ref{lem:ball} and to proceed then with a Harnack inequality
argument.\\
{\bf Step 1 (Touching $\partial \{ \chi=0\}$ with a smooth surface):}
\\
Rescaling
$u_{\rho}(x,t) := {u({\rho}x,{\rho}^2 t)\over {\rho}}, \chi_{\rho}(x,t) := \chi({\rho}x,{\rho}^2 t)$
we may assume that $\rho=1$.
Let
\begin{displaymath}
\eta(x',t)=\left\{\begin{array}{ll}
\exp(\frac{16(|x'|^2+|t-1|)}{1-16(|x'|^2+|t-1|)}),& |x'|^2+|t-1|<1/16,\\
0,& \textrm{ else}\end{array}\right.
\end{displaymath}
and let $s$ be the largest constant such that
\begin{displaymath}
Q_1\cap \{ u>0\} \subset \{ (x,t)\in Q_1:\; x_n<\sigma-s\eta(x',t) \}=: D.
\end{displaymath}
This implies that there exists a point
$(x_0,t_0):=Z\in \partial D\cap \partial \{ u>0\}\cap \{ t\ge 15/16\}$.
As $(0,1)$ is a free boundary point,
we know furthermore that $s\le \sigma$.
\\
Let us also define the barrier function $v$ by
\begin{displaymath}
\begin{array}{ll}
\Delta v - \partial_t v =0 & \textrm{ in } D \; ,\\
v=0 & \textrm{ on } \partial D \cap Q_1 \textrm{ and}\\
v=2\sigma -x_n & \textrm{ on } \partial D\cap\partial'Q_1.
\end{array}
\end{displaymath}
Note that this implies that $-\sigma\le v+x_n\le 2\sigma$.
\\
Since $|\nabla u|\le 1+\sigma$ we also obtain that $v\ge u$ on $\partial D$
and thus, by the maximum principle, that $v\ge u$ in $D$. As $\eta$
is a $C^2$-function, the assumptions of Lemma \ref{lem:ball} are satisfied at $Z$.
Therefore
\begin{equation}\label{eq:norest}
1\le
\limsup_{(x,t)\rightarrow Z}\frac{u(x,t)}{\textrm{\rm pardist}((x,t),B)}\le
-\partial_\nu v(Z),
\end{equation}
where $\nu$ is the outward space normal to $\partial D$ at $Z$.
In order to obtain an estimate from above we define
\begin{displaymath}
F(x,t)=2\sigma-x_n-v(x,t).
\end{displaymath}
$F$ is caloric in $D$ and satisfies $0\le F\le \sigma$. Since $D$ is a regular parabolic
domain, we know from standard regularity theory for
parabolic equations that $\sup_D |\nabla F|\le C_1\sigma$. Therefore
\begin{displaymath}
-\partial_n v(Z)=1+\partial_n F(Z)\le
1+C_1\sigma.
\end{displaymath}
By the flatness assumption
we know that $\nu$ is close to $e_n$. More precisely,
\begin{displaymath}
|\nu-e_n|=\big|
\frac{(-s\nabla \eta,1-\sqrt{s^2|\nabla \eta|^2+1})}{\sqrt{s^2|\nabla \eta|^2+1}}\big|
\le \sqrt{10} |\nabla \eta| s.
\end{displaymath}
Thus
\begin{displaymath}
-\partial_\nu v(Z)=-\nabla v(Z)\cdot(\nu-e_n) -
\partial_n v(Z) \le 1+C_1\sigma +\sqrt{10} |\nabla \eta| |\nabla v(Z)| s \le 1+ C_2 \sigma\; .
\end{displaymath}
From inequality (\ref{eq:norest}) we infer that
\begin{equation}\label{eq:bar}
1\le -\partial_\nu v(Z) \le 1+C_2 \sigma.
\end{equation}
\\
{\bf Step 2 (Harnack inequality argument):}\\
As we know already that $v$ is $\sigma$-close to $-x_n$, it is
sufficient to show that
$u$ is $\sigma$-close to $v$ on the set
$\{(x,t):\; x_n=-3/4,\; |x'|\le 1/2,\; t\le 3/4\}$. Once this is done,
we may integrate $u$
in the $x_n$-direction to establish the lemma.
\\
In order to prove the $\sigma$-closeness
we define
for $\xi=(\gamma,\tau)$, $\tau\in (-1,3/4)$, $|\gamma'|\le 1/2$ and
$\gamma_n=-3/4$
the function $\omega_\xi$ by
\begin{displaymath}
\begin{array}{ll}
\Delta \omega_\xi - \partial_t \omega_\xi= 0 &
\textrm{ in } D\cap \{t>\tau\} \\
\omega_\xi = -x_n & \textrm{ on } B_{1/8}(\gamma)\times \{t=\tau\} \\
\omega_\xi = 0 & \textrm{ on the remainder of the parabolic boundary of }
D\cap \{t>\tau\}.
\end{array}
\end{displaymath}
By the Hopf lemma we have
\begin{displaymath}
\partial_\nu \omega_\xi(Z)\le -\alpha <0
\end{displaymath}
uniformly in $\xi$.
\\
We would like to show that $u\ge v-C_4\sigma x_n$. The trick is to compare $u$
to $v-K\sigma \omega_\xi$ on the set $B_{1/8}(\gamma)\times \{t=\tau\}$ and to use
the information on the normal derivative of $u$ at $Z$ to prove that if $K$
is large, then $u>v-K\sigma \omega_\xi$ for at least one point in
$B_{1/8}(\gamma)\times \{t=\tau\}$. More precisely:
\\
Assume that $u\le v-K\sigma \omega_\xi$ in
$B_{1/8}(\gamma)\times \{t=\tau\}$. Then
$u\le v-K\sigma \omega_\xi$ in $D\cap \{t>\tau\}$. Consequently, we obtain from inequalities
(\ref{eq:norest}) and
(\ref{eq:bar}) that
\begin{displaymath}
1\le -\partial_\nu v(Z)+
K\sigma\partial_\nu \omega_\xi(Z)\le 1+
C_2\sigma-K\alpha\sigma.
\end{displaymath}
This yields a contradiction when $K$ is large enough, say $K = 2C_2/\alpha$. Thus
$u(X_\xi)>v(X_\xi)-K\sigma\omega_\xi(X_\xi)$ for at least one point
$X_\xi\in B_{1/8}(\gamma)\times \{t=\tau\}$.
\\
On the other hand, $v-u\ge 0$. Therefore we can apply the Harnack inequality and deduce
that
\begin{displaymath}
(v-u)(\tilde\xi)\le C_3\inf_{Q_{1/8}(\tilde\xi+(0,1/32))}(v-u)\le C_4\sigma,
\end{displaymath}
for every $\tilde\xi\in \{(x',-3/4,t):\; |x'|<1/2,\; -1\le t\le 1/2 \}$.
\\
This implies that $u(x',-3/4,t)\ge 3/4 - C_5\sigma$ in the above region.
Integrating in the $e_n$ direction and using the assumption $|\nabla u|\le 1+\sigma$
yields the estimate
\begin{displaymath}
u\ge -(x_n+C_6\sigma) \textrm{ in } \{ -3/4 \le x_n \le -\sigma\} \times Q'_{1/2}
\end{displaymath}
By our
initial assumption we also know that $u=0$ in $\{3/4 \ge x_n\ge \sigma\}\cap Q'_{1/2}$.
Translating $(u,\chi)$
in the $e_n$ direction so that the point $(0,1/4)\in \partial\{ u>0\}$
and using $\chi\ge \chi_{\{ u>0\}}$ of Definition \ref{solution} 1)
we obtain the statement of our
theorem.
\qed
\section{Inhomogeneous Blow-up}
In this section we consider inhomogeneous scaling of the solution
and the free boundary. The following lemma is our version
of \cite[Lemma 7.3]{AC}
\begin{lemma}\label{lem:nonhomogeneousblowup}
Suppose that $u_k \in F(\sigma_k,\sigma_k,\tau_k)$ in $Q_{\rho_k}$,
that $\sigma_k\rightarrow 0$ and that $\tau_k/\sigma_k^2\rightarrow 0$, and
define
\begin{displaymath}
f_k^+(x',t):=\sup\{h:\; \limsup_{r\to 0}r^{-n-2}\int_{Q_r(\rho_k x', \sigma_k\rho_k h, \rho_k^2 t)}\chi > 0\},
\end{displaymath}
\begin{displaymath}
f_k^-(x',t):=\inf\{h:\; \limsup_{r\to 0}r^{-n-2}\int_{Q_r(\rho_k x', \sigma_k\rho_k h, \rho_k^2 t)} \chi > 0\}.
\end{displaymath}
Then, as a subsequence $k\to\infty$,
$f_k^+$ and $f_k^-$ converge in $L^\infty_{\rm loc}(Q'_1)$ to some function $f$,
and $f$ is continuous in $Q'_1$.
\end{lemma}
\proof
Rescaling as before we may assume that $\rho_k=1$. Let
\[ D_k:=\{ (y',h,t):\; \limsup_{r\to 0}r^{-n-2}\int_{Q_r(y',\sigma_k h, t)} \chi > 0 \}\; .\]
We may assume -- passing if necessary to a subsequence --
that $D_k$ converges with respect to the usual (not the parabolic) Hausdorff distance
as $k\to\infty$.
Let us define \begin{displaymath}
f(x',t):=
\limsup_{(y',s)\rightarrow (x',t), k\rightarrow \infty} f_k^+(y',s),
\end{displaymath}
where we take the limit superior with respect to the above subsequence.
For every $(y_0',t_0)$ there exists then a sequence
$(y'_k,t_k)\rightarrow (y'_0,t_0)$ such that
$f^+_k(y_k',t_k)\rightarrow f(y'_0,t_0)$ as $k\to\infty$.
By definition $f$ is upper semi-continuous. Therefore we obtain for
$\varepsilon>0$ and sufficiently large $k$ that
\begin{displaymath}
\big( \overline{Q_{\varepsilon}'(y_k',t_k)}\times [f_k^+(y'_k,t_k)+\delta,\infty)\big) \cap \bar{D}_k=\emptyset.
\end{displaymath}
\\
Consequently $u_k\in F(\sigma_k\frac{\delta}{\varepsilon},1,\tau_k)$
in $Q_{\varepsilon}(y_k,\sigma_k f^+_k(y'_k,t_k),t_k)$. Applying Theorem
\ref{lem:flatabove} to $u_k$ we deduce that
\[u_k(x,t) \ge -(x_n +C\sigma_k\delta/2) \textrm{ for } (x,t)\in Q_{\varepsilon/2}(y_k',\sigma_k f^+_k(y'_k,t_k),t_k)\; .\]
In terms of $f^+_k$ and $f^-_k$ this yields
$f^-_k(y',t)\ge f^+_k(y'_k,t_k)-C\delta$ in $Q'_{\varepsilon/4}(y_k',t_k)$.
It follows that
$\lim_{k\to\infty}f^-_k(y',t)=f(y',t)$, that
$f^+_k$ and $f^-_k$ converge locally uniformly and that
$f$ is continuous.
\qed
\\
The next Proposition follows the lines of \cite[Lemma 5.7]{ACF}.
\begin{proposition}\label{lem:wcal} Suppose that the assumptions
of Lemma \ref{lem:nonhomogeneousblowup} are satisfied and that $k$ is the
subsequence of Lemma \ref{lem:nonhomogeneousblowup}.
Then
\begin{displaymath}
w_k(x',h,t)=\frac{u_k(\rho_k x',\rho_k h,\rho_k^2 t)+\rho_k h}{\sigma_k}
\end{displaymath}
is for each $\delta\in (0,1)$ bounded in
$Q_{1-\delta}\cap\{ x_n< 0\}$ (by a constant depending only on $\delta$ and $n$)
and
converges on compact subsets of $Q_1^-$ in $C^2$ to a caloric function $w$.\\
Moreover,
$w(x',h,t)$ is non-decreasing in the $h$-variable in
$Q_1^-$ and
\[ \lim_{Q_1^- \ni (y,s)\to (x',0,t)\in Q'_1,k\to\infty} w_k(y,s)=f(x',t)\> ;\]
here $f$ is the function defined
in Lemma \ref{lem:nonhomogeneousblowup}.
\end{proposition}
\proof
Rescaling as before we may assume that $\rho_k=1$.\\
The function $w_k$ is caloric in
$Q_1\cap \{h<-\sigma_k\}$.
Using Definition \ref{flatness} 3), we obtain that
\[ u_k\le -x_n +2\sigma_k \textrm{ in } Q_1\cap \{x_n \le 0\}\; ,\]
implying that $w_k\le 2$. From Theorem \ref{lem:flatabove}
and
Definition \ref{flatness} 3)
we infer
furthermore that
$u_k(x,t)\ge -(x_n + C_\delta\sigma_k)$ for
$(x',x_n,t)\in Q_{1-\delta}\cap \{ x_n\le 0\}$,
implying that $w_k \ge -C_\delta$ in $Q_{1-\delta}\cap \{ x_n\le 0\}$.
\\
By Definition \ref{flatness} 3) and the assumptions,
$|\nabla u_k|\le 1+o(\sigma_k^2)$. Consequently,
\begin{equation}\label{eq:hder}
-\partial_h w_k\le\frac{|\nabla u_k|-1}{\sigma_k}\le
\frac{\tau_k}{\sigma_k}\rightarrow 0\textrm{ as } k\to\infty\; .
\end{equation}
In the remainder of the proof we will show that $w$ attains the boundary data
$f$ as $h\to 0$.
First, we show that for fixed $L\in (1,+\infty)$
\begin{equation}\label{eq:interi}
w_k(x',\sigma_k h,t)-f_k^+(x',t)\rightarrow 0 \quad
\textrm{ uniformly in } Q'_{1-\delta}\times \{ -L\le h<0 \}
\end{equation}
as $k\to\infty$.
An estimate from above can be obtained easily from
inequality (\ref{eq:hder}):
\begin{displaymath}
w_k(x',h\sigma_k,t)-f^+_k(x',t)\le w_k(x',\sigma_k f^+_k(x',t),t)-f^+_k(x',t)+
(f^+_k(x',t)-h)\frac{\tau_k}{\sigma_k}
\end{displaymath}
\begin{displaymath}
\le (1+L)\frac{\tau_k}{\sigma_k}
\rightarrow 0 \textrm{ as } k\to\infty \; .
\end{displaymath}
This establishes an estimate from above. In order to derive an estimate
from below we use Theorem \ref{lem:flatabove}:
Consider a sequence of points $(x'_k,t_k)\in Q_{1-\delta}'$ and fixed $S\in (4,+\infty)$.
Then
\begin{displaymath}
u_k\in F(\tilde{\sigma}_k,1,\tau_k)
\textrm{ in } Q_{S\sigma_k}(x'_k,\sigma_k f_k^+(x'_k,t_k),t_k)
\end{displaymath}
for
\begin{displaymath}
\tilde{\sigma}_k=
\frac{1}{S}\sup_{(x',t)\in Q_{S\sigma_k}'}(f^+_k(x',t)-f^+_k(x'_k,t_k)).
\end{displaymath}
From the uniform convergence of $f^+_k$ to the continuous function
$f$, we infer that
$\tilde{\sigma}_k\rightarrow 0$ as $k\to\infty$. Now by Theorem \ref{lem:flatabove},
\begin{displaymath}
u_k\in F(C\bar{\sigma}_k,C\bar{\sigma}_k,\tau_k) \quad
\textrm{ in } Q_{S\sigma_k/2}(x'_k,\sigma_k f_k^+(x'_k,t_k)+CS\bar\sigma_k\theta/2,t_k),
\end{displaymath}
where $\bar{\sigma}_k=\max(\tilde{\sigma}_k, \tau_k)$ and $\theta \in [0,1]$.
\\
Thus for $h\in (\max(-L,-S/4),0)$
\begin{displaymath}
u_k(x_k+h\sigma_ke_n,t_k)\ge
-\sigma_k \left( h-f_k^+(x'_k,t_k)+C\bar\sigma_kS\right).
\end{displaymath}
Consequently
\begin{displaymath}
w_k(x_k+h\sigma_ke_n,t_k)=
\frac{u_k(x_k+h\sigma_ke_n,t_k)+h\sigma_k}{\sigma_k}\ge
f_k^+(x'_k,t_k)-C\bar{\sigma}_kS\; ,
\end{displaymath}
and (\ref{eq:interi}) holds.
\\
To establish $\lim_{Q_1^- \ni (y,s)\to (x',0,t)\in Q'_1,k\to\infty} w_k(y,s)=f(x',t)$, we need to extend the
convergence (\ref{eq:interi}) to larger values of $h$. To this end,
we define the barrier function $z_\varepsilon$ by
\begin{displaymath}
\begin{array}{ll}
\Delta z_\varepsilon -\partial_t z_\varepsilon = 0 &
\textrm{ in } Q^-_{1-\delta} \; ,\\
z_\varepsilon =g_\varepsilon & \textrm{ on } \partial' Q_{1-\delta}\cap \{ h=0 \}\; ,\\
z_\varepsilon = \inf_k \inf_{Q_{1-\delta}^-} w_k & \textrm{ on }
\partial' Q_{1-\delta}\cap \{ h < 0 \}\; ,
\end{array}
\end{displaymath}
where $g_\varepsilon \in C^{\infty}$ and
$f-2\varepsilon\le g_\varepsilon\le f-\varepsilon$. By (\ref{eq:interi}) we know
that $w_k\ge z_\varepsilon$ on $\partial'(Q_{1-\delta}\cap \{h\le -L\sigma_k\})$.
From the comparison principle it follows that $w_k\ge z_\varepsilon$ in
$Q_{1-\delta}^-\cap \{h\le -L\sigma_k\}$. Thus, by local boundary regularity for solutions of
the heat equation,
$\liminf_{Q_{1-2\delta}^- \ni (y,s)\to (x',0,t),k\to\infty} w_k(y,s)\ge g_\varepsilon(x',t)\ge f(x',t)-2\varepsilon$.\\
The opposite inequality follows from a similar argument, comparing $w_k$
to the upper barrier $\tilde z$ defined by
\begin{displaymath}
\begin{array}{ll}
\Delta \tilde z_\varepsilon -\partial_t \tilde z_\varepsilon = 0 &
\textrm{ in } Q^-_{1-\delta} \; ,\\
\tilde z_\varepsilon =\tilde g_\varepsilon & \textrm{ on } \partial' Q_{1-\delta}\cap \{ h=0 \}\; ,\\
\tilde z_\varepsilon = \sup_k \sup_{Q_{1-\delta}^-} w_k & \textrm{ on }
\partial' Q_{1-\delta}\cap \{ h < 0 \}\; ,
\end{array}
\end{displaymath}
where $\tilde g_\varepsilon \in C^{\infty}$ and
$f+2\varepsilon\ge \tilde g_\varepsilon\ge f+\varepsilon$.
\qed
\section{Scaling discrepancy and $C^\infty$-regularity of blow-up limits}\label{discrep}
In order to obtain ``better-than-Lipschitz''-regularity of the inhomogeneous blow-up limit $f$,
H.W. Alt-L.A. Caffarelli used the non-positive
mean curvature of $\partial\{ u>0\}$ at singularities.
The analogue of the non-positive
mean curvature property can still be proved in the time-dependent case,
however that path leads to problems in the sequel. Therefore
we replace it by a scaling discrepancy argument which gives hope to be applicable
in more general situations. We obtain $C^\infty$-regularity of $f$.
\begin{proposition}\label{discr}
Suppose that the assumptions
of Lemma \ref{lem:nonhomogeneousblowup} are satisfied and that $k$ is the
subsequence of Lemma \ref{lem:nonhomogeneousblowup}. Then
$\partial_n w=0$ on $Q'_{1/2}$ in the sense of distributions.
\end{proposition}
\proof
Rescaling as before we may assume that $\rho_k=1$.\\
In what follows, $g(x',t)=8(|x'|^2+ |t|)-4$. Note that $f\ge g$ in $Q'_{1/2}$.
Let us introduce the following notation:
$Z$ shall be the set $\{ (x',x_n,t): (x',t)\in Q'_1,x_n\in {\bf R}\}$.
Given a function $\phi: Q'_1\to {\bf R}$, we divide $Z$ into the three parts
\begin{displaymath}
Z^+(\phi)=\{ (x,t)\in Z: \; x_n> \phi(x',t) \},
\end{displaymath}
\begin{displaymath}
Z^-(\phi)=\{ (x,t)\in Z: \; x_n< \phi(x',t) \},
\end{displaymath}
\begin{displaymath}
Z^0(\phi)=\{ (x,t)\in Z: \; x_n = \phi(x',t) \}.
\end{displaymath}
Moreover let $\mu$ be defined by $\mu(A) := \int_{-\infty}^\infty {\mathcal H}^{n-1}(A \cap \{ s=t \})\> dt$
for any Borel set $A\subset {\bf R}^{n+1}$.
Adding an arbitrarily small constant to the function $g$, we may assume
that $\mu(Z^0(\sigma_k g)\cap R_k)=0$
for all $k$; here $R_k$ is the regular part of the
free boundary $\partial\{ u_k>0\}$ introduced in \cite[Proposition 9.1]{calc}, i.e.
$$R_k(t) :=\{ x\in \partial\{u_k(t)>0\}\> : \>
\textrm{there is } \nu_{R_k}(x,t)\in \partial B_1(0) \textrm{ such that }
v_r(y,s) = $$ $${u_k(x+ry,t+r^2s)\over r} \> \to \>
\max(-y\cdot \nu_{R_k}(x,t),0) \textrm{ locally uniformly in } (y,s)\in
{\bf R}^{n+1} $$ $$\textrm{ as } r \to 0\}\; .$$
Last, we define $E_k := \{ u_k >0\}\cap Z^-(\sigma_k g)$
and $\Sigma_k := \{ (x',t) : (x',\sigma_k g(x',t),t)\in \{ u_k>0\}\cap Z\}$.
By the choice of $g$ we know that the limit inferior of the sets $\Sigma_k$
contains $Q'_{1/2}$.
\\
We will deduce the result from the following three claims.\\
\textsl{Claim 1:}
\begin{displaymath}
\mu(Z^+(\sigma_k g)\cap {R_k})\le -\int_{{\Sigma_k}} (\partial_n u_k+1)dx'dt+
{\mathcal L}^n({\Sigma_k})+C_1\sigma_k^2.
\end{displaymath}
\textsl{Claim 2:}
$$
{\mathcal L}^n({\Sigma_k})-C_2\sigma_k^2 \le
\mu(Z^+(\sigma_k g)\cap R_k).
$$
\textsl{Claim 3:}
$$
\int_{{\Sigma_k}}|\partial_n w_k(x',\sigma_k g(x',t),t)| \; \to \; 0 \textrm{ as } k\to\infty\; .
$$
\textsl{Proof of Claim 1:} By the representation theorem
\cite[Lemma 11.3]{calc}
we know that for non-negative $\phi\in C^{\infty}_0$,
\begin{equation}\label{eq:claim1}
\int_{-\infty}^{\infty}\int_{{R_k}(t)}\phi \> d{\mathcal H}^{n-1}\> dt \le
-\int_{\{ u_k>0\}}\left(\nabla u_k \cdot \nabla \phi +
\partial_t u_k\phi \right)\> dx\> dt\; .
\end{equation}
Letting $\phi\rightarrow \chi_{Z^+(\sigma_k g)}\chi_{Q_2}$ the inequality (\ref{eq:claim1})
becomes
\begin{equation}\label{eq:troll}
\mu(Z^+(\sigma_k g)\cap {R_k})=
\int_{-\infty}^{\infty}\int_{{R_k}(t)\cap Z^+(\sigma_k g)} d{\mathcal H}^{n-1}\> dt
\end{equation}
\begin{displaymath}
\le \;
\int_{\{ u_k>0\}\cap Z^0(\sigma_k g)}\nabla u_k \cdot \nu \> dx\> dt -
\int_{\{ u_k>0\}\cap Z^+(\sigma_k g)}\partial_t u_k \> dx\> dt\; ,
\end{displaymath}
where $\nu$ is the outward unit space normal on $\partial Z^+(\sigma_k g)$.
Since
$$
\nu=\frac{1}{\sqrt{1+|\sigma_k \nabla' g|^2}}(\sigma_k\nabla' g,-1)\; ,
$$
we obtain
$$\mu(Z^+(\sigma_k g)\cap {R_k})\;
\le\;
\int_{{\Sigma_k}}(\nabla u_k)(x',\sigma_k g(x',t),t) \cdot (\sigma_k \nabla' g(x',t), -1) \> dx\> dt$$ $$-\>
\int_{\{ u_k>0\}\cap Z^+(\sigma_k g)}\partial_t u_k \> dx\> dt\; .
$$
Let us rewrite the integral
$$
\int_{{\Sigma_k}}(\nabla u_k)(x',\sigma_k g(x',t),t) \cdot (\sigma_k \nabla' g(x',t), -1) \> dx\> dt
$$
$$ = \;
\int_{{\Sigma_k}}\sigma_k(\nabla' u_k)(x',\sigma_k g(x',t),t) \cdot \nabla'g(x',t) -
(\partial_n u_k(x',\sigma_k g(x',t),t)+1)\> dx'\> dt \; + \; {\mathcal L}^n({\Sigma_k})
$$
$$
= \int_{{\Sigma_k}} -\sigma_k u_k(x',\sigma_k g(x',t),t) \Delta' g(x',t)
\> - \>{\sigma_k}^2 \partial_n u_k(x',\sigma_k g(x',t),t) |\nabla' g(x',t)|^2
$$
$$
- \> (\partial_n u_k(x',\sigma_k g(x',t),t)+1)\> dx'\> dt + {\mathcal L}^n({\Sigma_k})
$$
$$
+\> \int_{\partial {\Sigma_k}} \sigma_k u_k(x',\sigma_k g(x',t),t) \partial_\eta g(x',t)\> d{\mathcal H}^{n-2} \> dt\; ,
$$
where $\eta$ is the outward space normal on $\partial {\Sigma_k}$.
Since $u_k=0$ on $\partial {\Sigma_k}$, the last integral is $0$.
\\
Moreover, $\Delta' g= 16$ and $u_k \le C_3 \sigma_k$ on $(x',g(x',t),t)$,
implying that
$$
\int_{{\Sigma_k}}(\nabla u_k)(x',\sigma_k g(x',t),t) \cdot (\sigma_k \nabla' g(x',t), -1) \> dx\> dt
$$
$$ =\>
-\int_{{\Sigma_k}} (\partial_n u_k(x',\sigma_k g(x',t),t)+1)dx'dt+
{\mathcal L}^n({\Sigma_k})+C_4\sigma_k^2\; .
$$
By the definition of $w_k$ this tells us also that
\begin{equation}\label{later}
\int_{{\Sigma_k}}(\nabla w_k)(x',\sigma_k g(x',t),t) \cdot (\sigma_k \nabla' g(x',t), 0) \> dx\> dt
\; \to\; 0 \textrm{ as } k\to\infty\; ,
\end{equation}
a fact that will be used later on.\\
Last, integration by parts of the last term in (\ref{eq:troll})
with respect to the time variable yields
$$-
\int_{\{ u_k>0\}\cap Z^+(\sigma_k g)}\partial_t u_k \> dx\> dt\;
\le \;
C_5 {\sigma_k}^2\; .
$$
Combining the above estimates we obtain Claim 1.\\
\textsl{Proof of claim 2:}
With the outward space normal on the boundary of $Z^-(\sigma_k g)$
\begin{displaymath}
\nu_{g_k}=
\frac{1}{\sqrt{1+\sigma_k^2|\nabla' g|^2}}(-\sigma_k\nabla' g,1)
\end{displaymath}
and
with the outward space normal $\nu_{R_k}$ on the regular boundary of $E_k$ we compute
\begin{equation}\label{eq:eta}
\mu(Z^+(\sigma_k g) \cap R_k)\ge
\end{equation}
\begin{displaymath}
\int_{-1}^{1}\int_{Z^+(\sigma_k g)\cap
R_k(t)}
\nu_{g_k}\cdot \nu_{R_k}\> d{\mathcal H}^{n-1}\> dt
\end{displaymath}
\begin{displaymath}
=\; \int_{-1}^1 \int_{E_k\cap Z^+(\sigma_k g)} \textrm{\rm div }\nu_{g_k}\> d{\mathcal H}^{n-1}\> dt
\end{displaymath}
\begin{displaymath}
+\;
\int_{-1}^1\int_{\partial Z^+(\sigma_k g)\cap E_k} \nu_{g_k}\cdot
\nu_{g_k}d{\mathcal H}^{n-1}\> dt.
\end{displaymath}
The normal $\nu_{g_{k}}$ satisfies
\begin{displaymath}
\textrm{\rm div } \nu_{g_k}\ge
\frac{-\sigma_k \Delta g}{\sqrt{1+\sigma_k^2|\nabla' g|^2}}
\ge - C_6\sigma_k.
\end{displaymath}
Inserting this estimate for the divergence into (\ref{eq:eta}) yields
\begin{equation}\label{eq:est1}
\mu(Z^+(\sigma_k g)\cap R_k)\ge
\mu(\partial Z^+(\sigma_k g)\cap E_k)
\end{equation}
\begin{displaymath}-\;
\int_{-1}^1 \int_{E_k\cap Z^+(\sigma_k g)} C_6 \sigma_k \> d{\mathcal H}^{n-1}\> dt
\end{displaymath}
\begin{displaymath}
\ge \; \mu(\partial Z^+(\sigma_k g)\cap E_k)\> -\> C_7 \sigma_k^2\; ;
\end{displaymath}
the last inequality follows from the
fact that the width of the set $E_k$ is of order $O(\sigma_k)$.
As the area of $\partial Z^+(\sigma_k g)\cap E_k)$
is greater than that of $\Sigma_k$, the statement
of Claim 2 holds.\\
\textsl{Proof of Claim 3:} From \textsl{Claim 1} and \textsl{Claim 2} we
infer that
$$
-C_8 \sigma_k^2 \le -\int_{{\Sigma_k}} (\partial_n u_k(x',\sigma_k g(x',t),t)+1)\> dx'\> dt\; .
$$
But since $u_k\in F(\sigma_k,\sigma_k,\tau_k)$ and $\tau_k/\sigma_k^2\to 0$
as $k\to\infty$, it follows that
$$
\partial_n u_k+1\ge -|\nabla u_k|+1\;\ge \; -o(\sigma_k^2).
$$
Consequently
$$
\int_{{\Sigma_k}}|\partial_n w_k(x',\sigma_k g(x',t),t)|
\; = \; \int_{{\Sigma_k}}\left|\frac{\partial_n u_k(x',\sigma_k g(x',t),t)+1}{\sigma_k}\right|
$$
$$
\le \;
\int_{{\Sigma_k}}2\max\left(-\frac{\partial_n u_k(x',\sigma_k g(x',t),t)+1}{\sigma_k},0\right)
\> + \>
\int_{{\Sigma_k}}\frac{\partial_n u_k(x',\sigma_k g(x',t),t)+1}{\sigma_k}
$$
$$
\; \le
C_9 \sigma_k \to 0 \textrm{ as } k\to \infty\; ,
$$
and \textsl{Claim 3} is proved.\\
\textsl{Proof of the Proposition:}
Let $\zeta \in C^1_0(Q_{1/2})$. From Claim 3, from the fact that
$w_k$ is caloric in $Z^-(\sigma_k g)$, from (\ref{later})
and from a standard energy estimate for caloric functions
we infer now that
$$ o(1) \; = \; \int_{{\Sigma_k}}\zeta \>\partial_n w_k(x',\sigma_k g(x',t),t) \nu_n
$$
$$ = \; \int_{Z^-(\sigma_k g)} (\partial_n\zeta \> \partial_n w_k \> - \> \zeta \> \Delta' w_k\> + \> \zeta \> \partial_t w_k)
$$
$$
\; = \; o(1) \> + \> \int_{Z^-(\sigma_k g)} (\partial_n\zeta \> \partial_n w_k \> + \> \nabla'\zeta \cdot\nabla' w_k\> - \> w_k \> \partial_t \zeta)
$$
$$
\to \; \int_{Q^+_1} (\partial_n\zeta \> \partial_n w\> + \> \nabla'\zeta \cdot\nabla' w\> - \> w \> \partial_t \zeta)
\textrm{ as } k\to\infty\; ;
$$
here $\nu$ is the outward unit space normal on $\partial Z^-(\sigma_k g)$.
It follows that $\partial_n w=0$ on $Q'_{1/2}$ in the sense of distributions.
\qed
\begin{corollary}\label{cor:cinf} Suppose that the assumptions
of Lemma \ref{lem:nonhomogeneousblowup} are satisfied and that $k$ is the
subsequence of Lemma \ref{lem:nonhomogeneousblowup}. Then
$f\in C^{\infty}(Q_{1/2})$;
moreover,
$$
\Big| \frac{\partial^{\alpha+k} f}{\partial x^\alpha\partial t^k}\Big|\le C(n,|\alpha|,k)
$$
in $Q_{1/4}$
for any $k\in {\bf N}$ and multi-index $\alpha \in {\bf N}^n$.
\end{corollary}
\proof
Since
$\partial_n w=0$ on $Q'_{1/2}$ in the sense of distributions
we may reflect
$w$ to a caloric function in $Q_{1/2}$.
As $f=w|_{Q'_1}$
and $\Vert w \Vert_{L^\infty(Q_{3/4})}\le C(n)$
(see Proposition \ref{lem:wcal}),
the result follows from standard regularity theory
of caloric functions.\qed
\section{Flatness improvement and regularity}
Concluding regularity is then a standard procedure.
The following Lemma \ref{lem:flatbelow}, Lemma \ref{lem:improve}
and Theorem \ref{theo:main} extend Lemma 7.9, Lemma 7.10 and
Theorem 8.1 in \cite{AC}.
Finally, we apply Theorem \ref{theo:main} to regular free boundary points,
i.e. points in the set $R$
defined in \cite[Proposition 9.1]{calc} (or the proof of Proposition \ref{discr})
to obtain that $R$ is open relative to $\partial \{ u>0\}$.
\begin{lemma}\label{lem:flatbelow}
Let $\theta\in (0,1)$. Then
there exists a constant
$\sigma_{\theta}>0$
depending only on $\theta$ and the dimension $n$
such that if
$\sigma<\sigma_{\theta}$, $\tau\le \sigma_{\theta}\sigma^2$ and
$u\in F(\sigma,\sigma,\tau)$
in $Q_{\rho}$ in direction $\eta$, then
\begin{displaymath}
u\in F(\theta\sigma,1,\tau) \; \textrm{ in } Q_{c(n) \theta \rho}(\vartheta\eta,0)
\end{displaymath}
in direction $\overline{\eta}$ for some $\vartheta\in [-\sigma,\sigma]$ and some $\overline{\eta}$
satisfying
$|\overline{\eta}-\eta|\le {C(n)\sigma}$.
Here $c(n)>0$ and $C(n)<+\infty$ are constants depending only
on the dimension $n$.
\end{lemma}
\proof
We may rotate the coordinate system so $\eta=e_n$, and we may
assume that $\rho=1$.
By a contradiction argument, it is sufficient to prove the statement
of the lemma for $u_k$ as in Lemma
\ref{lem:nonhomogeneousblowup} and every large $k$.
\\
First, observe that by Corollary \ref{cor:cinf},
$$
f(x',t)\le f(0,0)+\ell\cdot x' + C(|x'|^2+|t|) \textrm{ in } Q'_{1/4},
$$
where $\ell$ is the space gradient of $f$, $|\ell|\le C$
and $C$ depends only on the dimension $n$.
Thus
$$
f(x',t)\le f(0,0)+\ell\cdot x' +\frac{\theta }{4}\frac{\theta}{4C} \textrm{ in }
Q_{\theta/(4C)}\; .
$$
It follows that for large $k$ the function $f^+_k$ in
Lemma \ref{lem:nonhomogeneousblowup} satisfies
\begin{displaymath}
f^+_k(x',t)\le f(0,0) + \ell\cdot x' +\theta \frac{\theta}{4C} \quad \textrm{ in } Q_{\theta/(4C)}\; .
\end{displaymath}
This means that $u_k \in F(\theta \sigma,1,\tau)$ in $Q_{\theta/(4C)}(0,f(0,0),0)$ in the
direction
$\bar{\eta}$, where
\begin{displaymath}
\bar{\eta}=\frac{(-\sigma_k\ell,1)}{\sqrt{1+|\sigma_k \ell|^2}}.
\end{displaymath}
The lemma follows. \qed
\begin{lemma}\label{gradient}
Let $u$ be a solution in the sense of Definition \ref{solution}. Then
$$ \max(|\nabla u|^2-1,0)(x,t) \to 0 \textrm{ as } 0< \textrm{\rm pardist}((x,t),\{ u=0\})\to 0\; .$$
\end{lemma}
\proof
Consider a sequence $\{ u>0\} \ni (x_k,t_k) \to (x_0,t_0)$ such that
$$1<\ell:=\limsup_{\{ u>0\} \ni (x,t)\to (x_0,t_0)} |\nabla u(x,t)|^2 = \lim_{k\to\infty} |\nabla u(x_k,t_k)|^2\; .$$
Setting $r_k := \textrm{\rm pardist}((x_k,t_k),\{ u=0\})$, the blow-up sequence
$$u_k(y,s) := {u(x_k+r_ky,t_k+{r_k}^2s) \over {r_k}},
\chi_k(y,s) := \chi(x_k+r_ky,t_k+{r_k}^2s)$$
converges to a solution
$(u_0,\chi_0)$
in the sense of Definition \ref{solution}
satisfying $u_0>0$ in $Q_1$, $|\nabla u_0|^2\le \ell$
and $|\nabla u_0(0)|^2=\ell$.
The strong maximum principle implies that
$u_0(y,s)=\ell \max(y\cdot e,0)$ in $\{ y\cdot e >0\}\cap \{ s<0\}$ for some $e\in \partial B_1$.
\\
From \cite[Theorem 11.1]{calc} we infer that
$$ \{ y\cdot e = 0\} \cap \{ s<0\}\subset \Sigma_{**}$$
up to a set of vanishing ${\mathcal L}^{n-1}$-measure,
where
$$\Sigma_{**}(t) :=\{ x\in \partial\{u_0(t)>0\}\> : \>
\textrm{there is } \theta(x,t)\in (0,1] \textrm{ and }
\xi(x,t)\in \partial B_1(0) \textrm{ such}$$
$$\textrm{that }{u_0(x+ry,t+r^2s)\over r} \> \to \> \theta(x,t)
\vert y\cdot \xi(x,t)\vert \textrm{ locally uniformly}$$
$$\textrm{in } (y,s)\in
{\bf R}^{n+1} \textrm{ as } r \to 0\}\; .$$
However $\theta(x,t)\in (0,1]$ contradicts $\ell>1$.
\qed
\begin{lemma}\label{lem:improve} For every $\theta\in (0,1)$ there exist $\sigma_{\theta}>0$
and $c_\theta\in (0,1/2)$ depending only on $\theta$ and the dimension $n$
such that if $u\in F(\sigma,1,\tau)$ in $Q_{\rho}$ in direction
$\eta$ with $\sigma\le \sigma_{\theta}$ and $\tau\le \sigma_{\theta}\sigma^2$
then $u\in F(\theta \sigma,\theta\sigma,\theta^2\tau)$ in
$Q_{c_\theta \rho}(\overline{y},0)$ in the direction $\overline{\eta}$ for some
$\overline{y},\overline{\eta}$ satisfying
$|\overline{\eta}-\eta|\le C(n) \sigma$ and $|\overline{y}|\le C(n) \sigma$.
Here $C(n)$ depends only on the dimension $n$.
\end{lemma}
\proof
We may
assume that $\rho=1$.
\\
From Lemma \ref{lem:flatabove}
we infer that $u\in F(C\sigma,C\sigma,\tau)$ in $Q_{1/2}(y,0)$ in direction
$\eta$
for some $y\in B_{C\sigma}$.
Consequently we may
apply Lemma \ref{lem:flatbelow} to deduce that
for some $\theta_1$ to be determined later,
$u\in F(C\theta_1\sigma,1,\tau)$
in $Q_{c(n)\theta_1}(\tilde y,0)$ in the direction $\bar{\eta}$ such that
$|\eta-\bar{\eta}|\le C\sigma$ and $|\tilde y-\bar{y}|\le (C+1)\sigma<1/2$,
provided that $\sigma_{\theta}$ has been chosen small enough in terms
of $\theta_1$.
\\
In order to be able to continue we need to show improvement with respect to the $\tau$-variable.
To that end, observe that
$U=\max(|\nabla u|-1,0)$ is by Lemma \ref{gradient} a continuous subcaloric function in $Q_1$
with boundary values less than
$\tau \chi_{\{ u>0\}}\le \tau \chi_{\{ x_n\le \sigma\}}$. We may therefore compare
$U$ to the caloric function with boundary values
$\tau \chi_{\{ x_n\le \sigma\}}$. It follows that $0\le U \le (1-c_1)\tau$
in $Q_{1/2}$ for some $c_1>0$ depending only on the dimension $n$.
Thus $u\in F(C\theta_1\sigma,1,(1-c_1)\tau)$ in $Q_{c(n)\theta_1}(\tilde y,0)$
in the direction $\bar{\eta}$. Choosing $\theta_0:=\sqrt{1-c_1}$
and $\theta_1 := \theta_0/C$ we obtain
$u\in F(\theta_0\sigma,1,\theta_0^2\tau)$ in $Q_{c_2\theta_0}(y,0)$ in the direction $\bar{\eta}$ such that
$|\eta-\bar{\eta}|\le C\sigma$, where $c_2\in (0,1)$ depends only on the dimension
$n$.
\\
Iterating this process we see that
$$
u\in F(\theta_0^m\sigma,1,\theta_0^{2m}\tau) \textrm{ in }
Q_{(c_2\theta_0)^m}(y_m,0)
\textrm{ in the direction }\bar{\eta}_m
$$
where
$|\eta-\bar{\eta}_m|\le C(n) \sigma\sum_{j=0}^{m-1} \theta_0^j$
and $|y_m| \le C(n) \sigma\sum_{j=0}^{m-1} (c_2\theta_0)^j$.
\\
Applying once more Lemma \ref{lem:flatabove}
and choosing $\theta_0 := \theta^{1\over m}/C$ we obtain the statement of the lemma.
\qed
\begin{theorem}\label{theo:main}
There exists a constant $\sigma_0>0$ such that
if $u\in F(\sigma,1,\tau)$ in $Q_\rho(t_0,x_0)$, $\sigma\le\sigma_0$ and $\tau \le \sigma_0\sigma^2$,
then the topological free boundary $\partial\{ u>0\}$ is in
$Q_{\rho/4}(t_0,x_0)$ the graph of a ${\bf C}^{1+\alpha,\alpha}$-function;
in particular the
space normal is H\"older continuous in $Q_{\rho/4}(t_0,x_0)$.
\end{theorem}
\proof
Using Lemma \ref{lem:improve} inductively we see that
\begin{equation}\label{eq:osccontroll}
\begin{array}{l}
u\in F(\theta^k\sigma,\theta^k\sigma,\theta^{2k}\tau) \textrm{ in }Q_{{c_{\theta \over 2}}^k\rho}(y,s)\textrm{ in the direction } \overline{\eta}^k \\
\textrm{where } |\overline{\eta}^k-\eta|\le C(n)\sigma \sum_{j=0}^{k-1}(2\theta)^j
\textrm{ and } |\overline{y}^k-y|\le C(n)\sigma\sum_{j=0}^{k-1}(2c_{\theta/2}\theta)^j\; ,
\end{array}
\end{equation}
provided that $(y,s)\in Q_{1/2}(t_0,x_0)\cap \partial\{ u>0\}$, $\theta < 1/4$ and $$\sigma_0 < \min(1/(4C(n)),\sigma_{\theta/2}/2)\; ;$$
here we sacrificed some flatness in order to keep the original free boundary
point $(y,s)$.
We obtain existence of the outward space normal $\nu$ on $Q_{1/2}(t_0,x_0)$.
Moreover, $\nu$ satisfies by
(\ref{eq:osccontroll})
$$
\textrm{osc}_{Q_{{c_{\theta/2}}^k\rho}(y,s)}\nu \le
C(n,\theta) \theta^k \sigma\; ,
$$
which implies H\"older-continuity of $\nu$.\qed
\begin{corollary}\label{regular}
For each point $(x_0,t_0)$ of the set $R$, the topological free boundary $\partial\{ u>0\}$
is in an open neighborhood
of $(x_0,t_0)$ the graph of a ${\bf C}^{1+\alpha,\alpha}$-function;
in particular, the
space normal is H\"older continuous in an open space-time neighborhood of $(x_0,t_0)$.
\end{corollary}
\proof
The Corollary follows from \cite[Proposition 9.1]{calc} and Lemma \ref{gradient}.
\qed
\bibliographystyle{plain}
| {
"timestamp": "2006-08-30T10:06:23",
"yymm": "0608",
"arxiv_id": "math/0608746",
"language": "en",
"url": "https://arxiv.org/abs/math/0608746",
"abstract": "Consider the parabolic free boundary problem $$ \\Delta u - \\partial_t u = 0 \\textrm{in} \\{u>0\\}, |\\nabla u|=1 \\textrm{on} \\partial\\{u>0\\} . $$ For a realistic class of solutions, containing for example {\\em all} limits of the singular perturbation problem $$\\Delta u_\\epsilon - \\partial_t u_\\epsilon = \\beta_\\epsilon(u_\\epsilon) \\textrm{as} \\epsilon\\to 0,$$ we prove that one-sided flatness of the free boundary implies regularity.In particular, we show that the topological free boundary $\\partial\\{u>0\\}$ can be decomposed into an {\\em open} regular set (relative to $\\partial\\{u>0\\}$) which is locally a surface with Hölder-continuous space normal, and a closed singular set.Our result extends the main theorem in the paper by H.W. Alt-L.A. Caffarelli (1981) to more general solutions as well as the time-dependent case. Our proof uses methods developed in H.W. Alt-L.A. Caffarelli (1981), however we replace the core of that paper, which relies on non-positive mean curvature at singular points, by an argument based on scaling discrepancies, which promises to be applicable to more general free boundary or free discontinuity problems.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "A parabolic free boundary problem with Bernoulli type condition on the free boundary",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575178175919,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7091542191622802
} |
https://arxiv.org/abs/1209.6044 | Bounded Geometry and Characterization of post-singularly Finite $(p,q)$-Exponential Maps | In this paper we define a topological class of branched covering maps of the plane called {\em topological exponential maps of type $(p,q)$} and denoted by $\TE_{p,q}$, where $p\geq 0$ and $q\geq 1$. We follow the framework given in \cite{Ji} to study the problem of combinatorially characterizing an entire map $P e^{Q}$, where $P$ is a polynomial of degree $p$ and $Q$ is a polynomial of degree $q$ using an {\em iteration scheme defined by Thurston} and a {\em bounded geometry condition}. We first show that an element $f \in {\TE}_{p,q}$ with finite post-singular set is combinatorially equivalent to an entire map $P e^{Q}$ if and only if it has bounded geometry with compactness. Thus to complete the characterization, we only need to check that the bounded geometry actually implies compactness. We show this for some $f\in \TE_{p,1}$, $p\geq 1$. Our main result in this paper is that a post-singularly finite map $f$ in $\TE_{0,1}$ or a post-singularly finite map $f$ in $\TE_{p,1}$, $p\geq 1$, with only one non-zero simple breanch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{\lambda z}$ or the form $ \alpha z^{p}e^{\lambda z}$, where $\alpha=(-\lambda/p)^{p}e^{- \lambda (-p/\lambda)^{p}}$, respectively, if and only if it has bounded geometry. This is the first result in this direction for a family of transcendental holomorphic maps with critical points. | \section{Introduction}
\label{sec:intro}
Thurston asked the question ``when can we realize a given branched covering map as a holomorphic map in such a way that the post-critical sets correspond?" and answered it
for post-critically finite degree $d$ branched covers of the sphere~\cite{T,DH}.
His theorem is that a postcritically finite degree $d\geq 2$ branched covering of
the sphere, with hyperbolic orbifold, is either combinatorially equivalent
to a rational map or there is a topological obstruction, now called a ``Thurston obstruction''.
The proof uses an iteration scheme defined for an appropriate Teichm\"uller space. The rational map, when it exists, is unique up to conjugation by a
M\"obius transformation.
Although Thurston's iteration scheme is well defined for transcendental maps, his theorem does not naturally extend to them because the proof uses the finiteness of both the degree and the post-critical set
in a crucial way.
In this paper, we study a class of entire maps: maps of the form $P e^{Q}$ where $P$ and $Q$ are polynomials of degrees $p\geq 0$ and $q\geq 0$ respectively.
This class, which we denote by $\mathcal E_{p,q}$ includes the exponential and polynomials. We call the topological family analogue to these analytic maps $\mathcal TE_{p,q}$ and define it below. Thurston's question makes sense for this family and our main theorem is an answer to this question.
Convergence of the iteration scheme depends on a compactness condition defined in section~\ref{sec:suff}. We approach the question of compactness, using the idea of ``bounded geometry''. This point of view was originally outlined in ~\cite{Ji} where the bounded geometry condition is an intermediate step that connects various topological obstructions with the characterization of rational maps. The introduction of this intermediate step makes understanding the characterization of rational maps relatively easier and the arguments are more straightforward (see~\cite{Ji,JZ,CJ}). It also gives insight into the characterization of entire and meromorphic maps.
In this paper we apply our techniques to characterize the class of post-singularly finite entire maps with exactly one asymptotic value and finitely many critical points, the model topological space $\mathcal TE_{p,q}$.
Our first result is
\medskip
\begin{theorem}~\label{main1}
A post-singularly finite map $f$ in $\mathcal TE_{p,q}$ is combinatorially equivalent to a post-singularly finite entire map of the form $Pe^{Q}$ if and only if it has bounded geometry and satisfies the compactness condition.
\end{theorem}
We next prove that the compactness hypothesis holds for a special family. Our main result is
\medskip
\begin{theorem}[Main Theorem]~\label{main2}
A post-singularly finite map $f$ in $\mathcal TE_{0,1}$ or a post-singularly finite map $f$ in $\mathcal TE_{p,1}$, $p\geq 1$, with only one non-zero simple branch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{\lambda z}$ or the form $ \alpha z^{p}e^{\lambda z}$, where $\alpha=(-\lambda/p)^{p}e^{- \lambda (-p/\lambda)^{p}}$, respectively, if and only if it has bounded geometry.
\end{theorem}
\medskip
Our techniques involve adapting the Thurston iteration scheme to our situation. We work with a fixed normalization. There are two important parts to the proof of the main theorem (Theorem~\ref{main2}). The first is the proof of sufficiency under the assumptions of both bounded geometry and compactness. This part shows that bounded geometry together with the compactness assumption implies the convergence of the iteration scheme to an entire function of the same type (see section \ref{sec:suff}). Its proof involves an analysis of quadratic differentials associated to the functions in the iteration scheme. The proof of the first part works for a more general post-singular finite map $f\in \mathcal TE_{p,q}$. The second part of the proof of the main theorem is in section~\ref{sec:proofmt} where we define a topological constraint. We prove in this part that the bounded geometry together with the topological constraint implies compactness.
\begin{remark}
Our main result is the first result to apply the Thurston iteration scheme to characterize a family of transcendental holomorphic maps with critical points. Post-singularly finite maps $f$ in $\mathcal TE_{0,1}$ were also studied by Hubbard, Schleicher, and Shishikura~\cite{HSS} who used them to characterize a family of holomorphic maps with one asymptotic value, that is, the exponential family. In their study, they used a different normalization; their functions take the form $\lambda e^{z}$ so that they all have period $2\pi i$. They study the limiting behavior of quadratic differentials associated to the exponential
functions with finite post-singular set. They use a Levy cycle condition (a special type of Thurston's topological condition) to characterize when it is possible to realize a given exponential type map with finite post-singular set as an exponential map by combinatorial equivalence. Their calculations involve hyperbolic geometry.
In this paper, by contrast, we normalize the maps in the Thurston iteration scheme by assuming they all fix $0, 1, \infty$. We can therefore use spherical geometry in most our calculations rather than hyperbolic geometry. We are able to consider the more general exponential maps in $\mathcal TE_{p,q}$. The characterization theorem for $f\in \mathcal TE_{p,1}$ in this paper, is completely new.
\end{remark}
\medskip
The paper is organized as follows. In \S2, we review the covering properties of $(p,q)$-exponential maps $E=Pe^{Q}$.
In \S3, we define the family $\mathcal TE_{p,q}$ of $(p,q)$-topological exponential maps $f$.
In \S4,
we define the combinatorial equivalence between post-singularly
finite $(p,q)$-topological exponential maps and prove there is a local quasiconformal $(p,q)$-topological exponential map in every combinatorial equivalence class of post-singularly finite $(p,q)$-topological exponential maps.
In \S5, we define the Teichm\"uller space $T_{f}$ for a post-singularly
finite $(p,q)$-topological exponential map $f$ and
in \S6, we introduce the induced map $\sigma_{f}$ from the Teichm\"uller space $T_{f}$ into itself; this is the crux of the Thurston iteration scheme.
In \S7, we define the concept of ``bounded geometry'' and in \S8 we prove the necessity of the bounded geometry condition.
In \S9, we give the proof of sufficiency assuming compactness.
The proofs we give in \S2-9 are for $(p,q)$-topological exponential maps $f$, $p\geq 0$ and $q\geq 0$.
In \S10.1, \S10.2, and \S10.3, we define a topological constraint for the maps in our main theorem; this involves defining markings and a winding number.
We prove that the winding numbers are unchanged during iteration of the map $\sigma_f.$ Furthermore, in \S10.4 and \S10.5, we prove that the bounded geometry together with the topological constraint implies the compactness. This completes the proof of our main result.
In \S11, we make some remarks about the relations between ``bounded geometry''
and ``canonical Thurston obstructions'' and between ``bounded geometry'' and ``Levy cycles'' in the context of $\mathcal TE_{p,q}$.
\medskip
{\bf Acknowledgement:} We would like to thank the referee whose careful reading and suggestions have much improved it.
The second and the third authors are partially supported by PSC-CUNY awards. The second author is also partially supported by the collaboration grant (\#199837) from the Simons Foundation, the CUNY collaborative incentive research grant (\#1861). This research is also partially supported by the collaboration grant (\#11171121) from the NSF of China
and a collaboration grant from Academy of Mathematics and Systems Science and the Morningside
Center of Mathematics at the Chinese Academy of Sciences.
\section{The space $\mathcal E_{p,q}$ of $(p,q)$-Exponential Maps}
\label{sec:epq}
\label{pq-exp maps}
We use the following notation:
${\mathbb C}$ is the complex plane, $\hat{\mathbb C}$ is the Riemann sphere and
${\mathbb C}^{*}$ is the complex plane punctured at the origin.
A {\em $(p,q)$-exponential map} is an entire function of the form $E =Pe^{Q}$ where $P$ and $Q$ are polynomials of degrees $p\geq 0$ and $q\geq 0$ respectively such that $p+q\geq 1$. We use the notation ${\mathcal E}_{p,q}$ for the set of $(p,q)$-exponential maps.
Note that if $P(z)=a_0 + a_1 z + \ldots a_pz^p$, $Q(z)=b_0 + b_1 z + \ldots b_q z^q$, $\widehat{P}(z)=e^{b_0} P(z)$ and $\widehat{Q}(z)=Q(z)-b_0$ then
$$P(z) e^{Q(z)} = \widehat{P}e^{\widehat{Q}(z)}.$$
To avoid this ambiguity we always assume $b_0=0$.
If $q=0$, then $E$ is a polynomial of degree $p$. Otherwise, $E$ is a transcendental entire function with essential singularity at infinity.
The growth rate of an entire function $f$ is defined as
$$
\limsup_{r\to \infty} \frac{\log \log M(r)}{\log r}
$$
where $M(r) =\sup_{|z|=r} |f(z)|$. It is easy to see that the growth rate of $E$ is $q$.
Recall that an asymptotic tract $V$ for an entire transcendental function $g$ is a simply connected unbounded domain such that $g(V) \subset \hat\mathbb C$ is conformally a punctured disk $D \setminus \{a\}$ and the map $g:V \rightarrow g(V)$ is a universal cover. The point $a$ is the asymptotic value corresponding to the tract. For functions $E$ in ${\mathcal E}_{p,q}$ we have
\begin{prop}
If $q\geq 1$, $E$ has $2q$ distinct asymptotic tracts that are separated by $2q$ rays. Each tract maps to a punctured neighborhood of either zero or infinity and these are the only asymptotic values.
\end{prop}
\begin{proof} From the growth rate of $E$ we see that for $|z|$ large, the behavior of the exponential dominates. Since $Q(z) = b_q z^q + \mbox{ lower order terms} $, in a neighborhood of infinity there are $2q$ branches of $\Re Q = 0$ asymptotic to equally spaced rays. In the $2q$ sectors defined by these rays the signs of $\Re Q$ alternate. If $\gamma(t)$ is a curve
such that $\lim_{t \to \infty} \gamma(t) = \infty$ and $\gamma(t)$ stays in one sector for all large $t$, then either $\lim_{t \to \infty}E(\gamma(t))=0$ or $\lim_{t \to \infty}E(\gamma(t))=\infty$, as $\Re Q$ is negative or positive in the sector. It follows that there are exactly $q$ sectors that are asymptotic tracts for $0$ and $q$ sectors that are asymptotic tracts for infinity. Because the complement of these tracts in a punctured neighborhood of infinity consists entirely of these rays, there can be no other asymptotic tracts. \end{proof}
\begin{remark} The directions dividing the asymptotic tracts are called {\em Julia rays} or {\em Julia directions} for $E$. If $\gamma(t)$ tends to infinity along a Julia ray, $E(\gamma(t))$ remains in a compact domain in the plane. It spirals infinitely often around the origin. \end{remark}
Two
$(p,q)$-exponential maps $E_{1}$ and $E_{2}$ are conformally equivalent if they are conjugate by a conformal automorphism $M$ of the Riemann sphere $\hat{\mathbb C}$, that is, $E_{1} =M\circ E_{2}\circ M^{-1}$. The automorphism $M$ must be a M\"obius transformation and it must fix both $0$ and $\infty$ so that it must be the affine stretch map $M(z)=az$, $a \neq 0$. We are interested in conformal equivalence classes of maps, so by abuse of notation, we treat conformally equivalent $(p,q)$-exponential maps $E_{1}$ and $E_{2}$ as the same.
The critical points of $E=Pe^{Q}$ are the roots of $P'+PQ'=0$. Therefore, $E$ has $p+q-1$ critical points counted with multiplicity which we denote by
$$
\Omega_{E}=\{ c_{1}, \cdots, c_{p+q-1}\}.
$$
Note that if $E(z)=0$ then $P(z)=0$. This in turn implies that if $c\in \Omega_E$ maps to $0$, then $c$ must also be a critical point of $P$.
Since $P$ has only $p-1$ critical points counted with multiplicity,
there must be at least $q$ points (counted with multiplicity) in $\Omega$ which are not mapped to $0$.
Denote by
$$
\Omega_{E,0}=\{ c_{1}, \cdots, c_{k}\}, \quad k\leq p-1,
$$
the (possibly empty) subset of $\Omega_E$ consisting of critical points such that $E(c_i)=0$. Denote its complement in $\Omega_E$ by
$$
\Omega_{E,1} =\Omega_{E}\setminus \Omega_{E,0}=\{ c_{k+1}, \cdots, c_{p+q-1}\}.
$$
When $q=0$, $E$ is a polynomial. The {\em post-singular set} in this special case is the same as the {\em post-critical set}. It is defined as
$$
P_{E} =\overline{\cup_{n\geq 1} E^{n}(\Omega_{E})}\cup\{\infty\}.
$$
To avoid trivial cases here we will assume that $\#(P_{E}) \geq 4$. Conjugating by an affine map $z \to az+b$ of the complex plane,
we normalize so that $0, 1\in P_{E}$.
When $q=1$ and $p=0$, $\Omega_{E}=\emptyset$ and $\mathcal E_{0,1}$ consists of exponential maps $\alpha e^{\lambda z}$, $\alpha, \lambda \in \mathbb C^*$.
The {\em post-singular set} in this special case is defined as
$$
P_{E} =\overline{\cup_{n\geq 0} E^{n}(0)} \cup \{\infty\}.
$$
Conjugating by an affine stretch $z \mapsto \alpha z$ of the complex plane,
we normalize so that $E(0)=1$. Note that after this normalization the family takes the form
$e^{\lambda z}$ , $\lambda \in \mathbb C^{*}$.
When $q\geq 2$ and $p=0$ or when $q\geq 1$ and $p\geq 1$, $\Omega_{E,1}$ is a non-empty set. Let
$$
{\mathcal V} =E(\Omega_{E,1}) =\{ v_{1}, \cdots, v_{m}\}
$$
denote the set of non-zero critical values of $E$.
The {\em post-singular set} for $E$ in the general case is now defined as
$$
P_{E} = \overline{\cup_{n\geq 0} E^{n}({\mathcal V}\cup \{ 0 \})} \cup \{\infty\}.
$$
We normalize as follows: \\
If $E$ does not fix $0$, which is always true if $q\geq 2$ and $p=0$,
we conjugate by an affine stretch $z \rightarrow az$ so that $E(0)=1$.
If $E(0)=0$, there is a critical point in $c_{k+1}$ in $\Omega_{E,1}$ with $c_{k+1}\not=0$ and $v_{1}=E(c_{k+1})\not= 0$.
In this case we normalize so that $v_{1}=1$. The family $\mathcal E_{1,1}$ consists of functions of the form $\alpha z e^{\lambda z}$.
After normalization they take the form
$$
-\lambda e ze^{\lambda z}.
$$
An important family we consider in this paper is the family in $\mathcal E_{p,1}$, $p\geq 1$, where each map in this family
has only one non-zero simple critical point. After normalization, the functions in this family take the form
$$
E(z)= \alpha z^{p} e^{\lambda z}, \quad \alpha=\Big( -\frac{\lambda}{p}\Big)^{p} e^{- \lambda \Big(-\frac{p}{\lambda}\Big)^{p}}.
$$
This is the main family we will study in this paper.
\section{Topological Exponential Maps of Type $(p,q)$}
\label{sec:Tpq}
We use the notation ${\mathbb R}^{2}$ for the Euclidean plane. We define the space ${\mathcal TE}_{p,q}$ of {\em topological exponential maps of type $(p,q)$} with $p+q\geq 1$. These are branched coverings with a single finite asymptotic value, normalized to be at zero, modeled on the maps in the holomorphic family $\mathcal E_{p,q}$. For a full discussion of the covering properties for this family see Zakeri~\cite{Z}, and for a more general discussion of maps with finitely many asymptotic and critical values see Nevanlinna~\cite{Nev}.
If $q=0$, then ${\mathcal TE}_{p,0}$ consists of all topological polynomials $P$ of degree $p$: these are degree $p$ branched coverings of the sphere such that $f^{-1}(\infty)=\{\infty\}$.
If $q=1$ and $p=0$, the space ${\mathcal TE}_{0,1}$ consists of universal covering maps $f: {\mathbb R}^{2}\to {\mathbb R}^{2}\setminus \{0\}$. These are discussed at length in \cite{HSS} where they are called topological exponential maps.
The polynomials $P$ and $Q$ contribute differently to the covering properties of maps in $\mathcal E_{p,q}$. As we saw, the degree of $Q$ controls the growth and behavior at infinity. Using maps $e^{Q}$ as our model we first define the space ${\mathcal TE}_{0,q}$.
\begin{definition}
If $q\geq 2$ and $p=0$, the space ${\mathcal TE}_{0,q}$ consists of topological branched covering maps $f: {\mathbb R}^{2}\to {\mathbb R}^{2}\setminus \{0\}$ satisfying the following conditions:
\begin{itemize}
\item[i)] The set of branch points, $\Omega_{f} =\{c\in {\mathbb R}^{2}\;|\; \deg_{c}f\geq 2\}$ consists of $q-1$ points counted with multiplicity.
\item[ii)] Let ${\mathcal V} =\{ v_{1}, \cdots, v_{m}\} =f(\Omega_{f})\subset {\mathbb R}^{2}\setminus \{0\}$ be the set
of distinct images of the branch points. For $i=1, \ldots, m$, let $L_{i}$ be a smooth topological ray in ${\mathbb R}^{2}\setminus \{0\}$ starting at $v_{i}$ and extending to $\infty$ such that the collection of rays
$\{L_{1}, \cdots, L_{m}\}$ are pairwise disjoint. Then
\begin{enumerate}
\item $f^{-1}(L_{i})$ consists of infinitely many rays starting at points in the preimage set $f^{-1}(v_{i})$. If $x\in f^{-1}(v_{i})\cap \Omega_{f}$, there are $d_{x}=\deg_{x} f$ rays meeting at $x$ called {\em critical rays}. If $x\in f^{-1}(v_{i})\setminus \Omega_{f}$, there is only one ray emanating from $x$; it is called a {\em non-critical ray}.
Set
$$
W=\mathbb R^2 \setminus ( \cup_{i=1}^m L_i \cup \{0\}).
$$
\item The set of critical rays meeting at points in $\Omega_{f}$ divides $f^{-1}(W)$ into $q=1+\sum_{c\in \Omega_{f}} (d_{c}-1)$ open unbounded connected components $W_{1}, \cdots, W_{q}$.
\item[(3)] $f: W_{i}\to W $ is a universal covering for each $1\leq i\leq q$.
\end{enumerate}
\end{itemize} \end{definition}
Note that the map restricted to each $W_i$ is a topological model for the exponential map $z\mapsto e^{z}$ and the local degree at the critical points determines the number of $W_i$ attached at the point.
We now define the space ${\mathcal TE}_{p,q}$ in full generality where we assume $p>0$ and there is additional behavior modeled on the role of the new critical points of $P e^{Q}$ introduced by the non-constant polynomial $P$.
\medskip
\begin{definition}~\label{topexpdef}
If $q\geq 1$ and $p\geq 1$, the space ${\mathcal TE}_{p,q}$ consists of topological branched covering maps $f: {\mathbb R}^{2}\to {\mathbb R}^{2}$ satisfying the following conditions:
\begin{itemize}
\item[i)] $f^{-1}(0)$ consists of $p$ points counted with multiplicity.
\item[ii)] The set of branch points, $\Omega_{f} =\{c\in {\mathbb R}^{2}\;|\; \deg_{c}f\geq 2\}$ consists of $p+q-1$ points counted with multiplicity.
\item[iii)] Let $\Omega_{f, 0} = \Omega_{f} \cap f^{-1}(0)$ be the $k<p$ branch points that map to $0$ and $\Omega_{f,1} =\Omega_{f}\setminus \Omega_{f,0}$ the $p+q-1-k$ branch points that do not. Note that $\Omega_{f,1}$ contains at least $q$ points and
${\mathcal V} =\{ v_{1}, \cdots, v_{m}\} =f(\Omega_{f,1})$ is contained in ${\mathbb R}^{2}\setminus \{0\}$. For $i=1, \ldots, m$, let $L_{i}$ be a smooth topological ray in ${\mathbb R}^{2}\setminus \{0\}$ starting at $v_{i}$ and extending to $\infty$ such that the collection of rays $\{L_{1}, \cdots, L_{m}\}$ are pairwise disjoint. Then
\begin{enumerate}
\item $f^{-1}(L_{i})$ consists of infinitely many rays starting at points in the pre-image set $f^{-1}(v_{i})$. If $x\in f^{-1}(v_{i}) \setminus \Omega_{f,1}$, there is only one ray emanating from $x$; this is a {\em non-critical ray}. If $x\in f^{-1}(v_{i})\cap \Omega_{f, 1}$, there are $d_{x}=\deg_{x} f$ {\em critical rays} meeting at $x$. Set $$W=\mathbb R^2 \setminus ( \cup_{i=1}^m(L_i) \cup \{0\}).$$
\item The collection of all critical rays meeting at points in $\Omega_{f,1}$ divides $f^{-1}(W)$ into $l=p+q-k=1+\sum_{c\in \Omega_{f,1}} (d_{c}-1)$ open unbounded connected components.
\item Set $f^{-1}(0) =\{ a_{i}\}_{i=1}^{p-k}$ where the $a_i$ are distinct. Each $a_i$ is contained in a distinct component of $f^{-1}(W)$; label these components $W_{i,0}$, $i=1, \ldots p-k$. Then the restriction $f: W_{i,0} \setminus \{a_i\} \rightarrow W$ is an unbranched covering map of degree $d_i=deg_{a_i}f$ where $d_i>1$ if $a_i \in \Omega_{f,0}$ and $d_i=1$ otherwise.
\item Label the remaining $q$ connected
components of $f^{-1}(W)$ by $W_{j,1}$, $j=1, \ldots, q$.
Then the restriction $f: {W_{j,1}} \rightarrow W$
is a universal covering map.
\end{enumerate}
\end{itemize}
\end{definition}
In section 3 of~\cite{ Z}, Zakeri proves that the $(p,q)$-exponential maps are topological exponential maps of type $(p,q)$.
The converse is also true.
\vspace*{5pt}
\begin{theorem}~\label{topexp}
Suppose $f\in {\mathcal TE}_{p,q}$ is analytic. Then $f=Pe^{Q}$ for two polynomials $P$ and $Q$ of degrees $p$ and $q$. That is, an analytic topological exponential map of type $(p,q)$ is a $(p,q)$-exponential map.
\end{theorem}
\begin{proof}
If $q=0$, then $f$ is a polynomial $P$ of degree $p$.
If $q\geq 1$, then $f$ is an entire function with $p$ roots, counted with multiplicity.
Every such function can be expressed as
$$
f (z)= P(z) e^{g(z)}
$$
where $P$ is a polynomial of degree $p$ and $g$ is some entire function (see~\cite[Section 2.3]{Al}).
Consider $$f'(z) = (P(z)g'(z)+P'(z))e^{g(z)}.$$
It is also an entire function, and by assumption it has $p+q-1$ roots so that $Pg'+P'$ is a polynomial of degree $p+q-1$. It follows that $g'$ is a polynomial of degree $q-1$ and $g=Q$ is a polynomial of degree $q$.
\end{proof}
Note that if $f \in {\mathcal TE}_{p,q}$, $ q \neq 0$, the origin plays a special role: it is the only point with no or finitely many pre-images. The conjugate of $f$ by $z \mapsto az$, $ a \in {\mathbb C}^*$, is also in ${\mathcal TE}_{p,q}$; conjugate maps are conformally equivalent.
For $f\in {\mathcal TE}_{p,q}$, we define the {\em post-singular set} as follows:
\begin{itemize}
\item[i)] When $q=0$, $E$ is a polynomial and, as mentioned in the introduction, is treated elsewhere. We therefore always assume $q \geq 1$.
\item[ii)] When $q=1$ and $p=0$, the {\em post-singular set} is
$$
P_{f} =\overline{\cup_{n\geq 0} f^{n}(0)}\cup\{\infty\}.
$$
We normalize so that $f(0)=1\in P_{f}$.
\item[iii)] When $q\geq 1$ and $p\geq 1$, the set of branch points is
$$\Omega_{f} =\{ c\in {\mathbb R}^{2}\;|\; \deg_{c} f \geq 2\}$$ and the {\em post-singular set} is
$$
P_{f} = \overline{\cup_{n\geq 0} f^{n}({\mathcal V}\cup \{ 0 \})} \cup \{\infty\}.
$$
If $q>1$ or if $q=1$ and $f(0) \neq 0$, we normalize so that $f(0)=1\in P_{f}$.
If $f(0)=0$, then, by the assumption $q \geq 1$, there is a branch point $c_{k+1}\not=0$ such that $v_{1}=f(c_{k+1})\not= 0$.
We normalize so that $v_{1}=1$.
\end{itemize}
To avoid trivial cases we assume that $\#(P_{f}) \geq 4$.
It is clear that, in any case, $P_{f}$ is forward invariant, that is,
$$
f(P_{f}\setminus \{\infty\})\cup \{\infty\} \subseteq P_{f}
$$
or equivalently,
$$
f^{-1} (P_{f} \setminus \{\infty\})\cup \{\infty\} \supset P_{f}.
$$
Note that since we assume $q\geq 1$, $f^{-1}(P_{f}\setminus \{\infty\}) \setminus (P_{f}\setminus \{\infty\})$
contains infinitely many points.
\begin{definition}
We call $f\in {\mathcal TE}_{p,q}$ {\em post-singularly finite} if $\#(P_{f})<\infty$.
\end{definition}
\section{Combinatorial Equivalence}
\label{sec:combequiv}
\begin{definition}
Suppose $f, g$ are two post-singularly finite maps in ${\mathcal TE}_{p,q}$. We say that they are {\em combinatorially equivalent} if they are topologically equivalent so that there are
homeomorphisms $\phi$ and $\psi$ of the sphere $S^{2}={\mathbb R}^{2}\cup\{\infty\}$ fixing $0$ and $\infty$ such that $\phi\circ f=g\circ \psi$ on ${\mathbb R}^{2}$ and if they satisfy the additional condition, $\phi^{-1}\circ \psi$ is isotopic to the identity of $S^{2}$ rel $P_{f}$.
\end{definition}
The commutative diagram for the above definition is
\begin{equation*}
\xymatrix{\mathbb{R}^2 \ar[d]^f\ar[r]^{\psi} & \mathbb{R}^2\ar[d]^{g}\\
\mathbb{R}^2\ar[r]^\phi & \mathbb{R}^2}
\end{equation*}
The isotopy condition says that $P_{g}= \phi(P_{f})$.
Consider
${\mathbb R}^{2}\cup \{\infty\}$ equipped with the standard conformal structure as the Riemann sphere. Then $f\in {\mathcal T}E_{p,q}$ is a map from $\hat{\mathbb C}$ into itself. We say $f\in {\mathcal TE}_{p,q}$ is {\em locally $K$-quasiconformal} for some $K>1$ if for any $z\in \hat{\mathbb{C}} \setminus \Omega_{f} \cup \{0\}$, there is a neighborhood $U$ of $z$ such that $f: U\to f(U)$ is $K$-quasiconformal. Since we are working with isotopies rel a finite set, the following lemma is standard.
\begin{lemma}
Any post-singularly finite $f\in {\mathcal TE}_{p,q}$ is combinatorially equivalent to some locally $K$-quasiconformal map $g\in {\mathcal TE}_{p,q}$.
\end{lemma}
\begin{proof}
Recall that $\Omega_{f}$ is the set of branch points of $f$ in $\mathbb C$ and $0$ is only asymptotic value in $\mathbb C$.
Consider the space $X=\mathbb C \setminus \Omega_{f}$. For every $p\in X$, let $U_{p}$ be a small neighborhood about $p$ such that $\phi_{p}=f|_{U}: U\to f(U)\subset \mathbb C$ is injective. Then $\alpha=\{ (U_{p}, \phi_{p})\}_{p\in X}$ defines an atlas on $X$ with charts $(U_{p}, \phi_{p})$. If $U_{p}\cap U_{q}\not=\emptyset$, then $\phi_{p}\circ \phi_{q}^{-1} (z)=z: \phi_{q}(U_{p}\cap U_{q}) \to \phi_{p} (U_{p}\cap U_{q})$. Thus all transition maps are conformal ($1-1$ and analytic) and the atlas $\alpha$ defines a Riemann surface structure on $X$ which we again denote by $\alpha$. Denote the Riemann surface by $S=(X, \alpha)$. From the uniformization theorem, $S$ is conformally equivalent to the Riemann surface $\mathbb C\setminus A$ with the standard complex structure induced by $\mathbb C$, where $A$ consists of $n=\#(\Omega_{f})+1$ points . The homeomorphism $h: \mathbb C\to \mathbb C$ with $h(0)=0$ and $h: S=(X, \alpha)\to \mathbb C\setminus A$ is conformal so that
$R=f\circ h^{-1}: \mathbb C\to \mathbb C$ is holomorphic with critical points at $h(\Omega_{f})$ and one asymptotic value at $0$. Since the set
$P_{f}$ is finite, following the standard procedure in quasiconformal mapping theory, there is a $K$-quasiconfornal homeomorphism $k: \hat{\mathbb C}\to \hat{\mathbb C}$ such that $h$ is isotopic to $k$ rel $P_{f}$. The map $g= R\circ k$ is a locally $K$-quasiconformal map in ${\mathcal TE}_{p,q}$ and combinatorially equivalent to $f$. This completes the proof of the lemma.
\end{proof}
Thus without loss of generality, in the rest of the paper, we will assume that any post-singularly finite $f \in {\mathcal TE}_{p,q}$ is locally $K$-quasiconformal for some $K\ge 1$.
\section{Teichm\"uller Space $T_{f}$}
\label{sec:teich sp}
Recall that we denote $\mathbb{R}^2 \cup \{\infty\}$ equipped with the standard conformal structure by $\widehat{\mathbb{C}}$. Let ${\mathcal M}=\{ \mu \in L^{\infty} (\widehat{\mathbb C})\;|\; \|\mu\|_{\infty}<1\}$ be the unit ball in the space of all measurable functions on the Riemann sphere. Each element $\mu\in {\mathcal M}$ is called a Beltrami coefficient.
For each Beltrami coefficient $\mu$, the Beltrami equation
$$
w_{\overline{z}}=\mu w_{z}
$$
has a unique quasiconformal solution $w^{\mu}$ which maps $\hat{\mathbb C}$ to itself fixing $0,1, \infty$.
Moreover, $w^{\mu}$ depends holomorphically on $\mu$.
Let $f$ be a post-singularly finite map in ${\mathcal TE}_{p,q}$ with post-singular set $P_f$. The Teichm\"uller space $T(\hat{\mathbb C}, P_f)$ is defined as follows. Given Beltrami differentials $\mu, \nu \in {\mathcal M}$ we say that $\mu$ and $\nu$ are equivalent in $\mathcal M$, and denote this by $\mu\sim \nu$, if $(w^{\nu})^{-1} \circ w^{\mu}$ is isotopic to the identity map of $\widehat{\mathbb C}$ rel $P_f$. The equivalence class of $\mu$ under $\sim$ is denoted by $[\mu]$.
We set
$$
T_f=T(\hat{\mathbb C}, P_f)= \mathcal M/ \sim.
$$
It is easy to see that $T_{f}$ is a finite-dimensional complex manifold and is equivalent to the classical Teichm\"uller space $Teich(\hat{\mathbb C}\setminus P_{f})$ of Riemann surfaces with basepoint $\hat{\mathbb C}\setminus P_{f}$. Therefore, the Teichm\"uller distance $d_{T}$ and the Kobayashi distance $d_{K}$ on $T_{f}$ coincide.
\section{Induced Holomorphic Map $\sigma_{f}$}
\label{sec:Thurston map}
For any post-singularly finite $f$ in ${\mathcal TE}_{p,q}$, there is an induced map $\sigma= \sigma_{f}$ from $T_{f}$ into itself given by
$$
\sigma([\mu]) =[f^{*}\mu],
$$
where
\begin{equation}~\label{pullbackformula}
f^{*}\mu(z) = \frac{\mu_f(z) + \mu_f((f(z))
\theta(z)}{1 + \overline{\mu_f (z)} \mu_f(f(z)) \theta(z)}, \quad \theta(z) =\frac{\bar{f}_{z}}{f_{z}}.
\end{equation}
It is a holomorphic map so that
\vspace*{5pt}
\begin{lemma}~\label{contractive}
For any two points $\tau$ and $\tilde\tau$ in $T_{f}$,
$$
d_{T}(\sigma(\tau), \sigma(\tilde\tau))\leq d_{T}(\tau, \tilde\tau).
$$
\end{lemma}
The next lemma follows directly from the definitions.
\vspace*{5pt}
\begin{lemma}
A post-singularly finite $f$ in ${\mathcal TE}_{p, q}$ is combinatorially equivalent to a $(p,q)$-exponential map $E=Pe^{Q}$ iff $\sigma$ has a fixed point in $T_{f}$.
\end{lemma}
\section{Bounded Geometry}
\label{sec:bounded geometry}
For any $\tau_{0}\in T_{f}$, let $\tau_{n}=\sigma^{n}(\tau_{0})$, $n\geq 1$. The iteration sequence $\tau_{n}=[\mu_{n}]$ determines a sequence of finite subsets
$$
P_{f,n} = w^{\mu_{n}}(P_{f}), \quad n=0, 1, 2, \cdots.
$$
Since all $w^{\mu_{n}}$ fix $0, 1, \infty$, it follows that $0, 1, \infty\in P_{f,n}$.
\begin{definition}[Spherical Version]
We say $f$ has {\em bounded geometry} if there is a constant $b>0$ and a point $\tau_{0}\in T_{f}$ such that
$$
d_{sp} (p_{n},q_{n}) \geq b
$$
for $p_{n}, q_{n}\in P_{f,n}$ and $n\geq 0$. Here
$$
d_{sp}(z,z')= \frac{|z-z'|}{\sqrt{1+|z|^{2}}\sqrt{1+|z'|^{2}}}
$$
is the spherical distance on $\hat{\mathbb C}$.
\end{definition}
Note that $d_{sp}(z, \infty) = \frac{|z|}{\sqrt{1+|z|^2}}$.
Away from infinity the spherical metric and Euclidean metric are equivalent.
Precisely, for any bounded $S \subset \mathbb C$,
there is a constant $C>0$ which depends only on $S$ such that
$$
C^{-1} d_{sp}(x,y) \leq |x-y|\leq Cd_{sp}(x,y)\quad \forall x,y \in S.
$$
Consider the hyperbolic Riemann surface $R=\hat{\mathbb C}\setminus P_{f}$
equipped with the standard complex structure as the basepoint $\tau_{0}=[0]\in T_{f}$.
A point $\tau$ in $T_{f}$ defines another complex structure $\tau$ on $R$.
Denote by $R_{\tau}$ the hyperbolic Riemann surface $R$ equipped with the complex structure $\tau$.
A simple closed curve $\gamma\subset R$ is called {\it non-peripheral} if each component of $\hat{\mathbb C}\setminus \gamma$ contains at least two points
of $P_{f}$. Let $\gamma$ be a non-peripheral simple closed curve in $R$.
For any $\tau=[\mu]\in T_{f}$, let $l_{\tau}(\gamma)$ be the hyperbolic length of the unique closed geodesic homotopic to $\gamma$ in $R_{\tau}$.
The bounded geometry property can be stated in terms of hyperbolic geometry as follows.
\begin{definition}[Hyperbolic version]
We say $f$ has {\em bounded geometry} if there is a constant $a>0$ and a point $\tau_{0}\in T_{f}$ such that
$l_{\tau_{n}}(\gamma)\geq a$ for all $n\geq 0$ and all non-peripheral simple closed curves $\gamma$ in $R$.
\end{definition}
The above definitions of bounded geometry are equivalent because of the following lemma and the fact that we have normalized so that $0,1,\infty$ always belong to $P_f$.
\vspace*{5pt}
\begin{lemma}~\label{sg} Consider the hyperbolic Riemann surface
$\hat{\mathbb C}\setminus X$, where $X$ is a finite subset of $\hat\mathbb C$ such that $0, 1, \infty \in X$, equipped with the standard complex structure.
Let $a>0$ be a constant. If every simple closed geodesic in $\hat{\mathbb C}
\setminus X$ has hyperbolic length greater than $a$, then the
spherical distance between any two distinct points in $X$ is bounded below by a bound $b>0$ which depends only on $a$ and $m=\#(X)$.
\end{lemma}
\begin{proof} If
$m=3$ there are no non-peripheral simple closed curves so in the following argument we may assume that $m\geq
4$. Let $X = \{x_{1}, \cdots, x_{m-1},x_{m} = \infty \}$ and let
$|\cdot|$ denote the Euclidean metric on ${\mathbb C}$.
Suppose $0=|x_{1}| \le \cdots \le |x_{m-1}|$. Let $M = |x_{m-1}|$.
Then $|x_{2}| \le 1$, and we have
$$
\prod_{2 \le i \le m-2} \frac{|x_{i+1}|}{|x_{i}|} =
\frac{|x_{m-1}|}{|x_{2}|} \ge M.
$$
Hence
$$
\max_{2 \le i \le m-2} \Big\{ \frac{|x_{i+1}|}{|x_{i}|}\Big\} \ge
M^{\frac{1}{m-3}}.
$$
Let
$$
A_{i} = \{z\in {\mathbb C} \quad \Big{|}\quad |x_{i}| < z <
|x_{i+1}|\}
$$
and let $\mod(A_{i}) = \frac{1}{2\pi} \log\frac{|x_{i+1}|}{ |{x_i}|}$ be its modulus. Then for some
integer $2\leq i_{0}\leq m-2$ it follows that
$$
\mod(A_{i_{0}}) \ge \frac{\log M}{2 \pi (m-3)}.
$$
Denote the extremal length of the core curve $\gamma_{i_0}$ in $A_{i_{0}} \subset \hat{\mathbb C}\setminus X$
by $\|\gamma_{i_{0}}\| $.
By properties of extremal length,
$$
\|\gamma_{i_{0}}\|= \frac{1}{\mod(A_{i_{0}})} \le \frac{2
\pi (m-3)}{\log M}.
$$
Since extremal length is defined by taking a supremum over all metrics and the area of $\hat{\mathbb C} \setminus X$ is $ 2\pi(m-2)$ for every hyperbolic metric,
$$
\|\gamma_{i_{0}}\| \geq \frac{ l_{\tau_n}^{2}(\gamma)}{2\pi(m-2)} \geq \frac{a^{2}}{2\pi(m-2)}.
$$
Setting $a'= \frac{a^{2}}{2\pi(m-2)}$, these inequalities imply
$$
M \le M_{0}=e^{\frac{2 \pi (m-3)}{a'}}.
$$
Thus the spherical distance between $\infty$ and any finite point in
$X$ has a positive lower bound $b$ which depends only on $a$ and $m$.
Next we show that the spherical distance between any two
finite points in $X$ has a positive lower bound depending only on
$a$ and $m$. By the equivalence of the spherical and Euclidean metrics in a bounded set in the plane,
it suffices to prove that $|x-y|$ is greater than a constant $b$ for any two finite points in $X$.
First consider the map $\alpha (z) =1/z$ which is a hyperbolic isometry from $X$ to $\alpha(X)$. It preserves the set $\{0, 1,\infty\}$ so that $0,1, \infty \in \alpha (X)$.
For any $2\leq i\leq m-1$, the above argument implies that $1/|x_{i}| \leq M_{0}$ and hence $|x_{i}|\geq 1/M_{0}$.
Similarly, for any $x_{i}\in X$, $2\leq i\leq m-1$, consider the map $\beta(z) =z/ (z-x_{i})$. It maps $\{0,\infty,x_i\}$ to $\{0,1,\infty \}$ so that $\beta(X)$ contains $\{0,1,\infty\}$ and it is also a hyperbolic isometry. For any $2\leq i\leq m-1$, the above argument implies that $|x_{j}|/|x_{j}-x_{i}| \leq M_{0}$
which in turn implies that $|x_{j}-x_{i}| \geq 1/M_{0}^{2}$ proving the lemma.
\end{proof}
\section{The Main Result--Necessity}
\label{sec:main result}
Our main result (Theorem~\ref{main2}) has two parts: necessity and sufficiency. The necessity is relatively easy and can be proved for the general case.
We prove the following statement.
\vspace*{5pt}
\begin{theorem}\label{necc}
A post-singularly finite topological exponential map $f\in {\mathcal TE}_{p,q}$ is combinatorially equivalent to a unique $(p,q)$-exponential map
$E=Pe^{Q}$ if and only if
$f$ has bounded geometry.
\end{theorem}
\begin{proof}[Proof of Necessity]
If $f$ is combinatorially equivalent to
$E=Pe^{Q}$, then $\sigma$ has a unique fixed point $\tau_{0}$ so that $\tau_{n}=\tau_{0}$ for all $n$. The complex structure on $\hat{\mathbb C} \setminus P_f$ defined by $\tau_0$ induces a hyperbolic metric on it. The shortest geodesic in this metric gives a lower bound on the lengths of all geodesics so that $f$ satisfies the hyperbolic definition of bounded geometry.
\end{proof}
\section{Sufficiency under Compactness}
\label{sec:suff}
The proof of sufficiency of our main theorem (Theorem~\ref{main2}) is more complicated and needs some preparatory material.
There are two parts: one is a compactness argument and the other is a fixed point argument.
Once one has compactness, the proof of the fixed point argument is quite standard (see~\cite{Ji}) and works for any $f\in {\mathcal TE}_{p,q}$.
This is the content of Theorem 1 which we prove in this section.
The normalized functions in ${\mathcal E}_{p,q}$ are determined by the $p+q+1$ coefficients of the polynomials $P$ and $Q$.
This identification defines an embedding into $\mathbb C^{p+q+1}$ and hence a topology on ${\mathcal E}_{p,q}$.
Given $f\in\mathcal TE_{p,q}$ and given any $\tau_{0}=[\mu_{0}]\in T_f$, let $\tau_{n}=\sigma^{n} (\tau_{0}) =[\mu_{n}]$
be the sequence generated by $\sigma$. Let $w^{\mu_{n}}$ be the normalized quasiconformal map with Beltrami coefficient $\mu_{n}$.
Then
$$
E_{n} = w^{\mu_{n}}\circ f\circ (w^{\mu_{n+1}})^{-1}\in {\mathcal E}_{p, q}
$$
since it preserves
$\mu_0$ and hence is holomorphic.
This gives a sequence $\{E_{n}\}_{n=0}^{\infty}$ of maps in ${\mathcal E}_{p,q}$ and a sequence of subsets
$P_{f,n}=w^{\mu_{n}} (P_{f})$.
Note that $P_{f,n}$ is not, in general, the post-singular set $P_{E_{n}}$ of $E_{n}$.
If it were, we would have a fixed point of $\sigma$.
\medskip
\noindent {\bf The compactness condition.} We say $f$ satisfies the compactness condition if the sequence $\{ E_n \}_{n=1}^{\infty}$ generated in the Thurston iteration scheme is contained in a compact subset of $ \mathcal E_{p,q}$.
\medskip
From a conceptual point of view, the compactness condition is very natural and simple. From a technical point of view, however, it is not at all obvious. We give a detailed proof showing how to get a fixed point from the condition of bounded geometry with compactness.
Suppose $f$ is a post-singularly finite topological exponential map in ${\mathcal TE}_{p,q}$.
For any $\tau=[\mu]\in T_{f}$, let $T_{\tau}T_{f}$ and $T^{*}_{\tau}T_{f}$ be the tangent space and the cotangent space of $T_{f}$ at $\tau$ respectively. Let $w^{\mu}$ be the corresponding normalized quasiconformal map fixing $0,1, \infty$.
Then $T^{*}_{\tau}T_{f}$ coincides with the space ${\mathcal Q}_{\mu}$ of integrable meromorphic quadratic differentials $q=\phi(z) dz^{2}$. Integrablility means that the norm of $q$, defined by
$$
||q|| =\int_{\hat{\mathbb C}} |\phi(z)| dzd\overline{z}
$$
is finite. This condition implies that the poles of $q$ must occur at points of $w^{\mu} (P_{f})$ and that these poles are simple.
Set $\tilde{\tau}=\sigma(\tau)=[\tilde{\mu}]$ and denote by $w^{\mu}$ and $w^{\tilde{\mu}}$ the corresponding normalized quasiconformal maps. We have the following commutative
diagram:
$$
\begin{array}{ccc} \hat{\mathbb C}\setminus f^{-1}(P_{f}) & {\buildrel w^{\tilde{{\mu}}} \over
\longrightarrow} & \hat{\mathbb C}\setminus w^{\tilde{{\mu}}} (f^{-1}(P_{f}))\cr \downarrow f
&&\downarrow E_{\mu,\tilde{\mu}}\cr \hat{\mathbb C}\setminus P_{f}& {\buildrel w^{\mu}
\over \longrightarrow} & \hat{\mathbb C}\setminus w^{\mu}(P_{f}).
\end{array}
$$
Note that in the diagram, by abuse of notation, we write $f^{-1}(P_{f})$ for $f^{-1}(P_{f}\setminus \{\infty\})\cup\{\infty\}$. Since by definition $\tilde{\mu} = f^*\mu$, the map
$E=E_{\mu,\tilde{\mu}}= w^{\mu} \circ f \circ (w^{\tilde{\mu}})^{-1}$ defined on ${\mathbb C}$
is analytic. By Theorem~\ref{topexp}, $E_{\mu,\tilde{\mu}}=P_{\tau, \tilde{\tau}}e^{Q_{\tau,\tilde{\tau}}}$ for a pair of polynomials $P=P_{\tau, \tilde{\tau}}$ and $Q=Q_{\tau,\tilde{\tau}}$ of respective degrees $p$ and $q$.
Let $\sigma_{*}: T_{\tau}T_{f}\to T_{\tilde{\tau}}T_{f}$ and $\sigma^{*}: T_{\tilde{\tau}}^*T_{f}\to T_{\tau}^*T_{f}$ be the tangent and co-tangent map of $\sigma$, respectively.
Let $\beta(t)=[\mu(t)]$ be a smooth path in $T_{f}$ passing though $\tau$ at $t=0$ and let $\eta=\beta'(0)$ be the corresponding tangent vector at $\tau$. Then the
pull-back $\tilde{\beta}(t)= [f^{*}\mu (t)]$ is a smooth path in $T_{f}$ passing though
$\tilde{\tau}$ at $t=0$ and $\tilde{\eta} = \sigma_{*} \eta=\tilde{\beta}'(t)$ is the corresponding tangent vector at $\tilde{\tau}$. We move these tangent vectors to the origin in $T_f$ obtaining the vectors $\xi, \tilde\xi$ using the maps
$$
\eta = (w^{\mu})^{*}\xi \quad \hbox{and}\quad \tilde{\eta} =
(w^{\tilde{\mu}})^{*} \tilde{\xi}.
$$
This gives us the following commutative diagram:
$$
\begin{array}{ccc} \tilde{\eta}& {\buildrel (w^{\tilde{\mu}})^{*} \over
\longleftarrow} & \tilde{\xi}\cr \hbox{\hskip17pt}\uparrow f^{*} &
& \hbox{\hskip17pt}\uparrow E^{*} \cr \eta & {\buildrel (w^{\mu})^{*} \over \longleftarrow}& \xi\cr
\end{array}
$$
Now suppose $\tilde{q}$ is a co-tangent vector in
$T^{*}_{\tilde{\tau}}$ and let
$q=\sigma^{*} \tilde{q}$ be the corresponding co-tangent vector in
$T^{*}_{\tau}$. Then
$\tilde{q}=\tilde{\phi} (w) dw^{2}$ is an integrable quadratic differential on $\hat{\mathbb C}$ that can have at worst simple poles along $w^{\tilde{\mu}}(P_{f})$
and $q=\phi (z) dz^{2}$ is an integrable quadratic differential on $\hat{\mathbb C}$ that can have at worst simple poles along
$w^{\mu}(P_{f})$. This implies that $q=\sigma_{*}\tilde{q}$
is also the push-forward integrable quadratic differential
$$
q=E_{*}\tilde{q} =\phi (z) dz^{2}
$$
of $\tilde{q}$ by $E$. To see this, recall from section~\ref{sec:Tpq} that $E$, and a choice of curves $L_i$ from the branch points, determine a finite set of domains $W_i$ on which $E$ is an unbranched covering to a domain homeomorphic to $\mathbb C^*$. Since $E$ restricted to each $W_i$ is either a topological model for $e^z$ or $z^k$, we may divide each $W_i$ into a collection of fundamental domains on which $E$ is bijective. Therefore the coefficient $\phi (z)$ of $q$ is given by the formula
\begin{equation}~\label{pushforwardformula}
\phi(z)= ({\mathcal L}\tilde{\phi}) (z) =\sum_{E(w) = z}
\frac{\tilde{\phi} (w)}{(E'(w))^{2}} = \frac{1}{z^{2}} \sum_{E(w)=z} \frac{\tilde{\phi} (w)}{(\frac{P'(w)}{P(w)} +Q'(w))^{2}}
\end{equation}
where ${\mathcal L}$ is the standard transfer operator and $\tilde{\phi}$ is the coefficient of $\tilde{q}$. Thus
\begin{equation}~\label{pushforwardformula1}
q= \phi(z) dz^{2}= \frac{dz^{2}}{z^{2}} \sum_{E(w)=z} \frac{\tilde{\phi} (w)}{(\frac{P'(w)}{P(w)} +Q'(w))^{2}}
\end{equation}
It is clear that as a quadratic differential defined on $\hat{\mathbb C}$, we have
$$
||q||\leq ||\tilde{q}||.
$$
Since $q$ is integrable and $0$ and $\infty$ are isolated singularities, it follows that $q$ has at worst possible simple poles at these points so that the inequality holds on all of $\hat{\mathbb C}$.
By formula~(\ref{pushforwardformula}), we have
$$
\langle \tilde{q} ,\tilde{\xi} \rangle = \langle q, \xi \rangle
$$
which implies
$$
\|\tilde{\xi}\| \leq \|\xi\|
$$
where this is the $L^{\infty}$ norm.
This gives another proof of Lemma~\ref{contractive}.
Furthermore, we have the following stronger assertion
\medskip
\begin{lemma}~\label{infstrongcon}
$$
||q||<||\tilde{q}||
$$
and
$$
\|\tilde{\xi}\| < \|\xi\|.
$$
\end{lemma}
\begin{proof}
Suppose there is a $\tilde{q}$ such that $||q||=||\tilde{q}||\not= 0$.
Using the change of variable $E(w)=z$ on each fundamental domain we get
$$
\int_{\hat{\mathbb C}} \Big| \sum_{E(w)=z} \frac{\tilde{\phi} (w)}{(E'(w))^{2}}\Big| dz \, d\overline{z}=
\int_{\hat{\mathbb C}} |\phi(z)|dz \, d\overline{z} =\int_{\hat{\mathbb C}} |\tilde{\phi} (w)| \, dw d\overline{w}
$$
$$
= \sum_{i} \int_{W_{i}} |\tilde{\phi} (w)| \, dw d\overline{w} = \int_{\hat{\mathbb C}} \sum_{i} \Big|\frac{\tilde{\phi} (w)}{(E'(w))^{2}}\Big| \, dz d\overline{z}.
$$
By the triangle inequality, all the factors $\frac{\tilde{\phi} (w)}{(E'(w))^{2}}$ in $\sum_{E(w)=z} \frac{\tilde{\phi} (w)}{(E'(w))^{2}}$ have the same argument. That is, there is a real number $a_{z}$ for every $z$ such that for any pair of points $w,w'$ with $E(w)=E(w')=z$,
$$
\frac{\tilde{\phi} (w)}{(E'(w))^{2}}=a_{z} \frac{\tilde{\phi} (w')}{(E'(w'))^{2}}.
$$
Now formula~(\ref{pushforwardformula}) implies
$\phi (z) =\infty$ which cannot be; this contradiction proves the lemma.
\end{proof}
\medskip
\begin{remark}
The real point here is that $E$ has infinite degree and any $q$ has finitely many poles. If there were a $\tilde{q}$ with $||q||=||\tilde{q}||\not= 0$ and if $Z$ is the set of poles of $\tilde{q}$, then
the poles of $q$ would be contained in the set $E(Z)\cup {\mathcal V}_{E}$, where ${\mathcal V}_{E}$ is the set of critical values of $E$.
Thus, by formula~(\ref{pushforwardformula}),
$$
E^{*} q =\phi (E(w)) dw^2 = d \tilde{q} (w),
$$
where $d$ is the degree of $E$. Furthermore,
$$
E^{-1} (E(Z)\cup {\mathcal{V}}_{E}) \subseteq Z \cup {\Omega}_{E}.
$$
Since $d$ is infinite, the last inclusion formula can not hold since the left hand side is infinite and the right hand side is finite.
\end{remark}
An immediate corollary is
\medskip
\begin{corollary}~\label{strongcon}
For any two points $\tau$ and $\tilde{\tau}$ in $T_{f}$,
$$
d_{T}\Big(\sigma(\tau), \sigma(\tilde{\tau})\Big)< d_{T}(\tau, \tilde{\tau}).
$$
\end{corollary}
Furthermore,
\medskip
\begin{lemma}~\label{fixedpt}
If $\sigma$ has a fixed point in $T_{f}$, then this fixed point must be unique. This is equivalent to saying that
a post-singularly finite $f$ in ${\mathcal TE}_{p, q}$ is combinatorially equivalent to at most one $(p,q)$-exponential map $E=Pe^{Q}$.
\end{lemma}
We can now finish the proof of the sufficiency in Theorem~\ref{main1}.
\begin{proof}[Proof of Theorem~\ref{main1}]
Suppose $f\in {\mathcal TE}_{p,q}$ has bounded geometry with compactness.
Recall that the map defined by
\begin{equation}\label{thu iter}
E_{n}=w^{\mu_{n}}\circ f\circ (w^{\mu_{n+1}})^{-1}
\end{equation}
is a $(p,q)$-exponential map.
If $q=0$, $E_n$ is a polynomial and the theorem follows from the arguments given in~\cite{CJ} and \cite{DH}.
Note that if $P_f = \{0,1,\infty\}$, then $f$ is a universal covering map of $\mathbb{C}^*$ and is therefore combinatorially equivalent to $e^{2 \pi i z}$.
Thus in the following argument, we assume that $\#(P_f) \geq 4$. Then, given our normalization conventions and the bounded geometry hypothesis we see that
the functions $E_{n}$, $n=0, 1, \ldots$ satisfy the following conditions:
\begin{itemize}
\item[1)] $m=\#(w^{\mu_{n}}(P_f))\geq 4$ is fixed.
\item[2)] $0, 1, \infty \in w^{\mu_{n}}(P_f)$.
\item[3)] $\Omega_{E_{n}}\cup \{ 0, 1, \infty\} \subseteq
E_{n}^{-1}(w^{\mu_{n}}(P_f))$.
\item[4)] there is a $b>0$ such that $d_{sp}(p_{n}, q_{n}) \geq b$ for any $p_{n}, q_{n}\in w^{\mu_{n}}(P_f)$.
\end{itemize}
As a consequence of the compactness, we have that in the sequence $\{ E_{n}\}_{n=1}^{\infty}$, there is a subsequence $\{ E_{n_{i}}\}_{i=1}^{\infty}$ converging to a map $E=Pe^{Q} \in \mathcal E_{p,q}$ where $P$ and $Q$ are polynomials of degrees $p$ and $q$ respectively.
Any integrable quadratic differential $q_{n} \in T^{*}_{\tau_{n}}{\mathcal T}_{f}$ has, at worst, simple poles in the finite set $P_{n+1,f}= w^{\mu_{n+1}}(P_f)$.
Since $T^{*}_{\tau_{n}}{\mathcal T}_{f}$ is a finite dimensional linear space, there is a quadratic differential $q_{n, max}\in T^{*}_{\tau_{n}}{\mathcal T}_{f}$
with $\| q_{n,max}\|=1$ such that
$$
0 \leq a_{n}=\sup_{||q_{n}||=1} \|(E_{n})_{*}q_{n}|| = \|(E_{n})_{*}q_{n,max}\| <1.
$$
Moreover, by the bounded geometry condition, the possible simple poles of $\{q_{n, max}\}_{n=1}^{\infty}$ lie in a
compact set and hence these quadratic differentials lie in a compact subset of the space of quadratic differentials on $\hat{\mathbb C}$ with, at worst, simples poles at $m=\#(P_{f})$ points.
Let
$$
a_{\tau_{0}} =\sup_{n\geq 0} a_{n}.
$$
Let $\{n_{i}\}$ be a sequence of integers such that the subsequence $a_{n_{i}}\to a_{\tau_{0}}$ as $i\to \infty$.
By compactness, $\{E_{n_{i}}\}_{i=0}^{\infty}$ has
a convergent subsequence, (for which we use the same notation)
that converges to a holomorphic map $E \in \mathcal E_{p,q}$.
Taking a further subsequence if necessary, we obtain a convergent sequence of sets $P_{n_{i},\tau_{0}} =w^{\mu_{n_{i}}}(P_{f})$ with limit set $X$.
By bounded geometry, $\#(X)=\#(P_{f})$ and $d_{sp}(x, y) \geq b$ for any $x,y\in X$.
Thus we can find a subsequence $\{ q_{n_{i}, max}\}$ converging to an integrable quadratic differential $q$ of norm $1$ whose only poles lie in $X$ and are simple.
Now by lemma~\ref{infstrongcon},
$$
a_{\tau_{0}} = ||E_{*} q|| <1.
$$
Thus we have proved that there is an $0< a_{\tau_{0}}< 1$, depending only on $b$ and $f$, such that
$$
\|\sigma_{*}\| \le
\|\sigma^{*}\| \le a_{\tau_{0}}.
$$
Let $l_{0}$ be a curve connecting $\tau_{0}$ and $\tau_{1}$ in $T_{f}$ and set $l_{n}=\sigma_{f}^{n}(l_{0})$ for $n\geq 1$. Then $l=\cup_{n=0}^{\infty}l_{n}$ is a curve in $T_f$
connecting all the points $\{\tau_{n}\}_{n=0}^{\infty}$. For each point $\tilde{\tau}_{0}\in l_{0}$, we have $a_{\tilde{\tau}_{0}} <1$. Taking the maximum gives a uniform $a<1$ for all points in $l_0$. Since $\sigma$ is holomorphic, $a$ is an upper bound for all points in $l$. Therefore,
$$
d_{T} (\tau_{n+1}, \tau_{n}) \leq a \, d_{T}(\tau_{n}, \tau_{n-1})
$$
for all $n\geq 1$.
Hence, $\{ \tau_{n}\}_{n=0}^{\infty}$ is a convergent sequence with a unique limit point $\tau_{\infty}$ in $T_{f}$ and $\tau_{\infty}$ is
a fixed point of $\sigma$. This combining with Lemma~\ref{fixedpt} completes the proof of the sufficiency of Theorem~\ref{main1}.
\end{proof}
From our proofs of Theorem~\ref{necc} and Theorem~\ref{main1}, the final step in the proof of the main theorem is to prove the compactness condition holds from bounded geometry. This is in contrast to the case of rational maps (see~\cite{Ji}) where the bounded geometry condition guarantees the compactness condition holds. In the case of $(p,q)$-exponential maps,
the bounded geometry condition must be combined with some topological constraints to guarantee the compactness. The topological constraints, together with the bounded geometry condition control the sizes of fundamental domains so that they are neither too small nor too big. Thus, before we prove the compactness condition holds, we will describe a topological constraint for the two types of map $f$ in the Main Theorem (Theorem~\ref{main2}).
\section{Proof of the Main Theorem (Theorem~\ref{main2}).}
\label{sec:proofmt}
In section~\ref{sec:Tpq} we defined two different normalizations for functions in $\mathcal TE_{p,q}$ that depend on whether or not $0$ is a fixed point of the map. The topological constraints
for post-singularly finite maps also follow this dichotomy.
\medskip
\subsection{A topological constraint for $f\in \mathcal TE_{0,1}$ satisfying the hypotheses of Theorem 2.}
\label{top1}
Any such $f$ has no branch points so $P_{f} =\cup_{k\geq 0} f^{k} (0) \cup\{\infty\}$ which is finite. Since $0$ is omitted, the orbit of $0$ is pre-periodic. Let $c_{k}=f^{k}(0)$ for $k\geq 0$. By the pre-periodicity, there are integers $k_{1}\geq 0$ and $l\geq 1$ such that
$f^{l}(c_{k_{1}+1}) = c_{k_{1}+1}$. This says that
$$
\{ c_{k_{1}+1}, \ldots, c_{k_{1}+l}\}
$$
is a periodic orbit of period $l$. Let $k_{2} =k_{1}+l$.
Let $\gamma$ be a continuous curve
connecting $c_{k_{1}}$ and $c_{k_{2}}$ in
${\mathbb R}^{2}$ disjoint from $P_f$, except for its endpoints. Because
$$
f(c_{k_{1}})=f(c_{k_{2}})=c_{k_{1}+1},
$$
the image curve $\delta= f(\gamma)$ is a closed curve.
\medskip
\subsection{A topological constraint for $f\in \mathcal TE_{p,1}$ satisfying the hypotheses of Theorem 2.}
\label{top2}
Any such $f$ has exactly one non-zero simple branch point which we denote by $c$;
$0$ is the only other branch point and it has multiplicity $p-1$. Then $f(0)=0$ and by our normalization, $f(c)=1$. In this case
$$
P_{f} =\cup_{k\geq 1} f^{k} (c) \cup\{0, \infty\}.
$$
Again by the hypothesis of Theorem~\ref{main2}, $P_{f}$ is finite. Set $c_{k}=f^{k}(c)$ for $k\geq 0$.
Suppose $c$ is not periodic. As above, there are integers $k_{1}\geq 0$ and $l\geq 1$ such that
$f^{l}(c_{k_{1}+1}) = c_{k_{1}+1}$. Again,
$$
\{ c_{k_{1}+1}, \ldots, c_{k_{1}+l}\}
$$
is a periodic orbit of period $l$. Let $k_{2} =k_{1}+l$.
As above, let $\gamma$ be a continuous curve
connecting $c_{k_{1}}$ and $c_{k_{2}}$ in
${\mathbb R}^{2}$ disjoint from $P_f$, except for its endpoints. Since
$$
f(c_{k_{1}})=f(c_{k_{2}})=c_{k_{1}+1},
$$
the image curve $\delta= f(\gamma)$ is a closed curve.
\subsection{Winding numbers}
\label{winding}
In each of the above cases, the {\em winding number} $\eta$ of the closed curve $\delta= f(\gamma)$ about $0$ essentially counts the number of fundamental domains
between $c_{k_{1}}$ and $c_{k_{2}}$ and defines the ``distance'' between the fundamental domains.
The following lemma is a crucial to proving that the compactness condition holds for each type of function in Theorem~\ref{main2}.
\medskip
\begin{lemma}\label{winding1}
The winding number $\eta$ is does not change under the Thurston iteration procedure.
\end{lemma}
\begin{proof}
Given $\tau_{0}=[\mu_{0}]\in T_f$, let $\tau_{n}=\sigma^{n} (\tau_{0}) =[\mu_{n}]$
be the sequence generated by $\sigma$. Let $w^{\mu_{n}}$ be the normalized quasiconformal map with Beltrami coefficient $\mu_{n}$.
Then in either of the above situations,
$$
E_{n} = w^{\mu_{n}}\circ f\circ (w^{\mu_{n+1}})^{-1}\in {\mathcal E}_{p, 1}
$$
since it preserves $\mu_0$ and is holomorphic. See the following diagram.
$$
\begin{array}{ccc} \hat{\mathbb C}& {\buildrel w^{\mu_{n+1}} \over
\longrightarrow} & \hat{\mathbb C}\cr
\downarrow f &&\downarrow E_{n}\cr
\hat{\mathbb C}& {\buildrel w^{\mu_{n}}
\over \longrightarrow} & \hat{\mathbb C}.
\end{array}
$$
Let $c_{k, n} =w^{\mu_{n}}(c_{k})$.
The continuous curve
$$
\gamma_{n+1}=w^{\mu_{n+1}} (\gamma)
$$
goes from $c_{k_{1}, n+1}$ to $c_{k_{2}, n+1}$.
The image curve is
$$
\delta_{n}=E_{n} (\gamma_{n+1}) = w^{\mu_{n}}(f((w^{\mu_{n+1}})^{-1} (\gamma_{n+1})))=w^{\mu_{n}} (f (\gamma)) = w^{\mu_{n}} (\delta).
$$
Note that $w^{\mu_{n}}$ fixes $0,1,\infty$. Thus $\delta_{n}$ is a closed curve through the point $c_{k_{1}+1, n}= w^{\mu_{n}} (c_{k_{1}+1})$ and it has winding number $\eta$ around $0$.
\end{proof}
The argument that this invariance plus bounded geometry implies the compactness is different in each of these two cases. We present these arguments in the two subsections below. Let set
$$
P_{n}=P_{f, n} =w^{\mu_{n}} (P_{f}), \quad n=0, 1, 2, \ldots.
$$
\subsection{The compactness condition for $f\in \mathcal TE_{0,1}$ satisfying the hypotheses of Theorem~\ref{main2}.}
\label{comp1}
In this case, all the functions in the Thurston iteration have the form $E_{n} (z) = e^{\lambda_{n} z}$. From our normalization, we have that
$$
0, \; 1=E_{n}(0),\; E_{n}(1) =e^{\lambda_{n}} \in P_{n+1}.
$$
Note that in this case $E_{n}(1) \neq1$. When $f$ has bounded geometry,
the spherical distance between $1$ and $E_{n}(1)$ is bounded away from zero.
That is, there is a constant $\kappa>0$ such that
$$
\kappa \leq |\lambda_{n}|, \;\; \forall n\geq 0.
$$
Now we prove that the sequence $\{|\lambda_{n}|\}$ is also bounded above.
We can compute
$$
\eta = \frac{1}{2\pi i} \oint_{\delta_{n}} \frac{1}{w} dw = \frac{1}{2\pi i} \int_{\gamma_{n+1}} \frac{E_{n}'(z)}{E_{n}(z)} dz= \frac{1}{2\pi i} \int_{\gamma_{n+1}} \lambda_{n}dz.
$$
The integral therefore depends only on the endpoints and we have
$$
\eta= \frac{1}{2\pi i} \int_{\gamma_{n+1}} \lambda_{n}dz=\frac{\lambda_{n}}{2\pi i} (c_{k_{2}, n+1}-c_{k_{1}, n+1}).
$$
Since $0$ is omitted, it can not be periodic. Therefore, both $c_{k_{2}, n+1}\not= c_{k_{1}, n+1}\in P_{n+1}$,
so by bounded geometry there is a positive constant which we again denote by $\kappa$ such that
$$
|c_{k_{2}, n+1}-c_{k_{1}, n+1}|\geq \kappa.
$$
This gives us the estimate
$$
|\lambda_{n}| \leq \frac{2\pi \eta}{|c_{k_{2}, n+1}-c_{k_{1}, n+1}|}\leq \frac{2\pi \eta}{\kappa}.
$$
which proves that $\{ E_{n}(z)\}_{n=0}^{\infty}$ is contained in a compact family in $\mathcal E_{0,1}$. This combined with Theorem~\ref{necc} and Theorem~\ref{main1} completes the proof of Theorem~\ref{main2} for $f\in \mathcal TE_{0,1}$.
\subsection{The compactness condition for $f\in TE_{p,1}$ satisfying the hypotheses in Theorem~\ref{main2}.}
\label{comp2}
For such a map, $f(0)=0$ and $0$ is a branch point of multiplicity $p-1$. And $f$ has exactly one non-zero branch point $c$ with $f(c)=1$. All the functions in the Thurston iteration have forms
$$
E_{n} (z) = \alpha_{n} z^{p} e^{\lambda_{n} z}, \quad
\alpha_{n}=e^{p} \Big( -\frac{\lambda_{n}}{p}\Big)^{p}.
$$
Note that $E_{n}(0)=0$ and $0$ is a critical point of multiplicity $p-1$. It is also the asymptotic value and hence it has no other pre-images.
Moreover, $E_{n}(z)$ has exactly one non-zero simple critical point
$$
c_{n}=-\frac{p}{\lambda_{n}}=w^{\mu_{n}}(c).
$$
and $\alpha_{n}$ is defined by the normalization condition $E_{n}(c_{n})=1$.
If $c$ is periodic, then $c\in P_{f}$. This implies that $c_{n} \not= 0, \infty\in P_{n}$ and thus its spherical distance from either $0$ or $\infty$ is bounded below. That is, there are two constants $0<\kappa<K<\infty$ such that
$$
\kappa\leq |\lambda_{n}|\leq K, \quad \;\forall n>0.
$$
This implies that the sequence $\{E_{n}\}_{n=1}^{\infty}$ is contained in a compact subset.
Now suppose $c$ is not periodic. By the the hypotheses in Theorem~\ref{main2}, $f(c)=1$ is also not periodic.
This implies that $k_{1}\geq 1$.
Recalling the notation $P_{n} =w^{\mu_{n}} (P_{f})$, we have
$$
0, \quad 1=E_{n}(c_{n}), \quad E_{n}(1) = e^{p} \Big( -\frac{\lambda_{n}}{p}\Big)^{p} e^{\lambda_{n}} \in P_{n}.
$$
Let $c_{k, n} =w^{\mu_{n}}(c_{k})$. Then $c_{k,n}\in P_{n}$ for all $k\geq 1$.
Let $\delta_{n} =w^{\mu_{n}}(\delta)$ and $\gamma_{n} =w^{\mu_{n}}(\gamma)$.
When $f$ has bounded geometry, since $E_{n}(1) \not=0$, its spherical distance from $0$ is bounded below. This implies that the sequence $\{|\lambda_{n}|\}$ is bounded below; that is, there is a constant $\kappa>0$ such that
$$
\kappa\leq |\lambda_{n}|, \;\; \forall n>0.
$$
By our hypothesis, $c_{k_{1},n+1}\not= c_{k_{2},n+1}$ both belong to $P_{n+1}$ and bounded geometry implies there are two constants, which we still denote by
$0<\kappa< K<\infty$, such that
$$
\kappa\leq |c_{k_{2}, n+1}|, \;\; |c_{k_{1}, n+1}|,\;\; |c_{k_{2}, n+1}-c_{k_{1},n+1}| \leq K, \quad \forall\; n\geq 1.
$$
Now we prove that the sequence $\{|\lambda_{n}|\}$ is also bounded above.
Recall that when we chose $\gamma$, we assumed it did not go through $0$ and thus by the normalization,
none of the $\gamma_{n+1}$ go through $0$ either. Therefore, for each $n$ we can find a simply connected domain
$D_{n+1} \subset \gamma_{n+1}$ that does not contain $0$.
As in the previous section we compute
$$
\eta= \frac{1}{2\pi i} \oint_{\delta_{n}} \frac{1}{w} dw = \frac{1}{2\pi i} \int_{\gamma_{n+1}} \frac{E_{n}'(z)}{E_{n}(z)} dz
= \frac{1}{2\pi i} \int_{\gamma_{n+1}} \Big(\frac{p}{z} +\lambda_{n}\Big)dz
$$
so that as above
$$
2 \pi i \eta = \int_{\gamma_{n+1}} \frac{p}{z} dz +\int_{\gamma_{n+1}} \lambda_{n} dz = \int_{\gamma_{n+1}} \frac{p}{z} dz + \lambda_{n} (c_{k_{2},n+1}-c_{k_{1},n+1}).
$$
Rewriting we have
$$
\lambda_{n} (c_{k_{2},n+1}-c_{k_{1},n+1}) =2 \pi i \eta - \int_{\gamma_{n+1}} \frac{p}{z} dz.
$$
This implies that
$$
\kappa |\lambda_{n}| \leq |\lambda_{n} (c_{k_{2},n+1}-c_{k_{1},n+1})| \leq 2 \pi \eta + \Big|\int_{\gamma_{n+1}}\frac{p}{z} dz\Big|
$$
so that if we can bound the integral on the right we will be done.
Notice that $\log z$ can be defined as an analytic function on the simply connected domain $D_{n+1}$ containing $\gamma_{n+1}$ that we chose above.
We take $\log z =\log |z| +2\pi i \arg (z)$ as the principal branch, with $0\leq \arg (z)<2\pi$ .
We then estimate
$$
\Big|\int_{\gamma_{n+1}}\frac{p}{z} dz\Big|=| \log c_{k_{2},n+1} - \log c_{k_{1},n+1}|
$$
$$
\leq |\log |c_{k_{2},n+1}|-\log |c_{k_{1},n+1}| | + |\arg(c_{k_{2},n+1}) - \arg (c_{k_{1},n+1})|
$$
$$
\leq 2\pi \eta + (\log K -\log \kappa) +4\pi .
$$
Finally we have
$$
|\lambda_{n}| \leq \frac{2\pi \eta +(\log K-\log \kappa) +4\pi}{\kappa}.
$$
which proves that $\{ E_{n}(z)\}_{n=0}^{\infty}$ is contained in a compact subset in $\mathcal E_{p,1}$.
This combined with Theorem~\ref{necc} and Theorem~\ref{main1} completes the proof of Theorem~\ref{main2} for $f\in \mathcal TE_{p,1}$.
\section{Some Remarks}
One can formally define a Thurston obstruction for a post-singularly finite $(p,q)$-topological exponential map $f$ with $p\geq 0$ and $q\geq 1$. Because such an $f$ is a branched covering of infinite degree, however, many arguments in the proof of the Thurston theorem that use the finiteness of the covering in an essential way do not apply (see~\cite{DH,JZ,Ji}) . Thus, how to define an analog of the Thurston obstruction to characterize a $(p,q)$-topological exponential map $f$ is not clear to us.
We can, however, define an analog of the {\em canonical} Thurston obstruction for a $(p,q)$-topological exponential map $f$ which depends on the hyperbolic lengths of curves (see to~\cite{Pi,CJ,Ji}).
Let $\sigma$ be the induced map on the Teichm\"uller space $T_{f}$. For any $\tau_{0}\in T_{f}$, and for $n\geq 1$,
let $\tau_{n}=\sigma^{n}(\tau_{0})$. Let $\gamma$ denote a simple closed non-peripheral curve
in ${\mathbb C}\setminus P_{f}$. Define
$$
\Gamma_{c}=\{ \gamma \;|\; \forall \tau_{0}\in T_{f}, \; l_{\tau_{n}}(\gamma) \to 0 \quad \hbox{as}\quad n \to \infty\}.
$$
We have
\begin{corollary}
If $\Gamma_{c}\not=\emptyset$, then $f$ has no bounded geometry and therefore, $f$ is not combinatorially equivalent to a $(p,q)$-exponential map.
\end{corollary}
The converse should also be true but we have no proof at this time. The difficulty is that in the characterization of post-critically finite rational maps, many arguments in the proof of the converse statement use the finiteness of the covering in an essential way (see~\cite{Pi,CJ}).
A {\em Levy cycle} is a special Thurston obstruction for rational maps. It can be defined for a $(p,q)$-topological exponential map $f$ as follows.
A set
$$
\Gamma=\{ \gamma_{1}, \cdots, \gamma_{n}\}
$$
of simple closed non-peripheral curves in ${\mathbb C}\setminus P_{f}$ is called a Levy cycle
if for any $\gamma_{i}\in \Gamma$, there is a simple closed non-peripheral curve component
$\gamma'$ of $f^{-1}(\gamma_{i})$ such that $\gamma'$ is homotopic to $\gamma_{i-1}$ (we identify $\gamma_0$ with $\gamma_n$) rel $P_{f}$
and $f: \gamma'\to \gamma_{i}$ is a homeomorphism. Following a result in~\cite{HSS}, we have
\begin{corollary}[\cite{HSS}]
Suppose $f$ is a $(0,1)$-topological exponential map with finite post-singular set. Then $f$ has no Levy cycle if and only if $f$ has bounded geometry.
\end{corollary}
We believe a similar result holds for all post-singularly finite maps in $\mathcal TE_{p,q}$ but we do not have a proof at this time.
\bigskip
| {
"timestamp": "2013-09-17T02:02:21",
"yymm": "1209",
"arxiv_id": "1209.6044",
"language": "en",
"url": "https://arxiv.org/abs/1209.6044",
"abstract": "In this paper we define a topological class of branched covering maps of the plane called {\\em topological exponential maps of type $(p,q)$} and denoted by $\\TE_{p,q}$, where $p\\geq 0$ and $q\\geq 1$. We follow the framework given in \\cite{Ji} to study the problem of combinatorially characterizing an entire map $P e^{Q}$, where $P$ is a polynomial of degree $p$ and $Q$ is a polynomial of degree $q$ using an {\\em iteration scheme defined by Thurston} and a {\\em bounded geometry condition}. We first show that an element $f \\in {\\TE}_{p,q}$ with finite post-singular set is combinatorially equivalent to an entire map $P e^{Q}$ if and only if it has bounded geometry with compactness. Thus to complete the characterization, we only need to check that the bounded geometry actually implies compactness. We show this for some $f\\in \\TE_{p,1}$, $p\\geq 1$. Our main result in this paper is that a post-singularly finite map $f$ in $\\TE_{0,1}$ or a post-singularly finite map $f$ in $\\TE_{p,1}$, $p\\geq 1$, with only one non-zero simple breanch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{\\lambda z}$ or the form $ \\alpha z^{p}e^{\\lambda z}$, where $\\alpha=(-\\lambda/p)^{p}e^{- \\lambda (-p/\\lambda)^{p}}$, respectively, if and only if it has bounded geometry. This is the first result in this direction for a family of transcendental holomorphic maps with critical points.",
"subjects": "Dynamical Systems (math.DS); Complex Variables (math.CV)",
"title": "Bounded Geometry and Characterization of post-singularly Finite $(p,q)$-Exponential Maps",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9825575167960731,
"lm_q2_score": 0.7217432122827968,
"lm_q1q2_score": 0.7091542184250058
} |
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